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Transductive Zero-Shot Learning with a Self-training dictionary approach Yunlong Yu, Zhong Ji, Xi Li, Jichang Guo, Zhongfei Zhang, Haibin Ling, Fei WuDecember 30, 2023 ========================================================================================== As an important and challenging problem in computer vision, zero-shot learning (ZSL) aims at automatically recognizing the instances from unseen object classes without training data. To address this problem, ZSL is usually carried out in the following two aspects: 1) capturing the domain distribution connections between seen classes data and unseen classes data; and 2) modeling the semantic interactions between the image feature space and the label embedding space. Motivated by these observations, we propose a bidirectional mapping based semantic relationship modeling scheme that seeks for cross-modal knowledge transfer by simultaneously projecting the image features and label embeddings into a common latent space. Namely, we have a bidirectional connection relationship that takes place from the image feature space to the latent space as well as from the label embedding space to the latent space. To deal with the domain shift problem, we further present a transductive learning approach that formulates the class prediction problem in an iterative refining process, where the object classification capacity is progressively reinforced through bootstrapping-based model updating over highly reliable instances. Experimental results on three benchmark datasets (AwA, CUB and SUN) demonstrate the effectiveness of the proposed approach against the state-of-the-art approaches.Zero-shot learning, transductive learning, bidirectional mapping, domain adaptation, bootstrapping. § INTRODUCTIONZero-shot learning (ZSL) <cit.> endows the computer vision system with the capability to recognize instances of a new class that has never seen before. A common framework to address this problem is to transfer the knowledge from the seen classes to unseen ones by resorting to a label embedding space where the semantic relatedness between different classes are measured. Commonly used semantic label embeddings include visual attributes <cit.> and word vectors <cit.>.In order to achieve the knowledge transfer, existing approaches fall into two main categories. The first one poses the seen classes as the mediators to connect the test instance and the unseen classes. It relies on learning a classification model for seen classes with the labeled instances, which is then used to compute the visual similarities between the test instance and seen classes. The prediction is implemented by matching the visual similarities and the semantic relatedness between the seen classes and the unseen classes, which is obtained with their label embeddings. In contrast, the approaches in the second category focus on modeling the semantic interactions between different modalities by directly learning a projection function either from the image feature space to the label embedding space <cit.>, or from a reverse direction <cit.>, and then predict the unseen instances in the label embedding space or image visual space.A common characteristic of existing ZSL approaches from both categories is that they all critically rely on the pre-defined label embeddings to compute the semantic relatedness between the seen and unseen classes. However, the noisy and uncertainty of the label embedding make it hard to characterize the semantic information explicitly, which will be blindly forced to the unseen data during the knowledge transfer. Besides, we only have a single sparse label semantic vector for each unseen class, which is insufficient to fully represent the data distribution of the class. Thus, the distribution connections between the seen domain and unseen domain are difficult to capture. Motivated by these observations, we propose a bidirectional mapping based semantic relationship modeling scheme that seeks for cross-modal knowledge transfer by simultaneously projecting the image features and label embeddings into a common latent space. In specific, the bidirectional connection relationship is formulated into a general dictionary framework, in which a common latent space is learned for preserving the semantic relatedness between different modalities. By projecting the label embeddings to the latent space where the embedding semantics are more suitably aligned, the influence of semantic gap across different modalities alleviates.As the seen classes and unseen classes are different and potentially unrelated, the projection function learned from the seen domain is usually biased on the unseen domain. To address this domain shift issue, many approaches focus on learning a more general projection function to bridge the semantic relationships between the image feature space and the label embedding space under a transductive setting <cit.>. The transductive setting means that the unlabeled unseen instances are used to improve the generalization accuracy. However, existing transductive approaches treat all unlabeled data equally and achieve the prediction in one pass, which makes the learned models difficult to relate the seen domain to the unseen domain. Based on this motivation, we further present a transductive learning approach that treats the unlabeled unseen instances in different levels by assessing their reliability and discriminability. Specifically, it formulates the class prediction problem in an iterative refining process, in which each iteration alternates between two paradigms, learning-to-predict and predicting-to-learn. In the learning-to-predict paradigm, the prediction is conducted on the unseen data with the current learned model to select reliable instances for the subsequent learning process; In the predicting-to-learn paradigm, the model is retrained with the feedback reliable instances for the next prediction. In this way, the object classification capacity is progressively reinforced through bootstrapping-based model updating over highly reliable instances.The flowchart of the proposed transductive ZSL approach is illustrated in Fig. 1. In conclusion, the main contributions of this paper are two folds: * To achieve the knowledge transfer from the seen classes data to the unseen classes data, we propose a general dictionary model to simultaneously project the image features and label embeddings into a common latent space, where the class semantic relatedness between different modalities are effectively preserved.* A novel transductive framework is developed for alleviating the domain shift problem in ZSL by formulating the class prediction step in an iterative refining process, in which the domain shift is gradually adapted by retraining a powerful classification model with highly reliable unseen instances. Experimental results show that the proposed transductive strategy can significantly improve the inductive classification model and outperform the state-of-the-art related approaches. The remaining sections are organized as follows. Section 2 describes the related work. Section 3 presents the proposed general dictionary model for achieving the cross-modal knowledge transfer and the transductive framework for addressing the domain shift problem in ZSL. Section 4 provides extensive experiments and evaluations, followed by the conclusion in Section 5.§ RELATED WORK §.§ Knowledge transfer for ZSLThe key idea of ZSL is transferring the knowledge from the seen domain to the unseen one. It relies on constructing a label semantic embedding space where each class can be represented as a vector and the semantic relationships among all classes can be precisely characterized. The most common label embeddings include visual attributes <cit.>, <cit.>, <cit.>, word vectors <cit.>, <cit.>, <cit.>, knowledge mined from the Web <cit.>, <cit.>. Visual attributes are a list of manually specified properties for categories, such as color, shape and presence or absence of a certain body part, which are shared across both the seen and unseen classes. In contrast, semantic word vectors are obtained from a large text corpus in an unsupervised way. With a language model, such as word2vec <cit.> and Glove <cit.>, each class name is embedded into the word vector space, where the class semantic information is defined. Given such label semantic embeddings, the existing approaches of ZSL focus on bridging the class semantic relationships between the instances and the categories with the help of label semantic embeddings. One of the pioneering studies is <cit.>, in which two probabilistic paradigms are proposed, i.e., directed attribute prediction (DAP) and indirected attribute prediction (IAP). DAP takes advantage of the class attributes as the middle layer between the input images and the output class labels. However, in IAP model, the seen classes are taken as the middle layer to connect the visual instances and the unseen classes, where the semantic relationships between seen classes and unseen classes are defined by their corresponding attributes.Considering that the visual instances and label embeddings are embedded in different spaces, recent work addresses ZSL by exploring the semantic relationships between the visual instances and the label embeddings, which has been widely explored in two ways: (1) learning a direct projection function by regressing from image feature space to the label embedding space with regressors <cit.> <cit.> or neural networks <cit.>; (2) projecting the visual features and label embeddings into a latent space, such as CCA <cit.>. Instead of learning two different mapping functions for image feature space and label embedding space, SJE <cit.> and DeViSE <cit.> combined the visual features and label embeddings into a bilinear model to represent the compatibility scores of different modalities and employ a ranking objective to enforce the correct class labels to be ranked higher than any of the other class labels. In order to improve the compatibility learning framework, <cit.> introduced a list of latent variables to learn a collection of mappings with the selection of the latent variable to match the current image-class pair. Taking the class labels into consideration, <cit.> proposed a simpler but more efficient method that associates the visual feature, label embedding and class label into an joint model. As an extension of <cit.>, Qiao et al. <cit.> proposed an ℓ_2,1-norm based objective function which can simultaneously suppress the noisy signal in the textual representation and learn a function to match the textual semantic vectors and visual features.Instead of projecting the visual features into the label embedding space, <cit.> showed that mapping label semantic vectors into the image feature space is desirable to suppress the emergence of hubs in the subsequent nearest neighbor search step. Analogously, <cit.> employed a dictionary learning scheme in which class attributes are considered to be coding coefficients which are used to reconstruct the visual instances. Based on the dictionary learning, Zhang et al. <cit.> proposed a latent probabilistic model to simultaneously project both the visual features and label embeddings into different latent spaces, and then learn a cross-domain similarity matrix for matching different modalities.§.§ Adaptation for domain shift problemDomain shift problem is a common issue in the situations where there are a lot training data in one domain but little to none in another. Traditional domain adaptation approaches are derived for both with <cit.>, <cit.> and without <cit.> requiring label information of the target domain. Since the label information of the unseen domain are not available in ZSL, thus the supervised domain adaptation approaches are not applicable for ZSL. Besides, different from the traditional domain shift problem <cit.>, <cit.>, the domain shift issue in ZSL is mainly due to the disjointness of the seen classes and unseen classes rather than the feature distribution shift. Recently, several work has proposed for mitigating domain shift problem in ZSL with methods ranging from subspace aligning <cit.>, data augmentation <cit.>, <cit.>, self-training <cit.> to hubness correction <cit.>. Transductive zero-shot learning was first considered by Fu et al. <cit.>, in which the unseen data attribute distribution is exploited by averaging the label prototype's k-nearest neighbours. In <cit.>, the domain shift problem was addressed by transductive multi-view hypergraph label propagation (TMV-HLP), in which the manifold structure of the unseen data is exploited to compensate for the the impoverished supervision available from the sparse semantic vector. By using graph-based label propagation to exploit the manifold structure of the unseen data, Rohrbach et al. <cit.> proposed a more elaborate transductive strategy for domain shift problem in ZSL. Different from these approaches, Xu et al. <cit.> proposed a data augmentation strategy by mitigating any available auxiliary dataset to the labeled seen data for training a general model for unseen data. Self-training adaptation <cit.> was a post-processing technique, which is based on adjusting the latent embeddings of unseen classes according to the distribution of all the test instance projections in the latent subspace.§ THE PROPOSED MODEL In this section, we focus on learning a specific classification model for recognizing the unlabeled unseen data. It consists of two parts: i) a general dictionary model is learned with the labeled seen data for initially predicting the unseen data, in which the semantic relatedness between different modalities are preserved by projecting the image features and label embeddings into a common latent space; ii) a transductive framework is presented for mitigating domain shift problem in ZSL by formulating the prediction step in an iterative refining precess, where the classification capacity is progressively reinforced through bootstrapping-based model updating over highly reliable instances.§.§ NotationsSuppose that we collect m labeled instances from M seen classes for training, and each class is associated with a vector embedded in the label embedding space. Denote 𝐗_s=[𝐱_1^s,...,𝐱_m^s]∈ℝ^p×m as the instances available at training stage, where p is the dimensionality of the image feature. And we use 𝐘_s∈{-1,1}^m×M and 𝐀_s∈ℝ^q×M to denote the corresponding ground truth label matrix and label embedding matrix for seen data, respectively. Here, each column of 𝐀_s represents a label embedding vector, and q is the dimensionality of the class label embedding. At testing stage, we are given n instances 𝐗_t=[𝐱_1^t,...,𝐱_n^t]∈ℝ^p×n from N unseen classes, which are disjoint from seen classes. Each unseen class is also associated with a label embedding vector. TABLE 1 shows the main notations used here in after.§.§ The Joint Embedding Dictionary Model (JEDM)For the labeled seen data, conventional dictionary learning models <cit.> aim at learning an effective data representation model from the input data 𝐗_s for classification tasks by exploiting the class label discriminative information of labeled data. Most existing dictionary learning approaches can be formulated under the following framework:{𝐃_s^*,𝐂_s^*,𝐖^*}=min_𝐃_s,𝐂_s,𝐖𝐗_s-𝐃_s𝐂_s_F^2+λ𝐂_s_p +Ψ(𝐖,𝐂_s,𝐘_s),where λ is a weight parameter, 𝐘_s denotes the class label matrix of the instances from input data matrix 𝐗_s, 𝐃_s is the dictionary matrix to be learned, and 𝐂_s is the representative coding matrix of 𝐗_s over 𝐃_s, and 𝐖=[𝐰_1,...,𝐰_M]is the classification matrix for seen classes, 𝐰_i is the classifier for class i, 1≤i≤M; 𝐂_s_p is the ℓ_p-norm regularizer on 𝐂_s, 𝐗_s-𝐃_s𝐂_s_F^2 is the reconstruction error term ensuring the representative ability of 𝐃_s, ·_F denotes the matrix Frobenious norm, and Ψ(𝐖,𝐂_s,𝐘_s) is a discriminative function, which ensures the discriminative ability of 𝐂_s.With Eq. (1), the shared dictionary matrix and classification matrix can be trained with labeled seen classes. However, no labeled data are available for unseen classes such that the classification parameters for unseen classes cannot be obtained directly. We thereby need to transfer the knowledge exploited from the labeled seen domain to the unseen domain. As previous work has indicated that the properties of a class can be well characterized by its corresponding label embedding, thus it is reasonable to assume that the classifier of a class can be derived from its label embedding. Thus, we replace the classification model 𝐰 with: 𝐕𝐚, where 𝐚 is the label embedding and 𝐕 is the compatibility matrix shared both the seen and unseen classes. Intuitively, the compatibility matrix aligns the semantic consistency between the visual instances and the label embeddings. Once obtaining the compatibility matrix 𝐕, the classification parameter 𝐰_i for unseen class i can be obtained by 𝐰_i = 𝐕𝐚_i. To this end, the remaining problem is to learn the compatibility matrix with the labeled seen data. Based on this idea, we propose to learn such a compatibility matrix together with the seen dictionary matrix. Formally, we get the Joint Embedding Dictionary Model (JEDM) for ZSL, {𝐃_s^*,𝐂_s^*,𝐕^*}=min_𝐃_s,𝐂_s,𝐕𝐗_s-𝐃_s𝐂_s_F^2+α𝐂_s^T𝐕𝐀_s-𝐘_s_F^2+β𝐕𝐀_s_F^2, d_i_2^2⩽1,where α and β are two parameters to trade-off different terms, which can be determined via the cross-validation.The first term of Eq. (2) is the reconstruction error, which compresses the visual features in a more representative latent space, and the second term incorporates the latent features, label embeddings and class labels into a joint framework for preserving the semantic relatedness across different modalities. By enforcing the visual latent features being close to the corresponding label embeddings while be far away from that of the other classes, this term is subject to exploit the semantic discriminant information across different modalities. The last term is a regularizer term.We next introduce the optimization process to solve the objective function in Eq. (2). Eq. (2) is not convex for 𝐃_s, 𝐂_s and 𝐕 simultaneously but is convex for each of them individually. Therefore, the optimization can be done alternatively between the following two steps.1). Fix 𝐃_s, 𝐕 and solve for 𝐂_s.𝐂_s^*=min_𝐂_s𝐗-𝐃_s𝐂_s_F^2+α𝐂_s^T𝐕𝐀_s-𝐘_s_F^2. This sub-problem is a standard least square problem; so we take the derivative of Eq. (3) with respect to 𝐂_s and make it equal to zero, which has the following closed-form solution:𝐂_s^*= (𝐃_s^T𝐃_s+α𝐕𝐀_s𝐀_s^T𝐕^T)^-1(𝐃_s^T𝐗_s+ α𝐕𝐀_s𝐘_s^T). 2). Fix 𝐂_s and solve 𝐕 and 𝐃_s. Since 𝐕 and 𝐃_s are independent, thus they can be solved separately,𝐕^*= min_𝐕α𝐂_s^T𝐕𝐀_s-𝐘_s_F^2+β𝐕𝐀_s_F^2. The closed-form solutions of 𝐕 can be obtained as:𝐕^*= (𝐂_s𝐂_s^T+γ𝐈)^-1(𝐂_s𝐘_s𝐀_s^T)(𝐀_s𝐀_s^T)^-1, β = αγ. The optimal 𝐃_s can be obtained by introducing a variable 𝐑: min_𝐃_s,𝐑𝐗_s-𝐃_s𝐂_s_F^2 s.t. 𝐃_s=𝐑, r_i_2^2⩽1. And the solution of Eq. (7) can be obtained by the alternating direction method of multipliers (ADMM) algorithm. In each iterative step, 𝐂_s and 𝐕 are obtained with closed-form solutions and the optimization of 𝐃_s is obtained with the ADMM algorithm, which converges rapidly. The iterative step stops when the difference between the variations in two adjacent iterations is less than a threshold.Once 𝐃_s^* and 𝐕^* are obtained, the compatibility score s(𝐱,𝐚_c) of a test instance 𝐱 over the unseen class c is estimated in the common latent space:s(𝐱,𝐚_c) = 𝐱^T𝐃_s^*𝐕^*𝐚_c,where 𝐚_c is the label embedding of the c-th unseen class, (𝐃_s^*)^T𝐱 is the approximate embedding of visual instance in the latent space, while 𝐕^*𝐚_c is the prototype of unseen class c, which is the latent embedding of 𝐚_c. Thus, ZSL is achieved by resorting to the largest compatibility score with respect to the unseen label embeddings.c^*=max_cs(𝐱,𝐚_c).§.§ Self-Labeled strategyLike most inductive ZSL approaches, the classification model which is learned only with the labeled seen data will generalize poorly on the unseen data due to that the class distribution of the seen domain is different from that of the unseen domain. To address this domain shift problem, we formulate the prediction step of ZSL in an iterative refining process, in which each iteration alternates between two paradigms, learning-to-predict and predicting-to-learn. With the model learned with the labeled seen data, the labels of the unseen data are previously predicted. This is the first learning-to-predict paradigm. Considering that the instances that have higher compatibility scores are more reliable to be correctly-predicted, it is reasonable to annotate these reliable instances as labeled data for unseen classes. With these feedback reliable instances, the unseen-specific model is retrained for the subsequent prediction step. This is a predicting-to-learn paradigm. Repeat this precess, the domain shift is progressively adapted in a confident way. The remaining problem is how to select reliable instances from unseen data. In this paper, we introduce a simple strategy to select instances from unseen data as labeled data. Specifically, for each unseen class, the test instances can be ranked according to the compatibility scores over their corresponding predicted unseen class. We then set a self-labeled rate δ to annotate the reliable instances as labeled data. For example, suppose that n_i instances are predicted into the unseen class y_i, [n_i×δ] instances are selected according to their ranking scores to the corresponding class, [·] is the rounding operation. Clearly, the self-labeled strategy is under a transductive setting.It should be noted that the self-labeled strategy can be seamlessly integrated into the various existing ZSL approaches. As shown in Fig. 2, the seen data are used for learning a previously classification model for initially predicting the unseen data, and then an iterative strategy is used for refining the learned model. At each iteration, only reliable instances from the unseen data are selected for refining the classification model. As more instances are selected, a powerful specific model is learned for unseen classes. §.§ Transductive Self-training Dictionary (TSTD) modelBy integrating the self-labeled strategy into the previously proposed JEDM, we obtain the final Transductive Self-Training Dictionary (TSTD) model. For the first learning-to-predict paradigm, the class labels of unseen data are previously predicted with the proposed JEDM. And then the classification model is retrained by the unseen data themselves. In each predicting-to-learn paradigm, two baselines are introduced to ensure that the refined model is more suitable for unseen classes. The first one is that the current learned dictionary model 𝐃_t is close to the previously optimal one 𝐃^*. Since the previously learned model is used to align different spaces, the currently learned model should refine the previous one by a fine step rather than adjusting with a large range. The other one is that the learned model ensures that the latent embeddings of the self-labeled instances are close to their predicted label prototypes in the latent space. Thus the objective function is defined as follows: {𝐃_t^*,𝐂^*}=min_𝐃_t,𝐂𝐗-𝐃_t𝐂_F^2+λ𝐕^*𝐀-𝐂_F^2 +μ𝐃_t-𝐃^*_F^2,where 𝐗 is the collected set which contains the selected self-labeled instances, 𝐕^* is the previous learned compatibility matrix shared both the seen domain and unseen domain. 𝐃_t is the currently learned dictionary matrix for unseen classes, 𝐂 is the latent embeddings of the self-labeled instances and 𝐀 is the predicted label embedding matrix that self-labeled instances correspond to. Since each unseen class is associated with a label semantic vector, 𝐀 is easily inferred by the predicted class labels. λ and μ are trade-off parameters. In our model, the latent embeddings of the input unseen data are enforced to be close to their corresponding predicted classes' label latent embedding in the common latent space, i.e., 𝐕^*𝐀-𝐂_F^2. In the following, we design an alternating optimization method to solve Eq. (10).When 𝐃_t is fixed, the optimization problem becomes:𝐂^* = min_𝐂𝐗-𝐃_t𝐂_F^2+λ𝐕^*𝐀-𝐂_F^2,which leads to a closed-form solution:𝐂^* =(𝐃_t^T𝐃_t+λ𝐈)^-1(𝐃_t^T𝐗+λ𝐕^*𝐀). With the fixed 𝐂, the optimal 𝐃_t^* can be easily solved by:𝐃_t^*=min_𝐃_t𝐗-𝐃_t𝐂_F^2+μ𝐃_t-𝐃^*_F^2. This is a standard least squares problem, and we have the optimization solution:𝐃_t^* =(𝐗𝐂^T+μ𝐃^*)(𝐂𝐂^T+μ𝐈)^-1. With the currently learned dictionary matrix 𝐃_t^*, the unseen data are revisited with Eq. (9). With the latest predicted results, we enlarge the self-labeled rate δ to incorporate more reliable instances for training. Repeat this refining process until all the unseen data are selected. Specifically, the values of self-labeled rate δ are successively selected from {0.4,0.6,0.8,1} in our experiments. The TSTD process is summarized in Algorithm 1.§.§ Further analysis With the learned dictionary matrix 𝐃_s^* and the compatibility matrix 𝐕^* from the seen data, the unseen instances and the label embeddings of unseen classes can be embedded into a latent space together. We visualize them with t-SNE approach, as illustrated in Fig. 3. We can observe that the projections of most visual instances from the same class are distributed around the corresponding class prototypes in the latent space. It is easy to conclude that the instances that are close to the corresponding class semantic prototypes tend to be classified correctly. In contrast, the instances that are farther away from the corresponding class prototypes tend to be classified into the wrong classes. Thus it is natural to annotate the instances that are close to the corresponding class prototypes as labeled data, which eliminates the issue that no training samples are available for unseen classes. Fixing the prototypes of unseen classes, the embeddings of unseen instances are gradually adjusted by retraining the embedding function with the reliable instances, and thus the domain shift issue in ZSL alleviates. The mechanism of the proposed transductive strategy is borrowing the knowledge from the seen classes to teach unseen data, and then learning a specific model with the unseen data by themselves in a word.§.§ Complexity AnalysisIn this section, we analyze the computational complexity of TSTD and the convergence of the proposed JEDM separately.Computational Complexity. In the training phase of JEDM, 𝐃_s, 𝐂_s and 𝐕 are updated alternatively. In each iteration, the time complexities of updating 𝐂_s and 𝐕 in Eq. (4) and Eq. (6) are O(mpd + d^3 + d^2m) and O(dmMq + q^3 + q^2dm + d^3 + d^2qM), respectively. As for the optimization of updating 𝐃_s, the time cost is about O(K(pmd + d^3 + d^2p + p^2d)), where K is the iteration number in ADMM algorithm. We have experimentally found that the ADMM algorithm converges with less than 20 iterations. In the domain adaptation phase of TSTD, 𝐂 and 𝐃_t are also updated alternatively. In each iteration, the time complexities of updating 𝐂 and 𝐃_t are O(npdδ + d^3 + d^2nδ) and O(dpnδ + d^3 + d^2p), respectively. Given that M≪d, q≪d, and m, p, n, d are in the same order of magnitude and our algorithm converges with a few iterations, the over time cost of our algorithm is O(d^3). It is worth noting that the dominant operation of our algorithm is matrix multiplication, which can greatly accelerate the training process. Convergence. We conduct empirical study on the convergence property using Animal with Attribute (AwA) with attributes as label semantic vectors. We set the hype-parameters α and β both as 0.1. The train/test split provided by the dataset is used accordingly. As Fig. 4 shows, the cost function of JEDM descends dramatically and converges with only 10 iterations, which clearly indicates the efficiency of the proposed JEDM. § EXPERIMENTSIn this section, we do a set of experiments to demonstrate the superiority of the proposed approaches. Firstly, we detail the datasets and settings for the experiments, and then compare the proposed JEDM with the state-of-the-art inductive ZSL approaches. Then, the effectiveness of the proposed self-training strategy is evaluated, followed by the comparison results about TSTD and the state-of-the-art transductive ZSL approaches.§.§ Datasets and Settings Datasets. To evaluate the effectiveness of the proposed approaches, we conduct extensive experiments on three benchmark datasets. (a). Animal with Attribute (AwA) <cit.> consists of 30,475 animal images from 50 different classes, and each class is associated with a 85-dimensional attribute vector. (b). Caltech-UCSD Bird2011 (CUB) <cit.> is a fine-grained dataset which contains 11,788 images from 200 bird subspecies, and a 312-dimensional attribute vector is provided for each class. (c). SUN Attribute <cit.> contains 717 scene categories annotated by 102 attributes, and each class has 20 images. For the seen/unseen class split, we use the standard 40/10 split setting for AwA dataset <cit.>. For CUB dataset, we follow the same 150/50 split in <cit.>. And for SUN dataset, we use 707 classes as the seen domain and 10 classes as the unseen domain, the same as that in <cit.>. The statistics for the three datasets are shown in TABLE 2.Visual representation. In our experiments, we use the vgg-verydeep-19 (denoted as VGG for short) features provided by those datasets for representing the visual instances.Label semantic embedding. In this paper, we explore the visual attributes andword vectors as label embedding space for AwA and CUB datasets. Meanwhile, only visual attributes are used for SUN dataset to be comparable with the existing practices in the literature.Besides, there are four hyper-parameters α, β, λ and μ in our proposed TSTD, α and β are two parameters in the JEDM and λ and μ are in the refining model. We select their best values with a 5-fold cross-validation (CV) strategy, where 20% of the seen classes are held out for validation and the remaining for training. Once the parameters are fixed, all seen classes are then trained together for the final model. All the parameters are selected from {0.01,0.1,1,10,100}. In all the experiments, the classification performances are evaluated with the average per-class top-1 accuracy. The average running time of our Matlab implementation is about 0.01ms per image on a desktop with an Intel Core i7-4790K processor and 32G RAM. §.§ Comparative results of JEDM In order to evaluate the effectiveness of the proposed JEDM, we conduct two experiments according to the types of label embedding space.We first take attributes as semantic vectors for classes. In this experiment, six state-of-the-art approaches are selected for comparison. For descriptive convenience, they are respectively referred to as DAP (Direct Attribute Prediction <cit.>), SJE (Structal Joint Embedding <cit.>), LatEm (Latent Embeddings <cit.>), ESZSL (Embarrassing Simple Zero-Shot Leaning <cit.>), SC (Synthesized Classifiers <cit.>) and JLSE (Joint Latent Similarity Embedding <cit.>). These selected competing methods are all inductive approaches. The results of the comparative methods are all from the original papers except <cit.>, which is obtained with the published codes under the same setting as ours. The results are summarized in TABLE 3, where `-' indicates that these methods were not tested on the datasets in their original work.From TABLE 3, we can observe that JEDM is comparable with the state-of-the-art approaches. More specifically, in the AwA dataset, JEDM achieves an improvement of 19.0% against the baseline method DAP <cit.> and beats the other competitors expect for JLSE <cit.>, which projects both modalities into different latent spaces. It is a more complicated model. For CUB dataset, our approach works better than others except for <cit.> and <cit.>. <cit.> tackles ZSL with exploiting the manifold structure to align the semantic space, which behaves robust for the fine-grained dataset. While <cit.> takes a more powerful visual feature as the input, which attributes to the fact that <cit.> works better than JEDM. Since the SUN dataset is less popular than the above two, only three recent approaches are selected for comparison. From the results, we can find that the proposed JEDM outperforms the previously published approaches by a large margin. Specifically, it outperforms DAP <cit.>, ESZSL <cit.> and <cit.> in 14%, 4% and 3.2% gains, respectively. Besides, it is found that classification performances of JEDM outperform that of ESZSL, which is similar to the proposed JEDM. The most difference between our method and ESZSL is that JEDM projects the visual features into a more discriminative latent space with a dictionary framework, while ESZSL uses the visual feature as input directly and designs an elaborated regularizer. The comparative results demonstrate the effectiveness of the dictionary representation. In the second experiment, the word vector space is taken as the label embedding space. Thanks to the recent advances in unsupervised neural language modeling <cit.> <cit.>, each word in a text corpus can be effectively embedded in a textual semantic space, where each word is represented as a semantic multi-dimensional vector. Specifically, we use word2vector model <cit.> to train a skip-gram language model on the latest Wikipedia corpus to extract 1000-dimensional word vector for each class from AwA and CUB datasets. Five wordvector-based approaches are selected for comparison, as illustrated in TABLE 4. From the results, we can find that JEDM has an impressive improvement in AwA dataset. Specially, JEDM outperforms CCA <cit.>, SJE <cit.>, LatEm <cit.> and ESZSL <cit.> in 5.9%, 20.3%, 10.4% and 4.1% gains, respectively. Meanwhile, it also achieves a competitive result in CUB dataset, which is only 0.9% lower than that of the previous best reported LatEm <cit.>.§.§ Evaluation of self-training strategyIn this section, we conduct a set of experiments on AwA and CUB datasets to demonstrate the generality and the effectiveness of the proposed self-training strategy. In specific, two typical ZSL approaches are selected for being integrated with the self-training strategy. These approaches are CCA and ESZSL, both of which have a closed-form solution. For descriptive convenience, we add a postfix -ST to the name of the approaches for representing the corresponding approaches with the self-training strategy. Specifically, the approach that JEDM integrates self-training strategy is called TSTD in this paper. In implementation, the baselines introduced in TSTD are also suitable for CCA-ST and ESZSL-ST. The comparative results are provided in TABLE 5.From the results, we can observe that the proposed transductive self-training strategy can not only improve the performance of the proposed JEDM with a large margin, but also boost other approaches substantially on different datasets with different semantic vectors. Specifically, on AwA dataset, the transductive self-training strategy helps JEDM improve 13.8% and 19.7% in gains with attribute and word vector as label embedding space, respectively. It should be noted that TSTD achieves 91.2% classification accuracy on AwA dataset with word vector as semantic space, which is even better than those attribute-based approaches. In contrast to AwA dataset, the improvement range of the transductive self-training strategy is smaller on CUB dataset. The reason is that the CUB dataset is a fine-grained dataset and its classification performance of JEDM is much lower than that of AwA dataset, such that the self-labeled set contains many fake instances that may spoil the classification model. Even so, the proposed transductive self-training strategy helps JEDM improve 10.6% and 3.0% absolute percentage points with visual attribute and word vector as semantic space, respectively.§.§ Comparison results of TSTD We also compare our TSTD approach with the state-of-the-art transductive ZSL approaches. TABLE 6 shows the comparison results. We can observe that the proposed TSTD has an overwhelming superiority to the competitors. Specifically, the proposed domain adaptation strategy on JEMD model achieves 90.3% classification accuracy on AwA dataset with visual attribute, which outperforms <cit.>, <cit.>, <cit.> and <cit.> in 9.8%, 11.8%, 14.7% and 2.4% gains, respectively. On CUB dataset, it achieves 58.2% classification accuracy, which improves 10.3%, 17.6% and 4.7% over <cit.>, <cit.> and <cit.>, respectively. Specifically, TMV-HLP and SMS are two transductive methods that integrate the seen data and unseen data together for training a general model for all classes. And <cit.> explores the label information of unseen data with an unsupervised cluster-based approach. However, <cit.> and our self-training strategy focus on re-training a suitable model for unseen classes. The main difference between these two strategies is that <cit.> uses an unsupervised model to exploit the structure information of the unseen domain while ours relies on a bootstrapping-based model updating over highly reliable instances to progressively reinforce the classification capacity.§.§ Evaluation of the self-labeled rate We next conduct a set of experiments to evaluate the influences of self-labeled rate δ to the maturity of the learned model. As illustrated in Fig. 5, we can observe that the performances increase steadily with the increase of δ and achieve their peaks when δ=0.8 on AwA dataset with different types of label semantic embeddings. This indicates that with the increase of δ, more correct self-labeled instances are selected for refining the classification model, thus the classification capacity is progressively reinforced. In contrary, on CUB dataset, the performances achieve their peaks when δ=0.6and δ=0.4 with attributes and word vector, respectively. And the performances decrease with increase of δ. This is due to the classification performances on CUB unseen data with the learned model are poor (47.6% and 30.9% with attribute and word vector respectively), and thus with increase of δ, more false instances are selected as self-labeled data, which may spoil the learned model. The curves of AwA dataset in Fig. 5 (a) also verify this explanation.§ CONCLUSIONS In this paper, we proposed a bidirectional mapping based scheme to address ZSL. It formulates the semantic interactions between image feature space and label embedding space in a general dictionary model by simultaneously projecting the image features and label embeddings into a common latent space. The experimental results demonstrated that the proposed approach achieves the state-of-the-art performance on three benchmark datasets. 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"authors": [
"Yunlong Yu",
"Zhong Ji",
"Xi Li",
"Jichang Guo",
"Zhongfei Zhang",
"Haibin Ling",
"Fei Wu"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170327013638",
"title": "Transductive Zero-Shot Learning with a Self-training dictionary approach"
} |
Existence of solutions for n^th-order nonlinear differential boundary value problems by means of new fixed point theorems Alberto Cabada andLorena Saavedra Instituto de Matemáticas,Facultade de Matemáticas, Universidade de Santiago de Compostela,Santiago de Compostela, Galicia, Spain [email protected], [email protected] December 30, 2023 ================================================================================================================================================================================================================================= This paper is devoted to prove the existence of one or multiple solutions of a wide range of nonlinear differential boundary value problems. To this end, we obtain some new fixed point theorems for a class of integral operators. We follow the well-known Krasnoselskiĭ's fixed point Theorem together with two fixed point results ofLeggett-Williams type. After obtaining a general existence result for a one parameter family of nonlinear differential equations, are proved, as particular cases, existence results for second and fourth order nonlinear boundary value problems. 2010 Mathematics Subject Classification: 34B15, 34B27, 34B18, 47H10Keywords: Green's functions, fixed point theorems, nonlinear boundary value problems § INTRODUCTION The use of different kind of fixed point theorems has been shown as a very useful tool to obtain the existence of solutions of nonlinear boundary value problems.For instance, in <cit.> it is proved that the following second order systemu_1”(t)+λ_1 f_1(u_1(t),u_2(t))=0 , t∈(0,1) ,u_2”(t)+λ_2 f_2(u_1(t),u_2(t))=0 , t∈(0,1) ,u_1'(0)=u_1(1)+u_1'(1)=0 ,u_2'(0)=u_2(1)+ε u_2(η)=0 , η , ε∈(0,1) ,has a solution for every λ_1, λ_2>0, by applying some previously obtained fixed point results on a related system of integral operators.In <cit.>, under suitable conditions for the functions f, g and λ∈ℝ, it is proved the existence of at least two strictly positive solutions of the second order boundary value problemu”(t)+λ g(t) f(u(t))=0 , t∈(0,1) ,u'(0)=σ u'(1)+u(η)=0 , η∈[0,1] , which models the behavior of a thermostat. Again, the existence result is obtained by applying a fixed point theorem.In <cit.>, it is studied a fourth order boundary value problem coupled with the cantilever beam boundary conditions:u^(4)(t)=f(t,u(t)) , t∈(0,1) ,u(0)=u'(0)=u”(1)=u”'(1)=0 . In such a work, two combined techniques on the existence of solutions are used: the critical and the fixed point theory. In <cit.>, it is obtained a result in the line of the Leggett-Williams fixed point theorem (see <cit.>), that guarantees the existence of at leasta positivefixed point on different sets defined by means of suitable functionals. As a direct application of this result, it is proved the existence of solution of the following second order nonlinear differential boundary value problem:u”(t)+f(u(t))=0 ,t∈(0,1) ,u(0)=u'(1)=0 . Moreover, in <cit.>, by means of a new fixed point theorem proved in that reference, a different existence result for this problem has been obtained.In <cit.> it is obtained an extension of the Leggett-Williams fixed point theorem given in <cit.> and, as an application, it is obtained a result which ensures the existence of multiple solutions of the following third order boundary value problem:u”'(t)+f(t,u(t),u'(t),u”(t))=0 , t∈(0,1) ,u(0)=∑_i=1^m-2k_i u(ξ_i) ,u'(0)=u'(1)=0 ,where 0<ξ_1<ξ_2<⋯<ξ_m-2<1, k_i∈ℝ^+ for i=1,…,m-2 and 0<∑_i=1^m-2k_i<1. In <cit.>, as an application of the Leggett-Williams fixed point Theorem, it is proved the existence of at least one solution for the problem:u^(4)(t)=f(u(t)) , t∈(0,1) ,u(0)=u'(0)=u(1)=u'(1)=0 ,under suitable conditions of f.In this paper, as an application of the results here presented, on Section 5 we prove existence and multiplicity results for a fourth order problem with the same clamped beam boundary conditions. However, in our case, we consider a non-autonomous nonlinear part f(t,u). Moreover, we prove, under suitable conditions of f, the existence of at least two or three solutions of the considered problem.In addition, in that section we study a family of second order problems couped with the Dirichlet boundary conditions:u”(t)+B u'(t)+f(t,u(t))=0 , t∈(0,1) ,u(0)=u(1)=0 ,where B∈ℝ.In fact, such results are included in a more general framework, which is delivered to ensure general existence and multiplicity results of a one parameter family of n^th-order boundary value problems given by(-1)^n-k T_n[M] u(t)=f(t,u(t)) , t∈ I≡ [a,b] ,coupled with the so-called (k,n-k) boundary conditionsu(a)=u'(a)=⋯=u^(k-1)(a) =0 ,u(b)=u'(b)=⋯=u^(n-k-1)(b) =0 ,where k∈{1,…,n-1} andT_n[M] u(t)=u^(n)(t)+p_1(t) u^(n-1)(t)+⋯+(p_n(t)+M) u(t) , t∈ I , It is well-known that the solutions of problem (<ref>)-(<ref>) are given as the fixed points of the integral operator ℒ_1u(t)=∫_a^b(-1)^n-kg_M(t,s) f(s,u(s)) ds , where g_M(t,s) is the associated Green's function of the operator T_n[M] in the space related to the given boundary conditions.Thus, in order to find the fixed points of operator ℒ_1, we previously study, in Sections 3 and 4, the existence of one or multiple positive fixed points of an integral operator defined as follows: [ ℒ C(I)⟶ C(I);u⟼ ℒ u(t):=∫_a^bG(t,s) f(s,u(s)) ds , ] where [f I×ℝ⟶ℝ;(t,u)⟼ f(t,u) , ] is a continuous function satisfying that f(t,u)⩾ 0 for all (t,u)∈ I×ℝ^+ and [ G I× I⟶ℝ;(t,s)⟼ G(t,s) , ] is an integral kernel. To this end, we impose the following hypothesis on the kernel G: (Pg_1) There exist , k_1 and k_2 continuous functions on I such that (s)>0 for all s∈(a,b), 0<k_1(t)⩽ k_2(t) for all t∈(a,b) and (s) k_1(t)⩽ G(t,s)⩽(s) k_2(t) ,∀(t,s)∈ I× I . It is important to note that(see for instance<cit.> and references therein) there are obtained several results for the existence of one or multiple fixed points of integral operators by imposing similar conditions to property (Pg_1) to the kernel G.In this paper, we use the well-known Krasnoselskiĭ's fixed point Theorem, collected in <cit.>. Moreover, in order to obtain the existence of two or three fixed points we use two results due to Avery and Henderson, <cit.>, and Avery, <cit.>, respectively. The structure whatwe follow is the one given in <cit.>, where these results are used to prove the existence of one or multiple solutions for a problem on time scales.Our final purpose is to prove the existence of solutions of the problem (<ref>)-(<ref>). Thus, we need to guarantee that the related Green's function satisfies the property (Pg_1). In <cit.>, by means of spectral theory, it has been obtained a characterization of the values of the parameter M for which the Green's function, related to the (k,n-k) boundary conditions, has constant sign. A fact which, in this case, also implies that such a function verifies the property (Pg_1). Hence, as an application of the previously obtained fixed point theorems, we will prove the existence of one or multiple positive solutions for (k,n-k) nonlinear boundary value problems.In addition, this property can be extended for many different boundary conditions. For instance, in <cit.>, a fourth order problem coupled with thesimply supported boundary conditions, u(a)=u(b)=u”(a)=u”(b)=0, is studied. Thus, the results here shown can also be applied in such a case. This paper is structured as follows: innext section, wedescribe the studied problem and show some preliminary results which are used along the paper. In Section 3, we obtain some results that ensure the existence of one or multiple fixed points by using the Krasnoseslkiĭ's fixed point Theorem given in <cit.>. Then, in Section 4, following the results ofAvery and Henderson, <cit.>, and Avery, <cit.>, we obtain the existence of at least two or three fixed points, respectively.Finally, in Section 5,as an application of these fixed point theorems, we deduce the existence and multiplicity of solutions of problem (<ref>)-(<ref>). Moreover, as particular cases, second and fourth order boundary value problems are considered. § DESCRIPTION OF THE PROBLEM AND SOME PREVIOUS FIXED POINT EXISTENCE RESULTS The aim of this paper is to study the existence of some fixed points of the integral operator described in (<ref>) in an appropriate cone. First, let us define the concept of cone. Let ℬ be a real Banach space. A nonempty closed convex set 𝒫⊂ℬ is called a cone if it satisfies the following two conditions: * λ x∈𝒫 for all x∈𝒫 and λ⩾ 0. * If x∈𝒫 and -x∈𝒫, then x=0. In the sequel, we describe the cone where the fixed points are found, as well as some constants which are used along the paper. Let us consider a subinterval I_1=[a_1,b_1]⊂I such that k_1(t)>0 for all t∈ I_1. Then, we denote:K_1=max_t∈ Ik_1(t)>0 , m_1=min_t∈ I_1k_1(t)>0 , K_2=max_t∈ Ik_2(t)>0 . Consider the Banach space ℬ=C(I) coupled with the norm u_∞=max_t∈ I|u(t)| ,and the cone𝒫={ u∈ℬ | u(t)⩾k_1(t)/K_2 u_∞ , t∈ I} .In the sequel, to make the paper more readable, we show some preliminary results which will be used along the paper. First, let us consider the Krasnoselskiĭ's fixed point Theorem, <cit.>: Let ℬ be a Banach space, 𝒫⊂ℬ be a cone, and suppose that Ω_1, Ω_2 are bounded open balls of ℬ centered at the origin, with Ω_1⊂Ω_2. Suppose further that ℒ𝒫∩(Ω_2\Ω_1)→𝒫 is a completely continuous operator such that either ℒu⩽ u , u∈𝒫∩∂ Ω_1 ,andℒu⩾ u , u∈𝒫∩∂ Ω_2 , or ℒu⩾ u , u∈𝒫∩∂ Ω_1 ,andℒu⩽ u , u∈𝒫∩∂ Ω_2 , holds. Then ℒ has a fixed point in 𝒫∩ (Ω_2\Ω_1).A map α is said to be a nonnegative continuous concave functional on a cone 𝒫 of a real Banach space ℬ if α𝒫 → [0,+∞) is continuous andα(t x+(1-t) y)⩾ t α(x)+(1-t) α(y) ,∀ x , y∈𝒫 , t∈[0,1] . Similarly, a map β is said to be a nonnegative continuous convex functional on a cone 𝒫 of a real Banach space ℬ if β𝒫 → [0,+∞) is continuous andβ(t x+(1-t) y)⩽ t β(x)+(1-t) β(y) ,∀ x , y∈𝒫 , t∈[0,1] .Now, let us consider β, γ and θ, nonnegative continuous convex functionals on the cone 𝒫, and α and ψ, nonnegative concave functionals on 𝒫.For nonnegative real numbers d, p and q, we define the following subspaces of the cone 𝒫:𝒫(γ, r) ={ u∈𝒫 | γ(u)<r} , 𝒫(γ,α,p,r) ={ u∈𝒫 | p⩽α(u) , γ(u)⩽ r} , 𝒬(γ,β,d,r) ={ u∈𝒫 | β(u)⩽ d , γ(u)⩽ r} , 𝒫(γ,θ,α,p,q,r) ={ u∈𝒫 | p⩽α(u) , θ(u)⩽ q , γ(u)⩽ r} . In the sequel we introduce a result, proved in <cit.>, which ensures the existence of two fixed points on the cone 𝒫. Let 𝒫 be a cone in a real Banach space ℬ. Let α and γ be increasing and nonnegative continuous functionals on 𝒫. Let θ be a nonnegative continuous functional on 𝒫 with θ(0)=0 such that, for some positive constants r and M, α(u)⩽θ(u)⩽γ(u)andu⩽ M α(u) ,for allu∈𝒫(α,r) . Assume that there exist two positive numbers p and q with p<q<r such that θ(λ u)⩽λ θ(u) ,for all0⩽λ⩽ 1and u∈∂𝒫(θ,q) . Suppose that ℒ𝒫(α,r)→𝒫 is a completely continuous operator satisfying * α(ℒ u)>r for all u∈∂𝒫(α,r), * θ(ℒ u)<q for all u∈∂𝒫(θ,q), * 𝒫 (γ,p)≠∅ and γ(ℒ u)>p for all u∈∂𝒫(γ,p). Then, ℒ has at least two fixed points u_1 and u_2 such thatp<γ(u_1) , withθ(u_1)<q ,andq<θ(u_2) , withα(u_2)<r .Finally, we introduce a result, proved in <cit.>, which warrants the existence of three fixed points of the operator ℒ on the cone 𝒫. Let 𝒫 be a cone in a real Banach space ℬ, and let r and M be positive numbers. Assume that α and ψ are nonnegative, continuous andconcave functionals on 𝒫, and γ, β and θ are nonnegative, continuous and convex functionals on 𝒫 with α(u)⩽β(u)andu⩽ M γ(u) ,∀u∈𝒫 (γ,r) . Suppose that ℒ𝒫(γ,r)→𝒫(γ,r) is a completely continuous operator and there exist nonnegative numbers h, d, p, q, with 0<d<p, such that: * { u∈𝒫(γ,θ,α,p,q,r) | α(u)>p}≠∅ and α(ℒ u)>p for u∈𝒫(γ,θ,α,p,q,r), * { u∈𝒫(γ,β,ψ,h,d,r) | β<d}≠∅ and β(ℒ u)<d for u∈𝒫(γ,β,ψ,h,d,r), * α(ℒ u)>p for u∈𝒫(γ,α,p,r) with θ(ℒ u)>q, * β(ℒ u)<d for u∈𝒬(γ,β,d,r) with Ψ(ℒ u)<h. Then ℒ has at least three fixed points u_1, u_2, u_3∈𝒫 (γ,r) such thatβ(u_1)<d , p<α(u_2)andd<β (u_3)withα(u_3)<p .§ EXISTENCE OF FIXED POINTS BY MEANS OF KRASNOSELSKIĬ FIXED POINT THEOREM The aim of this section consists on ensuring the existence of at least a fixed point of operator ℒ, defined in (<ref>). Such an existence will follow as an application of Theorem <ref>.Let us consider the following conditions on f: (H_1) There exists p>0 such that f(t,u)⩽pK_2 ∫_a^b(s) ds ,∀t∈ I , ∀u∈[0,p] . (H_2) There exists q>0 such that f(t,u)⩾K_2 uK_1 ∫_a_1^b_1k_1(s) (s) ds ,∀t∈ I_1 , ∀u∈[ m_1/K_2 q,q]. Following the same steps as in <cit.>, we prove the following result Suppose that there exist two positive numbers p≠ q such that condition (H_1) is satisfied with respect to p and condition (H_2) is satisfied with respect to q. Then, provided that the integral kernel G satisfies (Pg_1), operator ℒ, defined in (<ref>), has a fixed point, u∈𝒫, such that u_∞ lies between p and q. First let us see that ℒ(𝒫)⊂𝒫. Let u∈𝒫, we have: ℒ u(t):= ∫_a^b G(t,s) f(s,u(s)) ds⩾∫_a^b k_1(t) (s) f(s,u(s)) ds = k_1(t)K_2∫_a^bK_2 (s) f(s,u(s)) ds⩾k_1(t)K_2∫_a^bsup_t∈ I{ G(t,s)} f(s,u(s)) ds ≥ k_1(t)K_2sup_t∈ I{∫_a^bG(t,s) f(s,u(s)) ds} =k_1(t)K_2ℒ u_∞ ,∀ t∈ I . Thus,since f is a continuous function on I×ℝ, ℒ u∈𝒫 and ℒ𝒫→𝒫 is a completely continuous operator.Now, let us define the open balls centered at the origin as follows:Ω_p={ u∈ C(I) | u_∞<p}andΩ_q={ u∈ C(I) | u_∞<q} . From (Pg_1) and the positiveness of f for all u∈𝒫, we have that the following inequality is fulfilled.ℒ u_∞ =sup_t∈ I{∫_a^b G(t,s) f(s,u(s)) ds}⩽ K_2 ∫_a^b(s) f(s,u(s)) ds . On the other hand, for u∈𝒫∩∂Ω_p, we have that u_∞=p, so, from (<ref>) and (H_1), we have:ℒ u_∞⩽ K_2 ∫_a^b(s) f(s,u(s)) ds⩽ K_2∫_a^b(s)pK_2 ∫_a^b(s) ds ds=p=u_∞ . Thus, ℒ u_∞⩽u_∞ for all u∈𝒫∩∂Ω_p.Now, using(Pg_1) again, we have, for all u∈𝒫:ℒ u _∞ = sup_t∈ I{∫_a^b G(t,s) f(s,u(s)) ds}⩾sup_t∈ I{∫_a^bk_1(t) (s) f(s,u(s)) ds} = K_1∫_a^b(s) f(s,u(s)) ds. Let v∈𝒫∩∂Ω_q, thenmin_t∈ I_1v(t)⩾min_t∈ I_1k_1(t)K_2v_∞=m_1K_2 q .So, from (<ref>) and (H_2):ℒ v _∞≥K_1∫_a^b(s) f(s,v(s)) ds⩾ K_1∫_a_1^b_1(s) f(s,v(s)) ds ⩾ K_1 ∫_a_1^b_1(s) K_2 v(s)K_1 ∫_a_1^b_1k_1(s) (s) ds ds ⩾ ∫_a_1^b_1(s) K_2 k_1(s)K_2 v_∞∫_a_1^b_1k_1(s) (s) ds ds=v_∞=q . Hence, ℒ v_∞⩾v_∞ for all v∈𝒫∩∂Ω_q.Then, from Theorem <ref>, we conclude that ℒ has a fixed point in 𝒫 such that u_∞ lies between p and q. Definef_0^+(t):=lim sup_u→0^+f(t,u)u , f_∞^+(t):=lim sup_u→∞f(t,u)u ,f_0^-(t):=lim inf_u→0^+f(t,u)u , f_∞^-(t):=lim inf_u→∞f(t,u)u . Analogously to <cit.>, we have If G satisfies (Pg_1), then operator ℒ, defined in (<ref>),has a fixed point in 𝒫, provided that one of two following conditions hold (H_3) f_0^+(t)<1/K_2 ∫_a^b(s) ds for t∈ I and f_∞^-(t)>K_2^2/K_1 m_1∫_a_1^b_1k_1(s) (s) ds for t∈ I_1, (H_4) f_∞^+(t)<1/K_2 ∫_a^b(s) ds for t∈ I and f_0^-(t)>K_2^2/K_1 m_1∫_a_1^b_1k_1(s) (s) ds for t∈ I_1. In particular, there is one fixed point if f_0^+=0 and f_∞^-=∞ (f_∞^+=0 and f_0^-=∞). First, let us assume that (H_3) is fulfilled.Then, there exists p>0,small enough, and q>0, big enough, such that: * f(t,u)u⩽1K_2 ∫_a^b(s) ds for all t∈I and 0< u⩽ p. * f(t,u)u⩾K_2^2K_1 m_1∫_a_1^b_1k_1(s) (s) ds for all t∈ I_1 and u⩾ q. Thus, (H_1) and (H_2) are fulfilled for p>0 and K_2/m_1q>0, respectively. So, from Theorem <ref>, ℒ has a fixed point on 𝒫.Now, suppose that (H_4) is verified.Then, there exist 0<p<q, such that: * f(t,u)u⩾K_2^2K_1 m_1∫_a_1^b_1k_1(s) (s) ds for all t∈I_1 and 0< u⩽ p. * f(t,u)u⩽1K_2∫_a^b(s) ds for all t∈ I and u⩾ q. Hence, (H_2) is fulfilled for p>0.Let us see that (H_1) is also satisfied.First, let us assume that f(t,u) is a bounded function. That is, there exists N>0 such that 0⩽ f(t,u)⩽ N for all t∈ I and 0⩽ u<∞. Then, let us chooser⩾ N K_2 ∫_a^b(s) ds,such thatf(t,u)⩽ N⩽rK_2 ∫_a^b(s) ds ,∀ t∈ I , 0⩽ u⩽ r . Thus, (H_1) holds for this r>0.Now, suppose that f is not bounded. Then, there exists t_0∈ I and r̅⩾ q such that f(t,u)⩽ f(t_0,r̅) for all t∈ I and 0⩽ u⩽r̅. Thus, by using the hypothesis, we havef(t,u)⩽ f(t_0,r̅)⩽r̅K_2∫_a^b(s) ds ,∀ t∈ I , 0⩽ u⩽r̅ . Therefore, (H_1) is fulfilled for such r̅>0, and the result follows again from Theorem <ref>.To finish this section, we obtain the existence of at least two fixed points of the integral operator ℒ.Let us consider the following conditions on f: (H_1^*) There exists p>0 such that(H_1) is fulfilled and f(t,p)< pK_2 ∫_a^b(s) ds ,∀ t∈ I . (H_2^*) There exists q>0 such that (H_2) is fulfilled and f( t,q) > K_2 qK_1 ∫_a_1^b_1k_1(s) (s) ds ,∀ t∈ I_1. Realize that conditions (H_1^*) and (H_2^*) are a small restriction of (H_1) and (H_2), respectively. As we will see, this restriction allows us to prove that the two fixed points that we find are, in fact, different. If G satisfies (Pg_1), then the operator ℒ has at least two fixed points, u_1 and u_2∈𝒫, provided that f_0^-(t)=f_∞^-(t)=∞ for t∈ I and (H_1^*) holds. In such a case, 0<u_1_∞<p<u_2_∞ (p given in (H_1^*)). As in Corollary <ref>, we can see that f_0^-(t)=f_∞^-(t)=∞ imply that there exist positive numbers q_1 and q_2, with 0<q_1<p<q_2 such that (H_2) is fulfilled with respect to q_1 and q_2, respectively.Since (H_1^*) isa restriction of (H_1) we have that, in particular, (H_1) holds for p.Hence, from Theorem <ref>, we conclude that there exist two fixed points, u_1 and u_2, such that 0<q_1⩽u_1_∞⩽ p⩽u_2_∞⩽ q_2. To finish the proof, we need to ensure that u_1 and u_2 are, in fact, different. To this end, let us prove that if u∈𝒫∩∂Ω_p, then u cannot be a fixed point of ℒ.Let u∈𝒫∩∂Ω_p, that is, u_∞=p. From (H_1^*) and property (Pg_1), we have:ℒ u_∞ =sup_t∈ I∫_a^b G(t,s) f(s,u(s)) ds⩽sup_t∈ I∫_a^bk_2(t) (s) f(s,u(s)) ds=K_2 ∫_a^b(s) f(s,u(s)) ds⩽ K_2 ∫_a^b(s) pK_2 ∫_a^b(s) ds ds=p=u_∞ . Since u_∞=p we have that there exists t_0∈ I such that u(t_0)=p. Now, since f is a continuous function and u_∞=p, from (H_1^*), there exists a neighborhood of t_0 such that the strict inequality given in (H_1^*) is fulfilled, hence the last inequality in (<ref>) is strict andℒ u_∞<u_∞. Thus,u cannot be a fixed point. So, there exist two fixed points, u_1 and u_2, such that 0<q_1⩽u_1_∞< p<u_2_∞ and the result is proved. In an analogous way, we can prove the following “dual” result: If G satisfies (Pg_1), then the operator ℒ has at least two fixed points, u_1 and u_2∈𝒫, provided that f_0^+(t)=f_∞^+(t)=0 and (H_2^*) holds. In such a case, 0<u_1_∞<q<u_2_∞ (q given in (H_2^*)).§ EXISTENCE OF MULTIPLE FIXED POINTS In this section,by using Theorems <ref> and <ref>, we prove the existence of two or three non-trivial fixed points, respectively, of operator ℒ defined in (<ref>). We follow the steps given in <cit.>, however we impose slightlyweaker conditions on f than the ones given in that reference. Suppose that there exist positive numbers p, q and r such that 0<p<q<r, and assume that function f satisfies the following conditions: * f(t,u)⩾um_1∫_a_1^b_1(s) ds for all t∈ I_1 and u∈[ r,K_2m_1 r], being the inequality strict at u=r, * f(t,u)⩽qK_2∫_a^b(s) ds for all t∈ I and u∈[ 0,K_2m_1 q], being the inequality strict at u=q, * f(t,u)>K_2 uK_1∫_a_1^b_1k_1(s) (s) ds for all t∈ I_1 and u∈[ m_1K_2 p,p]. Then, if G satisfies (Pg_1), the operator ℒ has at least two fixed points, u_1 and u_2, such thatp<u_1_∞ ,max_t∈ I_1u_1(t)<q<max_t∈ I_1u_2(t) ,min_t∈ I_1u_2(t)<r . The proof is based on Theorem <ref>. Consider: α(u) :=min_t∈ I_1u(t) , θ(u) :=max_t∈ I_1u(t) , andγ(u) :=u_∞ . Clearly, α(u)⩽θ(u)⩽γ(u) for all u∈𝒫. Since u∈𝒫, then α(u)=min_t∈ I_1u(t)⩾min_t∈ I_1k_1(t)K_2 u_∞=m_1K_2 γ(u) , that is, γ(u)⩽K_2m_1 α(u) for all u∈𝒫. Moreover, for all λ∈ℝ and u∈𝒫, we have: θ(λ u)=max_t∈ I_1{λ u(t)} =λ max_t∈ I_1u(t)=λ θ(u) . As we have noticed, ℒ𝒫⟶𝒫 is a completely continuous operator. Moreover, let us consider u∈∂𝒫(α,r), i.e. min_t∈ I_1u(t)=r. we have thatα(u)=r⩾m_1K_2u_∞ . Thus, from (i), the following inequalities are verified:α(ℒ u) =min_t∈ I_1∫_a^bG(t,s) f(s,u(s)) ds⩾min_t∈ I_1∫_a^bk_1(t) (s) f(s,u(s)) ds⩾min_t∈ I_1 k_1(t)∫_a_1^b_1(s) f(s,u(s)) ds⩾ m_1∫_a_1^b_1(s) u(s)m_1 ∫_a_1^b_1(s) ds ds . On the other hand α(u)=r implies that there exists t_1∈ I_1 such that u(t_1)=r. Since f is a continuous function on I_1⊂ I, from (i), there exist I_0⊂ I_1 a nontrivial subinterval where the inequality given on (i)is strict. Thus, the last inequality of (<ref>) is also strict and we have for all u∈∂𝒫(α,r): α(ℒ u)> m_1∫_a_1^b_1(s) u(s)m_1 ∫_a_1^b_1(s) ds ds⩾ r . Now, for u∈∂𝒫(θ, q), i.e. max_t∈ I_1u(t)=q, we have thatγ(u)⩾θ (u)=q≥α(u)⩾m_1K_2γ(u) . Thus, q⩽u_∞⩽K_2m_1q and from (ii), we have: θ (ℒ u) =max_t∈ I_1∫_a^b G(t,s) f(s,u(s)) ds⩽max_t∈ I_1∫_a^bk_2(t) (s) f(s,u(s)) ds⩽ K_2 ∫_a^b(s) f(s,u(s)) ds⩽ K_2∫_a^b(s)qK_2∫_a^b(s) ds ds . Repeating the previous arguments, θ(u)=q implies that there exists t_1∈ I_1⊂ I such that u(t_1)=q. Since f is a continuous function on I, from (ii), there exist I_0⊂ I_1 a nontrivial subinterval where the inequality given on (ii)is strict. Thus, the last inequality of (<ref>) is also strict and we have for all u∈∂𝒫(θ,q): θ(ℒ u)< K_2∫_a^b(s)qK_2∫_a^b(s) ds ds=q . Finally, 𝒫(γ,p)={u∈𝒫 |u_∞<p}≠∅ and for all u∈∂𝒫(γ,p), we have that u_∞=p and α(u)⩾m_1/K_2 p. Thus, from (iii), we obtain: γ(ℒ u) =sup_t∈ I∫_a^bG(t,s) f(s,u(s)) ds⩾sup_t∈ I∫_a^bk_1(t) (s) f(s,u(s)) ds⩾ K_1∫_a_1^b_1(s) f(s,u(s)) ds⩾ K_1∫_a_1^b_1 (s) K_2 u(s)K_1 ∫_a_1^b_1k_1(s)φ(s) ds⩾∫_a_1^b_1(s) K_2 k_1(s)/K_2 u_∞∫_a_1^b_1k_1(s) (s) ds ds=p . So, we conclude that γ(ℒ u)>p for u∈∂𝒫(γ,p). Hence, all the hypotheses of Theorem <ref> are fulfilled. Thus, ℒ has at least two fixed points on 𝒫, u_1 and u_2, such that p<γ(u_1)=u_∞ and q>θ(u_1)=max_t∈ I_1u_1(t). Moreover, q<θ(u_2)=max_t∈ I_1u_2(t) and r>α(u_2)=min_t∈ I_1 u_2(t) and the result is proved. Realize that in the third item of Theorem <ref>, we cannot avoid the strict inequality on the whole interval, since u_∞=p implies that there exists t_1∈ I such that u(t_1)=p, however we cannot ensure that p∈ I_1. Even though, in most of cases it is fulfilled that the point t_1∈ I for which u(t_1)=p belongs to I_1. This is due because I_1 is built to avoid having zeros or small values of u. It is convenient to have I_1 as larger as possible to ensure that the obtained bounds are better, so t_1 is likely to be in I_1. In such a case, we can ask for the strict inequality only at u=p. Now, as an application of Theorem <ref>, we obtain the next result that ensuresthe existence of at least three critical points of operator ℒ. Let p, q and r be positive numbers satisfying the relation: 0<p<q<K_2m_1q⩽ r . Assume, moreover, that the function f satisfies the following conditions: * f(t,u)⩽rK_2∫_a^b(s) ds for all t∈ I and u∈[ 0,r], * f(t,u)<pK_2∫_a^b(s) ds for all t∈ I and u∈[ 0,p], * f(t,u)⩾um_1∫_a_1^b_1(s) ds for all t∈ I_1 and u∈[ q,K_2m_1q], being the inequality strict for u=q. Then, if G satisfies (Pg_1), the operator ℒ has at least three fixed points u_1, u_2, u_3∈{u∈𝒫 | u_∞⩽ r} such thatmax_t∈ I_1u_1(t)<p, q<min_t∈ I_1u_2(t) and p<max_t∈ I_1u_3(t) with min_t∈ I_1u_3(t)<q.Theproof follows from Theorem <ref>. We consider α, θ and γ as in (<ref>)–(<ref>) and, moreover Ψ(u)=α(u) and β(u)=θ(u). Clearly α and Ψ are concave and non-negative functionals in 𝒫. And β, θ and γ are convex and non-negative functionals in 𝒫.It is proved in Theorem <ref> that ℒ(𝒫)⊂𝒫. Let us see now that ℒ( 𝒫(γ,r)) ⊂𝒫(γ,r). Indeed, let u∈𝒫(γ,r) (i.e. u_∞⩽ r), from (a) we have: ℒ u _∞ =sup_t∈ I∫_a^bG(t,s) f(s,u(s)) ds⩽sup_t∈ I∫_a^bk_2(t) (s) f(s,u(s)) ds=K_2∫_a^b(s) f(s,u(s)) ds⩽ K_2∫_a^b(s) rK_2∫_a^b(s) ds ds=r ,thus, ℒ u∈𝒫(γ,r) and we conclude that ℒ( 𝒫(γ,r)) ⊂𝒫(γ,r).Obviously, α(u)⩽β(u) and γ(u)=u_∞.We consider u_q(t)=K_2m_1 q, it is obvious that u_q belongs to the following set { u∈𝒫( γ,θ,α,q,K_2m_1 q,r)|α(u)>q}, described as follows:{ u∈𝒫 | q<min_t∈ I_1u(t) , max_t∈ I_1u(t)⩽K_2m_1q , u_∞⩽ r}≠∅ . Let u∈𝒫( γ,θ,α,q,K_2m_1 q,r), then using (c), we obtain:α(ℒ u) =min_t∈ I_1∫_a^bG(t,s) f(s,u(s)) ds⩾min_t∈ I_1∫_a^bk_1(t) (s) f(s,u(s)) ds⩾ m_1∫_a_1^b_1(s) f(s,u(s)) ds⩾ m_1∫_a_1^b_1(s) u(s)m_1∫_a_1^b_1(s) ds ds . If there exists s_1∈ I_1 such that u(s_1)>q, then it must exist a nontrivial subinterval of I_1 where u(s)>q for all s in such a subinterval. Then, directly from (<ref>), we have α (ℒ u)>q.Now, if u(s)=q for all s∈ I_1, from (c) and (<ref>), we obtain:α(ℒ u)⩾ m_1∫_a_1^b_1(s) f(s,q) ds> m_1∫_a_1^b_1(s) qm_1∫_a_1^b_1(s) ds ds=q . So, the hypothesis a) on Theorem <ref> is fulfilled. Now, let us see b).If we consider the function u_p(t)=m_1K_2 p, it is clear that it belongs to theset { u∈𝒫( γ,β,Ψ,m_1K_2 p,p,r)|β(u)<p}, which is described as follows:{ u∈𝒫 |m_1K_2 p⩽min_t∈ I_1u(t) , max_t∈ I_1u(t)< p , u_∞⩽ r}≠∅ . Let u∈𝒫( γ,β,Ψ,m_1K_2 p,p,r), from (b), we have β(ℒ u) =max_t∈ I_1∫_a^bG(t,s) f(s,u(s)) ds⩽max_t∈ I_1∫_a^bK_2 (s) f(s,u(s)) ds<K_2∫_a^b(s) pK_2∫_a^b(s) ds ds=p . Thus, β(ℒ u)<p for all u∈𝒫( γ,β,Ψ,m_1K_2 p,p,r) and, as consequence, condition b) in Theorem <ref> is satisfied.Now, let u∈𝒫(γ,α,q,r) be such that θ (ℒ u)>q. Thenα(ℒ u) =min_t∈ I_1∫_a^b G(t,s) f(s,u(s)) ds⩾min_t∈ I_1∫_a^b k_1(t) (s) f(s,u(s)) ds=m_1K_2∫_a^bK_2 G(t,s) f(s,u(s)) ds⩾m_1K_2∫_a^bmax_t∈ I_1G(t,s) f(s,u(s) ds⩾m_1K_2max_t∈ I_1∫_a^bG(t,s) f(s,u(s)) ds=m_1K_2θ(ℒ u)>m_1K_2 q . Finally, take u∈𝒬(γ,β,p,r) such that Ψ(ℒ u)<m_1K_2p, thenβ(ℒ u) =max_t∈ I_1∫_a^b G(t,s) f(s,u(s)) ds⩽∫_a^b K_2 (s) f(s,u(s)) ds=K_2m_1∫_a^bm_1 (s) f(s,u(s)) ds⩽K_2m_1∫_a^bmin_t∈ I_1G(t,s) f(s,u(s) ds⩽K_2m_1min_t∈ I_1∫_a^bG(t,s) f(s,u(s)) ds=K_2m_1Ψ(ℒ u)<K_2m_1m_1K_2 p=p . Therefore, all the hypotheses of Theorem <ref> are fulfilled and we have ensured the existence of at least three critical points such that p>β(u_1)=max_t∈ I_1u_1(t), q<α(u_2)=min_t∈ I_1u_2(t) and p<β(u_3)=max_t∈ I_1u_3(t) with q>α(u_3)=min_t∈ I_1u_3(t). And the proof is complete. § EXISTENCE RESULTS FOR (K,N-K) PROBLEMS Let us consider the n^th-order differential operator introduced in (<ref>).As we have mentioned before, the existence of solution of the problem (<ref>)-(<ref>) is equivalent to the existence of a fixed point of the integral operator (<ref>). Let us denoteX_k={u∈ C^n(I) | u(a)=⋯=U^(k-1)(a)=u(b)=⋯=u^(n-k-1)(b)=0} ,the set of functions related to the boundary conditions (<ref>).Let us introduce a concept about the maximum number of zeros of the solutions of a linear differential equation. <cit.> Let p_k∈ C^n-k(I) for k=1,…,n. The n^th-order linear differential equation T_n[M̅] u(t)=0is said to be disconjugate on an interval I if every non trivial solution has, at most, n-1 zeros on I, multiple zeros being counted according to their multiplicity. In <cit.>, the main properties of te related Green's function, g_M(t,s), are studied.Now, we prove: Let M̅∈ℝ be such that equation T_n[M̅] u(t)=0is disconjugate on I. Then thefollowing properties are fulfilled: If n-k is even and 2⩽ k ≤ n-1, then g_M(t,s) verifies (Pg_1) if, and only if, M∈ (M̅-λ_1,M̅-λ_2], where: * λ_1>0 is the least positive eigenvalueof operator T_n[M̅] in X_k. * λ_2<0 is the maximum of: * λ_2'<0, the biggest negative eigenvalueof operator T_n[M̅] in X_k-1. * λ_2”<0, the biggest negative eigenvalue of operator T_n[M̅] in X_k+1. If k=1 and n is odd, theng_M(t,s) verifies (Pg_1) if, and only if, M∈ (M̅-λ_1,M̅-λ_2], where: * λ_1>0 is the least positive eigenvalueof operator T_n[M̅] in X_1. * λ_2<0 is the biggest negative eigenvalue of operator T_n[M̅] in X_2. If n-k is odd and 2⩽ k⩽ n-2, then-g_M(t,s) verifies (Pg_1) if, and only if, M∈ [M̅-λ_2,M̅-λ_1), where: * λ_1<0 is the biggest negative eigenvalue of operator T_n[M̅] in X_k. * λ_2>0 is the minimum of: * λ_2'>0, the least positive eigenvalueof operator T_n[M̅] in X_k-1. * λ_2”>0, the least positive eigenvalueof operator T_n[M̅] in X_k+1. If k=1 and n>2 is even , then-g_M(t,s) verifies (Pg_1) if, and only if, M∈ [M̅-λ_2,M̅-λ_1), where: * λ_1<0 is the biggest negative eigenvalue of operator T_n[M̅] in X_1. * λ_2>0 isthe least positive eigenvalueof operator T_n[M̅] in X_2. If k=n-1 and n>2, then-g_M(t,s) verifies (Pg_1) if, and only if,M∈ [M̅-λ_2,M̅-λ_1), where: * λ_1<0 is the biggest negative eigenvalue of operator T_n[M̅] in X_n-1. * λ_2>0 isthe least positive eigenvalueof operator T_n[M̅] in X_n-2. If n=2, then-g_M(t,s) verifies (Pg_1) if, and only if, M∈ (-∞,M̅-λ_1), where: * λ_1<0 is the biggest negative eigenvalue of operator T_2[M̅] in X_1. From the proof of <cit.> we obtain thatthe following inequalities are fulfilled for those Mwhich the result refers to. (∞>)(-1)^n-kg_M(t,s) >0 ,∀(t,s)∈ (a,b)×(a,b) , (∞>)(-1)^n-k∂^kg_M(t,s)∂ t^k_| t=a >0 ,∀s∈(a,b) , (∞>)(-1)^n-k∂^n-kg_M(t,s)∂ s^n-k_| s=a >0 ,∀t∈ (a,b) , (∞>)∂^n-kg_M(t,s)∂ t^n-k_| t=b >0 ,∀s∈ (a,b) , (∞>)(-1)^n∂^k g_M(t,s)∂ s^k_| s=b >0 ,∀t∈ (a,b) . Now, we construct the following function: (s)=(s-a)^n-k (b-s)^k>0 ,∀s∈(a,b) , and we have that u(t,s)=(-1)^n-kg_M(t,s)(s)>0 ,∀ (t,s)∈(a,b)×(a,b) . Moreover, the following limits are real and: lim_s→ a^+(-1)^n-kg_M(t,s)(s) =(-1)^n-k∂^n-kg_M(t,s)∂ s^n-k_| s=a(n-k)! (b-a)^k>0 ,∀ t∈(a,b) , lim_s→ b^-(-1)^n-kg_M(t,s)(s) =(-1)^n-k∂^kg_M(t,s)∂ s^k_| s=bk! (-1)^k(b-a)^n-k=(-1)^n∂^kg_M(t,s)∂ s^k_| s=bk!(b-a)^n-k>0 ,∀ t∈(a,b) . Hence, we can define ũ(t,s), as the continuous extension of u(t,s) to (a,b)× I, and we have k_1(t) :=min_s∈ Iũ(t,s)>0 ,∀ t∈(a,b) , k_2(t) :=max_s∈ Iũ(t,s)>0 ,∀ t∈(a,b) . Clearly, 0<k_1(t)<k_2(t) for all t∈(a,b) and (-1)^n-k g_M(t,s) fulfills condition (P_g1). Then, the result is proved. §.§ Particular casesThis section is devoted to apply previous results for some particular cases with fixed (k,n-k) boundary conditions. We distinguish the cases depending on the value of n. * Second order: n=2 and k=1.Let us study a second order operator with constant coefficientsT_2[B,M] u(t)=u”(t)+B u'(t)+M u(t) , t∈[0,1] ,for a fixed B∈ℝ.We consider the space of definition related to the boundary conditions (1,1). That is,X_1={ u∈ C^2([0,1]) | u(0)=u(1)=0} . As it is proved in<cit.>, the biggest negative eigenvalue of T_2[B,0] in X_1 is λ_B=-B^2+4 π^44. Moreover, T_2[B,0] u(t)=0 is a disconjugate equation on [0,1] for all B∈ℝ.Thus, from Theorem <ref>, -g_M(t,s), satisfies the property (Pg_1) if, and only if, M∈( -∞, B^2+4 π^24). In particular such a property is fulfilled for M=0. Hence, let us study the following problem:u”(t)+B u'(t)+f(t,u(t))=0 , t∈[0,1] , u(0)=u(1)=0 . Clearly, (s)=s(1-s). First, consider B=0. We have, (see <cit.>)-g(t,s)= s (1-t) , 0⩽ s⩽ t⩽1 (1-s) t , 0<t<s⩽1 .By direct calculations, we obtain k_2(t)=1=K_2 andk_1(t)=min_t∈[0,1]{1-t,t}= t , 0⩽ t⩽12 , 1-t , 12<t⩽ 1 ,and K_1=k_1( 1/2) =12.In Figure <ref>, it is represented the function ũ(t,s)=-g(t,s)(s), bounded from above by k_2(t) and from below by k_1(t).Moreover, in Figure <ref>, it is given the same representation considering the constant values t_0=2/3 and s_0=2/3, respectively. Let us consider I_1=[a_1,b_1]=[ 14,34], then m_1=14, and it is immediate to verify that∫_0^1(s) ds =16 , ∫_1/4^3/4(s) ds =1196 , ∫_1/4^3/4k_1(s) (s) ds =671536 . Thus, we are able to ensure the existence of at least two or three solutions of problem (<ref>) by means of the previously obtained fixed point theorems. First, let us write the corresponding assumptions (H_1) and (H_2) for this situation, (H_1) There exists p>0 such that f(t,u)⩽ 6 p for all t∈ [0,1] and u∈[0,p]. (H_2) There exists q>0 such that f(t,u)⩾307267u for all t∈[ 14,34] and u∈[ q4,q]. Finally,as a direct consequence of Theorems <ref> and <ref>, we obtain the following results: Let I=[0,1] and I_1=[ 1/4,3/4] and suppose that there exist positive numbers p, q and r such that 0<p<q<r, and assume that function f satisfies the following conditions: * f(t,u)⩾38411u for all t∈ I_1 and u∈[ r,4 r], being the inequality strict at u=r, * f(t,u)⩽ 6 q for all t∈ I and u∈[ 0,4 q], being the inequality strict at u=q, * f(t,u)>307267u for all t∈ I_1 and u∈[ p4,p]. Then, for B=0, problem (<ref>) has at least two positive solutions,u_1 and u_2, such thatp<u_1_∞ , max_t∈ I_1u_1(t)<q andq<max_t∈ I_1u_2(t) , min_t∈ I_1u_2(t)<r . Let I=[0,1] and I_1=[ 1/4,3/4] and suppose p, q and r are positive numbers such that 0<p<q<4 q⩽ r , such the function f satisfies the following conditions: * f(t,u)⩽ 6 r for all t∈ I and u∈[ 0,r], * f(t,u)<6 p for all s∈ I and u∈[ 0,p], * f(t,u)⩾307267u for all s∈ I_1 and u∈[ q,4 q], being the inequality strict at u=q. Then, for B=0, problem (<ref>) has at least three positive solutions,u_1, u_2, u_3∈{u∈𝒫 | u_∞⩽ r}, such thatmax_t∈ I_1u_1(t)<p, q<min_t∈ I_1u_2(t) and p<max_t∈ I_1u_3(t) with min_t∈ I_1u_3(t)<q. Now, let us see what happens for B>0. In this case, the expression of the Green's function (see <cit.>) is given by-g(t,s)=( e^B s-1) ( e^B (1-t)-1) B (e^B-1) , 0⩽ s⩽ t⩽ 1 ,( e^B-e^B s) ( 1-e^-B t) B (e^B-1) , 0<t<s⩽ 1 . After routine calculations we obtain, k_2(t)=(e^B-e^B t) (1-e^-B t)t (1-t) B (e^B-1)⩽ k_2(1)=1=K_2 .Moreover, we havek_1(t)=e^B-e^B (1-t)e^B-1 , 0⩽ t⩽ t_1=1-log( 1+e^B/2) B ,e^B (1-t)-1e^B-1 , t_1<t⩽ 1 . Thus, K_1=max_t∈[0,1]k_1(t)=k_1(t_1)=12.If we chooseI_1=[a_1,b_1]=[ 1-log( 1+3 e^B/4) B,1-log( 3+e^B/4) B] ,we obtain m_1=k_1(a_1)=k_1(b_1)=14. For B∈[0,2], both a_1 and b_1 are decreasing functions of B. Moreover, a_1∈[ 1-1/2log(1/4(1+3 e^2)),1/4] ⊂[ 3/25,1/4], and b_1∈[ 1-1/2log(1/4(3+e^2)),3/4] ⊂[ 13/25,3/4]. To attain the rest of the constants involved on the different results, we use approximations of the values of the integrals. To this end we need to fix the value of B∈[0,2]. The bounds shown above can be obtained for all B∈[0,2], in order to simplify the description of k_1, let us choose B such that t_1=13, that is,B=log(2+√(5))∈[0,2]. In such a case, we havea_1=log(√(5)-1)log(2+√(5))andb_1=log( 1+3/√(5)) log(2+√(5)) .Moreover, in this case, we obtain∫_a_1^b_1k_1(s) (s) ds ≊ 0.035872>3587100000 , ∫_a_1^b_1(s) ds ≊ 0.095719>95710000 . As for B=0, in Figure <ref>, we representfunction ũ(t,s), bounded from above by k_2(t) and from below by k_1(t) for B=log(2+√(5)).Moreover, in Figure <ref>, it is plotted the same representation considering the constant values t_0=2/3 and s_0=2/3, respectively. Thus, as in the case where B=0, we obtain the correspondent results of existence of solution for such a problem. First, let us write the related (H_1) and (H_2). Let I=[0,1] and I_1 be defined in (<ref>) for B=log (2+√(5)). (H_1) There exists p>0 such that f(t,u)⩽ 6 p for all t∈ I and u∈[0,p]. (H_2) There exists q>0 such that f(t,u)⩾2000003587u>u1/2∫_a_1^b_1k_1(s) (s) ds for all t∈ I_1 and u∈[ q4,q]. Finally, we can rewrite Theorems <ref> and <ref> as follows. Let I=[0,1] and I_1 be defined in (<ref>) for B=log (2+√(5)) and suppose that there exist positive numbers p, q and r such that 0<p<q<r, and assume that function f satisfies the following conditions: * f(t,u)⩾40000957u>u1/4∫_a_1^b_1(s) ds for all t∈ I_1 and u∈[ r,4 r], * f(t,u)⩽ 6 q for all t∈ I and u∈[ 0,4 q], being the inequality strict for u=q, * f(t,u⩾2000003587u for all t∈ I_1 and u∈[ p4,p]. Then, for B=log(2+√(5)),problem (<ref>) has at least two positive solutions, u_1 and u_2, such that p<u_1_∞, max_t∈ I_1u_1(t)<q<max_t∈ I_1u_2(t) and min_t∈ I_1u_2(t)<r. Let I=[0,1] and I_1 be defined in (<ref>) for B=log (2+√(5)) and suppose p, q and r are positive numbers such that 0<p<q<4 q⩽ r , such the function f satisfies the following conditions: * f(t,u)⩽ 6 r for all t∈ I and u∈[ 0,r], * f(t,u)<6 p for all t∈ I and u∈[ 0,p], * f(t,u)⩾40000957u for all s∈ I_1 and u∈[ q,4 q]. Then,for B=log(2+√(5)), problem (<ref>) has at least three positive solutions, u_1, u_2, u_3∈{u∈𝒫 | u_∞⩽ r}, such thatmax_t∈ I_1u_1(t)<p, q<min_t∈ I_1u_2(t) and p<max_t∈ I_1u_3(t) with min_t∈ I_1u_3(t)<q.Finally, repeating the same arguments for B<0, we obtain that for B∈[-2,0]:k_2(t) =( 1-e^B t)( e^-B-e^-B t) t (1-t) B (1-e^-B)⩽ 1=K_2 , k_1(t) =e^-B t-1e^-B-1 , 0⩽ t⩽ t_2=-log( e^-B+1/2) B ,e^-B-e^-B te^-B-1 , t_2<t⩽1 ,and K_1=max_t∈[0,1]k_1(t)=k_1(t_2)=12.In this case, if we chooseI_1=[a_1,b_1]=[ -log(e^-B+3/4)B,-log(3 e^-B+1/4)B] ,it is verified that m_1=k_1(a_1)=k_1(b_1)=14. For B∈[-2,0], both a_1 and b_1 are decreasing functions of B. Moreover, a_1∈[1/4,1/2log(1/4(3+e^2))] ⊂[1/4,12/25] , and b_1∈[3/4,1/2log(1/4(1+3 e^2))]⊂[3/4,22/25].Moreover, for the choice B=log(√(5)-2)∈[-2,0] and he correspondent interval I_1 defined in (<ref>), we attain the same bounds as for B=log(2+√(5)). Thus, (H_1) and (H_2) coincide in both cases and Theorems <ref> and <ref> remain valid for B=log(√(5)-2).It is important to recall that for all B∈[-2,2] any suitable bounds can be obtained by using the expression of k_1 and k_2 without any additional difficulty. If B∉[-2,2], the study is much more complicated. However the approach can also be done. For instance let us choose B=-2 π, we have-g(t,s)=(1- e^-2 π s) ( e^2 π-e^2π t) 2 π (e^2π-1) , 0⩽ s⩽ t⩽ 1 ,( e^2 π (1-s)-1) ( e^2 π t-1) 2 π (e^2 π-1) , 0<t<s⩽ 1 . In this case, we are not able to give the exact expressionof k_1(t), but we can obtain a lower bound for ũ(t,s) as follows: k_1(t)=e^2 π t-1e^2 π-1 , 0⩽ t⩽ t_3 ,25000062037( 1-e^-451/500 π) e^2 π-e^2 π t2 π (e^2 π-1) , t_3<t⩽ 1 ,where t_3=12 πlog( 125000( e^549/500 π-e^2 π) -62037 π125000( e^-451/500 π-1) -62037 π) ≊0.844992.Moreover, ũ(t,s)⩽ 1 for all (t,s)∈ [0,1]× [0,1].In this case, we conclude thatK_1=max_t∈[0,1]k_1(t)=k_1(t_3)≊ 0.37642>47125 . As in previous cases, in Figure <ref>, we represent the function ũ(t,s), bounded from above by K_2=1 and from below by k_1(t).Moreover, in Figure <ref>, it is plotted the same representation considering the constant values t_0=21/25 and s_0=21/25, respectively. If we choose,I_1=[a_1,b_1]=[ log(3+e^2 π/4) 2 π,0.9151] ⊂[0.78 ,0.9151],we have that m_1=k_1(a_1)=14 and∫_a_1^b_1(s) ds ≊ 0.0172072>432500 , ∫_a_1^b_1k_1(s) (s) ds ≊ 0.005393>539100000 . Thus, as in the previous cases, we deduce the correspondent results of existence of solution for this problem. Let I=[0,1] and I_1 be defined in (<ref>). First, let us write the related (H_1) and (H_2) as follows: (H_1) There exists p>0 such that f(t,u)⩽ 6 p for all t∈ I and u∈[0,p]. (H_2) There exists q>0 such that f(t,u)⩾12500000 u25333>u47/125∫_a_1^b_1k_1(s) (s) ds for all t∈ I_1 and u∈[ q4,q]. So, we can rewrite Theorems <ref> and <ref> as follows. Let I=[0,1] and I_1 be defined in (<ref>)and suppose that there exist positive numbers p, q and r such that 0<p<q<r, and suppose that function f satisfies the following conditions: * f(t,u)⩾1000043u>u1/4∫_a_1^b_1(s) ds for all t∈ I_1 and u∈[ r,4 r], * f(t,u)⩽6 q for all t∈ I and u∈[ 0,4 q], being the inequality strict for u=q, * f(t,u)⩾1250000025333u for all t∈ I_1 and u∈[ p4,p]. Then, for B=-2 π,problem (<ref>) has at least two positive solutions, u_1 and u_2, such that p<u_1_∞, max_t∈ I_1u_1(t)<q<max_t∈ I_1u_2(t) and min_t∈ I_1u_2(t)<r. Let us consider the following continuous function f(t,u)=(1007/88+225/88 t) 500000001190651u , u⩽128 ,(1007/88+225/88 t)312500058341899 u , 128<u⩽ 14 ,(1007/88+225/88 t)312500058341899 u+1000043(u-14) u ,u>14 . It is easy to verify that this function f satisfies the hypotheses of Theorem <ref>. * For the construction of f, (i) is trivially fulfilled for all u⩾ 15. * If q=72, then f(t,u)⩽ f( t,128) = (1007/88+225/88 t)500000001190651·128⩽250000001190651< 72· 6 for all t∈ I and u ∈ [0,14]. * If p=128, clearly f(t,u)⩾474·500000001190651u=1250000025333 u, for all u⩽ p and t∈ I_1 for all t∈[ 325,1], in particular for t∈ I_1 and u∈[ 1112,128]. Realize that, from Remarks <ref> and <ref>, the conditions imposed to f are stronger than those imposed in Theorems <ref> and <ref>, then the obtained f is also valid for these results. Thus, we can conclude that for B∈{-2 π,log (√(5)-2),0,log(√(5)+2)} problem (<ref>), with f defined in (<ref>), has at least two positive solutions, u_1 and u_2, such that 128<u_1_∞, max_t∈ I_1u_1(t)<72<max_t∈ I_1u_2(t), min_t∈ I_1u_2(t)<15. Let I=[0,1] and I_1 be defined in (<ref>)and suppose p, q and r are positive numbers such that 0<p<q<4 q⩽ r , such the function f satisfies the following conditions: * f(t,u)⩽ 6 r for all t∈ I and u∈[ 0,r], * f(t,u)<6 p for all s∈ I and u∈[ 0,p], * f(t,u)⩾1000043u for all s∈ I_1 and u∈[ q,4 q]. Then,for B=-2 π, problem (<ref>) has at least three positive solutions, u_1, u_2, u_3∈{u∈𝒫 | u_∞⩽ r}, such thatmax_t∈ I_1u_1(t)<p, q<min_t∈ I_1u_2(t) and p<max_t∈ I_1u_3(t) with min_t∈ I_1u_3(t)<q. Let us consider the following continuous functionf(t,u)= 12 ( 31/28t+25/28)u^3 , u⩽ 16 , 49152( 31/28t+25/28) , u>16 . It is easy to verify that f fulfills the hypotheses of Theorem <ref>: * If r=16384, then f(t,u)⩽ 98304=6·16384 for all t∈ I and u∈ [0,16384], * If p=12, then f(t,u)⩽12· 2·14·12=6·12, for all t∈ I and u∈[ 0,12], * If q=4, then f(t,u)⩾ 12·54 16u=240 u>1000043 u for all t∈[ 325,1], in particular for t∈ I_1 and u∈[4,16]. As in Remark <ref>, the imposed conditions on f are stronger than those in Theorems <ref> and <ref>. Thus, we can conclude that for B∈{-2 π,log (√(5)-2),0,log(√(5)+2)} problem (<ref>), with f defined in (<ref>), has at least three positive solutions, u_1, u_2, u_3∈{u∈𝒫 | u_∞⩽ 16384}, such thatmax_t∈ I_1u_1(t)<12, 4<min_t∈ I_1u_2(t) and 12<max_t∈ I_1u_3(t) with min_t∈ I_1u_3(t)<4. * Fourth order: n=4 and k=2. In <cit.> it is proved that the related Green's function of operator T_4[M] u(t)=u^(4)(t)+M u(t) on X_2,with I=[0,1], satisfies the property (Pg_1) if, and only if, M∈ (-λ_1^4,λ_2^4], where λ_1≊ 4.73 is the least positive solution ofcos (λ ) cosh (λ )=1 ,and λ_2≊ 5.55 is the least positive solution oftan( λ√(2)) =tanh( λ√(2)) .In particular, it is fulfilled for M=0. Thus, let us study the following problem:u^(4)(t)=f(t,u(t)) , t∈[0,1] , u(0)=u'(0)=u(1)=u'(1)=0 . The related Green's function (see <cit.>) is given byg(t,s)=s^26 (1-t)^2 (3 t-s-2 s t) , 0⩽ s⩽ t⩽ 1 ,(1-s)^26 t^2 (3 s-t-2 s t) , 0<t<s⩽ 1 . Thus, after tedious calculus, we can ensure that k_1(t)t^22 (1-t) , 0⩽ t⩽12 ,t2 (1-t)^2, 12<t⩽ 1 ,andk_2(t)=t2 (1-t)^2 , 0⩽ t⩽14 ,1-t24 (1+2 t)^2, 14<t⩽12 ,t24 (3-2 t)^2 , 12< t⩽34 ,t^22 (1-t), 34<t⩽ 1 . Hence, we haveK_2=max_t∈[0,1]k_2(t)=k_2( 12) =112andK_1=max_t∈[0,1]k_1(t)=k_1( 12) =116 . As in the second order case, in Figure <ref>, we represent the function ũ(t,s), bounded from above by k_2(t) and from below by k_1(t).Moreover, in Figure <ref>, it is shown the same representation considering the constant values t_0=3/4 and s_0=3/4, respectively. Let us choose I_1=[a_1,b_1]=[ 13,23], in this case we have m_1=min_t∈ I_1k_1(t)=k_1( 13) =127 and∫_0^1(s) ds =130 , ∫_a_1^b_1(s) ds =472430 , ∫_a_1^b_1k_1(s) (s) ds =462461470292480 . Hence, as in thesecond order case, we can obtain the correspondent (H_1) and (H_2): (H_1) There exists p>0 such that f(t,u)⩽ 360 p for all t∈ [0,1] and u∈[0,p]. (H_2) There exists q>0 such that f(t,u)⩾627056640 u426461 for all t∈[13,23] and u∈[ 49q,q]. Finally, we can rewrite Theorems <ref> and <ref> as follows. Suppose that there exist positive numbers p, q and r such that 0<p<q<r, and suppose that function f satisfies the following conditions: * f(t,u)⩾6561047u for all t∈[13,23] and u∈[ r,94r], being the inequality strict for u=r, * f(t,u)<360 q for all t∈[0,1] and u∈[ 0,94q], being the inequality strict for u=q, * f(t,u)>627056640426461u for all t∈[13,23] and u∈[ 49p,p]. Thenproblem (<ref>) has at least two positive solutions, u_1 and u_2, such that p<u_1_∞, max_t∈ I_1u_1(t)<q<max_t∈ I_1u_2(t) and min_t∈ I_1u_2(t)<r. Let us choose a particular function f that the satisfies the conditions imposed in previous result.f(t,u)= 1296 t , u⩽136 ,tu^2 , 136<u⩽334 ,64 t35937( u-294) ^5u , u>334 . Let us choose p=116, q=113 and r=27, we have: * For all t∈[13,23] and u∈[ 27,2434], f(t,u)=64 t35937( u-294)^5u⩾643 · 35937 ( 794) ^5 u=3077056399 u1724976>65610 u47. * For all t∈[0,1] and u∈[ 0,334], we have f(t,u)⩽1( 1/36) ^2=1296<113 · 360. * For all t∈[13,23] and u∈[ 136,116], f(t,u)=tu^3 u⩾1/3(1/16)^3 u=40963 u>627056640 u462461. Hence, for this function the problem (<ref>) has at least two positive solutions, u_1 and u_2, such that 116<u_1_∞, max_t∈ I_1u_1(t)<113<max_t∈ I_1u_2(t) and min_t∈ I_1u_2(t)<27. Now, Theorem <ref> reads as follows: Suppose p, q and r are positive numbers for which the following inequalities are fulfilled: 0<p<q<94q⩽ r , and that the function f satisfies the following conditions: * f(t,u)⩽ 360 r for all t∈ [0,1] and u∈[ 0,r], * f(t,u)<360 p for all s∈ [0,1] and u∈[ 0,p], * f(t,u)⩾6561047u for all s∈[13,23] and u∈[ q,94q], being the inequality strict for u=q. Then problem (<ref>) has at least three positive solutions, u_1, u_2, u_3∈{u∈𝒫 | u_∞⩽ r}, such thatmax_t∈ I_1u_1(t)<p, q<min_t∈ I_1u_2(t) and p<max_t∈ I_1u_3(t) with min_t∈ I_1u_3(t)<q. Let us see that the following function satisfies these hypotheses:f(t,u)= (2+3 t) u^2 , u⩽12 ,( u-12) u^4+(2+3 t) u^2 , 12<u⩽ 14 , 519204+588 t , u>14 . Let us choose p=12, q=569 and r=1444, we can check all the hypotheses of Theorem <ref>. * For all t∈ [0,1] and u∈[ 0,1444], f(t,u)⩽ 519204+588=519792<360 · 1444=519840 r. * For all s∈ [0,1] and u∈[ 0,12], then f(t,u)=(2+3 t) u^2⩽54<360 · 12=180. * For all s∈[13,23] and u∈[ 569,14], f(t,u)=( u-12) u^4+(2+3 t) u^2=( ( u-12) u^3+(2+3 t) u)u⩾( ( 569-12) ( 569) ^3+3 569) =9166696 u6561>65610 u47. Hence, for such a f the problem (<ref>) has at least three positive solutions, u_1, u_2 and u_3, such that max_t∈ I_1u_1(t)<12, 569<min_t∈ I_1u_2(t) and 12<max_t∈ I_1u_3(t) with min_t∈ I_1u_3(t)<569. 99 AndHof D. R. Anderson, J. Hoffacker, Existence of solutions for a cantilever beam problem, J. Math. Anal. Appl. 323 (2006) 958-973.AnAvHen D. R. Anderson, R. I. Avery, J. Henderson, Functional expansion-compression fixed point theorem of Leggett-Williams type, Electron. J. Differential Equations, Vol. 2010 (2010), No. 63, pp. 1-9.Av R. I. Avery, A generalization of the Leggett-Williams fixed point theorem, MSR Hot-Line 2 (1999) 9-14.AvAnHen R. I. Avery, D. R. Anderson, J. Henderson, An extension of the compression-expansion fixed point theorem of functional type, Electron. J. Differential Equations, Vol. 2016 (2016), No. 253, pp. 1-9.AvElHe R. I. Avery, P. Eloe, J. Henderson, A Leggett-Williams type theorem applied to a fourth order problem, Commun. Appl. Anal. 16 (2012) no. 4, 579-588.AvHe R. I. Avery, J. Henderson, Two positive fixed points of nonlinear operators on ordered Banach spaces, Comm. Appl. Nonlinear Anal. 8 (2001) 27-36.Cab A. Cabada, Green's Functions in the Theory of Ordinary Differential Equations, Springer Briefs in Mathematics, 2014.CaCiIn A. Cabada, J. A. Cid, G. Infante, A positive fixed point theorem with applications to systems of Hammerstein integral equations, Bound. Value Probl., 2014:254 (2014), 10pp. CaCiMa A. Cabada, J.A. Cid, B. Máquez-Villamarín, Computation of Green's functions for boundary value problems with Mathematica, Applied Mathematics and Computation 219 (2012) 1919-1936.CaPrSaTe A. Cabada, R. Precup, L. Saavedra, S. A. Tersian, Multiple solutions to a fourth-order boundary-value problem, Electron. J. Differential Equations, Vol. 2016 (2016), No. 254, pp. 1-18. CabSaa A. Cabada, L. Saavedra, The eigenvalue characterization for the constant sign Green's functions of (k,n-k) problems, Boundary Value Problems (2016, 44) 35ppCabSaa2 A. Cabada, L. Saavedra, Constant sign Green's function for simply supported beam equation, Advances in Differential Equations, ADE-C-1003, to appearCop W. A. Coppel, Disconjugacy. Lecture Notes in Mathematics, Vol. 220. Springer-Verlag, Berlin-New York, 1971.FrInPe D. Franco, G. Infante, J. Pern, A new criterion for the existence of multiple solutions in cones, Proc. Roy. Soc. Edinburgh, 142A (2012), 1043-1050.karlin S. Karlin, The Existence of Eigenvalues for Integral Operators. Transactions of the American Mathematical Society, 113, (1964),1- 17.Kra M. A. Krasnoselskiĭ, Positive solutions of operator equations, Noordhoof, Groningen, The Netherlands, 1964.Lan K. Q. Lan Multiple positive solutions of semilinear differential equations with singularities, J. London Math. Soc. (2) 63 (2001) 690-704. LeWi R. W. Leggett, L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J. 2 (1979), 237-253.WiLe L. R. Williams, R. W. Leggett, Unique and multiple solutions of a family of differential equations modeling chemical reactions, SIAM J. Math. Anal. Vol. 13, No. 1 (1982), 122-133.ZhSu G.Zhang, J. Sun, A generalization of the Leggett-Williams fixed point theorem and its application, J. Appl. Math. Comput. (2012) 39 : 385-399. | http://arxiv.org/abs/1703.09115v1 | {
"authors": [
"Alberto Cabada",
"Lorena Saavedra"
],
"categories": [
"math.CA",
"34B15, 34B27, 34B18, 47H10"
],
"primary_category": "math.CA",
"published": "20170327144203",
"title": "Existence of solutions for $n^\\mathrm{th}$-order nonlinear differential boundary value problems by means of new fixed point theorems"
} |
§ INTRODUCTIONThis contribution to LCWS2016 presents recent developments within the CLIC Detector and Physics Study Collaboration (CLICdp). The CLICdp collaboration currently consists of 28 institutes from 17 countries. Its main activities are detector R&D and optimisation studies for a detector at the Compact LInear Collider (CLIC), as well as the study of physics prospects at CLIC using full detector simulations including pile-up from beam-induced backgrounds.Recently CLICdp has published two important milestone papers; an updated baseline scenario for the staged operation of CLIC <cit.> and a Higgs physics overview paper <cit.>. Additionally a new detector concept has been defined <cit.>. § UPDATED CLIC STAGING BASELINE SCENARIOThe CLIC Conceptual Design Report (CDR) <cit.> was published in 2012, before the discovery of the Higgs boson and as such the Higgs mass was not fully taken into account in the optimisation of the different energy stages. Additionally, in the CDR the CLIC accelerator complex had been optimised for 3 TeV, while the two lower energy stages at 500 GeV and 1.4/1.5 TeV were at that time not fully optimised. Hence, after comprehensive studies of the CLIC performance, of cost and power optimisation and of further Higgs and top quark physics, an updated baseline staging scenario has been defined in which also the lower energy stages are optimised.In the new staging scenario CLIC will run at the following energies; 380 GeV, 1.5;3 TeV, collecting integrated luminosities of 500; 1500; 3000 , respectively.Additionally,a total of 100 of data will be collected in a threshold scan of energies around the top pair production threshold near 350GeV. <ref> shows the footprint of each of the CLIC energy stages in the vicinity of CERN. The CLIC accelerator for 380 GeV will be 11.4long and it will make use of one drive beam complex. The accelerating gradient will be lower than for the higher energy stages, namely 72 MV m. The higher energy stages of CLIC are defined based on the maximum energy attainable with the CLIC drive beam complex; with one complex this is 1.5 TeV and with two 3 TeV. The accelerator will be 29 km long for 1.5 TeV and 50 km for 3 TeV.For these higher energy stages modules with an accelerating gradient of 100 MV m will be used combined with the lower gradient modules of the initial 380 GeV energy stage.The full physics program will span 22 years, with 5 to 7 years of running at each stage.There will be upgrade periods of two years between the different energy stages.<ref> and <ref> show the luminosity and the integrated luminosity for the full staging scenario. For each stage the luminosity will be ramped up, as is illustrated in the figures. The assumption is made that CLIC will operate for the equivalent of 125 full days per year, that is 1.08e7 seconds per year. The motivations that led to the choice of the updated baseline staging scenario are described in detail in <cit.>.A short summary is given below. §.§ Lowest energy stage at 380 GeVThe lowest energy stage of CLIC is aimed at precision Standard Model Higgs and top quark physics. At this energy the Higgs mass can be determined with a statistical precision of about 100 MeV.The dominant Higgs production channel around this energy is Higgsstrahlung, →, as can be seen in <ref>.This process can be used to measure the coupling of the Higgs to the , g_, to a precision of 0.8% at 350 GeV. Combining this production channel with the WW-fusion channel, →, gives access to the Higgs total decay width and to the Higgs coupling to the , g_, via the ratio of the cross-sections. At lepton colliders a model independent determination of the Higgs couplings to fermions and bosons is possible and it depends on the absolute measurement of the Higgsstrahlung cross-section σ(). All other couplings are limited by the precision with which this coupling can be determined.Studies have shown that the best precision can be achieved around 350 GeV, despite the cross-section being higher around 250 GeV. The total Higgs decay width can be found by combining the decay channels →^* and →^*.Additionally, Higgs decays into invisible final states can be determined and used to constrain the invisible decay width of the Higgs.The most precise measurement of the top quark mass can be obtained via a threshold scan around the top pair production threshold at around 350 GeV, as illustrated in <ref>.A total uncertainty in the order of 50can be achieved <cit.>.The top form factors can be accessed via the forward-backward asymmetry combined with cross-section measurements, and can be determined at the percent accuracy, as illustrated in <ref>, where a comparison is made with HL-LHC and ILC. The light green bars include an extra conservative theory uncertainty of 3, as the exact calculations have not been done yet for 380 GeV.Studies are being pursued to determine the form factors above the production threshold, where an increased boost leads to better separation between the decay products of the two top quarks.The top quark is a promising candidate for the detection of physics processes that go beyond the Standard Model (BSM physics).BSM searches with a high statistical accuracy can be performed with top pair events recorded near the maximum of the top pair production cross-section at 420 GeV.However, theory uncertainties are larger near the production threshold, and comparisons with theory would benefit from measurements at higher energies.An energy of 380for the lowest energy CLIC stage follows from the physics described above; this energy is good for both Standard Model Higgs and top physics, as well as BSM physics. At this stage 15 of the running time will be dedicated to a threshold scan of energies around the top pair production threshold, in order to determine the top mass with high precision. §.§ Higher energy stagesThe higher energies offer extended potential in Higgs physics. Above 1 TeV WW-fusion and ZZ-fusion are the dominant Higgs production channels.The high luminosities planned for these energies, combined with high cross-sections, enable the determination of the couplings of the Higgs to fermions and bosons at the 1percent level.However, this precision can only be reached with essential input from the 380 GeV stage.The precision with which the couplings can be determined is illustrated in <ref> and <ref> for model-dependent and model-independent global fits to the data, respectively. The Higgs mass can be determined to 24 MeV through its decay → by combining data from 1.5; 3 TeV.For this measurement polarisation of the electron beam helps to increase the accuracy.Rare processes become available to study, e.g. the top Yukawa coupling can be determined from the process →.Accuracies of 4 5can be achieved (depending on the electron beam polarisation). Additionally, the Higgs tri-linear self-coupling λ can be determined to within 10 at the HHH vertex via the rare process →, by combining the 1.5; 3 TeV measurements. The Higgs self-coupling gives direct access to the coupling parameter of the Higgs potential; it is therefore an essential measurement to establish experimentally the Standard Model Higgs mechanism.Higher energies additionally offer extended assess to BSM physics.Direct searches for new particles can be performed with a 1 accuracy on their mass measurement up to approximately half the centre-of-mass energy. Additionally, for indirect searches of BSM physics CLIC reaches sensitivies beyond the centre-of-mass energy of the collider, through precision measurements of parameters and couplings in the Standard Model. For example, the process → is sensitive to the presence of a high mass boson.<ref> illustrated the 5σ discovery limit (in the minimal anomaly-free model) up to tens of TeV for the as a function of the integrated luminosity.In this study measurements of the cross-section, forward-backward asymmetry and the left-right asymmetry for opposite polarisations of the electron beam are used. As already mentioned in the previous section, BSM physics in the top quark sector suffers from a lower cross section (reduced statistical accuracy) compared to measurements at 380 GeV, but this is compensated by two effects: the top quark reconstruction efficiency is improved through the boost of the top quarks giving a better separation of the top quark decay products, and the relative BSM contribution in many BSM models is expected to increase with centre-of-mass energy.§ NEW CLIC DETECTOR MODEL: CLICDETThe CLIC CDR included two detectors concepts, each based on the detector models developed for the International Linear Collider, but adjusted to the CLIC environment: CLIC_SiD and CLIC_ILD.The beam-induced background levels at CLIC are particularly high in the multi-TeV region; there is a significant radiation of so-called beamstrahlung photons that leads to high rates of incoherent pairs and low p_t →hadrons events. The energy lost through beamstrahlung generates a long lower energy tail in the luminosity spectrum. The detector design is optimised to migitate the effects of the beam-induced backgrounds. The two detector models are being used in the CLICdp physics studies. Optimisation studies on these two models have led to a single new detector model for CLIC, called CLICdet, illustrated in <ref>. A detailed description of the new detector model is given in <cit.>. This model will be used in the next round of physics benchmark studies and has been fully implemented in the detector description toolkit DD4hep <cit.>. The main differences of CLICdet with respect to the CDR detector design are the following; there will be one single detector with an all silicon tracker and a magnetic field of 4T, the magnet return yoke has a smaller outer radius which is possible due to the less stringent requirements on the stray fields outside the magnet, and the last quadrupole magnet, the QD0, will now be placed outside of the detector.This new position of the QD0, provides a much better forward coverage of the hadronic calorimeter.Forward coverage is very beneficial for reconstructing particles at low polar angles, a requirement in a number of important physics scenarios, especially at high energies.While in this configuration CLIC will operate with a larger L* of 6 m, there is no significant loss in luminosity expected in the first energy stage, and for the higher energy stages a luminosity loss of only ∼ 10 is estimated.§.§ Vertex and tracking detectorsThe CLICdet vertex detector consists of a cylindrical barrel closed off in the forward directions by disks.Unlike in the CDR, these disks are arranged to form a spiral geometry allowing efficient air-flow cooling.A spatial resolution of 3 μm, timing at the 10 ns level, and a material budget of 0.2 of X_0 per layer is aimed for. This requires novel sensor techniques which are being investigated in an extensive R&D program. These activities are closely linked to the R&D for the silicon tracker. For the vertex detector thin active edge sensors are being investigated, while for the tracker SOI chips are being developed in parallel to a CMOS test chip based on the one used for the ALICE experiment. New larger chips are being designed, produced and tested and there have been systematic studies on capacitive coupling via finite element simulations, as well as tests of the alignment precision of glue assemblies.The tracker is larger than CLIC_SiD from the CDR.It offers 7 μm spatial resolution and 10 ns timing information. The new tracker layout is illustrated in <ref>. A larger tracker radius helps to improve the track momentum resolution, angular track resolution and jet energy resolution.Especially at the high energy stages an excellent tracker coverage also in the forward region is essential. The position of the support tube, that divides the tracker into an inner and an outer part, is placed at a larger radius, enabling tracking very close to the beam pipe, as is demonstrated in <ref>, which shows the number of hits that are measured per muon track as a function of its polar angle.The inner part consists of 3 barrel layers and 7 disks, while the outer layers also consist of 3 barrel layers and 4 disks. The largest challenge in the tracker design is to accommodate the large tracker volume with a material budget of only a little more than 1 of a radiation length per active layer. This is potentially realised in the thin sensors and hybrid systems under study, but also requires R&D on the support structure in order to achieve this low material budget. §.§ Calorimeters The electromagnetic calorimeter (ECAL) consists of 40 layers of tungsten absorbers interleaved with silicon sensors.It is based on a prototype from the CALICE Collaboration and has a high sensor granularity <cit.>. Its inner and outer radius are 1.5m and 1.7 m, respectively. The ECAL presented in the CDR was optimised considering the energy resolution of jets of particles.Since then the optimisation also takes into account the energy resolution of high energy photons, important in a large number of physics scenarios. A cell size of 5x5mm is chosen as it is optimal for jet energy resolution, as can be seen in <ref>. Although the jet energy resolution does not depend strongly on the number of layers, the photon energy resolution does, and 40 layers of 1.9 mm tungsten interleaved with silicon sensors are found to be optimal (see <ref>). Note that the ECAL depth of 22 X_0 is similar to the CDR design. The hadronic calorimeter (HCAL) is designed to achieve a high jet energy resolution. With respect to the CDR design the HCAL has a better forward coverage which improves the dijet invariant mass reconstruction as illustrated in <ref>. The HCAL consists of 60 layers of 19mm thick steel absorbers interleaved with polystyrene scintillator tiles of 30x30x3 mm individually read out by SiPMs, a design that also originates from the CALICE collaboration <cit.>. The total HCAL depth is 7.5 interaction lengths. Performance as a function of the number of layers and the cell size is shown in <ref> and <ref>, respectively.The muon system integrated in the magnet return yoke is smaller compared to the CDR due to the thinner yoke in CLICdet, but it is still sufficient for muon identification. There will be 6 sensitive layers, plus 1 additional layer in the barrel close to the magnet coil. The current design consists of resistive plate chambers (RPC) with 30x30 mm cell size, but alternatively scintillator strips are considered.The forward region of the detector houses additional calorimeters, LumiCal and BeamCal, originating from the FCAL collaboration <cit.>.Their position and radii, illustrated in <ref>, have only slightly changed with respect to the CDR. LumiCal (dark blue in the figure) will measure the luminosity via Bhabha scattering.It consists of 40 layers of tungsten and silicon sensors. BeamCal (light blue in the figure) is installed for beam monitoring and also consists of 40 tungsten layers interleaved with a radiation hard sensor material (e.g. GaAs or diamond). Additionally, these very forward calorimeters provide coverage for electrons and photons down to 10mrad.§ FURTHER CLICDP ACTIVITIESRecently CLICdp has finalised a detailed Higgs physics overview paper <cit.>. It demonstrates the CLIC Higgs physics reach at its 3 energy stages. This paper combines more than 25 independent analyses in a collaboration wide effort.<ref> shows the summary of the precisions obtainable in the Higgs observables for the first stage of CLIC and <ref> shows the precision for the higher energy stages, where rare processes are also accessible. The symbol '-' indicates that a measurement is not possible or relevant at this centre-of-mass energy, while the numbers marked with a '^*' were extrapolated from 1.4TeV to 3TeV. The collaboration is now preparing similarly detailed papers outlining the CLIC potential for top quark physics, and for physics beyond the standard model, that will be finalised in 2017/2018. In view of the next Update of the European Strategy for Particle Physics CLICdp is preparing a CLIC summary report based on the new staging scenario, the new detector model, CLIC technology R&D, and the physics potential overview papers of Higgs, top and BSM physics. This report aims to demonstrate that CLIC is an excellent option for a linear electron-positron collider and currently the only option for a multi-TeV lepton collider.If supported in the next European Strategy CLIC could be the next large collider facility at CERN.§ CONCLUSIONAfter extensive studies of the physics potential, performance and cost, an updated, optimised staging scenario for CLIC has been defined. At the lowest energy stage of 380 CLIC offers a rich physics program of precision tests of Standard Model Higgs and top quark physics, including sensitivity to BSM physics. The higher energy stages at 1.5; 3 TeV offer extended Higgs and BSM physics sensitivity. The potential for Higgs physics at CLIC has been extensively tested within the CLICdp collaboration and recently a paper summarising the full physics reach has been finalised.Similar extensive studies of top quark physics and BSM physics are ongoing.A new optimised single detector model at CLIC has been developed following optimisation studies on the previous detector models. A very active R&D program for novel vertex and tracker technologies is being pursued within CLICdp. Calorimeter R&D is performed in close collaboration with the CALICE and FCAL collaborations.The new detector model has been fully implemented in the simulation and reconstruction framework DD4hep and will be used for the next physics benchmark studies. The full program of physics studies at CLIC will demonstrate that the staged operation of CLIC offers a physics programme that reaches far beyond the HL-LHC, making CLIC an excellent option for a next facility at CERN. [title=References] | http://arxiv.org/abs/1703.08876v1 | {
"authors": [
"Naomi van der Kolk"
],
"categories": [
"physics.ins-det",
"hep-ex"
],
"primary_category": "physics.ins-det",
"published": "20170326222042",
"title": "CLIC Detector and Physics Status"
} |
APS/[email protected] Helmholtz-Zentrum Berlin fuer Materialien und Energie (HZB), Germany TRIUMF Canada's National Laboratory for Particle and Nuclear Physics, VancouverEuropean Organisation for Nuclear Research (Cern),Geneva SwitzerlandEuropean Organisation for Nuclear Research (Cern),Geneva SwitzerlandPaul Scherrer Institut (PSI), Villigen, Switzerland Argonne National Laboratory, USA Commissariat de l'énergie atomique et aux énergies renouvelables, FrancePaul Scherrer Institut (PSI), Villigen, SwitzerlandPaul Scherrer Institut (PSI), Villigen, Switzerland European Organisation for Nuclear Research (Cern),Geneva Switzerland Sheffield University, UKIllinois Institute of Technology, Chicago, USAPoint contact tunneling (PCT) and low energy muon spin rotation (LE-μSR) are used to probe, on the same samples, the surface superconducting properties of micrometer thick niobium films deposited onto copper substrates using different sputtering techniques: diode, dc magnetron (dcMS) and HIPIMS. The combined results are compared to radio-frequency tests performances of RF cavities made with the same processes. Degraded surface superconducting properties are found to correlate to lower quality factors and stronger Q slope. In addition, both techniques find evidence for surface paramagnetism on all samples and particularly on Nb films prepared by HIPIMS.A low energy muon spin rotation and point contact tunneling study of niobium films prepared for superconducting cavities J. Zasadzinski December 30, 2023 ======================================================================================================================== § CURRENT LIMITATIONS OF NIOBIUM ON COPPER CAVITIES Superconducting cavities prepared by coating a micrometer thick niobium film on a copper substrate enable a lower surface resistance compared to bulk niobium at [4.5]K the operation temperature of several accelerators using this technology, like the LHC or the HIE-Isolde at CERN. Additionally Nb/Cu cavities are more cost-effective and do not require magnetic shielding. Thermal stability is enhanced avoiding quenches <cit.>.Despite these advantages, this technology is currently not being considered for accelerators requiring highest accelerating gradient Eacc or lowest surface resistance RS at temperatures of [2]K and below. The reason for this is that RS increases strongly with Eacc. The origin of this field dependent surface resistance has been the subject of many past and recent studies <cit.>, but is still far from being fully understood. The film thickness of about [1.5]μm is large compared to the London penetration depth λL of about [32]nm. Differences in the performance (apart from thermal conductivity issues <cit.>) should therefore be correlated to the manufacturing procedure and the resulting surface structure. Since no single dominant source can be expected, several hypotheses need to be addressed individually to identify their origin and possibly mitigate their influence on the surface impedance and SRF cavity performances. §.§ Surface magnetism, a possible source of RF dissipation A possible source of dissipation under RF fields is surface magnetism in the oxide layer; magnetic impurities have a well known deleterious effect on superconductivity, causing inelastic scattering of the Cooper pairs <cit.> that increase the surface resistance RS and lower the quality factor Q∝ 1/RS of superconducting RF cavities. The presence of localized magnetic moments on bulk niobium samples was revealed for the first time by Casalbuoni et al. in 2005 <cit.> by SQUID magnetometry. The AC magnetic susceptibility was measured in an external magnetic field of [0.7]T on Nb samples prepared with standard SRF cavity treatments, i.e. buffered chemical polishing (BCP), electropolishing (EP) and low temperature baking at about 120 ^∘C. In all cases, the samples displayed a Curie-Weiss behavior below [50]K indicative of the presence of localized magnetic moments with antiferromagnetic correlation. Later, Proslier et al. <cit.> used the point contact tunneling spectroscopy (PCTS) technique to study samples prepared in a similar manner to SRF cavities. These samples exhibit a broadened density of states (DOS) compared to ideal superconductors with a finite zero bias conductance attributed to inelastic scattering processes. In addition, some tunnel junctions show large zero bias conductance peaks (ZBP) that survive in an external magnetic field above the second critical field H_C2 of Nb (≈[0.24]T). The temperature dependence of the ZBP measured under an external DC magnetic field of [0.55]T <cit.> was found to be consistent with Kondo tunneling which describes the interaction between tunneling electrons and localized magnetic moments present in the oxides or near the interface native oxide/bulk niobium. In light of these results, the broadened DOS was further examined and the tunneling conductance spectra were fitted using the Shiba theory that calculates the effects of diluted magnetic impurities on the superconducting DOS. The very good agreement between the theory and the experimental data provides a plausible microscopic origin for the inelastic scattering processes found in Nb samples.Further theoretical work was carried out to quantify the influence of magnetic impurities on the RF surface impedance <cit.>. The numerical simulations showed that magnetic impurities, present in small quantities (∼ few hundred ppm), cause the saturation of the surface impedance at low temperature, an effect called the 'residual resistance' which is observed experimentally on all cavities. More recently, PCT studies have been extended to cut-outs from cavities that showed pronounced medium field Q-slope. Some samples, labeled as 'hot spots', were cut out from regions that showed strong dissipation as measured by thermometry <cit.> and compared to samples cut out from regions that did not, labeled as 'cold pots'. The results show that tunneling spectra measured on hot spots have in average higher inelastic scattering parameters and lower gap values than cold spots. In addition, about [30]% of the junctions measured in hot spot samples showed zero bias peaks whereas almost none were measured in cold spot samples. These results further indicate a correlation between the presence of magnetic impurities and the dissipation measured in SRF cavities. Various explanations have been proposed to account for the presence of magnetic moments at surfaces. In general, several metal oxide layers and oxide nanoparticles <cit.> were found to develop magnetic properties in otherwise nonmagnetic oxides. It is interesting to note that only oxide nanoparticles with a size of about [10]nm showed magnetic moments whereas if the material was pressed into a bar and sintered the samples became diamagnetic. Besides, native niobium oxides are thermodynamically stable with substantial off-stoichiometry and Cava et al. <cit.> showed that oxygen deficient Nb_2O_5-δ samples develop magnetic moments associated with unpaired Nb4d electrons. Recent density functional theory (DFT) simulations <cit.> revealed that these local moments form 1d spin chains with weak antiferromagnetic correlations coupled to perpendicularly oriented conducting nano-wires that can be traced back to the NbO_6 octahedral structure.It has to be noted that although magnetic impurities can cause surface dissipation, other impurity phases with lower superconducting properties than niobium such as Nb hydrides <cit.> or carbon contamination <cit.> that have been found on cavity-grade niobium samples and cut outs can also be responsible for higher surface resistance. For Nb on Cu cavities, another dissipation mechanism extrinsic to superconductivity, the variable Nb-Cu interface thermal conductivity caused by inhomogeneous Nb film adhesion to the Cu substrate, has been proposed <cit.> to explain the strong Q-slope measured. § SAMPLE PREPARATION AND SURFACE CHARACTERIZATION Several coating techniques are presently exploited at CERN. Direct CurrentMagnetron Sputtering (dcMS) is the technique of choice for the 4-cell [352]MHz elliptical cavities previously used for the Large Electron-Positron (LEP) collider <cit.> and the single cell [400]MHz elliptical cavities currently installed in the LHC <cit.>. This technique is constantly being developed by making use of smaller scale test cavities operating at 1.3 or [1.5]GHz, which allow a faster turn-over and easier implementation of new coating and surface processing ideas. A dummy cavity is used for the preparation of the coupons, located at its equator, i.e. the cavity region with the largest diameter corresponding to the peak magnetic field. Typical dcMS sputtering conditions <cit.> require a DC power of [1]kW in Kr plasma, resulting in [1.5]μm thick films in [15]minutes coating time. Coatings are performed at [150]^∘C. High-power Impulse Magnetron Sputtering (HIPIMS) is also being developed on [1.3]GHz elliptical cavities as a potential technique to mitigate field dependent losses <cit.>. In HIPIMS the high current, high density Kr plasma ionises the Nb from the target thus allowing more dense films to be formed. The HIPIMS coating was performed on a coupon with the same identical setup as for dcMS, at similar average power of about [1]kW. The peak plasma current was [200]A, with [200]μs pulses at a repetition rate of about [100]Hz. The coating duration, owing to the lower efficiency of HIPIMS, is adapted in order to obtain a similar thickness as for dcMS.For the HIE-Isolde quarter wave [100]MHz cavities DC bias diode sputtering is used which is easier to apply to these more complex geometries <cit.>. A mock-up cavity has been designed as a sample holder to characterize the Nb layer as described in <cit.>. The samples measured in this study are taken from the top of the cavity, named i9 and TBi for for PCT and LE-μSR measurements respectively, as depicted in Fig. <ref>. This position corresponds to the peak RF magnetic field region of the cavity, which is one of the most critical parts form RF point of view. The thickness of the niobium layer at these position is [2.7]μm.The coating has been made in 15 successive runs of 23 minutes each at high temperature (from [300]^∘C to [620]^∘C) with [5.5]hours of cool-down in between each run resulting in a total time of 4 days. The setup consists of a cylindrical Nb cathode coaxially mounted around the inner conductor of the cavity and is used to coat inner and outer conductor. The sputtering is made at [8]kW with Ar.Different microstructures are obtained for the three sputtering techniques. In particular the grain size of the HIE-Isolde dc bias coated samples (≈[500-1000]nm) <cit.> is roughly 10 times larger compared to HIPIMS [50-100]nm and dcMS [100-200]nm <cit.>, see Fig. <ref> In this study one sample of each technique has been investigated by PCT and LE-μSR.§ EXPERIMENTAL RESULTS In this section results from point contact tunneling performed at Argonne National Laboratory and low energy muon spin roation (LE-μSR) carried out at the Paul Scherrer Institute are presented. The samples tested by these two methods were produced simultaneously at CERN. The results are being correlated to the RF performance of cavities produced using the same sputtering setups and parameters.§.§ RF cavity tests RF performance tests have been carried out on cavities prepared by all three coating techniques using the phase locked loop technique <cit.>. In these tests one usually obtains the unloaded quality factor Q_0 as a function of the accelerating gradient Eacc. Both parameters do not only depend on the material properties but also on the cavity geometry. To compare test results from differently shaped cavities in terms of their material properties it is more meaningful to plot the surface resistance RS as a function of the peak surface magnetic field Bp instead of Q_0 versus Eacc. While Eacc and Bp are directly proportional to each other, RS can be obtained from Q_0 using the geometry factor G, which has to be calculated by a simulation for realistic cavity shapes <cit.>. Assuming RS does not vary over the cavity surface it can be written as R_S=G/Q_0.Table <ref> lists the geometry factor and the ratio Bp/Eacc for all cavities used in this study. Note that the dcMS and HIPIMS coatings have been deposited on cavities of same shape but slightly different size. Therefore G and Bp/Eacc are identical and only f differs. To compare the RF performance of the DC biased diode coating deposited on the [100]MHz HIE-Isolde cavities to the dcMS and HIPIMS coatings, not only the cavity shape, but also the different RF frequency needs to be considered. Losses from thermally activated quasiparticles, so called 'BCS losses' scale exponentially with temperature and quadratically with frequency. A practical formula to quantify these is <cit.>RBCS=Aω^2 exp(-Δ(0)/kBT), where ω=2π f, Δ(0) is the superconducting energy gap at [0]K and A depends on material parameters. At a temperature of [4.5]K RBCS equals only a few nΩ at f=[100]MHz, while at [1.3]GHz it is at least several [100]nΩ in case of niobium. Operation of superconducting cavities at such high RS becomes uneconomical. Therefore the elliptical HIPIMS and dcMS cavities are designed for and usually measured at [1.8]K, while the operation temperature of the HIE-Isolde accelerator is [4.2]K. However test results of the best HIPIMS cavity at [4.2]K are also available. Additional to the BCS losses there are residual losses, which do not depend on temperature. The total surface resistance RS thus equalsRS=RBCS+Rres.The origin of the residual resistance Rres and how it scales with frequency is not completely understood. Experimentally it has been found that both RBCS and Rres depend on the applied RF magnetic field strength. Especially Nb/Cu cavities show a strong increase of Rres with applied RF field <cit.>.All cavity substrates were prepared using a polishing agent named SUBU, which is a mixture of sulfamic acid, hydrogen peroxide, n-butanol and ammonium citrate which is a standard technique for Nb/Cu cavities <cit.>. The elliptical cavities also received electropolishing (EP) prior to SUBU, which can provide a smoother substrate. RF test results from the following cavities have been chosen for this study: * HIPIMS M.2.3: Best performing HIPIMS cavity with EP+SUBU substrate (tested at 1.8 and [4.2]K), * dcMS H.6.8: Best performing dcMS cavity with EP+SUBU substrate (tested at [1.8]K only), * HIE-Isolde QP.1.4: Best performing HIE-Isolde cavity with SUBU substrate (tested at [4.2]K only). Figure <ref> shows RS as a function of Bp for these three cavities. The performance of the two elliptical cavities at [1.8]K is fairly similar. The HIE-Isolde cavity has the lowest RS at [60]mT even if this result is compared to the measurements at [1.8]K of the elliptical cavities. However if RS is quadratically scaled to [1.3]GHz as suggested by Eq. <ref> the performance is far weaker compared to the HIPIMS cavity at same temperature. This plot illustrates the difficulty to compare the performance of the elliptical HIPIMS and dcMS cavities to the HIE-Isolde quarter wave cavity due to different frequency and geometry. An heuristic approach to work around this issue is to compare each cavity type to similar ones made of bulk niobium and then in turn compare the performance differences to each other to get at least a rough idea about the RF performance independent of resonant frequency and cavity shape. Suitable devices for this comparison are the elliptical [1.3]GHz cavities and [100]MHz quarter wave resonators from TRIUMF, both manufactured from niobium sheets.In particular RF test results from these two cavities are taken: * PAV-2: A single-cell [1.3]GHz elliptical cavity produced as a prototype cavity for the ARIEL linac and tested at 1.8 and [4.2]K. The RF parameters are identical to the HIPIMS cavity. * ISAC-II-SCB5-3: One of the best performing [106]MHz cavities used in the ISAC-II accelerator. The RF parameters are listed in Tab. <ref> Figure <ref> shows RS for the two quarter wave cavities at [4.2]K. The residual resistance Rres at low Bp is significantly larger for the HIE-Isolde cavity while the increase of RS with field is similar. Figure <ref> shows RS for the three elliptical cavities. At [4.2]K the performance of the HIPIMS cavity exceeds the bulk niobium cavity especially at low field. This can be related to the shorter electron mean free path, resulting in a lowered RBCS <cit.>. At [1.8]K and low field RS is lowest for the HIPIMS cavity. However, at this temperature the well-known limitation of the niobium on copper technology becomes apparent; Rres increases strongly with the applied field strength. One might argue that the performance of the HIE-Isolde coating is weaker compared to the other two coatings, since it does not outperform bulk niobium at [4.2]K. However, it has to be noted that the performance of the [1.3]GHz cavities at [4.5]K is clearly dominated by BCS losses while in case of the [100]MHz quarter wave resonators residual losses are still relevant. Measurements at lower temperature could serve to quantify the extend of the two contributions. Unfortunately, such data is not available for the resonators being studied here. §.§ Point contact tunneling (PCT) PCT is a powerful technique that measures the surface density of states (DOS) and the related superconducting properties: the superconducting gap Δ, the critical temperature Tc and the phenomenological inelastic scattering parameter Γ using the native oxide layer as tunnel barrier <cit.>. These parameters play a crucial role in the SRF cavity performances; at low external magnetic field intensities (neglecting vortices dynamic) RS depends on the DOS at the fermi level ν (E_f) that in turn depends on Δ, Γ and the temperature for a superconductor. Qualitatively, in an ideal superconductor and for temperatures T ≪ Tc, ν (E_f) ∝ e^-Δ/k_B.T and lower Δ values increase the RS and therefore lower the quality factor Q_0. Inelastic scattering processes however yield a minimal ν (E_f) ∼Γ/Δ value independent of temperature that entails a saturation of the surface resistance at low temperature, Rres, and limits the cavity quality factor. The total surface resistance, RS, is a combination of these two effects.In order to understand the SRF cavity performances it is therefore crucial to measure the surface DOS for each new surface treatment or deposition technique. We present here the results obtained by PCT on three Nb on Cu films few microns thick and prepared with the different deposition techniques introduced above.The summary of the point contact tunneling results on the three samples measured at [1.6]K is displayed in Fig.<ref>. The gap, Δ, and the phenomenological inelastic scattering parameter, Γ values are extracted from fits of the conductance curves using the Blonder-Tinkham-Klapwijk (BTK) model <cit.>. For each sample, the mean values of the Δ and Γ/Δ distributions are shown. As apparent in Fig. <ref>, the HIPIMS (a) and the dcMS samples (b) have similar mean Δ (1.48 meV and 1.5 meV) and Γ/Δ (7.6% and 6.5%) values. These data suggest that the surface impedance of unprocessed Nb film on copper cavities made by HIPIMS and dcMS should be the same, in agreement with RF tests done on cavities as seen in Fig. <ref>. Furthermore, these results are very similar to previous measurements done on bulk niobium cut out from cavities treated with standard recipes and consistently, very close Rres values of ∼[10]nΩ at low field have been measured on both types of cavities at low RF field.The Nb/Cu sample made by diode sputtering Fig. <ref>(c) shows a much broader gap distribution than the other two samples with a very low average Δ of [1.3]meV and a Γ/Δ value of 4.6%. These results affect the RS in opposite way: lower average Δ should increase RS whereas lower Γ/Δ tends to decrease it. In the absence of RF tests done on cavities with the same frequency and temperature it is hard to correlate the PCT data obtained on the diode sputtered sample to the RF performances.All the samples show a majority of the tunneling spectra, between ∼ 70 % to ∼ 90 % depending on the samples, with high quality superconducting DOS characteristic of bulk Nb. However, the data also systematically reveals very low gap values ≤ 1.3 meV and sometimes even non-superconducting junctions (Δ=0 meV) such as the one displayed in Fig. <ref>(d). Investigating these tunneling spectrum in more detail, we found that for all the samples the low Δ junctions have large Γ/Δ values of ≥ 7 % up to 35 % indicative of some weak superconducting and potentially highly dissipative spots present on the surface. These spectra account for 4 to 8% of the junctions measured. Similar results have only been obtained previously on hot spots bulk niobium samples cut out from cavities with a strong medium field Q-slope <cit.>. Such correlation between strong Q-slope and very weak superconducting properties is not surprising; as the external magnetic field intensity is increased in the cavity, regions of the surface with lower gap values than bulk Nb (caused by proximity effect, magnetic or some non-magnetic impurities, deleterious phases such as hydrides or carbides) will serve as easy flux entry points and decrease the depairing critical current value. These effects combined suggest a runaway mechanism and weakened superconductivity generates dissipation exponentially dRS/dH ∝α RS, causing a medium field Q slope observed experimentally on some bulk Nb cavities and on all the Nb on Cu cavities (Fig. <ref>). What is more surprising is the fact that these regions do not seem to influence the value of the Q factor at very low field; bulk niobium cavities that do not have strong Q-slope or/and very low gap values as measured by PCT, show similar Q values at very low field ∼ 2.10^10 corresponding to a residual resistance of less than [10]nΩ. A possible explanation is that the good thermal conductivity of the surrounding clean Nb can remove efficiently the small amount of power generated by these spots at very low field.The microscopic origin for these degraded properties is unknown yet, there are however major structural and chemical differences between Nb films grown on Cu and bulk Nb samples: the average grain size ≈ [50-1000]nm is 2 to 3 orders of magnitude smaller than for fine grain bulk niobium and the impurity concentration <cit.> (mostly C and O) is about 10-100 times larger in Nb films. It is therefore possible that the low Δ junctions corresponds to grain boundaries regions with weakened superconductivity due to impurity segregation or porosities near the surface as seen on some thin Nb film cross sections <cit.>. Assuming a circular grain shape of size d with a weak superconducting ring of thickness 2ξ around the grain boundary, the probability, P of probing such a region with a constant steps size between junction measurement is ∝ 2ξ/d. Assuming ξ=[40]nm (≪ d) and d∼ [1000]nm, P∼ 8%. In this picture, the majority of the tunnel junctions with high quality superconducting DOS comparable to bulk niobium cavities are measured within the grains with gap values at T=[1.6]K consistent with bulk Nb of 1.5 to [1.6]meV and a typical BCS temperature dependence as shown on Fig. <ref>(c) with critical temperatures Tc from 9.25 to [9.4]K. While this simple model is consistent with the data measured on dc diode sputtered sample (10%), it fails to explain why the degraded properties are more often observed for this sample compared to the other two with smaller grain size, suggesting that for the dcMs and the HIPIMS samples grain boundaries do not systematically host low TC phases. Some tunnel spectra show very strong peaks in the conductance curves at a voltage ≥Δ, see Fig. <ref>(e) and (d). These features are present in all samples and particularly on the diode sputtered one. The peak positions are, in general, not symmetric with respect to the Fermi level and can hardly be attributed to charging effects. They are can be attributed however to localized defects in the tunnel barrier, here the Nb native oxide, with well defined energy levels within the oxide band gap. Coincidentally, the same sample also has the lowest mean average gap. It is therefore possible that these conductance peaks originate from impurities sometimes present within the tunnel barrier, sometimes underneath the oxide in the superconductor where they would be responsible for the lower gap values. To confirm this hypothesis more structural and chemical analysis are needed. These results have been reproduced on another diode sputtered sample made under similar conditions (not shown).Finally, some tunneling spectra such as the ones displayed in Fig. <ref>, show strong zero bias conductance peaks (ZBP) identical to spectra measured previously <cit.> on bulk Nb samples cut out from cavities with strong medium field Q-Slope and characteristic of the Kondo effect. Most zero bias peaks are found on the HIPIMS sample with ∼ 12% of the junctions measured, the HIE-Isolde sample has only a few ∼ 2%, and the dcMS sample has ∼ 5%. Assuming these ZBPs are caused by magnetic impurities located along grain boundaries, the probability of probing such a feature should be proportional to the grain size, which is consistent with the trend found in the measurements. These results confirm the presence of magnetic impurities on the surface of these niobium on copper samples, probably localized at grain boundaries, with the highest concentration on the HIPIMS one. Similar results have been reproduced on another HIPIMS sample (not shown). The latter sample also has the highest mean inelastic scattering parameter. Fitting the quasiparticle peaks symmetric with respect to the Fermi level one can estimate Γ/Δ and Δ values for each spectrum that shows a ZBP; the average Γ/Δ value is 12 % for the HIPIMS and 9.5 % for the dcMS sample, well above the overall average for both samples. This result indicates that indeed the presence of magnetic impurities tends to increase the average inelastic scattering values and hence the dissipation in cavities. The HIPIMS and dcMS cavities performances however are qualitatively similar (low field RS and Q-slope) indicating that magnetic impurities do not play a dominant role in dissipation processes for thin Nb films.§.§ LE-μSR measurementsMuon spin rotation relaxation and resonance (μSR) is an experimental technique providing information at an atomic level on thephysical properties of matter. It is based on the implantation of spin-polarized muons and the measurement of the evolution of their spin with time due to the magnetic field experienced by the particle <cit.>. The main applications of this technique are in magnetic materials and in superconductors. Muon beams of different energies and corresponding implantation depth are available at PSI, TRIUMF, J-PARC, ISIS and RIKEN-RAL. For the studies of the niobium surfaces we are interested in the outermost [100]nm of the sample. A suitable beam is currently only available at the Laboratory for Muon Spin Spectroscopy (LMU) at PSI <cit.>. §.§.§ Direct measurement of the London penetrationLow energy muon spin rotation (LE-μSR) enables the measurement of the magnetic field inside a sample as a function of depth. An external magnetic field with a value below Hc (type-I superconductor) or Hc1 (type-II superconductor) can be applied parallel to the sample surface to probe the London penetration depth. This has been done first in 2000 <cit.> and more recently for bulk niobium samples prepared for superconducting cavities <cit.>. In the latter study Romanenko et al. showed that there is a depth dependent mean free path for samples exhibiting low losses under RF fields after a [120]^∘C bakeout. These results gave the motivation to study niobium on copper samples to search for similar variations in the penetration profile. Figure <ref> shows the penetration profiles obtained from analyzing the LE-μSR data using the musrfit software <cit.>. Two fit models were used to calculate each data point. A simple Gaussian model based on London theory and a numerical time-domain model based on the non-local Pippard/BCS model <cit.>. The global Pippard fits have been performed to the whole raw data set. For the HIE-Isolde sample this fit did not converge. Therefore no global Pippard fit is displayed in Fig. <ref> for this sample. The deposition here is done in several runs with breaks in between. This might have resulted in an inhomogeneous penetration profile inconsistent with one set of global fit parameters. The penetration depth λ of the dcMS sample is so large that non-local effects become unimportant. The obtained λ is therefore consistent for the local and the non-local fit. For the HIPIMS sample, however, non-local effects are more relevant. Clear deviations from the results between local and non-local data analysis were found. Using [1.3]GHz superconducting cavities the penetration depth was also obtained by measuring the frequency shift as a function of temperature for the dcMS and the HIPIMS technique, see Appendix and <cit.>. These values are systematically higher than the values obtained by μSR, see Tab. <ref>. It should be noted that with RF one only measures the penetration depth change in the temperature range between about 4 and [9]K and obtains the penetration depth at [0]K, λ_0, as a fit parameter. Also this method relies on an exponential decay of the magnetic field. Comparing RF and LE-μSR results from the different samples we find that with HIPIMS and diode sputtering at high temperature films with a lower penetration depth and therefore lower impurity concentration can be produced. However the dcMS cavity, corresponding to the sample with the largest λ, performs best under RF fields of the three thin film cavities tested, see Fig. <ref>. The cause of the field dependent residual surface resistance must be related rather to localized impurities, possibly along grain boundaries or to other loss mechanism like the adhesion of the film on the substrate <cit.>.§.§.§ Zero field measurements A strength of μSR compared to e.g. NMR is the possibility to measure internal magnetic fields under zero applied field conditions. Here the implanted muon is measuring the local magnetic field at the muon site with high sensitivity. In case of a static random field distribution, e.g. nuclear dipole fields, and the absence of muon diffusion, the muon spin polarization is given by P_ ZF^ stat.(t) = 1/3 + 2/3[ 1 - (σ t)^2 ] exp[-1/2 (σ t)^2], where σ / γ_μ is the second moment of the magnetic field distribution at the muon site, and γ_μ is the muon gyromagnetic ratio. A point dipole calculation for the expected interstitial muon site in Nb results in σmag≃ 0.5 μ s^-1. If other static and randomly distributed sources of local magnetism, e.g. magnetic impurities, are present the total σ becomesσ^2 = σdip^2 + σmag^2,where σdip is the nuclear dipole contribution and σmag is originating of other sources, e.g. magnetic impurities. The static polarization function P_ ZF^ stat.(t) (Eq. (<ref>)) is modified in presence of fluctuating magnetic fields and/or muon diffusion. In a strong collision model the resulting muon spin polarization function is P(t)=P_ ZF^ stat.(t) exp(-ν t) ++ ν∫_0^t dt^'{ P(t-t^') P_ ZF^ stat.(t^') exp(-ν t^') }, where ν is the fluctuation rate (1/ν = τ being the correlation time). In the so called extreme motional narrowing limit ν≫σ, Eq.(<ref>) applied to Eq.(<ref>) results in P_ ZF^ fast(t) = exp(- λ_ mn t),λ_ mn = 2 σ^2/ν. Fast diffusion as well as fast fluctuating (nano to femto-second range) magneitc impurities will therefore yield an exponential depolarization function and in the extreme motional narrowing limit a constant depolarization function. For dilute impurities this function will be superimposed with the depolarization function of muons not sensing areas with magnetic impurities. Zero field measurements in order to search for hints of magnetic impurities were carried out at energies of 3.3 and [26.3]keV corresponding to mean muon stopping depths d of 17.3 and [104.8]nm for all three samples.Individual curves have been fitted to the dynamic Gaussian Kubo-Toyabe function, Eq. <ref>.The dcMS and the HIE-Isolde sample show only weak signs of dynamics and broadening beyond the expected nuclear dipole broadening, while the HIPIMS sample shows clear signs at both measured energies (see Tab. <ref>). For the HIPIMS sample σ is significantly exceeding the nuclear dipole contribution, [For comparison measurements on bulk niobium samples of different treatments (EP, EP+[120]^∘C, BCP, N doping) gave very similar ν values of around [0.4-0.48]Mhz for all samples <cit.>.] and the largest ν is found. This sample has the smallest grain size. Hopping should thus be strongly diminished <cit.>, indicating a different origin for the observed muon dynamics. Comparison with the PCT results suggests that magnetic impurities located at grain boundaries might be the cause.§.§.§ Longitudinal field studies of the HIPIMS sampleThe zero field measurements presented above indicate that surface magnetism is indeed present at least for the HIPIMS sample.However muon diffusion as the cause for these fluctuations cannot be ruled out. Also it has to be noted that the depolarization rate, σ, and the fluctuation rate, ν, are strongly correlated if both are used as individual fit parameters. Of the samples investigated above the HIPIMS sample showed the strongest hints of magnetic impurities. It was therefore decided to study the very same sample as investigated by PCT above in depth with LE-μSR. [Here it should be noted that the two studies were first started in parallel with different samples prepared simultaneously in the same deposition run. After completion of the PCT measurements these samples became available for the LE-μSR study presented in the following.] By applying different longitudinal fields and fitting the data simultaneously to a common σ value it is possible to get a better estimate for σ and ν. In order to further suppress possible systematic errors half of the data has been taken with the muon spin aligned in the direction of muon momentum and half of the data with the spin in opposite direction. Figure <ref> displays the asymmetry function versus time at [50]K for longitudinal fields of 0, 2 and [10]mT. The upper curves display data where the initial polarization was in the direction of the muon momentum. The hop rate,ν, as a function of temperature is displayed in Fig. <ref>. To understand the results they need to be compared with previous μSR studies on niobium. For a brief literature review refer to the Appendix. The most apparent feature is the minimum observed at [50]K. In studies where the muon has been implanted in the bulk this feature has also been observed and was correlated to interstitial impurities <cit.>. A more recent study on cavity grade bulk niobium <cit.> has also found a minimum at [50]K. It has to be noted that in this study the variation of the hopping rate with temperature was much stronger than what is observed here. Also in <cit.> ν was obtained from zero field measurements only. Samples prepared by sputter coating have impurity concentrations about 10-100 times larger than RRR niobium sheets. The total impurity concentration of the samples analyzed here can be estimated to be in the [1000]ppm range <cit.>. A sample with such a high interstitial impurity concentration has been analyzed in <cit.>. The depolarization rate found was similar as for material of higher purity at low temperature in accordance with the expectation for static muons.Unlike for samples of higher purity however σ and therefore ν did only weakly depend on temperature indicating stronger trapping [Note that in this study σ was taken as the only parameter to account for muon dynamics.]. The temperature dependence displayed in Fig. <ref> significantly differs from what has been observed in <cit.> and therefore suggests an additional cause for the muon dynamics in addition to hopping, which might be indeed paramagnetic impurities as suggested from the zero field LE-μSR and the PCT studies. §.§.§ Zero field studies of the HIPIMS sample with a N_2 overlayerThe following study is carried out to distinguish between magnetic and non-magnetic impurities. If these impurities are magnetic they would contribute to the depolarization in two ways. First they would, like nonmagnetic impurities, act as trapping sites. This effect would decrease the hopping rate ν. Second if the paramagnetic correlation time would be in the femto to nano-second range it would result in an increased ν, since the muon would see a time varying field in its lifetime. Unfortunately it is impossible to extract the influence of the latter mechanism from the data. The correlation time of paramagnetic moments in the niobium oxides is also unknown. To overcome this limitation a [180(20)]nm N_2 layer has been grown on top of the niobium sample and the muons have been stopped in this N_2 layer, close to the niobium. Figure <ref> displays the muon stopping profile obtained from the Monte-Carlo code[This program has been developed at MPI Garching by W. Eckstein <cit.> and adopted and experimentally verified for muons <cit.>.].From previous studies it is known that muons are static, that means they do not diffuse, in nitrogen as grown under the given conditions. Nevertheless an additional measurement has been performed to verify this important assumption. A [1.83]μm thick nitrogen layer has been grown on a Ni coated sample plate. The Ni coating has a thickness of about [2]μm. The N_2 layer thickness has been chosen since the static stray fields of the Ni plate increase the depolarization rate σ and therefore enhance the sensitivity in this cross-check experiment. The asymmetry function has been measured at [10]K in zero field, see Fig. <ref>. The data can be well described by a dynamic Kubo-Toyabe depolarization function with a very low hop rate or even a static Gaussian Kubo-Toyabe depolarization function, see Tab. <ref>. This clearly demonstrates that the muons are indeed static, i.e. not diffusing, in the N_2 layer. For the muons stopped in N_2 on top of the HIPIMS sample the static Gaussian Kubo-Tuyabe function cannot give a reasonable fit, since the asymmetry function does not relax to 1/3 of its initial value as expected for static muons. In fact the signal clearly shows a dynamic response, which further supports the presence of magnetic impurities in these films, since muon diffusion is fully suppressed here and hence the origin of the fluctuation rate can only be caused by magnetic fluctuations present in the niobium. Using a dynamic Kubo-Toyabe function instead gives an excellent fit to the data, see Fig. <ref> and Tab. <ref>. § SUMMARY AND CONCLUSION In this study niobium films for SRF application, sputter coated by three different techniques on copper substrates, have been investigated by point contact tunneling (PCT) and low energy muon spin rotation and relaxation (LE-μSR). The results from these material science techniques have been compared to RF measurements (surface resistance and penetration depth) of cavities prepared with the same setups and sputtering parameters. The goal was to get insight into the field dependent residual surface resistance currently limiting the application space of this technology to moderate accelerating gradients.For one sample (HIPIMS) the combined results give strong evidence for magnetic impurities, while there is little evidence found on the other two samples prepared by DC diode and DC magnetron sputtering (dcMS). The fraction of tunneling junctions exhibiting zero bias peaks, indicative of magnetic impurities, correlates inversely to the grain size. It is therefore likely that these impurities are located along grain boundaries, since the smaller the grain size the more likely it is to probe a grain boundary with PCT. Comparing this finding to the RF performance, it is however unlikely that pairbreaking caused by these magnetic impurities plays the dominant role for the field dependent residual surface resistance, since the dcMS and the HIPIMS cavities show a very similar RF performance. A comparison to the HIE-Isolde quarter wave resonator with the dc diode coating is difficult due to the lower resonance frequency and the unknown frequency dependence of the residual resistance. The dc diode sputtered sample showed smaller gap values Δ than the other two samples. Assuming that all RF losses scale quadratically with frequency, like BCS losses, this is in agreement with its weaker RF performance. Very low Δ values are found on all samples indicative of weak superconducting spots. These areas account for about 4-8% of all measured junctions. For bulk niobium such a feature has only been found for samples cut out from cavities with strong field dependent losses. It has been shown in <cit.> that such sub gaps states can be suppressed by high temperature baking. However, for Nb/Cu cavities such a treatment is not possible due to the lower melting temperature of copper. RF frequency shift and LE-μSR results show that with dc bias coating at higher temperatures and HIPIMS one can obtain a film with significantly longer electron mean free path and therefore lower impurity concentration compared to standard dcMS. However, neither of the two techniques yields an RF performance exceeding dcMS films, indicating that the electron mean free path does not play an important role for the field dependent residual surface resistance of Nb/Cu cavities either. The samples used in this study have been prepared in the same deposition chambers used for the actual cavities. However, it has to be noted that the RF losses have not been measured on the very same samples. It is well known that RF losses in superconducting cavities may not be evenly distributed but are often dominated by a few hot spots on the surface. Future work should then concentrate in testing hot and cold spots cut out from Nb/Cu cavities or samples measured in an RF sample test cavity by PCT and LE-μSR to directly correlate the RF performance to the superconducting properties of Nb/Cu samples.§ ACKNOWLEDGMENT The μSR measurements were performed at the Swiss Muon Source (SμS), at the Paul Scherrer Institute in Villigen, Switzerland. This work has been funded partly by a Marie Curie International Outgoing Fellowship and the EuCARD-2 project of the European Community’s 7th Programme and by the U.S. Department of Energy, Office of Sciences, Office of High Energy Physics, early Career Award FWP 50335 and Office of Science, Office of Basic Energy Sciences under Contract No. DE-AC02-06CH11357. We thank P. Kolb for providing the RF data of the PAV-2 cavity and Ignacio Aviles Santillana for providing the SEM images.§ APPENDIX I: STOPPING PROFILESFigure <ref> displays the muon stopping profiles obtained from the Monte-Carlo code . This program has been developed at MPI Garching by W. Eckstein <cit.>.and adopted and experimentally verified for muons <cit.>. § APPENDIX II: RF PENETRATION DEPTH MEASUREMENTS Figure <ref> displays the penetration depth change obtained from the RF frequency shift for a HIPIMS cavity <cit.>. The penetration depth at [0]K, λ_0 is extracted from a least squares fit to the Gorter-Casimir expression <cit.>λ(T)=λ_0/√(1-(T/Tc)^4), with the critical temperature Tc as a second independent fit parameter.The penetration depth, λ, is directly related to the electron mean free path l, a measure of the impurity concentration, via <cit.>λ(l)=λ(l→∞)√(1+πξ_0/2l),where the London penetration depth for infinite mean free path λ(l→∞) and the BCS coherence length, ξ_0, are material parameters for which literature values are available. In case of bulk niobium λ(l→∞)=[32]nm and ξ_0=[39]nm have been reported <cit.>, while LE-μSR results for rather clean films find λ_L=[27(3)]nm <cit.>. For dcMS sputter coated niobium films Benvenuti et al. derived λ=[29(3)]nm and ξ_0=[33]nm from RF measurements <cit.>.§ APPENDIX III: BRIEF LITERATURE REVIEW OF THE TEMPERATURE DEPENDENCE OF THE HOPPING RATE FOR NIOBIUM OF DIFFERENT PURITYThe depolarization of positve muons implanted in niobium has been extensively studied since the late 1970s <cit.>. In these publications muon diffusion and trapping is studied for samples covering a wide range in purity containing interstitial and substitutional impurities. While earliest studies where carried out in transverse field configuration it was in 1982 that Boekema et al. <cit.> pointed out that zero field measurements can unambiguously distinguish between static and dynamic muons, since the static muon will have its polarization value recover to 1/3 of its initial value for long times (see Eq.(<ref>)). It has to be noted that previous studies usually analyze their data in terms of a Gaussian or an exponential depolarization rate λ, only and not in terms of depolarization, σ, and fluctuation/hopping rate, ν, using the dynamic Gaussian Kubo-Toyabe depolarization function as given in Eq.(<ref>) and first derived in Ref. <cit.> in the context of Lapalce transfoms. In the former λ accounts for the muon dynamics. A low λ corresponds to a high hopping rate in the corresponding two parameter fit (see Eq.(<ref>)). This can be understood in terms of the concept of motional narrowing <cit.>, i.e. in case of fast diffusion different muons will in average sense almost identical fields during their lifetime leading to a small λ. For niobium with an impurity concentration of less than [1]ppm it has been found that ν does not depend on temperature <cit.>. In this case a depolarization rate significantly smaller than expected from the second moment of the nuclear dipole field is found indicating fast diffusion. Generally impurities give a more complex behavior of the depolarization or hopping rate as function of temperature. For moderately pure niobium with an impurity concentration of several [100]ppm of interstitial impurities a two plateau structure is generally observed and explained within a two trap model. At a temperature below about [16]K the muon rapidly finds an extended shallow trap and remains there during the observation time. As the temperature is increased the muon becomes thermally detrapped. If the temperature is further increased the muon is mobile enough to find deeper more localized traps. This yields a second minimum of the hopping rate around [60]K.It has to be noted that all these studies have been carried out with so called surface muons, implanted about [0.3]mm deep in the bulk of the samples. 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Hyperfine Interactions, 17(1-4):183–190, 1984.Hayano R. S. Hayano, Y. J. Uemura, J. Imazato, N. Nishida, T. Yamazaki, and R. Kubo. Zero- and low-field spin relaxation studied by positive muons. Phys. Rev. B, 20(3):850, 1979.Romanenko_pc A. Romanenko. Private communication. | http://arxiv.org/abs/1703.08635v3 | {
"authors": [
"T. Junginger",
"S. Calatroni",
"A. Sublet",
"G. Terenziani",
"T. Prokscha",
"Z. Salman",
"T. Proslier",
"J. Zasadzinski"
],
"categories": [
"cond-mat.supr-con"
],
"primary_category": "cond-mat.supr-con",
"published": "20170325011113",
"title": "A low energy muon spin rotation and point contact tunneling study of niobium films prepared for superconducting cavities"
} |
Institute of Photonics and Quantum Sciences, SUPA, Heriot-Watt University, Edinburgh, United Kingdom Department of Physics and Astronomy, Macquarie University, Sydney, NSW 2109, AustraliaSensors based on single spins can enable magnetic field detection with very high sensitivity and spatial resolution. Previous work has concentrated on sensing of a constant magnetic field or a periodic signal. Here, we instead investigate the problem of estimating a field with non-periodic variation described by a Wiener process. We propose and study, by numerical simulations, an adaptive tracking protocol based on Bayesian estimation. The tracking protocol updates the probability distribution for the magnetic field, based on measurement outcomes, and adapts the choice of sensing time and phase in real time. By taking the statistical properties of the signal into account, our protocol strongly reduces the required measurement time. This leads to a reduction of the error in the estimation of a time-varying signal by up to a factor 4 compared to protocols that do not take this information into account. Adaptive tracking of a time-varying field with a quantum sensor Dominic W. Berry December 30, 2023 =============================================================== § INTRODUCTIONSensors based on individual quantum objects, such as electrons or atoms, can enable measurements of physical quantities with very high spatial resolution <cit.>. Additionally, by exploiting quantum phenomena, one can reach a sensitivity beyond what possible with classical techniques <cit.>. In the past decades, the exciting scientific progress in the control ofquantum systems has led to the demonstration of quantum sensing protocols based on individual photons, electrons, etc. In this context, sensors based on single spins can map magnetic fields with nanometric spatial resolution, making them a revolutionary tool to study magnetic phenomena in nanoscale materials and biological processes <cit.>. The most prominent system in this field is the electronic spin associated to the nitrogen-vacancy (NV) centre in diamond. Due to the weak spin-orbit coupling and an almost spin-free ^12C environment, the NV centre spin preserves quantum coherence on timescales much longer than the manipulation time. Remarkably, the spin can be read out at ambient conditions by optically-detected magnetic resonance, making it a viable system for nanoscale magnetic sensing at ambient conditions.These properties have led to ground-breaking experiments in nanoscale sensing, achieving a spatial resolution down to 10-20 nm <cit.> and a sensitivity sufficient to detect individual electron spins <cit.> and nanoscale volumes of nuclear spins <cit.>, down to the individual nuclear spin level <cit.>. Remarkable experiments with NV centres include the application of nanoscale sensing to probe, for example, ballistic electron transport in a conductor <cit.>, topological magnetic defects <cit.>, spin waves <cit.> and vortices in superconducting materials <cit.>.The sensing capabilities of the NV electronic spin are not limited to magnetic fields but extend also to the measurement of other physical quantities such as temperature <cit.>, electric fields <cit.> and strain <cit.>. Additionally, other defects in different materials, such as silicon carbide, exhibit sensing properties complementary to those of the NV centre in diamond <cit.>.Quantum sensing experiments have mainly addressed the detection of constant (DC magnetometry) and periodic signals (AC magnetometry) <cit.>. DC magnetometry estimates a constant signal by detecting its effect on a coherent superposition, e.g. by measuring a spin rotation under an applied constant magnetic field. AC magnetometry deals with detecting the amplitude and/or phase of a signal composed of one or a few harmonic tones by applying echo sequences. AC magnetometry was extended to the reconstruction of non-periodic waveforms <cit.> by using a family of echo sequences that form a basis for the signal. Identical instances of the same signal are repeated and detected using each echo sequence in the family, which allows the retrieval of specific Fourier coefficients. Combining all Fourier coefficients, corresponding to all the echo sequences in the family, the wavefunction can be reconstructed up to an arbitrary precision. This waveform reconstruction technique enables the reconstruction of fast oscillating waveforms, but requires access to several identical instances of the same waveform in order to find the projection of the signal onto each echo sequence in the basis.Here, we focus on a different problem: the reconstruction of a single instance of a time-varying magnetic field of known statistical properties. We propose a novel tracking protocol, based on Bayesian estimation, that extends a DC-magnetometry protocol to the estimation of a time-varying stochastic signal.We study the protocol performance in the case of a Wiener process and show that our protocol reduces the estimation time by up to a factor 4 compared to known protocols in the literature, by taking the measurement history and the statistical properties of the signal into account. A different method for the related problem of measuring a phase that changes in discrete steps was presented in Ref. <cit.>.Our protocol may find applications for fast tracking of magnetic fields associated to diffusion processes, for example in biology or in chemical reactions, or to track the Brownian motion of trapped magnetic nanoparticles. Additionally, this work could provide a faster way to track the dynamics of the spin bath surrounding the quantum sensor in the material. This could lead to a narrowing of the magnetic fluctuations and an increase of the spin coherence time <cit.>.§ QUANTUM SENSING OVER A LARGE DYNAMIC RANGE §.§ Single-spin DC magnetometry In this subsection we summarize known techniques and results for measuring a constant frequency. A constant magnetic field B, along the quantization axis z, can be measured by detecting the rotation induced on a single spin (Ramsey experiment) in the xy plane. A spin initialized in the superposition state (|0 ⟩ + |1 ⟩)/√(2) evolves under B over timeas (|0 ⟩ + e^i γ B |1 ⟩)/√(2), where γ is the gyromagnetic ratio (γ∼ 28 MHz/mT for an electronic spin). Assuming perfect spin initialization and read-out and no decoherence, the probability to detect outcome μ∈{0,1} after timeis:p(μ|f_B) = 1 + (-1)^μcos (2 π f_B+ θ)/2where f_B = γ B/2π. The phase θ corresponds to the rotation angle of the spin read-out basis in the xy plane, relative to the initialization state. The goal of a sensing experiment is to retrieve the frequency f_B with the highest possible accuracy over the largest possible range of values.In realistic cases, the spin state associated with outcome μ can only be read out with finite fidelity ξ_μ, defined as the probability to detect μ given that the eigenstate corresponding to μ is prepared. Additionally, the coherence of the spin is limited by fluctuations of the magnetic environment, averaged over the sensing time. We include magnetic fluctuations induced by nuclear spins in the material as a Gaussian decoherence term described by the coherence time T_2^* <cit.>. In our following discussion, we assume T_2^* to be long (T_2^* ∼ 100 μs) so that we can focus on reconstructing the variation of the classical magnetic signal neglecting the fluctuations of the nuclear spin environment. Coherence times of several hundred microseconds have been experimentally demonstrated in isotopically-purified diamond samples <cit.>.Including finite read-out fidelity and decoherence, Eq. (<ref>) is modified to:p(μ=0| f_B) = 1+ξ_0 -ξ_1/2 ++ ξ_0 + ξ_1-1/2 e^- ( /T_2^*)^2cos (2 π f_B+ θ)and p(μ=1|f_B) = 1-p(μ=0|f_B). In the following, based on our previous experiment with resonant optical excitation of an NV centre at cryogenic temperature <cit.>, we assume good read-out fidelity for the outcome μ=1 (ξ_1 ∼ 1) and we only discuss the role of the read-out fidelity for outcome μ=0, which we simply denote ξ.One fundamental issue with frequency estimation by Ramsey measurements is the trade-off between sensitivity and measurement range. In other words, there is a limit on the dynamic range, defined as the ratio between the maximum measurable frequency before saturation (f_ max) and the smallest detectable frequency, described by the uncertainty σ_f.For a Ramsey experiment with sensing timerepeated many times for total time , the uncertainty σ_f decreases as 1/(2π√()). Therefore, the minimum uncertainty can be reached when measuring over the longest sensing timebefore decoherence becomes significant, ∼ T_2^*. On the other hand, the frequency range decreases withbecause the signal is periodic, creating ambiguity whenever 2π f_ max > π. As a result, the dynamic range is bounded as f_ max/σ_f < π√(/).Adaptive phase estimation protocols have been devised to overcome this limit. The basic idea is to probe the field with a combination of K+1 exponentially decreasing sensing times _ = 2^, whereis the smallest sensing time and =,…,0 <cit.>. In adaptive measurements, the phase θ is adjusted based on the results of previous measurements. Provided there are multiple measurements for each sensing time, the uncertainty in estimating a phase scales as 1/N, where N is the total number of applications of the phase shift <cit.>.Further developments also showed that adaptive feedback is not a strict requirement: non-adaptive protocols can reach 1/N scaling <cit.>. In the case of frequency estimation, this translates to an increase in the dynamic range to f_ max/σ_f ∼π (/_0) <cit.>. For frequency estimation with realistic read-out fidelity, it was initially found that non-adaptive protocols yielded the best results <cit.>, but later improvements were found using adaptive measurements <cit.>.These protocols use Bayesian estimation. The probability distribution P(f_B) for the frequency f_B is assumed to be uniform at the beginning of each estimation sequence, then is updated after every Ramsey experiment according Bayes' theorem. For the ℓ-th Ramsey in the estimation sequence, Bayes' theorem givesP(f_B|μ_1...μ_ℓ) ∝ P(f_B|μ_1...μ_ℓ-1) P (μ_ℓ|f_B)where P (μ_ℓ|f_B) is given by Eq. (<ref>). Although the frequency is not periodic, there are bounds to the range of possible values that will be considered, and the probability distribution for the frequency is periodic when multiples ofare used for the sensing time. It is therefore convenient to express the probability as a Fourier seriesP(f_B)=∑_j p_j e^i 2π j f_B .The coefficients {p_j} depend on the measurement results, but that dependence will not be shown for brevity.In the remainder of this subsection we summarize methods and results from Ref. <cit.>, except we replace the phase with 2π f_B. When quantifying the performance of measurement of a periodic quantity, it is convenient to use the Holevo variance <cit.>. A modification of the Holevo variance, analogous to the mean-square error, isV_H := ⟨cos( 2π (f_B-f̂_B) )⟩^-2 -1 ,where f̂_B is the estimate of the frequency, and the average is over the actual frequency f_B and measurement results. This measure is convenient for designing the feedback protocol, but in this work we evaluate the performance of the estimation by the usual mean-square error. The best estimate for the frequency, that minimizes V_H, is given byf̂_B = 1/2π⟨ e^i2π f_B ⟩,where f_B on the right-hand side is a dummy variable for the Bayesian phase distribution, and the expectation value is over that phase distribution. This estimate is very easily found from the Fourier coefficients asf̂_B = 1/2π| p_-1|.In addition, the value of ⟨cos( 2π (f_B-f̂_B))⟩ in the expression for V_H can be found by averaging over |⟨ e^i2π f_B ⟩|. Using the Fourier representation of the probability distribution, this means that V_H is given byV_H = (2π⟨| p_1|⟩)^-2-1,where the expectation is now over the actual frequency and measurement results, and we have used p_1=p_-1^*. This suggests using an adaptive sensing protocol, where the rotation of the spin read-out basis θ is selected to maximize the expected value of | p_1| after the next detection. That minimizes the value of V_H after the next detection.The situation is more complicated, because initially large multiples of the interaction timeare used, which means that p_1=p_-1=0. When the smallest interaction time that has been used so far is 2^, then one would instead replacewith 2^ in the above discussion. In the approach of Ref. <cit.>, one would choose θ to maximize | p_2^| after the next detection. If the smallest interaction time so far was 2^, and a measurement to be performed is with an interaction time of 2^-1, then it is better to minimize | p_2^-1| after the next detection. The appropriate value of θ to choose is thenθ = 1/2( p_-2^).The adaptive technique for realistic read-out fidelity in <cit.> uses this estimate in combination with a phase incrementdependent only on the last measurement outcome, obtained by numerically optimizing the final variance for the specific experimental parameters through a “swarm optimization” procedure <cit.>. In this work, we use that method for the measurements without tracking of the phase. In contrast, for the measurements with tracking of the phase, we always use Eq. (<ref>) to choose the controlled phase θ.In the following, we describe the estimation error for f_B as the standard deviation based on the Holevo variance V_H [σ_f = V_H^1/2/(2πτ_0)]. While here we follow the traditional approach of giving a point estimate and the variance, an alternative possibility would be to give credible intervals for the estimate <cit.>, for example the frequency values f_1 and f_2 such thatP(f_1< f_B< f_2| μ_1...μ_ℓ) = 0.95.Alternatively, in our Bayesian approach the probability distribution P(f_B) is available at all times after each measurement: neglecting memory and data processing considerations, giving P(f_B), as shownin Fig. <ref>(iv) and (vii), would provide the experimenter with the most complete information.Estimation error scaling as σ_f ∝ 2^-/ cannot be achieved by using only a single repetition for each of the exponentially-decreasing sensing times 2^. However, the required bound can be reached using a number of repetitions M_ = G + (-) F,where F, G are integers <cit.>. For the longest sensing time 2^K τ_0, a number G of repetitions is performed. The number of repetitions is then increased by F for each shorter sensing time. The additional number of repetitions for shorter sensing times removes the most detrimental errors, which correspond to measurements that make the largest distinction in frequency f_B. The total sensing time T for a single estimation sequence is=[(2^+1-1)G + (2^+1--2)F]which can be approximated by ∼ (G+F) 2^+1 for large . §.§ Estimating a time-varying signal Next we consider a frequency f_B(t) varying according to a Wiener processf_B ( + d) = f_B () +(),where (t) is an infinitesimal Wiener increment. It is defined such that the integral of 𝒲 (t) for time Δ has mean zero and variance Δ. It can be simulated by discretizing time to intervals of length Δ, and generating a normal distribution with variance Δ. The goal is to estimate f_B (t) in real time with the smallest possible error. That is, we wish to estimate f_B (t) at the current time using data up to the current time.The simplest way to estimate a time-varying field is to repeat the optimized adaptive protocol described in Section <ref> and experimentally demonstrated in <cit.>, which we will call the “non-tracking protocol”. The phase acquired during a Ramsey experiment, which for a constant f_B is simply φ = 2π f_B [as in Eq. (<ref>)], becomesφ = 2 π∫__0^_0+ f_B () d.As discussed in Section <ref>, for a constant f_B the minimum uncertainty is achieved by the longest sensing timeallowed by decoherence. In other words, the value ofshall be chosen so that the longest sensing time = 2^ is close to the spin coherence time T_2^*.This choice ofis not necessarily optimal for a time-varying signal. In this case, the optimalshould be chosen such that the estimation error is similar to the change in signal over the measurement sequence (provided it is less than T_2^*). By choosinglarger than this, the signal would fluctuate over the measurement time by more than the measurement accuracy, resulting in an unreliable outcome. On the other hand, ifis smaller than necessary, then we would be restricting ourselves to a reduced accuracy. Therefore, we assumeσ_f ∝ ^1/2where the total sensing time is ∼ (G+F) 2^+1.For a measurement scheme of this type the uncertainty σ_f should be inversely proportional to the total sensing time . More specifically, the scaling should be as σ_f∝ 1/(2^), but the constant of proportionality will depend on G and F. A rough approximation may be made by assuming that the constant of proportionality is 1/√(G). That is because the longest interaction time is repeated G times, and 1/√(G) is the scaling for measurements repeated G times. The measurements with shorter interaction times have more repetitions, but are primarily used for resolving ambiguities.That means we should have the scaling2^∝1/[G(G+F)]^1/3^2/3.That then yields an uncertainty scaling asσ_ no-tr∝(G+F)^1/3/G^1/6 ^2/3.The important part of this expression is the scaling with ^2/3, which is equivalent to the scaling that could be achieved for optical phase measurements with arbitrary squeezing <cit.>. We have performed numerical simulations of this non-tracking protocol for G=5, F=3, and the results are shown in Fig. <ref>. There is excellent agreement with theory, and fitting for the proportionality constant for Eq. (<ref>) yields σ_ no-tr =(1.033 ± 0.04 ) ^2/3.In the previous discussion we assumed that the estimation sequence only includes sensing time, i.e. all other operations, such as spin initialization, control and read-out are instantaneous. However, this is not true for realistic experiments, where all these operations contribute to an overhead time T_ OH. A measurement sequence with +1 different sensing times features a number of Ramsey experiments R_ = (+1)G + (+1) F/2 <cit.>, resulting in a total estimation time ∼ (G+F) 2^+1 + R_ T_ OH.§ ADAPTIVE TRACKING §.§ The algorithmFor a time-varying signal with known statistical properties, the available information can be exploited to shorten the estimation sequence. Given a frequency f_B() at time , the frequency at time + d is likely to be not too distant from f_B(). Using the known signal statistics, one can reasonably predict a narrower frequency range for the next estimation, so it is not necessary to explore the whole range of possible values. In other words, instead of starting each estimation sequence from a uniform probability distribution, one could start from the Bayesian probability distribution from the prior measurements, and take into account the variation of the frequency according to the signal statistics. This corresponds to tracking the time-varying signal.Updating the probability distribution. For simplicity, we will initially consider the case that the frequency is constant. We will also take the fidelities ξ_μ to be equal to 1. Then given an outcome μ_ℓ for the ℓ-th Ramsey experiment, featuring a sensing time _ = 2^, the update of the probability distribution corresponds to an update of the Fourier coefficients asp_j^(ℓ)= p_j^(ℓ-1)/2 + e^-(_/T_2^*)^2/4[ e^i(μ_ℓπ +θ) p_j-2^^(ℓ-1). . + e^-i(μ_ℓπ +θ) p_j+2^^(ℓ-1)].Over a small time interval δ t, the change in frequency δ f_B will have a normal distribution with variance ^2 δ:P_G (δ f_B) = 1/√(2πδ t) e^- (δ f_B)^2/(2^2 δ t).Ignoring any information from a measurement, the probability distribution for the frequency after a time δ t will be the convolution of the initial probability distribution with the Gaussian in Eq. (<ref>), givingP^(ℓ) (f_B)= ∫ P^(ℓ-1)(f_B - ν) P_G (ν)dν = ∑_j p_j e^-2(π j )^2 δ t e^i2π j f_B.Therefore the coefficients p_k^(ℓ) for the probability distribution after a time δ t (without measurement) can be calculated as:p^(ℓ)_j = p_j^(ℓ-1) e^-2(π j )^2 δ t.Provided the frequency does not vary significantly during an interaction time, the probability distribution may be approximated by using Eqs. (<ref>) and (<ref>) independently. Using this approach the simulations still accurately model measurements made using this technique. An exact calculation of the probability distribution could potentially result in more accurate estimates, but the method to perform such a calculation appears to be an open question.The adaptive tracking protocol. The tracking protocol is described by the pseudo-code in Algorithm 1. Each estimation sequence uses a probability distribution P(f_B) based on the previous measurements. As before, Ramsey experiments are performed starting from the longest sensing time 2^, updating the probability distribution P(f_B) according to Bayes' theorem. The difference is that now, instead of using all sensing times τ_k with G+(-)F repetitions, the protocol adaptively chooses the best sensing time for each estimation, starting from the longest sensing time 2^. To judge the accuracy of the estimate with sensing time τ_, a figure of merit ℱ is calculated based on the Bayesian probability distribution, and compared to a threshold ℱ^( thr)[]. If the figure of merit satisfies ℱ < ℱ^( thr)[] then the same sensing time τ_ is kept for the next estimation. If the estimate is not sufficiently accurate, then the sensing time for the next estimation is reduced to τ_-1. At that point, if the threshold is satisfied for the new estimation, the sensing time is increased back to τ_. Otherwise, if the threshold is not satisfied, then the sensing time is further reduced to τ_-2 and so on. While the non-tracking protocol requires a large number of Ramsey experiments for each estimation, the adaptive tracking protocol outputs an estimation of the time-varying field for each Ramsey experiment. We choose as a figure of merit ℱ an estimate of the standard deviation of the probability distribution, which can be retrieved from the Holevo variance as:ℱ = V_H^1/2/2πτ_0 =1/2πτ_0 [ (2π| p_-1|)^-2 - 1 ]^1/2.This expression only depends on the coefficient p_-1 and can easily be calculated in real time. The estimation error is expected to scale as σ_f ∝ 2^K/τ_0. Therefore, we set the threshold corresponding to the sensing time 2^τ_0 to be:ℱ^( thr)[] = α/2^τ_0.Numerical simulations suggest that the optimal value for the proportionality factor α is 0.15.This protocol is adaptive in two ways. First of all, it chooses in real-time the measurement phase according to Eq. (<ref>). Second, it adapts the sensing time at each step, based on the current estimated variance. As a consequence, while the non-tracking protocol requires an optimal choice of K for optimal performance (Section <ref>) and sub-optimal choices of K lead to large estimation errors, the tracking protocol is very robust and automatically selects the proper value of k at each step.Examples of reconstructions of time-varying fields with the tracking protocol, as compared with the non-tracking protocol described in Section <ref>, are shown in Fig. <ref>.The non-tracking protocol can successfully estimate time-varying parameters, with no previous knowledge of the properties of the signal. When the overhead is small [Figs. <ref>(a) and <ref>(c)], the performance of the non-tracking and tracking protocols appears to be similar. The advantage of the tracking protocol in following fast signal variations is clearly evident when overhead is large [Figs. <ref>(b) and <ref>(d), for T_ OH = 10 μs].§.§ Performance estimation The shortest estimation sequence for the tracking protocol would be to simply repeat the longest sensing time = 2^ once, so that the estimation error scales as σ_f ∝1/ 2^.Requiring that the variation of the frequency during a total sensing time of 2^Kτ_0 is comparable to the uncertainty leads to the scalingσ_ tr∝^2/3.If we assume the scaling constant is 1, the ratio η between the errors in the non-tracking and tracking cases is:η = σ_ no-tr/σ_ tr = (G+F)^1/3/G^1/6.For G=5, F=3, η∼ 1.5. The improvement expected for the tracking protocol is modest, and in simulations there is not a large difference, as illustrated by the time-domain waveforms on the left side of Fig. <ref>. The reason for this is that the primary mechanism for improving the performance is using the prior information to resolve ambiguities, instead of measurements with shorter sensing times. That means we can save the time used for the shorter measurements, and instead use the measurements with longer sensing times. However, the contribution to the total time from the measurements with shorter interaction times is not large. The longest sensing time 2^ is almost the same as the sum of all the other sensing times 2^-1… 2^0, which means that the tracking protocol does not dramatically shorten the total estimation time.A more consistent improvement can be expected when taking overhead into account. In the limit of large overhead, the major contribution to the estimation time is given by the overhead, while the sensing time can be neglected. The estimation time for the tracking protocol can be approximated by _ OH, as compared to ∼ [(+1)G + (+1) F/2] T_ OH for the non-tracking protocol. As an example, if T_ OH =100 μs, the tracking protocol delivers an estimation every 100 μs. In contrast, the non-tracking protocol performs one estimation every 1.24 ms (assuming G=5, F=3, K=7). Since the estimation error scales as the square root of the estimation time (Eq. <ref>), we expect an improvement on the order of η∼ (1.24 / 100 μ)^1/2, corresponding to a factor 3-4. §.§ Numerical simulationsThe performance of the two protocols has been tested by numerical simulations, for a range of parameter values. We select the minimum sensing time = 20 ns, corresponding to a frequency range f_B ∈ [-25, +25] MHz.An instance of a time-varying signal f_B () is produced according to the Wiener process in Eq. (<ref>), starting from a random value for f_B(0), with a temporal resolution of = 20 ns. In order to avoid values out of the[-25, +25] MHz range, the waveform is truncated if |f_B|> 24 MHz. The signal f_B (t) is reconstructed using either protocol, providing the reconstructed waveform f̂_B(). To evaluate the performance of the estimation, we use the mean-square errorε^2 = 1/T∫_0^T | f_B () - f̂_B() |^2 d. Simulation results for the limit of negligible overhead (T_ OH = 10 ns) are shown in Fig. <ref> (a). On the top plot, the tracking algorithm (blue downward-pointing triangles) exhibits a relatively small improvement compared to the non-tracking algorithm (red upward-pointing triangles), by a factor of η = 1.23 ± 0.09, similar to the theoretical prediction η∼ 1.5 in the previous section. The case of longer overhead is shown inFig. <ref> (b). As hinted by the reconstructed waveforms in Fig. <ref>, the advantage is here more significant, reaching η∼ 3-4. Given the different approximations used, the theoretical predictions based on Eq. (<ref>) agree surprisingly well with the results of the numerical simulations.The role of the overhead time is investigated in more detail by examining the protocols' performances for a fixed value of(= 2 MHz Hz^1/2), while sweeping the overhead time T_ OH between 0 and 300 μs. The results are plotted in Fig. <ref>. The improvement given by the tracking protocol (described by η) is small for small overhead, as already evidenced in Fig. <ref>. For larger overhead, η is larger, up to about a factor 3 and it is roughly independent of T_ OH, as predicted in Section <ref>.In the last set of numerical simulations (Fig. <ref>), we illustrate the effect of a reduced spin read-out fidelity ξ. We compare the waveform estimation error ε_f for the non-tracking and tracking protocols for ξ = 0.75 and ξ=0.88. The latter value corresponds to the fidelity of spin read-out for the experimental demonstration in Ref. <cit.>. Reduced read-out fidelity leads, as one can expect, to an increase in the estimation error. The ratio η, however, does not vary significantly.§ EXPERIMENTAL OUTLOOK Our analysis has been restricted to the case when the classical spin read-out noise is smaller than the spin projection noise, i.e. when “single-shot” read-out is available. Currently, single-shot read-out has only been experimentally demonstrated by exploiting resonant optical excitation of spin-dependent transitions at cryogenic temperature <cit.>. This technique features high-fidelity spin initialization (fidelity >0.99) andread-out (fidelities ξ_1 ∼ 0.99 and ξ_0 > 0.9). The requirement of cryogenic operation is, however, a serious restriction for applications to quantum sensing.Recent experiments have shown some promise towards the demonstration of single-shot read-out at room temperature. A first approach involves spin-to-charge conversion by optical ionization and successive detection of the defect charging state <cit.>. This approach may be further enhanced by the integration of electrical contacts to provide photo-electric spin read-out <cit.>. A second approach involves the storage of the electron spin population onto the nuclear spin, which can be read out several times through the electron spin itself <cit.>. In both cases, a large overhead is introduced, which provides further motivation for our analysis.Typically, the NV centre spin is read out at room temperature by spin-dependent photo-luminescence intensity, originating from spin-dependent decay rates through a metastable state under optical excitation (optically-detected magnetic resonance). In contrast to the “single-shot” readout case, the readout noise is here larger than the spin projection noise and spin read-out must be repeated several times for each sensing time. A large number of repetitions (∼ 10^5 for a typical experiment with NV centres) can be seen as a large overhead, suggesting that our tracking protocol will also be useful for such experiments.In this work, we discussed the specific case of a time-varying signal described by a Wiener process, assuming that the parameter κ is known. In case κ is not known, one could simply start by measuring the signal evolution with the non-tracking protocol, and retrieving an estimate of κ that can be used for tracking at a later stage. Additionally, our approach is quite general and can be easily extended to other kinds of stochastic processes. Particularly relevant is the Ornstein-Uhlenbeck process <cit.>, which describes the fluctuations of a nuclear spin bath in the semi-classical approximation.An interesting extension of this work would be to the case of the magnetic field induced by a quantum bath, such as the nuclear spin bath. If the correlation time of the bath dynamics is long enough, each estimation sequence provides a projective measurement of the magnetic field originating from the bath. The back-action of such projective measurement would narrow the probability distribution for the magnetic field induced by the bath, leading to an extension of the spin coherence time T_2^* <cit.>. By providing faster field estimation, our protocol could allow to partially relax the requirement of long bath correlation time and slow dynamics. Additionally, spin read-out by optical excitation can induce perturbations in the bath by causing unwanted electron spin flips that affect the bath through the hyperfine interaction. By reducing the number of read-outs required for each estimation, a reduction of unwanted bath perturbations induced by optical spin read-out can be expected.In addition to extending the sensor coherence time, the measurement back-action on the spin bath could also be used as a state preparation tool. For weakly-coupled nuclear spins in the bath, each Ramsey experiment in the estimation sequence acts as a weak spin measurement. It has been shown that a sequence of weak measurements with sensing times which are adapted in real time can be used as a tool for spin preparation <cit.>. It would be interesting to investigate whether the tracking protocol could be adapted to provide, at the same time, the preparation of pre-determined quantum states of the bath.A different research direction could be to extend this work to the spatial domain, to the field of microscopy. The acquisition of a bi-dimensional image requires considerable time, over which the system must be stable against fluctuations and drifts, posing a technological challenge. In this case, any a-priori knowledge of the statistical properties and correlation scales of the image could be used to speed up the measurement process <cit.>.§ CONCLUSIONSIn this work, we discussed the measurement of a single instance of a time-varying field, of known statistical properties, with a quantum sensor.We investigated the performance of a non-tracking protocol, previously considered only for constant fields, which does not use any information about the signal. through numerical simulations, we showed that the protocol can successfully track a time-varying field.We introduced a novel tracking protocol based on Bayesian estimation. By using the additional information about the statistical properties of the signal, the tracking protocol shortens the time required for each estimation, leading to a reduced measurement uncertainty in the estimation of a time-varying field. 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"authors": [
"Cristian Bonato",
"Dominic W. Berry"
],
"categories": [
"quant-ph",
"cond-mat.mes-hall"
],
"primary_category": "quant-ph",
"published": "20170327213413",
"title": "Adaptive tracking of a time-varying field with a quantum sensor"
} |
[Exploration–Exploitation in MDPs with Options Ronan Fruit Alessandro Lazaric Inria Lille - SequeL Team Inria Lille - SequeL Team ] While a large body of empirical results show that temporally-extended actions and options may significantly affect the learning performance of an agent, the theoretical understanding of how and when options can be beneficial in online reinforcement learning is relatively limited. In this paper, we derive an upper and lower bound on the regret of a variant of UCRL using options. While we first analyze the algorithm in the general case of semi-Markov decision processes (SMDPs), we show how these results can be translated to the specific case of MDPs with options and we illustrate simple scenarios in which the regret of learning with options can be provably much smaller than the regret suffered when learning with primitive actions.§ INTRODUCTION The option framework <cit.> is a simple yet powerful model to introduce temporally-extended actions and hierarchical structures in reinforcement learning (RL) <cit.>. An important feature of this framework is that Markov decision process (MDP) planning and learning algorithms can be easily extended to accommodate options, thus obtaining algorithms such as option value iteration and Q-learning <cit.>, LSTD <cit.>, and actor-critic <cit.>. Temporally extended actions are particularly useful in high dimensional problems that naturally decompose into a hierarchy of subtasks. For instance, <cit.> recently obtained promising results by combining options and deep learning for lifelong learning in the challenging domain of Minecraft. A large body of the literature has then focused on how to automatically construct options that are beneficial to the learning process within a single task or across similar tasks (see e.g., <cit.>). An alternative approach is to design an initial set of options and optimize it during the learning process itself (see e.g., interrupting optionsand options with exceptions ). Despite the empirical evidence of the effectiveness of most of these methods, it is well known that options may as well worsen the performance w.r.t. learning with primitive actions <cit.>. Moreover, most of the proposed methods are heuristic in nature and the theoretical understanding of the actual impact of options on the learning performance is still fairly limited. Notable exceptions are the recent results of <cit.> and <cit.>. Nonetheless, <cit.> rather focus on a batch setting and they derive a sample complexity analysis of approximate value iteration with options. Furthermore, the PAC-SMDP analysis of <cit.> describes the performance in SMDPs but it cannot be immediately translated into a sample complexity of learning with options in MDPs. In this paper, we consider the case where a fixed set of options is provided and we study their impact on the learning performance w.r.t. learning without options. In particular, we derive the first regret analysis of learning with options. Relying on the fact that using options in an MDP induces a semi-Markov decision process (SMDP), we first introduce a variant of the algorithm <cit.> for SMDPs and we upper- and lower-bound its regret (sections <ref> and <ref>). While this result is of independent interest for learning in SMDPs, its most interesting aspect is that it can be translated into a regret bound for learning with options in MDPs and it provides a first understanding on the conditions sufficient for a set of options to reduce the regret w.r.t. learning with primitive actions (Sect. <ref>). Finally, we provide an illustrative example where the empirical results support the theoretical findings (Sect. <ref>). § PRELIMINARIES MDPs and options. A finite MDP is a tuple M = {𝒮, 𝒜, p, r } where 𝒮 is a finite set of states, 𝒜 is a finite set of actions, p(s'|s,a) is the probability of transitioning from state s to state s' when action a is taken, r(s,a,s') is a distribution over rewards obtained when action a is taken in state s and the next state is s'. A stationary deterministic policy π : → maps states to actions. A (Markov) option is a tuple o = {ℐ_o, β_o , π_o } where ℐ_o ⊂𝒮 is the set of states where the option can be initiated, β_o: 𝒮→ [0,1] is the probability distribution that the option ends in a given state, and π_o : 𝒮→𝒜 is the policy followed until the option ends. Whenever the set of primitive actionsis replaced by a set of options Ø, the resulting decision process is no longer an MDP but it belongs to the family of semi-Markov decision processes (SMDP).[] For any MDP M and a set of options Ø, the resulting decision process is an SMDP M_Ø = {𝒮_Ø, 𝒪, p_Ø, r_Ø, τ_Ø}, where 𝒮_Ø⊆𝒮 is the set of states where options can start and end,_Ø = (∪_o∈Øℐ_o) ⋃(∪_o∈Ø{s: β_o(s) > 0}),Ø is the set of available actions, p_Ø(s,o,s') is the probability of transition from s to s' by taking the policy π_o associated to option o, i.e., p_Ø(s,o,s') = ∑_k=1^∞ p(s_k = s' | s,π_o) β_o(s'),where p(s_k = s' | s,π_o) is the probability of reaching state s' after exactly k steps following policy π_o, r_Ø(s,o,s') is the distribution of the cumulative reward obtained by executing option o from state s until interruption at s', and τ_Ø(s,o,s') is the distribution of the holding time (i.e., number of primitive steps executed to go from s to s' by following π_o). Throughout the rest of the paper, we only consider an “admissible” set of options Ø such that all options terminate in finite time with probability 1 and in all possible terminal states there exists at least one option that can start, i.e., ∪_o∈Ø{s: β_o(s) > 0}⊆∪_o∈Øℐ_o. We further assume that the resulting SMDP M_Ø is communicating i.e. has a finite diameter (see Def. <ref>). Finally, we notice that a stationary deterministic policy constructed on a set of options Ø may result into a non-stationary policy on the set of actions .Learning in SMDPs. Relying on this mapping, we first study the exploration-exploitation trade-off in a generic SMDP. A thorough discussion on the implications of the analysis of learning in SMDPs for the case of learning with options in MDPs is reported in Sect. <ref>. For any SMDP M = {𝒮, 𝒜, p, r, τ}, we denote by τ(s,a,s') (resp. r(s,a,s')) the expectation of τ(s,a,s') (resp. r(s,a,s')) and by τ(s,a) = ∑_s' ∈𝒮τ(s,a,s')p(s'|s,a) (resp. r(s,a) = ∑_s' ∈𝒮r(s,a,s')p(s'|s,a)) the expected holding time (resp. cumulative reward) of action a from state s. In the next proposition we define the average-reward performance criterion and we recall the properties of the optimal policy in SMDPs. Denote N(t) = sup{n ∈ℕ,∑_i=1^nτ_i≤ t } the number of decision steps that occurred before time t. For any policy π and s∈𝒮:ρ^π(s) def=lim sup_t → + ∞𝔼^π[ ∑_i=1^N(t)r_i/t| s_0 =s ]ρ^π(s) def=lim inf_t → + ∞𝔼^π[ ∑_i=1^N(t)r_i/t| s_0 =s ].If M is communicating and the expected holding times and reward are finite, there exists a stationary deterministic optimal policy π^* such that for all states s and policies π, ρ^π^*(s) ≥ρ^π(s) and ρ^π^*(s) = ρ^π^*(s) = ρ^*. Finally, we recall the average reward optimality equation for a communicating SMDPu^*(s) = max_a ∈𝒜{ r(s,a) - ρ^* τ(s,a) + ∑_s' ∈𝒮p(s'|s, a)u^*(s') },where u^* and ρ^* are the bias (up to a constant) and the gain of the optimal policy π^*.We are now ready to consider the learning problem. For any i ∈ℕ^*, a_i denotes the action taken by the agent at the i-th decision step[Notice that decision steps are discrete points in time in which an action is started, while the (possibly continuous) holding time is determined by the distribution τ.] and s_i denotes the state reached after a_i is taken, with s_0 being the initial state. We denote by (r_i(s,a,s'))_i ∈ℕ^* (resp. (τ_i(s,a,s'))_i ∈ℕ^*) a sequence of i.i.d. realizations from distribution r(s,a,s') (resp. τ(s,a,s')). When the learner explores the SMDP, it observes the sequence (s_0, …, s_i, a_i+1, r_i+1(s_i,a_i+1,s_i+1), τ_i+1(s_i,a_i+1,s_i+1), …). The performance of a learning algorithm is measured in terms of its cumulative regret. For any SMDP M, any starting state s ∈𝒮, and any number of decision steps n ≥ 1, let {τ_i}_i=1^n be the random holding times observed along the trajectory generated by a learning algorithm 𝔄. Then the total regret of 𝔄 is defined asΔ(M,𝔄,s,n) = ( ∑_i=1^nτ_i) ρ^*(M) - ∑_i=1^nr_i.We first notice that this definition reduces to the standard regret in MDPs for τ_i = 1 (i.e., primitive actions always terminate in one step). The regret measures the difference in cumulative reward obtained by the optimal policy and the learning algorithm. While the performance of the optimal policy is measured by its asymptotic average reward ρ^*, the total duration after n decision steps may vary depending on the policy. As a result, when comparing the performance of π^* after n decision steps, we multiply it by the length of the trajectory executed by the algorithm 𝔄. More formally, from the definition of ρ^* (Eq. <ref>) and Prop. <ref> we have[In this expectation, N(t) is a r.v. depending on π^*.]𝔼^π^*[ ∑_i=1^N(t)r_i| s_0 =s ] t → +∞∼ρ^*t + o(t).By introducing the total duration N(t) of 𝔄 we haveρ^*t + o(t) = ρ^*( ∑_i=1^N(t)τ_i) + ρ^*(t-∑_i=1^N(t)τ_i) + o(t).We observe that (t-∑_i=1^N(t)τ_i) = o(t) almost surely since (t-∑_i=1^N(t)τ_i) ≤τ_N(t)+1 and τ_N(t)+1 is bounded by an almost surely finite (a.s.) random variable since the expected holding time for all state-action pairs is bounded by assumption. So τ_N(t)+1/t t → +∞→ 0 a.s. and𝔼^π^*[ ∑_i=1^N(t)r_i| s_0 =s ] t → +∞∼ρ^*( ∑_i=1^N(t)τ_i) + o(t),which justifies the definition of the regret. § SMDP-UCRL In this section we introduce (Fig. <ref>), a variant of <cit.>. At each episode k, the set of plausible SMDPs ℳ_k is defined by the current estimates of the SMDP parameters and a set of constraints on the rewards, the holding times and the transition probabilities derived from the confidence intervals. Given _k, extended value iteration (EVI) finds an SMDP M_k ∈ℳ_k that maximizes ρ^*(M_k) and the corresponding optimal policy π_k^* is computed. To solve this problem, we note that it can be equivalently formulated as finding the optimal policy of an extended[In the MDP literature, the term Bounded Parameter MDPs (BPMDPs) <cit.> is often used for "extended" MDPs built using confidence intervals on rewards and transition probabilities.] SMDP M_k^+ obtained by combining all SMDPs in ℳ_k: M_k^+ has the same state space and an extended continuous action space 𝒜_k^+. Choosing an action a^+ ∈𝒜_k^+ amounts to choosing an action a ∈𝒜,a reward r_k(s,a), a holding time τ_k(s,a) and a transition probability p_k(| s,a) in the confidence intervals. When a^+ is executed in M_k^+, the probability, the expected reward and the expected holding time of the transition are respectively p_k(| s,a), r_k(s,a) and τ_k(s,a). Finally, π_k^* is executed until the number of samples for a state-action pair is doubled. Since the structure is similar to 's, we focus on the elements that need to be rederived for the specific SMDP case: the confidence intervals construction and the extended value iteration algorithm.Confidence intervals. Unlike in MDPs, we consider a slightly more general scenario where cumulative rewards and holding times are not bounded but are sub-exponential r.v. (see Lem. <ref>). As a result, the confidence intervals used at step 4 are defined as follows. For any state action pair (s,a) and for rewards, transition probabilities, and holding times we defineβ_k^r(s,a)=σ_r √(14 log( 2SA i_k/δ)/max{1, N_k(s,a)}), N_k(s,a) ≥2 b_r^2/σ_r^2log( 240SAi_k^7/δ)14 b_r log( 2SA i_k/δ)/max{1, N_k(s,a)}, β_k^p(s,a)= √(14 S log(2A i_k / δ)/max{1, N_k(s,a)}), β_k^τ(s,a) =σ_τ√(14 log( 2SA i_k/δ)/max{1, N_k(s,a)}), N_k(s,a) ≥2 b_τ^2/σ_τ^2log( 240SAi_k^7/δ)14 b_τlog( 2SA i_k/δ)/max{1, N_k(s,a)},where σ_r, b_r, σ_τ, b_r are suitable constants characterizing the sub-exponential distributions of rewards and holding times. As a result, the empirical estimates r̂_k, τ̂_k, and p̂_k are ±β_k^r(s,a), β_k^τ(s,a), β_k^p(s,a) away from the true values.Extended value iteration (EVI). We rely on a data-transformation (also called “uniformization”) that turns an SMDP M into an “equivalent” MDP M_ = {𝒮, 𝒜, p_, r_} with same state and action spaces and such that ∀ s,s' ∈𝒮, ∀ a ∈𝒜:r_(s,a)= r(s,a)/τ(s,a)p_(s' | s,a)= τ/τ(s,a)( p(s' | s,a) - δ_s,s') + δ_s,s'where δ_s,s' = 0 if s ≠ s' and δ_s,s' = 1 otherwise, and τ is an arbitrary non-negative real such that τ < τ_min. M_ enjoys the following equivalence property. If (v^*, g^*) is an optimal pair of bias and gain in M_ then (τ^-1v^*, g^*) is a solution to Eq. <ref>, i.e., it is an optimal pair of bias/gain for the original SMDP M.As a consequence of the equivalence stated in Prop. <ref>, computing the optimal policy of an SMDP amounts to computing the optimal policy of the MDP obtained after data transformation (see App. <ref> for more details). Thus, EVI is obtained by applying a value iteration scheme to an MDP M_k,eq^+ equivalent to the extended SMDP M_k^+. We denote the state values of the j-th iteration by u_j(s). We also use the vector notation u_j = (u_j(s))_s ∈𝒮. Similarly, we denote by p(·| s,a) = (p(s' | s,a))_s' ∈𝒮 the transition probability vector of state-action pair (s,a). The optimistic reward at episode k is fixed through the EVI iterations and it is obtained as r_j+1(s,a) = min{r̂_k(s,a) + β_k^r(s,a); R_maxτ_max}, i.e., by taking the largest possible value compatible with the confidence intervals. At iteration j, the optimistic transition model is obtained as p_j+1(| s,a) ∈_p() ∈𝒫_k(s,a){ p^⊺ u_j } and 𝒫_k(s,a) is the set of probability distributions included in the confidence interval defined by β_k^p(s,a). This optimization problem can be solved in O(S) operations using the same algorithm as in . Finally, the optimistic holding time depends on u_j and the optimistic transition model p_j+1 asτ_j+1(s,a) = min{τ_max; max{τ_min; τ̂_k(s,a) -sgn[ r_j+1(s,a) +τ(p_j+1(·|s,a)^⊺ u_j-u_j(s) ) ] β_k^τ(s,a) }},The min and max insure that τ_j+1 ranges between τ_min and τ_max. When the term r_j+1(s,a) + (p_j+1(·|s,a)^⊺ u_j-u_j(s) ) is positive (resp. negative), τ_j+1(s,a) is set to the minimum (resp. largest) possiblevalue compatible with its confidence intervals so as to maximize the right-hand side of Eq. <ref> below. As a result, for any τ∈]0,τ_min[, EVI is applied to an MDP equivalent to the extended SMDP M_k^+ generated over iterations asu_j+1 (s) = max_a ∈𝒜{r_j+1(s,a)/τ_j+1(s,a)+ τ/τ_j+1(s,a)( p_j+1(| s,a)^⊺ u_j - u_j(s) ) } + u_j(s)with arbitrary u_0. Finally, the stopping condition ismax_s ∈𝒮{ u_i+1(s) - u_i(s) }-min_s ∈𝒮{ u_i+1(s) - u_i(s) } < ϵ.We prove the following. If the stopping condition holds at iteration i of EVI, then the greedy policy w.r.t. u_i is ϵ-optimal w.r.t. extended SMDP M_k^+. The stopping condition is always reached in a finite number of steps. As a result, we can conclude that running EVI at each episode k with an accuracy parameter ϵ=R_max/√(i_k) guarantees that π_k is R_max/√(i_k)-optimal w.r.t. max_M' ∈ℳ_kρ^*(M').§ REGRET ANALYSIS In this section we report upper and lower bounds on the regret of . We first extend the notion of diameter to the case of SMDP as follows. For any SMDP M, we define the diameter D(M) byD(M) = max_s,s' ∈𝒮{min_π{𝔼^π[ T(s')|s_0 = s ]}}where T(s') is the first time in which s' is encountered, i.e., T(s') = inf{∑_i=1^nτ_i : n ∈ℕ,s_n = s'}. Note that the diameter of an SMDP corresponds to an average actual duration and not an average number of decision steps. However, if the SMDP is an MDP the two definitions of diameter coincides. Before reporting the main theoretical results about , we introduce a set of technical assumptions.For all s ∈𝒮 and a ∈𝒜, we assume that τ_max≥τ(s,a) ≥τ_min > 0 and max_s ∈𝒮,a ∈𝒜{r(s,a)/τ(s,a)}≤ R_max with τ_min, τ_max, and R_max known to the learning algorithm. Furthermore, we assume that the random variables (r(s,a,s'))_s,a,s' and (τ(s,a,s'))_s,a,s' are either 1) sub-Exponential with constants (σ_r, b_r) and (σ_τ , b_τ ), or 2) bounded in [0, R_maxT_max] and [T_min,T_max], with T_min>0. We also assume that the constants characterizing the distributions are known to the learning agent. We are now ready to introduce our main result. With probability of at least 1-δ, it holds that for any initial state s ∈𝒮 and any n>1, the regret of Δ(M,𝔄,s,n) is bounded by:O ( ( D √(S)+ 𝒞(M,n,δ) ) R_max√(S A n log( n/δ))),where 𝒞(M,n,δ) depends on which case of Asm. <ref> is considered [We denote max{a,b} = a ∨ b.]1) sub-Exponential𝒞(M,n,δ) = τ_max + (σ_r ∨ b_r/R_max + σ_τ∨ b_τ)√(log( n/δ)), 2) bounded𝒞(M,n,δ) = T_max + (T_max-T_min).Proof. The proof (App. <ref>) follows similar steps as in <cit.>. Apart from adapting the concentration inequalities to sub-exponential r.v. and deriving the guarantees about EVI applied to the equivalent MDP M_ (Lem. <ref>), one of the key aspects of the proof is to show that the learning complexity is actually determined by the diameter D(M) in Eq. <ref>. As for the analysis of EVI, we rely on the data-transformation and we show that the span of u_j (Eq. <ref>) can be bounded by the diameter of M_, which is related to the diameter of the original SMDP as D(M_) = D(M)/τ (Lem. <ref> in App. <ref>).The bound. The upper bound is a direct generalization of the result derived by <cit.> for in MDPs. In fact, whenever the SMDP reduces to an MDP (i.e., each action takes exactly one step to execute), then n=T and the regret, the diameter, and the bounds are the same as for . If we consider R_max=1 and bounded holding times, the regret scales as O(DS√(An) + T_max√(SAn)). The most interesting aspect of this bound is that the extra cost of having actions with random duration is only partially additive rather than multiplicative (as it happens e.g., with the per-step reward R_max). This shows that errors in estimating the holding times do not get amplified by the diameter D and number of states S as much as it happens for errors in reward and dynamics. This is confirmed in the following lower bound. For any algorithm 𝔄, any integers S, A ≥ 10, any reals T_max≥ 3T_min≥ 3, R_max>0, D > max{20 T_minlog_A(S), 12 T_min}, and for n≥max{ D, T_max} SA, there is an SMDP M with at most S states, A actions, and diameter D, with holding times in [T_min,T_max] and rewards in [0,1/2R_maxT_max] satisfying ∀ s ∈𝒮, ∀ a ∈𝒜_s, r(s,a) ≤ R_maxτ(s,a), such that for any initial state s ∈𝒮 the expected regret of 𝔄 after n decision steps is lower-bounded by:𝔼[Δ(M,𝔄,s,n)] = Ω( ( √(D) + √(T_max)) R_max√(S A n ))Proof. Similar to the upper bound, the proof (App. <ref>) is based on <cit.> but it requires to perturb transition probabilities and rewards at the same time to create a family of SMDPs with different optimal policies that are difficult to discriminate. The contributions of the two perturbations can be made independent. More precisely, the lower bound is obtained by designing SMDPs where learning to distinguish between “good” and “bad” transition probabilities and learning to distinguish between “good” and “bad” rewards are two independent problems, leading to two additive terms √(D) and √(T_max) in the lower bound.The bound. Similar to , this lower bound reveals a gap of √(DS) on the first term and √(T_max). While closing this gap remains a challenging open question, it is a problem beyond the scope of this paper. In the next section, we discuss how these results can be used to bound the regret of options in MDPs and what are the conditions that make the regret smaller than using on primitive actions.§ REGRET IN MDPS WITH OPTIONS Let M be an MDP and 𝒪 a set of options and let M_Ø be the corresponding SMDP obtained from Prop. <ref>. We index time steps (i.e., time at primitive action level) by t and decision steps (i.e., time at option level) by i. We denote by N(t) the total number of decision steps that occurred before time t. Given n decision steps, we denote by T_n = ∑_i=1^nτ_i the number of time steps elapsed after the execution of the n first options so that N(T_n) = n. Any SMDP-learning algorithm 𝔄_Ø applied to M_Ø can be interpreted as a learning algorithm 𝔄 on M so that at each time step t, 𝔄 selects an action of M based on the policy associated to the option started at decision step N(t). We can thus compare the performance of and when learning in M. We first need to relate the notion of average reward and regret used in the analysis of to the original counterparts in MDPs. Let M be an MDP, 𝒪 a set of options and M_Ø the corresponding SMDP. Let π_Ø be any stationary policy on M_Ø and π the equivalent policy on M (not necessarily stationary). For any state s ∈𝒮_Ø, any learning algorithm 𝔄, and any number of decision steps n we have ρ^π_Ø(M_Ø,s) = ρ^π(M,s) and Δ(M,𝔄,T_n) = Δ(M_Ø, 𝔄,n) + T_n (ρ^*(M) -ρ^*(M_Ø)).The linear regret term is due to the fact that the introduction of options amounts to constraining the space of policies that can be expressed in M. As a result, in general we have ρ^*(M) ≥ρ^*(M_Ø) = max_π_Øρ^π_Ø(M_Ø), where π_Ø is a stationary deterministic policy on M_Ø. Thm. <ref> also guarantees that the optimal policy computed in the SMDP M_Ø (i.e., the policy maximizing ρ^π_Ø(M_Ø,s)) is indeed the best in the subset of policies that can be expressed in M by using the set of options Ø.In order to use the regret analysis of Thm. <ref>, we still need to show that Asm. <ref> is verified.An MDP provided with a set of options is an SMDP where the holding times and rewards τ(s,o,s') and r(s,o,s') are distributed as sub-exponential random variables. Moreover, the holding time of an option is sub-Gaussian if and only if it is almost surely bounded. This result is based on the fact that once an option is executed, we obtain a Markov chain with absorbing states corresponding to the states with non-zero termination probability β_o(s) and the holding time is the number of visited states before reaching a terminal state. While in general this corresponds to a sub-exponential distribution, whenever the option has a zero probability to reach the same state twice before terminating (i.e., there is no cycle), then the holding times become bounded. Finally, we notice that no intermediate case between sub-exponential and bounded distributions is admissible (e.g., sub-Gaussian). Since these are the two cases considered in Thm. <ref>, we can directly apply it and obtain the following corollary. For any MDP M={,,p,r} with r(s,a,s') ∈ [0,R_max] and a set of options Ø, consider the resulting SMDP M_Ø={_Ø,_Ø,p_Ø,r_Ø,τ_Ø}. Then with probability of at least 1-δ, it holds that for any initial state s ∈𝒮 and any n>1, the regret of in the original MDP is bounded asO ( ( D_Ø√(S_Ø)+ 𝒞(M_Ø,n,δ) )R_max^_Ø√(S_Ø O n log( n/δ))) + T_n (ρ^*(M)-ρ^*(M_Ø)),where O is the number of options. We can also show that the lower bound holds for MDPs with options as well. More precisely, it is possible to create an MDP and a set of options such that the lower bound is slightly smaller than that of Thm. <ref>. Under the same assumptions as in Theorem <ref>, there exists an MDP with options such that the regret of any algorithm is lower-bounded asΩ( ( √(D_Ø) + √(T_max - T_min).. .. ) R_max^Ø√(S_Ø O n ))+ T_n (ρ^*(M)-ρ^*(M_Ø)).This shows that MDPs with options are slightly easier to learn than SMDPs. This is due to the fact that in SMDPs resulting from MDPs with options rewards and holding times are strictly correlated (i.e., r(s,o,s') ≤ R_maxτ(s,o,s') a.s. and not just in expectation for all (s,o,s')).We are now ready to proceed with the comparison of the bounds on the regret of learning with options and primitive actions. We recall that for Δ(M,,s,T_n) = O(DSR_max√(AT_n)). We first notice that[The largest per-step reward in the SMDP is defined as R_max^Ø≥max_s ∈𝒮,a ∈𝒜{r(s,a)/τ(s,a)}.] R_max^Ø≤ R_max and since 𝒮_Ø⊆𝒮 we have that S_Ø≤ S. Furthermore, we introduce the following simplifying conditions: 1) ρ^*(M)=ρ^*(M_Ø) (i.e., the options do not prevent from learning the optimal policy), 2) O ≤ A (i.e., the number of options is not larger than the number of primitive actions), 3) options have bounded holding time (case 2 in Asm. <ref>). While in general comparing upper bounds is potentially loose, we notice that both upper-bounds are derived using similar techniques and thus they would be “similarly” loose and they both have almost matching worst-case lower bounds.Let ℛ(M,n,δ) be the ratio between the regret upper bounds of using options Ø and , then we have (up to numerical constants)ℛ(M,n)≤(D_Ø√(S_Ø) + T_max) √(S_Ø On log(n/δ))/ DS√(AT_n log(T_n/δ))≤D_Ø√(S) + T_max/D√(S)√(n/T_n),where we used n≤ T_n to simplify the logarithmic terms. Since lim inf_n → +∞T_n/n≥τ_min, then the previous expression gives an (asymptotic) sufficient condition for reducing the regret when using options, that isD_Ø√(S) + T_max/D√(Sτ_min)≤ 1.In order to have a better grasp on the cases covered by this condition, let D_Ø = α D, with α≥ 1. This corresponds to the case when navigating through some states becomes more difficult with options than with primitive actions, thus causing an increase in the diameter. If options are such that T_max≤ D√(S) and τ_min > (1+α)^2, then it is easy to see that the condition in Eq. <ref> is satisfied. This shows that even when the introduction of options partially disrupt the structure of the original MDP (i.e., D_Ø≥ D), it is enough to choose options which are long enough (but not too much) to guarantee an improvement in the regret. Notice that while conditions 1) and 2) are indeed in favor of , S_Ø, O, and T_max are in general much smaller than S, A, D√(S) (S and D are large in most of interesting applications). Furthermore, τ_min is a very loose upper-bound on lim inf_n → +∞T_n/n and in practice the ratio T_n/n can take much larger values if τ_max is large and many options have a high expected holding time. As a result, the set of MDPs and options on which the regret comparison is in favor of is much wider than the one defined in Eq. <ref>. Nonetheless, as illustrated in Lem. <ref>, the case of options with bounded holding times is quite restrictive since it requires the absence of self-loops during the execution of an option. If we reproduce the same comparison in the general case of sub-exponential holding times, then the ratio between the regret upper bounds becomesℛ(M,n)≤D_Ø√(S) + 𝒞(M,n,δ)/D√(S)√(n/T_n),where 𝒞(M,n,δ) = O(√(log(n/δ))). As a result, as n increases, the ratio is always greater than 1, thus showing that in this case the regret of is asymptotically worse than . Whether this is an artefact of the proof or it is an intrinsic weakness of options, it remains an open question.§ ILLUSTRATIVE EXPERIMENT We consider the navigation problem in Fig. <ref>. In any of the d^2 states of the grid except the target, the four cardinal actions are available, each of them being successful with probability 1. If the agent hits a wall then it stays in its current position with probability 1. When the target state is reached, the state is reset to any other state with uniform probability. The reward of any transition is 0 except when the agent leaves the target in which case it equals R_max. The optimal policy simply takes the shortest path from any state to the target state. The diameter of the MDP is the longest shortest path in the grid, that is D = 2d-2. Let m be any non-negative integer smaller than d and in every state but the target we define four macro-actions: LEFT, RIGHT, UP and DOWN (blue arrows in the figure). When LEFT is taken, primitive action left is applied up to m times (similar for the other three options). For any state s' which is k≤ m steps on the left of the starting state s, we set β_o(s')=1/(m-k+1) so that the probability of the option to be interrupted after any k ≤ m steps is 1/m. If the starting state s is l steps close to the left border with l<m then we set β_o(s')=1/(l-k+1) for any state s' which is k≤ l steps on the left. As a result, for all options started m steps far from any wall, T_max = m and the expected duration is τ := τ(s,o) = (m+1)/2, which reduces to T_max = l and τ =(l+1)/2 for an option startedl < m step from the wall and moving towards it. More precisely, all options have an expected duration of τ(s,o) = τ in all but in m d states, which is small compared to the total number of d^2 states. The SMDP formed with this set of options preserves the number of state-action pairs S_Ø = S = d^2 and A'=A=4 and the optimal average reward ρ^*(M) = ρ^*(M'), while it slightly perturbs the diameter D_Ø≤ D + m(m+1) (see App. <ref> for further details). Thus, the two problems seem to be as hard to learn. However the (asymptotic) ratio between the regret upper bounds becomeslim_n →∞ ℛ(M,n) ≤(2d-2+ m^2 + m)d + m/(2d-2)d(lim_n →∞√(n/T_n))≤(1+ 2m^2/d)(lim_n →∞√(n/T_n)),where we assume m, d ≥ 2. While a rigorous analysis of the ratio between the number of option decision steps n and number of primitive actions T_n is difficult, we notice that as d increases w.r.t. m, the chance of executing options close to a wall decreases, since for any option only m d out of d^2 states will lead to a duration smaller than τ and thus we can conclude that n/T_n tends to 1/τ = 2/(m+1) as n and d grow. As a result, the ratio would reduce to (1+2m^2/d)√(2/(m+1)) that is smaller than 1 for a wide range of values for m and d. Finally, the ratio is (asymptotically in d) minimized by m ≈√(d), which gives ℛ(M,n) = O(d^-1/4), thus showing that as d increases there is always an appropriate choice of m for which learning with options becomes significantly better than learning with primitive actions. In Fig. <ref> we empirically validate this finding by studying the ratio between the actual regrets (and not their upper-bounds) as d and m (i.e., T_max) vary, and with a fixed value of T_n that is chosen big enough for every d. As expected, for a fixed value of d, the ratio ℛ first decreases as m increases, reaches a minimum and starts increasing to eventually exceed 1. As d increases, the value of the minimum decreases, while the optimal choice of m increases. This behaviour matches the theory, which suggests that the optimal choice for m increases as O(√(d)). In Fig. <ref> we report the cumulative regret and we observe that high values of T_max worsen the learning performances w.r.t. learning without options (T_max = 1, plotted in black). Finally, Fig. <ref> shows that, as n tends to infinity, T_n/n tends to converge to (m+1)/2 when m ≪ d, whereas it converges to slightly smaller values when m is close to d because of the truncations operated by walls.Discussion. Despite its simplicity, the most interesting aspect of this example is that the improvement on the regret is not obtained by trivially reducing the number of state-action pairs, but it is intrinsic in the way options change the dynamics of the exploration process. The two key elements in designing a successful set of options Ø is to preserve the average reward of the optimal policy and the diameter. The former is often a weaker condition than the latter. In this example, we achieved both conditions by designing a set Ø where the termination conditions allow any option to end after only one step. This preserves the diameter of the original MDP (up to an additive constant), since the agent can still navigate at the level of granularity of primitive actions. Consider a slightly different set of options Ø', where each option moves exactly by m steps (no intermediate interruption). The number of steps to the target remains unchanged from any state and thus we can achieve the optimal performance. Nonetheless, having π^* in the set of policies that can be represented with Ø' does not guarantee that the would be as efficient in learning the optimal policy as . In fact, the expected number of steps needed to go from a state s to an adjacent state s' may significantly increase. Despite being only one primitive action apart, there may be no sequence of options that allows to reach s' from s without relying on the random restart triggered by the target state. A careful analysis of this case shows that the diameter is as large as D_Ø' = D(1+m^2) and there exists no value of m that satisfies Eq. <ref> (see App. <ref>). § CONCLUSIONS We derived upper and lower bounds on the regret of learning in SMDPs and we showed how these results apply to learning with options in MDPs. Comparing the regret bounds of with , we provided sufficient conditions on the set of options and the MDP (i.e., similar diameter and average reward) to reduce the regret w.r.t. learning with primitive actions. To the best of our knowledge, this is the first attempt of explaining when and how options affect the learning performance. Nonetheless, we believe that this result leaves space for improvements. In fact, Prop. <ref> implies that the class of SMDPs is a strict superset of MDPs with options. This suggests that a more effective analysis could be done by leveraging the specific structure of MDPs with options rather than moving to the more general model of SMDPs. This may actually remove the additional √(log(n/δ)) factor appearing because of sub-exponential distributions in the regret. An interesting direction of research is to use this theoretical result to provide a more explicit and quantitative objective function for option discovery, in the line of what is done in <cit.>. 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In order to have a more rigorous treatment of SMDPs, we introduce further notations from <cit.>.A decision rule is a function d: ℋ→Δ(𝒜) where ℋ is the set of possible histories and Δ(𝒜) is the set of probability distributions over 𝒜. For an SMDP M, we will denote by D_M^HR the set of history-dependent randomized decision rules and D_M^MD the subset of Markovian deterministic decision rules (D_M^MD⊂ D_M^HR). A history-dependent randomized policy is a sequence of elements of D_M^HR indexed by the decision steps, i.e., π = (d_1, d_2, ...) ∈ (D_M^HR)^ℕ, while a stationary deterministic policy is a constant sequence of elements of D_M^MD: π = (d, d, ...) = d^∞. The set of history-dependent randomized policies will be denoted Π_M^HR and the subset of stationary deterministic policies will be denoted Π_M^SD: Π_M^SD⊂Π_M^HR. We also consider the more general case where the set of available actions may depend on the state, i.e., there exists a set _s for each s∈. §.§ Optimality criterion We start by defining the optimality criterion in SMDPs. Unlike for MDPs, where the average reward of a fixed policy is uniquely defined, in SMDPs there are three different definitions that are usually encountered in the literature (see , , and ).[Notice that the definition we provide in Eq. <ref> of Prop. <ref> is ρ_1.] Denote N(t) = sup{n : n ∈ℕ,∑_i=1^nτ_i≤ t } the number of decision steps that occured before time t. For any π∈Π_M^HR and s∈𝒮, we define:ρ_1^π(s) = lim sup_t → + ∞𝔼^π[ ∑_i=1^N(t)r_i/t| s_0 =s ], ρ_1^π(s) = lim inf_t → + ∞𝔼^π[ ∑_i=1^N(t)r_i/t| s_0 =s ] ρ_2^π(s) = lim sup_n → + ∞𝔼^π[ ∑_i=1^nr_i| s_0 =s ]/𝔼^π[ ∑_i=1^nτ_i| s_0 =s ], ρ_2^π(s) = lim inf_n → + ∞𝔼^π[∑_i=1^nr_i| s_0 =s ]/𝔼^π[ ∑_i=1^nτ_i| s_0 =s ]and for any d^∞∈Π_M^SD and s∈𝒮 we define:ρ_3^d^∞(s) = ∑_α = 1^ν(d)p^d(α|s) g_d(α), with g_d(α) = ∑_s ∈ R_α^dμ_α^d(s)r(s,d(s))/∑_s ∈ R_α^dμ_α^d(s)τ(s,d(s))where ν(d) is the number of positive recurrent classes under d^∞, p^d(α|s) is the probability of entering positive recurrent class α starting from s and following policy d^∞, R_α^d is the set of states of class α and μ_α^d is the stationary probability distribution of class α.§.§ Proof of Proposition <ref> We say that (d^*)^∞ is (3)-average-optimal if for all states s ∈𝒮 and decision rules d ∈ D_M^MD, ρ_3^(d^*)^∞(s) ≥ρ_3^d^∞(s). We say that π^* is (1)-average-optimal (respectively (2)-average-optimal) if for all states s ∈𝒮 and all π∈Π_M^HR, ρ_1^π^*(s) ≥ρ_1^π(s) (respectively ρ_2^π^*(s) ≥ρ_2^π(s)). We prove a slightly more general formulation than Proposition <ref>. If M is communicating and the expected holding times and rewards are finite, then * There exists a stationary policy π^* = (d^*)^∞ which is (1,2,3)-average-optimal.* All optimal values are equal and constant and we will denote this value by ρ^*:∀ s ∈𝒮,ρ_1^(d^*)^∞(s) = ρ_1^(d^*)^∞(s) = ρ_2^(d^*)^∞(s) = ρ_2^(d^*)^∞(s) = ρ_3^(d^*)^∞(s) = ρ^*.Step 1: Optimality equation of a communicating SMDP. We first recall the average reward optimality equations for a communicating SMDP (Eq. <ref>)∀ s ∈𝒮,u^*(s) = max_a ∈𝒜_s{r(s,a) - ρ^* τ(s,a) + ∑_s' ∈𝒮p(s'|s, a)u^*(s') }where u^* and ρ^* are the bias (up to additive constant) and average reward respectively. Since we need to analyse both the case where 𝒜_s is finite and the case where 𝒜_s is continuous, we verify that it is appropriate to consider the max instead of sup in the previous expression. For the original SMDP M, 𝒜_s is finite and the maximum is well-defined. For the extended SMDPs M_k^+ considered while computing the optimistic SMDP, 𝒜_k,s^+ is compact and r(s,a), τ(s,a) and p(| s, a) are continuous in 𝒜_k,s^+ by the very definition of M_k^+. The function r(s,a) - ρ^* τ(s,a) + ∑_s' ∈𝒮p(s'|s, a)u^*(s') is thus continuous on 𝒜_k,s^+ compact and by Weierstrass theorem, we know that the maximum is reached (i.e., there exists a maximizer). As a result, Eq. <ref> is well-defined and we can study the existence and properties of its solutions.Step 2: Data-transformation (uniformization) of an SMDP. The structure of EVI is based on a data-transformation (also called "uniformization") which turns the initial SMDP M into an “equivalent” MDP M_ = {𝒮, 𝒜, p_, r_} defined as in Eq. <ref>. As a result, we can just apply standard MDP theory to the equivalent MDP. The average optimality equation of M_ is <cit.>∀ s ∈𝒮,v^*(s) = max_a ∈𝒜_s{r(s,a)/τ(s,a) - g^* + τ/τ(s,a)∑_s' ∈𝒮p(s'|s, a)v^*(s') + ( 1 - τ/τ(s,a)) v^*(s) }Since τ < τ_min, every Markov Chain induced by a stationary deterministic policy on M_ is necessarily aperiodic (for any action, the probability of any state to loop on itself is strictly positive). Moreover, since M is assumed to be communicating, M_ is also communicating. The same holds for M_k,^+ (i.e., the MDP obtained from the extended SMDP M_k^+ after data transformation). Under these conditions, Eq. <ref> has a solution (v^*, g^*) where g^* is the optimal average reward of M_ (respectively M_k,^+) and the (stationary deterministic) greedy policy w.r.t. v^* is average-optimal. Moreover, standard value iteration is guaranteed to converge and it can be applied with the stopping condition in Eq. <ref> to obtain an ϵ-optimal policy in finitely many steps. This holds for both finite and compact 𝒜_s with continuous r_(s,a) and p_(s' | s,a) (see for exampleand ). It is easy to show that EVI in Eq. <ref> is exactly value iteration applied to M_k,^+. Finally, Lemma 2 of <cit.> (Prop. <ref>) shows the “equivalence” between M and M_ (respectively M_k^+ and M_k,^+): if (v^*, g^*) is a solution to Eq. <ref>, then (τ^ -1v^*, g^*) is a solution to Eq. <ref> and conversely. As a result, there exists a solution (u^*, ρ^*) to Eq. <ref> for both M and M_k^+.Step 3: Existence of deterministic stationary optimal policy. We are now ready to prove the existence of a deterministic stationary policy that is (1,2,3)-optimal and that the corresponding optimal value is constant and equal in all three cases. We consider the case of finite and continuous _s separately.Step 3a: For M (finite 𝒜_s). Since conditions (L), (F) and (R) of <cit.> hold, we can apply their main theorem and obtain that * Any greedy policy (d^* )^∞ w.r.t. u^* is such that ρ_1^(d^*)^∞(s) ≥ρ_1^π(s) for any π∈Π_M^HR and any s ∈𝒮,* ∀ s ∈𝒮,ρ_1^(d^*)^∞(s) = ρ^*,where (u^*, ρ^*) is a solution of Eq. <ref>. Furthermore, from renewal theory (see e.g., and <cit.>) we have that ∀ d^∞∈Π_M^SD, ρ_1^d^∞ = ρ_1^d^∞ =ρ_1^d^∞ (i.e., the limit exists for deterministic stationary policies), thus we can conclude that (d^* )^∞ is (1)-optimal. Furthermore, by Lemma 2.7 of <cit.>: ∀ d^∞∈Π_M^SD,∀ s ∈𝒮,ρ_1^d^∞(s) = ρ_3^d^∞(s) so d^* is also necessarily (3)-optimal. Finally, by Theorem 7.6 of <cit.>, since (u^*, ρ^*) is a solution of Eq. <ref>: * Any greedy policy (d^* )^∞ w.r.t. u^* is such that ρ_2^(d^*)^∞(s) ≥ρ_2^π(s) for any π∈Π_M^HR and any s ∈𝒮,* ∀ s ∈𝒮, ρ_2^(d^*)^∞(s) = ρ^*.By Theorem 11.4.1 of <cit.> we have ∀ d^∞∈Π_M^SD,ρ_2 = ρ_2 = ρ_2^d^∞ and thus d^* is also (2)-optimal. This concludes the proof for the finite case, which proves the statement of Prop. <ref>.Step 3b: For M_k^+ (compact 𝒜_k,s^+ with continuous rewards, holding times and transition probabilities). The proof is almost the same as with discrete action spaces. The only difference is that we can't apply the Theorem of <cit.> because conditions (R) and (C*) do not hold in general. However, we can use Propositions 5.4 and 5.5 of <cit.>and we have the same result as in the discrete case (assumptions (L), (C), (P) and (I) hold in our case and we know that the optimality equation <ref> admits a solution (u^*, ρ^*), see above). Since the state space is finite, the rest of the proof is rigorously the same (the Theorems and Lemmas still applies). This guarantees the same statement as Prop. <ref> but for the optimistic SMDP M_k^+. §.§ Proof of Lemma <ref>From the proof of Prop. <ref>, we already have that EVI converges towards the optimal average reward of M_k,^+, which is also the optimal average reward of M_k^+. We also know that the stopping criterion is met in a finite number of steps and that the greedy policy when the stopping criterion holds is ϵ-optimal in the equivalent M_k,^+. Then, in order to prove Lemma <ref>, we only need to prove that this policy is also ϵ-optimal in the optimistic SMDP M_k^+. <cit.> shows that for any stationary deterministic policy d^∞∈Π_M_k^+^SD, the (1)-average reward is the same in the SMDP and the MDP obtained by uniformization, that is∀ s ∈𝒮, ρ_1^d^∞(M_k^+) = ρ^d^∞(M_k,^+).Then it immediately follows that the policy returned by EVI is (1)-ϵ-optimal in M_k^+ and since ∀ d^∞∈Π_M^SD,ρ_1^d^∞ = ρ_3^d^∞, it is also (3)-ϵ-optimal. Note that for any deterministic stationary policy d ∈Π_M_k^+^SD defining a unichain Markov chain in M_k,^+ (or equivalently in M_k^+), we have: ∀ s ∈𝒮, ρ_1^d^∞(s) = ρ_2^d^∞(s) and this value is constant across states (see for example chapter 11 of <cit.>, Theorem 7.5 of <cit.> or <cit.>). However, in the general case, this equality does not hold (see Example 2.1 of <cit.>). Nevertheless, by Theorem 3.1 of <cit.> we have∀ d^∞∈Π_M_k^+^SD, ∀ s ∈𝒮, |ρ_2^d^∞(M_k^+, s) - ρ^d^∞(M_k,^+, s) | ≤ρ_max^d^∞(M_k,^+) - ρ_min^d^∞(M_k,^+)ρ_max^d^∞(M_k,^+) = max_s ∈𝒮ρ^d^∞(M_k,^+, s) ρ_min^d^∞(M_k,^+) = min_s ∈𝒮ρ^d^∞(M_k,^+, s).If we denote by d the policy returned by EVI and ρ^* the optimal gain of M_k,^+ and M_k^+ we obtain∀ s ∈𝒮,ρ^* - ρ_2^d^∞(M_k^+, s) = ρ^d^∞(M_k,^+, s)-ρ_2^d^∞(M_k^+, s) + ρ^ * - ρ^d^∞(M_k,^+, s) ≤ρ_max^d^∞(M_k,^+) - ρ_min^d^∞(M_k,^+) + ϵ = ρ_max^d^∞(M_k,^+) - ρ^* + ρ^* - ρ_min^d^∞(M_k,^+) + ϵ≤ 2ϵ.For the first inequality we used Eq. <ref> and the fact that d is ϵ-optimal in M_k,^+. For the second inequality, we used again that d is ϵ-optimal in M_k,^+ and we also used the fact that ρ_max^d^∞(M_k,^+) ≤ρ^*. In conclusion, the policy returned by EVI is (2)-2ϵ-optimal. The remaining part of Theorem <ref> is thus proved for all optimality criteria. By Theorem 8.3.2 of <cit.>, we know that there exists an optimal policy d^* of MDP M_k,^+ that yields a unichain Markov Chain (i.e., a Markov Chain with a single positive recurrent class). The Markov Chain induced by d^* in M_k^+ is thus also unichain and moreover: ρ_1^(d^*)^∞(M_k^+) = ρ^(d^*)^∞(M_k,^+) = ρ^*(M_k,^+) = ρ^*(M_k^+). We have seen that for any policy d ∈Π_M_k^+^SD yielding a unichain Markov Chain ρ_1^d^∞(M_k^+) = ρ_2^d^∞(M_k^+) and so in particular, it is true for d^*. Therefore, there exists a policy of M_k^+ which yields a unichain Markov Chain and which is (1)-optimal, (2)-optimal and (3)-optimal. This explains why the optimal gain is the same for criteria (1) and (2) but EVI must be run with a different accuracy to insure ϵ-accuracy (the Markov Chain induced by the policy returned by EVI is not necessarily unichain).§ ANALYSIS OF SMDP-UCRL (PROOF OF THEOREM <REF>) The proof follows the same steps as in <cit.>. Therefore, in the following we only emphasize the differences between SMDPs and MDPs and we refer to <cit.> for the parts of the proof which are similar. §.§ Splitting into Episodes We first recall the definition of sub-exponential random variables. A random variable X with mean μ < +∞ is said to be sub-exponential, if one of the following equivalent conditions is satisfied: * (Laplace transform condition) There exists (σ, b) ∈ℝ^+×ℝ^+* such that:𝔼 [ e^λ (X-μ) ] ≤ e^σ^2 λ^2/2 for all |λ| < 1/b.In this case, we say that X is sub-exponential of parameters σ, b and we denote it by X ∈(σ,b).* There exists c_0 >0 such that 𝔼 [ e^λ (X-μ) ] < + ∞for all|λ| ≤ c_0. In order to define the confidence intervals, we use the Bernstein concentration inequality for sub-exponential random variables.Let (X_i)_1 ≤ i ≤ n be a collection of independent sub-Exponential random variables s.t. ∀ i ∈{ 1, ..., n },X_i ∈(σ_i,b_i) and 𝔼[X_i] = μ_i. We have the following concentration inequalities:∀ t ≥ 0, ℙ( ∑_i=1^nX_i - ∑_i=1^nμ_i≥ t )≤ e^- t^2/2 n σ^2, 0 ≤ t ≤σ^2/b e^- t/2b, t > σ^2/b ℙ( ∑_i=1^nX_i + ∑_i=1^nμ_i≤ t )≤ e^- t^2/2 n σ^2, 0 ≤ t ≤σ^2/b e^- t/2b, t > σ^2/bwhere σ = √(∑_i=1^nσ_i^2/n) and b = max_1 ≤ i ≤ n{ b_i }. Denoting by N(s,a) the state-action counts we have∑_i=1^nr_i(s_i-1,a_i-1) = ∑_s ∈𝒮∑_a ∈𝒜_s∑_j=1^N(s,a)r_k_j(s,a).Conditionally on knowing (N(s,a))_s,a, the previous sum is equal (in distribution) to a sum of independent random variables with mean ∑_s ∈𝒮∑_a ∈𝒜_s N(s,a) r(s,a) and from Prop. <ref> we haveℙ( ∑_i=1^nr_i≤∑_s ∈𝒮∑_a ∈𝒜_s N(s,a) r(s,a) - σ_r √(5/2nlog( 13n/δ))| (N(s,a))_s,a)≤(δ/13n)^5/4≤δ/24n^5/4,n ≥5b_r^2/2σ_r^2log( 13n/δ) ℙ( ∑_i=1^nr_i≤∑_s ∈𝒮∑_a ∈𝒜_s N(s,a) r(s,a) - 5/2b_rlog( 13n/δ) | (N(s,a))_s,a)≤(δ/13n)^5/4≤δ/24n^5/4,n ≤5b_r^2/2σ_r^2log( 13n/δ)Similarly, the total holding time satisfiesℙ( ∑_i=1^nτ_i≥∑_s ∈𝒮∑_a ∈𝒜_s N(s,a) τ(s,a) + σ_τ√(5/2nlog( 13n/δ))| (N(s,a))_s,a)≤(δ/13n)^5/4≤δ/24n^5/4,n ≥5b_τ^2/2σ_τ^2log( 13n/δ) ℙ( ∑_i=1^nτ_i≥∑_s ∈𝒮∑_a ∈𝒜_s N(s,a) τ(s,a) + 5/2b_τlog( 13n/δ) | (N(s,a))_s,a)≤(δ/13n)^5/4≤δ/24n^5/4,n ≤5b_τ^2/2σ_τ^2log( 13n/δ)The optimal average reward can be bounded as follows: ρ^*(M) ≤max_s ∈𝒮,a ∈𝒜_s{r(s,a)/τ(s,a)}≤ R_max. In App. <ref> we prove that ρ^*(M) = ρ^*(M_) where ρ^*(M_) is the optimal average reward of an MDP M_ with same state and action spaces as SMDP M and with average rewards of the form r(s,a)/τ(s,a). All the rewards of M_ are thus bounded by R_max and so ρ^*(M_) is necessarily bounded by R_max as well and thus: ρ^*(M) ≤ R_max. We are now ready to split the regret over episodes. We define the per-episode regret asΔ_k = ∑_s ∈𝒮∑_a ∈𝒜_sν_k(s,a) ( τ(s,a)ρ^*- r(s,a) ).Setting γ_r = max{5/2b_r, √(5/2)σ_r } and γ_τ = max{5/2b_τ, √(5/2)σ_τ}, and using a union bound on the previous inequalities we have that with probability at least 1-δ/12n^5/4Δ(M,𝔄,s,n) ≤∑_k=1^nΔ_k + ( γ_r + γ_τ R_max) log( 13n/δ) √(n)§.§ Dealing with Failing Confidence RegionsFor any episode k ≥ 1, the probability that the true SMDP M is not contained in the set of plausible MDPs ℳ_k at step i is at most δ/15i_k^6, that is: ∀ k ≥ 1,ℙ( M ∉ℳ_k ) < δ/15i_k^6 This lemma is the SMDP-analogue of Lemma 17 in <cit.> and the proof is similar. Using an ℓ_1-concentration inequality for discrete probability distributions we obtainℙ(p(·|s,a) - p̂_k(·|s,a) _1 ≥β_k^p(s,a) )= ℙ(p(·|s,a) - p̂_k(·|s,a) _1 ≥√(14 S/nlog( 2A i_k/δ)))≤ℙ(p(·|s,a) - p̂_k(·|s,a) _1 ≥√(2/nlog( 2^S 20 S A i_k^7/δ)))≤ 2^S exp( -n/2×2/nlog( 2^S 20 S A i_k^7/δ) )= δ/20i_k^7SAIn the inequalities above, it is implicitly assumed that the value N_k(s,a) = n is fixed. To be more rigorous, we are bounding the probability of the intersection of event {p(·|s,a) - p̂_k(·|s,a) _1 ≥β_k^p(s,a) } with event { N_k(s,a) = n }but we omitted the latter to simplify notations, and we will also omit it in the next inequalities. Using Bernstein inequality (Prop. <ref>) and noting that 240≤ 2^7(SA/δ)^6 for S, A ≥ 2 and δ≤ 1, we have: * If n ≥2 b_r^2/σ_r^2log( 240SAi_k^7/δ):ℙ( | r(s,a) - r̂_k(s,a) | ≥β_k^r(s,a) )= ℙ( | r(s,a) - r̂_k(s,a) | ≥σ_r √(14/nlog( 2SA i_k/δ)))≤ℙ( | r(s,a) - r̂_k(s,a) | ≥σ_r √(2/nlog( 240 S A i_k^7/δ)))≤ 2 exp( -n/2 σ_r^2×2/nσ_r^2 log( 240 S A i_k^7/δ) )= δ/120i_k^7SA* If n < 2 b_r^2/σ_r^2log( 240SAi_k^7/δ):ℙ( | r(s,a) - r̂_k(s,a) | ≥β_k^r(s,a) )= ℙ( | r(s,a) - r̂_k(s,a) | ≥14b_r/nlog( 2SA i_k/δ) )≤ℙ( | r(s,a) - r̂_k(s,a) | ≥2b_r/nlog( 240SA i_k^7/δ))≤ 2 exp( -n/2 b_r×2/n b_r log( 240SA i_k^7/δ))= δ/120i_k^7SASimilarly for holding times we have:ℙ( | τ(s,a) - τ̂_k(s,a) | ≥β_k^τ(s,a) ) ≤δ/120i_k^7SA Note that when there hasn't been any observation, the confidence intervals trivially hold with probability 1. Moreover, N_k(s,a) < i_k by the stopping condition of an episode. Taking a union bound over all possible values of N_k(s,a) yields:ℙ( | τ(s,a) - τ̂_k(s,a) | ≥β_k^τ(s,a) ) ≤δ/120i_k^6SA ℙ( | r(s,a) - r̂_k(s,a) | ≥β_k^r(s,a) ) ≤δ/120i_k^6SA ℙ(p(·|s,a) - p̂_k(·|s,a) _1 ≥β_k^p(s,a) ) ≤δ/20i_k^6SASumming over all state-action pairs: ℙ( M ∉ℳ_k ) < δ/15i_k^6. We now consider the regret of episodes in which the set of plausible SMDPs ℳ_k does not contain the true SMDP M: ∑_k=1^mΔ_k 1_M ∉ℳ_k. By the stopping criterion for episode k (except for episodes where ν_k(s,a) = 1 and N_k(s,a) = 0 for which ∑_s ∈𝒮∑_a ∈𝒜_sν_k(s,a) = 1 ≤ i_k): ∑_s ∈𝒮∑_a ∈𝒜_sν_k(s,a) ≤∑_s ∈𝒮∑_a ∈𝒜_s N_k(s,a) = i_k - 1We can thus bound this part of the regret:∑_k=1^mΔ_k 1_M ∉ℳ_k ≤∑_k=1^m∑_s ∈𝒮∑_a ∈𝒜_sν_k(s,a) τ(s,a) ρ^* 1_M ∉ℳ_k≤τ_maxρ^*∑_k=1^m i_k 1_M ∉ℳ_k = τ_maxρ^* ∑_i=1^n i ∑_k=1^m1_i = i_k, M ∉ℳ_k≤τ_maxρ^* ( ∑_i=1^⌊ n^1/4⌋ i+∑_i=⌊ n^1/4⌋ +1^n i ∑_k=1^m1_i = i_k, M ∉ℳ_k) ≤τ_maxρ^* ( √(n) + ∑_i=⌊ n^1/4⌋ +1^n i ∑_k=1^m1_i = i_k, M ∉ℳ_k)where we defined: τ_max = max_s,aτ(s,a) < + ∞.By Lemma <ref>, the probability that the second term in the right hand side of the above inequality is non-zero is bounded by∑_i=⌊ n^1/4⌋^nδ/15i^6≤δ/15n^6/4 + ∫_n^1/4^+∞δ/15x^6 dx ≤δ/12n^5/4.In other words, with probability at least 1-δ/12n^5/4:∑_k=1^mΔ_k 1_M ∉ℳ_k≤τ_max R_max√(n).§.§ Episodes with M ∈ℳ_k Now we assume that M ∈ℳ_k and we start by analysing the regret of a single episode k. By construction, R_max≥ρ_k ≥ρ^* - R_max/√(i_k) hence:Δ_k = ∑_s ∈𝒮∑_a ∈𝒜_sν_k(s,a) ( τ(s,a)ρ^*- r(s,a) )≤∑_s ∈𝒮∑_a ∈𝒜_sν_k(s,a) ( τ(s,a)ρ_k- r(s,a) ) + R_max∑_s ∈𝒮∑_a ∈𝒜_sν_k(s,a)/√(i_k)τ(s,a) Δ_k≤∑_s ∈𝒮∑_a ∈𝒜_sν_k(s,a) ( τ_k(s,a)ρ_k- r(s,a) ) + R_max∑_s ∈𝒮∑_a ∈𝒜_sν_k(s,a) (τ(s,a)- τ_k(s,a) )+ R_maxτ_max∑_s ∈𝒮∑_a ∈𝒜_sν_k(s,a)/√(i_k) We now need two results about the extended value iteration algorithm.At any iteration i ≥ 0 of EVI (extended value iteration), the range of the state values is bounded as follows,∀ i ≥ 0, max_s∈𝒮 u_i(s) - min_s∈𝒮 u_i(s) ≤ R_max D(M)/τ,where R_max is an upper-bound on the per-step reward r(s,a)/τ(s,a), τ is the parameter used in the uniformization of the SMDP M and D(M) is its diameter (Def. <ref>).In Appendix <ref> we show that EVI is value iteration applied to the equivalent MDP M_k,^+ obtained by “uniformizing” the extended SMDP M_k^+. Thus, we focus on any SMDP M and its equivalent MDP M_. Using the same argument as in section 4.3.1 of <cit.>, we have that: ∀ i ≥ 0,max_s∈𝒮 u_i(s) - min_s∈𝒮 u_i(s) ≤R_max D(M_) since all rewards of M_ are bounded by R_max whenever the average reward in M is bounded by R_max. Thus we need to find a relationship between D(M) and D(M_). Let T(s') denote the first time at which state s' is reached in M or M_, that isM:T(s') = inf{∑_i=1^nτ_i : n ∈ℕ,s_n = s'} M_:T(s') = inf{ n : n ∈ℕ,s_n = s'}.We prove that ∀ s, s' ∈𝒮,∀π∈Π_M^SD = Π_M'^SD, 𝔼_M^π[T(s')|s_0 = s ] = τ𝔼_M'^π[T(s')|s_0 = s ]. We consider two cases: * If ℙ_M^π(T(s') = + ∞|s_0 = s ) > 0 then necessarily 𝔼_M^π[T(s')|s_0 = s ] = + ∞.Moreover: ℙ_M^π(T(s') = + ∞|s_0 = s ) > 0 ℙ_M_^π(T(s') = + ∞|s_0 = s ) > 0 and so 𝔼_M_^π[T(s')|s_0 = s ] = + ∞ =1/τ𝔼_M^π[T(s')|s_0 = s ].* Conversely: ℙ_M^π(T(s') = + ∞|s_0 = s ) =0 ℙ_M_^π(T(s') = + ∞|s_0 = s ) =0 in which case both expectations are finite. To prove they are equal up to factor τ, we see the holding time as a “reward” (the true rewards are ignored here). Note that any policy π induces Markov chains with different dynamics on M and M_ (different transition probabilities). We call these Markov chains MC and MC_ respectively. Suppose we modify MC as follows: all states that are not reachable from s are ignored, all other states are unchanged except s' that is assumed to be absorbing (i.e., π(s') is an action that loops on s' with probability 1). Furthermore, we build a Markov reward process MR with the same dynamics as MC and such that all transitions (s,π(s)) have an expected reward equal to τ(s,π(s)) except (s',π(s')) which has a reward of zero. The total expected reward of this Markov reward process (MRP denoted MR) starting from s trivially equals 𝔼_M^π[T(s')|s_0 = s ]. Since we assumed that 𝔼_M^π[T(s')|s_0 = s ] is finite, and because all states of MR are reachable from s (the other states were ignored), s' is reached with probability 1 no matter which starting state s of MR is chosen (or in other words, even though we ignored some states, the transition matrix of MR is stochastic -and not sub-stochastic- and has a single recurrent class consisting of the absorbing state s'). By <cit.>, the vector ( T(s) )_s∈𝒮 = ( 𝔼_M^π[T(s')|s_0 = s] )_s∈𝒮 is the unique solution to the system of equations∀s,T(s) = τ(s,d(s)) + ∑_sp(s|s,d(s)) T(s).Applying the same transformation to MC_ and assigning a reward of 1 to all transitions but (s',π(s')) (which has reward 0) in order to build MR_, we deduce that the vector ( T_(s) )_s∈𝒮 = ( 𝔼_M_^π[T(s')|s_0 = s] )_s∈𝒮 is the unique solution to the system of equations∀s,T_(s) = 1 + τ/τ(s,d(s))∑_sp(s|s,d(s)) T_(s) + (1 - τ/τ(s,d(s)))T_(s)∀s,(τT_(s))= τ(s,d(s)) + ∑_sp(s|s,d(s)) (τT_(s)).By uniqueness of the solution: τT_ = Tτ D(M_) = D(M).If the convergence criterion of EVI hold at iteration i, then:∀ s ∈𝒮, | u_i+1(s) - u_i(s) - ρ_k | ≤1/√(i_k) We introduce the following quantitiesM_i = max_s ∈𝒮{ u_i+1(s) - u_i(s) }, m_i = min_s ∈𝒮{ u_i+1(s) - u_i(s) }, ϵ = 1/√(i_k).Since EVI is just value iteration applied to MDP M_k', Theorem 8.5.6 of <cit.> hold and we have:1/2(M_i + m_i) ≥ρ_k - ϵ/2 m_i ≥ρ_k - ϵ/2 - 1/2(M_i - m_i)m_i ≥ρ_k - ϵ 1/2(M_i + m_i) - ρ_k ≤ϵ/2 M_i ≤ρ_k + ϵ/2 + 1/2(M_i - m_i)M_i ≤ρ_k + ϵ .In conclusion:∀ s ∈𝒮, -1/√(i_k)≤ u_i+1(s) - u_i(s) - ρ_k ≤1/√(i_k).Based on Lemma <ref>, Eq. <ref>, and optimiality equation Eq. <ref>, we have:∀ s ∈𝒮,| ( ρ_k - r_k(s,π_k(s))/τ_k(s,π_k(s))) - ( ∑_s' ∈𝒮p_k(s' | s, π_k(s)) u_i(s') - u_i(s) ) τ/τ_k(s,π_k(s))| ≤1/√(i_k) Setting r_k = ( r_k(s,π_k(s)) )_s ∈𝒮 to be the column vector of rewards under policy π_k, P_k = (p_k(s' | s, π_k(s)) )_s, s' ∈𝒮 the transition matrix and v_k = ( ν_k (s,π_k(s)) )_s ∈𝒮 the row vector of visit counts for each state and the corresponding action chosen by π_k. We will use the fact that a ≠π_k(s) ν_k(s,a) = 0. Δ_k ≤∑_s,aν_k(s,a) ( τ_k(s,a)ρ_k- r(s,a) ) + R_max∑_s,aν_k(s,a) (τ(s,a)- τ_k(s,a) ) + R_maxτ_max∑_s,aν_k(s,a)/√(i_k)= ∑_s,aν_k(s,a) ( τ_k(s,a)ρ_k- r_k(s,a) ) + ∑_s,aν_k(s,a) ( r_k(s,a)- r(s,a) ) + R_maxτ_max∑_s,aν_k(s,a)/√(i_k) + R_max∑_s,aν_k(s,a)(τ(s,a)- τ_k(s,a) )We will now upper-bound the four terms of the right-hand side of the above inequality. Setting c_r = max{ 14 b_r, √(14)σ_r } and c_τ = max{ 14 b_τ, √(14)σ_τ} we have:r_k(s,a) - r(s,a) ≤| r_k(s,a) - r̂_k(s,a) | + | r̂_k(s,a) - r(s,a) | ≤ 2 β_k^r(s,a) ≤ 2 c_r log(2S A i_k / δ)/√(max{ 1, N_k(s,a) }) τ(s,a) - τ_k(s,a) ≤| τ_k(s,a) - τ̂_k(s,a) | + | τ̂_k(s,a) - τ(s,a) | ≤ 2 β_k^τ(s,a) ≤ 2 c_τlog(2S A i_k / δ)/√(max{ 1, N_k(s,a) })Finally, using <ref> and noting that τ_k(s,a) ≤τ_max (by construction) we obtain:τ_k(s,a)ρ_k- r_k(s,a) ≤R_maxτ_max/√(i_k) + τ( ∑_s' ∈𝒮p_k(s' | s, π_k(s)) u_i(s') - u_i(s) ),a = π_k(s) ∑_s,aν_k(s,a) (τ_k(s,a)ρ_k- r_k(s,a) )≤R_maxτ_max∑_s,aν_k(s,a)/√(i_k) + τ(v_k ( P_k - I ) u_i )where i is the iteration at which the stopping condition of EVI holds. Defining the column vector w_k by:w_k(s) = u_i(s) - min_s ∈𝒮 u_i(s) + max_s ∈𝒮 u_i(s)/2and since the rows of P_k sum to one, we have: v_k ( P_k - I ) u_i = v_k ( P_k - I ) w_k. Moreover, by Lemma <ref>: w_k _∞≤ R_max D/2 τ. Noting that max{ 1, N_k(s,a) }≤ i_k ≤ n we get:Δ_k ≤τ(v_k ( P_k - I ) w_k ) + 2 ( R_maxτ_max + (c_r + R_maxc_τ) log(2S A n/δ) ) ∑_s,aν_k(s,a)/√(max{ 1, N_k(s,a) }) Using exactly the same arguments as in<cit.>, it is trivial to prove that with probability at least 1-δ/12n^5/4:∑_k=1^m v_k ( P_k - I ) w_k 1_M ∈ℳ_k≤R_max D/τ[ √(14S log( 2An/δ))∑_k=1^m∑_s,aν_k(s,a)/√(max{ 1, N_k(s,a) }) + √(5/2n log( 8n/δ)) +S A log_2 ( 8n/SA) ]Consider a sequence of positive reals (z_k)_k and define: ∀ k,Z_k = max{ 1, ∑_i=1^kz_i}. Assuming that 0 ≤ z_k ≤ Z_k-1 we have:∀ n ≥ 1,∑_k=1^nz_k/√(Z_k-1)≤( √(2) + 1 ) √(Z_n)See Appendix C.3 of <cit.>. Using Lemma <ref> we get:∑_k=1^m∑_s,aν_k(s,a)/√(max{ 1, N_k(s,a) })≤( √(2) + 1 ) ∑_s,a√(N(s,a))By Jensen's inequality we thus have:∑_k=1^m∑_s,aν_k(s,a)/√(max{ 1, N_k(s,a) })≤( √(2) + 1 ) √(SAn) In conclusion, when M ∈ℳ_k, with probability at least 1-δ/12n^5/4:∑_k=1^mΔ_k 1_M ∈ℳ_k≤ R_max D √(5/2n log( 8n/δ)) + R_max D S A log_2 ( 8n/SA) + (√(2)+1)[ 2 R_maxτ_max + 2 (c_r + R_maxc_τ) log( 2SAn/δ) + R_max D √(14S log( 2An/δ))] √(SAn)§.§ Computing the final boundGathering all previous inequalities, we have that with probability at least 1 - 3δ/12n^5/4 = 1 - δ/4n^5/4:Δ(M,𝔄,s,n)≤( γ_r + γ_τ R_max) log( 13n/δ) √(n) + τ_max R_max√(n) + R_max D √(5/2n log( 8n/δ))+ (√(2)+1)[ 2 τ_max + 2 (c_r + R_maxc_τ) log( 2SAn/δ) + R_max D √(14S log( 2An/δ))] √(SAn)+ R_max D S A log_2 ( 8n/SA) In <cit.> (see Appendix C.4), it is shown that when n > 34 A log( n/δ):DSAlog_2 ( 8n/SA) < 2/34DS√(Anlog( n/δ)), log( 2An/δ) ≤ 2 log( n/δ)and moreover if n > S log( n/δ) and A ≥ 2 (if A = 1 the regret is zero):n^2/δ^2≥n S log( n/δ)/δ≥n S/δn^2A^2/δ^2≥2SAn/δ 4 log( n/δ) ≥ 2 log( An/δ) ≥log( 2SAn/δ)Δ(M,𝔄,s,n) = O ( (D √(S) + τ_max + (C_r/R_max + C_τ) √(log( n/δ))) R_max√(S A n log( n/δ)))where C_r = max{ b_r , σ_r } and C_τ = max{ b_τ , σ_τ}.Note that if n ≤ 34 A log( n/δ) then we trivially have: ∑_k=1^mΔ_k≤τ_maxR_maxn = τ_maxR_max(√(n))^2 ≤ 34τ_maxR_max√(Anlog( n/δ))and if n ≤ S log( n/δ):∑_k=1^mΔ_k≤τ_maxR_maxn = τ_maxR_max(√(n))^2 ≤τ_maxR_max√(Snlog( n/δ))and thus the previous bound on the whole regret still holds. Taking a union bound over all possible values of n ≥ 1 we have that with probability at least 1-δ:∀ n ≥ 1,Δ(M,𝔄,s,n) = O ( ( D√(S) + τ_max + ( C_r/R_max + C_τ) √(log( n/δ))) R_max√(S A n log( n/δ))). The derivation for the case of bounded holding times is exactly the same with different concentration inequalities applied to estimates τ(s,a). Note that all the terms in the upper bound are very similar to those appearing in the derivation of the upper bound for MDPs, thus the constants in the big O are very close. This justifies the analysis of the ratio between the two upper bounds in Sect. <ref>.§ THE LOWER BOUND (THEOREM <REF>) I haven't checked this yet!!§.§ Lower Bound for SMDPs We will derive the lower bound by applying the same techniques as in the proof of the lower bound for MDPs (section 6 of <cit.>). We first consider the two-state SMDP M' depicted in Fig. <ref>. Since by assumption D/12> T_min and T_max/3> T_min, let τ∈]T_min,min{D/12, T_max/3}]. Define p = τ/T_max and δ = 4τ/D. By definition of τ we have: p,δ≤1/3. There are A' = ⌊A-1/2⌋ actions available in each state of M'. We assume that ∀ (s,a,s')∈𝒮×𝒜_s ×𝒮 and ∀ i ≥ 0, r_i(s,a,s') and τ_i(s,a,s') are independent. We also assume that ∀ i ≥ 0, τ_i(s_i-1,a_i,s_i) and r_i(s_i-1,a_i,s_i) are independent of the next state s_i and we write: τ_i(s_i-1,a_i) and r_i(s_i-1,a_i). For each action a in 𝒜_s_0, r(s_0, a)=0 and τ(s,a) ∼τ where τ is a r.v. defined in Table <ref>. Moreover, for all actions a but a specific action a_0^*, p(s_1|s_0,a) = δ whereas p(s_1|s_0,a_0^*) = δ + ϵ for some 0 < ϵ < δ specified later in the proof. For all actions a in 𝒜_s_1, p(s_0|s_1,a) = δ and τ(s_1,a) ∼τ. Finally, r(s_1, a) ∼ r for all actions a except a_1^* for which r(s_1, a_1^*) ∼ r^*, where r and r^* are r.v. defined in Table <ref> where 0 < η < p will be defined later in the proof. Note that since η < τ/T_max, we have: r≤τR_max and r^* ≤τR_max which satisfies the definition of R_max given in assumption <ref>. We denote 𝔼[·|s] the expectation conditionally on starting in state s.Let's define T(s_1) = inf{t: s_t = s_1} the first time in which s_1 is encountered. ∀ d^∞∈Π^SD_M' such that d(s_0) ≠ a_0^* we have:𝔼^d^∞[T(s_1)|s_0]= 𝔼^d^∞[∑_n=1^+ ∞[ ( ∑_i=1^nτ_i(s_i-1, a_i, s_i)) ( ∏_j=0^n-11_s_j = s_0) 1_s_n = s_1]| s_0 ] = ∑_n=1^+ ∞𝔼^d^∞[ ( ∑_i=1^nτ_i(s_0, d(s_0))) ( ∏_j=0^n-11_s_j = s_0) 1_s_n = s_1| s_0 ]= ∑_n=1^+ ∞𝔼^d^∞[ ∑_i=1^nτ_i(s_0, d(s_0))| s_0 ] 𝔼^d^∞[ ( ∏_j=0^n-11_s_j = s_0) 1_s_n = s_1| s_0 ]= ∑_n=1^+ ∞n τ×δ(1-δ)^n-1 = τ/δWe used the fact that τ_i is independent of the next state s_i and that τ = 𝔼[τ]. We can compute 𝔼^d^∞[T(s_1)|s_0] with d(s_0) = a_0^* and 𝔼^d^∞[T(s_0)|s_1] (for both d(s_0) ≠ a_0^* and d(s_0) = a_0^*) similarly. The diameter of SMDP M' is thus:D' = max{min{τ/δ, τ/δ + ϵ}, min{τ/δ, τ/δ}} = max{τ/δ + ϵ , τ/δ} = τ/δ Any policy d^∞∈Π^SD_M' induces a recurrent Markov Chain on M'. Let's denote by P_d^* the limiting matrix of this Markov Chain. We know (see <cit.>) that P_d^* = eμ_d^⊺ where μ_d = (1-p_d, p_d)^⊺ is the stationary distribution of the recurrent Markov Chain. The probability p_d can take only two different values: p_d =1/2 if and only if d(s_0) ≠ a_0^*, and p_d =δ + ϵ/2δ + ϵ if and only if d(s_0) = a_0^*. Using criterion 3 of Definition <ref>, the gain yielded by d^∞ has the form:ρ_d = p_d X/(1-p_d)τ + p_d τ = p_d X/τwhere X =r if and only if d(s_1) ≠ a_1^*, and X =r^* if and only if d(s_1) = a_1^*. Since r^*>r, the optimal decision rule d^* must satisfy: d^*(s_1) = a_1^*. Similarly, since δ + ϵ/2δ + ϵ > 1/2 we must have: d^*(s_0) = a_0^*. The optimal gain is thus:ρ^* = R_max/2×(δ + ϵ)(τ +η T_max)/(2δ + ϵ)τ The actual SMDP M that we will use to prove the lower bound is built by considering k = ⌊S/2⌋ copies of the two-state SMDP M', where only one of the copies has such "good" actions a_0^* and a_1^* (all the other copies have the exact same number of actions as M', but all actions are identical). A'+1 additional actions with deterministic transitions are introduced in every s_0-state. The reward for each of those actions is zero and the holding time is T_min. These actions connect the s_0-states of the k copies in a A'-ary tree structure on the s_0 states: one action goes toward the root and A' actions goes toward the leaves (the same structure is described in section 6 of <cit.>, see Figure 6 for a representation of the A'-ary tree). The diameter of such an SMDP M is at most 2(D/4 + T_min⌈log_A'k⌉) ≤ D. All holding times of M are in [T_min,T_max], and all rewards are in [0,1/2R_maxT_max]. For all (s,a) ∈𝒮×𝒜_s we have: r(s,a) ≤τ(s,a)R_max. Moreover, M has at most S states and A actions per state.For the analysis, we will study the simpler-to-learn SMDP M” where all s_0-states are merged together as well as all s_1-states. The "merged" s_0-state is set to be the initial state. M” is thus equivalent to the two-state SMDP M' with kA' available actions in both s_0 and s_1. Let's assume that the learning algorithm 𝔄 used is fixed. The probability distribution of the stochastic process (s_0,a_0,τ_0,r_0,s_1,...) is denoted: * ℙ_a_0,a_1when (a_0,a_1) are the best actions in respectively s_0 and s_1,* ℙ_∗ when the pair (a_0, a_1) identifying the best actions is first chosen uniformly at random from { 1, ..., kA' }×{ 1, ..., kA' } before algorithm 𝔄 starts,* ℙ_unif0,a1 when the pair a_1 is the best action in s_1 and ϵ =0 (no-optimal actions in s_0),* ℙ_a0,unif1 when the pair a_0 is the best action in s_0 and η =0 (no-optimal actions in s_1). By construction, it is trivial to see that: 𝔼_∗[Δ(M,𝔄,s,n)] ≥𝔼_∗[Δ(M”,𝔄,s,n)] (M” is easier to learn). We will show that 𝔼_∗[Δ(M”,𝔄,s_0,n)] = Ω((√(D') + √(T_max))R_max√(kA'n)) and the same result can be proved with initial state s_1 using similar arguments. This will necessarily imply that there exists at least one choice of pair (a_0,a_1) for which, for all states s we have: 𝔼_a_0,a_1[Δ(M,𝔄,s,n)] = Ω((√(D) + √(T_max))R_max√(SAn)). As already argued by <cit.>, for the analysis it is sufficient to consider algorithms with deterministic strategies for choosing actions.We assume algorithm 𝔄 is run for n decision steps, which means that n+1 states are visited in total (including the last state in which no action is taken). Let's denote by N_0 and N_1 the number of visits in states s_0 and s_1 respectively, last state excluded. For any (a, a) ∈𝒜_s_0×𝒜_s_1, let's denote by N_0^a and N_1^a the respective number of times actions a and a are taken. Finally, let's denote by N_0^* and N_1^* the respective number of times best actions in s_0 and s_1 are taken. The SMDP M” has the same transition probabilities as the MDP considered by <cit.> and we can use their proof to show that for any choice of best actions (a_0,a_1):𝔼_unif0,a_1[N_1] ≥n/2 - 1/2δ = n/2 - D'/2 τ𝔼_a_0,a_1[N_1] ≤n/2 + 𝔼_a_0,a_1[N_0^*]ϵ/δ + 1/2δ = n/2 + 𝔼_a_0,a_1[N_0^a_0]ϵ D'/2τ + D'/2τ The regret is defined as:𝔼_∗[Δ(M”,𝔄,s_0,n)] = 𝔼_∗[∑_i=1^nτ_i(s_i-1, a_i, s_i)] ρ^* - 𝔼_∗[∑_i=1^nr_i(s_i-1, a_i, s_i)]where the total duration is simply:𝔼_∗[∑_i=1^nτ_i(s_i-1, a_i, s_i)] = ∑_i=1^n𝔼_∗[τ_i(s_i-1, a_i) ] = n τand the cumulated reward is given by:𝔼_∗[∑_i=1^nr_i(s_i-1, a_i, s_i)]= ∑_i=1^n𝔼_∗[r_i(s_1, a_1^*)1_s_i-1=s_1, a_i = a_1^* + r_i(s_1, a_i ≠a_1^*)1_s_i-1=s_1,a_i ≠a_1^* ] =𝔼_∗[r^*(∑_i=0^n-11_s_i=s_1, a_i+1 = a_1^*) + r(∑_i=0^n-11_s_i=s_1,a_i+1 ≠a_1^*) ] =𝔼_∗[r(N_1 - N_1^*) + r^*N_1^* ] = 𝔼_∗[rN_1 + (r^*-r)N_1^* ] = R_max/2τ𝔼_∗[N_1] + ηR_max/2T_max𝔼_∗[N_1^*]hence the formula:𝔼_∗[Δ(M”,𝔄,s_0,n)] = τ(nρ^* - R_max/2𝔼_∗[N_1]) - ηR_max/2T_max𝔼_∗[N_1^*]Let f : { s_0, s_1 }^n+1×{ T_min, T_max}^n ×{ 0, 1/2R_maxT_max}^n → [0,M] be any function defined on state/reward sequence (s^n+1,τ^n,r^n)∈{ s_0, s_1 }^n+1×{ T_min, T_max}^n ×{ 0, 1/2R_maxT_max}^n observed in the SMDP M”. Then for any n ≥ 1, any 0 ≤δ≤1/2, any 0 ≤ϵ≤ 1-2δ, any 0 ≤ p ≤1/2, any 0 ≤η≤ 1-2p, and any (a_0,a_1) ∈{ 1, ..., kA' }×{ 1, ..., kA' }:|𝔼_a_0,a_1[ f(s^n+1,τ^n,r^n) ] - 𝔼_unif0,a_1[ f(s^n+1,τ^n,r^n) ] | ≤M/2ϵ/√(δ)√(2 𝔼_unif0,a_1[ N_0^a_0])|𝔼_a_0,a_1[ f(s^n+1,τ^n,r^n) ] - 𝔼_a_0,unif1[ f(s^n+1,τ^n,r^n) ] | ≤M/2η/√(p)√(2 𝔼_a_0,unif1[ N_1^a_1])We refer the reader to Appendix E of <cit.> where a similar Lemma is proved for the MDP-analogue of SMDP M”. In the following, we will only stress the main difference with the proof in <cit.>. We know from information theory that:|𝔼_a_0,a_1[ f(s^n+1,τ^n,r^n) ] - 𝔼_unif0,a_1[ f(s^n+1,τ^n,r^n) ] | ≤M/2√(2 log(2) KL( ℙ_unif0,a_1‖ℙ_a_0,a_1))|𝔼_a_0,a_1[ f(s^n+1,τ^n,r^n) ] - 𝔼_a_0,unif1[ f(s^n+1,τ^n,r^n) ] | ≤M/2√(2 log(2) KL(ℙ_a_0,unif1‖ℙ_a_0,a_1))By the chain rule of Kullback–Leibler divergences, it holds that:KL(ℙ_a_0,unif1‖ℙ_a_0,a_1) = ∑_i=1^n KL(ℙ_a_0,unif1(s_i,τ_i,r_i | s^i-1,τ^i-1,r^i-1) ‖ℙ_a_0,a_1(s_i,τ_i,r_i | s^i-1,τ^i-1,r^i-1) ) KL(ℙ(s_i,τ_i,r_i | s^i-1,τ^i-1,r^i-1) ‖ℚ(s_i,τ_i,r_i | s^i-1,τ^i-1,r^i-1) ) = ∑_s^i∈𝒮^i,τ^i∈𝒯, r^i∈ℛ^iℙ(s^i,τ^i-1,r^i)log_2(ℙ(s_i,τ_i,r_i|s^i-1,τ^i-1,r^i-1)/ℚ(s_i,τ_i,r_i|s^i-1,τ^i-1,r^i-1)) with 𝒮 = { s_0, s_1 }, 𝒯 = { T_min, T_max} and ℛ = { 0, 1/2 R_maxT_max}. The same holds for ℙ_unif0,a_1.Similarly to <cit.> and using the independence between s_i, τ_i and r_i, we obtain:KL (ℙ_unif0,a_1(s_i,τ_i,r_i | s^i-1,τ^i-1,r^i-1) ‖ℙ_a_0,a_1(s_i,τ_i,r_i | s^i-1,τ^i-1,r^i-1) )= ℙ_unif0,a_1(s_i-1 = s_0,a_i =a_0) ∑_ s'∈𝒮, τ' ∈𝒯, r' ∈ℛℙ_unif0,a_1(s',τ',r'|s_0, a_0)log_2(ℙ_unif0,a_1(s',τ',r'|s_0, a_0)/ℙ_a_0,a_1(s',τ',r'|s_0, a_0))= ℙ_unif0,a_1(s_i-1 = s_0,a_i =a_0) ∑_ s' ∈𝒮ℙ_unif0,a_1(s'|s_0, a_0)log_2(ℙ_unif0,a_1(s'|s_0, a_0)/ℙ_a_0,a_1(s'|s_0, a_0))= ℙ_unif0,a_1(s_i-1 = s_0,a_i =a_0) (δlog_2 ( δ/δ+ϵ) + (1-δ)log_2 ( 1-δ/1-δ-ϵ) ) KL (ℙ_a_0,unif1(s_i,τ_i,r_i | s^i-1,τ^i-1,r^i-1) ‖ℙ_a_0,a_1(s_i,τ_i,r_i | s^i-1,τ^i-1,r^i-1) )= ℙ_a_0,unif1(s_i-1 = s_1,a_i =a_1) ∑_ s'∈𝒮, τ' ∈𝒯, r' ∈ℛℙ_a_0,unif1(s',τ',r'|s_1, a_1)log_2(ℙ_a_0,unif1(s',τ',r'|s_1, a_1)/ℙ_a_0,a_1(s',τ',r'|s_1, a_1))= ℙ_a_0,unif1(s_i-1 = s_1,a_i =a_1) ∑_r' ∈ℛℙ_a_0,unif1(r'|s_1, a_1)log_2(ℙ_a_0,unif1(r'|s_1, a_1)/ℙ_a_0,a_1(r'|s_1, a_1))= ℙ_a_0,unif1(s_i-1 = s_1,a_i =a_1) (plog_2 ( p/p+η) + (1-p)log_2 ( 1-p/1-p-η) ) Using Lemma 20 of <cit.> we have that under conditions 0 ≤δ≤1/2, 0 ≤ϵ≤ 1-2δ, 0 ≤ p ≤1/2, and 0 ≤η≤ 1-2pthe following inequalities hold:δlog_2 ( δ/δ+ϵ) + (1-δ)log_2 ( 1-δ/1-δ-ϵ)≤ϵ^2/δlog(2) plog_2 ( p/p+η) + (1-p)log_2 ( 1-p/1-p-η) ≤η^2/p log(2)which concludes the proof. Note that by assumption: ϵ≤δ≤1/3≤ 1 - 2δ and η≤ p ≤1/3≤ 1 - 2p.We can bound 𝔼_∗[N_1] using Lemma <ref> as is done in <cit.>. This is because N_1 can be written as a function of (s^n+1,τ^n,r^n). Since the computations are rigorously the same except that δ = τ/D' instead of 1/D', we give the results without any further details:𝔼_∗[N_1] ≤n/2 + D'/2τ + ϵ n D'/2τkA' + ϵ D'^ 2/2τ^2 kA'+ ϵ^ 2 n D'/2τkA'√(D'kA'n/τ) + ϵ^ 2 n D'^2/2τ^2 kA'√(kA')Taking into account the fact that by assumption n ≥ DSA ≥ 16 D'kA' we get:𝔼_∗[N_1] ≤n/2 + D'/2τ + ϵ n D'/τ(1/2 k A' + 1/32τ k^2 A'^2) + ϵ^ 2 n D'/τkA'√(D'kA'n/τ)(1/2 + 1/8√(kA')) Given that δ≥ϵ≥ 0 we have the following inequality:ρ^* - R_max/4 = R_max/2×ϵτ + (δ + ϵ)η T_max/2 (2 δ + ϵ) τ≥R_max/2×( ϵ D'/6τ + η T_max/6 τ) Applying Lemma <ref> (N_1^* is a function of (s^n+1,τ^n,r^n)) and Jensen's inequality we get ∀ a_0 ∈{ 1,...,kA' }:𝔼_∗[N_1^*] = 1/kA'∑_a_1 = 1^kA'𝔼_a_0,a_1[N_1^a_1] ≤𝔼_a_0,unif1[N_1]/kA'+ n/2kA'η√(2kA'T_max/τ𝔼_a_0,unif1[N_1]) We will now derive an upper-bound on 𝔼_a_0,unif1[N_1]. Let (u_n)_n ∈ℕ∈ℝ^ℕ be any real sequence satisfying the following arithmetico-geometric recurrence relation:∀ n ∈ℕ,u_n+1≥ q u_n + r (q,r) ∈ℝ∖{ 1 }×ℝThen we have that:∀ n ∈ℕ, u_n≥( u_0 - r/1-q)α^n + r/1-q Defining sequence v_n = u_n - r/1-q, we have:∀ n ∈ℕ, v_n+1 = u_n+1 - r/1-q≥ q u_n + r - r/1-q = q (u_n - r/1-q) = q v_nBy trivial induction we get: ∀ n ∈ℕ, v_n≥ v_0 q^n. The result follows by replacing v_n by u_n - r/1-q. By the law of total probability and since ℙ_a_0,unif1(s_i = s_0) + ℙ_a_0,unif1(s_i = s_1) =1 we have:∀ i ≥ 0, ℙ_a_0,unif1(s_i+1 = s_0) =ℙ_a_0,unif1(s_i+1 = s_0 | s_i = s_0)ℙ_a_0,unif1(s_i = s_0) + ℙ_a_0,unif1(s_i+1 = s_0 | s_i = s_1)ℙ_a_0,unif1(s_i = s_1)=ℙ_a_0,unif1(s_i = s_0)( ℙ_a_0,unif1(s_i+1 = s_0 | s_i = s_0)- ℙ_a_0,unif1(s_i+1 = s_0 | s_i = s_1)) + ℙ_a_0,unif1(s_i+1 = s_0 | s_i = s_1) ≥ ℙ_a_0,unif1(s_i = s_0)(1-2δ-ϵ) + δSince the initial state is s_0 and 2δ+ϵ≤ 3δ we have by Lemma <ref>:∀ i ≥ 0,ℙ_a_0,unif1(s_i = s_0) ≥(1-δ/2δ+ϵ)(1-2δ-ϵ)^i + δ/2δ+ϵ≥1/3𝔼_a_0,unif1[N_0] = ∑_i=0^nℙ_a_0,unif1(s_i = s_0) ≥n/3𝔼_a_0,unif1[N_1]≤2n/3𝔼_∗[N_1^*] ≤2n/3kA' + nη/kA'√(T_maxnkA'/3τ)Hence the bound:η T_max𝔼_∗[N_1^*] ≤2nη T_max/3kA' + nη^2 T_max/kA'√(T_maxnkA'/3τ) By setting ϵ = c √(kA'/nD') and η = κ√(kA'/n T_max) and incorporating inequalities <ref>, <ref> and <ref> into inequality <ref> we obtain:𝔼_∗[Δ(M”,𝔄,s_0,n)] ≥ [ c/6 - c/2kA' - c/32k^2A'^2τ -c^2/2√(τ) -c^2/8√(kA'τ) -1/8kA']R_max/2√(D'kA'n)+ [ κ/6 - 2κ/3kA' - κ^2/√(3τ)]R_max/2√(T_maxkA'n)≥[ c/6 - c/2kA' - c/32k^2A'^2 -c^2/2 -c^2/8√(kA') -1/8kA']R_max/2√(D'kA'n)+ [ κ/6 - 2κ/3kA' - κ^2/√(3)]R_max/2√(T_maxkA'n)For the second inequality, we used the fact that τ > T_min≥ 1 (by assumption). If c and κ are sufficiently small, then the conditions of lemma <ref> are indeed satisfied and the above polynomials in c and κ are non-negative. For example, if c = κ = 1/5:n ≥ DSA ≥ 16D'kA' ϵ = 1/5√(kA'/n D')≤δ/20 < δ n ≥ T_maxSA ≥ 4T_maxkA' η = 1/5√(kA'/n T_max)≤p/10 < p𝔼_∗[Δ(M”,𝔄,s_0,n)] ≥ 0.0015 ×(√(D') + √(T_max)) R_max√(kA'n)§.§ Lower Bound for MDPs with options We first note that SMDP M' depicted in Fig. <ref> cannot be converted into an MDP with options. This is due to the fact that τ_i(s_i-1,a_i,s_i) and r_i(s_i-1,a_i,s_i) were assumed to be independent and the fact that ℙ(r^* = 1/2R_maxT_max) > ℙ(τ = T_max). However, it is possible to prove a slightly smaller lower bound for a family of SMDPs that can be transformed into an equivalent MDP with options. We will first present the SMDPs and the lower bound and then we will describe how to transform them into an MDP with options.The family of SMDPs is constructed in the same way as previously except that we use a slightly different SMDP M', represented on Fig. <ref> with the random variables given in Table <ref>. We take: p = 1/2(T_max - T_min) and δ = 4 τ/D. By assumption p<1/3 and δ <1/2. As before, we assume that for all i, τ_i(s_i-1,a_i,s_i) and r_i(s_i-1,a_i,s_i) are independent of the next state s_i. But the main difference with the previous lower bound is that we assume that r_i and τ_i are strongly correlated, namely: r_i(s_i-1,a_i,s_i) = R_max1_{s_i-1 = s_1 }τ_i(s_i-1,a_i). We assume ϵ≤δ and η≤ 1 - 2p. The optimal gain of M' is reached when a_0^* and a_1^* are chosen in s_0 and s_1 respectively and is equal to:ρ^* = R_max×(δ + ϵ)(τ +η (T_max-T_min) )/(2δ + ϵ)τ + (δ + ϵ) (T_max-T_min)By adapting the proof of the lower bound for general SMDPs, we can obtain the following result:𝔼_∗[Δ(M”,𝔄,s_0,n)] ≥ [ c/6 - c/2kA' - c/32k^2A'^2τ -c^2/2√(τ) -c^2/8√(kA'τ) -1/8kA']R_max√(D'kA'n)+ [ κ/6 - 2κ/3kA' - κ^2/√(3/2)]R_max√((T_max - T_min)kA'n)≥[ c/6 - c/2kA' - c/32k^2A'^2 -c^2/2 -c^2/8√(kA') -1/8kA']R_max√(D'kA'n)+ [ κ/6 - 2κ/3kA' - κ^2/√(3/2)]R_max√((T_max - T_min)kA'n)by setting ϵ = c √(kA'/nD') and η = κ√(kA'/n (T_max - T_min)). It is then possible to tune c and κ so that ϵ and η satisfy the constraints and:𝔼_∗[Δ(M”,𝔄,s_0,n)] = Ω((√(D') + √(T_max - T_min)) R_max√(kA'n)) SMDPs M' and M” can be transformed into equivalent MDPs with options. We illustrate this transformation on Fig. <ref> for an action a_1 ∈𝒜_s_1 different than a_1^*. The same method can be applied for the other actions. The idea consists in adding new states (blank states in Fig. <ref>) and primitive actions between those states. Note that the states added are just "hidden" states from which no option can be started. Thus, they should not be counted in the number of states for the lower bound. In our example it is sufficient to consider primitive actions with constant (i.e., deterministic) reward. On Fig. <ref> we give the probabilities of each primitive action when a_1 ∈𝒜_s_1∖{ a_1^* } is executed. § DISTRIBUTION OF THE HOLDING TIME AND REWARD OF A FINITE MARKOV OPTION (LEM. <REF>) We denote by ℝ^+ and ℝ^+* the set of positive and non-negative reals respectively. For the definition of sub-exponential random variables, we refer to Def. <ref>, while sub-Gaussian random variables are defined as follows. A random variable X with mean μ < +∞ is said to be sub-Gaussian if and only if there exists σ∈ℝ^+ such that:𝔼 [ e^λ (X-μ) ] ≤ e^σ^2 λ^2/2 for all λ∈ℝ.A finite[Note that if at least one option is not (almost surely) finite, the learning agent can potentially be stuck executing that option forever and the problem is ill-posed.] Markov option can be seen as an absorbing Markov Chain together with a reward process (i.e., a finite Markov option can be seen as an absorbing Markov Reward Process). To see this we add a new state s for every state s for which β_o(s)>0. We then add a transitions from s to s with probability β_o(s)>0 and reward 0, and we add a self-loop on s with probability 1 and reward 0 (s is an absorbing state). The Markov Reward Process obtained is indeed absorbing since we assumed the option to be a.s. finite, and it is equivalent to the original option (same reward and holding time). Let's denote by P the transition matrix of the Markov Chain. In canonical form we have:P = [ Q R; 0 I_r ]where r is the number of absorbing states, I_r is the identity matrix of dimension r, Q is the transition matrix between non-absorbing states and R the transition matrix from non-absorbing to absorbing states. If the option is a.s. finite then Q is necessarily (strictly) sub-stochastic (Qe ≤ e where e = (1,...,1)^⊺ and ∃ j s.t. (Qe)_j < 1) and irreducible (no recurrent class). It is well-known that such a matrix has a spectral radius strictly smaller than 1 (ρ(Q) <1) and thus I-Q is invertible (where I is the identity matrix). The holding time τ(s,o,s') of any option o is defined as the first time absorbing state s' is reached starting from state s: inf{ n≥ 1: s_n = s' s_0 = s} where (s_n)_n is the sequence of states in the absorbing Markov Chain defined by o. It is well-known in the literature <cit.> that this type of stopping times have Discrete Phase-Type distributions, with probability mass function given by:∀ k ∈ℕ^*, ℙ(τ(s,a,s')=k) = e_s^⊺ Q^k-1R e_s'where e_s = (0,0,... 0,1,0, ...,0)^⊺ is a vector of all zeros except in state s where it equals 1. These distributions generalize the geometric distribution (defined in dimension 1) to higher dimensions. The Laplace transform can be computed as follows (we simplify notations and denote: τ←τ(s,o,s') and τ←τ(s,o,s') = 𝔼[τ(s,o,s')]):𝔼[ e^λ (τ - τ)] = ∑_k=1^∞e^λ (k- τ)e_s^⊺ Q^k-1R e_s' = e^λ (1- τ)e_s^⊺[∑_k=0^∞( e^λ Q )^k]R e_s'The term ∑_k=0^∞( e^λ Q )^k is finite if and only if e^λρ(Q) <1, in which case we have:𝔼[ e^λ (τ - τ)] = e^λ (1- τ)e_s^⊺(I - e^λ Q )^-1 R e_s'and otherwise: 𝔼[ e^λ (τ - τ)] = +∞. Note that e^λρ(Q) <1 if and only if either λ < -log(ρ(Q) ) or ρ(Q) = 0. We will now analyse the two cases separately:* ρ(Q)=0 if and only if all the eigenvalues of Q in ℂ are 0, if and only if Q is nilpotent (∃ n >0 s.t. Q^n =0). This is because Q can always be triangularized in ℂ: Q=U T U^-1 where T is upper-triangular with the eigenvalues of Q on the diagonal that is, only zeros if ρ(Q)=0. This implies that ∃ n >0 s.t. T^n = U^-1 Q^n U =0Q^n = 0 hence Q is nilpotent. The reverse is obviously true: if Q is nilpotent then ρ(Q)=0, (otherwise there would exist λ≠ 0, v≠ 0 and n>0 s.t. Q^n =0 and Qv = λ vQ^nv = λ^n v = 0, which is absurd). By definition, matrix Q is nilpotent of order n if and only if the Markov Chain reaches an absorbing state in at most n steps (a.s.). In conclusion, ρ(Q)=0 if and only if the option is almost surely bounded. This happens if and only if there is no cycle in the option (with probability 1, every non-absorbing state is visited at most once).* In the case where ρ(Q)>0: it is clear that 𝔼[ e^λ (τ - τ)] can not be bounded by a function of the form λ→ e^σ^2 λ^2/2 for λ≥ -log(ρ(Q) ) so τ(s,o,s') is not sub-Gaussian (Definition <ref>). However, since ρ(Q)<1 we can choose 0< c_0 < -log(ρ(Q)) and we have 𝔼[ e^λ (τ - τ)]< +∞ for all |λ| < c_0, which implies that τ(s,o,s') is sub-exponential (Definition <ref>). In conclusion, either option o contains inner-loops (some states are visited several times with non-zero probability) in which case the distribution of τ(s,o,s') is sub-Exponential but not sub-Gaussian, or o has no inner-loop in which case o is bounded (and thus sub-Gaussian). There is no other alternative.The distribution of rewards r(s,o,s') is not as simple: the reward of an option is the sum of all micro-rewards obtained at every time step before the option ends, and every micro-reward earned at each time step can have a different distribution. The only constraint is that all micro-rewards should be (a.s.) bounded between 0 and R_max. As a result, if τ(s,o,s') is a.s. bounded (by let's say T_max) then r(s,o,s') is also a.s. bounded (by R_maxT_max). But if τ(s,o,s') is unbounded then r(s,o,s') may still be bounded if for example, all micro-rewards are 0. If however all micro-rewards are equal to R_max then r(s,o,s') has a discrete phase-type distribution just like τ(s,o,s'). r(s,o,s') can thus be unbounded (and even not sub-Gaussian). However, we will show that r(s,o,s') is always sub-Exponential. Using the law of total expectations and the fact that ℙ( r ≤ R_maxτ) = 1 we have:∀λ >0, 𝔼[ e^λ (r - r)] = ∑_k=1^∞𝔼[e^λ (r- r)| τ =k ] ℙ(τ = k)≤∑_k=1^∞𝔼[e^λ (R_maxτ- r)| τ =k ] ℙ(τ = k)= ∑_k=1^∞𝔼[e^λ (R_maxk- r)| τ =k ] ℙ(τ = k)= ∑_k=1^∞e^λ (R_maxk- r)ℙ(τ = k)= e^λ (R_max- τ)e_s^⊺[∑_k=0^∞( e^λ R_max Q )^k]R e_s'We can now conclude as we did for τ(s,o,s'): let 0< c_0 < -log(ρ(Q))/R_max, for all 0<λ < c_0 the quantity 𝔼[ e^λ (r - r)] is finite. Note that for λ≤ 0: 𝔼[ e^λ r] ≤ 1 so 𝔼[ e^λ (r - r)] < +∞. By Definition <ref>, r(s,o,s') is sub-Exponential.§ EQUIVALENT POLICIES IN AN MDP WITH OPTIONS AND THE INDUCED SMDP (LEM. <REF>) We consider the original MDP M and the SMDP M_Ø induced by the set of options Ø. By definition of M_Ø, the reward of an option is equal to the sum of the rewards of all the primitive actions taken until the option terminates (when the option is executed in M). Therefore ∑_i=1^nr_𝒪^i = ∑_i=1^N(T_n)r_𝒪^i = ∑_t=1^T_nr_t and:Δ(M,𝔄,s,T_n)= T_n ρ^*(M) - ∑_t=1^T_nr_t= T_n ρ^*(M') + T_n (ρ^*(M)-ρ^*(M_Ø)) - ∑_i=1^nr_𝒪^i = Δ(M_Ø,𝔄,s,n) + T_n (ρ^*(M)-ρ^*(M_Ø))The second part of Lem. <ref> is thus proved. We now define the (finite-time) average reward in the two processes∀ T ∈ℕ^*, ρ^π(M,s,T) = 𝔼_M^π[ ∑_t=1^Tr_t/T| s_0 = s ] ∀ T' ∈ℝ^+*, ρ^π_𝒪(M_Ø,s,T') = 𝔼_M_Ø^π_𝒪[ ∑_i=1^N(T')r_𝒪^i/T'| s_0 = s ].The limit lim_n→ +∞ T_n = + ∞ since the sequence (T_n)_n ∈ℕ^* is strictly increasing and unbounded (at least one primitive action is executed before the option ends: ∀ n ≥ 1, T_n+1≥ T_n + 1). Moreover, lim_T' → + ∞ρ^π_𝒪(M_Ø,s,T') exists since π_𝒪 is stationary and deterministic (see appendix <ref>) and by composition of the limit we havelim_n → + ∞ρ^π_𝒪(M_Ø,s,T_n) = lim_T' → + ∞ρ^π_𝒪(M_Ø,s,T') = ρ^π_𝒪(M_𝒪,s)The limit lim_T → + ∞ρ^π(M,s,T) also exists. To see this, we can build an augmented MDP equivalent to M where the state and actions encountered in two different options are duplicated (see section 3 of <cit.>). The equivalence between the original and augmented MDPs is in the strong sense: for any optional policy, the corresponding policy in the augmented MDP yields exactly the same reward for any finite horizon. In the augmented MDP, policy π is stationary deterministic and we know from MDP theory <cit.> that the corresponding average reward exists. We also have:∀n ≥1, 𝔼_M^π[ ∑_t=1^T_nr_t/T_n | s_0 = s ] = 𝔼_M_Ø^π_𝒪[ ∑_i=1^nr_𝒪^i/T_n | s_0 = s ]ρ^π_𝒪(M_Ø,s) = lim_n →+∞ρ^π(M,s,T_n) = lim_T →+∞ρ^π(M,s,T) = ρ^π(M,s) The first part of Lem. <ref> is thus proved. Finally, we know from the literature <cit.> that there exists a stationary deterministic optimal policy in the augmented MDP and thus there also exists a stationary deterministic optional policy (a policy using only options in Ø) in the original MDP. As a result, ρ^*(M_Ø) is the maximal average reward achievable in M using only options in Ø. In conclusion, the linear term T_n(ρ^*(M)-ρ^*(M_Ø)) in equation <ref> is the minimal asymptotic regret incurred if the learning agent decides to only use options. This linear loss is unavoidable.§ DETAILS OF THE ILLUSTRATIVE EXPERIMENTS In this section we will detail the experiments described in section <ref>. Terminating Condition Let's denote the current state by s_0 and for all k ∈{ 1 ... m }, denote by s_k the state which is k steps on the left to s_0. Assume option LEFT is taken in s_0. By definition, once s_k is reached, the probability of ending the option is given by β_o(s_k) = 1/(m-k+1). Since all transitions in the MDP have probability 1 (except at the target), the probability of ending in exactly k steps can be computed as follows: * If k=1: ℙ(τ = 1) = β_o(s_1) = 1/m * If k ≥ 1: ℙ(τ = k)= (∏_i = 1^k-1(1-β_o(s_i))) ×β_o(s_k) = (∏_i = 1^k-1(1-1/m-i+1)) ×1/m-k+1= (∏_i = 1^k-1(m-i/m-i+1)) ×1/m-k+1 = 1/m By symmetry, the other options (RIGHT, UP and DOWN) have the same holding time. Expected Holding Time Based on the previous result, we can easily compute the expected holding time:𝔼[τ] = ∑_k = 1^mk ·ℙ(τ = k) = 1/m∑_k = 1^mk = m+1/2Diameter Let s and s' be two distinct states in the grid. With the options defined above, the expected shortest path from s to s' is obtained if in each visited state on the way to s', we choose an option that goes in the direction of s'. For example, if s is the state located in the top left corner of the grid and s' is the target, the expected shortest path is obtained when either RIGHT or DOWN is taken in every state. With this policy, the expected time to get m-close to s' both horizontally and vertically is trivially bounded by D (red path on Fig. <ref>). Once we are m-close to s' (green square on Fig. <ref>, m = 3 on this example), we will potentially start cycling until we reach s'. On Fig. <ref>, we give an example (in one dimension) of a possible path before reaching s' once in an m-close state (the green arrows represent the successive transitions, and m=3 on this example). Since all options end after at most m time steps, once we are m-close to s', we stay m-close with the chosen policy. The expected time it takes to reach s' once we are m-close to it is m(m+1)/2 both horizontally and vertically. To prove this, we need to solve a linear system. For all i ∈{ 1 ... m-1 }, denote by τ_i the time it takes to go from s to the i-th state to the left (respectively right, up or down) when the option chosen is left (respectively right, up or down). The value is the same in all directions by symmetry. We can express the τ_i as follows:τ_1= 1/m + 1/m(2+τ_1) + … + 1/m(m+τ_m-1)τ_2= 1/m× 2 + 1/m(1+τ_1) + 1/m(3+τ_1) + … + 1/m(m+τ_m-2)τ_3= 1/m× 3 + 1/m(1+τ_2) + 1/m(2+τ_1) + 1/m(4+τ_1) + … + 1/m(m+τ_m-3) … τ_i = m+1/2 + 1/m∑_j=1^i-1τ_j + 1/m∑_j=1^m-iτ_jWith probability 1/m, the next state after executing the option is 1 step to the left of s and the value of τ_1 is then 1 . With probability 1/m the next state is 2 steps to the left of s and so s' is now located 1 step to the right of the new state: the value of τ_1 is thus 2+τ_1. With probability 1/m the next state is 3 steps to the left of s and and so s' is now located 2 steps to the right of the new state: the value of τ_1 is thus 3+τ_2. And so on and so forth. What we used here is basically the law of total expectations where the partition of events is the set of all possible states reached after executing the option only once. The same thing can be done for τ_2 …τ_m-1. It is trivial to verify that the only solution of the linear system in equation <ref> is: τ_i = m(m+1)/2, ∀ i ∈{ 1 ... m-1 }. This results is rather intuitive: m corresponds to the expected number of times the option needs to be executed to end up in the desired state s' whereas (m+1)/2 is the expected duration at every decision step. The simplicity of this result comes from the symmetry of the problem: every time an option is executed, we stay m-close to s' and the probability to exactly reach s' is always 1/m. So in this sense, we have i.i.d. Bernoulli trials where the probability of success is 1/m. The expected time to reach s' when we start in an m-close state both horizontally and vertically is thus 2 × m (m+1)/2 = m(m+1). Therefore, the expected time to go from s to s' is always bounded by D + m(m+1).On Fig. <ref> we illustrate what happens when the options are deterministic i.e., when they terminate after exactly m time steps. On this example we chose m = 3. If we start from state s_0, only the green states can be reached without resorting to the restart triggered by the target state, whereas if we start from s_1 only the blue states can be reached [Here we slightly simplified the problem. In reality, due to the truncation operated by the walls and if we assume s_0 to be the leftmost state, it is possible to go from s_1 to s_0 in one time step by executing LEFT. But for m ≥ 3 there will always be pairs of states that cannot be reached from each other without a restart. If s_0 is the leftmost state, this is the case for s_1 and the white state adjacent to it on Fig. <ref>. So the proof remains valid.] (and similarly for the white states). Let's assume that we want to go from a green state to a blue state. The only way to do so is to go to the target and "hope" to end up in one of the blue states after the restart (we recall that the restart state is chosen randomly with equi-probability). The shortest path to go from any state to the target is bounded by D and the probability to restart in a state with the desired colour is 1/m (1/m^2 in dimension 2). We can thus upper bound the diameter of the SMDP M_𝒪 by the expected time needed to go from s_0 to s_1 in the SMDP of Fig. <ref>, that is: D_𝒪≤ D(1+m^2). This bound is tight (up to a constant factor) since the average time to go from any state chosen at random with equi-probability to the target is exactly D/2 in the 2-dimensional grid. Optimality Since the target state is located in a corner of the grid, the shortest path to go from any state to the target is equally long in the original MDP and the MDP with options. As a result, the optimal average rewards are also equal (i.e., there exists an optimal policy using only options LEFT RIGHT, UP and DOWN which consists in applying only RIGHT or DOWN). Asymptotic behaviour We will now analyse the behaviour of the ratio n/T_n using results on martingales. Let X_1, ..., X_n be a martingale difference sequence w.r.t. a filtration ℱ_0, ℱ_1,..., ℱ_n and let A_1, ..., A_n be an increasing predictable sequence w.r.t. the same filtration with A_1 >0 and lim_n → + ∞ A_n = +∞ almost surely. If:∑_i=1^+∞𝔼[X_i^2 | ℱ_i-1]/A_i^2< + ∞then:1/A_n∑_i=1^nX_i 0Let's take X_i = τ_i - τ_i (where τ_i = τ_i(s_i-1,a_i)) and ℱ_i = σ(s_0, a_1, τ_1, r_1, ..., s_i, a_i+1). The sequence (X_i)_i≤ 1 is a martingale difference because 𝔼[X_i]<+∞ and 𝔼[X_i|ℱ_i-1] = 0. Since (τ(s,a,s'))_s,a,s' are sub-Exponential, all moments are finite and it is well known from the literature that the variance is bounded by the sub-Exponential constant σ_τ^2 hence: 𝔼[X_i^2 | ℱ_i-1]<σ_τ^2. If in addition we take A_i = i then the conditions of Theorem <ref> are satisfied and thus:T_n/n - T_n/n 0where T_n = 𝔼[T_n]. By definition: τ_maxn ≥T_n≥τ_minn hence: ∀ϵ>0,∃ N_ϵ >0s.t. ∀ n ≥ N_ϵ :| T_n/n - T_n/n| ≤ϵτ_min - ϵ≤T_n/n - ϵ≤T_n/n≤ϵ + T_n/n≤ϵ + τ_maxand so: lim inf_n → +∞T_n/n≥τ_min and lim sup_n → +∞T_n/n≤τ_max a.s. Finally:log{n/δ}/log{T_n/δ}= log{n/δ}/log{T_n - T_n/n + T_n/n} + log{n/δ}≤log{n/δ}/log{T_n - T_n/n + τ_min} + log{n/δ} 1In the general case of sub-Exponential rewards and holding times our results provide no theoretical evidence of the advantage of introducing options due to the fact that 𝒞(M',n,δ) scales as √(log(n)):lim_n → +∞ℛ(M,n,δ) =+∞but if the rewards and holding times are bounded we have:lim sup_n → +∞ℛ(M,n,δ) ≤1/√(τ_min)(1+ T_max/D√(S))Note that τ_min is a very loose upper-bound on lim inf_n → +∞T_n/n and in practice the ratio T_n/n can take much higher values if τ_max is big and many options have a high expected holding time. Tightness of the upper boundsOn Fig. <ref>, we plot the expected theoretical values taken by the ratio of the regrets according to our upper bounds (formula given in Sect. <ref>). On Fig. <ref> however, we plot the empirical values of the ratios in our experiments (same graph as on Fig. <ref>). We can see that the curves have similar shapes. In particular, they reach their respective minima for the same value of T_max and the value of these minimum is below 1 (meaning that learning with options is more efficient than with primitive actions in this case). Moreover, the theoretical ratios are upper-bounding the empirical ones for all values of T_max. We can conclude that the ratio of the upper bounds is a good proxy for the true ratio in this example. | http://arxiv.org/abs/1703.08667v2 | {
"authors": [
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"Alessandro Lazaric"
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"published": "20170325093031",
"title": "Exploration--Exploitation in MDPs with Options"
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C++,Pascallanguage=C++,basicstyle=,numbers=none,numberstyle=,stepnumber=1, numbersep=5pt, showspaces=false,showstringspaces=false,showtabs=false, keywordstyle=, commentstyle=,stringstyle=, captionpos=b,tabsize=2, frame=tb,breaklines=true, breakatwhitespace=false,escapechar=%, belowskip=0.3cm, morecomment=[s][]struct , morecomment=[s][]class , morecomment=[s][]public: , morecomment=[s][]public , morecomment=[s][]protected: , morecomment=[s][]private: , morecomment=[s][]operator+ , xleftmargin=0.1cm, op-tical net-works semi-conduc-tor Auto–paral-lelizing paral-lelizing Zahra Khatami Center for Computation and Technology, Louisiana State University The STE||AR Group, http://stellar-group.org [email protected] Hartmut Kaiser Center for Computation and Technology, Louisiana State University The STE||AR Group, http://stellar-group.org [email protected] J. Ramanujam Center for Computation and Technology, Louisiana State University [email protected] OP2 Compiler to Use HPX Runtime Asynchronous Techniques Zahra Khatami^1,2, Hartmut Kaiser^1,2, and J. Ramanujam^1^1Center for Computation and Technology, Louisiana State University ^2The STE||AR Group, http://stellar-group.orgReceived: date / Accepted: date =================================================================================================================================================================================Maximizing parallelism level in applications can be achieved by minimizing overheads due to load imbalances and waiting time due to memory latencies. Compiler optimization is one of the most effective solutions to tackle this problem. The compiler is able to detect the data dependencies in an application and is able to analyze the specific sections of code for parallelization potential. However, all of these techniques provided with a compiler are usually applied at compile time, so they rely on static analysis, which is insufficient for achieving maximum parallelism and producing desired application scalability. One solution to address this challenge is the use of runtime methods. This strategy can be implemented by delaying certain amount of code analysis to be done at runtime.In this research, we improve the parallel application performance generated by the OP2 compiler by leveraging HPX, a C++ runtime system, to provide runtime optimizations. These optimizations include asynchronous tasking, loop interleaving, dynamic chunk sizing, and data prefetching. The results of the research were evaluated using an Airfoil application which showed a 40-50% improvement in parallel performance. HPX, OP2, Asynchronous Task Execution, Interleaving Loops, Controlling Chunk Sizes, Prefetching Data.§ INTRODUCTION Unstructured grids are well studied and utilized in various application domains. OP2 provides a framework for the parallel execution of these unstructured grid applications on different multi-core/many-core hardware architectures <cit.>. The main goal of developing OP2 is to provide an abstraction level for users to parallelize their applications without having to worrying about architecture specific optimizations. This allows scientists to invest most of their time in understanding their domain problems, without learning details of new architectures, and still achieve efficient utilization of the available hardware. The framework is designed to achieve the near-optimal scaling on multi-core processors <cit.>. However, as the compiler only has a static and defined access pattern <cit.>, its analysis is not enough to obtain desired parallel scalability. In order to reach this goal, OP2 needs to be able to extract parallelism automatically at runtime. In this research, we propose different optimization methods that provide dynamic information for code generated by the OP2 compiler, including providing asynchronous task execution, interleaving different loops together, dynamically setting chunk sizes of different dependent loops based on each other, and prefetching data. These proposed techniques are implemented using HPX runtime system via redesigning the OP2 framework in a way that employs bothcompiler's static analysis and dynamic runtime information. HPX is a parallel C++ runtime system that facilitates distributed operations and enables fine-grained task parallelism resulting in a better load balancing <cit.>. It provides an efficient scalable parallelism by significantly reducing processor starvation and effective latencies while controlling overheads <cit.>. A closer analysis of unstructured applications reveals that synchronization is only required between small tasks. Prevalent parallelization paradigms, however, coerce users to join all tasks together before proceeding to the next step in the application. In HPX, we can utilize the future construct to allow every task to proceed as long as the values it depends on are ready <cit.>.This feature allows the HPX to relax the global barriers, enable flexibility, and improve the parallel performance of applications. In this research, HPX uses futures based techniques to develop a new task execution strategy for codes generated by the OP2 compiler which is the basis for asynchronous tasking and interleaving loops. In order to control the overheads introduced by the creation of each task, it is important to control the amount of work performed by each task. This amount of work is known as the chunk size <cit.>. In addition, to properly interleave loops it is important for each loop to have very similar execution times which allows the waiting time between the execution of each loop to be minimal. We propose to address these two obstacles by creating a new execution policy which will dynamically control the chunk sizes during the application's execution. In addition, we also propose to create a new cache prefetcher that aids in prefetching data for each time step to reduce memory accesses latencies. This method is implemented in such a way that data of the next iteration step is prefetched into the cache memory using a prefetching iterator called in each iteration within a loop. The main difference between this method and the other existing methods is that HPX implementation combines a thread based prefetching method with the asynchronous task execution, which results in having asynchronous execution while prefetching data of all the containers within a loop. To our knowledge, we present a first attempt of redesigning OP2 to utilize the runtime techniques for improving performance of the parallel unstructured grid applications. The combination of these proposed techniques should yield a more portable and performant software stack for unstructured grid applications and enable the applications to properly scale to a higher level of parallelism compared to the existing OP2 implementation. The results evaluated in Section <ref> show that the parallelization performances are improved by around 40-50% for an Airfoil application. The remainder of this paper is structured as follows: Section <ref> briefly introduces OP2; Section <ref> introduces a dataflow object in HPX; Section <ref> shows the details of the dataflow implementation with the new execution policy within OP2; Section <ref> presents the prefetching method implemented in one of the HPX parallel algorithms, and Section <ref> evaluates the the scaling speedup of the experimental tests. The conclusions and the future works can be found in Section <ref>. § OP2OP2 is an active library that provides a parallel execution framework for unstructured grid applications on different multi-core/many-core hardware architectures <cit.>. It utilizes a source-to-source translator for generating code which targets different hardware configurations <cit.>. The code can be transformed easily into different configurations such as serial, multi-threaded using OpenMP and CUDA, or heterogeneous which utilizes MPI, OpenMP, and CUDA <cit.>. In this section, we first walk through a simple OP2 code to show its implementation details and then we introduce the Airfoil application which is used as a case study for this research.§.§ Simple Code Implementation with OP2 This section generally shows how unstructured grids are defined with OP2. The OP2 API handles the data dependencies by providing mesh represented data layouts. The provided framework is defined based on sets, data on sets, mapping connectivity between the sets, and the computation on each set <cit.>. Sets can be nodes, edges or faces. In these unstructured grids, the connectivity information is used to specify different mesh topologies. Figure <ref> shows a mesh example that includes nodes and faces as sets. The value of data associated with each set is shown below each set and the mesh is represented by the connections between them. OP2 API for the mesh in figure <ref> is shown as follows, which is the C/C++ API and defines 12 edges and 9 nodes:[basicstyle=] op_set nodes; op_decl_set(9, nodes, "nodes"); op_set edges; op_decl_set(12, edges, "edges"); The mapping that declares the connection between 2 nodes is defined as follow:[basicstyle=] int edge_map[28]=0,1,1,2,2,5,5,4,4,3,3,6,6,7, 7,8,0,3,1,4,2,5,3,6,4,7,5,8op_map pedge; op_decl_map(edges,nodes,2,edge_map,pedge,"pedge")op_decl_map shows that each edge is mapped on two different nodes. The values of each node and face are assigned as follow:[basicstyle=] float valueFace[4]=0.123, 0.151, 0.420, 0.112; float valueNode[9]=5.3,1.2,0.2,3.4,5.4,6.2,3.2,2.5,0.9; op_dat data_face; op_decl_dat(face,1,"float",valueFace, data_face, "data_face"); op_dat data_node; op_decl_dat(node, 1, "float", valueNode, data_node, "data_node");These sets and meshes are used to define a loop over a given set. The more details about OP2 design and performance analysis can be found in <cit.> and <cit.>, which shows that allunstructured grid applications can be easily described with sets and meshes as shown in the above example. These methods place no restriction on the algorithm and they allow the programmer to choose unique operations on each set. §.§ Airfoil Application In this research, we study an Airfoil application, which is a standard unstructured mesh finite volume computational fluid dynamics (CFD) code, presented in <cit.>, for the turbomachinery simulation and consists of over 720K nodes and about 1.5 million edges. As described in <cit.> and <cit.>, it consists of five parallel loops: op_par_loop_save_soln, op_par_loop_adt_calc, op_par_loop_res_calc, op_par_loop_bres_calc, op_par_loop_update, shown in figure <ref>. All of the computations on each set are implemented within these loops by performing operations of the user's kernels defined in a header file for each loop: save_soln.h, adt_calc.h, res_calc.h, bres_calc.h and update.h. Each argument passed to each loop is generated based on data values used with op_arg_dat.Figure <ref> demonstrates op_par_loop_save_soln that applies save_soln on cells based on the arguments generated with op_arg_dat using p_q and p_qold data values. The function op_arg_dat creates an OP2 argument based on the information passed to it. These arguments explicitly indicate that how each of the underlying data can be accessed inside a loop: OP_READ (read only), OP_WRITE (write) or OP_INC (increment to avoid race conditions due to indirect data access) <cit.>. More details can be found in <cit.> and <cit.>. The loop parsed with OP2 in figure <ref> illustrates how each cell updates its data value by accessing blockId, offset_b, and nelem data elements. The arguments are passed to the save_soln user kernel subroutine, which does the computation for each iteration of an inner loop from offset_b to offset_b+nelem of each iteration of an outer loop from 0 to nblocks. Also it illustrates that OpenMP is used for the parallel processing within a node. It is important to note that the outputs of the computations shown in figure <ref> cannot be passed to the outside of the loop, therefore, the current OP2 design doesn't provide a method for interleaving loops together. This creates implicit global barrier after each loop as the threads inside the loop must wait to synchronize before exiting the loop <cit.>. Barriers, naturally, impede optimal parallelization by causing the parallel threads and processes to wait. In order to solve this problem, this research sets out to optimize the performance of code generated by the OP2 compiler using the HPX runtime. The source-to-source code translator of OP2 is written in Matlab and Python <cit.>. In this research, its Python source-to-source code translator is modified to automatically generate the parallel loops using HPX library calls. § HPXIn this research different dynamic optimizations are proposed for improving the performance of code generated by the OP2 compiler that are implemented using HPX runtime system, which has been developed to overcome limitations such as global barriers and poor latency hiding <cit.> by embracing new ways of coordinating parallel execution, controlling synchronization, and implementing latency hiding utilizing Local Control Objects (LCO) <cit.>. These objects have the ability to create, resume, or suspend a thread when triggered by one or more events. LCOs provide traditional concurrency control mechanisms such as various types of mutexes, semaphores, spinlocks, condition variables and barriers in HPX. These objects improve the efficiency of an application by permitting highly dynamic flow control as they organize the execution flow, omit global barriers, and enable thread execution to proceed as far as possible without waiting. More details about LCO design and its performance can be found in <cit.>.The two implementations of LCOs most relevant to this research are the future construct and the dataflow template. HPX provides a multi-threaded, message-driven, split-phase transaction, and distributed shared memory programming model using futures and dataflow based synchronization on the large distributed system architectures, which are explained in the following sections.§.§ Futurefuture is a computational result that is initially unknown but becomes available at a later time <cit.>. The goal of using future is to let every computation proceed as far as possible. Using future enablesthreads to continue their executions without waiting for the results of the previous steps to be completed, which eliminates the implicit global barrier at the end of the execution of an OpenMP parallel loop. future based parallelization provides the rich semantics for exploiting higher level parallelism available within each application that may significantly improve its scalability.Figure <ref> shows the scheme of the future performance with 2 localities, where a locality is a collection of processing units (PUs) that have access to the same main memory. It illustrates that the other threads do not stop their progress even if the thread, which waits for the value to be computed, is suspended. Threads access a future value by performing future.get(). When the result becomes available, the future resumes all HPX suspended threads waiting for that value. It can be seen that this process eliminates the global barrier synchronizations, as only those threads that depend on the future value are suspended. With this scheme, HPX allows asynchronous execution of the threads. §.§ Dataflow Object dataflow object provides a powerful mechanism for managing data dependencies without the use of global barriers <cit.>. Figure <ref> shows the schematic of a dataflow object, which encapsulates a function F(in_1,in_2,...,in_n) with n future or non-future inputs from different data resources. If an input is a future, then the invocation of the function will be delayed. Non-future inputs are passed through. A dataflow object waits for a set of futures to become ready and as soon as the last input argument has been received, the function F is scheduled for the execution <cit.>. Because the dataflow object returns a future, its result can be fed to other objects in the system including other dataflows. These chained futures, by their nature, represent a dependency tree that automatically generates an execution graph. This graph is executed by the runtime system as each nodes dependencies are meet. As a result, dataflow minimizes the total synchronization by scheduling new tasks as soon as they can be run instead of waiting for entire blocks of tasks to finish computation. § IMPLEMENTING DATAFLOW IN OP2 In this section, the new method is proposed for parallelizing loops generated with OP2, which is based on dataflow implementation that solves the current challenges of OP2. In this method, the OP2 API is modified in such a way that op_arg_dat used in each loop in figure <ref> produces an argument as a future for dataflow object inputs. Figure <ref> shows the modified op_arg_dat, in which data_a,... expressed at the last line of the code invokes a function only once all of them get ready. unwrapped is a helper function in HPX, which unwraps the futures and passes along the actual results. This implementation also generates an output argument as a future and as a result, all of the arguments of each loop in figure <ref> are passed as a future to the kernel function through op_par_loop.§.§ Parallelizing Loops Using for_eachParallelizing loops and controlling chunk sizes are implemented by using for_each algorithm and persistent_auto_chunk_size as an execution_policy respectively. In figure <ref>, dataflow is implemented with for_each for the loop in figure <ref>, that aids to parallelize the outer loop. for_each is one of the HPX parallel algorithms that is able to automatically control the chunk size during the execution by determining number of the iterations to be run on each HPX thread. Moreover, HPX is able to execute loops in sequential or in parallel by applying execution_policies, which are briefly described in Table <ref> <cit.>. The concept of the execution_policy developed in HPX is used to specify the execution restrictions of the work items, in which calling with a sequential execution policy makes the algorithm to be run sequentially and calling with a parallel execution policy allows the algorithm to be run in parallel <cit.>. | http://arxiv.org/abs/1703.09264v1 | {
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"published": "20170327185109",
"title": "Redesigning OP2 Compiler to Use HPX Runtime Asynchronous Techniques"
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Speech Enhancementusing a Deep Mixture of ExpertsShlomo E. Chazan, Jacob Goldberger and Sharon Gannot Senior Member, IEEE, Shlomo E. Chazan, Jacob Goldberger and Sharon Gannot are with the Faculty of Engineering, Bar-Ilan University, Ramat-Gan, 5290002, Israel (e-mail: [email protected]; [email protected]; [email protected]). Faculty of Engineering, Bar-Ilan University, Ramat-Gan, 5290002, Israel December 30, 2023 ================================================================================================================================================================================================================================================================================================================================================================================================== for Haim Goldberg (1939-2017)Using parametric analysis (curve fitting) we find a persistent temperature bump, coincident with World War 2 (WW2), in eight independent time series, four land- and four ocean-based. We fit the data with a Gaussian on a quadratic background. Six parameters (constant, linear and quadratic background terms and the amplitude, position and width of the Gaussian) are free to vary. The mean fitted Gaussian amplitude is 0.339±0.065 C, non-zero by 5.2σ and therefore not accidental. The area is 2.0±0.5 C yr. Temperature recovered to baseline rather quickly. Rather than coincidence, or systematic measuring error synchronized with WW2, we conjecture the bump is due to human activity, including the greatly increased combustion (relative to that era) of fossil and other fuels. Background surface temperature behavior, a byproduct of our study but largely independent of the WW2 bump, is far more consequential nowadays. The linear term, 0.747±0.023 C/century, agrees well with other findings, but the present-day rate of increase, ≈2.5 C/century, is far greater because of the quadratic term.Graphs of global surface temperature versus time often show a 0.2-0.3 C bump coincident with World War 2 (WW2). Taken at face value, that is direct evidence of the effect of human activity on global temperature. (There is ample indirect evidence.) The bump could test of climate models if the excess combustion during WW2 could be quantified. Here we show only that the bump is real. It is a persistent feature of the surface temperature record, appearing in eight independent time series, four land- and four ocean-based. A least-squares fit of a Gaussian on a quadratic background to each data set yields values and errors for the constant, linear and quadratic coefficients of the background as well as the magnitude, time of occurrence and duration of the bump. § METHOD We download twelve NOAA files of temperature anomaly <cit.>, half land- and half ocean-based. Each set covers the globe in six 30 zones of latitude and lists monthly surface temperature anomalies and variances from 1880 through 2016, 1644 measurements in all. We arrange and number them in contiguous zones N-S, 1-6 for land and 7-12 for ocean. We fit each zone separately using identical technique (starting values, convergence test …) and afterwards, find the weighted mean of seven with acceptable fits. (Data for the four polar zones are too sparse and zone 3 is discussed later.)Let t_i, i=1… M, be the time of the i^th measurement in years and T_i its temperature anomaly in C. The time range 1880-2017 is awkward for a polynomial fit because the coefficients vary by orders of magnitude and have rather obscure meanings: the first coefficient, for instance, is the temperature at year 0. Therefore, before fitting, we transform t to a dimensionless variable u ranging from -1 to 1 as t ranges from 1880 to 2017. Because 1 to any power is 1, the resulting polynomial coefficients are all of the same order of magnitude and easier to interpret. For instance, the first coefficient is simply the temperature anomaly at midrange, and T_M at the end of range (t=t_M, u=1) is simply the sum of the coefficients. The transformation and its inverse areu = B(t-A), t=A+(u/B) where in our caseA≡(t_M+t_1)/2=1948.5 yr, B≡ 2/(t_M-t_1)=0.014607 yr^-1The fit function isF(u) ≡ p_1+p_2u+p_3u^2+p_6 e^-.5((u-p_7)/p_8)^2with p_6, p_7 and p_8 the Gaussian amplitude, mean and σ respectively. That form was arrived at after some experimentation (note unused parameter numbers) where we found that a three-term (quadratic) polynomial was the most the data would support. The fit improves with more terms, but their error increases sharply, a sure sign of too many adjustable parameters.We use the standard Levenberg-Marquardt algorithm to minimizeχ^2 ≡ ∑_i=1^M (T_i-T(u_i;p⃗))^2/σ_i^2 = ∑_i=1^M w_i (T_i-T(u_i;p⃗))^2with respect to the six parameters p⃗. The weight w_i of the i^th point is the inverse of the variance σ_i^2 given in the NOAA tables. Our form of the algorithm, adapted from Numerical Recipes <cit.> allows us (with negligible effort) to hold selected parameters fixed at their initial values. Initial values for the Gaussian are set by hand in an input file and (along with all other program constants) are the same for all 12 data sets to avoid bias. Initial values for the polynomial coefficients are obtained from a preliminary linear least-squares fit without the Gaussian.The usual criterion for a good fit namely χ^2/(M-6)≈1is too much to hope for here. First, non statistical effects are present, for instance the El Niño Southern Oscillation (ENSO)in sveral ocean-based zones. Second, Eq. <ref> assumes accurate variances, very unlikely in this complicated a problem. The best we achieve for any zone is χ^2/(M-6)≈4. Of course, χ^2 still tells us whether one fit is better than another.After convergence, following <cit.>, we call Marquardt once more with λ≐0, whereupon the variance in each fitted parameter equals the corresponding diagonal term of the curvature matrix. We deliberately choose conditions to make the program look fairly hard for the χ^2 minimum: convergence requires <0.1% improvement in χ^2 and the initial Gaussian amplitude is set at ≈3× its final value. The program typically converges in 4-6 passes, except zone 3 which takes 12 passes to get the wrong answer, as we will see. Fitting all twelve zones takes about 1 s on an elderly laptop running Intel Visual Fortran.§ RESULTS§.§ Preliminary StudiesDetails, tables and graphs can be found in the long version of this report <cit.>. Raw land-based (atmospheric) data show much more scatter than ocean-based and for both, scatter decreases from poles to equator. Weights (inverse variances) have the same general pattern, varying by 3-4 orders of magnitude month-to-month at the poles. Repeating the final fit with all w_i≐100 shows that our results are insensitive to the weights, but we did use them for the results below.Smoothing the raw data by convolution with a Gaussian of σ=24 months reveals distinct WW2 bumps in the peri-equatorial ocean-based zones 9 and 10, and hints of bumps in some others. Smoothing can be a useful guide to where to look, but by itself leads to no numbers. After fitting, we studied the Gaussian compliance of fit residuals by examining their skewness and kurtosis, both ideally zero, and found some worse than others. ENSO, definitely a systematic effect, seems to be quasi random when averaged over many cycles. Finally, we replaced the WW2 Gaussian by a nearly square pulse also having three free parameters (amplitude, start and duration) and found it, too, in all eight zones. §.§ Final ResultsfitGAU10 shows the best fit, in zone 10 (zone 9 is similar). ENSO is apparent. multiyf summarizes all final fits. Left- and right-hand columns are land- and ocean-based respectively, with N at the top. The ordinate range ±1 Cis the same for all panels and the abscissa runs from 1880 through 2016 marked every two decades. The five zones that missed the final cut are cross-hatched. All six parameters are free to vary in all zones. Zone 3 diverts the three (Gaussian) degrees of freedom so as to improve the background. Jumping to an unsought nearby minimum of χ^2 is a common pathology of nonlinear least-squares fits, and one or another zone does this for every set of starting conditions we tried. multiyflock, however, which is the same fit with the Gaussian position and σ fixed but the amplitude still free to vary, shows that the bump is also present in zone 3 with a reasonable amplitude. Space precludes listing fit parameters by zone here (see <cit.>) but parms shows the trends. All fitsare shown including dropped zones. Error bars are 1σ; note scale factors and a few off-scale points. Starting from the left, the difference between L and O constant terms may result from slightly different definitions of temperature `anomaly'. It has no profound consequences. More interesting are the next two blocks showing linear and quadratic terms both larger for L than for O. Both the rate and the acceleration of surface temperature increase seem to be greater for atmospheric than oceanic measurements. Disequilibrium between the atmosphere and the oceans seems to be increasing with time. Relative errors in those numbers are small except for excluded zone 3.The Gaussian amplitude block confirms the impression (multiyf) that the WW2 bump is smaller in land-based measurements, but the final blocks show that the time of occurrence and duration are, within errors, the same. (We repeat that these were free to vary.) Table <ref> lists seven-zone weighted means of the fit parameters in two ways: in terms of the scaled time variable u used for the fits (left-hand column) and in conventional units (right-hand column). To compute the weighted fit for a given year convert to u using u and insert values from the left-hand column into F, or simply read it off the bold line in predicted below. The final two rows of Table <ref> list two derived quantities, the areas of the weighted Gaussian and of the weighted square pulse, in excellent agreement. Because each involves both the amplitude and a measure of duration, their relative errors are larger than in the fitted parameters themselves. Since the data do not support a more detailed model of the shape, saying that WW2 was equivalent to a 2.0±0.5 C yr impulse to global surface temperature is probably the most succinct formulation. § DISCUSSIONIt goes without saying that there are other systematic effects (ENSO being the most dramatic) in the temperature record and there may well be other one-shot events, linked to volcanic eruptions or other phenomena. We simply have not looked for them here.predicted encapsulates the present study, showing fitted results for three land-based and four ocean-based time series as well as their weighted average. We extended it about two decades beyond the present day. A similar figure in <cit.> corresponding to the fits in multiyflock looks cleaner but leads to the same conclusions. §.§ World War 2 BumpAccording to Table <ref> the amplitude p_6 of the WW2 bump differs from zero by 5.2 σ. From standard probability tables the chance of that happening at random is 0.2×10^-7. If we therefore accept that the effect is real, there remain three possibilities.First, a systematic measuring error might have come and gone coincident with WW2. That seems superficially plausible; after all, WW2 was a time of global upheaval. However, the systematic aggregation and analysis of surface temperature data cited here did not begin until decades later. Therefore it would have been necessary for a significant fraction of land- and ocean-based stations to change their protocols one way at the beginning of the war and back at the end. The nearest candidate for that is the well known `bucket cooling' effect <cit.>. However, that went in only one direction (after the war ended), it only affected ocean measurements, and ERSST.v4 attempts to correct for it <cit.>.Second, the bump might be pure coincidence: an unrelated climatic event happened to occur at that time, affecting both land- and ocean-based measurements. Logically speaking that argument can never be refuted.The third possibility is that the WW2 bump was caused by the increase (enormous relative to that era) in the combustion of fossil fuels by the world's armies, navies, air forces and factories, let alone the burning of oilfields, cities and the munitions themselves. Indeed, it would seem inconsistent to believe that burning fossil fuels promotes global warming and, simultaneously, that WW2 had no effect. §.§ Background BehaviorOur study was motivated by curiosity about the WW2 bump, but the background behavior of surface temperature which emerges as a byproduct is of infinitely greater consequence and, mathematically, depends very little on the existence or non-existence of the bump. The linear coefficient, 0.747±0.024 C/century (Table <ref>) agrees with 0.735±0.068 C/century from an independent fit to ERSST.v4 cited by Huang et al. <cit.>. Also, if in our analysis we omit the the quadratic term and the WW2 bump and simply fit everything with a straight line, the weighted average is 0.742±0.024 C/century, so that is all consistent.A positive quadratic coefficient is required by the mere fact that surface temperature exhibited a minimum around 1910 (as appears in many ocean-based data) or that it was level from (say) 1880-1920 and increased thereafter (as is more typical of land-based data). All we have done here is to quantify that as objectively as possible. The positive quadratic coefficient in turn guarantees that the present-day instantaneous rate of change is greater than the long-term average. Unfortunately, looking at predicted, it seems to be far greater: about 2.5 C/century. In summary, the good news is that surface temperature reverted to baseline very quickly after WW2. In that era at least, surface temperature was stable with respect to small changes. The bad news is that, because of the quadratic term, the present-day rate of temperature increase is very much larger than the average over the last 137 years.§ ACKNOWLEDGEMENTWe thank the Harvard University Physics Department for sustained and generous support. Opinions in this report are ours and not those of the University. 1ERSSTv4 https://www.ncdc.noaa.gov/data-access/marineocean-data/extended-reconstructed- sea-surface-temperature-ersst-v4.nr W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, “Numerical Recipes: the Art of Scientific Computing,” Cambridge University Press (1986).Gottschalk2017 Bernard Gottschalk, `Global surface temperature trends and the effect of World War II: a parametric analysis (long version),' arXiv:1703.06511v1 (2017).Thompson2008 David W.J. Thompson, John J. Kennedy, John M. Wallace and Phil D. Jones, `A large discontinuity in the mid-twentieth century in observed global-mean surface temperature,' Nature, 453 (2008) 646-649 doi:10.1038/nature06982.Huang2015 Boyin Huang, Viva F. Banzon, Eric Freeman, Jay Lawrimore, Wei Liu, Thomas C. Peterson, Thomas M. Smith, Peter W. Thorne, Scott D. Woodruff and Huai-Min Zhang, `Extended Reconstructed Sea Surface Temperature Version 4 (ERSST.v4). Part I: Upgrades and Intercomparisons,' J. Climate 28 (2015) 911-930. | http://arxiv.org/abs/1703.09281v1 | {
"authors": [
"Bernard Gottschalk"
],
"categories": [
"physics.ao-ph"
],
"primary_category": "physics.ao-ph",
"published": "20170327194109",
"title": "Global surface temperature trends and the effect of World War II"
} |
romanInformational Substitutes Yiling Chen and Bo WaggonerHarvard, University of Pennsylvania Updated: March 2017 ========================================================================== We propose definitions of substitutes and complements for pieces of information (“signals”) in the context of a decision or optimization problem, with game-theoretic and algorithmic applications. In a game-theoretic context, substitutes capture diminishing marginal value of information to a rational decision maker. We use the definitions to address the question of how and when information is aggregated in prediction markets. Substitutes characterize “best-possible” equilibria with immediate information aggregation, while complements characterize “worst-possible”, delayed aggregation. Game-theoretic applications also include settings such as crowdsourcing contests and Q&A forums. In an algorithmic context, where substitutes capture diminishing marginal improvement of information to an optimization problem, substitutes imply efficient approximation algorithms for a very general class of (adaptive) information acquisition problems.In tandem with these broad applications, we examine the structure and design of informational substitutes and complements. They have equivalent, intuitive definitions from disparate perspectives: submodularity, geometry, and information theory. We also consider the design of scoring rules or optimization problems so as to encourage substitutability or complementarity, with positive and negative results. Taken as a whole, the results give some evidence that, in parallel with substitutable items, informational substitutes play a natural conceptual and formal role in game theory and algorithms.arabic § INTRODUCTION AND OVERVIEW§.§ Motivation and challengeAn agent living in an uncertain world wishes to make some decision (whether to bring an umbrella on her commute, how to design her company's website, …).She can improve her expected utility by obtaining pieces of information about the world prior to acting (a weatherman's forecast or a barometer reading, market research or automated A/B testing, …). This naturally leads her to assign value to different pieces of information and combinations thereof. The value of information arises as the expected improvement it imparts to her optimization problem.We would like to generally understand, predict, or design algorithms to guide such agents in acquiring and using information. Consider the analogous case where the agent has value for items or goods, represented by a valuation function over subsets of items. A set of items are substitutes if, intuitively, each's value decreases given more of the others; they are complements if it increases. Here, we have rich game-theoretic and algorithmic theories leveraging the structure of substitutes and complements (S&C). For instance, in many settings, foundational work shows that substitutability captures positive results for existence of market equilibria, while complements capture negative results <cit.>. When substitutes are captured by submodular valuation functions <cit.>, algorithmic results show how to efficiently optimize (or approximately optimize) subject to constraints imposed by the environment (e.g. <cit.>). For example, an agent wishing to select from a set of costly items with a budget constraint has a (1-1/e)-approximation algorithm if her valuation function is submodular <cit.>.Can we obtain similar structural and algorithmic results for information? Here, a piece of information is modeled as a signal or random variable that is correlated in some way with the state of the world that the agent cares about (whether it will rain, how profitable are different website designs, …). Intuitively, one might often expect information to satisfy substitutable or complementary structure. For instance, a barometer reading and an observation of whether the sky is cloudy both yield valuable information about whether it will rain to an umbrella-toting commuter; but these are substitutable observations for our commuter in that each is probably worth less once one has observed the other. On the other hand, the dew point and the temperature tend to be complementary observations for our commuter: Rain may be only somewhat correlated with dew point and only somewhat correlated with temperature, but is highly correlated with cases where temperature and dew point are close (i.e. the relative humidity is high).Despite this appealing intuition, there are significant challenges to overcome in defining informational S&C. Pieces of information, unlike items, may have complex probabilistic structure and relationships. But on the other hand, this structure alone cannot capture the value of that information, which (again unlike items) seemingly must arise from the context in which it is used. Next, even given a measure of value, it is unclear how to formalize an intuition such as “diminishing marginal value”. Finally, it remains to demonstrate that the definitions are tractable and have game-theoretic and/or algorithmic applications. These challenges seem to have prevented a successful theory of informational S&C thus far. §.§ This paper: summary and contributionsThis paper has four components.1. We propose a definition of informational substitutes and complements (S&C).Beginning from the very general notion of value of information in the context of any specific decision or optimization problem, we define S&C in terms of diminishing (increasing) marginal value for that problem.This requires a definition of “marginal unit” of information.We consider a hierarchy of three kinds of marginal information: learning another signal, learning some deterministic function of another signal, and learning some randomized function (“garbling”) of another signal.These correspond respectively to weak, moderate, and strong versions of the definitions, formalized by strengthenings of submodularity and supermoduarity. We also investigate some useful tools and equivalent definitons.From an information-theoretic perspective, substitutes can be defined as signals that reveal a diminishing amount of information about a particular event, where the measure of information is some generalized entropy function.From a geometric perspective, substitutes can be defined using a measure of distance, namely some Bregman divergence, on the space of beliefs about the event; signals are substitutes if the average change in one's belief about the event is dimininishing in the amount of information already known.2. We give game-theoretic applications of these definitions, primarily on information aggregation in prediction markets.When strategic agents have heterogeneous, valuable information, we would like to understand when and how their information is revealed and aggregated in an equilibrium of strategic play.Prediction markets, which are toy models of financial markets, are possibly the simplest setting capturing the essence of this question. However, although the efficient market hypothesis states that information is quickly aggregated in financial markets <cit.>, despite much research on this question in economics (e.g. <cit.>) and computer science (e.g. <cit.>), very little was previously known about how quickly information is aggregated in markets except in very special cases. We address the main open question regarding strategic play in prediction markets: When and how is information aggregated?We show that informational substitutes imply that all equilibria are of the “best possible” form where information is aggregated immediately, while complements imply “worst possible” equilibria where aggregation is delayed as long as possible.Furthermore, the respective converses hold as well; e.g., if an information structure guarantees the “best possible” equilibria, then it must satisfy substitutes.[Informal] In a prediction where trader's signals are strict, strong substitutes with respect to the market scoring rule, in all equilibria, traders rush to reveal and aggregate information immediately.Conversely, if signals are not strong substitutes, then there are arrival orders for traders where no equilibrium has immediate aggregation.[Informal] In a prediction market where trader's signals are strict, strong complements with respect to the market scoring rule, in all equilibria, traders delay revealing and aggregating information as long as possible.Conversely, if signals are not strong complements, then there are arrival orders where no equilibrium exhibits full delaying.Informational S&C thus seem as fundamental to equilibria of (informational) markets as substitutable items are in markets for goods. We believe that informational S&C have the potential for broad applicability in other game-theoretic settings involving strategic information revelation, and toward this end, give some additional example applications.We show that S&C characterize analogous “rush/delay” equilibria in some models of machine-learning or crowdsourcing contests <cit.> and question-and-answer forums <cit.>.3. We give algorithmic applications, focusing on the complexity of approximately-optimal information acquisition.Namely, we define a very broad class of problems, termed /, in which a decision maker wishes to acquire information prior to making a decision, but has constraints on the acquisition process.For instance, a company wishes to purchase heterogeneous, pricey data sets subject to a budget constraint, or to place up to k sensors in an environment. We show that substitutes imply efficient approximation algorithms in many cases such as a budget constraint; this extends to an adaptive version of the problem as well.We also show that the problem is hard in general and in the complements case, even when signals are independent uniform bits.[Informal] For the / problem with e.g. cardinality or budget (“knapsack”) constraints, in the oracle model of input: If signals are weak substitutes, then polynomial-time 1-1/e approximation algorithms exist; but in general, or even if signals are assumed to be complements, no algorithm can achieve nonzero approximation with subexponentially many oracle queries. These results offer a unifying perspective on a variety of similar “submodularity-based” solutions in the literature <cit.>.4. We investigate the structure of informational S&C.We give a variety of tools and insights for both identifying substitutable structure and designing for it.For instance, we provide natural geometric and information-theoretic definitions of S&C and show they are equivalent to the submodularity-based definitions. We address two fundamental questions: Are there (nontrivial) signals that are substitutes for every decision problem?Second, given a set of signals, can we always design a decision problem for which they are substitutes?In the game-theoretic settings above, this corresponds to design of mechanisms for immediate aggregation, somewhat of a holy grail for prediction markets.In algorithmic settings, it has relevance for the design of submodular surrogates <cit.>.Unfortunately, we give quite general negative answers to both questions.Surprisingly, more positive results arise for complements.We give the geometric intuition behind these results and point toward heuristics for substitutable design in practice.In summary, the contributions of this paper are twofold: (a) in the definitions of informational S&C, along with a body of evidence that they are natural, tractable, and useful; and (b) in the applications, in which we resolve a major open problem on strategic information revelation as well as give a unifying and general framework for a broad algorithmic problem. Our results on structure and design of informational S&C points to potential for these very general definitions and results to have concrete applications.Taken all together, we believe these results give evidence that informational S&C, in analogy with the successful theories of substitutable goods, have a natural and useful role to play in game theory, algorithms, and in connecting the two.§.§ OutlineTwo sections have been placed in the appendix for convenience. Appendix <ref> gives a detailed survey of related work in a variety of areas. In Appendix <ref>, we overview and justify our general approach to defining informational S&C, including historical context, tradeoffs, and intuition. We particularly focus on a comparison to the proposed definitions of <cit.>, which we build on in this paper.In Section <ref>, we concisely and formally define informational S&C. We show that prediction problems, and the modern convex analysis understanding of them, can be used to analyze general decision problems. Leveraging these tools, we give three equivalent definitions from seemingly-disparate perspectives.In Section <ref>, we present game-theoretic applications. Primarily, we show that informational substitutes (complements) characterize best-case (worst-case) information aggregation in prediction markets.In Section <ref>, we present algorithmic applications. We define /, a class of information acquisition problems, and show that substitutes correspond to efficient approximation algorithms while there are strong hardness results in general.In Section <ref>, we investigate informational S&C themselves with an eye toward the previous applications. We give some results on general classes of S&C and on the design of prediction or optimization problems for which a given information structure is substitutable, with both game-theoretic and algorithmic implications.Section <ref> summarizes and discusses future work. § DEFINITIONS AND FOUNDATIONS§.§ Setting: information structure and decision problems We now formally present the setting and definitions. Motivation for the choices made and relation to prior work, particularly <cit.>, are described in depth in Section <ref>.§.§.§ Model of information and decision problems Information structure. We take a standard Bayesian model of probabilistic information. There is a random event E of interest to the decisionmaker, e.g. E ∈{rain,no rain}. There are also n “base signals” A_1,…,A_n, modeled as random events. These represent potential information obtained by a decision-maker, e.g. A_i ∈{cloudy, sunny}. An information structure is given by E, A_1,…,A_n, and a prior distribution P on outcomes (e,a_1,…,a_n). For simplicity, we assume that all A_i and E have a finite set of possible outcomes.In addition to the base signals, there will be other signals that intuitively represent combinations of base signals. Formally, there is a setof signals, with a generic signal usually denoted A or B. Any subscripted A_i always refers to a base signal, while A may in general be any signal in . We will describe howis generated from A_1,…,A_n momentarily, in Section <ref>.We will use lower-case p to refer to probability distributions on E, the event of interest. The notation p(e) refers to the probability that E=e, while p(a_i,e) = [A_i=a_i, E=e], and so on. The notation p(e | a_i) refers to the probability that E=e conditioned on A_i=a_i, obtained from the prior via a Bayesian update: p(e | a_i) = P(e,a_i) / P(a_i). We will sometimes use the shorthand notation p_a to refer to the posterior distribution on e conditioned on A = a, similarly for p_a,b when A=a and B=b, and so on. We will abuse notation and write E to represent a set of outcomes, so for instance we may write e ∈ E; similarly for signals. We also sometimes write _a∼ A[ …] for the expectation over outcomes a of A. Decision problems and value function. A single-agent decision problem consists of a set of event outcomes E, a decision space , and a utility function u: × E →ℝ, where u(d,e) is the utility for taking action d when the event's outcome is E=e. This decision problem, in the context of an information structure, will be how signals derive their value.Specifically, given the prior P, the decision that maximizes expected utility is . But now suppose a Bayesian, rational agent knows P and will first observe the signal A, then update to the posterior p_a on E, and then choose a decision maximizing expected utility for this posterior belief. In this case, her utility will be given by the following “value” function:^u,P(A) := _a [ max_d ∈ _e [ u(d,e) | A=a] ].We will useto denote a null signal, so that ^u,p() is the expected utility for deciding based only on the prior distribution. Where the decision problem and information structure are evident from context, we will omit the superscripts u,P.Intuitively, ^u,P is analogous to a valuation function v: 2^{1,…,n}→ℝ over subsets of items. However, inputs tomay not only represent subsets of A_1,…,A_n, but also signals that give partial information about them.§.§.§ Signal lattices We will consider three kinds of signal sets , leading to “weak”, “moderate”, and “strong” substitutes and complements. In each case, the set of signalswill form a lattice. A lattice (U,≼) is a set U together with a partial order ≼ on it such that for all A,B ∈ U, there are a meet AB and join AB in U satisfying:* AB ≼ A ≼ AB and AB ≼ B ≼ AB; and * the meet and join are the “highest” and “lowest” (respectively) elements in the order satisfying these inequalities.In a lattice,denotes the “bottom” element and ⊤ the “top” element, i.e. ≼ A ≼⊤ for all A ∈ U, if they exist. The following definition illustrates one very common lattice, that of subsets of a ground set. The subsets signal lattice generated by A_1,…,A_n consists of an element A_S for each subset S of {A_1,…,A_n}, where A_S is the signal conveying all realizations {A_i = a_i : i ∈ S}. Its partial order is A_S ≼ A_S' if and only if S ⊆ S'. Hence, its meet operation is given by set intersection and join by set union.The bottom elementof the subsets lattice exists and is a null signal corresponding to the empty set (we will use this notation somewhat often), while the top element also exists and corresponds to observing all signals. Also, the partial ordering ≼ denotes less informative. These facts will continue to hold for the other two signal lattices we define.For the other two lattices, we utilize the main idea from the classic model of information due to <cit.>. Let the set Γ⊆ A_1 ×⋯× A_n consist of all signal realizations (a_1,…,a_n) in the support of the prior distribution. Now, a partition is a collection of subsets of Γ such that each γ∈Γ is in exactly one subset. Each signal A_i corresponds to a partition of Γ with one subset for each outcome a_i, namely, the set of realizations γ = (⋯, a_i, ⋯). Example <ref>, given after the definition of discrete signal lattice, illustrates the partition model.As in Aumann's model, the partitions of Γ form a lattice, each partition corresponding to a possible signal. The partial ordering is that A ≼ B if the partition of A is “coarser” than that of B. One partition is coarser than another (which is finer) if each element of the former is partitioned by elements of the latter. The join of two partitions is the coarsest common refinement (the coarsest partition that is finer than each of the two), while the meet is the finest common coarsening. Example <ref>, given after the definition, illustrates coarsenings and refinements.The discrete signal lattice generated by A_1,…,A_n consists of all signals corresponding to partitions of Γ, where Γ is the subset of A_1 ×⋯× A_n with positive probability. Its partial order has A ≼ B if the partition associated to A is coarser than that of B. [Signals modeled as partitions](a) We have two independent uniform bits A_1 and A_2. In this case Γ = {(0,0), (0,1), (1,0), (1,1)}. Here A_1 is modeled as the partition consisting of two elements: {(0,0),(0,1)} and {(1,0),(1,1)}. The first element of the partition is the set of realizations where A_1 = 0, while the second is the set of realizations where A_1 = 1.(b) Now modify the example so that A_1 and A_2 are perfectly correlated: with probability 0.5, A_1 = A_2 = 0, and with probability 0.5, A_1 = A_2 = 1. Here, Γ = {(0,0), (1,1)} and A_1 corresponds to the partition consisting of {(0,0)} and {(1,1)}.(c) Now revisit the first case where A_1 and A_2 are independent. Imagine an agent who observes both base signals and wishes to reveal only the XOR A_1 ⊕ A_2 of the bits. This new signal released by the agent, call it A, may also be modeled as a partition of Γ, where the elements of the partition are {(0,0),(1,1)} and {(0,1),(1,0)}.[The order given by coarsenings](a) We have a single signal A_1 which is distributed uniformly on {1,2,3,4,5,6}. Then Γ = {1, 2, 3, 4, 5, 6} and A_1's partition contains these six subsets: {1},{2},{3},{4},{5},{6}.(b) Given the above information structure, suppose that an agent holding A_1 will commit to releasing some deterministic function of A_1. In terms of information revealed, the agent may map each realization a_1 ∈{1,…,6} to a different report – this is the same as just revealing a_1 – or she may map some realizations to the same report. Suppose that she reports “small” whenever a_1 ∈{1,2,3} and reports “large” whenever a_1 ∈{4,5,6}. The information revealed by this report is captured by a binary signal A corresponding to the partition with two elements: {1,2,3} and {4,5,6}. The partition of A is coarser than that of A_1, so A ≼ A_1 on the discrete lattice.(c) Given the same information structure, imagine that the agent will commit to releasing “even” whenever a_1 ∈{2,4,6} and “odd” whenever a_1 ∈{1,3,5}. This corresponds to a signal B whose partition has these two elements and is again coarser than that of A_1.(d) Consider the above two signals A and B. They are incomparable: Neither is coarser nor finer than the other. The meet AB will be the null signal[Formally, this is the signal whose partition contains a single element: all of Γ.] , intuitively because given A, one cannot guarantee anything about the outcome of B. The join AB intuitively corresponds to observing both signals. Let C = AB. The partition corresponding to C has the following four elements: {1,3}, {2}, {4,6}, {5}. These each correspond to a realization of the signal C; call the realizations respectively c_1,c_2,c_3,c_4. Here, when for example A = “small” and B = “even”, then C = c_2 and an observer of C would know that A_1 = 2. When A = “large” and B = “even”, then C = c_3 and an observer of C would know that A_1 ∈{4,6}, updating to a posterior on these possibilities.For the third and strongest notion, we extend the model by, intuitively, appending randomness to the signals on the discrete lattice. Given any signal A on the discrete lattice, a “garbling” of A can be captured by a randomized function of A; but this may be modeled as a deterministic function s(A,r) where r is a uniform [0,1] random variable[In some applications, it may be more desirable to use an infinite string of independent uniform bits.]. This observation allows us to “reduce” to the deterministic case, but where each possible signal carries extra information in the form of some independent randomness.Specifically, let Γ be defined as above (the subset of A_1 ×⋯× A_n with positive probability) and, for each partition Π of Γ, let R_Π∈ [0,1] drawn independently from the uniform distribution. Let Γ' = Γ×𝐑 where 𝐑 = _Π R_Π. Now, we proceed as before, but using Γ'.[To be precise, we should restrict to measurable subsets using the Lebesgue measure on [0,1]. We will omit this discussion for simplicity; if concerned, the reader may alternatively assume that each R_Π is drawn uniformly from a massive but finite set, with some tiny ϵ approximation carried through our results.] The continuous signal lattice consists of a signal corresponding to each partition of Γ'. Its partial order has A ≼ B if the partition associated to A is coarser than that of B. [Modeling garblings via the continuous lattice]Consider a uniformly random bit A_1 as the only base signal; the resulting Γ is {0, 1}.Now consider the garbling where, if A_1 = 0, then output “happy” with probability q_0 and “sad” otherwise; if it equals 1, then output “happy” with probability q_1 and “sad” otherwise. Call the output of the garbling A. Then A can be modeled as a partition of Γ× [0,1] with the following two subsets: {(0,x): 0 ≤ x ≤ q_0}∪{(1,x): 0 ≤ x ≤ q_1}, and {(0,x) : q_0 < x ≤ 1}∪{(1,x) : q_1 < x ≤ 1}. Here the first realization of A corresponds to the output “happy”, while the second corresponds to output “sad”. To see this, note for instance that the first realization contains all the elements of Γ× [0,1] where A_1 = 0 and the randomness variable x ≤ q_0. So when A_1 = 0, assuming x is drawn uniformly and independently from [0,1], then the outcome of A is “happy” with probability q_0. On the continuous lattice, A_1 corresponds to the partition of singletons such as {(0,0.35142)}, {(1,0.92241)}, and so on. That is, it corresponds to observing both the original binary bit as well as the random real number x. Because this partition is finer than that corresponding to A, we have A ≼ A_1 on the continuous lattice. The use of “happy” and “sad” for the outputs in the above example illustrates that it is not important, when considering the information conveyed by signal A, to consider what its realizations were named. All that matters is their distributions, e.g. the partitions they represent.[Modeling garblings, continued]Again let A_1 be a uniformly random bit, and now suppose A is obtained by adding to A_1 independent Gaussian noise with mean 0 and variance 1. In this case, intuitively, each outcome of A (say A=0.3) represents two possibilities (such as A_1 = 0 and the Gaussian is 0.3, or A_1 = 1 and the Gaussian is -0.7). A can be modeled as a partition of Γ× [0,1] where x ∈ [0,1] is interpreted as the quantile of the outcome of the Gaussian. Each member of the partition has two elements. These can be written (0,x_0) and (1,x_1) with x_0 = Φ(A) and x_1 = Φ(A-1), using the standard normal CDF Φ. Given the realization A=0, the posterior distribution on A_1 is given by a Bayesian update depending on the probability density of the Gaussian at 0 and at -1.§.§ The definitions of substitutes and complements We utilize a common notion of diminishing and increasing marginal value. For example, the idea of submodularity is that a lattice element B's marginal contribution to A should be smaller that to some A' ≼ A. For moderate and strong substitutes, we will use a strengthening of submodularity by requiring diminishing marginal value with respect to, respectively, all deterministic and randomized functions of a signal.[Previous versions of this paper used submodularity on the discrete and continuous signal lattices, which is a more restrictive definition. We only need this weaker definition for results in this paper, but in general both are interesting and the right choice may be context-dependent; or there could be other interesting variations.]Verbally, S&C capture that the more pieces of information one has, the less valuable (respectively, more valuable) B becomes. The levels of weak, moderate, and strong capture the senses in which “pieces of information” is interpreted. Weak substitutes satisfy diminishing marginal value whenever a whole signal is added to a subset of signals. However, they do not give guarantees about marginal value with respect to partial information about signals. Moderate and strong substitutes respectively satisfy diminishing marginal value when deterministic (randomized) partial information about a signal is revealed. Strong substitutes imply moderate substitutes, which imply weak substitutes. The same holds for complements. The respective lattices are supersets, i.e. the continuous signal lattice is a superset of the discrete lattice which is a superset of the subsets lattice, and the partial orderings agree. So each substitutes definition requires ^u,P to satisfy a set of inequalities at various points, and in going from weak to moderate to strong substitutes, we simply increase the set of required inequalities that signals must satisfy. [Substitutes]The event E is a uniformly random bit and the two signals A_1 = E and A_2 = E. That is, both signals are always equal to E. The decision problem is to predict the outcome of E by deciding either 0 or 1, with a payoff of 1 for correctness and 0 otherwise. In this case, one can immediately see that A_1 and A_2 are e.g. weak substitutes, as a second signal never gives marginal benefit over the first. [Complements]The event E and decision problem are the same as in Example <ref>, but this time A_1 and A_2 are uniformly random bits with E = A_1 ⊕ A_2, the XOR of A_1 and A_2. In this case, A_1 and A_2 are immediately seen to be e.g. weak complements, as a first signal never gives marginal benefit over the prior. [Weak vs moderate]Here is an example of weak substitutes that are not moderate substitutes. Intuitively, we will pair the previous two examples. The event E consists of a pair (E_b,E_c) of independent uniformly random bits. The decision problem is to predict both components of E, getting one point for each correct answer. Let the random variable B_1 = E_b and B_2 = E_b. Let the random variables C_1 and C_2 be uniformly random bits such that E_c = C_1 ⊕ C_2.Now, consider the signals A_1 = (B_1,C_1) and A_2 = (B_2,C_2). Intuitively, the first component of each signal completely determines E_b, while the second component gives no information about E_c until combined with the other signal. Hence these signals intuitively have both substitutable and complementary internal structure. Consider the subsets lattice {∅, {A_1}, {A_2}, {A_1,A_2}}. If we modify the decision problem such that predicting the first component of E is worth 1+ϵ points, then these signals are weak substitutes: Each alone is worth 1+ϵ points, while together they are worth 2+ϵ points. On the other hand, if we modify the decision problem such that the second component of E is worth 1+ϵ points, then these signals become weak complements for analogous reasons.On the other hand, these signals are neither moderate substitutes nor moderate complements. One way to see this is to consider “coarsening” A_1 into the signal B_1; this has diminishing marginal value when added to A_2. However, we could also coarsen A_1 into the signal C_1, which has increasing marginal value when added to A_2.§.§ Scoring rules and a revelation principle We now introduce proper scoring rules and prove a useful “revelation principle”.A scoring rule for an event E is a function S: Δ_E × E →ℝ, so that S(q̂,e) is the score assigned to a prediction (probability distribution) q̂ when the true outcome realized is E=e. Define the useful notation S(q̂;q) = _e∼ q S(q̂,e) for the expected score under true belief q for reporting q̂ to the scoring rule.The scoring rule is (strictly) proper if for all E,q, setting q̂ = q (uniquely) maximizes the expected score S(q̂;q). In other words, if E is distributed according to q, then truthfully reporting q to the scoring rule (uniquely) maximizes expected score.A fundamental characterization of scoring rules is as follows: [<cit.>]For every (strictly) proper scoring rule S, there exists a (strictly) convex function G: Δ_E → with (1) G(q) = S(q;q) and (2) S(q̂,e) = G(q̂) + ⟨ G'(q̂), δ_e - q̂⟩where G'(q̂) is a subgradient of G at q̂ and δ_e is the probability distribution on E putting probability 1 on e and 0 elsewhere.Furthermore, for every (strictly) convex function G: Δ_E →, there exists a (strictly) proper scoring rule S such that (1) and (2) hold. Given any (strictly) convex G, we first check that the induced S is (strictly) proper. Select a subgradient G'(p) at each point p. The expected score for reporting q̂ when E is distributed according to q isS(q̂;q)= _e∼ q S(q̂,e)= G(q̂) + ⟨ G'(q̂), q - q̂⟩≤ G(q) by convexity of G= S(q;q) .Note that the inequality follows simply because, for any convex G, if we take the linear approximation at some point q̂ and evaluate it at a different point q, this lies below G(q). Furthermore, if G is strictly convex, then this inequality is strict, implying strict properness.Now, given a (strictly) proper S, we show that it has the stated form. Define G(q) = S(q;q). Note that S(q̂;q) = _e∼ q S(q̂,e) is a linear function of q. By properness, each G(q) = S(q;q) = max_q̂ S(q̂;q). Since G(q) is a pointwise maximum over a set of linear functions of q, G is convex. If S was strictly proper, then G(q) was the unique maximum at every point, implying that G is strictly convex.Now we claim that S(q; ·) is a subtangent of G at q: it is linear, equal to G at q, and everywhere below G by definition of G. So in particular S(q,e) = S(q;δ_e) = G(q) + ⟨ G'(q̂), δ_e - q⟩, as promised. The log scoring rule is S(p,e) = log p(e), i.e. the logarithm (usually base 2) of the probability assigned to the realized event. The expected score function is ∑_e p(e) log p(e) = -H(p), where H is the Shannon entropy function. Notice that a scoring rule is a special case of a decision problem: The utility function is the scoring rule S, E is the event picked by nature, and the decision space = Δ_E. We now show that in a sense, scoring rules capture all decision problems. This is not surprising or difficult, and may have been observed prior to this work; but we formalize it because it captures a very nice and useful intuition. The idea of the proof, as suggested by the name, is simply for the agent to report her belief q about E to the scoring rule and for the scoring rule to simulate the optimal decision for this belief, paying the agent according to the utility derived from that decision. For a given distribution (“belief”) q on E, let d^*_q be the optimal decision, i.e. d^*_q = max_d∈_e∼ q u(d,e). Now, given u,,E, let S(q̂,e) = u(d^*_q̂, e).First let us show properness, i.e. S(q̂;q) ≤ S(q;q). We haveS(q̂;q)= _e∼ q S(q̂,e)= _e∼ q u(d^*_q̂, e)≤_e∼ q u(d^*_q, e)= _e∼ q S(q,e)= S(q;q)using the definition of d^*_q.Now let us check equivalence to the original problem. Let q_a be the distribution on E conditioned on A=a. We have^u,P(A)= _a max_d ∈_e∼ q_a u(d,e)= _a _e∼ q_a u(d^*_q_a, e)= _a _e∼ q_a S(q_a, e)= _a max_q̂_e ∼ q_a S(q̂, e) by properness= ^S,P(A) . This reduction is not necessarily computationally efficient, because the input q̂ to the scoring rule is a probability distribution over E which may have a large number of outcomes. We note two positives, however. First, the reduction does not necessarily need to be computationally efficient to be useful for proofs and analysis. Second, in any case where it seems reasonable to assume that the agent can solve her decision problem, which involves an expectation over possible outcomes of E, it seems reasonable to suppose that she can efficiently represent or query her beliefs. In this case we may often expect a computationally efficient reduction and construction of S. This is a direction for future work.The revelation principle (Theorem <ref>) and scoring rule characterization (Fact <ref>) together imply the following extremely useful fact about general decision problems. We do not claim originality for it; the idea can be found in <cit.> and similar ideas or statements are present in e.g. <cit.> and <cit.>. But it is worth emphasizing because we will put it to extensive use in this paper. As an example of usefulness, we provide a concise proof of the following classic theorem. [More information always helps]In any decision problem, for any signals A,B, (AB) ≥(A). In other words, more information always improves the expected utility of a decision problem. In other words,is a monotone increasing function on the signal lattices. Recall that we are using the notation p_a_1 for the distribution on E conditioned on A_1=a_1, and so on. In particular, p_a_1 is a vector, i.e. p_a_1 = (p(e_1 | A_1=a_1), …). By the revelation principle, for some convex G we have (A_1) = _a_1 G(p_a_1), andV(A_1A_2)= _a_1[ ∑_a_2 p(a_2|a_1) G(p_a_1a_2) ]≥_a_1 G(∑_a_2 p(a_2|a_1) p_a_1a_2) by Jensen's inequality= _a_1 G(p_a_1)= (A_1) .To obtain the last equality: Each term in the sum consists of the scalar p(a_2 | a_1) multiplied by the vector p_a_1a_2, and for each coordinate e of the vector, we have p(a_2|a_1)p(e|a_1,a_2) = p(e,a_2|a_1). Then ∑_a_2 p(e,a_2|a_1) = p(e|a_1). §.§ Characterizations In this section, we show how the substitutes and complements conditions can be phrased using the convexity connection just derived. We will leverage this structure to identify characterizations or alternative definitions of substitutes and complements. For brevity, we will focus on substitutes, but in all cases the extension to complements is immediate.From Corollary <ref>, we also get immediately the following characterization:To be clear, our use of parentheses here means that we are giving three definitions: one for weak substitutes, where the condition must hold for all A' on the subsets lattice; and analogously for moderate substitutes with the discrete lattice and strong substitutes with the continuous lattice.We view Definition <ref> mostly as a tool, although it may convey some intuition on its own as well. Definition <ref> will be pictured in Figure <ref> along with the final characterization.§.§.§ Generalized entropies Here, we seek an alternative interpretation of the definitions of S&C in terms of information and uncertainty. To this end, for any decision problem, consider the convex expected score function G and define h = -G. Then h is concave, and we interpret h as a generalized entropy or measure of information. The justification for this is as follows: Define the notation h(E | A) = _a∼ A h(p_a), where p_a is the distribution on E conditioned on A=a. Then concavity of h implies via Jensen's inequality that for all E,A, we have h(E) ≥ h(E|A). In other words, more information always decreases uncertainty/entropy.We propose that this is the critical axiom a generalized entropy must satisfy: If more information always decreases h, then in a sense it measures uncertainty, while if more information sometimes increases h, then it should not be considered a measure of uncertainty. However, admittedly, the appeal of this definition may increase by adding additional axioms as are common in the literature, such as maximization at the uniform distribution and value zero at degenerate distributions. Another very intriguing axiom would be a relaxation of the “chain rule” in either direction: h(E | A) is restricted to be either greater than or less than h(E,A) - h(A). (Note that the chain rule itself, h(E|A) = h(E,A) - h(A), along with concavity, uniquely characterizes Shannon entropy.) Such axioms may have interesting consequences for informational S&C. Examining the structure of S&C under such axioms represents an intriguing direction for future work.Under this interpretation, Definition <ref> can be restated:Intuitively, Definition <ref> says this: Consider the expected amount of information about E that is revealed upon learning B, given that some information will already be known. Use the generalized entropy h to measure this information gain. Then substitutes imply that, the more information one has, the less information B reveals. On the other hand, complements imply that, the more information one has, the more information B reveals.Revisiting Example <ref>, where E was a uniform bit and A_1=A_2=E, imagine predicting E against the log scoring rule. Our previous observations imply that here the generalized entropy function is Shannon entropy H(q) = ∑_e q(e) log1/q(e). We have H(p) = 1 and H(E|A_1) = H(E|A_2) = H(E|A_1,A_2) = 0, which already shows that A_1 and A_2 are weak substitutes.If we instead revisit Example <ref>, where A_1 ⊕ A_2 = E with A_1,A_2 uniformly random bits, and again consider predicting E according to the log scoring rule, then we see that H(E) = H(E|A_1) = H(E|A_2) = 1, while H(E|A_1,A_2) = 0, already proving that A_1 and A_2 are weak complements. §.§.§ Bregman divergences Given a convex function G, the Bregman divergence of G is defined asD_G(p,q) = G(p) - (G(q) + ⟨ G'(q), p-q⟩).In other words, it is the difference between G(p) and the linear approximation of G at q, evaluated at p. (See Figure <ref>.) Another interpretation is to consider the proper scoring rule S associated with G, by Fact <ref>, and note that D_G(p,q) = S(p;p) - S(q;p), the difference in expected score when reporting one's true belief p versus lying and reporting q. The defining property of a convex function is that this quantity is always nonnegative. This can be observed geometrically in Figure <ref> as well; there D_G(q,q̂) = G(q) - S(q̂;q).This notion is useful to us because, it turns out, all marginal values of information can be exactly characterized as Bregman divergences between beliefs.(AB) - (A)= _a,b G(p_ab) - _a G(p_a)= _a,b(G(p_ab) - G(p_a))= _a,b(D_G(p_ab, p_a) - ⟨ G'(p_a), p_ab - p_a ⟩)= _a,b D_G(p_ab, p_a) + _a ⟨ G'(p_a), ∑_b p(b|a) (p_ab - p_a ) ⟩= _a,b D_G(p_ab, p_a)because ∑_b p(e|a,b) p(b|a) = p(e|a), so ∑_b p(b|a) p_ab = p_a.This can be interpreted as a characterization of S&C where D_G serves as a distance measure of sorts (although it is not in general a distance metric). The characterization says that, if we look at how “far” the agent's beliefs move upon learning B, on average, then for substitutes this distance is decreasing in how much other information is available to the agent. But for complements, the more information the agent already has, the farther she expects her beliefs to move on average upon learning B.For the log scoring rule, D_G(p,q) is exactly the KL-divergence or relative entropy KL(p,q) between distributions on p and q. If we recall Example <ref>, in which E was a random bit and A_1 = A_2 = E, we can consider the decision problem of prediction E against the log scoring rule. In this case, the prior p = (1/2,1/2), while the posteriors p_a_1=0 = (1,0), p_a_1=1 = (0,1), and the same for A_2. Hence _a_1 KL(p_a_1,p) = 1. But the posteriors conditioned on both signals are the same, e.g. p_a_0=a_2=0 = (1,0) = p_a_0=0. Hence _a_1,a_2 KL(p_a_1,a_2,p_a_1) = 0.This already shows that A_1 and A_2 are weak substitutes. And in fact, if A_1 = A_2, then this argument extends to show that A_1 and A_2 are substitutes in any decision problem (as they should be), because given A_1, an update on A_2 moves the posterior belief a distance 0. § GAME-THEORETIC APPLICATIONS§.§ Prediction markets A prediction market is modeled as a Bayesian extensive-form game. The market's setting is specified by a strictly proper scoring rule S and an information structure with prior P and event E, and set of signals A_1,…,A_m. We assume these signals are “nontrivial” in that, given all signals but A_i, the distribution of E changes conditioned on A_i.An instantiation of the market is specified by a set of n traders, each trader i observing some subset of the signals (call the resulting signal B_i), and an order of trading i_1,…,i_T, where at each time step t=1,…,T, it is the turn of agent i_t ∈{1,…,n} to trade. We assume that no trader participates twice in a row (if they do, it is without loss to delete one of these trading opportunities).The market proceeds as follows. First, each trader i simultaneously and privately observes B_i, updating to a posterior belief. Then the market sets the initial prediction p^(0)∈Δ_E, which we assume to be the prior distribution p on E. We will also refer to a market prediction as the market prices. Then, for each t=1,…,T, trader i_t arrives, observes the current market prediction p^(t-1), and may update it to (“report”) any p^(t)∈Δ_E.After the last trade step T, the true outcome e of E is observed and each trader i receives payoff ∑_t: i=i_t S(p^(t), e) - S(p^(t-1), e). Thus, at each time t, trader i_t is paid according to the scoring rule applied to p^(t), but must pay the previous trader according to the scoring rule applied to p^(t-1). The total payment made by market “telescopes” into S(p^(T),e) - S(p^(0),e).At any given time step t, trader i_t is said to be reporting truthfully if she moves the market prediction to her current posterior belief on E. In other words, she makes the myopically optimal trade.The natural solution concept for Bayesian games is that they be in Bayes-Nash equilibrium, where for every player, her (randomized) strategy — specifying how to trade at each time step as a function of her signal and all past history of play — maximizes expected utility given the prior and others' strategies.Because this is a broad class of equilibria and can in general include undesirable equilibria involving “non-credible threats”, it is often of interest in extensive-form games to consider the refinement of perfect Bayesian equilibrium. Here, at each time step and for each past history, a player's strategy is required to maximize expected utility given her beliefs at that time and the strategies of the other players. (Note the difference to Bayes-Nash equilibrium in which this optimality is only required a priori rather than for every time step.) Here, at any time step and history of play, players' beliefs are required to be consistent with Bayesian updating wherever possible. (It may be that one player deviates to an action not in the support of her strategy; in this case other players may have arbitrary beliefs about the deviator's signal.)To be clear, every perfect Bayesian equilibrium is also a Bayes-Nash equilibrium. Hence, we note that an existence result is strongest if it guarantees existence of perfect Bayesian equilibrium. Meanwhile, a uniqueness or nonexistence result is strongest if it refers to Bayes-Nash equilibrium. Distinguishability criterion. For most of our results, we will need a condition on signals equivalent or similar to those used in prior works <cit.> in order to ensure that traders can correctly interpret others' reports. Formally, we say that signals are distinguishable if for all subsets S ⊆{1,…,m} and realizations {a_i : i ∈ S}, {a_i' : i ∈ S} of the signals {A_i : i ∈ S} such that, for some i ∈ S, a_i ≠ a_i', [e | a_i : i ∈ S] ≠[e | a_i' : i ∈ S] .We believe it may be possible to relax this criterion and/or interpret such criteria within the S&C framework, and this is a direction for future work. Our notation in prediction markets. Note that, for the proper scoring rule S with associated convex G, along with the prior P, we have the associated “signal value” function : (A) = _a,e S(p_a,e) = _a G(p_a) .In other words, (A) is the expected score for reporting the posterior distribution conditioned on the realization of A.A second key point is that, to a trader whose current information is captured by a signal A, the set of strategies available to that trader can be captured by the space of signals A' ≼ A on the continuous lattice. This follows because any strategy is a randomized function of her information, so the outcomes of the strategy can be labeled as outcomes of a signal A'.§.§.§ Substitutes and “all-rush”We now formally define an “all-rush” equilibrium and show that it corresponds to informational substitutes. The naive definition would be that each trader reports truthfully at their first opportunity, or (hence) at every opportunity. This turns out to be correct except for one subtlety. Consider, for example, the final trader to enter the market. Because all others have already revealed all information, this last trader will be indifferent between revealing immediately or delaying. Similarly, consider three traders i,j,k and the order of trading i,j,i,j,k. If trader i truthfully reports at time 1, then trader j is not strictly incentivized to report truthfully at time 2. She could also delay information revelation until time 4.An all-rush strategy profile in a prediction market is one where, if the traders are numbered 1,2,… in order of the first trading opportunity, then each trader i reports truthfully at some time prior to i+1's first trading opportunity (with the final trader reporting truthfully prior to the close of the market). Before presenting the main theorem of this section, we give the following useful lemma (which is quite well known): In every Bayes-Nash equilibrium, every trader reports truthfully at her final trading opportunity. Consider a time t at which trader i = i_t makes her final trade. Fix all strategies and any history of trades until time t; then i's total expected payoff from all previous time steps is fixed as well and cannot be changed by any subsequent activity. Meanwhile, i's unique utility-maximizing action at time t is to report truthfully, by the strict properness of the scoring rule. If i does not take this action, then her entire strategy is not a best response: She could take the same strategy until time t and modify this last report to obtain higher expected utility. Therefore, in Bayes-Nash equilibrium, i reports her posterior on her final trading opportunity. Let the traders be numbered in order of their first trading opportunity 1,2,…,n and let B_i be the signal of trader i. Before diving in, we develop a key idea. In equilibrium, we can view the market prediction p^(t) at time t as a random variable. Then, construct a “signal” C^(t) capturing the information contained in p^(t). This can be pictured as the information conveyed by p^(t) to an “outside observer” who knows the prior distribution and the strategy profile, but does not have any private information. Furthermore, if traders 1,…,k have participated thus far, then C^(t) is an element of the continuous signal lattice with C^(t)≼ B_1 ⋯ B_k, because p^(t) is a well-defined, possibly-randomized function of B_1 ⋯ B_k. Finally, if all participating traders have been truthful, then C^(t) is a member of the subsets signal lattice , as it exactly reveals the subset of the signals held by those traders.Now, let t_i^* be i's final trading opportunity prior to i+1's first trading opportunity. We prove by backward induction on t that, in BNE and for any participant i participating at some time t ≥ t_i, the following holds: if C^(t) is an element of the subsets lattice , then i reports truthfully at t. Now, suppose we have successfully proven this claim by backward induction; let us finish the proof. The claim implies that C^(t) really is inand i really is truthful at all such time steps for the following reason: C^(0) is the null signal and is an element of , so trader 1 participates truthfully, which implies that C^(t_2)∈, which implies that 2 participates truthfully, and so on. So in any BNE, all participants play all-rush strategies.Now let us prove the statement. For the base case t=T, the trader participating at the final time step is truthful by Lemma <ref>.Now for the inductive step, consider any t = t_i for some i. If t is i's final trading opportunity, then by Lemma <ref>, in BNE i reports truthfully at t. If t < t_i^*, then there is nothing to prove.Otherwise, let t' be i's next trading opportunity after t. By inductive hypothesis, i is truthful at time t' and thereafter. We compute i's expected utility for any strategy, and show that if i is not truthful at t, she can improve by deviating to the following strategy: Copy the previous strategy up until t, report truthfully at t, and make no subsequent updates.At t', i's strategy can be described as reporting truthfully according to C^(t') = C^(t'-1) B_i. For this trade, i obtains expected profit (C^(t')) - (C^(t'-1)), and i obtains no subsequent profit once her information is revealed. Meanwhile, consider i's strategy at time t, which induces some signal C^(t)≼ B_iC^(t-1). For this trade, i obtains expected profit at most (C^(t)) -G(p^(t-1)). This follows because a trade conveying signal C^(t) obtains at most (C^(t)).[Although we don't explicitly use it here, this implies that in equilibrium, every p^(t) = p_c^(t), that is, the price at time t equals the posterior distribution on E conditioned on all information that has been revealed so far, including at time t.]Let U be i's total expected utility at time t and greater. Once i reports truthfully at t', she expects to make no further profit in equilibrium. So U ≤(C^(t)) -G(p^(t-1)) + (C^(t'-1) B_i) - (C^(t'-1)) .Now suppose C^(t'-1) is inand i is not reporting truthfully at t. This implies that B_i ⋠C^(t'-1). By strong, strict substitutes, B_i gives higher marginal benefit to C^(t) than to C^(t'-1): (C^(t'-1) B_i) - (C^(t'-1)) < (C^(t) B_i) - (C^(t)) .So U < (C^(t) B_i) -G(p^(t-1)) .But i can achieve this by deviating to being truthful at time t, then not participating at any subsequent times. (This follows because if i is truthful at time t, she reveals the signal C^(t-1) B_i, which is exactly the same as C^(t) B_i.) This deviation does not affect i's utility from any previous times, so it is a strategy with higher total expected utility. So i's only BNE strategy can be to be truthful at t.The assumptions imply that there signals on the subsets lattice A,B and some A' on the continuous lattice with A' ≼ A and (A'B) - (A') < V(AB) - (A) .Then in particular, we can consider the scenario with two traders where “Alice” has signal A (i.e. she observes the corresponding subset of signals) and Bob has B, with a trading order Alice-Bob-Alice. In perfect Bayesian equilibrium (PBE), Bob must be truthful at his trading opportunity according to his beliefs even if Alice deviates from her strategy. By distinguishability, Alice can infer his signal from this truthful report, so in any PBE, Alice is truthful and correct in predicting p_ab at the second opportunity. Hence the two traders' expected utilities sum to the constant amount (AB) - (), even when Alice deviates. If Alice reports truthfully at her first opportunity (the all-rush strategy), then Bob's expected utility is (AB) - (A). But if Alice reports according to A' ≼ A, then Bob's expected utility is at most (A'B) - (A'), which by assumption of non-substitutes is strictly smaller. This implies that Alice prefers the deviation, so truthful reporting (and thus all-rush) could not have been an equilibrium. §.§.§ Complements and “all-delay”We begin by defining an “all-delay” strategy profile, analogous to all-rush.An all-delay strategy profile in a prediction market is one where, when the traders are numbered 1,…,n in order of their final trading opportunity, each trader i ≥ 2 reveals no information until after trader i-1's final trading opportunity. The ideas will be substantially the same as in Theorem <ref>, but the deviation argument is somewhat trickier. In the substitutes “rush” case, an agent could deviate to immediate truth-telling and ignore all subsequent consequences. Now, we will need agents to deviate to delaying all information revelation, relying on their opponents' response to ensure this becomes profitable later. This is also the reason that we restrict to perfect Bayesian equilibrium.The proof is by backward induction. We show that in any perfect Bayesian equilibrium: at each time t, if C^(t) is on the subsets lattice, then players play an all-delay strategy from time t onward. (As in the proof of Theorem <ref>, let C^(t) be the signal induced by the random variable p^(t) in equilibrium. Because in PBE strategies are well-defined in every subgame, C^(t) is also well-defined off the equilibrium path.) As in Theorem <ref>, this will prove the statement, because C^(0) is on the subsets lattice, corresponding to ∅, so the first trader plays all-delay at time 1, implying that C^(1) is on the subsets lattice, etc.For the base case, at t=T, this is the trader's final opportunity, so she is truthful by Lemma <ref>, which constitutes an all-delay strategy from T onward.Now consider any trading time t=T and participant i trading at time t.First suppose t is i's final trading opportunity; then by Lemma <ref>, she reports truthfully at this time. By induction, traders play all-delay at all times after t, so this shows that they play all-delay from time t onward.If t is not a trader's final trading opportunity, but is after i-1's final trading opportunity, then there is nothing to prove for this time step. So suppose trader i is trading at time t with i-1's final opportunity coming at some t_i-1 > t. The inductive assumption implies that in any subgame starting at time t+1 in PBE, i does not make any update until some t' > t_i-1. It also implies that no other trader participates between t_i-1 and t'. Finally, it implies that all traders participating between time t and t' (exclusive) report truthfully[In a trading order such as i,j,k,j,k, it is possible that j reports something nontrivial at time 2, then reports truthfully at time 4. But k does not participate at time 3, so j's multiple reports WLOG telescope into a single truthful report.]. Let B denote the join of their signals. Then i's total utility from time t onward is U = (C^(t)) -G(p^(t-1)) + (C^(t) BB_i) - (C^(t)B) .Now suppose for contradiction that i reveals some nontrivial information at time t, i.e. C^(t)≠ C^(t-1). Then i can deviate to revealing nothing at time t, reporting according to C^(t-1), and being truthful at time t'. In this case, by assumption of PBE, others continue to best-respond. By inductive assumption, in any subgame of a PBE (which itself must be in PBE), others who participate between t and t' therefore continue to report truthfully, implying that B is still revealed between time t and t'. Now, strong, strict complements imply (C^(t)) - (C^(t-1)) < (C^(t) B) - (C^(t-1) B) .So U < (C^(t-1)) -G(p^(t-1)) + (C^(t) BB_i) - (C^(t-1) B) .But this is i's utility for the deviation above (note that C_tBB_i = C_t-1 BB_i). Since the deviation is profitable, this gives a contradiction, implying that i (and all players) must play all-delay starting from time t in any PBE. Analogous to the substitutes case (Theorem <ref>). The assumptions imply that there are subsets-lattice signals A,B and continuous-lattice signal A' ≼ A such that (A'B) - (A') > V(AB) - (A) .Then in particular, we can consider the scenario where “Alice” has signal A and Bob has B, with a trading order Alice-Bob-Alice. In PBE, Bob is truthful even if Alice deviates. By distinguishability, Alice can infer his signal from this truthful report, so in any PBE, Alice is truthful and correct in predicting p_ab at the second opportunity. So utilities have the constant sum (AB) - () even when Alice deviates. If Alice reports nothing at her first opportunity, then Bob's expected utility is (B) - (). But if Alice deviates to reporting to A' ≼ A, then Bob's expected utility is at most (A'B) - (A'), which is strictly smaller. This implies that Alice prefers the deviation, so truthful reporting (and thus all-rush) could not have been an equilibrium.Hence, in perfect Bayesian equilibrium, trader 1 cannot play all-delay.Discussion. These results show that informational S&C are in a sense unavoidable in the study of settings such as prediction markets. However, the result raises many interesting questions for future work. Two major questions are: How can we identify structures that are substitutes or complements? and How can we design markets to encourage substitutability?We give some initial steps toward answering these questions in Section <ref>. §.§ Other game-theoretic applications We will now examine a few game-theoretic contexts in which our results have immediate applications or implications. Instead of developing full formal proofs and theorem statements, we focus on illustrating the intuition of how to extend our results and the conceptual lens of informational S&C to those settings.§.§.§ Crowdsourcing and contestsIn the study of crowdsourcing from a theoretical perspective, the “crowd” is a group of agents who hold valuable information and the goal is to design mechanisms that elicit this information. Specifically, here we are interested in “wisdom of the crowd” settings where the total information available to the crowd is greater than that of the most-informed individual, and the goal is to aggregate this information.Before describing how informational S&C apply in such settings, we would like to contrast with approaches to crowdsourcing that models each user's contribution as a monolithic submission that has some endogenous quality, such as <cit.> There, it is impossible to integrate or aggregate user contributions and the problem is to incentivize and select one of the highest possible quality. In these models, information plays no role and the model is equally well-suited to incentivizing production of a high-quality good of which only one is required; sometimes this is explicit in the motivation or model of the literature <cit.>.Here, we consider cases where users have heterogeneous information and we would like to aggregate it into a final form that is more useful than any one user. The question is how users behave strategically in revealing this information in the contest. Collaborative, market-based contests for machine learning. <cit.> proposes a mechanism for machine learning contests with a prediction market structure. This mechanism was later extended to elicit data points and to more general problems in <cit.>. While these mechanisms have appealing structure, seeming to align participants' incentives with finding optimal machine-learning hypotheses, the authors did not give results on equilibrium performance or behavior of strategic agents. Here, we briefly describe the framework of this mechanism and how our results can apply.In a machine learning problem, we are given a hypothesis class . There is some true underlying distribution e of data, which is initially unknown. In our setting we assume there is a prior belief on this distribution e, which distributed as a random variable E. The goal is to select a hypothesis d with minimum risk R(d,e) on the true data distribution. Here, R(d,e) = _z∼ eℓ(d,z) for a loss function ℓ(d,z) on hypothesis h and a datapoint z drawn from e.In the contest mechanism of <cit.>, the mechanism selects an initial market hypothesis d^(0). As in the prediction market model of Section <ref>, participants iteratively arrive and propose a new hypothesis d^(t) at each time t. At the end of the contest, the mechanism draws a test data point z ∼ e from the true distribution and rewards each participant by their improvement to the loss of the market hypothesis, i.e. if i updated the hypothesis at time t from d^(t-1) to d^(t), then i is rewarded ℓ(d^(t-1),z) - ℓ(d^(t),z) for that update.We observe that prediction markets are a special case of this framework: Each e corresponds to some fixed observation, for instance, for a given e the data point drawn is always the same z_e. The risk R(d,e) is therefore always equal to ℓ(d,z_e) = -S(d,e) for the proper scoring rule S used in the market. Thus, the above framework captures prediction markets as a special case. However, we now show that prediction markets capture the essential strategic features of this setting.Now, notice that in expectation over the test data z, this reward is equal to R(d^(t-1),e) - R(d^(t),e). Furthermore, we can define the utility of the designer to be u(d,e) = - R(d,e), that is, the negative of the risk of that hypothesis on that data distribution. Hence, by the revelation principle, there is some proper scoring rule S that is payoff-equivalent to u. A prediction market with proper scoring rule S is strategically identical to the above contest. Thus, with a few small caveats, our above results apply: in a Bayesian game setting where traders have signals A_1,…,A_n and a common prior on the distribution of signals and E, substitutes characterize “all-rush” equilibria with immediate aggregation, while complements characterize “all-delay” equilibria.The caveats are (1) that the scoring rule obtained by the revelation principle will not in general be strictly proper if two beliefs about E map to the same optimal hypothesis d; and (2) it is not guaranteed that traders can infer others' information from their trades without a condition analogous to the distinguishability criterion of Section <ref>. While these hurdles are surmountable, our purpose here is only to mention the key ideas for how the above results on prediction markets will generalize.This connection immediately presents several questions for future work: When, in a machine-learning setting, should we expect contest participants to have substitutable or complementary information? In particular, if agents hold data sets and the goal is to elicit these data sets using this structure (as explored in <cit.>), when should we expect data sets to be substitutable? Furthermore, how can we design loss functions so as to encourage substitutability and hence early participation? This last question is discussed in Section <ref>. Question-and-answer forums. <cit.> propose a model for analyzing strategic information revelation in the context of question-and-answer forums. Initially, some question is posed. Participants have private pieces of information A_1,…,A_n as in the prediction market model, and they arrive iteratively to post answers. Unlike in the prediction market model, rather than arriving multiple times, participants may only post a single answer; however, they may be strategic about when they post this answer. By waiting until later, a participant may be able to aggregate information from others' answers, allowing her to post a better response. Unlike in the prediction market setting, participants cannot “garble” their information. However these are not essential differences as compared to the substitutes or complements cases of prediction markets, where in equilibrium participants do not want to garble or participate multiple times, but instead fully reveal at the time that is optimal for them (as early or as late as possible, respectively).In the model of <cit.>, the asker of the question has a valuation function (S) over subsets of the pieces of information. <cit.> does not justify how such a valuation function may arise, but we can now justify this modeling decision because any decision problem faced by the asker gives rise to some such valuation function. In one case of <cit.>, the asker draws a uniform “stopping threshold” t on [(), (A_i ⋯ A_n)] (using our notation), and selects as the “winning answer” the one whose information raises her value above this threshold. For instance, if the first two users to post are i,j with signals A_i,A_j, and then the third user k posts with signal A_k, and we have (A_iA_j) < t ≤(A_iA_jA_k), then user k is declared the winner.From this model, the expected reward of a participant, which is an indicator for being declared the winner, is exactly proportional to the marginal value of her information to the information collected so far. Thus, substitutes imply that participants' dominant strategy is to rush to participate as early as possible, while complements imply that it is dominant to wait as long as possible. This follows from diminishing (increasing) marginal value of information.And indeed, <cit.> identify substitutes and complements conditions on the information which are exactly diminishing and increasing marginal returns, with the main result as stated above. (The paper also considers several other methods of selecting the winner with more complex features, which we will avoid discussing here for simplicity.)We would like to emphasize that, while the above discussion intentionally highlights the similarities between that work and this one, the authors do not provide any endogenous model of the information, e.g. whether it be probabilistic and if so how it is structured, nor of the utility of the asker of the question and how this utility might arise or be related to the structure of the information. Without such models, it does not justify why a structure might satisfy their substitutes or complements conditions (nor when/if one could expect the conditions to hold).Our work provides answers to all of these questions. Information may be modeled as Bayesian signals and the asker may face any decision problem. This gives rise to a valuation function over signals that can capture the model in <cit.>. Furthermore, under this model, that papers' substitutes and complements conditions used in <cit.> are subsumed by those proposed here. Hence, we are able to bring this work under the same umbrella as prediction markets, the crowdsourcing contests discussed above, and the algorithmic and structural results to be discussed later. An example of substitutes for any of these problems is an example for all of them.§ ALGORITHMIC APPLICATIONS Here, we investigate the implications of informational substitutes and complements on the construction and existence of efficient algorithms for information acquisition. We first define a very general class of problems, /, to model this problem, and show positive results corresponding to substitutes and negative results in general. These results are obtained by showing tight connections to maximization of (submodular) set functions. We also investigate an adaptive or online variant of the problem with similar results.To focus on the problem of information acquisition, we abstract out the complexity of interacting with the decision problem and prior. This differentiates these results from prior work on information acquisition, but the overall approach – utilizing submodular set functions – is common (even pervasive) in the literature (see <cit.>). So we view the contribution of these results as offering a unification or explanation for successful approaches in terms of informational substitutes.§.§ The / problemIn the problem /, one is given a decision problem u and information structure on n signals A_1,…,A_n; and also a family ℱ of feasible subsets of signals S ⊆{1,…,n}. The goal is to select an approximately optimal S ∈ℱ, i.e. to max_S ∈ℱ (_i ∈ S A_i ) . For instance, suppose that an agent has a budget constraint of B and each piece of information has a price tag; how to select the set that maximizes utility subject to the budget constraint? (This is also known as a knapsack constraint.)The / problem is not yet well-defined because we have not described how the input is represented. In general, the decision problem may be hard to optimize, and the prior distribution may have support 2^Ω(n). Because we want to abstract out the complexity of / independently of the difficulty of these problems, we will assume an efficiently-queryable input. The outline for our approach is as follows, pictured in Figure <ref>. * For positive results, we require an input representation that allows efficient computation of (_i∈ S A_i ) for some subset S of signals. In Section <ref>, we will discuss several such representations. The most natural of these we call the oracle model.* For negative results, we will reduce a hard problem – maximization of arbitrary monotone set functions given a value oracle – to /. The reduction will produce instances of / having a very concise and tractable input representation: The signals A_1,…,A_n will be independent uniformly random bits, the event E will be the vector (A_1,…,A_n), and the decision problem will be immediately computable, just requiring transparent calls to the value oracle of the original maximization problem. This implies that any algorithm that can solve /, under any “reasonable” model of input (particularly the oracle model and others we describe), can solve the instances produced by our reduction.Note that an oracle-based approach is very general because, if we do have an instance where the input is concise and given explicitly, for example, the decision-problem optimizer is given as a small circuit, then we can just run our algorithms treating this input as an oracle, evaluating it when necessary.We next discuss input representations for positive results, including describing the oracle model. We will then give our positive and negative results. Monotone set function maximization. Our results will involve relating complexity of / to that of maximizing some f: 2^N →ℝ where N = {1,…,n} is a finite ground set. The fact thatis increasing, i.e. more information always helps, implies that we restrict to monotone increasing f: If S ⊆ T, then f(S) ≤ f(T). Recall that submodular f correspond to substitutable items, while supermodular f correspond to complements.In set function maximization, the input is often given as a value oracle that, when given a subset S ⊆ N, returns in one time step f(S). When f is submodular, it is known that polynomial-time constant-factor approximation algorithms exist for many types of constraints. For instance, there are efficient (1-1/e)-approximation algorithms under the knapsack constraint described above <cit.> and more general matroid constraints <cit.>. On the other hand, in general set function maximization is known to be difficult information-theoretically, requiring exponentially-many oracle queries to obtain a nontrivial approximation factor even when restricting to monotone supermodular functions; we give an example in Proposition <ref>.§.§.§ The oracle model and computing V Here, we investigate the computation of , the value function. We restrict attention to evaluatingat a set of signals, i.e. the join _i∈ S A_i of a subset S of signals. The reason is that this case is sufficient for our positive results and is most compelling for /. Furthermore, no difficulty arises in how the input signal is represented, as it can always be given by a subset S of {1,…,n}.We begin with a case where the decision problem is specified by an oracle, but the prior p is given explicitly. Because p may be exponentially large in n, the number of signals, we will later introduce an oracle model for p as well. For any decision problem and set of signals A_1,…,A_n, given an oracle for computing the associated convex G, we can compute (_i ∈ S A_i) in time polynomial in n, ∏_i |Support(A_i)| and |Support(E)|. (This is the size of the problem in general, as the prior distribution ranges over this many outcomes (e,a_1,…,a_n).) We assume an oracle that computes G(q) for any distribution q on E. Note that G(q) = max_d [ u(d,e) | e ∼ q], so this is equivalent to assuming an oracle for the utility of the optimal decision for a given distribution on E.We need to calculate (_i∈ S A_i) = _{a_i : i ∈ S} G(p_a_i : i ∈ S) .Here, the expectation is over all realizations {a_i : i ∈ S} of the set of signals {A_i : i ∈ S}, and p_a_i : i ∈ S is the posterior distribution on E conditioned on that set of realizations.There are at most ∏_i∈ S |Support(A_i)| terms in the sum, and each posterior p_a_i : i ∈ S can be computed as follows: p(e | a_i : i ∈ S) = p(e, a_i : i ∈ S)/p(a_i : i ∈ S) ,for each e in the support of E. This can be computed in time polynomial in the products of the support sizes. In general, the running time of Proposition <ref> is exponential in n, the number of signals. This is unavoidable in general as the input itself may be this large (the prior distribution ranges over exponentially many events). Thus, it is natural to suppose that the input is given as an oracle in some fashion, making the input succinct. We define a natural model, give the associated positive result, and discuss possible variants or weakenings.In the oracle model for representing a decision problem u and prior p, one is given:* An oracle computing the prior probability of any realization of any subset of signals. * Access to independent samples from the prior distribution. * An oracle computing, for a distribution q on E, the expected optimal utility obtainable given belief q, namely G(q).In the oracle model, we can approximate (_i∈ S A_i) to arbitrary (additive) accuracy with arbitrarily high probability in time polynomial in n, ∑_i log |Support(A_i)|, and |Support(E)|. For any given S, we would like to approximate (_i∈ S A_i) = _{a_i : i ∈ S} G(p_a_i : i ∈ S)up to an additive ϵ error with probability 1-δ. We can abstract this problem as computing Z, with Z is distributed as G(p_a_i : i ∈ S) where {a_i : i ∈ S} is drawn from the prior. Letting K = max_q G(q) - min_q G(q) over all {q = p_a_i : i ∈ S : S ⊆{1,…,n}}, we can apply a standard Hoeffding bound: The average of m i.i.d. realizations of Z is within ϵ of the true average with probability 1-δ as long as m exceeds K^2ln(2/δ)/2ϵ^2.To see that we can in fact sample Z: We simply draw one sample from the prior, giving us {a_i : i ∈ S}. We compute the posterior conditioned on this sample as follows: For each outcome e of E, we have p_a_i : i ∈ S(e) = p(e, {a_i : i ∈ S})/p({a_i : i ∈ S}) .This requires two calls to the prior computation oracle; then a call to the oracle for G completes the calculation. The running time analysis simply observes that each signal realization a_i requires only log|Support(A_i)| bits to represent, and there are n of them; similarly for outcomes of E.One would like to make weaker assumptions. However, dropping either the assumption of independent samples or the oracle seems problematic. Without samples, evaluating G(p_a_i : i ∈ S) seems difficult because this is a sum over exponentially many terms, and we cannot a priori guess which terms are “large” or where to query the oracle for the prior.With independent samples but no oracle for marginal probabilities, naïve approaches break down because of the difficulty of accurately estimating conditional probabilities p(e | {a_i : i ∈ S}). For instance, it may be that each outcome {a_i : i ∈ S} is very unlikely, so that one cannot draw enough samples to accurately estimate the desired conditional probability using the ratio p(e, {a_i : i ∈ S}) / p({a_i : i ∈ S}). It is also of note that any distribution over the possible outcomes of the signals and of E, including the prior itself, in general has size |Support(E)| ·∏_i=1^n |Support(A_i)|, which is exponential in n. So one must avoid writing down such distributions; and any small sketch seems to quickly lose accuracy in estimating the conditional probability, which is computed from probabilities on events (and again, these probabilities may all be exponentially small even while conditional probabilities are large).We give two further examples of how to overcome this difficulty. The first is to assume that the prior is tractable in some way; in our case, sparse. The second is to push the difficulties mentioned into an oracle of a different sort. Suppose we are given access to an oracle for G explicitly given the prior p; and suppose that the prior is sparse, supported on k possible outcomes (e, a_1,…, a_n). Then we can compute (_i∈ S A_i) in time polynomial in n and k. We can now assume that the prior is explicitly given as part of the input. The expectation in the definition ofis now a sum that ranges over only at most k terms. For each term, corresponding to some subset {a_i : i ∈ S}, we can efficiently look up p(a_i : i ∈ S), as well as (for each e) p(e,{a_i : i ∈ S}), allowing us to compute the conditional probability p(e | a_i : i ∈ S). (Recall that our notation p_a_i : i ∈ S is simply the vector of these probabilities, ranging over outcomes e of E.) These are all the ingredients we need to evaluate each term in the sum, which is p(a_i : i ∈ S) · G(p_a_i : i ∈ S). Suppose we are given access to the following: * A decisionmaking oracle that, given a subset of signal realizations, returns an optimal decision d^*; and* an oracle for evaluating the utility u(d,e) of decision d when nature's event E=e; and* access to independent samples of (e,a_1,…,a_n) from the prior distribution.Suppose that u(d,e) is bounded. Then we can approximate (_i∈ S A_i) to arbitrary (additive) accuracy with arbitrarily high probability in time polynomial in n and ∑_i log |Support(A_i)| and log |Support(E)|. Again, given S, we wish to approximate f(S), which is equal to (_i∈ S A_i) = _e,a_1,…,a_n u(d^*(a_i : i ∈ S), e) ,where d^*(a_i : i ∈ S) is the optimal decision conditioned on observing signals {a_i : i ∈ S}. Again, assuming that u(d,e) lies in a bounded range of size K, the same Hoeffding bound applies: By drawing m i.i.d. samples from the prior and then calling the oracles to obtain d^* and u, we can approximate this expectation to arbitrary additive error with arbitrarily high probability, for a suitable (polynomial-sized) m.This range of results gives some evidence that in general, if the decision problem has some sort of succinct representation outside of the oracle model, then we may still hope to compute . In fact, the problem we will construct for our negative result, Theorem <ref>, will fit the model of Proposition <ref>, where it is trivial to find and evaluate the optimal decision.§.§.§ Positive and negative results via reductions Positive results. As a corollary of the above reductions, we are able to reduce / to set function maximization, netting positive results especially in the case of substitutes. Given an instance of / with signals A_1,…,A_n, we construct a monotone increasing set function f: 2^{1,…,n}→ℝ via f(S) = (_i∈ S A_i). If signals are weak substitutes, thenis submodular on the signal lattice, and hence f is submodular (as well as monotone). We can therefore apply known algorithms for submodular maximization, in particular, <cit.>. Now, there is a subtlety: Under some of the models we propose, particularly the oracle model, we do not compute f exactly but instead can only guarantee arbitrarily high accuracy with arbitrarily high probability. However, it is well known (e.g. <cit.>) and proven (e.g. <cit.>), that these algorithms still give guarantees when we have a high-probability, high-accuracy guarantee on evaluations of f. The key point is that we can still evaluate, with arbitrary accuracy, the gradient or marginal contributions of each element[Recent work (in preparation) considers the question of how much accuracy in evaluating f is required, showing negative results when (roughly) accuracy is worse than 1/√(n); but under the oracle model we can evaluate f with error an arbitrary inverse polynomial with only an exponentially-small probability of error (simply via a Hoeffding bound), in which regime it is well-known that this problem does not arise.].Discussion of approximate oracles. One would like a robustness guarantee of the following sort: Even if we do not have an oracle that exactly optimizes the decision problem, suppose we do have an oracle returning, say, a decision whose expected utility is within a constant factor of optimal. Then give a good algorithm for / in the substitutes case, with an appropriately-decreased guarantee.Unfortunately, it seems that this kind of result is unlikely without deeper investigation and further work. To see the challenge, imagine that an adversary is allowed to design the oracle subject to a constraint of some approximation ratio. Then our problem essentially reduces to submodular maximization with noisy or approximate value oracles, where the noise may be adversarially chosen subject to this approximation constraint. Unfortunately, recent work on these kinds of problems have shown them to be difficult in general <cit.>. This seems like a very difficult barrier, but perhaps future work can leverage the structure of decision problems in some way to make progress in this direction. Negative results. We show that in general, / is as difficult as optimizing general monotone set functions subject to constraints. In fact, this holds even for an easy special case of / where all signals are independent uniformly-random bits and the decision problem is trivial to optimize (the solution is essentially to list all of the outcomes of signals you have observed).To do so, we give a reduction in the opposite direction: Given f, we construct a decision problem and prior distribution such that (_i∈ S A_i) = f(S). Hence, any algorithm for / gives an algorithm for optimizing f. In terms of input representation, while f may require exponential space to represent explicitly, our reduction is essentially as useful as one could hope for in this respect, creating a trivial wrapper around a value oracle for f.It seems that any reasonable algorithm one might propose for optimally selecting sets of signals should be able to handle such a tractable input. Hence, this is a strong negative result that, in general, optimization over signal sets is just as hard as over item sets. Each signal A_i will be a uniform independent bit, and the event E of nature will consist of the vector of realized a_is (a binary string of length n). Intuitively, the idea is that having observed A_i, regardless of whether its realization is 0 or 1, corresponds to having item i in the set S, while not having observed A_i (hence having a uniform belief over A_i) corresponds to i ∉S.The decision problem will look like a scoring rule, but it will be for predicting the mean of E rather than predicting a probability distribution over E. This is good news in terms of the representation size: predicting the mean of E requires only reporting a vector in [0,1]^n, while a general probability distribution over all outcomes of E has support size up to 2^n.Formally, it turns out that the scoring rule characterization applies equally well to constructing proper rules for predicting the mean of a random variable such as E. Specifically, given any convex function F: ℝ^n →ℝ, one can construct a scoring rule R: ℝ^n ×ℝ^n →ℝ. Using the notation R(r;q) = _e∼ q R(r,e), we have the key scoring rule property that R(r;q) is maximized at r = _e∼ q e, where it equals F(_e∼ q e). A quick proof: Given F, define R(r,e) = F(r) + ⟨ F'(r), e-r ⟩ where F'(r) is a subgradient at r. We have R(_e∼ q e; q) = F(_e∼ q e) as desired. Now, note that D_F(_e∼ q e;r) = F(_e∼ q) - R(r;q), where D_F is the Bregman divergence of F; since Bregman divergences are nonnegative, this proves that R(r;q) is maximized at R(r;q) = R(_e∼ q; q) = F(_e∼ q). Hence, the roadmap is as follows.* Construct a function F: [0,1]^n →ℝ. * Verify that F is convex.This implies that there is a decision problem (namely, predicting the mean of E) where the expected utility for predicting μ when one's true expectation is μ equals F(μ). * Verify that, for any subset S of {1,…,n} and for any set of realizations {a_i : i∈ S}, we have F([e | a_i : i ∈ S]) = f(S). * Note this implies that, for any S, we have (_i ∈ S A_i) = f(S). * Check that, given a value oracle for f, we can efficiently compute any quantities of interest (the posterior distribution on E, the posterior expectation of E, F([ e | a_i : i ∈ S]), (_i ∈ S A_i)).(1) The construction of F is recursive on the dimension n. It is ugly, being discontinuous at its boundary. However, drawing some pictures should convince the reader that F can be “smoothed” to a more reasonable, continuous convex function. For a base case of n=1, on [0,1] we let F(r) = f(∅) on the interior where r ∈ (0,1), and F(0) = F(1) = f({1}). (That is, f evaluated at the set consisting of the item.) Note that, because f is monotone increasing, i.e. f({1}) ≥ f(∅), F is convex. The discontinuity implies that, for the associated proper scoring rule for the mean R, we must have R(μ,e) = -∞ whenever μ lies at an endpoint and e is in the interior. But again, any smoothing of F to be continuous will remove this property.On [0,1]^n, we let F(r) = f(∅) for r in the interior of the hypercube. That is, F is constant on its interior. Furthermore, we have F(_e∼ p e) = f(∅), where p is the prior, hence () = f(∅).Now we define F on its boundary. Consider any face of the n-dimensional hypercube. Each face corresponds to a particular setting of some A_i, either to 0 or 1, by the coordinate whose value is constant on that face. For instance, A_i=1 corresponds to the face consisting of the set of r ∈ [0,1]^n where r's ith coordinate equals one. On both of the faces corresponding to A_i, F(r) is defined to be F(r) = F_{i}(r_-i), where r_-i is r with the ith coordinate removed, and the function F_{i} is defined recursively as follows. Consider the set function f_{i}: 2^{1,…,n}∖{i}→ℝ with f_{i}(S) = f(S ∪{i}). Then let F_{i} be the result of our construction applied to f_{i}. This completes the definition of F.Verbally, for each face of the hypercube, we have fixed some i to be in the set S passed to f, and considered the resulting submodular function f_{i} on the remainder of {1,…,n}. The value of F on the interior of that face will be f({i}), by the recursive construction. To picture F and convince ourselves that it is well-defined, consider the intersection of the faces corresponding to, say, A_i = 1 and A_j = 0. This is a lower-dimensional face consisting of all points on the hypercube whose ith bit equals 1 and jth bit equals 0. On the interior of this face (i.e. no other bits are equal to 0 or 1), F has value f({i,j}). And so on all the way “out” to the corners r of the hypercube, where r ∈{0,1}^n; at all of these, F(r) = f({1,…,n}). (2) We prove F is convex on the hypercube by induction on n, with the base case n=1 already observed above. For the inductive step, note again the key point: by monotonicity of f, if r is on the boundary of the hypercube and s is in the interior then F(r) ≥ F(s). Consider any two points r,s ∈ [0,1]^n, and break into cases. If both points lie in the interior, then because F is constant there, F(α r + (1-α)s) = α F(r) + (1-α)F(s) for any 0 ≤α≤ 1. If one point lies in the interior and one on the boundary, then any convex combination of the two lies in the interior. F is constant in the interior and weakly larger on the boundary, so the convexity inequality is satisfied. If the points lie in different faces, then again any convex combination lies in the interior. Finally, if the points lie in the same face, then F coincides with some F_{i} on r and s, and F_{i} is convex by inductive hypothesis. (3) We now verify that F, when applied to the expected value of E given the realizations of signals corresponding to S, is equal to f(S). Consider any set of realizations {a_i : i ∈ S} and let μ = [e | a_i : i ∈ S]. If S = ∅, then by construction μ = (0.5,…,0.5) and F(μ) = f(∅). Otherwise, μ is the vector where each entry i is equal to a_i if i ∈ S and 0.5 otherwise; this follows from the i.i.d. distribution of the A_i. Hence, μ lies in the interior of the (low-dimensional) face of the hypercube corresponding to the realizations {a_i : i ∈ S}, hence F(μ) = F_S(μ_-S) = f_S(∅) = f(S). Here the notation f_S is the function obtained from f by fixing S; F_S is the corresponding recursively constructed function on [0,1]^n-|S|; and μ_-S is μ obtained by removing all coordinates. (4) Use the notation μ_a_i : i ∈ S = _e [ e | a_i : i ∈ S]. Since step (3) holds for all realizations of a given set of signals, in particular (where R is the scoring rule corresponding to F):(_i ∈ S A_i)= _a_i : i ∈ Smax_d ∈ [0,1]^n_e [ R(d,e) | a_i : i ∈ S ]= _a_i : i ∈ Smax_d ∈ [0,1]^n R(d ; p_a_i : i ∈ S) definition of notation R(d;q)= _a_i : i ∈ S R(μ_a_i : i ∈ S ; p_a_i : i ∈ S)properness of R= _a_i : i ∈ S F(μ_a_i : i ∈ S)construction of F and R= _a_i : i ∈ S f(S)step (3) of proof= f(S) .(5) Given any set of signals S, one can immediately compute (S) = f(S) by a call to the value oracle for f. Given their realizations {a_i : i ∈ S}, the posterior distribution is simply uniform on those A_j with j ∉S (and of course has each A_i = a_i with probability one for i ∈ S). This induces the posterior distribution over E (which can thus be concisely represented, even though there are 2^n possible outcomes in general), as well as the posterior expectation of E. Evaluating F(r) at an arbitrary point r can be done quickly: Let S be the subset of coordinates on which r is equal to either 0 or 1; then F(r) = f(S), as r lies on the interior of a face corresponding to S. This requires just a single call to the oracle for f. Evaluating R(d,e) can thus be done in time polynomial in n as well; the only additional step required is picking a subgradient of F at each point d ∈ [0,1]^n, and there is only one choice at each point (as noted above, unless F is smoothed, this construction does require R(μ,e) = -∞ if μ lies on a face and e in the interior). Finally, the optimal decision for a given expectation of e is just that expectation.Explicit negative results. First, our reduction has implications even for the substitutes case. Monotone submodular maximization to a better factor than 1-1/e, even under a simple cardinality constraint, requires an exponential number of value-oracle calls <cit.> and, even given a concise explicit input, is NP-hard <cit.>. Because our reduction preserves marginal values, a submodular set function reduces via Theorem <ref> to weak substitutes. This implies that our (1-1/e) approximation cannot be improved without a stronger assumption than substitutability (or e.g. a polynomial-time algorithm for SAT). Even when signals are weak substitutes with a cardinality constraint, achieving a strictly better approximation than 1-1/e to / requires an exponential number of oracle queries. Without the substitutes/submodularity condition, it is well known that maximizing general monotone set functions is difficult given only a value oracle, although such negative results are not always explicit in the literature. For instance and for concreteness, one can even restrict to monotone supermodular functions and the simple problem of maximizing f subject to a cardinality constraint (select any set S of at most k elements, for some given 0 ≤ k ≤ n). In this case, there does not seem to be a negative result in the literature despite this hardness being well-known (see <cit.>), so we give an explicit one here to illustrate the challenge. For any algorithm for monotone supermodular maximization subject to a cardinality constraint, if the algorithm makes a subexponential number of value queries, then it cannot have a nonzero approximation ratio. We construct a simple family of supermodular functions. Let k be the cardinality constraint, i.e. maximum cardinality of any feasible set. Let f(S) = 0 if |S| ≤ k and otherwise let f(S) = |S| - k. Pick one special set S^* of size k, uniformly at random from such sets, and let f(S^*) = 0.5. Hence the optimal solution to the problem is to pick S^* with solution value 0.5.This function is supermodular: Given any element i, the marginal contribution of i to S is 0 if |S| < k-1, then either 0.5 or 0 for |S|=k-1, then either 0.5 or 1 for |S|=k, then 1 for |S| > k. Because this marginal contribution is increasing, f is supermodular. Now any algorithm making subexponentially (in k) many queries cannot, except with vanishing probability, guess which set of size k is S^*, so it will with probability tending to 1 select some other set of size ≤ k, which has solution value 0. §.§.§ Adaptive /We now define an adaptive version of the problem and show that substitutability implies positive results here as well. This problem has already been studied in very similar settings with a very similar approach by <cit.>. There is a significant difference in that these works aimed to maximize a set function over items in tandem with observations about those items. Our model is a significant generalization in that it considers arbitrary kinds of observations and an arbitrary decision problem. However, the goal of this section is not to claim originality or generality of solution, but only to demonstrate that this problem can also be viewed through the lens of substitutability.In Adaptive /, one is given a decision problem u and information structure on n signals A_1,…,A_n; and also a family ℱ of feasible subsets of signals S ⊆{1,…,n}. An algorithm for this problem is a policy that first selects a signal i_1 to inspect; then observes the realization of A_i_1; then depending on that realization, selects a signal i_2 to inspect, observing the realization of A_i_2, and so on. The algorithm must guarantee that the total set S = {i_1,i_2,…} of signals selected is in ℱ. After ceasing all inspections, the algorithm outputs some decision d. The goal is to maximize [ u(d,e) ] over the distribution of signals and E as well as any randomness in the algorithm.We note that the definition of / abstracted out the process of deciding an action d given the realizations of the signals, as we assumed that it was known how to optimize the utility function. Here, it seems cleaner to integrate the choice of the decision with the algorithm; but this is not the fundamental difference between the two settings. The fundamental difference is adaptivity: In Adaptive /, the algorithm may observe the outcome of the first selected signal before deciding which signal to select next, and so on.Here we need a somewhat stronger substitutes condition. A set of signals on a subsets latticeare pointwise weak substitutes if, for each A' ≼ A and B in , and for each of the outcomes a' ∈ A',a ∈ A in the support of the prior, letting Q',Q be the posterior distributions conditioned on A'=a' and A=a respectively, ^u,Q'(B) - ^u,Q'() ≥^u,Q(B) - ^u,Q() .In other words, the left side is the marginal value of B given the observation a', and the right is its marginal value given observation a.Verbally, signals are pointwise substitutes if having observed more information never increases the expected marginal value of a signal. The key difference is that for “regular” substitutes, the condition can be written as follows: _a'[ ^u,Q'(B) - ^u,Q'() ] ≥_a [ ^u,Q(B) - ^u,Q() ].In other words, signals are substitutes if expecting to observe more information never increases the expected marginal value; they are pointwise substitutes if observing more information never increases expected marginal value.Our goal here is to illustrate that techniques from submodular maximization extend naturally. We focus on the simple cardinality constraint case and show that the greedy algorithm for monotone submodular maximization <cit.> also gives good guarantees in this adaptive setting, when we have a strong substitutes condition.When signals are pointwise weak substitutes, Algorithm <ref> (Adaptive Greedy) obtains a (1-1/e)-approximation algorithm for Adaptive / under cardinality constraints in the oracle model. The usual proof essentially goes through. We utilize notation from Algorithm <ref> (adaptive greedy). The expected performance of S_t, the set constructed at time t, is ^u,P(S_t) (we use this notation as shorthand for evaluation at the join of all signals in S_t).Let V^* be the expected performance of the optimal policy. We will upper-bound V^* for each t by the performance of a hypothetical algorithm that first selects S_t, then selects all k of the signals selected by the optimal policy. For a fixed S_t, such a hypothetical algorithm can obtain, in expectation, at most the performance of S_t plus k times the maximum expected marginal contribution of any signal to S_t. This follows from pointwise substitutes: as the algorithm acquires more information, the expected marginal contribution of a signal does not increase. So, taking the expectation over this condition, which holds for each realization of S_t: V^* ≤^u,P(S_t) + k (^u,P(S_t+1) - ^u,P(S_t)) . This is exactly the inequality that produces the approximation ratio in submodular set function case. The inequality implies by induction that ^u,P(S_k) ≥(1 - (1-1/k)^k )V^*, and this approximation ratio is always better than 1-1/e.§ STRUCTURE AND DESIGN In this section, we give some initial investigations into the structure of substitutes and complements. We focus on two kinds of questions: Understanding what types of information structures may be generally substitutes or complements for a broad class of decision problems; and understanding how to design a proper scoring rule or other decision problem so as to impose substitutability on a given signal structure.In terms of contributions of this paper to these problems and starting points for future work, it is also worth recalling the relatively diverse set of equivalent definitions of informational S&C derived in Section <ref>. There, we showed that substitutes can be defined in terms of submodularity, generalized entropies, or divergences. §.§ Universal substitutes and complements An ideal starting point would be characterizing information structures that are always substitutable or complementary; we term these “universal”. We note that <cit.> investigated a two-signal variant of this problem, with a definition essentially corresponding to weak substitutes on those two signals, with some results of the same flavor.Intuitively, there are “trivial” cases that satisfy this universality criterion, at least for weak S&C. An example of trivial weak substitutes is the case where A_1 = A_2 = ⋯ = A_n. Here, after observing any one signal, all of the others do not change the posterior belief at all. An example of trivial weak complements is the case where each A_i is an independent uniformly random bit and E = A_1 ⊕ A_2 ⊕⋯⊕ A_n, the XOR of all the bits. Here, any subset of n-1 signals does not change the posterior belief at all compared to the prior, but the final signal completely determines E. Given an information structure E,A_1,…,A_n with prior P, we term A_1,…,A_n universal weak substitutes if they are weak substitutes for every decision problem. Universal moderate/strong substitutes and weak/moderate/strong complements are defined analogously. We term the signals trivial substitutes if for every realization of a_1,…,a_n in the prior's support, p_a_i = p_a_1,…,a_n for all i and trivial complements if for all realizations a_1,…,a_n in the support, p_{a_j : j ≠ i} = p for all i. We term them somewhat trivial if the prior is a mixture distribution that is equal to a trivial structure with some probability, and some other arbitrary other structure with the remaining probability.If A_1,…,A_n are universal weak substitutes, then they are somewhat trivial. Furthermore, their “trivial” component is more informative than the nontrivial component, in the following sense. Let X_i ⊆Δ_E be the convex hull of {p_a_i : a_i ∈ A_i}, and let Y ⊆Δ_E be the convex hull of {p_a_1,…,a_n : a_1 ∈ A_1,…,a_n ∈ A_n}. If A_1,…,A_n are universal substitutes, then X_i = Y for all i. Consider any information structure that does have X_i = Y for all i. We give a decision problem for which it sometimes satisfies strictly increasing marginal value. To do so, by Corollary <ref>, it is enough to construct a convex function G with appropriate structure, for which we then know a decision problem (namely, a scoring rule, Fact <ref>) exists. The idea is pictured in Figure <ref>.Let X be the convex hull of {p_a_i : i=1,…,n; a_i ∈ A_i}, that is, of all possible posterior beliefs conditioned on one signal. Let G(q) be zero for q in the convex hull of that set of beliefs (note that the prior p must be in the convex hull) and G(q) be increasing outside of this convex hull. Then for any one signal A_i, we must have (A_i) = () = 0 as (A_i) = _a_i G(p_a_i). But by assumption, there exist posterior beliefs for multiple signals that fall outside this convex hull. This follows because each p_a_i = _a_j p_a_ia_j for any j ≠ i, so unless p_a_ia_j = p_a_i for all j, there are cases where some posterior belief falls outside the convex hull mentioned. (And if p_a_ia_j = p_a_i always, then repeat the argument for triples of signals a_i,a_j,a_k, and so on; by nontriviality, the argument will succeed at some point.)Now for these posterior beliefs falling outside the convex hull, they occur with positive probability and have a positive value of G, hence the marginal value of additional signals because strictly positive at some point, while the marginal value of the first signal was 0.To see that this implies a mixture containing trivial substitutes, note that the convex hulls X_i = X_j for all i,j, so in particular the corners of these convex hulls must be points where p_a_i is equal, in the case where A_i = a_i, to every posterior belief conditioned on any number of signals. All universal moderate substitutes are trivial. (Hence, the same holds for universal strong substitutes.) Any universal moderate substitutes must be universal weak substitutes as well; so we know via the proof of Proposition <ref> that any candidates must have a mixture with trivial substitutes. However, we can let A' be a signal determining whether that component of the mixture has occurred. Then one implication of moderate substitutes is that, conditioned on A', all signals are weak substitutes. Therefore, universal substitutes implies that even conditioned on A', the prior is a mixture containing a trivial component. Repeating the argument gives that the entire prior must consist only of trivial components. Consider a binary event E ∈{0,1} prior probability p = [E=1], and signals A_1,A_2 with posterior probabilities p_a_1 = [E=1|A_1=a_1], and so forth. Suppose that:* max_a_1p-p_a_1≤ r and max_a_2p-p_a_2≤ r; and * min_a_1,a_2p - p_a_1,a_2≥ 2r.Then {A_1,A_2} are universal moderate complements. The idea is that posterior beliefs on multiple signals usually tend to lie “farther” from the prior than those on single signals, because at least in some cases, more information leads to more certain (hence extreme) beliefs. Convex functions already tend to give larger marginal value to more extreme cases. Hence, convexity of the decision problem encourages complementarity; and it cannot discourage complementarity very much because a convex function can only be “so flat”. This is pictured in Figure <ref>. By moving the posteriors p_a_1,a_2 all very far from the prior, compared to the posteriors p_a_1,p_a_2, we get increasing marginal returns.Pick any convex G on [0,1] and let all probability distributions on E be represented as scalars q ∈ [0,1]. This includes the prior p and posteriors such as p_a_1. It suffices to show that G(p_a_1a_2) + G() ≥ G(p_a_1) +G(p_a_2). Let p^* be the minimizer of G on [0,1]. Then G(p_a_1a_2) ≥ G(p_a_1) + G'(p_a_1) (p_a_1a_2 - p_a_1). Now we claim that G'(p_a_1)(p_a_1a_2 - p_a_1) ≥ G(p_a_1) - G(p). This implies G(p_a_1a_2) + G(p) ≥ 2G(p_a_1), and after taking the analogous case reversing a_1 and a_2 and expectations, we get that the desired inequality must hold. To prove the claim, consider the case p_a_1a_2 > p_a_1. Then (p_a_1a_2 - p_a_1) ≥ r, and we want to show G'(p_a_1) ≥G(p_a_1)-G(p)/r. This holds by definition of a subgradient as the right side is at most the slope of a line connecting p and p_a_1. The analogous argument holds for the case p_a_1a_2 < p_a_1. An example of universal complements are the signals A_1,A_2 each independently equal to 1 with probability q ∈ [0.25,0.75] and E = A_1 ⊕ A_2 (the XOR operation).We note that these may be universal complements for larger ranges of q as well, and a very interesting question for future work is to characterize the set of universal complements. This seems to require more work particularly for E with a larger number of outcomes. For binary E which are equal to a deterministic function of A_1,…,A_n, it seems possible to relate complementarity to the sensitivity of a function in Boolean analysis. §.§ Identifying complementsOur main result here is to identify the following very broad class of complements. Independent signals are strong complements in any decision problem where G has a jointly convex Bregman divergence D_G(p,q). D_G(p,q) is termed jointly convex if it is a convex function on the domain Δ_E ×Δ_E (as opposed to the case where it is convex in each argument separately, for instance). Assume signals are independent; consider any A,B on the subsets signal lattice (incomparable to each other) and any A' on the continuous lattice with A' ≼ A. By Lemma <ref>, showing strong substitutes is equivalent to showing that _a',b D_G(p_a'b,p_a') ≤_a,b D_G(p_ab,p_a) .Use independence and rewrite to this convenient form: _b _a' D_G(p_a'b,p_a') ≤_b_a'_a|a' D_G(p_ab,p_a) .Note that if b were not independent of a,a', then the inner expectations would require conditioning the distributions of a and a' on the outcome of b. Now, it suffices to prove the following for all b,a': D_G(p_a'b,p_a') ≤_a|a' D_G(p_ab,p_a) .Now, since D_G is jointly convex, Jensen's inequality will imply this fact if we can just show that p_a'b = _a|a' p_ab and p_a' = _a|a' p_a. We prove the first equality; the second is exactly analogous but easier.p(e | a',b)= p(e,a',b)/p(a',b)= ∑_a p(e,a,a',b)/p(a',b)= ∑_a p(e|a,a',b) p(a,a',b)/p(a',b)= ∑_a p(e|a,b) p(a,a')p(b)/p(a')p(b)= ∑_a p(e|a,b) p(a|a') .We used Bayes rule and the law of total probability, then eventually used the fact that p(e|a,a',b) = p(e|a,b) (because a' and e are independent conditioned on a) and that a and a' are independent of b.A corollary is that independent signals are complements for the log scoring rule and the quadratic scoring rule, as their divergences (KL-divergence and L_2 distance squared) are jointly convex. On the other hand, some form of this restriction on the decision problem is needed:If D_G is not convex in its second argument, then independent signals are not necessarily complements. We consider a binary event E and G(q) where q ∈ [0,1] is a probability that E = 1. A counterexample is to let G(q) = 0 for q ≤ 0.75 and G(q) = q-0.75 for larger q. Consider any decision problem associated with this G (for instance, predicting against a proper scoring rule derived from G). The two signals A and B are independent uniform random bits, and E is equal to the binary OR of the bits. In this case, one can check that (A) = (B) = 1/8, but () = 0 and (AB) = 3/16. Thus in particular (A) + (B) ≥(AB) + (AB), so they are not complements. Note that G may be modified to be strictly convex while preserving the counterexample. Also note that the sharp “kink” in the graph of G at q=0.75 forces D_G to be non-convex in its second argument (one can take the first argument to be the prior on E, q=0.75, choosing the subgradient G'_0.75(q) = q-0.75). §.§ Identifying substitutes It was previously known <cit.> for prediction markets under the log scoring rule that conditionally independent signals induced players to reveal their information as early as possible. The result in a different guise was shown in an algorithmic context, namely, that for conditionally independent signals, . Both of these turn out to correspond to the fact that signals are always substitutable in the context of the log scoring rule: For the log scoring rule, signals that are conditionally independent upon the event E are strong substitutes. Suppose signals A_1,…,A_n are conditionally independent on E. Let A,B on the subsets signal lattice and A' on the continuous lattice with AB ≼ A' ≼ A. We are to show that (A'B) - (A') ≥(AB) - (A). Recall that, using the entropy characterization, (A) = -H(E|A) = H(A) - H(E,A) by the properties of the entropy function.(A'B) - (A')= H(A', B) - H(E, A', B) - H(A') + H(E, A') = H(B | E, A') - H(B | A').Similarly, (AB) - (A) = H(B | E,A) - H(B | A). Now because A' ≽ AB, by conditional independence, H(B|E,A') = H(B|E,A). Meanwhile, by conditional independence, H(B|A') ≤ H(B|A), which completes the proof. However, this fact is apparently quite special to the log scoring rule. Conditionally independent signals are not necessarily substitutes for the quadratic scoring rule (which has a jointly convex Bregman divergence). The quadratic scoring rule has expected score function G(p) = p_2^2, and therefore S(p,e) = 2p(e) - p_2^2. Let E be binary with p(E=1) = r > 0.5 and let each of A,B be i.i.d. conditioned on E, each equal to E with probability s > 0.5 and equal to the bit-flip of E otherwise.One can check that for cases where r is large compared to s, for instance r = 0.9 and s = 0.8, we have (A) + (B) ≤(AB) + (AB) (recall that (A) = _a G(p_a)). Hence substitutability is strongly violated. For instance, an agent observing A would prefer to report second in a prediction market after an agent observing B, even when both agents are constrained to report truthfully.Intuitively, what happens in this information structure is that neither the realization of A nor of B on its own is enough to change the rather strong prior on E. However, sometimes, observing both A and B (if they are both 0) can cause a large change in beliefs about E, which means that observing both can sometimes be very valuable.§.§ Designing to create substitutability We now briefly consider the question of designing a decision problem or scoring rule so as to enforce substitutability of information. In a game-theoretic setting such as prediction markets, one would like to design mechanisms where information is aggregated quickly; as we have seen, this is essentially equivalent to making information more substitutable.In an algorithmic setting, the decision problem with which one is faced may be difficult to optimize due to non-substitutability of information. One would like to construct a “surrogate” decision problem for which the information at hand is substitutable, then use algorithms for (Adaptive) / to approximately maximize that surrogate, with some guarantee for the original problem. This is the approach of submodular surrogates in the literature <cit.>. We do not directly consider this problem in this paper, but we hope that these techniques may yield insights or tools that are useful in these problems for future work.An important conclusion of this paper is that substitutability, or lack thereof, tends to arise from a combination of two factors:* The distances between beliefs due to the information structure.If updating on additional information tends to spread beliefs farther apart, then that information tends to be complementary.If it tends to make smaller changes in beliefs, it tends to be substitutable. * The curvature of the expected utility function G: Δ_E → associated to the decision problem.Highly-curved regions correspond to high marginal value of information to beliefs in those regions.We illustrate some of this intuition, with an eye toward design of substitute-encouraging G, in Figure <ref>. By designing a G that is highly curved close to the prior, then gradually less curved farther from the prior, one increases marginal value of information near the prior and decreases it (relatively) further away; this increases substitutability. A formalization of the problem. Given an information structure consisting of a prior E and signals A_1,…,A_n, (when) can we construct a decision problem u such that the signals are substitutes? Noting that a trivial decision problem, e.g. with one action, technically satisfies substitutes, we will seek decision problems that satisfy a “nontriviality property”. Substitutes are somewhat strict if the marginal value is always diminishing on the corresponding lattice, and sometimes strictly diminishing. Analogously, complements are somewhat strict if marginal value is always increasing and sometimes strictly increasing.Unfortunately, our results on universal S&C in Section <ref> essentially imply that this is not always achievable. For some information structures, it not possible to design a decision problem giving rise to weak, somewhat strict substitutes; the same holds for complements. However, if the information structure does not contain a mixture of trivial substitutes, then it is possible to design a decision problem under which signals are weak, somewhat strict complements. We showed in Corollary <ref> that nontrivial universal complements exist; for these structures, for every decision problem, they are weak complements, which implies that they cannot be somewhat strict substitutes.Meanwhile, in Proposition <ref>, we showed that unless the information structure contained a mixture with trivial substitutes, one can construct a decision problem satisfying weakly and sometimes strictly increasing marginal value.A crucial direction for future work is to better characterize for what information structures it is always possible to design for substitutability.. § DISCUSSION, CONCLUSION, AND FUTURE WORK§.§ Contributions and discussionThis paper makes two types of contributions. One type is to propose definitions of informational substitutes and complements and provide evidence that these definitions are natural, useful, and tractable. The other type is to prove concrete results for game-theoretic and algorithmic problems. These two contributions are interrelated: The concrete results rely on the definitions and characterizations of S&C, while evidence in favor of the definitions comes from their utility in proving the concrete results.We begin with a summary of the contributions in terms of concrete results, then summarize the evidence in favor of our proposed definitions of informational S&C. We then discuss future work in a variety of directions. Contributions independent of S&C The first application in this paper was to identify essentially necessary and sufficient conditions for the two types of prediction market equilibria that have been most studied: The “good” case where all traders rush to reveal information as soon as possible, and the “bad” case where all traders delay all revelation as long as possible. This result broadly generalizes previously-known special cases <cit.>. These conditions corresponded respectively to our definitions of informational substitutes and complements. Our other results regarding these definitions have implications and applications for the market setting. We gave some tools and approaches for identifying and designing substitutability, which (we showed) corresponds to the main goal of a market designer: encouraging immediate information revelation.We also gave some additional game-theoretic applications in other settings involving strategic information revelation and aggregation. We hope that future work can explore more connections.The second main application was to the algorithmic problem of information acquisition, which we formalized as the / problem. This is a very natural and general problem, capturing any sort of information acquisition scenario. We showed that substitutes imply efficient approximation algorithms via submodular maximization, offering a unifying lens or perspective on a variety of literature utilizing this approach. We also saw matching hardness in general when substitutability is not present. These results give hope that substitutable structure may be leveraged in future algorithmic investigations of related problems. Evidence for informational S&C Informational substitutes require much more background and work to define than their counterparts for goods, as discussed in Section <ref>. This leads to many ways in which the definition can “get it wrong”. However, there are several reasons to believe that the definitions proposed in this paper are substantially on the right track. The three goals we address are showing that they are natural, useful, and tractable. The evidence for these is summarized below; however, there is room for further investigation and discussion on the subtleties of the proposed definitions. As additional future applications are investigated, it may be found that tweaks in the definition improve it along these axes. Natural. The following evidence suggests that the definitions of informational S&C presented in this paper are natural.* They are defined in terms of sub- and super-modular lattice functions, sharing this property with widely accepted definitions of S&C for items.The weak, moderate, and strong versions of the definitions correspond to lattices have natural interpretations both game-theoretically – strategies that are binary, deterministic, or randomized – and algorithmically, optimizing over subsets, deterministic “poolings”, or randomized “garblings”. * They can be alternatively characterized information-theoretically, by diminishing (increasing) amount of information revealed by a signal. * They can be alternatively characterized geometrically, by the distance a belief is moved by a posterior update on one signal given the other. * S&C respectively characterize “all-rush” and “all-delay” equilibria in prediction markets.Hence, our definitions (or very close variants) seem unavoidable when studying equilibria of these basic models of strategic play. * The algorithmic complexity mirrors the now-familiar story in the case of items: approximately optimizing over substitutes can be done efficiently (at least in the case of weak substitutes), while for the general case or complements the problem is difficult.Useful. The usefulness of the definitions, so far, is reflected by the results summarized above. In particular, they allowed resolving an open problem on strategic aggregation in prediction markets that was previously solved only for particular cases. Tractable. Our definition of informational S&C refers to a very general setting of decision problems and information structures. It is not initially clear that a definition that broadly captures S&C can also be amenable to analysis or intuition. This particularly presents a challenge for our definitions because they depend on both the decision problem and the information structure, each of which can be a very complicated object.We presented a convexity-based approach, namely, studying the convex expected-utility function G, and showed that it both captures geometric intuitions for understanding S&C and gives useful analytic tools. One of these is prediction and the theory of proper scoring rules, which gives a way to construct a decision problem from any such convex G. We used these tools to give some example broad classes of informational S&C as well as positive and negative results on designing decision problems to encourage substitutability or complementarity. We also gave intuitive definitions of informational S&C from the perspectives of information theory (generalized entropies) and geometry (divergences). These found some formal applications already in this paper, e.g. a convexity condition on the divergence definition of substitutes implies that all independent signals are substitutes. §.§ Future work: game theoryAn immediate direction is to extend our results to broader models of financial markets. We believe that analogous results are likely to hold. It would also be ideal, albeit esoteric, to understand the distinction between when Bayes-Nash equilibria and perfect Bayesian equilibria exist or are essentially unique. In a Bayes-Nash equilibrium that is not perfect Bayesian, some trader essentially makes a “threat” that is “not credible”. Can such scenarios exist when signals are substitutes or complements?Many broader questions about S&C have a direct implication on markets via our results, and we outline some of these directions below.For more general Bayesian games, the implications of our results are not yet clear. With multiple players, for instance, it is no longer true that more information always helps. It may be most tractable to look at special cases such as auctions or signaling games, although those seem to have a significant component of strategic substitutes as well. One natural direction is common-value auctions, where in one case <cit.> informational substitutes have been explicitly used. §.§ Future work: algorithmsThere are many potential algorithmic questions raised by these definitions. One is to consider existing problems, such as algorithmic Bayesian persuasion <cit.> or signalling in auctions, through the lens of substitutability. However, such problems are already significantly more complex because of the interaction between the players, which is largely absent or abstracted out so far in this work. We hope that further investigation will uncover the connections to such settings. Meanwhile, one might ask whether, for instance, variants of / on the discrete or continuous lattice are well-motivated in any settings and, if so, whether they are tractable.More specifically and technically, one might ask whether the oracle model presented in this paper the best way to represent a decision problem, or if there is some other natural alternative beyond those we presented. Perhaps it is possible to obtain positive results with weaker or substantially different assumptions on the oracles.Another group of algorithmic questions relates to better understanding the definitions themselves. One concrete question is the following: Given a decision problem, perhaps in the oracle model (or some other well-justified model), determine whether signals are substitutes, complements, or neither. Or more strongly, given a decision problem and two signals A,B, find min_Q ≼ A(QB) - (Q) .An algorithm for this problem and multi-signal generalizations would help identify substitutes or complements, and would more generally solve e.g. the problem of how to trade in general prediction markets where signals are neither substitutes nor complements, but something in between. Many more possible algorithmic questions are likely to arise as investigation continues. §.§ Future work: structure of S&CA straightforward direction for future work is to identify further classes of substitutes and complements, including natural classes of decision problems and information structures that together produce S&C. This question also bears directly on the design of prediction markets.The second straightforward direction is how to “design for substitutability”. We believe that our results give a good start in this direction, but many questions remain. For instance, can we characterize all universal substitutes and complements? (This characterizes cases where we cannot design for substitutability or complementarity.) And, in cases where we can make signals strict substitutes, can we design some natural and computationally efficient method for doing so? We hope that the geometric intuitions in this paper provide a starting point for addressing such questions.Finally, one could be interested in exploring variants of the definitions taking the same approach but with some piece of the puzzle substantially altered. Concretely, one could imagine a very risk-averse agent with an adversarial view of nature and some more combinatorial representation of information. Can one formulate analogous definitions in cases like these? plainnat § OTHER RELATED WORK We first survey related work on substitutes and complements for information (and in general). Then, we discuss work relating to information aggregation in markets and other game-theoretic settings. Finally, we describe algorithmic work on information acquisition (particularly in “submodular” settings).§.§.§ Substitutes and complementsThe notion that pieces of information may exhibit substitutable or complementary features is certainly not a new intuition; but up until this work, it seems to have remained mainly an intuition. There seem to be few attempts at formalizing a general definition or even special cases. The only work we know of in this direction is <cit.>, which inspires our approach but also has significant drawbacks and limitations. We extensively discuss <cit.> in in Section <ref>, where we contrast it with our definitions.Two works in (algorithmic) game theory touching on informational S&C are <cit.> and <cit.>; however, these do not propose general definitions. They are discussed below.Somewhat related is the game-theoretic notion of strategic substitutes and complements <cit.>. Roughly, these concepts refer to cases where a change in action by one player in a game results in a response from another that is similar to the first player's (in the case of complements) or offsetting (in the case of substitutes). This notion seems relatively unrelated to our definitions of informational S&C. For one, our definitions focus on the case of a single decisionmaker or single optimization problem. Also, strategic S&C can be defined in complete-information games, where there are no signals or information of any kind. However, perhaps future work can discover classes of games in which the notions are more closely related. Valuations for items. In contrast to the lack of literature on informational S&C, in the case of valuation functions for items, substitutability and complementarity have been put on firm formal foundations, with strong connections between substitutes and existence of equilibria in markets for goods or matching markets <cit.>.In computer science, the literature on optimization has produced strong positive results leveraging substitutable structure. The main example of this is submodularity, which has been connected to computational tractability throughout theoretical computer science and machine learning <cit.>. Submodularity, in addition to having nice algorithmic properties, is also recognized as a natural model of substitutes in (algorithmic) game theory <cit.>.This paper draws parallels to such research because our definition of informational substitutes turns out to correspond formally to submodular valuation functions. We show market equilibrium results of a similar flavor, but for “information markets”; and we also show algorithmic results of a similar flavor for the problem we call /, which is the problem of selecting an optimal set of signals subject to constraints.However, we emphasize that informational S&C pose challenges that do not arise in the item setting:* Items are modeled as having an a priori innate value.Information is not; its value must arise from context. * Items are modeled as being atomic or indivisible, with no inner structure.In contrast, information, modeled as e.g. random variables, is defined by inner structure: the probability distributions from which it is drawn. * The relationship between items is completely determined by the valuation function in the context of interest.Concretely, when modeling a set function f:2^{1,…,n}→, it is usually not the case in the model that items 3, 7, and k have a special relationship that has an impact on allowable forms of f.In contrast, in a value-of-information setting, the value of observing a triple of signals cannot be completely arbitrary; it must depend somehow on correlations between these signals.We give an in-depth description of how our definitions overcome these challenges in Section <ref>.§.§.§ Information in marketsThe “efficient markets hypothesis” (EMH) refers to a large set of informal conjectures about how quickly information is revealed and incorporated into the prices of financial instruments in markets. For our purposes, a “financial instrument” may be formalized as a security with an uncertain value, which will be revealed after the close of the market; each share purchased of that security may then be redeemed for a payout equal to this revealed true value of the security. Concretely, one can picture a binary security that has value 1 if a certain event occurs and 0 otherwise.<cit.> discussed formalizations of the EMH with varying levels of strength. <cit.> defined a formal model of financial traders in markets, involving both informed traders and “noise” traders who are uninformed and essentially trade randomly; this is the current most common model of such trading in the economics literature. However, formal progress on this question was very slow until <cit.> showed that, in equilibrium of this model, information is always eventually aggregated under a certain condition on securities. Formally, this means that the price of a security converges to its ex post expected value conditioned on the information held by all traders. <cit.> considered both finite-round and infinite-round markets (with and without discounting), and considered prediction markets (described below) as well as Kyle's model. One subtlety that may be worth pointing out is that, in an infinite-round market without discounting, it is not known that a Bayes-Nash equilibrium always exists, while this is known for the other cases. <cit.> showed in all cases that aggregation occurs in any equilibrium, under his “separability” condition on securities. The condition essentially ensures that, if information is not yet aggregated, then some participant has information that can be used to make a “useful” trade, i.e. one that makes money and (therefore) intuitively makes “progress” toward aggregation.This showed that information is eventually aggregated; but how is it aggregated? It would be ideal if traders rush to reveal their information, but very bad if they “delay” as long as possible.Such questions are difficult to address in the economic model of financial markets of <cit.>. Research on the dynamics of strategic trading has made some progress in the model of prediction markets. These are simplified financial markets in which there are no uninformed “noise” traders and in which participants generally interact one-at-a-time with a centralized market maker, who sets prices via a transparent mechanism and subsidizes the market. Specifically, research on strategic play focuses, as does this paper, on the market scoring rule design of prediction markets <cit.>, which are based on proper scoring rules (see e.g. <cit.>). However, we note that market scoring rule markets are equivalent, in a strategic sense, to more traditional-looking “cost function” based prediction markets <cit.>.The following was known about equilibrium in such markets prior to this work, in addition to <cit.>. <cit.> studied the log scoring rule and a particular type of information structure among the traders, namely, that each trader's “signal” (information) was distributed independently of all others' conditional on the true value of the security. (For a simple example of such a structure, suppose that the true value of the security is distributed randomly in some way; then each trader observes this true value plus independent noise.) In this conditionally-independent case, <cit.> showed that the “ideal” outcome does indeed occur in equilibrium: Traders all rush to reveal their information as early as possible.Subsequently, <cit.> also studied the log scoring rule but considered other signal structures, particularly independent signals (that is, unconditionally independent).For a simple example, suppose each trader observes an i.i.d. random variable, and the true value of the security equals the sum of all the traders' observations. <cit.> showed that, in this case, the “ideal” outcome does not occur. However, when assuming discounting and an infinite number of trading rounds, <cit.> showed that information is “eventually” aggregated. This result was generalized to any scoring rule (not just log) by <cit.>. <cit.> and <cit.> were combined and extended in <cit.>.Then, <cit.> revisited the log scoring rule in finite-round markets and considered the information structure where all traders signals are unconditionally independent. In this case, <cit.> showed that the “worst possible” outcome occurs in equilibrium: Traders all delay as long as possible before making any trades based on their information. This casts doubt on the efficient markets hypothesis and suggests, taken in tandem with <cit.>, that structure of information is crucial to determining strategic behavior.§.§.§ Other game-theoretic settings.In almost any Bayesian extensive-form game, the question of information revelation is relevant. While we hope that future work may expand the set of topics to which informational substitutes and complements may apply, in this section, we will focus on the most closely related works, those that directly touch on S&C, or those for which we show results in Section <ref>.The model of prediction markets is in some sense the simplest model of strategic information revelation in dynamic settings. Thus, it is natural that settings such as crowdsourcing contests are closely related. One such model of crowdsourcing and machine-learning contests appears in <cit.>. In that framework, participants iteratively provide data sets or propose updates to a central machine-learning hypothesis, being rewarded for the improvement they make to performance on a test set of data. A prediction market can be seen as a special case. Those works did not address strategic equilibria of the mechanisms they proposed.Another related setting is the model of question-and-answer forums in <cit.>. That paper introduced a model where an asker has some value function for “pieces of information”, which are not modeled directly. Participants can strategically choose when to reveal information. <cit.> identified “substitutes” and “complements” cases in which participants rush (respectively, delay) to provide answers. However, <cit.> did not provide any endogenous model for asker utility or information; the information was modeled almost as discrete items without structure. Hence, it was not clear from that work under what circumstances (if any) pieces of information would satisfy their substitutes and complements conditions.In Section <ref>, we describe implications of our work for results in the above two settings.More broadly, there are large literatures dealing with signalling in games <cit.>, or the more recent Bayesian persuasion literature <cit.>. While the models and questions in this area are related, to our knowledge there is no immediate connection. It may be that future work uncovers connections of informational S&C to this field, but the literature on persuasion and signalling in games does not seem to have developed notions of informational substitutes nor tools for addressing the applications considered in this paper.Another significant line of work has considered signalling in auctions, e.g. <cit.>. The only paper in this literature that we know to explicitly formalize a notion of substitutable information is <cit.>, which considers a common-value auction with two bidders, one informed and one uninformed, with two signals, each a real-valued random variable with some positive affiliation with the item's value. The authors define a notion of substitutes specific to their context and show that it implies intuitive properties for this asymmetric-information auction setting. In future work, it would be interesting to see if there is a formal connection of our definitions to their setting.§.§.§ Algorithms for information acquisition.The value of information to a decision problem was formally introduced by <cit.> and is also closely related to the classic problem in statistics of Bayesian experimental design <cit.>. Given this perspective, it is natural to consider the problem of acquiring information under constraints. This problem has historically been investigated from many different angles, e.g. <cit.>. It is known to be very computationally difficult in some general settings <cit.>.A successful recent trend in this area is to leverage submodular structure to apply efficient approximation algorithms. For instance, an approximation ratio of (1-1/e) is obtained by the greedy algorithm for maximizing a monotone submodular function subject to a cardinality constraint. This algorithm or related submodular maximization algorithms were utilized by <cit.>, and a variety of literature since; see <cit.> for a survey. In cases where the information is acquired not in a batch but adaptively over time, based on the information observed so far, the problem (and/or solution) is known as adaptive submodularity <cit.>. § DEFINING S&C: INTUITION, CHALLENGES, AND HISTORICAL CONTEXT In this section, we describe the intuition, justification, and historical context behind our proposed definitions of informational substitutes and complements (S&C). The formal definitions are presented more concisely starting in Section <ref>, to which the reader may skip if uninterested in background. We focus on the substitutes case; the complements case, where not mentioned, is analogous.The only prior attempt at definitions of S&C of which we are aware is <cit.>, which will be introduced shortly. The setting and approach in that paper are very appealing, and they inspire our approach in this paper. However, there are also key drawbacks that motivate us to diverge from their approach in several ways. We will describe in context the drawbacks and our approach to overcoming them.§.§.§ Defining the value of informationDefining informational S&C turns out to be significantly more work than defining substitutes and complements for valuation functions over items, starting from the very beginning: How does value arise in the first place? It is generally accepted to model a valuation function over a set U of goods as some f: 2^U →ℝ, without justifying how f(S) arises (perhaps the items are yummy, shiny, or have other desirable qualities). However, outside of curiosity, it seems that information's innate value is more questionable; and furthermore, should depend on its probabilistic structure. For instance, two signals may be independent or may be highly correlated. How does this relate to their value?A solution arises from the observation that information's value is often determined by the use to which the information may be put. As in <cit.>, for us the value of information arises from its utility in the context of a decision problem. We consider a Bayesian model of information in which there is a prior distribution on the event E of interest and on the possible pieces of information, called signals.In a decision problem, the agent observes some signals and then makes a decision d ∈, after which the outcome E=e is revealed and the agent's utility is u(d,e). Thus (following <cit.>), we define a value function ^u,P on signals, for a given decision problem u and prior P, by the expected utility of the agent given that she first observes A, then takes the optimal action based on that information. Formally, ^u,P(A) = _a∼ A[ max_d∈_e∼ E[ u(d,e) | A=a] ] .Thus, one can compare, for instance, the value ^u,P(A_1) for observing signal A_1 alone versus the value ^u,P(A_2) for observing signal A_2 alone. Note that the marginal value for observing signal A_2, given that the agent will already observe signal A_1, is ^u,P(A_2A_1) - ^u,P(A_1), where the notation A_2A_1 means to observe both signals (this notation will be explained shortly).§.§.§ The approach of Börgers et al.<cit.> proposes the following definition: Given an event E, two signals A_1 and A_2 are substitutes if for every decision problem (and associated value function ^u,P), ^u,P(A_1) + ^u,P(A_2) ≥^u,P(A_1A_2) + ^u,P() ,where A_1A_2 denotes observing both signals, whiledenotes not observing any signal and deciding based on the prior.This definition has two properties that might seem attractive, but turn out to be fatal in many cases of interest: (a) it does not depend on the particular decision problem, but only on how A_1, A_2, and E are correlated; (b) it depends only on the values ^u,P(A_1), ^u,P(A_2), of both, and of neither, and does not depend on any internal structure of A_1 and A_2. We explain why these properties are problematic and how our definition will differ. a. Lack of dependence on the decision problem. The problem here is that in a majority of cases, two signals can turn out to be either substitutes or complements depending on the decision problem at hand. For example, whether two weather observations are substitutes or complements depends on what decision is being made. Temperature and dew point might be considered complements when deciding whether to bring one's umbrella.[Accepting the proposition that knowledge of both gives a much better prediction of rain than knowledge of either alone.] But when deciding, for instance, how warmly to dress, these two measurements might be considered substitutes since, given one of them, the other gives relatively less information about the comfort level of warm or cool clothing. For another example: To a trader deciding whether to invest in a technology index fund (that is, a stock whose value follows that of the general tech sector), the share prices of two given tech companies may be substitutable information, since each gives some indication of the current value of tech stocks. But to a trader deciding which of these two specific companies to invest in, these prices may be complements, since the decision can be made much more effectively with both pieces of information than with either alone.The definition of <cit.> cannot capture such scenarios because it requires two signals to be substitutes for every possible decision problem. Our solution is to define S&C relative to both the particular information structure and the particular decision problem. b. Lack of dependence on the internal structure of the signals. The other concern with the definition of <cit.> is that it only depends on “extreme” values: ^u,P(A_1), ^u,P(A_2), ^u,P(A_1A_2), and ^u,P(). Hence, it ignores the internal structure of A_1 and A_2, which can lead to incongruous predictions. For example, suppose that B_1 and B_2 are substitutes while C_1 and C_2 are complements. Now consider the signals A_1 = (B_1,C_1) and A_2 = (B_2,C_2). For some decision problems, it may be that the B signals are slightly more important and so A_1 and A_2 seem to be substitutes. For other decision problems, it may be that the C signals are slightly more important and A_1 and A_2 seem to be complements. This is formalized in Example <ref>.To see why this could be problematic for a predictive or useful theory, suppose that an agent will be able to observe A_1, and a seller wishes to sell to that agent the opportunity to observe A_2 as well. As just argued, one might have defined A and B to be “substitutes” or “complements” depending on very small fluctuations in the decision problem. But the seller, by “hiding” or “forgetting” either the B_2 or the C_2 component of his signal, can force the signals to become either substitutes or complements as she desires. A definition that does not account for internal structure may get such examples completely wrong, e.g. classifying the signals as substitutes when the seller can make them behave as complements.We will introduce definitions that account for the internal structure of signals.§.§.§ Our approach: dependence on context and internal structure Context. As mentioned above, we will allow the definitions of S&C to depend on the particular decision or value function ^u,P. That is, while <cit.> defined a particular information structure P to be substitutes on pairs of signals if ^u,P satisfied a condition for all u, we will define a pair (u,P) to be substitutes if ^u,P satisfies a similar condition. This will turn out to be crucial in all of our applications.The potential drawback is that it might be difficult to say anything general about when signals are substitutes or complements; it might seem that one must take things completely on a case-by-case basis.We make two counterpoints. First, a universal approach may be the wrong goal or “too much to hope for”. For instance, in the case of items, there is no such optimistic analogue; one does not consider items that are always substitutes for every valuation function. Despite this, there are many successful theories leveraging substitutable goods. These approaches start by assuming a context (e.g. valuation functions) for which the goods are substitutes; similarly, we can consider a set of signals and only those decision problems for which they are substitutes.Second, we later give some evidence that we need not take things completely case-by-case. We seek classes of signals that can be considered substitutes or complements in a broad class of decision problems. For example, we show that if signals are independent, then they are complements for any decision problem satisfying a smoothness condition. Our work also gives intuition for which kinds of signals are more likely to be substitutable or are substitutable in more contexts. And indeed, one of the exciting questions raised by our work is how the context of a decision problem and internal structure combine to produce substitutable or complementary features. Probabilistic structure. We will allow definitions of S&C to depend on the internal structure of signals. But how? Two signals may be related in complex ways by correlations with each other and with the event E of interest. Therefore, a more natural analogy than substitutability of two items may be substitutability of two subsets of items. Consider <cit.>, which studied valuation functions over sets of items. There, the authors identified a natural “no complementarities” condition where two sets of items, A_1 and A_2, could only be substitutes if all “pieces” of those subsets were substitutes: no subset of set A_1 could be complementary to the set A_2, and vice versa. This turned out to be exactly a requirement that the valuation function be submodular: that it exhibit diminishing marginal value.We would also like to capture diminishing marginal value. The challenge that arises is, what is a marginal “unit” of information? The answer actually may vary by application.* In some applications, a “marginal unit” may be an entire signal: Given the current subset of {A_1,…,A_n}, one can either add another A_i, or not.This would be appropriate for cases where our above arguments about internal structure may not apply.For example, perhaps the seller in an auction does not have the ability, for whatever reason, to process her signals in any way; she can just choose between allowing each of them to be either broadcast or kept private.In this paper, we utilize this notion, which will correspond to “weak substitutes”, in the context of discrete optimization problems where an algorithm must choose between acquiring different signals.In many contexts, it is impossible to acquire partial signals, so this is the natural marginal unit.While they may be useful, they also are subject to the criticisms given above; in many contexts that allow “pieces” of signals, they may not behave as substitutes or may even behave as complements. * Sometimes, a “marginal unit” may be some partial information about a signal, in the form of a “fact” about its realization.For instance, imagine a commuter learns something about the barometer reading but not the exact reading; e.g., whether it is above or below 30, or the measurement rounded to the nearest integer.This application arises when considering pure strategies in a game, or deterministic “post-processings” of a signal in algorithmic contexts.The effect of such processing is always to “coarsen” a signal by pooling multiple outcomes together under one announcement.In the barometer example, all realizations of the signal below 30 map to the same result, and all realizations above map to the other result; similarly when rounding to the nearest integer.If a set of signals exhibits diminishing marginal value with respect to this notion, we will term them “moderate substitutes”.We actually do not provide an application for moderate substitutes in this paper, but expect them to be useful in contexts such as those just described. * Finally, a “marginal unit” may be partial information in the form of a noisy “signal about the signal”.For instance, the commuter may learn the barometer reading plus Gaussian noise.To see that this notion may be much more fine-grained than the previous one, imagine starting from the binary barometer example, where the commuter learns whether or not the barometer is below 30; and now imagine that, with some probability p, this observation is “flipped” from the correct one.When p=0, the commuter can be certain that she learns correctly which outcome is the case (above or below 30).But as p →1/2, the commuter learns less from the signal.If a set of signals exhibits diminishing marginal value even with respect to such partial information – for instance, diminishing marginal value as p decreases from 1/2 to 0 – then we term them “strong substitutes”.In applications where, for instance, the barometer observation is controlled by a strategic agent whose strategy consists of a “garbling” of that observation, this will be a useful notion of marginal information.We formalize these marginal units of information using lattices of signals: sets of signals with a partial order ≼ corresponding to “informativeness” and satisfying some natural conditions. While our proposed uses for them here are quite new, the lattices we use, or closely related concepts, are relatively classical. For weak substitutes, we consider the lattice of subsets of signals, with partial order given by set containment; this corresponds directly to subsets of goods and the notion of substitutes is essentially the same.For moderate substitutes, we utilize a variant of Aumann's classic model of information in Bayesian games <cit.>, in which signals correspond to partitions on a ground state of the world. To our knowledge, although it is known that Aumann's signals form a lattice (because the space of partitions do), they have not been used to formalize marginal units of information. One difference in the variant we propose is that the ground states only determine the signals, not the event E or any other pieces of information; this makes our model much more useful for formalizing marginal pieces of information because the ground states only contain information about the signals.For strong substitutes, we extend Aumann's model to capture randomized “garblings” of signals. Although this is not the model normally used in that context, the idea and intuition is extremely similar to Blackwell's criterion or partial ordering on signals <cit.>. One major difference is that in our model, there is a particular event E of interest and signals are ordered according to informativeness about that event, rather than pure informativeness. Also, the use of Aumann's partition model allows our signals to form a lattice.§.§.§ Capturing “diminishing” marginal valueLuckily, once we have placed a lattice structure on signals, we can apply a now-classic criterion for diminishing marginal value: submodularity. Often, submodularity is a condition for functions on subsets, e.g. f: 2^{1,…,n}→, which is submodular if an element i's “marginal contribution” to S, f(S ∪{i}) - f(S), is decreasing as elements are added to S. This is a widely-used model for substitutability of discrete, indivisible items <cit.>. The same goes for supermodularity, increasing marginal value, and complementary items.The final piece of our puzzle will be to utilize submodularity and supermodularity, and extensions, to capture diminishing marginal value and increasing marginal value respectively. This is formalized in Section <ref>. | http://arxiv.org/abs/1703.08636v1 | {
"authors": [
"Yiling Chen",
"Bo Waggoner"
],
"categories": [
"cs.GT"
],
"primary_category": "cs.GT",
"published": "20170325013723",
"title": "Informational Substitutes"
} |
𝖬𝖨𝖧𝖺𝗌𝗁: Online Hashing with Mutual InformationFatih CakirFirst two authors contributed equally. Kun He[1] Sarah Adel Bargal Stan SclaroffDepartment of Computer ScienceBoston University, Boston, MA{fcakir,hekun,sbargal,sclaroff}@cs.bu.eduMarch 2017 =========================================================================================================================================================================================================================emptyLearning-based hashing methods are widely used for nearest neighbor retrieval, and recently, online hashing methods have demonstratedgood performance-complexity trade-offs by learning hash functions from streaming data. In this paper,we first address a key challenge for online hashing: the binary codes for indexed data must be recomputed to keep pace with updates to the hash functions. We propose an efficient quality measure for hash functions, based on an information-theoretic quantity, mutual information, and use it successfully as a criterion to eliminate unnecessary hash table updates. Next, we also show how to optimize the mutual information objective using stochastic gradient descent.We thus develop a novel hashing method, , that can be used in both online and batch settings. Experiments on image retrieval benchmarks (including a 2.5M image dataset) confirm the effectiveness of our formulation, both in reducing hash table recomputations and in learning high-quality hash functions. § INTRODUCTION Hashing is a widely used approach for practical nearest neighbor search in many computer vision applications.It has been observed that adaptive hashing methods that learn to hash from data generally outperform data-independent hashing methods such as Locality Sensitive Hashing<cit.>. In this paper, we focus on a relatively new family of adaptive hashing methods, namely online adaptive hashing methods <cit.>. These techniques employ online learning in the presence of streaming data, and are appealing due to their low computational complexity and their ability to adapt to changes in the data distribution.Despite recent progress, a key challenge has not been addressed in online hashing, which motivates this work: the computed binary representations, or the “hash table",may become outdated after a change in the hash mapping. To reflect the updates in the hash mapping, the hash table may need to be recomputed frequently, causing inefficiencies in the system such as successive disk I/O, especially when dealing with large datasets.We thus identify an important question for online adaptive hashing systems:when to update the hash table?Previous online hashing solutions do not address this question, as they usually update both the hash mapping and hash table concurrently.We make the observation that achieving high quality nearest neighbor search is an ultimate goal in hashing systems, and therefore any effort to limit computational complexity should preserve, if not improve, that quality. Therefore, another important question is: how to quantify quality? Here, we briefly describe our answer to this question, but first introduce some necessary notation. We would like to learn a hash mapping Φ from feature space 𝒳 to the b-dimensional Hamming space ℋ^b, whose outputs are b-bit binary codes. The goal of hashing is to preserve a neighborhood structure in 𝒳 after the mapping to ℋ^b.Given ∈𝒳, the neighborhood structure is usually given in terms of a set of its neighbors _, and a set of non-neighbors _. We discuss how to derive the neighborhood structure in Sec. <ref>. As shown in Fig. <ref>, the distributions of the Hamming distances fromto its neighbors and non-neighbors are histograms over {0,1,…,b}. Ideally, if there is no overlap between these two distributions, we can recover _ and _ by simply thresholding the Hamming distance. A nonzero overlap results in ambiguity, as observing the Hamming distance is no longer sufficient to determine neighbor relationships. Our discovery is that this overlap can be quantified using an information-theoretic quantity, mutual information, between two random variables induced by Φ. We then use mutual information to define a novel measure to quantify quality for hash functions in general. With a quality measure defined, we answer the motivating question of when to update the hash table.Wepropose a simple solution by restricting updates to times when there is an estimated improvement in hashing quality, based on an efficient estimation method in the presence of streaming data. Notably, since mutual information is a good general-purpose quality measure for hashing, this results in a general plug-in module for online hashing that does not require knowledge of the learning method. Inspired by this strong result, we next ask, can we optimize mutual information as an objective to learn hash functions?We propose a novel hashing method, , by deriving gradient descent rules on the mutual information objective, which can be applied in online stochastic optimization, as well as on deep architectures. The mutual information objective is free of tuning parameters, unlike others that may require thresholds, margins, . We conduct experiments on three image retrieval benchmarks, including the Places205 dataset <cit.> with 2.5M images. For four recent online hashing methods, our mutual information based update criterion consistently leads to over an order of magnitude reduction in hash table recomputations, while maintaining retrieval accuracy.Moreover, our novel method achieves state-of-the-art retrieval results, in both online and batch learning settings.§ RELATED WORKIn this section, we mainly review hashing methods that adaptively update the hash mapping with incoming data, given that our focus is on online adaptive hashing. For a more general survey on hashing, please refer to <cit.>.Huang <cit.> propose Online Kernel Hashing, where a stochastic environment is considered with pairs of points arriving sequentially. At each step, a number of hash functions are selected based on a Hamming loss measure and parameters are updated via stochastic gradient descent (SGD).Cakir and Sclaroff <cit.>argue that, in a stochastic setting, it is difficult to determine which hash functions to update as it is the collective effort of all the hash functions that yields a good hash mapping. Hamming loss is considered to infer the hash functions to be updated at each step and a squared error loss is minimized via SGD. In <cit.>, binary Error Correcting Output Codes (ECOCs) are employed in learning the hash functions.This work follows a more general two-step hashing framework <cit.>, where the set of ECOCs are generated beforehand and are assigned to labeled data as they appear, allowing the label space to grow with incoming data. Then, hash functions are learned to fit the binary ECOCs using Online Boosting.Inspired by the idea of “data sketching", Leng introduce Online Sketching Hashing <cit.> where a small fixed-size sketch of the incoming data is maintained in an online fashion.The sketch retains the Frobenius norm of the full data matrix, which offers space savings, and allows to apply certain batch-based hashing methods. A PCA-based batch learning method is applied on the sketch to obtain hash functions.None of the above online hashing methods offer a solution to decide whether or not the hash table shall be updated given a new hash mapping. However,such a solution is crucial in practice, as limiting the frequency of updates will alleviate the computational burden of keeping the hash table up-to-date. Although<cit.> and <cit.> include strategies to select individual hash functions to recompute, they still require computing on all indexed data instances.Recently, some methods employ deep neural networks to learn hash mappings, <cit.> and others.These methods use minibatch-based stochastic optimization, however, they usually require multiple passes over a given dataset to learn the hash mapping, and the hash table is only computedwhen the hash mapping has been learned.Therefore, current deep learning based hashing methods are essentially batch learning methods, which differ from the online hashing methods that we consider, methods that process streaming data to learn and update the hash mappings on-the-fly while continuously updating the hash table. Nevertheless, when evaluating our mutual information based hashing objective, we compare against state-of-the-art batch hashing formulations as well, by contrasting different objective functions on the same model architecture.Lastly, Ustinova <cit.> recently proposed a method to derive differentiation rules for objective functions that require histogram binning, and apply it in learning deep embeddings. When optimizing our mutual information objective, we utilize their differentiable histogram binning technique for deriving gradient-based optimization rules. Note that both our problem setup and objective function are quite different from <cit.>. § ONLINE HASHING WITH MUTUAL INFORMATION As mentioned in Sec. <ref>, the goal of hashing is to learn a hash mapping Φ: 𝒳→ℋ^b such that a desired neighborhood structure is preserved.We consider an online learning setup where Φ is continuously updated from input streaming data, and at time t, the current mapping Φ_t is learned from {𝐱_1,…,𝐱_t}. We follow the standard setup oflearning Φ from pairs of instances with similar/dissimilar labels <cit.>. These labels, along with the neighborhood structure, can be derived from a metric, two instances are labeled similar (neighbors of each other) if their Euclidean distance in 𝒳 is below a threshold. Such a setting is often called unsupervised hashing. On the other hand, in supervised hashing with labeled data, pair labels are derived from individual class labels: instances are similar if they are from the same class, and dissimilar otherwise.Below, we first derive the mutual information quality measure and discuss its use in determining when to update the hash table in Sec. <ref>. We then describe a gradient-based approach for optimizing the same quality measure, as an objective for learning hash mappings, in Sec. <ref>. Finally, we discuss the benefits of using mutual information in Sec. <ref>.§.§ MI as Update Criterion We revisit our motivating question: When to update the hash table in online hashing? During the online learning of Φ_t, we assume a retrieval set 𝒮⊆𝒳, which may include the streaming data after they are received. We define the hash table as the set of hashed binary codes: 𝒯(𝒮,Φ)={Φ(𝐱)|𝐱∈𝒮}. Given the adaptive nature of online hashing, 𝒯 may need to be recomputed often to keep pace with Φ_t; however,this is undesirable if 𝒮 is large or the change in Φ_t's quality does not justify the cost of an update. We propose to view the learning of Φ_t and computation of 𝒯 as separate events, which may happen at different rates.To this end, we introduce the notion of a snapshot, denoted Φ^s, which is occasionally taken of Φ_t and used to recompute 𝒯. Importantly, this happens only when the nearest neighbor retrieval quality of Φ_t has improved, and we now define the quality measure. Given hash mapping Φ:𝒳→{-1,+1}^b, Φ induces Hamming distance d_Φ:𝒳×𝒳→{0,1,…,b} asd_Φ(𝐱, ) = 1/2(b - Φ(𝐱)^⊤Φ()).Consider some instance 𝐱̂∈𝒳, and the sets containing neighbors and non-neighbors, _𝐱̂ and _𝐱̂. Φ induces two conditional distributions, P(d_Φ(𝐱,𝐱̂)|𝐱∈_𝐱̂) and P(d_Φ(𝐱,𝐱̂)|𝐱∈_𝐱̂) as seen in Fig. <ref>, and it is desirable to have low overlap between them. To formulate the idea, for Φ and 𝐱̂, define random variable 𝒟_𝐱̂,Φ:𝒳→{0,1,…,b}, ↦ d_Φ(𝐱, 𝐱̂), and let 𝒞_𝐱̂:𝒳→{0,1}be the membership indicator for _. The two conditional distributions can now be expressed asP(_,Φ|_=1) and P(_,Φ|_=0), and we can write the mutual information between 𝒟_𝐱̂,Φ and 𝒞_𝐱̂ asℐ(𝒟_𝐱̂,Φ; 𝒞_𝐱̂)= H(𝒞_𝐱̂) - H(𝒞_𝐱̂ | 𝒟_𝐱̂,Φ) = H(_,Φ)-H(_,Φ|_)where H denotes (conditional) entropy. In the following, for brevity we will drop subscripts Φ and 𝐱̂, and denote the two conditional distributions and the marginal P(_,Φ) as p_^+, p_^-, and p_, respectively.By definition, ℐ(;)measures the decrease in uncertainty in the neighborhood information 𝒞 when observing the Hamming distances 𝒟. We claim that ℐ(; ) also captures how well Φ preserves the neighborhood structure of 𝐱̂. If ℐ(;) attains a high value, which means 𝒞 can be determined with low uncertainty by observing 𝒟, then Φ must have achieved good separation (low overlap) between p_𝒟^+ and p_𝒟^-. ℐ is maximized when there is no overlap, and minimized when p_𝒟^+ and p_𝒟^- are exactly identical. Recall, however, that ℐ is defined with respect to a single instance ; therefore, for a general quality measure, we integrate ℐ overthe feature space: Q(Φ) = ∫_𝒳ℐ(𝒟_𝐱̂,Φ;C_𝐱̂)p(𝐱̂)d𝐱̂. Q(Φ) captures the expected amount of separation between p_𝒟^+ and p_𝒟^- achieved by Φ, over all instances in 𝒳.In the online setting, given the current hash mapping Φ_t and previous snapshot Φ^s,it is then straightforward to pose the update criterion asQ(Φ_t)-Q(Φ^s) > θ,where θ is a threshold; a straightforward choice is θ=0. However, Eq. <ref> is generally difficult to evaluate due to the intractable integral; in practice, we resort to Monte-Carlo approximations to this integral, as we describe next. §.§.§ Monte-Carlo Approximation by Reservoir Sampling We give a Monte-Carlo approximation of Eq. <ref>. Since we work with streaming data, we employ the Reservoir Sampling algorithm <cit.>, which enables sampling from a stream or sets of large/unknown cardinality.With reservoir sampling, we obtain a reservoir set ℛ≜{𝐱_1^r,…,𝐱^r_K} from the stream, which can be regarded as a finite sample from p(𝐱).We estimate the value of Q on ℛ as:Q_ℛ(Φ)=1/|ℛ|∑_𝐱^𝐫∈ℛℐ_ℛ(𝒟_𝐱^𝐫,Φ;𝒞_𝐱^𝐫).We use subscript ℛ to indicate that when computing the mutual information ℐ, the p_^+ and p_^- for a reservoir instance 𝐱^𝐫 are estimated from ℛ. This can be done in 𝒪(|ℛ|) time for each 𝐱^𝐫, as the discrete distributions can be estimated via histogram binning.Fig. <ref> summarizes our approach. We use the reservoir set to estimate the quality Q_ℛ, and “trigger" an update to the hash table only when Q_ℛ improves over a threshold.Notably, ourapproach provides a general plug-in module for online hashing techniques, in that it only needs access to streaming data and the hash mapping itself, independent of the hashing method's inner workings.§.§ MI as Learning Objective Having shown that mutual information is a suitable measure of neighborhood quality, we consider its use as a learning objective for hashing.Following the notation in Sec. <ref>, we define a loss ℒ with respect to ∈𝒳 and Φ asℒ(,Φ)=-ℐ(_,Φ; _).We model Φ as a collection of parameterized hash functions, each responsible for generating a single bit: Φ(𝐱) = [ϕ_1 (𝐱;W), ..., ϕ_b (𝐱;W)], where ϕ_i:𝒳→{-1,+1},∀ i, and W represents the model parameters. For example, linear hash functions can be written as ϕ_i() = sgn (w_i^⊤𝐱), and for deep neural networks the bits are generated by thresholding the activations of the output layer. Inspired by the online nature of the problem and recent advances in stochastic optimization, we derive gradient descent rules for ℒ. The entropy-based mutual information is differentiable with respect to the entries of p_, p_^+ and p_^-, and, as mentioned before, these discrete distributions can be estimated via histogram binning. However, it is not clear how to differentiate histogram binning to generate gradients for model parameters. We describe a differentiable histogram binning technique next.§.§.§ Differentiable Histogram BinningWe borrow ideas from <cit.> andestimate p_^+, p_^- and p_ using a differentiable histogram binning technique. For b-bit Hamming distances, we use (K+1)-bin normalized histograms with bin centers v_0 = 0, ..., v_K = b and uniform bin width Δ = b/K, where K=b by default. Consider, for example, the k-th entry in p^+_, denoted as p_𝒟,k^+. It can be estimated asp_𝒟,k^+ = 1/||∑_∈δ_,k,where δ_,k records the contribution ofto bin k.It is obtained by interpolating d_Φ(,) using a triangular kernel:δ_,k =(d_Φ(𝐱, ) - v_k-1)/Δ, d_Φ(𝐱, ) ∈ [v_k-1, v_k], (v_k+1 - d_Φ(𝐱, ))/Δ, d_Φ(𝐱, ) ∈ [v_k, v_k+1], 0,otherwise.This binning process admits subgradients:∂δ_, k/∂ d_Φ(𝐱, ) =1/Δ, d_Φ(𝐱, ) ∈ [v_k-1, v_k], -1/Δ, d_Φ(𝐱, ) ∈ [v_k, v_k+1], 0,otherwise.§.§.§ Gradients of MIWe now derive the gradient of ℐ with respect to the output of the hash mapping, Φ(). Using standard chain rule, we can first write∂ℐ/∂Φ()=∑_k=0^K[∂ℐ/∂ p^+_,k∂ p^+_,k/∂Φ() + ∂ℐ/∂ p^-_,k∂ p^-_,k/∂Φ()]. We focus on terms involving p_,k^+, and omit derivations for p_,k^- due to symmetry. For k=0,…,K, we have∂ℐ/∂ p^+_,k= -∂ H(|)/∂ p^+_,k+∂ H()/∂ p^+_,k = p^+(log p^+_,k+1)-(log p_,k+1)∂ p_,k/∂ p^+_,k = p^+(log p^+_,k-log p_,k),where we used the fact that p_,k=p^+p^+_,k+p^-p^-_,k, with p^+ and p^- being shorthands for the priors P(=1) and P(=0). We next tackle the term ∂ p^+_,k/∂Φ() in Eq. <ref>. From the definition of p_,k^+ in Eq.<ref>, we have∂ p_𝒟,k^+/∂Φ() = 1/||∑_𝐱∈∂δ_𝐱,k/∂Φ()= 1/||∑_𝐱∈∂δ_𝐱,k/∂ d_Φ(,)∂ d_Φ(,)/∂Φ()= 1/||∑_𝐱∈∂δ_𝐱,k/∂ d_Φ(,)-Φ()/2.Note that ∂δ_,k/∂ d_Φ(,) is already given in Eq. <ref>. For the last step, we used the definition of d_Φ in Eq. <ref>.Lastly, to back-propagate gradients to Φ's inputs and ultimately model parameters, we approximate the discontinuous sign function withsigmoid, which is a standard technique in hashing, <cit.>.§.§ Benefits of MI For monitoring the performance of hashing algorithms, it appears that one could directly usestandard ranking metrics, such as Average Precision (AP), Discounted Cumulative Gain (DCG), and Normalized DCG (NDCG) <cit.>. Here, we discuss the benefits of instead using mutual information. First, we note that there exist strong correlations between mutual information and standard ranking metrics. Fig. <ref> demonstrates the Pearson correlation coefficients between MI and AP, DCG, and NDCG, on three benchmarks. Although a theoretical analysis is beyond the scope of this work, empirically we find that MI serves as an efficient and general-purpose ranking surrogate.We also point out the lower computational complexity of mutual information. Let n be the reservoir set size. Computing Eq. <ref> involves estimating discrete distributions via histogram binning, and takes 𝒪(n) time for each reservoir item, sinceonly takes discrete values from {0,1,…,b}, In contrast, ranking measures such as AP and NDCG have 𝒪(nlog n) complexity due to sorting, which render them disadvantageous. Finally, Sec. <ref> showed that the mutual information objective is suitable for direct, gradient-based optimization. In contrast, optimizing metrics like AP and NDCG is much more challenging as they are non-differentiable,and existing works usually resort to optimizing their surrogates <cit.> rather than gradient-based optimization. Furthermore, mutual information itself is essentially parameter-free, whereas many other hashing formulationsrequire (and can be sensitive to) tuning parameters, such as thresholds or margins <cit.>, quantization strength <cit.>, . § EXPERIMENTS We evaluate our approach on three widely used image benchmarks.We first describe the datasets and experimental setup in Sec. <ref>. We evaluate the mutual information update criterion in Sec. <ref> andthe mutual information based objective function for learning hash mappings in Sec. <ref>.Our implementation is publicly available at <https://github.com/fcakir/mihash>. §.§ Datasets and Experimental Setup CIFAR-10 is a widely-used dataset for image classification and retrieval, containing 60K images from 10 different categories <cit.>. For feature representation, we use CNN features extracted from the fc7 layer of a VGG-16 network <cit.> pre-trained on ImageNet. Places205 is a subset of the large-scale Places dataset <cit.> for scene recognition. Places205 contains 2.5M images from 205 scene categories. This is a very challenging dataset due to its large size and number of categories, and it has not been studied in the hashing literature to our knowledge. We extract CNN features from the fc7 layer of an AlexNet <cit.> pre-trained on ImageNet, and reduce thedimensionality to 128 using PCA.LabelMe. The 22K LabelMe dataset <cit.> has 22,019 images represented as 512-dimensional GIST descriptors. This is an unsupervised dataset without labels, and standard practice uses the Euclidean distance to determine neighbor relationships. Specifically, 𝐱_i and 𝐱_j are considered neighbor pairs if their Euclidean distance is within the smallest 5% in the training set. For a query, the closest 5% examples are considered true neighbors. All datasets are randomly split into a retrieval set and a test set, and a subset from the retrieval set is used for learning hash functions. Specifically, for CIFAR-10, the test set has 1K images and the retrieval set has 59K. A random subset of 20Kimages from the retrieval set is used for learning, and the size of the reservoir is set to 1K.For Places205, we sample 20 images from each class to construct a test set of 4.1K images, and use the rest as the retrieval set. A random subset of 100K images is used to for learning, and the reservoir size is 5K.For LabelMe, the dataset is split into retrieval and test sets with 20K and 2K samples, respectively. Similar to CIFAR-10, we use a reservoir of size 1K.For online hashing experiments, we run three randomized trials for each experiment and report averaged results. To evaluate retrieval performances, we adopt the widely-used mean Average Precision (mAP). Due to the large size of Places205, mAP is very time-consuming to compute, and we compute mAP on the top 1000 retrieved examples (mAP@1000), as done in <cit.>.§.§ Evaluation: Update Criterion We evaluate our mutual information based update criterion, the Trigger Update module (TU). We apply TU to all existing online hashing methods known to us: Online Kernel Hashing (OKH) <cit.>, Online Supervised Hashing (OSH) <cit.>, Adaptive Hashing (AdaptHash) <cit.> and Online Sketching Hashing (SketchHash) <cit.>. We use publicly available implementations of all methods. The hash code length is fixed at 32 bits.As our work is the first in addressing the hash table update criterion for online hashing, we compare to a data-agnostic baseline, which updates the hash table at a fixed rate. The rate is controlled by a parameter U, referred to as the “update interval":after processing every U examples, the baseline unconditionally triggers an update, while TU makes a decision using the mutual information criterion. For each dataset,U is set such that the baseline updates 201 times in total. This ensures that the baseline is never too outdated, but updates are still fairly infrequent: in all cases, the smallest U is 100. Results for the Trigger Update module. Fig. <ref> depicts theretrieval mAP over time for all four online hashing methods considered, on three datasets, with and without incorporating .We can clearly observe a significant reduction in the number of hash table updates, between one and two orders of magnitude in all cases. For example, the number of hash table updates is reduced by a factor of 67 for the OKH method on LabelMe. The quality-based update criterion is particularly important for hashing methods that may yield inferior hash mappings due to noisy data and/or imperfect learning techniques. In other words, TU can be used to filter updates to the hash mapping with negative or small improvement. Thishas a stabilizing effect on the mAP curve,notably for OKH and AdaptHash.For OSH, which appears to stably improve over time, TU nevertheless significantly reduces revisits to the hash table while maintaining its performance. All results in Fig. <ref> are obtainedusing the default threshold parameter θ=0, defined in Eq. <ref>.We do not tune θ in order to show general applicability. We also discuss the impact of the reservoir set ℛ. There is a trade-off regarding the size of ℛ: a larger ℛ leads to better approximation but increases the overhead. Nevertheless, we observed robust and consistent results with |ℛ| not exceeding 5% of the size of the training stream.§.§ Evaluation: Learning Objective We evaluate the mutual information based hashing objective. We name our method ,and train it using stochastic gradient descent (SGD). This allows it to be applied to both the online setting and batch setting in learning hash functions.During minibatch-based SGD, to compute the mutual information objective in Eq. <ref> and its gradients, we need access to the sets _, _ for each considered , in order to estimate p_^+ and p_^-. For the online setting in Sec. <ref>, a standalone reservoir set ℛ is assumed as in the previous experiment, and we partition ℛ into {_,_} with respect to each incoming . In this case, even a batch size of 1 can be used. For the batch setting in Sec. <ref>,{_,_} are defined within the same minibatch as .§.§.§ Online SettingWe first consider an online setting that is the same as in Sec. <ref>.We compare against other online hashing methods: OKH, OSH, AdaptHash and SketchHash. All methods are equipped with the TU module with the default threshold θ=0, which has been demonstrated to work well. Results for Online Setting. We first show the mAP curve comparisons in Fig. <ref>. For competing online hashing methods, the curves are the same as the ones with TU in Fig. <ref>, and we remove markers to avoid clutter. clearly outperforms other online hashing methods on all three datasets, and shows potential for further improvement with more training data. The combination of TU and gives a complete online hashing system that enjoys a superior learning objective with a plug-in update criterion that improves efficiency.We next give insights into the distribution-separating effect from optimizing mutual information. In Fig. <ref>, we plot the conditional distributions p_^+ and p_^- averaged on the CIFAR-10 test set, before and after learning with the 20K training examples. Before learning, with a randomly initialized hash mapping, p_^+ and p_^- exhibit high overlap. After learning, achieves good separation between p_^+ and p_^-: the overlap reduces significantly, and the mass of p_^+ is pushed towards 0. This separation is reflected in the large improvement in mAP (0.68 vs. 0.22).In contrast with the other methods, the mutual information formulation is parameter-free. For instance, there is no threshold parameter that requires separating p_^+ and p_^- at a certain distance value. Likewise, there is no margin parameter that dictates the amount of separation in absolute terms. Such parameters usually need to be tuned to fit to data, whereas the optimization of mutual information is automatically guided by the data itself. §.§.§ Batch Setting To further demonstrate the potential of , we consider the batch learning setting, where the entire training set is available at once.We compare against state-of-the-art batch formulations,including: Supervised Hashing with Kernels (SHK) <cit.>, Fast Supervised Hashing with Decision Trees (FastHash) <cit.>, Supervised Discrete Hashing (SDH) <cit.>, Efficient Training of Very Deep Neural Networks (VDSH) <cit.>, Deep Supervised Hashing with Pairwise Labels (DPSH) <cit.> and Deep Supervised Hashing with Triplet Labels (DTSH) <cit.>.These competing methods have shown to outperform earlier and other work such as <cit.>. We focus on comparisons on the CIFAR-10 dataset, which is the canonical benchmark for supervised hashing.Similar to <cit.>, we consider two experimental settings, which we detail below.Setting 1:5K training examples are sampled for learning hash mappings, and 1K examples are used as the test set.All methods learn shallow models on top of fc7 features from a VGG-16 network <cit.> pretrained on ImageNet.For three gradient-based methods (DPSH, DTSH, and ), this means learning linear hash functions. Note that VDSH uses customized architectures consisting of only fully-connected layers, and it is unclear how to adapt it to use standard architectures; we used its full model with 16 layers and 1024 nodes per layer.Setting 2: We use the full training set of size 50K and test set of size 10K. We focus on comparing the end-to-end performance of against two recent leading methods: DPSH and DTSH, using the same VGG-F network architecture <cit.> that they are trained on. We use publicly available implementations for the compared methods, and exhaustively search parameter settings for them. For , the minibatch size is set to 100 and 250 in Settings 1 and 2, respectively. We use SGD with momentum, and decrease the learning rate when the training loss saturates. See supplementary material for more details.Results for Batch Setting. In Table <ref>, we list batch learning results for all methods. In Setting 1, outperforms all competing methods in terms of mAP, in some cases with only a single training epoch (against VDSH, DPSH). This suggests that mutual information is a more effective learning objective for hashing. learns a linear layer on the inputfeatures, while some other methods can learn non-linear hash functions: for instance, the closest competitor, FastHash, is a two-step hashing method based on sophisticated binary code inference and boosted trees.In Setting 2,with end-to-end finetuning, significantly outperforms DPSH and DTSH, the two most competitive deep hashing methods, and sets the current state-of-the-art for CIFAR-10.Again, note that has no tuning parameters in its objective function. In contrast, both DPSH and DTSH have parameters to control the quantization strength that need to be tuned.§ CONCLUSION We advance the state-of-the-art for online hashing in two aspects. In order to resolve the issue of hash table updates in online hashing, we define a quality measure using the mutual information between variables induced by the hash mapping. This measure is efficiently computable, correlates well with standard evaluation metrics, and leads to consistent computational savings for existing online hashing methods while maintaining their retrieval accuracy. Inspired by these strong results, we further propose a hashing method , by optimizing mutual information as an objective with stochastic gradient descent. In both online and batch settings, achieves superior performance compared to state-of-the-art hashing techniques. § ACKNOWLEDGEMENTSThis research was supported in part by a BU IGNITION award, US NSF grant 1029430, and gifts from NVIDIA. ieee§ APPENDIX§ IMPLEMENTATION DETAILS OF We discuss the implementation details of . In the online hashing experiments, for simplicity we model using linear hash functions, in the form of ϕ_i()=sgn(w_i^⊤)∈{-1,+1},i=1,…,b. The learning capacity of such a model is lower than the kernel-based OKH, and is the same as OSH, AdaptHash, and SketchHash, which use linear hash functions as well.For the batch hashing experiments, as mentioned in the paper,we similarly model using linear hash functions in the first setting, but perform end-to-end learning with the VGG-F network in the second setting.In this case, the hash functions become ϕ_i()=sgn(f_i(x;w))∈{-1,+1},i=1,…,b, where f_i are the logits produced by the previous layer in the network.We train using stochastic gradient descent. In Eq. 11 in the paper, we gave the gradients of the mutual information objective ℐ with respect to the outputs of the hash mapping, Φ(). Both ℐ and ∂ℐ/∂Φ() are parameter-free. In order to further back-propagate gradients to the inputs of Φ() and model parameters {w_i},we approximate the sgn function using the sigmoid function σ: ϕ_i()≈ 2σ(Aw_i^⊤)-1,where A>1 is a scaling parameter, used to increase the “sharpness" of the approximation. We find A from the set {10,20,30,40,50} in our experiments.We note that A is not a tuning parameter of the mutual information objective, but rather a parameter of the underlying hash functions. The design of the hash functionsis not coupled with the mutual information objective, thus can be separated. It will be an interesting topic to explore other methods of constructing hash functions, potentially in ways that are free of tuning parameters. § EXPERIMENTAL DETAILS §.§ The streaming scenarioWe set up a streaming scenario in our online hashing experiments. We run three randomized trials for each experiment. In each trial, we first randomly split the dataset into a retrieval set and a test set as described in Sec. 4.1 in the paper, and randomly sample the training subset from the retrieval set. The ordering of thetraining set isalso randomly permuted.The random seeds are fixed, so the baselines and methodswith the Trigger Update module observe the same training sequences.In a streaming setting, we also measure the cumulative retrieval performance during online hashing,as opposed to only the final results.To mimic real retrieval systems where queries arrive randomly, we set 50 randomized checkpoints during the online process. We first place the checkpoints with equal spacing, then add small random perturbations to their locations. We measure the instantaneous retrieval mAP at these checkpoints to get mAP vs. time curves (curves shown in Fig. 5 in the paper), and compute the area under curve (AUC). AUC gives a summary of the entire online learning process, which cannot be reflected by the final performance at the end. §.§ Parameters for online hashing methodsWe describe parameters used for online hashing methods in theonline experiments. Some of the competing methods require parameter tuning,therefore we sample a validation set from the training data and find the best performing parameters for each method. The size of the validation sets are 2K, 2K and 10K for CIFAR-10, LabelMe and Places205, respectively.Please refer to the respective papers for the descriptions of the parameters.* OSH: η is set to 0.1 for all datasets. The ECOC codebook C is populated the same way as in OSH. * AdaptHash: the tuple (α, λ, η) is set to (0.9, 0.01, 0.1), (0.1, 0.01, 0.001) and (0.9, 0.01, 0.1) for CIFAR-10, LabelMe and Places205, respectively.* OKH: the tuple (C, α) is set to (0.001, 0.3), (0.001, 0.3) and (0.0001, 0.7) for CIFAR-10, LabelMe and Places205, respectively.* SketchHash: the pair (sketch size, batch size) is set to (200, 50), (100, 50) and (100,50) for CIFAR-10, LabelMe and Places205, respectively.§.§ Parameters for batch hashing methods We use the publicly available implementations for the compared methods, and exhaustively search parameter settings, including the default parameters as provided by the authors.For DPSH and DTSH we found a combination that worked well for the first setting: the mini-batch size is set to the default value of 128, and the learning rate is initialized to 1 and decayed by a factor of 0.9 after every 20 epochs.Additionally, for DTSH, the margin parameter is set to b/4 where b is the hash code length. VDSH uses a heavily customized architecture with only fully-connected layers, and it is unclear how to adapt it to work with standard CNN architectures. In this sense, VDSH is more akin to nonlinear hashing methods such as FastHash and SHK. We used the full VDSH model with 16 layers and 1024 nodes per layer, and found the default parameters to perform the best, except that we increased the number of training iterations by an order of magnitude during finetuning.For , in the first setting we use a batch size of 100, and run SGD with initial learning rate of 0.1 and a decay factor of 0.5 every 10 epochs, for 100 epochs.For the second setting where we finetune the pretrained VGG-F network, batch size is 250, learning rate is initially set to 0.001 and decayed by half every 50 epochs.§ RUNNING TIME §.§ Online Setting: Trigger Update ModuleIn Table <ref> we report running time for all methods on the CIFAR-10 dataset with 20k training examples, including time spent in learning hash functions and the added processing time for maintaining the reservoir set and computing TU.Numbers are recorded on a 2.3GHz Intel Xeon E5-2650 CPU workstation with 128GB of DDR3 RAM.Most of the added time is due to maintaining the reservoir set, which is invoked in each training iteration;the mutual information update criterion is only checked after processing every U=100 examples.Methods with small batch sizes (OSH, batch size 1) therefore incur more overhead than methods with larger batches (SketchHash, batch size 50). Results for other datasets are similar.We note that in a real retrieval system with large-scale data, the bottleneck likely lies in recomputing the hash tables for indexed data, due to various factors such as scheduling and disk I/O. We reduce this bottleneck significantly by using TU.Compared to this bottleneck, the increase in training time is not significant. §.§ Batch SettingTable <ref> reports CPU times for learning 48-bit hash mappings in the first experimental setting on CIFAR-10 (5K training set). Retrieval mAP are replicated from Table 1 in the paper. For learning a single layer, our Matlab implementation of achieves 1.9 seconds per epoch on CPU. achieves competitive performance with a singleepoch,and has a total training time on par with FastHash, while yielding superior performance. § ADDITIONAL EXPERIMENTAL RESULTS §.§ Online Hashing: Other Code LengthsIn the online hashing experimentswe reported in the paper, all online hashing methods are compared inthe same setup with 32-bit hash codes. Additionally,we also present results using64-bit hash codes on all three datasets.The parameters for all methods are found through validation as described in <ref>.Similar to Sec 4.2 in the paper, we show the comparisonswithand without TU for existing online hashing methods in Fig. <ref>, and plot the mAP curves for all methods, including , in Fig. <ref>.The 64-bit results are uniformly better than 32-bit results for all methods in terms of mAP, but still follow the same patterns. Again, we can see that clearly outperforms all competing online hashing methods, and shows potential for improvement given more training data.§.§ Parameter Study: θWe present a parameter study on the parameter θ, the improvement threshold on the mutual information criterion in TU. In our previous experiments, we found the default θ=0 to work well, and did not specifically tune θ. However, tuning for a larger θ could lead to better trade-offs, since small improvements in the quality of the hash mapping may not justify the cost of a full hash table update.For this study, we vary parameter θ from -∞ to ∞ for all methods (with 32-bit hash codes).θ=-∞ reduces to the baseline. On the other hand, θ=∞ prevents any updates to the initial hash mapping and hash table, and results in only one hash table update (for the initial mapping) and typically low performance. The performance metricwe focus on in this study is the cumulative metric, AUC, since it better summarizes the entire online learning process than the final performance alone.We use a custom update schedule for SketchHash:we enforce hash table updates in the early iterations regardless of other criteria, until the number of observed examples reaches the specified size of the “data sketch", which SketchHash uses to perform a batch hashing algorithm. This was observed to be critical for the performance of SketchHash. Therefore, the number of hash table updates for SketchHash can be greater than1 even for θ=∞.We present full results in Tables <ref>, <ref>, <ref>. In all cases, we observe a substantial decrease in the number of hash table updates as θ increases. With reasonable θ values (typically around 0), the number of hash table updates can be reduced by over an order of magnitude with no loss in AUC. Note that the computation-performance trade-off achieved by the default θ=0 is always among the best, thereby in practice it can be usedwithout tuning. §.§ ParameterStudy: UWesimulate a data-agnostic baseline that updates hash tables at a constant rate, using the update interval parameter U. In the paper, U is set such that the baseline updates a total of 201 times for all datasets.This ensures that the baseline is never too outdated (compared to 50 checkpoints at which performance is evaluated), but is still fairly infrequent: the smallest U in this case is 100, which means the baselines process at least 100 training examples before recomputing the hash table. For completeness, here we present the results using different values of U, where all methods again use 32-bit hash codes and the default θ=0. We used a simple rule that avoids unnecessary hash table updates if the hash mapping itself does not change. Specifically, we do not update if Φ_t-Φ^s<10^-6, where Φ^s is the current snapshot and Φ_t is the new candidate. Some baseline entries have fewer updates because of this rule (AdaptHash on Places205). And as explained before, due to the custom update schedule, SketchHash may have more hash table updates than what is suggested by U.Please see Tables <ref>, <ref>, <ref> for the full results. In all experiments, we run three random trials and average the results as mentioned before, and the standard deviation of mAP and AUC scores are less than 0.01. Generally, using smaller U leads to more updates by both the baselines and methods with TU; recall that U is also a parameter of TU which specifies the frequency of checking the update criterion. However, methods with the TU module appear to be quite insensitive to the choice of U, the number of updates for SketchHash with TU on CIFAR-10 only increases by 2x while U is reduced by 20x, from 1000 to 50. We attribute this to the ability of TU to filter out unnecessary updates.Across different values of U, TU consistently brings computational savings while preserving/improving online hashing performance, as indicated by final mAP and AUC. | http://arxiv.org/abs/1703.08919v2 | {
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"title": "MIHash: Online Hashing with Mutual Information"
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[pages=1-last,linktodoc=true]2017_Dirscherl_et_al_WTS_chapter_translation.pdf | http://arxiv.org/abs/1703.08661v1 | {
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On pairs of geometric foliations on a cuspidal edge December 30, 2023Kentaro SajiWe study thetopological configurations of the lines of principal curvature, the asymptotic and characteristic curves on a cuspidal edge, in the domain of a parametrization of this surface as well as on the surface itself. Such configurations are determined by the 3-jets of a parametrization of the surface. We study thetopological configurations of the lines of principal curvature, the asymptotic and characteristic curves on a cuspidal edge, in the domain of a parametrization of this surface as well as on the surface itself. Such configurations are determined by the 3-jets of a parametrization of the surface.WPaxos: Wide Area Network Flexible ConsensusAilidani Ailijiang, Aleksey Charapko, Murat Demirbas and Tevfik KosarDepartment of Computer Science and EngineeringUniversity at Buffalo, SUNYEmail: {ailidani,acharapk,demirbas,tkosar}@buffalo.eduDecember 30, 2023 ================================================================================================================================================================================================================[Partly supported by the Japan Society for the Promotion of Science (JSPS) and the Coordenadoria de Aperfeiçoamento de Pessoal de Nível Superior under the Japan-Brazil research cooperative program and the Grant-in-Aid for Scientific Research (Young Scientists (B)) No. 26400087, from JSPS.] [ 2010 Mathematics Subject classification.Primary 53A05; Secondary 58K05, 57R45.] [Keywords and Phrases. Cuspidal edge, principal configuration, lines of curvature] § INTRODUCTION AND PRELIMINARIES ABOUT CUSPIDAL EDGES A singular point x of a map f:(^2,x)→(^3,0)is called a cuspidal edge if the map-germ f at x is 𝒜-equivalent to (u,v)↦(u,v^2,v^3) at 0. (Two map-germs f_1,f_2:(^n,0)→(^m,0) are 𝒜-equivalent if there exist diffeomorphisms S:(^n,0)→(^n,0) and T:(^m,0)→(^m,0) such that f_2∘ S=T∘ f_1.)If the singular point x of f is a cuspidal edge, then f at x is a front in the sense of <cit.> (see also <cit.>), and furthermore, they are one of two types of generic singularities of fronts (the other one is a swallowtail which is a singular point u of f satisfying that f at u is 𝒜-equivalent to (u,v)↦(u, u^2v + 3u^4, 2 uv + 4 u^3) at 0). It is shown in <cit.> that a cuspidal edge can locally be parametrized after smooth changes of coordinates in the source and isometries in the target byf(u,v) =(u,a_1(u)+v^22,b_2(u)+v^2b_3(u)+v^3b_4(u,v)), wherea_1(u),b_2(u),b_3(u),b_4(u,v) are C^∞-functions satisfying a_1(0)=a_1'(0)=b_2(0)=b_2'(0)=b_3(0)=0,b_2”(0)>0, b_4(0,0)0. Writinga_1(u)=a_20u^2/2+a_30u^3/6+u^4h_1(u), b_2(u)=b_20u^2/2+b_30u^3/6+u^4h_2(u), b_3(u)=b_12u/2+u^2h_3(u), b_4(u,v)=b_03/6+uh_4(u)+vh_5(u,v), we have[ f(u,v)= (u, a_20/2u^2+a_30/6u^3+v^2/2,; b_20/2u^2+b_30/6u^3+b_12/2uv^2 +b_03/6v^3) + h(u,v), ]where b_030, b_20≥0 andh(u,v)= ( 0,u^4h_1(u), u^4h_2(u)+u^2v^2h_3(u)+uv^3h_4(u)+v^4h_5(u,v) ),with h_1(u),h_2(u),h_3(u),h_4(u),h_5(u,v)smooth functions. Several differential geometric invariants of cuspidal edges are investigated (<cit.>),and coefficients of (<ref>) are such invariants. According to <cit.>, it is known that a_20 coincides with the singular curvature κ_s, b_20 coincides with the limiting normal curvature κ_ν, b_03 coincides with the cuspidal curvatureκ_c and b_12 coincides with the cusp-directional torsion κ_t at the origin. The singular curvature is the geodesic curvature of the singular set with sign, and the limiting normal curvature is the normal curvature of the singular set, and they relates to the shape of cuspidal edge (see <cit.>). The cuspidal curvature measures the wideness of the cusp, and the cusp-directional torsion measures the rotating ratio of the cusp along the singular set (see <cit.>).On the other hand, let U⊂^2 be an open subset and (u,v) a coordinate system on U. Letω = a(u,v) dv^2 + 2b(u,v) dudv + c(u,v) du^2 be a 2-tensor on U, where a,b,c are smooth functions, called the coefficients of ω.We call ω=0a binary differential equation (BDE)corresponding to ω. If b^2-ac>0 at x∈ U, then ω(x)=0 defines two directionsin T_xU, and integral curves of these directions for two smooth and transverse foliations, called foliations with respect to ω. If b^2-ac=0 at x∈ U, then generically ω(x)=0 defines a single direction, and the integral curves form in general a family of cusps. Thus we are mainly interested in behavior ofintegral curves of a BDE near a point where b^2-ac vanishes. We call discriminant of a BDE the set where b^2-ac=0. If the single direction is transverse tothe discriminant, thenthe BDE is equivalent to dv^2 +udu^2 = 0 (<cit.>). The normal form for the stable cases when the single direction is tangent to the discriminant is obtained in <cit.>. Topological classifications of generic families of BDEs are obtained in<cit.>. On the other hand, BDEs as geometric foliations on surfaces in three space is studied in <cit.>. See <cit.> for other approaches for geometric foliations.In this paper, following <cit.>, we stickto special BDEs from differential geometry of surface in ^3. There are three fundamental BDEs on a regular surface in ^3. Let f:(^2,0)→(^3,0) be a regular surface with a unit normal vector field ν. Letω_lc, ω_as andω_ch be 2-tensors defined by[ ω_lc= (EM-FL)dv^2+(EN-GL)dudv+(FN - GM)du^2,; ω_as= N dv^2+2M dudv+L du^2,; ω_ch=(2M(GM-FN) - N(GL-EN)) dv^2;+2(M(GL+EN) - 2FLN) dudv;+(L(GL-EN) -2M(FL-EM)) du^2, ]where (u,v) is a coordinate system on (^2,0), andE F F G= f_uf_u f_uf_v f_vf_u f_vf_v, L M M N= f_uuν f_uvν f_uvν f_vvν,wherestands for the standard inner product of ^3. Each integral curve of the foliations with respect to ω_lc is calleda line of curvature, each integral curve with respect to ω_as is called an asymptotic curve and each integral curve with respect to ω_ch is called a characteristic curve or harmonic mean curvature curve. Asymptotic curves appear only ona domain where the Gaussian curvature K of f is non-negative, andcharacteristic curves appear only ona domain where the Gaussian curvature K of f is non-positive. Since ω_ch=0 can be deformed as[ (NH-GK) dv^2+2(MH-FK) dudv+(LH-EK) du^2=0;⇔ N dv^2+2M dudv+L du^2G dv^2+2F dudv+E du^2=KH(= 2κ_1^-1+κ_2^-1), ]where K is the Gaussian curvature, H is the mean curvature, and κ_1, κ_2 are the principal curvatures of f, we see that along the characteristic curve, the normal curvature of it is equal to the harmonic mean of the principal curvatures (see <cit.>, for example). This paper is prepared while the author was visiting Luciana Martinsat IBILCE - UNESP. He would like to thank Luciana Martins for fruitful discussions. He would also like to thank Farid Tari for valuable comments. He would also like to thank the referee for careful reading and helpful suggestions. § PRELIMINARIES ON BDESIn this section, following<cit.>, we introduce a method to studythe configurations of the solution curves of a BDE. Let ω(u,v) bethe 2-tensor on (U;(u,v))⊂^2 as in (<ref>). If (a,b,c)(0,0,0) at x∈ U, thenω is called of Type 1 at x, and if (a,b,c)=(0,0,0) at x∈ U, thenω is called ofType 2 at x. If ω is of Type 2 at x, then δ=b^2-ac has a critical point at x. Since we are interested in local behavior of ω, we set x=(0,0). If ω is ofType 1, then ω defines a single direction at points on Δ, and if it is of Type 2, then all directions in the plane are solutions of ω=0 at that point. Moreover, if ω is of Type 2 at x, then Δ is not a smooth curve. We are interested inthe configurations of the foliationsof ω=0. We define the following equivalence.Two binary differential equationsω_1=0 and ω_2=0 are equivalent if there exist a diffeomorphism germ Φ:(^2,0)→(^2,0) and a non-zero function ρ:(^2,0)→ such that ρ(Φ^*ω_1)=ω_2 holds. If Φ is a homeomorphism such that Φ takes the integral curves of ω_1to those of ω_2, they are called topologically equivalent.If two binary differential equations are equivalent thenthe configurations of their foliations can be regarded the same. To obtain the topological configurations, we use the following method developed in<cit.>. We separate our consideration into the following three cases: * Case 1: (a(0),b(0),c(0))(0,0,0) and δ(0)0(Type 1).* Case 2: (a(0),b(0),c(0))(0,0,0) and δ(0)=0 (Type 1).* Case 3: (a(0),b(0),c(0))=(0,0,0) (Type 2).Consider the associated surface to ω={(u,v,[α,β])∈(^2,0)× P^1 |aβ^2+2bαβ+cα^2=0}.Thenis a smooth manifold if b^2-ac0,or if a_u+2b_up+c_up^2=0 and a_v+2b_vp+c_vp^2=0 do not have any common root. The second condition is equivalent toa_u 2b_u c_u 00 a_u 2b_u c_ua_v 2b_v c_v 00 a_v 2b_v c_v(0)0.Consider the projection π:→^2, π(u,v,[α:β])=(u,v). Then π^-1(u,v) consists of two points if b^2-ac>0, and is empty if b^2-ac<0. Let us set p=β/α(we need to consider the case q=α/β for some cases) and (u,v,p)=ap^2+2bp+c. If _p(0)0, then π is a local diffeomorphism, and if _p(0)=0 and _pp(0)0 hold,then π is a fold (a map-germ h:(^2,0)→(^2,0) is a fold ifh is -equivalent to (u,v)↦(u,v^2)). Let us consider the vector fieldξ(u,v,p)= p_p(u,v,p)∂_u+_p(u,v,p)∂_v -(p_u(u,v,p)+_v(u,v,p))∂ p.Then ξ is tangent to , and the projections dπ(ξ_1),dπ(ξ_2) satisfyω(dπ(ξ_i),dπ(ξ_i))=0(i=1,2),where ξ_i=ξ(u,v,p_i) andπ^-1(u,v) = {(u,v,p_1),(u,v,p_2)}, or π^-1(u,v) = {(u,v,p_1)}. To study geometric the foliations of ω, we use ξ on .§.§ Case 1We assume that b^2-ac>0 at 0. Then one can easily see that the BDE (<ref>)isequivalent toaBDE a dv^2+2b dudv+c du^2=0 which satisfies (a(0,0),b(0,0),c(0,0))=(1,0,-1). Furthermore,it holds that for any k>0, a BDE a dv^2+2b dudv+c du^2=0which satisfies (a(0,0),b(0,0),c(0,0))=(1,0,-1) is equivalent to a BDE a dv^2+2b dudv+c du^2=0 whose k-jet of (a,b,c) at (0,0) is(1,0,-1), moreover, it is equivalent to a BDE ω_reg=dv^2-du^2=0(<cit.>). Therefore the configuration is a pair oftransverse smooth foliations.§.§ Case 2We consider the case 2, namely(a(0),b(0),c(0))(0,0,0) and δ(0)=0. Weassumethat(a(0),b(0),c(0))(0,0,0)andδ(0) =0. If ω as in (<ref>) satisfiesj^1(a,b,c)(0,0)=(0,0,α_0) +(α_1v,α_2v, α_3u+α_4 v),then it is equivalent to aBDE a dv^2+2b dudv+c du^2=0 which satisfies (j^1a(0,0),j^1b(0,0),j^1c(0,0))=(1,0,u)(j^1a(0,0),j^1b(0,0),j^1c(0,0))=(1,0,u) when α_0α_10. We assume that α_0α_10. Consider u=V+β_1U^2+β_2V^2, v=U and R=1+β_3 U, where β_1,β_2,β_3∈. Then Rω(U,V) is given by[( α_1U+O(2)) dV^2 + (2(α_2+2α_0β_1)U+O(2)) dUdV; +(α_0+(α_4+α_0 β_3)U+ (α_3+4α_0 β_2)V+O(2) ) dU^2, ]where O(2) stands for remainders of order 2. Settingβ_1=-α_2/(2α_0),β_2=-α_3/(4α_0) and β_3=-α_4/α_0, and re-scaling, we get the desired result.Any BDE of the form (<ref>)with (j^1a(0,0),j^1b(0,0),j^1c(0,0))= (1,0,u) is smoothly equivalent toω_cusp=dv^2+u du^2=0(<cit.>, see also <cit.>,<cit.>). The solutions form a family of cusps.§.§ Case 3We assume (a(0),b(0),c(0))=(0,0,0). In this case, δ has a critical point at 0. We assumeis a smooth manifold. Since _p(0,0,p)=0 holds, ξ has a zero at (0,0,p)if and only if p_u(0,0,p)+_v(0,0,p)=0. This is a cubic equation for p. Set ϕ_ω(p)=ϕ(p)=p_u(0,0,p)+_v(0,0,p).Let D_ω=D denotes the discriminant of this equation. If D>0 then ϕ(p)=0 has three distinct real roots, and if D<0 then ϕ(p)=0 has one real and two distinct imaginary roots.When D>0, let p_1,p_2,p_3 be the solutions of ϕ(p)=0 and p_1<p_2<p_3. When D<0, let p_1 be the solution of ϕ(p)=0. If ϕ(0)=_v(0,0,0)=a_v(0)0 then p_i0 (i=1,2,3 or i=1) holds. We need to understand the singularity of ξ near p_i. If _u(0,0,p_i)0, thenis parameterized by v as (u(v,p),v,p) near (0,0,p_i). We denote the linear part of ξ byj^1ξ(0,0,p_i)= (ξ_11v+ξ_12(p-p_i))∂ v + (ξ_21v+ξ_22(p-p_i))∂ p.Also we remark that since (u(v,p),v,p)≡0, it holds that _uu_v+_v≡0, where ≡ means that the equality holds identically. On the other hand, p_i is a solution of ϕ(p)=0, and ϕ(p)=p_u+_v, we haveu_v(0,p_i)=p_i.Furthermore, _p=0 at (u,v)=(0,0), it holds that u_p=0 at (v,p)=(0,p_i). We have[ξ_11 =_upu_v+_pv;ξ_12 =_upu_p+_pp; =0( at (0,0,p_i));ξ_22 = -(_u+p_uuu_p+p_up+_uvu_p+_vp); ]Thus the eigenvalues of the linear part of ξ areα(p_i)= _upu_v+_pv and-ϕ'(p_i)= -(_u+p_uuu_p+p_up+_uvu_p+_vp).The configuration of the integral curves of ω is determined by these information. The following theorem is known. Let δ(0,0) be the determinant of the Hesse matrix of δ(u,v) at (0,0).<cit.> Let ωbe a 2-tensor as in (<ref>) and satisfies (a,b,c)(0)=(0,0,0), δ(0,0)<0, D0 and ϕ(p) and α(p) do not have any common roots. Then the BDE ω=0 is topologically equivalent to one of the following BDEs: * The case D>0 : (Then ϕ(p)=0 has 3 roots p_1,p_2,p_3.) * ω_3s=vdv^2+2ududv+vdu^2=0 (3 saddles) when-ϕ'(p_i)α(p_i) are negative for all i=1,2,3.* ω_2s1n=vdv^2+2(v-u)dudv+vdu^2=0 (2 saddles + 1 node) when -ϕ'(p_i)α(p_i) are two negative and one positive for i=1,2,3.* ω_1s2n=3vdv^2-4ududv+3vdu^2=0 (1 saddle + 2 nodes) when -ϕ'(p_i)α(p_i) are one negative and two positive for i=1,2,3. * The case D<0 :(Then ϕ(p)=0 has 1 root p_1.) * ω_1s=vdv^2+2ududv+vdu^2=0 (1 saddle) when-ϕ'(p_1)α(p_1) is positive.* ω_1n=2vdv^2-ududv+2vdu^2=0 (1 node) when-ϕ'(p_1)α(p_1) is negative.Note that in the case of D>0, all-ϕ'(p_i)α(p_i) i=1,2,3cannot be positive, see <cit.>. The integral curves of the aboveBDEs are in Figures <ref>, <ref>, <ref> which are taken from <cit.>.§ GEOMETRIC BINARY DIFFERENTIAL EQUATIONS ON A CUSPIDAL EDGELet f:(^2,0)→(^3,0) be a parametrization ofa cuspidal edge. We take f as in (<ref>). Then the coefficients of the first fundamental formof the cuspidal edge with respect to f are[E= 1+(a_20^2 + b_20^2)u^2+(a_20 a_30+b_20 b_30)u^3+b_12 b_20 uv^2; + 14(a_30^2 + b_30^2 + 2 a_20 p(0) +2 b_20 q_1(0)) u^4; +12(b_12 b_30+8 b_20q_2(0)) u^2v^2 +2b_20q_3(0) uv^3;+14b_12^2 v^4 + O(5), ] [F= a_20uv+ (12a_30+ b_12b_20)u^2v+ 12b_03b_20uv^2; + 14b_03 b_12 v^4+(12b_12^2+4 b_20 q_4(0,0)) u v^3; +14(b_03 b_30+12 b_20 q_3(0))u^2 v^2; +(12b_12 b_30+4p(0)+2 b_20 q_2(0)) u^3v+ O(5),;] [ G = v^2+ 14b_03^2 v^4+b_03 b_12 u v^3+b_12^2 u^2v^2 + O(5), ]where O(n) stands for remainders of order n(n=1,2,…). Since f is as in (<ref>), we see f_u× (f_v/v)0. We set ν_2=f_u× (f_v/v), andL_2=f_uuν_2,M_2=f_uvν_2,N_2=f_vvν_2. Then we have:[ L_2 =b_20+ (b_30- a_20 b_12) u -12a_20 b_03 v;-(a_30 b_12-12 q_1(0)+2 a_20 q_2(0)) u^2; -(a_30 b_03+6 a_20 q_3(0))u v; + (2 q_2(0)-4 a_20 q_4(0,0)) v^2 + O(3); M_2 =b_12 v+4q_2(0) u v + 3 q_3(0) v^2 + O(3); N_2 = 12b_03 v+3 q_3(0)uv+ 8 q_4(0,0) v^2 + O(3). ]We have L_2=L|ν_2|, M_2=M|ν_2|, N_2=N|ν_2|. It should be remarked that there exist C^∞-functions F,G,N,M such that G=v^2 G, F=v F, N_2=v N and M_2=v M holds. We set E=E, F=Fv, G=Gv^2, L=L_2, M=M_2v, N=N_2v. §.§ Lines of principal curvatureIn this subsection we consider the BDE ω_lc=0. Using (<ref>), ω_lc=0 is equivalent tov^2(FN - vGM)dv^2+v(EN-v^2GL)dudv +v(EM-FL)du^2=0.To determine the topological configuration of ω_lc=0, we factor out v and consider ω_lc=0, whereω_lc = v(FN - vGM)dv^2+(EN-v^2GL)dudv +(EM-FL)du^2.We have the following proposition.The BDE ω_lc=0 is equivalent tothe BDE ω_reg=0.This proposition implies that the lines of principal curvature of a cuspidal edge form a pair of smooth and transverse foliations in the domain of a parametrization.Set(a,b,c)=( vFN - v^2 GM, 12(EN -v GL), EM-FL).Then sinceb(0)=E(0)N(0), E(0)=f_uf_u(0)0 and N(0)=f_vvvν_2(0)=b_030 hold, ω_lc is as in Case 1. Moreover, since a(0)=0 and b(0)0, we see that b^2-ac>0 at 0. Hence ω_lc is equivalent to ω_reg=0 (See Section <ref>).The fact of the existence of the curvatureline coordinate system at cuspidal edge is also shown in <cit.>.An example of picture of this configuration onthe cuspidal edges is in Figures <ref>. Since one family of the integral curvesare tangent to the null direction on singular curve, one family of the integral curvesnear singular curve form the (2,3)-cusps. A map-germ (,0)→(^3,0) is an (2,3)-cusp if it is -equivalent to t↦(t^2,t^3).§.§ Asymptotic curvesIn this subsection we consider ω_as=0, and it is equivalent to ω_as=N_2 dv^2+2M_2 dudv+N_2 du^2=0. Since M_2(u,0)=0, and N_2(u,0)=0, the singular set i.e., the cuspidal edge curve is part of discriminant of ω_as=0, and, ∂_v is a solution to ω_as=0 on the singular set. By (<ref>),[ δ(u,v)= -12b_20b_03v +(1/4a_20 b_03^2 + b_12^2 - 8 b_20q_4(0,0)) v^2; + 12 b_03(a_20 b_12 - b_30-2b_20q_3(0))uv;+ (-b_03q_2(0) + 6 b_12 q_3(0);+ 6 a_20 b_03 q_4(0,0)-7b_20(q_4)_v(0,0))v^3; + ( 1/4a_30 b_03^2 + 3 a_20 b_03 q_3(0) - 8 b_30 q_4(0,0); + 8 b_12q_2(0)+ 8 a_20b_12 q_4(0,0) )uv^2; + (1/2a_30 b_03 b_12 - 6 b_03 q_1(0) + a_20 b_03 q_2(0); + 3 a_20 b_12 q_3(0) - 3 b_30 q_3(0))u^2v + O(4). ]Thus if b_200, thenthe BDE is in Case 2. In this case, by (<ref>), ω_as=0 is equivalent to ω_cusp=0 (see Subsection <ref>). By <cit.>,b_200 implies that the Gaussian curvatureis unbounded and changes sign betweenthe two sides of the cuspidal edge. This means that in this case, the singular set of cuspidal edge plays the same role as theparabolic curve on regular surfaces. Sinceb_200, thenthe BDE is ω_cusp=0, this implies that the folded saddle, the folded node, the folded focus in Davydov's classification <cit.> (see also <cit.>) does not appear not only cuspidal edges, but also all singularities written in the form (<ref>) (for instance, cuspidal cross cap).We assume now b_20=0.Then the BDE is in Case 3. We have the following. If a_20 b_12 - b_300, then δ is a Morse function near 0. We study the geometric foliation near 0as in Subsection <ref>. We consider(u,v,p)= N_2+2M_2p+L_2p^2.Then we have[ (_u,_v,_p)(u,v,p) = ( (N_2)_u+2 (M_2)_up+ (L_2)_up^2,;(N_2)_v+2 (M_2)_vp+ (L_2)_vp^2, 2 M_2+2 L_2p),; (_u,_v,_p)(0,0,p) =( (L_2)_up^2,(N_2)_v+2 (M_2)_vp+ (L_2)_vp^2, 2 Lp); = ((b_30-a_20 b_12) p^2, 12(b_03+4 b_12 p-a_20 b_03 p^2), 0). ]In this case, the left hand side of (<ref>) is (b_30-a_20b_12)^2b_03^2/4. Thus if b_30-a_20b_120 at 0, then ={=0} is a smooth manifold. We have ϕ_as=ϕ_ω_asand D_as=D_ω_asdefined in Subsection <ref> as follows[ϕ_as(p)=(b_30-a_20 b_12) p^3 -1/2 a_20 b_03 p^2 +2 b_12 p +1/2b_03,;4D_as= a_20^3 b_03^4 +13 a_20^2 b_03^2 b_12^2 -b_30 (128 b_12^3+27 b_03^2 b_30); +2 a_20 (64 b_12^4+9 b_03^2 b_12 b_30). ]Furthermore, α(p)is given byα(p)= 2 (b_30-a_20 b_12) p^2-a_20 b_03 p+2 b_12.If p_i is a solution of ϕ_as(p)=0 and pα(p)-2ϕ_as(p)=-b_03 - 2 b_12 p holds, then α(p_i)0 if and only if -b_03 - 2 b_12 p_i0. Assume that b_120. Substituting p=-b_03/(2 b_12) into ϕ_as(p), we getϕ_as(-b_032 b_12) = -18b_12^3 b_03(4 + b_03^2 b_30).If b_12=0, then pα(p)-2ϕ_as(p)=-b_030. Since we assume that b_30-a_20 b_120,if b_12=0 then b_300. Thus we get α(p_i)0 if and only if4b_12^3 + b_03^2 b_300.We can now use Theorem <ref> to obtain the following result.If b_200,then ω_as is equivalent to ω_cusp=0.If b_20=0, b_30-a_20 b_120, D_as0 and 4b_12^3 + b_03^2 b_300, then ω_as is topologically equivalent to one of the following: * The case D_as>0 :(Then ϕ_as(p)=0 has 3 roots p_1,p_2,p_3) * ω_3s (-ϕ_as'(p_i)α(p_i) are negative for all i=1,2,3).* ω_2s1n (-ϕ_as'(p_i)α(p_i) are two negative and one positive for i=1,2,3).* ω_1s2n (-ϕ_as'(p_i)α(p_i) are one negative and two positive for i=1,2,3). * The case D_as<0 :(Then ϕ_as(p)=0 has 1 root p_1) * ω_1s (-ϕ_as'(p_1)α(p_1) is negative).* ω_1n (-ϕ_as'(p_1)α(p_1) is positive). We observe that by the Proposition <ref>, b_20, b_30-a_20 b_12 and 4b_12^3 + b_03^2 b_30 have geometric meanings. In fact, b_20 is the limiting normal curvature and b_30-a_20 b_12 coincides with the derivationof limiting normal curvature (see <cit.>). The invariant 4b_12^3 + b_03^2 b_30 is related to the singularities of parallel surfaces of the cuspidal edge (see <cit.>). §.§ Characteristic curvesWe consider the BDE ω_ch=0. Using (<ref>), we show that ω_ch=0 is equivalent to v(a dv^2+2b dudv+c du^2)=0, where[a= v (EN^2 + (-GLN - 2 FMN)v+ 2 GM^2 v^2);b= v (-2 FLN + EMN + GLM v);c=-ELN+ (GL^2- 2 FLM +2 EM^2 ) v. ]We factor out v, so ω_ch=0 is topologically equivalent toω̃_ch=a dv^2+2b dudv+c du^2=0. Since a(u,0)=0, and b(u,0)=0, the singular set is a part of discriminant of ω̃_ch=0, and ∂_v is a solution to ω̃_ch=0 on the singular set. The function δ=b^2-ac is given by[ v[E^2 LN^3+(4 F^2 L^2-2 EGL^2 -4 EFLM -2 E^2 M^2) N^2v; + ( - G^2 L^3-4FGL^2 M -4F^2 LM^2 +6EGLM^2 +4EFM^3) Nv^2;+ ( GL^2 -4FLM+4EM^2) GM^2v^3]. ]When f is taken as in (<ref>), we have[ a =1/4b_03^2v+O(2),; b =1/2b_12b_03v+O(2),; c = -1/2b_20b_03 +1/2(a_20 b_12 b_03-b_30 b_03 -6 b_20 q_3(0))u; +1/4[ a_20 b_03^2+8 b_12^2+4b_20(b_20-8 q_4(0,0)) ]v +O(2),; δ = 18b_03^3 b_20v + 18b_03^2(-a_20 b_03 b_12 +b_03 b_30+18 b_20 q_3(0))u v; - 116b_03^2[a_20 b_03^2+4[b_12^2+2 b_20(b_20-12 q_4(0,0))]]v^2 +O(3). ]Since b_030,if b_200, then ω_ch is as in Case 2, and if b_20=0, then it is as in Case 3. In the following, we divide our consideration into these two cases. §.§.§ The case b_200 :By the argument in Subsection <ref>, ω_ch=0 is equivalent to ω_cusp=0. In this case, by (<ref>), ω_as=0 is equivalent to ω_cusp=0 (see Subsection <ref>). Like as the case of ω_as=0, the singular set of cuspidal edge plays the same role as theparabolic curve on regular surfaces. Moreover, the folded saddle, the folded node, the folded focus do not appear.§.§.§ The case b_20=0 :The left hand side of (<ref>) is b_03^6(a_20b_12-b_30)^2/64. Thusis a smooth manifold if a_20b_12-b_300 at 0. Furthermore,δ is of Morse type if and only if a_20 b_12-b_300. This is exactly the same conditionsas the case of asymptotic curves. We assume that a_20 b_12-b_300. We need to consider (u,v,p)=a+2bp+cp^2 and ϕ_ch(p)=(p_u+_v)(0,0,p). We have[4ϕ_ch(p) = b_03^2+ 4 b_03 b_12 p; + (a_20 b_03^2 + 8 b_12^2) p^2+ (2 a_20 b_03 b_12 - 2 b_03 b_30) p^3, ]and the discriminant D_ch=D_ω_ch of the cubic ϕ_ch isgiven by[D_ch =-b_03^264( a_20^3 b_03^6+11 a_20^2 b_03^4 b_12^2; -2 a_20 (16 b_03^2 b_12^4+9 b_03^4 b_12 b_30); +256 b_12^6 +160 b_03^2 b_12^3 b_30+27 b_03^4 b_30^2 ). ]Furthermore, 4α is given by4α(p)= 2b_03(a_20b_12-b_30)p^2+(a_20b_03^2+8b_12^2)p+2b_12b_03.Then we have[ 4ϕ_ch'(p)α(p)= 4 b_03^2 b_12^2 +4 b_03 b_12 (a_20 b_03^2+8 b_12^2) p; +(a_20^2 b_03^4+26 a_20 b_03^2 b_12^2+64 b_12^4-10 b_03^2 b_12 b_30) p^2; +5 b_03 (a_20 b_03^2+8 b_12^2)(a_20 b_12-b_30) p^3; +6 b_03^2 (-a_20 b_12+b_30)^2 p^4, ]and the condition that ϕ_ch(p) and α(p) do not have any common roots is given by 4b_12^3 + b_03^2 b_300. We summerize the above discussion in the following proposition.If b_200,then ω_ch is equivalent to ω_cusp=0.If b_20=0, b_30-a_20 b_120, D_ch0 and 4b_12^3 + b_03^2 b_300, then ω_ch is topologically equivalent to one of the following: * The case D_ch>0 :(Then ϕ_ch(p)=0 has 3 roots p_1,p_2,p_3) * ω_3s (-ϕ_ch'(p_i)α(p_i) are negative for all i=1,2,3).* ω_2s1n (-ϕ_ch'(p_i)α(p_i) are two negative and one positive for i=1,2,3).* ω_1s2n (-ϕ_ch'(p_i)α(p_i) are one negative and two positive for i=1,2,3). * The case D_ch<0 :(Then ϕ_ch(p)=0 has 1 root p_1) * ω_1s (-ϕ_ch'(p_1)α(p_1) is negative).* ω_1n (-ϕ_ch'(p_1)α(p_1) is positive). We remark that if b_12=0, then D_as=D_ch. Namely, the configrations of foliations with respect to ω_as and ω_ch are of the same type.Examples of pictures of these configrations onthe cuspidal edges are in Figures <ref>, <ref> and <ref>. Since the integral curves emanate from singular curve along the null direction, integral curves near the singular curve do not form the (2,3)-cusp but form the (3,4)-cusps (see Appendix <ref>). § GENERIC FOLIATIONSInPropositions <ref> and <ref>,all the conditions are written in terms of the 3-jet of (<ref>). We can state a genericity result for cuspidal edge. By (<ref>), we identify the set of jets of parametrization of cuspidal edges with (0,0)[ _k={(j^ka_1(0),j^kb_2(0),j^kb_3(0),j^kb_4(0,0)) ∈ J^k(1,1)^3× J^k(2,1)|; a_1(0)=a_1'(0)=b_2(0)=b_2'(0)=b_3(0)=0,b_2'(0)>0, b_4(0,0)0}, ]for k≥3. With notation as before, consider(a_20,a_30,…,b_20,b_30, …,b_12,…,b_03,…)as a coordinate system of _k (cf. (<ref>)). Define a subset of _k by[= {b_20=0, b_30-a_20b_12=0}∪{b_20=0, D_asy=0}; ∪{b_20=0, 4b_12^3-b_03^2b_30=0}∪{b_20=0, D_hmc=0}. ]Thenis an algebaric subset of _k of codimension 2. Since the singular set of a cuspidal edgeis a curve, generically it will avoid the set . This implies that for generic cuspidal edges the configuration of ω_lc,ω_as,ω_ch are those in Propositions <ref>,<ref>,<ref>. § CRITERIA FOR (3,4) AND (3,4,5)-CUSP In this section, we state criteria for (3,4) and (3,4,5)-cusp. Setc_1(t)=(t^3,t^4,0,…,0), c_2(t)=(t^3,t^4,t^5,0,…,0),and a map-germ γ:(,0)→(^n,0), where n≥2 (respectively, n≥ 3) is called (3,4)-cusp (respectively, (3,4,5)-cusp) if γ is -equivalent to the map-germc_1 (respectively, c_2) at 0.A map-germ γ:(,0)→(^n,0), where n≥3 (respectively, n≥ 2) is (3,4,5)-cusp (respectively, (3,4)-cusp) if and only if (i)γ'(0)=γ”(0)=0,(ii)γ^(3)(0),γ^(4)(0) and γ^(5)(0) are linearly independent (respectively, linearly dependent, and γ^(3)(0) and γ^(4)(0)are linearly independent) where ( )^(i)=d^i/dt^i. Although this proposition is known <cit.>, we give a sketch of proof for the readers who are not familiar with it.SinceΦ(γ(t))^(i)=dΦ_0(γ^(i)(0))(i=3,4,5)holds for a map Φ:(^n,0)→(^n,0) under the assumption γ'(0)=γ”(0)=0, it is obvious that the conditions do not depend on the parameter and the coordinate system on ^n.To show theproposition, it is enough to show that (t^3+a_1t^5,t^4+a_2t^5,a_3t^5,0,…,0)+O(6) is -equivalent to (t^3,t^4,a_3t^5,0,…,0)+O(6), where a_1,a_2,a_3∈. Considering the parameter change t↦ t-a_2t^2/4-(16 a_1 + 3 a_2^2)t^3/48, it can be proved.Using Proposition <ref>, we have the following:Let f:(^2,0)→(^3,0) be a cuspidal edge and γ:(,0)→(^2,0) an ordinary cusp such that df_0(γ”(0))=0. Then γ̂=f∘γ is a (3,4)-cusp at 0. Without loss of generality, one can assume that f is given by the form (<ref>), and γ(t)=(t^3+a_4t^4+a_5t^5+O(6),t^2) (a_4,a_5∈),because df_0(∂ v)=0. Then γ̂(t)=(t^3+a_4t^4+a_5t^5,t^4/2,0)+O(6). By Proposition <ref>, we have the conclusion.99 AGV V. 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Tari, Apparent contours in Minkowski 3-space andfirst order ordinary differential equations, Nonlinearity 26 (2013), no. 4, 911–932. ist S. Izumiya, K. Saji and N. Takeuchi, Flat surfaces along cuspidal edges, preprint. krsuy M. Kokubu, W. Rossman, K. Saji, M. Umehara and K. Yamada, Singularities of flat fronts in hyperbolic 3-space, Pacific J. Math. 221 (2005), no. 2, 303–351. MS L. F. Martins and K. Saji, Geometric invariants of cuspidal edges, Canad. J. Math. 68 (2016), 445–462.MSUY L. F. Martins, K. Saji, M. Umehara and K. Yamada,Behavior of Gaussian curvature andmean curvature near non-degenerate singularpoints on wave fronts,Geometry and Topology of Manifolds 154 Springer Proc. Math. and Statistics, 247–281.mu S. Murata and M. Umehara, Flat surfaces with singularities in Euclidean 3-space, J. Differential Geom. 82 (2009), no. 2, 279–316.nuy K. Naokawa, M. Umehara and K. Yamada, Isometric deformations of cuspidal edges, Tohoku Math. J. (2) 68 (2016), 73–90.joey J. M. Oliver, On pairs of foliations of a parabolic cross-cap, Qual. Theory Dyn. Syst. 10 (2011), 139–166.ot R. Oset Sinha and F. Tari, On the flat geometry of the cuspidal edge, preprint, arXiv:1610.08702.front K. Saji, M. Umehara, and K. Yamada, The geometry of fronts, Ann. of Math. 169 (2009), 491–529.faridtari F. Tari, On pairs of geometric foliations on a cross-cap,Tohoku Math. J. 59 (2007), no. 2, 233–258.faridsurvey F. Tari, Pairs of foliations on surfaces, Real and complex singularities,London Math. Soc. Lecture Note Ser. 380, 305–337,teramoto K. Teramoto, Parallel and dual surfaces of cuspidal edges, Differential Geom. Appl. 44 (2016), 52–62.t R. Thom, Sur les équations différentielles multiformes et leursintégrales singulières, Bol. Soc. Brasil. Mat. 3 (1972), no. 1, 1–11. Department of Mathematics,Graduate School of Science, Kobe University, Rokko, Nada, Kobe 657-8501, JapanE-mail: sajiOamath.kobe-u.ac.jp | http://arxiv.org/abs/1703.08903v1 | {
"authors": [
"Kentaro Saji"
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"math.GT",
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"primary_category": "math.GT",
"published": "20170327022756",
"title": "On pairs of geometric foliations on a cuspidal edge"
} |
Cosmic Equilibration: A Holographic No-Hair Theorem from the Generalized Second Law Sean M. Carroll and Aidan Chatwin-Davies December 30, 2023 ===================================================================================== * CNR-NANO, Via Campi 213a, 41125 Modena, Italy. * SISSA & CNR-IOM Democritos, Via Bonomea 265, 34136 Trieste, Italy. * CNR-ISM, Division of Ultrafast Processes in Materials (FLASHit),Area della Ricerca di Roma 1, 00016 Monterotondo Scalo, Italy. * Present address: Physics & Materials Science Research Unit,University of Luxembourg, 162a avenue de la Faïencerie,1511 Luxembourg, Luxembourg. * Present address: Dipartimento di Scienze Chimiche, Universitàdegli Studi di Padova, Via Marzolo 1, 35131 Padova, Italy. * Dipartimento di Scienze Fisiche, Informatiche e Matematiche (FIM), Università degli Studi di Modena e Reggio Emilia, 41125 Modena, Italy.Fifty years ago Walter Kohn speculated that a zero-gap semiconductor might be unstable against the spontaneous generation of excitons—electron-hole pairs bound together by Coulomb attraction. The reconstructed ground state would then open a gap breaking the symmetry of the underlying lattice, a genuine consequence of electronic correlations. Here we show that this excitonic insulator is realized in zero-gap carbon nanotubes by performing first-principles calculations through many-body perturbation theory as well as quantum Monte Carlo. The excitonic order modulates the charge between the two carbon sublattices opening an experimentally observable gap, which scales as the inverse of the tube radius and weakly depends on the axial magnetic field. Our findings call into question the Luttinger liquid paradigm for nanotubes and provide tests to experimentally discriminate between excitonic and Mott insulator.Long ago Walter Kohn speculated that grey tin—a zero-gapsemiconductor—could be unstable against the tendency of mutually attracting electrons and holes to form bound pairs,the excitons<cit.>. Being neutral bosoniclike particles, the excitons would spontaneously occupy the same macroscopic wave function, resulting in a reconstructed insulating ground state with a broken symmetry inherited from the exciton character<cit.>. This excitonic insulator (EI) would share intriguingsimilarities with the Bardeen-Cooper-Schrieffer (BCS) superconductor groundstate<cit.>, the excitons—akin to Cooper pairs—forming only below a critical temperature and collectively enforcing a quasiparticle gap.The EI was intensively sought afterin systems as diverse as mixed-valencesemiconductors and semimetals<cit.>,transition metal chalcogenides<cit.>,photoexcited semiconductors at quasi equilibrium<cit.>, unconventional ferroelectrics<cit.>, and, noticeably, semiconductor bilayers in thepresence of a strong magnetic field that quenches the kineticenergy of electrons<cit.>. Other candidates include electron-hole bilayers<cit.>, graphene<cit.> and related two dimensional structures<cit.>,where the underscreened Coulomb interactions might reach thecritical coupling strength stabilizing the EI. Overall, the observation of the EI remains elusive.Carbon nanotubes, which are rolled cylinders of graphene whose low-energy electrons are massless particles<cit.>,exhibit strong excitonic effects, due toineffective dielectric screening and enhanced interactionsresulting from one dimensionality<cit.>. As single tubes can be suspended to suppress the effects ofdisorder and screening by the nearby substrate orgates<cit.>, the field lines of Coulomb attraction between electron and hole mainly lie unscreened in the vacuum (Fig. 1a). Consequently, the interaction is truly long-ranged and in principle—even for zero gap—ableof binding electron-hole pairs close to the Dirac point in momentum space (Fig. 1b). If the binding energy is finite, then the ground state is unstable against the spontaneous generation of excitons having negative excitation energy,ε_u<0. This is the analogue of the Cooper instability that heralds the transition to the superconducting state—the excitons replacing the Cooper pairs. Here we focus on the armchair family of zero-gap carbon nanotubes, because symmetry prevents their gap from opening as an effect of curvature or bending<cit.>. In this paper we show that armchair tubes are predicted to be EIs by first-principles calculations. The problem is challenging, because the key quantitiescontrolling this phenomenon—energy band differences andexciton binding energies—involve many-body corrections beyonddensity functional theory that are of the order of a few meV,which is close to the limits of currently available methods.In turn, such weak exciton binding reflects in the extreme spatial extension of the exciton wave function, hence itslocalization in reciprocal space requires very high sampling accuracy.To address these problems, we perform state-of-the-art many-bodyperturbation theory calculations within the GW and Bethe-Salpeterschemes<cit.>. We find that bound excitons exist inthe (3,3) tube with finite negative excitation energies.We then perform unbiased quantum Monte Carlosimulations<cit.> to prove that the reconstructed ground state is the EI, its signature being the broken symmetry between inequivalent carbon sublattices—reminescent of the exciton polarization. Finally, to investigate the trend with the size of the system, which is not yet in reach of first-principles calculations,we introduce an effective-mass model, which showsthat both EI gap and critical temperature fall in the meV range and scale with the inverse of the tube radius.Our findings are in contrast with the widespread belief that electrons in undoped armchair tubes form a Mott insulator—astrongly correlated Luttinger liquid<cit.>. We discuss the physical origin of this conclusion and propose independent experimental tests to discriminate between excitonic and Mott insulator. § RESULTS § EXCITON BINDING AND INSTABILITY For the sake of computational convenience we focus on the smallest (3,3) armchair tube,which was investigated several times from first principles<cit.>.We first check whether the structural optimization of the tube might lead to deviations from the ideal cylindrical shape, affecting the electronic states.Full geometry relaxation (Methods) yields an equilibrium structure withnegligible corrugation. Thus, contrary to a previousclaim<cit.>, corrugation cannot be responsible of gap opening.We find that the average length of C-C bonds along the tube axis, 1.431 Å, is shorter than around the circumference, 1.438 Å, in perfect agreement with the literature<cit.>.We use density functional theory (DFT) to compute the band structure (solid lines in Fig. 2a), which provides the expected<cit.> zero gap at the Dirac point K.In addition, we adopt the G0W0 approximationfor the self-energy operator<cit.> to evaluatemany-body corrections to Kohn-Sham eigenvalues. The highest valence and lowest conduction bands are shown as dashed lines.The zoom near K (Fig. 2b) shows that electrons remain massless, with their bands stretched by ∼ 28% with respect to DFT(farther from K the stretching is ∼ 13%,as found previously<cit.>). Since electrons and holes in these bands have linear dispersion, they cannot form a conventional Wannier exciton, whose binding energy is proportionalto the effective mass. However, the screened e-h Coulomb interaction V(z) along the tube axis, projected onto thesame bands, has long range (Fig. 2c)—a remarkable effectof the topology of the tube holding even for vanishing gap. Consequently, V(q) exhibits a singularity in reciprocal space at q=0 (smoothed by numerical discretization in the inset of Fig. 2c),which eventually binds the exciton.We solve the Bethe-Salpeter equation over an ultradense grid of 1800 k-points, which is computationally very demanding but essentialfor convergence. We find several excitons with negative excitation energiesε_u,in the range of 1–10 meV (Table 1).The exciton spectral weight is concentratedin a tiny neighbourhood of K and K' points in reciprocalspace (Fig. 3b), hence the excitons are extremely shallow, spread over microns along the axis (Fig. 3c).Only e-h pairs with negative k in valley K andpositive k in valley K'contribute to the exciton wave function, which is overall symmetric under time reversal but not underaxis reflection within one valley, k→ -k, as shown in Fig. 3b (the axis origin is at Dirac point).On the contrary, the wavefunctions of excitons reported so far in nanotubes<cit.> are symmetric in k-space. The reason of this unusual behavior originates fromthe vanishing energy gap, since then e-h pairs cannot be backscattered by Coulomb interaction due to the orthogonality of initialand final states<cit.>.In addition, pair energies are not degenerate for k→ -k, as Dirac cones are slightly asymmetric (Supplementary Discussion and Supplementary Fig. 10).The exciton with the lowest negative ε_umakes the system unstable against the EI.The transition density, ϱ_tr(𝐫) =<u|ϱ̂ (𝐫)|0>, hints at the broken symmetryof the reconstructed ground state, as it connects the noninteracting ground state, |0>,to the exciton state, |u>,through the charge fluctuation operator ϱ̂ (Fig. 3d). Here we focus on the simpler charge order (spin singlet excitons) and neglect magnetic phenomena (spin triplet), as the onlyrelevant effect of spin-orbit coupling in realtubes<cit.> is to effectively mix both symmetries. Figure 3d may be regarded as a snapshotof the polarization charge oscillationinduced by the exciton, breaking the inversion symmetry between carbon sublattices A and B.Note that this originates from the opposite symmetries of |0> and |u> under A ↔ Binversion and not from the vanishing gap. This charge displacement between sublattices is the generic signature of the EI, asits ground state may be regarded as a BCS-like condensate ofexcitons |u>(see the formal demonstration in Supplementary Note 5).§ BROKEN SYMMETRY OF THE EXCITONIC INSULATOR We use quantum Monte Carlo to verify the excitonic nature of the many-body ground state, by defining an order parameter characteristic of the EI, ϱ_AB. In addition, we introduce an alternative order parameter,ϱ_Transl, peculiar to a dimerized charge density wave(CDW) similar to the Peierls CDW predicted by some authors<cit.> for the smallest armchair tubes.The EI order parameter measures the uniform charge displacement between A and B sublattices, ϱ_AB=( ∑_i∈A n_i -∑_i∈B n_i ) / N_atom, whereas ϱ_Transl detects any deviation from the periodicity of the undistorted structure by evaluating the charge displacement between adjacent cells, ϱ_Transl= ∑_in_i (-1)^i_z / N_atom (Figs. 4b-e). Here the undistorted structure is made of a unit cell of twelve C atoms repeated along the z direction with a period of 2.445 Å and labeled by the integer i_z, n_i is the operator counting theelectrons within a sphere of radius1.3 a.u. around the ith atom, and N_atom is the total number of atoms in the cluster.Both order parameters ϱ_AB and ϱ_Transl vanish in the symmetric ground state of the undistorted structure, which is invariant under sublattice-swapping inversionand translation symmetries. We then perform variational Monte Carlo (VMC),using a correlated Jastrow-Slater ansatz that has proved<cit.>to work well in 1D correlated systems (Methods), as well as it is able to recover theexcitonic correlations present in the mean-field EI wavefunction<cit.> (Supplementary Discussion). We plot VMC order parameters in Fig. 4a. Spontaneously broken symmetry occurs in the thermodynamic limitif the square order parameter,either ϱ_AB^2 or ϱ_Transl^2, scales as 1/N_atom and has a non vanishing limit value for N_atom→∞.This occurs for ϱ_AB^2 (black circles in Fig. 4a), confirming the prediction of the EI, whereas ϱ_Transl^2 vanishes (red squares), ruling out the CDW instability (see Supplementary Discussion as well as the theoretical literature<cit.> for the Peierls CDW case). We attribute the simultaneous breaking of sublattice symmetry and protection of pristine translation symmetryto the effect of long-range interaction. The vanishing of ϱ_Transl validates the ability of our finite-size scaling analysis to discriminate between kinds of order in the bulk. Though the value of ϱ_AB after extrapolation is small, ϱ_AB=0.0165 ± 0.0007, it is non zero within more than twenty standard deviations. Besides, the quality of the fit of Fig. 4a appears good, because the data for the five largest clusters are compatible with the linear extrapolations of both ϱ_AB^2 and ϱ_Transl^2 within an acceptable statistical error. The more accurate diffusion Monte Carlo (LRDMC) values(obtained with the lattice regularization),shown in Fig. 4a as blue circles, confirm the accuracy of the variational calculation.However, as their cost is on the vergeof present supercomputing capabilities, we were unable to treat clusters larger that N_atom=48, hence the statistical errors are too large to supporta meaningful non zero value in thethermodynamic limit. Nevertheless, we obtain a non zero LRDMC value smallerthan the one estimated by VMC but compatible with it within a few standard deviations. § TRENDS As the extension of ouranalysis to systems larger than the (3,3) tube is beyond reach, we design an effective-mass theoryto draw conclusions about trends in the armchair tube family, in agreement withfirst-principles findings. We solve the minimal Bethe-Salpeter equation for the masslessenergy bands ε(k)=±γ|k|(Fig. 2b and Supplementary Note 1) and thelong-range Coulomb interaction V(q), the latter diverging logarithmically in one dimension for small momentum transfer q,V(q)=(2e^2/Aκ_r)ln(|q|R)(inset of Fig. 2c and Supplementary Note 2). Here γ is graphene tight-binding parameter includingGW self-energy corrections,k is the wave vector along the axis, A is the tube length,R is the radius, and κ_raccounts for screening beyond the effective-mass approximation. By fitting the parameters γ=0.5449 eV·nm andκ_r=10 to our first-principles data, we obtain anumerical solution of Bethe-Salpeter equation recoveringapproximately 60% of the lowest exciton energy ε_u reported in Table 1 (Supplementary Note 3).Moreover, the wave function agrees with theone obtained from first principles (Fig. 3b, c). Importantly, ε_u smoothly converges in an energy range that—for screened interaction—is significantly smallerthan the extension of the Dirac cone, with no need of ultraviolet cutoff (Supplementary Fig. 9). Therefore,the exciton has an intrinsic length (binding energy),which scales like R (1/R).We adopt a mean-field theory of the EI as we expectthe long-range character of excitonic correlationsto mitigate the effects of quantum fluctuations. The EI wave function can be described as|Ψ_EI>=∏_σσ'τ k[u_τ k + χ_σσ' v_τ k e^i η ĉ^τ +_k,σv̂^τ_k,σ'] | 0 >.Here | 0 > is the zero-gap ground state with all valence states filled and conduction states empty, the operator ĉ^τ +_k,σ (v̂^τ +_k,σ) creates an electron in the conduction (valence) bandwith wave vector k, spin σ, valley τ= K or K', η is an arbitrary phase, and the 2× 2 matrix χ_σσ' discriminates between singlet and triplet spin symmetries of the e-h pair ĉ^τ +_k,σv̂^τ_k,σ'| 0 > (Fig. 1b). The positive variational quantities u_τ k and v_τ k are the population amplitudes of valence and conduction levels, respectively, with u_τ k^2 + v_τ k^2=1. Whereas in the zero-gap state u_τ k=1 and v_τ k=0, in the EI state both u_τ k and v_τ k are finiteand ruled by the EI order parameter Δ(τ k), according tou_τ k v_τ k=|Δ(τ k)|/2E(τ k), with E(τ k)=[ε(τ k)^2 + |Δ(τ k)|^2]^1/2. The parameter Δ(τ k) obeys the self-consistent equation|Δ(τ k)|=∑_τ' qV^ττ'(k,k+q)u_τ' k+q v_τ' k+q,which is solved numerically by recursive iteration (here V includes both long- and short-range interactionsas well as form factors, see Supplementary Note 4). As shown in Fig. 5a, in each valley |Δ(τ k)|is asymmetric around the Dirac point,a consequence of the peculiar character of the exciton wave function of Fig. 3b. The electrons or holes added to the neutral ground stateare gapped quasiparticle excitations of the EI, whose energy bands ± E(τ k) are shown in Fig. 5b. The order parameter at the Dirac point,|Δ (τ, k=0)|, is half the many-body gap. This gap is reminescent of the exciton binding energy, since in the ground state all electrons and holes are bound, so one needs to ionize an exciton-like collective stateto create a free electron-hole pair.The gap strongly depends on temperature,with a low-temperature plateau, a steepdescent approaching the critical temperature, and a milder tail (Fig. 5c). The gap approximately scales as 1/R for different tubes (circles in Fig. 5d):whereas at large R such scaling is exact (cf. dashed curve),at small R the gap is enhanced by short-range intervalley interaction (the decay of Δwill be mitigated if κ_r is sensitive to R).In experiments, many-body gaps are observed in undoped, ultraclean suspended tubes<cit.>, whereas Luttinger liquid signatures emerge in doped tubes<cit.>.Though it is difficult to compare with the measuredmany-body gaps<cit.>,as the chiralities of the tubes are unknown and the radii estimated indirectly,the measured range of 10–100 meV is at least one order of magnitude larger than our predictions. By doping the tube, we expect that the enhanced screening suppresses the EI order, quickly turning the system into a Luttinger liquid. We are confident thatadvances in electron spectroscopies will allow to test our theory. The broken symmetry associated with the EI ground state depends on the exciton spin<cit.>.For spin singlet(χ_σσ'=δ_σσ') and order parameter real (η=0,π), |Ψ_EI> breaks the charge symmetry between A and B carbon sublattices. The charge displacement per electron, Δ e/e,at each sublattice siteis Δ e/e = ±cosηa/A∑_τ k|Δ(τ k)|/2E(τ k),where the positive (negative) sign refers to the A (B) sublattice (Supplementary Note 6). For the (3,3) tube this amounts to ϱ_AB = 0.0068, which compares well with Monte Carlo estimates of 0.0067 and 0.0165 from LRDMC and VQMC, respectively. Note that assessing the energy difference between EI and zero-gap ground states is beyond the current capability of quantum Monte Carlo: the mean-field estimate of the difference is below 10^-6 Hartree per atom, which is less than the noise threshold of themethod (10^-5 Hartree per atom). § EFFECT OF MAGNETIC FIELD The EI is sensitive to the opening of a noninteracting gap, E_g,0, tuned by the magnetic field parallel to the tube axis, B. The ratio of the flux piercing the cross section, ϕ=π R^2B, to the fluxquantum, ϕ_0=ch/e, amountsto an Aharonov-Bohm phase displacing the position of the Dirac point along the transverse direction<cit.>, k_⊥=(ϕ/ϕ_0)R^-1.Consequently,E_g,0=2γ|k_⊥| is linear with ϕ/ϕ_0 (red line in Fig. 6a, c).Figure 6a shows the evolution oflow-lying singlet (blue lines) and triplet (black lines)excitons of the (3,3) tube. In addition, we have implemented a full first-principles description of B building ona previous method<cit.>.First-principles (circles) and model (solid lines) calculations show a fair agreement, which validates the effective-mass theorysince all free parameters have been fixed at zero field. Here we rescale energies by R/γ since we expect the plotto be universal, except for small corrections due to short-range interactions. Excitation energies obtained within the effective-mass model crossover from a low-field region, where ε_u is almost constant,to a high-field region, where ε_u increases linearly with ϕ/ϕ_0. Exciton wave functions are effectively squeezed by the field in real space (Fig. 6b), whereas in reciprocal space they loose their asymmetric character: the amplitudes become evenly distributedaround the Dirac points (Supplementary Discussion and Fig. 11)and similar to those reported inliterature<cit.>.At a critical flux ϕ_c/ϕ_0≈ 0.035the excitation energy ε_u becomes positive, hence the tube exits the EI phase and Δ vanishes in a BCS-like fashion.We point out that the critical field intensity, B_c≈ 460 T · (R [Å])^-2, is out of reach for the (3,3) tube but feasible for larger tubes. The total transport gap, E_g = (E_g,0^2 + 4|Δ|^2)^1/2, first scales with ϕ/ϕ_0 as E_g,0, then its slope decreases up to the critical threshold ϕ_c/ϕ_0,where the linear dependence on ϕ/ϕ_0 is restored (Fig. 6c).This behaviour is qualitatively similar to that observed by Coulomb blockade spectroscopy in narrow-gap tubes close to the`Dirac' value of B, which counteracts the effect of E_g,0 on the transport gap, fully suppressing the noninteractingcontribution<cit.>.§ DISCUSSION The observed<cit.> many-body gap of armchair tubeswas attributed to the Mott insulating state. The system was modeled as a strongly interacting Luttinger liquid with a gap enforced byshort-range interactions<cit.>,whereas the long tail of the interaction was cut offat an extrinsic, setup-dependentlength<cit.>. This model thus neglects the crucial effect of long-rangeinteraction, which was highlighted in Fig. 1: Were any cutoff length smaller than the intrinsic exciton length, which is micrometric and scales with R, excitons could not bind. Whereas armchair carbon nanotubes are regarded as quintessential realizations of the Luttinger liquid, since their low-energy properties are mapped into those of two-leg ladders<cit.>,we emphasize that this mapping is exact for short-range interactions only.Among e-h pair collective modes with totalmomentum q = 0, Luttinger liquid theoryroutinely describes plasmons<cit.> but not excitons. Contrary to conventional wisdom, armchair tubes are EIs.The excitonic and Mott insulators are qualitatively different.The EI exhibits long-range charge order, which does not affect the translational symmetry of the zero-gap tube. In the Mott insulator, charge and spin correlationsmay or may not decay, but always add a 2π/(2k_F) [or 2π/(4k_F)] periodicity to the pristine system, k_F being the Fermiwave vector<cit.>. The EI gap scales like 1/R (Fig. 5d),the Mott gap like 1/R^1/(1-g), withpredicted<cit.> values of g pointing to a faster decay, g<1. The EI order parameter is suppressed at high temperature(Fig. 5c) and strong magnetic field (Fig. 6c);the Mott gap is likely independent of both fields (the Aharonov-Bohm phase does not affect Hubbard-like Coulomb integrals). Importantly, the EI gap is very sensitive to the dielectric environment<cit.>, whereas the Mott gap is not. This could explain the dramatic variation of narrow transport gapsof suspended tubes submerged in different liquid dielectrics<cit.>. We anticipate that armchair tubes exhibitan optical absorption spectrum in the THz range dominated by excitons, which provides an independent test of the EI phase. Furthermore, we predict they behave as `chiral electronic ferroelectrics', displaying a permanent electric polarization 𝐏 of purely electronic origin<cit.>, whereas conventional ferroelectricity originates from ionic displacements. In fact, the volume average of 𝐏 is zero but its circulation along the tube circumference is finite. Therefore, a suitable time-dependent fieldexcites the ferroelectric resonance<cit.> associated with the oscillation of 𝐏. The special symmetry of armchair tubes<cit.> is expected to protect this collective (Goldstone) mode of oscillating electric dipoles from phase-locking mechanisms. The resulting soft mode—a displacement current along the tube circumference—is a manifestation ofthe long-debated<cit.> exciton superfluidity.In conclusion, our calculations demonstrated that an isolated armchair carbon nanotube at charge neutrality is an excitonic insulator, owing to the strong e-h binding in quasi-1D, and the almost unscreened long-rangeinteractions. The emergence of this exotic state of matter,predicted fifty years ago, does not fit the common picture of carbon nanotubes as Luttinger liquids. Our first-principles calculations provide tests to discriminate between the excitonic insulator and the Luttinger liquid at strong coupling, the Mott insulator state. We expect a wide family of narrow-gap carbon nanotubes to be excitonic insulators. Carbon nanotubes are thus invaluable systems for the experimental investigation of this phase of matter.§.§ Many-body perturbation theory from first principles. The ground-state calculations for the (3,3) carbon nanotube were performed by using a DFT approach, as implemented in the Quantum ESPRESSO package<cit.>. The generalized gradient approximation (GGA) PW91 parametrization<cit.> was adopted together with plane wave basis set and norm-conserving pseudopotentials to model the electron-ion interaction. The kinetic energy cutoff for the wave functions was set to 70 Ry. The Brillouin zone was sampled by using a 200 × 1 × 1 k-point grid. The supercell side perpendicular to the tube was set to 38 Bohr and checked to be large enough to avoid spurious interactions with its replica.Many-body perturbation theory<cit.> calculations were performed using the Yambo code<cit.>. Many-body corrections to the Kohn-Sham eigenvalues were calculated within the G0W0 approximation to the self-energy operator, where the dynamic dielectric function was obtained within the plasmon-pole approximation. The spectrum of excited states was then computed by solving the Bethe-Salpeter equation (BSE). The static screening in the direct term was calculated within the random-phase approximation with inclusion of local field effects; the Tamm-Dancoff approximation for the BSE Hamiltonian was employed after having verified that the correction introduced by coupling the resonant and antiresonant part was negligible. Converged excitation energies, ε_u, were obtained considering respectively 3 valence and 4 conduction bands in the BSE matrix. For the calculations of the GW band structure and the Bethe-Salpeter matrix the Brillouin zone was sampled with a 1793 × 1 × 1 k-point grid. A kinetic energy cutoff of 55 Ry was used for the evaluation of the exchange part of the self energy and 4 Ry for the screening matrix size. Eighty unoccupied bands were used in the integration of the self-energy.The effect of the magnetic field parallel to the axis on the electronic structure of the nanotube ground state (eigenvalues and eigenfunctions) was investigated following the method by Sangalli & Marini<cit.>. For each value of the field, the eigenvalues and eigenfunctions were considered to build the screening matrix and the corresponding excitonic Hamiltonian.To obtain the equilibrium structure, we first considered possible corrugation effects. We computed the total energy for a set of structures obtained by varying the relative positions of A and B carbon atoms belonging to different sublattices, so that they were displaced one from the other along the radial direction by the corrugation length Δ and formed two cylinders, as in Fig. 1(b) of Lu et al.<cit.>. Then, we fitted the total energy per carbon atom with an elliptic paraboloid in the two-dimensional parameter space spanned by Δ and the carbon bond length. In agreement with Lu et al.<cit.>, we find a corrugated structure with a bond length of 1.431 Å and a corrugation parameter Δ = 0.018 Å. Eventually, starting from this structure, we performed a full geometry relaxation of the whole system allowing all carbon positions to change until the forces acting on all atoms became less than 5· 10^-3 eV·Å^-1.After relaxation, the final structure presents a negligible corrugation (Δ < 10^-5 Å) and an average length of C-C bonds along the tube axis, 1.431 Å, slightly shorter than the C-C bonds around the tube circumference, 1.438 Å. The average radius and translation vector of the tube are respectively 2.101 Å and 2.462 Å, in perfect agreement with the literature<cit.>. The obtained equilibrium coordinates of C atoms in the unitary cell are shown in Supplementary Table 1. §.§ Quantum Monte Carlo method. We have applied the quantum Monte Carlo method to carbon nanotubes by using standard pseudopotentials for the 1s core electrons of the carbon atom<cit.>. We minimize the total energy expectation value of the first-principles Hamiltonian, within the Born-Oppheneimer approximation, by means of a correlated wave function, J |SD>. This is made of a Slater determinant, |SD>, defined in a localized GTO VDZ basis<cit.>(5s5p1d) contracted into six hybrid orbitals per carbon atom<cit.>,multiplied by a Jastrow term, J. The latter, J=J_1 J_2,is the product of two factors: a one-electron one term, J_1= ∏_iexp [u_1body(r_i) ], and a two-electron correlation factor, J_2 = ∏_i<jexp[u(r_i,r_j)]. The two-body Jastrow factor J_2 depends explicitly on the N_e electronic positions,{r_i }, and, parametrically, on the N_C carbon positions, R_I, I=1,⋯ N_C. The pseudopotential functions, u and u_1body, are written as:u( r, r^') =u_ee ( |r -r^'| )+ ∑_μ>0,ν>0 u_μν χ_μ(r)χ_ν(r^'), u_1body(r)=∑_μ>0 u_μ0 χ_μ(r),where u_ee= 2^-1r /( 1 + b_ee r) is a simple function, depending on the single variational parameter b_ee, which allows to satisfy the electron-electron cusp condition, and u_μν is a symmetric matrix of finite dimension. For non-null indices, μ,ν>0, the matrix u describes the variational freedom of J_2 in a certain finite atomic basis, χ_μ(r), which is localized around the atomic centers R_I(μ) and is made of 3s2p GTO orbitals per atom. Note that the one-body Jastrow term J_1 is expanded over the sameatomic basis and its variational freedom is determined by the first column of the matrix,u_μ0.We use an orthorombic unit cell L_x × L_y × L_z containing twelve atoms with L_x=L_y= 36 Å and L_z=2.445 Å. This cell is repeated along the z direction for n=1,2,3,4,5,6 times, up to 72 carbon atoms in the supercell. Periodic images in the xand y directions are far enough that their mutual interaction can be safely neglected. Conversely, in the z direction we apply twisted periodic boundary conditions and we integrate over that with a number n_θ of twists, n_θ=80,40,30,20,20,20 for n=1,2,3,4,5,6, respectively, large enough to have converged results for each supercell.The initial Slater determinant was taken by performing a standard LDA calculation. The molecular orbitals,namely their expansion coefficients in the GTO localized basis set, as well as the matrix u determining the Jastrow factor, were simultaneously optimized with well established methods developed inrecent years<cit.>,which allows us to consider up to 3000 independentvariational parameters in a very stable and efficient way. Note that the two-body Jastrow term J_2 can be chosen toexplicitly recover the EI mean-field wave function (<ref>), as shown in Supplementary Discussion.After the stochastic optimization the correlation functions / order parameters can be computed in a simple way within variational Monte Carlo (VMC).We also employ lattice regularized diffusion Monte Carlo (LRDMC) within the fixed-node approximation, using a lattice mesh of a_mesh=0.2 and a_mesh=0.4 a.u., respectively, in order to check the convergence for a_mesh→ 0. The fixed-node approximation is necessary for fermions for obtaining statistically meaningful ground-state properties. In this case the correlation functions / order parameters, depending only on local (i.e., diagonal in the basis) operators, such as the ones presented in this work, are computed with the forward walking technique<cit.>, which allows the computation of pure expectation values on the fixed-node ground state.§ CODE AVAILABILITY Many-body perturbation theory calculations were performed by means of the codes Yambo(http://www.yambo-code.org/) and Quantum ESPRESSO(http://www.quantum-espresso.org), which are bothopen source software. Quantum Monte Carlo calculations were based on TurboRVB code (http://trac.sissa.it/svn/TurboRVB), which is available from S.S. uponreasonable request (email: [email protected]).§ DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. § SUPPLEMENTARY NOTE 1§ EFFECTIVE-MASS THEORY OF ARMCHAIRCARBON NANOTUBESIn this Note we recall the effective-mass theory of electronic π-states in single-wall carbon nanotubes, focusing on the lowest conduction and highest valenceband of undoped armchair tubes<cit.>. Carbon nanotubes may be thought of as wrapped sheets of graphene, hence nanotube electronic states are built from those of graphene through a folding procedure, after quantizing the transverse wave vector. Low-energy graphene states belong to one of the two Dirac cones, whose apexes intersect thedegenerate K and K' points, respectively, at the corners of graphene first Brillouin zone. At these two points the energy gap is zero.Close to Brillouin zone corners τ=K,K', a nanotubestate ψ(𝐫) is the superposition of slowly-varying envelope functions F^τη(𝐫) multiplied by the Bloch states ψ_τη(𝐫), the latter having twoseparate components localized on sublattices η=A and η=B, respectively (cyan and red dots in Supplementary Fig. 1):ψ(𝐫)=∑_τ=K,K'∑_η=A,B F^τη(𝐫) ψ_τη(𝐫).The effective-mass approximation of SupplementaryEq. (<ref>) goesbeyond the usual one-valley treatment, as below we explicitly consider intervalley coupling due toCoulomb interaction. The relative phases of different Bloch state componentsψ_τη are fixed by symmetry considerations, as detailedin Supplementary Note 7.The envelope F^τη isa pseudospinor with respect to valley and sublattice indices, F≡(F^KA, F^KB,F^K'A,F^K'B)^T.In the valley-sublattice product space, F obeys the Dirac equation of graphene:γ[σ_x⊗1_τk̂_x+ σ_y⊗τ_z k̂_y ]F(𝐫)= ε F(𝐫).Here σ_x and σ_y are 2 × 2 Pauli matrices acting on the sublattice pseudospin, τ_z and the 2 × 2 identity matrix 1_τ act on the valley pseudospin, k̂_x = -i∂ /∂ x isis the wave vector operator along the circumference direction x and k̂_y = -i∂ /∂ yacts on the tube axis coordinate y, γ is graphene's band parameter, and ε is the single-particle energy. Furthermore, F obeysthe boundary condition along the tube circumference:F(𝐫+𝐋) = F(𝐫) exp( 2π iφ),where 𝐋 is the chiral vector in the circumference direction of the tube and |𝐋|=L=2π R is the circumference. A magnetic field may or may not be applied along the tube axis,with φ = ϕ / ϕ_0 being the ratio of the magnetic flux ϕ piercing the tube cross section to the magnetic flux quantum ϕ_0 = ch/e. Supplementary Eq. (<ref>) depends on thereference frame. Note that in our effective-mass treatmentthe xand y directions are parallel to the circumference and axis of the tube, respectively, as shown in Supplementary Fig. 1a, whereas in the main text as well as in the first-principles treatment the z axis is parallel to the tube.The energy bands are specified by the valley index τ,the valence index α = c, v denoting either the conduction (α = c)or the valence band (α = v), and the wave vector k in the axis direction.The wave functions in K and K' valleys are respectively F≡(F^K_α k(𝐫) ,0)^T and (0,F^K'_α k(𝐫))^T,with F^τ_α k(𝐫)≡(F^τ A_α k, F^τ B_α k)^T being a plane-wave pseudospinor in the sublattice space,F^τ_α k(𝐫)= ξ^τ_α k(x) 1/√(A)exp(iky) ,where A is the tube length and the wave functionξ^τ_α k(x) for the motion along the circumference direction isξ^τ_α k(x) = 1/√(L)exp(ik_⊥ x)F_τα k.The constant pseudospinor F_τα k is a unit vector with a k-dependent phase between the two sublattice components,F_Kα k= 1/√(2)b(k) s_α ,F_K' α k= 1/√(2)b^*(k) s_α,whereb(k) =k_⊥ -ik/√( k^2_⊥ + k^2 ) ,and s_α=± 1 for conduction and valence bands, respectively. In Supplementary Eqs. (<ref>) and (<ref>)the transverse wave vector k_⊥ is proportional to the magnetic flux φ,k_⊥ = φ/R.In each valley, the energy isε_α(k) = s_αγ√( k^2_⊥ + k^2 ) ,where the origin of the k axis is located at the Dirac point K (K').Figure 2a of main text shows the first-principles band structure of the (3,3)tube in a range of a few eV around the Dirac point,with k scanning half Brillouin zone, between the origin(k=0, Γ point) and k=π/a (a=2.46 Å is graphene lattice constant). The negative k axis, containing the K' point, is obtained by specular reflection.The DFT / GW location of the Dirac point is K = 0.289 (2π/a),whereas the effective-mass estimate is K = 1/3 (2π/a) (the discrepancy between DFT and tight-binding predictions is well documented in the literature<cit.>). As seen in Supplementary Fig. 2a, the GW bands are approximately linear in an energy range of at least ± 0.4 eV around the Dirac point, which validates the effective-mass model at low energy.Note that,in the absence of the magnetic field, electron states have a well definedchirality<cit.>, which is one ofthe two projections, C, of the sublattice pseudospin onto themomentum direction, expressed as the eigenvalues C=± 1of the operatorσ_y ⊗τ_z. The chirality index is highlighted by red (C=+1) and black (C=-1) colour in SupplementaryFig. 2a.§ SUPPLEMENTARY NOTE 2 § ELECTRON-ELECTRON INTERACTION:EFFECTIVE-MASS VS FIRST-PRINCIPLES DESCRIPTION Within the effective-mass framework, the Coulomb interaction v between two electrons on the carbon nanotube cylindrical surface located at 𝐫≡(x,y) and 𝐫'≡(x',y'),respectively, is<cit.>v(𝐫,𝐫')= ∑_q e^iq(y-y')2e^2/κ_r A K_0(2R|qsin(x-x'/2R)|),where κ_r is a static dielectric constant that takesinto account polarization effects due to the electrons not included in theeffective-mass description plus the contribution of the dielectric background. The interaction matrix element betweensingle-particle states is<cit.>V_(τ,α, k + q),(τ',β',k');(τ',α',k'+q) (τ,β,k) =∫ d𝐫∫ d𝐫' [F^τ_α k + q(𝐫)]^†·F^τ_β k (𝐫) v(𝐫,𝐫')[F^τ'_β' k'(𝐫')]^†·F^τ'_α' k'+q (𝐫') =1/A F_τα k + q^†·F_τβ kF_τ'β' k'^†·F_τ'α' k'+qv(q),where the one-dimensional effective interactionresolved in momentum space, v(q) = 2e^2/κ_r I_0(R|q|)K_0(R|q|),is modulated by a form factor given by overlap terms between sublattice pseudospinors[I_0(z) and K_0(z) are themodified Bessel functions of the first and second kind,respectively<cit.>]. The effect of screening due to the polarization of thoseelectrons that are treated within the effective-mass approximation is considered by replacing v(q) with w(q) = v(q)/ε(q)in the matrix element (<ref>), where ε(q) is the static dielectric function (to discriminate between screened and unscreened matrix elementswe use respectively `w' and `v' letters throughout the Supplementary Information). It may be shown that dynamical polarizationeffects are negligiblein the relevant range of small frequencies,which is comparable to exciton binding energies.Note that terms, similar to Supplementary Eq. (<ref>),that scatter electrons from one valley to the other are absent in the effective mass approximation. These small intervalley terms, as well as the interband exchange terms, which are both induced by the residual, short-range part of Coulomb interaction, are discussed in Supplementary Note 3.Effect of chiral symmetry. The chiralities of electron states,which is illustrated in Supplementary Fig. 3a(solid and dashed lines label C=+1 and C=-1, respectively), signficantly affects Coulomb interaction matrix elements. This occurs through the form factors of the type F^†·F appearing inSupplementary Eq. (<ref>), whichare overlap terms between sublattice pseudospinors.As apparent from their analytical structure,F_τα k + q^†·F_τβ k= 1/2[ sign(k) sign(k+q) + s_βs_α],the chiral symmetry of the states is conserved at each vertex of Coulomb scattering diagrams (see Supplementary Fig. 3b),hence initial and final states scattered within the same band must have the same momentum direction.This significantly affects the Bethe-Salpeter equation for excitons,as we show below.We are especially interested in the dominant long-range Coulomb matrixelement<cit.> that binds electrons and holes:V_(τ,c, k + q),(τ,v,k);(τ,v,k+q)(τ,c,k)≡Ṽ(k+q,k)/A.This term scatters electron-hole pairs from the initial pair state (c,k)(v,k) to the final state (c,k+q)(v,k+q) within the same valley τ. Throughout this Supplementary Information we use the tilde symbol for quantities whose dimension is an energy multipliedby a length, like V=Ṽ/A.In the first instance we neglect screening, since for low momentum transfer, q→ 0, polarization is suppressed hence ε(q)→ 1. In this limit Coulomb interaction diverges logarithmically,v(q) → -2e^2/κ_rln(R|q|),but this is harmless to the Bethe-Salpeter equation, since v(q) occurs only in the kernel of the scattering term, hence it is integrated over q for macroscopic lengths A,-1/A∑_q Ṽ(k+q,k)…→ -1/2π∫ dq Ṽ(k+q,k)…,which removes the divergence. Note that, throughout this Supplementary Information and opposite to theconvention of Fig. 2c of main text, we take V as a positive quantity. In detail,we discretize the momentum space axis, k→ k_i, where k_i=i2π/(Na),i=-N/2 + 1,…,0,1, …,N/2,N=A/a is the number of unitary cells, and Δ k = 2π/(Na) is the mesh used in the calculation. Hence, the regularized matrix element, integrated over the mesh, isV(k_j+q_i,k_j)=1/2π∫_q_i - Δ k^q_idq Ṽ(k_j+q,k_j). In Supplementary Fig. 4 we compare V (panel a, κ_r=10) with the modulus of the screened DFT matrix element W^DFTobtained for the (3,3) tube (panel b). The two plots are three-dimensional contour maps in a square domain (k,k') centered around the Dirac point,with K = 0.289(2π)/a and N=900. The two matrix elements agree almost quantitatively, as they both exhibit: (i) zero or very small values in the second and fourth quadrants, i.e., k > K and k'< K or k < K and k'> K(ii) a logarithmic spike on the domain diagonal, i.e., k'→ k.This behavior has a simple interpretation in terms of exciton scattering, as an electron-hole pair with zero center-of-mass momentum, (c,k)(v,k), has a well-definedchirality with respect to the noninteracting ground state, i.e.,ΔC = + 2 = 1 - (-1)for k> K (ΔC = - 2 for k< K).The chirality of the e-h pair is conserved during Coulomb scattering, i.e., as the pair changes its relative momentum from2k = k - (- k) to 2k'. Effect of electronic polarization. In order to appreciate the minor differencesbetween V(k,k') and W^DFT(k,k') it is convenient to compare the cuts of the maps of SupplementaryFig. 4 along a line k'=k_0,as shown in Supplementary Fig. 5 for k_0=0.289 (2π)/a (panel a) and 0.28 (2π)/a (panel b),respectively. For small momentum transfer, q=k-k_0≈ 0, V(k,k_0) (squares) exhibits a sharper spike than W^DFT(k,k_0) (filled circles). This is an effect of theregularization of the singularitity occurring in the DFT approach, as in the first-principles calculation the tube is actually three-dimensional. As |q| increases, V is systematically blushifted with respect to W^DFT since it does not take into account the effect of the RPA polarization, Π(q), which acquires a finite value. Within the effective-mass approximation, Π(q) enters the dressed matrix element W through the dielectricfunction<cit.>,ε(q) = 1 +2e^2/κ_r I_0(R|q|)K_0(R|q|) Π(q).Here we use the simple ansatzΠ(q) = A_ansatz(Rq)^2,as this choice makes the dressed Coulomb interactionscale like the three-dimensional bare Coulomb potential for large q (i.e., at short distances), W ∼ 1/q^2. In Supplementary Fig. 5a, b the dressed matrixelement W [empty circles, A_ansatz=50/(πγ), γ/a=1.783 eV] quantitatively agrees with its ab initio counterpart,W^DFT (filled circles), in the whole range of k in whichelectrons are massless (cf. Supplementary Fig. 2). Note that for k>K=0.289(2π)/a the effective-mass potentials are exactly zerowhereas W^DFT shows some numerical noise. Effect of the magnetic field. The magnetic field along the tube axis adds an Aharonov-Bohm phase to the transverse momentum, k_⊥. This breaks the chiral symmetry C of single-particle states, alters the form factors of Supplementary Eq. (<ref>)(see Ando<cit.>),and lifts the selection rule on k. This is apparent from the smearing of the maps of Supplementary Fig. 6 close to the frontiers of the quadrants, k,k'= K, wheres at the same locations in Supplementary Fig. 4 (no field)the plots exhibit sharp discontinuities. The cuts of Supplementary Fig. 6 along the line k'=k_0, as shown in Supplementary Figs. 7a and b for k_0=0.289 (2π)/a and 0.28 (2π)/a, respectively, confirm the good agreement between W(k,k_0) and W^DFT(k,k_0).§ SUPPLEMENTARY NOTE 3 § EFFECTIVE MASS: BETHE-SALPETER EQUATION In this Note we detail the calculation of low-lying excitons of armchair carbon nanotubes, |u>, within the effective mass theory. The analysis of the first-principles exciton wave functionfor the (3,3) tube shows that the lowest conduction and highest valence bands contribute more than 99.98% to the spectral weight of excitons. Therefore, according to conventional taxonomy, these excitonsare of the M_00 type.Within the effective-mass approximation, |u> is written as |u>=∑_σσ'τ kψ_τ(k) χ_σσ' ĉ^τ +_k,σ v̂^τ_k,σ'| 0 >,where | 0 > is the noninteracting ground state with all valence states filled and conduction states empty, and the operator ĉ^τ +_k,σ(v̂^τ +_k,σ) creates an electron in the conduction (valence) band labeled bywave vector k, spin σ, valley τ.The exciton |u> is a coherent superposition of electron-hole pairs having zero center-of-mass momentumand amplitude ψ_τ(k). The latter may be regarded as the exciton wave function in k space. The 2× 2 spin matrix χ_σσ' is the identity for singlet excitons, χ=1_s, whereas for triplet excitons χ=σ_s· 𝐧, where 𝐧 is the arbitrary direction of the spin polarization (|𝐧|=1) and σ_s is a vector made of the three Pauli matrices. Throughout this workwe ignore the small Zeeman term coupling the magnetic field with electron spin, hence triplet excitons exhibit three-folddegeneracy. Here we use the same notation,|u>, for both singlet and triplet excitons, as its meaning is clear from the context. The Bethe-Salpeter equation for the triplet exciton isE_eh(τ, k) ψ_τ(k)- 1/A∑_qW̃^τ( k+q,k )ψ_τ(k+q)- 1/A∑_τ'≠τ∑_qW̃^ττ'( k+q,k )ψ_τ'(k+q)= ε_u ψ_τ(k).The diagonal term E_eh(τ, k) is the energy cost tocreate a free electron-hole pair (τ,c,k)(τ,v,k),E_eh(τ, k) = 2γ√(k_⊥^2 + k^2 ) +Σ^τ(k),including the sum of self-energy corrections to electron and hole energies, Σ^τ(k), which may be evaluated e.g. within the GW approximation. This self-energy, which describes the dressing of electronsby means of the interaction with the other electrons present in the tube, is responsible for the smallasymmetry of the Dirac cone close to K, as shown by the GW dispersion of Supplementary Fig. 2a. Since this asymmetry appears already at the DFT level of theoryand is similar to the one predicted for the Dirac cones of graphene<cit.>, it necessarily originates from mean-field electron-electron interaction and it does not depend on R. We take into account the effect of Σ^τ(k) onto E_eh(τ, k) by explicitly consideringdifferent velocities (slopes ofthe linear dispersions) for respectively left- and right-moving fermions, according to:E_eh(K, k) =2γ[1 + α_sl sign(k)] √(k_⊥^2 + k^2 ),E_eh(K', k) =2γ[1 - α_sl sign(k)] √(k_⊥^2 + k^2 ).We infer the actual values of γ and slope mismatch parameterα_sl from the linear fitto the first-principles GW dispersion(in Supplementary Fig. 2b the solid linesare the fits and the dots the GW data), which provides γ=5.449 eV·Å and α_sl=0.05929.The second and third terms on the left hand side of Supplementary Eq. (<ref>) involve interband Coulomb matrix elements. The intravalley term W̃^τ is the dressed long-rangedinteraction discussed in the previous Note. The intervalley term W̃^KK' makes electron-hole pairs to hop between valleys. As illustrated by the DFT map ofW^KK'( k,k' )= W̃^KK'/A in (k,k') space (Supplementary Fig. 8),this term, almost constant in reciprocal space, is at least one order of magnitudesmaller than W, as seen by comparing the small range 0.18–0.3 meV of the energy axis ofSupplementary Fig. 8 with the range 0–9 meV of Supplementary Figs. 4 and 6. Therefore, W^KK' may be regarded as a weak contact interaction that couples the valleys, consistently with the model by Ando<cit.>,W̃^KK'( k,k' ) = Ω_0w_2/4π R,where Ω_0 = (√(3)/2)a^2 is the area of graphene unit cell and w_2>0 is the characteristic energy associated with short-range Coulomb interaction. We reasonably reproduce first-principles results takingw_2=2.6 eV—this would be a plane located at 0.24 meV inSupplementary Fig. 8.This estimate is not far from Ando's predictionw_2 = 4 eV. Note that the previous theory proposed by one ofus<cit.> relies on the scenario W^KK' > W, which isruled out by the present study. The Bethe-Salpeter equation for the singlet exciton is obtained from Supplementary Eq. (<ref>) by simply adding to the kernel the bare exchange term + Ω_0w_1/2π R A∑_τ'∑_q ψ_τ'(k+q),where w_1>0 is a characteristic exchange energy<cit.>. From first-principles results we estimatew_1=4.33 eV, whose magnitude is again comparable to that predicted by Ando<cit.>. Supplementary Eq. (<ref>),with or withoutthe exchange term, is solved numericallyby means of standard linear algebra routines.Minimal Bethe-Salpeter equation. The minimal Bethe-Salpeter equation illustrated in the main text includes only one valley (with α_sl = 0) and long-range Coulomb interaction. Within the effective-mass approximation, the Dirac cone indefinitely extends in momentum space, hence one has to introduce a cutoff onto allowed momenta, |k| ≤ k_c. Supplementary Fig. 9a shows the convergence of the lowest-exciton energy, ε_u, as a function of k_c.Reassuringly, ε_u smoothly converges well within the range in which GW bands are linear. This is especially true for the screened interaction W (black circles), whereas the convergence is slower for the unscreened interaction V (red circles), as it is obvious since W(q) dies faster with increasing q. This behavior implies that the energy scale associated withthe exciton is intrinsic to the tube and unrelated to the cutoff,as we further discuss below. In the reported calculations we took k_c = 0.05 (2π)/a as a good compromise between accuracy and computational burden (we expect that the maximum absolute error on ε_uis less than 0.1 meV).This corresponds to an energy cutoff of 1.4 eV for e-h pair excitations. Whereas for these calculations, as well as for the data of Supplementary Fig. 9a, the mesh Δ k in momentum space isfixed [Δ k = 1.43 · 10^-5(2π)/a], Supplementary Fig. 9b showsthe convergence of ε_u as afunction of the mesh, Δ k. Interestingly, ε_u smoothly decreases with Δ k only for a very fine mesh, whereas for larger values of Δ k the energy exhibits a non-monotonic behaviour. This is a consequence of the logarithmic spike of the Colulomb potentialat vanishing momentum, which requires a very fine mesh to be dealt with accurately.We refine the minimal effective-mass Bethe-Salpeter equation by including: (i) The short-range part of interaction, which couples the two valleys as well as lifts the degeneracy of spin singlet and triplet excitons. (ii) The tiny difference between the e-h pair excitation energies of left and right movers. This eventually leads to a quantitative agreement with exciton energies and wave functions obtained from first principles, as shown by Fig. 3b, cand Fig. 6a, b of main text.Scaling properties of the Bethe-Salpeter equation. If a well-defined (i.e., bound and normalizable)solution of the Bethe-Salpeter equation (<ref>) exists, then it must own a characteristiclength and energy scale—respectively the exciton Bohr radius and binding energy<cit.>. To check this, we introduce the scaling length ℓ to define the following dimensionless quantities: the wave vectorκ = k ℓ, theenergy E_u=ε_uℓ / γ, and the exciton wave function ξ_τ(κ)=ψ_τ(k)/√(ℓ). We also define the dimensionless intravalley interaction as Ω^τ(kR,k'R)=(κ_r/e^2)W̃^τ(k,k'),to highlight that the wave vector k appearing as an argument of the interaction is always multiplied by R. This is important for the exciton scaling behaviour.Neglecting the small corrections to the exciton binding energy due to intervalley scattering (w_2=0) and cone asymmetry (α_sl=0), the dimensionlessBethe-Salpeter equation for armchair tubes in the absence of a magnetic flux becomes2|κ|ξ_τ(κ) - α_graph/2π∫ dκ' Ω^τ[κ'(R/ℓ),κ(R/ℓ)] ξ_τ(κ') = E_u ξ_τ(κ),where α_graph = e^2/(κ_rγ) is graphene fine-structure constant,the scaled exciton wave function must satisfy the scale invariant normalization requirement,∑_τ∫ dκ|ξ_τ(κ) |^2 = 1, and the dielectric function entering Ω takes the dimensionless formε(κ) = 1 +2 A_ansatz/πα_graphκ^2 (R/ℓ)^2I_0(|κ|R/ℓ)K_0(|κ|ℓ/R). The only scaling length ℓ leavingSupplementary Eqs. (<ref>) and (<ref>) invariant is the tube radius, R, wich fixes the binding energy unit, γ/R. Supplementary Eq. (<ref>) shows that α_graph is the single parameter combination affecting the scale invariant solution, whereas solutions for different radius R are related via scaling,ε_u=E_0/R,with E_0 being calculated once for all for the (3,3) tube radius, R=2 Å. The same conclusion holds for finite coneasymmetry α_sl and dimensionless magnetic flux φ. Note that, for a fixed value of φ, the possible values of the magnetic field B scale like 1/R^2.The above demonstration relies on the assumption that the parameters κ_r and A_ansatz, whichcontrol the screening behavior of the carbon nanotube, do not depend significantly on R.On the other hand, one might expect to recover, for large R, the screening properties of graphene. This in turn would imply that κ_r would tend to smaller values and henceε_u would decay slower than 1/R. The first-principles investigation of this issue is left to future work.§ SUPPLEMENTARY NOTE 4 § SELF-CONSISTENT MEAN-FIELD THEORY OF THE EXCITONIC INSULATOR The ground-state wave function of the excitonic insulator,|Ψ_EI>, exhibits a BCS-like form,|Ψ_EI>=∏_σσ'τ k[u_τ k + χ_σσ' v_τ k e^i η ĉ^τ +_k,σv̂^τ_k,σ'] | 0 >,where η is the arbitrary phase of the condensate, the e-h pairs ĉ^τ +_k,σv̂^τ_k,σ'| 0 >replace the Cooper pairs (e.g. ĉ^K +_k,σĉ^K' +_-k,-σ| 0 >), and the 2× 2 matrix χ_σσ' discriminates between singlet and triplet spin symmetries.The positive variational quantities u_τ k and v_τ k are the population amplitudes of valence and conduction levels, respectively, which are determined at once with the excitonicorder parameter, Δ(τ k). Explicitly, one hasu_τ k^2=1/2(1 + E_eh(τ,k)/2/[ E^2_eh(τ,k)/4+|Δ(τ k)|^2]^1/2), v_τ k^2=1 - u_τ k^2,plus the self-consistent equation for Δ[equivalent to Eq. (2) of main text],|Δ(τ k)|=1/A∑_τ' k'W̃^ττ'(k,k')|Δ(τ' k')|/2[ E^2_eh(τ',k')/4+|Δ(τ' k')|^2]^1/2 .The symbol W̃^ττ'(k,k') in Supplementary Eq. (<ref>)is a shorthand for both intra and intervalley Coulomb interactionmatrix elements.For the spin-triplet EI [χ_σσ' = (σ_s· 𝐧)_σσ'], which is the absolute ground state, one has, for τ=τ',the long-range intravalley term,W̃^ττ(k,k')=W̃^τ(k,k'),and, for τ≠τ', theshort-range intervalley term,W̃^ττ'(k,k'). For the spin singlet(χ_σσ'= δ_σσ'),the unscreened direct term must be subtracted from the dressed interaction,W̃^ττ'(k,k') →W̃^ττ'(k,k') - Ω_0w_1/2π R.Supplementary Eq. (<ref>) allows for a scaling analysis similar to that for the exciton binding energy.If interaction matrix elements W̃ were constant,then Supplementary Eq. (<ref>)would turn into the familiargap equation of BCS theory,with Δ constant as well. Since the long-range part of interaction is singular,the dependence of Δ(τ k) on τ and k cannot be neglected and hence the solution is not obvious. It is convenient to rewrite SupplementaryEq. (<ref>) as a pseudo Bethe-Salpeter equation,2[ E^2_eh(τ,k)/4+|Δ(τ k)|^2]^1/2φ(τ k)-1/A∑_τ' k'W̃^ττ'(k,k') φ(τ' k') = 0,with the pseudo exciton wave function defined asφ(τ k) = |Δ(τ k)|/2[ E^2_eh(τ,k)/4+|Δ(τ k)|^2]^1/2.This shows that, at the onset of the EI phase, when Δ(τ k)is infinitesimal—at the critical magnetic field—theexciton wave function forε_u=0 is the same as φ apart from a constant, φ(τ k)∼ψ_τ(k). This observation suggests to use ψ_τ(k) at all values of the field as a good ansatz to start the self-consistent cycle ofSupplementary Eq. (<ref>), which is numerically implemented as a matrix product having the form φ_new = W·φ_old. Taking at the first iteration |Δ_old(τ k)| = 2^-1[E_eh(τ,k)-ε_u] |ψ_τ(k)/ψ_τ(0)| and building φ_old(τ k) according toSupplementary Eq. (<ref>), we obtain numerical convergence within a few cycles,φ_new = φ_old, with the number of iterations increasing with decreasing Δ.At finite temperatures, the self-consistent equation forΔ takes the form|Δ(τ k)| =1/A∑_τ' k'W̃^ττ'(k,k')|Δ(τ' k')|/2[ E^2_eh(τ',k')/4+|Δ(τ' k')|^2]^1/2× tanh{1/2k_BT[ E^2_eh(τ',k')/4+|Δ(τ' k')|^2]^1/2} ,where T is the temperature and k_B is Boltzmann constant.The quasiparticles of the EI arethe free electrons and holes.For example, in the simplest case of the spin-singlet EI(χ_σσ' = δ_σσ'),the electron quasiparticle wave function |Ψ_EI^τk↑>differs from the ground state |Ψ_EI> as the conduction electron state labeled by (τ, k,↑) is occupied with probability one as well as the corresponding valence state:|Ψ_EI^τ k↑> = ĉ^τ +_k,↑[u_τ k + v_τ k e^iη ĉ^τ +_k,↓v̂^τ_k,↓] ∏_στ' k'^'[u_τ' k' + v_τ' k' e^iη ĉ^τ' +_k',σv̂^τ' _k',σ] | 0 >.Here the symbol ∏^' means that the dummy indices τ' k' take all values but τ k. The quasiparticle energy dispersion is E(τ k)= √( E_eh^2(τ,k) /4+ | Δ(τ k) |^2,)with the reference chemical potential being zero, as for the noninteracting undoped ground state. E(τ k) isincreased quadratically by the amount |Δ(τ k)| with respect to the noninteracting energy, ε(τ k)=E_eh(τ,k)/2. This extra energy cost is a collective effect reminescent of the exciton binding energy, since nowthe exciton condensate must be ionized to unbind one e-h pair and hence have a free electron and hole. § SUPPLEMENTARY NOTE 5 § INVERSION SYMMETRY BREAKING IN THE EXCITONIC INSULATOR PHASE Carbon nanotubes inherit from graphene fundamental symmetries such as time reversal and spatial inversion. Time reversal 𝕋̂ swaps K and K' valleys whereas the inversion 𝕀̂ is a π rotationaround an axis perpendicular to the tube surface andlocated in the origin of one of the frames shown inSupplementary Fig. 1.This swaps the valleys as well as the A and B sublattices. Whereas the noninteracting ground state |0> is invariant under both inversion and time reversal, 𝕋̂|0>=|0> and 𝕀̂|0>=|0>, the EI ground state breaks the inversion symmetry<cit.>. Here we consider a spin-singlet exciton condensate (χ_σσ' =δ_σσ') with𝕋̂ |Ψ_EI>= |Ψ_EI>, hence the excitonic order parameteris real, η=0,π (otherwise the EI ground state would exhibit transverse orbital currents).To see that the inversion symmetry of the EI ground state is broken we use the following transformations(whose details are given in Supplementary Note 7):𝕀̂ v̂^τ_k,σ= -isign(k) v̂^-τ_-k,σ, 𝕀̂ ĉ^τ_k,σ= i sign(k) ĉ^-τ_-k,σ,where the shorthand -τ labels the valley other than τ. The transformed ground state is𝕀̂ |Ψ_EI> = ∏_στ k[u_τ k - v_τ k e^i η ĉ^τ +_k,σv̂^τ_k,σ] | 0 >,where we have used the fact that u_τ k=u_τ k^*=u_-τ -k and v_τ k=v_τ k^*=v_-τ -k, as a consequence of time reversal symmetry. The original and transformed ground states are orthogonal in the thermodynamic limit,< Ψ_EI | 𝕀̂|Ψ_EI> = 2∏_τ k(u_τ k^2 - v^2_τ k) → 0,since u^2-v^2<1. On the contrary, <0 | 𝕀̂|0>=1. Therefore, the symmetry of the EI ground state is lower than that of the noninteracting ground state so the EI phase has broken inversion symmetry, i.e., charge is displaced from A to B sublatticeor vice versa.§ SUPPLEMENTARY NOTE 6 § CHARGE DISPLACEMENT BETWEEN A AND B SUBLATTICES In this section we compute thecharge displacement between A and B carbon sublattices in the EI ground state. To this aim we must average over the ground state the space-resolved charge densityϱ(𝐫)=e ∑_i δ(𝐫-𝐫_i),where the sum runs over all electrons in the Dirac valleys. The explicit form of the charge density, in second quantization, isϱ̂(𝐫)=e ∑_τ kτ' k' σ[ φ^*_cτ k(𝐫)φ_cτ' k'(𝐫) ĉ^τ +_k,σĉ^τ'_k',σ + φ^*_vτ k(𝐫)φ_vτ' k'(𝐫) v̂^τ +_k,σv̂^τ'_k',σ +φ^*_cτ k(𝐫)φ_vτ' k'(𝐫) ĉ^τ +_k,σv̂^τ'_k',σ + φ^*_vτ k(𝐫)φ_cτ' k'(𝐫) v̂^τ +_k,σĉ^τ'_k',σ].We recall that the states of conduction (α=c) andvalence (α=v) bands appearing in Supplementary Eq. (<ref>), φ_ατ k(𝐫), are products of the envelope functions F times the Bloch states ψ_τ at Brillouin zone corners τ = K, K',φ_ατ k(𝐫) = F^τ A_α k(𝐫) ψ_τ A(𝐫) + F^τ B_α k(𝐫) ψ_τ B(𝐫),where ψ_τ A(𝐫) [ψ_τ B(𝐫)] is the component on the A (B) sublattice. Neglecting products of functions localized on different sublattices, like ψ^*_τ Aψ_τ B,as well as products of operators non diagonal in τ and k indices, which are immaterial when averaging over the ground state, one obtains:ϱ̂(𝐫)=e/2AL∑_τ[ |ψ_τ A(𝐫)|^2 + |ψ_τ B(𝐫)|^2 ]∑_k σ( v̂^τ +_k,σv̂^τ_k,σ+ ĉ^τ +_k,σĉ^τ_k,σ)+e/2AL∑_τ[ |ψ_τ A(𝐫)|^2 - |ψ_τ B(𝐫)|^2 ]∑_k σ( ĉ^τ +_k,σv̂^τ_k,σ+ v̂^τ +_k,σĉ^τ_k,σ) . The first and second line on the right hand side ofSupplementary Eq. (<ref>) arerespectively the intra and interband contribution to the charge density. Only the intraband contribution survives when averaging ϱ̂ over |0>, providing the noninteracting system with the uniform background charge density ϱ_0 (𝐫),ϱ_0(𝐫) = <0|ϱ̂(𝐫) |0> = e/aL∑_τ[ |ψ_τ A(𝐫)|^2 + |ψ_τ B(𝐫)|^2 ] ,with ∑_k 1=A/a. Since |ψ_K A(𝐫)| = |ψ_K' A(𝐫)| = |ψ_A(𝐫)|, and similarly for B, this expression may be further simplified asϱ_0(𝐫)= 2e/aL[ |ψ_A(𝐫)|^2 + |ψ_B(𝐫)|^2 ] .It is clear from this equation that ϱ_0 isobtained by localizing the two π-band electronsuniformly on each sublattice site. When averaging ϱ̂ over |Ψ_EI>, the charge density ϱ (𝐫)exhibits an additional interband contribution,ϱ (𝐫) =< Ψ_EI| ϱ̂(𝐫)|Ψ_EI> = ϱ_0(𝐫) +2ecosη/AL[ |ψ_A(𝐫)|^2 - |ψ_B(𝐫)|^2 ]∑_τ k u_τ kv_τ k, which is proportional to ∑_τ k u_τ kv_τ kand hence related to the EI order parameter. This term, whose origin is similar to that of the transition densityshown in Fig. 3d of main text, as it takes into account thepolarization charge fluctuation between |0>and a state with one or more e-h pairs excited, is driven from the long-range excitonic correlations.Importantly, the charge displacement is uniformamong all sites of a given sublattice and changes sign with sublattice, the sign depending on the phase of the exciton condensate, η. The charge displacement per electron, Δ e/e, on—say—each A site isΔ e/e = a cosη/A∑_τ k u_τ kv_τ k,which is the same as Eq. (3) of main text. In order to evaluate numerically Δ e/e,for the sake of simplicity we neglect the exchange termssplitting the triplet and singlet order parameters (i.e., we assume w_1=0).The quantum Monte Carlo order parameter ϱ_ABdefined in the main text is, in absolute value,twice |Δ e/e|as there are two relevant electrons per site.§ SUPPLEMENTARY NOTE 7 § REFERENCE FRAME AND SYMMETRY OPERATIONS The reference frame of the armchair carbon nanotubeshown in Supplementary Fig. 1ais obtained by rigidly translating the frame used byAndo in a series of papers<cit.>, recalled in Supplementary Fig. 1b.In Ando's frame the origin is placed on an atom of the B sublattice and the y axis is parallel to the tube axis, after a rotation by the chiral angle α with respect to the y^' axis of graphene.On the basis of primitive translation vectors of graphene 𝐚 and 𝐛 displayed in Supplementary Fig. 1b, the chiral vector takes the form 𝐋=-m𝐚 -(n+m)𝐛 when expressed in terms of the conventional<cit.>chiral indices (n,m). For an equivalent choice of 𝐋, one has α=π/6 for (n,n) armchair tubes and α=0 for (n,0) zigzag tubes. In the frame of Supplementary Fig. 1aused throughout this Supplementary Information,the vectors locating the sites of A and B sublattices are respectively𝐑^A_n_a,n_b = 𝐑_0^A + n_a 𝐚+ n_b 𝐛and 𝐑^B_n_a,n_b = 𝐑_0^B + n_a 𝐚 + n_b 𝐛,where (n_a,n_b) is a couple of integers and𝐑_0^A (𝐑_0^B) is the basis vector pointing tothe origin of the A (B) sublattice. Besides, one has 𝐚≡ a(√(3)/2,-1/2), 𝐛≡ a(0,1), 𝐑_0^A ≡ a(1/√(3),1/2), 𝐑_0^B ≡ a(1/(2√(3)),0), where a=2.46 Å is thelattice constant of graphene. Among the equivalent corners of graphene first Brillouin zone, we have chosen as Dirac points K≡2π/a (1/√(3), 1/3) and K' = - K. The corresponding Bloch states are:ψ_KA(𝐫) =1/√(N)∑_n_a,n_b e^iK·𝐑^A_n_a,n_b ϕ_π (𝐫 - 𝐑^A_n_a,n_b),ψ_KB(𝐫) = - e^iπ/6 ω1/√(N)∑_n_a,n_b e^iK·𝐑^B_n_a,n_b ϕ_π (𝐫 - 𝐑^B_n_a,n_b),ψ_K'A(𝐫) =e^iπ/6 ω^-11/√(N)∑_n_a,n_b e^iK'·𝐑^A_n_a,n_b ϕ_π (𝐫 - 𝐑^A_n_a,n_b),ψ_K'B(𝐫) =ω1/√(N)∑_n_a,n_b e^iK'·𝐑^B_n_a,n_b ϕ_π (𝐫 - 𝐑^B_n_a,n_b),where N is the number of sublattice sites, ϕ_π(𝐫) is the 2p_z carbon orbital perpendicular to the graphene plane, normalized as inSecchi & Rontani<cit.>, and ω=exp(i2π/3).The relative phase between the two sublattice components of Bloch states within each valley, shown inSupplementary Eq. (<ref>), is determined by symmetry considerations<cit.>. Specifically, the sublattice pseudospinortransforms as a valley-specificirreducible representation of the symmetry point group of the triangle, C_3v:F_Kα k∼x - iy x + iy ,F_K' α k∼x + iy -x + iy . In addition, the relative phase between Bloch states of different valleys is fixed by exploiting the additional C_2 symmetry. The latter consists of a rotation of a π angle around the axis perpendicularto the graphene plane and intercepting the frame origin. This rotation, which in the xy space is equivalent to theinversion 𝕀̂, swaps K andK' valleys as well as A and B sublattices. With the choice of phases explicited in Supplementary Eq. (<ref>) the inversion operator 𝕀̂ takes theform 𝕀̂ = - σ_y ⊗τ_y R̂,where R̂ is the inversion operator in the xy space. In contrast, the time-reversal operator 𝕋̂swaps valleys but not sublattices,𝕋̂ = σ_z ⊗τ_x K̂,where K̂ is the complex-conjugation operator. The orthogonal time-reversal ofSupplementary Eq. (<ref>) should not be confused with the symplectic transformation<cit.>, which does not exchange valleys. The magnetic field along the tube axis breaks both 𝕀̂ and 𝕋̂ symmetries. However, the reflection symmetry y→ -yalong the tube axis still swaps the valleys (but not sublattices), as it may be easily seen from a judicious choice of K and K' Dirac points. This protects the degeneracy of states belonging to different valleys in the presence of a magnetic field. § SUPPLEMENTARY DISCUSSION § EFFECTS OF DIRAC CONE ASYMMETRY AND MAGNETIC FIELD ON THE EXCITON WAVE FUNCTION The origin of theasymmetry of the exciton wave function in k space, illustratedby Fig. 3b of main text, may be understood within the effective massmodel applied to a single Dirac valley—say K. In the presence of a vanishing gap, electrons(and excitons) acquire a chiral quantum number, C, which was defined above. With reference to the noninteracting ground state, |0>, the e-h pairsĉ^K †_k,σv̂^K_k,σ'|0> have chiral quantum number ΔC=+2 for positive kand ΔC=-2 for negative k. Since long-rangeCoulomb interaction conserves chirality,we expect the wave function of a chiral exciton to live onlyon one semi-axis in k space, either ψ_K(k)=0 for k<0 and ΔC=+2, or ψ_K(k)=0 for k>0 and ΔC=-2.Supplementary Fig. 10 plots ψ_K(k) by comparingthe case of a perfectly symmetric Dirac cone (panel a,α_sl=0) with the case of a distorted cone, mimicking the first-principles GW band dispersion (panel b,α_sl=0.05929). This analysis is of course possible only within the effective mass model, as no free parameter such as α_sl may be changed in the first-principles calculation. In the symmetric case (Supplementary Fig. 10a) ψ_K(k) is even in k since nothing preventsthe numerical diagonalization routine from mixing the two degenerate amplitude distributions with ΔC=± 2. Hovever, as the Dirac cone symmetry under axis inversion, k→ -k,is lifted by energetically favoring e-h pairs withΔC=-2 (Supplementary Fig. 10b), the wave function weight collapses on the negative side of the axis. Therefore, the asymmetry of the exciton wave function is explained by the combined effects of chiral symmetry and cone distortion.As the chiral symmetry is destroyed by piercing the tube with a magnetic flux, the exciton wave function ψ becomes symmetrically distibuted around the origin of k axis. This is shown in Supplementary Fig. 11 from both first-principles (dots) and effective-mass (solid curves) calculations of |ψ_τ(k)|^2 for increasing values of the dimensionless magnetic flux ϕ/ϕ_0 (for panels a,b ϕ/ϕ_0= 0, for c, d ϕ/ϕ_0= 0.0015,for e, f ϕ/ϕ_0= 0.0091). As the gap increases with the field, the excitons becomes massive and more similar to the conventional Wannier excitons reported in the literature<cit.>:the weight distribution in k space is broader and its peak more rounded, with a Gaussian-like shape identical in both valleys (respectively valley K' in panels a, c, e and valley K in panels b, d, f). The agreement between first-principles (dots) andeffective-mass (solid lines) predictions is very good, further validating the model.However, at high field (panels e and f, ϕ/ϕ_0=0.0091), the effective-mass curve becomes discontinuous at the Dirac cone whereas the first-principles curve is smooth. This is an artefact of the effective-mass model as the high-field functional form of the distorted Dirac cone shown by Supplementary Eq. (<ref>) exhibits a step at k=0 that increases with k_⊥. This crude modelization may be cured rather simply: however, its drawbacks do not affect the results presented in this paper in anysignificant way.§ THE EI MEAN-FIELD WAVE FUNCTION AS SPECIALIZATION OFTHE QMC VARIATIONAL WAVE FUNCTION The QMC variational wave function,|Ψ_QMC>, is thezero-gap state, |0>, multiplied by the Jastrow factor, J=J_1J_2, which accounts for one- and two-body correlations encoding the variational degrees of freedom. In this section wefocus on a relevant specialization of the pair Jastrow factor, J_2=∏_i<jexp[u(𝐫_i,𝐫_j)],showing that a proper choice of the two-body term u(𝐫,𝐫') allows torecover the mean-field EI wave function to first order in u, i.e.,|Ψ_QMC>takes the form|Ψ_EI>=∏_k(u_k + v_k ĉ^†_kv̂_k ) | 0 >.Note that the first-order restriction is consistent with therange of validity of EI mean-field theory<cit.>.Throughout this section we take J_1=1 and suppress spin and valley indices, as they may be included straightforwardly in the derivation, as well as we assume positive order parameterfor the sake of clarity (η=0). To first order in the two-body factor u, the QMC wave function isΨ_QMC(𝐫_1,𝐫_2,…,𝐫_N_e) = [ 1 +∑_i<j u(𝐫_i,𝐫_j)]Φ_0(𝐫_1,𝐫_2,…,𝐫_N_e),where N_e is the number of electrons.The Slater determinant Φ_0 in real space is obtainedby projecting |0> ontoψ̂^†(𝐫_1)ψ̂^†(𝐫_2) …ψ̂^†(𝐫_N_e)|vac>,where |vac> is the vacuum with no electrons present. The Fermi field annihilation operator ψ̂ is spanned by the basis of conduction and valence band operators,ψ̂(𝐫) = ψ̂_c(𝐫)+ψ̂_v(𝐫),withψ̂_c(𝐫)=∑_kφ_ck(𝐫) ĉ_kandψ̂_v(𝐫)=∑_kφ_vk(𝐫) v̂_k,where the explicit effective-mass form of Bloch states φ_ck and φ_vk was given in Supplementary Note 1.Similarly, we work out the form of Ψ_EI in real space, Ψ_EI(𝐫_1,𝐫_2,…,𝐫_N_e) =<vac|ψ̂(𝐫_N_e) ψ̂(𝐫_N_e - 1)…ψ̂(𝐫_1)×∏_k u_k(1 + g_k ĉ^†_kv̂_k)v̂^†_k_1v̂^†_k_2…v̂^†_k_N_e| vac>,where the valence band states k_1, k_2, …, k_N_e,are filled up to the Dirac point in |0> andwe defined g_k = v_k / u_k. To first order in g_k, Ψ_EI readsΨ_EI(𝐫_1,𝐫_2,…,𝐫_N_e) = B Φ_0(𝐫_1,𝐫_2,…,𝐫_N_e) +B ∑_k g_k <vac|ψ̂(𝐫_N_e) ψ̂(𝐫_N_e - 1)…ψ̂(𝐫_1) ĉ^†_kv̂_k v̂^†_k_1v̂^†_k_2…v̂^†_k_N_e| vac>,where B=∏_k u_k is a constant.After expanding the field operators ψ̂ in the second row onto the basis of v̂ and ĉ [cf. (<ref>) and (<ref>)],we observe that the only non-vanishing contributions consist in products of N_e-1 operators v̂ times a single operator ĉ_k. Sinceĉ_k occurs N_e times in the ψ̂(𝐫_i)'s, with i=1,…,N_e, we may writeΨ_EI(𝐫_1,𝐫_2,…,𝐫_N_e) = B Φ_0(𝐫_1,𝐫_2,…,𝐫_N_e)+B ∑_k g_k ∑_i=1^N_e∑_k_1'⋯∑_k_i-1'∑_k_i+1'⋯∑_k_N_e' ×φ_ck(𝐫_i)φ_vk_1'(𝐫_1)…φ_vk_i-1'(𝐫_i-1) φ_vk_i+1'(𝐫_i+1) …φ_vk_N_e'(𝐫_N_e)×<vac|v̂_k_N_e'…v̂_k_i+1'ĉ_kv̂_k_i-1'…v̂_k_1' ĉ^†_kv̂_k v̂^†_k_1v̂^†_k_2…v̂^†_k_N_e| vac>. To make progress, we considerthe generic operator identityψ̂(𝐫) ψ̂^†(𝐫') + ψ̂^†(𝐫') ψ̂(𝐫) =δ(𝐫-𝐫').Since electrons are mainly localized at honeycomb lattice sites 𝐑 and there is—on the average—one electron per site (N_e=2N), this identity may approximately be expressed as ψ̂_v(𝐫_i) ψ̂_v^†(𝐫_j) + ψ̂_v^†(𝐫_j) ψ̂_v(𝐫_i) ≈δ_𝐫_i,𝐫_j/aLN,which provides a useful representation of the identity operator Î for any position of the ith electron:aLN ∑_j=1^N_e[ ψ̂_v(𝐫_i)ψ̂_v^†(𝐫_j) + ψ̂_v^†(𝐫_j) ψ̂_v(𝐫_i)] ≈Î.Furthermore, in the spectral representation of Î we single out the contribution of momentum k,Î≈aLN ∑_j=1^N_e[ φ_vk(𝐫_i) φ_vk^*(𝐫_j)( v̂_k v̂^†_k + v̂^†_k v̂_k ) + ∑_k'≠ kφ_vk'(𝐫_i) φ_vk'^*(𝐫_j) ( v̂_k'v̂^†_k' + v̂^†_k'v̂_k') ],which we plug into Supplementary Eq. (<ref>). Note that, unless 𝐫_i=𝐫_j, the contribution originating from the second addendum between square brackets of Supplementary Eq. (<ref>) is much smaller than the one linked to the first addendum because terms that are summed over k'cancel out as they have random phases, being proportional to exp[ik'(y_i - y_j)].The outcome is Ψ_EI(𝐫_1,𝐫_2,…,𝐫_N_e)=B Φ_0(𝐫_1,𝐫_2,…,𝐫_N_e)+B ∑_k g_k ∑_i,j=1^N_e∑_k_1'⋯∑_k_i-1'∑_k_i+1'⋯∑_k_N_e' × aLN φ_ck(𝐫_i) φ_vk^*(𝐫_j) φ_vk_1'(𝐫_1)…φ_vk_i-1'(𝐫_i-1)φ_vk(𝐫_i) φ_vk_i+1'(𝐫_i+1) …φ_vk_N_e'(𝐫_N_e)×<vac|v̂_k_N_e'…v̂_k_i+1' v̂_k v̂^†_k ĉ_k v̂_k_i-1'…v̂_k_1' ĉ^†_kv̂_k v̂^†_k_1v̂^†_k_2…v̂^†_k_N_e| vac>+(contact term),where the last contact term is negligible unless two electrons touch. Importantly, the e-h pair wave function φ_ck(𝐫) φ_vk^*(𝐫')= χ^cv_k( 𝐫 - 𝐫')occurring in the second row of (<ref>) depends on 𝐫 - 𝐫' only, which allows to decouple the sums over 𝐫_i and𝐫_i - 𝐫_j, respectively. Then Supplementary Eq. (<ref>) may be rearranged asΨ_EI(𝐫_1,𝐫_2,…,𝐫_N_e)=B Φ_0(𝐫_1,𝐫_2,…,𝐫_N_e)+B ∑_ℓ=1^N_e aLN ∑_k g_kχ^cv_k(𝐫_ℓ)×∑_k_1'⋯∑_k_i-1'∑_i=1^N_e∑_k_i+1'⋯∑_k_N_e' φ_vk_1'(𝐫_1)…φ_vk_i-1'(𝐫_i-1)φ_vk(𝐫_i) φ_vk_i+1'(𝐫_i+1) …φ_vk_N_e'(𝐫_N_e)×<vac|v̂_k_N_e'…v̂_k_i+1' v̂_kv̂_k_i-1'…v̂_k_1' v̂^†_k_1v̂^†_k_2…v̂^†_k_N_e| vac> +(contact term),where, among all addenda of the mixed sum over momenta k' and index i, the only non-vanishing contributions are those permutating the annihilation operators applied to |0> that belong to the set {v̂_k_1, v̂_k_2, …, v̂_k_N_e}.The final result isΨ_EI(𝐫_1,𝐫_2,…,𝐫_N_e)=B[ 1 +∑_ℓ= 1^N_eΦ_exc(𝐫_ℓ) ] Φ_0(𝐫_1,𝐫_2,…,𝐫_N_e) + (contact term),with the exciton wave function Φ_exc being defined asΦ_exc(𝐫) = aLN ∑_filled k g_k χ^vc_k(𝐫),where the sum over k is limited to those levels that are filled in|0> and 𝐫 is the electron-hole distance. Supplementary Eq. (<ref>) is a non trivial result, as it showsthat the EI wave function in real space is the product of the Slater determinant Φ_0—a conventional fermionic state—timesthe sum over ℓ of bosonic wave functionsΦ_exc(𝐫_ℓ)—the exciton wave functionintegrated over the whole range of possible e-h distances. The significance of Φ_exc relies on its Fourier transform in reciprocal space, g_k, which is the ratio of those variational factors that solve the gap equation, v_k and u_k. Thegap equation may be regarded as the many-exciton counterpart of the Bethe-Salpeter equation.Supplementary Eq. (<ref>) should be compared with Supplementary Eq. (<ref>): When no pairs of electrons are in contact, QMC and mean-field EIwave functions coincide apart from a normalization factor, provided that u(𝐫,𝐫')= 2Φ_exc(𝐫-𝐫')/N. When two electrons touch, say 𝐫_i=𝐫_j, a discrepancy arises, which is expected since the QMC wave function enforces the cusp condition whereasthe mean-field ansatz does not. § DETECTION OF PEIERLS CHARGE DENSITY WAVETHROUGH THE ORDER PARAMETER Ρ_TRANSL The QMC analysis of main text introduces the order parameter ϱ_Translas a measure of thecharge displacement between adjacent unitary cells along the tube axis. If the ground state is a charge density wave (CDW) withperiod 2a (the characteristic wave vector is q=π/a),then the quantum average of ϱ_Transl extrapolated to the thermodynamic limit is finite. In this section we discuss whether the order parameter ϱ_Translmay also detect a Peierls CDW with nesting vector q=2k_F,the Fermi wave vector being located at Dirac point K.A first issue is the commensurability of the QMC supercell with respect to the period of Peierls CDW. According to DFT calculation k_F = 0.289(2π)/a, hence q=2k_F=0.422 (2π)/a (folded back to first Brillouin zone)and the period is 2.37 a(Supplementary Fig. 12b).This implies that the size of the commensurate supercell exceeds our computational capability. On the other hand, the size of a smaller supercell may approximately match a multiple of the Peierls CDW period. This is the case e.g. of a supercell made of seven units,whose length compares with three times the period, 7.11a.The key issue is the finite-size scaling of ϱ_Transl averaged over the Peierls CDW ground state. To gain a better understanding, we introduce a simple model for a genericCDW. The charge density profile, n_q(z), is a sinusoidal modulation of wave vector q along the axis z,n_q(z) = n_modsinqz + n_0,where n_modis the modulation amplitude,n_0 is the homogeneous background, and we ignore the relaxation of the ground state occurring in a finite-size supercell.The order parameter ϱ^model_Transl that fits to the model (<ref>) isϱ^model_Transl = 1/N_cell∑_ℓ=1^N_cell (-1)^ℓ-1∫_a(ℓ-1)^aℓdz[ n_q(z) - n_0],whereN_cellis a number of unitary cells such that N_cella is approximately commensurate with the CDW period, 2π/q.If N_cell is even, then, except for a prefactor, ϱ^model_Transl is equivalent to ϱ_Transl as defined in the main text. The extrapolated value of ϱ^model_Transl in the thermodynamic limit, N_cell→∞, is trivial in two cases. For the undistorted structure, ϱ_Transl=0 as the integral ofn_q-n_0 over the unitary cell vanishes.This is illustrated in Supplementary Fig. 12a, where the blue (red) colour stands for positive (negative) charge deviation, n_q(z)-n_0. Second, for the dimerized CDW of period 2a, which is discussed in the main text,any cell with N_celleven is commensurateand hence ϱ^model_Transl = 2an_mod/π, the integral ofn_q - n_0 over the unitary cell exhibiting alternate signbetween adjacent cells (Supplementary Fig. 12c). In the following, in order to compare with the VQMC extrapolated order parameter ϱ_AB discussed in the main text, we take n_moda/π = ϱ_AB/2 = 0.00825.We now focus on the Peierls case of nesting vector q=2k_F (Supplementary Fig. 12b). We assume that N_cell takes only those integer values closest to (2.37) m, with m=1,2,…, which ensures that supercell and CDW periods are approximately commensurate. As illustrated by Supplementary Fig. 12d, (ϱ^model_Transl)^2exhibits a complex, non-monotonic dependence on the inverse number of atoms before vanishing as 1/N_atom→ 0 [here N_atom=12 N_cell as the (3,3) nanotube has twelve atoms per cell]. 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(eds.) booktitleMany-body physics, pages351–411 (publisherGordon and Breach, addressNew York, year1967).* This work was supported in part by European Union H2020-EINFRA-2015-1 programme under grant agreement No. 676598 project`MaX–Materials Design at the Exascale'.S.S. acknowledges computational resouces provided throughthe HPCI System Research Project No. hp160126 on the K computerat RIKEN Advanced Institute for Computational Science.D.V., E.M. & M.R. acknowledge PRACE for awarding them access toresource Marconi based in Italy at CINECA (Grant No. Pra14_3622). Contributions M.R. and E.M. initiated this project, D.V., E.M., and M.R. designed a comprehensive strategy to tackle the instability problem by means of different methods, D.V. developed the many-body perturbation theory calculations and analysis,D.V. and D.S. optimized the Yambo code for the calculation in the presence of the magnetic field, S.S. and M.B. developedthe quantum Monte Carlo calculations and analysis,M.R. developed the effective-masstheory and wrote the paper, all authors contributedto the analysis of data and critically discussed the paper. Competing Interests The authors declare that they have nocompeting financial interests. Correspondence Correspondence and requests for materialsshould be addressed to M.R. (email: [email protected]). | http://arxiv.org/abs/1703.09235v2 | {
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survey.bib theoremTheorem[section] definition[theorem]Definition corollary[theorem]Corollary proposition[theorem]Proposition lemma[theorem]Lemma statement[theorem]Statement remark[theorem]Remark conjecture[theorem]Conjecture example[theorem]Example | http://arxiv.org/abs/1703.08684v1 | {
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Comment on 'Fisher information for quasi-one-dimensional hydrogen atom' O. Olendski Department of Applied Physics and Astronomy, University of Sharjah, P.O. Box 27272, Sharjah, United Arab Emirates ================================================================================================================================== A further argument is provided in the discussion on the correct form of the bound-state momentum wave function of the quasi-one-dimensional hydrogen atom; namely, considering its behavior at the large quantum indices, it is reconfirmed that the complex expression from Olendski (2017) Eur. J. Phys. 38 038001 is a correct one. Groundlessness of other interpretations is also highlighted..7in Comment on 'Fisher information for quasi-one-dimensional hydrogen atom' O. Olendski Department of Applied Physics and Astronomy, University of Sharjah, P.O. Box 27272, Sharjah, United Arab Emirates ==================================================================================================================================A correct form of the bound-state momentum wave function Φ_n(p), n=1,2,…, of the quasi-one-dimensional (Q1D) hydrogen atom (i.e., a quantum structure whose potential V in position representation is given by V(x)={[ -α/x, x>0;∞,x≤0, ]. α>0) was derived in Ref. <cit.>. For doing this, its position counterpart Ψ_n(x) was found first as a solution of the corresponding Schrödinger equation, which yields the energy spectrum coinciding with the 3D hydrogen atom E_n=-1/2n^2, with the corresponding eigen functions represented as Ψ_n(x)=2x/n^5/2e^-x/nL_n-1^(1)(2x/n), with L_m^(β)(x), m=0,1,…, being a generalized Laguerre polynomial <cit.>. Starting from Eq. (<ref>), we use the Coulomb units where energies, distances and momenta are measured in terms of m_pα^2/ħ^2, ħ^2/(m_pα), and m_pα/ħ, respectively, m_p being a mass of the particle. Position and momentum waveforms are related through the Fourier transformation Φ_n(p)=1/√(2π)∫_0^∞ e^-ipxΨ_n(x)dx. As a result, one gets after simple but elegant calculation: Φ_n(p)=(-1)^n+1√(2n/π)(1-inp)^n-1/(1+inp)^n+1. Obviously, both Ψ_n(x) and Φ_n(p) satisfy orthonormality requirements:∫_0^∞Ψ_n'(x)Ψ_n(x)dx=∫_-∞^∞Φ_n'^∗(p)Φ_n(p)dp=δ_nn',δ_nn'={[1,n=n';0, n≠ n' ]. is a Kronecker delta, n'=1,2,…. Observe that since the position function Ψ_n(x) does not possess any symmetry, its momentum counterpart Φ_n(p) is essentially complex. One has to mention that the very similar form of the momentum waveform was derived for the motion on the whole x-axis in the potential V(x)=-|x|^-1 <cit.>.Comment <cit.> appeared as a reaction to the resent research <cit.> where it was stated that the momentum function of the same structure is real and given in an awkward form involving Chebyshev polynomials<cit.> with the argument (1-n^2p^2)/(1+n^2p^2). Authors of Ref. <cit.> did not agree with the conclusions leading to Eq. (<ref>), and in their latest Arxiv submission <cit.> they found that their momentum waveform from Ref. <cit.> is just an imaginary part of the function from Eq. (<ref>): Φ_n^STC(p)=(Φ_n(p))=(-1)^n√(2n/π)sin(2narctan(np))/1+n^2p^2. Inspired by this discovery, Saha, Talukdar and Chatterjee (STC) <cit.> analyze the graphs of (Φ_n(p)) and (Φ_n(p)), which they borrow from Ref. <cit.>, with their main conclusion (provided without any proof or relevant references) stating that since the position wave function Ψ_n(x) has (n-1) zeros on the real semi axis 0≤ x<∞, its Fourier transform Φ_n(p) should exhibit similar behavior as a function of p. Moving forward, they declare a function from Eq. (<ref>) 'physically inadmissible' since only its imaginary part vanishes at the origin while the real component, which, as was shown in Ref. <cit.>, is an even function of the momentum, has a nonzero value at p=0. Thus, STC state that only the function from Eq. (<ref>) "exhibitsthephysicallyacceptable number of nodesfor all values of n and represents the true momentum-space wavefunction for the Q1D hydrogen atom" <cit.>. Unfoundedness of such interpretation is evident and speaks for itself. To show a physical absurdity of these arguments, one has to recall that the most general physical meaning of the wave function is not determined by the number of its nodes and their correspondence to their counterparts in a conjugate space. The one and only fundamental physical meaning of the wave function is a square of its absolute value, which defines a probability of finding the quantum particle in the corresponding state. For the momentum density γ_n(p)=|Φ_n(p)|^2 one elementary finds for our system <cit.>: γ_n(p)=2n/π1/(1+n^2p^2)^2. Consider what will happen in the limit of the large quantum numbers, n→∞. First of all, from Eq. (<ref>) it follows that the corresponding energy turns to zero, E_n=∞=0. Next, this quasi classical approximation implies that the energy is so high that the particle does not 'see' the corresponding potential and its total energy equals to the kinetic component only, E_n=p^2, n→∞, which, obviously, becomes zero too: p→0at n→∞. But this is exactly what the probability density from Eq. (<ref>) tells us; namely, the corresponding function at the quantum index n tending to infinity transforms, as it is elementary to check, into the δ-function: γ_n(p)→δ(p), n→∞. Thus, in this limit all moments but p=0 are eliminated. Everything is physically logical and consistent! Evolution of the density γ_n(p) is shown in Fig. <ref> for several indices. It is seen that the bell-like shape of γ_n(p) at the larger n sharpens around p=0. In turn, the cornerstone of the STC theory was the fact that the momentum function (and, accordingly, the corresponding probability density) is an identical zero at the zero momentum what strictly forbids the states with p=0 at any quantum index. This contradiction of the unphysical and mathematically faulty STC arguments with the quasi-classical limit reconfirms the correctness of Eq. (<ref>).To make the function from Eq. (<ref>) normalizable, STC multiply it by two; as a result, their density becomes: γ_n^STC(p)=8n/πsin^2(2narctan(np))/(1+n^2p^2)^2. The point is that this density is normalizable for the non negative p only: ∫_0^∞γ_n^STC(p)dp=1. STC do not explain why they do arbitrarily forbid motion with the negative momenta but, doing so, they commit another blunder; namely, the function from Eq. (<ref>) violates Shannon entropy inequality <cit.>:S_ρ+S_γ≥1+lnπwith S_ρ=-∫Ψ^2(x)ln(Ψ^2(x))dxS_γ=-∫γ(p)ln(γ(p))dpand integrations carried out over all possible intervals. Calculating position Shannon entropy for the lowest level yields: S_ρ_1=2γ=1.1544 with γ being Euler's constant: γ=lim_n→∞(∑_i=1^n1/i-ln n)=0.5772. Momentum entropy withthe correct density from Eq. (<ref>) and with the correct limits -∞<p<∞ can be evaluated analytically for any quantum state: S_γ_n=-ln2n/π+4(ln2-1/2), yielding, in particular, for the ground level: S_γ_1=1.2242, S_ρ_1+S_γ_1=2.3786, satisfying, of course, Eq. (<ref>) since 1+lnπ=2.1447. At the same time, STC density, Eq. (<ref>), multiplied by its own logarithm and integrated over non negative momenta only gives: S_γ_1^STC=0.5575, S_ρ_1+S_γ_1^STC=1.7119, in obvious violation of Eq. (<ref>). This is not surprising since the rigorous derivation of this inequality <cit.> uses full Fourier transform, Eq. (<ref>), and not its arbitrary castrated sine counterpart.I would like to finish by citing again an except from Ref. <cit.>; this time, the phrase from its last sentence will be used; namely, let us hope that the analysis provided above will help "... to resolve any confusion regarding the choice of momentum-space wave function for the Q1D hydrogen atom". 99 Olendski1O. Olendski, Eur. J. Phys. 38, 038001 (2017). Abramowitz1M. Abramowitz and I. A. Stegun1964 Handbook of Mathematical Functions(New York: Dover) Nunez1H. N. Núñez-Yépez, A. L. Salas-Brito, and D. A. Solis, Phys. Rev. A 83, 064101 (2011). Saha1A. Saha, B. Talukdar, and S. Chatterjee, Eur. J. Phys. 38, 025103 (2017). Saha2A. Saha, B. Talukdar, and S. Chatterjee, arXiv:1703.03578. Bialynicki1I. Białynicki-Birula and J. Mycielski,44, 129 (1975). Beckner1W. Beckner,102, 159 (1975). | http://arxiv.org/abs/1703.10042v3 | {
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Count-ception: Counting by Fully Convolutional Redundant Counting Joseph Paul CohenMontreal Institute for Learning AlgorithmsUniversité of MontréalFriends of the Farlow FellowHarvard University Herbaria [email protected] Geneviève BoucherInstitute for Research in Immunology and CancerUniversité of Montré[email protected] Craig A. GlastonburyBig Data InstituteUniversity of [email protected] Z. LoDepartment of Computer ScienceUniversity of Massachusetts [email protected] BengioCIFAR Senior FellowMontreal Institute for Learning AlgorithmsUniversité of Montré[email protected] 30, 2023 ==============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================Counting objects in digital images is a process that should be replaced by machines. This tedious task is time consuming and prone to errors due to fatigue of human annotators. The goal is to have a system that takes as input an image and returns a count of the objects inside and justification for the prediction in the form of object localization. We repose a problem, originally posed by Lempitsky and Zisserman, to instead predict a count map which contains redundant counts based on the receptive field of a smaller regression network. The regression network predicts a count of the objects that exist inside this frame. By processing the image in a fully convolutional way each pixel is going to be accounted for some number of times, the number of windows which include it, which is the size of each window, (i.e., 32x32 = 1024). To recover the true count we take the average over the redundant predictions. Our contribution is redundant counting instead of predicting a density map in order to average over errors. We also propose a novel deep neural network architecture adapted from the Inception family of networks called the Count-ception network. Together our approach results in a 20% relative improvement (2.9 to 2.3 MAE) over the state of the art method by Xie, Noble, and Zisserman in 2016. § INTRODUCTION Counting objects in digital images is a process that is time consuming and prone to errors due to fatigue of human annotators. The goal of this research area is to have a system that takes as input an image and returns a count of the objects inside and justification for the prediction in the form of object localization.The classical approach to counting involves fine-tuning edge detectors to segment objects from the background <cit.> and counting each one.A large challenge here is dealing with overlapping objects which require methods such as the watershed transformation <cit.>. These approaches have many hyperparameters specifically for each task and are complicated to build. The core of modern approaches was described by Lempitsky and Zisserman in 2010 <cit.>. Given labels with point annotations of each object, they construct a density map of the image. Here, each object predicted takes up a density of 1, so a sum of the density map will reveal the total number of objects in the image. This method naturally accounts for overlapping objects We extend this idea and focus on two main areas:* We propose redundant counting instead of a density map approach in order to average over errors.* We propose a novel construction of networks and training that can apply to counting tasks with very complicated objects. We repose the problem of predicting a density map to instead predict a count map which contains redundant counts based on the receptive field of a smaller regression network. The regression network predicts a count of the objects that exist inside this frame as shown in Figure <ref>. By processing the image in a fully convolutional way <cit.> each pixel is going to be accounted for some number of times, the number of windows which include it, which is the size of each window, (i.e., 32 × 32 = 1024). To recover the true count we can take the average of all these predictions. Figure <ref> illustrates how this change in kernel makes more sense with respect to the receptive field of the network that must make predictions. Using the Gaussian density map forces the model to predict specific values based on how far the cell is from the center of the receptive field. This is a harder task than just predicting the existence of the cell in the receptive field.A comparison of these two types of count maps is shown in Figure <ref>.To perform this prediction we focus on a method using deep learning <cit.> and convolutional neural networks <cit.> like Xie <cit.> and Arteta <cit.> have. They utilized networks similar to FCN-8 <cit.> which form bottlenecks at the core of the network to capture complex relationships in different parts of the image. Instead, we pad the borders of the input image so that the receptive field of the regression network will redundantly count the correct number of times. This way we do not bottleneck the representation in any way. § RELATED WORK The idea of counting with a density map began with Lempitsky and Zisserman in 2010 <cit.> where they used dense SIFT features from the image as input to a linear regression to predict a density map. We predict redundant counts instead of a density map. Although a summation over the output of the model is taken over both causes, our method is explicitly designed to tolerate the errors when predictions are made.However, the density map of objects does count multiple times indirectly. It needs to properly predict a density map of objects which is generated from a small Gaussian with the mean at the point annotation. The values they need to predict vary as some are at the mean and some are not. It doesn't take into account the receptive field so the objects may be in view and the network has to suppress its prediction.Many approaches were introduced to predict a better density map. Fiaschi 2012 <cit.> used a regression forest instead of a linear model to make the density prediction based on BoW-SIFT features. Arteta 2014 <cit.> proposed an interactive counting algorithm which would extend this algorithm to more dynamically learn to count various concepts in the image. Xie 2016 <cit.> introduced deep neural networks to this problem. Their method built a network which would convolve a 100 × 100 region to a 100 × 100 density map. Once this network was trained it can be run in a fully convolutional way similar to our method. However, these approaches focus on predicting a density map which differentiates them from our work.Arteta 2016 <cit.> discuss new approaches past the density model. Their focus is different than our work. They tackle the problem of incorporating multiple point annotations from noisy crowd sourced data. They also utilize segmentation of the background to filter our erroneous predictions that may happen there.In Segui <cit.> their method takes the entire image as input and output a single count value using fully connected layers to break the spatial relationship. They discover that a network can learn to count and while doing this they learn features for identifying the objects such as MNIST digits. We use this idea in that the regression network is learning to count the 32 × 32 frame. But we expect it to produce errors so we perform this task redundantly.Xie in 2015 <cit.> presented an interesting idea similar to the direction we are going in. Their goal is to predict a proximity map which consists of cone shaped distributions over each cell which smooths each cell prediction using surrounding detections. This cone extended only 5 pixels from the point annotation which was the average size of the cell. However, this approach is more in line with a density map than a count map. § FULLY CONVOLUTIONAL REDUNDANT COUNTING§.§ Problem Statement We would like to obtain the count of objects in an input image I being given only a few training examples with point annotations of each object. The objects to count are often very small, and the overall image very large.Because counting is labor-intensive, there are often few labeled images in practice.§.§ Overview of Technique Motivation: We want to merge the idea of networks that count everything in their receptive field by Segui <cit.> with the density map of objects by Lempitsky and Zisserman <cit.> using fully convolutional processing like Xie <cit.> and Arteta <cit.>.Technique: Instead of using a CNN that takes the entire image as input and produces a single prediction for the number of objects we use a smaller network that is run over the image to produce an intermediate count map. This smaller network is trained to count the number of objects in its receptive field. More formally; we process the image I with this network in a fully convolutional way to produce a matrix F(I) that represents the counts of objects for a specific receptive field r × r of a sub-network that performs the counting. A high-level overview:* Pre-process image by padding* Process image in a fully convolutional way* Combine all counts together into total count for imageThe fully convolutional network processes an image by applying a network with a small receptive field on the entire image.This has two effects which reduce overfitting.First, by being small, the fully convolutional network has much fewer parameters than a network trained on the entire image.Second, by splitting up an image, the fully convolutional network has much more training data to fit parameters on. The following discussions will consider a receptive field of 32 for simplicity and in order to have concrete examples. This method can be used with any receptive field size. An overview of the process is shown in Figure <ref>.§.§ Input We want to count target objects in an image I.This image has multiple target objects that are labelled with single point labels L.Because the counting network only reduces the dimensions from (32 × 32) →(1 × 1) the input I must be padded in order to deal with objects that appear on the border. Objects on the border of the image will at most be in the receptive field of a network with only one column or row overlapping the input image. For r=32 a pixel in F(I) can only be 15 pixels from the border of I. F(I) is meant to align with the target T. It is important that these be aligned such that the receptive field of the network aligns with the proper regression target.§.§.§ Constructing the target image T The target image can be constructed from a point-annotated map L, the same size as the input image I, where each object is annotated by a single pixel. This is desirable because labeling with dots is much easier than drawing the boundaries for segmentation.Let R(x,y) be the set of pixel locations in the receptive field corresponding to T[x,y].Then we can construct the target image T: T[x,y]=∑_(x',y') ∈ R(x,y) L[x',y'] Here T[x,y] is the sum of cells contained in a region the size of the r × r receptive field. This will become the regression target for the r × r region of the image.§.§ Fully Convolutional Redundant Counting We use fully convolutional networks with a receptive field of 32 × 32.The output of the fully convolutional network on the entire 320 × 320 image is 287 × 287 pixels.This yields a fully convolutional network output image larger than the original input.Each pixel in the output will represent the count of targets in that receptive field.To perform this mapping we propose the Count-ception architecture which is adapted from the Inception family of networks by Szegedy et. al. <cit.>. Our proposed model is shown in Figure <ref>. At the core of the model Inception units are used to perform 1x1 (pad 0) and 3x3 (pad 1) convolutions at multiple layers without reducing the size of the tensor. After every convolution a Leaky ReLU activation is applied <cit.>. We notice an improvement of the regression predictions with the Leaky ReLU during training because the output can be pushed to zero and then recover to predict the correct count.Our modifications are in the down sampling layers. We removed the max pooling and stride=2 convolutions. They are replaced by large convolutions. This makes it easierto calculate the receptive field of the network because strides add a modulus to the calculation of the count map size.We perform this down sampling in two locations using large filters to greatly reduce the size of the tensor. A necessity in allowing the model to train is utilizing Batch Normalization layers <cit.> after every convolution.§.§ Loss Functions and Regularization We tried many combinations of loss functions and found L1 loss to perform the best.min||F(I) - T||_1 Xie found that the L2 penalty was too harsh to the network during training.We reached the same conclusion for our configuration and chose an L1 loss instead. We also tried to combine this basic pixel-wise loss with a loss based on the overall prediction in the entire image. We found this caused over-fitting and provided no assistance in training. The network would simply learn artifacts in each image in order to correctly predict the overall counts.§.§ Combining Sub-Image Counts The above loss is a surrogate objective to the real count that we want. We intentionally count each cell multiple times in order to average over possible errors.With a stride of 1, each target is counted once for each pixel in its receptive field. As the stride increases, the number of redundant counts decreases. # redundant counts = (r/s)^2 In order to recover the true count we divide the sum of all pixels by the number of redundant counts. # true counts =∑_x,y F(I)[x,y]/# redundant counts There are many benefits to using redundant counts. If the pixel label is not exactly at the center of the cell, or even outside the cell, the network can still learn because on average the cell will appear in the receptive field. §.§ Limitations With this approach we sacrifice the ability to localize each cell exactly with x,y coordinates. Viewing the predicted count map can localize where the detection came from (shown in Figure <ref>) but not to a specific coordinate. For many applications accurate counting is more important than exact localization. Another issue with this approach is that a correct overall count may not come from correctly identifying cells and could be the network adapting to the average prediction for each regression. One common example is if the training data contains many images without cells the network may predict 0 in order to minimize the loss. A solution similar to Curriculum Learning <cit.> is to first train on a more balanced set of examples and then take well performing networks and train them on more sparse datasets. § DATASETSVGG Cells: To compare with the state of the art we first use the standard benchmark dataset which was introduced by Lempitsky and Zisserman in 2010 <cit.>. There are 200 images with a 256x256 resolution that contain simulated bacterial cells from fluorescence-light microscopy created by <cit.>. Each image contains 174 ± 64 cells which overlap and are at various focal distances simulating real life imaging with a microscope. MBM Cells: We also use a real dataset based on the BM dataset introduced by Kainz et al. in 2015 <cit.> which consists of eleven 1,200 × 1,200 resolution images of bone marrow from height healthy individuals. The standard staining procedure used depicts in blue the nuclei of the various cell types present whereas the other cell constituents appear in various shades of pink and red. We modified this dataset in two ways to create the MBM dataset (Modified BM). First the 1,200 × 1,200 images were cropped to 600 × 600 in order to process the images in memory on the GPU and also to smooth out evaluation errors during training for a better comparison. This yields a total of 44 images containing 126 ± 33 cells (identified nuclei).In addition, the ground truth annotations were updated after visual inspection to capture a number of unlabeled nuclei with the help of domain experts.Adipocyte Cells:Our final dataset is a human subcutaneous adipose tissue dataset obtained from the Genotype Tissue Expression Consortium (GTEx) <cit.>. 200 Regions Of Interest (ROI) representing adipocyte cells were sampled from high resolution histology slides by using a sliding window of 1700 × 1700. Images were then down sampledto 150 × 150, representing a suitable scale in which cells could be counted using a 32 × 32 receptive field. The average cell count across all images is 165±44.2. Adipocytes can vary in size dramatically (20-200μ) <cit.> and given they are densely packed adjoining cells with few gaps, they represent a difficult test-case for automated cell counting procedures. § EXPERIMENTS First, we compare the overall performance of our proposed model to existing approaches in Table <ref> for each dataset. For each dataset we follow the evaluation protocol used by Lempitsky and Zisserman in 2010 that has been used by all future papers. In this evaluation protocol, training, validation, and testing subsets are used. The held-out testing set size is fixed for all experiments while training and validation sizes (N) are varied to simulate lower or higher numbers of labeled examples. The algorithm trains on the training set only while being able to early stop by evaluating its performance on the validation set. The size of the training and validation sets are varied together for simplicity. The results of the algorithm using at least 10 random splits are computed and we present the mean and standard deviation. The testing set size remains constant in order to provide constant evaluation. If the testing set were chosen to be all remaining examples (|Testing| =|Total|-2N) instead of a fixed size then smaller N values would be less impacted by difficult examples in the test set because examples are not sampled with replacement.As a practitioner baseline comparison we compare our results to Cell Profiler's <cit.> which uses segmentation to perform object identification and counting. This is representative of how cells are typically counted in biology laboratories.To do so, we designed two main different pipelines and evaluated the error on 10 splits of 100 randomly chosen images for the synthetic dataset (VGG Cells) and on 10 splits of 10 images for the bone marrow dataset (MBM Cells) to mimic the experimental setup in place since Cell Profiler does not use a training set.For the MBM Cells, we report the performance using the same pipeline (single) for all images and using three slightly modified versions of the pipeline (multiple) where a parameter was adjusted to account for color differences seen in 8 of the 44 images.Among other methods we compare with Xie's FCRN-A network <cit.>. Only Xie's and our method (Count-ception) are neural network based approaches. Our network is sufficiently deeper than the Xie's FCRN-A network and that representational power together with our redundant counting we are able to perform significantly better. We show in <ref> that the performance of our model matches that of Xie's when the redundant counting is disabled by changing the stride to eliminate redundant counting.§.§ Training In order to train the network we used the Adam optimization technique <cit.> with a learning rate of 0.005 and a batch size of 4 images. The training runs for 1000 epochs and the best model based on the validation set error is evaluated on the test set. The weights of the network were initialized using the Glorot initialization method <cit.> adjusted for ReLU gain. §.§ Redundant CountingWe claim redundant counting is significant to the success of the method. By increasing the stride we can reduce double counting until there is none. We present the reader Table <ref> which indicates that a stride of 1, meaning the maximum amount of redundant counting patch_size^2, is the optimal choice. As we increase the stride to equal the patch size where no redundant counting is occurring the accuracy is reduced.The power of this algorithm is in the redundant counting. However, increasing the redundant count is complicated. The receptive field could be increased but this will add more parameters which cause the network to overfit the training data. We explored a receptive field of 64x64 and found that it did not perform better. Another approach could be to use dilated convolutions <cit.> which would be equivalent to scaling up the input image resolution. §.§ Runtime and Implementation The run-time of this algorithm is not trivial. We explored models with less parameters and found they could not achieve the same performance. Shorter models (fewer layers) or narrower models (less filters per layer) tended to not have enough representational power to count correctly. Making the network wider would cause the model to overfit. The complexity of the Inception modules were significant to the performance of the model.The network was implemented in lasagne (version 0.2.dev1) <cit.> and Theano (version 0.9.0rc2.dev) <cit.> using the libgpuarray backend. The source code and data will be made available online[<https://github.com/ieee8023/countception>].§ CONCLUSIONIn this work we rethink the density map method by Lempitsky and Zisserman <cit.> and instead predict counts in a redundant fashion in order to average over errors and reduce overfitting. This redundant counting approach merges ideas by Segui <cit.> of networks that count everything in their receptive field with ideas by Lempitsky and Zisserman of using the density map of objects together with ideas by Xie <cit.> and Arteta <cit.> of using fully convolutional processing.We call our new approach Count-ception because our approach utilizes a counting network internally to perform the redundant counting. We demonstrate that this approach outperforms existing approaches and can also perform well with very complicated cell structure even where the cell walls adjoin other cells. This approach is promising for tasks with different sizes of objects which have complicated structure. However, the method has some limitations. Although the count map can be used for localization it cannot easily provide x,y locations of objects. § ACKNOWLEDGMENTS This work is partially funded by a grant from the U.S. National Science Foundation Graduate Research Fellowship Program (grant number: DGE-1356104) and the Institut de valorisation des données (IVADO). This work utilized the supercomputing facilities managed by the Montreal Institute for Learning Algorithms, NSERC, Compute Canada, and Calcul Quebéc. We also thank NVIDIA for donating a DGX-1 computer used in this work. ieee | http://arxiv.org/abs/1703.08710v2 | {
"authors": [
"Joseph Paul Cohen",
"Genevieve Boucher",
"Craig A. Glastonbury",
"Henry Z. Lo",
"Yoshua Bengio"
],
"categories": [
"cs.CV",
"cs.LG",
"stat.ML"
],
"primary_category": "cs.CV",
"published": "20170325164903",
"title": "Count-ception: Counting by Fully Convolutional Redundant Counting"
} |
Department of Physics and Astronomy, University of Nevada, Las Vegas, Las Vegas, NV 89154, [email protected] The repeating FRB 121102 (the “repeater”) shows repetitive bursting activities and was localized in a host galaxy at z=0.193. On the other hand, despite dozens of hours oftelescope time spent on follow-up observations, no other FRBs have been observed to repeat. Yet, it has been speculated that the repeater is the prototype of FRBs, and that other FRBs should show similar repeating patterns. Using the published data, we compare the repeater with other FRBs in the observed time interval (Δ t) - flux ratio (S_i / S_i+1) plane.We find that whereas other FRBs occupy the upper (large S_i / S_i+1) and right (large Δ t) regions of the plane due to the non-detections of other bursts, some of the repeater bursts fall into the lower-left region of the plot (short interval and small flux ratio) excluded by the non-detection data of other FRBs. The trend also exists even if one only selects those bursts detectable by the Parkes radio telescope. If other FRBs were similar to the repeater, our simulations suggest that the probability that none of them have been detected to repeat with the current searches would be ∼ (10^-4-10^-3). We suggest that the repeater is not representative of the entire FRB population, and that there is strong evidence of more than one population of FRBs.§ INTRODUCTION A decade ago a mysterious new class of millisecond-duration radio transients called Fast Radio Bursts (FRBs) were discovered <cit.>. These are highly dispersed, bright (∼ Jy) bursts detected from high galactic latitudes <cit.>. Since the first discovery, twenty-six FRBs have been discovered and twenty two have been published[http://www.astronomy.swin.edu.au/pulsar/frbcat/]. The large dispersion measure (DM), if interpreted as due to dispersion of free electrons in the intergalactic medium, implies that the sources are at cosmological distances, e.g. in the redshift range 0.2-1.5. The isotropic energy release of FRBs is estimated to be in the range of ∼ 10^40-41 erg <cit.>.Out of these detected FRBs, one source, FRB 121102 (also known as the “repeater”), clearly repeats <cit.>. Thanks to its repeating behavior, a compact, steady radio counterpart associated with the bursting source was detected<cit.> and a star formation host galaxy at z=0.193 was identified <cit.>, which provides solid evidence of the cosmological origin of at least this FRB source. Most other FRBs have been re-observed after their discoveries, but so far no positive detection of a repeating burst has been reported <cit.>. One possible exception might be the FRB 110220 / FRB 140514 pair, which are in the same 14.4 arcmin beam and are 9 arcmin apart. <cit.> hypothesized that they may originate from the same neutron star embedded within a supernova remnant (SNR) that provides an evolving DM as the ejecta expands. A detection of a third burst with an even smaller DM from this locationwould support this hypothesis. In any case, it was speculated by some authors<cit.> that other FRBs are not much different from the repeater, and the non-detection of the repeating bursts from those sources might be due to the lower sensitivity of the Parkes telescope (which detected most of the FRBs published so far) than the Arecibo 300-m telescope (which detected many bursts from the repeater). On the other hand, it has been also suggested <cit.> that there might be more than one class of FRBs: e.g. repeating and non-repeating ones. The non-repeating FRBs are found to be usually unresolved, whereas the repeating bursts of FRB 121102 are all resolved with a temporal structure.Many FRB progenitor models have been proposed in the literature. Some models invoke non-catastrophic events such as giant magnetar flares <cit.>, giant pulses from young pulsars <cit.> and young rapidly-spinning magnetars <cit.>, repeated captures of asteroids by a neutron star <cit.>, and repeated “cosmic combing” events when an astrophysical stream interacts with the magnetosphere of an old neutron star <cit.>. Alternatively, other classes of models invoke catastrophic events, such as the collapse of supra-massive neutron stars (the so-called “blitzars") <cit.> and mergers of compact stars (NS-NS, NS-BH, BH-BH) <cit.>. The FRBs produced from these systems would not repeat.In this paper, we use the published detection data of the repeating FRB 121102 and the non-detection data of other FRBs to investigate whether the FRB 121102 source is representative of all FRBs in terms of its repeating behavior. § FRB REPEATING TIME INTERVAL - FLUX RATIO DISTRIBUTION If other FRBs also repeat but no repeating bursts are detected, it could be due to either of the following two reasons: (1) The repeating time interval (duty cycle) is long, and the current observation time is not long enough to catch another burst yet; (2) Other repeating bursts might have occurred, but they are too faint to be detected by the searching telescopes (e.g. Parkes). In order to quantify these two effects, we introduce two parameters: the observed time interval between bursts, Δ t, and the peak flux ratio between two adjacent bursts, S_i/S_i+1. The former addresses the duty cycle of the repeating bursts, whereas the latter addresses the probability that the additional bursts are too faint to be detected. §.§ The Raw Δ t - (S_i/S_i+1) Plot For both repeating FRB 121102 and other non-repeating FRBs, observations were initially carried out in a certain time interval. Later the sources were re-observed multiple times. For each source, there are several time gaps during which no telescope was observing the source. As a result, it is impossible to precisely define the intrinsic time intervals between two adjacent bursts. In our analysis, we define an observed time interval Δ t as the time interval between bursts during the observational time span only, so that the time gaps during which no observation was carried out are removed. In order to test the validity of using Δ t, we perform a set of Monte Carlo simulations by assuming different (uniform, power law, Gaussian, and log-Gaussian) distributions of the intrinsic time interval Δ t_ intr between adjacent bursts. By randomly choosing observational time intervals following a uniform distribution, we find that regardless of the Δ t_ intr distribution, the observed Δ t distribution essentially follows the Δ t_ intr distribution if the number of observational windows is large enough. As an example, we show in Fig.<ref> the simulated Δ t distribution (red) against a log-Gaussian Δ t_ intr distribution (black), with the number of observations comparable to that for FRB 121102.We therefore use the Δ t data of FRB 121102 to derive the Δ t_ intr distribution of FRBs in general (assuming that they are similar to each other). For FRB 121102,the first burst was discovered in the PALFA survey at 1.4 GHz <cit.>. Later an extensive follow-up observation using Arecibo resulted in the detection of 10 more bursts <cit.>. Another six bursts from this source were later detected and published: five from Green Bank telescope (GBT) at 2.0 GHz and one at 1.4 GHz with the Arecibo Observatory (AO). Later, 14 more bursts from FRB121102 were detected and published: ten from GBT at 2.0 GHz and four from AO at 1.4 GHz <cit.>. In our analysis, we include a total of 40 bursts based on the published data, even though by now > 140 bursts from this source have been detected, but are not published yet (L. Spitler, 2017, talk presented at the Aspen FRB conference, http://aspen17.phys.wvu.edu/Spitler.pdf). Including all >140 bursts in our analysis will only strengthen the conclusion of this paper. Bursts from this source are irregular and clustered in time <cit.>. The total telescope time spent on FRB121102 was over 177 hours for the observations of the 40 bursts. Among the 40 bursts, six bursts were found within a 10 min of observing period, and four were detected in a 20-min observing window <cit.>. Recently, <cit.> reported two bursts that are 0.038 seconds apart. The total observing time is spread over telescopes with different sensitivities, and the observations were done in a range of radio frequencies. For completeness, we include all the bursts regardless of the observing telescope and frequency[Ignoring repeating bursts detected by telescopes other than Arecibo would introduce a bias of the repeating duty cycle of the source. It would be more appropriate to include all the available information in this analysis, even though the data are not uniform. An attempt of removing the non-uniformity of the data is presented below in Section <ref>.]. The observed time intervals Δ t are calculated for each pair of adjacent bursts defined by the sequence numbers i and i+1[The definition of the burst number is arbitrary. Given two adjacent bursts defined, one can derive S_i/S_i+1 and Δ t. When a higher detection threshold is defined (e.g. for Parkes-threshold bursts only, see Sect. <ref>), the burst numbers are re-ordered, and S_i/S_i+1 and Δ t are re-defined.]. Table <ref> presents Δ t of all adjacent pairs of the bursts, which are also presented in Figure <ref>.Other FRBs (detected by Parkes and GBT) have been extensively followed up on a variety range of timescales, most with dozens of hours of observing times (Table <ref> and references therein). No additional bursts have been detected. We include all the observing times after the detection of the original burst and define it as the lower limit of Δ t for the bursts whose flux reaches the flux threshold. The data are shown in Table <ref> and presented in Figure <ref>.The flux ratio is defined as S_i/S_i+1, where S_i and S_i+1 are the peak fluxes of i-th and (i+1)-th burst. It is greater/less than one if the next FRB is fainter/brighter.The peak fluxes and pulse widths for each burst from FRB 121102 are different. For completeness, we include all the bursts to define S_i/S_i+1, most of which were detected at 1.4 GHz, but some of which were detected at 2 GHz. They are presented in Figure <ref>.For other FRBs, since no repeating bursts have been detected, any possible burst during the observation time frame, if any, must have a peak flux that is faint enough to evade detection.For these FRBs, we can only plot the lower limit of S_i/S_i+1, which is simply S_1/S_th, where S_th is the 10σ detection flux with respect the receiver noise level. Such a threshold has been widely adopted to claim the detection of an FRB. The values of S_1/S_th are adopted as the lower limit of S_i/S_i+1 as shown in Table <ref>. The raw Δ t - S_i/S_i+1 plot is shown in Figure <ref>a. The orange stars denote the repeating bursts from FRB 121102, while for other bursts, the allowed regions are above the horizontal line and to the right of the vertical line for each burst (as denoted by the right and upper arrows). One can immediately see that some bursts from the repeating FRB 121102 are located in the region (lower left region) in the Δ t - S_i/S_i+1 plane that is excluded by the data of other FRBs. Compared with other FRBs, for the repeater the time intervals between bursts are much shorter, and the flux ratios are smaller, some of them being even less than unity (suggesting that the later burst is brighter than the earlier one).Assuming that FRB 110220 and FRB 140514 belong to the same source, as argued by <cit.>, we plot another point (purple square) based on the detected values. This point sits between the region of FRB 121102 and the region of other FRBs. On the other hand, the non-detection of other bursts following FRB 140514 again places the second point of the source to the region of other FRBs. cccccc The observational information of the 40 bursts detected from the repeater FRB 121102. Column 1 gives the burst number, column 2 gives the receiver and telescope name, column 3 gives the observed time interval Δ t between the burst i and i+1, column 4 gives the peak flux density for each burst, column 5 gives the flux ratio S_i/S_i+1, and column 6 gives the references. Burst Telescope/Δ tPeakFlux RatioRef Number Receiver (s)Flux (Jy)(S_i/S_i+1) 1 AO/ALFA 128 0.04 0.00 [1,2] 2 AO/ALFA 11694 0.03 1.33 [2,3] 3 AO/ALFA 512 0.03 1.00 [2,3] 4 AO/ALFA 1444 0.04 0.75 [2,3] 5 AO/ALFA 569 0.02 2.00 [2,3] 6 AO/ALFA 419 0.02 1.00 [2,3] 7 AO/ALFA 23 0.14 0.142 [2,3] 8 AO/ALFA 58 0.05 2.80[2,3] 9 AO/ALFA 186 0.05 1.00 [2,3] 10 AO/ALFA 169 0.05 1.00 [2,3] 11 AO/ALFA 58 0.31 0.16 [2,3] 12 GBT/S-band 22400 0.04 7.75[3] 13 GBT/S-band 28033 0.06 0.66 [3] 14 GBT/S-band 414 0.04 1.50 [3] 15 GBT/S-band 442 0.02 2.00[3] 16 GBT/S-band 416 0.09 0.22 [3] 17 AO/L-WIDE 53871 0.03 3.00[2,3] 18VLA/3GHz 302382 0.12 0.25[4,5] 19 VLA/3GHz 10543 0.67 0.18[4,5] 20 VLA/3GHz 13210.02724.81 [4,5] 21 VLA/3GHz 65560.0630.43[4,5] 22 VLA/3GHz 28736 0.3280.19[4,5] 23 VLA/3GHz 11701 0.0398.41[4,5] 24 VLA/3GHz 81850.0500.78[4,5] 25 GBT/S-band 60490.36 0.14[6] 26 GBT/S-band 0.0380.08 4.5 [6] 27 VLA/3GHz 281430.086 0.93[4,5] 28 GBT/S-band 2292 0.36 0.24 [6] 29 GBT/S-band 3836 0.29 1.24[6] 30 VLA/3GHz 105000.1581.83 [6] 31 GBT/S-band 776530.17 0.92[6] 32 GBT/S-band 1319 0.38 0.44[6] 33 GBT/S-band 367 0.20 1.9[6] 34 GBT/S-band 2745 0.09 2.22[6] 35 GBT/S-band 1723 0.56 0.16 [6] 36 GBT/S-band 1358 0.11 5.09 [6] 37 AO/1.4GHz7636 0.09 1.22 [6] 38 AO/1.4GHz 6780.02 4.5[6] 39 AO/1.4GHz 1045 0.02 1.0[6] 40 AO/1.4GHz 1949 0.03 0.66 [6] [1] <cit.>; [2] <cit.>; [3] <cit.>; [4] <cit.>; [5] <cit.>: [6] <cit.> cccc The follow-up observation information of other FRBs that do not show repeating bursts, including the total number of observation hours and the lower limit of S_i/S_i+1, which is defined by S_1/S_th. FRB Name Total hours Flux Ratio Reference (yymmdd) (hours) S_i/S_i+1FRB01062115.5>1.63[6,11]FRB01072440>2.30[7]FRB09062533.65>3.00 [1,3]FRB1102201.75>4.90[1,2]FRB1105236.6>4.20[15]FRB11062611.25>1.10[1,2]FRB11070310.1>1.60[1,2]FRB1201275.5>1.10[1,2]FRB12100210.25>1.60[1,3]FRB1306269.5>2.10 [1,3]FRB1306289.0>2.90[1,3]FRB13072910>1.40[1,3]FRB13110478>3.00[8]FRB14051419.2>1.60[4]FRB15021517.5>1.90[14]FRB15041813.0>3.90[9]FRB150807215>5.00 [10]FRB16031730.0>1.30 [12]FRB16041030.0>1.30 [12]FRB16060830.0>1.20 [12]FRB17010726.1>1.60[13] † - Follow up hours are not published [1]<cit.>; [2]<cit.> [3] <cit.> [4] <cit.> [5] <cit.> [6] <cit.> [7]<cit.> [8] <cit.> [9] <cit.> [10]<cit.>[11] <cit.> [12] <cit.> [13] <cit.> [14] <cit.> §.§ Correcting to the sensitivity limit of Parkes Since Arecibo is more sensitive than Parkes, a more rigorous treatment of the repeating FRB 121102 should include only those bursts that are above the Parkes sensitivity threshold for a fair comparison with the results of other FRBs. The peak flux densities (for pulse widths ranging from 1.28 - 8.192 ms using a signal-to-noise ratio of 10.0) of Parkes FRBs are in the range0.11 - 0.28 Jy. Out of the 40 repeating bursts from FRB 121102, ten bursts (burst numbers 7, 11, 19, 22, 25, 28, 30, 32, 33, 35) would have been above the Parkes S/N cutoff limit at 1.4 GHz.For example, both bursts 7 and 11 were originally detected by Arecibo within a 10 min observation window <cit.>. Bursts 8, 9, and 10 would not be detected (for a flat spectrum) by Parkes, we therefore pretend that they were not detected and redefine Δ t and S_i/S_i+1 and plot the points in Figure <ref>. Out of the 15 bursts detected by GBT at 2 GHz, five of them (number 25, 28, 32, 33, 35) would have been detected by Parkes for a flat spectrum[Burst numbers 19, 22, 33 where detected by Very Large Array at 3 GHz, and would be also detected by Parkes for a flat spectrum. The spectral indices of the repeating bursts of FRB 121102 vary significantly from burst to burst <cit.>, so it is hard to reliably extrapolate the fluxes from 2GHz (GBT) and 3 GHz (VLA) to 1.4 GHz. In any case, for indicative purposes we assume a flat spectrum for all the bursts and include all the bursts detected by GBT and VLA in our analysis. Introducing more precise spectral indices would modify some data points presented in Figs. <ref> and <ref>, but would not change the conclusion of this paper.]. The results are presented in Figure <ref>.Ten bursts can define nine points in Figure <ref>. One can see that although most of the points still fall in the regime allowed by the non-detection limits of other FRBs, two points lie in the lower-left corner of the plot, a region excluded by the non-detection data of other FRBs. Since the total observing hours of other FRBs are usually long (two of them, FRB131104 and FRB150807, each has follow-up hours more than 70 hours), at least some repeating bursts should have been detected if all other FRBs are similar to FRB 121102. The non-detection of any repeating bursts from any of those FRBs therefore suggests that FRB 121102 is likely not representative of the FRB population.§.§ Monte Carlo simulationsIn order to quantify the probability of non-detection any burst if all other FRBs are similar to FRB 121102, we perform Monte Carlo simulations of typical repeating FRBs based on the Δ t andS_i/S_i+1 distributions as observed in FRB 121102. The distributions are presented as the histograms in the upper and right panels of Figure <ref>b. They are fitted with logarithmic Gaussian distributions with Cash statistics <cit.>. The best fits are presented as solid curves overlapping with the histograms, with median values and standard deviations labelled. There were hundreds of pointings on FRB 121102. The intervals between these pointings (in units of seconds) generally follow a logarithmic Gaussian distribution, with a median value 4.9 and standard deviation 0.9. Each pointing has a duration from hundreds to tens of thousands of seconds, with the logarithmic median value 3.6 and standard deviation 0.4. Following these realistic observational properties, we simulate 500 pointings with logarithmic Gaussian distributed durations and intervals. Tens of bursts are “detected”, and their observed time intervals follow what was observed.In order to test whether other non-repeating FRBs are similar to FRB 121102, we simulate other FRBs with the sameΔ t and S_i/S_i+1 distributions as FRB 121102. Non-repeating FRBs were typically observed for tens of hours. We simulate the observational intervals and durations of these FRBs similar to FRB 121102, and keep thetotal observational time as the true observational time of each FRB. We notice that the flux range of FRB 121102 is by a factor 0.67/0.02 ∼ 34. We therefore require that the simulated fluxes of other bursts from the non-repeating FRBs have a flux distribution by a factor of 40. One important parameter to perform the simulations is the normalization of the flux distribution. We consider two cases for the flux distribution: Case I: The flux range is (0.05 - 2) times of the detected flux. This assumes that the detected burst belongs to one of the brightest among the bursts, which is possible due to the observational selection effect (brighter ones are easier to detect). Notice that this is the most conservative case in terms of detecting repeating bursts, since mostother bursts are fainter than this one and more likely to be below the sensitivity threshold. Case II: This possibility assumes that the detected burst is anormal one, so that the flux range is (0.16 - 6.3) times of the detected flux.Based on the detected flux of each FRB, we randomly choose S_i/S_i+1 and Δ t from their respective distributions. We check whether the resultant flux is within the defined flux range. If so, we move to simulate next burst. If not, we discard the S_i/S_i+1 value and draw another one. We repeat the process until the simulated flux is within the pre-defined flux range. This simulation effectively takes into account the flux distribution of FRB 121102. Indeed, after simulations of multiple bursts, our simulated burst flux distribution is consistent with the flux distribution of FRB 121102 (aside from the difference in the normalization factor). For each non-repeating FRB, we simulate the source based on the above procedure for a duration of the total observing time scale for the source. For each burst that is generated from the simulation, we compare it with the flux sensitivity threshold based on the S/N of the detected burst and the relative flux between the simulated burst and the initial detected burst. Only those bursts that are detectable by Parkes are kept. We focus on the most conservative Case I. Figure <ref> upper panel shows one realization of the simulations, which is the histogram of the detected burst number besides the already detected one. One can see that for this realization, even though most sources (14 altogether) are consistent with having zero detected bursts (consistent with observations), other 7 sources all have extra detected bursts. One source even has 8 detectable bursts. This is inconsistent with the observations.To quantify the probability of non-detection of any repeating burst from all 21 non-repeating FRBs, we perform 10,000 simulations and draw the probability distribution of the expected detectable bursts from all 21 sources. This is shown in the black histogram in Figure <ref> lower panel. One can see the most probable case is to have 7 bursts detected from these sources. The probability of having no detection is p ∼ 1.2× 10^-3. If one considers the less conservative normalization (Case II) by assuming that the detected bursts are typical for FRBs, the peak distribution of the detectable bursts shifts to 9 (blue dotted histogram), and the non-detection probability is dropped to p ∼8.9 × 10^-5. We therefore conclude that the assumption that all FRBs are similar to FRB 121102 has a very low probability to be consistent with the data.There is strong evidence of more than one population of FRBs.§ CONCLUSIONS AND DISCUSSION Using the published data, we compare the repeating FRB 121102 and other FRBs in the Δ t - (S_i/S_i+1) plane. We find that some bursts of the repeater are well separated from other FRBs in the plane. In particular, they occupy a region that has a much smaller duty cycle and a smaller flux ratio than other bursts. The conclusion remains valid if one only chooses the bursts that are detectable with the Parkes radio telescope. Dozens of hours of telescope time have been spent on the follow-up observations of other FRBs, yet none of them have been observed to repeat. Our Monte Carlo simulations suggest that the probability of non-detection of any burst in the 21 FRB sources is ∼ (10^-4-10^-3). We therefore draw the conclusion that the repeating FRB 121102 is not representative of the entire FRB population, and that more than one population of FRBs is needed to account for the current data.Since only the published data are used to perform this study, our conclusion is very conservative. Many more bursts have been detected from the repeating FRB 121102 (L. Spitler, 2017, http://aspen17.phys.wvu.edu/Spitler.pdf), and many more hours have been spent on other FRBs without a detection of repeating FRBs. This suggests that FRB 121102 and other FRBs should be even more separated in the Δ t - (S_i/S_i+1) plane, which further strengthens our conclusion. One possibility is that the unusually active behavior of FRB 121102 is a result of a special amplification effect. For example, some propagation effects, such as scintillation-induced intermittency or plasma lensing <cit.>, may play a role of facilitating the detections of the bursts from FRB 121102, since its Galactic latitude is relatively low. Even though such a possibility is plausible, we notice that if other FRBs have a similar S_i/S_i+1 distribution as FRB 121102, still more than one burst should have been detected even if they were not amplified. This may be seen by artificially raising the detecting threshold for the FRB 121102 bursts (e.g. Fig. <ref>).We therefore suggest that the propagation effect cannot explain the dichotomy of the repeating behaviors of FRB 121102 and other FRBs.The difference between FRB 121102 and other repeating FRBs is likely physical. One may consider two possibilities. First, maybe all FRBs share the same progenitor system, but FRB 121102 is unusually active in terms of producing repeating bursts. Within the spindown and magnetic powered models <cit.>, a more active central engine should correspond to a younger age. According to this picture, other FRBs should have older ages than the repeater. However, the essentially zero evolution of DM for the repeater <cit.> places a lower limit on the age of the putative supernova remnant associated with the pulsar <cit.>. This is in apparent conflict with the putative FRB 110220/140514 association <cit.>, which has a much more rapid DM evolution but less frequent bursts.A more likely possibility is that there are more than one population of FRBs. The repeating FRB 121102 has a distinct progenitor system than (at least some of) other FRBs. The current data already showed strong evidence of such a possibility. It is possible that (at least some of) other FRBs are produced in catastrophic events so that they are intrinsically non-repeating FRBs. Blitzars <cit.> and merger of compact stars <cit.> remain attractive candidate progenitors for these non-repeating FRBs. A future coincident detection of an FRB with a catastrophic event (e.g. a gravitational wave chirp signal) would unambiguously establish such a non-repeating FRB population.We thank an anonymous referee for very helpful suggestions. This work is partially supported by NASA through grants NNX15AK85G and NNX14AF85G to UNLV. 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"authors": [
"Divya Palaniswamy",
"Ye Li",
"Bing Zhang"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170327180005",
"title": "Are there multiple populations of Fast Radio Bursts?"
} |
[footnoteinfo]The material in this paper was not presented at any conference.First]Huong [email protected], First]James S. [email protected], Second]Mazen [email protected],[First]School of Electrical Engineering and Computer Science, The University of Newcastle, Australia [Second]Gipsa-lab, Control Systems Department. University of Grenoble, France continuous time identification;time delay, useful redundancy, filtering. This paper proposes an algorithm to estimate the parameters, including time delay, of continuous time systems based on instrumental variable identification methods. To overcome the multiple local minima of the cost function associated with the estimation of a time delay system, we utilise the useful redundancy technique. Specifically, the cost function is filtered through a set of low-pass filters to improve convexity with the useful redundancy technique exploited to achieve convergence to the global minimum of the optimization problem. Numerical examples are presented to demonstrate the effectiveness of the proposed algorithm.§ INTRODUCTIONThe goal of system identification is to estimate the parameters of a model in order to analyse, simulate and/or control a system. In the time domain, there are two typical approaches to identify a system, i.e. discrete-time (DT) identification and continuous-time (CT) identification. For several decades, DT identification has been dominant due to the strong development of the digital computer. More recently, estimation using continuous-time identification methods has received much attention due to advantages such as providing insights to the physical system and being independent of the sampling time <cit.> <cit.> <cit.>. For example, with irregular sampling time, the DT model becomes time-varying; hence the DT system identification problem becomes more difficult while for CT identification, the system is still time-invariant. In reality, irregular sampling occurs in many cases, e.g. when the sampling is event-triggered, when the measurement is manual and also in the case of missing data <cit.>.In some CT identification problems, one wants to estimate the system parameters and any unknown time delay, as there exist many practical examples including, chemical processes, economic systems, biological systems, that possess time delays. It is important to estimate the delay accurately since a poor estimate can result in poor model order selection and inaccurate estimates of the system parameters. There are many approaches to estimate a system time delay <cit.>. A simple approach is to consider the impulse response data, e.g. estimate the time delay by finding where the impulse response becomes nonzero <cit.> or by noting the delay where the correlation between input and output is maximum <cit.> <cit.>. Another approach is to model the delay by a rational polynomial transfer function using a Padé or similar approximation and then estimate the time delay as part of the system parameters <cit.> <cit.> <cit.>. In <cit.> <cit.> <cit.>, the time delay and system parameters of a Multiple Input Single Output CT system are estimated in a separable way using an iterative global nonlinear least-squares or instrumental variable method.Recently, a method of estimating the parameters and time delay of CT systems has been suggested in <cit.> which is based on a gradient technique. The parameters and the time delay are estimated separately, i.e. when one is estimated, the other is fixed which is then repeated in an iterative manner. In this approach, the Simplified Refined Instrumental Variable (SRIVC) method is used to estimate the parameters whilst the time delay is estimated using the Gauss-Newton method. In addition, due to the effects of multiple minima in the cost function to be minimized for the time delay <cit.> <cit.> <cit.>, a low-pass filter is employed to increase convexity. As shown in <cit.> <cit.> <cit.>, a suitable low-pass filtering operation on the estimation data can help to extend the global convergence region of the cost function, hence improve the accuracy of the time delay estimate.In this paper, we adopt the idea of using a low-pass filter where instead of using only one filter, we suggest to use multiple low-pass filters and incorporate the useful redundancy technique <cit.> <cit.>. Useful redundancy is a technique that was developed to avoid local minima when solving a nonlinear inverse problem. The concept is to generate a family of cost functions that have different local minima, but share the same global minimum with the original cost function of the optimization problem. Whenever the algorithm is trapped in a minimum, the solver path is switched to another solver path in a way such that the minimum found using the new solver path corresponds to a decrease in the original cost function. This allows the algorithm to cross local minima and converge to the global minimum, hence improving the accuracy of the estimated parameters. For the algorithm described in this paper, the multiple cost functions are generated by filtering the original time delay cost function through a number of low-pass filters with different cut-off frequencies that span the system bandwidth.The paper is organized as follows. Section 2 describes the model setting and Section 3 recalls both the SRIVC algorithm and the SRIVC-based time delay estimation using a low-pass filter. Section 4 formulates the algorithm of the new method and provides analysis on the effectiveness of the proposed method. Section 5 presents numerical experiments and results for both regular and irregular sampling schemes. Finally, the conclusion will be drawn in Section 6. § MODEL SETTINGConsider a continuous-time linear, time invariant, single input single output system,x(t) = G_0(p)u(t-τ_0) =B(p)/A(p)u(t-τ_0),withB(p)= b_0p^m+b_1p^m-1+...+b_m, A(p)= p^n+a_1p^n-1+...+a_n, n≥ mwhere τ_0 is the time delay, u(t), x(t) are the input and deterministic output of the system respectively and p is the differential operator, i.e. p^(i)x(t)= d^ix(t)/dt^i. In addition, the following assumptions are made:Polynomials B(p) and A(p) are coprime. The system is asymptotically stable.The high frequency gain of G_0(p) is 0, i.e. G_0(p) is strictly proper. The deterministic output x(t_k) is measured as y(t_k) in the presence of noise, i.e.y(t_k) = x(t_k) + e(t_k). Furthermore, we consider the sampling time of the input u(t_k) and output data y(t_k) as either regular or irregular. The time-varying sampling interval is denoted as,h_k = t_k+1-t_k,k = 1,2, ..., N-1,where N is the length of the data. The objective of a CT system identification problem is to estimate the time delay, τ_0, and the parameters a_1, a_2, ..., a_n, b_0, b_1, ..., b_m of the CT model in (<ref>), using the measured input and output data, u(t_k) and y(t_k)^N_k=1 respectively.§ PARAMETER AND TIME DELAY IDENTIFICATION OF CONTINUOUS-TIME MODELS WITH SRIVC AND FILTERINGThere exists a large number of continuous time identification methods see, for example, <cit.> <cit.>. In this paper, we consider the Simplified Refined Instrumental Variable Continuous Time method (SRIVC), which is developed in the literature by Young and Garnier <cit.><cit.><cit.>. We begin with a basic description of the SRIVC method. §.§ Traditional SRIVC method To simplify the description we first assume the time delay τ_0 is known. The SRIVC method can then be summarized as follows.From (<ref>) and (<ref>),A(p)/A(p)y(t_k) = B(p)/A(p)u(t_k-τ_0) + e(t_k),which becomes,A(p)y_A(t_k) = B(p)u_A(t_k-τ_0) + e(t_k),withy_A(t_k) = 1A(p)y(t_k), u_A(t_k-τ_0) = 1A(p)u(t_k-τ_0). From (<ref>), a linear regression model can be formed as y_A^(n)(t_k) = φ_N(t_k,τ_0)θ + e(t_k)where φ_N(t_k) = [-y_A^(n-1)(t_k),-y_A^(n-2)(t_k),... , -y_A(t_k), u_A^(m)(t_k-τ_0),u_A^(m-1)(t_k-τ_0) ...u_A(t_k-τ_0)], θ = [a_1, a_2, ..., a_n, b_0, b_1, ..., b_m]^T,y_A^(i)(t_k) = d^iy_A(t_k)dt^i, u_A^(i)(t_k-τ_0) = d^iu_A(t_k-τ_0)dt^i. To solve the linear regression problem in (<ref>), the SRIVC algorithm uses an instrumental variable (IV) method. The instruments in the SRIVC method are chosen as the estimated noise free outputs, i.e. the regressor becomes,φ̂_N(t_k) = [-x̂_A^(n-1)(t_k),-x̂_A^(n-2)(t_k),... , -x̂_A(t_k),u_A^(m)(t_k-τ_0),u_A^(m-1)(t_k-τ_0) ...u_A(t_k-τ_0)]where x̂_A(t_k) = 1/Â(p)x̂(t_k); x̂(t_k)= B̂(p)/Â(p)u(t_k-τ_0),with B̂(p) and Â(p) estimates of B(p) and A(p). §.§ Implementation of the CT filtering operation for irregular sampled data and arbitrary time delay The SRIVC method can be used to estimate system parameters from regular sampled data as well as from irregular sampled data. However, one difficulty in implementing the SRIVC method for irregular sampled data in the presence of an arbitrary time delay is the CT filtering operation, e.g., u_A(t_k-τ) = 1A(p)u(t_k-τ),when τ is an arbitrary time delay.There are two reasons for this difficulty: * u(t_k-τ) is not available from the measured data,* The digital simulation of u(t_k-τ) is generally performed in state-space form hence, an equivalent discrete-time state space representation of the CT state space model needs to be computed. For irregular data, as the sampling interval h(t_k) is time-varying, the computational load using standard methods, e.g. expm in Matlab, to compute the transformation matrices will be large. As suggested in <cit.>, the two problems mentioned above can be solved as follows,* u(t_k-τ) can be constructed from the neighbouring data based on the inter-sample behaviour, e.g. zero-order-hold (ZOH), first-order-hold (FOH), etc.* To reduce the computational load, the time-varying sampling interval is divided into two intervals: the first interval is a multiple of a constant sampling period, the second interval is the residual. The discretization matrix of the first interval is pre-computed using a standard method and stored in an array. The discretization matrix for the second interval is computed by a fast approximation, e.g. the 4th order Runge-Kutta (RK4) method. The final discretization matrix of the sampling interval h_k is a product of the two matrices.Specifically, convert the transfer function (TF) model in (<ref>) to the equivalent CT state-space model,ż(t-τ) = F_cz(t-τ)+G_cu(t-τ) u_A(t-τ) = H_cz(t-τ).The equivalent DT state-space model will be,z(t_k+1-τ) = F(h_k)z(t_k-τ)+M(t_k) u_A(t_k-τ) = H_cz(t_k-τ)where F(h_k), M(t_k) are computed as follows, * Denote t_k,1, t_k,n_k (n_k ≥ 2) as t_k-τ, t_k+1-τ respectively; t_k,i, i = 2, ..., n_k-1 as the transition time-instants of u(t) between t_k-τ and t_k+1-τ; and h_k,i = t_k,i+1-t_k,i.* Let Δ be a constant sampling period; m_k,i be a positive integer; δ_k,i≥ 0 such that: h_k,i=m_k,iΔ+δ_k,i<(m_k,i+1)Δ.* Compute F(mΔ), G(mΔ).* Compute F(δ_k,i), G(δ_k,i) using, e.g. RK4.* Finally, compute F(h_k), M(t_k) as follows,F(h_k) = e^F_ch_k, M(t_k) = ∑_i=1^n_k-1∏_j=i+1^n_k-1F(h_k,j)G(h_k,i) u(t_k,i),withF(h_k,i) = F(m_k,iΔ)F(δ_k,i), G(h_k,i) = F(m_k,iΔ)G(δ_k,i) + G(m_k,iΔ). Now that we can generate filtered signals of the irregular sampled data, we describe the traditional SRIVC algorithm <cit.><cit.><cit.> in Algorithm I.ALGORITHM I Step 1. Initialization* Create the stable state variable filter (SVF),F(p) = 1(p+ω_c^SVF)^n,where ω_c^SVF is chosen to be larger than or equal to the bandwidth of the system.* Filter y(t_k) and u(t_k-τ_0) via the SVF to generate derivatives of the signals, i.e. y_A^(n)(t_k), y_A^(n-1)(t_k), ..., y_A(t_k) and u_A^(m)(t_k-τ_0), u_A^(m-1)(t_k-τ_0), ..., u_A(t_k-τ_0) by using the method described in <cit.>.* Use the least squares method to estimate the initial parameters, θ̂_0 = [Φ_NΦ_N^T]^-1Φ_NY_N.withΦ_N = [φ_N^T(t_1) ... φ_N^T(t_N)],Y_N = [y_A^(n)(t_1) y_A^(n)(t_2) ... y_A^(n)(t_N)]^T,where φ_N is defined in (<ref>). Step 2. Iterative estimationfor j = 1:convergence[Convergence requires the relative error between the estimated parameters of two consecutive iterations to be less than an ϵ>0 where ϵ is a small value chosen to achieve the desired accuracy.] * If the estimated TF model is unstable, reflect the right half plane zeros of Â(p,θ̂^j-1) into the left half plane and construct the new estimate. * Generate the instrumental variable,x̂(t_k) = B̂(p,θ̂^j-1)Â(p,θ̂^j-1)u(t_k-τ_0). * Filter the input u(t_k-τ_0), output y(t_k) and the instrumental variable x̂(t_k) by the continuous time filterF_c(p) = 1Â(p,θ̂^j-1),to generate the derivatives of these signals.* Using the prefiltered data, generate an estimate θ̂^j using the IV method,θ̂^j = [Ψ_NΦ_N^T]^-1Ψ_NY_N.where Ψ_N is the IV matrix generated by the instrumental variables x̂(t_k) and Φ_N is the regression matrix,Φ_N = [φ_N^T(t_1) ... φ_N^T(t_N)], Ψ_N = [φ̂_N^T(t_1) ... φ̂_N^T(t_N)],where φ̂_N is defined in (<ref>). end§.§ SRIVC-based time delay estimation with filteringA recently developed method <cit.> to estimate both the time delay and parameters for the CT model (<ref>) considers it as a separable nonlinear least squares problem. The SRIVC algorithm and the Gauss-Newton method are used in <cit.> to estimate the system parameters and the time delay respectively. In this problem the cost function for the time delay estimation has multiple minima <cit.>, hence a low-pass filter is utilized to extend the global convergence region <cit.> <cit.> <cit.>.Let L(p) be a CT low-pass filter with cut off frequency ω_c^L. Applying the filter L(p) to the linear regression (<ref>), we obtainy̅_A^(n)(ρ) = φ̅_N(ρ)θ + ϵ̅(ρ),where ρ = [θ^T, τ] and .̅ represents the signal y,u or e filtered by L(p).When the estimated parameter θ̂ is a function of τ, i.e.θ̂(τ) = [Ψ̅_N(ρ)Φ̅_N^T(ρ)]^-1Ψ̅_N(ρ)Y̅_N,then the time delay τ can be estimated as <cit.><cit.>,τ̂ = τarg minJ̃̅̃_̃̅̃Ñ̅̃(τ) = τarg minJ̅_̅N̅ (ρ) |_θ=θ̂(τ),withJ̅_N(ρ) = 12(N-s+1)ϵ̅^T(ρ)ϵ̅(ρ),where s is chosen to guarantee that at sample s, t_s ≥τ.By using the Gauss-Newton method, the system parameters and the time delay can be iteratively estimated,τ̂^j+1 = τ̂^j-μ^j [∇^2J̃̅̃_̃̅̃Ñ̅̃(τ̂^j)] ^-1∇J̃̅̃_̃̅̃Ñ̅̃(τ̂^j)θ̂^j+1 = [Ψ̅_N(θ̂^j,τ̂^j+1)Φ̅_N^T(θ̂^j,τ̂^j+1)]^-1Ψ̅_N(θ̂^j,τ̂^j+1)Y̅_N(θ̂^j),where μ^j is the step size and∇J̃̅̃_̃̅̃Ñ̅̃(τ̂^j) = ϵ̅^T_τϵ̅(ρ), ∇^2J̃̅̃_̃̅̃Ñ̅̃(τ̂^j) = ϵ̅^T_τϵ̅_τ-ϵ̅^T_τϵ̅_̅θ̅(ϵ̅^T_θϵ̅_̅θ̅)^-1ϵ̅^T_θϵ̅_̅τ̅, ϵ_τ = ∂ϵ(ρ)∂τ|_ρ=ρ̂^j = pG(p,θ̂^j)u(τ̂^j), ϵ_θ = ∂ϵ(ρ)∂θ|_ρ=ρ̂^j = -Ψ̅_N^T(θ̂^j,τ̂^j). The SRIVC-based time delay estimation with filtering <cit.> is summarized in Algorithm II. ALGORITHM II Step 1. Initialization* Set the initial value τ̂^0, the boundaries[τ_min and τ_max are boundaries for the delay, which are known a priori, i.e. τ_min≤τ_0 ≤τ_max] τ_min, τ_max, Δτ_min[Δτ_min is the limit value of the step size of the time delay estimate using Gauss-Newton method.], the cut-off frequency ω_c^L of L(p), the cut-off frequency of F(p), ω_c^SVF, and a small positive ϵ to indicate convergence.* Based on the initial value τ̂^0, use the SRIVC method to compute θ̂^0.* Compute ∇^2J̃̅̃_̃̅̃Ñ̅̃(τ̂^0) and ∇J̃̅̃_̃̅̃Ñ̅̃(τ̂^0) from (<ref>).Step 2. Iterative estimationfor j = 1:convergence * Compute Δτ̂^j using the filtered input/output data, equation (<ref>) andΔτ̂^j = - [∇^2J̃̅̃_̃̅̃Ñ̅̃(τ̂^j)] ^-1∇J̃̅̃_̃̅̃Ñ̅̃(τ̂^j). * * Compute τ̂^j+1 = τ̂^j + Δτ̂^j If τ̂^j+1∉[τ_min,τ_max], let Δτ̂^j = Δτ̂^j/2 and repeat this step.If |Δτ̂^j |≤Δτ_min, break.* Estimate θ̂^j+1 using the SRIVC method and time delay τ̂^j+1 from the filtered input/output data.* Compute J̃̅̃_̃̅̃Ñ̅̃(τ̂^j+1). If J̃̅̃_̃̅̃Ñ̅̃(τ̂^j+1) ≥J̃̅̃_̃̅̃Ñ̅̃(τ̂^j), let Δτ̂^j = Δτ̂^j/2 and return to (a). * If J̃̅̃_̃̅̃Ñ̅̃(τ̂^j)-J̃̅̃_̃̅̃Ñ̅̃(τ̂^j+1) ≥ϵ, go to Step 1, else break.end Step 3. Refined parameter estimationRepeat Step 2 with low-pass filter L(p) = 1. The smaller the cut-off frequency ω_c^L, the larger the global convergence region. However, when ω_c^L is chosen too small, information from the data is lost, which affects the accuracy of the parameter estimation <cit.>. The suggestion in <cit.> is, ω_c^L ≥110bw,where bw is the bandwidth of the system G_0(p) (rad/sec).§ SRIVC-BASED TIME DELAY ESTIMATION WITH MULTIPLE FILTERING AND REDUNDANCY METHODIn this section, we propose a method that improves convergence to the global minimum of the time delay optimization problem using the useful redundancy technique <cit.><cit.>. The useful redundancy technique was developed to avoid local minima when solving a non-linear inverse problem. As stated in the previous section, the main problem in time delay estimation is that the cost function J_N(ρ) possesses many local minima <cit.>. The filtering operation <cit.> described in Section 3.3 can be employed in order to increase the global convergence region of the cost function associated with the time delay estimation. However, when the data is very noisy or the initial value, τ̂^0, is located far from the global minimum, using only one filter does not guarantee that the solution of the optimization problem converges to the global minimum. To demonstrate how useful redundancy can be utilized in our estimator, we first describe the useful redundancy technique. §.§ The useful redundancy technique We define the useful redundancy technique by quoting directly from <cit.>.Definition 1 <cit.>. Consider anoptimization problem,ρminJ_0(ρ).Then it is called M-safely redundant if and only if the following conditions hold: * There exists a finite M cost functions J_i sharing the same global minimum ρ^* ∈ℝ^n.* There exists a solver (or an iterative scheme) ℓ and a finite number of iterations r^* ∈ℕ such that for some γ∈ [0,1) and all ρ∈ℙ the following inequality holds,Δ^γ_N(ρ) = i ∈{ 0,...,M }min [J_0(ℓ^(r^*)(ρ,J_i))-γ J_0(ρ)] ≤ 0where ℓ^(r^*)(ρ,J_i) is the candidate solution obtained after r^* iterations of ℓ using the cost function J_i and starting from the initial guess ρ. □The solver path (ρ, J_i) is defined as the sequence of iterates ℓ^(j)(ρ,J_i) for the solver ℓ when the cost function J_i starts from an initial guess ρ. Condition 2 means that for any initial ρ, there always exists a solver path ℓ that corresponds to a decrease in the original cost J_0 after at most r^* iterations. It is proven in <cit.> that if an optimization problem is M-safely redundant following from Definition 1, that convergence to a global minimum is guaranteed.An algorithm that describes the useful redundancy technique is given in Algorithm III.ALGORITHM III Step 1. InitializationChoose an initial value ρ^(0). Step 2. Iterative estimation for k=0:converge * Set i=0 and failure = true* While failuredo * Use the solver path (ρ, J_i) starting from ρ^(k), find the corresponding minima ξ^(k,i).* Compute failure = (J_0(ξ^(k,i)) > (1-ϵ)J_0(ρ^(k)) ) [ϵ is a predefined small value that is chosen based on the desired accuracy.]* If failure then i = (i+1) mod M, Else set k = k+1, ρ^(k) = ξ^(k,i).End whileend§.§ Theoretical results related to the time delay cost function To construct an M-safely redundant optimization problem for the estimation of the time delay, we need a cost function J_0 and multiple solver paths that satisfy the two conditions in Definition 1. Here, the solver paths are generated by filtering the time delay estimation error using a set of low-pass filters with different cut-off frequencies. The cost function J_0 is formulated from these filtered cost functions such that there always exists a solver path that corresponds to a decrease in J_0 after a finite number of iterations. Next we describe the set of filters and the cost function J_0 required to satisfy the conditions in Definition 1.Consider a continuous-time, linear, time-invariant SISO system described byy(t) = G_0(p)u(t-τ_0) + e(t).We make a further assumption on the noise e(t).e(t) is white random process uncorrelated with u having intensity λ. For an estimate G(p,θ) and τ, the estimation error ϵ(t,θ,τ) can be computed as,ϵ(t,θ,τ) = y(t) - G(p,θ)u(t-τ). As mentioned in Section 3, the delay can be estimated by minimizing the cost function J(θ,τ), where J(θ,τ) = ∫_-∞^∞ϵ(t,θ,τ)^2dt. If the estimation error is filtered by the low-pass filter L(p), then an estimate of θ and τ can be computed by,(θ̂,τ̂) = arg min J̅(θ,τ),whereJ̅(θ,τ)= ∫_-∞^∞ϵ̅(t,θ,τ)^2dt = ∫_-∞^∞{ L(p)ϵ(t,θ,τ) }^2dt= ∫_-∞^∞{ L(p)[y(t)-G(p,θ)u(t-τ)] }^2dtwhich by Parseval's theorem is equivalent to,J̅(θ,τ)= 12π∫_-∞^∞[|G_0(jω)e^-jωτ_0-G(jω,θ)e^-jωτ|^2 ×Ψ_u(ω) +Ψ_v(ω)] | L(jω)|^2dω,with Ψ_u(ω) the spectral density of u(t) and the spectral density Ψ_v(ω) = λ follows from Assumption <ref>. If the transfer function G_0(p) is known; the input signal white noise, i.e. Ψ_u(ω) = 1; and δτ = τ-τ_0, the cost function J̅(θ,τ) can be written as,J̅(δτ) = 12π∫_-∞^∞[ | 1-e^-jωδτ|^2 | G_0(jω)e^-jωτ_0|^2×Ψ_u(ω) +λ] | L(jω)|^2dω = 1π∫_-∞^∞[ (1-cos(ωδτ)) | G_0(jω)|^2 +λ2] ×| L(jω)|^2dω.Recall that we are concerned with how to choose a set of filters and the cost function J_0 such that the two conditions of an M-safely redundant problem are satisfied. To satisfy the first condition we need to ensure all the cost functions share the same global minimum. The second condition requires the filter set to be constructed such that for any initial value of δτ, there always exists a cost function whose solver path corresponds to a decrease in the cost function J_0 after a finite number of iterations. Note that these conditions can be checked if we know the locations of the global minimum and the extrema, i.e. the minima and maxima of J̅(δτ). First we establish a result for the location of the global minimum of the (non)filtered delay cost function. Consider the system G_0(p) as described in (<ref>). When λ = 0, ∀δτ∈ℝ, ∀ low-pass filters L(p) ≠ 0, such that J̅(δτ) ≥J̅(0), the equality J̅(δτ) = J̅(0) occurs if and only if δτ = 0.Proof. The proof of Theorem <ref> is provided in Appendix A.1.Theorem <ref> provides a result for the locations of the extrema of the filtered cost function J̅(δτ).If L(p) is selected such that L(p)G_0(p) is an ideal low-pass filter with cut-off frequency ω_c rad/sec, then the locations of the i^th positive extrema of the time delay cost function J̅(δτ) can be approximated by,δ̃τ_i ≃2i+14T_c - T_c(2i+1)π^2,where T_c = 2π/ω_c and i ≥ 1. When i is even, the corresponding extremum is a minimum and when i is odd, it is a maximum. Proof. The proof of Theorem <ref> is provided in Appendix A.2.From Theorem <ref>, the locations of the extrema for the filtered cost function are known, hence the global convergence region is, |δτ| < 3π2ω_c-23πω_c,corresponding to the distance from the maxima closest to the global minimum.The result of Remark <ref> can be seen in Fig. <ref>, which provides the plot of J̅(δτ) when ω_c=2π (the constant λ of the integral in (<ref>) is set to 0 here for simplicity).§.§ Time delay estimation with the useful redundancy technique Now we can specify a filter set that satisfies the two conditions in Definition 1.Condition 1: From Theorem <ref>, the first condition is satisfied for any set of low-pass filters as the global minimum of all the filtered time delay cost functions occurs when δτ = 0.Condition 2: The second condition depends critically on the choice of filters and cost function J_0. The filter set needs to be chosen such that for any initial value of δτ, there always exists a cost function whose solver path corresponds to a decrease in the cost function J_0 after a finite number of iterations. To verify that condition 2 is satisfied, consider the necessary and sufficient conditions. To satisfy the necessary condition, we need to show that for any initial value δτ_0, there always exists a path that has a minima location, δτ_1, closer to the global minimum w.r.t. δτ_0. For the sufficient condition, we need to establish that J_0(δτ_1)<J_0(δτ_0).First, we check the necessary condition. A simple choice is a filter set L_k(p), k = 1,n_f where the cut-off frequencies, ω_c,k, of L_k(p)G_0(p) span from 1/β to 1 of the system bandwidth (10 ≥β > 1)[From Remark <ref>, β is chosen ≤10.]. The cut-off frequencies are chosen based on the corresponding periods, T_c,k, being linearly spaced, i.e.,T_c,k = β T_bw + (k-1)(1-β)T_bwn_f-1, k = 1,n_f,where T_bw = 2π/bw, bw is the bandwidth of the system G_0(p) (rad/sec). This is due to the local minima of the filtered cost function being linearly dependant on the period of the cut-off frequency (Theorem <ref>). Finally we need to prove the existence of such a filter set that satisfies the necessary condition of Condition 2 in Definition 1.There exists a filter set L_k(p), k=1,n_f chosen such that, ∀ β≥5/4-1/(5π^2)3/4-1/(3π^2), where L_k(p)G_0(p) are ideal low-pass filters chosen with linearly spaced periods such that the cut-off frequencies, ω_c,k, spanning from 1/β to 1 of the system bandwidth, i.e.T_c,k = β T_bw - (k-1)(β-1)T_bwn_f-1, k = 1,n_f,where T_bw = 2π/bw and T_c,k = 2π/ω_c,k. Then,∀δτ_0 ≠ 0, ∃ L_q(p): |ξ(L_q(p),δτ_0)| < |δτ_0|,where ξ(L_q(p),δτ_0) is the corresponding minimum found using the filtered time delay cost function generated by L_q(p) with the initial delay δτ_0.Proof. The proof of Theorem <ref> is provided in Appendix A.3. From Theorem <ref>, we see that with the choice of,β≥5/4-1/(5π^2)3/4-1/(3π^2),there always exists an n_f, that can be computed by (<ref>) in Appendix A.3, that ensures the filter set defined in Theorem <ref> satisfies the necessary part of Definition 1, Condition 2. Note that (<ref>) provides a loose lower bound. It is possible to have a smaller value of n_f and still obtain a filter set that satisfies the necessary condition. If G_0(p) does not have resonant peaks then it doesn't matter if G_0(p) is known or unknown, as any ideal low pass filter L_k(p) with bandwidth smaller than the bandwidth of G_0(p) will allow L_k(p)G_0 (p) to approximate the desired low pass filter behaviour. If G_0(p) has resonant peaks and the bandwidth of L_k(p) is chosen significantly smaller than that of G_0(p) then L_k(p)G_0 (p) will approximate the desired low pass filter behaviour.An empirical method can also be utilized to determine the number of filters. For example, to check if a set of n_f filters with the ratio β satisfies the necessary condition, we can plot n_f cost functions that follow from (<ref>) in Appendix A.2 and check the minima locations using (<ref>) to see that for any initial delay, δτ_0, there always exists a cost function where the corresponding minima δτ_i is found closer to the global minimum with respect to δτ_0. Fig. <ref> shows a set of six cost functions with the ratio β = 2 for a system bandwidth of 2π rad/sec, i.e. the cut-off frequencies span from 1/2 to 1 (Hz). From the figure, we can see that for any initial value of the time delay less than 15 sec, there always exists a cost function where the minima is found closer to the global minimum with respect to the initial delay. This can be confirmed by computing the minima of all the cost functions using (<ref>) and observing the path to the global minimum. Note that in practice, as G_0(p) is not known a priori, and with the difficulty of implementing an ideal low-pass filter with irregular data, the filters L_k(p) are implemented as Butterworth filters with cut-off frequencies ω_c,k. Due to this, the extrema locations are not exactly those specified in (<ref>). We discuss the effect of this difference by first citing the following theorem from <cit.>, as a lemma. <cit.> Consider a system with transfer function F(s) = L(s)G_0(s)e^-sτ_0, let u be the input signal and w the corresponding output signal of the system. Then we have the following relationship,J̅(δτ) = 2[γ_w(0)-γ_w(δτ)] = 2γ_w(0)[1-ρ_w(δτ)],where γ_w(t) is the autocorrelation function of w(t) and ρ_w(δτ) = γ_w(δτ)/γ_w(0).Proof. See proof in <cit.>.With respect to the model setting in (<ref>), γ_w(δτ) in Lemma <ref> is the autocorrelation of the output of the system. Hence from Lemma <ref>, for an input signal of Gaussian distributed white noise, the filtered delay cost function J̅(δτ) will have the form,J̅(δτ) = ∑_k=1^n_1 a_ke^-b_kδτ + ∑_k=1^n_2 h_ke^-c_kδτcos(d_kδτ+ψ_k),where b_k is the absolute value of the k^th real pole of L(s)G_0(s); c_k and d_k are the absolute values of the real and imaginary parts of the k^th complex pole of L(s)G_0(s); and weighting coefficients h_k are related to the magnitude of the frequency response at the corresponding frequency. Augmenting a low-pass filter L(s) to G_0(s) obviously introduces slow poles into the system and by choosing L(s) to be a Butterworth filter, it inherently introduces complex poles to the system. Hence, from (<ref>), the filtered delay cost function will exhibit a slightly underdamped response. It can then be seen that the locations of the extrema of the cost function are associated with the underdamped response of the Butterworth filter. Hence, the augmentation of L(s) with G_0(s) when L(s) is implemented as a Butterworth filter provides a cost function similar in nature to Fig. <ref>.Next, to establish the sufficient condition, we need to derive a cost function J_0 that shares the same global minimum with the filtered cost functions where the minimum path corresponds to a decrease in J_0. To analyse this condition, we take into consideration the model error that is generated in the iterations when estimating the rational part of the system due to an inaccurate estimate of the delay. Consider we have an estimate for the delay, τ̂, then when estimating the rational component of (<ref>), we have the following relationship in the Laplace domain,Ĝ(s) = G_0(s)e^-(τ_0-τ̂)s,where Ĝ(s) is the estimate for the rational part of the system. It is obvious that when τ̂≠τ_0, Ĝ(s)≠ G_0(s). Hence there always exists model error in estimating the rational part of the system if τ̂≠τ_0.We now quantify the model error to understand the impact of it on the cost function. Here, a model reduction technique based on an identification method <cit.> is used to find a rational transfer function G_r(s). The idea is to generate a noise-free data set from the system G_0(s)e^-(τ_0-τ̂)s, then use the SRIVC method to obtain a rational system Ĝ_r(s) that has the same model order as the true system and satisfies (<ref>). For example, Fig. <ref> shows the graphs of ten filtered delay cost functions after including the model error for the system G(s) = 2e^-8s/(0.25s^2+0.7s+1).We now define the cost function J_0 as,J_0(δτ) = 1n_f∑_k=1^n_fJ̅̅̅_k(δτ),where J̅̅̅_k(δτ) = 100∥ y(t)-x̂̅̂_k(t) ∥_2∥ y(t) - 𝔼{y(t)}∥_2, x̂̅̂_k(t) is the estimated system output when the filter L_k(s) is used. Note that J̅̅̅_k(δτ) is the normalized version of the filtered cost function, J̅_k(δτ).The averaging is used to obtain a smooth decreasing cost function J_0 as there are always errors in the estimation due to noise and model error. Fig. <ref> shows the average normalized mean square error for the ten filtered delay cost functions shown in Fig. <ref>. It can be seen that J_0 has the same global minimum as the ten filtered cost functions. Also shown in Fig. <ref> is the minimum path through the filtered cost function that corresponds to a decrease in J_0. Note that the `∗' in Fig. <ref> corresponds to a switch between cost functions J_k(δτ).Therefore, in summary, the M-safely redundant optimization problem for the time delay estimation is defined as, τminJ_0(τ),with J_0(τ) defined as in (<ref>). The filter set consists of n_f Butterworth filters with cut-off frequencies, chosen such that the corresponding periods are linearly spaced, that span from 1/β to 1 of the system bandwidth where β≥5/4-1/(5π^2)3/4-1/(3π^2) and n_f is chosen using the methodology discussed in Section 4.3.§.§ AlgorithmThe proposed algorithm utilizing useful redundancy with multiple filters is described in Algorithm IV. ALGORITHM IV Step 1. Initialization* Select a set of low-pass filters as suggested in Section 4.3.* Set the boundaries τ_min, τ_max, Δτ_min, Δτ_max[Note that Δτ_max is used to constrain the increment Δτ̂ in the time delay estimation, i.e. when Δτ̂^j > Δτ_max, set Δτ̂^j = Δτ_max. This is to ensure the Wolfe conditions are satisfied at each iteration in the Gauss-Newton technique <cit.>.], and the SVF cut-off frequency ω_c^SVF. * Choose an initial value τ̂_0 of the time delay.Step 2. Iterative estimationfor i=1:converge * for k=1:n_f * Choose the low-pass filter L_k(p).* Set the initial delay τ̂^0_i,k as τ̂_i-1* Use the SRIVC algorithm and Gauss-Newton method as described in Algorithm II (without Step 3) to estimate the time delay τ̂_i,k. * Compute J_0(τ̂_i,k) (follows (<ref>)).end* Choose τ̂_i = τ̂_i,kargminJ_0(τ̂_i,k).* If |(τ̂_i-τ̂_i-1)/τ̂_i | ≥ϵ, go to Step 1, else break. [ϵ is a small value selected to obtain the desired accuracy.] end Step 3. Refine parameter estimationRepeat Step 2 with the low-pass filter L(p) = 1 .§ NUMERICAL EXAMPLESIn this section, we demonstrate the effectiveness of the algorithm through numerical examples. These examples are commonly used in the literature <cit.> <cit.> <cit.> and act as defacto benchmarks to compare the performance between different CT system identification algorithms. §.§ Case 1Case 1 considers the second order system used in <cit.>, i.e.G(s) = 2e^-8s0.25s^2+0.7s+1.In this example, we consider two sampling schemes for the input and output data.* Regular (uniform) sampling: The excitation input signal is a PRBS (pseudo-random binary sequence) of maximum length. The sampling time is 50ms, the number of stages in the shift register is 10 and the number of samples, N = 4000.* Irregular (nonuniform) sampling: The input excitation signal is a PRBS of maximum length. The number of stages in the shift register is 10 and the clock period is 0.5s. The input and output data are sampled at an irregular time instant t_k, with a sampling interval h_k uniformly distributed as, h_k ∼ U[0.01, 0.09] (s). The realization of h_k is randomized for each run. The number of samples, N=4000. The additive output disturbance is Gaussian distributed white noise with zero mean designed to give Signal to Noise Ratios (SNR) of 5dB and 15dB. The SNR is defined as,SNR = 10logP_xP_e,with P_x and P_e the average power of the noise-free output x(t_k) and the disturbance noise e(t_k) respectively.The system order is assumed known a priori (order 2). The frequency ω_c^SVF is chosen as 1 (rad/sec) which is same as the value used in <cit.>. To evaluate the algorithm, the time delay is initialized from different values τ̂^0=0,1,3,5,7,9 (similar to the experiment in <cit.>) and also from a random value drawn from a uniform distribution U[0, 9] (s).For the two sampling schemes, 100 different data sets are generated for each noise level. The system parameters and time delay are estimated using both the algorithm described in <cit.> and the algorithm proposed in this paper: * First we consider the algorithm developed in <cit.>: The cut off frequency of the filter is chosen as 1/10 of the system bandwidth as suggested in <cit.>. For regular data, the command `tdsrivc' from the CONTSID toolbox <cit.> is used to estimate the time delay and the system parameters. * For the multiple filtering algorithm utilizing the useful redundancy technique as proposed in this paper, the number of filters is set to 10 and the cut off frequencies, chosen based on linearly spaced periods, of the filters span from 1/10 to 1 of the system bandwidth. The order of the Butterworth filter is set to 10. For both algorithms, the time delay boundaries τ_min, τ_max are set to 0 and 15 seconds respectively. The maximum number of iterations for the Gauss-Newton method (Step 2 in Algorithm II and IV) is set to 10. The threshold, ϵ, to determine convergence is set to 10^-3. We consider the estimated time delay to be the global minimum when the relative error, ϵ_r, is less than 1%, i.e. ϵ_r = | τ̂-τ_0 |τ_0× 100%,where τ̂ and τ_0 are the estimated delay and the true system delay respectively.Fig. <ref> contains plots of the 10 cost functions corresponding to the filters used in the useful redundancy method. It can be seen that they all share the same global minimum, however they all possess different local minima. Fig. <ref> presents a plot of J_0, for a single dataset, showing the trajectory of the time delay estimate when the sampled data is regular and the SNR is 15dB. The algorithm is started with the initial value of the time delay, τ̂^0 = 0. The `*' in Fig. <ref> represents the algorithm switching to another solver path. The algorithm stops when the condition in Algorithm IV is satisfied for the threshold ϵ = 10^-3, i.e. when the relative error in the time delay between consecutive iterations is 10^-3. Next we demonstrate the effectiveness of the proposed algorithm by showing the global convergence of 100 datasets with different initial values for the delay. The results are provided in Tables <ref> and <ref> where they are also compared to results from the existing method <cit.>. Note that the results presented here for the existing method are different to those achieved in <cit.> as we chose a larger bound on the delay. In <cit.>, τ_max was set to 10 seconds while here, it is set to 15 seconds.The results presented in Tables <ref> and <ref> show that the proposed method, which utilizes the useful redundancy technique, performs much better than the existing method <cit.> irrespective of the initial delay. When the initial delays are close to the true system delay, the global convergence percentage is high, i.e. approximately 100% for both algorithms. However, when the initial delays are poorly selected, the global convergence percentage is low for the existing method, e.g. with the initial delay = 0, the percentage convergence for the existing method <cit.> is less than 1% while the proposed method achieves 100% global percentage convergence for both SNRs and the two sampling schemes. §.§ Case 2: Rao-Garnier test system In this section, we consider a system based on the Rao-Garnier continuous time benchmark <cit.>, G(s) = (-6400s+1600)e^-8ss^4+5s^3+408s^2+416s+1600. This system is linear, non-minimum phase with complex poles and a time delay. As with case 1, the experiment is conducted using both regular and irregular sampling schemes. For each sampling scheme, the input excitation signal is a PRBS of maximum-length. In the regular sampling case, the sample time is 10ms, the number of stages in the shift register is 10, the number of samples, N = 8000. For the irregular sampling, the number of stages in the shift register is 10 and the clock period is 0.5s. The input and output data are sampled at an irregular time instant t_k, where the sampling interval h_k is uniformly distributed as, h_k ∼ U[0.01, 0.05] (s). The number of samples, N=4500.We consider two SNR levels: 5dB and 15dB. The frequency ω_c^SVF is chosen as 25 rad/sec, which is approximately the bandwidth of the system. Again, the initial values for the time delay, τ̂^0, are selected as 0, 1, 3, 5, 7, 9 as well as a random value from the uniform distribution U[0, 9] (s). The delay boundaries τ_min, τ_max are set to 0 and 15 sec respectively. For the existing method, the cut off frequency of the filter is chosen as 1/10 of the system bandwidth. For the proposed method, the cut off frequencies of the filters, chosen based on linearly spaced periods, span from 1/10 to 1 of the system bandwidth and the number of filters used is 15. The order of the Butterworth filters is set to 10. The maximum number of iterations for the Gauss-Newton method (Step 2 in Algorithm II and IV) is 10. The threshold, ϵ, for convergence is 10^-3.For comparison, 100 different data sets are generated for each noise level of each sampling scheme.A graph showing the trajectory of a time delay estimate is plotted in Fig. <ref> for a single data set. Switching between cost functions is indicated by '*'. Note that, if only one filter were to be used, there is a high chance of convergence to a local minima. However, by using the useful redundancy technique, the proposed algorithm can avoid the local minima and converge to the global minimum.The numerical results of the experiments are provided in Table <ref> and <ref>. We can see that for both sampling schemes, the proposed method based on the useful redundancy technique performs much better as compared to the existing method. For all the initial values of delay used in this experiment, the existing method <cit.> never convergences 100% to the global minimum. However the proposed method utilizing useful redundancy still achieves a very high global convergence percentage, i.e. mostly 100% with any initial delay for both SNRs and sampling schemes.§ CONCLUSIONThe paper presents a new algorithm to estimate the parameters and time delay of a continuous-time system from regularly and irregularly sampled data. The idea is based on Instrumental Variable methods and employing the useful redundancy technique to enhance the global convergence by generating multiple cost functions by filtering the data several times. The paper also develops theoretical results related to the minima locations of the filtered delay cost function and the choice of filters to ensure the algorithm can converge to a global minimum. Numerical results show a significant improvement in the global convergence rate of the time delay estimation as compared to existing methods irrespective of the SNR. plain § APPENDIX§.§ Proof of Theorem 1 From (<ref>), when λ=0, J̅(δτ) becomes,J̅(δτ) = 1π∫_-∞^∞ (1-cos(ωδτ)) | G_0(jω) L(jω)|^2dω.∀δτ∈ℝ, ω≥ 0, we have1-cos(ωδτ) ≥ 0, ∀δτ, | G_0(jω) L(jω)|^2 ≥ 0, ∀ω.Therefore, J̅(δτ) ≥ 0, ∀δτ∈ℝ, ω≥ 0. Note that J̅(0) = 0, hence we have J̅(δτ) ≥J̅(0), ∀δτ∈ℝ, ω≥ 0. The equality occurs when L(jω) = 0, ∀ω or G_0(jω) = 0, ∀ω or cos(ωδτ)) = 1, ∀ω. As the system G_0(p) and the filter L(p) are not 0 (condition of Theorem 1), the equality only occurs when cos(ωδτ)) = 1, ∀ω, hence δτ = 0. □§.§ Proof of Theorem 2 If L(s) is selected such that L(s)G_0(s) is an ideal low-pass filter with cut-off frequency ω_c, then J̅(δτ) becomes,J̅(δτ) = 12π∫_-ω_c^ω_c| 1-e^-jωδτ|^2 dω + C = 2ω_cπ-2sin(ω_cδτ)πδτ + C,where C = 12π∫_-∞^∞λ| L(jω)|^2 dω.The minima and maxima of J̅(δτ) occur at the roots of,= dJ̅(δτ)dδτ = -2πω_ccos(ω_cδτ)δτ-sin(ω_cδτ)δτ^2.From (<ref>), we can see that the extrema of J̅(δτ) are also the extrema of the function sinc(ω_cδτ). However, the minima of J̅(δτ) will be the maxima of sinc(ω_cδτ) and the maxima of J̅(δτ) will be the minima of sinc(ω_cδτ). Note that for the function sinc(ω_cδτ), the locations of the i^th (i ≥ 1) positive extremum δ̃τ can be approximated by,δ̃τ_i≃ (i+12)πω_c - 1(i+12)πω_c≃2i+14T_c - T_c(2i+1)π^2.where T_c is the corresponding period of the cut-off frequency ω_c. The minima occurs when i is odd and the maxima occurs when i is even. Therefore, for J̅(δτ), the positive extrema can also be approximated using (<ref>), the only difference is that the minima of J̅(δτ) occurs when i is even and the maxima occurs when i is odd. □§.§ Proof of Theorem 3 Here we use the positive extrema locations of the time delay cost function in (<ref>). Since the filtered time delay cost function J̅(δτ) is an even function, we only need to prove the theorem for δτ_0 > 0.Denote δ̃τ_i,k^min as the i^th positive minimum of the filtered time delay cost function J̅(δτ) generated by the filter L_k(p); δ̃τ_i,k^max as the i^th positive maximum of the filtered time delay cost function J̅(δτ) generated by the filter L_k(p). From Theorem <ref>, they can be computed by,δ̃τ_i,k+1^min= 4i+14T_c,k+1 - T_c,k+1(4i+1)π^2,= (4i+14 - 1(4i+1)π^2)(β-kβ-1n_f-1)T_bw δ̃τ_i,k+1^max= 4i-14T_c,k+1 - T_c,k+1(4i-1)π^2,= (4i-14 - 1(4i-1)π^2)(β-kβ-1n_f-1)T_bw.From (<ref>), for any filter L_k(p), we have the following property,δ̃τ_i,k^max < δ̃τ_i,k^min < δ̃τ_i+1,k^max, ∀ i ∈ℤ^+. Consider the two following cases: Case 1: δτ_0 ≤δ̃τ_1,1^max, then from (<ref>), we have, ξ(L_1(p),δτ_0) = 0, which is smaller than δτ_0 as δτ_0 ≠ 0. Therefore, ξ(L_1(p),δτ_0) < δτ_0. □Case 2: δτ_0 > δ̃τ_1,1^max. Denote i_0 such that,δ̃τ_i_0-1,1^max < δτ_0 ≤δ̃τ_i_0,1^max, i_0 ∈ℕ, i_0 ≥ 2.Now consider a filter set L_k(p), k = 1,n_f defined as in Theorem <ref>, with,n_f ≥ (1/M+β-2)/(1/M-1),n_f ∈ℕ,where M = 4i_0-3-4/((4i_0-3)π^2)4i_0-1-4/((4i_0-1)π^2).We need to prove the filter set satisfies the condition in (<ref>). First, we prove that the filter set L_k(p), k = 1,n_f has the following property,δ̃τ_i+1,k+1^max≥δ̃τ_i,k^min,where i = 1,i_0-1, k = 1,n_f-1. From (<ref>), n_f ≥1/M+β-21/M-1⇒M ≤ 1 - (β-1)/(n_f-1)β - (n_f-2)(β-1)/(n_f-1).Note that, as β > 1,1 - (β-1)/(n_f-1)β> 1 - (β-1)/(n_f-1)β - (β-1)/(n_f-1) > 1 - (β-1)/(n_f-1)β - (n_f-2)(β-1)/(n_f-1).From (<ref>) and (<ref>), 1 - (β-1)/(n_f-1)β - (k-1)(β-1)/(n_f-1)≥ M,where k = 1,n_f-1, hence, combining (<ref>) and (<ref>),β-k(β-1)/(n_f-1)β - (k-1)(β-1)/(n_f-1)≥4i_0-3-4/((4i_0-3)π^2)4i_0-1-4/((4i_0-1)π^2),where k = 1,n_f-1. Next, consider the function f(x) = 4x+1-4/((4x+1)π^2)4x+3-4/((4x+3)π^2).Taking the derivative w.r.t. x of (<ref>), ∂ f(x)∂ x = 8+16(4x+1)π^2+16(4x+3)(4x+1)^2π^2+64/π^4(4x+1)(4x+3)^2((4x+3)-4/((4x+3)π^2))^2- 16(4x+3)π^2+16(4x+1)(4x+3)^2π^2+64/π^4(4x+1)^2(4x+3)((4x+3)-4/((4x+3)π^2))^2Now ∀ x ≥ 1,∂ f(x)∂ x > 8 - 16(4x+3)π^2 - 16(4x+1)(4x+3)^2π^2 - 64/π^4(4x+1)^2(4x+3)((4x+3)-4/((4x+3)π^2))^2 > 8 - 16(4x+3)π^2 - 16(4x+3)π^2 - 64/π^4(4x+1)^2(4x+3)((4x+3)-4/((4x+3)π^2))^2 > 8 - 2167π^2 - 64175π^4((4x+3)-4/((4x+3)π^2))^2 > 0.Hence, f(x) is an increasing function for x ≥ 1, so we have, f(i_0-1) ≥ f(i), i=1,i_0-1, or, 4i_0-3-4/((4i_0-3)π^2)4i_0-1-4/((4i_0-1)π^2)≥4i+1-4/((4i+1)π^2)4i+3-4/((4i+3)π^2), i=1,i_0-1.From (<ref>) and (<ref>),β-k(β-1)/(n_f-1)β - (k-1)(β-1)/(n_f-1)≥4i+1-4/((4i+1)π^2)4i+3-4/((4i+3)π^2),where i = 1,i_0-1, k = 1,n_f-1, which can be rewritten as, (4i+34-1(4i+3)π^2) (β-kβ-1n_f-1) ≥(4i+14-1(4i+1)π^2) (β-(k-1)β-1n_f-1),for i = 1,i_0-1, k = 1,n_f-1. From (<ref>) and (<ref>), we conclude,δ̃τ_i+1,k+1^max≥δ̃τ_i,k^min, i = 1,i_0-1, k = 1,n_f-1. Now, we will prove that δ̃τ_i,1^max≥δ̃τ_i,n_f^min, i = 1,i_0. Similar to the previous analysis, it's trivial to show that,5-4/(5π^2)3-4/(3π^2)≥4i+1-4/((4i+1)π^2)4i-1-4/((4i-1)π^2), ∀ i ∈ℤ^+As β≥5-4/(5π^2)3-4/(3π^2), therefore, for ∀ i ∈ℤ^+,(4i-1-4(4i-1)π^2)β≥(4i+1-4(4i+1)π^2).Then by multiplying with T_bw, we have,δ̃τ_i,1^max≥δ̃τ_i,n_f^min, ∀ i ∈ℤ^+.Lastly, we show that for any filter set L_k(p), k = 1,n_f defined as in Theorem <ref> with n_f ≥ (1/M+β-2)/(1/M-1), n_f ∈ℕ,∀δτ_0 ≠ 0, ∃ L_q(p): ξ(L_q(p),δτ_0) < δτ_0.Case 2-A: δ̃τ_i_0-1,1^min < δτ_0 ≤δ̃τ_i_0,1^maxFrom (<ref>) and (<ref>), it is obvious that ξ(L_1(p),δτ_0) = δ̃τ_i_0-1,1^min, and δ̃τ_i_0-1,1^min < δτ_0, hence ξ(L_1(p),δτ_0) < δτ_0. □ Case 2-B: δ̃τ_i_0-1,1^max < δτ_0 ≤δ̃τ_i_0-1,1^min.From (<ref>) we see that, ∀ i ∈ℤ^+,δ̃τ_i,n_f^min < δ̃τ_i,n_f-1^min < ... < δ̃τ_i,2^min < δ̃τ_i,1^min,hence,δ̃τ_i_0-1,n_f^min < δ̃τ_i_0-1,n_f-1^min < ... < δ̃τ_i_0-1,1^min.Now combining (<ref>) and (<ref>),δ̃τ_i_0-1,n_f^min≤δ̃τ_i_0-1,1^max, δ̃τ_i_0-1,1^max < δτ_0≤δ̃τ_i_0-1,1^min,we have,δ̃τ_i_0-1,n_f^min < δτ_0 < δ̃τ_i_0-1,1^min.Considering (<ref>) and (<ref>), there always exists a value q≥ 2 such that δ̃τ_i_0-1,q^min < δτ_0 < δ̃τ_i_0-1,q-1^min. Following from (<ref>), we have δ̃τ_i_0-1,q-1^min≤δ̃τ_i_0,q^max. Therefore, δ̃τ_i_0-1,q^min < δτ_0< δ̃τ_i_0,q^max.Combining (<ref>) and (<ref>),ξ(L_q(p),δτ_0) = δ̃τ_i_0-1,q^min,therefore, ξ(L_q(p),δτ_0) < δτ_0. □ | http://arxiv.org/abs/1703.09344v1 | {
"authors": [
"Huong Ha",
"James S. Welsh",
"Mazen Alamir"
],
"categories": [
"math.ST",
"stat.TH"
],
"primary_category": "math.ST",
"published": "20170327233746",
"title": "Useful redundancy in parameter and time delay estimation for continuous-time models"
} |
AIP/123-QED [email protected] School of Computer Science, Fudan University, Shanghai 200433, ChinaShanghai Key Lab of Intelligent Information Processing, Fudan University, Shanghai 200433, ChinaIt has been recently reported that the reciprocity of real-life weighted networks is very pronounced, however its impact on dynamical processes is poorly understood. In this paper, we study random walks in a scale-free directed weighted network with a trap at the central hub node, where the weight of each directed edge is dominated by a parameter controlling the extent of network reciprocity. We derive the mean first passage time (MFPT) to the trap, by using two different techniques, the results of which agree well with each other. We also analytically determine all the eigenvalues as well as their multiplicities for the fundamental matrix of the dynamical process, and show that the largest eigenvalue has an identical dominant scaling as that of the MFPT. We find that the weight parameter has a substantial effect on the MFPT, which behaves as a power-law function of the system size with the power exponent dependent on the parameter, signaling the crucial role of reciprocity in random walks occurring in weighted networks. 36.20.-r, 05.40.Fb, 05.60.CdEffects of reciprocity on random walks in weighted networks Yibin Sheng December 30, 2023 =========================================================== § INTRODUCTIONAs an emerging science, complex networks have witnessed substantial progress in the past years <cit.>. One of the ultimate goals in the study of complex networks is to uncover the influences of various structural properties on the function or dynamical processes taking place on them. Among different dynamical processes, random walks lie at the core, since they are a fundamental mechanism for a wealth of other dynamic processes, such as navigation <cit.>, search <cit.>, and cooperative control <cit.>. Except for the importance in the area of network science, random walks also provide a paradigmatic model for analyzing and understanding a large variety of real-world phenomena, for example, animal <cit.> and human <cit.> mobility. Thus far, random walks have found numerous applications <cit.> in many aspects of interdisciplinary sciences, including image segmentation <cit.>, community detection <cit.>, collaborative recommendation <cit.>, and signal propagation in proteins <cit.> to name a few.A highly desirable quantity for random walks is first passage time (FPT) <cit.>, defined as the expected time for a random walker going from a starting node to a given target. The mean of FPTs over all starting nodes to the target is called mean first passage time (MFPT), which is an important characteristic of random walks due to the first encounter properties in numerous realistic situations. In the past years, the study of MFPT has triggered an increasing attention from the scientific community <cit.>. One focus of theoretical activity is to develop general methods to efficiently compute MFPT <cit.>. Another direction is to unveil how the behavior of MFPT is affected by different structural properties of the underlying systems, such as heterogeneity of degree <cit.> or strength <cit.>, fractality <cit.>, and modularity <cit.>. Previous studies proposed several frameworks for evaluating MFPT and uncovered the discernible effects of some nontrivial structural aspects on the target search efficiency measured by MFPT. However, most existing works ignoring the impact of link reciprocity, the tendency of node pairs to form mutual connection in directed networks, on the behavior of random walks, despite the fact that reciprocity is a common characteristic of many realistic networks <cit.>, such as the World Wide Web <cit.>, e-mail networks <cit.>, and World Trade Web <cit.>. In addition to binary networks, the nontrivial pattern reciprocity is also ubiquitous in real-life systems described by weighted networks <cit.>. It has been shown the ubiquitous link reciprocity strongly affects dynamical processes in binary networks, for example, spread of computer viruses <cit.> or information <cit.>, and percolation <cit.>. By contrast, the influence of reciprocity on dynamical processes in weighted networks has attracted much less attention, although it is suggested that reciprocity could play a crucial role in network dynamics. In particular, the lack of analytical results in this field limits our understanding of the impact of weight reciprocity on the function of weighted networks <cit.>.In this paper, we propose a weighted directed scale-free network by replacing each edge in the previous binary network <cit.> by double links with opposite directions and different weights. In the weighted network, the link weights are adjusted by a parameter characterizing the weight reciprocity of network. We then study random walks in the weighted network in the presence of a perfect trap at the central large-degree node. During the process of random walks, the transition probability is dependent on the weight parameter. We derive the MFPT to the target by using two disparate approaches, the results of which completely agree with each other. We also determine all the eigenvalues and their multiplicities of the fundamental matrix characterizing the random-walk process, and show that the largest eigenvalue has the same leading scaling as that of the MFPT. The obtained results demonstrate that the behavior of MFPT to the trap depends on the weighted parameter, signalling a drastic influence of theweight reciprocity on random walks defining on weighted networks. § NETWORK MODELS AND PROPERTIES Before introducing the weighted directed network with scale-free fractal properties. We first give a brief introduction to a binary scale-free fractal network, which has the same topology as the weighted network. §.§ Model and properties of binary network. The binary treelike network is constructed in an iterative way <cit.>. Let F_g (g ≥ 0) represent the network after g iterations (generations). For g=0, F_0 is an edge linked by two nodes. In each successive iteration g ≥ 1, F_g is constructed from F_g-1 by performing the following operations on every existing edge in F_g-1 as shown in Fig. <ref>: two new nodes (called external nodes) are firstly created and attached, respectively, to bothendpoints of the edge; then, the edge is broken, another new node (referred to as an internal node) is placed in its middle and linked to both endpoints of the original edge. Figure <ref> illustrates the first several construction processes of the network. The structure of F_g is enciphered in its adjacency matrix A_g, the entries A_g(i,j) of which are defined by A_g(i,j)=1 if two nodes i and j are adjacent in F_g, or A_g(i,j)=0 otherwise. The particular construction of the network allows to calculate exactly its relevant properties. At each generation g_i (g_i≥ 1), the number of newly created nodes is Υ_g_i=3·4^g_i-1. Let Λ_g_i be the set of nodes generated at iteration g_i, then Λ_g_i can be further classified into two sets Λ_g_i,ext and Λ_g_i,int satisfying Λ_g_i = Λ_g_i,ext∪Λ_g_i,int, among whichΛ_g_i,extis the set of external nodes and Λ_g_i,int is the set of internal nodes. We use |Ω| to stand for the cardinality of a set Ω. Because |Λ_g_i,ext| =2|Λ_g_i,in|, it is easy to derive|Λ_g_i,int| = 4^g_i-1 and |Λ_g_i,ext| = 2·4^g_i-1. We represent the set of nodes in F_g as Λ_g. Hence, the number of nodes and edges in F_g is N_g=|Λ_g|=∑_g_i=0^gΥ_g_i=4^g+1 and E_g=N_g-1=4^g, respectively. Let k_i(g) denote the degree of an arbitrary node i in F_g that was generated at generation g_i (g_i≥ 0), then k_i(g+1)=2 k_i(g). Hence, after each new iteration the degree of every node doubles.This resultant network displays the remarkable scale-free <cit.> and fractal <cit.> features as observed in diverse real-life systems. It has a power law degree distribution with an exponent 3, and its fractal dimension is 2. §.§ Model and properties of weighted directed network. The above introduced binary network F_g can be extended to a weighted directed network with nonnegative and asymmetrical edge weights. Let F⃗_g denote the weighted directed network corresponding to F_g. BothF⃗_g and F_g have an identical topological structure. The only difference between F⃗_g and F_g is that every undirectededge in F_g is replaced by two directed edges with opposite directions and distinct positive weights. We use W_g to represent the nonnegative and asymmetrical weight matrix for F⃗_g such that W_ij(g)>0 if and only if there is a directed edge (arc) pointing to node j from node i. The weight of each arc in the weighted directed network is defined recursively in the following way. When g=0, F⃗_0 has two nodes, denoted by a and b, and the weights of arcs e⃗(a,b) and e⃗(b,a) are defined to be W_ab(0)=W_ba(0)=1. When g ≥ 1, by construction, F_g is obtained from F_g-1 by substituting each undirected edge e(u,v) in F_g-1 with two undirected edges e(u,w) and e(w,v), and generating two additional nodes, x and y, attaching to u and v, respectively. The weights of resultant arcs in F⃗_g are defined as: W_uw(g)=W_uv(g-1), W_vw(g)=W_vu(g-1), W_wu(g)=W_wv(g)=1, W_xu(g)=1, W_yv(g)=1, W_ux(g)=θW_uv(g-1), and W_vy(g)=θW_vu(g-1). Here θ is a tunable positive real number, that is, θ>0. The weight parameter is of paramount importance since it characterizes the weight reciprocity of network F⃗_g. Whenθ=1, F⃗_g reduces to F_g, and the weights in two directions between any pair of adjacent nodes are completely reciprocated; when θ≠ 1, the weights are non-reciprocated: the larger the deviation of θ from 1, the smaller the level of weight reciprocity. In undirected weighted networks <cit.>, node strength is a key quantity characterizing the property of a node. Here we extend the definition of strength of a nodeto the directed weighted network F⃗_g by defining the out-strength and in-strength of node i in F⃗_g as s_i^+(g)=∑_j=1^N_gW_ij(g) and s_i^-(g)=∑_j=1^N_gW_ji(g), respectively. For F⃗_g, we can obtain the out-strength for an arbitrary node i that entered the network at generation g_i (g_i≥ 0). If i was an external node when it entered the network, s_i^+(g)=(θ+1)^g-g_i; otherwise, if i was an internal node when it was born, s_i^+(g)=2(θ+1)^g-g_i. Therefore, after each new iteration, the out-strength of a node increases by a factor of θ. It is easy to obtain the node out-strength in F⃗_g obeys a distribution ofpower law form with the exponent being 1+2ln 2/ln (θ+1). Note that in some realistic networks, the node strength also display a broad distribution <cit.>.§ FORMULATION OF BIASED WALKS IN THE WEIGHTED DIRECTED NETWORK After introducing the construction and property of the weighted directed network F⃗_g, we now define and study biased discrete-time random walks performing F⃗_g. Let r_ij(g)=W_ij(g)/s_i^+(g) denote the transition probability that a particle jumps from node i to its neighboring node j per time step. Note that r_ij(g) constitutes an entry of transition matrix R_g=(S_g)^-1W_g, where S_g is the diagonal out-strength matrix of F⃗_g, with the ith diagonal entry of S_g being s_i^+(g).In this paper, we focus on a specific case of biased random walks, often called trapping problem, in F⃗_g in the presence of a trap placed at the central hub node, i.e., the internal node generated at the first iteration. To facilitate the description of the following text, all N_g nodes in F⃗_g are labeled sequentially as 1,2,…, N_g -1,N_g as follows. For F⃗_1 , the newly generated internal node is labeled 1, the initial two nodes in F⃗_0 are labeled as 2 and 3, while the two new external nodes are labeled by 4 and 5. For each new iteration g_i >1, we label consecutively the new nodes born at this iteration from N_g_i-1 + 1 to N_g_i,while we keep the labels of those nodes created before iteration g_i unchanged.For the trapping problem, what we are concerned with are the trapping time and the average trapping time. Let T_i^(g) represent the trapping time for a particle initially placed at node i (i ≠ 1) in F⃗_g to arrive at the trap node for the first time, which is equal to the FPT from the i to the trap. The average trapping time, ⟨ T ⟩_g, is actually the MFPT to the trap, defined as the mean of T_i^(g) over all non-trap initial nodes in network F_g:⟨ T ⟩_g=1/N_g-1∑_i=2^N_g T_i^(g) .Below we will show how to computethe two quantities T_i^(g) and ⟨ T ⟩_g. For T_i^(g), it obeys therelationT_i^(g)=1 +∑_j=2^N_g r_ij T_j^(g) ,which can be recast in matrix form as:T = e + R̅_gT,where T = (T_2^(g),T_3^(g),…,T_N_g^(g))^⊤ is an (N_g-1)-dimensional vector, e=(1,1,...,1)^⊤ is the (N_g-1)-dimensional vector of all ones, and R̅_g is a matrix of order N_g-1, which a submatrix of R_g and obtained fromR_g by deleting the first row and the first column corresponding to the trap. From Eq. (<ref>) we haveT=(I-R̅_g)^-1e=K_g e ,where I is the (N_g-1)×(N_g-1) identity matrix. Matrix K_g=(I-R̅_g)^-1 is the fundamental matrix <cit.> of the addressed trapping problem. Equation (<ref>) impliesT_i^(g)=∑_j=2^N_g K_g(i,j) ,where K_g(i,j) is the ijth entry ofmatrix K_g, representing the expected number of visitations to node j by a particle starting from node i before being absorbed by the trap. Plugging Eq. (<ref>) into Eq. (<ref>) yields⟨ T ⟩_g=1/N_g-1∑_i=2^N_g∑_j=2^N_g K_g(i,j) . Equation (<ref>) indicates that the computation of MFPT ⟨ T ⟩_g can be reduced to finding the sum of all entries of the corresponding fundamental matrix. A disadvantageof this method is that it demands a large computational effort when the network is very large. However, Eq. (<ref>) provides exact results for ⟨ T ⟩_g that can be applied to check the results for MFPT obtained using other techniques. Next we analytically determine the closed-form expression for MFPT ⟨ T ⟩_g using an alternative approach, the results of which are consistent with those of Eq. (<ref>).§ EXACT SOLUTION TO THE MFPT ⟨ T ⟩_G The particular selection of trap location and the specific network structure allow to determine exactly the MFPT ⟨ T ⟩_g for arbitrary g. In order to obtain a close-form expression for ⟨ T ⟩_g, we first establish the dependence of T_i^(g) on iteration g. For a node i inF⃗_g, at iteration g+1, its degree doubles, increasing from k_i(g) to 2k_i(g). All these 2k_i(g) neighboring nodes are created at iteration g+1, among which one half are external nodes with a single degree, and the other half are internal nodes with degree 2.We now consider the trapping problem in F⃗_g+1. Let A be the FPT for a particle starting from node i to any of its k_i(g) old neighbors, that is, those nodes adjacent to i at iteration g; let B (resp. C) be the FPT for a particle staring from any of the k_i(g) internal (resp. external) neighbors of i to one of its k_i(g) old neighbors. Then the FPTs obey relations:{[ A = θ/θ+1(1+C) + 1/θ+1(1+B) ,; B =1/2 + 1/2(1+A) ,; C = 1+A . ].Eliminating B and C in Eq. (<ref>), we obtain A=4(θ+1). Therefore, when the network grows from iteration g to iteration g+1, the FPT from any node i (i ∈F⃗_g) to another node j (j ∈F⃗_g) increases by a factor of 4(θ+1). Hence,T_i^(g+1)=4(θ+1) T_i^(g) hold for any g, which is a useful for deriving the exact expression for MFPT.Having obtained the scaling dominating the evolution for FPTs, we continuedetermining the MFPT ⟨ T ⟩_g. For this purpose, we introduce two intermediary quantities for any n ≤ g: T_n, tot^(g)=∑_i∈Λ_n T_i^(g) and T_n, tot^(g)=∑_i∈Λ_n T_i^(g). Then,T_g,tot^(g) = T_g - 1,tot^(g) +T_g,tot^(g)= (4θ+4) T_g - 1,tot^(g-1) + T_g,tot^(g) .By definition, ⟨ T ⟩_g=1/N_g-1T_g,tot^(g). To find T_g,tot^(g), it is necessary to explicitly determine the quantity T_g,tot^(g). To this end, we define two additional quantities for n ≤ g: T_n,int^(g) = ∑_i ∈Λ_n,intT_i^(g) and T_n,ext^(g) = ∑_i ∈Λ_n,ext T_i^(g). Obviously, T_g,tot^(g) = T_g,int^(g) + T_g,ext^(g). Thus, in order to find T_g,tot^(g), one may alternatively evaluate T_n,int^(g) and T_n,ext^(g).We first establish the relationship between T_n,int^(g) and T_n,ext^(g). By construction (see Fig. <ref>), at a given generation, each edge connecting two nodes u and vwill give rise three new nodes (w, x, and y) in the next generation. The two external nodes x and y are separately attached to u and v, while the only internal node w is linked simultaneously to u and v. For any iteration g, the FPTs for the three new nodes satisfy: T^(g)_x=1 +T^(g)_u, T^(g)_y=1 +T^(g)_v, and T^(g)_w=1/2[1+T^(g)_u]+1/2[1+T^(g)_v]. Therefore, T^(g)_x+T^(g)_y = 2 T^(g)_w. Summing this relation over all old edges at the generation before growth, we find that for all n ≤ g, T_n,ext^(g) = 2 T_n,int^(g) always holds. In this way, issue of determining T_g,tot^(g) is reduced to finding T_g,ext^(g) that can be obtained as follows. For an arbitrary external node i_ ext in F⃗_g, which is created at generation g and attached to an old node i, we have T_i_ ext^(g) = 1 + T_i^(g), a relation valid for any node pair containing an old node and one of its new external adjacent nodes. By applying relation T_i_ ext^(g) = 1 + T_i^(g) to two sum (the first one is over a given old node and all its new adjacent external nodes, the other is summing the first one over all old nodes), we obtainT_g,ext^(g) = |Λ_g,ext|+∑_i ∈Λ_g-1(k_i(g-1)× T_i^(g)) = |Λ_g,ext|+(T_g-1,ext^(g)+2T_g-1,int^(g)) +(2T_g-2,ext^(g)+4T_g-2,int^(g)) + ⋯+(2^g-2T_1,ext^(g)+2^g-1T_1,int^(g)) =2×4^g-1+2T_g-1,ext^(g)+4T_g-2,ext^(g)+⋯+2^g-1T_1,ext^(g)From Eq. (<ref>), one can derive the recursive relationT_g+1,ext^(g+1) =16(θ+1) T_g,ext^(g) - (4θ+2)4^g .Considering the initial condition T_2,ext^(2) = 48θ^2+80θ+40, Eq. (<ref>) is solved to yieldT_g,ext^(g) = 12θ^2+17θ+7/(θ+1)(4θ+3)2^4g-4(θ+1)^g + 2θ+1/4θ+32^2g-1 .Because T_g,tot^(g) = T_g,int^(g) + T_g,ext^(g) and T_g,ext^(g)=2 T_g,int^(g), we haveT_g,tot^(g)= 36θ^2+51θ+21/(θ+1)(4θ+3)2^4g-5(θ+1)^g +6θ+3/4θ+32^2g-2 .Inserting Eq. (<ref>) into Eq. (<ref>) leads toT_g,tot^(g) =(4θ+4) T_g-1,tot^(g-1)+6θ+3/4θ+32^2g-2+36θ^2+51θ+21/(θ+1)(4θ+3)2^4g-5(θ+1)^g .Using T_1,tot^(1) = 8θ+6, Eq. (<ref>) is solved to getT_g,tot^(g) = 12θ^3+17θ^2+7θ/θ(θ+1)(4θ+3)2^4g-3(θ+1)^g+16θ^3+28θ^2+20θ+6/θ(θ+1)(4θ+3)2^2g-3(θ+1)^g -3(θ+1)(2θ+1)/θ(θ+1)(4θ+3)2^2g-2 .Then, the rigorous expression for the MFPT ⟨ T ⟩_g of the weighted directed network F⃗_g is⟨ T ⟩_g= 12θ^3+17θ^2+7θ/8θ(θ+1)(4θ+3)2^2g(θ+1)^g+16θ^3+28θ^2+20θ+6/8θ(θ+1)(4θ+3)(θ+1)^g -3(θ+1)(2θ+1)/4θ(θ+1)(4θ+3) . We have checked the analytical solution in Eq. (<ref>) against extensive numerical results obtained from Eq. (<ref>), see Fig. <ref>. For different θ and g, both the analytical and numerical results are in full agreementwith each other, indicating that the explicit expression in Eq. (<ref>) is correct. In addition,for the particular caseθ=1, the network F⃗_g is reduced to F_g, and Eq. (<ref>) recovers the result <cit.> previously obtained for F_g. This also validates Eq. (<ref>). We proceed to express ⟨ T ⟩_g in terms of the network size N_g, in order to uncover how ⟨ T ⟩_g scales with N_g. From N_g=4^g+1, we have g=log_4(N_g-1). Then,⟨ T ⟩_g= 12θ^3+17θ^2+7θ/8θ(θ+1)(4θ+3)(N_g-1)^1+log_4(θ+1)+16θ^3+28θ^2+20θ+6/8θ(θ+1)(4θ+3)(N_g-1)^log_4(θ+1)-3(θ+1)(2θ+1)/4θ(θ+1)(4θ+3) .For a very large network (i.e., N_g →∞), the leading term of ⟨ T ⟩_g can be represented as:⟨ T ⟩_g∼ (N_g)^1+log_4(θ+1) .Equation (<ref>) shows that for thedirected weighted network F⃗_g, the MFPT ⟨ T ⟩_g behaves as a power-law function of the network size N_g, with the exponent η(θ)=1+log_4(θ+1) increasing with the weight parameter θ. Thus, the weight reciprocity has an essential effect on the efficiency on the trapping problem, measured by the MFPT. § EIGENVALUES OF THE FUNDAMENTAL MATRIX We now study the eigenvalues of the fundamental matrix K_n of the trapping problem addressed above. We will determine all the eigenvalues of the fundamental matrix aswell as their multiplicities. Moreover, we will show that the largest eigenvalue has the same leading scaling as that of ⟨ T⟩_g. To attain this goal, we introduce matrix P_g defined by P_g=K_g^-1. Let λ_i(g) and σ_i(g), where i=1,2,…,N_n-1, denote the eigenvalues of P_g and K_g, such that λ_1(g)≤λ_2(g)≤λ_3(g) …≤λ_N_g-1(g) andσ_1(g)≥σ_2(g) ≥σ_3(g) ≥…≥σ_N_n-1(g). Then, the one-to-one relation λ_i(g)=1/σ_i(g) holds. Thus, to compute the eigenvalues of matrix K_n, we can alternatively determine the eigenvalues for P_g. In the sequel, we will use the decimation method <cit.> to find all the eigenvalues of matrix P_g.§.§ Full spectrum of fundamental matrix The decimation procedure <cit.> makes it possible to obtain the eigenvalues for related matrix of current iteration from those of the previous iteration.We now consider the eigenvalue problem for matrix P_g+1. Let α denote the set of nodes in network F⃗_g, and β the set of nodes created at iteration g+1. Suppose that λ_i(g+1) is an eigenvalue of P_g+1, and u=(u_α,u_β)^⊤ is an eigenvector associated with λ_i(g+1), where u_α and u_β correspond to nodes belonging to sets α and β, respectively. Then, eigenvalue equation for matrix P_g+1 can be represented in a block form:[[P_α,α P_α, β;P_β,α P_β, β ]] [[ u_α; u_β ]]=λ_i(g+1) [[ u_α; u_β ]] ,where P_α,α and P_β, β are the identity matrix.Equation (<ref>) can be expressed as two equations:P_α,αu_α+P_α, βu_β=λ_i(g+1)u_α , P_β,αu_α+P_β, βu_β=λ_i(g+1)u_β . Equation (<ref>) impliesu_β=1/λ_i(g+1)-1P_β,αu_α ,provided that λ_i(g+1) ≠ 1. Inserting Eq. (<ref>) into Eq. (<ref>) yieldsP_α, βP_β,αu_α=[λ_i(g+1)-1]^2 u_α.In this way, we reduce the problem of determining the eigenvalue λ_i(g+1) for matrix P_g+1 of order 4^g+1 to finding the eigenvalue problem of matrix P_α, βP_β,α with a smaller order 4^g.We can prove (see Methods) thatP_α, βP_β,α=I_g-1/2θ+2P_g,where I_g is the identity matrix of order 4^g, identical to that of P_g. Equation (<ref>) relates the product matrix P_α, βP_β,α to matrix P_g. Therefore, the eigenvalues of matrix P_g+1 can be expressed in terms of those of matrix P_g.We next show how to obtain the eigenvalues of P_g+1 through the eigenvalues of P_g. According to Eqs. (<ref>) and (<ref>), we can deriveP_g u_α=-(2θ+2)[λ^2_i(g+1)-2λ_i(g+1)]u_α .Hence, if λ_i(g) is an eigenvalue of P_g associated with eigenvector u_a, Eq. (<ref>) indicatesλ_i(g)=-(2θ+2)[λ_i(g+1)^2-2λ_i(g+1)].Solving the above quadratic equation in the variable λ_i(g+1) given by Eq. (<ref>), one obtains the two roots:λ_i,1(g+1)=1-√(1-λ_i(g)/2θ+2) ,andλ_i,2(g+1)=1+√(1-λ_i(g)/2θ+2) .Equations (<ref>) and (<ref>) relate λ_i(g+1) to λ_i(g), with each eigenvalue λ_i(g) of P_g giving rise two different eigenvalues of P_g+1. As a matter of fact, all eigenvalues of the P_g+1 can be obtained by these two recursive relations. In Methods, we determine the multiplicity of each eigenvalue and show that all the eigenvalues can be found by Eqs. (<ref>) and (<ref>).Since there is a one-to-one relation between the eigenvalues of P_g and the fundamental matrix K_g, we thus have also found all theeigenvalues of K_g.§.§ The largest eigenvalue of fundamental matrix and MFPT In the above, we have determined all eigenvalues for the inverse P_g of the fundamental matrix K_g and thus all eigenvalues of K_g. Here we continue to estimate the greatest eigenvalue σ_ max(g) of the fundamental matrix K_g, whichactually equals the reciprocal of the smallest eigenvalue for matrix P_g, denoted by λ_ min(g). Below we will show that in a large network the leading behavior of the MFPT ⟨ T ⟩_g for trapping in F⃗_g and the reciprocal of λ_ min(g) is identical, that is, ⟨ T ⟩_g ∼ 1/λ_ min(g)=σ_ max(g).We begin by providing some useful properties of eigenvalues for matrix P_g. Assume that Δ_g is the set of the 4^g eigenvalues of matrix P_g, namely, Δ_g={λ_1(g),λ_2(g),λ_3(g),⋯, λ_4^g(g)}.According to the above analysis, Δ_g can be categorized into two subsets Δ_g^(1) and Δ_g^(2) satisfying Δ_g=Δ_g^(1)∪Δ_g^(2), where Δ_g^(1) consists of all eigenvalues 1, while Δ_g^(2) contains the rest eigenvalues. Thus,Δ_g^(1)= { 1,1,1,…,1,1}_2×4^g - 1 .These 2×4^g - 1 eigenvalues are labeled sequentially by λ_4^g-1+1(g), λ_4^g-1+2(g), ⋯, λ_3×4^g-1(g), since they provide a natural increasing order of all eigenvalues for P_g, as will been shown.The remaining 2× 4^g-1 eigenvalues in set Δ_g^(2) are all determined by Eqs. (<ref>) and (<ref>). Let λ_1(g-1), λ_2(g-1), λ_3(g-1), ⋯, λ_4^g-1(g-1) be the 4^g-1 eigenvalues of matrix P_g-1, arranged in an increasing order λ_1(g-1)≤λ_2(g-1) ≤λ_3(g-1) ≤…≤λ_4^g-1(g-1). Then, for each eigenvalue λ_i(g-1) in P_g-1, Eqs. (<ref>) and (<ref>) produce two eigenvalues of P_g, which are labeled as λ_i(g) and λ_4^g-i+1(g):λ_i(g)= 1-√(1-λ_i(g-1)/2θ+2)andλ_4^g-i+1(g)=1+√(1-λ_i(g-1)/2θ+2) .Plugging each eigenvalue of P_g - 1 into Eqs. (<ref>) and (<ref>) generates all eigenvalues in Δ_g^(2).It is easy to see that λ_i(g) given by Eq. (<ref>) monotonously increases with λ_i(g-1) and belongs to interval (0,1), while λ_4^g-i+1(g) provided by Eq. (<ref>) monotonously decreases with λ_i(g-1) and lies in interval (1,2). Thus, λ_1(g),λ_2(g),λ_3(g),⋯, λ_4^g(g) provide an increasing order of all eigenvalues for matrix P_g.We continue to estimate λ_ min(g) of matrix P_g. From the above arguments, the smallest eigenvalue λ_ min(g) must be the one generated from λ_ min(g-1) through Eq. (<ref>):λ_ min(g)= 1-√(1-λ_ min(g-1)/2θ+2) .Using Taylor's formula, we haveλ_ min(g) ≈ 1-[1-λ_ min(g-1)/4θ+4]=λ_ min(g-1)/4θ+4 .Considering λ_ min(1)=1-√(1-1/θ+1), Eq. (<ref>) is solved to yieldλ_ min(g) ≈( 1-√(1-1/θ+1))(4θ+4)^1-g .Thus,1/λ_ min(g)≈1/4 (θ+1)( 1+√(1+1/θ+1))4^g(θ+1)^g ,which, together with Eq. (<ref>), means that 1/λ_ min(g) and ⟨ T ⟩_g have the same dominating term and thus identical leading scaling.§ CONCLUSIONS Real-life weighted networks exhibit a rich and diverse reciprocity structure. In this paper, we have proposed a scale-free weighted directed network with asymmetric edge weights, which are controlled by a parameter characterizing the network reciprocity. We then studied random walks performed on the network with a trap fixed at the central hub node. Applying two different approaches, we have evaluated the MFPT to the trap. Moreover, based on the self-similar architecture of the network, we have found all the eigenvalues and their multiplicities of the fundamental matrix describing the random-walk process, the largest one of which has the same leading scaling as that of the MFPT. The obtained results indicate thatthe MFPT scales as a power-law function of thethe system size, with the power exponent increasing with the weight parameter, revealing that the reciprocity has a significant impact on dynamical processes running on weighted networks. This work deepens the understanding of random-walk dynamics in complex systems and opens a novel avenue to control random walks in a weighted network by changing its reciprocity.The authors thank Bin Wu for his assistance in preparing this manuscript. This work was supported by the National Natural Science Foundation of China under Grant No. 11275049.§ PROOF OF EQ. (<REF>)In order to prove Eq. (<ref>), we rewrite P_α,β and P_β,α in the block form asP_α,β=(U_1,U_2,⋯,U_E_g)andP_β,α=( [ V_1; V_2; ⋮; V_E_g; ]) ,respectively. In Eqs. (<ref>) and (<ref>), E_g=4^g is the number of edges in F_g; U_i (1 ≤ i ≤ E_g) is a 4^g × 3 matrixdescribing the transition probability from the 4^g non-trap nodes of F_g to the three nodes newly generated by the ith edge of F_g; similarly, V_i (1 ≤ i ≤ E_g) is a 3× 4^g matrix indicating the transition probability from the three new nodes created by the ith edge to those 4^g old non-trap nodes belonging to F_g. Then, P_α, βP_β, α=∑_i=1^E_g U_iV_i = ∑_i=1^E_g( a_i/θ+1ε_l_i + b_i/θ+1ε_r_i, θ/θ+1a_iε_l_i, θ/θ+1b_iε_r_i)× ([ -ε_l_i^⊤+ε_r_i^⊤/2; ; -ε_l_i^⊤; ; -ε_r_i^⊤; ;])= -1/2θ+2× ∑_i=1^E_g[(2θ+1)(a_i ε_l_iε_l_i^⊤+ b_i ε_r_iε_r_i^⊤)+ a_i ε_l_iε_r_i^⊤ + b_i ε_r_iε_l_i^⊤]= I_g-1/2θ+2P_g , which completes the proof of Eq. (<ref>). Note that in Eq. (<ref>), l_i and r_i are the two endpoints of the ith edge of F_g; ε_i is a vector having only one nonzero element 1 at ith entry with other entries being zeros; a_i and b_i are two entries of P_g corresponding to edges (l_i,r_i) and (r_i,l_i), respectively.§ ALTERNATIVE PROOF OF EQ. (<REF>)Equation (<ref>) can also be proved using another technique. Assume that R_g=P_α,βP_β,α and Q_g=I_g-1/2θ+2P_g. In order to prove P_α,βP_β,α=I_g-1/2θ+2P_g, it suffices to show that the entries of R_g are equal to their counterparts of Q_g. For matrix Q_g, it is easy to see that its entries are:Q_g(i,i)=2θ+1/2θ+2 for i= j and Q_g(i,j)=-1/2θ+2P_g(i,j) otherwise. If P_g+1(i,j) denotes the (i,j) entry of matrix P_g+1, the entries of R_g(i,j) of matrix R_g can be evaluated by distinguishing two cases: i=j and i≠ j.For the case of i=j, the diagonal element of R_g isR_g(i,i)= = ∑_z ∈β P_g+1(i,z)P_g+1(z,i)=∑_z ∈βW_iz(g+1)/s^+_i(g+1)W_z i(g+1)/s^+_z(g+1)= 1/2∑_z ∈β, i z k_z(g+1) = 2 W_iz(g+1)/s^+_i(g+1) + ∑_z ∈β, i z k_z(g+1) = 1 W_iz(g+1)/s^+_i(g+1)= 1/2s^+_i(g)/s^+_i(g+1)+ θ s^+_i(g)/s^+_i(g+1)= 2θ+1/2θ + 2=Q_g(i,i),where the relation s^+_i(g+1)=(θ+1) s^+_i(g) is used. In Eq. (<ref>), i zindicates that two nodes i and z are adjacentin network F_g+1.For the other case of i ≠ j, the non-diagonal element of R_g isR_g(i,j) = ∑_z ∈β P_g+1(i,z)P_g+1(z,j) = ∑_A_g+1(i,z) = 1 A_g+1(z,j) = 1 W_iz(g+1)/s^+_i(g+1)W_z j(g+1)/s^+_z(g+1)= 1/2W_ij(g)/s^+_i(g+1)=-1/2θ+2P_g(i,j)= Q_g(i,j),which, together with (<ref>) provesEq. (<ref>). § MULTIPLICITIES OF EIGENVALUESBy numerically computing the eigenvalues for the first several iterations, we can observe some important phenomena and properties about the structure of the eigenvalues. When g=1, the eigenvalues of P_1 are 1-√(1-1/θ+1) and 1+√(1-1/θ+1), both of which have a multiplicity of 2. When g=2, P_2 have 16 eigenvalues:eigenvalue 1 with degeneracy 8 and 4 two-fold other eigenvalues generated by 1-√(1-1/θ+1) and 1+√(1-1/θ+1). When g ≥ 3, all the eigenvalues P_g can be put into two classes. The first class includeseigenvalue 1 and those generated by 1, which display the following feature that each eigenvalue appearing at a given iteration g_i will continue to appear at all subsequent generations greater than g_i. The second class contains those eigenvalues generated by the two 1-√(1-1/θ+1) and 1+√(1-1/θ+1) in P_1. Each eigenvalue in this class is two-fold, andeach eigenvalue of a given iteration g_i does not appear at any of subsequent iterations larger than g_i. For the two eigenvalue classes,each eigenvalue (other than 1) of current generation keeps the multiplicity of its father of the previous generation. Using the above-observed properties of the eigenvalue structure, we can determine the multiplicities of all eigenvalues. Let M_g(λ) denote the multiplicity of eigenvalue λ of matrix P_g. We first determine the number of eigenvalue 1 for P_g. To this end, let r(X) denote the rank of matrix X. ThenM_g(λ=1)=4^g-r(P_g-1× I_g) .For g=1, M_1(1)=0; for g=1, M_2(1)=8. For g≥ 2, it is obvious that r(P_g+1-I_g+1)=r(P_α,β) + r(P_β,α), where r(P_α,β) and r(P_β,α) can be determined in the following way.We first show that P_β,α is a full column rank matrix. Letϕ=(ϕ_1,ϕ_2,⋯ ,ϕ_3× 4^g)^⊤ = ∑_i∈αi ≠ 1 k_iM_i,where M_i is the column vector of P_β,α representing the ith column of P_β,α. Let M_i = (M_1,i,M_2,i,⋯,M_3× 4^g,i)^⊤. Suppose that ϕ=0. Then, we can prove that for an arbitrary k_i, k_i=0 always holds. By construction, for any old node i ∈α, there exists a new leaf node l ∈β attached to i. Then, for ϕ_l=k_1M_1,l+k_2M_2,l,⋯,+k_3× 4^gM_3× 4^g,l, only M_i,l≠ 0 but all M_x,l=0 for x≠ i. From ϕ_l=0, we have k_i=0. Therefore, r(P_β,α)=4^g. Analogously, we can verify that P_α,β isa full row rank matrix and r(P_α,β)=4^g.Combining the above results,the multiplicityof eigenvalue 1 of P_g isM_g(λ=1)= 0,g=1, 2× 4^g-1,g ⩾ 2.We continue to compute the multiplicities of other eigenvalues generated by 1 that are in the first eigenvalue class. Since every eigenvalue at a given iteration keeps the multiplicity of its father at the preceding iteration, for matrix P_g, the multiplicity of each first-generation descendant of eigenvalue 1 is 2× 4^g-2, the multiplicity of each second-generation descendant of eigenvalue 1 is 2× 4^g-3, and the multiplicity of each (g-2)nd generation descendant of eigenvalue 1 is 2×4. Moreover, we can derive that that the number of the ith (0≤ i≤ g-2) generation distinct descendants of eigenvalue is 2^i, where 0th generation descendants refer to the 2× 4^g-1 eigenvalues 1 themselves. Finally, it is easy to verify that the number of all the eigenvalues in the second eigenvalue class is 4× 2^g-1. Hence, the total number ofeigenvalues of matrix P_g is∑_i=0^g-2[(2× 4^g-1-i)× 2^i]+4× 2^g-1= 4^g ,indicating that all the eigenvalues of P_g are successfully found.10 url<#>1urlprefixURLNe03 authorNewman, M. E. 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"authors": [
"Zhongzhi Zhang",
"Huan Li",
"Yibin Sheng"
],
"categories": [
"cs.SI"
],
"primary_category": "cs.SI",
"published": "20170327144354",
"title": "Effects of reciprocity on random walks in weighted networks"
} |
Regular variation, Generalized contents, and fractal strings]Regularly Varying Functions, Generalized contents, and the spectrum of fractal stringsTechnische Universität Berlin, Service-centric Networking, Telekom Innovation Laboratories, Ernst-Reuter-Platz 7, 10587 Berlin, GermanyKarlsruhe Institute of Technology, Institute of Stochastics, Englerstr.2, 76131 Karlsruhe, Germany[2000]35P20, 28A80We revisit the problem of characterizing the eigenvalue distribution of the Dirichlet-Laplacian on bounded open sets Ω⊂ with fractal boundaries. It is well-known from the results of Lapidus and Pomerance <cit.> that the asymptotic second term of the eigenvalue counting function can be described in terms of the Minkowski content of the boundary of Ω provided it exists. He and Lapidus <cit.> discussed a remarkable extension of this characterization to sets Ω with boundaries that are not necessarily Minkowski measurable. They employed so-called generalized Minkowski contents given in terms of gauge functions more general than the usual power functions.The class of valid gauge functions in their theory is characterized by some technical conditions,the geometric meaning and necessity of which is not obvious. Therefore, it is not completely clear how general the approach is and which sets Ω are covered.Here we revisit these results and put them in the context of regularly varying functions. Using Karamata theory, it is possible to get rid of most of the technical conditions and simplify the proofs given by He and Lapidus, revealing thus even more of the beauty of their results. Further simplifications arise from characterization results for Minkowski contents obtained in <cit.>. We hope our new point of view on these spectral problems will initiate some further investigations of this beautiful theory. [ Steffen Winter December 30, 2023 =====================§ INTRODUCTION Given a bounded open set Ω⊂^d with boundary F:=Ω, consider the eigenvalue problem[ -Δ u= λ uin Ω,;u= 0 onF, ]where Δ = ∑_i=1^d∂^2/∂ x_i^2 denotes the Laplace operator. Recall that λ∈ is called an eigenvalue of (<ref>), if there exists a function u ≠ 0 in H_0^1(Ω) (the closure of C_0^∞(Ω), the space of smooth functions with compact support contained in Ω, in the Sobolev space H_1(Ω)) satisfying -Δ u = λ u in the distributional sense.It is well-known that the spectrum of Δ is positive and discrete, i.e. the eigenvalues of (<ref>) form an increasing sequence (λ_i)_i∈of strictly positive numbers with λ_i→∞, as i→∞. If Ω is interpreted as a vibrating membrane held fixed along its boundary,the reciprocals of the eigenvalues can be interpreted as the natural frequencies and the corresponding eigenfunctions as the natural vibrations (overtones) of the system. Much work has been devoted to the question how much geometric information about Ω can be recovered just from listening to its sound. The famous Weyl's law <cit.> describes the growth of the eigenvalue counting function N, defined byN(λ) := #{ i ∈ :λ_i ≤λ}, λ>0.It states (originally for sufficiently smooth domains, but nowadays known to hold for arbitrary bounded open sets Ω⊂^d, see <cit.>) that N is asymptotically equivalent to the so-called Weyl term φgiven byφ(λ) := (2π)^-dω_d|Ω|_d λ^d/2.Here ω_d denotes the volume of the d-dimensional unit ball and |·|_d is the Lebesgue measure in ^d.Since the dimension d of Ω as well as its volume appear in the Weyl term, we are not only able to infer the dimension of Ω from the growth of N but also its volume. For sets with sufficiently smooth boundaries, Weyl conjectured in <cit.> the existence of a second additive term of the order (d-1)/2 in the asymptotic expansion of N with a prefactor being determined by the surface area of the boundary F of Ω:N(λ) = φ(λ) + c_d-1 ^d-1(F)λ^(d-1)/2 + o(λ^(d-1)/2),as λ→∞,with a constant c_d-1 solely depending on the dimension d-1 of F. Many people have contributed to the verification and generalization of Weyl's conjecture, culminating in the results of Seeley <cit.>, Pham The Lai <cit.> (who established that for all bounded open sets Ω with C^∞-boundary the order d-1/2 of the second term is correct) and Ivrii <cit.>, who proved the correctness of the prefactor under some additional assumption (roughly, that the set of multiply reflected periodic geodesics in Ω is of measure zero). Therefore, for sets with smooth boundaries, one can `hear' not only the volume of the set but also the dimension and measure of its boundary.And yet, what if the boundary is non-smooth? Motivated by experiments on wave scattering in porous media, the physicist M. V. Berry conjectured in <cit.> that for general bounded open sets Ω the second term in (<ref>) should be replaced by c_n,H^H(F)λ^H/2 involving Hausdorff dimension H and Hausdorff measure ^H(F) of the boundary F of Ω. This was disproved by Brossard and Carmona <cit.>, who suggested to use instead Minkowski content and dimension in the second term. More precisely, these notions need to considered relative to the set Ω. Recall that for any bounded set Ω⊂^d, F:=Ω and s≥0, the s-dimensional Minkowski content of F (relative to Ω) is definedby^s(F) = lim_r↘ 0|F_r∩Ω|_d /r^d-s,provided the limit exists. HereF_r:={x∈^d: d(x,F)≤ r} is the r-parallel set of F. We write ^s(F) and ^s(F) for the corresponding lower and upper limits. The numbers dim_M F:=inf{s≥ 0: ^s(F)=0} and dim_M F:=inf{s≥ 0: ^s(F)=0} are the upper and lower Minkowski dimension of F, and in case these numbers coincide, the common value is known as the Minkowski dimension _M F of F (relative to Ω). If ^s(F) is both positive and finite (for some s), we say that F is (s-dimensional) Minkowski measurable, and if 0<^s(F)≤^s(F)<∞, then F is called (s-dimensional) Minkowski nondegenerate. Note that in this case, necessarily _M F=s. (In the above notation and also in most of the further discussion, we suppress the dependence on Ω, but of course it should be kept in mind that all these notions are considered relative to Ω here.) Lapidus <cit.> established that indeed the (upper) Minkowski dimension D of F=Ω gives the correct order of growth for the second term (provided ^D(F)<∞), andformulated the so-called Modified Weyl-Berryconjecture (MWB) under the assumption that F is Minkowski measurable:N(λ) = φ(λ)-c_d,D^D(F) λ^D/2 +o(λ^D/2), as λ→∞.Here c_d,D is a constant depending only on d and the Minkowski dimension D = _M(F) of F.Lapidus and Pomerance showed in <cit.> that the MWB holds in dimension d=1, and in <cit.> they constructed counterexamples, which disprove MWB in any dimension d≥ 2. In <cit.>, they also established the following more general result, which characterizes the second order asymptotics of N for sets Ω⊂ with a Minkowski nondegenerate boundary F.Recall that (for some a∈ [0,∞]) two positive functions h_1, h_2 defined in some neighborhood of aare called asymptotically similar as y→ a, in symbols h_1(y)≍ h_2(y), as y→ a, if and only if there are positive constants c, c such that ch_1(y) ≤ h_2(y) ≤ch_1(y) for all y close enough to a. Moreover, h_1 and h_2 are asymptotically equal, as y→ a,h_1(y)∼ h_2(y), as y→ a, if and only if lim_y→ a h_1(y)/h_2(y)=1. Let Ω⊂ be a bounded open set and F:=Ω. Let D ∈ (0,1) andlet =(l_j)_j∈ denote the associated fractal string. Then the following assertions are equivalent: (i) 0 < ^D(F) ≤^D(F) <∞, (ii) l_j≍ j^-1/D, as j →∞, (iii) ∑_j=1^∞{l_jx}≍ x^D, as x →∞, (iv) φ(λ) - N(λ) ≍λ^D/2, as λ→∞. Here and throughout, {y} := y- [y] denotes the fractional part and [y] the integer part of a number y ∈. (The connection between (iii) and (iv)is due to the general relation (<ref>), see also Remark <ref>.)Given a bounded open set Ω⊂^d, unfortunately, very often its boundary F=Ω is neither Minkowski measurable nor Minkowski nondegenerate. Its volume function V_F(r):=|F_r∩Ω|_d, r>0 may exhibit a growth behavior which differs significantly from the behavior of the functions r^1-D, as r↘ 0 captured by the Minkowski contents. Substituting power functions with some well-chosen more general gauge functions h:(0,∞)→(0,∞), it may be possible to understand much better the behavior of V_F at the origin. This leads to the notion of generalized Minkowski contents: For any functionh:(0,∞)→(0,∞), the h-Minkowski content of F (relative to Ω), is defined by(h;F) := lim_r↘ 0 |F_r∩Ω|_d/h(r) ,provided the limit exists. We denote by (h;F) and (h;F) the corresponding lower and upper limits. Moreover, we call F h-Minkowski measurable, if and only if 0<(h;F)<∞, and h-Minkowski nondegenerate, if 0<(h;F)≤(h;F)<∞. (Note that again all these notions are relative to the set Ω.)Given some Ω⊂^d with boundary F and some h such that the h-Minkowski content (h;F) exists, the natural question arises, whether the second order asymptotic term of the eigenvalue counting function N is still determined by the asymptotic behavior of the volume functionV_F, now described by (h;F).A partial answer to this question has been given by He and Lapidus <cit.>. Introducing a class _s of gauge functions (for each s∈ (0,1); cf. Definition <ref>), they generalized the results in <cit.> employing h-Minkowski contents with gauge functions h∈_s. In particular, they obtained a generalization of (<ref>) to h-Minkowski measurable boundaries as well as the following generalization of Theorem <ref> to sets with h-Minkowski nondegenerate boundaries: Let Ω⊂ be a bounded open set and F:=Ω. Denote by ℒ =(l_j)_j∈ the associated fractal string and let h∈_1-D for some D∈(0,1). Then the following assertions are equivalent: (i) 0 < (h;F) ≤(h;F)<∞, (ii) l_j≍ g(j), as j →∞, (iii) ∑_j=1^∞{l_jx}≍ f(x), as x →∞, (iv) φ(λ) - N(λ) ≍ f(√(λ)), as λ→∞, where g(x) := H^-1(1/x) with H(x):=x/h(x), and f(x):=1/h(1/x).The classes _s of gauge functions appearing in Theorem <ref> are defined as follows. [<cit.>]Let s∈ (0,1). A function h:(0,∞) → (0,∞) is in _s, if and only if the following three conditions are satisfied: (H1) h is a continuous, strictly increasing function with lim_x↘ 0h(x)=0, lim_x↘ 0h(x)/x=∞ and lim_x→∞h(x) = ∞. (H2) For any t>0, lim_x↘ 0h(tx)/h(x) = t^s, where the convergence is uniform in t on any compact subset of (0,∞). (H3) There exist some constants τ∈ (0,1), m>0, x_0, t_0 ∈ (0,1] such that h(tx)/h(x)≥ mt^τ, for all 0<x≤ x_0, 0<t≤ t_0.From the definition of the classes _s it is not clear at all, how restrictive they are with respect to the family of sets covered by this approach. Given Ω, it is not easy to decide, whether some suitable gauge function h exists in _s (for some s) or not, such that the boundary F=Ω is h-Minkowski nondegenerate or even h-Minkowski measurable. It is also not clear whether all of the conditions in the definition of _s are really necessary to establish the derived connections between the eigenvalue counting function N and the volume function V_F. On the other hand, there might be a simpler class of gauge functions capable of serving the same purpose.In the present paper we aim at shedding some light on these problems. We will introduce (for each ϱ∈) the class _ϱ of C^1-smooth gauge functions h that are regularly varying with index ϱ, i.e. which satisfy lim_x↘ 0h(tx)/h(x) = t^ϱ for any t>0 (see Definition <ref> and compare with condition (H2) above). Using results from Karamata theory, the theory of regularly varying functions, we will establish results for the classes _ϱ completely analogous to those obtained in <cit.>. The new classes _ϱ of gauge functions have several advantages compared to the classes _s in He and Lapidus <cit.>. First, their definition is simpler involving only two conditions - regular variation and smoothness (and the latter will turn out to be a convenience rather than a restriction). Second, the new analogues of Theorem <ref> and (<ref>) for _ϱ, stated in Theorem <ref>, hold for all Ω⊂ covered by the old result. (We were hoping to extend the family of sets Ω covered, but, in fact, we will show that exactly the same family of sets is addressed by our results.)Third, using regularly varying gauge functions allows some structural insights, e.g. concerning (generalized) Minkowski measurability. Last but not least, the proofs simplify significantly when using the new class. Employing Karamata theory, allows to separate very clearly the main geometric (and algebraic) arguments from pure `asymptotic calculus'. Moreover, the differentiability allows amongst others to apply some characterization results of Minkowski contents in terms of S-contents from <cit.>.S-contents describe the asymptotic behavior of the parallel surface areas of the given set F. They have been used to simplify some parts of the proof of Theorem <ref>, see <cit.>. In analogy to this approach, we add a criterion in terms of generalized S-contents, which will admit further simplification of the proofs also in this setting of general gauge functions.For these reasons, the new classes _ϱ are certainly helpful in understanding which sets are covered by the results of He and Lapidus and which are not and whether the theory can be extended further. We see our discussion as a starting point for further investigations, paving the road for a further generalization of the theory.§ REGULARLY VARYING FUNCTIONS AND STATEMENT OF THE MAIN RESULTS We will now define the classes of gauge functions that we are going to use and state our main results. We start with regular variation, a notion dating back to Jovan Karamata in the 1930'ies. Nowadays, the theory of such functions is known as Karamata theory. While the classic theory is formulated for regular variation at ∞, see also Definition <ref> below, we will require regular variation at 0. Let ϱ∈. A function h:(0,y_0)→ (0,∞) (with y_0>0, possibly ∞) is called regularly varying (at 0) of index ϱ, if and only if h is measurable and satisfies for all t>0lim_y↘ 0h(ty)/h(y) = t^ϱ.We write _ϱ for the family of these functions.Functions h∈_0 are also called slowly varying (at 0).Furthermore, a regularly varying function h∈_ϱ is in the class _ϱ of smoothly varying functions (at 0) of index ϱ, if and only if h is C^1-smooth in a neighborhood of 0, i.e. h ∈ C^1(0,y_1], for some y_1>0.We first discuss the relation of the classes _ϱ, _ϱ and _ϱ. It turns out that they are asymptotically equivalent, by which we mean the following: Two classes of functions are asymptotically equivalent, if for any function h in one class there is an asymptotically equivalent function h in the other class, i.e. one such that h(y)∼ h(y), as y↘ 0, and vice versa. (It is easy to see that this is an equivalence relation on classes of functions, justifying the notion.)For any ϱ∈(0,1), the three classes _ϱ, _ϱ and _ϱ areasymptotically equivalent.It is rather obvious thatany h∈_ϱ is in _ϱ (because of condition (H2)), similarly we have _ϱ⊂_ϱ, but in both cases a reverse relation is not obvious. A proof is given in Section <ref>, when we discuss regularly varying functions in more detail (see Corollary <ref> and Theorem <ref>).Theorem <ref> is not particularly deep but very useful. It clarifies that we have some freedom in choosing gauge functions. Regular variation is the basis of all three classes, but we may impose some additional properties for our convenience without loosing generality. This is exemplified by the following simple statement saying that generalized Minkowski contents are invariant with respect to asymptotic equivalence. Let F⊂^d be compact and let h,h̃:(0,∞) → (0,∞)be gauge functions such thath(y) ∼h̃(y), as y↘ 0. Then F is h-Minkowski nondegenerate if and only if F is h̃-Minkowski nondegenerate. More precisely, one even has 0< M(h̃;F) =M(h;F)<∞ and 0< M(h̃;F) = M(h;F) < ∞. In particular, F is h-Minkowski measurable if and only if F is h̃-Minkowski measurable. In connection with Proposition <ref>, Theorem <ref> clarifies that when searching for a suitable gauge function h (for a given set F) within the class of regularly varying functions _ϱ (such that F becomes h-Minkowski measurable/nondegenerate), then we can as well restrict to one of the classes _ϱ or _ϱ. Note also that the existence of such an h∈_ϱ necessarily implies that _M F exists and equals d-ϱ, cf. Remark <ref>.From the above considerationsone might get the impression that not much can be gained from reproving the results of He and Lapidus, obtained for gauge functions in _ϱ, in the context of the classes _ϱ and _ϱ. Indeed, the above results make very clear that not a single additional set Ω will be covered by the new statements. However, even though our original intention was an extension of the theory to more general sets Ω, our approach using regularly varying functions is still very useful, as it provides a more structural point of view on the studied classes of gauge functions and, more importantly, as it clarifies and simplifies many of the arguments of the long and technical proofs in <cit.>. It shows that regular variation is the essential property which makes things work.As an interesting side result, we mention that generalized Minkowski measurability with respect to regularly varying functions can be characterized without referring to any particular gauge function:Let Ω⊂^d be a bounded open set and F=Ω. There exists a gauge function h∈_ϱ for F such that F is h-Minkowski measurable, if and only if the volume function V_F of F (i.e., the function r↦ |F_r∩Ω|_d, r>0) is regularly varying with index ϱ, i.e. if and only if V_F∈_ϱ. This means that the assumed regular variation of the gauge function reflects an essential property of the volume function itself. In particular, if a set is h-Minkowski measurable for some h∈_ϱ, then it cannot be g-Minkowski measurable for some gauge function g outside the class _ϱ and vice versa. A proof of Theorem <ref> is given in Section <ref>, see page proof:mink. Before we formulate our main result, we recall the notion of (generalized) S-content.For any gauge function h:(0,y_0)→(0,∞), the upper and lower h-S-contents of F:=Ω (relative to Ω) are defined by(h;F) := lim sup_r ↘ 0H^d-1(∂ F_r∩Ω)/h(r) and (h;F) = lim inf_r ↘ 0H^d-1(∂ F_r∩Ω)/h(r),respectively. If (h;F)=(h;F), then the common value (h;F) is the h-S-content of F. There are close general relations between (generalized) Minkowski and S-contents of a set F. In particular, for differentiable gauge functions h the h-Minkowski content (h;F) and the h'-S-content (h';F) coincide whenever one of these contents exists. We refer to <cit.> and Section <ref> for more details. Moreover, in dimension 1 there are close relations between the (generalized) S-content of a set and the so-called string counting function of the associated fractal string, see Section <ref> for details. Another advantage of smooth gauge functions is that, for any h∈_1-D, D∈(0,1), the following two auxiliary functions f and g are well-defined for x large enough (i.e. for all x∈[x_0,∞) for some x_0≥ 0):f(x) = xh(1/x)andg(x) = H^-1(1/x),where H(x) = x/h(x), x∈(h) and H^-1 is the inverse of H, we refer to Proposition <ref> for details. g and f take the same role as the corresponding functions in Theorem <ref>. They characterize the decay of the lengths ofthe associated string (l_j), as j→∞ and of the packing defect δ(x):= ∑_j=1^∞{l_jx}, as x→∞, respectively. [Main Theorem] Let Ω⊂ be a bounded open set and F:=Ω.Assume _MF = D ∈ (0,1) anddenote by ℒ =(l_j)_j∈ the associated fractal string.Let h ∈_1-D and let f and g be given as in (<ref>). I. (Two-sided bounds.) The following assertions are equivalent: * 0< (h;F) ≤(h;F) < ∞, * 0< (h';F) ≤(h';F) < ∞, * l_j ≍ g(j), as j→∞, * ∑_j=1^∞{l_jx}≍ f(x), as x→∞, * φ(λ) - N(λ) ≍ f(√(λ)), as λ→∞. II. (Minkowski measurability.) The following assertions are equivalent: (vi) F is h-Minkowski measurable, (vii) F is h'-S measurable, (viii) l_j ∼ L g(j), as j→∞ for some positive L>0. Under these latter assertions, h-Minkowski content, h'-S-content of F and the constant L are connected by the relation (h;F) = (h';F) = 2^1-DL^D/1-D.Moreover, these assertions implyN(λ)=φ(λ)- c_1,D(h; F) f(√(λ)) + o(f(√(λ))),as λ→∞,where c_1,D=2^D-1π^-D(1-D)(-ζ(D)) with ζ being the Riemann zeta function. Note that Theorem <ref> comprises the classical results of Lapidus and Pomerance <cit.> for sets Ω with Minkowski measurable/nondegenerate boundaries as a special case (by choosing the gauge function h(y)=y^1-D). It parallels the results of He and Lapidus in <cit.> obtained for gauge functions h of the classes _1-D. Our proofs bring the generalized theory of He and Lapidus in some parts closer back to the original arguments in the proofs of <cit.> (in some other parts, however, we will use results on S-contents from <cit.>). One may wonder, whether also a converse to the last assertion in Theorem <ref> holds, i.e. whether the existence of an asymptotic second term for N as in (<ref>) (assumed to hold for some f given in terms of some h as in (<ref>)) implies the h-Minkowski measurability of the boundary F. He and Lapidus have addressed this question (for h∈_1-D). In <cit.> they showed in particular that such a converse fails for all dimensions D, for which the Riemann zeta-function has a zero on the line (s)=D, generalizing thus earlier results of Lapidus and Maier <cit.> and showing a connection of this question to the Riemann hypothesis. More precisely, the converse fails (for the above mentioned D) for any h∈_1-D which is differentiable and satisfies y h'(y)/h(y)≥μ for some μ>0 (and all y). We point out that the differentiablity implies h∈_1-D and that, by Lemma <ref> below, we have thus lim_y↘ 0y h'(y)/h(y)=1-D, implying in particular that y h'(y)/h(y)≥ (1-D)/2=:μ for all y in a suitable neighborhood of 0. Hence <cit.> directly applies to any h∈_1-D. We are optimistic that our methods might also help to simplify the proof of this statement, but we have not yet tried to do this.In the present paper, we have restricted ourselves to subsets Ω of . In <cit.>, also some results in the higher dimensional case are obtained, i.e. for bounded open sets Ω∈^d, d>1. It is known that the MWB conjecture fails in this case and therefore these results are necessarily of a weaker nature. More precisely, under the assumption that the upper h-Minkowski content of F:=Ω is finite (for some suitable gauge function h), an upper bound for the second asymptotic term of N is derived in terms of the function f (determined by h as in (<ref>)). Although, the assumptions on the gauge function h are slightly different in these higher dimensional results, it may be worth to revisit them in the light of the methods used here. Although we have clarified significantly the (one-dimensional) theory of He and Lapidus by showing that regular variation is the driving force behind these results (while the other conditions can be omitted or follow automatically), we think that this is not the end of the story. Several examples of sets Ω and suitable associated gauge functions h have been discussed in <cit.>, to which the results apply. Since all occurring gauge functions h are differentiable and thus in the classes _1-D, they can also serve as examples for Theorem <ref> (and to apply the theorem, it is not necessary to check any of the technical conditions (H1)-(H3), only the regular variation). However, there might exist sets Ω, for which regularly varying gauge functions are not suitable but for which nevertheless results similar to Theorem <ref> hold. It would be desirable to have such an example or to prove that it does not exist. In either case, we believe that the concept of regular variation – and its various generalizations studied in Karamata theory (e.g. de Haan theory), see e.g. the monograph <cit.> – will be useful for such an attempt. The results of Lapidus and Pomerance <cit.> and in particular the discovered connections to the Riemann zeta function, have initiated an extensive study of fractal strings, which led to the theory of complex dimensions, see <cit.> and the many references therein. Nowadays these spectral problems and many related questions, including Minkowski measurability, are studied with the help of various zeta functions, which allow to derive explicit formulas for functions such as the the eigenvalue counting function or the tube volume. We have not made any efforts to relate our results to this theory. It seems, however, that sets that are generalized Minkowski measurable/nondegenerate (but not Minkowski measurable/nondegenerate) are not covered so far by the results in this theory. In the last years, even extensions (of some part of the theory) to higher dimensions are being discussed, see <cit.>. In the language of <cit.>, where a classification scheme for sets ^d is proposed in terms of the behaviour of their tube functions, the boundaries F studied here and in <cit.> fall into the class of weakly degenerate sets. The remaining parts of the paper are organized as follows. After some preliminary considerations concerning asymptotic similarity in Section <ref>, we recall in Section <ref> some results from Karamata theory and discuss various useful consequences. On the way, we will prove the asymptotic equivalence of the classes _ϱ, _ϱ and_ϱ (Theorem <ref>) as well as Theorem <ref>. Sections <ref> and <ref> are devoted to the proof of Theorem <ref>. In Section <ref>, we will treat the `geometric part' and establish the connection between Minkowski contents, S-contents and the growth behaviour of the lengths of the associated fractal string, while in Section <ref>, we finally establish the connection between the geometry of Ω and its spectral properties. An overview over the various steps of the proof is given in the diagrams on page diagrams. § SOME PRELIMINARIES ON ASYMPTOTIC EQUIVALENCE AND SIMILARITYIn this section we collect some basic (and well-known) facts about asymptotic similarity ≍ and asymptotic equivalence ∼ of functions, recall the definitions from page def:asymp. Here we formulate all facts for asymptotic similarity/equivalence at a=0, but for the case a=∞ they hold analogously.Consider the familyof real valued positive functions f defined on a right neighborhood of 0, i.e. functions f:(0,y_0)→(0,∞) for some y_0>0. Observe that ∼ and ≍ are equivalence relations on . In particular, both relations are transitive, i.e. if f_1≍ f_2 and f_2≍ f_3 then f_1≍ f_3, as y↘ 0 and similarly for ∼. Moreover, since asymptotic equivalence implies asymptotic similarity, we also havethe following implication:f_1(y)≍ f_2(y) andf_2(y)∼ f_3(y)implies f_1(y)≍ f_3(y),asy↘ 0.We collect some useful rules for ≍ and ∼. Some further rules will be stated later on for functions with extra properties such as regular variation or smoothness (see e.g. Lemma <ref> or <ref>). Let f_1,f_2,h, g_1, g_2∈. Then, as y↘ 0, (i) f_1∼ f_2/h, if and only if h· f_1∼ f_2;(ii) if f_1∼ f_2 and g_1∼ g_2, then f_1· g_1 ∼ f_2· g_2 and f_1/ g_1 ∼ f_2/ g_2;(iii) if f_1∼ f_2∼ g_1∼ g_2, then f_1+f_2∼g_1+g_2.The assertions hold analogously with ∼ replaced by ≍. As a first simple application, we provide a proof of Proposition <ref>. Note that a statement similar to Proposition <ref> could be formulated for S-contents. The first assertion is a direct consequence of (<ref>) applied to f_1=V_F, f_2=h and f_3=h̃. To see the equality of the upper and lower generalized Minkowski contents, let ϵ∈ (0,1). Then there is a y_0>0 such that (1-ϵ) h(y)<h̃(y) < (1+ϵ) h(y) for all y∈ (0,y_0]. Hence we obtain 0< V_F(y)/(1+ϵ)h(y)≤V_F(y)/h̃(y)≤V_F(y)/(1-ϵ)h(y)<∞and the assertion follows by taking upper and lower limits in this relation as y↘ 0 and thenletting ϵ↘ 0. The last assertion is a direct consequence of the transitivity of ∼. § REGULARLY VARYING FUNCTIONS AND KARAMATA THEORYIn this section we discuss functions of regular variation in more detail. In our exposition we concentrate on those results which turned out to be useful for our purposes and which prepare the proof of our main result. On the way we will also prove Theorems <ref> and <ref>.While the classic theory is formulated for regular variation at ∞ (see Definition <ref> below), we will focus on regular variation at 0 (cf. Definition <ref>). We will reformulate some of the classic results of Karamata theory for this case (which is an easy exercise due to Remark <ref>). For the versions at ∞ (which we will also use occasionally) and further details we refer to the monograph <cit.>. Let ϱ∈. A function f :(x_0,∞) → (0,∞) (with x_0≥ 0) is called regularly varying (at ∞) of index ϱ, if and only if f is measurable and satisfies for all t>0lim_x→∞f(tx)/f(x) = t^ϱ.We write _ϱ[∞] for the family of all such functions. f∈_0[∞] are also called slowly varying (at ∞). Note that, by Definition <ref>, the domain (h) of any h∈_ϱ is an interval of the form (0,y_0) (a right neighborhood of zero). h can easily be extended to a function on (0,∞)without affecting the regular variation property at 0. It is easy to see that one can even extend h to (0,∞) in such a way that properties like continuity, differentiability or strict monotonicity (which h may have) are preserved. (Of course, if lim_y→ y_0h(y)=∞, then one has to redefine h near y_0 for this extension.) Similarly, any function f∈_ϱ[∞] (with domain (f)=(x_0,∞)) can be extended to the whole positive axis.As pointed out above, regularly varying functions (at ∞) are well-studied, see e.g. <cit.>. The following simple observation allows to transfer virtually all results known for regular variation at ∞ to regular variation at 0. h∈_ϱ if and only if the function h̃ defined by h̃(x)=h(1/x), 1/x∈(h) is in _-ϱ[∞]. Indeed, if the domain of h∈_ϱ is (0,y_0) then the domain of h̃ is the interval (1/y_0,∞) and for any t>0, one obtainslim_x→∞h̃(tx)/h̃(x) = lim_x→∞h(1/(tx))/h(1/x)=lim_y↘ 0h(t^-1y)/h(y)=t^-ϱ,implying h̃∈_-ϱ[∞].Observe that condition (H2) in the definition of the classes _ϱ is regular variation at 0 plus an extra uniformity requirement in the variable t. We will see below from Theorem <ref> that this local uniformity is automatically satisfied for regularly varying functions.In <cit.>, property (H2) was called a homogeneity property. Indeed, any regularly varying function h∈_ϱ is asymptotically homogeneous of degree ϱ in the following sense: for any t>0, h(ty) = h(y) (t^ϱ+o(1)),asy↘ 0.This relationis equivalently described by h(ty) ∼ t^ϱ h(y), as y↘ 0. The following simple observation characterizes regular variation in terms of slow variation, compare also with the Characterization Theorem <cit.>:Let ϱ∈. For any h∈_ϱ, there is a slowly varying function ℓ∈_0 such that h(y)=y^ϱℓ(y) for any y∈(h).Defineℓ by ℓ(y):=h(y)/y^ϱ, y∈(h) and observe that, for any t>0, ℓ(ty)/ℓ(y)→ 1, as y↘ 0 implying ℓ∈_0. Our next observation is that the classes _ϱ are stable with respect to asymptotic equivalence. Any function gasymptotically equivalent to some regularly varying function h is already regularly varying itself. In contrast, asymptotic similarity ≍ does not preserve regular variation. Let h∈_ϱ and let g:(0,y_0)→(0,∞) be a measurable function such thath(y) ∼ g(y), as y↘ 0. Then g∈_ϱ, i.e. g is regularly varying at 0 with the same index ϱ. The function z defined by z(y) := g(y)/h(y) for all y∈(h)∩(g) is measurable and converges to 1 as y↘ 0. Therefore, g(ty)/g(y)= h(ty)/h(y) z(ty)/z(y) → t^ϱ,asy↘ 0, for all t>0. Hence g is regularly varying of index ϱ. This simple observation will be used frequently in the proofs later on. It is also the key to Theorem <ref>, which we prove now.By the assumptions, there exists some h∈_ϱ and some M>0, such that (h;F)=M, which means that lim_r↘ 0V_F(r)/Mh(r)=1 or, equivalently V_F(r)∼ Mh(r), as r↘ 0. Since any (positive) constant multiple of a function in _ϱ is still in _ϱ, it follows from Lemma <ref>, that V_F∈_ϱ. The Uniform Convergence Theorem (UCT) is one of the central results of Karamata theory. We formulate a version for functions h∈_ϱ. It can easily be derived from the corresponding statement for functions in _ϱ[∞], cf. Remark <ref>.[UCT, cf. <cit.> for a version at ∞] Let ϱ∈ and h ∈_ϱ. Then h(ty)/h(y)→ t^ϱ, as y↘ 0 converges uniformly in t on any compact subset of (0,∞).Theorem <ref> implies in particular that condition (H2) is satisfied for any regularly varying function h∈_ϱ. As a further consequence of Theorem <ref>, we may now prove the following result on asymptotic similarity/equivalence of regularly varying functions. It will be extremely useful in the proofs later on. Let κ∈ and g∈_κ. Let f_1,f_2:(0,∞)→(0,∞) such that lim_y↘ 0 f_2(y)=0 andassume f_1(y) ≍ f_2(y), as y↘ 0. Then g(f_1(y)) ≍ g(f_2(y)),as y↘ 0.Moreover, if g∈_0or f_1(y) ∼ f_2(y), as y↘ 0, theng(f_1(y)) ∼ g(f_2(y)),as y↘ 0.Completely analogous relations hold, if lim_y↘ 0 f_2(y)=∞ andg∈_κ[∞], or if f_1(x)≍ f_2(x), as x→∞. By assumption, there are positive constants y_0, c_1, c_2 such that c(y) :=f_1(y)/f_2(y)∈ [c_1,c_2] for all y∈ (0,y_0). Assume first that κ≥ 0 and fix some ∈(0,c_1^κ).By the Uniform Convergence Theorem <ref>, the convergence g(tx)/g(x)→ t^κ (as x↘ 0) is uniform in t on the compact interval [c_1,c_2], i.e. there exists some δ>0 such that |g(tx)/g(x)-t^κ|≤ for each 0<x<δ and each t∈[c_1,c_2]. Therefore, g(f_1(y))/g(f_2(y)) = g(c(y)f_2(y))/g(f_2(y))≤ c(y)^κ+≤ c_2^κ+=:c_2',for each y∈(0,y_0) such that f_2(y)<δ. Observe that the hypothesis on f_2 implies there exists some y_1=y_1(δ)>0 such that the f_2(y)<δ is true for all 0<y<y_1. In a similar way one obtains that the expression g(f_1(y))/g(f_2(y)) is bounded from below by c_1':=c_1^κ- for all y∈(0,y_1).This shows (<ref>) for the case κ≥ 0.For κ<0, the only difference is that now the function t↦ t^κ is decreasing. Fixing some ∈(0,c_2^κ), a similar argument as in the previous case gives the upper bound c_2':=c_1^κ+ and the lower bound c_1':=c_2^κ- for the expressiong(f_1(y))/g(f_2(y)) for all y∈(0,y_1). This completes the proof of the first assertion.In case κ=0 the constants used above are c_1'=1- and c_2'=1+, where the >0 was fixed. Observing thatcan be chosen as close to zero as we wish, assertion (<ref>) follows for g∈_0. Assertion (<ref>) is also valid under the hypothesis f_1∼ f_2, since in this case there is for any >0 some y_0 such that (1-)≤ f_1(y)/f_2(y)≤ (1+) for all y∈(0,y_0]. This impliesg((1-)f_2(y))/g(f_2(y))≤g(f_1(y))/g(f_2(y))≤g((1+)f_2(y))/g(f_2(y))→ (1+)^κ,as y↘ 0. Similarly, the left expression converges to (1-)^κ, as y↘ 0. Now (<ref>) follows by lettingtend to 0.The arguments for the case lim_y↘ 0 f_2(y)=∞ and g∈_κ[∞], and for f_1(x)≍ f_2(x), as x→∞ are completely analogous.A second central result of Karamata theory is the following Representation Theorem, which provides an integral representation for slowly varying functions. Again, we formulate it for slow variation at 0.[Representation Theorem, cf.<cit.>] A function ℓ is slowly varying at 0, if and only if it has a representation ℓ(y) = c(y) exp( ∫_y^aε(u)/udu ),y ∈ (0,a), for some a>0, where c and ε are measurable functions withlim_y↘ 0 c(y) = C ∈ (0,∞) andlim_y↘ 0ε(y) = 0. Applying <cit.> to the function ℓ̃∈_0[∞] given by ℓ̃(x):=ℓ(1/x), yields the existence of some positive constants ã and C and some functions c̃ andwith c̃(x)→ C and (x)→ 0 as x→∞ such thatℓ̃(x) = c̃(x) exp( ∫_ã^x(u)/udu ),x≥ã.Therefore, we get for any 0<y<a:=1/ã,ℓ(y)=ℓ̃(1/y) = c̃(1/y) exp( ∫_ã^1/y(u)/udu )=c(y)exp( ∫_1/a^1/y(u)/udu ),where c(y):=c̃(1/y) satisfies obviously c(y)→ C as y↘ 0. The representation (<ref>) follows now with the substitution u↦ 1/u and setting (y):=(1/y). Note that the representation of ℓ given by Theorem <ref> is not unique, for we may add to ε any function δ with lim_y↘ 0δ(y)=0 (and adjust c accordingly).Moreover, the upper bound a of the integration interval in (<ref>) may be shifted arbitrarily in (ℓ), again by adjusting the function c. If a>1, for instance, we haveℓ(y)= c(y)exp( ∫_y^a ε(u)/udu ) = c̃(y)exp( ∫_y^1 ε(u)/udu),wherec̃(y) := c(y) exp( ∫_1^aε(u)/udu ) . Combining Lemma <ref> with the Representation Theorem <ref>, we obtain an analogous representation for regularly varying functions:Since for h∈_ϱ, ϱ≠ 0, there is some ℓ∈_0 such that h(y)=y^ϱℓ(y), there must existfunctions c and(with c(y)→ C∈(0,∞) and (y)→ 0, as y↘ 0) such thath(y) = y^ϱℓ(y) = c(y) exp( ϱ·log y + ∫_y^1ε(u) du/u) = c(y) exp(∫_y^1 -ϱ + ε(u) du/u)From this representation it is easy to conclude the following asymptotic behaviour. [cf. [BGT, Proposition 1.5.1]] Let ϱ≠ 0 and h∈_ϱ. Then h(y) →{[0,if ϱ>0;∞,if ϱ<0 ].,asy↘ 0.Let Ω⊂^d be a bounded open set and F=Ω. Assume that F is h-Minkowski nondegenerate for some h∈_d-D, i.e. 0<(h;F)≤(h;F)<∞. Then the Minkowski dimension of F exists and equals D. Indeed, by Lemma <ref>, we can write h as h(r)=r^d-Dℓ(r) for some ℓ∈_0 and so,for any γ>0,^D+γ(F)=lim_r ↘ 0V_F(r)/r^d-D-γ =lim_r ↘ 0V_F(r)/r^d-Dℓ(r) r^γℓ(r) ≤(h;F) lim_r ↘ 0r^γℓ(r).Since r↦ r^γℓ(r) is regularly varying with index γ>0, by Proposition <ref>, the last limit vanishes,implying ^D+γ(F)=0 and thus _MF≤ D.Similarly, we get ^D+γ(F)≥(h;F) lim_r ↘ 0r^γℓ(r), where the last limit is now +∞ for γ<0, implying _M F≥ D. This shows _M F=D. Therefore, clearly, the index of any suitable regularly varying gauge function h for F is necessarily equal to its (Minkowski) co-dimension d-D. We point out that this does not imply that for every F with dimension _M F =D there exists a suitable gauge function h∈_d-D.The following fact will be useful in the sequel, e.g. in Theorem <ref> below. It is another immediate consequence of the Representation Theorem. Let ϱ∈ and h∈_ϱ. Then there is x_0>0 such that the functions h and 1/h are locally bounded and locally integrable on (0,x_0]. The Representation Theorem allows to show that any regularly varying function h ∈_ϱ of indexϱ∈(0,1) already satisfies condition (H3) (cf. Definition <ref>). This is another important step towards the asymptotic equivalence of the classes _D and _D. Let ϱ∈ (0,1) and h ∈_ϱ. Then h satisfies hypothesis (H3), i.e. there are constants m>0, τ∈ (0,1), y_0, t_0 ∈ (0,1] such that h(ty)/h(y)≥ m t^τ, for all 0<y ≤ y_0 and 0< t ≤ t_0. By Lemma <ref>, h(y)=y^ϱℓ(y) for some ℓ∈_0, and, by the Representation Theorem <ref>, we can write ℓ as in equation (<ref>). Therefore, for any t,y∈(0,1),ℓ(y)/ℓ(ty) =c(y)·exp( ∫^1_yε(u)/udu)/c(ty) ·exp( ∫^1_tyε(u)/udu ) =c(y)/c(ty)exp( -∫_ty^yε(u)du/u),where c, are measurable functions such that c(y)→ C (for some C∈(0,∞)) and (y)→ 0 as y↘ 0. Let δ>0 such that C-δ>0 and let γ>0 such that ϱ+γ<1. Since, c(y)→ C as y↘ 0, there is y_0∈(0,1) such that |c(y)-C|< δ for all y ∈ (0,y_0). Moreover, since (y)→ 0, there is also some y_1∈(0,y_0] such that |(y)|<γ for any y∈(0,y_1).Therefore, we obtain for any y∈(0,y_1),ℓ(y)/ℓ(ty) ≤C+δ/C-δexp( ∫_ty^y |ε(u)|du/u)≤ M exp( ∫_ty^yγdu/u)=M t^-γ, where we have set M := C+δ/C-δ. This implies ℓ(ty)/ℓ(y)≥ M^-1 t^γ and thus h(ty)/h(y)=t^ϱℓ(ty)/ℓ(y)≥ M^-1 t^ϱ+γ, for all y∈(0,y_1) and all t∈(0,1). Thus the assertion follows for m:=M^-1 and τ:=ϱ+γ, which by the choice of γ is strictly less than 1. Up to now, we have seen that any regularly varying function h∈_ϱ with ϱ∈(0,1) satisfies conditions (H2) and (H3), by Theorem <ref> and Proposition <ref>, respectively. Concerning condition (H1), it is easy to see from Proposition <ref> that such h also satisfy lim_y↘ 0h(y)=0 and lim_y↘ 0h(y)/y=∞. The condition lim_y→∞h(y) = ∞ is not always satisfied but it is not very relevant. One can easily extend or redefine h on some interval [x_0,∞) bounded away from zero to meet this condition without affecting the asymptotic properties of h at 0.On the other hand, regularly varying functions need neither be continuous nor strictly increasing. It will be our aim now to clarify that for each h∈_ϱ there is an asymptotically equivalent function with these two properties. It turns out that there are even smooth representatives for h.§.§ Smoothly varying functions. We will now discuss the classes _ϱ in more detail. We will show in particular the asymptotic equivalence of the classes _ϱ (cf. Definition <ref>) and _ϱ, see Theorem <ref>. Recall that for any ϱ∈, a regularly varying function h∈_ϱ is in the class _ϱ, if and only if there is some y_0>0 such that h ∈ C^1(0,y_0]. Similarly, we write _ϱ[∞] for the class of functions that are smoothly varying at ∞. By the following result, we may assume slowly varying functions ℓ∈_0 to be smooth modulo asymptotic equivalence, implying that the classes _0 and _0 are asymptotically equivalent.[cf. <cit.>] Let ℓ∈_0. Then there is a smoothly varying function ℓ_1∈_0 asymptotically equivalent to ℓ(i.e. ℓ_1(y)∼ℓ(y), as y↘ 0). ℓ_1 can be chosen in such a way that all derivatives of the function p_1 defined by p_1(x) := log(ℓ_1(e^-x)),e^-x∈(ℓ_1) vanish, i.e., for all n∈, p_1^(n)(x)→ 0,asx→∞. Combining Theorem <ref>with Lemma <ref>, we obtain the following result, which establishes in particular that the classes _ϱ and _ϱ are asymptotically equivalent for any ϱ∈. Let ϱ∈ and h ∈_ϱ. Then there is a smoothly varying function h_1 ∈_ϱ asymptotically equivalent to h (i.e. with h_1(y)∼ h(y), as y↘ 0). h_1 can be chosen in such a way that the function p_1 defined by p_1(x) := log(h_1(e^-x)), x∈(h_1) satisfieslim_x→∞ p_1'(x)=-ϱ and lim_x→∞ p_1^(n)(x)=0 for eachn∈, n≥ 2.By Lemma <ref>,the functionh can be written as h(y) = y^ϱℓ(y) for some ℓ∈_0. Theorem <ref> implies the existence of a smoothly varying function ℓ_1∈_0 asymptotically equivalent to ℓ. Defining h_1 by h_1(y) := y^ϱℓ_1(y), y>0, we obtainp_1(x) =log(h_1(e^-x)) = log(e^-ϱ xℓ_1(e^-x)) = -ϱ x +log (ℓ_1(e^-x)).By Theorem <ref>, all derivatives of the function x↦log (ℓ_1(e^-x)), x>0 tend to zero, as x→∞, which yields the assertion.The next statement characterizes the asymptotic behavior of the the derivative of a smoothly varying function. It is not only needed in the proof of the subsequent proposition but it will also be extremely useful in the proofs in Section <ref>. The expression E_h(y):=yh'(y)/h(y) is known as the elasticity of a function h at y.For any ϱ∈ and any function h∈_ϱ, the elasticity of h at y converges to ϱ as y↘ 0, i.e.lim_y↘ 0y h'(y)/h(y) =ϱ.For ϱ≠ 0, this can be rephrased as h'(y)∼ϱh(y)/y, as y↘ 0. It is enough to show the assertion for slowly varying functions. Indeed, for h∈_ϱ, there is, by Lemma <ref>, some function ℓ∈_0 such that h(y)=y^ϱℓ(y) on (h) and thereforeE_h(y)=y h'(y)/h(y)=y (y^ϱℓ(y))'/y^ϱℓ(y)=ϱ y^ϱℓ(y) +y^ϱ+1ℓ'(y) /y^ϱℓ(y)=ϱ+y ℓ'(y)/ℓ(y)=ϱ+ E_ℓ(y).That is, the elasticities of h and ℓ differ pointwise by ϱ and so in particular, as y↘ 0, E_h(y)→ϱ if and only if E_ℓ(y)→ 0.So let h∈_0.For any fixed y∈(h), we apply the linear substitution u(t)=yt-y in the differential quotient of h at y and obtainlim_y↘ 0lim_u↘ 0h(y+u)-h(y)/uy/h(y) =lim_y↘ 0lim_t→ 1h(ty)-h(y)/y(t-1)h(y) y= lim_t→ 11/t-1lim_y↘ 0h(ty)-h(y)/h(y)=0,where in the last expression we have changed the order of the limits. This is justified, since, by the Uniform Convergence Theorem <ref>, the inner limit in this last expression vanishes uniformly in t on any compact interval. (So choose one containing 1 in its interior.) It follows that also the outer limit (as t→ 1) of this expression exists (and equals 0), implying that it is safe to interchange the order of the limits.Our next observation is that smoothly varying functions are monotone near zero. One of the consequences is that the derivatives of smoothly varying functions are again regularly varying.Let ϱ≠ 0 and h ∈_ϱ.If ϱ>0, then h is strictly increasing in some right neighborhood of 0, and if ϱ<0, then h is strictly decreasing in some right neighborhood of 0. In particular, the function ϱ^-1 h' (when restricted to this neighborhood) is in the class _ϱ-1.Let ϱ≠ 0 and h∈_ϱ. Fix some ϵ∈(0,1) and let (0,y_0) be an interval in which h is differentiable. Then (<ref>) in Lemma <ref> implies that there is some y_1∈(0,y_0] such that for each y∈(0,y_1), (1-ϵ)ϱ h(y)/y<h'(y)<(1+ϵ)ϱ h(y)/y. For ϱ>0, the expression on the left is strictly positive, which means in particular that h' is strictly positive in the interval (0,y_1). Hence the function h is strictly increasing in (0,y_0). For ϱ<0, the expression on the right is strictly negative, which implies that h' is strictly negative in(0,y_1). Hence h is strictly decreasing in (0,y_1).For the last assertion note that ϱ^-1 h' is positive in the interval (0,y_1). Moreover, by Lemma <ref>, we have ϱ^-1 h'∼ h(y)/y, as y↘ 0. Therefore, ϱ^-1 h'∈_ϱ-1 follows from Lemma <ref>, since the function y↦ h(y)/y is in _ϱ- 1.This completes the proof. We have gathered all the results necessary in order to show the asymptotic equivalence of the classes _ϱ and _ϱ. Let ϱ∈ (0,1). For any function h∈_ϱ, there is a smoothly varying function h̃∈_ϱ asymptotically equivalent to h,and vice versa. Hence, the classes _ϱ and _ϱ are asymptotically equivalent. Let ϱ∈ (0,1) and h∈_ϱ. By condition (H2), h is regularly varying (at 0), and therefore, by Corollary <ref>, there exists some function h̃∈_ϱ such that h̃(y)∼ h(y), as y↘ 0.To show the converse, let h̃∈_ϱ.Then h̃ satisfies hypothesis (H2) by the Uniform Convergence Theorem <ref>. Furthermore, h̃ satisfies (H3) by Proposition <ref>. It remains to show that the function h̃satisfies hypothesis (H1) modulo asymptotic equivalence.The function h̃ is by definition C^1-smooth and therefore in particular continuous on some interval (0,y_0] for some y_0>0. By Proposition <ref>, h̃ is also strictly increasing on the interval (0,y_1], for some y_1∈(0,a]. Therefore, h̃ can now easily be extended or redefined on the interval (y_1,∞) in such a way that it becomes continuous and strictly increasing on (0,∞) and satisfies lim_y→∞h̃(y)=∞ (e.g. by setting h(y) := (y-y_1) + h̃(y_1) for y>y_1 and h(y):=h̃(y) on (0,y_1]). Note that this will not affect the asymptotic properties at 0, i.e. for the new function h, we haveh(y)∼h̃(y) as y↘ 0 and h∈_ϱ. In particular, h still satisfies (H1) and (H2). By the Lemma <ref>, we have the representation h(y) = y^ϱℓ(y) for someℓ∈_0and so it is easy to see from Proposition <ref> thatlim_y↘ 0 h(y)/y = lim_y↘ 0 y^ϱ-1ℓ(y) = ∞ and lim_y↘ 0 h(y) = 0,since ϱ∈(0,1). Therefore, we have found a function h, asymptotically equivalent to h̃, satisfying all the conditions of the class _ϱ. This completes the proof. Combining Corollary <ref> and Theorem <ref>, we conclude that for any ϱ∈(0,1) the three classes _ϱ, _ϱ and _ϱ are asymptotically equivalent, hence Theorem <ref> is proved. Regular variation is indeed the essential property which makes things work.Therefore, it is only natural to expect that in Theorem <ref> the class _1-Dcan be substituted with the class _1-D. However, in order to do this, one needs to clarify that the auxiliary functions f and g associated in Theorem <ref> to any h∈_1-D have a well defined counterpart for any h∈_1-D. This could be achieved by using the concept of generalized asymptotic inverses (see e.g. <cit.>), which, however, would impose additional technical difficulties. The latter can be avoided by restricting to smoothly varying functions, which, due to the asymptotic equivalence of the classes _1-D and _1-D, is rather a convenience than a restriction. In fact, we will not only see that the functions f and g are well-defined for any h∈_1-D, but also that they inherit from h regular variation and smoothness. For this purpose we let D∈(0,1) and h∈_1-D (with 1-D∈(0,1)). We define the function H by H(y):= y/h(y), y∈(h) and observe thatH ∈_D. By Proposition <ref>, H is strictly increasing on some interval (0,y_0) and therefore it can be inverted on this interval. The inverse function H^-1 is then well-defined on the interval (0,H(y_0)) (and may be extended beyond this interval in some arbitrary way, if needed). H^-1 inherits the properties of smoothness and strict monotonicity from H. Moreover, observing that for any z∈(0,H(y_0)) there is a unique y∈(0,y_0) such that z=H(y) (and that z depends continuously on y), we obtain for any t>0, lim_z↘ 0H^-1(tz)/H^-1(z)=lim_y↘ 0H^-1(t H(y))/H^-1(H(y))=lim_y↘ 0H^-1(H(t^1/Dy))/H^-1(H(y))=t^1/D, where we have used for the second equality that H∈_D. Hence, we have shown that H^-1 is smoothly varying with index 1/D, i.e. H^-1∈_1/D. We are now ready to define the functions f and g.Let D∈(0,1) and h∈_1-D. Then there is some x_0≥ 0 such that the functions g and f are well-definedfor any x∈[x_0,∞) byg(x):=H^-1(1/x)andf(x):=1/H(1/x)=x h(1/x),and C^1-smooth on this interval. Moreover, g∈_-1/D[∞] andf∈_D[∞]. By the considerations above, H^-1 is well-defined on (0,H(y_0)], where y_0 is chosen such that h (and thus H) is strictly increasing and C^1-smooth on (0,y_0]. Letting x_g:= 1/H(y_0), we have 1/x∈ (0, H(y_0)] for any x∈ [x_g,∞), and so H^-1(1/x) and thus g(x) are well-defined and smooth on [x_g,∞). Since H^-1∈_1/D, it follows directly from Remark <ref> that g∈_-1/D[∞].Similarly, f is well-defined and C^1 on [x_f,∞) for x_f:=1/y_0 whenever H is C^1 on (0,y_0]. Moreover, y↦ 1/H(y) is regularly varying (at 0) with index -D, since H has index D. Thus, again by Remark <ref>, we obtain f∈_D[∞]. (The actual assertion of the statement is satisfied for x_0:=max{x_g,x_f}, however, we will not need a common interval for g and f in our applications.) In the proof of Theorem <ref> given in <cit.>, hypothesis (H3) is a technical assumption in order to leisurely apply Lebesgue dominated convergence. Karamata theory, however, allows to circumvent this kind of reasoning due to, among others, the following powerful result, known as Karamata's Theorem. As we will only need a version for functions regularly varying at ∞, we directly restate the corresponding result in <cit.> for this class of functions.[Karamata's Theorem; direct half <cit.>] Let ϱ∈ and f ∈_ϱ[∞]. Let furtherX>0 such that f is locally bounded in [X,∞), cf. Cor. <ref>. Then (i) For any σ≥ -(ϱ+1) x^σ+1f(x)/∫_X^xu^σ f(u)du →σ+ϱ+1 , asx→∞. . (ii)For any σ < -(ϱ+1) (and for σ = -(ϱ+1) if ∫_·^∞ u^-(ϱ+1)f(u)du <∞) x^σ+1 f(x)/∫_x^∞u^σ f(u)du→ -(σ+ϱ+1) , asx→∞. . One of the most interesting consequences of Karamata's Theorem <ref> in the context of regularly varying gauge functions is that we may take slowly varying functions ℓ∈_0[∞] out ofintegrals in the following fashion: ∫_x^∞ u^ϱℓ(u)du∼ℓ(x)∫_x^∞u^ϱ du,asx→∞,whenever ϱ<-1 (and this is the case we will need). Indeed, this follows easily from part (ii) of Karamata's Theorem. Moreover, this theorem provides the following useful relations. We point out that statements similar to (i) and (ii) below have been proved and used by He and Lapidus for functions related to the classes _ϱ, see <cit.>. Let ϱ<-1 and g∈_ϱ[∞]. Then (i) ∫_x^∞g(u)du∼-1/ϱ+1xg(x), asx→∞; (ii) for any t>0, ∫_tx^∞g(u)du/∫_x^∞g(u)du→ t^ϱ+1,asx→∞, where the convergence is uniform in t on any compact subset of (0,∞); (iii) ∑_j=k^∞ g(j)∼ -1/ϱ+1kg(k),ask →∞. Assertion (i) follows directly from Karamata's Theorem <ref> (ii) for σ =0.For a proof of (ii) observe that the function x↦ x g(x), x∈(g) is regularly varying (at ∞) withindex ϱ+1. Hence, by Lemma <ref> andassertion (i), the function G(x):=∫_x^∞g(u)du, x∈(g) is in _ϱ+1[∞]. Therefore, the convergence G(tx)/G(x)→ t^ϱ+1, as x→∞ is obvious and the uniformity follows from the Uniform Convergence Theorem <ref> (version at ∞).It remains to prove (iii). By Lemma <ref>, there is some ℓ∈_0[∞] such that g(x)=x^ϱ ℓ(x), x∈(g). Define g̃ by g̃(x):=g(j), for x∈[j,j+1)∩(g) and j∈,and ℓ̃ by ℓ̃(x):=g̃(x)/x^ϱ, x∈(g). We claim thatg̃∈_ϱ[∞] or, equivalently, ℓ̃∈_0[∞]. To prove this, by Lemma <ref>, it suffices to show that ℓ̃(x)∼ℓ(x), as x→∞. First observe that, for any x∈[j,j+1)∩(g), ℓ̃(x)=g(j)/x^ϱ and thus, since x↦ x^ϱ is decreasing,ℓ(j)=g(j)/j^ϱ≤ℓ̃(x)≤g(j)/(j+1)^ϱ=(j/j+1)^ϱℓ(j).Since j=[x] and thus j/(j+1)→ 1 as x→∞, we conclude ℓ̃(x)∼ℓ([x]), as x→∞. Finally, we note that ℓ([x])∼ℓ(x), as x→∞, since ℓ∈_0[∞]. Indeed, letting t_x:=[x]/x and noting that 1/2≤ 1-1/x≤ t_x≤ 1 for any x≥ 2, the Uniform Convergence Theorem<ref> (applied to ℓ for t on the compact interval [1/2,1]) yieldslim_x→∞ℓ([x])/ℓ(x)=lim_x→∞ℓ(t_x x)/ℓ(x)=1.This completes the proof of our claim that g̃∈_ϱ[∞]. Now assertion (iii) follows directly by applying (i) to g̃, since, for any k∈∩(g),∑_j=k^∞ g(j) =∑_j=k^∞g̃(j)=∫_k^∞g̃ (u) du.§ THE GEOMETRIC PART OF THE PROOFIn this section we discuss the equivalence of the first three assertions in Theorem <ref> as well as that of assertions (vi),(vii) and (viii). The direct proofs of (i) ⇔ (iii) and (vi) ⇔ (viii) given in <cit.> are rather technical, cf. <cit.>. Using characterization results for Minkowski contents in terms of S-contents from <cit.>, does not only allow to add another equivalent criterion to each of the two parts of Theorem <ref>, but also to simplify the proofs significantly. Instead of proving (i) ⇔ (iii) directly, we will establish (i) ⇔ (ii) and (ii) ⇔ (iii) separately. Similarly we will show (vi) ⇔ (vii) and (vii) ⇔ (viii).The (generalized) S-contents, which describe the behavior of the boundary measure of the parallel sets, provide thus an extremely useful connecting link between Minkowski contents (volume of the parallel sets) and the growth of the lengths in the associated fractal string.§.§ S-contents vs. Minkowski contents.Our first aim is to verify the equivalences (i) ⇔ (ii) and (vi) ⇔ (vii) in Theorem <ref>. It turns out that they can be derived essentially from the results in <cit.>.Therefore, we will not reprove the equivalence here, but rather explain the minor modifications necessary in the relevant results of <cit.> to cover our present situation. In fact, we can establish this equivalence for any bounded open set Ω⊂^d and any gauge function h∈_d-D,D∈(0,d). There is no need to restrict to subsets offor this result.Let Ω⊂^d be bounded and F=Ω. Let h∈_d-D for some D∈[0,d). Then the following assertions are equivalent: (i) 0< (h;F) ≤(h;F) < ∞,(ii)0< (h';F) ≤(h';F) < ∞.In particular, these assertions imply _M F=D.Recall that the (generalized) Minkowski contents and S-contents appearing in the statement above as well as in Theorem <ref> below are defined relative to the set Ω, cf. (<ref>) and (<ref>). The results from <cit.> that we are going to use in the proofs are formulated for the `full' contents (with the set Ω in the definitions (<ref>) and (<ref>) omitted). However, due the fact that Ω is metrically associated with its boundary F and that therefore the volume function r↦ V_F(r)=|F_r∩Ω|_d is a Kneser function, all the results in <cit.> hold literally for the relative contents used here. For more details we refer to the discussion of relative contents in<cit.>.In view of Remark <ref>, the stated equivalence follows essentially by combining <cit.> (where the easier implication (ii) ⇒ (i) is established) with <cit.> (where the reverse implication is obtained). While in <cit.> h is assumed to be differentiable with derivative h' being non-zero in some right-neighborhood of 0, an assumption met for any h∈_d-D due to Proposition <ref>,there are additional assumptions in <cit.>: h is assumed to be of the form h(y)=y^d-D g(y) with g being non-decreasing andlim sup_y↘ 0y h'(y)/h(y) <∞.The latter assumption is satisfied due to Lemma <ref>, which says that the above limit exists and equals d-Dfor h∈_d-D.Moreover, by Lemma <ref>, h is clearly of the form h(y)=y^d-Dg(y) for some g∈_0. But the slowly varying factor g is not necessarily non-decreasing. However, inspecting the proof of <cit.> it is rather easy to see that `non-decreasing' can be replaced by `slowly varying' in this statement. The monotonicity of g is only used once in the proof of <cit.> to ensure that g(ar) is bounded from below by g(r) for some a>1 and all sufficiently small r>0. If g∈_0 and ϵ>0, then we can certainly find some r_1>0, such that g(ar)≥ (1-ϵ)g(r) for any r∈(0,r_1) and this suffices to extend the argument in the proof of <cit.> to functions g∈_0. Since <cit.> is essentially a direct application of <cit.>, also this statement extends to functions g∈_0. For the assertion _M F = D see Remark <ref>. This completes the proof.Also the following statement is a direct consequence of a result in <cit.> and the elasticity properties derived in Lemma <ref>. It establishes the equivalence (vi) ⇔ (vii) in Theorem <ref>, i.e. the equality of the h-Minkowski content and the h'-S-content.[cf. <cit.>] Let Ω⊂^d be bounded and F=Ω. Let h∈_d-D for some D∈[0,d). Suppose M>0. Then M(h;F) = M,if and only if S(h';F)=M. Moreover, in this case _M F = D. By Lemma <ref>, h can be written in the form h(y)=g(y)y^d-D for some g∈_0 and by Lemma <ref>, we have lim_y→ 0 g'(y)y/g(y)=0. Therefore, taking into account Remark <ref>, the hypothesis of <cit.> is satisfied except that the function g is now slowly varying and not necessarily non-decreasing.However, we can argue similarly as in the proof of Theorem <ref> above that the forward implication in <cit.> follows essentially from <cit.>, which is a refinement of <cit.> and in which the estimates can easily be modified to work for slowly varying g. The reverse implication in <cit.> (which is the reverse implication in Theorem <ref>) is a direct consequence of <cit.>, which can be applied since, by Proposition <ref>, h' is non-zero in some right neighborhood of 0. Finally, for a proof of _M F = D see Remark <ref>. §.§ S-Contents and fractal strings.Recall that in dimension d=1, we can associate to any bounded open set Ω⊂, its fractal string =(l_j)_j∈, encoding the lengths l_j of the connected components I_j of Ω. They appear inin non-increasing order, i.e. l_1≥ l_2≥ l_3≥…, and according to their multiplicities. We will discuss now the relations between S-contents and the asymptotic growth of the lengths l_j, and establish in particular the equivalences (ii)⇔(iii) and (vii)⇔(viii) of Theorem <ref>. Note that (and with F=Ω), the`surface area' of F_∩Ω in the definition of the S-content reduces in dimension d=1 to the counting measure ^0. Therefore, the key idea behind these two relations is the following simple geometric observation (everything else is `asymptotic calculus'): All the intervals I_j of F with length l_j≤2 are completely covered by F_ and do not contribute to the boundary F_, while each of the remaining ones contributes exactly two points.That is, for any j∈ and ∈[l_j/2,l_j-1/2), we have^0( F_∩Ω)=2(j-1).Recall from Proposition <ref> that for D∈(0,1) and h∈_1-D, the function g is given by g(x) := H^-1(1/x), where H^-1 is the inverse of H(y) := y/h(y). Let Ω⊂ be open and bounded, F:=Ω and=(l_j)_j∈ the associated fractal string. Let D∈ (0,1) and h∈_1-D. Then the following assertions are equivalent: (i)0< S(h';F) ≤S(h';F) <∞,(ii) l_j ≍ g(j), as j →∞.We first reformulate both assertions using `asymptotic calculus'. Then their equivalence will follow easily from (<ref>).On the one hand, assertion (i) can be rewritten as ^0( F_∩Ω)≍ h'(), as ↘ 0, which, by Lemma <ref>, is equivalent to ^0( F_∩Ω)≍ h()/, as ↘ 0.On the other hand, since g(x)=H^-1(1/x) and H∈_D, we can apply H to both sides of (ii) to see that, by Lemma <ref>, (ii) is equivalent to H(l_j)=l_j/h(l_j)≍ 1/j,asj→∞.By Lemma <ref>, this can also be written as j≍ h(l_j)/l_j, as j→∞. Using the asymptotic homogeneity of h∈_1-D (which implies h(y)∼ 2^1-Dh(y/2), as y↘ 0, cf. Remark <ref>) and recalling that positive constants do not matter in `≍'-relations, this is equivalently given by 2j≍h(l_j/2)/(l_j/2), asj→∞. Since (j-1)/j→ 1, as j→∞, we can replace j by j-1 on the left. Therefore, the statement of the theorem is equivalent to^0( F_∩Ω)≍h()/,as ↘ 0,iff2(j-1)≍h(r_j)/r_j,asj→∞,where r_j:=l_j/2, j∈. (r_j is the `inradius' of an interval of length l_j.)The forward implication in (<ref>) is obvious: since (<ref>) implies in particular that ^0( F_r_j∩Ω)=2(j-1) for each j∈, we just need to plug in the sequence (r_j)_j∈ forin the left assertion.For a proof of the reverse implication in (<ref>), assume the right hand side holds, which means, there are constants c_1, c_2 such that c_1≤ 2(j-1)/(h(r_j)/r_j)≤ c_2 for each j. Since the function ↦ h()/ is in _-D, it is strictly decreasing in some right neighborhood (0,_0) of 0, cf. Proposition <ref>. Moreover, lim_↘ 0 h()/=+∞. Therefore, for any sufficiently large j∈ (such that r_j-1<_0) and any ∈ [r_j,r_j-1),the quotient ^0( F_∩Ω)/h()/ is bounded from above and below byc_1≤2(j-1)/h(r_j)/r_j≤^0( F_∩Ω)/h()/≤2(j-1)/h(r_j-1)/r_j-1≤ c_2+2/h(r_j-1)/r_j-1,where the last summand on the right vanishes as j→∞. But this implies the left assertion in (<ref>), completing the proof of Theorem <ref>. A very similar argument allows to establish the equivalence of generalized S-measurability of F=Ω and generalized `L-measurability' of the associated fractal string, i.e. the equivalence of the assertions (vii) and (viii) in Theorem <ref>. Let F⊂ be a compact set and let =(l_j)_j∈ be the associated fractal string. Let D∈(0,1) and h∈_1-D. Let L>0. Then the following assertions are equivalent: * F is h'-S measurable (i.e. 0< S(h',F) <∞) with S(h',F)=S:=2^1-D L^D/1-D,* l_j ∼ L· g(j),as j→∞. We follow the line of the argument in the proof of Theorem <ref> and start by reformulating the statement.In view of (<ref>) and using again r_j:=l_j/2, our first claim is that the relation (i)⇔(ii) is equivalent to the following equivalence: ^0( F_∩Ω)∼ 2^1-D L^D h()/,as ↘ 0, iff 2(j-1)∼ 2^1-DL^D h(r_j)/r_j,asj→∞. Indeed, assertion (i) in Theorem <ref> means ^0( F_∩Ω)∼ S· h'(), as ↘ 0 and, by Lemma <ref>, h'() can be replaced by (1-D)h()/ (since h∈_1-D), showing the equivalence of (i) and the left assertion in (<ref>). (Recall that (1-D)S=2^1-D L^D.) Similarly, by applying H to both sides of assertion (ii) and recalling that g(x)=H^-1(1/x), we can infer from Lemma <ref> that (ii) is equivalent to H(L^-1 l_j)=L^-1l_j/h(L^-1l_j)∼ 1/j, as j→∞. Taking into account the homogeneity of h (cf. Remark <ref>) and Lemma <ref>, this can be rephrased asj∼ L^D h(l_j)/l_j, as j→∞, which is easily seen to be equivalent to the right assertion in (<ref>), using again the homogeneity of h. This completes the proof of the above claim. It is therefore sufficient to prove the equivalence in (<ref>). The forward implication in (<ref>) is again obvious from (<ref>), by plugging in the sequence (r_j)_j∈ forin the left assertion. The reverse implication in (<ref>) requires now a slightly refined argument. Assume the right hand side holds. Then, for each δ>0, there is some j_0=j_0(δ) such that1-δ≤2(j-1)/c· h(r_j)/r_j≤ 1+δfor each j≥ j_0, where c:=2^1-DL^D. Since the function ↦ h()/ is in _-D, it is strictly decreasing in some right neighborhood (0,_0) of 0, cf. Proposition <ref>.Therefore, for any sufficiently large j∈ (such that j≥ j_0 and r_j-1/2<_0) and any ∈ [r_j,r_j-1),the quotient ^0( F_∩Ω)/c· h()/ is bounded from above and below by1-δ≤2(j-1)/c· h(r_j)/r_j≤^0( F_∩Ω)/c· h()/ ≤2(j-1)/c· h(r_j-1)/r_j-1≤ 1+δ +2/c · h(r_j-1)/r_j-1,where again the last summand on the right vanishes as j→∞. Since the argument works for any δ>0, the left assertion in (<ref>) follows, completing the proof of Theorem <ref>.§ THE `SPECTRAL' PART OF THE PROOF OF THEOREM <REF> In this section, we will finally establish the connection between the geometric and the spectral properties of Ω, i.e. in particular the equivalence of the assertions (iii) and (iv) in part I ofTheorem <ref> and the validity of (<ref>) in part II. Because of the results in the previous section, we can use a combination of the assertions (i)-(iii) of Theorem <ref> to conclude the validity of (iv) – we will use (ii) and (iii).In contrast, it is enough to show that (iv) implies at least one of the assertions (i)-(iii).It will be convenient now to use the string counting function J definedbyJ(ε) := max{ j | l_j > ε},for any fractal string =(l_j)_j∈, counting the number of lengths l_j inthat are strictly larger than ε. Since2J(2)=^0( F_∩Ω), for any >0, it is easy to see that the asymptotics of ^0( F_∩Ω) (described by the S-content) determines the asymptotics of J(), and vice versa.Under the hypothesis of Theorem <ref>, any of the assertions (i),(ii) and (iii) in Theorem <ref>is equivalent toJ()≍ h()/,as ↘ 0.Similarly, any of the assertions (vi), (vii) or (viii) in Theorem <ref>is equivalent toJ()∼ L^D h()/,as ↘ 0,where L is the constant in (viii). In particular, this implies J∈_-D.Due to the results of the previous section, it suffices to relate (<ref>) to assertion (ii) and (<ref>) to (vii), which is easy to do with the help of (<ref>). For the first equivalence recall from the proof of Theorem <ref> (see (<ref>)) that assertion (ii) in Theorem <ref> is equivalent to ^0( F_∩Ω)≍ h()/,as ↘ 0 and apply (<ref>).For the second stated equivalence, recall from the proof of Theorem <ref> (see (<ref>)) that assertion (vii) is equivalent to ^0( F_∩Ω)∼ 2^1-D L^D h()/ as ↘0.Therefore, by (<ref>), 2J(2)∼ 2^1-D L^D h()/, as ↘0. Substituting 2 byand taking into account the asymptotic homogeneity of h (of degree 1-D), we obtainJ()∼ 2^-D L^D h(/2)//2∼ 2^1-D L^D h() (1/2)^1-D/=L^Dh()/,as ↘ 0,which completes the proof of (<ref>). Finally, since ↦ h()/ is regularly varying with index -D,J∈_-D follows from (<ref>) by Lemma <ref>.The following observation will turn out to be very useful in the proof of the implication (iii)⇒(iv). Assume that assertion (iii) of Theorem <ref> holds. Then∑_j> J(2) l_j≍ h(),as ↘ 0. Due to the assumption l_j≍ g(j) as j→∞, we find positive constants j_0,α,α such that l_j/g(j)∈[α,α] for all j≥ j_0. Fix _0>0 small enough such that J(2_0)≥ j_0. Thenwe have for any 0<≤_0α∑_j> J(2) g(j)≤∑_j> J(2) l_j≤α∑_j> J(2) g(j).This shows∑_j> J(2) l_j≍∑_j> J(2) g(j),as ↘ 0.Since J(2)→∞ as ↘ 0 and g∈_-1/D (with -1/D<-1), we infer from Proposition <ref> (iii) that∑_j> J(2) g(j)∼D/1-D J(2) g(J(2)),as ↘ 0. Using (<ref>), it follows from Lemma <ref> and the definition of g that the right hand side is asymptotically similar to h()/ g(h()/)=h(), as ↘ 0. Combining this with (<ref>), the assertion of the lemma follows. Now we are ready to reformulate and prove the implication (iii) ⇒ (iv) in Theorem <ref>. It is convenient to employ the functionδ̃(2):=δ(1/2)=∑_j=1^∞{l_j/2}, >0. Let D∈ (0,1) and h ∈_1-D. Let (l_j)_j∈ be a fracal string withl_j ≍ g(j), as j→∞. Then δ̃(2)≍h()/,as ↘ 0,or, equivalently, δ(x)≍ f(x),asx→∞, where f and g are given as in Proposition <ref>. The equivalence of the two assertions is obvious from the substitution x=1/2 and the definition of f. Therefore, it suffices to prove (<ref>). We splitδ̃ as followsδ̃(2) = ∑_j=1^∞{l_j/2} = ∑_j>J(2){l_j/2} + ∑_j≤ J(2){l_j/2}.Now observe that for j>J(2), we have l_j/2<1 and thus {l_j/2}=l_j/2. Therefore, we can employ Lemma <ref> to the first sum on the right and infer that∑_j>J(2){l_j/2}=1/2∑_j>J(2)l_j≍h()/2,as ↘ 0.Since 0≤{y}<1, we infer for the second sum on the right of (<ref>) that, for any >0,0≤∑_j≤ J(2){l_j/2}≤ J(2),and by (<ref>) in Proposition <ref>, J(2) is bounded above by c·h()/ for some c>0.Combining the estimates of both sums, we conclude that δ̃(2) is bounded from above and below by some constant multiple of h()/ for all sufficiently small >0, proving assertion (<ref>).Our next aim is to show that h-Minkowski measurability (i.e. any of the assertions (vi), (vii), (viii) in Theorem <ref>) implies the exact asymptotic second term of the eigenvalue counting function as stated in (<ref>). For the main argument we follow closely the idea of the proof in <cit.> for the case of Minkowski measurable sets (which is also used in <cit.>) to split the sum δ(x) in a very special way into three summands and then estimate each of them separately. In the estimation part Karamata theory turns out to be very useful again, allowing a simpler argument as in <cit.>.Our first step is a refinement and generalization of Lemma <ref> above.Assume that assertion (viii) of Theorem <ref> holds. Then, for any k∈,∑_j> J(k) l_j∼D L^D/1-D k^1-Dh(),as ↘ 0. Due to the assumption l_j∼ Lg(j) as j→∞, we find for any γ >0 some j_0>0 such that l_j/(L g(j))∈[1-γ,1+γ] for all j≥ j_0. Given k∈, fix _0=_0(k)>0 small enough such that J(k_0)≥ j_0. Then we have for any 0<≤_0(1-γ)L∑_j> J(k) g(j)≤∑_j> J(k) l_j≤ (1+γ)L∑_j> J(2) g(j).Since we can find such j_0 and _0 for any γ>0, we infer∑_j> J(k) l_j∼ L ∑_j> J(k) g(j),as ↘ 0,for any fixed k∈. Since J(k)→∞ as ↘ 0 and g∈_-1/D (with -1/D<-1), we infer from Proposition <ref> (iii) that, for any k∈,∑_j> J(k) g(j)∼D/1-D J(k) g(J(k)),as ↘ 0. Applying (<ref>) and taking into account the definition of g (cf. Proposition <ref>) and Lemma <ref>, we inferJ(k) g(J(k))∼ L^D h(k)/k g(L^D h(k)/k) ∼ L^D-1h(k)/k g(h(k)/k)=L^D-1 h(k),as↘ 0. Combining the last three estimates and using the homogeneity of h, the assertion of the lemma follows. Now we are ready for the main step in the proof of an exact asymptotic second term of N. We will show that (viii) in Theorem <ref> implies an exact estimate for the packing defect δ. Let D∈ (0,1) and h ∈_1-D. Let (l_j)_j∈ be a fractal string withl_j ∼ L g(j), as j→∞. Then δ(x)∼ -ζ(D)L^D f(x),asx→ 0,where g and f are given as in Proposition <ref>. For fixed k∈, we splitδ as followsδ(x) = ∑_j>J(1/x){l_jx} + ∑_q=2^k∑_j= J(q/x)+1^J((q-1)/x){l_jx}+∑_j≤ J(k/x){l_j x}Now observe that J(q/x)<j≤ J((q-1)/x) implies [l_jx]=q-1, and therefore the second sum equals ∑_q=2^k∑_j= J(q/x)+1^J((q-1)/x) (l_jx- (q-1)) =∑_j=J(k/x)+1^J(1/x) l_jx -∑_q=2^k(q-1)(J(q-1/x)-J(q/x))=x ∑_j=J(k/x)+1^J(1/x) l_j -∑_q=1^k-1J(q/x)+(k-1)J(k/x).Combining the first sum of this last expression with the first sum in (<ref>), in which all terms satisfy l_jx<1 implying{l_jx}=l_jx, we obtainδ(x) = x ∑_j>J(k/x)l_j + (kJ(k/x)-∑_q=1^k-1J(q/x)) +∑_j≤ J(k/x){l_j x}- J(k/x),for each k∈ and each x>0. We will see in a moment, that the first two termsA:= x ∑_j>J(k/x)l_jandB:= kJ(k/x)-∑_q=1^k-1J(q/x)contribute asymptotically to δ(x) as x→∞ for any k∈, while the remainder termC:=∑_j≤ J(k/x){l_j x}- J(k/x)=∑_j≤ J(k/x)({l_j x}-1)will vanish as k→∞. (Note that A, B, C depend on x and k.) Up to here we followed the argument of He and Lapidus in the proof of <cit.> (which is similar to the one in the proof of <cit.>), the estimates are now derived in an easier way using Karamata theory.First, by Lemma <ref>, we get for A:A∼D L^D/1-D k^1-Dx h(1/x)= L^D D/1-D k^1-Df(x),asx→∞.Second, since J∈_-D by Proposition <ref>, the homogeneity implies in particular that J(q/x)∼ q^-D J(1/x), as x→∞, for any q>0 and together with (<ref>) we infer thatB∼(k^1-D -∑_q=1^k-1q^-D)J(1/x) ∼ L^D (k^1-D -∑_q=1^k-1q^-D)f(x),asx→∞.Combining the estimates for A and B we conclude that, for each k∈,A+B/L^D f(x)→1 /1-D k^1-D -∑_q=1^k-1q^-D= w_k(D)+1/1-D asx→∞,where the function w_k is defined for each s∈ byw_k(s):=∫_1^k(t^-s-[t]^-s) dt (=-1/1-s +1/1-sk^1-s-∑_q=1^k-1 q^-s, s≠ 1).The functions w_k are entire and, as k→∞, they converge (uniformly on any compact subset of (s)>0) to the functionw(s):=∫_1^∞(t^-s-[t]^-s) dt.w is analytic in (s)>0 and known to satisfy the relation w(s)=-1/(1-s)-ζ(s) for(s)>0, see e.g. <cit.>. In particular, this impliesw_k(D)+1/1-D→ -ζ(D),ask→∞.It remains to estimate C. The obvious relation -1≤l_j x-1<0 implies -J(k/x)≤ C≤ 0 for any k∈. Using again Proposition <ref>, we infer that J(k/x)∼ k^-D J(1/x)∼ L^D k^-D f(x), as x→∞. Hence the expression -L^D k^-D f(x) is essentially a lower bound for C. More precisely, - k^-D= lim_x→∞-J(k/x)/L^D f(x)≤lim inf_x→∞C/L^D f(x)≤lim sup_x→∞C/L^D f(x)≤ 0.Combining now the estimates for C in (<ref>) and A+B in (<ref>), and recalling that δ(x)=A+B+C, we infer that, for each k∈,w_k(D)+1/1-D- k^-D≤lim inf_x→∞δ(x)/L^D f(x)≤lim sup_x→∞δ(x)/L^D f(x)= w_k(D)+1/1-DLetting now k→∞, the left and right expression both converge to -ζ(D), which completes the proof of the theorem. It remains to show that assertion (iv) in Theorem <ref> implies at least one of the assertions (i), (ii), (iii) (and therefore all of them by the results in Section <ref>). Due to the equivalence of (<ref>) with (ii), the following statement is essentially the implication (iv) ⇒ (ii). Let D ∈ (0,1) and h∈_1-D. Let =(l_j)_j∈ be a fractal string such that δ(x) ≍ f(x), as x→∞.Then J(1/x) ≍ f(x), as x→∞.By assumption, there are positive constants a_1, a_2 and x_0 such thata_1 f(x) ≤δ(x) ≤ a_2 f(x),for all x≥ x_0.Part 1: We first derive a lower bound for J. Fix some small ϵ_0>0 and choose some integer k ≥ 2withk > ( 2 a_2(1+ϵ_0)/a_1)^1/1-D,i.e. such thatk^D<a_1/2a_2(1+ϵ_0)k.Since f∈_D[∞], there is an x_1≥ x_0>0 such thatk^D(1-ϵ_0) ≤f(kx)/f(x)≤ k^D (1+ϵ_0) < a_1/2a_2k,for all x≥ x_1, where the last inequality is due to the choice of k.We split the packing defect δ(x) into two sums as followsδ(x) = ∑_{l_jx} < k^-1{l_j x}+ ∑_{l_jx}≥ k^-1{l_jx}=:U(x)+V(x).Observe that for any γ>0 such that k {γ}<1, we have k{γ} = { kγ}. Indeed, since k [γ]∈, we get1 > k{γ}= { k {γ}} = { k ( γ - [γ]) } = { kγ -k [γ]} = { k γ}.Applying this to the sum U, we obtain for x≥ x_1,kU(x)=∑_{l_jx} < k^-1{ l_jkx}< δ(kx)(<ref>)≤ a_2 f(kx) (<ref>)≤a_1/2kf(x) (<ref>)≤1/2kδ(x).Thus U(x)≤1/2δ(x), which together with U(x)+V(x) = δ(x) implies that V(x)≥1/2δ(x)for all x≥ x_1. On the other hand, since every summand in V is less than 1, we haveV(x)≤∑_{l_jx}≥ k^-1 1≤#{j:{l_jx}≥ k^-1}≤#{j:l_jx ≥ k^-1}=J(1/kx).Combining both estimates, we obtain for x≥ x_1J(1/kx)≥1/2δ(x)≥1/2a_1f(x),which provides a lower bound on J(1/x) in terms of f(x) as desired. Substituting kx by x and using again the homogeneity (<ref>), we conclude thatJ(1/x)≥a_1/2f(x/k)≥a_1/2k^-D(1-ϵ_0)f(x),for all x ≥ kx_1.Part 2: In order to derive an upper bound for J, we first prove the following estimate: there are positive constants c, x_2 such that, for each x≥ x_2,J(1/x)-J(2/x) ≤ cf(x). For a proof of (<ref>) consider the interval I:=(J(2/x),J(1/x)]. Let J_σ⊂ I denote the subset of integers j in Iwith {l_jx}≥1/2, and J_κ⊂ I denote the set of those integers j in I with {l_jx} < 1/2. We write σ := #J_σ and κ := #J_κ for the corresponding numbers of elements. Then σ + κ = J(1/x)-J(2/x). On the one hand, we see thatδ(x) = ∑_j∈{l_jx}≥∑_j ∈ J_σ{l_jx}≥∑_j∈ J_σ1/2= 1/2σ.On the other hand,J(2/x) < j ≤ J(1/x) implies 2/x > l_j ≥1/x, or equivalently 2> l_jx≥ 1. Yet for j∈ J_κ, we have {l_jx}≤1/2, which together yields l_jx ∈ [1,3/2]. Hence {l_jx}≤{l_jx/2}∈ [1/2,3/4]. Therefore, we obtainδ(x/2) = ∑_j∈{l_jx/2}≥∑_j ∈ J_κ{l_jx}≥∑_j∈ J_κ1/2= 1/2κ.Combining inequalities (<ref>) and (<ref>), and taking into account (<ref>), we obtainJ(1/x)-J(2/x) = σ + κ≤ 2δ(x) + 2 δ(x/2)≤ 2a_2f(x) + 2 a_2 f(x/2).Since f∈_D[∞], the homogeneity property implies in particular that we can find positive constants c_2,x_2 (with c_2<1!) such thatf(x/2) <c_2 f(x),for all x≥ x_2. (Without loss of generality, we may assume that x_2≥ x_0.) Applying this to (<ref>)yieldsJ(1/x)-J(2/x)≤ 2a_2(1+c_2) f(x),for each x≥ x_2, proving the assertion (<ref>) for the constant c:=2a_2(1+c_2).Now we use (<ref>), to derive an upper bound for J. For x≥ 2x_2 let m = m(x) be the unique integer such that 2^m x_2< x ≤ 2^m+1 x_2.Note that, by the choice of m, we have 2^m/x≥1/2x_2, which implies J(2^m/x)≤ J(1/2x_2)=:j_0. Writing J(1/x) as a telescope sum and applying (<ref>), we infer J(1/x) = ∑_k=0^m-1( J(2^k/x) - J(2^k-1/x)) + J(2^m/x) ≤∑_k=0^m-1 c f(x/2^k) + j_0≤ c ∑_k=0^m-1c_2^k f(x) + j_0 ≤ c f(x) ∑_k=0^∞c_2^k + j_0 = c/1-c_2 f(x) + j_0,for any x≥ 2x_2, where the convergence of the geometric series is ensured, since c_2 <1. This gives an upper bound on J(1/x) in terms of f(x) and completes the proof. (Note that f(x)→∞ as x→∞.)To complete the proof of Theorem <ref>, we recall that there is a direct connection between the packing defect δ and the string counting function N valid for any bounded open set Ω⊂ independent of any additional assumptions on the growth behavior. It is given by the equationφ(λ)-N(λ)=δ(√(λ)/π), λ >0,cf. e.g. <cit.> and see also Remark <ref> below.It implies the equivalence (iv) ⇔ (v) in part I of Theorem <ref> as well as the equivalence of assertion (<ref>) in Theorem <ref> with assertion (<ref>) in part II of Theorem <ref>. Therefore, by Theorem <ref>, the assertions (vi)-(viii) imply indeed (<ref>). This completes the proof.Equation (<ref>) is seen as follows. For an open interval I=(a,b) of length l=b-a, consider the Laplace operator —d^2/dy^2. Under Dirichlet boundary conditions (i.e. u(a)=u(b)=0) the eigenvalues are λ_k=(π/l)^2 k^2, k∈. Hence, for the eigenvalue counting function N(I;λ) of I, we haveN(I;λ)=#{k∈: λ_k≤λ}=#{k∈:k≤ l√(λ)/π}=[l√(λ)/π].Hence, we get for the eigenvalue counting function N ofΩ (consisting of the disjoint open intervals I_j of lengths l_j, j∈)N(λ)=∑_j=1^∞ N(I_j;λ)=∑_j=1^∞ [l_j √(λ)/π]=∑_j=1^∞ [l_j x],where x:=√(λ)/π. Since |Ω|_1=∑_j=1^∞ l_j, it follows in particular thatφ(λ)-N(λ)=√(λ)/π∑_j=1^∞ l_j-∑_j=1^∞ [l_j x]=∑_j=1^∞ l_jx-∑_j=1^∞ [l_j x]=δ(x).The following diagram gives an overview over the various steps of the proof of part I of Theorem <ref>. It also shows the central role of the S-content (or the string counting function J) for the proof. (i)@<=>[r]^^ Thm <ref>(ii) @<=>[r]^^ Prop <ref> (<ref>)@<=[r]^^ Thm <ref>@/_/[r](iv) @<=>[r]^^(<ref>) (v) (iii) @<=>[u]_ Thm <ref>@-[rru]_ Thm <ref> Our proof of part II of Theorem <ref> has a similar structure: (vi)@<=>[r]^^ Thm <ref>(vii) @<=>[r]^^ Prop <ref> (<ref>)@/_/[r](<ref>)@<=>[r]^^(<ref>) (<ref>) (viii) @<=>[u]_ Thm <ref>@-[rru]_ Thm <ref> Relation (<ref>) between the contents and L is obvious from the definition of the constant S in Theorem <ref> (and Theorem <ref>). AcknowledgementsWe are grateful to Günter Last for pointing us to the theory of regularly varying functions. Part of the results are based on the master's thesis of the first author. abbrv | http://arxiv.org/abs/1703.09140v1 | {
"authors": [
"Tobias Eichinger",
"Steffen Winter"
],
"categories": [
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"primary_category": "math.MG",
"published": "20170327151546",
"title": "Regularly Varying Functions, Generalized contents, and the spectrum of fractal strings"
} |
#1 | http://arxiv.org/abs/1703.08823v1 | {
"authors": [
"Na Li",
"Quan-Lin Li",
"Zhe George Zhang"
],
"categories": [
"cs.PF",
"68M20, 90B22, 90B18",
"C.4; C.2.3"
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"published": "20170326151336",
"title": "Groups of Repairmen and Repair-based Load Balancing in Supermarket Models with Repairable Servers"
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APS/[email protected] School of Chemistry, The University of Sydney, Sydney 2006, Australia Australian Centre for Neutron Scattering, Australian Nuclear Science and Technology Organisation, Menai 2234, Australia Australian Centre for Neutron Scattering, Australian Nuclear Science and Technology Organisation, Menai 2234, Australia Neutron Group, National Synchrotron Radiation Research Center, Hsinchu 30077, Taiwan Physics Department and Centre for the Physics of Materials, McGill University, 3600 University Street, Montreal, Quebec, H3A 2T8, Canada Institut Laue Langevin, 71 avenue des Martyrs, Grenoble 38042, France School of Chemical Sciences, University of Auckland, Auckland 1142, New Zealand We have experimentally identified a new magnetic ground state for the kagomé lattice, in the perfectly hexagonal Fe^2+ (3d^6, S=2) compound Fe4Si2Sn7O16. Representational symmetry analysis of neutron diffraction data shows that below T_N=3.5 K, the spins on 2/3 of the magnetic ions order into canted antiferromagnetic chains, separated by the remaining 1/3 which are geometrically frustrated and show no long-range order down to at least T=0.1 K. Mößbauer spectroscopy confirms that there is no static order on the latter 1/3 of the magnetic ions – i.e., they are in a liquid-like rather than a frozen state – down to at least 1.65 K. A heavily Mn-doped sample Fe_1.45Mn_2.55Si_2Sn_7O_16 has the same magnetic structure. Although the propagation vector q=(0,1/2,1/2) breaks hexagonal symmetry, we see no evidence for magnetostriction in the form of a lattice distortion within the resolution of our data. We discuss the relationship to partially frustrated magnetic order on the pyrochlore lattice of Gd2Ti2O7, and to theoretical models that predict symmetry breaking ground states for perfect kagomé lattices. PACS numbers 75.10.Jm, 75.25.-j, 75.47.Lx, 75.50.Ee, 76.80.+yStriped Magnetic Ground State of the Kagomé Lattice in Fe4Si2Sn7O16 T. Söhnel December 30, 2023 ===================================================================Ever since the first published consideration of the ground state of a triangular lattice of Ising spins, <cit.> the pursuit of materials with geometrically frustrated magnetic (GFM) lattices has been an important driver in experimental condensed matter physics. <cit.> Perfect GFM lattices are proving grounds for a number of predicted exotic states of matter. The most famous of these is the quantum spin liquid (QSL), in which there is effectively no energy barrier between macroscopically degenerate ground states for S=1/2 spins, which can therefore continue to fluctuate down to T=0 K. <cit.> The simplest GFM case is a triangular lattice, followed by the expanded triangular network known as the kagomé lattice. Undistorted (perfectly hexagonal) magnetic kagomé lattices are rare – the most studied examples are naturally occurring minerals or synthetic versions thereof, notably the jarosites AB_3(SO4)2(OH)6 where B^3+ can be Fe^3+ (S=5/2), <cit.> Cr^3+ (S=3/2), <cit.> or V^3+ (S=1), <cit.> which generally still undergo Néel ordering due to Dzyaloshinkii-Moriya anisotropy. A series of quinternary oxalates studies by Lhotel et al., <cit.> which contain Fe^2+ (S=2) lattices equivalent to kagomés (for 1st neighbor interactions only), all freeze into a q=0 Néel state below ∼3.2 K. More recently, the Cu^2+ (S=1/2) undistorted kagomé compound herbertsmithite ZnCu3(OH)6Cl2 <cit.> has attracted a great deal of attention as arguably the most promising QSL candidate so far discovered (for a recent review see ref. <cit.>). Fe4Si2Sn7O16 <cit.> is a synthetic compound that incorporates an undistorted kagomé lattice of high-spin (HS) Fe^2+ (3d^6, S=2) magnetic ions on the 3f Wyckoff sites of its hexagonal (trigonal space group P3̅m1, #164) structure. The kagomé lattice is located in layers of edge-sharing FeO6 and SnO6 octahedra (hereafter called the oxide layer), which alternate with layers of oxygen-linked FeSn6 octahedra (the stannide layer). The layers are separated by SiO4 tetrahedra (see Fig. <ref>). A triangular lattice of low-spin (LS) Fe^2+ (3d^6, S = 0) on the 1a Wyckoff site in the stannide layer is magnetically inactive. We recently reported a long-range antiferromagnetic (AFM) Néel ordering transition in Fe4Si2Sn7O16 at T_N=3.0 K and its Mn-doped (in the oxide layer) analogue Fe_1.45Mn_2.55Si_2Sn_7O_16 at T_N=2.5 K. <cit.> There was no evidence of spin-glass behavior or a ferromagnetic (FM) component to the ground state. Given their perfectly hexagonal lattices above T_N, the ordered magnetic ground state was expected to be either the conventional q=0 or (√(3)×√(3)) solution, which preserve hexagonal symmetry. <cit.> In the work reported in this Letter, we set out to test this by collecting low-temperature neutron powder diffraction (NPD) and Mößbauer spectroscopy data above and below T_N. Contrary to expectation, we found that the ground state is a striped AFM structure in which 2/3 of the magnetic sites are completely ordered, while the other 1/3 are frustrated and remain completely disordered down to at least 0.1 K. To the best of our knowledge, this state has no precedent experimentally and has not been explicitly predicted theoretically.New magnetometry data collected for the present study were identical to those in ref. <cit.> apart from slightly revised values for Fe4Si2Sn7O16 of T_N=3.5 K, μ_eff=5.45 μ_B per HS Fe^2+ and θ=-12.7 K, which corresponds to a modest frustration index <cit.> f=|θ/T_N|=3.6. [See Figs. 1–3 of Supplemental Material at [URL will be inserted by publisher]] Note that the orbital angular momentum L is not completely quenched (μ_eff spin-only=4.90 μ_B), as is commonly observed in Fe^2+ oxides. High-resolution NPD data were collected on the instrument Echidna <cit.> at the OPAL reactor, Lucas Heights, Australia. Samples were placed in 6 mm diameter vanadium cans using neutrons of wavelength λ = 2.4395 Å, over the range 2.75-162^∘ 2θ with a step size of 0.125^∘ 2θ. Low-temperature data were collected to 1.6 K in a cryostat and 0.1 K in a dilution fridge for Fe4Si2Sn7O16, and to 1.8 K for Fe_1.45Mn_2.55Si_2Sn_7O_16. Rietveld-refinements of the nuclear structure above T_N were consistent with our previous work using single-crystal X-ray diffraction <cit.> and a combination of synchrotron X-ray and neutron powder diffraction.<cit.> New low-angle Bragg peaks emerge for Fe4Si2Sn7O16 at 1.6 K, below T_N, indicative of 3D long-range ordered magnetism (Fig. <ref>). The same peaks are observed at 0.1 K, i.e., the magnetic structure shows no further change down to at least this temperature; and at 1.8 K for Fe_1.45Mn_2.55Si_2Sn_7O_16, i.e., the magnetic structure is not fundamentally changed by almost complete substitution of Mn^2+ for Fe^2+. [See Fig. 2 of Supplemental Material at [URL will be inserted by publisher]] All peaks could be indexed to a propagation vector q=(0,1/2,1/2), which breaks hexagonal symmetry. However, we do not observe any peak splitting or broadening within the resolution of our NPD data, indicating that magnetostriction is very small. We solved the magnetic structure by representational symmetry analysis using the BasIreps routine in the program Fullprof. <cit.> The propagation vector q=(0,1/2,1/2) acting on the space group P3̅m1 splits the magnetic HS Fe^2+ on the 3f site in the oxide layer into two orbits, Fe-3f(1) and Fe-3f(2), in a 1:2 ratio. The irreducible representations (irreps) for each orbit decompose in terms of two 1D irreps for Fe-3f(1), Γ_mag=2Γ_2+Γ_4, and another two for Fe-3f(2), Γ_mag=3Γ_1+3Γ_3. The basis vectors are given in Table 1 of Supplementary Information. We first tested these four possible representations independently (i.e., with only the Fe-3f(1) or Fe-3f(2) site magnetically active) by Rietveld refinement against 1.6 K data using Fullprof. We found that the Γ_1 representation for Fe-3f(2) gave by far the best fit. We tried adding the Γ_2 and Γ_4 representations for Fe-3f(1) to the refinement, but in neither case was the fit improved, and the moments on Fe-3f(1) refined to zero within error. The component of the moment on Fe-3f(2) along the z axis also refined to zero within error. The final refinement, for which the fit is shown in Fig. <ref>, was therefore carried out using only the x and y axis components of the Γ_1 representation for Fe-3f(2). This structure is equivalent to the Shubnikov magnetic space group (Opechowski-Guccione setting, C_2c2/m #12.6.71).Fig. <ref> shows the final refined magnetic structure at 1.6 K, with an ordered moment of 2.52(6) μ_B (μ_x=0.44(6), μ_y=2.71(5) μ_B). The refinement at 0.1 K yieldedincreased moment of 3.2(2) μ_B (μ_x=0.3(2), μ_y=3.3(2) μ_B). The slight reduction compared to the total spin-only moment of 4 μ_B for HS Fe^2+ may be an effect of crystal field and spin-orbit coupling. The most striking feature is the absence of long-range magnetic order on the Fe-3f(1) sites, which sit on the kagomé legs between magnetically ordered rows of Fe-3f(2) sites along the x direction, despite all Fe-3f sites being HS Fe^2+ (d^6, S=2) at all temperatures. Note that although the spin orientation in the x-y plane refined robustly, NPD cannot distinguish between the model shown in Fig. <ref> and an alternative version in which the ordered rows of spins are shifted by 1/2a, to point approximately towards/away from the non-magnetically ordered Fe-3f(1) site rather than towards/away from the center of the hexagon. It is clear from Fig. <ref> that the Fe-3f(1) site is geometrically frustrated by its four nearest-neighbor Fe-3f(2) sites, regardless of the sign of magnetic exchange with those sites. This is consistent with the zero refined moment for Fe-3f(1), and the fact that because the propagation vector splits the Fe sites into two independent orbits, the molecular field created by one sublattice on the other one is zero, i.e., the 3f(1) magnetic moments can only couple to each other through J_d interactions. However, NPD cannot determine whether Fe-3f(1) is locally ordered but long-range disordered, or completely locally disordered. More importantly, it cannot distinguish between the model shown in Fig. <ref> and a multi-q structure with three arms of the propagation vector star q_1=(0,1/2,1/2), q_2=(1/2,0,1/2), q_3=(1/2,1/2,1/2). Such multi-q structures could preserve trigonal symmetry without requiring stripes of disordered Fe^2+ ions. The anisotropy of Fe^2+ could be an important parameter in the Hamiltonian to stabilize such a structure. We therefore tested all 1-, 2-, and 3-q symmetry allowed models, with two of the 3-q models (Shubnikov groups P_C2/m, #10.8.56 and P_2c3̅m1, #164.6.1320 in Opechowski-Guccione settings) giving comparable fits to the striped 1-q model discussed above. To resolve the single-q vs. multi-q question, we conducted a Mößbauer spectroscopy experiment. The Mößbauer spectra above and below T_N were obtained on a conventional spectrometer operated in sine-mode with both the sample and ^57CoRh source cooled by flowing He gas. The system was calibrated using α-Fe metal at room temperature and the spectra were fitted to a sum of Lorentzians with positions and intensities derived from a full solution to the nuclear Hamiltonian. <cit.> The spectrum above T_N at 5 K shown in Fig. <ref>, which is effectively identical to the data published in ref. <cit.>, shows that the subspectra from the Fe in the 1a and 3f sites are clearly resolved, being distinct in both spectral area (1:3 as expected from site multiplicities) and hyperfine parameters (as expected from the different spin configurations). The LS Fe^2+ on the 1a site gives an isomer shift δ=0.33(1) mm/s with a quadrupole splitting Δ=0.48(1) mm/s, while the HS Fe^2+ on the 3f site gives δ=1.19(1) mm/s and Δ = 2.41(1) mm/s. Cooling through T_N to 1.8 K leads to remarkably limited changes. The subspectrum from the Fe-1a site is completely unchanged. There is no evidence for magnetic splitting at this site and therefore no magnetic order, consistent with its LS d^6 (S = 0) configuration. The well-split doublet from the Fe-3f site also persists unchanged, but its intensity is greatly reduced to equal that of the 1a subspectrum. This non-magnetically ordered subspectrum therefore now reflects only 1/3 of HS Fe^2+ on the 3f site, consistent with the Fe-3f(1) site in our single-q striped model. The remaining contribution to the 1.8 K spectrum in Fig. <ref> comes from the 2/3 of HS Fe^2+ 3f sites that do order, consistent with Fe-3f(2) in our striped model. The Fe-3f(2) subspectrum can be fitted assuming the same values for δ and Δ as for Fe-3f(1), but with a small additional hyperfine magnetic field (B_hf). The magnetic component could not be fitted as a single subspectrum, and required further splitting into two equal (within error) area sub-components. This is very weak effect (the fields are only 7.5 T and 4.3 T) that does not change the main result. It may indicate that the moments on the two Fe-3f sub-sites do not make exactly the same angle with the electric field gradient axes in the kagomé plane, as a result of which they are not truly equivalent, producing some small difference in orbital contribution. Crucially, the splitting of the Mößbauer spectra is not consistent with any of the multi-q models, including the two mentioned above that fit our NPD fits as well as the striped model does. Finally, the observed hyperfine fields are remarkably small (7.5(3) T and 4.3(3) T), and further cooling to 1.65 K did not lead to a significant increase and so these values appear to be close to their T=0 limits. The simplest explanation for such small fields is highly dynamic spins, but the dynamic component would have to be very fast (>100 MHz) because we not see any line broadening due to slower dynamics.We note here that while the Mößbauer analysis fully and independently confirms the striped model from NPD in which the Fe-1a site does not order and only 2/3 of the Fe-3f site orders, it allows us to go further. For moments on a crystallographic site to contribute to a (Bragg) diffraction peak they must be long-range-ordered. However, for a non-zero hyperfine field to be observed in a Mößbauer spectrum, the moments need only be static on a time scale of ∼0.1 μs – they just need to have a non-zero time average over a relatively short time. Thus, our observation that B_hf is zero for 1/3 of the Fe-3f sites means that we can rule out any “frozen” random spin configuration that does not give magnetic Bragg peaks in NPD, as well as the multi-q state. There is no static order at 1/3 of these Fe-3f sites, at least down to 1.65 K. Furthermore, since there are no significant changes in either δ or Δ, we can also rule out changes in the electronic configuration of some or all of the Fe ions (e.g., a HS → LS transition making them non-magnetic).A further striking aspect of the striped state of Fe4Si2Sn7O16 is that it breaks hexagonal symmetry. Zorko et al. <cit.> recently presented experimental evidence for symmetry breaking in herbertsmithite, but this appears to be related to the presence of significant (5-8%) disorder on its otherwise “perfect” kagomé lattice, which our diffraction data rule out in the present case. A number of theoretical models predict symmetry breaking on S=1/2 kagomé lattices, notably the valence bond crystal (VBC) state <cit.> (with the help of magnetoelastic coupling) and the striped spin-liquid crystal state. <cit.> However, these models are based on the resonance valence bond (RVB) picture of paired-up S=1/2 spins, so their relevance to the present S=2 case is unclear. Similarly, a type of striped order was predicted by Ballou <cit.> in itinerant electron kagomé systems (the disordered sites being really non-magnetic in this case), but the mechanism should be different to the present insulating case. The magnetic structure of Fe4Si2Sn7O16 – and the fact that it is not altered by substituting Mn^2+ (HS d^5, S=5/2) for Fe^2+ (HS d^6, S=2) – are important new experimental observations against which to test theoretical models of the large-S kagomé lattice. In the present case, the fact that the angle between ordered spins along the x axis is very close to 120^∘ (the value at 0.1 K is 129^∘) is consistent with the magnetic moments being confined along the two-fold axis, which suggests that magnetocrystalline anisotropy may play an important role.Preliminary Hubbard-corrected density functional theory (DFT+U) calculations [We used the projector augmented wave method and GGA-PBE implemented in the Vienna ab initio simulations package (VASP), and considered different U_eff=(U-J) values to include electron correlation effects stemming from interactions in the Fe[3d] shell <cit.>. Electronic and magnetic states were stabilized for U_eff > 4 eV, with a self-consistent field energy convergence breakdown condition set to 10^-5 eV.] for the non-collinear striped ground state of Fe4Si2Sn7O16 reproduced the zero net moment on the Fe-3f(1) sites. However, the q=0 state in the same 1× 2× 2 supercell was still found to be energetically lower. Noting that even the definition of a “ground state” in this system is problematic, dedicated detailed theoretical treatments are clearly required. In this context we note that Iqbal et al. <cit.> and Gong et al. <cit.> recently treated the S=1/2 kagomé lattice using high-level renormalization group theory. They identified the dominant magnetic interaction as AFM exchange through the long diagonals of the hexagons, labeled J_d in Fig. <ref>; the ground state then depends on the balance between nearest-neighbor (J_1) and second-nearest-neighbor (J_2) exchange. For |J_1|<|J_2|, they obtain the “cuboc1” phase, which is consistent with the vertical components of the spin vectors as shown in Fig. <ref>; while for |J_1|>|J_2|, they obtain the “cuboc2” phase, which is consistent with the horizontal components. The experimental magnetic structure of Fe4Si2Sn7O16 can thus be described as a linear combination of cuboc1 and cuboc2. Although this solution was not explicitly predicted for |J_1|≈|J_2|, our experimental case has additional features, notably much bigger spins which make a VBC state highly unlikely, and the presence of ∼90^∘ Fe–O–Fe superexchange pathways in addition to Fe–Fe direct exchange. A comparably detailed theoretical study is beyond the scope of the present work, but may represent a productive way forward.Finally, we note an intriguing experimental comparison to Gd2Ti2O7. The topology of the magnetic Gd^3+ lattice in this pyrochlore-type compound can be described as four sets of inter-penetrating kagomé planes, the triangles of which meet to form tetrahedra. Below T_N=1.1 K, it adopts the partially ordered “1-k” structure, in which one of those four sets of kagomé planes (involving 3/4 of the spins) is q=0 long-range AFM ordered, while the remaining 1/4 of the spins between those planes remain frustrated. <cit.> If the spins in Fig. <ref> are collectively rotated about the x-axis, the magnetic structure of Fe4Si2Sn7O16 becomes equivalent to one of the three other kagomé planes in Gd2Ti2O7, which cut through the frustrated spin. Below T'=0.7 K, the frustrated spin in Gd2Ti2O7 may order weakly into the “4-k” structure, <cit.> although this has been disputed <cit.> and muon-spin relaxation (μSR) data show that fluctuations continue down to at least 20 mK. <cit.> Future neutron diffuse scattering and/or μSR experiments at dilution temperatures might therefore provide similar insights into Fe4Si2Sn7O16, and the reasons for which it adopts the partially ordered striped state in preference to the fully ordered q=0 one. The authors received financial support from the Australian Research Council (DP150102863), the School of Chemical Sciences, University of Auckland (FRDF Project 3704173), the Natural Sciences and Engineering Research Council of Canada, and the Fonds Québécois de la Recherche sur la Nature et les Technologies. | http://arxiv.org/abs/1703.08637v3 | {
"authors": [
"C. D. Ling",
"M. C. Allison",
"S. Schmid",
"M. Adveev",
"J. S. Gardner",
"C. -W. Wang",
"D. H. Ryan",
"M. Zbiri",
"T. Soehnel"
],
"categories": [
"cond-mat.str-el"
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"primary_category": "cond-mat.str-el",
"published": "20170325015422",
"title": "Striped Magnetic Ground State of the Kagome Lattice in Fe4Si2Sn7O16"
} |
Department of Physics, University of Tennessee, Knoxville, Tennessee 37996-1200, [email protected] quenching is observed at both RHIC and LHC energies.This suggeststhatpartons lose energy as they traverse the medium.When the trigger jet isstudied relative to the event plane, the path length dependence of mediummodifications can be studied.We present measurements of the angular correlations relative to theevent plane between reconstructed jets and charged hadrons in Pb–Pb collisions at√(s_NN)=2.76 TeV in ALICE.A newly implemented, robust background subtraction methodto remove the complex, flow dominated, combinatorial background is used in this analysis.§ INTRODUCTIONJets are ideal probes of the Quark Gluon Plasma (QGP) becausethey originate from hard-scattered partons created early in the collision, prior to the formationof the medium.These partons are modified in the presence of a medium through collisional energyloss and induced gluon radiation.This modification is observed at both LHC and RHIC energies viathe suppression of high-momentum particles <cit.>. A suppression is also seen for high p_T di-hadroncorrelations <cit.>.By reconstructing jets, ideally, a more complete picture of how lost energy is redistributed and howthe effects of jet quenching emerge can be obtained.This article will discuss thestatus of azimuthal correlations of reconstructed jets binned relative to the event planewith charged hadrons in ALICE. § EXPERIMENTClusters measured in the Electromagnetic Calorimeter (EMCal) and charged hadrons measured with thecentral tracking system allow ALICE to study fully reconstructed jets <cit.>. Charged hadrons are reconstructed from their tracks using information from the TimeProjection Chamber (TPC) and the Inner Tracking System (ITS).Tracks are reconstructed over the full azimuthal range and come from mid-rapidity in a range of|η_lab|<0.7 when reconstructed in a jet, and |η_lab|<0.9 for the associated charged hadrons. The EMCal has an acceptance window of |η_lab|<0.7and |Δϕ|=107.For a complete description of the ALICE detector, see<cit.>. § JET-HADRON CORRELATIONS§.§ Data sampleThe data used for the correlation analysis was collected by the ALICE Experiment in 2011 duringthe 2.76 TeV Pb–Pb collision data taking.This work uses events which fired the gamma trigger usedby the ALICE EMCal <cit.>.In addition, a cluster constituent of the trigger jet was requiredto be matched to the fired trigger patch of the event. §.§ Jet Reconstruction Jets are reconstructed with a resolution parameter of R=0.2 using the anti-k_T jet-findingalgorithm from the FastJet package <cit.>.Jets reconstructed in this analysisrequire a leading cluster to have E_T>6.0 GeV, which exceeds the trigger threshold. The jets are reconstructed with constituent tracks ofp_T>3.0 and clusters of E_T>3.0 GeV.Since the primary goal of jet findingis to reconstruct the jet axis accurately, this high constituent cut limits the influence ofbackground on jet finding. §.§ MeasurementWe define the jet-hadron correlation function in heavy-ion collisions by Eq. <ref>.1/N_trigd^2N_assoc,jet/dΔϕ dΔη = 1/ϵ a N_trigd^2 N^same_pairs/dΔϕ dΔη-b_0(1+ ∑ v^trig_n v^assoc_n cos(nΔϕ))Here the first term represents the same event pairs which are divided by an acceptancecorrection, a, provided by mixed events.Mixed events are usedto correct for the limited detector efficiency and acceptance for track pairs. The single track reconstruction efficiency of associated particles is denoted by ϵ. The second term of Eq. <ref> is the combinatorial heavy-ionbackground where b_0 is the background level and the v_n terms are the Fouriercoefficients of the trigger jet and associated particles. The trigger jets in this analysis are binned in angle relative to the event plane toexplore the path length dependence of medium modifications.The orientations are definedsuch that in-plane is 0 < | Δφ | < π/6, mid-planeis π/6 < | Δφ | < π/3, and out-of-planeis π/3 < | Δφ | < π/2, where Δφdenotes the angular difference between the trigger jet and the reconstructed event plane. §.§ Background subtractionWhen the trigger jet is restricted relative to the event plane, both the backgroundlevel and v_n are modified and will contain a dependence onthe reaction plane resolution .The derivation of the event plane dependent backgroundequations are given in <cit.>.The reaction planeresolution corrects for the differencebetween the reconstructed event plane and the underlying symmetry plane, Ψ_n.Various background subtraction methods were developed in<cit.> and applied to the azimuthal correlation functions.The primarymethod is the reaction plane fit (RPF).TheRPF method works under the assumption that the signal is negligiblein the large Δη and small Δϕ region.The 2D correlations are projected over 0.8<|Δη|<1.2to define the background dominated region, while the signal+background region is defined to be |Δη|<0.6. To ensure as much information as possible is going into the fit by constraining the shape and level of the background,the in-plane, mid-plane, and out-of-plane orientations are simultaneously fit up to fourth order in v_nand required to have the same fit parameters.The background dominated region is fit over the region |Δϕ|<π/2,shown in Fig. <ref>.We see from the blueband that the RPF fit models the data quite well, even at low p_T where the background is large. The higher order Fourier coefficents clearly matter, asthe background for trigger jets mid-plane has four peaks, consistent with a v_4 dependence. This method does not require independent measurements of v_n and is able to extract the signal with smallererrors while requiring fewer assumptions and less bias than prior subtraction methods.This is especiallyuseful since there are currently no v^jet_3 or v^jet_4 measurements.§.§ ResultsWe extract the signal by subtracting the large correlated background from the correlation function. Figure <ref> shows the signal for associated particles of 2.0-3.0 . The uncertainties are dominated by statistics.With higher statistics, the uncertaintiescould be vastly reduced to allow for a more precise measurement.We define the yields by Eq. <ref>.Y = 1/N_trig∫_c^d∫_a^bd(N_meas - N_bkgd)/dΔϕdΔϕ dΔη Where the integration limits a and b correspond to Δϕ values of -1.047 and +1.047 on thenear-side and +2.094 and +4.189 on the away-side respectively.In addition, the integration limits c and d correspondto Δη values of -0.6 and 0.6 for both the near-side and away-side. The near-side (left) and away-side (right) jet yields for 1.0<p^assoc_T<10.0 areshown in Fig. <ref>.The near-side yield is consistent with little or no modification. There is no clear dependence of the away-side peaks on orientation relative to the event plane. There are competing effects across different p_T ranges.Jet quenching could cause a decrease in yieldgoing from in-plane to out-of-plane, while gluon radiation could cause an increase. § SUMMARY AND OUTLOOKThe jet-hadron correlation results were seen to have their uncertainties dominated by statistics. No significant event plane dependence was seen to within the current uncertainties on the extractedjet yield.The RPF background subtraction method applied to data was seen to have various advantages in that it doesnot require independent v_n measurements, has less bias and fewer assumptions than prior backgroundsubtraction methods, and can extract the signal with great precision. The current results are consistent with the re-analysis of STAR data seen in <cit.>. § REFERENCES 10 url<#>1#1urlprefixURL Adare:2007vu Adare A et al. (PHENIX Collaboration) 2008 Phys.Rev. C77 011901 (Preprint 0705.3238)Adams:2003kv Adams J et al. (STAR Collaboration) 2003 Phys. Rev. Lett. 91 172302 (Preprint nucl-ex/0305015)Adare:2010mq Adare A et al. (PHENIX Collaboration) 2011 Phys.Rev. C84 024904 (Preprint 1010.1521)Adam:2015ewa Adam J et al. (ALICE Collaboration) 2015 Phys. Lett. B746 1–14 (Preprint 1502.01689)PhysRevLett.95.152301 Adams J et al. (STAR Collaboration) 2005 Phys. Rev. Lett. 95(15) 152301PhysRevC.80.064912 Abelev B I et al. (STAR Collaboration) 2009 Phys. Rev. C 80(6) 064912PhysRevC.85.014903 Agakishiev G et al. (STAR Collaboration) 2012 Phys. Rev. C 85(1) 014903PhysRevC.82.024912 Aggarwal M M et al. (STAR Collaboration) 2010 Phys. Rev. C 82(2) 024912Abelev:2014ffa Abelev B B et al. (ALICE Collaboration) 2014 Int. J. Mod. Phys. A29 1430044 (Preprint 1402.4476)Abeysekara:2010ze Abeysekara U et al. (ALICE EMCal Collaboration) 2010 (Preprint 1008.0413)Abelevetal:2014dna Abelev B et al. (ALICE Collaboration) 2014 J. Phys. G41 087002Alme2010316 Alme J et al. 2010 Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 622 316 – 367 ISSN 0168-9002Cacciari:2011ma Cacciari M, Salam G P and Soyez G 2012 Eur.Phys.J. C72 1896 (Preprint 1111.6097)Bielcikova:2003ku Bielcikova J, Esumi S, Filimonov K, Voloshin S and Wurm J 2004 Phys.Rev. C69 021901 (Preprint nucl-ex/0311007)Sharma:2015qra Sharma N, Mazer J, Stuart M and Nattrass C 2016 Phys. Rev. C93 044915 (Preprint 1509.04732)Nattrass:2016cln Nattrass C, Sharma N, Mazer J, Stuart M and Bejnood A 2016 Phys. Rev. C94 011901 (Preprint 1606.00677) | http://arxiv.org/abs/1703.09287v1 | {
"authors": [
"Joel Mazer"
],
"categories": [
"nucl-ex"
],
"primary_category": "nucl-ex",
"published": "20170327195333",
"title": "Jet-hadron correlations relative to the event plane at the LHC with ALICE"
} |
State Key Laboratory of Surface Physics, Department of Physics, and Laboratory of Advanced Materials, Fudan University, Shanghai 200433, People's Republic of ChinaState Key Laboratory of Surface Physics, Department of Physics, and Laboratory of Advanced Materials, Fudan University, Shanghai 200433, People's Republic of China Collaborative Innovation Center of Advanced Microstructures, Nanjing, 210093, China Materials Genome Institute, Shanghai University, Shanghai 200444, People's Republic of ChinaUVSOR Synchrotron Facility, Institute for Molecular Science, National Institutes of Natural Science, Myodaiji, Okazaki 444-8585, JapanUVSOR Synchrotron Facility, Institute for Molecular Science, Myodaiji, Okazaki 444-8585, Japan [email protected] State Key Laboratory of Surface Physics, Department of Physics, and Laboratory of Advanced Materials, Fudan University, Shanghai 200433, People's Republic of China Collaborative Innovation Center of Advanced Microstructures, Nanjing, 210093, China [email protected] State Key Laboratory of Surface Physics, Department of Physics, and Laboratory of Advanced Materials, Fudan University, Shanghai 200433, People's Republic of China Collaborative Innovation Center of Advanced Microstructures, Nanjing, 210093, China [email protected] Materials Genome Institute, Shanghai University, Shanghai 200444, People's Republic of ChinaHere we report the electronic structure of FeS, a recently identified iron-based superconductor. Ourhigh-resolution angle-resolved photoemission spectroscopy studies show two hole-like (α andβ) and two electron-like (η and δ) Fermi pockets around the Brillouin zone center and corner, respectively, all of which exhibit moderate dispersion along k_z. However, a third hole-like band (γ) is not observed, which is expected around the zone center from band calculations and is common in iron-based superconductors. Since this band has the highest renormalization factor and is known to be the most vulnerable to defects, its absence in our data is likely due to defect scattering — and yet superconductivity can exist without coherent quasiparticles in the γ band. This may help resolve the current controversy on the superconducting gap structure of FeS. Moreover, by comparing the β bandwidths of various iron chalcogenides, including FeS, FeSe_1-xS_x, FeSe, and FeSe_1-xTe_x, we find that the β bandwidth of FeS is the broadest. However, the band renormalization factor of FeS is still quite large, when compared with the band calculations, which indicates sizable electron correlations. This explains why the unconventional superconductivity can persist over such a broad range of isovalent substitution inFeSe_1-xTe_x and FeSe_1-xS_x.74.25.Jb, 74.70.-b, 79.60.-i, 71.20.-bElectronic structure of FeS Fei Chen December 30, 2023 ===========================§ I. INTRODUCTION Among all the iron-based superconductors, the iron-chalcogenide FeSe has the simplest layered structure yet extraordinarily rich physics <cit.>. It undergoes a structural transition around 90 K <cit.>, then becomes superconducting near 8 K. Under high pressure, its superconducting temperature (T_c)can be enhanced to 37 K <cit.>. When FeSe was intercalated with potassium by liquid gating <cit.> or dosed with potassium on the surface <cit.>,the T_c could reach 46 K. Most remarkably, FeSe monolayer film grown on SrTiO_3(001) substrate even exhibits T_c as high as 65 K <cit.>.Isovalent substitution is a particularly effective way of tuning the properties of FeSe. The T_c of FeSe_1-xTe_x can reach 14 K <cit.>, whilethat of FeSe_1-xS_x is first enhanced by dilute S substitution, then decreases upon further substitution <cit.>. Recently, it was found that FeS, in which Se is totally substituted by S, is stillsuperconducting with a T_c of about 4.5 K <cit.>. During this process, thenematic (orthorhombic) phase transition temperature(T_N) in FeSe is suppressed, andthere is no structural or nematic transition in FeS <cit.>.The presence of superconductivity in FeS is quite unexpected, since S substitution is expected to significantly reduce the electron correlations, which are crucial for the unconventional superconductivity <cit.>. For example, it has been shown that S substitution enhances the bandwidth, and thus reduces the correlation effects in K_xFe_2-ySe_2-zS_z, which is not superconducting for z≥ 1.4 <cit.>. To address this issue, it is important to examine the electronic structure of FeS. Moreover, the gap structure of FeS is currently under debate.Muon spin rotation (μSR) measurements favor the fully-gapped superconductivity coexisting with a low-moment magnetic state at low temperature in polycrystalline FeS <cit.>. On the contrary,thermal conductivity and specific heat measurements support the existence of a nodal gap <cit.>, while scanning tunneling spectroscopy indicates a highly anisotropic or nodalsuperconducting gap structure <cit.>. Furthermore, a theoretical calculation predicted d_x^2-y^2pairing symmetry in FeS <cit.>. A thorough understanding of the electronic structure of FeS will contribute to these discussions. In this paper, we study the electronic structure of single-crystalline FeS by performing high-resolution angle-resolved photoemission spectroscopy (ARPES). There are two electron-like bands around the Brillouin zone (BZ) corner and two hole-like bands (α and β) around the BZ center. The Fermi surface exhibits modest k_z variation. Moreover, we deduce that the absence of the third hole-like band with the d_xy orbital character, which was discovered in many other iron-based compounds <cit.>, is likely due to strongdefect scattering. Compared with density functional theory (DFT) calculations<cit.>, the α and β bands near the Fermi energy (E_F)are renormalized by factors of3 and 2.5, respectively, indicating sizable electron correlations. By studying the relation between T_c and the β bandwidth of variousisovalently substituted FeSe compounds, we find that the overall evolution of T_cis closely related to the strength of electron correlations. § II. EXPERIMENT AND RESULTS So far, FeS single crystals could not be synthesized from elemental Fe and S as for FeTe and FeSe <cit.>.Instead, the FeS single crystals used in our studies were synthesized by de-intercalation of K from K_0.8Fe_1.6S_2 precursor with a hydrothermal method <cit.>. Since the starting material is K_0.8Fe_1.6S_2, whose Fe vacancy concentration is known to be quite high, the hydrothermal de-intercalation will likely make samples with Fe_1-xS_1-y. Here, the x and y should be quiteclose, since the Fe:S ratio is almost 1:1 within the experimental error ±1 % according to the chemical analysis <cit.>. Moreover, our samples exhibit quite high residual resistivity ratios (RRR) of about 40 <cit.>, which is better than the typical RRR of FeCh (Ch = Te, Se, S) and close to the one as reported in Refs. <cit.>. Therefore,the x and y should be quitesmall. This is also consistent with the sizableT_c of the sample,since a few percent of defects at the Fe sites would normally kill the superconductivity. Taken FeSe as an example, only 2 % Cu doping on the Fe site would fully suppress the superconductivity <cit.>. Therefore for simplicity, we still use the nominal chemical formula of FeS hereafter. The photoemission data were collected with a Scienta R4000 electronanalyzer at Beamline 5-4 of the Stanford Synchrotron Radiation Laboratory (SSRL) and an MBS A-1 analyzer at Beamline 7U of the ultraviolet synchrotron orbital radiation facility (UVSOR). Both beamlines are equipped with an elliptically-polarized undulator which can switch the photon polarization among horizontal, vertical, and circular modes. The overall energy resolution is set to be 15 meV or better, and the typical angular resolution is 0.3 degrees. The samples were all cleaved in situ and measured around 10 K under ultra-high vacuum better than 3×10^-11 torr. Sample aging effects were carefully monitored to ensure they did not cause any artifacts in our analyses and conclusions.The photoemission intensity distribution of FeS around E_F is shown in Fig. 1(a). The spectral weight is mainly located around the BZ center and corner. To clearly resolve the band structure near E_F, we measured along the two cuts #1 and #2 crossing the BZ center and corner with both s- and p-polarized photons as shown in Fig. 1(b), and these two polarizations correspond to odd and even symmetries with respect to the mirror plane defined by the incoming light and outgoing photoelectron, respectively. According to the momentum distribution curves (MDCs) and the energy distribution curves (EDCs) in Figs. 1(c) and 1(d), two hole-like bands (α and β) are located around the BZ center, and two electron-like bands (η and δ) are located around the BZ corner. Based on the polarization of incident photons, α and η exhibit even symmetry, while β and δ exhibit odd symmetry. Based on the identified band dispersions, two circular hole Fermi pockets around the BZ centerare determined as shown in the lower panel of Fig. 1(a). Fermi crossings of the η and δ bands are too close to be distinguished, but they would contribute to two nearly degenerate electron Fermi pocketsaround the BZ corner that are mutually orthogonal, as predicted by the band calculations on FeS <cit.>.Since the electronic structures of iron-chalcogenides usually contain three hole-like bands around the BZ center <cit.>, the so-called γ band having d_xy orbital symmetry is missing here. Among the five bands near E_F in the iron-based superconductors,the γ band is by far the most sensitive to defects <cit.>, and its renormalization factor in FeSe is around 9 <cit.>, which is much higher than those of the other bands. With increased Te or Co concentrations, the γ bands ofFeSe_1-xTe_x, NaFe_1-xCo_xAs, and LiFe_1-xCo_xAs all quickly broaden, indicating the γ band becomes more incoherent, and even become featurelessin some cases<cit.>. For sulfur-substituted FeSe, γ `disappears' for sulfur content higher than 15 % <cit.>, i.e, its intensitysubmerges into the background, and becomes undetectable.Furthermore, the defects oniron sites would cause even stronger scatterings than those on pnictogen/chalcogen sites. As the hydrothermal de-intercalation process in thesynthesis mightretain some of the Fe vacancies in the FeS layer,the `absence' of the γ band is highly likely caused by the scattering of the defects in both Fe and S sites. They could severelybroaden the γ band, so that no feature could be resolved.A rather flat band around 170 meV below E_F is observed around the BZ center taken with s-polarized photons in Fig. 1(b). This flat band in iron-based superconductors usually exhibits d_z^2 orbital character and should be observed with p-polarized photons <cit.>. The observation of this flat band here is due to the particular experimental geometry at the Beamline 5-4 end-station at SSRL, where s polarized photons contain a polarization component along the z axis <cit.>. To illustrate the k_z dependence of the Fermi crossings, Fig. 2(a) plots the MDCs along the Γ-M direction for these four bands near E_F, taken with both s- and p-polarized light and many different photon energies. The Fermi surface cross-section in the Γ-Z-A-M plane can be traced by the peak positions of the MDCs [Fig. 2(b)]. The Fermi momentum (k_F) of α first shrinks and then expands along the Γ-Z direction,in contrast to that of the β band. Meanwhile, the k_Fs of both η and δ shrink along the M-A direction, and they almost always coincide with each other within the experimental resolution. In general, the Fermi surface of FeS exhibits a quasi-two-dimensional behavior, consistent with quantum oscillation data<cit.>. Compared with BaFe_2(As_0.7P_0.3)_2 and Ba_1-xK_xFe_2As_2, FeS is more two dimensional<cit.>. Based on Luttinger's theorem, we estimate the electron concentrationto be 0.12 electrons per unit cell through the volume of the three-dimensional Fermi surface of the four detected bands <cit.>. Since FeS should have electron-hole balanced Fermi surface volumes like FeSe <cit.> and the compositions ratio of Fe : S here is almost 1:1, the electron and hole pockets should normally have the same total volume. Therefore, there should be a missing Fermi pocket with 0.12 holes, which can be naturally attributed to the γ band.As shown in Fig. 1(c), the dispersion of β exhibits parabolic behavior. By fitting its dispersion below E_F by a quadratic curve [Fig. 3(a)], we estimate the position of the β band top to beabout 10 meV above E_F, and its effective band mass to be -1.74 m_e (m_e is the free electron mass). Meanwhile, the β band bottom overlaps with the ϵ band as shown in both photoemission intensity distribution [Fig. 3(a)] and the EDCs [Fig. 3(b)]. The overlap occurs around 180 meV below E_F. If we take the full width at half maximum of the EDC peaks of ϵ as the error bar, and take the overlapping region as the β bandbottom, the β bandwidth is about 190±25 meV.Similarly, we estimate the effective band mass of the α band is about -1.47 m_e.Fig. 3(c) reproducesthe band calculationsof FeS from Ref.<cit.>. Althoughthere aresubtle differences, the observed band dispersions qualitatively agree with the calculations, for example, α and β are almost degenerate around the zone center in both the data and the calculations. By comparing theband masses obtained in both the data and the calculations, we obtain a renormalization factor of about 2.5 for the β band, and 3 for α, which indicate sizeble electronic correlations in FeS.To position FeS in the bigger picture of the so-called`11' series of bulk iron chalcogenide superconductors,photoemission intensity distributions of FeSe_1-xTe_x and FeSe_1-xS_x along the Γ-M direction and their corresponding EDCs for various substitutions are collected in Fig. 4.All the data here weretaken with s-polarized photons, to emphasize β band.Since the β bandwidth can be easily estimated, it is taken as a characterization of the correlations in these materials. Figure 4 shows that the bandwidth of β increases monotonically from the tellurium endin FeSe_1-xTe_x to the sulfur end ofFeSe_1-xS_x, as expected from the decreasing bond length<cit.>.Consistent with the evolution of the bandwidth, the effective mass of β near the BZ center, obtained by fitting the parabolic curves in Fig. 4(a), decreases monotonically from the tellurium endin FeSe_1-xTe_x to the sulfur end ofFeSe_1-xS_x.In Fig. 5, the β bandwidths are plotted onto the phase diagram of FeSe_1-xTe_x and FeSe_1-xS_x. Although more data points are required to make a comprehensive case, the existing data fall on a line, as a function of substitution. From the FeTe end, bulk superconductivity emerges whenthe bandwidth exceeds a certain value, and persists all the way to the FeS end. The T_c of FeSe_1-xTe_x and FeSe_1-xS_x is enhanced at first, and then generally weakened with increased bandwidth. However, the relation between T_c and substitutionis not monotonic. The superconductivity strengthens in the lightly S-substituted regime, likely due to the enhanced (π,0) spin fluctuations related to the nematic order or to subtle Fermi surface topology effects <cit.>. § III. DISCUSSIONS It has been shown for almost all the iron-based superconductors that their phase diagrams can be understood from the bandwidth perspective <cit.>. In particular, since isovalent substitution does not alter carrier density or the Fermi surface <cit.>, the strength of electron correlations, which can be represented by the inverse of bandwidth, would control the superconductivity of FeSe_1-xTe_x and FeSe_1-xS_x. Themeasured β bandwidth of FeS is about 40% larger than that of FeSe, which suggestsweaker correlation effects in FeS than in FeSe. On the other hand, the sizable renormalization factors of FeS bands indicate that the electrons in FeS still experience significant interactions amongst themselves and with bosonic excitations, such as magnons and phonons. This might explain the robustness of superconductivity in FeSe_1-xS_x. This is reminiscent of Ba_1-xK_xFe_2As_2, in which strong correlations exist throughout the entire doping range, allowing superconductivity to persist <cit.>.It is also worthwhile to point out thatin both FeSe_1-xS_x and K_xFe_2Se_2-yS_y, T_c decreases with S substitution, although these two families have different Fermi surface topologies. The superconducting rangeof β bandwidth is 100∼200 meV for Rb_xFe_2Se_2-zTe_zandK_xFe_2Se_2-yS_y <cit.>. Superconductivity in FeSe_1-xTe_x and FeSe_1-xS_x also emerges for a bandwidth of ∼100 meV, but isstillnot suppressed at 190 meV (FeS end member).Our data clearly imply that the superconductivity in FeS can survive without coherent quasiparticles in the γ band. However, the incoherent spectral weight of γ mightstill contribute to the zero-energy excitations at low temperatures. This could explainthe nodal-gap-like behavior observed in thermal conductivity and specific heatmeasurements<cit.>. On the other hand,because the incoherent γ band would not contribute to the superfluid response, if the superconducting gaps in the other bands are nodeless,μSR could still observe an overall nodeless-gap behavior for FeS <cit.>. Our resultsmay thus help resolve the contradictory reports by these techniques on the FeS gap structure. § IV. CONCLUSIONS In summary, we have studied the electronic structure of superconducting FeS. Two hole-like bands and two electron-like bands around the BZ center and corner, respectively, have been resolved. The third hole-like band near Γ(the γ band)may be too strongly scattered by defects to be observed, which may help resolve the current debate on the gap structure. The k_z dispersions of the Fermi surfaces of FeS exhibit quasi-two-dimensional behavior, and the two electron-like bands are almost degenerate around E_F within our resolution. Using the β bandwidth as an indication of correlation strength, we illustrate the evolution of T_c with electron correlation in FeSe_1-xTe_x and FeSe_1-xS_x, and explain the robustness of superconductivity which still exists in the end member FeS. The observed electronic structure of FeS establishes a concrete foundation for further theoretical calculations and will help understand its superconducting properties.§ ACKNOWLEDGMENTS The authors thank Dr. D. H. Lu for the experimental assistance at SSRL. 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Pham, L. Spinu, Y. Qiu, V. Thampy, A. T. Savici, J. A. Rodriguez, and C. Broholm, Nat. Mater. 9, 718 (2010). PetrovicTP A. F. Wang, L. J. Wu, V. N. Ivanovski, J. B. Warren, J. J. Tian, Y. M. Zhu, and C. Petrovic, Phys. Rev. B 94, 094506 (2016). | http://arxiv.org/abs/1703.08682v1 | {
"authors": [
"J. Miao",
"X. H. Niu",
"D. F. Xu",
"Q. Yao",
"Q. Y. Chen",
"T. P. Ying",
"S. Y. Li",
"Y. F. Fang",
"J. C. Zhang",
"S. Ideta",
"K. Tanaka",
"B. P. Xie",
"D. L. Feng",
"Fei Chen"
],
"categories": [
"cond-mat.supr-con"
],
"primary_category": "cond-mat.supr-con",
"published": "20170325122141",
"title": "Electronic structure of FeS"
} |
École polytechnique fédérale de Lausanne The weighted stable matching problem Linda FarczadiThis research was supported by the The Swiss National Science Foundation, with grant ref. FNS 513835, Natália Guričanová December 30, 2023 ============================================================================================================================================ We study the stable matching problem in non-bipartite graphs with incomplete but strict preference lists, where the edges have weights and the goal is to compute a stable matching of minimum or maximum weight. This problem is known to be NP-hard in general. Our contribution is two fold: a polyhedral characterization and an approximation algorithm. Previously Chen et al. have shown that the stable matching polytope is integral if and only if the subgraph obtained after running phase one of Irving's algorithm is bipartite. We improve upon this result by showing that there are instances where this subgraph might not be bipartite but one can further eliminate some edges and arrive at a bipartite subgraph. Our elimination procedure ensures that the set of stable matchings remains the same, and thus the stable matching polytope of the final subgraph contains the incidence vectors of all stable matchings of our original graph. This allows us to characterize a larger class of instances for which the weighted stable matching problem is polynomial-time solvable. We also show that our edge elimination procedure is best possible, meaning that if the subgraph we arrive at is not bipartite, then there is no bipartite subgraph that has the same set of stable matchings as the original graph. We complement these results with a 2-approximation algorithm for the minimum weight stable matching problem for instances where each agent has at most two possible partners in any stable matching. This is the first approximation result for any class of instances withgeneral weights. § INTRODUCTION An instance of the Stable Matching Problem (SMP) is a pair G, where G = (V,E) is a graph and ≺= {≺_v}_v ∈ V is a set of preference lists, where for each v ∈ V, ≺_v is a strict linear order on δ(v), the neighbors of v in G. The vertices of G represent the set of agents and the edges correspond to mutually acceptable pairs. If an agent v ∈ V has two neighbors u,w in G such that uv w then we say that v prefers u over w.Let M ⊆ E be a matching of G. For each vertex v ∈ V, we denote M(v) its partner in M, or let M(v) = v if v is unmatched in M. The edge uv ∈ E is a blocking edge for M if u is unmatched or prefers v to its current partner and if, at the same time, v is unmatched or prefers u to its current partner. A matching M is stable if no edge e ∈ E is a blocking edge for M. In the Weighted Stable Matching Problem (wSMP), in addition to G, a weight w(e) ≥ 0 is assigned to each edge e ∈ E and we wish to find a stable matching M of maximum or minimum weight. When G is bipartite, we obtain the well-studied Stable Marriage Problem first introduced by Gale and Shapley <cit.>. In their seminal work, Gale and Shapley showed that every instance of SMP admits a solution and such a solution can be computed efficiently using the so-called deferred acceptance algorithm. In fact, both SMP and wSMP are solvable in polynomial time. In particular, there exists a system of linear inequalities, known as the stable matching polytope, that describes the convex hull of the incidence vectors of stable matchings. Since its introduction, the stable marriage problem has become one of the most popular combinatorial problems with several books being dedicated to its study <cit.>, <cit.> and more recently <cit.>.The popularity of this model arises not only from its nice theoretical properties but also from its many applications. In particular, a wide array of allocation problems from many diverse fields can be analysed within its context. Some well known examples include the labour market for medical interns, auction markets, the college admissions market and the organ donor-recipient pair market <cit.>. If G is non-bipartite we have what is known as the Stable Roommates Problem. This problem was also proposed by Gale and Shapley <cit.>, however its properties are quite different from the bipartite case. To begin with, not every instance admits a stable matching. Irving provided a polynomial time algorithm that either finds a stable matching or reports that none exist. Unlike in the bipartite case, the stable matching polytope is no longer integral, and the wSMP problem becomes NP-hard in general.§.§ Our contribution and results.We study instances where G is non-bipartite. Our main result is to characterize a new and larger class of instances for which the wSMP can be solved in polynomial time. We say that an instance G is bipartite reducible if there exists a subgraph H of G such that H is bipartite and has the same set of stable matchings as G. Our first result is polynomial time edge-elimination procedure that either finds such a bipartite subgraph or determines that the original instance is not bipartite reducible. There exists a polynomial time algorithm that either finds abipartite subgraph H of G whose set of stable matchings is the same as that of G, or determines that no such subgraph exists.Since the stable matching polytope is integral for bipartite graphs, this implies that we can optimize a linear function over the set of stable matchings of any bipartite reducible instance. Previously, this was known only for instances where the subgraph obtained after running phase one of Irving's algorithm is bipartite. We show that our result is a strict generalization. wSMP is polynomially solvable for all instances G that are bipartite reducible. Moreover, the class of bipartite reducible instances is a strict superset of the class of instances for which phase one of Irving's produces a bipartite graph. We then consider approximation algorithms for the minimum weight stable matching problem. Previous research on this topic has focused on special classes of weight functions such as the egalitarian stable matching and U-shaped weights. However, no approximation algorithm was previously known for any class of instances under general weights. We provide a first result of this kind by considering instances where each agent is matched to one of two possible partners in any stable matchings.There exists a 2-approximation algorithm for the minimum-weight stable matching problem for instances where each agent has at most two possible partners in any stable matching. §.§ Related work. Gale and Shapley <cit.> showed that a stable matching always exists if G is bipartite, and gave a polynomial time algorithm known as the deferred acceptance algorithm for finding such a matching. They also observed that if G is non-bipartite, a stable matching does not always exist. Irving <cit.> gave the first polynomial-time algorithm that either finds a stable matching when G is non-bipartite, or determines that no such matching exists. His algorithm was originally for instances where the underlying graph G is the complete graph, however it can be easily generalized for any graph G.Vande Vate <cit.> first characterized incidence vectors of perfect stable matchings as vertices of a certain polytope in the case when G is a complete bipartite graph, thus showing that wSMP is polynomial-time solvable in this case. Later, Rothblum <cit.> extended this result to general bipartite graphs. Feder <cit.> showed that wSMP is NP-hard in general. A particular case of weight function is the egalitarian weight function, where for each edge e = uv its weight w(e) is given by the sum of ranks of this particular edge in its endpoints' preference lists. Feder <cit.>,<cit.> gave a 2-approximation for finding a minimum-weight stable matching with egalitarian weights and showed that there exists an α-approximation for minimum-weight SMP if and only if there exists an α-approximation for the Minimum Vertex Cover. Later, it was showed that, assuming the Unique Games Conjecture, Minimum Vertex Cover cannot be approximated within 2 - ϵ for any ϵ > 0 <cit.>. Teo and Sethuraman <cit.> constructed a 2-aproximation algorithm for the minimum-weight SMP for a larger class of weight functions than the egalitarian weight functions, namely weight functions where w(uv) = w_u(v) + w_v(u) for each uv ∈ E, and the functions {w_u} satisfy the so-called U-shape condition. Recently, Cseh et al. <cit.> considered a class of minimum-weight SMP problems with egalitarian weights where all preference lists are of length at most d. For d = 2, they gave a polynomial time algorithm for solving this problem while they showed that it is NP-hard even for d = 3. Moreover, for d ∈{3,4,5}, they gave a 2d + 3/7-approximation algorithm for minimum-weight SMP with egalitarian weights, improving the results of Feder. § PRELIMINARIES Let G be an instance of SMP. If e = uv and f = wv are two edges in E with uv w, we can write this equivalently as ev f, and we say that e dominates f at v. For each e ∈ E, let ϕ(e) be the set containing e and all the edges that dominate e at one of the endpoints. That is,ϕ(e) = {f ∈ E: ∃v ∈ V: fv e}∪{e}Note that under this definition, a matching is stable if and only if |M ∩ϕ(e)| ≥ 1 for each e ∈ E. For each u,v s.t. uv ∈ E, we let rank_u(v) be the rank of v in u's preference list. We say that a matching M is perfect if every agent is matched in M; that is M(v) ≠ v, for all v ∈ V. For each subgraph K of G, we let ≺^K be the restriction of ≺ to K. For simplicity, we will denote the SMP instance (K, ≺^K) by (K, ≺). For each v ∈ V, we denote the most and least preferred partner of v in K by f_K(v) and l_K(v), respectively. We denote E(K) the set of edges of K and V(K) the set of vertices of K. For an instance G of SMP, we let the Stable matching polytope SMG be the convex hull of incidence vectors of its stable matchings. If x = χ^M is an incidence vector of a stable matching M, it satisfies the following inequalities: (i) matching: x(δ(v)) ≤ 1∀ v ∈ V (ii) stability: x(ϕ(e)) ≥ 1∀ e ∈ E (iii) non-negativity: x ≥ 0 For an instance G of SMP the Fractional stable matching polytope is given by FSMG = { x ∈^E : [x(δ(v)) ≤ 1∀ v ∈ V; x (ϕ(e)) ≥ 1∀ e ∈ E;x ≥ 0]}, The Fractional stable matching polytope was first studied by Abeledo and Rothblum <cit.>, who proved that the incidence vectors of stable matchings are precisely the integral vertices of FSMG. They also showed that this polytope is integral if G is bipartite, and that all the vertices of FSMG are half-integral in general, meaning that x_e ∈{0,1/2,1} for each e ∈ E for each vertex x. Irving <cit.> gave a two-phase polynomial-time algorithm for finding a stable matching of an instance G or determining that no such matching exists. Phase one of this algorithm consists of a proposal sequence that results in a unique subgraph of G that has the same set of stable matchings as G. For an instance G of SMP let G_I be the subgraph of G obtained by applying phase one of Irving's algorithm to G. Chen et al. <cit.> show that FSMG is integral if and only if the graph G_I is bipartite.Note that, for bipartite graphs G, the integrality of FSMG follows easily from this result. The following are equivalent. * FSMG is integral. * G_I is a bipartite graph. We will use the following well known property of stable matchings, which says that whenever an edge is in the support of some fractional stable matching then the corresponding stability constraint for this edge is satisfied with equality. Let x ∈ FSMG. Then for any e ∈ E, x_e > 0⇒ x(ϕ(e)) = 1. In the same paper, Albedo and Rothblum show that the set of matched vertices is the same for every stable matching M of G. Given G, the vertex set V can be partitioned into two sets V^0 and V^1 such that V^0 = {v ∈ V: vis not matched in any stable matchingMofG}V^1 = {v ∈ V: vis matched in every stable matchingMofG}. Moreover, for each j = 0,1 and v ∈ V^j, it holds that x(δ(v)) = j, ∀ x ∈ FSMG. For our purposes, it will be easier to work with instance where stable matchings are perfect, meaning that V_0 = ∅. We show that we can focus on such instances without any loss of generality. First, we can use Irving's algorithm to decide if stable matchings exists, and if so, find the bipartition of the agents into V_0 and V_1 as in Lemma <ref>. Now, consider the subgraph G[V_1] of G. Clearly, any stable matching of G is a stable matching of G[V_1]. Moreover, a stable matching of G[V_1] will be a stable matching of G as long as none of the edges with an endpoint in V_0 are blocking, meaning that any vertex u which has a neighbor w ∈ V_0 is matched to someone they prefer over w. We thus obtain the following observation. Given G, let V^0 and V^1 be as in Lemma <ref>. Let E^1 = { uv ∈ E:u,v∈ V^1 ∀w ∈ V^0,uw ∉Eorvu w ∀w ∈ V^0,vw ∉Eoruv w } andG^1 = (V^1,E^1). Then M is a stable matching of G if and only if it is a perfect stable matching of G^1. Observe that, by Lemma <ref>, if an instance G has a perfect stable matching, then all stable matchings of G are perfect. Thus, from now on, we consider without loss of generality only instances which have at least one stable matching, and all the stable matchings are perfect. In our analysis we will rely on a useful characterization of the vertices of FSMGintroduced by Chen et al. <cit.>. This characterization is based on the notion of semi-stable partitions, which we define below. A cycle C=v_0v_1 ⋯ v_k-1 v_0 of G is said to have a cyclic preference if v_i-1≺_v_i v_i+1 for all 0 ≤ i ≤ k-1 or v_i+1≺_v_i v_i-1 for all 0 ≤ i ≤ k-1, where all indices are taken modulo k. Given G let 𝒞 = {C_1,…,C_k} be a set where * each C_i is an edge of G or a cycle in G with cyclic preferences in G, * each vertex v ∈ V is contained in precisely one C_i, and let x_𝒞∈^E be defined as (x_𝒞)_e = 1 ifeis an edge in 𝒞 1/2ifeis a part of a cycleCin 𝒞 0 otherwiseThen𝒞 is a semi-stable partition if x_𝒞∈ G. In other words, for 𝒞 to be a semi-stable partition, 𝒞 must be a collection of edges and cycles with cyclic preferences, such that the point x assigning value 1 to the single edges and 1/2 to the edges on the cycles is a feasible point of FSMG. Chen et al. <cit.> show that if x is a vertex of FSMG, then x = x_𝒞 for some semi-stable partition 𝒞 . § IDENTIFYING REDUNDANT EDGES Chen et al. <cit.> prove that the instances G, for which the polytope FSMG is integral, are precisely those, for which the graph G_I obtained from phase one of Irving's algorithm is bipartite. Thus, the polytope FSMG can be used to solve the weighted stable matching problem only for this class of instances. The edges deleted in phase one are, in a sense, redundant for the instance. By this we mean that none of these edges appear in any stable matching of G, and also by deleting them, the set of stable matchings of the instance stays unchanged, i.e. G and G_I have the same set of stable matchings. However, it could still be the case that G_I contains edges that never appear in any stable matching of G. This suggests the one could improve upon phase one of Irving's algorithm by further searching for edges that never appear in a stable matching and whose removal preserves the set of stable matchings. In general, when removing an edge e from a graph G that is never a part of a stable matching, the set of stable matchings either stays the same, or gets larger. All the original stable matchings of G are still stable matchings of (G ∖ e,≺); however, it can happen that a matching M' is stable in (G ∖ e,≺), but is not stable in G, because e is a blocking edge for M' in G. Thus our goal is to identify edges e that are not in any stable matching, and whose removal does not introduce any extra stable matchings. As a motivation for this approach, consider the following example. Let the preferences be as follows, where each line v|v_1 … v_k means v_1v … v v_k 1 3 4 5 2 2 1 4 3 5 6 3 5 6 1 2 4 5 6 1 2 5 1 2 3 4 6 2 3 4 During phase one, only edges 12, 23 and 45 get eliminated. Figure <ref> depicts the original graph G on the left and the graph G_I in the middle. The red numbers represent the ranks of the edges by their respective endpoints.It is easy to check that FSM G = conv{x,y}, where x_e=1 e ∈{ 14, 25, 36 }0 otherwise andy_e = 1/2e ∈{ 13, 35, 15, 24, 46, 26 }0 otherwise The edge 12 is never a part of any stable matching, and, at the same time, the worst (remaining) option for agent 1. By removing this edge we don't introduce any new stable matchings, since any stable matching M in G ∖{12} is perfect, and 1 has a strictly better partner in M than 2, thus M is stable in G, too. By repeating the above argument, we can keep reducing our graph by eliminating edges 12, 26, 32, 42, 54, 64, 15, 31 and 53 in turn, while always preserving the same set of stable matchings. The resulting graph H consists of precisely the three edges 14, 25, 36 that form a stable matching - the only stable matching of the original instance G.This example illustrates that there are cases where even though G_I is not bipartite and FSMG is not integral, one can still identify redundant edges in order to arrive at a bipartite subgraph, whose set of stable matchings is the same as the original instance meaning that wSMP can be solved in polynomial time. Our goal is to develop a polynomial-time algorithm that finds such a subgraph whenever it exists. We start by showing that it is possible to identify the set of edges that appear in a stable matching of G in polynomial time using Irving's algorithm. Given G, letE_M = {e ∈ E: ∃ stable matchingMs.t.e ∈ M}be the set of edges of G that ever appear in a stable matching of G. Further, let H = (V, E_M) be the graph consisting only of the edges in E_M. The graph H is a subgraph of the graph G_I, and it shares the following useful property with G_I. Let G be given anduv ∈ E_M. Then u = f_H(v) ⇔ v = l_H (u). We remark however, that while (G_I, ≺) keeps the same stable matchings as (G, ≺), the set of stable matchings of (H, ≺) can be a strict superset of the set of stable matchings of (G, ≺). The following lemma shows that we can decide in polynomial time whether a given edge e ∈ E belongs to the set E_M. Given an instance G and an edge uv ∈ G, there exists a stable matching containing uv if, and only if, G' = (V', E') contains a stable matching, where V'=V ∖{u,v} E_u= {wy ∈ G[V']: uw ∈ E, w u v, uw y } E_v= {wy ∈ G[V']: vw ∈ E, wv u, vw y } E'=E(G[V']) ∖ E_u ∖ E_v. In particular, this implies that the set of edges that are part of a stable matching can be identified in polynomial time. [ A more general version of the problem of deciding whether a given set of edges is contained in a stable matching was considered by Dias et al <cit.>.] As already mentioned, by removing an edge e ∈ E ∖ E_M from a graph G, we can introduce new stable matchings. The following lemma tells us that we can check whether this is the case or not. Given G, let E_M be defined as above and let e ∈ E ∖ E_M. Then it can be checked in polynomial time whether the set of stable matchings of G and the set of stable matchings of G ∖ e are the same. To see why Lemma <ref> is true, observe that for a matching M' to be stable in G ∖{uv}, but not in G,both u and v must prefer each other over their respective partners in M'. For each such an assignment of partners for u and v, we can check whether it can be extended to a perfect stable matching in G ∖{uv} as in <cit.>. Note in particular, that if an edge e is the worst choice for one of its endpoints, then it can never be blocking for any perfect matching. If in addition e ∈ E ∖ E_M then removing it keeps the set of stable matchings the same. We thus have the following lemma, whose proof is in the Appendix. Let G be given and let an edge e = uv ∈ E ∖ E_M be such that u = l_G(v). Then G and (G ∖ e,≺) have the same set of stable matchings. The proofs of all the lemmas in this section are deferred to the Appendix. §.§ Algorithm Instead of applying phase one of Irving's algorithm to a given instance G, we will now use Lemma <ref> to keep identifying and removing edges from the graph while preserving the set of stable matchings. Given G, let H be the graph obtained from G by applying the Algorithm 1 to it. Denote E_H the set of its edges. First thing to observe is that the graph we obtain from the Algorithm 1 is independent of the order of deletion of the edges. In other words, H is well-defined. The graph H is well-defined. The main idea behind the proof of Lemma <ref> is that once an edge e can be removed from G in Algorithm 1, it can also be removed from any graph G' otained from G by a sequence of edge removals as in Algorithm 1. Thus e will be removed from G no matter what the order of deletion of the edges is. The full proof of Lemma <ref> can be found in the Appendix. We now enumerate several properties of H. The proof is found in the Appendix. Given G, let E_M and H. Then * E_M ⊆ H. * G and H have the same set of stable matchings. * H is a subgraph of G_I. * H = H_I, where H_I is obtained from H by applying phase one to H. * Let uv ∈ E_H. Then u = f_H(v) ⇔ v = l_H(u). * Let v ∈ V. Then vf_H(v) ∈ E_M. * Let uv ∈ E_H ∖ E_M. Then f_H(v)v uv l_H(v) and f_H(u)u vu l_H(u). Note in particular that, for each agent, his worst and best remaining options are in E_M. That means that, in the algorithm, we were deleting edges that were worse than edges in E_M for one of the endpoints, but by doing this, all the edges that were better for one of the endpoints than edges in E_M got removed as well. Since G and H have the same set of stable matchings by construction of H, the polytopes FSMG and FSMH have exactly the same set of integral vertices. Moreover, if H is bipartite, then Theorem <ref> implies that FSMH is integral. In general, it can happen that FSMG is not integral, but FSMH is, which happens precisely when G_I is not bipartite, but H is. If this is the case, then there exists at least one semi-stable partition 𝒞 such that x_𝒞 is a fractional vertex of FSMG, and supp(x) ⊈ E_H. Moreover, no semi-stable partition 𝒞 corresponding to fractional vertex of FSMG satisfies E(𝒞) ⊆ E_H. § CHARACTERIZING BIPARTITE REDUCIBLE GRAPHS In the Algorithm 1, we removed some edges such that their deletion didn't affect the set of stable matchings. However, it is not guaranteed that this deletion couldn't continue further, or that some other method of eliminating edges from G to obtain a smaller subgraph of G with the same set of stable matching wouldn't yield a graph K whose polytope FSMK is integral. In this section, we will show that if some order of deletion would yield such a K, or, in fact, if any other method of obtaining such a K worked, then H would be bipartite and the corresponding polytope would be integral.Our goal is to prove the following theorem. Let G be given. Then if H is not bipartite, then neither is any subgraph K of G with the property that K and G have the same set of stable matchings. In other words, H is bipartite if and only if G is bipartite reducible. §.§ Modifying the FSM polytope We start by having a closer look at the fractional stable matching polytope FSMG. It is easy to see that each edge e ∈ E ∖ E(G_I) satisfies x_e = 0 for all points x ∈ FSMG, but it is in general not true that e ∉E_M ⇒ x_e = 0, ∀ x ∈ FSMG. Thus we start by adding the constraintx_e = 0 for each edge e ∈ E ∖ E_M to FSMG. Given a system G with E_M as above, let FSM'G = { x ∈^E :[x_e = 0∀ e ∈ E ∖ E_M;x(δ(v) ∩ E_M) ≤ 1∀ v ∈ V; x (ϕ(e) ∩ E_M) ≥ 1∀ e ∈ E;x ≥ 0]}. FSM G = { x ∈^E_M : [x(δ(v) ∩ E_M) ≤ 1∀ v ∈ V; x (ϕ(e) ∩ E_M) ≥ 1∀ e ∈ E;x ≥ 0]}. We observe that FSM'G is the intersection of FSMG with some of its facets - namely the facets {x_e = 0}_e ∈ E ∖ E_M. Further, FSM G is just a restriction of FSM'G to the edges in E_M,where we drop coordinates in E ∖ E_M, which satisfy x_e = 0∀ x ∈ FSM'G. Thus, the following holds: For an arbitrary instance G, * each vertex of FSM' G is also a vertex of FSMG, * all vertices of FSM' G are half-integral, * the integral vertices of FSM' G are precisely the integral vertices of FSMG, * there is a 1-to-1 correspondence between the vertices of FSM G and those of FSM' G. Each vertex of FSM G is a vertex of FSM'G projected to E_M. Following these remarks, we can observe that the integral vertices of FSM G correspond to stable matchings of G. Moreover, it can be the case that by intersecting the polytope FSMG with its facets {x_e = 0} for all e ∈ E ∖ E_M we cut off all its fractional vertices, and the polytope FSM G is integral. Let us revisit Lemma <ref> from the previous section in the light of the polytope FSM G. We defer the proof to the Appendix. Let G be given and let an edge e = uv ∈ E ∖ E_M be such that u = l_G(v). Then * G and (G ∖ e,≺) have the same set of stable matchings. * FSM G = FSMG ∖ e. In other words, the inequality x(ϕ(e) ∩ E_M) ≥ 1 is redundant in FSM G. Note that the removal of an edge that is the worst remaining choice for one of its endpoints does not change the polytope FSM G. This leads us to the following result. For any G, we have that FSM G = FSM H. §.§ Conditions for integrality of FSM G We now give sufficient and necessary conditions for integrality of FSM G. For the rest of this section, we will consider semi-stable partitions with respect to the polytope FSM G (note that in Definition <ref> they were defined with respect to FSMG).Analogously to our original definition, we will say that 𝒞 is a semi-stable partition w.r.t FSM G, if and only if 𝒞 is a collection of vertex-disjoint edges and cycles with cyclic preferences in E_M, such that the point x_𝒞 assigning value 1 to the single edges and 1/2 to the edges on the cycles is a feasible point of FSM G. We let E(𝒞) be the edges in 𝒞, and we use the notation of Chen et al. <cit.> to define the set of edgesE_𝒞 = { uv ∈ E_H: ∃e_1,e_2,f_1,f_2 ∈ E(𝒞): e_1 ≼_u uv ≼_u e_2andf_1 ≼_v uv ≼_v f_2}. Observe that E_𝒞 consists of all edges in 𝒞 and of all so-called intermediate edges, which are edges with both endpoints on a cycles from 𝒞 that fit in between the edges in 𝒞 in both endpoint's preference lists. We then let H_𝒞 be the subgraph of H induced by the edges E_𝒞. The following lemma, whose proof is in the Appendix, characterizes the vertices of FSM G. Let x be a vertex of FSM G. Then x = x_𝒞 for some semi-stable partition 𝒞 w.r.t. FSM G. With this, we obtain the following equivalent conditions for the integrality of FSM G. The following are equivalent. * FSM G is integral, * H is bipartite, * H_𝒞 is bipartite for any semi-stable partition 𝒞 w.r.t. FSM G. Moreover, if x is a vertex of FSM G, then x is half-integral. We note that both Lemma <ref> and Theorem <ref> are analogous to the results proven by Chen et al. <cit.> for FSMG. Note that, in particular, if H_𝒞 is not bipartite for some semi-stable partition 𝒞 w.r.t FSM G, then FSM G is not integral. On the other hand, if FSM G is integral, then all the graphs H_𝒞 have the same bipartition, namely the bipartition of H. To make the proof more readable, we will refer to semi-stable partitions w.r.t. FSM G, simply as semi-stable partitions. Proof of Theorem <ref>. (2) ⇒ (1). If H is bipartite, then FSM H is integral, and so is FSM H, by Remark <ref>. By Corollary <ref>, FSM G = FSM H. (3) ⇒ (2). To see this, note that the set F = {vf_H(v): v ∈ V} of all the edges most preferred in H by one of its endpoints is a semi-stable partition. Moreover, H_F = H. (1) ⇒ (3). Assume FSM G is integral. By Lemma <ref>, FSM G is integral if and only if for each semi-stable partition 𝒞, x_𝒞 is a convex combination of integral points in FSM G. Fix a semi-stable partition 𝒞. Then we can write x_𝒞 = ∑_i = 1^k λ_i y^i, where k ≥ 1, λ_i > 0, ∑_i = 1^k λ_i = 1 and y^i's are all integral points in FSM G. For each i ∈ [k], denote by M^i the perfect stable matching consisting of edges in supp(y^i). For simplicity, we will denote x_𝒞 by x. If k = 1, then x = y^1, thus 𝒞 is a matching. Then E(H_𝒞) = 𝒞, and H_𝒞 is bipartite. Hence we can assume k ≥ 2. Let {e_i = u_iv_i}_i ∈ [m] be the set of single edges in 𝒞, and let W = V ∖{u_i, v_i}_i ∈ [m] be the vertices on cycles in 𝒞. For each i ∈ [k] we have x_e = 0 ⇒ y^i_e = 0 and x_e = 1 ⇒ y^i_e = 1, so in particular supp(y^i) ⊆ supp(x). Since for each i ∈ [k], M^i must be a perfect matching whose edges are a subset of E(𝒞), we immediately have that all the cycles in 𝒞 must be even. Further note that each v ∈ W is matched to one of only two possible u's in any of the M^i's, namely to one of its two neighbors on the cycle in 𝒞 it is a part of. Let denote these two partners v_+ and v_-, where v_+v v_-. Then let W_1 = {v ∈ W: vv_+∈ M^1} be the set of agents that are matched to their more preferred neighbor in M^1. Let W_2 = W ∖ W_1. Claim: (W_1 ∪{u_i},W_2 ∪{v_i}) is a bipartition of H_𝒞. Proof: Note that u = v_+ ⇔ v = u_- and, in particular, the only edges of H_𝒞 that could lie in W_1 × W_1 or in W_2 × W_2 are those in E(H_𝒞) ∖ E(𝒞). Further, for each uv in E(H_𝒞) ∖ E(𝒞), (ϕ(uv) ∩ supp(x)) = {uu_+, vv_+}. If there was an edge uv ∈ W_2 × W_2, then y^1(ϕ(uv)) = 0 since neither uu_+ nor vv_+ gets value 1 in y^1, so the stability condition at uv would be violated. Finally, suppose there was an edge uv ∈ W_1 × W_1. Let z = ∑_i = 2^k λ_i/(1 - λ_1) y^i. Then z ∈FSM G and x = λ_1 y^1 + (1 - λ_1)z. Thus z = 1/1 - λ_1(x - λ_1 y) and z(ϕ(uv)) = 1/1 - λ_1( (x_uu_+ - λ_1 y_uu_+) + (x_vv_+ - λ_1 y_vv_+) ) = 2(1/2 - λ_1) /1 - λ_1 = 1 - 2 λ_1/1 - λ_1 < 1. This finishes the proof of the claim and of the theorem.§.§ Proof of Theorem <ref> We are now ready to prove Theorem <ref>. Proof of Theorem <ref>. Suppose that K is a bipartite subgraph of G such that G and K have the same set of stable matchings. Let K' be the graph obtained from K by applying Algorithm 1 to it. Since K' is bipartite, by Theorem <ref> applied to K, FSM K is integral, and so is FSMK' by Corollary <ref>. Since in K', for each agent we have that its best and worst remaining edge must be in E_M, K' contains no edges in E ∖ E_H, thus E(K') ⊆ E_H. But then all the inequalities from FSMK' are in FSM H, meaning that FSMK'⊇FSM H. Further, H and K' have the same set of stable matchings, so the integral vertices of FSM H are the integral vertices of FSMK' and vice versa. Hence FSM H is integral. By Corollary <ref> and Theorem <ref>, H is bipartite. § APPROXIMATION ALGORITHM In this section, we study instances where each agent has at most two possible partners in any stable matching, meaning that H consists only of single edges and disjoint cycles with cyclic preferences. Note that even in these instances H being bipartite is not enough to ensure that FSM G is integral. However, we show that we can obtain a 2-approximation algorithm for the minimum-weight stable matching in this case. Suppose G is such that H consists only of single edges and vertex-disjoint cycles. Then, for any weight function w: E →_≥ 0, we can find, in polynomial time, a stable matching M^⋆ such that w(M^⋆) ≤ 2 w(M_OPT), where M_OPT is a stable matching of G of minimum weight. Suppose that H consists of vertex-disjoint cycles C_1,…, C_k and single edges e_1,…, e_m. Then, by definition of H, these cycles must all have even length (otherwise it would not be true that each edge in H is a part of a stable matching) and also that the preferences on the cycles are cyclic (following from Theorem <ref>). Let M be a stable matching of G. Then e_i ∈ M∀ i ∈ [m] and also |C_k ∩ M| = 1/2|C_k|,∀ i ∈ [k]. Hence, for any cycle C_i = v_i^1,…, v_i^2r_i we have either that {v_i^2jv_i^2j+1}_j ∈ [r_i]⊆ M or {v_i^2j-1v_i^2j}_j ∈ [r_i]⊆ M. For each i ∈ [k] letw_i^+ = ∑_j ∈ [r_i] w(v_i^2j-1v_i^2j) andw_i^- = ∑_j ∈ [r_i] w(v_i^2jv_i^2j+1) and suppose without loss of generality that w_i^+ ≥ w_i^-. We construct a weight function w̃ as follows: For each i ∈ [k] we set w̃ (v_i^1v_i^2) = w_i^+ - w_i^- and we set the weights of all the other edges to zero. Intuitively, for each stable matching M and for each cycle C_i we must pick half of the edges of C_i to be in M, and there are only two ways of doing it. Hence the individual weights of the edges on the cycles do not play any role. In the new weight function we construct, all edges have weight 0, apart from one on each cycle C_i whose weight we set to the difference between w_i^+ and w_i^-. Then, for any stable matching M of G, we have thatw(M) = w̃(M) + W^-,whereW^- = ∑_i ∈ [k] w_i^- + ∑_i ∈ [m] w(e_i). In particular, M is optimal for w̃ if and only if it is optimal for w. Now, ovserve that for each agent v ∈ V, at most one of its neighboring edges has non-zero weight in w̃, and this edge would have to be either the most or the least preferred by V in H. This implies that the weight function w̃ has the so-called U-shaped property of Teo and Sethuraman <cit.>. Thus we can find, in polynomial time, a stable matching M^⋆ such thatw̃(M^⋆) ≤ 2 w̃(M_OPT) meaning thatw(M^⋆) - W^- ≤ 2 (w(M_OPT) - W^-)and so in factw(M^⋆) ≤ 2w(M_OPT) - W^- ≤ 2w(M_OPT). splncs03 § APPENDIX §.§.§ Proof of Lemma <ref>. (⇒): Suppose that u = f_H(v). If there was any partner w for u such that uw ∈ E_M and vu w, then in a (perfect) stable matching M containing uw, v would have a different partner than u, in particular uv M(v) and the edge uv would be blocking. Hence v = l_H(u). (⇐): Every agent w is the most preferred in H by precisely one agent, as otherwise there would be two agents x, y such that w = f_H (x) = f_H (y), implying x = l_H (w) = y. For each agent w, denote by w^- the unique agent for which w = f_H(w^-). Then, for each w, l_H(w) = w^- by the above implication. In particular, for every agent w we have x = l_H (w) ⇔ x = w^- ⇔ w = f_H(x). §.§.§ Proof of Lemma <ref>.Let G and e = uv be given. There exists a stable matching M of G s.t. uv ∈ M if and only if there exists a perfect stable matching M' of G[V']not containing any of the edges of G[V'] which would make the matching M' ∪{e} blocking in G. If M' is a matching in G[V'] containing an edge wy such that uw ∈ E, w u v, and uw y, then the edge uw is blocking for the matching M' ∪{uv}.Thus any matching M' for which the matching M' ∪{uv} is stable in G excludes all edges in E_u ∪ E_v.On the other hand, any perfect matching M' that is stable in G' is stable in G[V'],since none of the edges in E_u ∪ E_v is blocking for M' in G'.Thus e appears in a stable matching of G if and only if there exists a perfect stable matchingin G'.G' can be constructed in polynomial time and it can be decided whether G' contains a perfect stable matching using Irving's algorithm in polynomial time. §.§.§ Proof of Lemma <ref>. Suppose we want to check whether an edge e = uv ∈ E ∖ E_M can be removed without affecting the set of stable matchings. Let ψ_u(e) = { f ∈ E: eu f} and ψ_v(e) ={f ∈ E: ev f} be the sets of edges strictly dominated by the edge e at the endpoints u and v, respectively. Then a matching M is stable in G ∖ e, but not in G if and only if M is a (perfect) stable matching of G ∖ e such that |M ∩ψ_v(e)| = |M ∩ψ_u(e)| = 1. Since |ψ_u(e)|,|ψ_v(e)| ≤ n, we have to check at most 𝒪(n^2) combinations of edges, checking for each pair (f', f”) ∈ψ_u(e) ×ψ_v(e) whether it can be extended to a perfect stable matching M of G ∖ e. For each such a pair, the check can be carried in polynomial time <cit.>. §.§.§ Proof of Lemma <ref>. If there was a matching M' that was stable in G∖ e, but not stable in G, then e would be the only edge blocking M'. Since u = l_G(v) and all stable matchings of G and G ∖ e are perfect, we must have M'(v)v u, a contradiction. §.§.§ Proof of Lemma <ref>. Suppose for contradiction that two executions of the algorithm, 𝒯_1 and 𝒯_2, yield two different graphs H_1 and H_2, respectively. Suppose that e ∈ E(H_1) ∖ E(H_2) is an edge present in H_1 that is not in H_2. Denote E_0 the set of edges that got deleted in 𝒯_2, prior to the deletion of e. Without loss of generality suppose that E_0 ∩ E(H_1) = ∅, so in other words, all the edges that were deleted in 𝒯_2 prior to e were deleted in 𝒯_1 too. Then, since in G_0 = G[E ∖ E_0], we removed e as the next edge in the 𝒯_2, e is the worst remaining option for one of it's endpoints in G_0. Then e is also the worst remaining option for one of its endpoints in H_1 as E(H_1) ⊂ (E ∖ E_0) and so e can be deleted from H_1, a contradiction. §.§.§ Proof of Theorem <ref>. * This is clear as edges in E_M never get removed from the graph. Since each removal preserves the set of stable matchings, E_M does not change as the algorithm goes on. * Similarly, the edges are eliminated so that the set of stable matchings stays unchanged. * Edges removed in phase one of Irving's algorithm are not in E_M. The set of edges removed in phase one is E_I = { uv ∈ E: l_G_I(u)u v}, i.e. these are precisely those edges that are worse for one of the endpoints then its worst remaining choice in the resulting graph G_I. Consequently, each of the edges in E_I at some point becomes an edge as in Lemma <ref> in the current remaining subgraph of G. Thus all the edges that get eliminated in phase one get also eliminated during our procedure. Hence H ⊆ G_I. * If this was not true than phase one applied to Hwould eliminate at least one edge. There would be at least one eliminated edge that was the worst remaining choice for one of its endpoints. * This is a property of any graph after applying phase one of Irving's algorithm to it. * Since vf_H(v) = f_H(v)l_H(f_H(v)), if it was not in E_M, it would get eliminated. * An edge in E_H ∖ E_M must be neither the last, thus nor the best remaining option for both of its endpoints. §.§.§ Proof of Lemma <ref>. We have already proved part (i) as Lemma <ref>. For part (ii), it is enough to show that (ϕ(e) ∩ E_M) ⊇ (ϕ(f) ∩ E_M) for some edge f ∈ G ∖ e. Since u is v's worst remaining choice, we have that (δ(v) ∩ E_M) ⊆ (ϕ(e) ∩ E_M). In other words, all edges in E_M adjacent to v are contained in ϕ(e) ∩ E_M. Let w be such that w = l_H (v) and let f = wv. Note that such an f must exists and also that f ≠ e. Since w = l_H (v), by Lemma <ref> we have that v = f_H (w) and thus (ϕ(f) ∩ E_M) = (δ(v) ∩ E_M). Hence (ϕ(f) ∩ E_M) ⊆ (ϕ(e) ∩ E_M) and the inequality x (ϕ(e) ∩ E_M) ≥ 1 is dominated by the inequality x(ϕ(f) ∩ E_M) ≥ 1. §.§.§ Proof of Lemma <ref>. By Remark <ref>, all the vertices of FSM G are projections of the vertices of FSMG. By Theorem <ref> and by the matching inequality (<ref>), which is satisfied with equality by each v ∈ V by Lemma <ref>, we see that, for each v ∈ V, we have either that precisely one of its adjacent edges has value 1 and all the others zero, or precisely two of its adjacent edges have value 1/2 and all the others have value zero. This immediately gives us that supp(x) consists of single edges and vertex-disjoint cycles. Suppose that C is such a a cycle and w.l.o.g suppose C = {v_1,…, v_m}. Then, since the stability inequality (<ref>) must be satisfied at the edge v_1v_2 and also by Lemma <ref>, we have that precisely one of the edges v_2v_3 and v_1v_m dominates v_1v_2 and one is dominated by it. Say, w.l.o.g., that v_3 v_2 v_1 and v_2 v_1 v_m. Then, since v_3 v_2 v_1, Lemma <ref> applied to edge v_2v_3 implies that v_4 v_3 v_2. Inductively, we see that v_i+1v_i v_i-1 for all i = 1,…, m. | http://arxiv.org/abs/1703.09083v1 | {
"authors": [
"Linda Farczadi",
"Natália Guričanová"
],
"categories": [
"cs.GT",
"cs.DS"
],
"primary_category": "cs.GT",
"published": "20170327135731",
"title": "The weighted stable matching problem"
} |
shapes,shadows,arrows#1##1#1ALG@line-1 .11em plus .33em minus .07emOutput: for each S[FOR]ForEach[1] #1 definitionDefinition mylemmaLemma mytheoremTheorem propositionProposition corollaryCorollary breakablealgorithmalgorithmheight.8pt depth0pt 2pt 2pt [email protected]@[email protected]@gmail.com [email protected][cor2]This research was done while the author was a graduate student at Department of Information Technology, NIT Durgapur, India. [label1]Department of Computer science and Engineering, NIT, Durgapur, India [label2]Department of Computer science, Polytechnic University of Catalonia, Barcelona, Catalonia, Spain [label3]Applications Engineer, Oracle, Bangalore, IndiaVery often in some censorious healthcare scenario, there may be a need to have some expert consultancies (especially by doctors) that are not available in-house to the hospitals. Earlier, this interesting healthcare scenario of hiring the ECs (mainly doctors) from outside of the hospitals had been studied with the robust concepts of mechanism design with or without money. In this paper, we explore the more realistic two sided matching in our set-up, where the members of the two participating communities, namely patients and doctors are revealing the strict preference ordering over all the members of the opposite community for a stipulated amount of time. We assume that patients and doctors are strategic in nature. With the theoretical analysis, we demonstrate that the TOMHECs, that results in stable allocation of doctors to the patients is strategy-proof (or truthful) and optimal. The proposed mechanisms are also validated with exhaustive experiments.E-Healthcare hiring ECs DSIC mechanism design stable allocation.§ INTRODUCTION The expert advices or consultancies provided by the expert consultants (ECs) mainly doctors can be thought of as one of the most indispensable events that occurs in the hospital(s) or medical unit(s) on a regular basis. Over the past few years, there had been a perplexing growth in the demand of ECs (especially doctors) during some critical surgical processes (or operations) that are taking place in the operation theatres (OTs) of the hospitals. The unprecedented growth in the demand of the ECs, has made ECs busy and scarce in nature. It is to be noted that, this unique nature (i.e. busy and scarce) of ECs in the healthcare lobby provides an edge to the research community in the healthcare domain to think of: How to manage or schedule these limited (or scarce) ECs in the OTs of the hospitals, during some censorious healthcare situation? In order to answer the above coined question, previously, there had been a spate of research work in the direction of handling the issues of scheduling the in-house ECs especially doctors <cit.> and nurses <cit.> in an efficient and effective manner. In <cit.> different techniques are discussed and presented to schedule the physicians that are in-house to the hospitals in an efficient way for some critical operations that are taking place in the OTs of that hospitals. In past, the work had been also done in the direction of managing the OTs and the hospitals during the patient congestion scenario. The work in <cit.> focuses on the question of: how to effectively and efficiently plan and schedule the OTs? In <cit.> the work has been done for allocating OTs on time to increase operating room efficiency.More importantly, in healthcare domain, one scenario that may be thought of as a challenging issue is, say; in certain critical medical cases, there may be a requirement of some external manpower in the form of ECs (mainly doctors) that are not available in-house to the hospitals. Now, the immediate natural question that came in the mind is that, how to have some external expertise mainly in the form of doctors that are not available in-house to the hospitals? Surprisingly, literature is very limited for this problem in healthcare domain. This interesting situation of taking expert consultancy from outside of the in-house medical unit during some censorious medical scenario (mainly surgical process) was taken care by Starren et. al. <cit.>. Moreover, the introduction of such a pragmatic field of study in the healthcare domain by Starren et. al. <cit.> has given rise to several open questions for the researchers, such as: (a) which ECs are to be considered as the possible expertise provider in the consultancy arena? (b) What incentives policies in the form of perks and facilities are to be presented in-front of the ECs, so as to drag as many ECs as possible in the consultancy arena?In <cit.>, the problem of hiring one or more doctors for a patient from outside of the admitted hospital for some critical operation under monetary environment (experts are charging for their services) with the infinite budget are addressed. With the consideration that, ECs are having some social connections in real life, Singh et. al. <cit.> considered a budgeted setting of the problem in <cit.> motivated by <cit.> for hiring k doctors out of the available n doctors (k < n), such that the total payment made to the ECs do not exceed the total budget of a patient. As opposed to the money involved hiring of ECs as mentioned in <cit.>, another market of hiring ECs can be thought of where the ECs are providing theirexpertise free of cost. Recently, Singh et. al. <cit.> have addressed this idea, where the expert services are distributed free of cost. For hiring ECs in money free environment (i.e. money is not involved in any sense) they have utilized the idea of one sided strict preference (in this case strict preference from patient side) over the available doctors in the consultancy arena.In this paper, we have tried to model the ECs hiring problem as a two sided preference market in healthcare domain motivated by <cit.>. The idea behind studying the ECs hiring problem as a more appealing two sided preference market is that, in this environment, the members present in two different communities have the privilege to provide the strict preference ordering over all the available members of the opposite community. For example, in our case, we have two communities (or parties) in the consultancy arena: (a) Patient party (b) Doctor party. So, the members of the patient party provide strict preference ordering over all the available members (or the subset of available members) in the doctor party and vice versa.§.§ Our ContributionsThe main contributions of our work are as follows.∙ We have tried to model the ECs hiring problem as a two sided matching problem in healthcare domain. ∙ We propose two mechanisms: a naive approach i.e. randomized mechanism for hiring expert consultants (RAMHECs) and a truthful and optimal mechanism; namely truthful optimal mechanism for hiring expert consultants (TOMHECs).∙ We have also proved that for any instance of n patients and n doctors the allocation done by TOMHECs results in stable, truthful, and optimal allocation for requesting party.∙ TOMHECs establish an upper bound of O(kn^2) on the number of iterations required to determine a stable allocation for any instance of n patients and n doctors. ∙ A substantial amount of analysis and simulation are done to validate the performance of RAMHECs and TOMHECs via optimal allocation measure.The remainder of the paper is structured as follows. Section 2 describes our proposed model. Some required definitions are discussed in section 3. Section 4 illustrates the proposed mechanisms. Further analytic-based analysis of the mechanisms are carried out in section 5. A detailed analysis of the experimental results is carried out in section 6. Finally, conclusions are drawn and some future directions are depicted in section 7.§ SYSTEM MODEL We consider the scenario, where there are multiple hospitals say n given as ħ = {ħ_1, ħ_2, …, ħ_n}. In each hospital ħ_i ∈ħ, there exists several patients with different diseases (in our case patients and doctors are categorized based on the diseases and areas of expertise respectively.) belonging to different income group that requires somewhat partial or complete expert consultancies from outside of the admitted hospitals. By partial expert consultancies it is meant that, the part of expertise from the overall required expert consultancies. The set of k different categories is given as: 𝒞 = {c_1, c_2, …, c_k}. The set of all the admitted patients in different categories to different hospitals is given as: 𝒫 = ⋃_ħ_k ∈ħ⋃_c_i ∈𝒞⋃_j=1^ħ_k^i p_i(j)^ħ_kwhere p_i(j)^ħ_k is the patient j belonging to c_i category admitted to ħ_k hospital. The expression ħ_k^i in term p_i(ħ_k^i)^ħ_k indicates the total number of patients in hospital ħ_k belonging to c_i category. The patients who need consultancy may belong to different income bars. So, in this scenario, each hospital tries to select the patient from the lowest income bar in a particular category (say c_i category) who will get the free consultation. On the other hand, there are several doctors having different expertise associated with different hospitals say ℋ = {ℋ_1, ℋ_2, …, ℋ_n}. The set of all the available doctors in different categories associated with different hospitals is given as: 𝒟 = ⋃_ℋ_k ∈ℋ⋃_c_j ∈𝒞⋃_i=1^ℋ_k^i d_j(i)^ℋ_k where d_i(j)^ℋ_k is the doctor j belonging to c_i category associated to ℋ_k hospital. The expression ℋ_k^j in term d_j(ℋ_k^j)^ℋ_k indicates the total number of doctors associated with hospital ℋ_k in c_j category. Our model captures only a single category say c_i but it is to be noted that our proposed model works well for the system considering multiple categories simultaneously. Only thing is that we have to repeat the process k times as k categories are existing. For simplicity purpose, in any category c_i we have considered the number of patients and number of doctors are same i.e. n along with an extra constraint that each of the members of the participating parties provides a strict preference ordering over all the available members of the opposite party. But, one can think of the situation where there are n number of patients and m number of doctors in a category such that m ≠ n (m>n or m<n). Moreover, the condition that every members of the participating party is providing the strict preference ordering over all the available members of the opposite community is not essential and can be relaxed for all the three cases (i.e. m=n, m<n, and m>n).By relaxation, it is meant that the members of the participating parties may provide the strict preference ordering over the subset of the members of the opposite party. At a particular time several doctors (> n) are providing their willingness to impart free consultancy to some patients present in the consultancy arena in a particular category as shown in the right side of the Figure <ref>. In the schematic diagram shown in Figure <ref>, for representation purpose one doctor is selected from all the interested doctors from each hospital belonging to a particular category c_i.But in general one can think of the situation where multiple doctors can be selected from the available doctors from a particular hospital in a particular category c_i such that |𝒫_i| = ∑_ℋ_j ∈ℋℋ̇_j^i; where 0 ≤ℋ̇_j^i≤ n is the number of doctors selected from hospital ℋ_j in c_i category and placed into the consultancy arena. Following the above discussed criteria, the third party selects n doctors out of all the doctors in a particular category c_i as a possible expert consultant and is given as 𝒟_i = {d_i(1)^ℋ_1, d_i(2)^ℋ_2, …, d_i(n)^ℋ_n} and a set of selected patients from c_i category is given as 𝒫_i = {p_i(1)^ħ_1, p_i(2)^ħ_2, …, p_i(n)^ħ_n}. If not specified explicitly, n denotes the total number of patients and the total number of doctors that are participating in the consultancy arena in any category c_i. For placing n doctors in the consultancy arena from the available doctors, the third party can take the help of the qualification of the doctors and number of successful operations or consultancies given so far by that doctor.Each patient p_i(j)^ħ_k reveals a strict preference ordering over the participating set of doctors 𝒟_i in a category c_i and also each doctor d_i(t)^ℋ_k provides the strict preference ordering over the set of participating patients of category c_i in the consultancy arena. The strict preference ordering of the patient p_i(k)^ħ_j over the set of doctors 𝒟_i is denoted by ≻_k^i. More formally, the significance of d_i(ℓ)^ℋ_j≻_k^i d_i(m)^ℋ_k is that the patient p_i(k)^ħ_t ranks doctor d_i(ℓ)^ℋ_j above the doctor d_i(m)^ℋ_k. The preference profile of all the patients for k different categories is denoted as ≻ = {≻^1, ≻^2, …, ≻^k}, where ≻^i denotes the preference profile of all the patients in category c_i over all the doctors in set 𝒟_i represented as ≻^i = {≻_1^i, ≻_2^i, …, ≻_n^i}. The preference profile of all the patients in c_i category except the patient r is given as ≻_-r^i = {≻_1^i, ≻_2^i, …, ≻_r-1^i, ≻_r+1^i, …≻_n^i}. On the other hand, the doctors may give preferences based on the location where he/she (henceforth he) and the patients are located. The strict preference ordering of the doctor d_j(t)^ℋ_k is denoted by ⋟_j^t over the set of patients 𝒫_j, where p_j(ℓ)^ħ_k⋟_j^t p_j(m)^ħ_i means that doctor d_j(t)^ℋ_k ranks p_j(ℓ)^ħ_k above p_j(m)^ħ_i. The set of preferences of all the doctors in k different categories is denoted as ⋟ = {⋟_1, ⋟_2, …, ⋟_k}, where ⋟_j contains the strict preference ordering of all the doctors in c_j category over all the patients in set 𝒫_j represented as ⋟_j = {⋟_j^1, ⋟_j^2, …, ⋟_j^n}. The strict preference ordering of all the doctors in c_j category except the doctor s is represented as ⋟_j^-s = {⋟_j^1, ⋟_j^2, …, ⋟_j^s-1, ⋟_j^s+1, …, ⋟_j^n}. It is to be noted that the allocation of the doctors to the patients for category c_i under consideration is captured by the allocation function 𝒜_i: ≻ × ⋟ → 𝒫_i × 𝒟_i. The resulting allocation vector is given as𝒜 = {𝒜_1, 𝒜_2, …, 𝒜_k}. Each allocation vector 𝒜_i∈𝒜 denotes the patient-doctor pairs belonging to the c_i category denoted as 𝒜_i = ⋃_j=1;k=1^n;n a_jk^i, where each a^i_lm∈𝒜_i is a pair {p_i(l)^ħ_k, d_i(m)^ℋ_j}. The matching between the patients and doctors for any category c_i is captured by the mapping function ℳ: 𝒫_i ∪𝒟_i →𝒟_i ∪𝒫_i. More formally, ℳ(p_i(j)^ħ_k) = d_i(k)^ℋ_ℓ means that patient p_i(j)^ħ_k is matched to d_i(k)^ℋ_ℓ doctor andℳ(d_i(ℓ)^ℋ_k) = p_i(t)^ħ_ℓ means that doctor d_i(ℓ)^ℋ_k is matched to p_i(t)^ħ_ℓ. § REQUIRED DEFINITIONSBlocking pair: Fix a category c_k. We say that a pair p_k(i)^ħ_t and d_k(j)^ℋ_ℓ form a blocking pair for matching ℳ, if the following three conditions holds: (i) ℳ(p_k(i)^ħ_t) ≠ d_k(j)^ℋ_ℓ, (ii) d_k(j)^ℋ_ℓ ≻_i^k ℳ(p_k(i)^ħ_t), and (iii) p_k(i)^ħ_t ⋟_k^j ℳ(d_k(j)^ℋ_ℓ). Stable matching: Fix a category c_k. A matching ℳ is stable if there is no pair p_k(i)^ħ_t and d_k(j)^ℋ_ℓ such that it satisfies the conditions mentioned in (i)-(iii) in Definition 1.Perfect matching: Fix a category c_k. A matching ℳ is perfect matching if there exists one-to-one matching between the members of 𝒫_k and 𝒟_k.Patient-optimal stable allocation: Fix a category c_k. A matching ℳ is patient optimal, if there exists no stable matching ℳ' such that ℳ'(p_k(j)^ħ_t) ≻_j^kℳ(p_k(j)^ħ_t) or ℳ'(p_k(j)^ħ_t) =_j^kℳ(p_k(j)^ħ_t) for at least one p_i(j)^ħ_t∈𝒫_i. Similar is the situation for doctor-optimal stable allocation. Strategy-proof for requesting party: Fix a category c_k. Given the preference profile ≻^k and ⋟_k of the patients and doctors in c_k category, a mechanism 𝕄 is strategy-proof (truthful) for the requesting party if for each members of the requesting party 𝒜_k is preferred over 𝒜̂_k; where 𝒜̂_k is the allocation when at least one member in requesting party is misreporting. § PROPOSED MECHANISMSThe idea behind proposing randomized mechanismi.e. RAMHECs is to better understand the more robust and philosophically strong optimal mechanism TOMHECs. The further illustration of the mechanisms are done under the consideration that patient party is requesting. Moreover, one can utilize the same road map of the mechanisms by considering doctors as the requesting party. This can easily be done by just interchanging their respective roles in the mechanisms. §.§ RAMHECsThe idea lies behind the construction of initialization phase is to handle the system consisting k different categories. The algorithm is depicted in Algorithm 1. §.§.§ Upper bound analysis of RAMHECsThe overall running time of RAMHECs is O(1) + O(kn) = O(kn). §.§.§ Correctness of RAMHECsThe correctness of RAMHECs is proved with the loop invariant technique <cit.>. The loop invariant that we have to prove is that at the end of the i^th iteration each of the patients in c_j category gets one distinct doctor allocated. We must show three things for the loop invariant technique to be true.Initialization: It is true prior to the first iteration of the while loop. Just before the first iteration of the while loop 𝒜_j ←ϕ. This confirms that 𝒜_j contains no patient-doctor pair prior to the first iteration of the while loop.Maintenance: The loop invariant to be true, we have to show that if it is true before each iteration of while loop, it remains true before the next iteration. The body of the while loop allocates a doctor to a patient in a particular category i.e. each time 𝒜_j is incremented by 1. Just before the i^th iteration, the 𝒜_j data structure contains (i-1) number of patient-doctor pairs. After the i^th iteration, the 𝒜_j data structure contains i patient-doctor pairs. This way at the end of the i^th iteration all the i patients gets a distinct doctor and the patient-doctor pairs are stored in 𝒜_j [1i].Termination: The third property is to check, what happens when the while loop terminates. The condition causing the while loop to terminate is that, for any category c_j, each of the patients are allocated with one distinct doctor leading to n patient-doctor pairs in 𝒜_j data structure. Because each loop iteration increments 𝒜_j by 1, we must have |𝒜_j| = n when all n patients are already processed. So, when the loop terminates we have a data structure 𝒜_j [1n] that is already processed and it consists of n patient-doctor pairs.If the RAMHECs is true for a particular category c_j ∈𝒞 it will remain true when all category in 𝒞 taken simultaneously. Hence, the RAMHECs is correct. §.§.§ Illustrative exampleFor understanding purpose, let the category be c_3 (say eye surgery). The set of patient from 4 different hospitals ħ = {ħ_1, ħ_2, ħ_3, ħ_4} is given as: 𝒫_3 = {p_3(1)^ħ_2, p_3(2)^ħ_3, p_3(3)^ħ_4, p_3(4)^ħ_1}. The set of available doctors engaged to 4 different hospitals ℋ = {ℋ_1, ℋ_2, ℋ_3, ℋ_4} is given as: 𝒟_3 = {d_3(1)^ℋ_3, d_3(2)^ℋ_1, d_3(3)^ℋ_4, d_3(4)^ℋ_2}. The preference profile of patient set 𝒫_3 is given as: p_3(1)^ħ_2 = [d_3(4)^ℋ_2≻_1^3 d_3(3)^ℋ_4≻_1^3 d_3(1)^ℋ_3≻_1^3 d_3(2)^ℋ_1], p_3(2)^ħ_3 = [d_3(3)^ℋ_4≻_2^3 d_3(4)^ℋ_2≻_2^3 d_3(2)^ℋ_1≻_2^3 d_3(1)^ℋ_3], p_3(3)^ħ_4 = [d_3(4)^ℋ_2≻_3^3 d_3(2)^ℋ_1≻_3^3 d_3(1)^ℋ_3≻_3^3 d_3(3)^ℋ_4], p_3(4)^ħ_1 = [d_3(2)^ℋ_1≻_4^3 d_3(3)^ℋ_4≻_4^3 d_3(4)^ℋ_2≻_4^3 d_3(1)^ℋ_3]. Similarly, the preference profile of doctor set 𝒟_3 is given as: d_3(1)^ℋ_3 = [p_3(1)^ħ_2⋟_3^1 p_3(2)^ħ_3⋟_3^1 p_3(4)^ħ_1⋟_3^1 p_3(3)^ħ_4], d_3(2)^ℋ_1 = [p_3(2)^ħ_3⋟_3^2 p_3(4)^ħ_1⋟_3^2 p_3(1)^ħ_2⋟_3^2 p_3(3)^ħ_4], d_3(3)^ℋ_4 = [p_3(3)^ħ_4⋟_3^3 p_3(1)^ħ_2⋟_3^3 p_3(2)^ħ_3⋟_3^3 p_3(4)^ħ_1], d_3(4)^ℋ_2 = [p_3(4)^ħ_1⋟_3^4 p_3(3)^ħ_4⋟_3^4 p_3(1)^ħ_2⋟_3^4 p_3(2)^ħ_3]. Given the preference profiles,while loop in line 9-17 randomly selects patient p_3(3)^ħ_4 from the available patients list 𝒫_3. Line 12 of the RAMHECs randomly selects doctor d_3(4)^ℋ_2 from the available preference ordering of p_3(3)^ħ_4. At the end of first iteration of the while loop, the RAMHECscaptures (p_3(3)^ħ_4, d_3(4)^ℋ_2) pair in the 𝒜_3 data structure. In the similar fashion, the remaining allocation is done. The final patient-doctor allocation pair done by the mechanism is 𝒜_3 = {(p_3(3)^ħ_4, d_3(4)^ℋ_2), (p_3(2)^ħ_3, d_3(3)^ℋ_4), (p_3(4)^ħ_1, d_3(1)^ℋ_3), (p_3(1)^ħ_2, d_3(2)^ℋ_1)}. §.§ TOMHECsOur main focus is to propose a mechanism that satisfy the two important economic properties: truthfulness, and optimality .The TOMHECs is illustrated in Algorithm 2. §.§.§ Running timeThe total running time of TOMHECs is given as: T(n)= ∑_i=1^k(O(1) + (∑_i=1^n O(n)))= O(kn^2)TOMHECs (𝒟, 𝒫, 𝒞, ≻, ⋟) §.§.§ Correctness of the TOMHECsThe correctness of the TOMHECs is proved with the loop invariant technique <cit.>.The loop invariant: Fix a category c_i. At the start of ℓ^th iteration of the while loop, the number of temporarily processed patient-doctor pairs or in other words the number of patient-doctor pairs held by 𝒜_i is given as: |∪_j=1^ℓ-1𝒜'_j|, where 𝒜'_j is the net patient-doctor pairs temporarily maintained in the set 𝒜'_j at the j^th iteration. So, on an average the number of patients or doctors (whomsoever is greater) that are to be explored in further iterations are n - |∪_j=1^ℓ-1𝒜'_j|. From the construction of the TOMECs it is clear that after any ℓ^th iteration this condition holds: 0 ≤ n - |∪_i=1^ℓ𝒜'_j| ≤ n; where 1 ≤ℓ≤ n^2. The net minimum number of patient-doctor pairs that can be processed temporarily at any iteration may be zero. Hence, inequality 0 ≤ n - |∪_i=1^ℓ𝒜'_j| ≤ n is always true. We must show three things for this loop invariant to be true.Initialization: It is true prior to the first iteration of the while loop. Just before the first iteration of the while loop, in TOMHECs the inequality 0 ≤ n - |∪_i=1^ℓ𝒜'_j| ≤ n blows down to 0 ≤ n - 0 ≤ n ⇒ 0 ≤ n ≤ n i.e. no patient-doctor pair is temporarily added to 𝒜_i prior to the first iteration ofwhile loop. This confirms that 𝒜_i contains no patient-doctor pair.Maintenance: For the loop invariant to be true, if it is true before each iteration of the while loop, it remains true before the next iteration of the while loop. The body of while loop allocates doctor(s) to the patient(s) with each doctor is allocated to a patient; i.e. each time the cardinality of 𝒜_i is either incremented by some amount or remains similar to previous iteration. Just before the ℓ^th iteration the patient-doctor pairs temporarily added to A_i are ∪_i=1^ℓ-1𝒜'_j. So, one can conclude from here that the number of patient-doctor pairs that are left is given by inequality: 0 ≤ n - |∪_i=1^ℓ-1𝒜'_j| ≤ n. After the (ℓ-1)^th iteration, the available number of patient-doctor pair n - |∪_i=1^ℓ-1𝒜'_j| ≥ 0 can be captured under two cases:Case 1: If |𝒜_i| = n: This case will lead to exhaust all the remaining patient-doctor pair in the current ℓ^th iteration and no patient-doctor pair is left for the next iteration. The inequality n - (|∪_i=1^ℓ-1𝒜'_i ∪𝒜'_ℓ|) = n - (|∪_i=1^ℓ𝒜'_i |) = n - |𝒜_i| = 0. Hence, it means that all the remaining patient-doctor is absorbed in this iteration and no patient-doctor pair is left for processing.Case 2: If |𝒜_i| < n:This case captures the possibility that there may be the scenario when few patient-doctor pairs from the remaining patient-doctor pairs may still left out; leaving behind some of the pairs for further iterations. So, the inequality n - (|∪_i=1^ℓ-1𝒜'_i ∪𝒜'_ℓ|) > 0 ⇒n > n - (|∪_i=1^ℓ𝒜'_i |) > 0 is satisfied.From Case 1 and Case 2, at the end of ℓ^th iteration the loop invariant is satisfied. Termination: It is clear that in each iteration the cardinality of output data structure i.e. 𝒜_ieither incremented by some amount or remains as theprevious iteration. This indicates that at some ℓ^th iteration the loop terminates by dissatisfying the condition of the while loop |𝒜_i| ≠ n at line 12. When the loop terminates it is for sure that |𝒜_i| = n. We can say n - |∪_i=1^ℓ𝒜'_i| = 0 ⇒ 0 ≤ n. Thus, this inequality indicates that all the n patient and doctors in c_i category are processed and each patient allocated a best possible doctor when the loop terminates.If the TOMHECs is true for the c_i ∈𝒞 category it will remain true when all category in 𝒞 taken simultaneously. Hence, the TOMHECs is correct. §.§.§ ExampleConsidering the initial set-up discussed in section 4.1.3. According to line 13-16 of TOMHECs each of the patients p_3(1)^ħ_2, p_3(2)^ħ_3, p_3(3)^ħ_4, and p_3(4)^ħ_1 are requesting to the most preferred doctor from their respective preference list i.e. d_3(4)^ℋ_2, d_3(3)^ℋ_4, d_3(4)^ℋ_2, and d_3(2)^ℋ_1 respectively. In the next step, we will check if any requested doctor among d_3(1)^ℋ_3, d_3(2)^ℋ_1, d_3(3)^ℋ_4, and d_3(4)^ℋ_2 has got the multiple request from the patients in 𝒫_3. Now, it can be seen that, in the first iteration of TOMHECs doctor d_3(4)^ℋ_2 have got requests from patients p_3(1)^ħ_2, and p_3(3)^ħ_4. As each doctor can be assigned to only one patient, so this competitive environment between patient p_3(1)^ħ_2, and p_3(3)^ħ_4 can be resolved by considering the strict preference ordering of doctor d_3(4)^ℋ_2 over the available patients in 𝒫_3. From the strict preference ordering of doctor d_3(4)^ℋ_2 it is clear that patient p_3(3)^ħ_4 is preferred over patient p_3(1)^ħ_2. Hence, patient p_3(1)^ħ_2 is rejected. So, for the meanwhile p_3(2)^ħ_3 gets a doctor d_3(3)^ℋ_4, p_3(3)^ħ_4 gets a doctor d_3(4)^ℋ_2, and p_3(4)^ħ_1 gets a doctor d_3(2)^ℋ_1. Now, as the patient p_3(1)^ħ_2 do not get his/her (henceforth his) most preferred doctor i.e. d_3(4)^ℋ_2 from his preference list. So, he will request the second best doctor i.e. d_3(3)^ℋ_4 from his preference list. As doctor d_3(3)^ℋ_4 is already been requested by p_3(2)^ħ_3, the similar situation now occurs in case of doctor d_3(3)^ℋ_4 where patients p_3(1)^ħ_2 and p_3(2)^ħ_3 are simultaneously requesting to doctor d_3(3)^ℋ_4. Looking at the preference list of d_3(3)^ℋ_4, we get, patient p_3(1)^ħ_2 is preferred over p_3(2)^ħ_3. So, patient p_3(2)^ħ_3 is rejected. Now, p_3(2)^ħ_3 request the second best doctor i.e. d_3(4)^ℋ_2 from his preference list. In the similar fashion, the remaining allocation is done. The final allocation is:{(p_3(1)^ħ_2, d_3(3)^ℋ_4), (p_3(2)^ħ_3, d_3(1)^ℋ_3), (p_3(3)^ħ_4, d_3(4)^ℋ_2), (p_3(4)^ħ_1, d_3(2)^ℋ_1)}. §.§ Several propertiesThe proposed TOMHECs has several compelling properties. These properties are discussed next.The matching computed by the Gale-Shapley mechanism <cit.> results in a stable matching.A stable matching computed by Gale-Shapley mechanism <cit.> is requesting party optimal.Gale-Shapley mechanism <cit.> is truthful for the requesting party. Following the above mentioned propositions and motivated by <cit.> we have proved that the TOMHECs results in stable, optimal, and truthful allocation when all the k different categories are taken simultaneously. TOMHECs results in a stable allocation for the requesting party (patient party or doctor party). Fix a category c_i ∈𝒞. Let us suppose for the sake of contradiction there exists a blocking pair (p_i(j)^ħ_k, d_i(j)^ℋ_l) that results in an unstable matching ℳ for the requesting party. As their exists a blocking pair (p_i(j)^ħ_k, d_i(j)^ℋ_l) it may be due to the case that (p_i(j)^ħ_k, d_i(k)^ℋ_j) and (p_i(k)^ħ_j, d_i(j)^ℋ_l) are their in the resultant matching ℳ. This situation will arise only when d_i(j)^ℋ_l≻_j^i d_i(k)^ℋ_j i.e. in the strict preference ordering of patient p_i(j)^ħ_k doctor d_i(j)^ℋ_l is preferred over doctor d_i(k)^ℋ_j. From the matching result ℳ obtained, it can be seen that in-spite the fact that d_i(j)^ℋ_l≻_j^i d_i(k)^ℋ_j; d_i(j)^ℋ_l is not matched with p_i(j)^ħ_k by the TOMHECs. So, this upset may happen only when doctor d_i(j)^ℋ_l received a proposal from a patient p_i(k)^ħ_j to whom d_i(j)^ℋ_l prefers over p_i(j)^ħ_k i.e. p_i(k)^ħ_j⋟_i^j p_i(j)^ħ_k. Hence, this contradicts the fact that the (p_i(j)^ħ_k, d_i(j)^ℋ_l) is a blocking pair. As their exists no blocking pair, it can be said that the resultant matching by TOMHECs is stable.From our claim that, the TOMHECs results in a stable matching in a particular category c_i, it must be true for any category. Hence, it must be true for the system considering the k categories simultaneously. A stable allocation resulted by TOMHECs is requesting party (patient or doctor) optimal. Fix a category c_i. Let us suppose for the sake of contradiction that the allocation set ℳ obtained using TOMHECs is not an optimal allocation for requesting party (say patient party). Then, from Lemma <ref> thereexists a stable allocation ℳ' such that ℳ'(p_i(j)^ħ_k) ≻_j^iℳ(p_i(j)^ħ_k) or ℳ'(p_i(j)^ħ_k) =_j^iℳ(p_i(j)^ħ_k) for at least one patient p_i(j)^ħ_k ∈ 𝒫_i. Therefore, it must be the case that, some patient p_i(j)^ħ_k proposes to ℳ'(p_i(j)^ħ_k) before ℳ(p_i(j)^ħ_k) since ℳ'(p_i(j)^ħ_k) ≻_i^jℳ(p_i(j)^ħ_k) and is rejected by ℳ'(p_i(j)^ħ_k). Since doctor ℳ'(p_i(j)^ħ_k) rejects patient p_i(j)^ħ_k, the doctor ℳ'(p_i(j)^ħ_k) must have received a better proposal from a patient p_i(k)^ħ_j to whom doctor ℳ'(p_i(j)^ħ_k) prefers over p_i(j)^ħ_k i.e. p_i(k)^ħ_j⋟_i^j p_i(j)^ħ_k. Since, this is the first iteration at which a doctor rejects a patient under ℳ'. It follows that the allocation ℳ is preferred over allocation ℳ' for the patient p_i(j)^ħ_k. Hence, this contradicts the fact that the allocation set ℳ obtained using TOMHECs is not an optimal allocation. As their exists an optimal allocation ℳ.Form our claim that, the TOMHECs results in optimal allocation in a particular category c_i, it must be true for any category. Hence, it must be true for the system considering the k categories simultaneously.A stable allocation resulted by TOMHECs is requesting party (patient or doctor) truthful. Fix a category c_i. Let us suppose for the sake of contradiction that the matching set ℳ obtained using TOMHECs is not a truthful allocation for requesting party (say patient party). The TOMHECs results in stable matching ℳ when all the members of the proposing party reports their true preferences. Now, let's say a patient p_i(j)^ħ_k misreport his preference list ≻_j^i and getting better off in the resultant matching ℳ'. Let 𝒫'_i be the set of patients who are getting better off in ℳ' as against ℳ. Let 𝒟'_i be the set of doctors matched to patients in 𝒫'_i in matching ℳ'. Let d_i(k)^ℋ_ℓ be the doctor that p_i(j)^ħ_k gets in ℳ'. Since ℳ is stable, we know that d_i(k)^ℋ_ℓ cannot prefer p_i(j)^ħ_k to the patient got in ℳ, because this would make (p_i(j)^ħ_k, d_i(k)^ℋ_ℓ) a blocking pair in ℳ (see Lemma 1). In other words, doctor ℳ(d_i(k)^ℋ_ℓ) ⋟_i^k p_i(j)^ħ_k. Now, if ℳ(d_i(k)^ℋ_ℓ) patient would not improve in ℳ' then ℳ(d_i(k)^ℋ_ℓ) ⋟_i^k p_i(j)^ħ_k. Hence, d_i(k)^ℋ_ℓ can not be matched with p_i(j)^ħ_k in ℳ', a contradiction. Therefore, patient in ℳ also improves in ℳ'. That is, 𝒟'_i is not the only set of doctors in ℳ' of those patient who are getting better off in ℳ; but also the set of doctors where patient inℳ improve in ℳ'. In other words, each doctor in 𝒟_i is matched to two different patient from 𝒫_i in match ℳ and ℳ', being better off in ℳ than in ℳ'. It can also be proved using Lemma 1 that ℳ' is not stable; a contradiction that terminates the proof.From our claim that, the TOMHECs results in a truthful matching in a particular category c_i, it must be true for any category. Hence, it must be true for the system considering the k categories simultaneously. § FURTHER ANALYTICS-BASED ANALYSISIn order to provide sufficient reasoning to our simulation results presented in section 6, the two proposed mechanisms are in general analyzed on the ground of the expected distance of allocation done by the mechanisms from the top most preference. As a warm up, first the the analysis is done for any patient j, to estimate the expected distance of allocation from the top most preference. After that the analysis is extended to more general setting where all the patients present in the system are considered. It is to be noted that the results revealed by the simulations can easily be verified by the lemmas below. The allocation resulted by RAMHECs for any patient (or doctor) j being considered first is on an average n/2 distance away from its most preferred doctor (or patient) i.e. E[Z] ≃n/2; where Z is the random variable measuring the distance from the top most preference. Fix a category c_i ∈𝒞, and an arbitrary patient j being considered first. In RAMHECs, for any arbitrary patient (AP) being considered first are allotted a random doctor from his preference list. The index position of the doctor in the preference list is decided by k, where k = 1, 2, …, n. Now, when a doctor is selected randomly from the preference list any of these k (1 ≤ k ≤ n) may be selected. So any index k could be the outcome of the experiment (allocation of a doctor) and it is to be noted that selection of any such k is equally likely. Therefore, for each k such that 1 ≤ k ≤ n any k^th doctor can be selected with probability 1/n. For k = 1, 2, …, n, we define indicator random variable X_k whereX_k = I{k^th doctor selected from patients' preference list}X_k = 1,if k^th doctor is selected 0,otherwise Taking expectation both side, we get;E[X_k] = E[I{k^th doctor selected from patients' preference list}] As always with the indicator random variable, the expectation is just the probability of the corresponding event <cit.>:E[X_k]= 1 · Pr{X_k=1} + 0 · Pr{X_k=0} = 1 · Pr{X_k=1} = 1/nFor a given call to RAMHECs, the indicator random variable X_k has the value 1 for exactly one value of k, and it is 0 for all other k. For X_k=1, we can measure the distance of k^th allocated doctor from the most preferred doctor in the patient j's preference list. So, let d_k be the distance of k^th allocation from the best preference. More formally, it can be represented in the case analytic form as:Z = d_0:If 1^st agent is selected from the preference list (k=1)d_1:If 2^nd agent is selected from the preference list (k=2) [=]⋮ [=]⋮d_n-1:If n^th agent is selected from the preference list (k=n)Where Z is the random variable measuring the distance of the allocation from the patient's top most preference. Here, d_0=0, d_1=1, d_2=2, …, d_n-1=n-1.It is to be observed that, once the doctor k is selected from the patient j's preference list, the value calculation of d_k is no way dependent on k. Now,observe that the random variable Z that we really care about can be formulated as:Z = ∑_k=1^nX_k · d_k-1Taking expectation both side. We get;E[Z] = E[∑_k=1^nX_k · d_k-1] = ∑_k=1^nE[X_k · d_k-1] (by linearity of expectation) = ∑_k=1^nE[X_k] · E[d_k-1] (X_k and d_k-1 are independent) = ∑_k=1^n1/n· E[d_k-1] = 1/n∑_k=1^n E[d_k-1] = 1/n∑_k=1^n d_k-1 (once k is fixed d_k-1 becomes constant) = 1/n·(n-1)(n)/2 =(n-1)/2≃n/2,as claimed.In RAMHECs, E[D] ≃n^2/16; where D is the total distance of all the patients in the system from the top most preference. Fix a category c_i ∈𝒞. We are analyzing, the expected distance of the allocations done to the patients by RAMHECs from the top most preferences. For this purpose, as there are n patients, the index of these patients are captured by i such that i = 1, 2, …, n. Without loss of generality, the patients are considered in some order. The index position of the doctor in any patient j's preference list is decided by k, where k = 1, 2, …, n. For any patient i (1≤ i≤ n) selected first, when a doctor is selected randomly from the preference list any of the available k (1≤ k ≤ n) doctors can be selected. So, any index k could be the outcome of the experiment (allocation of doctor) and any such k is equally likely. But what could the case, if instead of considering the patient in the first place, say a patient is selected in i^th iteration. In that case, from the construction of RAMHECs the length of the preference list of the patient under consideration would be n-i+1. So, when a doctor is selected randomly from the preference list, any of the (n-i+1) doctors may be selected. It is to be noted that the selection of any of the (n-i+1) doctors is equally likely. Therefore, for a patient under consideration in i^th iteration, for each k such that 1 ≤ k ≤ n-i+1 any k^th doctor can be selected with probability 1/n-i+1. Here, we are assuming that each agent's top preferences are still remaining when that agent is considered by the RAMHECs. To get the lower bound this is the best possible setting. If an agent is not provided that list, he will be further away from his top most preference. For each patient i and for k = 1, 2, …, n, we define indicator random variable X_ik whereX_ik = I{k^th doctor selected from patient i's preference list}X_ik = 1,if k^th doctor is selected from patient i's preference list 0,otherwise Taking expectation both side, we get;E[X_ik] = E[I{k^th doctor selected from patient i preference list}] As always with the indicator random variable, the expectation is just the probability of the corresponding event:E[X_ik]= 1 · Pr{X_ik=1} + 0 · Pr{X_ik=0} = 1 · Pr{X_ik=1} = 1/n-i+1For a given call to RAMHECs, the indicator random variable X_ik has the value 1 for exactly one value of k, and it is 0 for all other k. For X_ik=1, we can measure the distance of k^th allocated doctor from the most preferred doctor in the patient j's preference list. So, let d_ik be the distance of k^th allocation from the best preference. More formally, it can be represented in the case analytic form as:D = d_i0:If 1^st agent is selected from the preference list (k=1)d_i1:If 2^nd agent is selected from the preference list (k=2) [=]⋮ [=]⋮d_i(n-1):If n^th agent is selected from the preference list (k=n)Where D is the total distance of all the patients in the system from the top most preference. It is to be observed that, once the doctor k is selected from the patient j's preference list, the value calculation of d_k is no way dependent on k. Now,observe that the random variable D that we really care about is given as:D ≥∑_i=1^n∑_k=1^n-i+1X_ik· d_ikTaking expectation both side. We get;E[D] ≥ E[∑_i=1^n∑_k=1^n-i+1X_ik· d_ik] = ∑_i=1^n∑_k=1^n-i+1E[X_ik· d_ik] (by linearity of expectation) = ∑_i=1^n∑_k=1^n-i+1E[X_ik] · E[d_ik] (X_ik and d_ik are independent) = ∑_i=1^n∑_k=1^n-i+11/n-i+1· d_ik ≥∑_i=1^n∑_k=1^n-i+11/n· d_ik = 1/n∑_i=1^n∑_k=1^n-i+1d_ik =1/n[∑_i=1^n/2∑_k=1^n-i+1d_ik + ∑_i=n/2^n∑_k=1^n-i+1d_ik] ≥1/n[∑_i=1^n/2∑_k=1^n-i+1d_ik] + [∑_i=n/2^n∑_k=1^n-i+10] (discarding the lower order terms) ≥1/n[∑_i=1^n/2∑_k=n/2^n-i+1d_ik] ≥1/n[∑_i=1^n/2∑_k=n/2^n-i+1d_in/2] (replacing each term of the series by its first term) = 1/n[∑_i=1^n/2∑_k=n/2^n-i+1n/2] = 1/2[∑_i=1^n/2∑_k=n/2^n-i+1 1] ≥(1/2∑_j=1^n/2 j) -1 =1/2[n/2(n/2+1)/2]-1 =n^2+2n-16/16≃n^2/16as claimed. It is to be observed that for each agent, the expected distance of allocation done by RAMHECs from the top preference in an amortized sense is n/16.The expected number of rejections for any arbitrary patient (or doctor) j resulted by TOMHECs is constant. If the probability of any k length rejection is considered as 1/2 i.e. Pr{Y_k=1}= 1/2 then E[Y] = 2; where Y is the random variable measuring the total number of rejections made to the patient (or doctor) under consideration.Fix a category c_i ∈𝒞, and an arbitrary patient j. To analyze the expected number of rejections suffered by the patient under consideration in case of TOMHECs, we capture the total number of rejections done to any patient j by a random variable Y. So, the expected number of rejections suffered by any patient j is given as E[Y]. It is considered that the rejection by any member k = 0, …, n-1, present on the patients' j preference list is an independent experiment. It means that, the m length rejections suffered by an arbitrary patient j is no way dependent on any of the previous m-1 rejections. Let us suppose for each 0 ≤ k ≤ n-1, the probability of rejection by any k^th doctor be 1/2 (it can be any value between 0 and 1 depending on the scenario). For k=0,…, n-1, we define indicator random variable Y_k whereY_k = I{k length rejection}Y_k = 1,if k length rejection 0,otherwise Taking expectation both side, we get;E[Y_k] = E[I{k length rejection}] As always with the indicator random variable, the expectation is just the probability of the corresponding event:E[Y_k]= 1 · Pr{Y_k=1} + 0 · Pr{Y_k=0} = 1 · Pr{Y_k=1} = (1/2)^k Observe that the random variable Y that we really care about is given as,Y = ∑_k=0^n-1Y_kTaking expectation both side. We get;E[Y] = E[∑_k=0^n-1Y_k ] = ∑_k=0^n-1E[Y_k] (by linearity of expectation) = ∑_k=0^n-1(1/2)^k < ∑_k=0^∞(1/2)^k= 1/1-(1/2) = 2as claimed. Moreover, if we consider the probability of k^th rejection as 2/3 then, the expected number of rejections will be given as 3 i.e E[Y]=3. Similarly, E[Y]=10 if the probability of k^th rejection is taken as 9/10. It means that, even with the high probability of rejection to any arbitrary patient j by the members of the proposed party, there is a chance that after constant number of rejections patient j will be allocated a good doctor according to his choice. Hence, we can say that each agent's allocation is not far away from his top most preference. In TOMHECs, E[R] = 2n, where R is the random variable measuring the total number of rejections made to all the patients. Fix a category c_i ∈𝒞. We are analyzing the total number of rejections suffered by all the patients in expectation. For this purpose, as there are n patients, the index of these patients are captured by i such that i = 1, 2, …, n. The index position of the doctor in any patient j's preference list is decided by k, where k=1, 2, …, n. We capture the total number of rejections done to all patients by a random variable R. So, the expected number of rejections suffered by all the patients is given as E[R]. It is considered that the rejection by any member k = 1, …, n-1, present on the patients' i preference list is an independent experiment. It means that, the m length rejections suffered by an arbitrary patient i is no way dependent on any of the previous m-1 rejections. Let us suppose for each patient i and for each 1 ≤ k ≤ n-1, the probability of rejection by any k^th doctor be 1/2 (it can be any value between 0 and 1 depending on the scenario). For k=1,…, n-1, we define indicator random variable R_ik whereR_ik = I{k length rejection of i^th patient}R_ik = 1,if k length rejection of i^th patient 0,otherwise Taking expectation both side, we get;E[R_ik] = E[I{k length rejection of i^th patient}] As always with the indicator random variable, the expectation is just the probability of the corresponding event:E[R_ik]= 1 · Pr{R_ik=1} + 0 · Pr{R_ik=0} = 1 · Pr{R_ik=1} = (1/2)^k Observe that the random variable R that we really care about is given as,R = ∑_i=1^n∑_k=1^n-iR_ikTaking expectation both side. We get;E[R] = E[∑_i=1^n∑_k=1^n-iR_ik] = ∑_i=1^n∑_k=1^n-iE[R_ik] (by linearity of expectation) = ∑_i=1^n∑_k=1^n-i(1/2)^k < ∑_i=1^n∑_k=0^∞(1/2)^k = ∑_k=0^n-11/1-(1/2) = 2nas claimed. It is to be observed that for each patient, the expected number of rejections in case of TOMHECs in an amortized sense is O(1). As we have shown that for all n agents, the expected number of rejection are O(n).§ EXPERIMENTAL FINDINGSThe experiments are carried out in this section to compare the efficacy of the TOMHECs based on the preference lists of the doctors and patients generated randomly using Random library in Python. RAMHECs is considered as the benchmark mechanism.§.§ Simulation setupFor creating a real world healthcare scenario we have considered 10 different categories of patients and doctors for our simulation purpose. One of the scenarios that is taken into consideration is, say there are equal number of patients and doctors present in each of the categories along with the assumption that each of the patients are providing strict preference ordering (generated randomly) over all the available doctors and also each of the doctors are providing strict preference ordering over all the available patients. Second scenario is the case where, there are equal number of patients and doctors are present in the market. Each of the members in the respective parties are providing the strict preference ordering over the subset of the members of the opposite community. The other two scenarios i.e. m<n and m>n with partial preference are not shown due to page limit.§.§ Performance metricsThe efficacy of TOMHECs is measured under the banner of two important parameters: (a) Satisfaction level (η_ℓ): It is defined as the sum over the difference between the index of the doctor (patient) allocated from the patient's (doctor's) preference list to the index of the most preferred doctor (patient) by the patient (doctor) from his/her preference list. Considering the requesting party, the η_ℓ^j for c_j category is defined as: η_ℓ^j = ∑_i=1^n(ξ_i - ξ_i ); where, ξ_i is the index of the doctor (patient) allocated from the initially provided preference list of the patients (doctors) i, and ξ_i is the index of the most preferred doctor (patient) in the initially provided preference list of patient (doctor) i. For k categories, η_ℓ = ∑_j=1^k∑_i=1^n(ξ_i - ξ_i ). It is to be noted that lesser the value of satisfaction level higher will be the satisfaction of patients or doctors. (b) Number of preferable allocation (ζ): The term "preferable allocation" refers to the allocation of most preferred doctor or patient from the revealed preference lists by the patients or the doctors respectively. For a particular patient or doctor the preferable allocation is captured by the function f: 𝒫_i→{0, 1}. For the category c_i, the number of preferable allocation (NPA) is defined as the number of patients (doctors) getting their first choice from the initially provided preference list. So, ζ_i = ∑_j=1^n f(p_i(j)^ħ_ℓ). For k categories ζ = ∑_i=1^k∑_j=1^n f(p_i(j)^ħ_ℓ).§.§ Simulation directions The three directions are seen for measuring the performance of TOMHECs, they are: (1) All the patients and doctors are reporting their true preference list. (2) When fraction of total available membersof the requesting party are misreporting their preference lists. (3) When fraction of total available members of the requested party are misreporting their preference lists.§.§ Result analysis In this section, the result is simulated for the above mentioned three cases and discussed. *Expected amount of patients/doctors deviating The following analysis motivated by <cit.> justifies the idea of choosing the parameters of variation. Let χ_j be the random variable associated with the event in which j^th patient in c_i category varies its true preference ordering. Thus, χ_j = {j^th patient varies preference ordering}.χ = ∑_j =1^nχ_j. We can write E[χ] = ∑_j =1^n E[χ_j] = ∑_j=1^n 1/8 = n/8. Here,Pr{j^th patient varies preference ordering} is the probability that given a patient whether he will vary histrue preference ordering. The probability of that is taken as 1/8 (small variation).Our result analysis is broadly classified into four categories:∙ Case 1a: Requesting party with full preference (FP) In Figure <ref> and Figure <ref>, it can be seen that the satisfaction level of the requesting party in case of TOMHECs is more as compared to RAMHECs. As TOMHECs always allocates the most preferred member from the preference list. Under the manipulative environment of the requesting party, it can be seen in Figure <ref> and Figure <ref> that, the satisfaction level of the system in case of TOMHECs with large variation is less than the satisfaction level of the system in case of TOMHECs with medium variation is less than the satisfaction level of the system in case of TOMHECs with small variation is less than the satisfaction level of the system in case of TOMHECs. It is natural from the construction of TOMHECs. Considering the second parameter i.e. number of preferable allocation, it can be seen in Figure <ref> and Figure <ref> that the NPA of the requesting party in case of TOMHECs is more as compared to RAMHECs. Under the manipulative environment of the requesting party, it can be seen in Figure <ref> and Figure <ref> that, the NPA of the system in case of TOMHECs with large variation is less than the NPA of the system in case of TOMHECs with medium variation is less than the NPA of the system in case of TOMHECs with small variation is less than the NPA of the system in case of TOMHECs. It is natural from the construction of TOMHECs. ∙ Case 1b: Requesting party with partial preference (PP) In Figure <ref> and Figure <ref>, it can be seen that the satisfaction level of the requesting party in case of TOMHECs is more as compared to RAMHECs. As TOMHECs always allocates the most preferred member from the preference list. Under the manipulative environment of the requesting party, it can be seen in Figure <ref> and Figure <ref> that, the satisfaction level of the system in case of TOMHECs with large variation is less than the satisfaction level of the system in case of TOMHECs with medium variation and even less than RAMHECs is less than the satisfaction level of the system in case of TOMHECs with small variation even less than RAMHECs is less than the satisfaction level of the system in case of TOMHECs. It is natural from the construction of TOMHECs. Considering the second parameter i.e. number of preferable allocation, it can be seen in Figure <ref> and Figure <ref> that the NPA of the requesting party in case of TOMHEcs is more as compared to RAMHECs. Under the manipulative environment of the requesting party, it can be seen in Figure <ref> and Figure <ref> that, the NPA of the system in case of TOMHECs with large variation is less than the NPA of the system in case of TOMHECs with medium variation is less than the NPA of the system in case of TOMHECs with small variation is less than the NPA of the system in case of TOMHECs.∙ Case 2a: Requested party with full preference (FP) In Figure <ref>, Figure <ref> and Figure <ref>, Figure <ref>, it can be seen that the satisfaction level and the NPA respectively of the requested party in case of TOMHECs is more as compared to RAMHECs. It can be seen from Figure <ref>-<ref> and Figure <ref>-<ref> that the TOMHECs is requesting party optimal.It is natural from the construction of TOMHECs.∙ Case 2b: Requested party with partial preference (PP) In Figure <ref>, Figure <ref> and Figure <ref>, Figure <ref>, it can be seen that the satisfaction level and the NPA respectively of the requested party in case of TOMHECs is more as compared to RAMHECs.§ CONCLUSIONS AND FUTURE WORKSWe have tried to model the ECs hiring problem as a two sided matching problem in healthcare domain. This paper proposed an optimal and truthful mechanism, namely TOMHECs to allocate the ECs to the patients. The more general settings are of n patients and m doctors (m ≠ n or m == n) with the constraint that members of the patient party and doctor party can provide the preference ordering (not necessarily strict) over the subset of the members of the opposite party can be thought of as our future work.§ ACKNOWLEDGEMENTWe would like to thank Prof. Y. Narahari and members of the Game Theory Lab. at Department of CSA, IISc Bangalore for their useful advices. We would like to thank the faculty members, and PhD research scholars of the department for their valuable suggestions. We highly acknowledge the effort undertaken by Ministry of Electronics and Information Technology, Media Lab Asia, Government of India through the Visvesvaraya scheme.elsarticle-num | http://arxiv.org/abs/1703.08698v2 | {
"authors": [
"Vikash Kumar Singh",
"Sajal Mukhopadhyay",
"Aniruddh Sharma",
"Arpan Roy"
],
"categories": [
"cs.GT"
],
"primary_category": "cs.GT",
"published": "20170325143940",
"title": "Hiring Expert Consultants in E-Healthcare: A Two Sided Matching Approach"
} |
twocolumn,aps,prd,amsmath,showpacs,amssymb,floatfix,nofootinbib]revtex4-1 #1 #1#1 #1ansätze αεM_⊙ρ_B GBgbsn[][email protected] School of Mathematics and Physics, China University of Geosciences, Lumo Road 388, 430074 Wuhan, China INFN Sezione di Catania, Dipartimento di Fisica, Universitá di Catania, Via Santa Sofia 64, 95123 Catania, Italy We study rapidly rotating hybrid stars with the Dyson-Schwinger model for quark matter and the Brueckner-Hartree-Fock many-body theory with realistic two-body and three-body forces for nuclear matter. We determine the maximum gravitational mass, equatorial radius, and rotation frequency of stable stellar configurations by considering the constraints of the Keplerian limit and the secular axisymmetric instability, and compare with observational data. We also discuss the rotational evolution for constant baryonic mass, and find a spinup phenomenon for supramassive stars before they collapse to black holes. Rotating hybrid stars with the Dyson-Schwinger quark model H.-J. Schulze December 30, 2023 ========================================================== § INTRODUCTION Neutron stars (NS) are among the densest objects known in the Universe. They contain an extreme environment shaped by the effects of the four fundamental interactions. NSs have the typical mass M ∼ 1.4 and radius R∼10km. Therefore, the mean particle density can reach (2–3)ρ_0, and the core density (10–20)ρ_0<cit.>, where ρ_0=0.17 fm^-3 is the so-called nuclear saturation density. At this density, the nucleons might undergo a phase transition to quark matter (QM), and a hybrid NS (HNS) with a QM core is formed. This makes NS ideal astrophysical laboratories to study hadronic interactions over a wide range of densities <cit.>.Unfortunately, as a key ingredient of the investigation of NS, the equation of state (EOS) remains uncertain. The microscopic theory of the nucleonic EOS has reached a high degree of sophistication <cit.>, but the QM EOS is poorly known at zero temperature and at the high baryonic density appropriate for NS, because it is difficult to perform first-principle calculations of QM.Therefore one can presently only resort to more or less phenomenological models for describing QM, such as the MIT Bag model <cit.>, the Nambu-Jona-Lasino model <cit.>, or the quasi-particle model <cit.>. In Ref. <cit.> we developed a Dyson-Schwinger quark model (DSM) for deconfined QM, which provides a continuum approach to QCD that can simultaneously address both confinement and dynamical chiral symmetry breaking <cit.>. In that work, we considered static and spherical symmetric HNSs, whereas in this paper we include the effects of rotation.Rotation is a common property of NS. Of the thousands of currently observed pulsars, the fastest one has been discovered in the globular cluster Terzan 5 with a frequency of 716 Hz <cit.>. At this rapid rotation, a NS would be flattened by the centrifugal force, and the Tolman-Oppenheimer-Volkoff equation, suitable for a static and spherically symmetric situation, cannot describe correctly the rotating stellar structure. In the present paper we approximate the NS as a axisymmetric and rigid rotating body, and resort to Einstein's theory of general relativity for a rapidly rotating star. Numerical methods for (axisymmetric) rotating stellar structure have been advanced by several groups <cit.>. In this work we utilize the KEH method <cit.> to obtain the properties of rapidly rotating HNSs.This paper is organized as follows. In Sec. II we briefly discuss the construction of the EOS of a HNS. In Sec. III we present the rotation effects on the HNS; the allowed ranges of gravitational mass, equatorial radius, and Kepler frequency are discussed in this section and compared with observational data. The rotational evolution for a constant baryonic mass is also analyzed. Sec. IV contains our conclusions.§ THE EQUATION OF STATE§.§ Nuclear matter For nuclear matter we resort to the Brueckner-Hartree-Fock (BHF) many-body theory with realistic two-body and three-body nucleonic forces, which has been extensively discussed in Ref. <cit.>. We recall that this theory has also been extended with the inclusion of hyperons, which might appear in the core of a NS. The hyperonic EOS in this theory turns out to be very soft, and this results in too low NS maximum masses <cit.>,well below the current observational limit of about two solar masses <cit.>. The presence of strange baryonic matter often inhibits the appearance of QM. In this work we do not discuss this aspect, but limit ourselves to consider only nucleons and leptons in the hadronic phase.In the BHF theory the energy per nucleon of nuclear matter is given by BA= 3 5k^2_F2m+ 1 2ρ∑_k,k'<k_F⟨ kk' | G[e(k)+e(k');ρ] | kk' ⟩_A,where G[E;ρ] is the solution of the Bethe-Goldstone equation G[E;ρ] = V + ∑_k_a,k_b>k_F V| k_a,k_b ⟩ Q ⟨ k_a,k_b |E-e(k_a)-e(k_b)G[E;ρ], V is the bare nucleon-nucleon (NN) interaction, ρ is the nucleon number density, and E the starting energy. The single-particle energy e(k) = e(k;ρ) = k^2 2m + U(k;ρ) and the Pauli operator Q determine the propagation of intermediate baryon pairs. The BHF approximation for the single-particle potential using thecontinuous choice is U(k;ρ) = ∑ _k'≤ k_F⟨ k k' | G[e(k)+e(k'); ρ] | k k' ⟩_A.Due to the occurrence of U(k) in Eq. (<ref>), the above equations constitute a coupled system that has to be solved in a self-consistent manner for several momenta of the particles involved, at the considered densities. The only input quantities of the calculation are the NN two-body potentials. In this work we present results obtained with the Bonn-B (BOB) potential <cit.> as input, supplemented with compatible three-body forces <cit.>. The associated EOS yields fairly large maximum masses of about 2.5 for purely nucleonic NS (NNS).For the calculation of the energy per nucleon of asymmetric nuclear matter, we use the so-called parabolic approximation <cit.>BA(ρ,x) = BA(ρ,x=0.5)+ (1-2x)^2 E_sym(ρ),where x=ρ_p/ρ is the proton fraction and E_sym(ρ) is the symmetry energy, which can be expressed in terms of the difference of the energy per nucleon of pure neutron matter (x=0) and symmetric matter (x=0.5): E_sym(ρ) = BA(ρ,x=0) - BA(ρ,x=0.5).The parametrized results of pure neutron and symmetric matter with different interactions can be found in Ref. <cit.>. The energy density of baryon/lepton matter as a function of the different partial densities is then (ρ_n,ρ_p,ρ_e,ρ_μ)= (ρ_n m_n +ρ_p m_p)+ (ρ_n+ρ_p) B/A(ρ_n,ρ_p)+ _e(ρ_e) + _μ(ρ_μ),where _e(ρ_e) and _μ(ρ_μ) are the energy densities of electrons and muons. Once the energy density is known, the chemical composition of the beta-equilibrated matter can be calculated and finally the EOS, P = ρ^2 d dρ({ρ_i(ρ)})ρ= ρd dρ -. §.§ Quark matter The quark propagator based on the Dyson-Schwinger equation at finite chemical potential μ≡μ_q = μ_B/3 assumes a general form with rotational covariance, S(p;μ)^-1 = iγ p + i γ_4 (p_4+iμ) + m_q + Σ(p;μ) ≡ i γ p A(p^2,p· u) + B(p^2,p· u)+ i γ_4(p_4+iμ)C(p^2,p· u),where m_q is the current quark mass, u=(0,iμ), and possibilities of other structures, e.g., color superconductivity <cit.>, are disregarded. The quark self-energy can be obtained from the gap equation, Σ(p;μ)= ∫d^4q/(2π)^4 g^2(μ) D_ρσ(p-q;μ)×λ^a/2γ_ρ S(q;μ)λ^a/2Γ_σ(q,p;μ),where λ^a are the Gell-Mann matrices, g(μ) is the coupling strength, D_ρσ(k;μ) the dressed gluon propagator, and Γ_σ(q,p;μ) the dressed quark-gluon vertex at finite chemical potential.For the quark-gluon vertex and the gluon propagator we employ the widely-used "rainbow approximation" <cit.>Γ_σ(q,p;μ) = γ_σ ,and assume the Landau gauge form for the gluon propagator, with an infrared-dominant interaction modified by the chemical potential <cit.>g^2(μ) D_ρσ(k,μ) =4π^2 d k^2/ω^6 e^-k^2+μ^2/ω^2(δ_ρσ-k_ρ k_σ/k^2).The various parameters can be obtained by fitting meson properties and chiral condensate in vacuum <cit.>, and we use ω=0.5 GeV, d=1 GeV^2. The phenomenological parameterrepresents a reduction of the effective interaction with increasing chemical potential. This parameter cannot yet be fixed independently and its value has been amply discussed in previous works <cit.>.Knowing the quark propagator, the EOS of cold QM can be obtained via the momentum distribution <cit.>, f_q(| p|;μ)= 14π∫_-∞^∞dp_4 tr_D[-γ_4S_q(p;μ)],ρ_q(μ)=6∫d^3p(2π)^3 f_q(| p|;μ),P_q(μ_q)=P_q(μ_q,0) +∫_μ_q,0^μ_q dμρ_q(μ).The total density and pressure for pure QM are given by summing the contributions of all flavors. In addition, we define the phenomenological bag constantB_DS≡ -∑_q=u,d,s P_q(μ_q,0).In this work we set the value as B_DS=90MeV fm^-3, see the discussion in <cit.>. §.§ Construction of the hybrid star EOS In order to study the properties of a rapidly rotating HNS, we should first construct the EOS of the star. We assume that the hadron-quark phase transition is of first order, and perform the Gibbs construction, thus imposing that nuclear matter and QM are betastable and globally charge neutral. This is at variance with the Maxwell construction, where the two phases must be separately charge neutral.In the purely nucleonic phase, which consists of baryons (n,p) and leptons (e,μ), the conditions ofbeta stability and charge neutrality can be expressed as μ_n-μ_p = μ_e = μ_μ ,ρ_p = ρ_e + ρ_μ ,where μ_i are the chemical potentials and ρ_i the particle number densities. Similarly the pure QM phase, which contains three-flavor quarks (u,d,s) and leptons (e,μ), should satisfy the constraints of beta stability and charge neutrality μ_d = μ_u+μ_e = μ_u+μ_μ = μ_s,2ρ_u-ρ_d-ρ_s 3 - ρ_e - ρ_μ = 0. According to the Gibbs construction, there is a mixed phase where the hadron and quark phases coexist, and both phases are in equilibrium with each other <cit.>. This can be expressed as μ_i = b_i μ_B - q_i μ_e,p_H = p_Q = p_M.where b_i and q_i denote baryon number and charge of the particle species i=n,p,u,d,s,e,μ in the mixed phase. To solve those equations, we also need the global charge neutrality condition χρ_c^Q + (1-χ)ρ_c^H = 0,where ρ_c^Q and ρ_c^H are the charge densities of quark and nuclear matter, and χ is the volume fraction occupied by QM in the mixed phase. From these equations, we can derive the energy density _M and the baryon density ρ_M of the mixed phase as _M= χ_Q + (1-χ)_H,ρ_M= χρ_Q + (1-χ)ρ_H. In the upper panel of Fig. <ref> we show the pressure versus baryon chemical potential μ_B = μ_n = μ_u+2μ_d. The solid black curve represents the calculation for beta-stable and asymmetric nuclear matter with BOB EOS; the curves labeled DSα are for pure QM with several choices of the phenomenological parameter α. In the lower panel the complete EOSs of HNSs are shown, i.e., pressure vs. baryon density. We can see that the EOS contains three sections: a pure hadronic phase at low density, followed by a mixed phase, and a pure quark phase at high density. We note that the onset of the phase transition is determined by the value of the parameter α; larger α produces an increasingly softer QM EOS with a lower phase transition onset density. For high values of α we find that QM appears quite early, e.g., for α = 10 at a baryon density ρ≈ρ_0.For completeness, we mention that for the calculation of the stellar structure we use the EOSs by Feynman-Metropolis-Teller <cit.> and Baym-Pethick-Sutherland <cit.> for the outer and inner crusts, respectively.§ RESULTS AND DISCUSSION The structure of a rapidly rotating NS is different from the static one, since the rotation can strongly deform the star. We assume NS are steadily rotating and have axisymmetric structure. Therefore the space-time metric used to model a rotating star can be expressed asds^2 = -e^γ+ρdt^2 + e^2β( dr^2+r^2dθ^2 ) + e^γ-ρ r^2sin^2θ( dϕ-ω dt )^2,where the potentials γ,ρ,β,ω are functions of r and θ only. The matter inside the star is approximated by a perfect fluid and the energy-momentum tensor is given byT^μν = (+p)u^μu^ν - p g^μν ,where , p, and u^μ are the energy density, pressure, and four-velocity, respectively. In order to solve Einstein's field equation for the potentials γ,ρ,β,ω, we adopt the KEH method and use the public RNS code <cit.> for calculating the properties of a rotating star. §.§ Keplerian limit The rotational frequency is a directly measurable quantity of pulsars, and the Keplerian (mass-shedding) frequency f_K is one of the most-studied physical quantities for rotating stars <cit.>. In Fig. <ref> we show the gravitational NS mass as a function of the central baryon density (left panel) and of the equatorial radius (right panel), using the EOSs displayed in Fig. <ref>. Results are plotted for both the static configurations (thin curves) and for the ones rapidly rotating at Keplerian frequency (bold curves).In all cases the maximum masses of HNSs are lower than those of NNSs, because the appearance of QM in the core of the star results in a softening of the very hard nucleonic EOS. Comparing Keplerian and static sequences, rotations increase the maximum mass and equatorial radius substantially. The maximum masses of the static and Keplerian sequences with various EOSs, as well as the corresponding central densities, are listed in Table <ref>. The maximum masses increase by about 20% from the static to the Keplerian sequence. According to the current observations of massive pulsars <cit.>, the DSM EOSs with α≳2 are ruled out.In Fig. <ref> we present the Keplerian frequency as a function of gravitational mass for some selected EOSs. We observe that it increases monotonically both for NNSs and HNSs.The Keplerian frequency of HNSs increases more rapidly after QM onset, and is larger than the one of a NNS with the same gravitational mass, because the stellar radius is smaller in the former case due to the presence of a very dense QM core. However, due to the lower maximum mass of HNSs, the maximum Keplerian frequency of HNSs is lower than the one of NNSs, as also listed in Table <ref> for the various EOSs discussed above. Our results satisfy the constraint from the observed fast-rotating pulsar PSR J1748-2446ad with 716 Hz<cit.>, or the even more severe constraint from XTE J1739-285 with 1122 Hz<cit.>, which has not been confirmed, however.We compare our results with the empirical formula f_K = f_0 ( M/)^1/2( R_s/10km)^-3/2 ,proposed in <cit.>, where M is the gravitational mass of the Keplerian configuration, R_s is the radius of the nonrotating configuration of mass M, and f_0 is a constant, which does not depend on the EOS. In Ref. <cit.> an optimal prefactor f_0=1080 Hz in the range 0.5< M < 0.9 M_max^static was obtained. Rotating HNSs with masses in that range are characterized by a purely nucleonic phase, and therefore the empirical formula cannot be applied. This is at variance with NNS configurations. As displayed in Fig. <ref>, our results for NNSs below 2.1 can be fitted well with the same parameter f_0=1080 Hz, as shown by the thin curve. §.§ Stability analysis In order to complete the description of Figs. <ref> and <ref>, one should pay attention to the stability criteria of stars. It is well known that the onset of the instability of the static sequence is determined by the condition dM/dρ_c = 0, i.e., the curve should stop at its mathematical maximum, which thus gives the maximum mass of the static stable sequence. In the rotating case, the above criterium has to be generalized, i.e., a stellar configuration is stable if its mass M increases with growing central density for a fixed angular momentum J<cit.>. Therefore the onset of the instability, which is called secular axisymmetric instability (SAI), is expressed by . ∂ M/∂ρ_c|_J = 0. The configurations in the Keplerian sequences shown in Fig. <ref> have different angular momenta, and thus the curves do not stop at the mathematical maximum. In the upper panel of Fig. <ref> we show, for some selected EOSs, the gravitational mass for the Keplerian sequence vs. the angular momentum (solid black curves), along with the SAI condition, Eq. (<ref>), represented by the dashed red curves. Thus the Keplerian sequence should stop at the intersection with the SAI curves, which is indicated by an open circle. This constraint determines the corresponding endpoints of the curves in Figs. <ref> and <ref>.Some enlarged details are shown in the insets of Fig. <ref>. For a given mass M, there are two possible values of angular momentum J, which correspond to two possible values of radius R in Fig. <ref>. In the case of NNSs with the BOB EOS, the branch with the lower R has a larger values of J∼ M R^2 f_K, because the Kepler frequency f_K increases faster than R^2 diminishes on the Keplerian sequence. In the case of HNSs, the situation is opposite: the branch with the lower R has also a lower value of J. Therefore for NNSs the Kepler curve meets the SAI at large R, before it reaches the mathematical maximum of the mass. This is different from the case of HNSs, whose curves extend a little further on the unstable branch after they reach their mathematical maximum, before meeting the SAI, and thus the maximum mass of the stable configurations coincides with the mathematical maximum value. The maximum mass and maximum angular momentum, as well as the end point given by the SAI constraint, are obtained with different stellar configurations, and are labelled by the open squares, triangles, and circles, respectively. The discussed effects are however very small, of the order of 0.01 at most.In order to visualize better the intricate relations between M, R, and f_K, we present in Fig. <ref> the mass-radius relations of NS with EOS BOB (upper panel) and DS2 (lower panel) at various fixed rotation frequencies (dash-dotted olive curves). The stable configurations are constrained by the Kepler and SAI conditions at large and small radius, respectively. At a low frequency (f=796Hz for HNSs), the lower boundary of M is fixed by the Kepler condition and the upper boundary by the SAI condition. As the frequency increases (f=1082Hz), the SAI mark point moves to the left side of the mathematical maximum (MaM), and the upper boundary of M is now fixed by the MaM, but not anymore by the SAI condition. This is indicated by the dotted blue curve that passes through the MaMs for fixed frequency. As the frequency increases further (f=1194/1273Hz), the lower (upper) boundary values of M are fixed by the SAI (MaM/Kepler) conditions. Finally, at the maximum frequency the Kepler and SAI conditions meet at the same point. In Fig. <ref> we present the allowed domain of NNSs and HNSs in the R_eq–f plane (upper panel) and the M–f plane (lower panel), together with some observational data. We use the same conventions as in Fig. <ref>, i.e., dotted blue curves, dashed green curves, and short-dashed red curves represent MaM, mass-shedding, and SAI limits. The allowed region of HNSs with the DS2 EOS is the grey area delimited by the dash-dotted lila curve (PT), which represents the onset of the phase transition. One interesting feature we should mention here is that at high rotation frequency the mass range is small, while the range of radii is still large, corresponding to a flat top of the M(R) curves in Fig. <ref>. This means the radii are very sensitive to the mass at high rotating frequency.As discussed above, the current observations on pulsar masses constrain our parameter to α<2, hence we present the results of HNSs with the EOS DS2. For smaller α the corresponding (shaded) area of HNSs will shrink and move towards the lower (upper) boundary of NNSs in the upper (lower) panel. The minimum (maximum) mass of HNSs with EOS DS2 is 1.68 (2.02) in the static sequence, and increases as the rotation frequency increases, while the range concentrates to a single value 2.47 at the maximum frequency f=1.4 kHz. Therefore, in the lower panel of Fig. <ref>, the three stars with lower masses should be conventional NS, and the others could be HNSs in our DS2 model.The observational data of the radius still suffer large uncertainties. In the upper panel we include the sources 4U1820-30 and SAXJ1808.4-3658, whose mass, radius, and spin are available. One can see that according to their small radii both sources should preferably be high-mass compact HNSs in our model, whereas their masses in the lower panel identify them as preferably “low-mass” NNSs. This can also be seen in Fig. <ref>, where the same data points are reproduced. However, within the large error bars, both data are still consistent with our model. We expect more accurate observations to constrain our parameters or rule out the model. §.§ Phase transition caused by rotational evolution The possibility of a phase transition to QM caused by rotational evolution has been widely discussed in literature <cit.>. For a constant baryonic mass, a rotating star loses its rotation energy by magnetic dipole radiation, which makes the star spin down and the central density increase. When the central density of a NNS reaches a critical value, the phase transition from hadronic matter to QM will take place, and the star converts to a HNS. As the star continues spinning down and the central density continues increasing, more and more QM appears in the core of the HNS.This is clearly shown in Fig. <ref>, where we display the change of the number density of all particle species with rotational frequency in the interior of a star with baryonic mass M_B=2.0 for the DS2 EOS (corresponding to M=1.74 in the static sequence and M=1.80 at the Kepler frequency f_K=1018Hz, see the lower panel of Fig. <ref>). One notes that this star at Keplerian frequency has no QM core, but as it spins down, it is compressed to a smaller volume, which enhances the central density, and the star is converted into a HNS. As the frequency decreases further, the QM mixed phase extends outward from the core and the region occupied by the pure hadron phase gets narrower. At the same time, the radius of the star is decreasing.In Fig. <ref> we present the stellar models with DS2 EOS in the f–M_B plane, where the same labels as in Fig. <ref> are used, i.e., the dash-dotted lila curve represents the onset of conversion from a NNS to a HNS. It can be seen that the conversion is possible only in the baryon mass range 1.84<M_B/<2.37. Examples could be the pulsars J1903+0327 and 4U1820-30, located at the edge of the phase transition boundary in Fig. <ref>. Above that range, even the fastest rotating stars are already HNSs. In addition, when the star's baryonic mass is larger than 2.35, the static configuration is unstable, and the star will collapse to a black hole as it loses angular momentum and meets the SAI borderline (dashed red curve). These aresupramassive stars <cit.> that will be discussed in more detail in the following. The maximum baryonic mass for the DS2 EOS is 2.87. The various limits are indicated by vertical lines in Fig. <ref>.For further illustration, we show in the upper panel of Fig. <ref> the fraction of QM in HNSs as function of the rotation frequency for several choices of fixed baryonic mass with the DS2 EOS. The trajectories in the M–R_eq plane for the same values of M_B are reported in Fig. <ref>. Usually the QM fraction increases with decreasing frequency due to the increasing density and extension of the QM domain in the star, see Fig. <ref>. The maximum value of 8.39% is reached for the heaviest possible static NS with M_B=2.35, see Fig. <ref>. This value can be increased by choosing larger values ofin the DSM, but then the maximum HNS mass falls below two solar masses. Supramassive HNSs (M_B>2.35) have no static limit and collapse when reaching the (dashed red) SAI line. Their QM fraction remains below the maximum static value.In the lower panel of Fig. <ref> we show the angular momentum as a function of rotation frequency for NNSs and HNSs. The conversion points between NNSs and HNSs on the PT line are indicated by markers in some cases. Normal HNSs (M_B<2.35) are spinning down when losing angular momentum in the evolution, whereas supramassive stars spin up close to the collapse <cit.>. A similar backbending phenomenon is often related to the onset of the phase transition from hadronic matter to QM <cit.>, but here it occurs for both HNSs and NNSs in supramassive configurations, in the case of NNSs for 3.10<M_B/<3.59, see Fig. <ref>.In more detail, for example for the M_B=2.6 trajectory in Fig. <ref>, Fig. <ref>, and in the inset of Fig. <ref>, the HNS spins down until it reaches the minimum of the fixed rotation frequency curve (f=1082 Hz). Then it spins up until the final SAI point. In fact, in the evolution the maximum angular momentum is given at the Kepler sequence and the minimum angular momentum at the static sequence or the SAI line. Therefore, if the lower boundary of the frequency in Fig. <ref> is not at the static sequence or the SAI line, there must be a spinup with loss of angular momentum.Quantitatively, the difference of angular momentum between NNSs and HNSs with equal baryonic mass is slight at lower baryon mass (M_B<2.35), but becomes important for larger masses, where the QM content increases and only HNSs exhibit the spinup phenomenon.§ CONCLUSION We have investigated the properties of rotating HNSs, employing an EOS constructed with the BHF approach for nucleonic matter and the DSM for QM, and assuming the phase transition under the Gibbs construction. We computed the properties of HNSs in the Keplerian sequence, respecting the SAI constraint. HNSs are more compact and have lower maximum masses and maximum Kepler frequencies than NNSs. 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"authors": [
"J. -B. Wei",
"H. Chen",
"G. F. Burgio",
"H. -J. Schulze"
],
"categories": [
"nucl-th",
"astro-ph.HE"
],
"primary_category": "nucl-th",
"published": "20170327070002",
"title": "Rotating hybrid stars with the Dyson-Schwinger quark model"
} |
[ Hung Cong Tran December 30, 2023 ===================== Knowing the correct skull conductivity is crucial for the accuracy of EEG source imaging, but unfortunately, its true value, which is inter- and intra-individually varying, is difficult to determine. In this paper, we propose a statistical method based on the Bayesian approximation error approach to compensate for source imaging errors related to erronous skull conductivity. We demonstrate the potential of the approach by simulating EEG data of focal source activity and using the dipole scan algorithm and a sparsity promoting prior to reconstruct the underlying sources. The results suggest that the greatest improvements with the proposed method can be achieved when the focal sources are close to the skull.Electroencephalography, Bayesian modelling, inverse problems, skull conductivity§ INTRODUCTION Source reconstruction from electroencephalography (EEG) data is an ill-posed inverse problem, and the solution depends strongly on the accuracy of the discretized head model <cit.>. The essential head features are the geometry and the electrical properties of the tissues of which the geometry can be extracted, to some extent, using auxiliary imaging tools <cit.>. However, the accurate determination of the electrical conductivities of the head tissues is an open question. Especially the accurate modelling of the skull is essential for source reconstruction. There are only few techniques available and they can be used to calibrate only few tissue conductivity parameters either by using well defined somatosensory evoked potentials / fields in combination with EEG <cit.>, combined EEG/MEG <cit.> or electrical impedance tomography (EIT) -based techniques <cit.>. Since all these methods rely on auxiliary measurements, it would be highly beneficial to find methods that are purely computational for this purpose.In this paper, we propose to use the Bayesian approximation error (BAE) approach to compensate for the imaging artefacts caused by the approximately modelled skull conductivity. The main idea of BAE is to use an approximate model (e.g. because the accurate model is unknown or computationally unfeasible) and to take statistically into account the induced modelling errors <cit.>. In our case, we first choose an observation model with a standard skull conductivity value and then estimate statistics of the errors between this approximate model and the set of accurate models that is constructed based on the postulated distribution of possible skull conductivity values. This results in an additive error term that is marginalized using a Gaussian approximation. Previously, BAE has been successfully used in EIT <cit.>, optical tomography <cit.> and recently also in EEG source imaging <cit.> for geometry related modelling errors. We demonstrate here by using a specific BAE model, namely the enhanced error model (EEM), that the imaging errors caused by the erroneous skull conductivity can be alleviated. The work was carried out by using state-of-the-art finite element (FE) -based head models, skull conductivity values based on the known literature and appropriate sparsity priors to recover focal source configurations.§ THEORY§.§ Bayesian framework with linear forward model The computational domain is denoted with Ω and its material properties with σ(x) where x ∈Ω. For numerical implementations, the domain is discretized and the observation model is written as v = A(σ) d + e,where v ∈ℝ^m are the measurements, m is the number of measurements, A(σ) ∈ℝ^m × 3n is the lead field matrix that depends on electric conductivity σ, d∈ℝ^3n is the distributed dipole source configuration and e ∼𝒩(e_*,Γ_e) is the measurement noise.Note that the model A(σ) assumes that the accurate values of electric conductivities are known (which in practice without additional effort is almost never the case).In the Bayesian framework, the inverse solution is the posterior density of the Bayes’ formulaπ(d|v) ∝π(v|d)π(d),where π(v|d) is the likelihood and π(d) the prior. The likelihood model can be written asπ(v|d) ∝exp(-1/2(v-Ad-e_*)^TΓ_e^-1(v-Ad-e_*)).§.§ Enhanced error model In BAE, we replace the accurate lead field matrix, A, with an approximate lead field, A_0, in which we employ standard fixed values for the electric conductivity, σ_0. We can writev= A_0d+ε+ewhere ε = Ad-A_0d is the induced approximation error, ε∈ℝ^m.In the enhanced error model (EEM), we further approximate that ε∼𝒩(ε_*,Γ_ε) and d are independent, i.e., ε is considered as another random additive error term. Even though, in practice, ε and d are usually correlated, this approximation often leads to very similar inverse solutions <cit.>. In our case, since we assume that the true source activity d is focal (sparse), the cross-covariances with ε will be negligible. Based on these approximations, we formulate the probability distribution of the likelihood as π(v | d) ∝exp(-1/2(v-A_0d-ε_*-e_*)^T(Γ_ε+Γ_e)^-1(v-A_0d-ε_*-e_*)).§.§ Prior model In this paper, we consider sparse focal source models and we employ an L_1,2 norm inducedprior model (group lasso) π(d)∝exp(-α/2∑_i=1^nw_id_i_2)where d_i=(d_ix,d_iy,d_iz) and d_i^1_2=√(d_ix^2+d_iy^2+d_iz^2) is the strength of the source at location i, and α is a scaling constantthat fixes the distribution of the source strengths. The depth weighting factors w_i are used to reduce the bias of the prior to favor superficial source distributions <cit.>. §.§ Maximum a posteriori estimate In this paper, we compute maximum a posteriori (MAP) estimates of the posterior. Based on the likelihood model (<ref>), the MAP estimate of the source configuration is d̂_ACC=min_dL_e(v-Ad-e_*)_2^2 + α∑_i=1^nw_id_i_2 ,where L_e comes from the Cholesky factorization Γ_e^-1=L_e^T L_e. The subscript "ACC" refers to the assumption that the electrical conductivities are accurately known.If we, however, compute the solution using the fixed standard electrical conductivity values, σ_0, we getd̂_STAN=min_dL_e(v-A_0d-e_*)_2^2 + α∑_i=1^nw_id_i_2. From the likelihood of the EEM (<ref>), the source configuration can be estimated asd̂_EEM=min_dL_ε+e(v-A_0d-ε_*-e_*)_2^2 + α∑_i=1^nw_id_i_2where (Γ_ε+Γ_e)^-1=L_ε+e^T L_ε+e. Note that the EEM result is computed using the same lead field matrix as in Equation (<ref>).The resulting non-linear convex minimization problem is solved using the truncated Newton interior point method <cit.>.§ MATERIALS AND METHODS§.§ Head modelsThe geometry of the head was constructed based on T1- and T2-weighted magnetic resonance images of a healthy subject measured with a 3 T MR scanner. The scalp, eyes, skull compacta, skull spongiosa, cerebro spinal fluid (CSF), gray matter (GM) and white matter (WM) were segmented, for more details see <cit.>. In the simulation set-up, 74 measurement electrodes were attached to the sculp.The electric conductivities (in S/m) of the different tissues were 0.43 for the scalp <cit.>, 0.505 for the eyes <cit.>, 1.79 for the CSF <cit.>, 0.14 for the WM <cit.> and 0.33 for the GM <cit.>. The skull conductivities of the different head models were the following:First, we created 200 head models with skull conductivity drawn from a Gaussian distribution with mean 0.01855 and standard deviation 0.007225. This distribution was set in such a way that the two standard deviation lower and upper values were 0.0041 <cit.> and 0.033 <cit.>, respectively. We refer to these head models as sample head model. We also created a standard head model with the mean skull conductivity 0.01855. The skull spongiosa conductivities in all the models were selected based on the spongiosa:compacta conductivity ratio 3.6:1 <cit.>.The head geometry was discretized using tetrahedral FEs. For the forward simulations, a source space that covered the GM was constructed with 30,105 nodes on a regular grid with grid size 2 mm. For the inverse computations, two source spaces that covered the GM were used: the first one consisted of 10,782 source locations on a regular grid with grid size 3 mm and the second 2,249 source locations with grid size 5 mm. The forward grid was chosen in such a way that it did not contain the same coordinate points as the inverse grids. The lead field matrices used in the simulations were computed using standard piecewise linear FE basis functions with the Saint Venant source modelling approach <cit.>.§.§ Computation of the approximation error statistics The approximation error statistics was created by first choosing randomly one of the sample head models, evaluating the model with randomly chosen source configuration, and finally calculating the approximation error by evaluating the standard model with the same source configuration, ε^(j) = A(σ^(j))d^(j) - A_0d^(j),where A(σ^(j)) is one of the sample models, d^(j) random source configuration and A_0 is the standard model.The procedure was repeated J=200,000 times, and these simulated error samples were used to calculate the sample mean, ε_*, and the sample covariance, Γ_ε, of the approximation error.§ RESULTS AND DISCUSSION§.§ Dipole scan with noiseless data To demonstrate the potential of BAE, we first carried out reconstructions from noiseless data with the dipole scan algorithm. The noiseless data was computed using the accurate model that had skull conductivity 0.0041 S/m <cit.>. This value was chosen to showcase a rather large difference compared to the skull conductivity in the standard model, 0.01855 S/m. Single sources in GM with orientation that was normal to the surface of the cortex were used. For the dipole scan, the 3 mm regular grid was used and two results were computed: one with the standard model and another with the EEM. From the results, we calculated the localization error (in milli meters) between the actual and the reconstructed source for both models, X_STAN and X_EEM, respectively. Figure <ref> shows the differences Δ = X_STAN - X_EEM which are positive (red Δ signs) if EEM has improved the source localization, close to zero (white Δ signs) if both models give the same localization error, and negative (blue ∇ signs) if EEM performs worse than the standard model.It can be seen that the largest improvements occur for sources that are close to the skull. These are also the sources that usually have the largest localization errors due to erronous skull conductivity. The deeper in the brain the sources are the less improvement can be seen, and for some deep locations the EEM model actually performs worse than the standard model. §.§ L_1,2 prior with noisy data To demonstrate the feasibility with a sparsity prior, we present a result computed from noisy boundary data (SNR = 40 dB) by using the L_1,2 prior described in Section II-<ref>. Here, the results were computed by evaluating Equations (<ref>)-(<ref>), respectively. The computations were carried out in the regular 5 mm grid. The first result in Figure <ref> shows the correct location of the source (black cone) and the reconstruction when the accurate skull conductivity, 0.0041 S/m, is known (green cones). The second result shows the reconstruction when the standard model is used (blue cones) with skull conductivity 0.01855, and the third result the reconstruction with the EEM (red cones). As expected, the best results is achieved when the accurate skull conductivity is known. However, if the accurate conductivity is not know, then EEM gives a better estimate than the standard model.§ CONCLUSION AND FUTURE WORK We have demonstrated that with the help of EEM it is possible to reduce source localization errors caused by erroneous skull conductivity. The EEM showed the greatest improvement for sources close to the skull. In the future, the statistical modelling of conductivity errors can be combined with various geometry related approximations.§ CONFLICT OF INTEREST The authors declare that they have no conflict of interest. | http://arxiv.org/abs/1703.09031v1 | {
"authors": [
"Ville Rimpiläinen",
"Alexandra Koulouri",
"Felix Lucka",
"Jari P Kaipio",
"Carsten H Wolters"
],
"categories": [
"physics.med-ph",
"physics.comp-ph"
],
"primary_category": "physics.med-ph",
"published": "20170327123354",
"title": "Bayesian Modelling of Skull Conductivity Uncertainties in EEG Source Imaging"
} |
𝐱 ŁL Fei Jiang^1, Xiao-Yang Liu^1,2, Hongtao Lu^1, Ruimin Shen^1 ^1Department of Computer Science and Engineering, Shanghai Jiao Tong University ^2Department of Electrical Engineering, Columbia University Graph Regularized Tensor Sparse Coding for Image Representation Herbert Van de Sompel December 30, 2023 =============================================================== Sparse coding (SC) is an unsupervised learning scheme that has received an increasing amount of interests in recent years. However, conventional SC vectorizes the input images, which destructs the intrinsic spatial structures of the images. In this paper, we propose a novel graph regularized tensor sparse coding (GTSC) for image representation. GTSC preserves the local proximity of elementary structures in the image by adopting the newly proposed tubal-tensor representation. Simultaneously, it considers the intrinsic geometric properties by imposing graph regularization that has been successfully applied to uncover the geometric distribution for the image data. Moreover, the returned sparse representations by GTSC have better physical explanations as the key operation (i.e., circular convolution) in the tubal-tensor model preserves the shifting invariance property. Experimental results on image clustering demonstrate the effectiveness of the proposed scheme.Sparse Coding, Tensor Representation, Manifold Learning, Tensor-linear Combination § INTRODUCTION Sparse coding (SC), which encode the images using only a few active coefficients, has been successfully applied to many areas across computer vision and pattern recognition <cit.>, since it is computationally efficient and has physical interpretations.However, conventional SC <cit.> for image representations suffers from the following two major problems: (i) the vectorization preprocess breaks apart the local proximity of pixels and destructs the object structures of images; and (ii) the geometric distributions of the image space are ignored, while such information can significantly enhance the learning performance.Two different kinds of sparse coding models have been proposed to preserve the intrinsic spatial structures of images: tensor sparse coding (TenSR) <cit.> and convolutional sparse coding (CSC) <cit.>. For TenSR models <cit.>, tensors are exploited for the image representation and a series of separable dictionaries are used to approximate the structures in each mode of images. Though the spatial structures are preserved by tensor representations, the relationships between the learned sparse coefficients and dictionaries are more complicated, which will cause the encoding (sparse coefficients) hard to interpret. For CSC models <cit.>, images are represented as the summation of the convolutions ofthe filters that capture the local patterns of images and the corresponding feature maps. Each feature map has nearly the same size as the image, which significantly increases the computational complexity for further analyses of images, such as image classification and clustering.And Figure <ref> shows the fundamental theoretical differences of the above-mentioned sparse coding models. Moreover, thosetwo kinds of models do not consider the geometric structures of the image space.Several sparse coding models incorporating the geometrical structures of the images space have been proposed. They are based on the locally invariant idea, which assumes that two close points in the original space are likely to have similar encodings. It has been shown <cit.> that the learning performance can be significantly enhanced if the geometrical structure is exploited and the local invariance is considered. However, these sparse coding models ignore the spatial structure of the images due to the vectorization preprocess.Motivated by the progress in tubal-tensor representation <cit.>, in this paper, we propose a novel graph regularized tensor sparse coding (GTSC) scheme for image representation that simultaneously considers the spatial structures of images and geometrical distributions of the image space. Firstly, we propose a novel tensor sparse coding model based on the tensor-product operation, which preserves the spatial structures of images by tensor representation. Unlike TenSR <cit.>, the learned coefficients by our model have better physical explanations, which show the contributions of corresponding bases and their shifted versions, as shown in Figure <ref>. Then we incorporate the geometric distributions of the image space byusing the graph Laplacian as a smooth regularizer to preserve the local geometrical structures. By perserving the locally invariant property, GTSC has better discriminating power than the conventional SC <cit.>.The rest of this paper is organized as follows: Section <ref> introduces the proposed tubal-tensor sparse representation of images. Section <ref> presents the proposed GTSC model and the alternating minimization algorithm for GTSC. The experimental results on image clustering are presented in Section <ref>. Finally, we conclude the paper in Section <ref>. § TUBAL-TENSOR SPARSE REPRESENTATION§.§ NotationA third-order tensor of size m× n× k is denoted as 𝒳∈ℝ^m× n× k. 𝒳^(ℓ) represents the ℓ-th frontal slice 𝒳(:,:,ℓ) which is a matrix, 𝒳∈ℝ^mk× nk represents the expansion of 𝒳 along the third dimension where 𝒳=[𝒳^(1);𝒳^(2);⋯;𝒳^(k)],𝒳 is the discrete Fourier transform (DFT) along the third dimension of 𝒳, and 𝒳^† represents the transpose of 𝒳 where 𝒳^†^(1)=𝒳^(1)^T and 𝒳^†^(ℓ)=𝒳^(k+1-ℓ)^T, 2≤ℓ≤ k. The superscript “T" denotes the transpose of matrices.For convenience, tensor spaces ℝ^1×1× k, ℝ^m×1× k, ℝ^m× n× k are denoted as 𝕂, 𝕂^m, and 𝕂^m× n, respectively. [k] denotes the set {1,2,⋯,k}. The ℓ_1 and Frobenius norms of tensors are denoted as 𝒳_1=∑_i,j,k|𝒳(i,j,k)|, and 𝒳_F^2=(∑_i,j,k𝒳(i,j,k)^2)^1/2, respectively.§.§ Tensor-linear CombinationA two-dimensional image of size m× k is represented by a third-order tensor 𝒳∈𝕂^m, which can be approximated by the tensor-productbetween 𝒟∈𝕂^m× r and ℬ∈𝕂^r as𝒳 = 𝒟 * ℬ,where * denotes tensor-product introduced in <cit.>.Note that, 𝒳 can be rewritten as a tensor-linear combination of tensor bases {D(:,j,:)}_j=1^r⊂𝕂^m with corresponding tensor representations {ℬ(j,1,:)}_j=1^r⊂𝕂:𝒳 =𝒟*ℬ=𝒟(:,1,:)*ℬ(1,1,:)+⋯+𝒟(:,r,:)*ℬ(r,1,:).Equation (<ref>) is quite similar to linear combinations as tubes ℬ(r,1,:) play the same role as scalars in the matrix representation.§.§ Tubal-tensor Sparse Representation Given n images of size m× k, we present them asa third-order tensor 𝒳∈𝕂^m× n. Let 𝒟∈𝕂^m× r be the tensor dictionary, where each lateral slice 𝒟(:,j,:) represents a tensor basis, and 𝒮∈𝕂^r× n be the tensor corresponding representations. Each image 𝒳(:,j,:) is approximated by a sparse tensor-linear combination of those tensor bases. Our tubal-tensor sparse coding (TubSC) model can be formulated as:min_𝒟,ℬ1/2𝒳 - 𝒟 * ℬ_F^2 + βℬ_1 s.t.𝒟(:,j,:)_F^2 ≤ 1, j∈[r],where β is the sparsity regularizer.Conventional SC is a special case of TubSC.§.§ Explanations of Tensor Representations To explain the tensor representations, we introduce Lemma <ref> which bridges the tensor-product with the matrix-product. <cit.>The tensor-product 𝒳=𝒟*ℬ has an equivalentmatrix-product as:𝒳 = 𝒟^cℬ,where 𝒟^c is the circular matrix of 𝒟 defined as follows:D^c = ([ 𝒟^(1) 𝒟^(k) ⋯ 𝒟^(2); 𝒟^(2) 𝒟^(1) ⋯ ⋯; ⋯ ⋯ ⋯ 𝒟^(k); 𝒟^(k) 𝒟^(k-1) ⋯ 𝒟^(1); ]).Assuming the vectorization formulations of tensor bases are 𝐃=[d_1,⋯,d_r]∈ℝ^mk× r, where d_j=𝒟(:,j,:)(:), j∈[r], then 𝒟^c(:,1:r) is actually 𝐃. Moreover, 𝒟^c are the set of shifted versions of 𝐃, which can be denoted as 𝒟^c=[𝐃,𝐃_1,⋯,𝐃_k-1].If we rewrite ℬ=[b;b_1;⋯;b_k-1]∈ℝ^rk with corresponding to the shifted versions of 𝐃, the tensor-product (<ref>) isfurther transformed into linear combination as follows:𝒳 = 𝒟^cℬ=𝐃b+𝐃_1b_1+⋯+𝐃_k-1b_k-1.From (<ref>), we can see the explicit meanings of tensor representations ℬ, which display the reconstruction contributions of thecorresponding original bases and the shifted versions of bases simultaneously. Figure <ref> shows the shifted versions of an image basis. It can be seen that the shifted versions of the basis are used for image reconstruction without storing them.§ GRAPH REGULARIZED TENSOR SPARSE CODING In this section, we present our graph regularized tensor sparse coding (GTSC) model which simultaneously takes into account the spatial structures of images and local geometric information of the image space. §.§ Problem FormulationFrom one aspect, TubSC model defined by (<ref>) can preserve the spatial structures of images based on the tensor-linear combinations in (<ref>). From another aspect, one might further hope that the learned tensor dictionary can respect the intrinsic geometrical information of the image space. A natural assumption is to keep local invariance wherethe learned sparse representations of two close points in the original space are also close to each other. This assumption is usually referred to as manifold learning <cit.>, which can significantly enhance the learning performance. Given a set of n images {𝐗_j}_j=1^n of size m× k, a q-nearest neighbor graph𝐖∈ℝ^n is constructed. Considering the problem of mapping the weighted graph to the tensor sparse representation ℬ, we first make an expansion of ℬ along the third dimension as ℬ∈ℝ^mk× n , where each column is a sparse representation of an image. A reasonable criterion for choosing a better mapping is to minimize the following objective function:1/2∑_ijℬ(:,i)-ℬ(:,j)_F^2𝐖_ij=Tr(ℬ𝐋ℬ^T),where Tr(·) represents the trace of the matrix, and 𝐋=𝐄-𝐖 is the Laplacian matrix, where 𝐄=diag(e_1,⋯,e_n), and e_i=∑_j=1^n𝐖_ij.By incorporating the Laplacian regularizer (<ref>) into the TubSC model (<ref>), we propose a novel model named graph regularized tensor sparse coding (GTSC) as:min_𝒟,ℬ1/2𝒳 - 𝒟 * ℬ_F^2 +αTr(ℬ𝐋ℬ^T) + βℬ_1 s.t.𝒟(:,j,:)_F^2 ≤ 1, j∈[r],where α≥0 is the graph regularizer, and β≥0 is the sparsity regularizer.Problem (<ref>) is quite challenging due to the non-convex objective function and the convolutional operation. Instead of transforming (<ref>) into conventional graph regularized SC formulation based on Lemma <ref>, we propose a much more efficient algorithm by alternatively optimizing 𝒟 and ℬ directly in the tensor space. §.§ Graph Regularized Tensor Sparse Representations ℬIn this subsection, we discuss how to solve (<ref>) by fixing the tensor dictionary 𝒟. Problem (<ref>) becomes:min_ℬ∈𝕂^r1/2𝒳-𝒟*ℬ_F^2 + αTr(ℬ𝐋ℬ^T) + βℬ_1By Lemma 1, (<ref>) is equivalent to:min_S∈ℝ^rk1/2𝒳-𝒟^cℬ_F^2 + αTr(ℬ𝐋ℬ^T) + βℬ_1.The size of the dictionary 𝒟^c∈ℝ^mk× rk in (<ref>) will be significantly increased for high dimensional images, which will need more computational resources.To alleviate the above-mentioned problem, we propose a novel Iterative Shrinkage Thresholding algorithm based on the tensor presentation (ISTT) to solve (<ref>) directly, which is rewritten as:min_ℬf(ℬ)+β g(ℬ),where f(ℬ) stands for 1/2𝒳-𝒟*ℬ_F^2 + αTr(ℬ𝐋ℬ^T), and g(ℬ) stands for the sparsity constraint ℬ_1.Then, the iterative shrinkage function is constructed by the linearized function around the previous estimation of ℬ_p with the proximal regularization and the nonsmooth regularization. Thus, at the p+1-th iteration, ℬ_p+1 is updated by:ℬ_p+1 = min_ℬ f(ℬ_p) + ⟨∇ f(ℬ_p),ℬ-ℬ_p⟩+ L_p+1/2ℬ-ℬ_p_F^2 + β g(ℬ),To solve (<ref>), we firstly show ∇ f(ℬ) w.r.t. the data reconstruction term 1/2𝒳-𝒟*ℬ_F^2:∇ f(ℬ) = 𝒟^†*𝒟*ℬ - 𝒟^†*𝒳 + 2αℬ*ℒ,where ℒ∈𝕂^n× n with the first frontal slice ℒ(:,:,1)=𝐋 and the other slices ℒ(:,:,ℓ)=0, ℓ∈{2,⋯,k}.Secondly, we discuss how to determine the Lipschitz constant Lip_p+1 in (<ref>). For every ℬ and 𝒞, we have∇ f(ℬ) - ∇ f(𝒞)_F=𝒟^†*𝒟*(ℬ-𝒞)+2α(ℬ-𝒞)*ℒ_F =∑_j=1^n(𝒟^†*𝒟)^c(ℬ_j-𝒞_j)+2α(ℬ_j-𝒞_j)𝐋_2≤((𝒟^†*𝒟)^c_2+2α𝐋_2)∑_j=1^nℬ_j-𝒞_j_2 ≤(∑_ℓ=1^k𝒟^(ℓ)^H𝒟^(ℓ)_F^2+2α𝐋_2)ℬ - 𝒞_F,where the superscript “H" represents conjugate transpose, and ℬ_j=ℬ(:,j,:).Thus the Lipschitz constant of f(ℬ) used in our algorithm is Lip(f)=∑_ℓ=1^k𝒟̂^(ℓ)^H𝒟̂^(ℓ)_F^2+2α𝐋_2.Lastly, (<ref>) can be solved by the proximal operator Prox_β/L_p+1(ℬ_p-1/L_p+1∇ f(ℬ_p)), where Prox_τ is the soft-thresholding operator Prox_τ(·)→sign(·)max(|·| - τ,0).To speed up the convergence of the iterative shrinkage algorithm, an extrapolation operator is adopted <cit.>. Algorithm <ref> summarizes our proposed Iterative Shrinkage Thresholding algorithm based on tensor presentation (ISTT). §.§ Tensor Dictionary Learning 𝒟For learning the dictionary ℬ while fixed 𝒮, the optimization problem is:min_𝒟1/2𝒳-𝒟*ℬ_F^2 s.t.𝒟(:,j,:)_F^2≤ 1, j∈[r],where atoms are coupled together due to the circular convolution operation. Therefore, we firstly decompose (<ref>) into k nearly-independent problems (that are coupled only through the norm constraint) by DFT as follows:min_𝒟^(ℓ),ℓ∈[k]∑_ℓ=1^k𝒳^(ℓ)-𝒟^(ℓ)ℬ^(ℓ)_F^2 s.t. ∑_ℓ=1^k𝒟^(ℓ)(:,j)_F^2≤ k, j∈[r]. Then, we adopt the Lagrange dual <cit.> for solving the dual variables by Newton's algorithm. Another advantage of Lagrange dual is that the number of optimization variables is r, which is much smaller than mkr of the primal problem for solving 𝒟.To use the Lagrange dual algorithm, firstly, we consider the Lagrangian of (<ref>):Lag(𝒟,Λ) = ∑_ℓ=1^k𝒳^(ℓ)-𝒟^(ℓ)ℬ^(ℓ)_F^2+ ∑_j=1^rλ_j(∑_ℓ=1^k𝒟^(ℓ)(:,j)_F^2-k),where λ_j≥0, j∈[r] is a dual variable, and Λ=diag(λ).Secondly, minimizing over 𝒟 analytically, we obtain the optimal formulation of 𝒟: 𝒟^(ℓ) = (𝒳^(ℓ)ℬ^(ℓ)^H)(ℬ^(ℓ)ℬ^(ℓ)^H+Λ)^-1, ℓ∈[k].Substituting this expression into the Lagrangian ℒ(𝒟,Λ), we obtain the Lagrange dual function 𝒟(Λ), and the optimal dual variables by using Newton's method.𝒟(Λ)=-∑_ℓ=1^kTr(𝒟^(ℓ) ^H𝒳^(ℓ)ℬ^(ℓ)^H)-k∑_j=1^rλ_j.Once getting the dual variables, the dictionary can be recovered using Equation (<ref>).The algorithm we proposed for GTSC is shown in Algorithm <ref> §.§ Complexity AnalysisGiven n images of size m× k, the numbers of bases r and q nearest neighbors, the computational complexities of GTSC and GraphSC <cit.> are as follows:For sparse representation learning, GTSC is based on an iterative shrinkage thresholding algorithm in the tensor space, and GraphSC is based on a feature-sign algorithm. The computational complexity for GTSC is O(rmkn+klogk) and for GraphSC<cit.> is O(r0^3 + r0^2mkn), where r0 is the number of non-zero coefficients.For dictionary learning, both GTSC and GraphSC<cit.> are based on Lagrange-dual algorithms, and the computational complexities are O(r^2n). For GTSC, the optimal dictionary is obtained slice by slice, and the computational complexity for each slice is O(r^3+r^2n+rmn). For GraphSC, the computational complexity is O(r^3+r^2n+rmnk).Overall, the computational complexity of GTSC is less than GraphSC<cit.>, especially for high dimensional data.§ EVALUATION We apply our proposed algorithms, TubSC in (<ref>) and GTSC in (<ref>) models, to image clustering tasks on four image databases: COIL20[http://www1.cs.columbia.edu/CAVE/software/softlib/coil-100.php], USPS[http://www.cad.zju.edu.cn/home/dengcai/Data/MLData.html], ORL[http://www.uk.research.att.com/facedatabase.html], Yale[http://www.cad.zju.edu.cn/home/dengcai/Data/FaceData.html]. The important statistics of these datasets are sumarized in Table <ref>. Two metrics, theaccuracy (ACC) and the normalized mutual information (NMI), are used for evaluations. ACC measures the percentage of correct labels obtained by an algorithm and NMI measures how similar two clusters are. The details of these two metrics can refer to <cit.>.§.§ Compared Algorithms To evaluate the clustering performances, we compare our proposed methods against the conventional SC <cit.> and GraphSC <cit.>. The performance scores are obtained by averaging over the 10 tests. For each test, we first apply the compared methods to learn new representations for images, and then apply K-means in the new representation space. For SC <cit.> and GraphSC <cit.> methods, PCA is used to reduce the data dimensionality by retaining 98% of the variance. The numbers of bases for USPS and YALE are set to 128, and those for COIL20 and ORL are set to 256, respectively. For our methods, we do not need to reduce the data dimensionality. Moreover, the numbers of bases are much smaller than those used in SC <cit.> and GraphSC <cit.>, due to the powerful representation generated from the tensor-product. For our methods, the numbers of bases are set to 45 for all data sets except YALE, which is set to 80. Based on the physical explanations of tensor sparse representations, we use C as the final image representation, which is defined as𝐂(i,j) = √(∑_ℓ^k𝒮(i,j,ℓ)^2).For GraphSC<cit.> and GTSC, we empirically set the graph regularization parameter alpha to 1 and the number of nearest neighbors to 3.§.§ Clustering ResultsTable <ref> shows the clustering results in terms of ACC and NMI. As can be seen, our GTSC algorithm performs the best in all the cases. TubSC performs much better than conventional SC, which indicate that by considering spatial proximity information of images, the learning performance can be significantly enhanced. Moreover, GraphSC outperforms SC, which shows that by encoding geometrical distribution information of the image space, the learning performance can also be improved.We would like to point out that we use much smaller sizes of dictionaries in our proposed models than SC and GraphSC, but without any dimensionality reduction preprocessing.For another aspect, we do not compare the clustering performances with TenSR <cit.> and CSC <cit.>, which also consider the spatial structures of images. Without dimensionality reduction, the representations learned from TenSR and CSC are larger than the original images, which significantly increase the computational complexity of clustering. § CONCLUDE In this paper, we propose a novel graph regularized tensor sparse coding (GTSC) model for image presentation, which explicitly considers both the spatial proximity information of images and geometric structures of the image space. GTSC is based on a novel tubal-tensor sparse coding (TubSC) model where the tensor encodings of TubSC have richer explanations than conventional sparse coding. The experimental results on image clustering have demonstrated that our proposed algorithm can have better representation power and significantly enhance the clustering performance. § ACKONWLEDGEMENTSThis work is supported by NSFC (No.61671290) in China, the Key Program for International S&T Cooperation Project (No.2016YFE0129500) of Chinaand partially supported by the Basic Research Project of Innovation Action Plan (No. 16JC1402800) of Shanghai Science and Technology Committee. IEEEbib | http://arxiv.org/abs/1703.09342v1 | {
"authors": [
"Fei Jiang",
"Xiao-Yang Liu",
"Hongtao Lu",
"Ruimin Shen"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170327233403",
"title": "Graph Regularized Tensor Sparse Coding for Image Representation"
} |
Tracing the Use of Practices through Networks of Collaboration Rahmtin Rotabi Department of Computer Science Cornell University Ithaca, NY, 14853Cristian Danescu-Niculescu-MizilDepartment of Information Science Cornell University Ithaca, NY, 14853Jon KleinbergDepartment of Computer Science Cornell University Ithaca, NY, 14853December 30, 2023 =========================================================================================================================================================================================================================================================================================================== An active line of research has used on-line data to study the ways in which discrete units of information—including messages, photos, product recommendations, group invitations—spread through social networks. There is relatively little understanding, however, of how on-line data might help in studying the diffusion of more complex practices—roughly, routines or styles of work that are generally handed down from one person to another through collaboration or mentorship. In this work, we propose a framework together with a novel type of dataanalysis that seeks to study the spread of such practices by trackingtheir syntactic signatures in large document collections. Central to this framework is the notion of an inheritance graph that represents how people pass the practice on to others through collaboration. Our analysis of these inheritance graphs demonstrates that we can trace a significant number of practices over long time-spans, and we show that the structure of these graphs can help in predicting the longevity of collaborations within a field, as well as the fitness of the practices themselves. § INTRODUCTIONOn-line domains have provided a rich collection of settings in which to observe how new ideas and innovations spread through social networks. A growing line of research has discovered principles for both the local mechanisms and global properties involved in the spread of pieces of information such as messages, quotes, links, news stories, and photos <cit.>, the diffusion of new products through viral marketing <cit.>,and the cascading recruitment to on-line groups <cit.>.A common feature in these approaches has been to trace some discrete “unit of transmission” that can be feasibly tracked through the underlyingsystem: a piece of text, a link, a product, or membership in a group. This is natural: the power of on-line data for analyzing diffusion comes in part through the large scale and fine-grained resolution with which we can observe things flowingthrough a network; therefore, to harness this power it is crucial for those things to be algorithmically recognizable and trackable.As a result, certain types of social diffusion have been particularly difficult to approach using on-line data—notably,a broad set of cascading behaviors that we could refer to as practices, which are a collection of styles or routineswithin acommunity that are passed down between people over many years, often through direct collaboration, mentorship or instruction. Particular stylistic elements involved in writing software,or performing music, or playing football, might all be examples of such practices in their respective fields. While complex practices are one of the primary modes studied by qualitative research in diffusion <cit.>, the challenge for large-scale quantitative analysis has been both to recognize when someone has begun to adopt a practice, and also to identify how it was transmitted to them.Tracking the Spread of Practices. A natural approach to tracking the spread of a practice is tofind a concretely recognizable “tag” that tends to travel with the practice as it is handed down from one person to another,rendering its use and transmission easily visible. A beautiful instance of this strategy was carried out byDavid Kaiser in his analysis of the use of Feynman diagrams in physics <cit.>. Feynman diagrams were proposed by Richard Feynman as a way toorganize complex physics calculations, and due to the technicalsophistication involved in their use, the initial spread of Feynman diagrams within the physics community proceeded in muchthe style described above, with young researchers adopting the practice through collaboration with colleagues who had already used it. In contrast to many comparable practices, Feynman diagrams had a distinctive syntactic format that made it easy to tell when they were being used.As a result, their spread could be very accurately tracked through the physics literature of the mid-20th-century. The result, in Kaiser's analysis, was a detailed map of how an idea spread through the field via networks of mentorship. As he writes:The story of the spread of Feynman diagrams reveals the work required to craft both research tools and the tool users who will put them to work. The great majority of physicists who used the diagrams during the decade after their introduction did so only after working closely with a member of the diagrammatic network. Postdocs circulated through the Institute for Advanced Study, participating in intense study sessions and collaborative calculations while there. Then they took jobs throughout the United States (and elsewhere) and began to drill their own students in how to use the diagrams. To an overwhelming degree, physicists who remained outside this rapidly expanding network did not pick up the diagrams for their research. Personal contact and individual mentoring remained the diagrams' predominant means of circulation even years after explicit instructions for the diagrams' use had been in print.<cit.> The Feynman diagram thus functions in two roles in this analysis: as an important technical innovation, and as a “tracking device” formapping pathways of mentorship and collaboration. If we want to bring this idea to a setting with large-scale data,we must deal with the following question: where can we find arich collection of such tracking devices with which to perform this type of analysis?We do not expect most objects in this collection to be technical advances comparable to the Feynman diagram, but we need a large supply of them, and we needto be able to mechanically recognize both their use and their spread. The present work: Diffusion of practices in academic writing. In this paper, we describe a framework for tracking the spread of practices as they are passed down through networks of collaboration, and we demonstrate a number of ways in which our analysis has predictive value for the underlying system. We make use of a setting where practices have the recognizability that we need—a novel dataset of latex macros in the e-print arXivrecently developed by Rotabi et al <cit.>. While the earlier work that developed this dataset used macros for other purposes (specifically, treating macros names as instances of naming conventions),macros in our context have a number of the key properties we need. First, a latex macro is something whose presence can be tracked as it spreads through the papers in the arXiv collection; we can thus see when an author first uses it,and when their co-authors use it. Second, while an arbitrary macro clearly does notcorrespond in general to an important technical innovation, a sufficiently complex macro often does encode some non-trivial technical shorthand within a concrete sub-field, and hence its use signifies the corresponding use of some technical practice within the field. And finally, there are several hundred thousand latex macros inpapers on the arXiv, and so we have the ability to track a huge number of such diffusion events, and to make comparative statements about their properties.If we want to use macros to trace the diffusion of practices between collaborators, we first need to establish whether macrosindeed spread via “inheritance” from co-authors: as with the Feynman diagram,can most of the initial set of uses of a macro trace a path back toa single early use through a chain of co-authorship? We find that this is true for a significant fraction of macros, by using an inheritance graph for each macro that records how each author's first use can be imputed to a co-authorship with an earlier user of the macro. Specifically, for each macro we can build a graph on the set of authors who have used it, and we include a directed inheritance edge from author u to author v if (i) u used the macro before v did, and (ii) v's first use of the macro is in a paper with u. We find that many of these inheritance graphs contain giant directed subtrees rootedat a single early use of the macro, indicating that a significant fraction of the users of the macro can indeedtrace a direct path back to a single shared early ancestor under this inheritance relation.These structures represent interesting instances of diffusion for several reasons.First, they are “organic” in a way that the spread of many on-line memes are not: when we studyon-line diffusion in settings where a user's exposure to content is governed by a recommendation system or ranking algorithm, there is the added complexity that part of the diffusion process is being guided by the internals of the algorithms underlying the system. With macros in arXiv papers, on the other hand, while authors may use automated tools to format the source of their papers, there is relatively little influence from automated recommendations or rankings in the actual decisions to include specific macros. Second, we are studying processes here that play out overyears and even decades; among other findings about the structure of our inheritance graphs, we observe that their diameters can take multiple years to increase even by one hop. We are thus observing effects that are taking place over multiple academic generations.The present work: Estimating fitness. If these inheritance graphs—obtaining by tracing simple syntactic signatures in the source files of papers—are telling us something about the spread of practices through the underlying community, then theirstructural properties may contain latent signals about the outcomes of authors, topics, and relationships. In the latter part of the paper, we show that this is the case, by identifying such signals built from the inheritance structures, and showing that they have predictive value.As one instance, suppose we wish to estimate the future longevity of a collaboration between two authors u and v—that is, controlling for the number of papers they have written thus far, we ask how many papers they will write in the future. If (u,v) is an edge of the inheritance graph for some macro, does this help in performing such an estimate? One might posit that since this edge represents something concrete that u passed on to v in their collaboration, we shouldincrease our estimate of the strength of the relationship and hence its future longevity. This intuition turns out not to be correct on its own: the existence of a (u,v) edge by itself doesn't significantly modifythe estimate. However, we find that something close to this intuition does apply. First, we note that since a (u,v) edge only means that a macro used by u showed up subsequently in a paper that u co-authored with v, it is providing only very weak information about v's role in the interaction. We would have a stronger signal if (u,v) were an internal edge of some inheritance graph, meaning that v has at least one outgoing edge; in this case, v was part of a paper that subsequently passed the macro on to a third party w. We find that if (u,v) is an internal edge of an inheritance graph, this does in fact provide a non-trivial predictive signal forincreased longevity of the u-v collaboration; informally, it is not enough that u passed something on to v, but that v subsequently was part of the process of passing it on to a third party w. In fact, we find something more: when (u,v) is an edge that is not internal (so that u's passing on of the macro “ends” at v), it in fact provides a weak predictive signal that the collaboration will actually have slightly lower longevity than an arbitrary collaboration between two co-authors (again controlling for the number of joint papers up to the point of observation).In what follows, we formalize this analysis and its conclusion. We also develop analyses through which macro inheritance can be used to help estimate the future longevity of an author—how many paperswill they write in the future? —and the fitness of an individual macroitself—how many authors will use it in the future?The remainder of the paper is organized into three main sections. We first briefly describe the structure of the data and how it is used in our analyses. We then formally define the inheritance graphs and survey some of their basic properties. Finally, we analyze the relation between these inheritance structures and the longevity of co-authorships, authors, and macros. § DATA DESCRIPTIONThe dataset we study contains the macros used in over one million papers submitted to the e-print arXiv from its inception in 1991 throughNovember 2015. The arXiv is a repository of scientific pre-prints in different formats, primarily in .Macros have two major components, the and the . Whenever the author uses \the compiler replaces it with the and compiles the text. In our study the serves as the “tracking device” discussed in the introduction, for studying how a macro is passed between collaborators over time. In general, when we refer to a “macro”, we mean a macro unless specified otherwise. For our study we use macro that have length greater than 20characters, and which have been used by at least 30 different authors. We apply the length filter so that we can focus on macros that are distinctive enough that we expect them to move primarily through copying and transmission, rather than independent invention. Further information can be found in <cit.>, which introduces this dataset.§ METHOD§ INHERITANCE GRAPHSDefining inheritance graphs.We begin by formally defining the inheritance graphs described in the introduction. For each macro m we create a graph (V_m, E_m) where V_m is the set of authors who have used macro m in at least one of their papers.We add a directed edge (u,v) to the edge set E_m ifthere is a paper that uses m with u and v as co-authors, such that (i) this is v's first use of m, but(ii) u has used m in at least one previous paper. This is the formal sense in whichm is being passed from u to v:v's first use of m occurs in collaboration with u, a prior user of m. Note that there can be multiple edges leading into a single node. For instance take a paper with authors u,v and z that uses macro m, andassume that u and v have used m before but z is using itfor the first time; then both the edges (u,z) and (v,z) are in the graph.Now, if all authors of a paper p are using m for the first time, then the nodes corresponding to these authors will not have any incoming edges. (Nodes of this form are the only ones with no incoming edges.)For each such paper p, we replace the nodes corresponding to the authors of p with a single supernode corresponding to p. We will refer to this as a source node, and to the authors of p as source authors.The resulting graph, with supernodes for papers where no author has used the macro before, and with author nodes for all others, is the inheritance graph G_m for the macro m. Because the process of inheritance, as defined, goes forward in time, G_m is necessarily a directed acyclic graph (DAG). Using these graphs we should be able to trace back a macro's life to its inception and to the authors who first used it.Note that there might be multiple source papers, and hence several groups of co-authors who independently serve as “origins” for the macro. For portions of the analysis where we are interested in looking at the number of authors who all follow from a single source paper, we will identify the source paper that has directed paths to the largest number of nodes in thegraph G_m.We will refer to this as the seed paper, and to the set of authors of this paper as the seed authors. (Note that the seed paper might not be the chronologically earliest paper to use the macro m; it is simply the one that can reach the most other nodes.)Analyzing the inheritance graphs. Our dataset contains several hundred thousand different macros, and as a first step we analyze the properties of the graphs G_m that they produce.In Figure <ref> we take three sample macros and show subsets of the breadth-first search (BFS) trees that are obtained starting from theseed paper. For example in Figure <ref> the graph is created on the macro, and the seed node is the paper astroph/9405052 with authors Xavier Barcons and Maria Teresa Ceballos.The seed paper used this macro in 1994, and some of the nodes at depth 6 in the BFS tree are from 2014—a 20-year time span to reach a depth of 6 in the cascading adoption of the macro. This reinforces the sense in which we are studying cascades that play out on a multi-generational time scale of decades, rather than the time scale of minutes or hours that characterizes many on-line cascades. The seed nodeof Figure <ref> is the paper hep-th/0106008 with authors Selena Ng and Malcolm Perry, and the seed node of Figure <ref> is the paper hep-ph/9302234 with authors Jose R Lopez et al.Since all other nodes in these BFS trees have incoming edges, they all correspond to individual authors who enter the graph at their first adoption of the macro, whereas the root node corresponds to a single paper and to the contracted set of authors of this paper.We now consider some of the basic properties of these inheritance graphs. First, each source paper has a reachable set in G_m—the set of nodes it can reach by directed paths—and recall that we defined the seed paper to be the source paper with the largest reachable set. In Figure <ref> we observe that a non-trivial fraction of the macros have a seed paper whose reachable set is a large fraction of all the authors who eventually adopt the macro. This provides a first concrete sense in which the inheritance patterns contained in G_m represent a global structure that spans much of the use of the macro m. In Figure <ref> and <ref> we show the properties of the graphs and nodes grouped based on the maximum depth of the BFS tree and thedepth of the individual nodes. Figure <ref> shows the average time it takes for the macro to get from the root to the nodes in each depth grouped by the maximum depth of the tree. This figure shows how these cascades can take multiple years to add a single level of depth to the tree, and a decade or more to reach their eventual maximum depth. In Figure <ref> we show the median width (number of nodes) of trees at each depth, again grouped by the maximum depth of the tree.Based on this plot we see that most of these trees have are narrow in their top and bottom layers, with fewer nodes, and are wider in the middle.The plots thus far have been concerned with the global structure of the inheritance graph and its shortest paths as represented bybreadth-first search trees. Now we take a deeper look at the properties of individual edges in the graph. For this we will first define the notions of local and global experience, and we will use these two terms throughout the paper.At time t the global experience of an author is the number of papers the respective author has written. At time t the local experience of an author is defined with respect to a macro m and is the number of papers up to time t in which the author has used m. This is a version of the notion of local experience relative to an arbitrary term, as used in <cit.>. Consider an edge (u,v) in the inheritance graph for a macro m. At the moment when the macro is passed from u to v, the local experience of v with respect to m is 0 by definition, and the local experience of u with respect to m is greater than 0. What do we expect about the global experience of these two nodes? To the extent that passing on a macro is a form of “teaching” from one person to another, we may expect the global experience of u (the “teacher”) to be higher than the global experience of v (the “learner”).On the other hand, there is a history ofsociological work in the diffusion of innovations suggesting that innovations often originate with outsiderswho come from the periphery of the system <cit.>, which would be consistent with v having higher global experience than u. Figure <ref> addresses this question by showing the cumulative distribution of the global experience difference between u and v. The median experience difference is clearly shifted in the positive direction, consistent with the “teacher” node u having the higher global experience in general.§ FITNESSNow that we have some insight into how the information diffusion process unfolds in our data, we investigate whether these inheritance structures can provide predictive signal for the outcomes of co-authorships, authors, and the macros themselves. In all cases we will think in terms of the fitness of the object in question—the extent to which it survives for a long period of time and/or produces many descendants. §.§ Fitness of collaborations We start by considering the fitness of collaborations—given two authors u and v who have written a certain number of papersup to a given point in time, or perhaps who have not yet collaborated, can we use anything in the structure of macro inheritance to help predict how many more papers they will write in the future?A natural hypothesis is that if v inherits macros from u, then this indicates a certain strength to the relationship (following the teacher-learner intuition above), and this may be predictive of a longer future history of collaboration. To examine this hypothesis, we perform the following computational test as a controlled paired comparison. We find pairs of co-authorships u-v and u'-v' with properties that (i) neither pair has collaborated before, (ii) their first co-authorship happens in the same month, (iii) (u,v) is an edge in an inheritance graph, and (iv) (u',v') is not. (Note that since we are looking at pairs of co-authorships, we are looking at four authors in total for each instance: u, v, u', and v'.) Now we can ask, aggregating over many such pairs of co-authorships, whether there is a significant difference in the future number of papers that these pairs of authors write together. (Since their initial co-authorships took place in the same month, they have a comparable future time span in which to write further papers.)In fact, we find that there isn't a significant difference, at odds with our initial hypothesis about macro inheritance. However, there is more going on in the inheritance structure that we can take advantage of. We divide the edges of the inheritance graphs into two sets: internal edges (u,v), where the node v has at least one outgoing edge, and terminal edges (u,v), where the node v has no outgoing edge. Internal edges add extra structural information, since theyindicate that not only u passed the macro m to v, but that v was then part of the process of passing m in a collaboration subsequent to the one in which they originally inherited it. We find that the fitness of u-v co-authorships is significantly higher when (u,v) forms an internal edge, in contrast to the lack of effect when (u,v) is an arbitrary edge. We evaluate this using an extension of our previous paired comparison: in conditions (i)-(iv) above for forming pairs of co-authorships, we replace conditions (iii) and (iv) with the following: * Internal edge vs. arbitrary co-authorship: (iii) (u,v) is an internal edge and (iv) (u',v') is not an edge.* Internal edge vs. terminal edge: (iii) (u,v) is an internal edge and (iv) (u',v') is a terminal edge.* Terminal edge vs. arbitrary co-authorship: (iii) (u,v) is a terminal edge and (iv) (u',v') is not an edge.In each of these three settings, we look at the fraction of times that one of the categories produced the co-authorship with morefuture papers.In our paired setting, if we were to draw two co-authorships uniformly at random over all possible co-authorships (without regard to the type of the edge), there is a 50% chance that the first would produce the higher number of future papers. Thus,we can calibrate each of the three comparisons listed above using this 50% baseline. Figure <ref> shows these results, grouped into two-year bins: we find that internal edges win a large fraction of the comparisons against each of the other two categories,whereas there is little difference between terminal edges and arbitrary co-authorships. §.§ Fitness of authors We now consider the fitness of the authors themselves; we will show that the way authors use macros can provide a weak but non-trivial signal about how many papers they will eventually write, a quantity that we refer to as the fitness of the author.The particular property we consider is a type of “stability” in the usage of the macro. For a given macro , there are many possible that can be used for it, and authors differ in the extent to whichtheir papers preserve a relatively stable choice of names for the same macro body: some almost always use the same name, while for other authors the name changes frequently. (For example, an author who almost always uses the namefor the macro body ,versus an author whose papers alternate between using , , , , and others, all for this same macro body.) We could think of the first type of author as exerting more control over the source of her papers than the second type of author,and this distinction between the two types of authors—based on their behavior with respect to macros—naturally raises thequestion whether the stability of macro names could provide predictive value for author fitness.Here is how we formally define this measure. For a particular author a, we say they change the name of macro m on paper p if the previous time they used m's macro body, the name was different. Then, for a set of authors A and a set of macros M, we define f(A,M,x) to be the probability of an author in A changing the name of a macro in M the x^ th timethey use it. We considerthis name-change probability, f(A,M,x) for x ∈ [0,40] and different groups of authors and macros. In particular we look at groups of authors that have more than θ papers in the entire corpus. We set θ to be 40, 50, …, 130 and we let Mrange over three possible sets: the set of all macros;the set of wide-spread macros (more than 250 authors use the macro body); and the set of narrow-spread macros(at least 20 authors used it and at most 250). One source of variability in this analysis is that even once we fix the minimum number of papers θ written by an author a, as well as the usage number x of the macro m that we are considering, it is still possible that author a's x^ th use of the macro might come toward the end of their professional lifetime or early in their professional lifetime. (It must come at the x^ th paper they write or later, since they need time to have used the macro m a total of x times, but this is all we know.) It is easy to believe that authors who use a macro in their early life stages might exhibit different phenomena from those who use it in a later life stage. Therefore, in addition to the measures defined so far, we also consider analyses involving only the set of macro uses thatcome early in the authors' professional lifetime—specifically only macro uses that happen in the author's first 40 papers.The results for all these settings are shown in Figure <ref>: the threesets of macros (all macros, wide-spread macros, and narrow-spread macros);for each of these sets, we consider both the authors' full lifetimes and just their early life stages. In each case,the x-axis shows the number of macro uses (i.e. the authors' local experience with respect to the macro), and the different curves represent authors grouped by different values of the minimum number of papers θ. This suggests that overtime authors build a certain “loyalty” to the names they have used consistently; this is consistent with our previous findings regarding the competition between macro-naming conventions<cit.>.But we also find something else: that (eventually) more prolific authors (larger θ) have a lower name-change probability (compare ordering of curves ineach subplot of Figure <ref>. This suggests that the macro name change probability might be a signal with predictive value for author fitness (which, again, we define as the number of papers the author will eventually write).To test this idea, we set up an author fitness prediction task as follows. For a given minimum number of papers θ we consider the low-fitness authors to be the ones with fitness below the 20^th percentile and high-fitness authors to be those above the 80^th percentile. We then see whether simply using the frequency with which an author changes macro names in the first θ papers can serve as a predictor for this two-class problem: whether an author's fitness is below the 20th percentile or above the 80th percentile.By using the probability of macro name changes, we are able to predict which of these two classes an author belongs to with a performance that exceeds the random baseline of 50% by a small but statistically significant amount. Figure <ref> shows the performance for different values of θ. We emphasize that predicting an author's fitness is a challenging task for which one doesn't expect strong performance even from rich feature sets; this makes it all the more striking that one can obtain non-trivial performance fromthe frequency of macro name changes, a very low-level property about the production of the papers themselves.Moreover, for settings involving large values of θ the name-change probability is more predictive than an arguably more natural structural feature: the author's total number of co-authors (Figure <ref>).We also note that in such settings the name-change feature also outperforms other more direct macro-based features, such as the total number of macros used, or total number of distinct macro bodies used. §.§ Fitness of macros Finally, we consider the fitness of the macros themselves. We define the fitness of a macro to be the total number of authors who eventually use the macro body, and investigate which features are predictive of this variable.We set up a prediction task as follows. We first find all macros that get adopted by at least k authors.Each of these macros has a fitness (of at least k), and we define σ(k) to be the median of this multiset of fitness values: of all macros that reach at least k authors, half of them have a fitness of at most σ(k), and half of themhave a fitness of at least σ(k). In table <ref> we report σ(k) and the number of macro instances for different values of k.We can thus use σ(k) to construct a balanced prediction task, in the style of the cascade prediction analyses from <cit.>. For a given macro that reaches at least k authors,we observe all the information on the papers and authors up to the point at which the k^ th author adopts the macro,and the task is then to predict if this macro will eventually reach σ(k) authors. We learn a logistic regression model for different values of k and report the accuracy in Figure <ref> on an 80-20 train-test split.[We can achieve a 1% to 4% better accuracy by using a non-linear classifier such as decision trees, but we opt to use the more interpretable model.]We use the following features. * Features related to the diffusion speed of the macro:the number of papers that the macro needs in order to reach k/2 and k distinct authors; and the number of months that the macro needs in order to reach k/2 and k distinct authors.* Experience of the macro users: the average usage experience of the first k authors who adopted it.* Structural features of the macro users: the local and global clustering coefficients of the co-authorship graph on the first k authors to use the macro.* Structural features of the macro body: the length of the macro body, the number of dollar signs in the macro (generally used for mathematical notation), the number of non-alphanumerical characters, and the maximum depth of nested curly brackets. In Figure <ref> we show the predictionperformance for different subsets of these features, as a function of k; note that performance increases with increasing k. As observed above, predicting macro fitness is a problem whose syntactic form is closely analogous to the prediction of cascade size for memes in social media <cit.>; given this, and the fact that the spread of macros plays out over so much longer time scales, and without the role of ranking or recommendation algorithms, it is interesting to notethe similarities and contrasts in the prediction results. One of the most intriguing contrasts is in the role of diffusion speed features: for cascade prediction in social media, the speed features alone yielded performance almost matching that of the full feature set, and significantly outperforming the set of all non-speed features<cit.>. For our domain, on the other hand, the speed features perform 5-10% worse than the full feature set; they also perform worse for most values of k than the set of all non-speed features. This suggests that for macro fitness, the speed featuresare considerably less powerful than they are in the social media context, indicating that there may be more to be gained from the synthesis of a much broader set of features.§ CONCLUSIONS The spread of practices between collaborators is a challenging form of diffusion to track, since one needs to be able torecognize when someone has begun using a practice, and how it was conveyed to them.Motivated by work that used the Feynman diagram as an easily recognizable “tracer” of a complex practice <cit.>, we track the spread of several hundred thousand macros through the papers of the e-print arXiv over a 25-year period. Long macros often serve as technical shorthand within a defined sub-field, and their syntactic precision makes it easy to follow their flow through the collaboration network. We construct inheritance graphs showing how the macrospread between collaborators, andwe find that many macros have a clear “seed set” of authors with the property that a large fraction of the subsequent users of the macro can trace a direct inheritance path back to this seed set. The resulting diffusion patterns are intriguing, in that they span multiple academic generations and several decades, and unlike cascades in social media, the spread of these macros takesplace with very little influence from ranking or recommendation algorithms.We also find that properties of macro inheritance provide signals that are predictive for larger-scale properties that have nothing to do with macros.These include predictions about the longevity ofcollaborations and the number of papers that an author will write over their professional lifetime on the arXiv.Our work suggests a number of directions for future research. First, it would be interesting to develop a comparative analysis between the structure of our inheritance graphs and thecorresponding structures for the diffusion of on-line memes. Are there systematic ways in which the two types of diffusion patterns differ, and can these be connected to differences in the underlying mechanisms? Second, we believe that there may well be additional links between inheritance structures and prediction problems for the trajectory of the overall system; for example, can we evaluate the future course of larger sub-areas based on the inheritance patterns that exhibit? And finally, identifying “tracers” for complex practices is a style of analysis that can applied in other domains as well; as we broaden the set of contexts in which we can perform this type of analysis, we may better understand the ways in which the flow of practices helps reinforce and illuminate our understanding of large collaborative communities.AcknowledgmentsWe are grateful to Paul Ginsparg for his valuable advice and for his help with the arXiv dataset. We would like to thank Peter Lepage for introducing us to David Kaiser's analysis of the diffusion of Feynman diagrams, whichformed the basis for our thinking about the approach in this paper. This work was supported in part by ARO, Facebook, Google, the SimonsFoundation, and a Discovery and Innovation Research Seed Award from Cornell's OVPR. abbrv | http://arxiv.org/abs/1703.09315v1 | {
"authors": [
"Rahmtin Rotabi",
"Cristian Danescu-Niculescu-Mizil",
"Jon Kleinberg"
],
"categories": [
"cs.SI",
"physics.soc-ph",
"H.2.8"
],
"primary_category": "cs.SI",
"published": "20170327213059",
"title": "Tracing the Use of Practices through Networks of Collaboration"
} |
Distributions of a particle's position and their asymptotics in the q-deformed totally asymmetric zero range process with site dependent jumping rates Eunghyun LeeDepartment of mathematics, Nazarbayev University, Astana, 010000 Kazakhstan, <[email protected]>, Supported by the Social Policy Grant of Nazarbayev University.Dong WangDepartment of Mathematics, National University of Singapore, 119076 Singapore, <[email protected]>. Support partially by the Singapore AcRF Tier 1 grant R-146-000-217-112. December 30, 2023 ========================================================================================================================================================================================================================================================================================================================================================================== In this paper we study the probability distribution of the position of a tagged particle in the q-deformed Totally Asymmetric Zero Range Process (q-TAZRP) with site dependent jumping rates. For a finite particle system, it is derived from the transition probability previously obtained by Wang and Waugh. We also provide the probability distribution formula for a tagged particle in the q-TAZRP with the so-called step initial condition in which infinitely many particles occupy one single site and all other sites are unoccupied. For the q-TAZRP with step initial condition, we provide a Fredholm determinant representation for the probability distribution function of the position of a tagged particle, and moreover we obtain the limiting distribution function as the time goes to infinity. Our asymptotic result for q-TAZRP with step initial condition is comparable to the limiting distribution function obtained by Tracy and Widom for the k-th leftmost particle in the asymmetric simple exclusion process with step initial condition (Theorem 2 in Commun. Math. Phys. 290, 129–154 (2009)). § INTRODUCTION The Zero Range Process (ZRP) is the system of identical particles on a countable set S whose dynamics is described by the following rules (if the sites are homogeneous): if a site x∈ S is occupied by k particles, then one of k particles leaves x after an exponential waiting time with rate g(k). The particle leaving x chooses a target site y with probability p(x,y) and then it immediately moves to y.As a special case of the ZRP, the system of N particles on S=ℤ with p(x,x+1) = 1 and g(k) = (1-q^k)(1-q) with q∈ (0,1), is an integrable model originated from the q-boson model introduced by Sasamoto and Wadati <cit.>. In this paper, we will call this special case of the ZRP the (spatial-homogeneous) q-deformed Totally Asymmetric Zero Range Process (q-TAZRP). The integrability of the q-TAZRP was studied by Povolotsky <cit.> and its transition probability was obtained by Korhonen and Lee <cit.>. The q-TAZRP is also a special case of the zero-range chipping model introduced by Povolotsky <cit.> in which multiple particles are allowed to jump at the same time and a jumping rate depends on the number of particles jumping as well as thenumber of particles at the departing site. The q-TAZRP can be generalized to let the jumping rates depend on not only the number of particles at the departure site, but also the site x itself, so that the jumping rate at x occupied by k particles is given byg (k; x) = a_x(1-q^k).The finite system with the rate function (<ref>) is well defined if a_x > 0, and we assume that all a_x are in a compact subset of (0,∞). The transition probability of this model with finitely many particles was obtained by Wang and Waugh <cit.> as an extension of Korhonen and Lee's work <cit.>. We note that the transition probability can also be derived as a degeneration of the higher spin stochastic six vertex model studied by Borodin and Petrov <cit.>. Although the q-TAZRP with spatial-inhomogeneous jumping rate (that we refer to simply as q-TAZRP throughout the paper) is not much studied in literature, its dual model, the q-deformed Totally Asymmetric Simple Exclusion Process (q-TASEP) with particle-dependent jumping rate, has been extensively studied in <cit.>,<cit.>,and <cit.>.The aim of this paper is two-fold: First for an arbitrary initial condition in the N-particle q-TAZRP, we find the explicit distribution function for the position of a tagged particle (as the n-th rightmost or leftmost particle) at time t > 0 if the jumping rates are given by (<ref>). In <cit.> the transition probability foran N-particle system was obtained from which we derive the distribution of a tagged particle's position as amarginal probability. The result is given in Theorem <ref>. We also find the distribution function for a tagged particle's position under the so-called step initial condition which is such that initially infinitely many particles occupy a single site and all other sites are empty, as shown in Theorem <ref>. Under the step initial condition, the distribution of a tagged particle's position is represented by a contour integral of the Fredholm determinant of a certain trace class operator. We also study the asymptotic behaviour of this Fredholm determinant representation when t goes to infinity.This result is given in Theorem <ref>. We note that although the distribution of the m-th rightmost particle at time t in the limit that both m and t go to infinity can be obtained by the known results for the q-TASEP, as discussed in Appendix <ref>, the limiting distribution when m is fixed is new. Since our limiting distribution agrees with that for the Asymmetric Simple Exclusion Process (ASEP) (<cit.>), these results may describe a new universal behaviour of interacting particle systems. §.§ Definition of the ModelFirst, we consider a system that consists of N identical particles on ℤ.We label particles 1,…,N from the rightmost particle. We assume that if more than one particle occupy the same position, they are ordered vertically such that the label of a particle at a higher position is smaller than the label of a particle at a lower position. We interpret the particle labelled m as the m-th rightmost particle. We denote by x_m(t) the position of the m-th rightmost particle at time t, and t may be omitted if it is not necessary to specify the time. By abuse of notation, we also use x_m to denote the m-th particle itself. The state of an N-particle system is represented byX=(x_1, …, x_N) ∈ := {(i_1,…, i_N) ∈ℤ^N | i_1 ≥ i_2 ≥…≥ i_N }and a state at time t is denoted by X(t) = (x_1(t),…, x_N(t)). Alternatively, a state can be specified by the number of particles at each site. We denote the number of particles at site k (at time t) by n_k (or n_k(t)).Hence, a state X(t) is equivalently expressed as (n_k(t))_k ∈, which satisfies ∑^∞_k = -∞ n_k(t) = N.The rules of the model are a specialization of the rules for the general ZRP explained in the beginning of this paper. But since we label the otherwise identical particles, the rules can be described more concretely. We fix q ∈ (0,1). If a site x is occupied by k particles, then the particle at the top of the particles (the particle itself if a site is occupied by only one particle) jumps to the bottom of the particles at x+1 (if there are other particles at x + 1) after an exponential time with rate a_x(1-q^k) where a_x>0 for all x ∈ℤ.In this procedure only the particle at the top may jump after an exponential time and other particles do not respond to the exponential clock. If the local state at x is changed, that is, if the number of particles at x is changed from k to k', then the exponential clock is reset to have the new rate a_x(1-q^k') and the particle at the top jumps after a new exponential time.Also, we assume that all waiting times are independent. We denote by _Y the probability measure with an initial state X(0) = Y = (y_1,…, y_N) of the process.We also consider an infinite system as a limiting case of N-particle system. An infinite particle q-TAZRP is well defined by <cit.> if we let the transition rules there be specified as P(x, y) in <cit.> is equal to δ_x + 1, y and φ(m) in <cit.> depends on x in the way φ(m; x) = a_x(1 - q^m)/m. (In <cit.> the ZRP is assumed to be spatial-homogeneous, but it is straightforward to generalize the proof to our spatial-inhomogeneous case.) In plain words, if site x is occupied by m < ∞ particles, a particle jumps out of site x at rate a_x(1 - q^m), which is the Γ(m) defined in <cit.>, a particle jumps out of site x to site x + 1, and If site x is occupied by infinitely many particles, the rate for one particle to jump out of x to x + 1 is naturally defined, corresponding to the Γ(∞) defined in <cit.>, as a_x(1 - q^∞) = a_x. In all q-TAZRP models considered in our paper, the infinite particles are labelled by consecutive integers. We assume that at any time t ≥ 0, among all particles at site x, particles are stacked vertically with x_m above x_n if m < n, and further assume that the particle on the top (that with the smallest label) exists. Furthermore, we assume that this top particle jumps out of x to x + 1, and stays below all existing particles there. Hence at all time t ≥ 0, x_m(t) ≥ x_n(t) if m < n. The q-TAZRP with step initial condition, which is the infinite particle model analyzed in this paper, is defined in the following subsection. §.§ Relationship between q-TASEP and q-TAZRP, and the height functionIn general, a ZRP onhas a particle-spacing duality with a simple exclusion process on<cit.>. In particular, the q-TAZRP considered in our paper is dual to the q-TASEP introduced by Borodin and Corwin <cit.>. The intuitive meaning of this duality is simple: we interpret the number of particles at site k in the q-TAZRP as the spacing between the k-th and (k - 1)-th particles in the q-TASEP. Below we describe the duality in precise terms and focus on the construction of the q-TASEP from the q-TAZRP. The inverse construction can be done in the same way, and we omit it since it is not relevant in our paper.It is better to consider the duality between q-TAZRP and q-TASEP with infinitely many particles, rather than the finite-particle system defined above in our paper. Suppose that initially no site is occupied by infinitely many particles andthere are infinitely many particles to the left and to the right of x, respectively, for each x ∈ℤ. In this case it is impossible to say the “rightmost” particle and the “leftmost” particle, and in probability 1 no site is occupied by infinitely many particles at any finite time t ≥ 0. Let us label the particles at the initial time t = 0 as follows. We find the closest occupied site on (-∞,0] to the origin, and call a particle at this site x_1. If there are more than one particle at this site, since we assume that they are vertically stacked and the top particle exists, we call the particle at the top x_1. By the ordering method introduced in section <ref> we label the other particles at the same position as that of x_1 and the particles to the left of x_1 as x_2, x_3, … consecutively, and label the particles to the right of x_1 as x_0, x_-1, … consecutively. Hence, at any time t ≥ 0, we express a state X(t)=(…, x_-1(t), x_0(t), x_1(t), x_2(t), x_3(t), …) as a bi-infinite sequence whose terms are in weakly decreasing order, and x_n(t) →∓∞ as n →±∞. There is an alternative unique parametrization (n_k(t))_k ∈ with n_k(t) = #(particles at site k at time t) < ∞ and ∑^∞_k = -∞ n_k(t) = ∞. Let X(t) and (n_k(t))_k ∈denote the state of the particles in the q-TAZRP described above. Then we define a bi-infinite strictly decreasing sequence Y(t) = (y_k(t))_k ∈ recursively byy_k - 1(t) - y_k(t) = n_k(t) + 1,y_0(0) = k_0 for some k_0 ∈,and for t > 0y_0(t) = k_0 + #(particles in the q-TAZRP X(t) that move from 0 to 1 during time (0, t)).If we interpret y_k(t) as the position of the particle labelled k in an infinite particle system onat time t, the dynamics of this particle system can be described as follows: Each particle has an exponential clock that rings independently such that the clock for theparticle labelled k has ratea_k (1 - q^n_k(t)) = a_k (1 - q^y_k - 1(t) - y_k(t) - 1). When the clock rings, the particle moves one step to the right, see Figure <ref>. (If the right neighbour site is blocked by another particle, we have that the exponential clock rings at rate 0, and the particle does not move.) This is exactly the q-TASEP model defined in <cit.> with infinitely many particles without the leftmost paritcle or the rightmost particle. (In <cit.>, the rightmost particle exists, though the particle number can be infinity.)If the q-TAZRP model has infinitely many particles and the rightmost particle, which is denoted by x_1(t), exists, then X(t) = (x_k(t))_k ∈_+ = (x_1(t), x_2(t), …) is uniquely described by (n_k(t))_k ∈. We also assume that n_k(t) < ∞ for all k ∈ and t ≥ 0as before. The corresponding q-TASEP is described by (<ref>) and (<ref>), and has the feature that sites of ℤ far to the right are densely packed with particles at any given time t, since n_k(t) = 0 for k ≫ 0. Moreover, if there areN < ∞ particlesin the q-TAZRP so that we may denote by x_1(t) and by x_N(t) the positions of the rightmost particle and the leftmost particle, respectively, then again the corresponding q-TASEP is described by (<ref>) and (<ref>). In this case the corresponding q-TASEP has infinitely many particles, and sites of ℤ far to either side are densely packed with particles at any given time t.We note that in all the forms of the correspondence between q-TAZRP and q-TASEP described in the above, thenumber of particles in the q-TASEP is infinite, and we cannot specifythe rightmost particle and the leftmostparticle. But, if in the q-TAZRP a unique site, say site 0, is occupied by infinitely many particles and all sites to its left are empty, then the corresponding q-TASEP has the rightmost particle. This correspondence is realized by the q-TAZRP with step initial condition, and we explain it below.Suppose Y(t) gives a q-TASEP model. Let the so-called height function h(x; t) associated to Y(t) be a continuous and piecewise linear function in x such that it takes integer values at k + 1/2 for k ∈, and linearly interpolate between k - 1/2 and k + 1/2. h(x; t) is defined uniquely by Y(t) thath(-1/2; t) = #(particles that passes from -1 to 0 between time 0 and t in the q-TASEP),and for any integer k we haveh(k + 1/2) - h(k - 1/2) =-1if site k is occupied at time t in the q-TASEP, 1if site k is empty at time t in the q-TASEP.The graph y = h(x; t) in the xy plane is a Dyck path, see Figure <ref>. It is clear that the Dyck path has a 1-1 correspondence with the q-TASEP model, and as time t increases, the Dyck path monotonically moves upward. Actually the move of the Dyck path is in the KPZ universality class, and has been studied by <cit.>, <cit.>, <cit.>, <cit.> and <cit.>.A special state of the q-TASEP, which can be realized only as an initial condition at time t = 0, is that infinitely many particles are labelled as y_0, y_1, y_2, … such that y_i+1 = y_i-1. The Dyck path corresponding to this configuration of q-TASEP has the shape like a step, so this configuration is called the step initial condition for q-TASEP. Since this configuration of q-TASEP is constructed by the configuration of q-TAZRP that consists particles x_1, x_2, x_3, … with x_k = 0 for all k, we call this configuration also the step initial condition for q-TAZRP, especially if it is at the initial time t = 0.In the q-TASEP with the step initial condition, the generating function 𝔼(q^y_m(t)) of the m-th rightmost particle's position where the probability distribution of y_m(t) is encoded was obtained in<cit.> and <cit.>. Furthermore, it was shown that an e_q-Laplace transform of q^y_m(t) is the Fredholm determinant of an integral operator so that the distribution of y_m(t) is obtained from the inversion formula of the e_q-Laplace transform. In the q-TAZRP, we directly obtain the probability distribution of x_m(t) based on Wang and Waugh's transition probability <cit.> as in Tracy and Widom's approach in <cit.> and <cit.>. In the end of this subsection we consider the relation between the distribution of particles in the q-TAZRP with step initial condition x_k(0) = 0 for all k ≥ 1, and the distribution of particles in the corresponding q-TASEP with step initial condition y_k(0) = -k for all k ≥ 0. At any time t > 0, we have that for all integers x ≥ 0,∑^∞_k = x + 1 n_k(t) = y_x(t) + x.Since x_m(t) > x is equivalent to ∑^∞_k = x + 1 n_k(t) ≥ m, we have that x_m(t) > x if and only if y_x(t) + x ≥ m. We conclude that_0^∞(x_m(t) > x) = ^_(y_x(t) ≥ m - x),where the two sides are the probabilities that x_m(t) > x and y_x(t) ≥ m - x respectively in the q-TAZRP/q-TASEP with step initial condition given above. §.§ Main results First we state the result on the distribution function for a single particle x_n(t) in the N-particle q-TAZRP, under the initial condition that Y = (y_1, …, y_N). This result is based on transition probability P_Y(X;t) obtained by Wang and Waugh <cit.>, which is in turn a generalization of the result by Korhonen and Lee <cit.> in the homogeneous case. We first define notations to be used later in this paper.First, throughout this paper, we letb_x = (1 - q)a_xfor all x ∈. For a state X = (x_1, …, x_N) expressed in terms of (n_k)_k ∈, letW(X) = ∏_k ∈[n_k]_q!,where [m]_q! is the q-deformed factorial, defined by[m]_q! = [1]_q[2]_q⋯ [m]_q = ∏^m_k = 11-q^k/1-q.Note that all but finitely many n_k are 0, and the factor [0]_q! = 1, so the infinite product in (<ref>) is well defined.For a permutation σ∈ S_N, an inversion of σ is an ordered pair (σ(i),σ(j)) such that i<j and σ(i)>σ(j). Let S(w_α,w_β) = -qw_β - w_α/qw_α - w_βwhere w_α and w_β are complex variabls andA_σ(w_1,…, w_N) = ∏_(β,α) is an inversion of σ S(w_α,w_β).We define A_σ(w_1,…, w_N) = 1 if σ is the identity permutation. For notational simplicity we define ∏^' as an extension of ∏. If f(k) is defined for all k ∈, then'∏_k=m^n f(k) = ∏_k=m^nf(k)if n ≥ m,1if n = m - 1, ∏_k=n+1^m-11/f(k) if n < m - 1.The integral signis a shorthand for 1/2 π i∮. <cit.> Given the notations in the above, the transition probability of the q-TAZRP with rates (<ref>) is given byP_Y(X;t) = 1/W(X)∏^N_k=1-1/b_x_k∑_σ∈ S_N_C dw_1 ⋯_C dw_N A_σ(w_1, …, w_N)×∏^N_j=1[ ^x_j_k=y_σ(j)( b_k/b_k - w_σ(j)) e^-w_jt],where C is a counter-clockwise circle centered at 0 with sufficiently large radius enclosing all singularities b_k. The first main result of this paper is the marginal distribution for the n-th particle's position at time t > 0. We provide the probability in two forms which are reminiscent of Tracy and Widom's results in <cit.> for the ASEP. First define for any r variables w_1, …, w_rB_r(w_1, …, w_r) = ∏_1 ≤ i < j ≤ rw_i - w_j/qw_i - w_j.Then for the subset S={s_1,…, s_r}⊆{1,2,…, N} with | S | = r, we define for n ≤ rc_S(n) = (-1)^n q^n(n - 1)/2 - nr + ∑^r_i = 1 s_ir - 1n - 1,c̃_S(n) = (-1)^n q^n(n - 1)/2 + r(r - 1)/2 + ∑^r_i = 1 s_ir - 1n - 1,where q-binomial coefficient nk is defined bynk = [n]_q!/[k]_q![n-k]_q!.LetI(w_s_1, w_s_2, …, w_s_r;M, t) = ∏_1 ≤ i < j ≤ r B_r(w_s_1, …, w_s_r) ∏^r_i=1[ ^M_k = y_s_i( b_k/b_k - w_s_i) e^-w_s_it].Given the notations in the above,_Y(x_n(t) > M) = ∑^N_r = n (-1)^r×∑_S = { s_1 , …, s_r }⊆{ 1, …, N } and | S | = r (-1)^r c_S(n) _Cdw_s_1/w_s_1⋯_Cdw_s_r/w_s_r I(w_s_1, …, w_s_r;M, t),where the contour C is a positively oriented circle centred at 0 and enclosing all b_k._Y(x_N - n + 1(t)≤ M) = ∑^N_r = n q^-rN ×∑_S = { s_1 , …, s_r }⊆{ 1, …, N } and | S | = rc̃_S(n) _C̃_1dw_s_1/w_s_1⋯_C̃_rdw_s_r/w_s_r I(w_s_1, …, w_s_r;M, t),where the contour C̃_j is a positively oriented simple closed curve which encloses all b_k but does not enclose 0, and C̃_j encloses qC̃_i if j>i.We note that the n = N case of (<ref>), which is stated separately in Proposition <ref>, is similar to the q-monent formulas for the q-TASEP and more generally the q-Whittaker processes, see <cit.> and <cit.>. The n = N case of (<ref>), which is stated separately in Proposition <ref>, generalizes <cit.> that is the homogeneous case of Proposition <ref>.If the initial condition Y isY = 0^N = (0, …, 0), then it is clear that the formulas (<ref>) and (<ref>) can be simplified. Moreover, if we consider the limiting case of 0^N = (0, …, 0), that is, the step initial condition, Y = (y_1, y_2, …) = 0^∞ := (0, 0, …), then the probability _0^∞(x_m(t) > M) can be expressed by a contour integral of a Fredholm determinant. To this end, we define the integral operator K_M, t from L^2(Γ) to L^2(Γ) where Γ is a large enough contour enclosing 0, such that(K_M,tf)(w) = 1/2π i∮_Γ K_M,t(w,w')f(w') dw'with kernelK_M,t(w,w') = e^-wt/qw'-w∏_k=0^M ( b_k/b_k - w)whose poles b_0, b_1, …, b_M are enclosed in Γ, and the scaled down contour qΓ is enclosed in Γ. It is clear that K_M, t is a trace class operator and the Fredholm determinant (I + ζ K_M, t) is well defined for all ζ∈. Here we do not need to specify the shape of Γ, since later we only need the Fredholm determinant (I + ζ K_M, t) that is invariant when Γ is continuously deformed, as shown in <cit.>.We note that the integral operator K_M, t appears also in the study of q-TASEP in the so-called Cauchy-type determinant, see <cit.>, and see <cit.> for an analogous “large contour formula” for the q-Whittaker processes. For the q-TAZRP,_0^∞(x_m(t) > M) = 1/2π i∮_C dζ/ζ(I + ζ K_M,t)/(1- ζ)(1-qζ)⋯ (1-q^m-1ζ),where C is a positively oriented circle centred at 0 large enough to enclose that all singularities 1, q^-1, …, q^1 - m.Here we remark that the probability _0^∞(x_m(t) > M) depends on the values of b_k with 0 ≤ k ≤ M, but not the order of them. This symmetry is not obvious from the definition of the model.Below we give the asymptotics of the probability _0^∞(x_m(t) > M) ast →∞ when m is fixed and M is scaled accordingly. We consider only a special case that all but a fixed number of b_k are equal. Since _0^∞(x_m(t) > M) is invariant if b_k and b_j (0 ≤ k, j ≤ M) are swapped, we may assume the spiked b_k are b_0, b_1, …, b_l. This setting is analogous to the q-TASEP model with all but a fixed number of particles having the same speed parameter, as considered in <cit.>. SupposeM = n + l + 1,t = n - τ√(n) b_k = β_k/√(n)for k = 0, 1, …, land b_k = 1 for all k > l. Let l, β_0, …, β_l, τ be fixed. Then, as n →∞, we have uniformly for ζ in any compact subset of ,lim_n →∞. (1 + ζ K_M, t) |_M = n + l + 1andt = n - τ√(n) = (I + ζ),where K_M, t is defined in (<ref>) and (<ref>) with parameters specified in (<ref>), andis the integral operator on the infinite vertical contour Γ_∞ = { -1 - yi | y ∈}, oriented downward, with kernel(z, w) = _τ; β_0, …, β_l(z, w) =e^w^2/2 + τ w/w - qzγ(w), whereγ(w) = ∏^l_k = 0β_k/β_k - w.Hencelim_n →∞. _0^∞(x_m(t) > M) |_M = n + l + 1andt = n - τ√(n) = 1/2π i∮_C dζ/ζ(I + ζ)/(1- ζ)(1-qζ)⋯ (1-q^m-1ζ),where C is the same as in Theorem <ref>. To include the important special case that all b_k are identically 1 into Theorem <ref>, we simply let l = -1, and then γ(w) = 1. In this case, we denote the limiting kernel (z, w) = _τ; -(z, w). Here we remark that unlike the dual q-TASEP process with step initial condition in which the limiting distribution of x_m(t) is simple for small m, the limiting distribution of x_m(t) is non-trivial even for small m in the q-TAZRP with step initial condition. For example, by applying the residue theorem to (<ref>), we havelim_n →∞_0^∞(x_1(t) ≤ M) = lim_n →∞.(I + K_M,t) |_M = n + l + 1andt = n - τ√(n) = (I + ).It shows that the limiting distribution of x_1(t) is non-Gaussian, on the other hand the right-most particle in the q-TASEP behaves as a free particle and hence the fluctuation of the first particle's position is Gaussian at large time.In the study of 1-dimensional interacting particle systems like the Totally Asymmetric Exclusion Process (TASEP) and its variations, usually the focus is on the behaviour of particles in the bulk rather than those at the edge, partly because the limiting behaviour of bulk particles often shows the feature of the Kadar–Parisi–Zhang (KPZ) universality class, partly because the limiting behaviour of the edge particles is often trivial. An exception is the recent paper on the facilitated TASEP by Baik, Barraquand, Corwin and Suidan <cit.>, but the limiting behaviour of edge particles in their model is quite different from that in Theorem <ref>. Also nontrivial limiting behaviours of the edge particles in the ASEP model are found by Tracy and Widom <cit.>, <cit.>. It turns out that the limiting distributions for the rightmost particles in the q-TAZRP with the step initial condition are the same as those for the leftmost particles in the ASEP with step initial condition given in <cit.> when all b_k are identical. Hence the limiting behaviour of the edge particles in the q-TAZRP and ASEP may be in a new universal class and worth more investigation.*The specialization ofin Theorem <ref> with l = -1 as in Remark <ref> satisfies for all ζ∈,(I + ζ_τ; -) = (I + ζK̂χ_(τ(1 + q)/(1 - q), ∞)),where K̂ is the operator on L^2(), defined in <cit.> with the parameters p and q replaced by q/(1 + q) and 1/(1 + q) respectively, and has the kernelK̂(z, z') = 1/√(2π)(1 + q) e^-1 + q^2/(1 + q)^2z^2 + z'^2/4 + q/(1 + q)^2 zz'. *The limiting distribution of the m-th rightmost particle in our q-TAZRP with step initial condition and all b_k = 1 is the same as the limiting distribution of the m-th leftmost particle in the ASEP with step initial condition and the left and right jumping rates equal to q/(1 + q) and 1/(1 + q) respectively, under proper scaling.From the exact contour integral formula (<ref>), we can also derive the limiting distribution of x_m(t) when both m and t approach ∞ and t/m is a constant. The asymptotic analysis is similar to that for <cit.>, but can be simplified, due to the reason mentioned in Remark <ref>. However, we omit the asymptotic analysis, since the limit of _0^∞(x_m(t) > M) can be expressed directly as a corollary of (<ref>) and the results in <cit.> and <cit.>. See Appendix <ref> for details. §.§ Organization of the paper In Section <ref> we prove Theorems <ref> and <ref>, and in Section <ref> we prove Theorem <ref> and Corollary <ref>. In Appendix <ref> we discuss the limiting behaviour of particles in the bulk based on known results on q-TASEP. §.§ Acknowledgements We thank the helpful discussion with Guillaume Barraquand.§ DISTRIBUTION OF THE POSITION OF A TAGGED PARTICLE §.§ Finite system with an arbitrary initial condition Y Let us consider an N-particle system with an arbitrary initial condition X(0) = (x_1(0), …, x_N(0)) = Y = (y_1, …, y_N)∈. We find the distribution functions of x_n(t) and x_N - n + 1(t) given by n-fold contour integrals. We first solve the problem for n = N (the leftmost particle) and N - n + 1 = 1 (the rightmost particle), and then solve for general n based on the two special cases.In this section, we need the following notations and identities. First, recall the notations of A_σ defined in (<ref>) and B_n defined in (<ref>). We have a formula that is proved in <cit.>:∑_σ∈ S_n A_σ(w_1, …, w_n) = [n]_q! B_n(w_1, …, w_n).Also later we use the elementary symmetric functione_l(w_1, …, w_n) = ∑_1 ≤ i_1 < i_2 < … < i_l ≤ n w_i_1 w_i_2… w_i_l.and the Vandermonde determinantΔ_n(w_1, …, w_n) = (w^k - 1_j)^n_j, k = 1 = ∏_1 ≤ i < j ≤ n (w_j - w_i). §.§.§ Positions of the leftmost particle and the rightmost particleLeftmost particleThe distribution of x_N(t) is given by the following proposition: For the N-particle q-TAZRP model with initial condition Y = (y_1, …, y_N), the leftmost particle x_N satisfiesℙ_Y(x_N(t) >M) = ∫_Cdw_1/w_1⋯∫_Cdw_N/w_N B_N(w_1, …, w_N) ∏_j=1^N [ _k=y_j^M ( b_k/b_k - w_j) e^-w_j t],where the contour C is specified as in Theorem <ref>.The proof of Proposition <ref> relies on the following technical result:Let q ∈ (0,1), Y = (y_1, …, y_N) ∈^N, b_k be in a compact subset of (0,∞) for all k ∈ℤ, and F(w_1, …, w_N) be a meromorphic function in w_1, …, w_N. Let C be a positively oriented, large enough circle centered at 0 and enclosing all b_k, and F(w_1, …, w_N) be continuous if w_k ∈ C. DenotingI(x_1, …, x_N) = 1/W(X)( ∏_k=1^N-1/b_x_k) _C dw_1 …_C dw_N F(w_1, …, w_N)×∑_σ∈ S_N A_σ(w_1, …, w_N) [ ∏_j=1^N ( _k=y_σ(j)^x_jb_k/b_k - w_σ(j)) e^-w_j t]for X = (x_1, …, x_N) ∈⊆, where A_σ is defined in (<ref>), then we have∑_X = (x_1, …, x_N)withM < x_N ≤…≤ x_1 I(x_1, …, x_N) = _Cdw_1⋯_Cdw_N F(w_1, …, w_N) B_N(w_1, …, w_N) ∏_j=1^N [ e^-w_j t/w_j_k=y_j^Mb_k/b_k - w_j].Using the expressionℙ_Y(x_N(t) > M) = ∑_X = (x_1,…, x_N)withM<x_N ≤…≤ x_1 P_Y(X;t)and the contour integral formulafor P_Y(X; t) in Proposition <ref>, we have that Proposition <ref> is the special case of Proposition <ref> with F(w_1, …, w_N) = 1. The identity (<ref>) is a generalization of <cit.>. The symmetrization identity in the q-Hahn TASEP <cit.>) with two parameters is also a generalization of <cit.> in a different direction. Similar to <cit.>, the proof of (<ref>) is by a straightforward mathematical induction, while the proof of <cit.> is based on a result of the spectral Plancherel theorem, see <cit.> for more details.We prove (<ref>) by induction on N. The N = 1 case of (<ref>) is reduced to∑^∞_x_1 = M+1_C-1/b_x_1 F(w_1)e^-w_1t'∏^x_1_k = y_1b_k/b_k - w_1 dw_1 = _Ce^-w_1t/w_1F(w_1) '∏^M_k = y_1b_k/b_k - w_1 dw_1.Suppose without loss of generality that M≥ y_1. Since the radius of C islarge enough and all b_k are in a compact subset of (0,∞),∑^∞_x_1 = M + 1-1/b_x_1 F(w_1)∏^x_1_k = y_1b_k/b_k - w_1converges on C uniformly. Hence it suffices to show that∑^∞_x_1 = M + 1-1/b_x_1∏^x_1_k = M+1b_k/b_k - w_1 = 1/w_1.This is done by the telescoping trick, noting thatw_1 ( -1/b_x_1∏^x_1_k = M + 1b_k/b_k - w_1) = -∏^x_1_k = M + 1b_k/b_k - w_1 + ∏^x_1 - 1_k = M + 1b_k/b_k - w_1. Now we consider the general N case inductively, assuming that (<ref>) holds for 1, 2, …, N - 1. We write the left-hand side of (<ref>) as∑^∞_m = M + 1∑_l=1^N I^(l)(m), where I^(l)(m) = ∑_m<x_N-l≤…≤ x_1 I(x_1,…, x_N - l, m, …, m_x_N =… = x_N-l+1=m).Then with the index U = (u_1, …, u_N - l) ∈[N - l],I^(l)(m) =(-1/b_m)^l/[l]_q!∑_ m < u_N - l≤…≤ u_11/W(U)( ∏^N - l_k = 1-1/b_u_k) _C dw_1 …_C dw_N F(w_1, …, w_N) ×∑_σ∈ S_NA_σ(w_1, …, w_N) ∏^N_j = N - l + 1[ ^m_k = y_σ(j)( b_k/b_k - w_σ(j)) e^-w_σ(j) t] ×∏^N - l_j = 1[ ^u_j_k = y_σ(j)( b_k/b_k - w_σ(j)) e^-w_σ(j) t],such that I^(N)(m) degenerates to I(m, …, m) in (<ref>). To evaluate I^(l)(m), we decompose the index set S_N into Nl disjoint subsets S_N(I_l) where I_l ⊂{1, 2, …, N } has l elements andS_N(I_l) = {σ∈ S_N |σ({N - l + 1, …, N - 1, N }) = I_l }.In the subsequent part of the paper, we take the notational convention that I_l = { i_1, …, i_l } and I^c_l = { 1, …, N }∖ I_l = { j_1, …, j_N-l} such that i_k and j_k are in ascending order. Let us define a bijective mappingφ_I_l(·|·) : S_l × S_N - l⟶ S_N(I_l) byφ_I_l(λ|τ)(k) =j_τ(k) if k ≤ N - l, i_λ(k + l - N) if k > N - l.Then, for each σ∈ S_N(I_l), there is a unique pair λ∈ S_l and τ∈ S_N-l such that φ_I_l(λ|τ) = σ by (<ref>), and thenA_σ(w_1, …, w_N) = G_I_l(w_1, …, w_N) A_λ(w_i_1, …, w_i_l) A_τ(w_j_1, …, w_j_N - l),whereG_I_l(w_1, …, w_N) = ∏_i ∈ I_l, j ∈{ 1, …, N }∖ I_l i < j -qw_j - w_i/qw_i - w_j.Dividing the summation over S_N in (<ref>) into Nl terms where each term is the summation over an S_N(I_l), and using (<ref>) to decompose the A_σ factor, we haveI^(l)(m) = (-1/b_m)^l/[l]_q!∑_I_l ⊆{ 1, …, N }∑_λ∈ S_l_C dw_i_1…_C dw_i_l A_λ(w_i_1, …, w_i_l)×∏^l_r = 1[ ^m_k = y_i_λ(r)( b_k/b_k - w_i_λ(r)) e^-w_i_λ(r) t]×(∑_ m < u_N - l≤…≤ u_11/W(U)( ∏^N - l_k = 1-1/b_u_k) _C dw_j_1…_C dw_j_N - l F(w_1, …, w_N) × G_I_l(w_1, …, w_N) ∑_τ∈ S_N - l A_τ(w_j_1, …, w_j_N - l) ∏^N - l_r = 1^u_r_k = y_j_τ(r)( b_k/b_k - w_j_τ(r)) e^-w_j_τ(r) t).Applying the induction hypothesis to (<ref>), we simplify it asI^(l)(m) =(-1/b_m)^l/[l]_q!∑_I_l ⊆{ 1, …, N }∑_λ∈ S_l_C dw_i_1…_C dw_i_l A_λ(w_i_1, …, w_i_l)×∏^l_r = 1[ ^m_k = y_i_λ(r)( b_k/b_k - w_i_λ(r)) e^-w_i_λ(r) t]×(_Cdw_j_1/w_j_1…_Cdw_j_N - l/w_j_N - l F(w_1, …, w_N) G_I_l(w_1, …, w_N) × B_N - l(w_j_1, …, w_j_N - l) ∏^N - l_r = 1[ ^m_k = y_j_r( b_k/b_k - w_j_r) e^-w_j_r t] ).Next, by (<ref>), we further haveI^(l)(m) = ( -1/b_m)^l_C dw_1 …_C dw_N Γ(w_1, …, w_N; l) F(w_1, …, w_N) ∏^N_r = 1[ ^m_k = y_r( b_k/b_k - w_r) e^-w_r t],whereΓ(w_1, …, w_N; l) = ∑_I_l ⊆{ 1, …, N }G_I(w_1, …, w_N)/∏^N - l_k = 1 w_j_k B_l(w_i_1, w_i_2…, w_i_l) B_N - l(w_j_1, …, w_j_N - l).LetΓ_m(w_1, …, w_N) = ∑^N_l = 1( -1/b_m)^lΓ(w_1, …, w_N; l).Since the left-hand side of (<ref>) is given by (<ref>), we plug (<ref>) into (<ref>), and have that the left-hand side of (<ref>) can be expressed as∑^∞_m = M + 1_C dw_1 …_C dw_N Γ_m(w_1, …, w_N) ∏^N_r = 1[ ^m_k = y_r( b_k/b_k - w_r) e^-w_r t]F(w_1, …, w_N). To complete the proof, we need to show that (<ref>) is equal to the right-hand side of (<ref>). To this end, we need to simplify Γ(w_1, …, w_N; l) and Γ_m(w_1, …, w_N). Let us considerq^(N - l)(N - l - 1)/2 (-1)^N(N - 1)/2∏^N_k = 1 w_k ∏_1 ≤ i < j ≤ N(qw_i - w_j) Γ(w_1, …, w_N; l) = ∑_I_l ⊆{ 1, …, N } (-1)^(i_1 + … + i_l) - l(l + 1)/2∏^l_k = 1 w_i_k∏_1 ≤ k < r ≤ l (w_i_r - w_i_k) ∏_1 ≤ k < r ≤ N - l (qw_j_r - qw_j_k)×∏^l_k = 1∏^N - l_r = 1 (qw_j_r - w_i_k).For each I_l, the summand on the right-hand side of (<ref>) is equal to the Vandermonde-like determinant( f^I_l_j, k(w_k) )^N_j, k = 1, where f^I_l_j, k(x) =x^jif k ∈ I_l, (qx)^j - 1 if k ∈{ 1, …, N }∖ I_l.Summing up all the Nl terms on the right-hand side of (<ref>) with each term expressed by (<ref>), we have that the result is the z^l coefficient of the polynomial in z( (zw_k + q^j - 1)w^j - 1_k )^N_j, k = 1.Using elementary row operations, we can simplify the determinant in (<ref>), and write it as( (zw_k + q^j - 1)w^j - 1_k )^N_j, k = 1 =∑^N_j = 0 q^(N - l)(N - l - 1)/2( g^(l)_j(w_k) )^N_j, k = 1 z^l =∑^N_j = 0 q^(N - l)(N - l - 1)/2 e_l(w_1, …, w_N) Δ_N(w_1, …, w_N) z^l,where e_l and Δ_N are defined in (<ref>) and (<ref>), andg^(l)_j(x) =x^j - 1 if j ≤ N - l, x^jotherwise.Thus we conclude thatΓ(w_1, …, w_N; l) = e_l(w_1, …, w_N) B_N(w_1, …, w_N) ∏^N_k = 11/w_k.Hence by (<ref>) and the identity ∑^N_l = 1 e_l(w_1, …, w_N) z^l = ∏^N_k = 1 (1 + zw_k) - 1,Γ_m(w_1, …, w_N) = [ ∏^N_k = 1( b_m - w_k/b_m) - 1 ] B_N(w_1, …, w_N) ∏^N_k = 11/w_k.Finally, substituting (<ref>) into (<ref>), and using the telescoping trick, we find that the left-hand side of (<ref>) is∑^∞_m = M + 1_Cdw_1/w_1…_Cdw_N/w_N[ ∏^N_k = 1( b_m - w_k/b_m) - 1 ] B_N(w_1, …, w_N) ×∏^N_r = 1[ ^m_k = y_r( b_k/b_k - w_r) e^-w_r t] F(w_1, …, w_N) =_Cdw_1/w_1…_Cdw_N/w_N B_N(w_1, …, w_N) ∏^N_r = 1[ ^M_k = y_r( b_k/b_k - w_r) e^-w_r t] F(w_1, …, w_N).This completes the proof, and the convergence of (<ref>) is due to that the radius of C is larger than sup_k ∈(b_k).Rightmost particleSimilarly, we can find the distribution of x_1(t), the rightmost particle. The argument is parallel, and we stress the difference between the two derivations.For the N-particle q-TAZRP model with initial condition Y = (y_1, …, y_N), the rightmost particle x_1 satisfiesℙ_Y(x_1(t) ≤ M) = (-1)^N q^N(N - 1)/2_C̃_1dw_1/w_1…_C̃_Ndw_N/w_N B_N(w_1, …, w_N)×∏^N_j = 1[ ^M_k = y_j( b_k/b_k - w_j) e^-w_j t],where the contours C̃_1, …, C̃_N are specified as in Theorem <ref>.The proof of this proposition is by expressing, similar to (<ref>),ℙ_Y(x_1(t) ≤ M) = ∑_X = (x_1,…, x_N)with-M̃≤ x_N ≤⋯≤ x_1≤ M P_Y(X;t)where -M̃ is a small enough integer, say, -M̃ < y_N, and then using the following technical result with F(w_1, …, w_N) = 1. Let q ∈ (0,1), Y = (y_1, …, y_N) ∈^N, positive numbers b_k for k ∈ℤ and meromorphic function F(w_1, …, w_N) be defined as in Proposition <ref>. Let C be a positively oriented contour containing all b_k, and we further require that F(w_1, …, w_N) is continuous if all w_k are on or within C. Moreover, let C̃_1, …, C̃_N be positively oriented contours such that each of them encloses all b_k, none of them encloses 0, and C̃_j enclose q ·C̃_i if j > i. We also require that as the contours for w_k deform gradually from C to C̃_k, the meromorphic function F(w_1, …, w_N) does not meet any singularity. Suppose I(x_1, …, x_N) are defined as in (<ref>) of Proposition <ref> with the contour C specified in this proposition. We have∑_X = (x_1,…, x_N)with-M̃≤ x_N ≤⋯≤ x_1≤ M I(x_1, …, x_N) = (-1)^N q^N(N - 1)/2_C̃_1dw_1/w_1…_C̃_Ndw_N/w_N × F(w_1, …, w_N) B_N(w_1, …, w_N) ∏^N_j = 1[ ^M_k = y_j( b_k/b_k - w_j) e^-w_j t],where -M̃ is a small enough integer, say, -M̃ < min(y_1, …, y_N).We prove by the induction on N. For the N = 1 case, we use the telescoping trick as for (<ref>), and have∑^M_x_1 = -M̃( -1/b_x_1^x_1_k = y_1b_k/b_k - w_1) =1/w^-M̃ - 1_k = y_1b_k/b_k - w_1 - 1/w^M_k = y_1b_k/b_k - w_1=1/w∏^y_1 - 1_k = -M̃b_k - w_1/b_k - 1/w^M_k = y_1b_k/b_k - w_1,where we use that -M̃ < y_1. Since by assumption, F(w_1) has no pole for w_1 on or within the contour C, we have∑^M_x_1 = -M̃ I(x_1) =_C dw_1 F(w_1) [ ∑^M_x_1 = -M̃_1( -1/b_x_1^x_1_k = y_1b_k/b_k - w_1) ] e^-w_1 t=_C̃_1 dw_1 F(w_1) [ ∑^M_x_1 = -M̃_1( -1/b_x_1^x_1_k = y_1b_k/b_k - w_1) ] e^-w_1 t=_C̃_1dw_1/w_1 F(w_1) ( ∏^y_1 - 1_k = -M̃b_k - w_1/b_k) e^-w_1 t - _C̃_1dw_1/w_1 F(w_1) ( ^M_k = y_1b_k/b_k - w_1) e^-w_1 t= - _C̃_1dw_1/w_1 F(w_1) ( ^M_k = y_1b_k/b_k - w_1) e^-w_1 t.Hence the N = 1 case of (<ref>) is proved.Assuming that (<ref>) holds if N is replaced by1, 2, …, N - 1, analogous to (<ref>) we write the left-hand side of (<ref>) as∑^M - 1_m = -M̃ - 1∑^N - 1_l = 0 J^(l)(m), where J^(l)(m) = ∑_M̃≤ x_N ≤…≤ x_N - l + 1≤ m I(m + 1, …, m + 1_x_1 =… = x_N-l = m + 1, x_N - l + 1, …, x_N).Then analogous to (<ref>), we have with the index U = (u_1, …, u_l) ∈[l],J^(l)(m) =(-1/b_m + 1)^N - l/[N - l]_q!∑_ -M̃≤ u_l ≤…≤ u_1 ≤ m1/W(U)( ∏^l_k = 1-1/b_u_k) _C dw_1 …_C dw_N F(w_1, …, w_N) ×∑_σ∈ S_N A_σ(w_1, …, w_N) ∏^N - l_j = 1[ ^m + 1_k = y_σ(j)( b_k/b_k - w_σ(j)) e^-w_σ(j) t] ×∏^l_j = 1[ ^u_j_k = y_σ(j)( b_k/b_k - w_σ(N - l + j)) e^-w_σ(N - l + j) t],such that J^(0)(m) degenerates to I(m + 1, …, m + 1) in (<ref>). To evaluate J^(l)(m), we also decompose S_N into Nl disjoint subsets S_N(I_l) that are defined in (<ref>). Similar to (<ref>), we haveJ^(l)(m) = (-1/b_m + 1)^N - l/[N - l]_q!∑_I_l ⊆{ 1, …, N }∑_τ∈ S_N - l_C dw_j_1…_C dw_j_N - l A_τ(w_j_1, …, w_j_N - l)×∏^N - l_r = 1^m + 1_k = y_j_τ(r)( b_k/b_k - w_j_τ(r)) e^-w_j_τ(r) t ×(∑_ -M̃≤ u_l ≤…≤ u_1 ≤ m1/W(U)( ∏^l_k = 1-1/b_u_k) _C dw_i_1…_C dw_i_l F(w_1, …, w_N) × G_I_l(w_1, …, w_N) ∑_λ∈ S_l A_λ(w_i_1, …, w_i_l) ∏^l_r = 1^u_r_k = y_j_λ(r)( b_k/b_k - w_i_λ(r)) e^-w_i_λ(r) t),where the notations I_l = { i_1, …, i_l } and { 1, …, N }∖ I_l = { j_1, …, j_N - l} are the same as in (<ref>), and the decompositions of σ into λ and τ, and of A_σ into the product of G_I_l A_λ A_τ are given by (<ref>), (<ref>) and (<ref>).Applying the induction hypothesis to (<ref>), we haveJ^(l)(m) = (-1/b_m + 1)^N - l/[N - l]_q!∑_I_l ⊆{ 1, …, N }∑_τ∈ S_N - l_C dw_j_1…_C dw_j_N - l A_τ(w_j_1, …, w_j_N - l)×∏^N - l_r = 1^m + 1_k = y_j_τ(r)( b_k/b_k - w_j_τ(r)) e^-w_j_τ(r) t ×((-1)^l q^l(l - 1)/2_C̃_i_1dw_i_1/w_i_1…_C̃_i_ldw_i_l/w_i_lF(w_1, …, w_N) G_I_l(w_1, …, w_N) B_l(w_i_1, …, w_i_l) ×∏^l_r = 1[ ^m_k = y_i_r( b_k/b_k - w_i_r) e^-w_i_r t] ).By the property of the contours C and C̃_j, we have that the integral contours for w_j_r in (<ref>) can be deformed from C to C̃_j_r for each r = 1, …, N - l. Then by (<ref>), we further haveJ^(l)(m) = ( -1/b_m + 1)^N - l (-1)^l q^l(l - 1)/2_C̃_1 dw_1 …_C̃_N dw_N Σ_m(w_1, …, w_N; l)× F(w_1, …, w_N) ∏^N_r = 1[ ^m + 1_k = y_r( b_k/b_k - w_r) e^-w_r t],where, analogous to Γ(w_1, …, w_N; l) in (<ref>), we defineΣ_m(w_1, …, w_N; l) = ∑_I_l ⊆{ 1, …, N } G_I(w_1, …, w_N)∏^l_k = 1(b_m + 1 - w_i_k)/b_m + 1 w_i_k × B_l(w_i_1, w_i_2…, w_i_l) B_N - l(w_j_1, …, w_j_N - l). Analogous to Γ_m(w_1, …, w_N) in (<ref>), we defineΣ_m(w_1, …, w_N) = ∑^N_l = 0( -1/b_m + 1)^N - l (-1)^l q^l(l - 1)/2Σ_m(w_1, …, w_N; l),and then similar to (<ref>), we have that the left-hand side of (<ref>) is expressed as∑^M - 1_m = -M̃ - 1_C̃ dw_1 …_C̃_N dw_N ( Σ_m(w_1, …, w_N) - (-1)^N q^N(N - 1)/2∏^N_k = 1b_m + 1 - w_k/b_m + 1 w_k B_N(w_1, …, w_N) )×∏^N_r = 1[ ^m + 1_k = y_r( b_k/b_k - w_r) e^-w_r t]F(w_1, …, w_N).(Formally, (<ref>) has an extra term comparing with (<ref>), but this difference results from that Σ_m(w_1, …, w_N) in (<ref>) is the sum of N + 1 terms, while Γ_m(w_1, …, w_N) in (<ref>) is the sum of N terms.) Similar to (<ref>), we haveq^(N - l)(N - l - 1)/2 (-1)^N(N - 1)/2∏^N_k = 1 w_k ∏_1 ≤ i < j ≤ N(qw_i - w_j) Σ_m(w_1, …, w_N; l) = ∑_I_l ⊆{ 1, …, N } (-1)^(i_1 + … + i_l) - l(l + 1)/2∏^l_k = 1 (1 - w_i_k/b_m + 1) ∏^N - l_k = 1 w_j_k ×∏_1 ≤ k < r ≤ l (w_i_r - w_i_k) ∏_1 ≤ k < r ≤ N - l (qw_j_r - qw_j_k) ∏^l_k = 1∏^N - l_r = 1 (qw_j_r - w_i_k).It is not hard to check that for any intex set I_l, the term in the sum (<ref>) is equal to( f^I_l_j, k(w_k) )^N_j, k = 1, where f^I_l_j, k(x) =(1 - x/b_m + 1) x^jif k ∈ I_l, x (qx)^j - 1 if k ∈{ 1, …, N }∖ I_l.Then by summing up all the Nl terms, we have that the left-hand side of (<ref>) is the z^l coefficient of the following polynomial in z( [z(1 - w_k/b_m + 1) + q^j - 1w_k] w^j - 1_k )^N_j, k = 1.Hence(-q)^N(N - 1)/2∏^N_k = 1 w_k ∏_1 ≤ i < j ≤ N(qw_i - w_j) Σ_m(w_1, …, w_N) =∑^N_l = 0( -1/b_m + 1)^N b^l_m + 1 q^(N - 1)l( z^l coefficient of ( [z(1 - w_k/b_m + 1) + q^j - 1w_k] w^j - 1_k )^N_j, k = 1) =( -1/b_m + 1)^n . ( [z(1 - w_k/b_m + 1) + q^j - 1w_k] w^j - 1_k )^N_j, k = 1|_z = -q^N - 1 b_m + 1= (-1)^N q^N(N - 1)Δ_N(w_1, …, w_N).Then by (<ref>) we can express Σ_m(w_1, …, w_N) in terms of B_N(w_1, …, w_N), and by plugging this expression into (<ref>) we have that the left-hand side of (<ref>) is(-1)^N q^N(N - 1)/2∑^M - 1_m = -M̃ - 1_C̃_1dw_1/w_1…_C̃_Ndw_N/w_N[ 1 - ( ∏^N_k = 1b_m + 1 - w_k/b_m + 1) ] B_N(w_1, …, w_N)×∏^N_r = 1[ ^m + 1_k = y_r( b_k/b_k - w_r) e^-w_r t]F(w_1, …, w_N) = (-1)^N q^N(N - 1)/2( _C̃_1dw_1/w_1…_C̃_Ndw_N/w_N∏^N_r = 1[ ^M_k = y_r( b_k/b_k - w_r) e^-w_r t]F(w_1, …, w_N) . - . _C̃_1dw_1/w_1…_C̃_Ndw_N/w_N∏^N_r = 1[ ^-M̃ - 1_k = y_r( b_k/b_k - w_r) e^-w_r t]F(w_1, …, w_N) ),where we use the telescoping trick for the identity. Hence we prove Theorem <ref> by noting that_C̃_1dw_1/w_1…_C̃_Ndw_N/w_N∏^N_r = 1[ ^-M̃ - 1_k = y_r( b_k/b_k - w_r) e^-w_r t]F(w_1, …, w_N) = 0,which is due to that M̃ < min(y_1, …, y_N) and F(w_1, …, w_N) has no singularity if all w_k are on or within C̃_k. §.§.§ Relation between the two types of nested contour integrals. Using the residue formula inductively, we have that if the contours C and C̃_1, …, C̃_N are defined as in Theorem <ref>, then_C dw_1/w_1…_C dw_N/w_N B_N(w_1, …, w_N) ∏^N_j = 1[ ^M_k = y_j( b_k/b_k - w_j) e^-w_j t] =∑^N_l = 0∑_I_l ⊆{ 1, …, N } q^(i_1 + … + i_l) - (2N -l + 1)l/2 ×_C̃_i_1dw_i_1/w_i_1…_C̃_i_ldw_i_l/w_i_l B_l(w_i_1, …, w_i_l) ∏^l_r = 1[ ^M_k = y_i_r( b_k/b_k - w_i_r) e^-w_i_r t],where I_l = { i_1, …, i_l } with i_k in ascending order, and the term corresponding to l = 0 and I_l = ∅ is taken as 1. On the other hand, we have_C̃_1dw_1/w_1…_C̃_Ndw_N/w_N B_N(w_1, …, w_N) ∏^N_j = 1[ ^M_k = y_j( b_k/b_k - w_j) e^-w_j t] =∑^N_l = 0∑_I_l ⊆{ 1, …, N } (-1)^N - l q^(i_1 + … + i_l) - l - N(N - 1)/2 ×_C dw_i_1/w_i_1…_C dw_i_l/w_i_l B_l(w_i_1, …, w_i_l) ∏^l_r = 1[ ^M_k = y_i_r( b_k/b_k - w_i_r) e^-w_i_r t],where the term corresponding to l = 0 and I_l = ∅ is taken as q^-N(N - 1)/2.We remark that relation (<ref>) is analogous to the expression of μ̃_k as a linear combination of μ_j in <cit.>.§.§.§ Proof of Theorem <ref> In this subsection we compute the distribution function _Y(x_n(t) ≤ M) and _Y(x_N - n + 1(t) > M), where n = 1, 2, …, N. Let us write_Y(x_n(t) ≤ M) = ∑^n - 1_l = 0 P^(l)_Y(M), _Y(x_N - n + 1(t) > M) = ∑^N_l = N - n + 1 P^(l)_Y(M),whereP^(0)_Y(M) = _Y(x_1(t) ≤ M),P^(N)_Y(M) = _Y(x_N > M),and for l = 1, …, N - 1,P^(l)_Y(M) = ℙ_Y(x_l + 1(t) ≤ Mandx_l(t) > M).Recall the notations I_l = { i_1, …, i_l }⊆{ 1, 2, …, N } with i_1 < … < i_l, the complement set I^c_l = {1, …, N }∖ I_l = { j_1, …, j_N - l} with j_1 < … < j_N - l, the subset S_N(I_l) defined in (<ref>) of the permutation group S_N, the mapping φ_I_l(·|·) defined in (<ref>) that maps S_l × S_N - l bijectively to S_N(I_l). We decomposeP^(N - l)_Y(M) = ∑_I_l ⊆{ 1, …, N } P^(N - l)_Y(M; I_l),where, with indices U = (u_1, …, u_N - l) ∈[N - l] and V = (v_1, …, v_l) ∈[l],P^(N - l)_Y(M; I_l) = ∑_-M̃≤ v_l ≤…≤ v_1 ≤ M < u_N - l≤…≤ u_11/W(U)∏^N - l_j = 1( -1/b_u_j) 1/W(V)∏^l_j = 1( -1/b_v_j)×_C dw_1 …_C dw_N ∑_λ∈ S_l, τ∈ S_N - l A_φ_I_l(λ|τ)(w_1, …, w_N)×∏^N - l_r = 1[ ∏^u_r_k = y_j_τ(r)( b_k/b_k - w_j_τ(r)) e^-w_j_τ(r) t] ∏^l_r = 1[ ∏^v_r_k = y_i_λ(r)( b_k/b_k - w_i_λ(r)) e^-w_i_λ(r) t],and P^(N - l)_Y(M; I_l) degenerates into (<ref>) if l = N and I_l = { 1, …, N }, or degenerates into (<ref>) if l = 0 and I_l = ∅. In (<ref>) we assume the contour C and constant -M̃ are specified in Proposition <ref>. Recall that A_φ_I_l(λ|τ)(w_1, …, w_N) can be decomposed by (<ref>). Now we apply Proposition <ref> to the right-hand side of (<ref>), with X replaced by U, the N variables w_1, …, w_N replaced by the (N - l) variables w_j_1, …, w_j_N - l, and the function F(w_j_1, …, w_j_N - l) specialized byF(w_j_1, …, w_j_n - l) = ∑_-M̃≤ v_l ≤…≤ v_1 ≤ M1/W(V)∏^l_j = 1( -1/b_v_j) _C dw_i_1…_C dw_i_l × G_I_l(w_1, …, w_N) A_λ(w_i_1, …, w_i_l) ∏^l_r = 1[ ∏^v_r_k = y_i_λ(r)( b_k/b_k - w_i_λ(r)) e^-w_i_λ(r) t],where G_I_l(w_1, …, w_N) is defined in (<ref>). We then rewrite (<ref>) asP^(N - l)_Y(M; I_l) = _C dw_j_1/w_j_1…_C dw_j_N - l/w_j_N - l F(w_j_1, …, w_j_N - l)× B_N - l(w_j_1, …, w_j_N - l) ∏^N - l_r = 1[ ∏^M_k = y_j_r( b_k/b_k - w_j_r) e^-w_j_r t].Now we note that F(w_j_1, …, w_j_N - l) can be simplified by Proposition <ref>, with X replaced by V, the N variables w_1, …, w_N replaced by the l variables w_i_1, …, w_i_l, and the function F(w_i_1, …, w_i_l) specialized by G_I_l(w_1, …, w_N) in which w_i_1, …, w_i_l are variables and w_j_1, …, w_j_N - l are parameters. Thus (<ref>) is further simplified intoP^(N - l)_Y(M; I_l) = (-1)^l q^l(l - 1)/2_C dw_j_1/w_j_1…_C dw_j_N - l/w_j_N - l_C̃_i_1dw_i_1/w_i_1…_C̃_i_ldw_i_l/w_i_l × G_I_l(w_1, …, w_N) B_l(w_i_1, …, w_i_l) B_N - l(w_j_1, …, w_j_N - l) ∏^N_j = 1[ ∏^M_k = y_j( b_k/b_k - w_j) e^-w_j t].Up to here P^(N - l)_Y(M; I_l) is expressed by a single, but heterotypic contour integral. We can use the residue theorem to express it into a sum of multiple integrals over C, or a sum of multiple integrals over C̃_1, …, C̃_N. The former approach leads to the proof of (<ref>), and the latter approach leads to the proof of (<ref>). The n-th rightmost particleWe first prove (<ref>). For any pair of disjoint subsets I' = { i'_1, …, i'_l } and J' = { j'_1, …, j'_m } of { 1, …, N } with i'_1 < … < i'_l and j'_1 < … < j'_m, we denoteC_I', J' = ∏_i < i' ∈ I'w_i - w_i'/qw_i - w_i'∏_j < j' ∈ J'w_j - w_j'/qw_j - w_j'∏_i ∈ I', j ∈ J',andi < j -qw_j - w_i/qw_i - w_j,so that if we take I' = I_l = { i_1, …, i_l } and J = I^c_l = { j_1, …, j_N - l} as above (<ref>), thenC_I_l, I^c_l = G_I_l(w_1, …, w_N) B_l(w_i_1, …, w_i_l) B_N - l(w_j_1, …, w_j_N - l).Using the residue theorem to (<ref>) and deform the contours for w_j_r (r = 1, …, N - l) from C to C̃_j_r one by one, we have (| I' | means the number of elements in set I')P^(N - l)_Y(M; I_l) = (-1)^l q^l(l - 1)/2∑^N_r = l∑_I' = { i'_1, …, i'_r }⊆{ 1, …, N } | I' | = randI' ⊇ I_l q^(i'_1 + … + i'_r) - (2N - r + 1)r/2 ×_C̃_i'_1dw_i'_1/w_i'_1…_C̃_i'_rdw_i'_r/w_i'_r C_I_l, I' ∖ I_l∏^r_j = 1[ ^M_k = y_i'_j( b_k/b_k - w_i'_j) e^-w_i'_j t].First we use (<ref>) and (<ref>) to deriveP^(N - l)_Y(M) = (-1)^l q^l(l - 1)/2∑^N_r = l∑_I' = { i'_1, …, i'_r }I' ∈{ 1, …, N } and | I' | = r q^(i'_1 + … + i'_r) - (2N - r + 1)r/2 ×_C̃_i'_1dw_i'_1/w_i'_1…_C̃_i'_rdw_i'_r/w_i'_r( ∑_I”⊆ I'and | I”| = l C_I”, I' ∖ I”) ∏^r_j = 1[ ^M_k = y_i'_j( b_k/b_k - w_i'_j) e^-w_i'_j t]. Below we compute the sum ∑_I”⊆ I'and | I”| = l C_I”, I' ∖ I”. For notational simplicity, we denote u_j = w_i'_j for j = 1, …, r. Then we have that for any I”⊆ I', analogous to (<ref>),(-1)^r(r - 1)/2 C_I”, I' ∖ I” = q^-l(l - 1)/2/∏_1 ≤ i < j ≤ r (qu_i - u_j)( h^I”_j, k(u_k) )^r_j, k = 1,whereh^I”_j, k(x) =(qx)^j - 1 if i'_j ∈ I”, x^j - 1 otherwisewhich is analogous to (<ref>). Then we have∑^r_l = 0[ ∑_I”⊆ I'and | I”| = l( h^I”_j, k(u_k) )^r_j, k = 1] t^l =( (1 + q^j - 1 t) u^j - 1_k )^r_j, k = 1=( u^j - 1_k )^r_j, k = 1∏^r_j = 1 (1 + q^j - 1 t) =∏_1 ≤ i < j ≤ r (u_i - u_j) ∑^r_l = 0 q^l(l - 1)/2rl t^l,where the last identity is by <cit.>. Hence by using (<ref>) and taking the t^l coefficient in (<ref>), we have∑_I”⊆ I'and | I”| = l C_I”, I' ∖ I” =q^-l(l - 1)/2/∏_1 ≤ i < j ≤ r (qu_i - u_j)∑_I”⊆ I'and | I”| = l (-1)^r(r - 1)/2( h^I”_j, k(u_k) )^r_j, k = 1=q^-l(l - 1)/2/∏_1 ≤ i < j ≤ r (qu_i - u_j) q^l(l - 1)/2rl∏_1 ≤ i < j ≤ r (u_i - u_j) =rl B_r(u_1, …, u_r). Thus we have by (<ref>), (<ref>) and (<ref>) thatP^(N - l)_Y(M) = (-1)^l q^l(l - 1)/2∑^N_r = l∑_I' = { i'_1, …, i'_r }I' ⊆{ 1, …, N } and | I' | = r q^(i'_1 + … + i'_r) - (2N - r + 1)r/2rl ×_C̃_i'_1dw_i'_1/w_i'_1…_C̃_i'_rdw_i'_r/w_i'_r B_r(w_i'_1, …, w_i'_r) ∏^r_j = 1[ ^M_k = y_i'_j( b_k/b_k - w_i'_j) e^-w_i'_j t]for l = 0, 1, …, N. Hence by (<ref>), we haveP_Y(x_n(t) ≤ M) = ∑^N_r = N - n + 1∑_I' = { i'_1, …, i'_r }I ⊆{ 1, …, N } and | I' | = r q^(i'_1 + … + i'_r) - (2N - r + 1)r/2∑^r_l = N - n + 1 (-1)^l q^l(l - 1)/2rl ×_C̃_i'_1dw_i'_1/w_i'_1…_C̃_i'_rdw_i'_r/w_i'_r B_r(w_i'_1, …, w_i'_r) ∏^r_j = 1[ ^M_k = y_i_j( b_k/b_k - w_i'_j) e^-w_i'_j t].Furthermore, using <cit.> and then <cit.>, we have∑^r_l = N - n + 1 (-1)^l q^l(l - 1)/2rl = -∑^N - n_l = 0 (-1)^l q^l(l - 1)/2rl= (-1)^N - n + 1 q^(N - n)(N - n + 1)/2r - 1N - n.Hence we have_Y(x_n(t) ≤ M) = (-1)^N - n + 1 q^(N - n)(N - n + 1)/2∑^N_r = N - n + 1r - 1N - n ×∑_I' = { i'_1, …, i'_r }I ⊆{ 1, …, N } and | I' | = r q^(i'_1 + … + i'_r) - (2N - r + 1)r/2 ×_C̃_i'_1dw_i'_1/w_i'_1…_C̃_i'_rdw_i'_r/w_i'_r B_r(w_i'_1, …, w_i'_r) ∏^r_j = 1[ ^M_k = y_i_j( b_k/b_k - w_i'_j) e^-w_i'_j t],and prove (<ref>). The n-th leftmost particleNext we prove (<ref>). Similar to (<ref>), using the residue theorem to (<ref>) and deform the contours for w_i_r (r = 1, …, l) from C̃_i_r to C one by one, we haveP^(N - l)_Y(M; I_l) = (-1)^l q^l(l - 1)/2∑^N_r = N - l∑_J' = { j'_1, …, j'_r }⊆{ 1, …, N } | J' | = randJ' ⊇ I^c_l (-1)^N - r q^(j'_1 + … + j'_r) - r - N(N - 1)/2 + (N - l)(N - r) ×_C dw_j'_1/w_j'_1…_C dw_j'_r/w_j'_r C_J' ∖ I^c_l, I^c_l∏^r_i = 1[ ^M_k = y_j'_i( b_k/b_k - w_j'_i) e^-w_j'_i t].Using (<ref>) we can derive, analogous to (<ref>),P^(N - l)_Y(M) = (-1)^l q^l(l - 1)/2∑^N_r = N - l∑_J' = { j'_1, …, j'_r }J' ⊆{ 1, …, N } and | J' | = r (-1)^N - r q^(j'_1 + … + j'_r) - r - N(N - 1)/2 + (N - l)(N - r) ×_C_j'_1dw_j'_1/w_j'_1…_C_j'_rdw_j'_r/w_j'_r( ∑_J”⊆ J'and | J”| = N - l C_J' ∖ J”, J”) ∏^r_i = 1[ ^M_k = y_j'_i( b_k/b_k - w_j'_i) e^-w_j'_i t].Similar to (<ref>), (<ref>) and (<ref>), we have that if we denote v_i = w_j'_i for i = 1, …, r, then∑_J”⊆ J'and | J”| = N - l C_J' ∖ J”, J”=q^-(r - N + l)(r - N + r - 1)/2/∏_1 ≤ i < j ≤ r (qv_i - v_j) (-1)^r(r - 1)/2( h^J' ∖ J”_j, k(v_k) )^r_j, k = 1=q^-(r - N + l)(r - N + r - 1)/2/∏_1 ≤ i < j ≤ r (qv_i - v_j) q^(r - N + l)(r - N + l - 1)/2rr - N + l∏_1 ≤ i < j ≤ r (v_i - v_j) =rN - l B_r(v_1, …, v_r),whereh^J' ∖ J”_j, k(x) =(qx)^j - 1 if j'_j ∈ J' ∖ J”, x^j - 1 otherwise. Thus we have by (<ref>), (<ref>) and (<ref>) thatP^(N - l)_Y(M) = (-1)^l q^l(l - 1)/2∑^N_r = N - l∑_J' = { j'_1, …, j'_r }J' ⊆{ 1, …, N } and | J' | = r (-1)^N - r q^(j'_1 + … + j'_r) - r - N(N - 1)/2 + (N - l)(N - r) ×rN - l_C dw_j'_1/w_j'_1…_C dw_j'_r/w_j'_r B_r(w_j'_1, …, w_j'_r) ∏^r_i = 1[ ^M_k = y_j'_i( b_k/b_k - w_j'_i) e^-w_j'_i t]for l = 0, 1, …, N. Hence similar to (<ref>), we have by (<ref>) thatP_Y(x_N - n + 1(t) > M) = ∑^N_r = N - n + 1∑_J' = { j'_1, …, j'_r }J' ⊆{ 1, …, N } and | J' | = r (-1)^N - r q^(j'_1 + … + j'_r) - r - N(N - 1)/2 ×∑^r_l = N - n + 1 (-1)^N - l q^(N - l)(N - l - 1)/2 + l(N - r)rl ×_C dw_j'_1/w_j'_1…_C dw_j'_r/w_j'_r B_r(w_j'_1, …, w_j'_r) ∏^r_i = 1[ ^M_k = y_j_i( b_k/b_k - w_j'_i) e^-w_j'_i t].Using (<ref>) and the relation rl = rr - l, we have∑^r_l = N - n + 1 (-1)^N - l q^(N - l)(N - l - 1)/2 + l(N - r)rl= (-1)^N - r q^(N + r - 1)(N - r)/2∑^r - (N - n + 1)_l = 0 (-1)^l q^l(l - 1)/2rl= (-1)^N - r q^(N + r - 1)(N - r)/2 (-1)^r - (N - n + 1) q^[r - (N - n + 1)][r - (N - n)]/2r - 1r - (N - n + 1)= (-1)^n - 1 q^(N - r)(N - n) + n(n - 1)/2r - 1N - n.Plugging this into (<ref>), we haveP_Y(x_N - n + 1(t) > M) = (-1)^N - n + 1 q^(N - n + 1)(N - n)/2∑^N_r = N - n + 1 (-1)^r r - 1N - n ×∑_J' = { j'_1, …, j'_r }J ⊆{ 1, …, N } and | J' | = r q^(j'_1 + … + j'_r) - (N - n + 1)r ×_C dw_j'_1/w_j'_1…_C dw_j'_r/w_j'_r B_r(w_j'_1, …, w_j'_r) ∏^r_i = 1[ ^M_k = y_j_i( b_k/b_k - w_j'_i) e^-w_j'_i t],and prove (<ref>). §.§ Infinite system with step initial condition In this subsection we prove Theorem <ref>. First consider the q-TAZRP model with N particles starting at Y = 0^N = (0, 0, …, 0). By (<ref>), for a nonnegative integer M,ℙ_0^N(x_m(t) > M) = (-1)^m q^m(m - 1)/2∑^N_r = m S_N(M, t; r, m),whereS_N(M, t; r, m) = (-1)^r s_N(r, m) _Γd z_1/z_1…_Γd z_r/z_r B_r(z_1, …, z_r) ∏^r_j = 1[ ∏^M_k = 0( b_k/b_k - z_j) e^-z_j t] = (-1)^r s_N(r, m) _0^r(x_r(t) > M),such that Γ is the contour in (<ref>), ands_N(r, m) = ( ∑_J' = { j'_1, …, j'_r }⊆{ 1, …, N } and | J' | = r q^(j'_1 + … + j'_r) - mr) r - 1m - 1. To prove Theorem <ref>, we take the N →∞ limit of (<ref>). Since _0^r(x_r(t) > M) is a number between 0 and 1 that is independent of N, we only need to take the limit of s_N(r, m) as N →∞. Since s_N(r, m) > 0 for all N, r, m, the convergence of the series as N →∞ is absolute, so we can freely change the order of summation.We consider first the m = 1 case of Theorem <ref>, and have in that cases_N(r, 1) = ∑_J' = { j'_1, …, j'_r }⊆{ 1, …, N } and | J' | = r q^(j'_1 + … + j'_r) - r.As N →∞, we have that for all rlim_N →∞ s_N(r, 1) = ∑_1≤ j'_1 < j'_2 < … < j'_r < ∞ q^(j'_1 + … + j'_r) - r = q^r(r + 1)/2 - r/(1 - q)(1 - q^2) … (1 - q^r) = q^r(r - 1)/2/(1 - q)^r [r]_q!.Thus we have that for any fixed M and r, as N →∞, we have_0^∞(x_1(t) > M) = lim_N →∞_0^N(x_1(t) > M) = -∑^∞_r = 1 (-1)^r q^r(r - 1)/2/(1 - q)^r [r]_q!_0^r(x_r(t) > M).Next, we use <cit.> and express(-1)^r q^r(r - 1)/2_0^r(x_r(t) > M) = [r]_q!/r! (1 - q^-1)^r _Γ dz_1 …_Γ dz_r ( 1/z_j/q - z_k)^r_j, k = 1∏^r_j = 1[ ∏^M_k = 0( b_k/b_k - z_j) e^-z_j t],and then use <cit.> to further express∑^∞_r = 0 (-1)^r q^r(r - 1)/2/(1 - q)^r [r]_q!_0^r(x_r(t) > M) =∑^∞_r = 01/r!_Γ dz_1 …_Γ dz_r ( -q^-1/z_j/q - z_k∏^M_k = 0( b_k/b_k - z_j) e^-z_j t)^r_j, k = 1=(I + K_M, t),where K_M, t is the integral operator, from L^2(C) to L^2(C), defined in (<ref>). Hence by (<ref>) we conclude that_0^∞(x_1(t) ≤ M) = 1 - lim_N →∞_0^N(x_1(t) > M) = (I + K_M, t),which is the m = 1 case of (<ref>).By the same method, we can prove Theorem <ref> for general m. Similar to (<ref>), we havelim_N →∞ s_N(r, m) =( ∑_J' = { j'_1, …, j'_r }⊆{ 1, …, N } and | J' | = r q^(j'_1 + … + j'_r) - mr) r - 1m - 1=q^r(r - 1)/2/(1 - q)^r [r]_q! q^-(m - 1)rr - 1m - 1.By the q-binomial theorem (See <cit.>)∑^∞_r = m q^-(m - 1)rr - 1m - 1ξ^r =q^-m(m - 1)ξ^m 1/(1 - ξ)(1 - q^-1ξ) … (1 - q^-(m - 1)ξ),where the convergence is for |ξ| < q^m - 1, so we have that_0^∞(x_m(t) > M) =lim_N →∞_0^N(x_m(t) > M) = (-1)^m q^m(m - 1)/2∑^∞_r = m (-1)^r q^r(r - 1)/2/(1 - q)^r [r]_q! q^-(m - 1)rr - 1m - 1_0^r(x_r(t) > M) =(-1)^m q^-m(m - 1)/2/2π i∮_0 dξ/ξ[ ∑^∞_r = 0ξ^-r/r!_Γ dz_1 …_Γ dz_r ( -q^-1/z_j/q - z_k∏^M_k = 1( b_k/b_k - z_j) e^-z_j t)^r_j, k = 1] (-1)^m q^-m(m - 1)/2/2π i∮_0×ξ^m/(1 - ξ)(1 - q^-1ξ) … (1 - q^-(m - 1)ξ)=(-1)^m q^-m(m - 1)/2/2π i∮_0 dξ/ξ(I + ξ^-1 K_M, t) ξ^m/(1 - ξ)(1 - q^-1ξ) … (1 - q^-(m - 1)ξ),where the ξ contour lies in the region |ξ| < q^m - 1. Finally we take the change of variables ζ = ξ^-1, and have (<ref>) for general m, where the ζ contour lies out of the region |ζ| > q^1 - m. We complete the proof by noting that the ξ contour in (<ref>) can be taken that the corresponding ζ contour is the contour C given in Theorem <ref>.§ ASYMPTOTICSThe main part of this section is the proof of Theorem <ref>, and in the end of this section we prove Corollary <ref>.To prove Theorem <ref>, we assume that M, t and b_k are specified as in (<ref>), depending on constants τ, l, β_0, …, β_l, and n →∞. Then we follow the approach in <cit.> and <cit.>, and prove the following two limiting properties of the integral operator K_M, t in Theorem <ref>: * ( ( . K_M, t|_M = n + l + 1, t = n - τ√(n))^k ) →^kfor all k = 1, 2, …, where = _τ; β_0, …, β_l whose kernel is defined in (<ref>).*. (I + ζ K_M, t) |_M= n + l + 1,andt = n - τ√(n) is uniformly bounded for large n on compact ζ sets.If we have the two properties above, then it is straightforward to check that the convergence in (<ref>) holds uniformly in λ, and then the contour integral on the right-hand side of (<ref>) converges to the contour integral on the right-hand side of (<ref>), if the contour C is specified as in Theorem <ref>. Hence Theorem <ref> is proved. Below we prove the two properties. We note thatis a trace class operator, so its Fredholm determinant is well defined. We denotef(w) = w + log(1 - w) and g(w) = ∏^l_k = 0β_k/β_k - w n^1/2.Note that log(1 - w) is not well defined on the whole Γ, and we take the branch cut of log(1 - w) at [1, +∞) and assume its imaginary part is between -π and π. Nevertheless, the exponential of f(w) is holomorphic on Γ, andK_M, t(w, w') = exp[ -n ( f(w) - τ/√(n) w ) ]/qw' - w g(w).It is not hard to see that w = 0 is the unique critical point for f(w). Inspired by this result and owing to the flexibility of the shape of the contour Γ, we deform Γ into the rectangle with vertices -n^-1/2 + i, 2 + i, 2 - i and -n^-1/2 -i, and the four edges denoted by Γ_1, Γ_2, Γ_3, Γ_4, as shown in Figure <ref>. Here we note that the orientation of Γ_1 is downward.It is easy to check that f(w) attains its minimum at w = -n^-1/2 over Γ. To see it, we only need to consider η∈Γ∩_+ since f(w̅) = f(w). Using that [label=(*)]* as w = -n^-1/2 + iy ∈Γ_1 ∩_+, f(w) = -n^-1/2 + log| 1 - w | is a decreasing function of y,* as w = x + i ∈Γ_4, f(w) is an increasing function of x for d/dx f(w) = 1 + ((1 - w)^-1) > 0, and* as w = 2 + iy ∈Γ_3 ∩_+, f(w) is an decreasing function of y with minimum at f(2) >f(-n^-1/2),we verify that -n^-1/2 is the minimum of f(w) on Γ.Next, we denoteΓ_ = { w ∈Γ_1 || w | < n^-2/5}, Γ_ = Γ∖Γ_.By the discussion on f(w) above, it is straightforward to see that( f(w) - τ/√(n) w ) ≥ f(-n^-1/2 + i n^-2/5) > -n^-1/2 + c n^-4/5, for all w ∈Γ_,where c is a small enough positive number, say, c = 1/10. Thus the kernel K_M, t(w, w'), although defined on the whole Γ, concentrates locally at the middle of Γ_1, around 0.Let w = n^-1/2 u and w' = n^-1/2 v. Then for w, w' = (n^-2/5), u, v = (n^1/10). We have that for all w ∈g(w) = γ(u),where γ is defined in (<ref>). On the other hand, sincef(0) = f'(0) = 0, and f”(0) = -1,we have that for w ∈Γ_,-n ( f(w) - τ/√(n) w ) = u^2/2 + τ u^2 + n (w^3) =u^2/2 + τ u^2 + (n^-1/5). Now we are ready to prove (<ref>). For notational simplicity, we give the detail for the k = 2 case, and then explain how to extend the argument to general k.We have by (<ref>) and <cit.>( (K_M, t)^2 ) = ∫_Γ dw_1 ∫_Γ dw_2 K_M, t(w_1, w_2) K_M, t(w_2, w_1) = I_,+ I_,+ I_,+ I_, ,whereI_*, ⋆ = ∫_Γ_* dw_1 ∫_Γ_⋆ dw_2 exp[ -n ( f(w_1) + f(w_2) - τ/√(n) (w_1 + w_2) ) ]/(w_1 - qw_2)(w_2 - qw_1) g(w_1) g(w_2),, ⋆ =or .By (<ref>), we haveI_,=∫^-1 - n^1/10 i_-1 + n^1/10 i du ∫^-1 - n^1/10 i_-1 + n^1/10 i dv e^(u^2 + v^2)/2 + τ(u + v)/(u - qv)(v - qu)γ(u) γ(v) (1 + (n^-1/5)) =∫_Γ_∞ du ∫_Γ_∞ dv e^(u^2 + v^2)/2 + τ(u + v)/(u - qv)(v - qu)γ(u) γ(v) + (n^-1/5),where Γ_∞ = { -1 - iy | y ∈} is defined in Theorem <ref>. By (<ref>), we have the estimateI_,= o(e^-2c n^1/5),and using (<ref>) and (<ref>) together, we derive thatI_,= o(e^-c n^1/5),I_,= o(e^-c n^-1/5).On the other hand, by the definition ofin (<ref>),^2 = ∫_Γ_∞ dz ∫_Γ_∞ dw e^(z^2 + w^2)/2 + τ(z + w)/(z - qw)(w - qz)γ(z) γ(w).Comparing (<ref>) with (<ref>), (<ref>) and (<ref>), we prove the k = 2 case of (<ref>). For general k, we need to replace the two-fold integral in (<ref>) by a k-fold integral as in <cit.>, and then similarly decompose the k-fold integral into the sum of 2^k terms, such that each term is a k-fold iterated integral with each integral domain being Γ_ or Γ_. Then the argument for the k = 2 case works for the general k case.We note that the integral operator K_M, t is a trace class operator as well as a Hilbert–Schmidt operator, so we can write(I - λ K_M, t) = (1 - λ K_M, t) e^ K_M, t,whereis defined in <cit.>. First, by <cit.>, we haveK_M, t = 1/2π i∫_Γ K_M, t(w, w) dw.Thus as M = n + l + 1, t = n - τ√(n) and n →∞, we have, like the computation of ((K_M, t)^2) in the proof of property <ref>,K_M, t =1/2π i∫_Γexp[ -n ( f(w) - τ/√(n) w ) ]/(1 - q)w g(w) dw =1/2π i∫_Γ_exp[ -n ( f(w) - τ/√(n) w ) ]/(1 - q)w g(w) dw + o(e^-c n^1/5) =1/2π i∫^-1 - n^1/10 i_-1 + n^1/10ie^u^2/2 + τ u/(1 - q)uγ(u) (1 + (n^-1/5)) du + o(e^-c n^1/5) = 1/2π i∫_Γ_∞e^u^2/2 + τ u/(1 - q)uγ(u) du + (n^-1/5) = C + (n^-1/5),where c is the same as in (<ref>) and the C is equal to the contour integral on Γ_∞, and we do not compute it explicitly.On the other hand, by <cit.>, we have|(1 - λ K_M, t) |≤exp( |λ|^2 ‖ K_M, t‖^2_2),where ‖·‖_2 is the Hilbert–Schmidt norm. (The constant Γ_n in <cit.> can be taken as 1.) Thus we only need to show that ‖ K_M, t‖_2 is bounded as M = n + l + 1, t = n - τ√(n) and n →∞. To this end, we use <cit.>, and have‖ K_M, t‖^2_2 = 1/4π^2∫_Γ| dw' |∫_Γ| dw ||φ(w)/qw' - w g(w') |^2.As n →∞, we have, by the concentration of K_M, t(w, w') over Γ around 0,‖ K_M, t‖^2_2 =1/4π^2∫_Γ| dw' |∫_Γ| dw ||exp[ -n ( f(w) - τ/√(n) w ) ]/qw' - w g(w) |^2=1/4π^2∫_Γ_| dw' |∫_Γ_| dw ||exp[ -n ( f(w) - τ/√(n) w ) ]/qw' - w g(w) |^2 + o(e^-cn^1/5)=1/4π^2∫^-1 - n^1/10 i_-1 + n^1/10 i| dv |∫^-1 - n^1/10 i_-1 + n^1/10 i| du ||e^u^2/2 + τ u/qv - uγ(u) |^2 (1 + (n^-1/5)) + o(e^-cn^1/5)=1/4π^2∫_Γ_∞| dv |∫_Γ_∞| du ||e^u^2/2 + τ u/qv - uγ(u) |^2 + (n^-1/5).We are not going to evaluate the integral in (<ref>) explicitly, but it is clear that ‖ K_M, t‖^2_2 is bounded as M, t, b_k are specified in (<ref>) and n →∞. Hence we finish the proof. Our proof to Theorem <ref> is similar to the proof of <cit.> that obtains analogous limiting distributions for the leftmost eigenvalues in the ASEP with step initial condition. The argument in our proof is simpler than that in <cit.>, because our integral operator K_M, t is on L^2(Γ) such that Γ encloses the pole 0, while the counterpart integral operator K_2 defined in <cit.> is defined on L^2(γ) such that the contour γ does not enclose the pole 0. Analogously, one can compute the limiting distribution of x_m(t) as both m, t →∞ by the method in the proof of <cit.>, and some steps can be simplified since our contour Γ is more friendly to asymptotic analysis than the γ in <cit.>.We only need to prove part <ref> by checking (<ref>), and then part <ref> follows as we plug the Fredholm determinants (I + ζ_τ; -) and (I + ζK̂χ_(τ(1 + q)/(1 - q), ∞)) into the countour integral formulas for the m-th right-most particle in q-TAZRP (the right-hand side of our (<ref>)) and for the m-th left-most particle in ASEP (the right-hand side of <cit.>) respectively.To check (<ref>), we note that for all z, w ∈Γ_∞, since (w - qz) = q - 1 < 0,1/w - qz = -∫^∞_0 e^(w - qz)ξ dξ,and then_τ; -(z, w) = -∫^∞_0 e^-qz ξ e^w^2/2 + w(ξ + τ) dξ.Hence we define the integral operators A: L^2(Γ_∞) → L^2(_+) and B: L^2(_+) → L^2(Γ_∞), with kernelsA(ξ, w) = e^w^2/2 + w(ξ + τ),B(z, ξ) = e^-qz ξ.We have that _τ; - = -AB. If we define _τ; - = -BA, an operator from L^2(_+) to L^2(_+), then (I + _τ; -) = (I + _τ; -). Here _τ; - is an integral operator with kernel_τ; -(ξ, η) = -1/2π i∫_Γ_∞ e^z^2/2 + z(ξ + τ) e^-qz η dz = 1/√(2π) e^-1/2 (ξ - qη + τ)^2.By change of variables, we hav3 that(I + K_τ; -) =(I + K̃_τ; -) =( I - χ_(0, ∞)(ξ) 1/√(2π) e^-1/2 (ξ - qη + τ)^2χ_(0, ∞)(η) ) =( I - χ_(τ/(1 - q), ∞)(ξ) 1/√(2π) e^-1/2 (ξ - qη)^2χ_(τ/(1 - q), ∞)(η) ) =( I - χ_(τ(1 + q)/(1 - q), ∞)(ξ) 1/√(2π)(1 + q) e^-(ξ - qη)^2/2(1 + q)^2χ_(τ(1 + q)/(1 - q), ∞)(η) ).Noting that1/√(2π)(1 + q) e^-(ξ - qη)^2/2 = e^(1 - q^2)η^2/4/e^(1 - q^2)ξ^2/4K̂(ξ, η),where K̂ is defined in (<ref>), and that Fredholm determinant is invariant under conjugation, we prove (<ref>).§ LIMITING DISTRIBUTIONS OF X_M(T) AS M, T →∞ AND T/M → CWe first fix some notations following <cit.>. LetΨ_q(z) = ∂/∂ zlogΓ_q(z), whereΓ_q(z) = (1 - q)^1 - z(q; q)_∞/(q^z; q)_∞.Then for θ > 0, letκ= κ(q, θ) =Ψ'_q(θ)/(log q)^2 q^θ, f = f(q, θ) =Ψ'_q(θ)/(log q)^2 - Ψ_q(θ)/log q - log(1 - q)/log q,χ = χ(q, θ) =1/2(Ψ'_q(θ) log q - Ψ”_q(θ)).Furthermore, fix α∈ (0, 1], we defineg = g(q, θ) =Ψ'_q(θ)/(log q)^2α/q^θ - Ψ_q(log_q α)/log q - log(1 - q)/log q,σ = σ(q, θ) =Ψ'_q(θ) α/q^θ - Ψ'_q(log_q α).We note that as θ runs over (0, +∞), χ(q, θ) > 0, and κ(q, θ) decreases monotonically from +∞ to (1 - q)^-1. Recall that in (<ref>) we introduced the notation ^_(y_n(t) ≥ x) to represent the distribution of the n-th particle in the q-TASEP with step initial condition.<cit.>Suppose b_k = 1 for all k > m, α = min(b_1, …, b_m) ∈ (0, 1], and k out of b_1, …, b_m are equal to α. Let c ∈ be a constant. * For θ∈ (log_q α, +∞), we havelim_n →∞^_( y_n ( (1 - q)κ n + (1 - q) q^-θ c n^2/3) .≥. (f - 1)n + cn^2/3 - c^2 (log q)^3/4χ n^1/3 + χ^1/3/log qξ n^1/3) = F_(ξ), * For θ = log_q α, we havelim_n →∞^_( y_n ( (1 - q)κ n + (1 - q) q^-θ c n^2/3) .≥. (f - 1)n + cn^2/3 - c^2 (log q)^3/4χ n^1/3 + χ^1/3/log qξ n^1/3) = F_, k, 𝐛(ξ),where 𝐛 = (b, …, b) with b = c(log q)^2/(2χ^2/3).* If θ∈ (0, log_q α), we havelim_n →∞^_( y_n ( (1 - q)κ n + (1 - q) α^-1 c n^1/2) ≥ (g - 1)n + c n^1/2 + σ^1/2/log qξ n^1/3)= G_k(ξ). Here F_ is the GUE Tracy–Widom distribution, F_, k, 𝐛 is the BBP (Baik–Ben Arous–) distribution with rank k, and G_k is the distribution of the largest eigenvalues of a k × k GUE. Their precise formulas are given in <cit.>. The phase transition of the three distributions in <cit.> was studied first in the spiked Wishart ensemble in random matrix theory <cit.>. Hence by relation (<ref>), we have the corresponding results for the q-TAZRP with step initial condition.Suppose b_k = 1 for all k > m, α = min(b_1, …, b_m) ∈ (0, 1], and k out of b_1, …, b_m are equal to α. Let c ∈ be a constant. * For θ∈ (log_q α, +∞), we havelim_n →∞_0^∞( x_⌈ (f - 1)n + cn^2/3 - c^2 (log q)^3/4χ n^1/3 + χ^1/3/log qξ n^1/3⌉( (1 - q)κ n + (1 - q) q^-θ c n^2/3) > n ) = F_(ξ), * For θ = log_q α, we havelim_n →∞_0^∞( y_⌈ fn + cn^2/3 - c^2 (log q)^3/4χ n^1/3 + χ^1/3/log qξ n^1/3⌉( (1 - q)κ n + (1 - q) q^-θ c n^2/3) > n ) = F_, k, 𝐛(ξ),where 𝐛 = (b, …, b) with b = c(log q)^2/(2χ^2/3).* If θ∈ (0, log_q α), we havelim_n →∞_0^∞( y_⌈ gn + c n^1/2 + σ^1/2/log qξ n^1/3⌉( (1 - q)κ n + (1 - q) α^-1 c n^1/2) > n ) = G_k(ξ).10Andrews-Askey-Roy99 G. E. Andrews, R. Askey, and R. Roy. Special functions, volume 71 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1999.Baik-Barraquand-Corwin-Suidan16 J. Baik, G. Barraquand, I. Corwin, and T. Suidan. Facilitated exclusion process and Pfaffian Schur processes, 2016. arXiv:1606.00525.Baik-Ben_Arous-Peche05 J. Baik, G. Ben Arous, and S. Péché. Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab., 33(5):1643–1697, 2005.Barraquand15 G. Barraquand. A phase transition for q-TASEP with a few slower particles. Stochastic Process. Appl., 125(7):2674–2699, 2015.Borodin-Corwin13 A. Borodin and I. Corwin. Macdonald processes. Probab. Theory Related Fields, 158(1-2):225–400, 2014.Borodin-Corwin-Petrov-Sasamoto15 A. Borodin, I. Corwin, L. Petrov, and T. Sasamoto. Spectral theory for the q-Boson particle system. Compos. Math., 151(1):1–67, 2015.Borodin-Corwin-Sasamoto14 A. Borodin, I. Corwin, and T. Sasamoto. From duality to determinants for q-TASEP and ASEP. Ann. Probab., 42(6):2314–2382, 2014.Borodin-Petrov16 A. Borodin and L. Petrov. Higher spin six vertex model and symmetric rational functions, 2016. arXiv:1601.05770, to appear in Selecta Math. (N.S.).Ferrari-Veto15 P. L. Ferrari and B. Vető. Tracy-Widom asymptotics for q-TASEP. Ann. Inst. Henri Poincaré Probab. Stat., 51(4):1465–1485, 2015.Holley70 R. Holley. A class of interactions in an infinite particle system. Advances in Math., 5:291–309 (1970), 1970.Imamura-Sasamoto17 T. Imamura and T. Sasamoto. Fluctuations for stationary q-TASEP, 2017. arXiv:1701.05991.Korhonen-Lee14 M. Korhonen and E. Lee. The transition probability and the probability for the left-most particle's position of the q-totally asymmetric zero range process. J. Math. Phys., 55(1):013301, 15, 2014.Povolotsky04 A. M. Povolotsky. Bethe ansatz solution of zero-range process with nonuniform stationary state. Phys. Rev. E (3), 69(6):061109, 7, 2004.Povolotsky13 A. M. Povolotsky. On the integrability of zero-range chipping models with factorized steady states. J. Phys. A, 46(46):465205, 25, 2013.Sasamoto-Wadati98 T. Sasamoto and M. Wadati. Exact results for one-dimensional totally asymmetric diffusion models. J. Phys. A, 31(28):6057–6071, 1998.Simon05 B. Simon. Trace ideals and their applications, volume 120 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, second edition, 2005.Spitzer70 F. Spitzer. Interaction of Markov processes. Advances in Math., 5:246–290 (1970), 1970.Tracy-Widom08a C. A. Tracy and H. Widom. A Fredholm determinant representation in ASEP. J. Stat. Phys., 132(2):291–300, 2008.Tracy-Widom08 C. A. Tracy and H. Widom. Integral formulas for the asymmetric simple exclusion process. Comm. Math. Phys., 279(3):815–844, 2008.Tracy-Widom09 C. A. Tracy and H. Widom. Asymptotics in ASEP with step initial condition. Comm. Math. Phys., 290(1):129–154, 2009.Tracy-Widom09a C. A. Tracy and H. Widom. On ASEP with step Bernoulli initial condition. J. Stat. Phys., 137(5-6):825–838, 2009.Wang-Waugh16 D. Wang and D. Waugh. The transition probability of the q-TAZRP (q-Bosons) with inhomogeneous jump rates, 2015. | http://arxiv.org/abs/1703.08839v1 | {
"authors": [
"Eunghyun Lee",
"Dong Wang"
],
"categories": [
"math.PR",
"math-ph",
"math.MP",
"nlin.SI",
"82C22, 82B23"
],
"primary_category": "math.PR",
"published": "20170326161941",
"title": "Distributions of a particle's position and their asymptotics in the $q$-deformed totally asymmetric zero range process with site dependent jumping rates"
} |
In their precedent work, the authors constructed closed oriented hyperbolic surfaces with pseudo-Anosov homeomorphisms from certain class of integral matrices. In this paper, we present a very simple algorithmto compute the Teichmüller polynomial corresponding to those surface homeomorphisms by first constructing an invariant track whose first homology group canbe naturally identified with the first homology group of the surface, andcomputing its Alexander polynomial. Hyungryul Baik, Chenxi Wu Appendix by KyeongRo Kim and TaeHyouk Jo December 30, 2023 ========================================================================== § INTRODUCTIONFor orientable 3-manifolds, Thurston defined a norm, so-called Thurston norm, on the second homology group. In this paper, by Thurston norm, we always meanthe dual norm on the first cohomology group. The unit norm ball for Thurston norm is a rational polytope. Thurston showed that for a top-dimensional face F of the unit norm ball,if an integral point in the cone F ·_+ is represented by a fibration of the 3-manifold over S^1, then all integral points in the same cone are also represented by fibrations.In this case, F is called a fibered face, and F ·_+ is called a fibered cone. In <cit.>, McMullen defined a polynomial invariant for fibered faces, so-called Teichmüller polynomial.If we are given a surface S with a pseudo-Anosov monodromy ϕ,we will simply say the Teichmüller polynomial for the pair (S, ϕ) to denote the Teichmüller polynomial for the fibered cone which contains a cohomology corresponding to thefibration defined by S, ϕ. There have been a lot of interesting applications of Teichmüller polynomialsby many authors which we do not even attempt to make a complete list here.One of the most noteworthy application of Teichmüller polynomial lies in the study of minimal dilatation of pseudo-Anosov surface homeomorphisms. See, for instance, <cit.>. There are also other polynomial invariants developed in the literature as analogues to Teichmüller polynomial. For instance, Dowdall, Kapovich, and Leininger defined the poynomial invariant, which they call a McMullen polynomial, for the cohomology classes of free-by-cyclic groups in <cit.> based on <cit.>, which is in a sense a generalization of the Teichmüller polynomial.A similar result was obtained by Algom-Kfir, Hironaka, and Rafi in <cit.>.We also note that in <cit.>, McMullen used so-called clique polynomials to give a sharp lower bound on the spectral radius of a reciprocal Perron-Frobenius matrix with a given size.The first step in McMullen's construction of Teichmüller polynomial is to define a module from the 2-dimensional lamination (i.e. the suspension of the stable lamination of the pseudo-Anosov monodromy ofa fibration). This 2-dimensional lamination (up to isotopy) does not depend on the choice of the fibration of the manifold. One can replace the stable lamination by the invariant train track, and then the 2-dimensional lamination is replaced by a branched surface. The definition of Teichmüller polynomial in <cit.> is through an algorithm of computing them based on these invariant train tracks. However, due to the importance of this invariant other more efficient algorithms of computation in specific cases have been developed, e.g. <cit.>. In this paper, we present a simple algorithm to compute Teichmüller polynomial for so-called odd-block surfaces constructed from some {0,1}-matrices in a precedent work of the authors <cit.>.The definition and construction of such surfaces will be recalled in Section <ref>.Odd-block surfaces form a large class of translation surfaces which come with pseudo-Anosov self-homeomorphisms.We observe that for this special case,the Teichmüller polynomial is just the Alexander polynomial of an associated finitely presented group, and the latter is fairly easy to understand and compute. This reduction of the computation of the Teichmüller polynomial to the computation of the Alexander polynomial is described in Proposition <ref>. As explained in the proof of Proposition <ref>, what makes our algorithm to work for odd-block surface is thatthe matrix we start with in our construction indicates the transition matrix of a Markov partition of the surface under this pseudo-Anosov map, and the rectangular blocks of this Markov partition is ordered in a very regular way which allows us to analyze its lift to the abelian cover easily. In particular, one gets a natural identification between the first homology of the surface and the first homology of the train track we construct in the algorithm (see Step I in Section <ref>), and this allows us to compute the Teichmüller polynomial via the Alexander polynomial of the branched surface which is the suspension of the train track we deal with.Also, it is essential that the train track in our case is orientableas explained in the proof of Proposition <ref>. After we recall the set up in Section <ref>, we present our algorithm in Section <ref> using a running example, and a give a proof of that our algorithm works in Section <ref>.§CONSTRUCTION OF ODD-BLOCK SURFACESIn this section, we quickly recall the construction of surfaces given in <cit.>. We start with an aperiodic non-singular n × n matrix M with only entires 0 and 1 satisfying so-called odd-block condition. Namely, M is said to be an odd-block matrix if the following two conditions are satisfied:(i) In each column of A, the non-zero entries form one consecutive block; and(ii) There is a map ϕ:{0,1,...,n}→{0,1,...,n} such that the entry A_ij is odd if and only if min{ϕ(j-1),ϕ(j)} < i ≤max{ϕ(j-1),ϕ(j)}. In each column, inside the consecutive block of non-zero entries, there is a consecutive sub-block of odd entries. Moreover, the "final position" of the odd-block in a column is the "initial position" of the odd-block inthe next column where "final position" and "initial position" are just the values of the function ϕ in the definition.Hence the odd-blocks form a snake-shape. Note that the odd-block in some column could be empty. The concept of odd-block matrices are originally introduced in <cit.>, but the name was coined in <cit.> where the properties of odd-block matrices were further investigated. Let us use the following running example: M = [ 0 0 0 0 0 0 1 1; 0 0 0 0 1 1 1 1; 0 0 0 0 1 1 1 0; 0 0 1 1 1 1 1 0; 0 0 1 1 1 0 0 0; 1 1 1 1 1 0 0 0; 1 1 1 0 0 0 0 0; 0 1 1 0 0 0 0 0; ] In the rest of the section, we will follow the recipe given in <cit.> to construct a surface of finite type and an orientation-preserving pseudo-Anosov homeomorphism from the above matrix M. In fact, one could construct an orientation-reserving homeomorphism but since we are going to use only orientation-preserving homeomorphisms for the rest of the paper,we only recall the process of constructing orientation-preserving pseudo-Anosov homeomorphisms. The eigenvalue v⃗ of M^T for the leading eigenvalue normalized so that the L^1-norm is 1 is (approximately) {0.0740679, 0.0874795, 0.187546, 0.134026, 0.21327, 0.117861,0.139202, 0.0465469 }.Let v_i denote the ith entry of v⃗. Then one can use v⃗ to get a partition {x_0, x_1, …, x_8} of [0,1] so that x_i - x_i-1 = v_i for each i = 1, …, 8.Let's consider the 8 × 8 grid diagramso that the boxes corresponds to the entries of M flipped upside down.Identify each side of this grid diagram with the closed interval [0,1] so that that diagram represents the region [0,1]× [0,1]. We adjusts the heights and widths of rows and columns of this diagram so that the ith column has width v_i and(n-i)th row has height v_i for each i. In each box with entry 1, we draw a line segment connecting the top and the bottom of the box.By arranging them appropriately, one can always draw a graph of a continuous piecewise-linear map h_M as in Figure <ref>. As shown in <cit.>, for the map h_M, M is an extended incident matrix, the set {x_0, …, x_8} is contained in the post-critical set of h_M, and the leading eigenvalue of M is the absolute value of the slope in each piece. Say λ_M is the leading eigenvalue. The red dots are { (x_i, h_M(x_i) ) : i = 1,…, 7}. One define so-called alignment function α: {1, …, 7}→{-1, 1} as follows; at each i, consider the horizontal line passing through(x_i, h_M(x_i)). If the horizontal line meets the graph of h_M on the right side of x_i, then defineα(i) = 1, and if it meets the graph on the left side, define α(i) = -1. If the horizontal line meetsthe graph of h_M meets on both sides of x_i, then the construction fails. If the horizontal line does not meet the graph at all, we leave α undefined at that point. In our running example,α(1) = α(3) = α(5) = 1, α(4) = α(6) = -1, and α(2), α(7) are undefined. At the moment, the alignment function α is defined only on a proper subset of {1, …, 7}.As defined in <cit.>, we say M satisfies the alignment condition if α can be extended to theentire {1, …, 7} while satisfying the following conditions;(a) For critical x_i, α(i) =-1 if x_i is a local max.1 if x_i is a local min.(b) For noncritical x_i,α(i) = α(ϕ_i) if h'(x_i)>0.- α(ϕ_i) if h'(x_i)<0.In our example, the condition (b) is vacuously satisfied, since there is no non-critical x_i.Setting α(2) = -1 and α(7) = 1, we get an alignment function α defined at every x_i. Now let's go back to the grid diagram above but flip it along the horizontal axis once more so that the ith row hasheight v_i. We consider rectangles R_i, i=1, …, 8, for the rows of this grid diagram. More precisely,R_i is obtained from putting the boxes in ith row which are labeled with 1 side by side. In our example, all the1's in each row are consecutive, but it is not necessarily the case (see, for instance, Figure 3 of <cit.>).From the same grid diagram, we define another set of rectangles C_i, i=1, …, 8, for columns of the diagram. Namely, C_i is just the non-zero block in the ith column. We make a polygonal region P_0 by putting R_i+1 right below R_i for each i with the following rule;align R_i, R_i+1 on the left or right according to whether α(i) = -1 or 1 respectively. Note that P_0 is the union of R_i's but at the same time the union of C_i's. We define a piecewise-affine map f from P_o to P_o as follows; we map each R_i to C_i via an affine map which stretches R_i vertically by the factor of λ_M and compresses horizontally by the factor of 1/λ_M. When h_M has negative derivative on [x_i-1, x_i], we compose this affine map with 180 degree rotation. Then we obtain a piecewise-affine map on P_0 which is well-defined in the interior of each R_i. For our example, see Figure <ref>. Note that f is not well-defined on line segments shared by R_i and R_i+1. For each such line segment, f has two images. To make f well-defined, we need to identify those two images. By doing this for every such line segment, one gets well-defined f. On the other hand, this process still leaves a problem of ill-definedness of f^2, so the images of the parts of P_0 which are identified in the previous step need to be identified again. Repeating this process, one gets infinitely many gluing information on the boundary of P_0. By definition of f, these gluing information are all on the horizontal edges of P_0. To get gluing information on the vertical edges of P_0, one just repeat the same process with f^-1 instead of f, and focus on the line segments shared by C_i's. What the authors showed in <cit.> is that the quotient of P_0 with these gluing information on the boundary given by the map f is a closed surface equipped with a singular Euclidean metric with finitely many singular points. By construction, f is automatically a pseudo-Anosov map on this surface.Hence, we just obtained a translation surface with a pseudo-Anosov homeomorphism on it. Let us call a translation surface obtained in above process a odd-block surface.Even though one can produce a large class of examples using the constructed described above, it seems to be not so easy to characterize the odd-block surfaces with their intrinsic properties. We propose the following open problem for future research.Find an interesting chracterization of odd-block surfaces. In the rest of the paper, we will focus on the case when n is even and any two consecutive entries of the eigenvector corresponding to the leading eigenvalue are different. As shown in <cit.>(very last part of Section 5), a careful analysis of the gluing information on the boundary of P_0 reveals that the resulting translation surface has n+1 cone points Q_i's with cone angle π and a single cone point Q with cone angle (n-1)π. The assumption that two consecutive entries of the eigenvector guarantees that the neighboring rectangles all have different heights hence none of the Q_i would disappear.Taking the double cover of the surface which are ramified at Q and Q_i's, one gets a surface S_g where the genus g of the surface is exactly n/2. In this case, the curves connecting Q_i and Q_i+1 in P_0 lift to loops which form a basis of H_1(S_g). It is also observed in <cit.> that with respect to this basis, M represents the action of the lift of the pseudo-Anosov map constructed above on H_1(S_g). § AN ALGORITHM TO COMPUTE TEICHMÜLLER POLYNOMIAL FOR ODD-BLOCK SURFACES Let n be an even number and M be a n × n non-singular, aperiodic, odd-block {0,1}-matrix, such that any two consecutive entries of the eigenvector corresponding to the leading eigenvalue are different. Say S' is an odd-block surface and ψ' is an orientation-preserving pseudo-Anosov homeomorphism constructed as in the previous section. As we remarked at the end of the last section, one can take a branched double cover S of S' whose genus is exactly n/2, and ψ' lifts to a pseudo-Anosov homeomorphism ψ.Consider the mapping torus N_ψ = S × [0,1] / (x, 1) ∼ (ψ(x), 0). In H^1(N_ψ, ℝ), there exists a fibered cone containing an integral cohomology class corresponding to the fibration with fiber S and monodromy ψ. In this case, we present an algorithm which computes the Teichmüller polynomial associated with this fibered cone. As a running example, again we use M defined in the previous section; M = [ 0 0 0 0 0 0 1 1; 0 0 0 0 1 1 1 1; 0 0 0 0 1 1 1 0; 0 0 1 1 1 1 1 0; 0 0 1 1 1 0 0 0; 1 1 1 1 1 0 0 0; 1 1 1 0 0 0 0 0; 0 1 1 0 0 0 0 0; ] Step I. Compute the eigenvectors v⃗^j of M, and then for each non-zero entry v_i^j of v⃗^j, write ∑_j v_i^js_j as a superscript of every entry of i-th column of M. In our example, the only eigenvector v⃗^t is (1, 0, 0, 0, -1, 0, 1, 0). Hence the result of Step I for our example is [0000001^s1;0000 1^-s11^s1;0000 1^-s11^s0;0011 1^-s11^s0;0011 1^-s000;1^s111 1^-s000;1^s1100000;01100000;] This is an example where the first betti number of the mapping cylinder is 2. The case when the first betti number is 1 is trivial in that the Teichmüller polynomial, as well as the Alexander polynomial of the suspended train track (c.f. Proposition 1), would both be identical to the characteristic polynomial of the original matrix. Note that in this case the Steps II-IV would not change the matrix, hence this is consistent with the result of our algorithm. Step II. We push all superscripts ∑ n^js_j to the right in each row. In this process,the entries of M are multiplied by t_j or t^-1_j by applying the following rules repeatedly; * ^s_j 1 ↦ t_j^s_j* ^-s_j 1 ↦(t_j^-1)^-s_jThe result of Step II for our example is [0000001t^s;000011/t1/t1;000011/t1/t0;001111/t1/t0;0011 1^-s000;1tttt000;1tt^s00000;01100000;] Step III. Let r be some row with superscript ∑_j n^j s_j at the end. Now look at the rows above r which corresponds to decreasing piece in the piecewise-linear map (see the previous section for details).Say r_1, …, r_k are such rows where each row r_i has superscript ∑_jn^j_i s_j at the end. Then replace the row r by r ·∏_j t_j^-Σ_i n^j_i + ∏_j (t_j^n^j -1) ·Σ_l=1^k(∏_j t_j^-Σ_i=1^l n^j_i) r_l .and add a entry ∏_jt_j^n_j-1 to the end.The final result for our example is [0000001tt-1;000011/t1/t10;000011/t1/t00;001111/t1/t00;001/t1/t2/t-1 2/t(1/t-1) 2/t(1/t-1)1/t-11/t-1;1tttt0000;tt^2t^2+t-1t^2-1t^2+t-22-2/t2-2/tt-1t-1;011000000;] Step IV. Let M be the resulting matrix from Step III, and let N be the matrix of the same size as M obtained by adding the zero column to the right of the identity matrix. And take the greatest common divisor of the largest minors of the matrix M-xN.In our example, a direct computation shows that the gcd of the largest minors is the following polynomial; x^7 -x^6 - (2t+5+2/t)x^5 + x^4 - x^3 +(2t+5+2/t)x^2 + x -1.We shall show in the next section, that:The polynomial obtained from steps I-IV is the Teichmüller polynomial of the pseudo-Anosov map on S. We can verify the correctness of our algorithm in this example by using McMullen's algorithm. The “thickening” construction in the previous section gives us an Markov partition of the flat surface into 16 rectangular regions. This induces an invariant train track by associating an edge with each rectangle and a vertex with the each connected component of the union of the left edges (c.f. the proof section later). Hence, by section 3 of <cit.>, we have:Θ(u,t)= (u𝐈-([ 0 0 0 0 0 0 1 t 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1; 0 0 0 0 1 1/t 1/t 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0; 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0; 1 t t 0 0 0 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1; 0 0 0 0 1 1/t 1/t 1 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0; 0 0 1 1 1 1/t 1/t 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0; 1 t t t t 0 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0; 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ]))(u𝐈-( [ 0 0 0 0 0 0 1 0 0; 0 0 0 0 0 0 0 0 1; 0 0 0 0 1 0 0 0 0; 0 0 0 0 0 0 0 1 0; 0 0 1 0 0 0 0 0 0; 0 0 0 0 0 1 0 0 0; 1 0 0 0 0 0 0 0 0; 0 0 0 1 0 0 0 0 0; 0 1 0 0 0 0 0 0 0 ])) =x^8 - 2x^7 + (-2t - 2 t - 4)x^6 + (2t + 2 t+ 6)x^5 - 2x^4 + (2t +2 t+6)x^3 +(-2t - 2 t - 4)x^2 - 2x + 1x-1 =x^7 -x^6 - (2t+5+2/t)x^5 + x^4 - x^3 +(2t+5+2/t)x^2 + x -1.In the case when n is odd, or more generally, when the genus of the surface constructed via the procedure <cit.> is of a genus smaller than n/2, the above procedure can still be used to calculate the Teichmüller polynomial so long as Step I above is modified to only consider those eigenvectors corresponding to (absolute) cohomology classes of the resulting surface.§ PROOF OF THE MAIN THEOREM Step I.We describe the invariant train track and the train track map. As we explained in Section <ref>, an odd-block surfaces is built by gluing blocks where each block corresponds to a column of the given matrix, say M, and then we take a ramified double cover so that theinvariant foliation is orientable. We call a block to be “in the front” if it is in the polygon P_0 described in Section <ref>, “in the back” if otherwise.We call the part of the surface consisting of the original blocks the “front” and the part consisting of the other blocks the “back”. Let (S_g, ψ) be the pair of the surface an the psuedo-Anosov map we obtain. Now we can build an invariant (topological) train track as follows:each block corresponds to an edge and each connected component of the union of vertical edges of the blocks corresponds to a vertex.See Figure <ref>. We call an edge “in the front” if the corresponding block is in the front, “in the back” if the corresponding block is in the back.On this graph, call it Γ, we get a map, call it ψ_∗, induced by the surface map ψ.Choose the vertex corresponding to the left-edge of the left-most rectangular block as the base point v. Then the invariant train track λ is as in Figure <ref>Here, α_i are the edges in the front, and β_i are the edges in the back. The fundamental group of λ: π_1(λ,v) is a free group generated by γ_i=∏_j∈{1,2,… i-1}β_jα_iβ_i^-1(∏_j∈{1,2,… i-1}β_j)^-1 . Here, if S is an ordered set of indices, ∏_i∈ S g_i is the product of elements g_i in the order determined by the order of S. Step II. Next, we describe an isotopic train track map that fixes a base point. Let the “left-most” vertex, v, of Γ be the base point. For each loop γ in Γ based at v, ψ_∗(γ) is a loop conjugate to a loop based at v. This conjugacy can be written uniquely as follows:for each loop, we conjugate it to a based loop through an embedded path consisting of such edges.Let ψ_1 be the the map on Γ which maps each based loop γ to the conjugate based loop of ψ_∗(γ) as described above. Then this induces a homomorphism ψ_1_∗ from π_1(Γ, v) to itself.We can now write down the mapψ_1_∗ under the generating set γ_i using 1-dimensional PCF map f we started with, whose incidence matrix is M. Recall that the post-critical set decomposes the interval into n segments, which we label I_1,… I_n. Let ℐ be the index set of segments on which f is increasing and 𝒟 be the index set on which f is decreasing, and let w_i be the word (also seen as an ordered set) consisting of the indices of the edges in the image of I_i. Then, by the “thickening” construction inSection 2, we know that the image of α_i under ψ is ∏_j∈ w_iα_j if i∈ℐ and∏_j∈ w_iβ_j if i∈𝒟, and the image of β_i is s ∏_j∈ w_iα_j if i∈ℐ and∏_j∈ w_iβ_j if i∈𝒟. Hence, we have: ψ_1_∗(γ_1)=∏_j∈ w_1γ_j ψ_1_∗(γ_i)=∏_k∈𝒟∩{1,… i-1}(∏_j∈ w_kγ_j)∏_j∈ w_iγ_j(∏_k∈𝒟∩{1,… i-1}(∏_j∈ w_kγ_j))^-1 Step III. We now describe the infinite cover of the train track graph as well as the map in II. In particular, we explain what choices have been made in the previous steps.Note that the matrix M describes the induced action of ψ on the first homology of S_g. For simplicity, suppose the dimension of the eigenspace corresponding to the eigenvalue 1 of M is 1. Everything works exactly the same in the case the eigenspace has higher dimension. Let x⃗ be the eigenvector corresponding to the eigenvalue 1 of M.Now cut each edge of Γ in the front near the right endpoint of such an edge. Consider the -copies of this cut Γ, and enumerate as Γ_i with i ∈.On each copy of Γ, the edge corresponds to the i-th block of the polygon P_0 in Section 2 is called i-th edge.Now glue i-th edge of Γ_j to the i-th edge of Γ_j+x_i for all i and j where x_i denotes the i-th entry of the vector x⃗.Then we call the resulting infinite graph Γ_∞ which is -fold cover of λ.Let ψ_∞ be the lift of ψ_1_∗ to Γ_∞ so that all the lifts of the based point v are fixed. Step IV. Finally we obtain the algorithm from the map in III. Let G be the HNN extension of π_1(λ) with respect to the map we obtained in the previous step, i.e.,G = <π_1(λ), u : uγ u^-1 = ψ_1_∗ (γ), ∀γ∈π_1(λ) >.We can now compute the Alexander polynomial of G by Fox Calculus <cit.> (also c.f. <cit.>) and the Alexander Matrix is of size (2g+1)× 2g because the presentation above has 2g+1 generators (γ_1,…, γ_2g, u) and 2g relations R_i=ψ_1_∗(γ_i)uγ_i^-1u^-1, i=1,2,… 2g, and is of the form (M-[xI_2g,0])^T where M is the 2g× (2g+1) matrix obtained by our algorithm and x is the variable corresponding to the generator u.Lastly, we show that the Alexander polynomial of G is the Teichmüller polynomial of S.The Teichmüller polynomial for the pair (S, ψ) of odd-block surface and a pseudo-Anosov homeomorphism (obtained as in Section 2) coincides with the Alexander polynomial for the corresponding mapping torus of 𝒯. Let A = H_1(M_ψ, )/, and let  = Hom(A, ^*). Also, let H = Hom(H^1(S, )^ψ, ) ≅^b be the dual of the ψ-invariant cohomology of S.The we have the splitting A = H ⊕, and (t, u) be the coordinates on  adapted to the splitting. Note for our construction of τ in Step I, there is an identification between H_1(S) and H_1(τ) (see also <cit.>).Hence H is also the maximum subgroup of H_1(τ) that is invariant under ψ.The rest of the proof is basically just copying and pasting the proof of Theorem 7.1 of <cit.>.Let P(t) and Q(t) denote the action of ψ on Z_1(τ, _t) and H_1(τ, _t) respectively. Here the twisted coefficient _t takes value in [Ĥ] and corresponds to the abelian covering defined by Ĥ. Then by definition, the Alexander polynomial for 𝒯 is (uI - Q(t)), and the Teichmüller polynomial for (S, ψ) is (uI - P(t)). On the other hand, since τ is 1-dimensional and orientable, we have Z_1(τ, _t) ≅ H_1(τ, _t)for any character t ∈Ĥ. Note that the orientability of τ is essential here to get this identification between Z_1(τ, _t) and H_1(τ, _t). For instance, by Corollary 2.4 of <cit.>, when τ is non-orientable, the rank of Z_1(τ, _t) is strictly smaller, so it cannot be isomorphic to H_1(τ, _t). Note that in <cit.>, Q(t) denotes the action on H_1(S, _t) not on H_1(τ, _t), and the Alexander polynomial there means the Alexander polynomial of the 3-manifold. In that case, the Alexander polynomial is just a factor of Teichmüller polynomial. This is mainly due to the fact that the map in (7.1) of <cit.> is a mere surjection not an isomorphism. On the other hand, in our case, Q(t) is defined using H_1(τ, _t) not H_1(S, _t), and we use the Alexander polynomial of the branched surface which is the suspension of the train track.so the range of the map π in(7.1) of <cit.> becomes the same group as the domain, and we have the isomorphism. This shows that P(t) = Q(t), and the proposition follows.This concludes the proof of the main theorem. Remark. In general, H_1(S) is a quotient of H_1(τ, _t), hence the Teichmüller polynomial is a specialization of the Alexander polynomial for 𝒯 when τ is orientable. In our case, due to the identification betweenH_1(S) and H_1(τ), this specialization is not needed.Remark. In Step IV, say the eigenspace had dimension k, and we have the eigenvectors x⃗_1, …, x⃗_k.Then we consider the ^k-copies of the cut Γ. So now each copy has k coordinates, so write Γ_t_1, …, t_k with (t_1, …, t_k) ∈^k. Then for each i-th edge of Γ_t_1, …, t_k is glued to the i-th edge ofΓ_t_1 + (x⃗_1)_i, t_2 + (x⃗_2)_i, …, t_k + (x⃗_k)_iwhere (x⃗_j)_l denotes the l-th entry of the vector x⃗_j.This defines a ^k-fold cover of Γ we need to compute the Teichmüller polynomial.§ ACKNOWLEDGEMENTS We greatly appreciate Ahmad Rafiqi for many helpful discussions and comments. In particular, we are indebted to Ahmad for the running example in the paper whose Teichmüller polynomial was confirmed by his computation. We also thank Erwan Lanneau for a lot of inspiring discussions. Finally we thank the anonymous referee for helpful comments which greatly improved the readability of our paper. The first author was partially supported by the ERC Grant Nb. 10160104.§ ODD-BLOCK MATRICES WITH ENTRIES BIGGER THAN 1In the main text of the paper, the authors provided an algorithm to compute the Teichmüller polynomial for odd-block surfaces, and the odd-block surfaces are constructed as described in Section <ref> following <cit.>. One of the limitations of the construction of odd-block surfaces is that the given odd-block matrix is assumed to have only 0, 1 entires. The main purpose of this assumption is to guarantee the uniqueness of the corresponding piecewise-linear map h_M. See Figure <ref> for an example of an odd-block matrix with entries bigger than 1 where h_M is not unique. On the other hand, for any given odd-block matrix, one can obtain an odd-block matrix with {0,1}-entries with essentially the same information. More precisely, we prove the following theorem. Let M be an n × n non-singular, aperiodic, odd-block, nonnegative integral matrix. For each choice of the piecewise-linear map h_M, then there exists an aperiodic, odd-block matrix N with only {0, 1}-entries such that h_N coincides with h_M. Furthermore, the leading eigenvalue of N is the leading eigenvalue of M. Let M be a matrix as in Theorem <ref>. For instance, one can consider an example shown in the left part of Figure <ref>. We show that there is a canonical way to convert M into another aperiodic odd-block matrix M with only entries 0 and 1, having the same leading eigenvalue (say λ). Let v be the L^1-normalized eigenvector of M^T for the leading eigenvalue. As we did in section 2, we get a partition P={ x_0 , x_1, … , x_n } of [0,1] so that v_i=x_i-x_i-1 for each i=1,…,n. Let's consider the n× n grid diagram on [0,1]× [0,1] generated by the partition P. Each (i,j) box corresponds to M_ij so that M is flipped upside down. Because M is odd-block, it is always possible to draw a graph of piecewise-linear map h_M so that the number of line segments of h_M in each box is the same with the corresponding entry of M. As we draw on the grid diagram, the slopes of h_M are either λ or -λ. There may be few possible graphs with that property, but one can choose any of them. Note that the conversion depends on the choice of the graph.Now we are in a position to convert M. Let P={ y_0 , y_1, … , y_n } be the union of P andthe set of all critical points of h_M. Then since the post-critical set of h_M is contained in P, h_M(P)⊂P and so P is invariant under h_M. The extended incident matrix, say M, of h_M associated with P is the desired converted matrix. See the right part of Figure <ref> for a resulting matrix M of this process. Note that M is inevitably singular because of duplicated rows. Each entry of M is 0 or 1. Since P includes all critical points of h_M. The vector w=(w_1,…,w_m), w_i=y_i-y_i-1 for each i=1,…,m is the L^1-normalized eigenvector of M^T for λ.This is because the equation (M^Tw)_i=λ w_i simply represents the length relation between [y_i-1,y_i] and h_M([y_i-1,y_i]). To prove M is aperiodic, we use following fact:For positive integer p, i-th column of M^p is positive if and only if h_M^p([y_i-1,y_i])=[0,1]. Since M is aperiodic and h_M([y_i-1,y_i])=[x_j, x_k] for some j<k, h_M^p([y_i-1,y_i]) eventually covers [0,1] as p grows. Thus M is aperiodic. Finally, from the Perron-Frobenius theorem, we conclude that λ is the leading eigenvalue of M as it is the associated eigenvalue of the positive eigenvector w of the aperiodic matrix M. 9 AKHR15 Y. Algom-Kfir, E. Hironaka, K. Rafi. (2015) Digraphs and cycle polynomials for free-by-cyclic groups. Geom. Topol. 19 (2015), no. 2, 1111–1154.BRW16 H. Baik, A. Rafiqi and C. Wu. (2016). Constructing pseudo-Anosov maps with given dilatations, Geom. Dedicata. 180(1): 39–48.DKL15 S. Dowdall, I. Kapovich, C. Leininger. (2015) Dynamics on free-by-cyclic-groups. Geom. Topol., 19(5):2801–2899. DKL16 S. Dowdall, I. Kapovich, C. Leininger. McMullen polynomials and Lipschitz flows for free-by-cyclic groups. arXiv:1310.7481FLM11 B. Farb, C. Leininger, D. Margalit. (2011) Small dilatation pseudo-Anosov homeomorphisms and 3-manifolds. Adv. Math. 228, no. 3, 1466–1502.F53 R. Fox. (1953) Free differential calculus. I: Derivation in the free group ring. Ann. Math.: 547–560. H97 E. Hironaka. (1997) Alexander stratifications of character varieties. Annales de l'institut Fourier. Vol. 47. No. 2. Hiro10 E. Hironaka. (2010) Small dilatation mapping classes coming from the simplest hyperbolic braid. Algebr. Geom. Topol. 10, no. 4, 2041–2060.Hiro14 E. Hironaka. (2014) Penner sequences and asymptotics of minimum dilatations for subfamilies of the mapping class group. Topology Proc. 44, 315–324. KKT13 E. Kin, S. Kojima, M. Takasawa. (2013) Minimal dilatations of pseudo-Anosovs generated by the magic 3-manifold and their asymptotic behavior. Algebr. Geom. Topol. 13, no. 6, 3537–3602.LV14 E. Lanneau, F. Valdes. (2014) Computing the Teichmüller polynomial. arXiv:1412.3983 McM00 C. McMullen. (2000) Polynomial invariants for fibered 3-manifolds and Teichmüller geodesics for foliations. Ann. Sci. Éc. Norm. Supér, 33(4): 519–560.McM02 C. McMullen. (2002) The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology. Ann. Sci. Éc. Norm. Supér. 35(2): 153–171.McM15 C. McMullen. (2015) Entropy and the clique polynomial. J Topol. 8 (1): 184–212. Sun15 H. Sun. (2015) A transcendental invariant of pseudo-Anosov maps. J. Topol. 8 (2015), no. 3, 711–743.ThursEnt W. Thurston. (2014) Entropy in dimension one. arXiv:1402.2008. Department of Mathematical SciencesKAIST291 Daehak-ro, Yuseong-guDaejeon 34141, South KoreaE-mail: [email protected] of MathematicsRutgers UniversityHill Center - Busch Campus110 Frelinghuysen RoadPiscataway, NJ 08854-8019, USAE-mail: [email protected] Kim:Department of Mathematical SciencesKAIST291 Daehak-ro, Yuseong-guDaejeon 34141, South KoreaE-mail: [email protected] TaeHyouk Jo:Department of Mathematical SciencesKAIST291 Daehak-ro, Yuseong-guDaejeon 34141, South KoreaE-mail: [email protected] | http://arxiv.org/abs/1703.09089v4 | {
"authors": [
"Hyungryul Baik",
"Chenxi Wu"
],
"categories": [
"math.GT",
"math.DS",
"57M27, 37E30, 57M50"
],
"primary_category": "math.GT",
"published": "20170327140756",
"title": "An algorithm to compute the Teichmueller polynomial from matrices"
} |
* §.§ 0pt2.5ex0.5ex *§ 0pt3.5ex1.0ex C>c<Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy ETH Zürich, Laboratorium für Physikalische Chemie, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland [Corresponding author: ][email protected] Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy [Corresponding author: ][email protected] Zürich, Laboratorium für Physikalische Chemie, Vladimir-Prelog-Weg 2, 8093 Zürich, SwitzerlandVariational approaches for the calculation of vibrational wave functions and energies are a natural route to obtain highly accurate results with controllable errors. Here, we demonstrate how the density matrix renormalization group (DMRG) can be exploited to optimize vibrational wave functions (vDMRG) expressed as matrix product states. We study the convergence of these calculations with respect to the size of the local basis of each mode, the number of renormalized block states, and the number of DMRG sweeps required. We demonstrate the high accuracy achieved by vDMRG for small molecules that were intensively studied in the literature. We then proceed to show that the complete fingerprint region of the sarcosyn-glycin dipeptide can be calculated with vDMRG.Vibrational Density Matrix Renormalization Group Markus Reiher December 30, 2023 ================================================ § INTRODUCTION Vibrational spectroscopy is a valuable tool for the characterization of molecular systems.Different techniques— ranging from standard infrared absorption,<cit.> to more intricate spectroscopies such as vibrational circular dichroism,<cit.> Raman,<cit.> and Raman Optical Activity<cit.>— allow one to record vibrational spectra for a detailed characterization of chemical systems, that span a wide range from molecules of astrochemical interest<cit.> to large biomolecules.<cit.> In order to decode the detailed information contained in an experimental spectrum of a complex system, simple selection rules based on semi-empirical Hamiltonians are insufficient, and ab initio calculations are required.Computational approaches for the calculation of vibrational properties of molecular systems can be assigned to two main classes,namely variational and perturbative approaches.In variational calculations, the vibrational energies and wave functions are obtained by diagonalization of a vibrational Hamiltonian in a given basis set such as the harmonic-oscillator eigenfunctions<cit.> or eigenfunctions from a vibrational self-consistent field (VSCF) calculation.<cit.>The fraction of vibrational correlation energy that is missing in the VSCF mean-field approach can then be captured by vibrational configuration interaction<cit.> or vibrational coupled cluster<cit.> methods. With variational approaches it is possible to obtain fully converged results for a given Hamiltonian by systematically increasing the basis set. The Hamiltonian in turn depends on the electronic potential that is calculated with quantum-chemical ab initio electronic structure methods and remains the main source of error.Unfortunately, the computational cost of these variational approaches grows exponentially with the size of the system and limits the range of application to rather small molecules (up to 20 atoms).Perturbative approaches on the other hand, such as the most commonly applied vibrational second-order perturbation theory (VPT2)<cit.> and vibrational Møller-Plesset perturbation theory,<cit.> are computationally more feasible and can be applied to systems with up to 100 atoms. However, although reliable results can be obtained with VPT2 for semi-rigid systems, this approach certainly fails for molecules with shallow potential energy surfaces (PESs) and corresponding highly anharmonic, large-amplitude modes.Several techniques have been developed and successfully applied to reduce the computational effort of variational approaches. Basis pruning algorithms<cit.>were developed with the aim to include only a limited number of basis functions in the variational calculation.An alternative approach is based on so-called contracted basis techniques, in which basis functions are obtained by diagonalizing sub-blocks of the full Hamiltonian, involving only strongly interacting coordinates.<cit.>The computational cost of variational calculations can also be reduced with local mode techniques<cit.> instead of normal coordinates to reduce the number of off diagonal anharmonic couplings. A further alternative, whose analog is widely applied in electronic structure theory, but hardly explored in vibrational calculations, is the parameterization of the vibrational wave function in tensor formats, such as canonical decomposition<cit.> and matrix product states (MPS). These wave function representations must then be optimized with efficient algorithms such as the density matrix renormalization group (DMRG) algorithm<cit.> for the optimization of MPS wave functions. Here, we develop a DMRG optimization of MPS representations for vibrational wave functions and energies.We denote our approach as vDMRG.Recently, the eigenfunctions of a vibrational Hamiltonian were expressed in a tensor train format<cit.>, with a discrete variable representation (DVR) basis set<cit.>.While the tensor train format is algebraically equivalent to the MPS format, the optimization protocol proposed in Ref. Rakhuba2016_TensorTrain is different from the two-site DMRG algorithm chosen for our vDMRG approach. Moreover, our implementation of vDMRG expresses both the wave function and the Hamiltonian in tensor format, as an MPS and matrix product operator (MPO), respectively. Our paper is organized as follows.Section <ref> describes the underlying theory of vDMRG. After this brief description of the computational details of the implementation, the application of vDMRG to several molecules of varying size is discussed.First, we demonstrate the reliability of vDMRG at the example of a triatomic molecule, ClO_2, for which fully converged variational energies can beeasily calculated. Then, two medium-sized molecules (CH_3CN and C_2H_4) are studied in detail.For CH_3CN, we chose a quartic PES from density functional theory calculations<cit.> and for C_2H_4 a sextic PES from accurate coupled cluster calculations.<cit.>Results are compared to experimental data.<cit.> Finally, the vibrational properties of the sarcosyn-glycin dipeptide (SarGly^+) are calculated to assess the reliability of vDMRG for large systems, for which standard variational calculations are generally unfeasible. § VDMRG THEORY The vibrational wave function |Ψ_k⟩ of a molecule in the k-th vibrational state with L degrees of freedom can be expressed by afull configuration interaction (FCI) expansion, |Ψ_k⟩ = ∑_σ_1,...,σ_LC_σ_1, … , σ_L^(k)|σ_1,…,σ_L⟩, where the occupation number vectors are built from a basis of one-dimensional functions for each vibrational degree of freedom. Whereas for an electronic wave function, due to Fermi-Dirac statistics, the occupation number of each orbital cannot exceed 1,it is unbounded for a mode in a bosonic vibrational wave function. To limit the size of the basis set in Eq. (<ref>), an upper bound (N_max^i) for the occupation number of each mode has to be defined.For a total vibrational state k expressed in terms of occupation number vectors |σ_1,...,σ_L⟩, the MPS N-body wave function |Ψ_k⟩reads|Ψ_k⟩ = ∑_σ_1,...,σ_L∑_a_1,...,a_L-1^m M_1,a_1^(k)σ_1M_a_1,a_2^(k)σ_2...M_a_L-1,1^(k)σ_L|σ_1,...,σ_L⟩expressed in a basis set of L one-dimensional basis functions.The coefficients of the linear expansion in Eq. (<ref>) are decomposed as products of L matricesM^(k)σ_i={M_a_i-1,a_i^(k)σ_i} with maximum dimension m(with the exception of M^(k)σ_1 and M^(k)σ_L, which are row and column vectors, respectively, of that dimension).By restricting the maximum dimension of the individual matrices M^(k)σ_i to m, the DMRG algorithm achieves a reduction of the scaling from exponential to polynomial. This is a significant computational advantage over the FCI expansion, where the exponential scaling<cit.> limits applications to small systems.Hence, the CI coefficients are encoded in MPS form, C_σ_1, … , σ_L^(k) = ∑_a_1, … , a_L-1^mM_1,a_1^(k) σ_1M_a_1,a_2^(k) σ_2…M_a_L-1,1^(k) σ_L,which is an approximation for finite choices of m, which we denote the 'number of renormalized block states'. The product structure of Eq. (<ref>) implies a one-dimensional ordering of the vibrational modes i that are now associated with the matrices M^(k)σ_i. We refer to these modes as sites and denote the sequence of these sites as a lattice for consistency with the general DMRG nomenclature.In vDMRG, the matrices M^(k)σ_i are calculated variationally by minimizing the expectation value of the vibrational Hamiltonian ℋ_vib over the state |Ψ_k⟩. In this work,an approximate form of the Watson Hamiltonian<cit.>, in which only the second-order Coriolis terms are included,<cit.> and higher-order terms in the expansion of the inertia tensor are neglected, ℋ_vib =1/2∑_i=1^Lω_i ( p̂_i^2 + q̂_i^2 )+ 1/6∑_ijk=1^LΦ_ijkq̂_i q̂_j q̂_k + 1/24∑_ijkl=1^LΦ_ijklq̂_i q̂_j q̂_k q̂_l+∑_ijkl=1^L∑_τ^x,y,z B^τζ_ij^τζ_kl^τ√(_j _l/_i _k)q̂_i p̂_j q̂_k p̂_l,is applied for the calculation of vibrational energies. Here, q̂_i is the position operator associated with the i-th normal mode, and p̂_i is its conjugate momentum.In a fourth-order (quartic) Taylor expansion of the potential, ω_i are the harmonic frequencies and Φ_ijk and Φ_ijkl are the third- and fourth-order reduced force constants, respectively, which can be defined in terms of the third- and fourth-order partial derivatives, k_ijk and k_ijkl, of the PES, Φ_ijk= k_ijk/√(_i _j _k)andΦ_ijkl = k_ijkl/√(_i _j _k _l).In Eq. (<ref>), B^τ are the rotational constants and ζ_ij^τ the Coriolis coupling constants. As the generalization of Eq. (<ref>) to support also higher-order terms is straightforward, calculations with fifth- and sixth-order potentials are also presented in this work.To exploit the DMRG formalism, a second-quantized Hamiltonian is required. The second-quantized form of the vibrational Hamiltonian ℋ_vib can be obtained by the following substitution:p̂_i= 1/√(2)( b̂_i^+ - b̂_i ) ,q̂_i= 1/√(2)( b̂_i^+ + b̂_i ) ,where b̂_i and b̂_i^+ are the bosonic annihilation and creation operators for the i-th vibrational mode, obeying the following rules:b̂_i^+ |σ_1,... ,σ_i,...,σ_L⟩= √(σ_i+1)|σ_1,...,σ_i+1,...,σ_L⟩ , b̂_i |σ_1,... ,σ_i,...,σ_L⟩= {[ √(σ_i)|σ_1,...,σ_i-1,...,σ_L⟩if σ_i > 0; 0if σ_i = 0 ]. . The second-quantized form of the Watson Hamiltonian can be obtained by substitution of Eqs. (<ref>) and (<ref>) in Eq. (<ref>) and reads:<cit.>. ℋ_vib = ∑_i=1^L_i ( b̂_i^+b̂_i + 1/2)+ 1/12√(2)∑_ijk=1^LΦ_ijk ( b̂_i^+ + b̂_i ) ( b̂_j^+ + b̂_j ) ( b̂_k^+ + b̂_k )+ 1/96∑_ijkl=1^LΦ_ijkl ( b̂_i^+ + b̂_i ) ( b̂_j^+ + b̂_j ) ( b̂_k^+ + b̂_k ) ( b̂_l^+ + b̂_l )+ 1/4∑_ijkl=1^L∑_τ^x,y,z B^τζ_ij^τζ_kl^τ√(_j _l/_i _k) ( b̂_i^+ + b̂_i ) ( b̂_j^+ - b̂_j ) × ( b̂_k^+ + b̂_k ) ( b̂_l^+ - b̂_l )Hence, the third-order potential term, for example, can be written as a sum of all possible products of bosonic creation or annihilation operators localized on sites i, j, and k (such as b̂_i b̂_j^+ b̂_k).A similar result is obtained for the fourth-order potential term, but in this case 16 terms are present. More attention must be paid to the evaluation of the sign associated to a Coriolis term due to the presence of the momentum operator.At variance with the electronic Hamiltonian, the second-quantized form of the vibrational Hamiltonian contains operator strings with differentnumbers of creation and annihilation operators and, as a consequence, a computational scheme such as DMRG that generates explicit matrix representations of these elementary operators, will face significant challenges that can only be well met by a strictly modular, general, and object-oriented implementation.An operator Ŵ can be written as an MPO,Ŵ = ∑_σσ'∑_b_1,…,b_L-1 W_1 b_1^σ_1,σ_1'… W_b_l-1 b_l^σ_l,σ_l'… W_b_L-1 1^σ_L,σ_L'|σ⟩⟨σ'|,where |σ⟩ is a compact notation for |σ_1,…,σ_L⟩. Compared to the MPS matrices in Eq. (<ref>), the MPO matrices W^σ_i,σ_i'(the summation over the b_i indices is equivalent to a matrix-matrix multiplication) have two superscripts that specifyelements of the bra-ket vectors |σ⟩⟨σ'|. While the dimension of the MPS matrices M^(k)σ_i is adapted to a maximum value during the DMRG optimization,the dimensions of the MPO matrices W^σ_i,σ_i', and therefore the b_i indices, are fixed for each site by theparticular form of the operator encoded in MPO format.The MPO can now be constructed from matrix representations of the elementary operators b̂_i^+ and b̂_i. To understand the formulation of the Watson Hamiltonian in Eq. (<ref>) in MPO format, we write explicitly the construction of the first (harmonic) term for mode i that we denote Ŵ_harm.^Watson(i):Ŵ_harm.^Watson(i) =ω_i ·Î_1 ⊗…⊗b̂_i^+ ·b̂_i ⊗…⊗Î_L ,with unit matrices Î on all sites but i. Therefore, the construction of the Hamiltonian scales with L^4 for quartic force fields(as in Eq. (<ref>)) and L^6 for sextic force fields in a naive MPO construction.In the DMRG algorithm, the MPS optimization requires the repeated application of exact diagonalization and singular value decomposition. This procedure is called a sweep algorithm because the MPS site matrices are subsequently optimized from the first to the L-th site(one sweep) before they are optimized in reverse order from the L-th to the first site (next sweep). This is an optimization of reduced dimensional basis states contributing to the quantum state in a least-squares sense. It is a peculiarity of the algorithm that parts of the total vibrational MPS wave function, entering the energy expectation value,are not known in the first sweep. Therefore, an initial guess with random numbers as MPS entries or an MPS solution of an MPO with only harmonic contributions can be applied.While a certain excitation rank has to be fixed in standard VCI calculations, this is not the case for vDMRG. vDMRG approximates the correct wave function in a least-squares sense and optimizes the coefficients of the configurations with the largest weight in the total wave function. Hence, the algorithm can be understood as a sophisticated way of selecting the most important configurations and is therefore capable of approximating wave functions of arbitrary degree of multi-configurational character given that the number of renormalized block states required to converge the calculation is low enough to render the calculation affordable. Importantly, such situations are detectable within vDMRG itself through well-established truncation errors.We note that in Refs. anda more general second-quantized form of the vibrational Hamiltonian was proposed, where a general one-dimensional basis set is applied for each mode, the eigenfunctions of the harmonic-oscillator Hamiltonianbeing only one specific choice. Although these more general formulations can, in principle, be exploited within vDMRG, they would increase its computational cost significantly.The reason is that, when a general basis set is employed for each mode, excitations of a mode are represented by different basis functions (modals) with one pair of creation and annihilation operators to be introduced for each modal, and then the total number of local second-quantized elementary operators scales as L × N_max, where N_max is the maximum number of modals for each mode. However, when Hermite polynomials are employed as local basis set (as chosen for this work), the number of local elementary operators grows as L, because one pair of creation and annihilation operators is needed for each mode only. The more general basis functions are usually accompanied by the n-mode representation of the potential (see e.g., Refs. and , which is more flexible than the Taylor expansion in Eq. (4). The total number of terms in the Hamiltonian will be approximately the same for both representations. However, the number of sites on the vDMRG lattice will equal the number of modals in the n-mode representation with a general basis, while it equals the number of modes in the Taylor expansion of the potential with harmonic oscillator basis functions. The vDMRG algorithm and our implementation allow one to work with any type of basis functions and potential.Here, we focus on the Taylor expansion of the potential with a harmonic oscillator basis in this initial implementation and extend this to a more general formulation in future work, as this extension would be beyond the scope of this introduction of the vDMRG algorithm. To conclude, we note that the n-mode representation of the potential would in principle allow one to develop a state-specific version of vDMRG, where different basis functions are used for each mode. Therefore it is possible to target directly highly-excited vibrational states when combined with an advanced diagonalization algorithm (vide infra), where a specific energy range can be selected. § IMPLEMENTATION AND METHODOLOGY With the second-quantized form of the Watson Hamiltonian, Eq. (<ref>), the implementation of the MPO-based DMRG algorithm presented in Ref. can be extended to the calculation of vibrational properties.We implemented this Hamiltonian into our QCMaquis-V program<cit.> that is based on an MPS library developed for spin Hamiltonians<cit.> and extends the QCMaquis program to the calculation of vibrational energies. QCMaquis provides an implemention of the full electronicHamiltonian with up to four fermionic elementary creation and annihilitation operators. For the generalization of this implementation, we needed to 1) implement the commutation symmetry of bosonic creation and annihilation operators (to replace the anticommutation symmetry of the fermionic creation and annihiliation operators expressed in terms of F_s operators in Ref. ), 2) allow for arbitrary occupations σ_i, and implement MPOs for sequences of more than four elementary creation and annihilation operators. All of this was comparatively easy to accommodate by virtue of the modular and object-oriented structure of QCMaquis-V. The input data for our vDMRG calculations (i.e., harmonic frequencies, anharmonic force constants, Coriolis coupling constants) were either taken from the literature data or, for the dipeptide SarGly^+, calculated with the Gaussian program.<cit.> Detailed information on the electronic structure methods applied for the generation of the PESs is specified in the respective sections below.We emphasize that the variational optimization of the MPS provides the anharmonic zero-point vibrational energy (ZPVE).To also obtain transition frequencies, the energy of vibrationally excited states must be determined. This is accomplished by an optimization in the space orthogonal to the one spanned by the lower-energy vibrational states through an appropriate projection operation as described in Ref. . Transition energies hν_k are then calculated ashν_k = E_k-ZPVE.§ RESULTS§.§ Three-atomic molecule: ClO_2 To validate our implementation of vDMRG, we calculated the vibrational states of the triatomic molecule ClO_2 in its electronic ground stateof X^2B_1 symmetry because an accurate PES obtained from multi-reference configuration interaction (MRCI) calculations is available in the literature.<cit.> Fully converged vibrational energies up to 3300 cm^-1 were calculated from this PES with variational approaches.<cit.> The high accuracy of these calculated frequencies was demonstrated by comparison to high-resolution experimental data.<cit.>These theoretical results<cit.> will be our reference in the following to assess the convergence of our vDMRG calculations.Before discussing the results of our calculations, we compare the vDMRG approach with the variational approaches of the reference papers. In the study by Peterson,<cit.> the PES is expressed as a Taylor expansion in terms of the internal coordinates (the two bond lengths and the bond angle) of the molecule. The variational calculations of the vibrational problem are then carried out directly in these coordinates, with Morse oscillator eigenfunctions as a basis set for the two stretching coordinates and a DVR for the bond angle.<cit.>In a more recent work,<cit.> the quartic force field in internal coordinates is first transformed to a sixth-order force field in Cartesian normal coordinates, by a direct tensor transformation.<cit.> The variational calculation is then carried out in a basis of harmonic-oscillator wave functions. In this work, we employ the latter representation of the PES in Cartesian normal coordinates, because the second-quantized expression of the vibrational Hamiltonian given in Eq. (<ref>)holds for a basis ofharmonic-oscillator wave functions. The two representations are, in principle, different because internal coordinates are non-linear functions of the Cartesian normal modes, but they result in equal fundamental frequencies<cit.> and are therefore equivalent close to the equilibrium structure.Although more recent results are reported in Ref. , the data reported in Ref. will serve as our reference, because the authors of Ref. also included overtones and combination bands in their discussion. As has already been pointed out above, two parameters control the convergence of the vDMRG vibrational wave function and energy. The first, N_max, corresponds to the number of basis functions that describe each mode and its value must be increased with increasing anharmonicity of a given mode. The second parameter, the number of renormalized block states m, controls the degree of dimension reduction of the final MPS wave function and has to be increased with increasing anharmonic mode coupling. These two parameters are, in general, independent and the convergence with respect to both has to be analyzed. vDMRG vibrational energies up to 3000 cm^-1 calculated with different values forand m are reported in Table <ref>, together with the harmonic results and theoretical reference data taken from Ref. The number of vDMRG sweeps was set to 40 for all states and convergence with respect to the number of sweeps was found in all cases. With ten basis functions per mode (=10), converged results are obtained for energies up to 2000 cm^-1 with as little as two renormalized states, and variations below 0.1 cm^-1 are obtained by increasing m to 5 and 10. The vDMRG results are in good agreement with the theoretical reference values and confirm that a sixth-order Taylor expansion in Cartesian normal modes is sufficient to obtain accurate fundamental frequencies. However, for some of the higher-energy transitions, two renormalized states are not sufficient to reach convergence of energies below 1 cm^-1.This is particularly evident for the 2ν_1+ν_2, 3ν_1 and 2ν_1+ν_3 transitions, where results obtained with m=2 and m=10 deviate by more than 2 cm^-1.It should be noted that these are coupled vibrational modes with the highest energy among those considered in this table, and therefore show the largest anharmonic corrections. For this reason, a higher number of renormalized states is required to recover accurate variational energies.By further increasing the dimension of the local basis to twelve, no further variations occur and the energies can therefore be considered converged with respect to the basis set size. In Table S1 of the Supporting Information, the results obtained with a quartic potential are reported.To conclude, the results obtained for ClO_2 show the robustness of our vDMRG implementation.§.§ Parameter dependence: CH_3CN To investigate the convergence of the vDMRG energies and wave functions with respect to m and N_max in more detail, we calculated the 33 lowest vibrational states of acetonitrile in its electronic ground state for various choices of these parameters. A quartic force field for acetonitrile was reported in Ref. and has become a classical benchmark for newly developed variational approaches.<cit.> We derived the full quartic force field with the symmetry relations given in Refs. . From a footnote in Ref. , which refers to a private communication, it appears that there is apparently some confusion in the literature about the application of these symmetry relations. Therefore, we included the explicit expressions for the symmetry relations that we applied in the Supporting Information in order to provide unambiguous information about the underlying force field of our vDMRG calculations. Note that in Ref. only force constants larger than 7 cm^-1 are reported, and therefore minor deviations of our data from the theoretical reference data in Ref. avil11 are to be expected.Table <ref> contains the vDMRG results for m=20,50,100 and N_max = 3,6,9 for the 11 lowest vibrational states along with their deviation from the results of Ref. . We applied 50 DMRG sweeps for each state and ensured that the energy was converged with respect to the number of sweeps. If that was not the case, we had to include more sweeps until the energy was converged to less than 0.001 cm^-1. The assignment and the symmetry were identified by comparison of the energies to the results of Ref. . Since the coefficients of the configurations are not as directly available for the MPS wave function as they are in case of standard VCI calculations, the description of the character and symmetry of a given state would require an additional computational step. It has been demonstrated for the electronic structure problem (Refs. and ) that a sampling of the most important configurations yields an approximate expression of the wave function in determinant basis. A similar strategy will be exploited in future work to facilitate the determination of the character and symmetry of the vibrational vDMRG states. Apparently, the small value of m=20 is sufficient to guarantee convergence to less than 0.1 cm^-1 for the states considered in Table <ref> when a local basis size of at least N_max=6 is employed. This is encouraging because the calculation of a single vibrational state with m=20 takes only around 15 % of the time of the same calculation with m=100. Since this factor is approximately the same for all states, the savings are huge when a large number of vibrational states is to be calculated, which isobviously the case for large molecules. Figure <ref> shows the error of a given set of parameters with respect to the largest calculation with m=100 and N_max=9. Clearly, if the local basis size is too small, the errors can become very large (>100 cm^-1), especially for higher excited states because the harmonic wave functions for the overtones are simply not included in the basis set. The deviations increase for higher excited states since each excited state is optimized in the space orthogonal to all lower-lying states. Of course, the optimization will become unreliable if the wave functions of the lower-lying states are not converged. However, for both, N_max=6 and 9, the deviations do not exceed 1.3 cm^-1 and are most often below 0.2 cm^-1. In the light of these results and those obtained in the previous section, we conclude that a local basis set size of at least N_max = 6 is required, whereas the number of renormalized block states m can be as small as 10 or 20.Some vibrational states are certainly more sensitive to m (red vertical lines in Figure <ref>) than others. In the first four cases (states 5, 11, 19, and 22), it is the A_1/2 component of a vibrational state that has almost the same energy as its corresponding degenerate component of E symmetry. The fifth state with a discrepancy of up to 0.5 cm^-1 is, however, one component of an E state (corresponding to 4 ν_11). We conclude that special care must be taken to fully converge states that are close in energy or even degenerate if an accuracy below one reciprocal centimeter is desired. In regions with a high density of states (such as the C-H stretch region in organic molecules) this becomes especially challenging. The convergence with respect to all DMRG specific parameters has to be monitored such that each individual state is fully converged. If this is not ensured, the calculation of higher excited states can become inaccurate because these states are optimized in the space orthogonal to only partially converged lower states and errors might accumulate. This limitation can be in principle overcome with modified diagonalization algorithms. Among them, Lanczos (Refs. and ) and modified Davidson (see Refs.for example of application both to electronic and vibrational structure problems) algorithms are particularly appealing in this respect. Additionally, subspace diagonalization algorithms, that have been applied both to electronic structure calculations<cit.> and to vibrational problems,<cit.> would improve the efficiency of vDMRG, and work is in progress in this respect.Deviations with respect to the reference values on the order of 1 cm^-1 remain even for our most accurate calculations (see Table <ref>). Since our calculations are converged with respect to the number of sweeps and size of the local basis set, the discrepancy could bea result of the symmetry relations applied to construct the full quartic force field from the force constants of Ref. .§.§ Higher-order expansion of the potential energy surface: Ethylene The third example selected to demonstrate the reliability of vDMRG is ethylene (C_2H_4).Here, we adapt a highly accurate ab initio PES from the literature<cit.> and compare the results with previous calculations and experimental data.<cit.> Two decades ago, an accurate PES for ethylene was constructed from CCSD(T) calculations by Martin and coworkers.<cit.>With this PES, vibrational energies and spectra were determined with various variational approaches.<cit.>Recently, a new PES was constructed from CCSD(T) calculations with a large quadruple-ζ atomic orbital basis, from which more accurate vibrational properties could be calculated.<cit.>We applied the latter PES in our vDMRG calculations. The analytical expression in terms of internal coordinates can be found in Ref. . Since our implementation of vDMRG is based on Cartesian normal coordinates as a reference coordinate system, we converted the PES to a Taylor series expansion in terms of these coordinates with the procedure discussed in Refs. and. Fourth- and sixth-order Taylor series expansion of the potential, with and without the inclusion of Coriolis effects, were considered in the vDMRG calculations with 20 sweeps for each state. The order of the Taylor expansion was chosen based on the variational results reported in Ref. , where it was shown that a sixth-order expansion is sufficient to reach convergence within 1 cm^-1.In the theoretical reference work,<cit.> Coriolis terms were included only for two transitions, and therefore they have been neglected also in our vDMRG calculations. Vibrational energies for the ground and the first two excited states of C_2H_4 are reported in Tables S2, S3, and S4 of the Supporting Information. The data reported in Table S2 highlight the effect of each of the two parameters m and N_max on the ZPVE.First, we discuss the dependence of the ZPVE on m for a given order of the Taylor expansion of the potential (a fixed Hamiltonian) and then for a given value of N_max (a fixed local basis set). The data are also shown in Figure <ref>. With five renormalized states, inaccurate results with deviations larger than 10 cm^-1 with respect to the converged values, are obtained.Convergence (with deviations < 1 cm^-1) is reached with ten renormalized block states for all Hamiltonians (i.e., witha quartic and a sextic force field). This indicates that the further inclusion of high-order terms in the potential will not affect significantly the structure of the Hamiltonian, i.e., it will not introduce further long-range correlations that would require a larger number of renormalized block states and DMRG sweeps to converge the wave function.Regarding the convergence of the ZPVE with respect to the size of the local basis , fully converged results are obtained for =6, and with higher values (8 and 10), the change in the ZPVE is below 0.01 cm^-1.For C_2H_4, neither highly anharmonic, large-amplitude vibrations are present nor resonances that usually cause strong variational corrections.Therefore, a reliable representation for the vibrational wave function can be obtained from a relatively small number of harmonic-oscillator eigenfunctions as local basis functions.As noted above for ClO_2, a virtually infinite-order Taylor expansion in Cartesian normal coordinates would be required to obtain a potential that is equivalent to the one reported in Ref , and therefore convergence with respect to the order of the Taylor series expansion of the potential must be studied. With a fourth-order PES, the ZPVE is underestimated by ≈ 7 cm^-1 for all choices of m and N_max, and the further inclusion of fifth- and sixth-order terms reduces the error to less than 1 cm^-1, therefore providing satisfactory convergence.This result is in line with the analysis reported in Refs. and , where results converged within 1 cm^-1 were obtained from a full sextic potential without including Coriolis couplings. The vDMRG ZPVE obtained with the sextic potential and with =6 and m=20 is 11016.15 cm^-1, and agrees well with the theoretical reference value of 11014.91 cm^-1,<cit.> which was calculated without Coriolis couplings.In Tables S3 and S4, a similar analysis is carried out for the first two fundamental bands of ethylene.Again, converged results were obtained with =6, and only minor changes are observed with =8.Once again, the convergence with respect to the number of renormalized block states m is slower than for the ZPVE and discrepancies of more than 1 cm^-1 can be observed for m=10.A complete list of the 18 lowest vibrational frequencies (including both overtones and combination bands) of C_2H_4 calculated with vDMRG is collected in Table <ref>. In view of these results, all calculations were carried out with = 6, with both a quartic and a sextic potential andwith 10 and 20 renormalized block states m. These results confirm the trend already found for the three lowest-energy states.First of all, the comparison of the results shows that ten renormalized states are sufficient to achieve an accuracy of 1 cm^-1 for the lowest energy states. However, the difference between the results obtained with m=10 and m=20 increases for higher excited states but remains below 10 cm^-1, even for states up to 1800 cm^-1 above the ZPVE. In analogy with acetonitrile, the largest deviations are usually obtained for states involved in resonances, as for example the 2ν_7 and ν_7+ν_8 states in Fermi resonance. It has already been widely discussed in the literature that variational corrections are particularly relevant for resonant states.<cit.> For this reason, a higher number of renormalized block states m is required to obtain converged energies. These observations hold for both the quartic and the sextic potential. Moreover, for the latter potential, a slower convergence with respect to m is observed for vibrational states with higher energy.The frequencies we calculate from the sextic potential agree significantly better with the theoretical reference data than the frequencies we obtained from the quartic potential, especially for higher frequencies, above 1500 cm^-1. Finally, we calculated vibrational wave functions and energies from a sextic potential with Coriolis coupling. Rotational effects were also considered in the theoretical reference work<cit.> for two, high-energy fundamental transitions (ν_11 and ν_9, with harmonic frequencies of 3140.91 cm^-1 and 3248.71 cm^-1, respectively) by fitting the energy of more than 50 calculated rovibrational states.For all other transitions, rotational effects were neglected.The rotational correction amounted to approximately 3 cm^-1, which is about the same order of magnitude as the discrepancy between vDMRG and experimental data.The results obtained including Coriolis terms for the rotational ground state (reported in Table <ref> as "Sextic + C") confirm that rotational corrections are relevant for several bands (e.g., ν_7 and ν_8), for which this effect amounts to up to 9 cm^-1.In general, the inclusion of rotational corrections improves the agreement between vDMRG and experimental data.It should be noted that, in some cases (as, e.g., for the ν_4 fundamental band), the discrepancy between vDMRG and theoreticalreference results without rotational correction is significant (above 5 cm^-1).The inclusion of a rotational correction however enhances the agreement with the experimental data and indicates a better agreement of the variational calculations than the theoretical reference work.<cit.>§.§ Application to a large molecule: The SarGly^+ dipeptide Our last example is the protonated sarcosine-glycine dipeptide (referred to as SarGly^+ in the following, whose Lewis structure is shown in Figure <ref>). Computational studies of the anharmonic vibrational properties of medium-size biomolecules have been limited mostly to VSCF<cit.> and VPT2<cit.> approaches, and only recently vibrational CI (VCI) studies were published mainly based on local mode approaches to reduce the computational effort.<cit.>In this section, the convergence of the vibrational energies of SarGly^+ is studied for a varying number of renormalized block states m. In fact, if the value of m required to reach convergence turns out to be small and largely independent of the system size, vDMRG can be a valuable alternative to local mode approaches for large systems. In this section we assess the performance of vDMRG for large systems and do not aim to accurately reproduce experimental frequencies. In fact, a quartic force field is usually insufficient for variational calculations and terms up to sixth order<cit.> have to be included to obtain results that are close to their experimental counterparts. For vDMRG, a higher-order force-field simply requires longer strings of creation and annihilation operators and therefore increasing the scaling. However, such terms of the Hamiltonian can easily be provided in the input to our QCMaquis-V program. Furthermore, a more accurate electronic-structure method than the one employed here (B3LYP/6-311+G(d,p)) is necessary to reproduce experimental frequencies.Due to the size of SarGly^+, theoretical results with fully-converged VCI calculations are not available. However, a theoretical study of the vibrational properties of SarGly^+ was performed recently with VSCF and a local mode ansatz based on a B3LYP/6-311+G(d,p) quartic force-field.<cit.> For the sake of comparability, the same electronic-structure approach was applied here for the structure optimization and the computation of the semi-quartic force field, where the quartic force constants k_ijkl with four different indices were neglected together with those smaller than 1 cm^-1. In view of the results of the previous sections,was set to 6 and values of m range from 5 to 20 in all calculations.The harmonic vibrational wave function served as the initial guess for the MPS.To limit the computational cost, the reduced-dimensionality (RD) scheme presented in Refs. andwas exploited for treating all modes below 900 cm^-1 as harmonic ones, therefore neglecting all couplings between these modes and the fully anharmonically treated modes. Such a selection of the modes might seem, in general, crude and prone to arbitrariness, and for larger systems a more accurate definition of the reduced-dimensionality model is certainly required.<cit.> However, the aim of this section is to demonstrate the efficiency of vDMRG for large systems, not the accurate reproduction of experimental results that will also be limited by the electronic structure approach. Therefore, an RD potential, consisting of 35 modes (corresponding to 35 DMRG sites), represents a viable setup for vDMRG.A comparison of the RD and a full-dimensional scheme is presented in Table S6 for generalized vibrational second-order perturbation theory (GVPT2, further details in Refs. and ) calculations on SarGly^+ and revealed that the energies deviate by less than 5 cm^-1 for most modes under 2000 cm^-1. This indicates that the RD scheme is valid for this molecule within the desired accuracy.Obviously, the RD scheme largely improves the efficiency of the vDMRG calculation.In fact, the full-dimensional Hamiltonian would have a large number of low-frequency eigenvalues, corresponding to fundamentals, overtones, and combination bands of low-frequency modes, below the fingerprint region (≈ 900-1700 cm^-1). 1.25The vDMRG fundamental frequencies of SarGly^+ in the fingerprint region are collected in Table <ref>. Localized VSCF (l-VSCF) from Ref. are given together with GVPT2 results obtained with the same RD potential as the one in the vDMRG calculations.Experimental frequencies from Ref. are included where available.The convergence of the energies with respect to the number of renormalized block states m is similar to that observed for smaller systems as analyzed in the previous sections.Almost fully converged results were obtained already with m=10 and corrections below 2 cm^-1 were observed upon increasing m to 20. The ZPVE converges more slowly with respect to m than all the excitation frequencies, with a difference of 8 cm^-1 between results obtained from m=10 and m=20. As shown in Figure S1 of the Supporting Information, converged results can be obtained with m=40, and a further increase of m do not lead to any significant modifications. Due to error compensation for the transition energies, the convergence of the ZPVE appears to be slower. In the harmonic approximation, the contribution of high-energy modes to state energies is the same and equal to half of the fundamental frequency, and therefore this contribution cancels out when calculating excitation energies as states energy differences.In Figure <ref>, the spectrum obtained from vDMRG anharmonic frequencies of SarGly^+ with =6 and m=20 and harmonic intensities is compared to the experimental spectrum.<cit.>As expected, the inclusion of anharmonic effects leads to an overall red shift of the frequencies, providing a considerably better agreement with the experimental data.This is pronounced for the band at 1147 cm^-1 (C-O-H bending), whose energy is overestimated by approximately 50 cm^-1 with harmonic calculations, whereas it is correctly reproduced by vDMRG.Similarly, the pattern recorded between 1350 and 1450 cm^-1, that is composed by three nearly equidistant bands with comparable intensity, is reproduced more accurately by vDMRG, whereas two of the three bands have nearly the same energy in the harmonic approximation.Therefore, although a potential higher than fourth order is usually required to obtain reliable variational energies, anharmonic variational calculations from a quartic potential considerably improve the description of this system compared to purely harmonic calculations.These vDMRG calculations on SarGly^+ with the largest set of parameters among the ones studied here (=6, m=20, 10 sweeps)took, on average, 17.500 seconds per state (approximately 4 hours and 48 minutes).This corresponds to an overall computational time of less than four days for the lowest 20 excited states on an Intel Xeon E5-2670 @2.6 GHz with 2x8 central processing units and a 64 GB RAM node.Therefore, systems with more than 30 modes can be studied with manageable computational effort. As already noted above, an additional factor determining the efficiency of the vDMRG calculation is the iterative optimization algorithm. In fact, the number of renormalized block states m and the dimension of the local basisdetermine the computational cost of each sweep.However, in order to make vDMRG applicable to large molecules, the number of sweeps required to reach convergence should be largely independent of the system size.In Figure <ref>, the energies of the first three vibrational states of SarGly^+ (lower panel) and C_2H_4 (upper panel) are reported as a function of the number of sweeps. The vDMRG calculations were performed with quartic force-fields (semi-diagonal for SarGly^+) with =6 and m=10. For ethylene, the convergence of the optimization algorithm is fast, with deviations below 1 cm^-1 from the converged value (obtained with 20 sweeps) already within six sweeps for both the ground and the excited states.The efficiency of the procedure is only slightly lower for SarGly^+, even if, to obtain a ground-state energy with an accuracy within 1 cm^-1, at least eight sweeps are required (eight for the first excited state). In Table S5 of the Supporting Information, the previous analysis has been extended to all vibrational energies below 1600 cm^-1.A graphical representation of the difference with respect to fully converged results as a function of the number of sweeps is reported in Figure <ref>.With only four sweeps, most of the vibrational frequencies are converged within 1 cm^-1, with the exception of only five frequencies. With eight sweeps, only three frequencies were not converged which demonstrates the efficiency of the optimization procedure also for highly-excited states. It is interesting to note that the frequencies that need the largest number of sweeps to converge, ν_36 and ν_37, are involved in a Darling-Dennison 1-1 resonance (as can be tested by performing the test described, for example, in Ref. ) and, as noted above, anharmonic corrections are usually relevant for resonant states. This indicates once more that the number of sweeps required to reach convergence increases with the magnitude of anharmonic corrections. In order to increase the efficiency of the optimization algorithm in such cases, the GVPT2 vibrational states could be exploited as an initial guess of the optimization procedure, but this analysis is beyond the scope of the present work. § CONCLUSIONSIn this work, we presented the first theoretical formulation and implementation of an MPO-based vibrational DMRG algorithm for the calculation of vibrational properties of molecular systems. The second-quantized Watson Hamiltonian was employed with a Taylor expansion up to sixth-order of the PES and Cartesian normal coordinates as the reference coordinate system. This Hamiltonian was compactly represented as an MPO.We demonstrated that highly-accurate, converged results can be obtained with vDMRG and a moderate number of renormalized block states. However, the main advantage of vDMRG lies in the fact that results with an accuracy of 1-2 cm^-1 can be obtained with a very low number of renormalized block states also for large systems, with 30-40 normal modes. Therefore, vDMRG represents a reliable method for the calculation of vibrational energies for large systems. In fact, very large variational space can be sampled without any restriction to a predefined excitation hierarchy. Furthermore, vDMRG could benefit from most of the strategies used for large-scale VCI calculations, such as state-selection algorithms and localized modes Hamiltonians.Future work will focus on increasing the efficiency of vDMRG.First of all, the present implementation supports only the harmonic-oscillator wave functions as a local basis set for the individual modes. However, it has already been shown for established approaches that other choices of the local basis set may provide a faster convergence of the variational expansion. For example, basis functions arising from VSCF calculations are usually better suited for VCI calculations,<cit.> and the extension of our vDMRG implementation to support a VSCF reference is possible with the second-quantized Hamiltonian presented in Ref. . Being a one-dimensional algorithm, DMRG is most efficient for Hamiltonians with mainly short-range (or local) interactions.For this reason, the extension to localized modes,<cit.> possibly curvilinear,<cit.> is particularly appealing, especially for large systems.<cit.> Furthermore, it might be beneficial to place resonant modes close to each other on the DMRG lattice and ordering methods based on quantum entanglement can be of value here. An adaptation of the basis set size to the entanglement of a given mode with all other modes can additionally help to select a local basis with favorable convergence properties.In the present work, vibrational frequencies were assigned to specific molecular vibrations by comparing the energies to reference data or to VPT2 results. However, this procedure lacks robustness, especially for large systems with pronounced variational corrections.In order to build a more robust assignment algorithm, a sampling procedure already applied in electronic structure calculations to express an MPS in terms of a FCI basis set could be employed.<cit.>Finally, the full Hamiltonian is included in the variational treatment so far. However, it has been shown<cit.> that more efficient approaches can be devised, where only some terms of the Hamiltonian are included in the variational treatment, and the remaining terms are treated perturbatively. The coupling of the vDMRG approach presented here with a perturbative treatment is possible in analogy to approaches already proposed for electronic structure methods<cit.> and would reduce the size of the Hamiltonian that is treated variationally, and consequently also the computational cost.§ ACKNOWLEDGMENTSThis work was supported by the Schweizerischer Nationalfonds (No. 20020_169120).C.J.S. gratefully acknowledges a Kékule fellowship from the Fonds der Chemischen Industrie. A.B. and V.B. acknowledge funding from the European Research Council under the European Unions Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement n. [320951] and the Italian MIUR (PRIN 2015 Grant Number 2015F59J3). § SUPPORTING INFORMATION AVAILABLEAdditional information and results are collected in the supporting information. This information is available free of charge via the Internet at http://pubs.acs.org/.§ REFERENCES 100IRBook Rijs, A. M.;Oomens, J. 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"authors": [
"Alberto Baiardi",
"Christopher J. Stein",
"Vincenzo Barone",
"Markus Reiher"
],
"categories": [
"physics.chem-ph",
"physics.comp-ph"
],
"primary_category": "physics.chem-ph",
"published": "20170327211736",
"title": "Vibrational Density Matrix Renormalization Group"
} |
=17pt On positive-definite ternary quadratic forms with the same representations overRyoko Oishi-TomiyasuGraduate School of Science and Engineering,Yamagata University / JST PRESTO 990-8560 1-4-12, Kojirakawa-cho, Yamagata-shi, Yamagata, JapanE-mail: [email protected][Current affiliation: Institute of Mathematics for Industry (IMI), Kyushu University] ===================================================================================================================================================================================================================================================================================================[2010 Mathematics Subject Classification: Primary 11E25; Secondary 11E20.][Key words and phrases:quadratic forms, pair of quadratic forms, representations, simultaneous representations, quartic rings.]Kaplansky conjectured thatif two positive-definite ternary quadratic forms have perfectly identical representations over , they are equivalent overor constant multiples of regular forms, oris included in either of two families parameterized by ^2.Our results aim to clarify the limitations imposed to such a pair by computational and theoretical approaches. Firstly, the result of an exhaustive search for such pairs of integral quadratic formsis presented, in order to provide a concrete version of the Kaplansky conjecture. The obtained list contains a small number of non-regular forms that were confirmed to have the identical representations up to 3,000,000 by computation. However, a strong limitation on the existence of such pairs is still observed, regardless of whether the coefficient field isor . Secondly, we prove that if two pairs of ternary quadratic forms have the identical simultaneous representations over , their constant multiples are equivalent over . This was motivated by the question why the other families were not detected in the search. In the proof, the parametrization of quartic rings and their resolvent rings by Bhargava is used to discuss pairs of ternary quadratic forms. § INTRODUCTION The aim of this article is to investigate the pairs of ternary positive-definite quadratic forms f and g with perfectly identical representations over .If such a pair also has the identical multiplicities (theta series),it is known that f and g are equivalent over<cit.>.Therefore, “identical representations” means q_(f) = q_(g) herein, where q_(f) := { f(x) : 0 ≠ x ∈^3 } is the set of representations over . The same problem was also discussed in <cit.>, and crystallography as mentioned in the following. For any N-ary quadratic forms f and g with real coefficients, we will use the notation f ∼ g whenf and g are equivalent over , f(𝐱 w) = g(𝐱) for some w ∈ GL_N(). With regard to the binary case, it was proved by a number of mathematicians that if two positive-definite quadratic forms f ≁g have the identical representations over , then such a pair is provided by d (x_1^2 - x_1 x_2 + x_2^2) and d (x_1^2 + 3 x_2^2) for some d > 0 <cit.>. All the indefinite binary cases were also determined in <cit.>, <cit.>.As an immediate consequence, it is not difficult to verify that f and g in each of the following families,satisfy q_(f) = q_(g), regardless of the values of c, d (see Section <ref>): * { f, g } = { c(x_1^2 - x_1 x_2 + x_2^2) + d x_3^2, c(x_1^2 + 3 x_2^2) + d x_3^2 }, * { f, g } = { c(x_1^2 - x_1 x_2 + x_2^2) + d (x_1 + x_2 + 3 x_3)^2, c(x_1^2 + 3 x_2^2) + d (x_1 + 3 x_3)^2 }.Kaplansky conjectured in his letter to Schiemann in 1997 thatall the pairs of non-regular forms f, g satisfy f ≁g and q_(f) = q_(g), as long as they belong to either of the above (i), (ii). A quadratic form f overis said to be regular, ifany m ∈ that is represented by f over _v for any primes v, is also represented by f over , where v = ∞ is also included, and _∞ =.It was proved in <cit.> that the conjecture holds if only diagonal quadratic forms are considered (all such cases are provided by No.34 and No.50 in Tables <ref>–<ref>). In order to obtain more detailed information about this problem,an exhaustive search for such f, g with integral quadratic coefficients were carried out. It can be proved that if f, g oversatisfy q_(f) = q_(g), infinitely many f_2, g_2 overwith q_(f_2) = q_(g_2) are generated from these f, g (Lemma <ref>). Hence, the search also provides information about the case of real forms.The result is presented in Tables <ref>–<ref> in Section <ref>,which indicates that the existence of such pairs is rather limited as conjectured by Kaplansky, although the current list includes some non-regular cases.If the quadratic forms contained in the above (i), (ii) are excluded,our exhaustive search finds only 151 equivalence classes of quadratic formsthat have perfectly identical representations overas another class. Among the 151 classes, 36 are not regular. In addition, the list includes a case that has been proved to be regular only under the generalized Riemann hypothesis <cit.>.In what follows, { f, g }∼{ f_2, g_2 } means that either of f ∼ f_2, g ∼ g_2 or f ∼ g_2, g ∼ f_2 holds. The following is suggested from the computational result: Kaplansky conjecture (modified version): if two ternary positive-definite quadratic forms f ≁g satisfy q_(f) = q_(g), one of the following holds: * { f, g }∼{ c ( x_1^2 - x_1 x_2 + x_2^2 ) + d x_3^2,c ( x_1^2 + 3 x_2^2) + d x_3^2 } for some c, d ∈, * { f, g }∼{ c ( x_1^2 - x_1 x_2 + x_2^2 ) + d (x_1 + x_2 + 3 x_3)^2,c ( x_1^2 + 3 x_2^2) + d (x_1 + 3 x_3)^2 } for some c, d ∈, * { f, g }∼{ c f_2, c g_2 } for some c ∈ and f_2, g_2 contained in either of the No.1–53 in Tables <ref>–<ref>.Although the non-regular cases newly found in our search are also included in the above,it should be noted thatthey were just confirmed to have the identical set of representations up to 3,000,000, by computation. With regard to regular quadratic forms, it is not difficult to confirm that all of their integral representations are perfectly identical.Before proceeding to our theoretical results motivated by the Kaplansky conjecture,first we provide the following proposition; for any field k of characteristic 2,the set of all the n-ary quadratic forms over k is denoted by Sym^2 (k^n)^*, and the set of all the s-tuples of such formsis denoted by Sym^2 (k^n)^* ⊗_k k^s. For any subring R ⊂ k and f_1, …, f_s ∈ Sym^2 (k^n)^*, the elements of q_R(f_1, …, f_s) := { (f_1(v), …, f_s(v)) : 0 ≠ v ∈ R^n } are called simultaneous representations of f_1, …, f_s over R. With regard to positive-definite f ∈ Sym^2 (^3)^* not contained in ^× Sym^2 (^3)^*, the Kaplansky conjecture is trueif and only if the following (*) is true: (*) If both of (A_i, B_i) ∈ Sym^2 (^3)^* ⊗_^2 (i = 1, 2) satisfy * A_i and B_i are linearly independent over ,* c A_i + d B_i is positive-definite for some c, d ∈ (d-pencil),* q_(A_1, B_1) = q_(A_2, B_2), then (A_1, B_1) = (w, 1) · (A_2, B_2) holds for some w ∈ GL_3(), or otherwise, { (A_1, B_1), (A_2, B_2) } equals either of the following as a set, for some (w_i, v) ∈ GL_3() × GL_2() (i = 1, 2): * { (w_1, v) · ( x_1^2 - x_1 x_2 + x_2^2, x_3^2 ), (w_2, v) · ( x_1^2 + 3 x_2^2, x_3^2 ) }, * { (w_1, v) · ( x_1^2 - x_1 x_2 + x_2^2, (x_1 + x_2 + 3 x_3)^2 ), (w_2, v) · ( x_1^2 + 3 x_2^2, (x_1 + 3 x_3)^2 ) }, where GL_3() × GL_2() acts on Sym^2 (^3)^* ⊗_^2 by(w,[ r s; t u ]) · (A, B) = (r A(𝐱 w) + s B(𝐱 w), t A(𝐱 w) + u B(𝐱 w)). We shall say that a pair (A, B) ∈ Sym^2 (k^n)^* ⊗_k k^2 is singular, if (A x - B y) = 0 as a polynomial in k[x, y], and non-singular otherwise. A non-singular (A, B) is said to be anisotropic over k if A(x) = B(x) = 0 does not hold for any 0 ≠ x ∈ k^n. It should be noted that any pair (A, B)∈ Sym^2 (k^n)^* ⊗_k k^2 with n ≥ 3 is a d-pencil if and only if (A, B) is non-singular and anisotropic over<cit.> (<cit.>).It can be proved without difficulty that(*) holds true in the following case:If (A_i x - B_i y) = 0 has a multiple root (for at least one of i = 1, 2),the above (*) holds true.Motivated by the Kaplansky conjecture and Proposition <ref>, the following is proved in this article. We assume that (A_1, B_1), (A_2, B_2) ∈ Sym^2 (^3)^* ⊗_^2 satisfy *A_i and B_i are linearly independent over.(b')(A_i, B_i) is non-singular and anisotropic over .(c')q_(A_1, B_1) = q_(A_2, B_2).In this case,(r_1 A_1, r_1 B_1) is equivalent to (r_2 A_2, r_2 B_2) by the action of GL_3() ×{ 1 } for any integers r_1, r_2 that satisfy r_1^-1(A_1 x - B_1) = r_2^-1(A_2 x - B_2). Since (b') is obtained by replacingin (b) with ,Theorem <ref> handles a more general case than Proposition <ref>. In the proof, one-to-one correspondence between the set of pairs of quadratic forms and the set of quartic rings and its resolvent cubic rings<cit.>, is used.We shall explain the outline of the proof of Theorem <ref>; as proved in Proposition <ref>, the situation of Theorem <ref>leads to (A_1 x - B_1 y) = c (A_2 x - B_2 y) for some c ∈^×. This and q_(A_1, B_1) = q_(A_2, B_2) imply thatboth (A_i, B_i) can be transformed to another pairs (Ã_i, B̃_i) (i = 1, 2) that correspond to quartic -algebras [x]/(f_i(x)),by the action of some (W_i, V) ∈ GL_3() × GL_2() (Lemma <ref>). Proposition <ref> also implies thatif the resolvent cubic polynomial of f_i(x) is denoted by f_i^res(x), [x]/(f_i^res(x)) (i = 1, 2) are isomorphic as k-algebras. Since the above V is common, (Ã_i, B̃_i) also satisfy (a), (b') and (c'). In particular, (b') implies that f_i(x) = 0 (i = 1, 2) have a root in _pand [x]/(f_i^res(x)) completely splits over _p with regard to the same set of primes p (Corollaries <ref>, <ref>). Thus, [x]/(f_i(x)) (i = 1, 2) are isomorphic as k-algebras (Lemma <ref>). Theorem <ref> and some relation formula between f_1(x) and f_2(x) (Proposition <ref>) are obtained as a result.In crystallography, in order to determine the crystal lattice (the equivalence class overof a positive-definite ternary quadratic form f with real coefficients) from information about q_(f) that is extracted from the experimental data, it has been recognized thatsome f ≁g have the perfectly identical representations over(<cit.>, <cit.>). A three-dimensional lattice is hexagonal if and only ifit has a basis v_1, v_2, v_3 satisfying ( v_i · v_j )_1 ≤ i, j ≤ 3 = [ c -c /2 0;-c/2 c 0; 0 0 d; ], for some c, d ∈. A three-dimensional lattice is rhombohedral if and only ifit has a basis satisfying ( v_i · v_j )_1 ≤ i, j ≤ 3 = [c+d -c/2+d -c/2+d; -c/2+dc+d -c/2+d; -c/2+d -c/2+dc+d ]= [100;010; -1 -11 ][c+d -c/2+d3 d; -c/2+dc+d3 d;3 d3 d9 d ][10 -1;01 -1;001 ], for some c, d ∈. Hence, (i) c (x_1^2 - x_1 x_2 + x_2^2) + d x_3^2 and (ii) c (x_1^2 - x_1 x_2 + x_2^2) + d (x_1 + x_2 + 3 x_3 )^2(c, d ∈) parameterize all hexagonal and rhombohedral lattices, respectively. § NOTATION AND SYMBOLS Throughout this paper,a quadratic form ∑_1 ≤ i ≤ j ≤ n s_ij x_i x_j is always identified with the symmetric matrix (s_ij) with s_ii in the (i, i)-entry and s_ij/2 in the (i, j)-entry. For any quadratic forms f(x_1, …, x_m), g(x_1, …, x_n), their direct sum is the (n+m)-ary quadratic form f(x_1, …, x_m) + g(x_m+1, …, x_m+n), and denoted by f ⊥ g. A quadratic form ∑_i=1^n c_i x_i^2 is represented as a diagonal matrix or [c_1, …, c_n]. In particular, [c] means the unary quadratic form c x^2.In Tables <ref>–<ref> presenting the search result,the ternary quadratic forms are reduced, in the sense that they are Minkowski-reduced Eq.(<ref>) and satisfy the boundary conditions C1–5 provided by Eisenstein <cit.>: * s_12, s_13, s_23 > 0 or s_12, s_13, s_23≤ 0, * s_11 = s_22⟹s_23≤s_13, * s_22 = s_33⟹s_13≤s_12. * case of positive Eisenstein forms (s_12, s_13, s_23> 0): for any distinct 1 ≤ i, j, k ≤ 3, s_ii = 2 s_ij⇒s_i k≤ 2s_j k. * case of non-positive Eisenstein forms (s_12, s_13, s_23≤ 0): for any distinct 1 ≤ i, j, k ≤ 3, s_ii = 2 s_ij⇒ s_i k = 0. In addition, s_11 + s_22 = 2s_12 + s_13 + s_23⇒ s_11≤ s_12 + 2 s_13. With regard to the notation for pairs of quadratic forms and quartic rings,those of <cit.>, in addition to <cit.> are adopted herein. For any unitary commutative ring R,(R^n)^* :=Hom(R^n, R) has the structure of a finitely-generated free R-module. For any R-module,Sym^i M is the i-th symmetric power of M,and Λ^i M is the i-th exterior power of M. If char R2, Sym^2(R^n)^* and Sym^2(R^n)^* ⊗_R R^s can be naturally identified with the set of all the n-ary quadratic forms over R, and the set of all the s-tuples of n-ary quadratic forms over R, respectively.It is said that f, g ∈ Sym^2 (R^n)^* are equivalent over R, and denoted by f ∼_R g, if there exists w ∈ GL_n(R) such that f(𝐱 w) = g(𝐱). For any subring R_2 ⊂ R and f ∈ Sym^2 (R^n)^*, the elements of q_R_2(f) := { f(v) : 0 ≠ v ∈ R_2^n } are called the representations of f over R_2.Throughout this paper, k is a global field with char k2,although some statements hold true for any fields. The algebraic closure of k is always denoted by k̅. As in the theory of prehomogeneous vector spaces, Sym^2 (k^3)^* ⊗_k k^2 and GL_3(k) × GL_2(k)are denoted by V_k and G_k, respectively.The discriminant Disc(P(x, y)) of a polynomial P(x, y) = ∏_j=1^n (α_j x - β_j y) is defined by Disc(P(x, y)) = ∏_1 ≤ j < k ≤ n (α_j β_k - α_k β_j)^2. The resolvent of a quartic polynomial f(x) := x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 is the cubic polynomial f^res(x) :=x^3 - a_2 x^2 + (a_1 a_3- 4 a_0) x+ (4 a_0 a_2 - a_1^2 - a_0 a_3^2).For any (A, B) ∈ V_k, Disc(4(A x - B y)) is denoted by Disc(A, B). (kAB, ⟨ 1, ξ_1, ξ_2, ξ_3⟩)and (kAB, ⟨ 1, ω_1, ω_2⟩) denote the quartic k-algebra and its resolvent cubic algebra with their bases, assigned to (A, B) by Eq.(<ref>), (<ref>) and (<ref>). § A TABLE OF QUADRATIC FORMS WITH THE SAME REPRESENTATIONS OVER The algorithm used to exhaustively search for sets of positive-definite ternary quadratic forms with the identical representations is presented in Table <ref> of Section <ref>,with all the discussions to prove that the algorithm actually works.The following P1–3 describe the searched region;for each quadratic form ∑_1 ≤ i ≤ j ≤ 3 s_ij x_i x_j that satisfies the following conditions, all the representations less than a given threshold are computed as a sorted set Λ = ⟨ q_1, …, q_t ⟩, and passed to the algorithm as an input.During the execution of the algorithm, the computer program always checks if the threshold is large enoughto output all the forms with the required properties. * all of s_ij are integral and their greatest common divisor is 1. * The form is reduced, satisfies Eq.(<ref>) and C1– 5 in Section <ref>.* s_33≤ 115. All quadratic forms over(or their scalar multiples) can be contained in the searched region by increasing the number 115 in (P3).The algorithmwas also applied to all (possibly) regular quadratic forms in the tables of <cit.>, in order to check that all the pairs of regular forms are contained in the output.The results are presented in Tables <ref>–<ref>.By using a computer, the quadratic forms in each set have been confirmed to have the identical representations overup to 3,000,000.The set contained in either of the hexagonal and rhombohedral families were removed from the output.Overall, 53 sets consisting of 151 quadratic forms were obtained as the candidates that may have the completely identical representations as a non-equivalent form. It should be noted that all of themcan be obtained if the algorithm in Section <ref> is carried out under the constraint s_33≤ 48. Since the number is small compared with the upper bound 115 of the searched region, we expect that the 53 cases, in addition to those in the hexagonal and rhombohedral families, provide all the searched quadratic forms, up to the action of GL_3() and constant multiple.The tables also have information about the regularity of each quadratic form. This is based on the tables of Jagy et al. <cit.> and <cit.>. If a form in Tables <ref>–<ref> is regular, it is marked with ** or *. In the tables, 38 out of the 53 cases consist of regular (or possibly regular) quadratic forms. The others are neither regular nor spinor-regular from the result in <cit.>. In general, it is difficult to exactly determine the set of integral representations for a ternary quadratic form (<cit.>).Although it is occasionally possible to prove that the forms have the same representations, without providing the exact set of their representations, as done for the hexagonal and the rhombohedral families, the author could not do this for the non-regular forms in the tables. When two positive-definite quadratic forms f, g ∈ Sym^2 (^3)^* satisfy q_(f) = q_(g), it can be proved that they are equivalent overif and only if f /g is a square in ^×.§ PRELIMINARIES FOR THE MAIN THEOREMSome basic properties on simultaneous representations that will be used in the proof of Theorem <ref> are presented herein.Herein, k is a field with char k ≠ 2 as defined in Section <ref>. The following lemmas and Corollary <ref> are repeatedly used to prove the theorem. Lemma <ref> can be seen as a generalization of the well-known fact that any pairs of positive-definite quadratic forms are simultaneously diagonalized over :For any (A, B) ∈ Sym^2 (k^n)^* ⊗_k k^2 with A ≠ 0, let K ⊂k̅ (resp., K_2 ⊂k̅) be the field generated by all the roots (resp., all the roots of multiplicity > 1) of (Ax - B) = 0 in k̅ over k.If (A, B) is anisotropic over K_2, then A and B are simultaneously diagonalized by the action of GL_3(K). Consequently, for any α∈ K,the rank of A α - B equals n minus the multiplicity of α as a root of (A x - B) = 0.Let w ∈ GL_n(K) be the matrix that provides a Jordan decomposition of S := A^-1 B. Since B is symmetric, we have w B w = (w A w) (w^-1 S w) = (w^-1 S w) (w A w). Hence, if w^-1 S w is diagonal,(w A w, w B w) is a simultaneous block-diagonalization of (A, B) and each block corresponds to an eigenspace of S. If [A_1, ⋯, A_m] and [B_1, ⋯, B_m] are the blocks of A and B, then at least one of A_i, B_i is a constant multiple of the other. Hence, (A, B) can be simultaneously diagonalized.We next assume that S has a Jordan block of size m > 1 that corresponds to a multiple root α∈ K_2 of (A x - B) =A (I x - S) = 0. Let S_2, A_2, B_2 be the m × m blocks of w^-1 S w, w A w, w B w, respectively, that correspond to the eigenspace of α. Since we have B_2 = A_2 S_2 = S_2 A_2, the (1, 1)-entries of A_2 and B_2 equal 0. This is impossible since (A, B) is anisotropic over K_2. Hence, all the Jordan blocks in S must have size 1. The lemma is proved. In particular, K_2 = k always holds for n=3. Therefore, any (A, B) ∈ V_k anisotropic over k, is simultaneously diagonalized by the action of GL_3(K).We assume that (A, B) ∈ V_k is linearly independent, non-singular and anisotropic over k, and satisfies A ≠ 0, Disc(A, B) ≠ 0. Let α_i ∈k̅ (i = 1, 2, 3) be the roots of (A x - B) = 0 and K_i be the Galois closure of k_i = k(α_i) over k. We choose C_i ∈ Sym^2 (k_i^2)^* satisfying A α_i - B ∼_k_i C_i ⊥ [ 0 ]. Then,C_i is anisotropic over K_i for at least two of i = 1, 2, 3 and all of i with k_i ≠ k. Since Disc(A, B) ≠ 0,the vector 0 ≠ v_i ∈ k_i^3 with (A α_i - B) v_i = 0 is uniquely determined, up to constant multiple of k_i^×. From (A α_i - B) v_i = 0 and (A α_j - B) v_j = 0, we have v_i A v_j =v_i B v_j = 0 for any distinct i, j = 1, 2, 3. Hence, if w equals the matrix (v_1 v_2 v_3), then w A w and w B w are diagonal. It may be assumed that [d_1, d_2, d_3] and [d_1 α_1, d_2 α_2, d_3 α_3] are diagonal entries of A and B. For any i, j = 1, 2, 3,if k_i, k_j ≠ k are conjugate over k, it may be assumed thatthe isomorphism σ_ij : k_i → k_j over k induced by α_i ↦α_j maps v_i to v_j. Then, σ_ij maps d_i to d_j. We now have the following: ( w, [ α_1-1; α_2-1 ]) · (A, B) = ( [ 0 0 0; 0 d_2 (α_1 - α_2) 0; 0 0 d_3 (α_1 - α_3) ], [ d_1 (α_2 - α_1) 0 0; 0 0 0; 0 0 d_3 (α_2 - α_3) ]). In what follows, the pair in the right-hand side is denoted by (Ã, B̃). For the proof, we may assume one of the following: * k_1 = k_2 = k_3 = k, * k_1, k_2 are quadratic over k, * k_1, k_2, k_3 are cubic over k.We shall show that the assumption that the diagonal D_1 = [d_2 (α_1 - α_2), d_3 (α_1 - α_3)]is isotropic over K_1 and D_2 = [ d_2 (α_1 - α_2), d_3 (α_1 - α_3) ] is isotropic over K_2,leads to a contradiction. In fact, the assumption holds if and only ifsome β_1 ∈ K_1, β_2 ∈ K_2 satisfyd_2 d_3 (α_1 - α_2)(α_1 - α_3) = -β_1^2 andd_1 d_3 (α_2 - α_1)(α_2 - α_3) = -β_2^2. Then, (Ã, B̃)(𝐱) = 0 holds, if we put 𝐱 := (-β_2 / d_1 (α_1 - α_2), β_1 / d_2(α_1 - α_2), ± 1). Hence,(A, B) is isotropic over k in case (i). The same is true in case (ii),since(A, B)(𝐱 w) = (Ã, B̃)(𝐱) = 0 and 𝐱 w ∈ k^3,if we choose β_2 such that β_2 = σ_12(β_1). In case (iii), A α_i - B is isotropic over K_1 = K_2 = K_3 for all i = 1, 2, 3 due to conjugacy. There exist β_i ∈ k_i (i = 1, 2, 3) such that either of the following holds for every distinct 1 ≤ h, i, j ≤ 3, considering that d_i d_j (α_h - α_i)(α_h - α_j) belongs to k_h: * d_i d_j (α_h - α_i)(α_h - α_j) = -β_h^2, * d_i d_j (α_h - α_i)(α_h - α_j) = -β_h^2 (α_1 - α_2)^2 (α_2 - α_3)^2 (α_3 - α_1)^2.It is possible to choose β_i so that β_1, β_2, β_3 are conjugate. If we put 𝐱_2 := (1/β_1, 1/β_2, 1/β_3), we have (Ã, B̃)(𝐱_2) = (A, B)(𝐱_2 w) = 0 and 𝐱_2 w ∈ k^3. Thus, the lemma is proved.In general, for any global field k, prime v of k, and (A_i, B_i) ∈ V_k (i = 1, 2), if we start from q_k(A_1, B_1) = q_k(A_2, B_2),the following are immediately obtained: * If both (A_1, B_1) and (A_2, B_2) are anisotropic over k_v, q_k_v (A_1, B_1) = q_k_v(A_2, B_2), since the topological closure of q_k(A_i, B_i) in k_v^2 equals { (0, 0) }∪ q_k_v(A_i, B_i). * If both (A_1, B_1) and (A_2, B_2) are isotropic over k_v, q_k_v (A_1 π - B_1) = q_k_v(A_2 π - B_2) = k_v for any π∈ k_v. * If (A_1, B_1) is anisotropic and (A_2, B_2) is isotropic over k_v, A_1 π - B_1 is isotropic over k_v for any π∈ k_v.As a corollary of Lemma <ref>, (C3) is excluded in the following case:Let v be a prime of k. We assume that (A_1, B_1), (A_2, B_2) ∈ V_ksatisfy q_k(A_1, B_1) = q_k(A_2, B_2), and (A_1 x - B_1 y) completely splits in k_v. In this case, either of the above (C1) or (C2) is true, hence q_k_v (A_1 π - B_1) = q_k_v(A_2 π - B_2) for any π∈ k_v. In particular, (A_1, B_1) is isotropic over k_v if and only if so is (A_2, B_2). For understanding of the following sections, we need to recall thata one-to-one correspondence between the elements of V_ :=Sym^2 (^3)^* ⊗^2and all pairs of a quartic ring and its resolvent ring was recently proved in <cit.>. The result was generalized to the case of any base scheme S <cit.>. Herein,the generalization for Dedekind domains of <cit.> is adopted.In the following of this section, as in <cit.>, R is always a Dedekind domain, a Noetherian, integrally closed integral domain that has the property that every nonzero prime ideal is maximal. Therefore, any fields are also a Dedekind domain. A finitely generated, torsion-free R-module is called a lattice over R. If M is a lattice over R and k is the field of fractions of R, the rank of M is defined as the dimension of M ⊗_R k over k. A unitary commutative associative R-algebra is called a quartic ring (resp. cubic ring), if it has rank 4 (resp. 3) as a lattice over R. When we put L := Q / R, M := C/R and L^* :=Hom(L, R), a quadratic map means an element of Sym^2 L^* ⊗_R M. Let R be a Dedekind domain. For any quartic ring Q,its cubic resolvent ring C (also called numerical resolvent in <cit.>) is defined as the R-algebra with the following properties: * It is equipped with an R-module isomorphism θ: Λ^2 (C/R) →Λ^3 (Q/R) and a quadratic map ϕ: Q/R → C/R such that x ∧ y ∧ x y = θ(ϕ(x) ∧ϕ(y)) for anyx, y ∈ Q/R. * The multiplicative structure of C is determined by x ∧ x^2 ∧ x^3 = θ(ϕ(x) ∧ϕ(x)^2)for anyx ∈ Q/R, where ϕ(x)^2 is the square ϕ̃(x)^2 of any lift ϕ̃ : Q → C of ϕ.The quadratic map ϕ is called the resolvent map of (Q, C). If R is a field, and Q = R[α] the quartic extension by α, the classical resolvent map from Q to R is defined by α↦αα^' + α^''α^''',by using the conjugates α^', α^'', α^''' of α over R. The following explains a generalization of the Bhargava correspondence for quartic ringsto the case over a Dedekind domain: Let R be a Dedekind domain. There is a canonical bijection between * Isomorphism classes of pairs (Q, C), where Q is a quartic ring and C is a cubic resolvent ring of Q. * Quadruples (L, M, θ, ϕ), where L and M are lattices of ranks 3 and 2 over R, respectively, θ: Λ^2 M →Λ^3 L is an isomorphism and ϕ: L → M is a quadratic map. Under this bijection, the identifications Q/R ≅ L and C/R ≅ M are obtained. Any quartic ring Q has a cubic resolvent, and if Q is Dedekind, the resolvent is unique. In what follows, we explain how the pair of Q and R in (i), is associated with(L, M, θ, ϕ) in (ii). Since R is a Dedekind domain, the lattice Q/R is isomorphic to 𝔞_1 ξ̅_1 ⊕𝔞_2 ξ̅_2 ⊕𝔞_3 ξ̅_3 for some ξ̅_1, ξ̅_2, ξ̅_3 ∈ Q/R and ideals 𝔞_1, 𝔞_2, 𝔞_3 of R. Similarly, C/R is isomorphic to 𝔟_1 ω̅_1 ⊕𝔟_2 ω̅_2 for some ω̅_1, ω̅_2 ∈ C/R and ideals 𝔟_1, 𝔟_2 of R. If these 𝔞_i, ξ̅_i, 𝔟_j, ω̅_j are fixed,the quadratic map ϕ : Q/R → C/Ris uniquely associated with a pair of ternary quadratic forms (A, B) over the fraction field k, by ϕ(x_1 ξ̅_1 + x_2 ξ̅_2 + x_3 ξ̅_3) = B(x_1, x_2, x_3) ω̅_1 + A(x_1, x_2, x_3) ω̅_2. If we put L := Q/R, Q is isomorphic to R ⊕ L as an R-module. Let ξ_i ∈ Q (1 ≤ i ≤ 3) be the element corresponding to ξ̅_i in Q / R. We first note that Eq.(<ref>) implies ( ∑_i=1^3 x_i ξ̅_i ) ∧( ∑_i=1^3 y_i ξ̅_i ) ∧( ∑_i=1^3 ∑_j=1^3 x_i y_j ξ̅_i ξ̅_j ) = θ( ϕ( ∑_i=1^3 x_i ξ̅_i ) ∧ϕ( ∑_i=1^3 y_i ξ̅_i ) ).The multiplicative structure of Q, all c_ij^k ∈ R of the equalities ξ_i ξ_j = c_ij^0 + ∑_i=1^3 c_ij^k ξ_k, is determined from Eq.(<ref>) as follows; we denote the coefficients of A and B by A(x_1, x_2, x_3) = ∑_1 ≤ i ≤ j ≤ 3 a_ij x_i x_j, B(x_1, x_2, x_3) = ∑_1 ≤ i ≤ j ≤ 3 b_ij x_i x_j. We then have (ξ_1 ∧ξ_2 ∧ξ_3) x_1 y_1∑_i, j=1^3 c_ij^1 x_i y_j x_2 y_2∑_i, j=1^3 c_ij^2 x_i y_j x_3 y_3∑_i, j=1^3 c_ij^3 x_i y_j = - θ( ω̅_1 ∧ω̅_2 ) ∑_1 ≤ i ≤ j ≤ 3∑_1 ≤ k ≤ l ≤ 3 x_i x_j y_k y_l a_ijb_ij a_klb_kl . Replacing each ξ_i by -ξ_i if necessary,we can fix the sign as follows: x_1 y_1∑_i, j=1^3 c_ij^1 x_i y_i x_2 y_2∑_i, j=1^3 c_ij^2 x_i y_i x_3 y_3∑_i, j=1^3 c_ij^3 x_i y_i = - ∑_1 ≤ i ≤ j ≤ 3∑_1 ≤ k ≤ l ≤ 3 x_i x_j y_k y_l a_ijb_ij a_klb_kl . Comparing the coefficients of each term, the following equations are obtained: c^j_ii = ϵλ^ii_ik, c^k_ij = ϵλ^jj_ii, c^j_ij - c^k_ik = ϵλ^jk_ii, c^i_ii - c^j_ij - c^k_ik = ϵλ^ij_ik, λ^ij_kl :=a_ijb_ija_klb_kl,where (i, j, k) denotes any permutation of (1, 2, 3) and ϵ = ± 1 is its sign. The c^0_ij are determined from the associative law of Q. Consequently, allc_ij^k ∈ R, are uniquely determined, up to the transformations given byc^j_ij↦ c^j_ij + a, c^k_ij↦ c^k_ij + a, c^i_ii↦ c^i_ii + 2a (a ∈ Q) which corresponds to the replacement of ξ_i by ξ_i + a. In order to uniquely determine c_ij^k, the following constraints were applied in <cit.>:c^1_12 = c^2_12 = c^1_13 = 0. If we put f_det(x, y) := 4 (A x - B y),the following equation holds for any x = ∑_i=1^3 x_i ξ̅_i ∈ Q/R.x ∧ x^2 ∧ x^3 = f_det(B(x_1, x_2, x_3), A(x_1, x_2, x_3)) ξ̅_1 ∧ξ̅_2 ∧ξ̅_3. Hence, the ring structure of C determined by Eq.(<ref>), is same as that provided by 4 (A x - B y) under the Delone-Faddeev-Gan-Gross-Savin correspondence (<cit.>, <cit.>). If the coefficients are denoted by 4 (A x - B y) = a x^3 + b x^2 y + c x y^2 + d y^3, the basis ⟨ω̅_1, ω̅_2 ⟩ of C/R as an R-algebra is lifted to a basis ⟨ 1, ω_1, ω_2 ⟩ of C that satisfies:ω_1^2=-ac + b ω_1 - a ω_2,ω_1 ω_2=-ad,ω_2^2=-bd + d ω_1 - c ω_2. § A CANONICAL FORM FOR ELEMENTS OF V_KHerein, the method to obtain a canonical form of (A, B) ∈ V_k is discussed, assuming that char k ≠ 2, 3.For any h_0 ∈ k and h := (h_1, h_2, h_3) ∈ k^3, if we put α := h_0 + ∑_j=1^3 h_j ξ_j∈kAB, the following holds:1, α, α^2, α^3is a basis of kAB overk ⟺(B(h) A - A(h) B) ≠ 0.When (A, B) is non-singular and anisotropic over k, kAB contains such an α if and only if A, B are linearly independent over k.For any element α∈kAB,we denote the image of α by the natural epimorphism kAB↠kAB / k · 1 by α. By Eq.(<ref>), the following matrix M satisfies M = 4 (B(h) A - A(h) B):M [ ξ_1; ξ_2; ξ_3 ]= [ α; α^2; α^3 ].The first statement is obtained from this. With regard to the second one,the only-if part immediately follows from the fact that kAB is isomorphic to k[x, y, z] / (x^2, y^2, z^2, xy, yz, xz), if A, B are linearly dependent. We now prove the if part. If (A x - B y) = 0 does not have roots [x : y] ∈ℙ^1(k), then(B(h) A - A(h) B) = 0 ⇔ A(h) = B(h) = 0. However, none of 0 ≠ h ∈ k^3 satisfies this, owing to the anisotropy of (A, B). If ∅Π⊂ℙ^1(k) is the set of all the roots of (A x - B y) = 0, then, (B(h) A - A(h) B) = 0 ⟺ u A (h) = v B (h)for some[u : v] ∈Π.Since u A - v B is a non-zero quadratic form,if the cardinality of Π is 1, some h ∈ k^3 does not satisfy u A (h) = v B (h). Otherwise, all roots of (A x - B y) = 0 belong to ℙ^1(k), hence, (A, B) is simultaneously diagonalized over k by Lemma <ref>. Thus, it is easily seen that this case is eliminated as well. In the following, for fixed (A, B) ∈ V_k, we will take α as in Lemma <ref> and put a x^3 + b x^2 y + c x y^2 + d y^3 := 4(A x - B y). Let ch_α(x) := x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 be the characteristic polynomial of α∈kAB.From Eq.(<ref>), these a_i (0 ≤ i ≤ 3) are represented as an integral polynomial of h_i (0 ≤ i ≤ 3) and the coefficients of A, B.It was pointed out in <cit.> that the image of α∈ Q_(A, B) by the resolvent map equals z + B(h) ω_1 + A(h) ω_2 for some z ∈.In general, it can be verified by direct calculation that the following equality holds as a polynomial of h_i and the coefficients of A, B.ch_α^res(x) =(x-z)^3 + ( c A(h) - b B(h) ) (x-z)^2 + { b d A(h)^2 + (3 a d - b c) A(h) B(h)+ a c B(h)^2 } (x-z) + a d^2 A(h)^3 - (b^2 d - 2 a c d) A(h)^2 B(h)+ (a c^2 - 2 a b d) A(h) B(h)^2 - a^2 d B(h)^3 = ∏_i=1^3 ( x - z + a B(h) u^(i)/v^(i) - d A(h) v^(i)/u^(i)),where [u^(i) : v^(i)] ∈ℙ^1(k̅) (1 ≤ i ≤ 3) are the roots of 4(A x - B y) = 0. The z equals (a_2 + c A(h) - b B(h)) / 3, which is obtained by comparing the coefficients of x^2.We assume that (A, B) ∈ V_k is linearly independent, non-singular and anisotropic over k. As in Lemma <ref>, α∈kAB and h ∈ k^3 that satisfies ( B(h) A - A(h) B) ≠ 0 are fixed. When ch_α(x) = x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 is the characteristic polynomial of α, let Λ := ⟨ 1, α, α^2 + a_3α + a_2, α^3 + a_3α^2 + a_2α + a_1⟩ be a basis of k[α]. Let (R, Φ) be the cubic k-algebra corresponding to the resolvent polynomial ch_α^res(x+a_2/3). Then, (k[α], Λ) and (R, Φ) coincide with the quartic ring kÃB̃ and the resolvent cubic kÃB̃ of the following (Ã, B̃).à = [ 0 0 1/2; 0-1 0; 1/2 0 0 ],B̃ = - [1a_3/2a_2/6;a_3/2 2a_2/3a_1/2;a_2/6a_1/2a_0 ]. It is straightforward to verify by direct computation that Λ satisfies Eq.(<ref>), and (Ã, B̃) corresponds to the pair of (k[α], Λ) and (R, Φ). In particular, we have 4 ( à x- B̃) =ch_α^res(x + a_2/3). We shall construct (W, V) ∈ G_k such that(Ã, B̃) = (W, V) · (A, B) holds; when ⟨ 1, ξ_1, ξ_2, ξ_3 ⟩ is the basis of kAB, let W be the matrix uniquely determined by:W [ ξ_1; ξ_2; ξ_3 ]= [ 1 0 0; a_3 1 0; a_2 a_3 1 ][ α; α^2; α^3 ].The determinant is given by W = 4 (B(h) A - A(h) B) ≠ 0. We define V ∈ GL_2(k) byV :=W^-1[ 1 0; (c A(h) - b B(h))/3 1 ][B(h) -A(h); -c A(h) B(h) - d A(h)^2 -a B(h)^2 - b A(h) B(h) ],which is derived from the following equiation obtained by using Eq.(<ref>) and 4 ( Ax - B y ) = a ∏_i=1^3 (x - (u^(i) / v^(i)) y):ch_α^res( x +a_2 + c A(h) - b B(h) /3) =4 (B(h) A - A(h) B) / Wch_α^res( x +a_2 + c A(h) - b B(h) /3) =a / W ∏_i=1^3 ( B(h) - u^(i)/v^(i) A(h) )( x + a B(h) u^(i)/v^(i) - d A(h) v^(i)/u^(i)) = a/ W∏_i=1^3 { B(h) x + c A(h) B(h) + d A(h)^2 - u^(i)/v^(i) (A(h) x - a B(h)^2 - b A(h) B(h)) }=( W^-1) 4 ( (B(h) x + c A(h) B(h) + d A(h)^2) A + (-A(h) x + a B(h)^2 + b A(h) B(h)) B ). As a result of Lemma <ref>, the following corollary is immediately obtained:We assume that (A, B) and α are taken as in Lemma <ref>. For any field k ⊂ K ⊂k̅, (A, B) is isotropic over K if and only ifch_α(x) = 0 has a root in K. Using the action of G_k, we may replace (A, B) with (Ã, B̃) in Eq.(<ref>). This is proved as follows:(Ã, B̃)(𝐱) = 0for some0 ≠𝐱∈ K^3⇔(Ã, B̃)(s^2, st, t^2) = 0for some[s : t] ∈ℙ^1(K) ⇔ch_α(u) = 0for someu ∈ K.As proved in Section <ref>, any (A_1, B_1), (A_2, B_2) ∈ V_k can be simultaneously transformed to pairs of the form of Eq.(<ref>), if (A_1 x - B_1 y) = c(A_2 x - B_2 y) for some c ∈ k^×, and some (q_A, q_B) ∈ q_k(A_1, B_1) ∩ q_k(A_2, B_2) satisfies (q_B A_i - q_A B_i) ≠ 0.§ AUTOMORPHISMS OF CUBIC POLYNOMIALSThe purpose of this section is to prove Lemma <ref>which is used in the proof of Theorem <ref>. The assumption char k ≠ 2, 3 is also used herein. If a cubic polynomial f (x, y) = a x^3 + b x^2 y + c x y^2 + d y^3 ∈ k[x, y] hasno multiple roots and satisfies ( V)^-1 f ( (x, y) V ) = u f(x, y) for some V ∈ GL_2(k) and u ∈ k, thenV^n = u^n I for at least one of n = 1, 2, 3.Let [p_i : q_i] ∈ℙ^1(k̅) (i = 1, 2, 3) be the roots of f(x, y) = 0. V ∈ GL_2(k) exchanges [p_i : q_i]. Hence, V^n = v I for some n = 1, 2, 3 and v ∈ k. From ( V^n)^-1 f((x, y)V^n) = v f(x, y) = u^n f(x, y), v=u^n is obtained. In what follows, p_i, q_i that satisfy f(x, y) = ∏_i=1^3 (q_i x - p_i y) are fixed. The following examples list all the cases in which V ≠ I.[Case V^2 = u^2 I] It may be assumed that V swaps [p_1 : q_1] and [p_2 : q_2], and fixes [p_3 : q_3].For some 0 ≠ r_1, r_2 ∈k̅,V=u [ p_1 q_1; p_2 q_2 ]^-1[ 0 r_2; r_1 0 ][ p_1 q_1; p_2 q_2 ].From V^2 = u^2 I, r_1 r_2 = 1 is obtained. V [q_3; -p_3 ] = -u [q_3; -p_3 ] is also obtainedfrom ( V)^-1 f((x, y)V) = - u^-2 f((x, y)V) = u f(x, y). Hence, r_1 = r_2^-1 = -(p_2 q_3 - q_2 p_3) / (p_1 q_3 - q_1 p_3). Thus,-(p_2 q_3 - q_2 p_3) (p_1 q_3 - q_1 p_3) V = u/ p_1 q_2 - p_2 q_1 [q_2 -q_1; -p_2p_1 ][ 0 (q_1 p_3 - p_1 q_3)^2; (q_2 p_3 - p_2 q_3)^2 0 ][ p_1 q_1; p_2 q_2 ]=u [p_1 p_2 q_3^2 - p_3^2 q_1 q_2 (p_1 q_2+ p_2 q_1) q_3^2 - 2 p_3 q_1 q_2 q_3;(p_1 q_2 + p_2 q_1) p_3^2 - 2 p_1 p_2 p_3 q_3 -p_1 p_2 q_3^2 + p_3^2 q_1 q_2 ].Therefore, if we putC := -1/(p_2 q_3 - q_2 p_3) (p_1 q_3 - q_1 p_3) p_3 / ∂ f/∂ y (p_3, q_3),V=u C [b p_3 + c q_3 -3 a p_3 - b q_3;c p_3 + 3 d q_3 -b p_3 - c q_3 ].If [p_3 : q_3] ∉ℙ^1(k), V ∈ GL_2(k) implies that3 a c = b^2 and 3 b d = c^2, then f(x, y) = (3b c)^-1 (b x + c y)^3.Since f is assumed to have no multiple roots, we obtain [p_3 : q_3] ∈ℙ^1(k).[Case V^3 = u^3 I] It may be assumed thatV maps [p_i : q_i] to [p_j : q_j] for any (i, j) = (1, 2), (2, 3), (3, 1). Since V^3 = u^3 I, for some c_i ∈k̅ with c_1 c_2 c_3 = u^3, we have [ p_1 q_1; p_2 q_2; p_3 q_3; ] V = [ 0 c_2 0; 0 0 c_3; c_1 0 0 ][ p_1 q_1; p_2 q_2; p_3 q_3; ]. We have V = c_i_2 c_i_3 (p_i_2 q_i_3 - p_i_3 q_i_2) / (p_i_1 q_i_2 - p_i_2 q_i_1) for any (i_1, i_2, i_3) = (1, 2, 3), (2, 3, 1), (3, 1, 2). Hence, [ c_1; c_2 ] =c_3 p_3 q_1 - p_1 q_3/p_1 q_2 - p_2 q_1[ p_2 q_3 - p_3 q_2/p_1 q_2 - p_2 q_1; p_3 q_1 - p_1 q_3/p_2 q_3 - p_3 q_2 ]. From c_1 c_2 c_3 = u^3,some ζ∈k̅ with ζ^3 = 1 satisfies c_3 = u ζ (p_1 q_2 - p_2 q_1)/(p_3 q_1 - p_1 q_3). Thus, V=u ζ[ p_1 q_1; p_2 q_2 ]^-1[ p_3 q_1 - p_1 q_3/p_2 q_3 - p_3 q_2 0; 0 p_1 q_2 - p_2 q_1/p_3 q_1 - p_1 q_3 ][ p_2 q_2; p_3 q_3 ]. From V = u^2 ζ^2 ∈ k, ζ∈ k is obtained.Hence, u may be replaced with u ζ^-1. If we put Δ = (p_1 q_2 - p_2 q_1)(p_2 q_3 - p_3 q_2)(p_3 q_1 - p_1 q_3), we have Disc(f(x, 1)) = Δ^2 and Δ =-2 (p_1^2 p_2 q_2 q_3^2 + p_2^2 p_3 q_1^2 q_3 + p_1 p_3^2 q_1 q_2^2) - b c + 3 a d =2 (p_1 p_2^2 q_3^2 q_1 + p_3 p_1^2 q_2^2 q_3 + p_2 p_3^2 q_1^2 q_2) + b c - 3 a d, V = u/Δ[(9 a d - b c - Δ)/2b^2 - 3 a c; -c^2 + 3 b d -(9 a d - b c + Δ)/2 ].Since V ∈ GL_2(k), we have Δ∈ k. Therefore, f(x, y) = 0 completely splits over k, ork[x]/(f(x, 1)) is a Galois cubic field over k. In the following lemma, 4(A x - B y) is denoted by f_det(x, y). We assume that (A, B) ∈ V_k with Disc(A, B) ≠ 0 (hence, linearly independent), is non-singular and anisotropic over k.We further assume that there are u ∈ k and a matrix u IV ∈ GL_2(k) such that q_k(A, B) = q_k((I, V) · (A, B)) and ( V)^-1 f_det((x, y) Ṽ) = u f_det(x, y), Ṽ := [10;0 -1;] V [10;0 -1;].Then,there existsW ∈ GL_3(k) such that (W, -u^-1 V) · (A, B) = (A, B). Using the action of GL_2(k), we may assume A ≠ 0. Let α_1, α_2, α_3 ∈k̅ be the roots of (A x - B) = 0, and k_i be the field k(α_i). We fix 0 ≠ v_i ∈ k_i^3, d_i ∈ k_i (1 ≤ i ≤ 3) and w = (v_1 v_2 v_3) as in the first paragraph in the proof of Lemma <ref>. (A, B) is transformed as in Eq.(<ref>).From Ṽ u I,there are the following two cases: * (Ṽ^2 = V^2 = u^2 I) It may be assumed that V exchanges [α_1 : -1] and [α_2 : -1]. We then have k_1 = k_2 and α_3 ∈ k. Furthermore, V=u [ α_1-1; α_2-1 ]^-1[0 -α_1 - α_3/α_2 - α_3; -α_2 - α_3/α_1 - α_30 ][ α_1-1; α_2-1 ].Hence, ( w, [ α_1-1; α_2-1 ] V ) · (A, B) = ( I, u [0 -α_1 - α_3/α_2 - α_3; -α_2 - α_3/α_1 - α_30 ]) ·( [ 0 0 0; 0 d_2 (α_1 - α_2) 0; 0 0 d_3 (α_1 - α_3) ], [ d_1 (α_2 - α_1) 0 0; 0 0 0; 0 0 d_3 (α_2 - α_3) ]) = ( u [ d_1(α_1 - α_3)/d_2(α_2 - α_3)· d_2 (α_1 - α_2)00;000;00 -d_3 (α_1 - α_3) ], u [000;0 d_2(α_2 - α_3)/d_1(α_1 - α_3)· d_1 (α_2 - α_1)0;00 -d_3 (α_2 - α_3) ]). Owing to q_k(A, B) = q_k((I, V) · (A, B)), for any primes 𝔭 of k that completely splits in k_1 = k_2, we must haveq_k_𝔭( [ d_2 (α_1 - α_2) 0; 0 d_3 (α_1 - α_3) ])=q_k_𝔭( u [ d_1(α_1 - α_3)/d_2(α_2 - α_3)· d_2 (α_1 - α_2)0;0 -d_3 (α_1 - α_3) ]), q_k_𝔭( [ d_1 (α_2 - α_1) 0; 0 d_3 (α_2 - α_3) ])=q_k_𝔭( u [ d_2(α_2 - α_3)/d_1(α_1 - α_3)· d_1 (α_2 - α_1)0;0 -d_3 (α_2 - α_3) ]).This implies that there exists β∈ k_i such that - d_1(α_1-α_3)/d_2(α_2 - α_3) = β^2 and ( W, -u^-1 V ) · (A, B) = (A, B) if we put:W = w^-1[ 0 β 0; ± 1/β 0 0; 0 0 1 ] w. If k_1 = k_2 = k, then W ∈ GL_3(k) has the required property. Otherwise, k_1 = k_2 is quadratic over k.If the signature of ± 1/β is chosen so that β and ± 1/β are conjugate over k,W ∈ GL_3(k) is obtained. * (Ṽ^3 = V^3 = u^3 I) In this case, k_1 = k_2 = k_3 = k or a Galois cubic field over k. As shown in the above example,V=u [ α_1-1; α_2-1 ]^-1[ α_3 - α_1/α_2 - α_3 0; 0 α_1 - α_2/α_3 - α_1 ][ α_2-1; α_3-1 ], =u [ α_1-1; α_2-1 ]^-1[0α_3 - α_1/α_2 - α_3; -α_2 - α_3/α_3 - α_1 -1 ][ α_1-1; α_2-1 ].Hence, ( w, [ α_1-1; α_2-1 ] V ) · (A, B) = ( I, u [0α_3 - α_1/α_2 - α_3; -α_2 - α_3/α_3 - α_1 -1 ]) ·( [ 0 0 0; 0 d_2 (α_1 - α_2) 0; 0 0 d_3 (α_1 - α_3) ], [ d_1 (α_2 - α_1) 0 0; 0 0 0; 0 0 d_3 (α_2 - α_3) ]) = ( -u [ d_1(α_3 - α_1)/d_2(α_2 - α_3)· d_2 (α_1 - α_2)00;000;00d_3 (α_1 - α_3) ], -u [d_1 (α_2 - α_1)00;0 d_2(α_1 - α_2)/d_3(α_3 - α_1)· d_3 (α_2 - α_3)0;000 ]).Then, there exits β_i_1∈ k_1 = k_2 = k_3 such that d_i_2 (α_i_1 - α_i_2) / d_i_3 (α_i_1 - α_i_3) = β_i_1^2for every (i_1, i_2, i_3) = (1, 2, 3), (2, 3, 1), (3, 1, 2). ( W, -u^-1 V ) · (A, B) = (A, B) and W ∈ GL_3(k) hold, if we put:W = w^-1[ 0 β_3 0; 0 0 β_1; β_2 0 0 ] w.Consequently, the lemma is proved. § PROOF OF THEOREM <REF>In what follows, we assume that k =.In order to prove the theorem, we will first show that q_(A_1, B_1) = q_(A_2, B_2) implies that either of the determinants is a constant multiple of the other. If (A_i, B_i) ∈ V_ (i = 1, 2) satisfy the conditions (<ref>), (b') and (c') of Theorem <ref>,there are coprime integers r_1, r_2 such that r_1^-1 (A_1 x - B_1 y) = r_2^-1 (A_2 x - B_2 y).Using the action of GL_2() on V_, we may assume that A_1 ≠ 0. If Disc(A_1, B_1) = 0, let α∈ be the multiple root of(A_1 x - B_1) = 0. In this case, A_1 α - B_1 has rank ≤ 1 (Lemma <ref>). Owing to q_(A_1 α - B_1) = q_(A_2 α - B_2), A_2 α - B_2 has the same rank.Therefore, α is a multiple root of (A_2 x - B_2) = 0. Let β≠α be another root of (A_1 x - B_1) = 0. Since (A_1, B_1) is anisotropic over , C ∈ Sym^2 (^2)^* with A_1 β - B_1 ∼_ C ⊥ [ 0 ] must be anisotropic over .Hence, A_2 β - B_2 must have rank 2, and (A_2 β - B_2) = 0. Thus, the proposition is proved if Disc(A_1, B_1)=0. The same holds if Disc(A_2, B_2)=0.We next assume that Disc(A_i, B_i) ≠ 0 (i = 1, 2). Let α∈ be a root of (A_1 x - B_1) = 0 and K be the Galois closure of k(α) over .If C ∈ Sym^2 (K^2)^* with A_1 α - B_1 ∼_K C ⊥ [ 0 ] is anisotropic over K, then there are a finite prime p ofand an embedding ι: K ↪_p such that ι(C) is anisotropic over _p. In this case, as a result of Corollary <ref>, (A_2 α - B_2) = 0 is proved by q__p(A_2 ι(α) - B_2) = q__p(A_1 ι(α) - B_1)= { 0 }∪ q__p(ι(C)).Thus, by Lemma <ref>, the equations (A_i x - B_i) = 0 (i = 1, 2) have at least two common roots α≠β. In addition, if there exist distinct γ_1 ≠γ_2 such that (A_i γ_i - B_i) = 0 (i = 1, 2), both γ_i belong toand both A_i γ_i - B_i are isotropic over . Thus, by using the action of G_, we may assume the following:(A_1, B_1)= ( [ a_11 a_120; a_12 a_220;00 a_33 ], [ 0 c_1 0; c_1 0 0; 0 0 0 ]), (A_2, B_2)= ( [ 0 c_2 0; c_2 0 0; 0 0 0 ], [ b_11 b_120; b_12 b_220;00 b_33 ]).Furthermore, by using the action of G_, we may assume that a_11 = b_11 = 1 and c_1 = c_2 = -1/2. For the anisotropy of (A_i, B_i) over , -a_33, -b_33, -a_22 a_33, -b_22 b_33∉ (^×)^2 is required. In this case, if we put g(x) := (x - α)(x - β) ∈[x], then4( A_1 x - B_1 )=-{ 4 (a_12^2 - a_22) x^2 + 4 x a_12 + 1 } a_33 x = -4 (a_12^2 - a_22) a_33 x g(x), 4( A_2 x -B_2 )= { x^2 + 4 x b_12 + 4 (b_12^2 - b_22) } b_33 = b_33 g(x).In particular, we have 4 (a_12^2 - a_22) b_12 = a_12 and 4 (b_12^2 - b_22) a_12 = b_12. From this, we obtain (a_22 - a_12^2)/(b_22 - b_12^2) = a_22 / b_22 = a_12^2 / b_12^2 ∈ (^×)^2. Furthermore, for any 0 ≠ x ∈ we have A_1 x - B_1 ∼_ [ x 0 0; 0 -4 (a_12^2 - a_22) g(x) / x 0; 0 0a_33 x ],A_2 x - B_2 ∼_ [-1 0 0; 0g(x) 0; 0 0 -b_33 ]. Let P_0 be the set of odd primes p ofsuch that p completely splits in [x] / (g(x)) (a_22, b_22∈ (_p^×)^2). For a fixed p ∈ P_0, we will denote the roots of g(x) = 0 in _p by α_p and β_p. Let P ⊂ P_0 be the subset consisting of all p ∈ P with4 b_12∈_p anda_33, b_33, 4 (a_12^2 - a_22), 4(b_12^2 - b_22) ∈_p^×. In this case, for any p ∈ P, α_p and β_p belong to _p^×. By setting z to an element of _p close to α_p (resp. β_p), g(z) ∈ p ^× and 4 (a_12^2 - a_22) / z ∈β_p^-1 (_p^×)^2 (resp. α_p^-1 (_p^×)^2) can be assumed. ( -a_33/p) = ( -b_33/p) is then obtained from q__p(A_1 x - B_1) = q__p(A_2 x - B_2). In addition, either of the following holds: * ( -a_33/p) = ( -b_33/p) = 1. * (-α_p /p) = ( -β_p /p) = 1.Let K ⊂ be the extension ofobtained by attaching √(b_22) to, the splitting field of g(x) over . If K(√(-b_33)) is denoted by F_1, we have F_1 ⊋ K, since -b_33, -b_22 b_33∉ (^×)^2. Let F_2 be the extensions of K that are obtained by attaching the roots of g(-x^2) = 0 to K. F_2 = K is proved as follows; if F_2 ⊋ K, let F_3 be the composition F_1 F_2. Let Q_i (i = 1, 2, 3) be the set of all primes of K that completely splits in F_i.Since the extension F_i / K is Galois, the Kronecker densityd_i := lim_s ↓ 1∑_p ∈ Q_i p^-s/log (1/(s - 1)) equals 1 / [F_i : K] (Theorem 8.41 (2), <cit.>). However, from p ∈ P ⇒ (i) or (ii), we obtain d_1 + d_2 ≥ 1. Owing to d_1 = 1/2, in order that d_3 > 0, d_2 = 1, K = F_2 is required.As a result, -α = δ_1^2, -β = δ_2^2 for some δ_1, δ_2 ∈ K^×, and δ_1+δ_2, δ_1δ_2 ∈ may be assumed. We also have the following:a_12 = δ_1^-2 + δ_2^-2/4,a_22 = (δ_1^-2 - δ_2^-2/4)^2,b_12 = δ_1^2 + δ_2^2/4,b_22 = (δ_1^2 - δ_2^2/4)^2. We shall show that a_22, b_22∈ (^×)^2, and therefore, δ_1, δ_2 ∈, owing to K = F_2 =; if we put c := (δ_1^-1 + δ_2^-1)/2 andd := δ_1^-1δ_2^-1/2, then c, d are rational, and satisfy d^2 = a_12^2 - a_22, a_12 + d = c^2 and a_12 - d = a_22/c^2. Furthermore, A_1(𝐱) = 0 if and only if 𝐱 = (x_1, x_2, x_3) ∈ satisfies (x_1 + (a_12 + d) x_2)(x_1 + (a_12 - d) x_2) = -a_33 x_3^2.Hence, if x_1 + (a_12 + d) x_20, there are s, 0t ∈ such that-2d x_1 = { a_12 - d + (a_12 + d) a_33 s^2 } t,2d x_2 = (1 + a_33 s^2) t and x_3 = st. Thus, for a field F ⊃, if T_1, F, T_2, F are defined as follows, T_1,= T_2, is obtained by considering the representations (0, *): T_1, F := {( 1 + a_33 s^2 ) (a_12 - d + (a_12 + d) a_33 s^2 ) t^2: s, t ∈ F, t ≠ 0}= {( 1 + a_33 s^2 ) ( a_22/c^2 + a_33 c^2 s^2 ) t^2: s, t ∈ F, t ≠ 0}, T_2, F :=q_F(g_1) ∪ q_F(g_2).whereg_1, g_2 are binary quadratic forms defined by g_1(x, y) := x^2 + b_33 y^2, g_2(x, y) := b_22 x^2 + b_33 y^2. From ( -a_33/p) = ( -b_33/p) for any p ∈ P, we also have a_33 / b_33∈ (K^×)^2, hence, a_33 / b_33∈ (^×)^2 or a_33 / b_22 b_33∈ (^×)^2.T_1, F⊂ T_2, F implies that for any s ∈, either of the following holds: * (1 + a_33 s^2) ( a_22 / c^2 + a_33 c^2 s^2 ) ∈ q_(g_1), * (1 + a_33 s^2) ( a_22 / c^2 + a_33 c^2 s^2 ) ∈ q_(g_2).In case of a_33 / b_33∈ (^×)^2, each item implies the following, which is obtained by using the composition of binary quadratic forms: * a_22 / c^2 + a_33 c^2 s^2 ∈ q_(g_1), * 1 + a_33 s^2 ∈ q_(g_2).However, this is impossible if a_22∉ (^×)^2,because there are finite primes p_1p_2such that q__p_1(g_1) ⊄q__p_1(g_2) and q__p_2(g_2) ⊄q__p_2(g_1), and it is possible to choose (s_1, s_2) ∈^2 with the following properties (hence s_10), by using the Chinese remainder theorem: * s_1^2 + a_33 s_2^2 ∈ q__p_1(g_1) ∖ q__p_1(g_2),* a_22 s_1^2 / c^2 + a_33 c^2 s_2^2 ∈ q__p_2(g_2) ∖ q__p_2(g_1).Therefore, a_22∈ (^×)^2 must hold, which is similarly proved in case of a_33 / b_22 b_33∈ (^×)^2. Therefore, the following (A_i δ_1^2 + B_i, A_i δ_2^2 + B_i) ∈ Sym^2 (^3)^* ⊗_^2 (i = 1, 2) have the identical simultaneous representations over . (A_1 δ_1^2 + B_1, A_1 δ_2^2 + B_1) = ( δ_1^2 [ 1-δ_1^-2 - δ_2^-2/4 0;-δ_1^-2 - δ_2^-2/4 (δ_1^-2 - δ_2^-2/4)^2 0; 0 0a_33 ], δ_2^2 [ 1 δ_1^-2 - δ_2^-2/4 0; δ_1^-2 - δ_2^-2/4 (δ_1^-2 - δ_2^-2/4)^2 0; 0 0a_33 ]), (A_2 δ_1^2 + B_2, A_2 δ_2^2 + B_2) = ( [ 1-δ_1^2 - δ_2^2/4 0;-δ_1^2 - δ_2^2/4 (δ_1^2 - δ_2^2/4)^2 0; 0 0a_33 ], [ 1 δ_1^2 - δ_2^2/4 0; δ_1^2 - δ_2^2/4 (δ_1^2 - δ_2^2/4)^2 0; 0 0a_33 ]). Let D be the square free integer in a_33 (^×)^2. By using the action of GL_3(), (A_i δ_1^2 + B_i, A_i δ_2^2 + B_i) (i = 1, 2) are transformed into the following (Ã_i, B̃_i), respectively: (Ã_1, B̃_1):= ( [ 1 0 0; 0 0 0; 0 0 D ], (δ_2 / δ_1)^2 [ 0 0 0; 0 1 0; 0 0 D ]), (Ã_2, B̃_2):= ( [ 1 0 0; 0 0 0; 0 0 D ], [ 0 0 0; 0 1 0; 0 0 D ]).From q_(Ã_1, B̃_1) = q_(Ã_2, B̃_2), for any (q, q) ∈ q_(Ã_i, B̃_i) and k ∈,(q, (δ_1/δ_2)^2k q) ∈ q_(Ã_i, B̃_i) (i = 1, 2) holds.Equivalently, for any 0(u_1, u_3) ∈^2 and k ∈, there exists u_2 ∈ such that u_1^2 + D u_3^2 = (δ_2/δ_1)^2k (u_2^2 + D u_3^2). Hence, some ζ∈(√(-D)) with the norm 1 satisfies ζ (u_1 + √(-D) u_3) = (δ_2/δ_1)^k(u_2 + √(-D) u_3). As a result of Hilbert's theorem 90, some 0(s, t) ∈^2 satisfies ζ = (s + √(-D) t)/(s - √(-D) t). Therefore, { u_3 (s^2 - D t^2) + 2 u_1 s t }/(s^2 + D t^2) = (δ_2/δ_1)^k u_3. Now (δ_2/δ_1)^21, because otherwise a_22 = 0. If η_1, η_2 are coprime integers with δ_2/δ_1 = η_2/η_1, some prime p divides either of η_1, η_2. Hence, if k is chosen so that (δ_2/δ_1)^k is sufficiently close to 0 in _p,Eq.(<ref>) implies that the quadratic form f_u_1, u_3(x_1, x_2) := u_3 (x_1^2 - D x_2^2) + 2 u_1 x_1 x_2 is isotropic over _p. A contradiction is obtained by choosing 0(u_1, u_3) ∈^2 with u_1^2 + D u_3^2 ∉ (_p^×)^2. We can now proceed to the proof of Theorem <ref>. As in the previous section, we denote the quartic and cubic -algebras assigned to (A_i, B_i) by(Q_(A_i, B_i), ⟨ 1, ξ_i,1, ξ_i,2, ξ_i,3⟩) and(R_(A_i, B_i), ⟨ 1, ω_i,1, ω_i,2⟩). We fix (A_i, B_i) ∈ V_ (i =1, 2) as stated in Theorem <ref>. From Proposition <ref>,we have r_1^-1 (A_1 x + B_1 y) = r_2^-1 (A_2 x + B_2 y) for some coprime integers r_1, r_2. In what follows, we denote 4(A_i x - B_i y) / r_i by f̃_det(x, y) = a x^3 + b x^2 y + c x y^2 + d y^3 (a, b, c, d ∈).From Lemma <ref>,some h_1 := (h_1,1, h_1,2, h_1,3) ∈^3 satisfies f̃_det( B_1(h_1), A_1(h_1) )≠ 0. From the assumption (c'),there exists h_2 := (h_2,1, h_2,2, h_2,3) ∈^3 such that (A_1, B_1)(h_1) = (A_2, B_2)(h_2), hence, f̃_det( B_2(h_2), A_2(h_2) ) ≠ 0. Usingthese h_1, h_2 and arbitrarily chosen h_1,0, h_2,0∈, we define α_i := h_i,0 + ∑_j=1^3 h_i,jξ_i,j∈ Q_(A_i, B_i). If we put q_A := A_1(h_1) = A_2(h_2) and q_B := B_1(h_1) = B_2(h_2), then α_ihas the characteristic polynomial ch_α_i(x) := x^4 + a_i,3 x^3 + a_i,2 x^2 + a_i,1 x + a_i,0 as follows:ch_α_i^res(x +a_i,2 + r_i (c q_A - b q_B) /3) =x^3 + r_i (c q_A - b q_B) x^2 + r_i^2 { b d q_A^2 + (3 a d - b c) q_A q_B+ a c q_B^2 } x+ r_i^3 { a d^2 q_A^3 - (b^2 d - 2 a c d) q_A^2 q_B + (a c^2 - 2 a b d) q_A q_B^2 - a^2 d q_B^3 }.Hence, r_1^-3 ch_α_1^res( r_1 x +a_1,2/3) = r_2^-3 ch_α_2^res( r_2 x +a_2,2/3).Let W_1, W_2 and V_i be the rational matrix determined byW_i [ ξ_1; ξ_2; ξ_3 ] = [ 1 0 0; a_i,3 1 0; a_i,2 a_i,3 1 ][ α_i; α_i^2; α_i^3 ], V_i := 1/ r_i f̃_det(q_B, -q_A)[ 1 0; r_i(c A(h) - b B(h)) /3 1 ][ q_B-q_A;-r_i (c q_A q_B + d q_A^2) - r_i (a q_B^2 + b q_A q_B) ]. Since[ r_1 0; 0 1 ] V_1 = [ r_2 0; 0 1 ] V_2, the following (Ã_i, B̃_i) (i = 1, 2) have the identical sets of simultaneous representations overowing to the (c'):(Ã_i, B̃_i):=( W_i,[ r_i 0; 0 1 ] V_i ) (A_i, B_i).From 4(Ã_i x - B̃_i) =ch_α_i^res( r_i x + a_i,2 / 3 ), (Ã_1 x - B̃_1) = (r_1/r_2)^3 (Ã_2 x - B̃_2) is obtained. Even if α_i is replaced by α_i - α_i/4 (Tr is the trace function), all the above hold. The proof of the theorem is completed by the following Proposition <ref>.We assume that r_1, r_2 are coprime integers and the following (A_i, B_i) ∈ V_ (i = 1, 2) satisfythe assumptions (b') and (c') of Theorem <ref> and (A_1 x - B_1) = (r_1/r_2)^3 (A_2 x - B_2): (A_i, B_i) := ( [ 0 0 r_i/2; 0 - r_i 0; r_i/2 0 0 ], - [ 1 0 a_i,2 / 6; 0 2 a_i,2 / 3 a_i,1/2; a_i,2 / 6 a_i,1/2 a_i,0 ]). Then, either of the following holds: * there exist coprime integers u_1, u_2 such that r_i = u_i^2 (i = 1, 2), and(A_1, B_1) = (w, I) · (A_2, B_2)for the following w ∈ GL_3():w = [ 1 0 0; 0 u_1/u_2 0; 0 0 (u_1/u_2)^2 ].* There exist s_1, s_2 ∈ such that (r_2 / r_1) (s_1^4 + a_1,2 s_1^2 + a_1,1 s_1 + a_1,0) = (r_1 / r_2) (s_2^4 + a_2,2 s_2^2 + a_2,1 s_2 + a_2,0) = c^2 for some c ∈^×, and (w, (r_2/r_1) I) · (A_1, B_1) = (A_2, B_2) for the following w ∈ GL_3():w = [ 1 0 0; 2 s_2 1 0; s_2^2 s_2 1 ]^-1[ 0 0 1/c; 0 1 0; c 0 0 ][ 1 0 0; 2 s_1 1 0; s_1^2 s_1 1 ].The above 1. always holds, if Disc(A_i, B_i) = 0 for (either of) i = 1, 2.For the proof, the following lemma is used. Let k be a global field with char k ≠ 2. Assume that f_1(x), f_2(x) ∈ k[x] are monomial quartic polynomialswith no roots in k, no multiple roots in k̅, and k[x] / (f_1^res(x)) and k[x] / (f_2^res(x)) are isomorphic as k-algebras. Furthermore, assume that f_1(x), f_2(x) have a root in k_𝔭 with respect to the same 𝔭∈ P, where P is the set of all the primes of k that completely splits in k[x] / (f_i^res(x)).In this case, k[x] / (f_1(x)) and k[x] / (f_2(x))are isomorphic as k-algebras.It may be assumed that the coefficient of x^3 in f_i(x) equals 0.Fix a prime 𝔭∈ P so that both of f_1(x) and f_2(x) have a root in k_𝔭. Let α_i, 𝔭∈ k_𝔭 (i = 1, 2) be the root. If the roots of g_i, 𝔭(x) := f_i(x) / (x - α_i, 𝔭) ∈ k_𝔭[x] are denoted byβ_i,1, β_i,2, β_i,3∈ k_𝔭, thenα_i, 𝔭β_i,1 + β_i,2β_i,3 =-α_i, 𝔭 (α_i, 𝔭 + β_i,2 + β_i,3) + β_i,2β_i,3=-2 α_i, 𝔭^2 + (α_i, 𝔭 - β_i, 2) (α_i, 𝔭 - β_i, 3) =-2 α_i, 𝔭^2 - f_i^'(α_i, 𝔭) /(β_i, 1 - α_i, 𝔭),wheref_i^'(x) is the first derivative of f_i(x) with respect to x. Thus, there exists 0 ≠ C ∈ k_𝔭 such thatg_i, 𝔭(x) = C (x - α_i, 𝔭)^3 f_i^res( -f_i^'(α_i, 𝔭) / x - α_i, 𝔭 - 2 α_i, 𝔭^2 ).From the assumption about f_1^res(x) and f_2^res(x),𝔭 completely splits in k[x] / (f_1(x)) if and only if it does in k[x] / (f_2(x)).Let f_i(x) = ∏_j=1^m g_ij(x) (g_ij∈ k[x]) be a factorization in k. By assumption, k[x] / (f_i(x)) is a direct sum of k[x]/(g_ij(x)) with degree 2 or 4 over k. For each g_ij(x), wefix an embedding ι_ij : k[x]/(g_ij(x)) ↪k̅ and let K_i be the composite field of ι_ij (k[x]/(g_ij(x))) (j = 1, ⋯, m). K_1, K_2 are quadratic or quartic fields over k, and any prime 𝔭 of k completely splits in K_1 if and only if it does over K_2. Hence, K_1, K_2 have the identical Galois closure over k (Theorem 8.8, <cit.>).If both of K_i are Galois over k, then K_1 = K_2. If either of K_1, K_2 is not Galois over k, both must be a quartic field not Galois over k. Even in this case, K_1 and K_2 are isomorphic over k. This can be seen as follows; first suppose that f_i^res(x) = 0 has a root u_i in k. In this case,f_i(x) := x^4 + a_i, 2 x^2 + a_i, 1 x + a_i, 0 is decomposedas follows:f_i(x) = (x^2 + u_i/2)^2 - (u_i^2 - 4 a_i,0) (a_i, 1/ u_i^2 - 4 a_i,0 x - 1/2 )^2.Even if u_i^2 = 4 a_i,0, a_i,1=0 follows from f_i^res(x) = (x - a_i, 2)(x^2 - 4 a_i, 0) - a_i, 1^2.Therefore K_1, K_2 are quadratic extensions of a quadratic field over k. Since they have the same Galois closure, the quadratic field is common,and K_1, K_2 are conjugate over k. Next, suppose that f_i^res(x) does not have a root in k. In this case, the Galois closure F of K_1, K_2 contains a cubic field isomorphic to k[x]/(f_i^res(x)).Since Gal(F/k) is isomorphic to a subgroup of S_4, this happens only when Gal(F / k) ≅ S_4 or A_4. Since all the subgroups of S_4 (resp. A_4) of index 4are conjugate to S_3 = ⟨ (1, 2, 3), (1, 2) ⟩ (resp. A_3= ⟨ (1, 2, 3) ⟩), K_1, K_2 are conjugate over k. Consequently, if k[x] / (g_ij(x)) = K_i for all 1 ≤ j ≤ m, k[x] / (f_1(x)), k[x] / (f_2(x)) are isomorphic as k-algebras. It remains to verify the case ofK_1 = F_1 · F_2 and K_2 = F_1 ⊕ F_2,where F_1, F_2 are distinct quadratic fields. They correspond to the G_k-orbitscontaining * k(√(d_1), √(d_2)),(A, B) := ( [ -d_100;010;000;][ d_2 0 0; 0 0 0; 0 0-1; ]).* k(√(d_1)) ⊕ k(√(d_2)),(A, B) := ( [ -d_100;0 -d_20;001;][0 -1/20; -1/200;000;]). However, (A x - B y) of the former has only rational roots, whereas that of the latter has irrational roots. Therefore, f_1^res(x) = f_2^res(x) cannot happen. Hence, this case can be eliminated. We first put f_i(x) := x^4 + a_i, 2 x^2 + a_i, 1 x + a_i, 0. It follows from (A_1 x - B_1y) = (r_1/r_2)^3 (A_2 x - B_2 y) that(1/r_1)^2 (4 a_1,0 + a_1,2^2/3)=(1/r_2)^2 (4 a_2,0 + a_2,2^2/3), (1/r_1)^3 (a_1,1^2 - 8 a_1,2 a_1,0/3 + 2 a_1,2^3 / 27)=(1/r_2)^3 (a_2,1^2 - 8 a_2,2 a_2,0/3 + 2 a_2,2^3 / 27).Hence, r_1^-1 a_1,2 = r_2^-1 a_2,2 implies that r_1^-2 a_1,0 = r_2^-2 a_2,0 and r_1^-3 a_1,1^2 = r_2^-3 a_2,1^2. Thus, in this case, case 1. occurs.Therefore, in what follows we assume that r_1^-1 a_1,2≠ r_2^-1 a_2,2.In case of Disc(f_1^res) =Disc(f_2^res) = 0, the multiple root of f_i^res(x) =(A_i x - B_i) for i = 1, 2,equals x = 2 a_1, 2 / 3 r_1 = 2 a_2, 2 / 3 r_2, which is seen by checking when A_i x - B_i in Eq.(<ref>) is rank 1.In particular, r_1^-1 a_1,2 = r_2^-1 a_2,2 follows in this case.Since Disc(f_1) = 0 implies Disc(f_1^res) = 0, we may now assume that both f_i and f_i^res have no multiple roots. By Corollary <ref>,[x] / (f_i(x)) is a direct sum of number fields of degree greater than 1 over . Let p be a finite prime that completely splits in [x] / (f_i^res(x)). (A_i, B_i) is isotropic over _pif and only if _p contains a root of f_i(x) := x^4 + a_i, 2 x^2 + a_i, 1 x + a_i, 0 = 0. Hence,f_1(x) = 0 has a root in _p if and only if f_2(x) = 0 does. By Lemma <ref>, [x] / (f_1(x)) and [x] / (f_2(x)) are isomorphic as -algebras. Thus, there exists x_2 ∈[x] / (f_1(x)) such that x ↦ x_2 provides an isomorphism [x] / (f_2(x)) →[x] / (f_1(x)).Now A_1B_1 is a -algebra isomorphic to [x] / (f_1(x)) with the basis ⟨ 1, r_1 x, r_1 (x^2 + a_1,2), r_1 (x^3 + a_1,2 x + a_1,1) ⟩. We assume that x_2 is represented as h_0 + r_1 { h_1 x + h_2 x^2 + h_3 (x^3 + a_1,2 x) } in A_1B_1,using some h_0 ∈ and h := (h_1, h_2, h_3) ∈^3. If we put b_0 = -(1/r_i)^3 (a_i,1^2 - 8 a_i,2 a_i,0/3 + 2 a_i,2^3 / 27) and b_1 = - (1/r_i)^2 (4 a_i,0 + a_i,2^2/3),then f̃_det(x, y) := 4 r_i^-3 (A_i x - B_i y) = x^3 + b_1 x y^2 + b_0 y.In this case, by using the formulas given in the proof of Lemma <ref>, (W, V) ∈ G_ satisfying (W, V) · (A_1, B_1) = (A_2, B_2) is obtained as follows (herein, x̅, x̅_̅2̅ are the classes of x, x_2 in A_1B_1 /):W [r_1 x;r_1 x^2; r_1(x^3 + a_1,2 x) ] = [ x_2; x_2^2; x_2^3 + a_2,2 x_2 ], V :=( W)^-1[ r_2 0; r_1^3 b_1 A_1(h) /3 1 ][B_1(h) -A_1(h); -r_1^3 ( b_1 A_1(h) B_1(h) + b_0 A_1(h)^2 ) -r_1^3 B_1(h)^2 ]=( W)^-1[r_2 B_1(h)- r_2 A_1(h); -r_1^3 ( 2/3 b_1 A_1(h) B_1(h) + b_0 A_1(h)^2 )- r_1^3 ( 1/3 b_1 A_1(h)^2 + B_1(h)^2) ].These matrices have the determinants W = r_1^3 ( B_1(h)^3 + b_1 A_1(h)^2 B_1(h) + b_0 A_1(h)^3 ) and V = -r_2(W)^-1.From q_(A_1, B_1) = q_(A_2, B_2), q_(A_1, B_1) = q_((I, V) · (A_1, B_1)) is obtained. In addition, V satisfies:( W)^2 f̃_det((x, y)Ṽ) = (r_2/r_1)^3 f̃_det(x, y), Ṽ := [10;0 -1;] V [10;0 -1;].By Lemma <ref>,Ṽ^n = u^n I holds for u :=(r_2/r_1)^3 / ( VW^2) = r_2V/r_1^3 and either of n = 1, 2, 3. If n1, by Lemma <ref>, there exists W_2 ∈ GL_3() such that(W_2, -u^-1 V) · (A_1, B_1) = (A_1, B_1). Thus, we may assume that V is a scalar multiple of I, by replacing (W, V) with (W, V) (W_2, -u^-1 V),and the above x_2 by another element of [x]/(f_1(x)) with the characteristic polynomial f_2(x).Thus, A_1(h) = 0, B_1(h) = - r_1^-3 r_2, W = - r_1^-6 r_2^3,and V = r_1^3 r_2^-1 I. Therefore,(r_1 W, (r_1/ r_2) I) also maps (A_1, B_1) to (A_2, B_2).Since A_1(h) = 0,there are 0 ≠ C ∈ and 0 ≠ (s, t) ∈ such that h = C (s^2, s t, t^2). If we put f_i(X, Y) := Y^4 f_i(X/Y) (i = 1, 2), then C^2 f_1(s, t) = r_1^-3 r_2 follows from B_1(h) = - r_1^-3 r_2. Since x, x_2 satisfies x^2 = - 2 a_1,2, x^3 = - 3 a_1,1 and x = x_2 = 0 in [x] / (f_1(x)), we obtainx_2 = C r_1 ( s^2 x + s t (x^2 + a_1, 2/2) + t^2 (x^3 + a_1,2 x + 3 a_1,1/4)). Hence, (t x - s){ t x_2 + C r_1 (s^3 + a_1,2 s t^2/2 + a_1,1 t^3 / 4) }=C r_1 (t x - s) { s^3 + s^2 t x + s t^2 (x^2 + a_1, 2) + t^3 (x^3 + a_1,2 x + a_1,1) }=-C r_1 { s^4 - t^4 x^4 + a_1, 2 t^2 (s^2 - t^2 x^2) + a_1,1 t^3 (s - t x) }=-C r_1 f_1(s, t).If we put C̃ := -C r_1 f_1(s, t),C̃^2 = (r_2 / r_1) f_1(s, t) follows from C^2 f_1(s, t) = r_1^-3 r_2. Hence, if t = 0, then r_1 / r_2 ∈ (^×)^2, and x_2 = c x holds for c := C r_1 s^2, which satisfies c^2 = C^2 r_1^2 s^4 = r_2/r_1. As a result, a_2, 2 = c^2 a_1, 2, a_2, 1 = c^3 a_1, 1 and a_2, 0 = c^4 a_1, 0 are obtained, which implies the case 1.If t ≠ 0,the characteristic polynomials of x̃ := t x - s, x̃_2 := C̃ (t x - s)^-1are as follows:ch_x̃(X) =f_1(X + s, t) = X^4 + 4 s X^3 + (6 s^2 + a_1, 2 t^2) X^2 + (4 s^3 + 2 a_1, 2 s t^2 + a_1, 1 t^3) X + f_1(s, t), ch_x̃_2(X) =f_1(s, t)^-1 X^4ch_x̃(C̃ / X) =f_1(s, t)^-1{C̃^4+ 4 C̃^3 s X + C̃^2 (6 s^2 + a_1, 2 t^2) X^2 + C̃ (4 s^3 + 2 a_1, 2 s t^2 + a_1, 1 t^3) X^3 + f_1(s, t) X^4 }. Therefore, in this case,Eq.(<ref>) is obtainedby putting s_1 = s, s_2 = C̃ (s^3 + a_1,2 s t^2 / 2 + a_1,1 t^3 / 4) / f_1(s, t) and c = C̃. Thus, the proposition is proved. § CASE OF DISC(A_I, B_I) = 0 (PROOFS OF PROPOSITIONS <REF>, <REF>)From the known result in the binary case, it is immediately obtained thatq_(x_1^2 - x_1 x_2 + x_2^2, x_3^2) = q_(x_1^2 + 3x_2^2, x_3^2). In the case of (ii), if we put (A_1, B_1) = (x_1^2 - x_1 x_2 + x_2^2, (x_1 + x_2 + 3 x_3)^2) and (A_2, B_2) = (x_1^2 + 3x_2^2, (x_1 + 3 x_3)^2), we have(A_2, B_2)(x_1, x_2, x_3)=(A_1, B_1)(x_1 + x_2, 2 x_2, -x_2 + x_3) =(A_1, B_1)(2 x_1, x_1 + x_2, -x_1 + x_3) =(A_1, B_1)(x_1 + x_2, x_1-x_2,-x_1 - x_3). Hence, q_(A_2, B_2) ⊂ q_(A_1, B_1),and the converse is also true, since any (y_1, y_2) ∈^2can be represented in either of the ways (x_1 + x_2, 2 x_2), (2 x_1, x_1 + x_2) or (x_1 + x_2, x_1-x_2) for some x_1, x_2 ∈.Proposition <ref> can be also proved in an elementary way. As proved in Proposition <ref>, q_(A_1, B_1) = q_(A_2, B_2)implies that all the roots of (A_1 x - B_1 y) and (A_2 x - B_2 y) are common. In particular, Disc(A_1, B_1) = 0 leads to Disc(A_2, B_2) = 0.By the action of G_, (A_1, B_1) can be transformed into:(A_1, B_1) = (M, 1) ·( [ a_11 a_120; a_12 a_220;000 ], [ 0 0 0; 0 0 0; 0 0 1 ]),M :=[ 1 0 m_1; 0 1 m_2; 0 0 m_3 ],where m_1, m_2 and 0 ≠ m_3 ∈ may be assumed to have the greatest common divisor 1.By comparing the representations (0, *) of (A_1, B_1) and (A_2, B_2), it is seen that (A_2,B_2) can be simultaneously transformed into:(A_2, B_2) = (M̃, 1) ·( [ ã_11 ã_120; ã_12 ã_220;000 ], [ 0 0 0; 0 0 0; 0 0 1 ]),M̃ :=[10 m̃_1;01 m̃_2;00m_3 ].The greatest common divisor of m̃_1, m̃_2, 0 ≠ m_3 ∈ is 1, owing to q_(B_1) = q_(B_2). Furthermore,either of the following may be assumed owing to q_(A_1) = q_(A_2): * [ a_11 a_12; a_12 a_22 ] = [1 -1/2; -1/21 ],[ ã_11 ã_12; ã_12 ã_22 ] = [ 1 0; 0 3 ]. * à := [ a_11 a_12; a_12 a_22 ] = [ ã_11 ã_12; ã_12 ã_22 ],0 < a_11≤ a_22, 0 ≤ - 2 a_12≤ a_11, à is reduced. For any m ≠ 0 and n ∈,the class of n in / m is denoted by nm. In case (I),the following is proved byconsidering the representations (A_i(h), B_i(h)) = (1, *), (3, *):{±m_1m_3, ±m_2m_3, ±m_1+m_2m_3} = {±m̃_1m_3},{±m_1 - m_2m_3, ±m_1 + 2 m_2m_3, ±2 m_1 + m_2m_3} = {±m̃_2m_3}.The above can hold only when either of the following holds: * m_1m_3 = m_2m_3 = m̃_̃1̃m_3 = m̃_̃2̃m_3 = 0m_3 or * m_3 = 3, m_13 = m_23 = ±m̃_̃1̃3, m̃_̃2̃3 = 03.We note that each corresponds to the cases (i), (ii), respectively. Thus, the theorem is proved in case (I). In case (II),m_1m_3 = m̃_1m_3, m_2m_3 = ±m̃_2m_3 may be assumed, by changing the basis of ^3 without losing the property (II).In what follows, we assume that m_1m_3≠ -m_1m_3 and m_2m_3 = -m̃_2m_3≠ -m_2m_3, because otherwise, (A_1, B_1) and (A_2, B_2) are equivalent by the action of GL_3(). Furthermore,the multiplicity of the representation a_11 + a_22 + 2a_12 of à must be greater than 1, otherwise we would have m_1 + m_2m_3 = m̃_1 + m̃_2m_3,and m_2m_3 = m̃_2m_3. This can happen only if a_12 = 0 or a_12 = -a_11/2. If a_12 = 0, then (A_1, B_1) and (A_2, B_2) are equivalent by the action of GL_3(). If a_12 = -a_11/2,then we see from the representations (A_i(h), B_i(h)) = (a_11 + a_22 + 2a_12, *) that either of the following is required: * {±m_2m_3, ±m_1 + m_2m_3} = {±m_2m_3, ±m_1 - m_2m_3}. * a_11 = a_22 = - 2 a_12 and {±m_1m_3, ±m_2m_3, ±m_1 + m_2m_3} = {±m_1m_3, ±m_2m_3, ±m_1 - m_2m_3}.If m_1 + m_2m_3 = ±m_1 - m_2m_3, thenm_1m_3 = -m_1m_3 or m_2m_3 = -m_2m_3 must hold, which contradicts with the above assumption. However, in the former of the above cases, m_1 + m_2m_3 = ±m_1 - m_2m_3 always holds. In the latter, m_1 + m_2m_3≠±m_1 - m_2m_3 implies that m_1 + m_2m_3 = ±m_im_3 and m_1 - m_2m_3 = ±m_jm_3 hold for either (i, j) = (1, 2) or (i,j)=(2, 1). From m_1m_3≠ -m_1m_3 and m_2m_3≠ -m_2m_3, the following is obtained:(m_1 + m_2m_3, m_1 - m_2m_3) = (-m_1m_3, m_2m_3)or(-m_2m_3, -m_1m_3). Comparing the representation (a_11 + a_22 - 2 a_12, *), we have{±m_1 - m_2m_3, ±m_1 + 2 m_2m_3, ±2 m_1 + m_2m_3}= {±m_1 + m_2m_3, ±m_1 - 2 m_2m_3, ±2 m_1 - m_2m_3}. If (m_1 + m_2m_3, m_1 - m_2m_3) = (-m_1m_3, m_2m_3),the following is obtained fromm_1m_3 = 2 m_2m_3 and m_2m_3 = -2 m_1m_3:{±m_2m_3, 0m_3} = {±m_1m_3, 0m_3}. Even if (m_1 + m_2m_3, m_1 - m_2m_3) = (-m_2m_3, -m_1m_3),Eq.(<ref>) is obtained fromm_1m_3 = -2 m_2m_3 and m_2m_3 = 2 m_1m_3. Eq.(<ref>) implies m_1m_3 = ±m_2m_3, hence this case is impossible. Proposition <ref> is proved in the remaining part. Since the “only if” part is clear, we shall prove the “if” part; for any positive-definite f_1, f_2 ∈ Sym^2 (^3)^* with q_(f_1) = q_(f_2), take λ_1, …, λ_s∈ linearly independent overandpositive-definite A_1j, A_2j∈ Sym^2 (^3)^* (1 ≤ j ≤ s) satisfying f_1 = ∑_j=1^s λ_j A_1j, f_2 = ∑_j=1^s λ_j A_2jas in Lemma <ref>. Furthermore,In this case, q_(A_11, …, A_1s) = q_(A_21, …, A_2s) holds. Therefore, the proposition is obtained by proving the following (**): (**) If both of (A_i1, A_i2, …, A_is) ∈ Sym^2 (^3)^* ⊗_^s (i = 1, 2) satisfy all of * A_i1, …, A_is span a linear space of dimension more than 1 over ,* ∑_j = 1^s c_j A_ij is positive definite for some c_1, …, c_s ∈,* q_(A_11, …, A_1s) = q_(A_21, …, A_2s),there exists w ∈ GL_3() such that (A_11, …, A_1s) = (w, 1) · (A_21, …, A_2s), or there exist w_1, w_2 ∈ GL_3() and v ∈ GL_s() such that { (A_11, …, A_1s), (A_21, …, A_2s) } equals either of the following: * { (w_1, v) · ( 0, ⋯, 0, x_1^2 - x_1 x_2 + x_2^2, x_3^2 ), (w_2, v) · ( 0, ⋯, 0, x_1^2 + 3 x_2^2, x_3^2 ) }, * { (w_1, v) · ( 0, ⋯, 0, x_1^2 - x_1 x_2 + x_2^2,(x_1 + x_2 + 3 x_3)^2 ), (w_2, v) · ( 0, ⋯, 0, x_1^2 + 3 x_2^2, (x_1 + 3 x_3)^2 ) },where 0 is the ternary quadratic form that maps all the 𝐱∈^3 to 0, and GL_3() × GL_s() ∋ (w, v) acts on Sym^2 (^3)^* ⊗_^s by (w, (v_ij) ) · (A_1 … A_s) =( ∑_j=1^s v_1j A_j(𝐱 w), ∑_j=1^s v_2j A_j(𝐱 w), …, ∑_j=1^s v_sj A_j(𝐱 w) ).The case of s = 2 is equivalent to the assumption (*). In order to prove (**) by induction, we assume (**) is true for s = 2, …, nfor some n ≥ 2. If A_11, …, A_1 n+1 span a linear space of dimension m < n+1, by using the action of { 1 }× GL_n + 1(), (A_11, …, A_1 n+1) is mapped to (Ã_1, …, Ã_m, 0 …, 0) for some Ã_j ∈ Sym^2 (^3)^*. From q_(A_11, …, A_1s) = q_(A_21, …, A_2s), (A_21, …, A_2s) is simultaneously mapped to (B̃_1, …, B̃_m, 0 …, 0) for some B̃_j ∈ Sym^2 (^3)^*. The proposition is proved by induction in this case. Thus we assume the dimension of the space spanned by A_11, …, A_1 n+1 is exactly n+1. By using the action of GL_3() × GL_n+1(), either of the following may be assumed: * A_1j = A_2j (1 ≤ j ≤ n) and ∑_j=1^n c_j A_1j≻ 0 for some c_j ∈, * n = 2 and (A_11, A_12) = (x_1^2 - x_1 x_2 + x_2^2, x_3^2), (A_21, A_22) = (x_1^2 + 3 x_2^2, x_3^2), * n = 2 and (A_11, A_12) = (x_1^2 - x_1 x_2 + x_2^2, (x_1 + x_2 + 3 x_3)^2), (A_21, A_22) = (x_1^2 + 3 x_2^2, (x_1 + 3 x_3)^2).In the case (I),define H := { g ∈ GL_3() : A_ij = g · A_ij for all1 ≤ j ≤ n }.For any c ∈,owing toq_(A_11, …, A_1 n-1, A_1 n + c A_1 n+1) = q_(A_21, …, A_2 n-1, A_2 n + c A_2 n+1) and the assumption of induction,there existsh ∈ H such that (h, 1) · (A_11, …, A_1 n-1, A_1 n + c A_1 n+1) = (A_21, …, A_2 n-1, A_2 n + c A_2 n+1). Since H is a finite group, some h ∈ H satisfies h · (A_1 n + c_i A_1 n+1)= A_2 n + c_i A_2 n+1 for some distinct c_1 ≠ c_2.Hence h · A_1j = A_2j holds for all 1 ≤ j ≤ n + 1.In the cases (II) and (III), owing to A_11≁A_21, Disc(A_11, A_12 + d A_13) =Disc(A_21, A_22 + d A_23) = 0is required for any d ∈. Therefore A_i1, A_i2, A_i3 are linearly dependent overfor each i = 1, 2. Hence, (**) is true in any case.Acknowledgments The author would like to extend her gratitude to Professor T. Kamiyama of KEK.He gave her an opportunity to apply the theory of quadratic forms to a real scientific problem. She also appreciatesProfessor A. Earnest of Southern Illinois University for sending her his lecture slides containing some references and a copy of Kaplansky's letter to Schiemann, Professor T. Oda of the university of Tokyo, Dr. S. Harashita of Yokohama National University and Dr. K. Gunji ofChiba Institute of Technology for discussion during a regular seminar. For various checks of our program and computation of genera and spinor genera, we used the algebra system Magma <cit.>.§ THE ALGORITHM TO OBTAIN ALL THE POSITIVE-DEFINITE QUADRATIC FORMS WITH A GIVEN SET OF REPRESENTATIONSIn this section, the method to obtain all the positive-definite quadratic forms with the identical set of representations overis explained.For such a quadratic form f and a given M > 0,all the elements q_1, …, q_t of q_(f) less than M can be computed <cit.>. Therefore, the problem is reduced to enumeration of all the quadratic forms with the property q_(f) ∩ [0, M] = { q_1, …, q_t }. The methoddescribed in Table <ref> can be applied to any N-ary positive-definite quadratic forms as long as N ≤ 4, which equals the theoretical optimum, because infinitely many solutions may exist if N > 4.In what follows, a set of vectors v_1, ⋯, v_n of ^Nis said to be primitive if it is a subset of some basis of ^N. An N-ary quadratic form S is Minkowski-reduced if the following holds for any 1 ≤ n ≤ N:S (𝐞_n) = min{ S(v) : v ∈^Nsuch that ⟨𝐞_1, …, 𝐞_n-1, v ⟩is a primitive set of ^N },where 𝐞_n is a vector with 1 in its n-th component and 0 in the remaining components. It is known that if N ≤ 4, then S = (s_ij) is Minkowski-reduced if and only if the following inequalities hold (Lemma 1.2 of chap. 12, Cassels (1978)Cassels78):0 < s_11≤⋯≤ s_NN, s_jj≤ S(v)for any1 ≤ j ≤ Nand vectorsvwith the entries v_i = -1, 0, 1 (1 ≤ i < j),v_j = 1, v_k = 0 (j < k ≤ N). The following is frequently added to the definition of Minkowski reduction:s_i i+1≤ 0 (1 ≤ i < N).In what follows, Eq.(<ref>), (<ref>) are adopted as the inequalities of the Minkowski reduction for N ≤ 4.Table <ref> presents a recursive procedure for generating all candidates of N-ary quadratic forms from a sorted set Λ := ⟨ q_1, ⋯, q_t ⟩⊂_> 0. If the recursive procedure begins with the arguments m=n=1, q_min = q_max = q_1, all positive-definite quadratic forms satisfying the following in addition to Eq.(<ref>), (<ref>) are enumerated in the output array Ans:anyq ∈Λ withq ≤ s_NN belongs toq_(S), s_nn, s_mm + s_nn + 2 s_mn∈Λ for any1 ≤ m, n ≤ N. After the execution of the algorithm, it is possible to reduce the number of candidate solutions in Ansby * checking if Λ = q_(S) ∩ [0, q_t] holds,* removing either S or S_2 from Ans, if they are equivalent over . As a consequence, if q_t is sufficiently large, all the N-ary positive-definite quadratic form S that satisfy Λ = q_(S) ∩ [0, q_t] are contained in the output array. If we set Λ := q_(S_0) ∩ [0, q_t] for some S_0,all the N-ary quadratic forms S with q_(S) = q_(S_0) are in the output. After the execution of the algorithm, it is possible to determine whether or not q_t is large enough to obtain all such S, just by checking if t_2 < t in line 13 and p_max≤ q_t in line 17 hold. If both are true, then all the quadratic forms S satisfying Λ = q_(S) ∩ [0, q_t] are contained in Ans. The algorithm is completed in a finite number of steps, even if q_(S_0) of some S_0 is used instead of the finite set Λ (in fact, this can be programmed by adding elements of q_(S_0) in Λ when they are necessary). Even though t = ∞ in such cases, t_2 in line 13 of Table <ref> is always finite,as a consequence of the following proposition: If S, S_2 are positive-definite quadratic forms overof degree N and N_2, respectively,with1 ≤ N_2 < min{ 4, N }, thenq_(S) ⊄q_(S_2), hence q_(S) ⊄q_(S_2).It may be assumed that S and S_2 have rational entries, since due to Lemma <ref>, they are simultaneously represented as finite sums S = ∑_j=1^2 λ_j T_j, S_2 = ∑_j=1^s λ_j T_2j,where λ_1, …, λ_s ∈ are linearly independent overand every T_j, T_2j is rational and positive-definite. In the rational case, the proposition follows from Lemma <ref>.Let S_i (1 ≤ i ≤ m) be N_i-ary positive-definite quadratic forms over . There are λ_1, …, λ_s∈ linearly independent overand N_i-ary positive-definite quadratic forms T_ij over(1 ≤ i ≤ m, 1 ≤ j ≤ s) such that every S_i is represented as a finite sum S_i = ∑_j=1^s λ_j T_ij. If the condition that T_ij is positive-definite is removed, it is clear that such λ_j and T_ij exist. Since ∑_j=1^s λ_j T_ij is positive-definite for all of i = 1, …, m, there exists ϵ > 0 such that∑_j=1^s c_j T_ij satisfies the same property for any (c_1, …, c_s) ∈^s with c_j - λ_j< ϵ (j = 1, …, s). Hence, if we choose U = (u_j_1 j_2) ∈ GL_s() so that u_j_1 j_2 - λ_j_2 < ϵ holdsfor all j_1, j_2 = 1, …, s, the new (λ_1, …, λ_s) and T_ij replaced by(λ_1, …, λ_s) U^-1 and ∑_j_2=1^s u_j j_2 T_i j_2,satisfy all the required properties.For any N-ary quadratic form S over a field k, S is singular, if the determinant of the corresponding symmetric matrix is 0. S is isotropic over k, if 0 ∈ q_k(S), Otherwise S is anisotropic over k. For any positive integers N, N_2 with 1 ≤ N_2 < min{ 4, N }, we assume that an N-ary rational quadratic form S is non-singular and an N_2-ary rational quadratic form S_2 is anisotropic over . Then, q_(S) ⊄q_(S_2). We may assume that N_2 + 1 = N = 4, as the other cases easily follow from this.Since S is not singular, it satisfies q__p(S) ⊃_p^× for any p ≠∞. In addition, there exists a finite prime p such that q__p(S_2) ⊅_p^× (Corollary 2 of Theorem 4.1 in Chapter 6, Cassels (1978)). If q_(S) ⊂ q_(S_2),then q__p(S) ⊂ q__p(S_2) is required for any p. This is a contradiction. In line 13 of Table <ref>, s_n+1 n+1≤ q_t_2 is assumed, which is proved as follows; with regard to the symmetric matrix T defined in line 12, q_t_2∈Λ does not belong to q_(T). Hence, if S is an extension of T searched for, there exists v ∈^N such that 𝐞_1, …, 𝐞_n, v are linearly independent and S(v) = q_t_2. Let us recall that the n-th successive minimum λ_n of S is defined as followsλ_n := min{max{ S(v_i) : 1 ≤ i ≤ n } : v_1, …, v_n ∈^Nare linearly independent over }.If 1 ≤ n ≤ N ≤ 4,the above v_1, …, v_n can be chosen from a primitive set of ^N (Wan der Waerden (1956)Waerden56).Therefore, s_n+1 n+1≤ q_t_2 holds in line 13. plain | http://arxiv.org/abs/1703.08854v11 | {
"authors": [
"Ryoko Oishi-Tomiyasu"
],
"categories": [
"math.NT"
],
"primary_category": "math.NT",
"published": "20170326171941",
"title": "On ternary positive-definite quadratic forms with the same representations over Z"
} |
[ GPU Activity Prediction using Representation LearningAswin Raghavan, Mohamed Amer, Timothy Shields, David Zhang, Sek [email protected] SRI International, 201 Washington Rd. Princeton, NJ08540Representation learning; Time-Series analysis; Cognitive Architecture0.3in ] GPU activity prediction is an important and complex problem. This is due to the high level of contention among thousands of parallel threads. This problem was mostly addressed using heuristics. We propose a representation learning approach to address this problem. We model any performance metric as a temporal function of the executed instructions with the intuition that the flow of instructions can be identified as distinct activities of the code. Our experiments show high accuracy and non-trivial predictive power of representation learning on a benchmark. § INTRODUCTION The performance of a computing system relies on the sustained operational throughput. Sustained operation is becoming harder to achieve as computation workloads become more complex. At the same time, with the end of Dennard scaling <cit.>, and the increasing abundance of Big Data, it is imperative to minimize wasted processor effort in order to achieve processor reliability and scalability <cit.>.The goal of this paper is to demonstrate the efficacy of machine learning to designing computing systems. We anticipate that classical problems such as branch predictions and cache management can be re-evaluated such that heuristically based approaches <cit.> can be replaced with a machine learning approach <cit.>. It is well understood in the computer architecture community that processor behavior is highly complex and data dependent. Processor data is widely available in the form of benchmarks <cit.>, and algorithms are extensively compared using these benchmarks <cit.>.We choose a well-understood and well-defined problem of predicting GPU Cache Misses <cit.> for this paper. General Purpose GPU (GPGPU) achieve high throughput execution via a high level of parallelism. Predicting GPU Cache Misses is complex due to the high level of contention among thousands of threads. Cache contention is a bottleneck for parallel execution when many threads are waiting for cache operation, causing the addition of more threads (or cores) to be detrimental. Predicting whether a cache miss is about to occur is useful for better cache management such as cache bypassing <cit.>, pre-fetching <cit.>, prioritized allocation <cit.> etc. Further, cache misses indirectly cause increased energy and power usage <cit.> because of second order effects beyond memory latency. In principle, our approach is amenable to predict these higher order events (such as voltage scaling <cit.> and faults <cit.>) either directly <cit.> or via hierarchical modeling. We propose a new model that can predict key processor events that limit processor throughput.We propose a new variant of the Conditional Restricted Boltzmann Machines (CRBMs) <cit.> to directly address system performance and reliability.CRBMs efficiently model short-term temporal phenomenon. Prior work used a perceptron to predict cache misses <cit.>. Unlike their approach, our model accounts for time-series and count data. Our approach assumes the availability of a simulator for CUDA <cit.> that can generate a dataset for training our model. In principle, this approach can be used in real-time by incrementally augmenting the dataset. Multiple repeated executions can even lead to increased predictive power because more data is available for machine learning.Our predictor is naturally agnostic to the hardware and architecture as it relies on execution traces. Our contributions: * Prediction of processor events as temporally-extended activities in a stream of instructions. * Using Discriminative Conditional Restrictive Boltzmann Machines (DCRBM) to learn processor states. § LITERATURE REVIEWCache Miss Prediction: There is a large body of research on branch prediction to improve cache performance. Simple static solutions can achieve 80% correct prediction by analyzing control-flow and static heuristics <cit.>. Dynamic solutions <cit.> are more complicated as they are implemented with counters and tables to store branch history based on branch memory address. Other approaches that are data-driven use perceptrons <cit.> and feed-forward neural networks <cit.>. Representation Learning: Restricted Boltzmann Machines (RBMs) form the building blocks in energy based deep networks <cit.>. Recently, temporal models based on deep networks have been proposed, capable of modeling a more temporally rich set of problems. These include Conditional RBMs (CRBMs) <cit.> and Temporal RBMs (TRBMs) <cit.>. CRBMs have been used in both visual <cit.> and audio <cit.>. In addition to efficiently modeling time-series data, RBMs were formulated to be trained discriminatively for classification <cit.>, and model word-count vectors from a large set of documents <cit.>.§ MODEL The input to our model (called visible units) is an instruction mix per time step, ie. the histogram of counts of instructions being excuted, obtained from the GPU simulator. The labels are any chosen performance metric also output by the simulator.We discuss a sequence of models, gradually increasing in complexity, so that the different components of our model can be understood in isolation. We start with the basic CRBM model, then we extend to the discriminative DCRBM, and finally CountDCRBM. Conditional Restricted Boltzmann Machines: CRBMs <cit.>, are a natural extension of RBMs for modeling short term temporal dependencies. A CRBM is an RBM which takes into account history from the previous time instances t-N,,t-1 at time t. This is done by treating the previous time instances as additional inputs. Doing so does not complicate inference. v is a vector of visible nodes, h is a vector of hidden nodes, and v_<t is the visible vectors from the previous N time instances, which influences the current visible and hidden vectors. E_C is the energy function, and Z is the partition function. The parameters θ to be learned are a and b the biases for v and h respectively and the weights W. A and B are matrices of concatenated vectors of previous time instances of a and b. The CRBM is fully connected between layers, with no lateral connections.This architecture implies that v and h are factorial given one of the two vectors.This allows for the exact computation of p_C( v| h, v_<t) and p_R( h| v, v_<t).Some approximations have been made to facilitate efficient training and inference, more details are available in <cit.>. A CRBM defines a probability distribution p_C as a Gibbs distribution (<ref>).[ p_C( v_t, h_t| v_<t)=exp[-E_C( v_t, h_t| v_<t)]/Z(θ). ] The energy function E_C( v_t, h_t| v_<t) in (<ref>) is defined in a manner similar to that of the RBM.0.9![ E_C-Real( v_t, h_t| v_<t)= -∑_i (c_i-v_i,t)^2/2 - ∑_j d_j h_j,t;- ∑_i,j v_i,t w_i,j h_j,t,; E_C-Binary( v_t, h_t| v_<t)=-∑_i c_iv_i,t - ∑_j d_j h_j,t; -∑_i,j v_i,t w_i,j h_j,t,;E_C-Count( v_t, h_t| v_<t)=-∑_i (c_iv_i,t - log(v_i,t!));- ∑_j d_j h_j,t - ∑_i,j v_i,t w_i,j h_j,t, ] The probability distributions for the visible nodes are defined in (<ref>),0.9![p_C-Real(v_i,t| h_t, v_<t) = 𝒩(c_i+ ∑_j h_j,tw_i,j,1),;p_C-Binary(v_i,t = 1| h_t, v_<t) = σ(c_i+ ∑_j h_j,tw_i,j),; p_C-Count(v_i,t| h_t, v_<t) = 𝒫(m, exp(c_i + ∑_j h_jw_ij)), ] where, 𝒩 is a normal distribution, σ is a sigmoid distribution, and 𝒫 is a Poisson distribution. The hidden nodes is defined in (<ref>), p_C(h_j,t = 1 | v_t, v_<t) = σ(d_j + ∑_i v_i,t w_i,j).c_i = a_i + ∑_pA_p,i v_p,<t,d_j = b_j + ∑_pB_p,j v_p,<t. Discriminative CRBMs: DCRBMs are based on the model in <cit.>, generalized to account for temporal phenomenon using CRBMs. DCRBMs are a simpler version of the Factored Conditional Restricted Boltzmann Machines <cit.> and Gated Restricted Boltzmann Machines <cit.>. Both these models incorporate labels in learning representations, however, they use a more complicated potential which involves three way connections into factors. DCRBMs define the probability distribution p_DC as a Gibbs distribution (<ref>).0.9!p_DC( y_t, v_t, h_t| v_<t;θ)=1/Z(θ)exp[-E_DC(y_t, v_t, h_t| v_<t)] The hidden layer h is defined as a function of the labels y and the visible nodes v. A new probability distribution for the classifier is defined to relate the label y to the hidden nodes h as in (<ref>),as well as relate h to y as in (<ref>). The new energy function E_DC is shown in (<ref>). 0.9!p_DC(h_j,t = 1 |y_t, v_t, v_<t)= σ(d_j+ u_j,k + ∑_i v_i,t w_ij), p_DC(y_l,t| h_t)=exp[s_l+∑_j u_j,lh_j,t]/∑_l^*exp[s_l^*+∑_j u_j,l^*h_j,t] 0.9!E_DC( y_t, v_t, h_t| v_<t)= E_C( v_t, h_t| v_<t)_Generative - ∑_j,l h_j,t u_jl y_l,t- ∑_l s_l y_l,t_Discriminative Count-DCRBMs: We extend the DCRBM to CountDCRBM Figure <ref>. Count-DCRBMs are based on the model in <cit.>, generalized to account for temporal phenomenon using CRBMs, and discriminative classification. Count-DCRBMs are used to model time varying histograms of counts. The probability distribution over the visible layer will follow a constrained Poisson distribution, p_C-Count(v_i,t| h_t, v_<t) defined in (<ref>), the hidden layer follows (<ref>) and the label layer follows (<ref>) and the energy function E_C-Count( v_t, h_t| v_<t) defined in (<ref>).§ INFERENCE AND LEARNING Inference: to perform classification at time t in the CountDCRBM given v_<t and v_t we use a bottom-up approach, computing a cost for each possible label y_t then choosing the label with least cost. We compute the cost for label y_t to be the free energy -log p_DC( y_t, v_t| v_<t) computed by marginalizing over h_<t and h_t. Then, the cost associated with the candidate label is the free energy in the CountDCRBM, namely -log p_DC( y_t, h_t| h_<t) is tractable, because the sum over exponentially many terms can be algebraically eliminated.Learning: the parameters our model could be learned using Contrastive Divergence (CD) <cit.>, where ⟨·⟩ _data is the expectation with respect to the data distribution and ⟨·⟩ _recon is the expectation with respect to the reconstructed data. The learning is done using two steps a bottom-up pass and a top-down pass using sampling equations from (<ref>), (<ref>), and (<ref>). Bottom-up: the reconstruction is generated by first sampling the hidden layer p(h_t,j=1| v_t, v_<t,y_l) for all the hidden nodes in parallel. Top-down: This is followed by sampling the visible nodes p(v_i,t| h_t, v_<t) and p(y_l,t| h_t,h_<t) for all the visible nodes in parallel.§ EXPERIMENTS We used the open-source simulator GPGPU-Sim <cit.> to generate data to validate our approach.The simulator has been verified rigorously for accuracy against on a suite of 80 microbenchmarks <cit.>.We used the BACKProp problem from the RODINIA benchmark <cit.>, and simulate a NVIDIA GTX480 GPU with the default configurations for GPGPU-Sim. This benchmark CUDAprogram trains a feedforward neural network with one hidden layer consisting of 4096 units. To generate our dataset, we modified GPGPU-Sim to retrieve the time-indexed list of instruction mix, ie. for each time cycle the number of different instruction types based on opcode. These are the visible units in our model.We tested our approach on three different caches (Instruction (IC), Data Read (D_R), Data Write (D_W))localized within one core of the GPU.For each cache, GPGPU-Sim outputs a list of time-indexed binary labels corresponding to whether a cache miss occured.Since we want to predict a cache miss ahead of time, we aggregated the labels over 128 cycles so a label of y(t)=1 means that a cache miss occurred in cycles [t,t+128].The Count-DCRBM was trained on a Tesla K20C GPU using Contrastive Divergence with a constant learning rate of 10^-5. Table <ref> shows the final accuracies of a model with 15 hidden nodes and varying temporal history available for DCRBM. The second and third columns are metrics that describe predictive power, taking into account false positives and negatives. We observe high accuracy and predictive power of the model for all three caches. We also observe that increased history generally leads to better performance despite the increased model complexity. Our baseline is an SVM that uses the raw instruction mix as features without any temporal history. Model accuracy can be misleading because cache miss events are rare (e.g. about 10% for IC). Figure <ref> (Top) shows these metrics over training epochs for data write cache. Note that the initial model accuracy is already about 70% where the model predicts that cache miss never occurs, with a corresponding metric F1 and Mathew Correlation Coefficient (MCC) value of zero. As training epochs increase, we note a sharp increase in predictive power around 5000 epochs. We also show the reconstruction error in Figure <ref> (Middle), the objective value for training, over epochs for the data write cache.We observe that the reconstruction error significantly drops in the first 20k epochs. Figure <ref> (Bottom) shows a measure of the classification error, measured in terms of the binary cross entropy between the true and predicted labels. We also observe that the classification error continues to drop steadily even though the reconstruction error has converged, showing that the model accounts for label information. Figure <ref> shows the prediction using a history of 10 cycles, in comparison with the ground truth.Future work includes validating our approach across microbenchmarks. § CONCLUSIONS Our approach has significant implications for the GPU revolution of computing. A data driven approach can potentially identify mix of instructions that cause performance bottlenecks. Although we focused on cache misses, any statistic of interest to the computer architecture community such as power consumption and voltage can potentially be predicted. The extension to an online embedded setting is straightforward and could potentially save computation time.Prediction of performance bottlenecks is a step towards a cognitive processor architecture.§ ACKNOWLEDGMENTSThis research is partially funded under NSF #1526399, the Defense Advanced Research Projects Agency (DARPA) and the Air Force Research Laboratory (AFRL). The views, opinions and/or findings expressed are those of the authors and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government.icml2016 | http://arxiv.org/abs/1703.09146v1 | {
"authors": [
"Aswin Raghavan",
"Mohamed Amer",
"Timothy Shields",
"David Zhang",
"Sek Chai"
],
"categories": [
"cs.LG"
],
"primary_category": "cs.LG",
"published": "20170327153102",
"title": "GPU Activity Prediction using Representation Learning"
} |
Transductive Zero-Shot Learning with Adaptive Structural Embedding Yunlong Yu, Zhong Ji, Member, IEEE, Jichang Guo, and Yanwei Pang, Senior Member, IEEEDecember 30, 2023 ======================================================================================================================Zero-shot learning (ZSL) endows the computer vision system with the inferential capability to recognize instances of a new category that has never seen before. Two fundamental challenges in it are visual-semantic embedding and domain adaptation in cross-modality learning and unseen class prediction steps, respectively. To address both challenges, this paper presents two corresponding methods named Adaptive STructural Embedding (ASTE) and Self-PAsed Selective Strategy (SPASS), respectively. Specifically, ASTE formulates the visual-semantic interactions in a latent structural SVM framework to adaptively adjust the slack variables to embody the different reliableness among training instances. In this way, the reliable instances are imposed with small punishments, wheras the less reliable instances are imposed with more severe punishments. Thus, it ensures a more discriminative embedding. On the other hand, SPASS offers a framework to alleviate the domain shift problem in ZSL, which exploits the unseen data in an easy to hard fashion. Particularly, SPASS borrows the idea from self-paced learning by iteratively selecting the unseen instances from reliable to less reliable to gradually adapt the knowledge from the seen domain to the unseen domain. Subsequently, by combining SPASS and ASTE, we present a self-paced Transductive ASTE (TASTE) method to progressively reinforce the classification capacity. Extensive experiments on three benchmark datasets (i.e., AwA, CUB, and aPY) demonstrate the superiorities of ASTE and TASTE. Furthermore, we also propose a fast training (FT) strategy to improve the efficiency of most of existing ZSL methods. The FT strategy is surprisingly simple and general enough, which can speed up the training time of most existing methods by 4300 times while holding the previous performance.Zero-shot learning, transductive learning, adaptive structural embedding, domain shift.§ INTRODUCTION Recently, image classification has made tremendous improvements due to the prosperous progress of deep learning and the availability of large scale annotated databases <cit.>, <cit.>. However, in many applications, it is impractical to obtain adequate labeled object categories <cit.>, <cit.>. To tackle this limitation, zero-shot learning (ZSL) <cit.> is proposed to recognize the unseen categories that no labeled data are available for training, i.e., the categories in training and testing are disjoint. It is inspired by the human beings' inferential ability that can recognize a novel class without seeing its visual instances and has received increasing attention in recent years.The emergence of semantic vectors enables the realization of ZSL. By representing the class labels with semantic vectors, a semantic space is built to associate the semantic relationships between the seen and unseen categories.Thus, the knowledge from the seen categories can be transferred to the unseen categories within this space. Two popular semantic vectors used in ZSL are attributes <cit.> and word vectors <cit.>. Particularly, attributes define a few properties of objects, such as shape, color and the presence or absence of a certain body part, which are manually defined <cit.> or discriminatively learned <cit.>. Word vectors represent class names with vectors based on a distributed language representation technique, such as Word2Vec <cit.> and Glove <cit.>.With the class semantic vectors, the cross-modality relationships between the image visual features and the semantic vectors can be exploited by a visual-semantic embedding either fromthe visual space to the semantic space <cit.>, or vice versa <cit.> or a shared common space <cit.>. Many effective methods have been proposed to build the visual-semantic embedding, including linear-based <cit.>, nonlinear-based <cit.>, bilinear-based <cit.> and max margin-based <cit.> approaches. Specifically, the max margin-based methods employ a ranking function to measure the compatibility scores between the images and the class semantic vectors, in which a compatibility matrix is derived by enforcing the correct label to be ranked higher than any of the other labels. However, in such models, the seen instances are typically treated without counting for their different reliableness during training, in which the structural information of the seen data may be undermined. To address this problem, we formulate the visual-semantic embedding in a latent structural SVM framework to adaptively adjust the slack variables to distinguish the training data, where the reliable instances are imposed with small punishments while the less reliable instances are imposed with more severe punishments. In this way, the structural information in the seen data are effectively exploited by assessing their reliability and discriminability.After the embedding step, the label of an unseen instance can be determined by performing a nearest neighbor (NN) search to match its visual feature against the candidate unseen categories in the embedding space. However, since the seen and unseen categories are different and potentially unrelated, the embedding is biased when it is directly applied to the unseen data. This is the well-known projection domain shift problem in ZSL <cit.>. To alleviate this bias, many transductive ZSL approaches have been developed to aggregate the unseen data together with the seen data to learn a more general visual-semantic embedding to improve classification performance <cit.>. However, such approaches mainly focus on exploiting the structural information in the unseen data and the potential label information is disregarded or underestimated. Actually, although the unseen data are unlabeled, we can predicted their potential labels with the knowledge learned from the unseen data. To this end, we propose to exploit the potential unseen label information in an easy to hard fashion, which includes two steps: (1) learning the visual-semantic embedding with the labeled seen data; (2) gradually refining the visual-semantic embedding with the seen data and unseen data in an iterative way. At each iteration, the unseen data is firstly predicted with the current visual-semantic embedding and the reliable unseen instances are selected as pseudo labeled data with the self-paced selective strategy, and then the pseudo labeled data are added into the labeled data set to refine the visual-semantic embedding. In this way, the knowledge is adapted progressively from the seen domain to the unseen domain. Meanwhile, the potential labeled information of unseen data is exploited in a confident way, thus the domain shift problem can be readily addressed. Figure 1 illustrates the proposed transductive framework. The contributions of our work can be summarized as: * A novel Adaptive Structural Embedding (ASTE) method for visual-semantic embedding in ZSL is proposed, which formulates the embedding problem in a latent structural SVM framework to embody the different reliableness by adaptively adjusting the slack variables. Extensive experiments show it achieves comparable or better performance than state-of-the-art inductive methods.* To alleviate the domain shift problem in ZSL, a Self-PAced Selective Strategy (SPASS) is presented, which iteratively selects a set of pseudo labeled instances from the unseen data to gradually refine the previously learned model. By combining this strategy and ASTE, we develop a Transductive ASTE (TASTE) method to progressively reinforce the classification capacity. TASTE outperforms most state-of-the-art transductive ZSL methods on three benchmark datasets: AwA, aPY, and CUB.* Furthermore, to improve the training efficiency, we present a surprisingly simple but effective Fast Training (FT) strategy. It is based on an audacious idea that representing the visual features in each training category with their visual pattern. Extensive experiments demonstrate that it is a general strategy suitable to many existing ZSL methods, and can greatly increase the training speed of most existing methods by 4 to 300 times while hold their previous performances. § RELATED WORK §.§ Visual-semantic embedding for ZSLVisual-semantic embedding is the key technique in ZSL, which bridges the relationships between the visual features and the semantic representations. It is learned from the seen data, while is applied to recognize the unseen data. Actually, it can be considered as a process of knowledge transfer or inference from the seen domain to the unseen domain.In recent years, several significant progresses have been made in the research of the visual-semantic embedding methods. One of the pioneering studies is <cit.>, where two fundamental visual-attributes embedding paradigms, i.e., Direct Attribute Prediction (DAP) and Indirect Attribute Prediction (IAP) are presented. Both of them use a probabilistic model to match the attribute prediction with the unseen categories. Specifically, IAP employs the attributes to connect the seen categories in a lower layer and the unseen categories in a higher layer, while DAP exploits the seen and unseen categories in one layer and connects them with attributes directly. In contrast, Socher et al. <cit.> and Frome et al. <cit.> are among the first researchers to establish the visual-word vector embedding. Specifically, Socher et al. <cit.> developed a two-layer neural networks to build a regression model for constructing interactional relationships between the visual space and the word vector space. It is a nonlinear method, which has not an explicit embedding matrix. Different from <cit.>, DeViSE <cit.> trains a linear mapping to link the image representation with the word vector using a combination of dot-product similarity and hinge rank loss, which is trained to produce a higher dot-product similarity between the visual feature and the word vector representation of the correct label than that of the visual feature and any other labels. Afterwards, several bilinear methods have also been proposed. For example, Structured Joint Embedding (SJE) <cit.> relates the input embedding and output embedding through a compatibility function, and implements ZSL by finding the label corresponding to the highest joint compatibility score. Further, LatEm <cit.> employs a bilinear compatibility model to learn a collection of maps as latent variables for the current image-class pair. The model is trained with a ranking based objective function that penalizes incorrect rankings of the true class for a given image.The studies similar to our ASTE method are <cit.> and <cit.>. In <cit.>, an embarrassingly simple embedding method (ESZSL) is presented. It constructs a general framework to model the relationships between visual features, class attributes and class labels with a bilinear model, and the closed-form solution makes it efficient. As similar bilinear formulation is used as <cit.>, the proposed ASTE not only considers these relationships, but also captures the discriminantive inter-class information by penalizing the incorrect predictions with a max-margin model. In <cit.>, the authors treat ZSL as a standard semi-supervised learning problem over the seen data and unsupervised clustering problem over the unseen data and integrate both parts in a latent max-margin multi-classification framework. Although using a similar max-margin multi-classification framework, the proposed ASTE learns a more discriminative embedding by distinguishing the different reliableness of the training data in a latent structural SVM framework. Furthermore, in our transductive framework, the potential label information of unseen data is gradually exploited by constructing the interaction between the seen and unseen data rather than exploiting the structural information of unseen data with an unsupervised clustering method. §.§ Domain shift in ZSLZSL can be viewed as a special case of transductive domain adaptation where the training and testing data have non-overlapping labels <cit.>. Since the embedding function learned from the seen data is biased when directly applied to the disjoined unseen data, the domain shift problem occurs <cit.>. It will decrease the classification performance on the unseen data. Different from the conventional domain shift problem <cit.>, the domain shift problem in ZSL is mainly due to the projection shift rather than the feature distribution shift. Many successful attempts have been made to address the domain shift problem in ZSL. A simple way is enlarging the seen data by employing a large amount of additional data with more categories and instances <cit.>. The more abundant categories and data ensure the visual-semantic embedding better generalize to the unseen data. Another effective way is importance weighting, which is borrowed from the field of transfer learning <cit.>. For example, Xu et al. <cit.> selectively re-weighted the relevant seen data to minimize the discrepancy between the marginal distributions of the seen and unseen data. The idea behind it is to augment the impacts of those data relevant to the unseen data to expect a better embedding generalization ability.Recently, there are considerable interests on developing elaborately transductive ZSL to rectify the domain shift problem. For example, Fu et al. <cit.> proposed a novel transductive multi-view hypergraph label propagation (TMV-HLP) model, in which the manifold structure of the unseen data is exploited to compensate for the impoverished supervision available from the sparse semantic vector. In this way, ZSL is achieved by semi-supervised label propagation from the semantic vector to the unseen data points within and across the graphs. With the idea that the instances of each class are condensed in a cluster in the deep feature space, <cit.> first trained a linear transformation to map the class semantic vectors to the deep visual space, and then used a clustering algorithm to assign labels to instances of unseen classes. Under the transductive setting, <cit.> used the label embeddings of the unseen data to regularize the learned unseen domain projection under a sparse coding framework. Different from them, TASTE firstly uses the learned visual-semantic embedding from the seen data to predict the labels of unseen data, and then refines the learned visual-semantic embedding in an iterative way, where the classification capacity is progressively reinforced with highly reliable instances.§ PROPOSED APPROACHIn this section, we first propose an Adaptive STructural Embedding (ASTE) model with the labeled seen data to learn the visual-semantic embedding, and then a Self-PAced selective Strategy (SPASS) is presented to alleviate the domain shift problem by gradually exploiting the potential label information in the unseen data in an easy to hard fashion. Finally, a Transductive ASTE (TASTE) method is developed to reinforce the discriminant capacity by combining the SPASS and ASTE.§.§ ASTE for ZSL Let 𝒮={(𝐱_n,𝐲_n),n = 1,...,N} denote the seen data from K seen categories, where 𝐱_n∈𝒳 is the input visual feature, 𝐲_n∈𝒴 is its label, N is the number of the seen data. If 𝐱_n belongs to a class k, its corresponding class label is 𝐲_n=1_k (1_k denotes a column vector of length K with all zeros except a single 1 at its k-th entry). We aim at learning f: 𝒳→𝒴 between the input visual space 𝒳 and the output structural label space 𝒴 by minimizing the empirical risk on the seen data 1/N∑_n=1^Nℓ(𝐲_n,f(𝐱_n)), where ℓ(𝐲_n,f(𝐱_n)) is the cost of predicting f(𝐱_n) when its true label vector is 𝐲_n. TABLE 1 shows the main notations used in this paper.We define a compatibility function F : 𝒳×𝒴→ℝ to measure how compatible the pair (𝐱_n,𝐲_n) are and formulate it as:F(𝐱_n,𝐲_n; 𝐖) = g(𝐱_n;𝐖)∙𝐲_n, where the operation sign ∙ denotes inner product, g(𝐱_n;𝐖) = 𝐖^T𝐱_n is the predicted label embedding of 𝐱_n in the seen label space, where 𝐖 = [𝐰_1,...,𝐰_K] is the parameter matrix. The parameter vector of category k is 𝐰_k, which maps the input visual feature 𝐱_n to its compatibility score over class k.In this way, the function F can be written as a bilinear form, i.e.,F(𝐱_n,𝐲_n;𝐖) = 𝐱_n^T𝐖𝐲_n. Here the value of F represents the compatibility score between the input visual feature and the output class label. The larger the value is, the more confidently that 𝐱_n belongs to class 𝐲_n. Thus, the label prediction for an instance 𝐱_n is achieved by maximizing F over the seen classes: f(𝐱_n;𝐖) = max_𝐲_n∈𝒴F(𝐱_n,𝐲_n;𝐖),where f(𝐱_n;𝐖) is the predicted label of 𝐱_n. It is the class label with the largest compatibility score.To achieve the knowledge transfer, we assume that the parameter vector of each category can be derived from its class semantic vector since it provides the corresponding class' properties. Based on this assumption, a function that independent from the categories is employed for obtaining the parameter vector 𝐰 from the corresponding class semantic vector 𝐚, i.e., 𝐰=𝐕𝐚, where 𝐕∈ℝ^p×q is the compatibility matrix shared by both the seen and unseen categories. To this end, the compatibility function for seen categories is written: F(𝐱_n,𝐲_n;𝐖_s) = 𝐱_n^T𝐖_s𝐲_n=𝐱_n^T𝐕𝐀_s𝐲_n,where 𝐖_s = 𝐕𝐀_s is the parameter matrix for seen categories and 𝐀_s=[𝐚_1,...,𝐚_K]∈ℝ^q×K is the class semantic matrix for seen categories. Thus, f(𝐱_n;𝐖_s) and g(𝐱_n;𝐖_s) can also be written as f(𝐱_n,𝐀_s;𝐕) and g(𝐱_n,𝐀_s;𝐕), respectively.Now, the key problem is to learn the compatibility matrix 𝐕 with the labeled seen data. In ASTE, the compatibility matrix is learned by enforcing the compatibility score of each seen instance over its correct class label to be ranked higher than that of the other classes, which can be formulated as follows: max_𝐲∈𝒴\𝐲_nF(𝐱_n,𝐲;𝐖_s)<F(𝐱_n,𝐲_n;𝐖_s), ∀n∈[1,...,N]. This formulation is closely related to <cit.> and <cit.>. Particularly, by formulating the visual feature and class semantic vector in a bilinear function, ALE <cit.> and DeVISE <cit.> use a pairwise ranking objective and SJE <cit.> employs the unregularized structured SVM formulation. In contrast, ASTE enforces the compatibility score of each seen instance over its correct class to be ranked higher than the closest runner-up in a multiclass objective.According to the maximum-margin principle employed in SVM, we select the compatibility matrix 𝐕 that makes the separation margin γ (the minimal difference between the score of the correct label and the closest runner-up) be maximal. Restricting the ℓ_2 norm of 𝐰 to make the problem well-posed leads to the following optimization problem:max_γ,𝐰_l(𝐱_n): 𝐰_l(𝐱_n)=1γs.t. 𝐲∈𝒴\𝐲_n:F(𝐱_n,𝐲_n;𝐖_s)-F(𝐱_n,𝐲;𝐖_s)>γ, ∀n∈[1,...,N],where 𝐰_l(𝐱_n) is the parameter vector of the class that 𝐱_n belongs to, γ = F(𝐱_n,𝐲_n;𝐖_s)-max_𝐲∈𝒴\𝐲_nF(𝐱_n,𝐲;𝐖_s).The problem can be equivalently expressed as a convex quadratic problem in the standard form: min1/2𝐰_l(𝐱_n)^2 s.t. F(𝐱_n,𝐲_n;𝐖_s)-max_𝐲∈𝒴\𝐲_nF(𝐱_n,𝐲;𝐖_s)>1, ∀n∈[1,...,N]. To allow the errors in the seen data, we add a penalty term to the objective function and relax the constraints:minC/2𝐕𝐀_s_F^2+∑_n^Nξ_n s.t. ξ_n≥1+F(𝐱_n,f(𝐱_n,𝐀_s;𝐕))-F(𝐱_n,𝐲_n;𝐖_s),∀n∈[1,...,N], ξ_n≥0,where C>0 is a constant that controls the trade-off between the training error minimization and the margin maximization terms, ξ_n is the slack variable, which can be viewed as the punishment for the instance 𝐱_n when it violates the constraint in Eq. (5). For 𝐱_n, if its predicted class label is the ground-truth label, i.e., f(𝐱_n,𝐀_s;𝐕) = 𝐲_n, its cost is a constant 1, otherwise, its cost is 1+F(𝐱_n,f(𝐱_n,𝐀_s;𝐕))-F(𝐱_n,𝐲_n;𝐖_s). We can observe that two different correctly-predicted instances are under equal punishments. Meanwhile, the incorrectly-predicted instances are under same punishment level against those correctly-predicted instances if the value of F(𝐱_n,f(𝐱_n,𝐀_s;𝐕))-F(𝐱_n,𝐲_n;𝐖_s) is small. Clearly, this is unfair and cannot reflect the instances' differences. Intuitively, the correctly-predicted instances with higher compatibility scores should be imposed smaller punishments than those with smaller compatibility scores. Similarily, the incorrectly-predicted instances should be imposed more severe punishments than the correctly-predicted instances. Based on this assumption, we propose an adaptive function Δ: 𝒴×𝒴→ℝ for differing the reliableness of the prediction. Specifically, the Euclidean distance between the predicted vector of an instance in the label space and its ground truth label is applied to define the adaptive function: Δ(𝐲_n,g(𝐱_n, 𝐀_s;𝐕)) = g(𝐱_n, 𝐀_s;𝐕)-𝐲_n_2^2,where g(𝐱_n, 𝐀_s;𝐕)=(𝐕𝐀_s)^T𝐱_n is the predicted vector of 𝐱_n in the label space, Δ(𝐲_n,g(𝐱_n, 𝐀_s;𝐕)) quantifies the loss associated with g(𝐱_n, 𝐀_s;𝐕) and its ground truth label. For descriptive convenience, we represent Δ(𝐲_n,g(𝐱_n, 𝐀_s;𝐕)) as Δ for short. The smaller the value of Δ is, the more confidently that the instance is predicted. An example is provided to explain the principle of cost function in ASTE in Fig. 2. Replacing the fixed margin in Eq. (8) with Δ, the final objective function of ASTE is obtained:minC/2𝐕𝐀_s_F^2+∑_n^Nξ_n s.t. ξ_n≥Δ+F(𝐱_n,f(𝐱_n,𝐀_s;𝐕))-F(𝐱_n,𝐲_n;𝐖_s), ∀n∈[1,...,N], ξ_n≥0. This problem can be equivalently expressed as a concave-convex program in the standard form:minC/2𝐕𝐀_s_F^2+∑_n^NΔ+F(𝐱_n,f(𝐱_n,𝐀_s;𝐕))-F(𝐱_n,𝐲_n), ∀n∈[1,...,N]. This objective function is similar to the latent structural SVM formulation <cit.>, which can be viewed as minimizing the sum of a convex and a concave function. It has been shown to converge to a local minimum or a saddle point solution <cit.>. The optimal compatibility matrix 𝐕^* can be obtained by a concave-convex procedure (CCCP), which consists in pair (𝐱_n,𝐲_n) at each iteration and searches for class label f(𝐱_n,𝐀_s;𝐕) that achieves the highest compatibility score for 𝐱_n. Specifically, if f(𝐱_n,𝐀_s;𝐕) = 𝐲_n, the cost function for 𝐱_n is:ℓ(𝐱_n) = C/2N𝐕𝐀_s_F^2+Δ. It is equivalent to the variation of ESZSL <cit.>, of which Δ is the loss function, and 𝐕𝐀_s_F^2 is one regularizer. In this situation, ESZSL <cit.> can be seen as a special case of ASTE. If f(𝐱_n,𝐀_s;𝐕) ≠𝐲_n, the cost function for 𝐱_n is:ℓ(𝐱_n) = C/2N𝐕𝐀_s_F^2+Δ+ F(𝐱_n,f(𝐱_n,𝐀_s;𝐕))-F(𝐱_n,𝐲_n;𝐖_s). The compatibility matrix 𝐕 is updated by the stochastic gradient descent approach as follows: 𝐕^(t) = 𝐕^(t-1)-η_t∂ℓ(𝐱_n)/∂𝐕^(t-1),where η_t is the learning step size at iteration t. The whole optimization procedure for Eq. (11) is outlined in Algorithm 1.Once the optimal compatibility matrix 𝐕^* is obtained, the class label of an unseen instance 𝐱_m can be predicted by resorting to the largest compatibility score that corresponds to one of the unseen categories:𝐳^∗ =max_𝐳∈𝒵𝐱_m^T𝐕𝐀_t𝐳,where 𝒵 is the label space of unseen classes that is disjoint from that of the seen class 𝒴, i.e., 𝒴∩𝒵 = ∅, and 𝐀_t is the unseen class semantic matrix in which each column denotes a class' semantic vector. The flowchart of the proposed ASTE is illustrated in Fig. 3.§.§ The Self-PAced Selective Strategy (SPASS) Most inductive ZSL approaches directly apply the learned model from the seen data to recognize the unseen data. However, the different distribution between the seen data and the unseen data will lead the learned embedding model to be biased on the unseen data. In order to address this domian shift issue, we take the ZSL as a special case of transductive learning. Considering that the primary cause of domain shift problem in ZSL lies in the label absence of the unseen data, a natural idea is to select some pseudo labeled instances from unseen data in a transductive learning way to tackle it. Thus, the challenge is how to choose the reliable pseudo labeled data. To address this issue, we propose a selective strategy to gradually select the unseen data in an easy to hard fashion. As defined in <cit.>, the process of gradually added training instances is called a curriculum. A straightforward way to design a curriculum is to select the instances based on certain heuristical “easiness" measurements. In the curriculum, the easy instances are selected for training. Based on this idea, the previously predicted unseen data are ranked according to the compatibility scores. The higher compatibility score is, the more reliable of an instance to be correctly predicted. In this paper, the easiness of an instance is defined as: an instance is easy if it is reliable to be predicted correctly.To this end, we present a Self-PAced Selective Strategy (SPASS) to iteratively select the reliable pseudo labeled unseen instances. Inspired by self-paced learning <cit.>, a binary variable u_m is introduced to indicate whether the m-th instance is easy or not. Then, the selective process is defined by:min_𝐮∑_m=1^Mu_mf(𝐱_m;𝐖_t) - g(𝐱_m;𝐖_t)_F^2-θu_m,where M is the instance number of the unseen data, f(𝐱_m;𝐖_t) is the predicted class label of 𝐱_m by Eq. (3),and g(𝐱_m;𝐖_t)=𝐖_t^T𝐱_m is the embedding of the test instance 𝐱_m in the unseen label space, 𝐖_t=𝐕𝐀_t is the parameter matrix for unseen classes; 𝐮=[u_1,...,u_M]∈[0,1]^M is the indicate vector, and θ is a threshold parameter for controlling the number of instances to be selected. f(𝐱_m;𝐖_t) - g(𝐱_m;𝐖_t)_F^2 is the potential loss for unseen instance 𝐱_m. The smaller it is, the more reliable an instance is correctly predicted. In this way, the instances whose losses are smaller than a certain threshold θ are taken as “easy" instances, and will be selected as the pseudo labeled data, otherwise unselected. When θ is small, only “easy" instances are selected. As θ grows, more instances with larger losses will be gradually aggregated with the labeled set for a more suitable model. The selective process stops when all unseen data are selected. §.§ Transductive ASTE (TESTE) In the situation of both the labeled seen data and the unlabeled unseen data are off-the-shelf, a transductive ZSL approach can be developed by combining the SPASS and ASTE. We call it Transductive ASTE (TASTE). The united framework of TASTE is:min_𝐕∑_n^Nℒ(𝐲_n,f(𝐱_n,𝐀_s;𝐕))+∑_m^Mℒ(f(𝐱_m,𝐀_t;𝐕),g(𝐱_m,𝐀_t;𝐕))+λℛ,where ℒ() is the loss function, ℛ is a regularization function, and λ is the trade-off weight. Specifically, ℒ(𝐲_n,f(𝐱_n,𝐀_s;𝐕))=Δ(𝐱_n,f(𝐱_n,𝐀_s;𝐕)) +F(𝐱_n,f(𝐱_n,𝐀_s;𝐕))-F(𝐱_n,𝐲_n), ℒ(f(𝐱_m,𝐀_t;𝐕),g(𝐱_m,𝐀_t;𝐕))= u_mf(𝐱_m,𝐀_t;𝐕)-g(𝐱_m,𝐀_t;𝐕)_2^2,ℛ = 𝐕𝐀_s_F^2+𝐕𝐀_t_F^2.The solution to Eq. (17) is addressed by the Alternative Convex Search (ACS) method <cit.> in which the variables to be optimized (i.e., 𝐮 and 𝐕) are divided into two disjoint blocks. When 𝐮 is fixed, the problem converts a convex and concave function. Thus, 𝐕 can be obtained by the CCCP algorithm. More specifically, for labeled pair (𝐱_n,𝐲_n), its loss is obtained by Eq. (13). As for the unseen instance 𝐱_m, the predicted label f(𝐱_m,𝐀_t;𝐕) is taken as its pseudo label, of which the easiness is controlled by the value of u_m. The loss for it is:ℓ(𝐱_m) = C/2M𝐕𝐀_t_F^2+ u_mf(𝐱_m,𝐀_t;𝐕) - g(𝐱_m,𝐀_t;𝐕)_2^2. Minimizing this loss function means to enforce the embedding of unseen instance to be close to its predicted class label. In this way, the 𝐕 is updated by Eq. (14).On the other hand, fixing 𝐕, then 𝐮 can be easily updated by:u_m = {[ 1, f(𝐱_m,𝐀_t;𝐕) - g(𝐱_m,𝐀_t;𝐕)_2^2 < θ,; 0, otherwise. ].It means that a specific instance 𝐱_m whose loss is smaller than a certain threshold θ is selected as “easy" instance, and its pseudo label is f(𝐱_m,𝐀_t;𝐕); otherwise, it is unselected.As θ grows, more reliable instances will be gradually added into the labeled set, thus the classification capacity is progressively reinforced.§ A NOVEL FAST TRAINING (FT) STRATEGYThe approaches with closed-form solution, such as ESZSL <cit.> and Linear Regression (LR) <cit.>, generally have a relatively fast training speed. In contrast, the gradient descent based approaches, such as SJE <cit.>, DeViSE <cit.>,ASTE and TASTE, are slow due to the heavy iterative computational burdens. In order to improve the training efficiency, we propose a surprisingly simple fast training (FT) strategy by applying the visual pattern of each seen class to represent the corresponding class. In this paper, the visual pattern of a class is defined as the average visual features in a class. The idea behind it is that one class can be well represented as a semantic vector, thus we can try to represent each class with its visual pattern in the visual space. In this way, the FT strategy simply takes the visual pattern of each class as training data which greatly alleviates the computational burdens especially for those gradient descent based approaches.Simple as FT strategy is, it is effective. The experiments show it can speed up the running time of most existing methods by 4300 times while holding the previous performance. However, it is hard to explain this interesting phenomenon, i.e., the performance is almost steady with and without FT strategy. An intuitive explanation is that the FT strategy can suppress the noise and particularly possible outliers during the adaptation. Besides, we also try to provide its theoretical analysis in the risk bounds for domain adaptation in the following.Generally, the objective functions of different domain adaptation approaches on the seen data can be expressed as:min∑_𝐱_n∈𝒮ℒ(𝐲_n;f(𝐱_n; 𝐖_s))+ℛ,where 𝐲_n and f(𝐱_n; 𝐖_s) are ground truth class label and the predicted class label, respectively. The expected error in the seen data 𝒮 and the unseen data 𝒰 are defined as:ϵ_s(f) = 𝔼_𝐱_n𝒮[|f(𝐱_n;𝐖_s)-𝐲_n|],ϵ_t(f) = 𝔼_𝐱_m𝒰[|f(𝐱_m;𝐖_t)-𝐲_m|]. According to the Theorem 1 in <cit.>, given a hypothesis space ℋ of VC-dimension d̅ in 𝒰 that contains m̅ instances, then with probability at least 1-δ, for every f∈ℋ, its expected error in 𝒰 is:ϵ_t(f) ≤ϵ_s(f)+ 4√(2d̅/m̅(log2m̅/d̅+log4/δ)) +λ+d_ℋ(𝒮,𝒰),where λ is an upper-bound of inf_f∈ℋ[ϵ_s(f)+ϵ_t(f)], d_ℋ(𝒮,𝒰) is the distribution distance between 𝒮 and 𝒰.Here, the FT strategy mainly brings the variation on the first term ϵ_s(f) in Eq. (25), which is the expected error in the seen data. It is obtained by summing the expected error of each instance. With FT strategy, only the expected error of the visual pattern in each seen class is considered. The value of ϵ_s(f) of the approaches with the FT strategy is smaller than those without FT strategy. This conclusion will be verified in the following. Then, the error bound ϵ_t(f) in Eq. (25) is smaller than those approaches without FT strategy, which indicates the information is easier to transfer from the seen domain to the unseen domain. Therefore, the performances will be hold when applying the FT strategy to most existing methods.Without loss of generality, we take the Linear Regression (LR) as an example to verify the above conclusion since LR is simple and has a closed-form solution. Given the model parameter 𝐖, the empirical loss of LR for N seen instances from K classes is:ϵ_s = 1/N∑_i=1^K∑_j=1^k_i𝐱_ij^T𝐖-y_i_2^2,where 𝐱_ij is the j-th instance from i-th class, k_i is the instance number of class i. In contrast, the empirical loss in LR with FT strategy is:ϵ̅_s = 1/K∑_i=1^K𝐱̅_i^T𝐖-y_i_2^2,where 𝐱̅_i= 1/k_i∑_j=1^k_i𝐱_ij is the visual pattern for class i. In Eq. (26), the operation ∙ is calculated with N times, while it is calculated with K times in Eq. (27), in which K≪N. Furthermore, we have proposition 1.Proposition 1. For a specific class i, we can conclude𝐱̅_i𝐱̅_i^T≤1/k_i∑_j=1^k_i𝐱_ij𝐱_ij^T. Proof: For a specific matrix 𝐗 = [𝐱_1,...,𝐱_m], its variance is:D(𝐗) = E(𝐱_i^2)-[E(𝐗)]^2,where E(𝐗)=1/m∑_i^m𝐱_i is the average of 𝐗 and E(𝐱_i^2) = 1/m∑_i^m𝐱_i𝐱_i^T. It is easy to verify Eq. (28) because the variance value D(𝐗) is non-negative.According to Proposition 1, we can conclude that the value of ϵ̅_s is smaller than that of ϵ_s for LR. Replace the ϵ_s(f) with ϵ̅_s(f) in Eq. (25), the upper bound ϵ_t of the unseen domain decreases correspondingly, which verifies the conclusion that the expected error with FT strategy is smaller than that without FT strategy.§ EXPERIMENTSTo evaluate the effectiveness and efficiency of the proposed approaches, we conduct extensive experiments on three benchmark zero-shot learning datasets. We first detail the datasets and the experimental settings, then present the experimental results of ASTE, followed by the comparative results of TASTE. Finally, the evaluations of the FT strategy on ASTE, TASTE and other ZSL approaches are provided.§.§ Datasets and experimental settingsDatasets. The proposed approaches are evaluated on three benchmark datasets: Animal with Attributes (AwA) <cit.>, Caltech UCSD Birds (CUB) <cit.>, and aPascal-aYahoo (aPY) <cit.>. AwA is a standard attribute dataset for ZSL; CUB is a fine-grained dataset with little variations among different classes; aPY is a combined dataset of aPascal and aYahoo, in which aYahoo dataset is collected from Yahoo image search that is different from the ones in aPascal. These datasets contain diverse categories such as animals, birds and objects. More specifically, AwA contains 30,475 images from 50 different animals, paired with a set of human provided 85 continuous attributes for each class. We follow the standard seen/unseen split <cit.>, where 40 classes with 24,295 images are taken as the seen data and the remaining 10 classes with 6180 images are adopted as the unseen data. CUB contains 11,788 images from 200 bird species with 312 associated attributes. In this dataset, we use the same zero-shot split as <cit.> with 150 classes for seen data and 50 disjoint classes for the unseen data. For aPY, it contains 2,644 images from 32 classes, in which each image is annotated by 64 binary attributes. To represent each class with an attribute vector, we average the attributes of the images in each class. In the experiment, the aPascal is used as the seen data, and the aYahoo is used as the unseen data.Experimental settings. To make our approaches easily compared with the previous approaches, we use the popular VGG-VeryDeep-19 <cit.> model that pre-trained on imageNet to extract the visual features. In specific, we use the fully connected layer (FC7 layer of VGG-VeryDeep-19) for representing the image. We denote it with VGG in the following.There are two parameters in TASTE (the weight C and the threshold θ) and one parameter in ASTE (the weight C),which can be determined via cross validation. However, in our experiments, we found the performance is not sensitive to the weight C, thus we fix C = 0.1. As for the threshold θ, we initiate it with the half value of the maximal loss in the unseen data as the initial threshold, then gradually increase its value until it equals to the maximal loss in the unseen data. In this paper, θ is gradually selected from (0.5δ, 0.7δ, 0.9δ, δ), where δ is the maximal value of the loss function.The optimization of 𝐕 is performed with stochastic gradient descent approach, which is initialized randomly with normal distribution. The size of the mini-batch is 50, the updating rate η_t is selected from {1, 0.1, 0.01} successively with 50 times as a round. As the proposed ASTE is a non-convex problem, thus different initialization will lead to different local minimum. For a fair comparison, we perform 5 trials and report the mean and the standard variance as the final performance. For all datasets, we apply the popular average per-class top-1 accuracy to evaluate the performance <cit.>. The average testing time in our Matlab implementation is about 1ms per unseen instance in a desktop computer with an Intel Core i7-4790K processor and 32G RAM. §.§ Experimental results of ASTE Competitors. Ten state-of-the-art inductive ZSL approaches are selected for comparison with ASTE, ranging from linear-based <cit.>, nonlinear-based <cit.>, bilinear-based <cit.> and max margin-based <cit.>. The performance results of the selected approaches are all from the original papers except <cit.> and <cit.>. As for SJE <cit.> and ESZSL <cit.>, we implement them with VGG features by ourselves. Specifically, the settings in SJE is similar to ASTE, and the parameters in ESZSL are picked via cross validation.Comparative Results. The comparison results are outlined in Table 2, which shows that ASTE achieves the state-of-the-art performance on three datasets. It is highlighted that the ASTE performs the best on AwA dataset. Specifically, with the same VGG features, ASTE outperforms DAP <cit.>, KDICA <cit.>, SJE <cit.>, ESZSL <cit.>, SSE-ReLU <cit.>, Zhang et al. <cit.>, Bucher et al. <cit.> in 23.77, 7.2, 0.9, 6.6, 4.67, 0.54 and 3.68 absolute percentage point on AwA dataset, respectively. On CUB dataset, ASTE achieves the third best performance, which is inferior to <cit.> and <cit.> in 4.5% and 1.6%, respectively, and superior to KDICA <cit.>, SJE <cit.>, ESZSL <cit.>, SSE-ReLU <cit.>, Zhang et al. <cit.>, Bucher et al. <cit.>, in 6.5, 3.9, 3.4, 19.79, 8.09 and6.91 absolute percentage point, respectively. On aPY dataset, ASTE achieves the second best performance, which has 9.84%, 3.5%, 6.2%, 1.77%, 9.06% gains against DAP <cit.>, SJE <cit.>, ESZSL <cit.>, SSE-ReLU <cit.>, Zhang et al. <cit.>, respectively.§.§ Experimental results of TASTE First, we compare in Fig. 4 the performance of ASTE and TASTE. It can be observed that the usage of SPASS improves the performance of ASTE significantly. Specifically, the performances have 8.74% and 17.33% improvements in AwA and aPY datasets, respectively. In contrast, the performance has a smaller improvement of 4.02% in CUB dataset. This is mainly due to that the CUB dataset is a fine-grained dataset which is more difficult to be classified, thus SPASS may select some less reliable instances that provide little positive information for refining the model.Next, the performance superiority of the TASTE is evaluated against five state-of-the-art transductive ZSL methods, as shown in Table 3. It can be observed that TASTE outperforms the others on both AwA and aPY datasets, and is comparable to the others on CUB dataset. Specifically, the absolute performance gains of TASTE against the others are from 1.1% to 16.54% in AwA dataset. It is more significant on aPY dataset, which has noticeable improvements of 26.3% and 15.56% against SMS <cit.> and <cit.>, respectively. On CUB dataset, TASTE is inferior to <cit.> in 4.55%, but outperforms the others significantly. Overall, the results on the three benchmark datasets demonstrate that the effectiveness of SPASS and TASTE.§.§ Evaluation of the FT strategyTo confirm the efficacy and efficiency of the proposed FT strategy, we conduct a set of experiments on the three benchmark datasets. Meanwhile, three additionally typical methods are chosen to indicate the generality of the FT strategy. These methods are LR <cit.>, ESZSL <cit.> and SJE <cit.>, representing the linear approach, bilinear approaches with and without closed-form solutions, respectively. To facilitate the introduction, we add a prefix Fast- to the name of these methods, and call them fast methods. Specifically, for Fast-SJE, Fast-ASTE, and Fast-TASTE, we initialize the compatibility matrix 𝐕 with the optimal parameters in ESZSL to reach the convergence quickly, since ESZSL has a closed-form solution.Table 4 depicts the performance comparisons of those with and without the FT strategy. It can be observed that the performances of the fast methods are close to that of their corresponding methods on all three datasets. In particular, all the fast methods outperform their corresponding ones in aPY dataset. In AwA dataset, the FT strategy improves those closed-form expression approaches LR and ESZSL andslightly lower than those without closed-form expression approaches.For CUB dataset, except that Fast-ESZSL and Fast-TASTE are slightly lower than ESZSL and TASTE, the other methods also have an obvious performance improvement. In addition, from the perspective of the fast methods, all of them have performance improvements on at least two datasets except for TASTE. Therefore, the FT strategy is effective in holding the original methods' performance. On the other hand, almost all the standard variances of the fast methods are smaller than their corresponding original approaches, which indicates the fast methods are more robust to noises and particularly possible outliers.Table 5 verifies the efficiency of the FT strategy. It can be seen that the fast methods speed up the run time significantly. Concretely, for those methods with closed-form solutions, i.e., Fast-LR and Fast-ESZSL, the speed up ratios are about 4. More impressively, the speed up ratios are about 230310 for those methods without closed-form solutions, including Fast-SJE, Fast-ASTE, and Fast-TASTE.What's more, we also evaluate the impact of the amount of training data to the visual-semantic embedding models. To this end, we first randomly select one instance from each class for training the model, and then vary the percent of each training class number from 10% to 100% in intervals of 10% of the corresponding class. All results are reported with the average value of 5 trials. As Fig. 5 shows, with the increase of the training instances, the classification performances steadily increase for all approaches when the rate of each training class number is lower than 0.4. In contrast, when the percent of each training class number is above 0.4, the performances of different approaches reach a plateau, possibly because there is no further margin to improve <cit.>. Meanwhile, we can observe that the performance with randomly selective strategy (i.e., randomly choose one instance from each class) is worse than that of the FT strategy, which suggests that the success of FT strategy is due to the representative power of the all input data in each class. § CONCLUSIONIn this paper, we mainly addressed the visual-semantic embedding and domain adaptation in ZSL. For the first one, we proposed an adaptive structural embedding (ASTE) method by formulating the visual-semantic embedding in an adaptively latent structural SVM framework where the reliability and discriminability of the training instances are exploited. For the second one, we presented a self-paced selective strategy (SPASS) to iteratively select the unseen instances from reliable to less reliable to gradually transfer the knowledge from the seen domain to the unseen domain. Then, we combined ASTE and SPASS to develop a transductive ZSL approach named TASTE to progressively reinforce the discriminant capacity. Extensive experiments on three benchmark datasets have verified the superiorities of these proposed methods. Specifically, ASTE achieved the best performance on AwA dataset, and TASTE performs the best on AwA and aPY datasets. Besides, we also present a simple but effective fast training (FT) strategy to speed up the training speed of ZSL by employing the visual pattern of each class as input training data. The speed up rations are about 4 for the methods with closed-form solutions and hundreds for those without closed-form solutions. 1copr16:PangY. Pang, M. Sun, X. Jiang, and X. Li, “Convolution in Convolution for Network in Network," CoRR, 1603.06759, 2016.tnnls14:ShaoL. Shao, L. Liu, X. Li, “Feature Learning for Image Classification Via Multiobjective Genetic Programming," IEEE Trans. Neural Netw. Learn. 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Tuytelaars, “Unsupervised visual domain adaptation using subspace alignment," in Int. Conf. Comput. Visi., Sydney, Australia, Dec. 2013, pp. 2960-2967. cvpr09:Duan L. Duan, I. W. Tsang, D. Xu, and S. J. Maybank,“Domain transfer svm for video concept detection," in Proc. Comput. Visi. Pattern Recognit., Miami, USA, Jun 2009, pp. 1375-1381. pami14:Xu J. Xu, S. Ramos, D. Vzquez, A. Lpez, “Domain adaptation of deformable part-based models,"IEEE Trans. Pattern Anal. Mach. Intell., vol. 36, no. 12, pp. 2367-2380, 2014.tnnls12:Duan L. Duan, D. Xu, I. W. Tsang, “Domain adaptation from multiple sources: A domain-dependent regularization approach." IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 3, pp. 504-518, 2012. Technical11:Wah C. Wah, S. Branson, P.Welinder, P. Perona, and S. Belongie, “The Caltech-UCSD Birds-200-2011 Dataset," Technical report, 2011. | http://arxiv.org/abs/1703.08897v1 | {
"authors": [
"Yunlong Yu",
"Zhong Ji",
"Jichang Guo",
"Yanwei Pang"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170327014441",
"title": "Transductive Zero-Shot Learning with Adaptive Structural Embedding"
} |
In this paper we prove that translation structures for which the corresponding vertical translation flows is weakly mixing and disjoint with its inverse, form a G_-dense set in every non-hyperelliptic connected component of the moduli space ℳ. This is in contrast to hyperelliptic case, where for every translation structure the associated vertical flow is isomorphic to its inverse. To prove the main result, we studylimits of the off-diagonal 3-joinings of special representations of vertical translation flows. Moreover, we construct a locally defined continuous embedding of the moduli space into the space of measure-preserving flows to obtain the G_-condition. [ N Krauter and F Stefani===========================§ INTRODUCTIONLet M be an orientable compact connected topological surface, and Σ be a finite set of singular points. On M we can consider a translation structure ζ, i.e. an atlas on M∖Σ such that every transition transformationis a translation. Every translation surface can be viewed as a polygon with pairwise parallel sides of the same length which are glued together (gluing is made by a translation). Parameters given by the sides of such polygons establish a parametrization of the so-called moduli space ℳ and yield a topology on ℳ. To each translation structure ζ weassociate the corresponding Lebesgue measure _ζ on M. Moreover, for every direction we consider the flow which acts by translation in that direction with unit speed.Such translation flows preserve _ζ. In this paper we are interested in vertical translation flows. It is worth to mention that the study of directional flows on translation surfaces originates fromproblems concerning billiard flows on rational polygons (see <cit.>,<cit.>).In <cit.> the authors give a complete characterisation of connected components of the moduli space; all of them are orbifolds. On each connected component C we consider an action of SL_2() which is derived form the linear action of SL_2() on polygons. Moreover, there is a Lebesgue measure ν_C on C which is invariant under this action. Let ℳ_1 be the set of ζ∈ℳ such that _ζ(M)=1 and let C_1:=C∩ℳ_1. We also consider a measure ν_C_1 on C_1 which is a projectivization of ν_C. This measure is finite and invariant under the action of SL_2() on C_1. In fact, the action of SL_2() is ergodic with respect to this measure (see <cit.> and <cit.>). This gives an opportunity to use the ergodic theory to study dynamical properties of vertical flows on almost all translations structures. In particular, it was used to prove that the sets of translation structures for which the vertical translation flow is ergodic (see <cit.>), and further is weakly mixing (see <cit.>) are of full measure in both C and C_1. At the same time, there are no mixing translation flows (see <cit.>). In this paper we are interested in translation structures for which the corresponding vertical translation flow is disjoint with its inverse, which is a stronger notion than being not reversible. Recall that a measure preserving flow {T_t}_t∈ on (X,μ) is reversible, if there exists an involution θ:X→ X which preserves μ andθ∘ T_-t=T_t∘θ for all t∈.Our result concerns the topological typicality of the desired property rather than measure-theoretical. As a by-product, we give a method to show that the set of translation structures for which the associated vertical translation flows satisfy any property which is G_ in the space of measure preserving flows, is also a G_-set. Among these properties are for instance weak mixing, ergodicity and rigidity (see <cit.>).In the classification of connected components given in <cit.> we distinguish so called hyperelliptic components. For every hyperelliptic component C there exists an involution θ:M→ M such that for every ζ∈ C it is given in local coordinates by the formula z↦ -z+c for some c∈ℂ. In particular, the vertical flow on (M,ζ) is reversible; it is isomorphic with its inverse by the involution θ (see remark <ref>). In contrast, in this paper we show that on non-hyperelliptic components of the moduli space the set of translation structures for which the vertical flow is disjoint with its inverse is topologically large. It is expressed by the following theorem. Let C be a non-hyperelliptic connected component of the moduli space of translation structures. Then the set of translation structures whose vertical flow is weakly mixing and disjoint with its inverse is a G_-dense set in C.It is also worth to mention that on non-hyperelliptic components we can also find a non-trivial set of translation structures for whichthe vertical flow is reversible. Let C be a non-hyperelliptic connected component of the moduli space of translation structures. Then the set of translation structures whose vertical flow is reversible isdense in C. Recall that a measure-preserving flow {T_t}_t∈ on a standard Borel probability space (X,ℬ,μ) is disjoint with its inverse if the only (T_t× T_-t) - invariant probability measure on X× X, which projects on each coordinate as μ is the product measure μ⊗μ. In <cit.> the authors developed techniques to prove non-isomorphism of a flow T^f to its inverse that arebased on studying the weak closure of off-diagonal 3-self-joinings. Moreover, in <cit.> the authors improved those techniques to show that a large class of special flows over interval exchange transformations and under piecewise absolutely continuous functionshave the property of being non-isomorphic withtheir inverse. The idea of detecting non-isomorphism of a dynamical system and its inverse by studying the weak closure of off-diagonal 3-self-joinings was introduced by Ryzhikov in <cit.>. In this paper we prove that techniques mentioned earlier can be used to detect disjointness of a vertical flow with its inverse.To prove the G_ condition, we use the result of Danilenko and Rhyzhikov from <cit.> (which derives from a version for automorphisms given in <cit.>), where they proved that the flows with the property of being disjoint with their inverse form a G_δ-dense set in the space of measure preserving flows. To use their result we construct a locally defined continuous embedding of the moduli space into the space of measure preserving flows. To show the density condition, we largely rely on the proof of Lemma 14 in <cit.>.In Section <ref> we givea general background concerning joinings, interval exchange transformations, space of measure preserving flows, translation flows and moduli spaces. In particular, we give some tools needed to prove the continuity of a map with values in the space of flows and we state some connections between the moduli space and interval exchange transformations.In Section <ref> we introduce a criterion of disjointness of two flows by researching the weak limits of certain 3-self-joinings. This is a direct improvement ofthe criteria stated in <cit.> and <cit.> as we show that these are actually criteria of two weakly mixing flows being disjoint. Furthermore, we state a criterion of a flow being weakly mixing, which also uses weak limits of 3-self-joinings as a tool.In Section <ref> we state combinatorial conditions on translation structures which are later used in proving the density condition in our main theorem. To be precise, we show that our results apply to every non-hyperelliptic component ofthe moduli space.In Section <ref> we show that on a given translation surface (M,ζ) there exists _ζ>0 such that for every absolutely continuous measure μ on M which has density f satisfying ∫_M|f(x)-1|dx<_ζ there exists a homeomorphism H:M→ M such that H_*μ is the Lebesgue measure with H depending continuously on f. The results of section <ref> were inspired by the works of Moser in <cit.> and Goffman, Pedrick in <cit.>.In Section <ref> we use the results presented inthe previous section to construct a continuous embedding of each connected component of the moduli space into the space of measure-preserving flows. The embedding is defined locally, but we also show that this is enough to transfer the G_δ condition.Finally, in Section <ref> we first state a result which is a conclusion from the previoussections, that in every connected component ofthe moduli space the set of translation structures whose associated vertical translation flow is disjoint with its inverse, is a G_ set. In the remainder of this section we use the results from <cit.> to show that the criteria introduced in Section <ref> can be used for a dense set of translation structures in every non-hyperelliptic component, which leads to theproof of Theorem <ref>. As a by-product we get the proof of Proposition <ref>. § PRELIMINARIESWe will now give some details regarding interval exchange transformations, joinings of dynamical systems and some basic information about moduli space.§.§ JoiningsIn this subsection we give some definitions which are stated for standard Borel probability spaces. However these definitions can be easily extended to standard Borel spaces with finite measures. Though here we state definitions for probability spaces, in the remainder of this paper we will freely use them in case whenthe measure is finite and not necessarily a probability. In particular we say that the measure preserving flows 𝒯={T_t}_t∈ on (X,ℬ,μ) and 𝒮={S_t}_t∈ on (Y,𝒞,ν) are isomorphic, if there exists a measurable F:(X,ℬ)→(Y,𝒞) such thatT_t=F^-1∘ S_t∘ F for t∈and F_*μ=μ(X)/ν(Y)ν.Let K>0 be a natural number and for 1≤ i≤ K let 𝒯^i={T^i_t}_t∈ be a measure preserving flow acting on a standard Borel probability space (X^i,ℬ^i,μ^i) . We say that a measureon (X^1×…× X^K,ℬ^1⊗…⊗ℬ^K) is a K- joining if it is𝒯^1×…×𝒯^K-invariant and it projects on X^i as μ^i for each i=1,…,K. We denote by J(𝒯^1,…,𝒯^K) the set of all joinings of 𝒯^i for i=1,…,K and by J^e(𝒯^1,…,𝒯^K) the subset of ergodic joinings. If for i=1,…,K, (X^i,ℬ^i,μ^i,𝒯^i) are copies of the same flow, then we say thatis a K-self joining. We denote the set of K-self joinings of a flow 𝒯 by J_K(𝒯) and ergodic K-self joinings by J^e_K(𝒯). If 𝒯^1,…,𝒯^K are ergodic, then the following remarks hold. J(𝒯^1,…,𝒯^K) is a compact simplex in the space of all 𝒯^1×…×𝒯^K-invariant measures and its set of extremal points (J(𝒯^1,…,𝒯^K)) equals J^e(𝒯^1,…,𝒯^K).[Ergodic decomposition] For each ∈ J(𝒯^1,…,𝒯^K) there exists a unique probability measure κ on J^e(𝒯^1,…,𝒯^K) such that =∫_J^e(𝒯^1,…,𝒯^K)ρ dκ(ρ).Assume that K=2. Note that μ^1⊗μ^2∈ J(𝒯^1,𝒯^2). We say thatthe flows 𝒯^1 and 𝒯^2 are disjoint in the sense of Furstenberg (or simply disjoint) if the product measure is the only joining between them. If two flows are disjoint, then they have no common factor. In particular, they are not isomorphic.Thenotions of joinings and disjointness can be rewritten for automorphisms instead of flows. Then we also have the following well-known observation. If (X,ℬ,μ,T) is an ergodic automorphism and (Y,𝒞,ν,Id) is the identity then T and Id are disjoint. Let ϕ:(X^1,ℬ^1,μ^1,𝒯^1)→(X^2,ℬ^2,μ^2,𝒯^2) be an isomorphism. It is easy to see that μ_ϕ^1:=(Id×ϕ)_*μ^1 is a joining of 𝒯^1 and 𝒯^2. We say that μ_ϕ^1 is a graph joining. We have the following remark. Let ∈ J(𝒯^1,𝒯^2) and let Π⊆ℬ^1 be a family of measurable sets. Let ϕ:(X^1,ℬ^1,μ^1,𝒯^1)→(X^2,ℬ^2,μ^2,𝒯^2) be an isomorphism. Then the following are equivalent: (1) (A× B)=μ^1(A∩ϕ^-1(B)) for all A∈Π and B∈ϕ(Π); (2) (A× XX×ϕ A)=0 for every A∈Π; (3) (A×ϕ A^c)=(A^c×ϕ A)=0 for every A∈Π. Consider graph joinings between two identical flows (X,ℬ,μ,𝒯). If ϕ=T_-t for some t∈ then we say that μ_ϕ is a 2-off-diagonal joining and we denote it by μ_t. In other words for A,B∈ℬ we haveμ_t(A× B)=μ(A∩ T_t B)=μ(T_-tA∩ B).This definition is easily extended to higher dimensions, namely a K-off-diagonal joining μ_t_1,…,t_K-1 is a K-joining given by the formulaμ_t_1,…,t_K-1(A_1×…× A_K)=μ(T_-t_1A_1∩…∩ T_-t_K-1A_K-1∩ A_K),for all A_1,…,A_K∈ℬ.Let 𝒫(^K-1) be the set of Borel probability measures on^K-1. For every P∈𝒫(^K-1) we considerthe K-integral joining given by(∫_^K-1μ_t_1,…,t_K-1 dP(t_1,…,t_K-1))(A_1×…× A_K):=∫_^K-1μ_t_1,…,t_K-1(A_1×…× A_K)dP(t_1,…,t_K-1), where A_1,…,A_K∈ℬ. A Markov operator Ψ:L^2(X,ℬ,μ)→ L^2(X,ℬ,μ) is a linear operator which satisfies * it is a contraction, that is Ψ f_2≤f_2 for every f∈ L^2(X,ℬ,μ); * f≥ 0⇒Ψ(f)≥ 0; * Ψ(1)=1=Ψ^*(1). With every 2-self joining ∈ J_2(𝒯), we can associate a Markov operator Ψ():L^2(X,ℬ,μ)→ L^2(X,ℬ,μ) such that(A× B)=∫_XΨ()(χ_A)χ_Bdμ for any A,B∈ℬ.Denote by 𝒥(𝒯) the set of all Markov operators which commute with the Koopmanoperator associated with 𝒯. It appears that if we consider weak-⁎ topology on 𝒥(𝒯), then (<ref>) defines an affine homeomorphism Ψ:J_2(𝒯)→𝒥(𝒯).For more information about joinings and Markov operators we refer to <cit.>.Considerthe affine continuous map Π_1,3:J_3(𝒯)→ J_2(𝒯) given byΠ_1,3()(A× B):=(A× X× B) for any A,B∈ℬ.In other words Π_1,3() is the projection of the joiningon the first and thirdcoordinates. Analogously, we define Π_2,3, the projection on the second and thirdcoordinates. Since J_2(𝒯) and 𝒥(𝒯) are affinely homeomorphic, we can considerthe affine continuous maps Ψ∘Π_i,3:J_3(𝒯)→𝒥(𝒯) for i=1,2. Note that for any t,s∈ we haveΨ∘Π_1,3(μ_t,s)=T_-t, and Ψ∘Π_2,3(μ_t,s)=T_-s.For i∈{1,2} let σ_i:^2→ be the projection onthe i-th coordinate. Then for every P∈𝒫(^2), we also haveΠ_i,3(∫_^2μ_-t,-s dP(t,s))=∫_^2Π_i,3(μ_-t,-s) dP(t,s) andΨ∘ Π_i,3(∫_^2μ_-t,-s dP(t,s))=∫_T_td((σ_i)_*P)(t),for i=1,2.§.§ Special flows. Let (X,ℬ,μ) be a standard Borel probability space. Let T:X→ X be an ergodicμ-preserving automorphism. Let f∈ L^1([0,1)) be positive and for any n∈ considerf^(n)(x):= ∑_i=0^n-1f(T^ix)if n≥ 10if n=0-∑_i=n^-1f(T^ix)if n≤ -1.Define X^f:=((x,r);x∈ X, 0≤ r<f(x)) and on X^f considerthe measure μ⊗Leb(|_X^f.The special flow T^f={T^f_t}_t∈ isthe measure preserving flow acting on X^f by the formulaT^f_t(x,r):=(T^nx,r+t-f^(n)(x)),where n∈ is unique, such that f^(n)(x)≤ r+t<f^(n+1)(x). We say that f is the roof function and T is the base ofthe special flow. In view of Ambrose Representation Theorem (see <cit.>), every ergodic flow is measure theoretically isomorphic to a special flow. Such a special flow is called a special representation of the flow. In this paper we deal with special flows whose roof functions are piecewise continuous andwhose bases are interval exchange transformations. We always assume that roof functions are right-continuous and that the left limits exist. If a piecewise continuous bounded function f has a discontinuity at x, thenthe jump at x is the number d:=f(x)-lim_y→ x^-f(x).§.§ Space of flowsLet (X,ℬ(X),μ) be a standard Borel probability space. By Flow(X) we denote the set of all measure preserving flows on X. Let 𝒯={T_t}_t∈∈Flow(X), A∈ℬ(X) and >0. LetU(𝒯,A,):={𝒮={S_t}_t∈∈Flow(X);sup_t∈[-1,1]μ(T_tA S_tA)<}.It appears that the family of sets of the above form gives a subbase of a topology, and Flow(X) endowed with this topology is a Polish space.Let (Y,d) be a metric space. It follows that a map F:Y→Flow(X) is continuous if for any y∈ Y and A∈ℬ(X) we havefor any >0 there exists >0 such thatd(y,z)<⇒ F(z)∈ U(F(y),A,).By using the fact that for any A_1,B_1,A_2,B_2∈ℬ(X) we haveA_1 B_1=A_1^c B_1^c and (A_1∪ B_1)(A_2∪ B_2)⊆(A_1 A_2)∪(B_1 B_2),we can prove thatthe set of all A∈ℬ(X), for which for every >0 there exists _A such that (<ref>) is satisfied, form an algebra.By using the triangle inequalityμ(A B)≤μ(A C)+μ(B C) for A,B,C∈ℬ(X), we can prove that this algebra is closed under taking the countable union of increasing family of sets and thus, it is a σ-algebra. Hence it is enough to check (<ref>) for a family of sets which generates ℬ(X).All non-atomic standard Borel probability spaces are measure theoretically isomorphic (see Theorem 3.4.23 in <cit.>). Let (X_1,ℬ(X_1),μ_1) and (X_2,ℬ(X_2),μ_2) bestandard Borel non-atomic probability spaces and letH:X_1→ X_2 besome isomorphism. Then Flow(X_1) and Flow(X_2) can be identified by a homeomorphism ϕ:Flow(X_1)→Flow(X_2) given by the formulaϕ(𝒯):=H∘𝒯∘ H^-1. To prove that F:(Y,d)→Flow(X_1) is continuous, we can instead prove that ϕ∘ F:(Y,d)→Flow(X_2) is continuous. In other words, we need to prove that for every y∈ Y and A∈𝒟⊂ℬ(X_2), where 𝒟 generates ℬ(X_2), we have for any >0 there exists >0 such thatd(y,z)<⇒ϕ∘ F(z)∈ U(ϕ∘ F(y),A,). §.§ Interval exchange transformationsLet 𝒜 be an alphabet of d elements. Let now ϵ∈{0,1} and let π_ϵ:𝒜→{1,…,d} be bijections. We will now considera permutation π as a pair {π_0,π_1} where π_0() corresponds to the position of letterbefore permutation, while π_1() defines the position ofafter permutation. We say that apermutation π is irreducible if there is no 1≤ k<d such thatπ_1∘π_0^-1({1,…,k})={1,…,k}.In this paper we will only deal with irreducible permutations, so this assumption will usually be omitted. We say that the permutation is symmetric ifπ_1()=d+1-π_0() for every ∈𝒜.Note that a symmetric permutation is always irreducible.Theintervals that we will now consider are always left-side closed and right-side open unless told otherwise. Let I be an interval equipped withits Borel σ-algebra and Lebesgue measure Leb(.Without losing generality, we can assume that the left endpoint of I is 0. Let {I_}_∈𝒜 be a partition of I into d intervals, where I_ has length _≥ 0. We will denote :={_}_∈𝒜 the length vector and obviously we have ||:=∑_∈𝒜=Leb(I).The interval exchange transformation (IET) T_π,:I→ I isthe automorphism which permutes intervals I_ according to the permutation π. Let now Ω_π:=[(Ω_π)_β]_,β∈𝒜 bethe d× d matrix given by the following formula(Ω_π)_β:=+1if π_0()<π_0(β) and π_1()>π_1(β);-1if π_0()>π_0(β) and π_1()<π_1(β);0 otherwise.We will say that Ω_π isthe translation matrix of T^π,. The name of the matrix is derived from the fact that T_π, acts on an interval I_ as a translation by number ∑_β∈𝒜(Ω_π)_β_β.Let ∂ I_ be the left endpoint of I_. We say thatthe IET T_π, satisfies Keane's condition ifT^m_π,(∂ I_)=∂ I_β for m>0 implies =π_1^-1(1),β=π_0^-1(1) and m=1.It is easy to see that T_π, satisfies Keane's conditionwheneveris a rationally independent vector (that is there is no nontrivial integer linear combination of numbers _, which will givea rational number).Denote by S_𝒜^0 the set of all irreducible permutations of𝒜. We may consider the space S_𝒜^0×_≥0^𝒜 of all IETs of d intervals. Definethe operator R:S_𝒜^0×_≥0^𝒜→ S_𝒜^0×_≥0^𝒜, such that R(π,)=R(T_π,) isthe first return map of T_π, to the interval [0,||-min{_π_0^-1(d),_π_1^-1(d)}). The operator R is called the Rauzy-Veech induction (or righthand side Rauzy-Veech induction). The Rauzy classes are the minimal subsets of S_𝒜^0 which are invariant under the induced action of R on S_𝒜^0. Letl:{1,…,d}→{1,…,d} be given by l(i)=d+1-i. The function l acts on S_𝒜^0 by mapping {π_0,π_1} onto {l∘π_0,l∘π_1}. The extended Rauzy classes are minimal subsets of S_𝒜^0 which are invariant under R and action of l. We have the following result. [Rauzy] Any Rauzy class of permutations of d≥ 2 elements contains at least one permutation π such that π_1∘π_0^-1(1)=dandπ_1∘π_0^-1(d)=1.§.§ Translation surfaces andmoduli space.Let M be an orientable compact and connected topological surface of genus g≥ 1. Let Σ:={A_1,…,A_s} be a finite subset of singular points in M. Let κ:=(κ_1,…,κ_s) be a vector of positive integers satisfying ∑_i=1^sκ_i=2g-2. A translation structure on M is a maximal atlas ζ on M∖Σ of charts by open sets of ℂ such that any coordinate change between charts is a translation and for each 1≤ i≤ s there exists a neighbourhood V_i⊂ M of A_i, a neigbourhood W_i⊂ℂ of 0 and a ramified covering π:(V_i,A_i)→(W_i,0) of degree κ_i+1 such that each injective restriction of π is a chart of ζ. On (M,ζ) we can consider a holomorphic 1-form which in the local coordinates can be written as dz. We will denote this form also by ζ. We identify the associated 2-form i/2ζ∧ζ̅ with the Lebesgue measure _ζ on M. Moreover, the quadratic form |ζ|^2 yields a Riemannian metric (M,ζ). By d_ζ we denote the distance given by this metric. We also consider on (M,ζ) a vertical translation flow, denoted by 𝒯^ζ={𝒯^ζ_t}_t∈, which inlocal coordinates is a unit speed flow in the vertical direction. The flow 𝒯^ζ preserves _ζ and thus can be viewed as an element of Flow(M,_ζ). In the set of all translation structures on M we identifythe structures ζ_1 and ζ_2 if there exists a homeomorphism H:M→ M which sends singular points of ζ_1 onto singular points of ζ_2 and ζ_1=H^*ζ_2. In terms of local coordinates, H is locally a translation. This is an equivalence relation and its equivalence classes form the moduli space denoted by Mod(M). The moduli space can be divided into subsets called strata ℳ(M,Σ,κ)=ℳ(κ), for which the vector of degrees of singularities is given by κ. It appears that each such stratum ℳ(M,Σ,κ) is a complex orbifold (see <cit.>) and has a finite number of connected components (see <cit.>). On ℳ we can consider an action of SL(2,). It is given by composing the charts of a translation surface witha given linear map. The strata are invariant under the action of SL(2,). It is worth noting that in particular for every θ∈/ we can apply the rotation r_θ by θ to the translation structure and almost every angle θ yields no saddle connection of a vertical flow. Let π=(π_0,π_1) be a permutation of the alphabet 𝒜 of d>1 elements and let ∈^𝒜_≥0. Consider alsoτ∈^𝒜 such that for each 1≤ k<d we have ∑_{;π_0()≤ k}τ_>0 and ∑_{;π_1()≤ k}τ_<0.Moreover we require thatπ_i()=π_i(β)+1 ∧ _=_β=0 ⇒ τ_·τ_β>0 for all i=0,1 and ,β∈𝒜.For a fixed permutation π, we denote by Θ_π the set of triples (π,,τ) satisfying the above conditions.Consider thepolygonal curve in ℂ obtained by connecting the points 0 and ∑_i≤ k(_π^-1_0(i)+iτ_π^-1_0(i)) for k=1,…,d, using theline segments. Analogously we can consider thepolygonal curve obtained by connecting the points 0 and ∑_i≤ k(_π^-1_1(i)+iτ_π^-1_1(i)) for k=1,…,d. These twopolygonal curves define a polygon with d pairs of parallel sides. By identifying those sides we obtain a translation surface M, with Σ being the set of vertices of the polygon (some of them may be identified). We denote by M(π,,τ) the translation structure given by (π,,τ).It appears that whenever 𝒯^ζ admits no saddle-connections, ζ can be viewed as M(π,,τ) for some (π,,τ)∈Θ_π, with π being some permutation (seee.g. <cit.>). Moreover we can consider (π,,τ)∈Θ_π as local coordinates in the neighbourhood of such ζ in the corresponding stratum. Since almost every rotation yields no saddle-connections, to obtain local coordinates in the neighbourhood of ζ for which 𝒯^ζ has a saddle connection, we can use the rotation to obtain local coordinates around rotated form and then rotate it back. Kontsevich and Zorich in <cit.> gave a complete characterization of connected components of strata in the moduli space.In particular, they showed that each stratum ℳ(2g-2) and ℳ(g-1,g-1), where g is the genus of the surface, contains exactly one so-called hyperelliptic connected component, which we denote by ℳ^hyp(2g-2) and ℳ^hyp(g-1,g-1) respectively.For every hyperelliptic component C⊂ℳ, there exists an involution ϕ:M→ M such that for every ζ∈ C we have ϕ^*ζ=-ζ. In particular we have the following remark.For every hyperelliptic connected component C⊂ℳ and for every ζ∈ G, the vertical flow on (M,ζ) is isomorphic with its inverse. It appears that the connected components of the moduli space can be described by the Rauzy classes of permutations.Let us recall firstthe notion of non-degenericity, as introduced by Veech.We say that a permutation π={π_0,π_1} of 𝒜 is degenerate if one of the following conditions is satisfied: π_1∘π_0^-1(j+1)=π_1∘π_0^-1(j)+1 for some1≤ j< d;π_1∘π_0^-1(π_0∘π_1^-1(d)+1)=π_1∘π_0^-1(d)+1π_0∘π_1^-1(1)-1=π_0∘π_1^-1(π_1∘π_0^-1(1)-1)π_0∘π_1^-1(d)=π_0∘π_1^-1(1)-1 and π_1∘π_0^-1(d)=π_1∘π_0^-1(1)-1. Otherwise the permutation is called non-degenerate. The property of non-degenericity is invariant under the action of the Rauzy-Veech induction.The importance of this notion is given by the following theorem.[Veech] The extended Rauzy classes of nondegenerate permutations are in one-to-one correspondence with the connected components of the strata in the moduli space. In view of the above theorem, for each genus g≥ 2, the hyperelliptic components ℳ^hyp(2g-2) and ℳ^hyp(g-1,g-1) correspond to the extended Rauzy classes of symmetric permutations of 2g and 2g-1 elements respectively.Connected components which are associated with extended Rauzy graphs of permutations of d≤ 5 elements are hyperelliptic. Fora given extended Rauzy class ℛ, let C_ℛ be its associated connected component of the moduli space.Then for any π∈ℛ the map M:Θ_π→ C_ℛ given by (π,,τ)↦ M(π,,τ) is continuous and the range of the map M is dense in C_ℛ. Moreover, recall that, due to Theorem <ref>, for every connected component of the moduli space we can find a permutation π̅ belonging to the corresponding extended Rauzy class, satisfyingπ̅_1∘π̅_0^-1(d)=1 andπ̅_0∘π̅_1^-1(d)=1. Hence, to prove that some condition is satisfied for a dense set of translation structures in C_ℛ, it is enough to prove that it holds for translation structures, whose associated polygonal parameters belong to a dense subsetof Θ_π̅. Let ℛ be any extended Rauzy class.Let usconsider a transformation R̃:⋃_π∈ℛΘ_π↦⋃_π∈ℛΘ_π called apolygonal Rauzy Veech induction (or righthand side polygonal Rauzy Veech induction) which yields different parameters of a translation surface.Let π∈ℛ and let (π,,τ)∈Θ_π. Assume that _π_0^-1(d)≠_π_1^-1(d). If _π_0^-1(d)<_π_1^-1(d), then for any a∈𝒜 defineπ̃_0(a) :=π_0(a)if π_0(a)≤π_0(π_1^-1(d)); π_0(π_1^-1(d))+1if π_0(a)=d; π_0(a)+1if π_0(π_1^-1(d))<π_0(a)≤ d-1,π̃_1(a) :=π_1(a), _a :=_π_1^-1(d)-_π_0^-1(d)if π_1(a)=d; _aotherwise,τ̃_a :=τ_π_1^-1(d)-τ_π_0^-1(d)if π_1(a)=d; τ_aotherwise.Analogously, if _π_0^-1(d)>_π_1^-1(d), we defineπ̃_0(a) :=π_0(a),π̃_1(a) :=π_1(a)if π_1(a)≤π_1(π_0^-1(d)); π_1(π_0^-1(d))+1if π_1(a)=d; π_1(a)+1if π_1(π_0^-1(d))<π_1(a)≤ d-1,_a :=_π_0^-1(d)-_π_1^-1(d)if π_0(a)=d; _aotherwise,τ̃_a :=τ_π_0^-1(d)-τ_π_1^-1(d)if π_0(a)=d; τ_aotherwise.We define R̃ by setting R̃(π,,τ):=(π̃,,τ̃). It is defined almost everywhere on ⋃_π∈ℛΘ_π and if M(π,,τ) admits no saddle connection, it can be iterated indefinitely. Similarly, we can also define a left hand side polygonal Rauzy Veech induction. Note that the polygons derived from (π,,τ) and (π̃,,τ̃) represent the same translation surface, i.e. M(π,,τ)=M(π̃,,τ̃). Indeed, the latter is obtained from (π,,τ) by cutting out the triangle formed by the last top side and the last bottom side and gluing it to a side of a polygon which is identified with one of the two sides forming the triangle.Every ζ∈ C_π which does not have vertical saddle-connections can be represented as M(π,,τ), for some (π,,τ)∈Θ_π. We can consider the metric on the neighbourhood of M(π,,τ) on ℳ(M,Σ,κ) given by d_Mod((π,',τ'),(π,”,τ”)):=∑_a∈𝒜(|_a'-_a”|+|τ_a'-τ_a”|). If ζ admits vertical saddle-connections, we can apply r_θ for some θ∈/, so that r_θ^*ζ does not have vertical saddle-connections and then define a metric in the neighbourhood of ζ.For any ζ=M(π,,τ)∈ℳ(M,Σ,κ) we can consider a special representation of the vertical flow on (M,ζ). The basis of this special flow is the IET T_π, and the roof function h is positive and constant over exchanged intervals. Hence h can be considered as a vector (h_a)_a∈𝒜∈_>0^𝒜, where h_a is the value of h over the exchanged interval labelled by a. The vector h is given by the formulah=-Ω_πτ,where Ω_π isthe translation matrix of (π,). This gives rise tonew local coordinates of the moduli space. In particular, the polygonal Rauzy-Veech induction receives a new form. Namely, if _π_0^-1(d)≠_π_1^-1(d) then R̃(π,,h)=(π̃,,h̃), where the formulas for π̃ andremain unchangedand for any a∈𝒜 we takeh̃_a:= h_π_0^-1(d)+h_π_1^-1(d) if a=π_0^-1(d) and _π_0^-1(d)<_π_1^-1(d);h_π_0^-1(d)+h_π_1^-1(d) if a=π_1^-1(d) and _π_0^-1(d)>_π_1^-1(d);h_a otherwise.§ CONSEQUENCES OF LIMIT JOININGSIn this section, we formulate a criterion for two flows to be disjoint, and a criterion for a flow to be weakly mixing. Both criteria rely on the properties of the weak limit ofsome sequence of 3-off diagonal joinings. For every measure ∈𝒫(X× Y), we denote by |_X and |_Y the projections ofon X and Y respectively, that is for every measurable subsets A⊆ X and B⊆ Y we have|_X(A)=(A× Y)and|_Y(B)=(X× B). Let 𝒯={T_t}_t∈ and 𝒮={S_t}_t∈ be weakly mixing flows acting on standard Borel spaces (X,ℬ,μ) and (Y,𝒞,ν) respectively.Let ∈ J^e(𝒯, 𝒮). Let ρ∈ J^e_2(𝒯×𝒮,), which is defined on X_1× Y_1× X_2× Y_2 with X_1=X_2=X and Y_1=Y_2=Y. Assume that for some r,r'∈ we have ρ|_X_1× X_2=μ_T_r and ρ|_Y_1× Y_2=ν_S_r'. If r≠ r' then =μ⊗ν.First we prove that =(T_r× S_r')_*. We show that (3) inRemark <ref> is satisfied for the π-system of product setsand the isomorphism ϕ:=T_-r× S_-r' between (X_1× Y_1,) and (X_2× Y_2,). In other words, for every A∈ℬ and B∈𝒞 we have ρ(A× B× (T_-r× S_-r')(A× B)^c)=ρ((A× B)^c× (T_-r× S_-r')(A× B))=0.Indeed, recall that μ_r and ν_r' are graph joinings of 𝒯 and 𝒮 given by T_-r and S_-r' respectively. By Remark <ref> this implies that for every A∈ℬ and B∈𝒞 we haveμ_r(A× T_-rA^c)=0andν_r'(B× T_-rB^c)=0.Thus we obtainρ(A× B × (T_-r× S_-r')(A× B)^c)=ρ(A× B× T_-rA^c× S_-r'B)+ρ(A× B× T_-rA^c× S_-r'B^c) +ρ(A× B× T_-rA× S_-r'B^c)≤2ρ(A× Y× T_-rA^c× Y)+ρ(X× B× X× S_-r'B^c)=2μ_r(A× T_-rA^c)+ν_r'(B× S_-r'B^c)=0andρ((A× B)^c × T_-rA× S_-r'B)=ρ(A^c× B× T_-rA× S_-r'B)+ρ(A^c× B^c× T_-rA× S_-r'B) +ρ(A× B^c× T_-rA× S_-r'B)≤2ρ(A^c× Y× T_-rA× Y)+ρ(X× B^c× X× S_-r'B)=2μ_r(A^c× T_-rA)+ν_r'(B^c× S_-r'B)=0.Hence we have proved that (3) in Remark <ref> is satisfied forthe π-system of product sets. Since ρ∈ J^e_2(𝒯×𝒮,), in view of (2) in Remark <ref> we get (A× B) =ρ(A× B× X× Y)=ρ(X× Y× T_-rA× S_-r'B) =(T_-rA× S_-r'B)=(T_r× S_r')_*(A× B),for all A∈ℬ and B∈𝒞. Since the π-system of product sets generates ℬ⊗𝒞, we get that the measuresand (T_r× S_r')_* are equal. By the (𝒯×𝒮)-invariance of , we have thatis (Id× S_r-r')-invariant. By weak mixing of 𝒮, S_r-r' is ergodic whenever r≠ r'. Since Id is disjoint with every ergodic transformation (see Remark <ref>), we get =μ⊗ν. Assume that for some real sequences (a_n)_n∈ and (b_n)_n∈ we haveμ_a_n,b_n→(1-)∫_^2μ_-t,-udP(t,u)+ξ_1,andν_a_n,b_n→(1-')∫_^2ν_-t,-udQ(t,u)+'ξ_2,for some 0≤,'< 1, measures P,Q∈𝒫(^2) and ξ_1∈ J_3(𝒯), ξ_2∈ J_3(𝒮). Assume moreover, that there exists a set B∈ℬ(^2), such that(1-)P(B)-(1-')Q(B)>'.Then 𝒯 and 𝒮 are disjoint.The above proposition can be also proven in higher dimensional case, that is when we consider limits of joinings of higher rank. Let ξ_1=∫_J_3^e(𝒯)ρ^𝒯 d κ_1(ρ^𝒯) and ξ_2= ∫_J_3^e(𝒮)ρ^𝒮 dκ_2(ρ^𝒮) be the ergodic decompositions of ξ_1 and ξ_2 respectively. Let also 𝒜_1 be the set of 3-off-diagonal joinings in J_3^e(𝒯) and 𝒜_2 be the set of 3-off-diagonal joinings in J_3^e(𝒮). In view of Souslin theorem the sets 𝒜_1 and 𝒜_2 are measurable. We can assume that κ_1(𝒜_1)=κ_2(𝒜_2)=0. Indeed, let β:=1-κ_1(𝒜_1)≥ 0 and β':=1-κ_2(𝒜_2)≥ 0. Then ξ_1=(1-β)∫_^2μ_-t,-udP'(t,u)+βξ_1'andξ_2=(1-β')∫_^2ν_-t,-udQ'(t,u)+β'ξ_2',where ξ_1'∈ J_3(𝒯) and ξ_2'∈ J_3(𝒮) do not have 3-off-diagonal joinings in their ergodic decomposition. Thenμ_a_n,b_n →(1-)∫_^2μ_-t,-udP(t,u)+((1-β)∫_^2μ_-t,-udP'(t,u)+βξ_1')=(1-β)∫_^2μ_-t,-ud(1-/1-βP+(1-β)/1-βP')+βξ_1'=(1-β)∫_^2μ_-t,-udP̅+βξ_1',where P̅=1-/1-βP+(1-β)/1-βP'. Analogouslyν_a_n,b_n→(1-'β')∫_^2ν_-t,-udQ̅+'β'ξ_2',where Q̅=1-'/1-'β'Q+'(1-β')/1-'β'Q'. Then for the set B satisfying (<ref>) we have(1 -β)P̅(B)-(1-'β')Q̅(B)=(1-)P(B)+(1-β)P'(B)-(1-')Q(B)-'(1-β')Q'(B)>'+(1-β)P'(B)-'(1-β')Q'(B)≥'-'(1-β')='β'.It is enough then, to replace P,Q by P̅,Q̅ and ,' by β,'β' respectively.Let ∈ J^e(𝒯,𝒮). We show that =μ⊗ν. Consider the sequence {_a_n,b_n}_n∈ in J_3^e(𝒯×𝒮, ). By the compactness of J_3(𝒯×𝒮, ) we have that _a_n,b_n→η weakly in J_3(𝒯×𝒮, ), up to taking a subsequence. Moreover, by assumptions we haveη|_X_1× X_2× X_3=(1-)∫_^2μ_-t,-udP(t,u)+ξ_1andη|_Y_1× Y_2× Y_3=(1-')∫_^2ν_-t,-udQ(t,u)+'ξ_2.Let h^𝒯:^2→𝒜_1 and h^𝒮:^2→𝒜_2 be given byh^𝒯(t,u) :=μ_-t,-u and h^𝒮(t,u):=ν_-t,-u. Thenη|_X_1× X_2× X_3=∫_J_3^e(𝒯) ρ^𝒯d((1-)h^𝒯_*P+κ_1)(ρ^𝒯),andη|_Y_1× Y_2× Y_3=∫_J_3^e(𝒮)ρ^𝒮d((1-') h^𝒮_*Q+'κ_2)(ρ^𝒮).Let now η=∫_J_3^e(𝒯×𝒮,)ψ dκ(ψ) be the ergodic decomposition of η. Then we haveη|_X_1× X_2× X_3=∫_J_3^e(𝒯×𝒮,)ψ|_X_1× X_2× X_3 dκ(ψ),andη|_Y_1× Y_2× Y_3=∫_J_3^e(𝒯×𝒮,)ψ|_Y_1× Y_2× Y_3 dκ(ψ).Since ψ∈ J_3^e(𝒯×𝒮), we have ψ|_X_1× X_2× X_3∈ J_3^e(𝒯) and ψ|_Y_1× Y_2× Y_3∈ J_3^e(𝒮). Consider Ω^𝒯:J_3^e(𝒯×𝒮,)→ J_3^e(𝒯) and Ω^𝒮:J_3^e(𝒯×𝒮,)→ J_3^e(𝒮) given byΩ^𝒯(ψ)=ψ|_X_1× X_2× X_3 and Ω^𝒮(ψ)=ψ|_Y_1× Y_2× Y_3.We haveη|_X_1× X_2× X_3=∫_J_3^e(𝒯)ρ^𝒯 d(Ω_*^𝒯κ)(ρ^𝒯),andη|_Y_1× Y_2× Y_3=∫_J_3^e(𝒮)ρ^𝒮 d(Ω_*^𝒮κ)(ρ^𝒮).By comparing this with (<ref>) and (<ref>) and using the uniqueness of ergodic decomposition we obtain thatΩ_*^𝒯κ=(1-)h_*^𝒯 P+κ_1 and Ω_*^𝒮κ=(1-')h_*^𝒮 Q+'κ_2. Let now𝒜:={ψ∈ J_3^e(𝒯×𝒮,): ∃ t,u,t',u'∈, (t,u)≠(t',u'),ψ|_X_1× X_2× X_3=μ_-t,-u,ψ|_Y_1× Y_2× Y_3=ν_-t',-u'}. We now show that κ(𝒜)>0. For any measurable subsets C⊂ J_3^e(𝒯) and D⊂ J_3^e(𝒮) denote by C×̅D the set of all ψ∈ J_3^e(𝒯×𝒮,) such that ψ|_X_1× X_2× X_3∈ C and ψ|_Y_1× Y_2× Y_3∈ D.Assume that κ(𝒜)=0. Let B be the set satisfying (<ref>). If (t,u)∈ B then by the definition of h^𝒯 and h^𝒮 we have μ_-t,-u∈ h^𝒯(B) and ν_-t,-u∈ h^𝒮(B). Moreover κ(𝒜)=0 and h^𝒯(B)×̅(𝒜_2∖ h^𝒮(B)) ⊂𝒜 yieldκ(h^𝒯(B)×̅𝒜_2)=κ(h^𝒯(B)×̅h^𝒮(B)). Note that κ_1(h^𝒯(B))≤κ_1(𝒜_1)=0. Hence, (<ref>) and (<ref>) implies(1-)P(B) =(1-)h_*^𝒯 P(h^𝒯(B))=[(1-)h_*^𝒯P+κ_1](h^𝒯(B))=Ω_*^𝒯κ(h^𝒯(B))=κ(h^𝒯(B)×̅J_3^e(𝒮))=κ(h^𝒯(B)×̅𝒜_2)+κ(h^𝒯(B)×̅(J_3^e(𝒮)∖𝒜_2))=κ(h^𝒯(B)×̅h^𝒮(B))+κ(h^𝒯(B)×̅(J_3^e(𝒮)∖𝒜_2)).Analogously wealso obtain(1-') Q(B)=(1-')h_*^𝒮 Q(h^𝒮(B))=κ(h^𝒯(B)×̅h^𝒮(B))+κ((J_3^e(𝒯)∖𝒜_1)×̅h^𝒮(B)). Moreover, in view of (<ref>) we getκ(h^𝒯(B)×̅(J_3^e(𝒮)∖𝒜_2)) ≤κ(J_3^e(𝒯)×̅(J_3^e(𝒮)∖𝒜_2))=Ω_*^𝒮κ(J_3^e(𝒮)∖𝒜_2)='κ_2(J_3^e(𝒮)∖𝒜_2)= '. Since (1-)P(B)-(1-')Q(B)>', by substracting (<ref>) and (<ref>) we obtain' <(1-)h_*^𝒯 P(h^𝒯(B))-(1-')h_*^𝒯 Q(h^𝒮(B))=(κ(h^𝒯(B)×̅h^𝒮(B))+κ(h^𝒯(B)×̅(J_3^e(𝒮)∖𝒜_2))) -(κ(h^𝒯(B)×̅h^𝒮(B))+κ((J_3^e(𝒯)∖𝒜_1)×̅h^𝒮(B)))=κ(h^𝒯(B)×̅(J_3^e(𝒮)∖𝒜_2))-κ((J_3^e(𝒯)∖𝒜_1)×̅h^𝒮(B))≤',which is a contradiction. This yields κ(𝒜)>0 and hence 𝒜 is non-empty. Therefore, there exists ψ∈𝒜⊂ J_3^e(𝒯×𝒮,) such that ψ|_X_1× X_2× X_3=μ_t,u and ψ|_Y_1× Y_2× Y_3=ν_t',u' with (t,u)≠(t',u'). Assume that t≠ t' (the case when u≠ u' is analogous).Then ϕ:=Π_1,3(ψ)∈ J_2^e(𝒯×𝒮,) satisfiesϕ|_X_1× X_3=μ_t and ϕ|_Y_1× Y_3=ν_t'.Thus, by Lemma <ref>, =μ⊗ν. The above criterion strengthens the results obtained in <cit.>, that is the flows described in this paper are not only non-isomorphic with their inverses, but also disjoint. To prove the main result of this paper, we use the following simplified version ofProposition <ref>. Let 𝒯={T_t}_t∈ and 𝒮={S_t}_t∈ be weakly mixing flows acting on the standard Borel spaces (X,ℬ,μ) and (Y,𝒞,ν) respectively. Assume that for some real sequences (a_n)_n∈ and (b_n)_n∈ we have μ_a_n,b_n→∫_^2μ_-t,-udP(t,u)andν_a_n,b_n→∫_^2ν_-t,-udQ(t,u), for some measures P,Q∈𝒫(^2). If P≠ Q, then 𝒯 and 𝒮 are disjoint.Let ξ:^2→ be given by ξ(t,u):=t-2u. The following result givesa condition on limit joinings which imply weak mixing of a flow. Let 𝒯={T_t}_t∈ be an ergodic flow on (X,ℬ,μ) and assume that there exists a real increasing sequence {b_n}_n∈, a real number ρ∈[0,1) and a probability measure P∈𝒫(^2) such that μ_2b_n,b_n→(1-ρ)∫_^2μ_-t,-udP(t,u)+ρψ, for some ψ∈ J_3(𝒯). If P is not supported on an affine lattice in ^2 then 𝒯 is weakly mixing. In particular, if there exist two rationally independent real numbers d_1 and d_2 such that d_1,d_2 and 0 are atoms of ξ_*P, then the flow 𝒯 is weakly mixing. Assume that P is not supported on an affine lattice and the flow 𝒯 is not weakly mixing. Then there exists a non-zero function f∈ L^2(X,μ) and a∈∖{0} such that ∀ t∈, f∘ T_t=e^-2π iatf. Recall that σ_1:^2→ denotes the projection on the first coordinate. By applying Ψ∘Π_1,3 (see (<ref>)) to (<ref>) and using (<ref>) and (<ref>), we obtain T_2b_n→ (1-ρ)∫_ T_t dP_1(t)+ρΨ_1, where P_1:=(σ_1)_*P and Ψ_1 is a Markov operator. Let ⟨·,·⟩ be the scalar product on L^2(X,μ). By (<ref>), we get f^2=|⟨ f,f⟩|=|⟨ f,e^-2π iatf⟩|=|⟨ f,f∘ T_t ⟩|=|⟨ f,f∘ T_2b_n⟩| for every n∈. As n→∞, we get f^2=|⟨ f,f∘ T_2b_n⟩|=|⟨ f,(1-ρ)∫_ f∘ T_t dP_1(t)+ρΨ_1(f) ⟩|. On the other hand by the fact that Markov operator is a contraction, we get |⟨f,(1-ρ)∫_ f∘ T_t dP_1(t)+ρΨ_1(f) ⟩|≤(1-ρ)|⟨ f,∫_ f∘ T_t dP_1(t)⟩|+ρ|⟨ f,Ψ_1(f) ⟩|≤ (1-ρ)|∫_⟨ f,f∘ T_t⟩ dP_1(t)|+ρf^2=(1-ρ)f^2|∫_ e^-2π iat dP_1(t)|+ρf^2 Thus we get |∫_ e^-2π iatdP_1(t)|=1 that is ∫_ e^-2π iatdP_1(t)=e^-2π ibfor some b∈. It follows that ∫_ e^-2π i(at-b)dP_1(t)=1. This implies P_1({t∈; at-b∈})=1. Consider now P_2:=(σ_2)_*P. Analogously, by applying Ψ∘Π_2,3 to (<ref>), we get P_2({u∈; au-c∈})=1for some c∈. Combining the two above results, we finally obtain P({(t,u)∈^2; a(t,u)-(b,c)∈^2 })=1, which is a contradiction with our assumption. Thus if P is not supported on an affine lattice then the flow 𝒯 is weakly mixing. Suppose now that ξ_*P has atoms at points 0,d_1 and d_2. Assume again that 𝒯 is not weakly mixing and that e^2π i a, a≠ 0, is an eigenvalue. By the definition of ξ, the lines (x,1/2(x-d_i)) for i=1,2 and (x,1/2x) have positive measure P. This together with (<ref>) yields x_0,x_1,x_2∈, such that a(x_0,12x_0)-(b,c)∈^2,a(x_1,12(x_1-d_1))-(b,c)∈^2,a(x_2,12(x_2-d_2))-(b,c)∈^2. This implies a(x_1-x_0,12(x_1-x_0)-12d_1)∈^2,a(x_2-x_0,12(x_2-x_0)-12d_2)∈^2. By applying ξ to the above, we get that ad_1∈ and ad_2∈. Since a,d_1,d_2≠0, we get that here (ad_1)d_2-(ad_2)d_1=0 is a non-trivial integer combination of d_1 and d_2. By the rational independence of d_1 and d_2 this yields a=0. This is a contradiction, hence 𝒯 is weakly mixing. § ACCEPTABLE PERMUTATIONSIn this section, we establish a technical result concerning a particular non-degenerate permutation, which plays a key role in proving that our main result applies to all non-hyperelliptic connected components ofthe moduli space. In particular, in view of the Remark <ref>, we assume that the alphabet we consider has d≥ 6 elements. Recall that in every Rauzy class we can fix a non-degenerate permutationπ={π_0,π_1} satisfying π_1∘π_0^-1(1)=dandπ_1∘π_0^-1(d)=1. We have the following theorem. In every Rauzy class corresponding to a non-hyperelliptic connected component ofthe moduli space ℳ, there exists a permutation π={π_0,π_1} satisfying (<ref>) such that there exist distinct symbols _1,_2,_1,_2∈𝒜∖{π_0^-1(1),π_0^-1(d)} satisfyingthe three following properties Ω__1_2=Ω__2_1=0, Ω__1_2Ω__2_1= 0 Ω__1_1Ω__2_2≠0,where Ω:=Ω_π isthe associated translation matrix.Let π={π_0,π_1} be a non-degenerate permutation satisfying (<ref>) that belongs to a Rauzy class associated with a non-hyperelliptic connected component. Then it is not symmetric,henceits translation matrix Ω contains zero entries outside the diagonal. Indeed, assume contrary to our claim thatπ_0()<π_0(β)⇔π_1()>π_1(β) for all ,β∈𝒜.Then for every ∈𝒜π_1()=#{β∈𝒜; π_1(β)<π_1()}+1=#{β∈𝒜; π_0(β)>π_0()}+1=d-π_0()+1.Hence π is a symmetric permutation. We need to consider two cases separately. Case 1. Assume first that there exists a symbol ∈𝒜 such that for all symbols β∈𝒜 with 1<π_0(β)<d we haveπ_0()< π_0()⇔π_1()< π_1()that isΩ_β=0for allβ∈𝒜∖{π_0^-1(1),π_0^-1(d)}.Since π is non-degenerate, there exist symbols _1,_1 such that1<π_0(_1)<π_0(_1)<π_0()and 1<π_1(_1)<π_1(_1)<π_1().Otherwise, π satisfies (<ref>) and hence, it is degenerate. Similarly, there exist symbols _2,_2 such thatd>π_0(_2)>π_0(_2)>π_0()and d>π_1(_2)>π_1(_2)>π_1().Otherwise, π satisfies (<ref>) and it is again degenerate. Thus we haveΩ__1_2=Ω__2_1=Ω__1_2=Ω__2_1=0andΩ__1_1=1andΩ__2_2=-1,which is the desired property. Hence _1,_2,_1,_2 are the desired symbols.Case 2. Assume now, that there are no symbols satisfying (<ref>). Since there are zeroes outside the diagonal in Ω_π, there exist two distinct symbols _1,_2∈𝒜 such that Ω__1_2=Ω__2_1=0.Case 2a. Suppose first that the rows of Ω_π corresponding to _1 and _2 are not identical. Then there existsa symbolsuch that Ω__1≠ 0 and Ω__2=0 or Ω__2≠ 0 and Ω__1=0. Assume that the first case holds (the second is done analogously) and set _1:=.Note that _1∈𝒜∖{π_0^-1(1),π_0^-1(d)}. Since _2 does not satisfy (<ref>), there exist two _2∈𝒜∖{π_0^-1(1),π_0^-1(d)}, _2≠_1, such that Ω__2_2≠ 0. Thus we obtain (<ref>). Case 2b. Suppose that the rows ofthe matrix Ω corresponding to indices _1,_2 are identical. Then there are no indices _1,_2 such that _1,_2,_1,_2 satisfy (<ref>). We show that there is a different set of symbols satisfying (<ref>).Note that all symbols β such thatπ_0(_1)<π_0(β)<π_0(_2)satisfyπ_1(_1)<π_1(β)<π_1(_2).Otherwise only one of the entries Ω__1β and Ω__2β would be non-zero. In other words all symbols β∈𝒜 satisfying (<ref>) satisfy Ω__1β=Ω__2β=0. Observe that there exist two different symbols _1,_1∈𝒜 satisfying(<ref>) such that Ω__1_1≠ 0. Otherwise the permutation π satisfies(<ref>) and it is degenerate. Since _1,_2 do not satisfy (<ref>) and the corresponding rows are identical, there existsa symbol _2∈𝒜 such that Ω__1_2=Ω__2_2≠ 0. It follows that Ω__1_2=Ω__2_2≠ 0. We haveΩ__1_2=Ω__2_1=Ω__2_1=0andΩ__1_1≠ 0andΩ__2_2≠ 0.Hence, _1,_2,_1,_2 are the desired symbols. If π is a nonsymmetric and nondegenerate permutationsatisfying (<ref>), and τ∈^𝒜 is a rationally independent vector, then there exist _1,_2∈𝒜 such that Ω__1_2=Ω__2_1=0 andfor each i=1,2 the numbers(Ωτ)__2-(Ωτ)__1, and(Ωτ)__i-((Ωτ)_π_0^-1(1)+(Ωτ)_π_0^-1(d))arerationally independent.We prove the case when i=1. If i=2, the proof goes along the same lines. Consider symbols _1,_2,_1,_2 given byProposition <ref>. We have Ω__1_2=Ω__2_1=0. Assume that there existintegers p and q such that 0 =p((Ωτ)__2-(Ωτ)__1)+q( (Ωτ)__1-((Ωτ)_π_0^-1(1)+(Ωτ)_π_0^-1(d)))=∑_β∈𝒜(-qΩ_π_0^-1(1)β-qΩ_π_0^-1(d)β+(q-p)Ω__1β+pΩ__2β)τ_β. By rational independence of τ, this yields -qΩ_π_0^-1(1)β-qΩ_π_0^-1(d)β+(q-p)Ω__1β+qΩ__2β=0, for every β∈𝒜. Since Ω_π_0^-1(1)β=1 for β∈𝒜∖{π_0^-1(1)} and Ω_π_0^-1(d)β=-1 for β∈𝒜∖{π_0^-1(d)}, we have (q-p)Ω__1β+qΩ__2β=0 for β∈𝒜∖{π_0^-1(1),π_0^-1(d) } Since by (<ref>) the matrix [ Ω__1_1 Ω__1_2; Ω__2_1 Ω__2_2 ] is triangular and has non-zero entries on the main diagonal, it follows that p=q=0, which proves the rational independence. § THE MEASURES ON THE SURFACEIn this section we will deal with measures on a given surface (M,Σ)which are absolutely continuouswith respect to the Lebesgue measure. We want to prove that, if the density of such measure is bounded and close enough tothe constant function 1 in L^1, then there isan explicit way to construct a homeomorphism which pushes this measure tothe Lebesgue measure. The computation given below is partially inspired by the paper of Moser <cit.>. We will need the following auxiliary lemma.Let x,y∈^2 be two points onthe plane and let xy bethe segment with endpoints at x and y. Let H_1,H_2 be two affine transformations on ^2. If H_1(x)=H_2(x) and H_1(y)=H_2(y), then H_1|_xy=H_2|_xy. Moreover, for each noncollinear triples x_1,x_2,x_3∈^2 and y_1,y_2,y_3∈^2 there exists a unique invertible affine transformation H such that H(x_i)=y_i for i=1,2,3.We will now prove lemmas which give a construction of a homeomorphism of an isosceles right triangle which pushes forward any given absolutely continuous measure whose density satisfies some conditions to the Lebesgue measure. For any affine transformation G^2→^2 define lin(G)=DG as the matrix determining its linear part, D denotes the derivative. Moreover for any real 2× 2 matrix M=([ a b; c d ]) we have the following formula for the operator normM:=√(a^2+b^2+c^2+d^2+√((a^2+b^2+c^2+d^2)^2-4((M))^2)/2).In particular, lin(G) is a Lipschitz constant of G. On the space of homeomorphisms Hom(X)of a compact metric space X, we will considerthe standard metric d_Hom(H,G):=max{sup_x∈ Xd(H(x),G(x)),sup_x∈ Xd(H^-1(x),G^-1(x))}, where d isthe metric on X.Throughout this section we will heavily depend on the following construction.Let0<a<1, and let V be the triangle in ^2 with vertices in points (0,a),(a,0),(0,-a). Let V_1 and V_2 be the triangles whose vertices are (0,a),(a,0),(0,0) and (0,0),(a,0),(0,-a) respectively. Let 0≤ h<1 and 0<ϵ<1. Let y(h):=(ϵ a(1-h),ha). Consider the triangles given by the followingsets of vertices: * C_1=C_1(h,ϵ) given by {(0,0),(0,a),y(h,ϵ)}; * C_2=C_2(h,ϵ) given by {(a,0),(0,a),y(h,ϵ)}; * C_3=C_3(h,ϵ) given by {(0,0),(ϵ a,0),y(h,ϵ)}; * C_4=C_4(h,ϵ) given by {(a,0),(ϵ a,0),y(h,ϵ)}; * C_5=C_5(h,ϵ) given by {(0,0),(0,-a),(ϵ a,0)}; * C_6=C_6(h,ϵ) given by {(a,0),(0,-a),(ϵ a,0)}. Let ĥ:=h/h+1≥ 0. Consider the point ŷ(h,ϵ)=(ϵ a-ĥϵ a,-ĥa). Consider also the triangles * Ĉ_1=Ĉ_1(h,ϵ) given by {(0,0),(0,a),(ϵ a,0)}; * Ĉ_2=Ĉ_2(h,ϵ) given by {(a,0),(0,a),(ϵ a,0)}; * Ĉ_3=Ĉ_3(h,ϵ) given by {(0,0),(ϵ a,0),ŷ(h,ϵ)}; * Ĉ_4=Ĉ_4(h,ϵ) given by {(a,0),(ϵ a,0),ŷ(h,ϵ)}; * Ĉ_5=Ĉ_5(h,ϵ) given by {(0,0),(0,-a),ŷ(h,ϵ)}; * Ĉ_6=Ĉ_6(h,ϵ) given by {(a,0),(0,-a),ŷ(h,ϵ)}. By the definition of h and ĥ we have Leb(C_1)/Leb(Ĉ_1)=Leb(C_2)/Leb(Ĉ_2)=1-hand Leb(C_3)/Leb(Ĉ_3) =Leb(C_4)/Leb(Ĉ_4)=Leb(C_5)/Leb(Ĉ_5)=Leb(C_6)/Leb(Ĉ_6)=1+h. Define H(h,ϵ):V→ V as apiecewise affine homeomorphism such that(i) H(h,ϵ)(C_i)=Ĉ_i, H(h,ϵ)|_C_i is affine fori=1,…,6; (ii) H(h,ϵ) fixes (0,0),(0,a),(0,-a),(a,0), (iii) H(h,ϵ)(y)=(aϵ,0) andH(h,ϵ)(aϵ,0)=ŷ. Note that by Lemma <ref>, H(h,ϵ) is well defined everywhere on V and also H(h,ϵ)|_∂ V=Id|_∂ V. Moreover lin(H(h,ϵ)|_C_1):= [1+h/1-h0; -h/ϵ (1-h)1 ]; lin(H(h,ϵ)|_C_2):= [ 1-ϵ h/(1-h)(1-ϵ )-ϵ h/(1-h)(1-ϵ ); h/(1-h)(1-ϵ ) 1+h/(1-h)(1-ϵ ) ]; lin(H(h,ϵ)|_C_3):= [1-h/1+h 2ϵ/1+h; -h/ϵ (1+h) 1-2h/1+h ]; lin(H(h,ϵ)|_C_4):= [1+ϵ h/(1+h)(1-ϵ ) ϵ (2+h-2ϵ)/(1+h)(1-ϵ );h/(1+h)(1-ϵ ) 1-h(1-2ϵ )/(1+h)(1-ϵ ) ]; lin(H(h,ϵ)|_C_5):= [1-h/1+h0; -h/ϵ (1+h)1 ]; lin(H(h,ϵ)|_C_6):= [ 1+ϵ h/(1+h)(1-ϵ ) -hϵ/(1+h)(1-ϵ ); h/(1+h)(1-ϵ ) 1-h/(1+h)(1-ϵ ) ]. By (<ref>) we have (lin(H(h,ϵ)|_C_1))=(lin(H(h,ϵ)|_C_2))=1/1-h≥ 1, and (lin(H(h,ϵ)|_C_3))=(lin(H(h,ϵ)|_C_4)) =(lin(H(h,ϵ)|_C_5))=(lin(H(h,ϵ)|_C_6))=1/1+h≤ 1. It is also worth noting that (0,0) is fixed by the affine maps H(h,ϵ)|_C_1, H(h,ϵ)|_C_3 and H(h,ϵ)|_C_5, while (a,0) is a common fixed point for the transformations H(h,ϵ)|_C_2, H(h,ϵ)|_C_4 and H(h,ϵ)|_C_6. We can also define H(h,ϵ):V→ V for -1<h≤ 0. Let J:^2→^2 be the reflection acrossthe x-axis. Note that JV=V, JV_1=V_2 and JV_2=V_1. Now define ĥ:=h/1+|h|,C_i(h,ϵ):=J(C_i(-h,ϵ)), Ĉ_i(h,ϵ):=J(Ĉ_i(-h,ϵ)) andH(h,ϵ):=J∘ H(-h,ϵ)∘ J.For i=1,…,6 we havelin(H(h,ϵ)|_C_i=J∘lin(H(-h,ϵ)|_C_i∘ J.Since J is an isometry, we also obtain(lin(H(h,ϵ)|_C_1))=(lin(H(h,ϵ)|_C_2))=1/1+h≥ 1,and(lin(H(h,ϵ)|_C_3))=(lin(H(h,ϵ)|_C_4)) =(lin(H(h,ϵ)|_C_5))=(lin(H(h,ϵ)|_C_6))=1/1-h≤ 1.Hence in general for -1<h<1 we have (lin(H(h,ϵ)|_C_1))=(lin(H(h,ϵ)|_C_2))=1/1-|h|≥ 1, and (lin(H(h,ϵ)|_C_3))=(lin(H(h,ϵ)|_C_4)) =(lin(H(h,ϵ)|_C_5))=(lin(H(h,ϵ)|_C_6))=1/1+|h|≤ 1. For any fixed ϵ>0 and for every h_1,h_2∈(-1/2,1/2) we have d_Hom(H(h_1,ϵ),H(h_2,ϵ))≤20a/ϵ|h_2-h_1|. We first prove thatH(h_1,ϵ)(x)-H(h_2,ϵ)(x)≤20a/ϵ|h_2-h_1|,for every x∈ V. Indeed, assume that h_1 and h_2 are non-negative numbers and h_1≥ h_2. Consider the triangle W_1 with vertices (0,0),y(h_1,ϵ),y(h_2,ϵ) andthe triangle W_2 given bythe points(a,0),y(h_1,ϵ), y(h_2,ϵ).Assume that x∈ V∖(W_1∪ W_2). Then x∈ C_i(h_1,ϵ)⇔ x∈ C_i(h_2,ϵ). This implies that both H(h_1,ϵ) and H(h_2,ϵ) act on x byaffine transformations whose linear parts are of the same form. Since for i=1,…, 6 the affine maps H(h_1,ϵ)|_C_i(h_1,ϵ) and H(h_2,ϵ)|_C_i(h_2,ϵ) share a common fixed point,we get thatH(h_1,ϵ)(x)-H(h_2,ϵ)(x)=lin(H(h_1,ϵ)|_C_i(h_1,ϵ))x-lin(H(h_2,ϵ)|_C_i(h_2,ϵ))x. By using the formula (<ref>) for each i=1,…,6 we getlin(H(h_1,ϵ)|_C_i(h_1,ϵ))-lin(H(h_1,ϵ)|_C_i(h_2,ϵ))≤10/ϵ(h_1-h_2). Since the above norm is the operator norm for H(h_1,ϵ)-H(h_2,ϵ) (which is a linear transformation), this implies thatsup_x∈ V∖(W_1∪ W_2)H(h_1,ϵ)(x)-H(h_2,ϵ)(x)≤10/ϵ(h_1-h_2)x<20a/ϵ(h_1-h_2). Next note that W_1=C_3(h_1,ϵ)∩ C_1(h_2,ϵ) and W_2=C_4(h_1,ϵ)∩ C_2(h_2,ϵ).We now prove that for x∈ W_1∪ W_2 we also have H(h_1,ϵ)(x)-H(h_1,ϵ)(x)≤20a/ϵ(h_1-h_2). Suppose that x∈ W_1; the proof for x∈ W_2 is analogous. Considerthe segment I_x⊂ W_1 with endpoints onthe segments (0,0),y(h_1,ϵ) and (0,0), y(h_2,ϵ) such that x∈ I_x andwhich is parallel to y(h_1,ϵ), y(h_2,ϵ). Then|I_x|≤y(h_1,ϵ)-y(h_2,ϵ)=a√(1+ϵ) (h_1-h_2)<2a(h_1-h_2).Note that I_x dividesthe intervals (0,0),y(h_1,ϵ) and (0,0), y(h_2,ϵ) with the same ratio. Since affine transformations do not change the ratio of the lengths ofcollinear segments andH(h_1,ϵ)((0,0),y(h_1,ϵ))= H(h_2,ϵ)((0,0), y(h_2,ϵ))= (0,0),(0, ϵ a),it followsthatthe segments H(h_1,ϵ)(I_x) and H(h_2,ϵ)(I_x) share a common endpoint in (0,0),(0,ϵ a).Using again the conservation of the ratio of the lengths of collinear segments by affine transformations, we get|H(h_1,ϵ)(I_x)|/|I_x|=|H(h_1,ϵ)(y(h_1,ϵ),y(h_2,ϵ))|/|y(h_1,ϵ),y(h_2,ϵ)|=|H(h_1,ϵ)(y(h_1,ϵ),(ϵ a,0))|/|y(h_1,ϵ),(ϵ a,0)|=1/1+h_1≤ 1,and|H(h_2,ϵ)(I_x)|/|I_x|=|H(h_2,ϵ)(y(h_1,ϵ),y(h_2,ϵ))|/|y(h_1,ϵ),y(h_2,ϵ)|=|H(h_2,ϵ)((0,a),y(h_2,ϵ))|/| (0,a),y(h_2,ϵ)|=1/1-h_2≤ 2.As H(h_1,ϵ)(x)∈ H(h_1,ϵ)(I_x) and H(h_2,ϵ)(x)∈ H(h_2,ϵ)(I_x), we obtainH(h_1,ϵ)(x)-H(h_2,ϵ)(x) ≤ |H(h_1,ϵ)(I_x)|+|H(h_2,ϵ)(I_x)|≤3|I_x|< 6a(h_2-h_1)<20/ϵa(h_1-h_2). By proceeding analogously for h_2≥ h_1 we prove thatH(h_1,ϵ)(x)-H(h_2,ϵ)(x)≤20/ϵa(h_2-h_1). The case when h_1 and h_2 are non-positive is analogous. To provethe similar inequality when h_1 and h_2 are of different sign, let h_0:=0. Then H(h_0,ϵ)=Id. Using the previous case we show thatH(h_2,ϵ)(x)-x≤20/ϵa|h_0-h_2|,andH(h_1,ϵ)(x)-x≤20/ϵa|h_1-h_0|.Since h_1, h_2 have different sign, the numbers h_0-h_2, h_1-h_0 are of the same sign. It follows thatH(h_1,ϵ)(x)-H(h_2,ϵ)(x)≤H(h_1,ϵ)(x)-x+H(h_2,ϵ)(x)-x≤20/ϵa|h_2-h_1|. By proceedingas in the proof of (<ref>) and replacing H(h_i,ϵ) by H^-1(h_i,ϵ) for i=1,2,we can prove that for every x∈ V we also haveH^-1(h_2,ϵ)(x)-H^-1(h_1,ϵ)(x)≤20/ϵa|h_2-h_1|,which concludes the proof of the lemma. Let V, V_1 and V_2 be the triangles definedabove.Let0<<10^-8 , and let κ>0. Suppose that f∈ L^1(V) satisfiesf>κ ;1/1+<f or f<1/1-;∫_Vf(x)dx=Leb(V).Let μ_f:=f dx. Then there existsa piecewise affine homeomorphism H_f:V→ V such that (i) (H_f)_*μ_f(V_i)=Leb(V_i) for i=1,2; (ii) H_f|_∂ V=Id|_∂ V; (iii) there exists -<h_f< such that(DH_f^-1) is constant on eachV_i and is equal to 1± h_f; (iv) the Lipschitz constants of H_f and H^-1_f are less than 5/4; (v) the maps f↦ H_f∈ Hom(V) and f↦(DH_f^-1)∈ L^∞(V)are continuous on the set of f∈L^1(V) satisfying (<ref>) for a given κ.Since μ_f is an absolutely continuous measure with respect to Leb(, there are no segments of positive measure μ_f in V. Hence there exists a unique -1<h_f< 1 and a point y=y_f=(√()a(1-|h_f|),h_fa) such that thequadrilateral with vertices {(0,a),(0,0),(a,0),y} and thequadrilateral with vertices {(0,-a),(0,0),(a,0),y} have the same measure μ_f equal to 1/2Leb(V). Consider the triangles C_i=C_i^f:=C_i(h_f,√()) for i=1,…,6. By the definition of h_f we have μ_f(C_1∪ C_2)=μ_f(C_3∪ C_4∪ C_5∪ C_6)=1/2Leb(V). We now evaluate the bounds on h_f. Assume that f>1/1+. Since Leb(V)=a^2 we have 1/2a^2 =μ_f(C_3∪ C_4∪ C_5∪ C_6)=∫_C_3∪ C_4∪ C_5∪ C_6f(x)dx>1/1+Leb(C_3∪ C_4∪ C_5∪ C_6)=1/1+(1/2(a+|h_f|a)a). Hence f>1/1+⇒ |h_f|<. Now assume that f<1/1-. Then we have 1/2a^2 =μ_f(C_1∪ C_2)=∫_C_1∪ C_2f(x)dx<1/(1-)Leb(C_1∪ C_2)=1/1-(1/2(a-|h_f|a)a). This shows f<1/1-⇒ |h_f|<. Definition of H_f. Define H_f:=H(h_f,√()), a piecewise affine homeomorphism on V. Note that by definition we have H_f|_∂ V=Id|_∂ V. Moreover (H_f)_*μ_f(V_1)=μ_f(C_1∪ C_2)=1/2Leb(V)=Leb(V_1) and (H_f)_*μ_f(V_2)=μ_f(C_3∪ C_4∪ C_5∪ C_6)=1/2Leb(V)=Leb(V_2). Hence H_f satisfies points (i) and (ii). Furthermore, by (<ref>) and (<ref>) we have that (lin(H_f|_C_1))=(lin(H_f|_C_2))=1/1-|h_f|≥ 1, and (lin(H_f|_C_3))=(lin(H_f|_C_4)) (lin(H_f|_C_5))=(lin(H_f|_C_6))=1/1+|h_f|≤ 1. Note that V_1=Ĉ_1∪Ĉ_2 and V_2=Ĉ_3∪Ĉ_4∪Ĉ_5∪Ĉ_6 for h_f≥ 0 and V_1=Ĉ_3∪Ĉ_4∪Ĉ_5∪Ĉ_6 and V_2=Ĉ_1∪Ĉ_2 for h_f≤ 0. Hence by (<ref>) and (<ref>) we have (lin(H_f^-1|_V_1))=1-h_f and (lin(H_f^-1|_V_2))=1+h_f Thus H_f satisfies (iii). The norm of the linear part. We will now prove that lin(H_f)|_C_i)<5/4 for i=1,…,6. Note that each of the matrices lin(H_f)|_C_i is of the form M=([ 1+b c; d 1+e ]), where |b|,|c|,|d|,|e|<3√(). Hence, using the formula (<ref>) and the fact that <10^-8, we obtain that lin (H_f)|_C_i<<√(2(1+3√())+2· 3√()+√((2(1+3√())+2· 3√())^2-4((lin(H_f)|_Ĉ_i))^2)/2)<√(2+12√()+36+√((2+12√()+36)^2-4(1/1+)^2)/2)<√(1+5√())<5/4. In the same way we prove that lin(H_f)^-1|_Ĉ_i)<5/4. Thus H_f satisfies (iv). Continuity of f↦ H_f.Suppose that f,g∈ L^1(V) satisfy (<ref>). By Lemma <ref>, we already know thatd_Hom(H_f,H_g)≤20/√()a|h_f-h_g|.We prove that |h_f-h_g|≤ Cf-g_L^1,for some constant C>0 depending only on a and κ. Let :=f-g_L^1.Case h_f and h_g have the same sign. Assume that h_f≥ h_g≥ 0 or 0≥ h_g≥ h_f. Then 0 =μ_f(C^f_1∪ C^f_2)-μ_g(C_1^g∪ C_2^g)=∫_C^f_1∪ C^f_2f(x)dx-∫_C^g_1∪ C^g_2g(x)dx=∫_C^g_1∪ C^g_2(f-g)(x)dx- ∫_(C^g_1∪ C^g_2)∖(C^f_1∪ C^f_2)g(x)dx≤-|h_f-h_g|aκ/2, and hence |h_f-h_g|≤2/aκ. Thus (<ref>) holds withC=2/aκ.Case of h_f, h_g with different sign. Suppose that h_f≥ 0≥ h_g. Then we have 0 =μ_f(C^f_1∪ C^f_2)-μ_g(C_3^g∪ C_4^g∪ C_5^g∪ C_6^g)=∫_C^f_1∪ C^f_2f(x)dx-∫_C_3^g∪ C_4^g∪ C_5^g∪ C_6^gg(x)dx=∫_C^f_1∪ C^f_2(f-g)(x)dx-∫_(C_3^g∪ C_4^g∪ C_5^g∪ C_6^g)∖(C^f_1∪ C^f_2)g(x)dx≤-|h_f-h_g|aκ/2. Thus we have 0≤ h_f-h_g≤2/aκ,which completes the proof of (<ref>).By combining (<ref>) and (<ref>) we obtaind_Hom(H_f,H_g)≤10/√()|h_f-h_g|<10/√()Cf-g_L^1.This concludes the proof of the continuity of f↦ H_f. By the formula given in (<ref>), the continuity of the map f↦ h_f also implies the continuity of the map f↦(DH_f^-1)∈ L^∞(V). Thus (v) is proved. Let (X,μ) be a standard metric probability space. For 0<s_1<s_2, define 𝒲(X,s_1,s_2)⊂ L^1(X,μ) by 𝒲(X,s_1,s_2):={f∈ L^1(X); s_1< f< s_2; ∫_Xf dμ(x)=μ(X)}. Let V be the triangle with vertices (0,-a),(0,a),(a,0), equipped with the (normalized) 2-dimensional Lebesgue measure. We need the following lemma. Let H:𝒲(V,s_1,s_2)→Hom(V) be a continuous map. Assume that there exists ℓ>0 such that, for every f∈𝒲(V,s_1,s_2), the homeomorphism H(f)^-1 is Lipschitz with constant ℓ. Then the transformation W(s_1,s_2)∋ f↦ f∘ H(f)∈ L^1(V) is continuous. Take f∈𝒲(V,s_1,s_2) and ϵ>0. Let f_ϵ:V→ be a uniformly continuous function such that f_ϵ-f_L^1<ϵ. Let 0<<ϵ be such that x-y<δ⇒ |f_ϵ(x)-f_ϵ(y)|<ϵ. Consider 0<'<ϵ such that for every g∈ W(V,s_1,s_2) we have f-g_L^1<'⇒ d_Hom(H(f),H(g))<, and let g∈ W(V,s_1,s_2) be such that f-g_L^1<'. Since H(g)^-1 is Lipschitz with constant ℓ, H(g)_*Leb is an absolutely continuous measure with density bounded by ℓ^2. Hence for every h∈ L^1(V) we have h∘ H(g)_L^1=∫_V|h∘ H(g)(x)|dx≤∫_Vℓ^2|h(x)|dx=ℓ^2h_L^1. Then f∘ H(f)-g∘ H(g)_L^1≤f∘ H(f)-f∘ H(g)_L^1+f∘ H(g)-g∘ H(g)_L^1 and, by (<ref>), f∘ H(g)-g∘ H(g)_L^1≤ℓ^2f-g_L^1. Moreover f∘ H(f)-f∘ H(g)_L^1≤ f∘ H(f)-f_ϵ∘ H(f)_L^1+f_ϵ∘ H(f)-f_ϵ∘ H(g)_L^1+f_ϵ∘ H(g)-f∘ H(g)_L^1 ≤ 2ℓ^2f-f_ϵ_L^1+f_ϵ∘ H(f)-f_ϵ∘ H(g)_L^1, where the last inequality comes from (<ref>). By (<ref>) and (<ref>), we have f_ϵ∘ H(f)-f_ϵ∘ H(g)_L^1<ϵ. To sum up we obtain f∘ H(f)-g∘ H(g)_L^1≤ℓ^2f-g_L^1+2ℓ^2f-f_ϵ_L^1+ϵ≤(3ℓ^2+1)ϵ, which proves the assertion. The statement of Lemma <ref> remains valid if we replace V with any 2-dimensional Riemannian surface M. Let 0<<10^-8.Let f∈𝒲(V,1/1+,1/1-) andμ_f:=f dx. Then there exists a homeomorphism H_f:V→ V, depending continuously on f, such that (H_f)_*μ_f=Leb and H_f|_∂ V=Id|_∂ V. We assume that a=1. The prove for a≠ 1 goes along the same lines. Let f∈𝒲(V,1/1+,1/1-). Denote by V_1^1 and V_2^1 the two halves of V which are both isosceles right triangles with diam(V_1^1)=diam(V_2^1)=√(2). Inductively, for n∈ define the family {V_i^n}_i=1^2^n of congruent right isosceles triangles which divide V, V_i^n=V_2i-1^n+1∪ V_2i^n+1 for i=1,…,2^n and they satisfydiam(V_n^i)=1/√(2)^n-2. We will construct H_f inductively as a limit of piecewise affine transformations.In the first step, using Lemma <ref>, we obtain a piecewise affine homeomorphism H^1_f:V→ V such that(H^1_f)_*μ_f(V_1^1)=(H^1_f)_*μ_f(V_2^1)=1/2Leb(V) and H^1_f|_∂ V=Id|_∂ V.Moreover (D(H_f^1)^-1) is constant on eachV_1^1 and V_2^1.Suppose now that for some n∈ we have constructed piecewise affine homeomorphisms H^j_f:V→ V for j=1,…,n such that for all i=1,…,2^n we have(H^n_f∘…∘ H^1_f)_*μ_f(V_i^n)=1/2^nLeb(V)=Leb(V_i^n) and H^j_f|_∂ V=Id|_∂ V.Moreover, suppose that (D(H^n_f∘…∘ H^1_f)^-1) is constant on each V_i^n and equals d_i^n>0.With these assumptions the measure (H^n_f∘…∘ H^1_f)_*μ_f isabsolutely continuousand its density f_n:V→_>0 satisfiesf_n(x)=d_i^n · f( (H^n_f∘…∘ H^1_f)^-1x) if x∈ V_i^n,and by (<ref>)∫_V_i^nf_n(x)dx=(H^n_f∘…∘ H^1_f)_*μ_f(V_i^n)=Leb(V_i^n). Take any 1≤ i≤ 2^n. In view of (<ref>), ifd_i^n<1 thenf_n(x)<d_i^n/1-<1/1- for all x∈ V_i^nand if d_i^n≥ 1 thenf_n(x)>d_i^n/1+≥1/1+ for all x∈ V_i^n.It follows that f_n:V_i^n→_>0 is a positive density satisfying (<ref>) with κ=d_i^n/1+.Hence we can apply Lemma <ref> to the triangle V_i^n and the density functionf_n:V_i^n→_>0, thus obtaining a piecewise affine homeomorphism H_f^n+1,i:V_i^n→ V_i^n such that(H_f^n+1,i)_*(μ_f_n|_V_i^n)(V_2i-1^n+1)=(H_f^n+1,i)_*(μ_f_n|_V_i^n)(V_2i^n+1)=1/2Leb(V_i^n)=1/2^n+1Leb(V), H^n+1,i_f|_∂ V_i^n=Id|_∂ V_i^n,andD((H_f^n+1,i)^-1) is constant on both V_2i-1^n+1 and V_2i^n+1.Finally we define a piecewise affine homeomorphism H^n+1_f:V→ V given byH^n+1_f(x):=H^n+1,i_f(x) whenever x∈ V_i^n.Then H^n+1_f(V^n_i)=V^n_i and, by (<ref>), we haveH^n+1_f|_∂ V=Id|_∂ V. Moreover, by (<ref>),(H^n+1_f∘…∘ H^1_f)_*μ_f(V_2i-1^n+1)=(H^n+1_f∘…∘ H^1_f)_*μ_f(V_2i^n+1)=1/2^n+1Leb(V).In view of (<ref>),(D(H^n+1_f)^-1) is constant on each V^n+1_j for j=1,…,2^n+1 andthen sois (D(H^n+1_f∘…∘ H^1_f)^-1). Thus, we have proved that H_f^n+1 satisfies the assumptions of the induction. Notethat, by (iii) in Lemma <ref>, we have 1-<(D(H^n_f)^-1)<1+ for every n∈, and since (H^j_f)^-1 are piecewise linear homeomorphisms, it follows that(1-)^n≤(D(H^n_f∘…∘ H^1_f)^-1)≤(1+)^n almost everywhereand the above inequalities do not depend on f.We now show thatH_f(x):=lim_n→∞H_f^n∘…∘ H_f^1(x)yields a homeomorphism H_f:V→ V.First note thatH_f^m(V_i^n)=V_i^n for i=1,…,2^nand m>n.Moreover, by (<ref>) we have max_i=1,…,2^ndiam(V_i^n)→ 0 for n→∞.This implies that {H_f^m∘…∘ H_f^1}_n∈ is a Cauchy sequence. Indeed, for any ϵ>0 by (<ref>) we can choose N∈ such that max_i=1,…,2^Ndiam(V_i^N)<ϵ. Moreover, by (<ref>),for all m,n≥ N we haveH_f^n∘…∘ H_f^1(x)∈ V_i^N⟺ H_f^m∘…∘ H_f^1(x)∈ V_i^N.Hence H_f^n∘…∘ H_f^1(x)-H_f^m∘…∘ H_f^1(x)<ϵ for all x∈ V. It follows that the map H_f:V→ V given by (<ref>) is well defined and the convergence in (<ref>) is uniform. This implies that H_f is continuous. Since H_f^n|_∂ V=Id|_∂ V for all n∈, wealso have H_f|_∂ V=Id|_∂ V.Set W_i^n:=(H_f^n∘…∘ H_f^1)^-1(V^n_i). In view of (<ref>),W_i^n=(H_f^m∘…∘ H_f^1)^-1(V^n_i) for m>n.Therefore,(H_f^n∘…∘ H_f^1)^-1(x)∈ W_i^N⟺ (H_f^m∘…∘ H_f^1)^-1(x)∈ W_i^N if m,n≥ N.By (iv) in Lemma <ref>, (H_f^n)^-1 is a Lipschitz automorphism with a Lipschitz constant 5/4. Thus, by (<ref>), we havediam(W_i^n)<diam(V_i^n)(5/4)^n=2(5/4√(2))^n,somax_i=1,…,2^ndiam(W_i^n)→ 0 for n→∞.Using (<ref>) and (<ref>) andrepeating the same arguments as for H_f by replacing V_i^n with W_i^n, we obtain that themap G_f:V→ V given byG_f(x):=lim_n→∞(H_f^n∘…∘ H_f^1)^-1(x)is well defined and continuous. Wenow show that H_f∘ G_f=Id and G_f∘ H_f=Id. First note that in view of (<ref>) and the compactness of V^n_i and W^n_i we have H_f(W_i^n)=V_i^n and G_f(V_i^n)=W_i^n. Hence H_f∘ G_f(V_i^n)=V_i^n and G_f∘ H_f(W_i^n)=W_i^n. Let ϵ>0 and N∈ be such thatmax_i=1,…,2^Ndiam(V_i^N)<ϵ and max_i=1,…,2^Ndiam(W_i^N)<ϵ.This implies thatH_f(G_f(x))-x<ϵ and G_f(H_f(x))-x<ϵ for every x∈ V.Since ϵ was arbitrary, this shows that H_f∘ G_f=Id and G_f∘ H_f=Id. ThusH_f is a homeomorphism.Note that the family of sets {V_i^n;n∈,i=1,…,2^n} generates the Borel σ-algebra on V. Since (H_f)^-1(V_i^n)=W_i^n=(H_f^n∘…∘ H_f^1)^-1(V^n_i), by (<ref>), we have(H_f)_*μ_f(V_i^n)=(H_f^n∘…∘ H_f^1)_*μ_f(V^n_i)=Leb(V_i^n).It follows that (H_f)_*μ_f=Leb.In the reminder of the proof we will show that H_f depends continuously on f. Fix ϵ>0 and then choose m∈ such that 2(5/4√(2))^m<ϵ/3.Thenmax_i=1,…,2^mdiam(V_i^m)<ϵ/3 and max_i=1,…,2^mdiam(W_i^m)<ϵ/3.Since (H_f)^-1(V_i^m)=W_i^m=(H_f^m∘…∘ H_f^1)^-1(V^m_i), it follows thatsup_x∈ VH_f(x)-H_f^m∘…∘ H_f^1(x)<ϵ/3 and sup_x∈ V(H_f)^-1(x)-(H_f^m∘…∘ H_f^1)^-1(x)<ϵ/3.Therefored_Hom(H_f,H_f^m∘…∘ H_f^1)<ϵ/3 for every f∈𝒲(V,11+,11-).By (v) in Lemma <ref>, the maps f↦ H_f^1∈ Hom(V) and f↦ D(H_f^1)^-1∈ L^∞(V) are continuous. Suppose now that fork≥ 1 we proved thatf↦ H_f^k∘…∘ H_f^1∈ Hom(V)andf↦ D(H_f^k∘…∘ H_f^1)^-1∈ L^∞(V)are continuous. We now prove thatf↦ H_f^k+1∘…∘ H_f^1andf↦ D(H_f^k+1∘…∘ H_f^1)^-1are also continuous. Since for every i=1,…,k, H_f^i and (H_f^i)^-1 are Lipschitz homeomorphisms with constant 5/4, we getH_f^k∘… H_f^1 and (H_f^k∘… H_f^1)^-1 are Lipschitz with constant (54)^k.Moreover by (<ref>) we havef_k= D(H_f^k∘…∘ H_f^1)^-1·(f∘(H_f^k∘…∘ H_f^1)^-1)∈𝒲(V,(1-)^k1+,(1+)^k1-).In view of Lemma <ref>, H_f^k+1 depends continuously on f_k. By (<ref>) and (<ref>), Lemma <ref> implies that f↦ f∘(H_f^k∘…∘ H_f^1)^-1∈ L^1(V) is continuous. Together with (<ref>) this gives the continuity off↦ f_k =f∘(H_f^k∘…∘ H_f^1)^-1· D(H_f^k∘…∘ H_f^1)^-1∈ L^1(V).It follows that H_f^k+1 depends continuously on f.Again, since H_f^i and (H_f^i)^-1 are Lipschitz with constant 5/4,for any x∈ V and f,g∈𝒲(V,1/1+,1/1-)we have H^k+1_f (H_f^k∘…∘ H_f^1(x))-H^k+1_g(H_g^k∘…∘ H_g^1(x)) ≤H^k+1_f(H_f^k∘…∘ H_f^1(x))-H^k+1_g(H_f^k∘…∘ H_f^1(x)) +H^k+1_g(H_f^k∘…∘ H_f^1(x))-H^k+1_g(H_g^k∘…∘ H_g^1(x))≤ d_Hom(H^k+1_f,H^k+1_g)+5/4d_Hom(H_f^k∘…∘ H_f^1,H_g^k∘…∘ H_g^1)and similarly(H_f^k ∘…∘ H_f^1)^-1(H^k+1_f)^-1(x)-(H_g^k∘…∘ H_g^1)^-1(H^k+1_g)^-1(x)≤(5/4)^kd_Hom(H^k+1_f,H^k+1_g)+d_Hom(H_f^k∘…∘ H_f^1,(H_g^k∘…∘ H_g^1)).This proves the continuous dependence of H^k+1_f∘…∘ H^1_f on f. Finally, since H_f^i are piecewise linear homeomorphisms, we have D(H_f^k+1∘…∘ H_f^1)^-1=D(H_f^k+1)^-1D(H_f^k∘…∘ H_f^1)^-1 almost everywhere.By (v) in Lemma <ref>, f_k↦ D(H_f^k+1)^-1∈ L^∞(V) depends continuously on f_k. Since f↦ f_k∈ L^1(V) is continuous,it follows that f↦ D(H_f^k+1)^-1∈ L^∞(V) is also continuous. The uniform boundaries in (<ref>) and in (<ref>), together with (<ref>), yield the continuity of f↦ D(H_f^k+1∘…∘ H_f^1)^-1∈ L^∞(V).Fix any f∈𝒲(V,1/1+,1/1-). Then there exists δ>0 such that for any g∈𝒲(V,1/1+,1/1-) with f-g_L^1<δ we haved_Hom(H_f^m∘…∘ H_f^1,H_g^m∘…∘ H_g^1)<ϵ/3.Combining this with (<ref>) we obtaind_Hom(H_f,H_g)<ϵ,which concludes the proof ofthe continuity of f↦ H_f.Note that theabove lemma isalso valid for any triangle, since every two triangles are conjugated by an affine map (althoughthe restriction onmay vary). The following theorem is the main result of this section.Let (M,Σ,ζ) be a translation surface. There exists _ζ=>0 such that, for allf∈𝒲(M,11+,11-), there exists a homeomorphism ℋ_f:(M,Σ)→(M,Σ) such that (ℋ_f)_*μ_f=_ζ, where μ_f:=f _ζ. Moreover, ℋ_f depends continuously on f.On (M,Σ) consider a triangulation of m+1 triangles such that the set of vertices of this triangulation contains the set Σ.By connectedness,there is an ordering {U_i:0≤ i≤ m} of the triangles such that, for every i=1,…,m, the triangle U_i has a common edge with some U_k(i) for 0≤ k(i)<i. Indeed, choose any triangle U_0 from the triangulation. Next choose any neighbouring triangle as U_1 and set k(1)=0. Now suppose that for some 1≤ℓ≤ m we have chosen triangles {U_i:0≤ i≤ℓ}such that k(i)<i for 1≤ i≤ℓ. If ℓ=m then the process is over. If ℓ< m then choose as U_ℓ+1 any triangle that hasa common boundary with ⋃_i=0^ℓU_i. This triangle exists by connectedness. Finally, let 0≤ k(ℓ+1)≤ l be such that U_ℓ+1 hasa common edge with U_k(ℓ+1).For every i=1,…,m consider a small isosceles right triangle W_i⊂ U_i∪ U_k(i) such that its shortest height lies on the common edge of U_i and U_k(i) and W_i∩ W_j=∅ whenever i≠ j. Furthermore, we assume that each of the triangles W_i is of the same size and we choose a parametrization such that W_i has vertices in points (0,-a),(0,a) and (a,0), with (0,-a)∈ U_i for each i=1,…,m (see Fig. 2). Let B=[b_ij]_i,j∈{1,…,m} bethe matrix given byb_ij= 1if i=j;-1if i=k(j);0otherwise.For each positivef∈𝒲(M,11+,11-), denote by v^f∈^m the solution of the following system of linearequationsBv^f=[_ζ(U_i)-μ_f(U_i)]_i=1,… m.Then f↦ v^f is continuous. Let >0 be such that Lemma <ref> can be applied to any triangle U_i (it exists due to Remark <ref>). Observe that if f is constant equal to 1, then v^f=(0,…,0). By the continuity of f↦ v^f we canchoose 0<</3 such that |v_i^f|<a^2/12for everyf∈𝒲(M,11+,11-).Letf∈𝒲(M,11+,11-). We now construct afamily of piecewise affine homeomorphisms G_f^i:W_i→ W_i,i=1,…,m, such thatG_f^i depends continuously on f,G^i_f|_∂ W_i=Id,1-/3<(D(G_f^i)^-1(x))<1+/3 whenever D(G_f^i)^-1(x) is well defined,and finally we require thatthe homeomorphism 𝒢_f:M→ M definedby𝒢_f(x):=G_f^i(x)if x∈ W_i for some i=1,…,m,xotherwise,satisfies(𝒢_f)_*μ_f(U_j)=_ζ(U_j) for all 0≤ j≤ m. First note that for each i=1,…,m we can choose -1<h_f^i<1 such that the quadrilateral Q_f^i⊂ W_i with vertices in points (0,0)(0,-a),(a,0) and y^i_f:=(√()a(1-|h_f^i|),h_f^ia) has measure μ_f equal to μ_f(W_i∩ U_i)+v_i^f. Indeed, μ_f(W_i∩ U_i)≥1/1+_ζ(W_i∩ U_i)=a^2/2(1+)≥a^2/4>|v_i^f| and analogously μ_f(W_i∩ U_k(i))>|v_i^f|.Thus0<μ_f(W_i∩ U_i)+v_i^f<μ_f(W_i),which, together with the absolute continuity of μ_f, yields the existence of h_f^i for i=1,…,m.We now estimate |h_f^i|. Since |v_i^f| is the μ_f measureof the triangle with vertices (0,0),(a,0) and y_f^i in W_i and a|h_f^i| is its height, we have|v_i^f|>1/1+a^2|h_f^i|/2.Hence, by (<ref>),|h_f^i|<2(1+)|v_i^f|/a^2<(1+)/6</3. Let G_f^i:W_i→ W_i be given by G_f^i:=H(h_f^i,√()) for i=1,…,m as in (<ref>). Since(G_f^i)^-1(W_i∩ U_i)=Q_f^i,we have(𝒢_f)_*μ_f(W_i∩ U_i)=μ_f(Q_f^i)=μ_f(W_i∩ U_i)+v_i.Analogously(𝒢_f)_*μ_f(W_i∩ U_k(i))=μ_f(W_i∖ Q_f^i)=μ_f(W_i)-(μ_f(W_i∩ U_i)+v_i)=μ_f(W_i∩ U_k(i))-v_i.By the definition of v_i^f and by the fact that W_i and W_j are disjoint for i≠ j, we havefor i=1,…,m(𝒢_f )_*(μ_f)(U_i)= (𝒢_f)_*(μ_f)(U_i∖ (W_i∪⋃_{j;k(j)=i}W_j))+(𝒢_f)_*(μ_f)(W_i∩ U_i)+∑_{j;k(j)=i}(𝒢_f)_*(μ_f)(W_j∩ U_i)=μ_f(U_i∖ (W_i∪⋃_{j;k(j)=i}W_j))+(μ_f(W_i∩ U_i)+v^f_i)+∑_{j;k(j)=i}(μ_f(W_j∩ U_i)-v^f_j)=μ_f(U_i)+v_i^f-∑_{j;k(j)=i}v_j^f=μ_f(U_i)+(Bv^f)_i=_ζ(U_i). . Since (G_f^i)_*(μ_f)(W_i)=μ_f(W_i), we also have(𝒢_f)_*(μ_f)(M)=μ_f(M)=_ζ(M),and thus(𝒢_f)_*(μ_f)(U_0)=_ζ(U_0).Hence (<ref>) is satisfied. Moreover (<ref>) together with (<ref>) and (<ref>) yield (<ref>). What is left to prove is that G_f^i depends continuously on f. By Lemma <ref>, we only need to prove that h_f^i depends continuously on f.Takef∈𝒲(M,11+,11-), and let >0. Since v^g depends continuously on g, there exists 0<δ'≤δ such that,for any g∈𝒲(M,11+,11-),f-g_L^1<δ'⟹max_i=1,…,m|v_i^f-v_i^g|<.We now evaluate the differencebetween the respective Lebesgue measures of the quadrilaterals given by(0,0), (0,-a), (a,0), y_f^i andby (0,0), (0,-a), (a,0), y_g^i:a^2|h_f^i-h_g^i|/2< (1+)|μ_f(W_i∩ U_i)+v_i^f-μ_g(W_i∩ U_i)-v_i^g+∫_W_i|f-g|d_ζ|≤(1+)(|μ_f(W_i∩ U_i)-μ_g(W_i∩ U_i)|+|v_i^f-v_i^g|+∫_W_i|f-g|d_ζ)<(1+)(∫_W_i∩ U_i|f-g|d_ζ++∫_W_i|f-g|d_ζ|)≤(1+)(2f-g_L^1+).Hence, for everyg∈𝒲(M,11+,11-) such thatf-g_L^1<δ', we have|h_f^i-h_g^i|<6(1+)/a^2, which implies that h_f^i depends continuously on f.Since G_f^idepends continuously on f, by the definition 𝒢_f also depends continuously on f. Sincef∈𝒲(M,11+,11-), and </3,by (<ref>), (𝒢_f)_*μ_f is an absolutely continuous measure with density f̂=(f∘𝒢_f^-1)· D(𝒢_f^-1) satisfying1/1+<1-/3/1+/3<f̂<1+/3/1-/3<1/1-and∫_U_if̂d_ζ=(𝒢_f)_*μ_f(U_i)=_ζ(U_i) for every i=0,…,m.Therefore, on each U_i the density f̂ satisfies the assumptions of Lemma <ref>. Hence, for each i=0,…,m, there exists a homeomorphism H^i_f̂:U_i→ U_i which transportsthe measure (𝒢_f)_*μ_f|_U_i to _ζ|_U_i, andsuch that H^i_f̂|_∂ U_i=Id. Therefore, we can define a homeomorphism H_f̂:M→ Msuch thatH_f̂(x):= H^i_f̂(x) whenever x∈ U_i.Then(H_f̂∘𝒢_f)_*μ_f=( H_f̂)_*( (𝒢_f)_*μ_f)=_ζ. Let ℋ_f:=H_f̂∘𝒢_f. What is left to prove is that f↦ℋ_f is continuous.By Lemma <ref> H_f̂^i depend continuously on f̂ and hence f̂↦ H_f̂ is continuous. Moreover, (v) in Lemma <ref> implies that f↦(D𝒢_f^-1)∈ L^∞(M) is continuous. Furthermore, by (iv) in Lemma <ref>, the homeomorphism 𝒢_f^-1:M→ M is Lipschitz with constant 5/4. Thus, by Lemma <ref> and Remark <ref>, f↦ f∘𝒢_f^-1 is continuous. Hence𝒲(M,11+,11-)∋ f↦f̂=(f∘𝒢_f^-1)· D(𝒢_f^-1)∈ L^1(M)is continuous andthis implies the continuity of f↦ H_f̂. Now consider any f∈𝒲(M,11+,11-). Since H_f̂:M→ Mis uniformly continuous, for any η>0 we can find 0< such thatd_M(x,y)<⇒ d_M( H_f̂(x), H_f̂(y))<η.Then, for every x∈ M and anyg∈𝒲(M,11+,11-) such that d_Hom( H_f̂, H_ĝ)< η and d_Hom(𝒢_f,𝒢_g)<, we have d_M(H_f̂∘𝒢_f(x), H_ĝ∘𝒢_g(x))≤ d_M(H_f̂∘𝒢_f(x), H_f̂∘𝒢_g(x))+ d_M(H_f̂∘𝒢_g(x),H_ĝ∘𝒢_g(x))<2 η.Analogously,d_M(( H_f̂∘𝒢_f)^-1(x),( H_ĝ∘𝒢_g)^-1(x))<2 η.This concludes the proof of the continuity of f↦ℋ_f and the proof of the whole theorem.§ LOCAL CONTINUOUS EMBEDDING OF THE MODULI SPACEIn this section, we finalize the construction of a continuous mappings on open subsets of a connected component of any stratum, which is needed to prove the main result of this paper. We do it in two steps.Firstly, for each ζ∈ℳ(M,Σ,κ) we constructa neighborhood 𝒰_ζ⊂ℳ(M,Σ,κ)of ζ, so that for every ω∈𝒰_ζ there exists a piecewise affine homeomorphism 𝔥_ω:M→ M such that(𝔥_ω)_*_ω=f_ωλ_ζwith 1/1+_ζ<f_ω<1/1-_ζ, where _ζ>0 is given by Theorem <ref>. We will also require that ω↦ f_ω∈ L^1(M) is continuous.Secondly, we use the results of the previous sections to show the existence of a homeomorphism ℋ_ω:M→ M such that (ℋ_ω∘𝔥_ω)_*_ω=_ζ. Moreover, we show that these homeomorphisms yield the existence of a continuous mapping 𝔖:𝒰_ζ→Flow(M,ζ) such that 𝔖(ω) is isomorphic by a homeomorphism to 𝒯^ω - the vertical translation flow on (M,ω). Let ζ∈ℳ(M,Σ,κ). There existsa neigbourhood 𝒰_ζ⊂ M(M,Σ,κ) such that, for every ω∈𝒰_ζ, the following holds: (i) there exists a triangulation 𝒴(ω) and a piecewise affine homeomorphism 𝔥_ω:(M,ω)→(M,ζ) which is affine on elements of 𝒴(ω), fixes Σ and is Lipschitz with constant 11/10, (ii) (𝔥_ω)_*_ω is an absolutely continuous measure with respect to _ζ with piecewise constant density f_ω satisfying 1/1+_ζ<f_ω<1/1-_ζ, (iii) the mapping U_ζ∋ω↦ f_ω∈ L^1(M,_ζ) is continuous. Moreover, for given ϵ>0, there exists >0 such that (iv) for any ω̅∈𝒰_ζ, if d_Mod(ω,ω̅)< then 𝔥_ω^-1∘𝔥_ω̅:(M,ω̅)→(M,ω) is affine on elements of 𝒴(ω̅), 𝔥_ω̅^-1∘𝔥_ω:(M,ω)→(M,ω̅) is affine on elements of 𝒴(ω), they are both Lipschitz with constant 1+ϵ, Id-D(𝔥_ω^-1∘𝔥_ω̅)|_A<ϵ for every A∈𝒴(ω̅) and Id-D(𝔥_ω̅^-1∘𝔥_ω)|_B<ϵ for every B∈𝒴(ω), (v) for any ω̅∈𝒰_ζ such thatd_Mod(ω,ω̅)< and for the set M̃(ω):={x∈ M;inf_σ∈Σd_ω(𝒯_t^ω(x),σ)>4ϵ for all t∈[-1,1]}, we have _ω(M̃(ω))>1-Kϵ, where K>0 depends only on stratum, and for x∈M̃(ω) we have d_ω(𝒯^ω_t(x),𝔥_ω^-1∘𝔥_ω̅∘𝒯^ω̅_t∘𝔥_ω̅^-1∘𝔥_ω(x))<ϵ for any t∈[-1,1]. and for every σ∈Σ d_ω(σ,𝔥_ω^-1∘𝔥_ω̅∘𝒯^ω̅_t∘𝔥_ω̅^-1∘𝔥_ω(x))>3ϵ for any t∈[-1,1]. Let π be a permutation of the alphabet 𝒜 with d elements which belongs to a Rauzy class corresponding to ℳ(M,Σ,κ). We can assume that ζ has no vertical saddle-connections and thus there is a polygonal representation (π,^ζ,τ^ζ) of ζ. Otherwise we can rotate ζ to obtain a form ζ' which does not admit vertical saddle connections, construct a triangulation 𝒴(ζ') and rotate it back together with this triangulation to obtain a triangulation 𝒴(ζ) (note that a rotation is an isometry and that it acts continuously on ℳ(M,Σ,κ), see <cit.>). Let ω∈ℳ(M,Σ,κ), and assume that ω=M(π,^ω,τ^ω) for some ^ω,τ^ω. Let 𝒫(ω)⊂ℂ be the polygon corresponding to ω, whose vertices R_0(ω),R_1(ω),…,R_d(ω),R_1'(ω),…,R_d-1'(ω) are given by R_i(ω):=∑_{;π_0()≤ i}(_^ω+iτ_^ω) andR_i'(ω):=∑_{;π_1()≤ i}(_^ω+iτ_^ω) for i=0,…,d. Note that R_0(ω)=R_0'(ω)=0, R_d(ω)=R_d'(ω). For i=1,…,d-1 consider the vertical segments connecting R_i(ω) and R_i'(ω) with the opposite side of 𝒫(ω). Denote the other endpoints of those segments by Q_i(ω) and Q_i'(ω) respectively (see Fig. 3). Since each side on the upper half of the polygon is identified with one of the sides on the lower half of the polygon, there exist representations of Q_i(ω) and Q_i'(ω) on the opposite half of the polygon which we denote by S_i(ω) and S_i'(ω) respectively. Note that Re(S_i(ω))=T_π,^ω^-1(Re(R_i(ω)))andRe(S_i'(ω))=T_π,^ω(Re(R_i'(ω))), where T_π,^ω is the IET given by (π,). Let 𝒱(ω):={R_0(ω),…,R_d(ω),R_1'(ω),…,R_d-1'(ω),S_1(ω),…,S_d-1(ω),S_1'(ω),…,S_d-1'(ω)}. 𝒱(ω) is fully determined by (^ω,τ^ω). If ω has no vertical saddle connections, then for all distinct x,y∈𝒱(ω) we have Re(x)≠Re(y). Consider the sequence {V_j(ω)}_j=0^4d-2, which is an ordering of 𝒱(ω), such that {Re(V_j)}_j=0^4d-2 be an increasing sequence. Let _1>0 be such that for every ω∈ℳ(M,Σ,κ) satisfying d_Mod(ζ,ω)<_1, the orderings of the sets 𝒱(ζ) and 𝒱(ω) are the same, that is V_j(ω)=R_i(ω) iffV_j(ζ)=R_i(ζ); V_j(ω)=R_i'(ω) iff V_j(ζ)=R_i'(ζ), V_j(ω)=S_i(ω) iffV_j(ζ)=S_i(ζ)and V_j(ω)=S_i'(ω) iff V_j(ζ)=S_i'(ζ), for every j=0,…,2d, and {Re(V_i(ω))}_i=0^d is strictly increasing. For every ω∈ℳ(M,Σ,κ) with d_Mod(ζ,ω)<_1, we now construct a triangulation𝒴(ω) of 𝒫(ω). (We abuse the word “triangulation” sincethe edges may connect vertices of triangulation which are actually the same points.) Let {r(k)}_0≤ k≤ 2dbe the strictly increasing sequence in {0,…,4d-2} such that V_r(0),…,V_r(2d) be the vertices of 𝒫(ω). Let Ṽ_r(k)(ω):=Q_i(ω) whenever V_r(k)(ω)=R_i(ω) and analogously let Ṽ_r(k)(ω):=Q_i'(ω) whenever V_r(k)(ω)=R_i'(ω). Consider the triangle given by the points V_r(0)=0,V_r(1),Ṽ_r(1). If r(1)=1, then this triangle belongs to 𝒴(ω). If r(1)≠ 1 and Im(V_1(ω))>0, then we connect by segments all V_i(ω) such that i≤ r(1) and Im(V_i(ω))<0 with V_1(ω), and for all i≤ r(1) such that Im(V_i(ω))>0, we connect V_i(ω) with the point V_r(1)(ω) or Ṽ_r(1)(ω), whichever has the negative imaginary part. If r(1)≠ 1 and Im(V_1(ω))<0, then we proceed symmetrically. The triangles obtained by using the above segments are elements of 𝒴(ω). By applying vertical reflection, we use the same construction for the triangle given by points V_r(2d-1)(ω),Ṽ_r(2d-1)(ω),V_r(2d)(ω). For every k=1,…,2d-2, consider the trapezoid given by V_r(k)(ω), V_r(k+1)(ω), Ṽ_r(k)(ω) and Ṽ_r(k+1)(ω). In each of those trapezoids, we take the diagonal connecting the top-left vertex with the bottom-right vertex. If r(k+1)=r(k)+1, then the two resulting triangles belong to 𝒴(ω). If r(k+1)≠ r(k)+1, then for every r(k)<i<r(k+1), we connect V_i(ω) with the bottom-right vertex if Im(V_i(ω))>0,and with the top-left vertex if Im(V_i(ω))<0. We include the resulting triangles into 𝒴(ω). In this way we get a triangulation 𝒴(ω) of 𝒫(ω) into triangles which have vertices in 𝒱(ω) (see Fig. 4). Define 𝔥_ω:(M,ω)→ (M,ζ) as the piecewise affine transformation, such that 𝔥_ω(V_i(ω))=V_i(ζ) for every i=0,…,4d-2 and sending affinely each triangle from 𝒴(ω) with vertices V_j(ω), V_k(ω), V_ℓ(ω) onto the triangle with vertices V_j(ζ), V_k(ζ), V_ℓ(ζ). Note that the map 𝔥_ω is uniquely determined by the points in 𝒱(ω). Moreover, since Σ⊂𝒱(ω), 𝔥_ω fixes Σ. Let 0<_2<_1 be such that, defining 𝒰̃_ζ:={ω∈ℳ(M,Σ,κ),d_Mod(ζ,ω)<_2 }, we have ω∈𝒰̃_ζ⇒ 1-_ζ<_ζ(A)/_ω(𝔥_ω^-1(A))<1+_ζ for every A∈𝒴(ζ). This implies that (𝔥_ω)_*_ω is absolutely continuous with respect to λ_ζ and has a piecewise constant density f_ω given by f_ω(x)=_ω(𝔥_ω^-1(A))/_ζ(A) for every x∈ A where A∈𝒴(ζ). Hence 1/1+_ζ<f_ω<1/1-_ζ. Moreover, note that for every A∈𝒴(ζ), the mapping ω→_ω(𝔥_ω^-1(A)) is continuous. Hence, the formula (<ref>) implies the continuity of ω↦ f_ω∈ L^1(M,λ_ζ). Let ω∈𝒰̃_ζ. Then for any ω̅∈𝒰̃_ζ, 𝔥_ω̅^-1∘𝔥_ω is a continuous piecewise affine homoeomorphism which is affine on the elements of 𝒴(ω) and 𝔥_ω̅^-1∘𝔥_ω(V_i(ω))=V_i(ω̅). Take any A∈𝒴(ζ) and let V_j(ζ),V_k(ζ),V_ℓ(ζ)∈𝒱(ζ) be its vertices. Then lin(𝔥_ω̅^-1∘𝔥_ω|_𝔥_ω^-1A) is given by a matrix B_A(ω,ω̅)=[b_ij(ω,ω̅)]_i,j=1,2, where b_1,1(ω,ω̅)=Re(V_k(ω̅)-V_j(ω̅))Im(V_ℓ(ω)-V_j(ω))-Re(V_ℓ(ω̅)-V_j(ω̅))Im(V_k(ω)-V_j(ω))/Re(V_k(ω)-V_j(ω))Im(V_ℓ(ω)-V_j(ω))-Re(V_ℓ(ω)-V_j(ω))Im(V_k(ω)-V_j(ω)), b_1,2(ω,ω̅)=Re(V_ℓ(ω̅)-V_j(ω̅))Re(V_k(ω)-V_j(ω))-Re(V_k(ω̅)-V_j(ω̅))Re(V_ℓ(ω)-V_j(ω))/Re(V_k(ω)-V_j(ω))Im(V_ℓ(ω)-V_j(ω))-Re(V_ℓ(ω)-V_j(ω))Im(V_k(ω)-V_j(ω)), b_2,1(ω,ω̅)=Im(V_k(ω̅)-V_j(ω̅))Im(V_ℓ(ω)-V_j(ω))-Im(V_ℓ(ω̅)-V_j(ω̅))Im(V_k(ω)-V_j(ω))/Re(V_k(ω)-V_j(ω))Im(V_ℓ(ω)-V_j(ω))-Re(V_ℓ(ω)-V_j(ω))Im(V_k(ω)-V_j(ω)), and b_2,2(ω,ω̅)=Re(V_k(ω)-V_j(ω))Im(V_ℓ(ω̅)-V_j(ω̅))-ImRe(V_ℓ(ω)-V_j(ω))(V_k(ω̅)-V_j(ω̅))/Re(V_k(ω)-V_j(ω))Im(V_ℓ(ω)-V_j(ω))-Re(V_ℓ(ω)-V_j(ω))Im(V_k(ω)-V_j(ω)). Note that, to obtain a formula for B_A(ω̅,ω)=lin(𝔥_ω^-1∘𝔥_ω̅|_𝔥_ω̅^-1A), it is enough to switch ω with ω̅. Observe that B_A(ω,ω)=Id. Since all coefficients depend continuously on the elements of 𝒱(ω̅) and 𝔥_ζ=Id, by taking ω̅=ζ we can find 0<_3≤_2 such that for all ω∈𝒰_ζ:={ω∈ℳ(M,Σ,κ),d_Mod(ζ,ω)<_3 }, 𝔥_ω and 𝔥_ω^-1 are Lipschitz with constant 11/10. Moreover, for any ϵ>0 and any ω∈𝒰_ζ we can find >0 such that for all A∈𝒴(ζ) and for all ω̅ satisfying d_Mod(ω,ω̅)<, we have that 𝔥_ω^-1∘𝔥_ω̅ and 𝔥_ω̅^-1∘𝔥_ω are Lipschitz with constant 1+ϵ and Id -B_A(ω,ω̅)<ϵ and Id-B_A(ω̅,ω)<ϵ. To prove (v), notethat the set M̃(ω) does not contain points which are in the 4ϵ-neighbourhood of ingoing and outgoing vertical separatrix segments of length 1 starting from singular points. That is the complement M̃(ω)^c is of measure _ω at most (1+4ϵ)8ϵ times the number of ingoing and outgoing separatrices, which determines the value of K. For every ω̅∈𝒰_ζ denote by X^ω̅:M→^2 the unit constant vertical vector field on (M,ω̅) defined on M∖Σ which generates 𝒯^ω̅. Then by (iv) we have that X^ω(x)-D(𝔥_ω^-1∘𝔥_ω̅)_h_ω̅^-1∘𝔥_ω(x)X^ω̅(𝔥_ω̅^-1∘𝔥_ω(x))<ϵ. Note that the vector field D(𝔥_ω^-1∘𝔥_ω̅)_h_ω̅^-1∘𝔥_ω(x)X^ω̅(h_ω̅^-1∘𝔥_ω(x)) is well defined everywhere except on the edges of the triangulation 𝒴(ω). Since this vector field is constant on the interiors of the elements of the triangulation 𝒴(ω), we can define it on M∖Σ by choosing on each edge of 𝒴(ω) one of the vectors derived from one of the two triangles forming this edge. It is worth to mention that, if the direction of an edge of 𝒴(ω̅) given by some triangles 𝔥_ω̅^-1(A) and 𝔥_ω̅^-1(B)∈𝒴(ω̅) coincides with the direction of the flow 𝒯^ω̅, then the vector field D(𝔥_ω^-1∘𝔥_ω̅)_h_ω̅^-1∘𝔥_ω(x)X^ω̅(h_ω̅^-1∘𝔥_ω(x)) takes the same values on the triangles 𝔥_ω^-1(A),𝔥_ω^-1(B)∈𝒴(ω). Hence the flow induced by this vector field is well defined and (<ref>) holds everywhere on M∖Σ. Thus, in view of (<ref>), for any t∈[-1,1] we have |∫_0^tX^ω(𝒯_s^ω(x))-D(𝔥_ω^-1∘𝔥_ω̅)_𝒯_s^ω̅∘𝔥_ω̅^-1∘𝔥_ω(x)X^ω̅(𝒯_s^ω̅∘𝔥_ω̅^-1∘𝔥_ω(x))ds|=|∫_0^tX^ω(𝔥_ω^-1∘𝔥_ω̅∘𝒯_s^ω̅∘𝔥_ω̅^-1∘𝔥_ω(x))-D(𝔥_ω^-1∘𝔥_ω̅)_𝒯_s^ω̅∘𝔥_ω̅^-1∘𝔥_ω(x)X^ω̅(𝒯_s^ω̅∘𝔥_ω̅^-1∘𝔥_ω(x))ds|<|t|ϵ≤ϵ. Since for every x∈M̃(ω) we have 𝔥_ω^-1∘𝔥_ω̅∘𝒯_t^ω̅∘𝔥_ω̅^-1∘𝔥_ω(x)=x+∫_0^tD(𝔥_ω^-1∘𝔥_ω̅)_𝒯_s^ω̅∘𝔥_ω̅^-1∘𝔥_ω(x)X^ω̅(𝒯_s^ω̅(𝔥_ω̅^-1∘𝔥_ω(x)))ds in local coordinates, we deduce that d_ω (𝒯_t^ω(x),𝔥_ω^-1∘𝔥_ω̅∘𝒯_t^ω̅∘𝔥_ω̅^-1∘𝔥_ω(x))≤|∫_0^tX^ω(𝒯_s^ω(x))-D(𝔥_ω^-1∘𝔥_ω̅)_𝒯_s^ω̅∘𝔥_ω̅^-1∘𝔥_ω(x)X^ω̅(𝒯_s^ω̅(𝔥_ω̅^-1∘𝔥_ω(x)))ds| ≤ϵ. Since x∈M̃(ω), this also implies that for every σ∈Σ we have d_ω(σ,𝔥_ω^-1∘𝔥_ω̅∘𝒯_t^ω̅∘𝔥_ω̅^-1∘𝔥_ω(x)) >3ϵ. This concludes the proof of (v) and thus the proof of the whole lemma. Let ω∈ℳ(M,Σ,κ) and let D be a rectangle in (M,ω). For every >0 there exists >0 such that, for every _ω-measure preserving F:D→(M,ω) satisfying sup_x∈ Dd_ω(x,F(x))<, we have _ω(D)-_ω(D∩ F(D))<. Let >0. We assume that <_ω(D), otherwise the result is obvious. Choose >0 such that the set D̂:={x∈ D; ∀y∈ M, d_ω(x,y)<⇒ y∈ D } has measure _ω(D̂)>_ω(D)-. Then, for any _ω-measure preserving F:D→(M,ω) satisfying (<ref>), we have F(D̂)⊂ D. Hence _ω(D∩ F(D))≥_ω(F(D̂))=_ω(D̂)>_ω(D)-. Let ζ∈ℳ(M,Σ,κ) and let 𝒰_ζ bethe neighbourhood given by Lemma <ref>. There exists a continuous mapping 𝔖:𝒰_ζ→Flow(M,_ζ) such that for every ω∈𝒰_ζ the vertical flow on (M,ω) is measure-theoretically isomorphic by a homeomorphism to the measure-preserving flow 𝔖(ω) on (M,_ζ). By (i) and (ii) in Lemma <ref>, for every ω∈𝒰_ζ there exists a homeomorphism 𝔥_ω:M→ M fixing Σ and such that (𝔥_ω)_*_ω=f_ω_ζ , where f_ω satisfies 1/1+_ζ<f_ω<1/1-_ζ. Hence we can apply Theorem <ref> to obtain a homeomorphism ℋ_ω:=ℋ_f_ω:M→ M, which depends continuously on f_ω and(ℋ_f_ω)_*(f_ω_ζ)=_ζ. By (iii) in Lemma <ref>, it follows that the map ω↦ f_ω is continuous. Hence ω↦ℋ_ω, as a composition of two continuous mappings, is also continuous. Now define a homeomorphism of M𝒮_ω:=ℋ_ω∘𝔥_ω.Note that (𝒮_ω)_*_ω=_ζ and the flow 𝒮_ω∘𝒯^ω∘𝒮_ω^-1 is _ζ-measure preserving. To conclude the proof we show now that the mapping 𝒰_ζ∋ω↦𝒮_ω∘𝒯^ω∘𝒮_ω^-1=:𝔖(ω)∈Flow(M,_ζ) is continuous. Fix ω∈𝒰_ζ. We now prove the continuity of 𝔖 in ω. On (M,ω) choose a family 𝒬 of open rectangles,with vertical and horizontal sides, that generates the Borelσ-algebra on M. We may assume that for every Q∈𝒬 we have _ω(Q)≤1/4.Note that 𝒮_ω^-1:(M,ζ)→(M,ω) is a measure-theoretic isomorphism. Hence, in view of Remark <ref>, to prove that 𝔖 is continuous, it is sufficient to prove that the map 𝒰_ζ∋ω̅↦𝒮_ω^-1∘𝔖(ω̅)_t∘𝒮_ω∈Flow(X,_ω) is continuous. That is for every >0 and Q∈𝒬 there exists >0 such thatd_Mod(ω,ω̅)<⇒sup_t∈[-1,1]_ω(𝒯^ω_t Q𝒮_ω^-1∘𝔖(ω̅)_t∘𝒮_ω Q)<. Fix Q∈𝒬 and >0. We now prove (<ref>) for Q. Denote by k the number of times the ingoing and outgoing vertical separatrix segments of length 2 starting from the singular points σ∈Σ intersect with Q. By extending those segments if necessary, we obtain segments v_j⊂ Q, for j=1,…,k, such that the endpoints of v_j lie on a horizontal sides of Q. Let 0<ϵ_Q<, and consider the subset Q̃⊆ Q obtained by cutting out from Q all rectangles of which the segments v_j are vertical sides and whose width is 4ϵ_Q. Assume that ϵ_Q is small enough so that_ω(Q̃)>(1-)_ω(Q). Note that Q̃ is a union of l≤ k+1 rectangles D_j for j=1,…,l. By Lemma <ref>, there exists γ>0 such that for every j=1,…,l and every _ω-preserving transformation F:D_j→ M satisfying sup_x∈ D_jd_ω(x,F(x))<4γ we have_ω(D_j∩ F(D_j))>(1-)_ω(D_j). Take 0<ϵ<min{γ,ϵ_Q}. Since ω̅↦ℋ_ω̅ is continuous, we can choose >0 such that for every ω̅∈ U_ζd_Mod(ω,ω̅)<⟹(sup_x∈ Md_ζ(ℋ_ω^-1∘ℋ_ω̅(x),x)<ϵ∧sup_x∈ Md_ζ(ℋ_ω̅^-1∘ℋ_ω(x),x)<ϵ).Moreover, by applying (iv) from Lemma <ref> for ϵ and taking smallerif necessary, we get that 𝔥_ω^-1∘𝔥_ω̅:(M,ω̅)→(M,ω) and 𝔥_ω̅^-1∘𝔥_ω:(M,ω)→(M,ω̅) are Lipschitz piecewise affine homeomorphisms with constant 1+ϵ. Furthermore, (v) in Lemma <ref> gives us that the set M̃(ω) satisfies_ω(M̃(ω))>1-Kϵ and for x∈M̃(ω) we haved_ω(𝒯^ω_t(x),𝔥_ω^-1∘𝔥_ω̅∘𝒯^ω̅_t∘𝔥_ω̅^-1∘𝔥_ω(x))<ϵ for any t∈[-1,1].It also implies that, for every σ∈Σ, d_ω(σ,𝔥_ω^-1∘𝔥_ω̅∘𝒯^ω̅_t∘𝔥_ω̅^-1∘𝔥_ω(x))>3ϵ for any t∈[-1,1].Since ϵ<ϵ_Q, we have Q̃⊂M̃(ω). We now estimate the distance between the orbits of the flows 𝔥_ω^-1∘𝔥_ω̅∘𝒯_t^ω̅∘𝔥_ω̅^-1∘𝔥_ω and𝒮_ω^-1∘𝒮_ω̅∘𝒯_t^ω̅∘𝒮_ω̅^-1∘𝒮_ω=𝔥_ω^-1∘ℋ_ω^-1∘ℋ_ω̅∘𝔥_ω̅∘ T^ω̅_t∘ h_ω̅^-1∘ℋ_ω̅^-1∘ℋ_ω∘ h_ω. By (<ref>) we have thatd_ζ(𝔥_ω(x), ℋ_ω̅^-1∘ℋ_ω∘𝔥_ω (x))<ϵfor every x∈ M. By (i) in Lemma <ref>, 𝔥_ω̅^-1:(M,ζ)→(M,ω) is Lipschitz with constant 11/10. Thus we haved_ω̅(𝔥_ω̅^-1∘𝔥_ω(x), 𝔥_ω̅^-1∘ℋ_ω̅^-1∘ℋ_ω∘𝔥_ω (x))<11/10ϵ.Since 𝔥_ω^-1∘𝔥_ω̅ is Lipschitz with constant 1+ϵ andfixes Σ, (<ref>) implies thatmin_σ∈Σinf_t∈[-1,1]d_ω̅(𝒯_t^ω̅∘𝔥_ω̅^-1∘𝔥_ω(x),σ)>3ϵ/1+ϵ>2ϵfor every x∈M̃(ω). Hence on the 2ϵ-neighbourhood of 𝔥_ω̅^-1∘𝔥_ω(x), {𝒯^ω̅_t}_t∈[-1,1] acts isometrically. Thusd_ω̅(𝒯^ω̅_t(𝔥_ω̅^-1∘𝔥_ω(x)), 𝒯^ω̅_t∘𝔥_ω̅^-1∘ℋ_ω̅^-1∘ℋ_ω∘𝔥_ω (x))<11/10ϵ,for t∈[-1,1] and for every x∈M̃(ω).Since 𝔥_ω̅:(M,ω̅)→(M,ζ) is Lipschitz with constant 11/10, this implies thatd_ζ(𝔥_ω̅∘𝒯^ω̅_t∘𝔥_ω̅^-1∘𝔥_ω(x),𝔥_ω̅∘𝒯^ω̅_t∘𝔥_ω̅^-1∘ℋ_ω̅^-1∘ℋ_ω∘𝔥_ω (x))<121/100ϵ.Again by using (<ref>) we obtain thatd_ζ(𝔥_ω̅∘𝒯^ω̅_t∘𝔥_ω̅^-1∘𝔥_ω(x),ℋ_ω^-1∘ℋ_ω̅∘𝔥_ω̅∘𝒯^ω̅_t∘𝔥_ω̅^-1∘ℋ_ω̅^-1∘ℋ_ω∘𝔥_ω (x))<221/100ϵ.Finally, since 𝔥_ω^-1 is also Lipschitz with constant 11/10, we obtain thatd_ω(𝔥_ω^-1∘𝔥_ω̅∘𝒯^ω̅_t∘ h_ω̅^-1∘ h_ω(x),𝔥_ω^-1∘ℋ_ω^-1∘ℋ_ω̅∘𝔥_ω̅∘𝒯^ω̅_t∘ h_ω̅^-1∘ℋ_ω̅^-1∘ℋ_ω∘ h_ω (x))<2431/1000ϵ.By combining this with (<ref>) we obtain that for every x∈M̃(ω) we haved_ω(𝒯_t^ω(x),𝒮_ω^-1∘𝔖(ω̅)_t∘𝒮_ω(x))<3431/1000ϵ<4ϵ.By the definition of M̃(ω), {𝒯^ω_t}_t∈[-1,1] acts isometrically onthe 4ϵ-neighbourhood of x∈M̃(ω). Henced_ω(x,𝒯_-t^ω∘𝒮_ω^-1∘𝔖(ω̅)_t∘𝒮_ω(x))<4ϵ. Since D_j⊆Q̃⊂M̃(ω), (<ref>) is satisfied for all x∈ D_j. ConsiderF:= 𝒯_-t^ω∘𝒮_ω^-1∘𝔖(ω̅)_t∘𝒮_ω.Note that F is _ω-measure preserving. Thus, by (<ref>), we get_ω(D_j∩𝒯_-t^ω∘𝒮_ω^-1∘𝔖(ω̅)_t∘𝒮_ω(D_j))>(1-)_ω(D_j).Together with 𝒯^ω-invariance of _ω, this yields_ω(𝒯_t^ω(D_j)∩𝒮_ω^-1∘𝔖(ω̅)_t∘𝒮_ω (D_j))>(1-)_ω(𝒯_t^ω(D_j)),for every t∈[-1,1]. By summing up over j=1,…,l we get_ω(𝒯_t^ω(Q̃)∩𝒮_ω^-1∘𝔖(ω̅)_t∘𝒮_ω(Q̃))>(1-)_ω(𝒯_t^ω(Q̃))and by (<ref>) this yields_ω(𝒯_t^ω(Q)∩𝒮_ω^-1∘𝔖(ω̅)_t∘𝒮_ω (Q))>(1-2)_ω(𝒯_t^ω(Q)).Since _ω(Q)<1/4, we have_ω(𝒯_t^ω(Q)𝒮_ω^-1∘𝔖(ω̅)_t∘𝒮_ω (Q))<4_ω(𝒯_t^ω(Q))≤. Thus we get (<ref>), and this concludes the proof of thetheorem. Since the construction above is local, we need to show that this suffices to transport the G_-condition from the space of flows to the moduli space. Let X be a metric topological space. Let {U_i}_i∈ be a sequence of open subsets such that ⋃_i∈U_i=X. If V⊆ X is such that V∩ U_i is a G_-set for each i∈, then V is a G_-set. Note that V=⋂_i∈(V∩ U_i)∪ U_i^c. Since X is metrizable, every closed set is a G_-set. To finish the proof it is enough to observe that the union of two G_-sets is a G_-set. § TRANSLATION FLOWS DISJOINT WITH THEIR INVERSES ARE G_-DENSEWe have the following result which follows from the proof of Corollary 3.3 in <cit.>. Let (X,ℬ(X),μ) be a nonatomic standard Borel probability space, and let Flow(X) be the space of μ-invariant flows on X. The set of flows which are weakly mixing and disjoint with their inverse is G_-dense in Flow(X). The following result allows us to transfer the G_ condition onto any connected component of the moduli space. Let 𝔓𝔯𝔬𝔭 be a property of a measure-preserving flow such that the set of elements having this property is a G_ subset of the space Flow(X). Then in every connected component C of ℳ, the set of translation structures for which the vertical flow has the property 𝔓𝔯𝔬𝔭 is a G_ set in C. Let C be a connected component of ℳ. In view of Theorem <ref> for every ζ∈ C there exists an open neighbourhood 𝒰_ζ of ζ and a continuous mapping 𝔖_ζ:𝒰_ζ→ Flow(M,_ζ) such that for every ω∈𝒰_ζ the vertical flow 𝒯^ω is measure-theoretically isomorphic to 𝔖_ζ(ω). Since C is a topological manifold, it is σ-compact. Thus there exists a sequence {ζ_n}_n∈ of translation structures such that ⋃_n∈𝒰_ζ_n=C. For each n∈ we have that 𝒴_ζ_n: ={ω∈𝒰_ζ_n; 𝒯^ω satisfies 𝔓𝔯𝔬𝔭}={ω∈𝒰_ζ_n; 𝔖_ζ_n(ω) satisfies 𝔓𝔯𝔬𝔭}=𝔖_ζ_n^-1{𝒯∈ Flow(M,_ζ_n); 𝒯 satisfies 𝔓𝔯𝔬𝔭} is a G_ set in 𝒰_ζ_n. By Lemma <ref>, this gives that the set of ω∈ C such that 𝒯^ω satisfies 𝔓𝔯𝔬𝔭 is a G_ set in C. By combining Theorem <ref> and Proposition <ref> we get the following result. The set of translation structures ζ such that the vertical flow on (M,ζ) is weakly mixing and disjoint with its inverse is a G_ set in every connected component C of the moduli space. Throughout this section we use the following notation. Let C⊂ℳ be a non-hyperelliptic connected component of the moduli space, i.e. C is not of the form ℳ^hyp(2g-2) or ℳ^hyp(g-1,g-1) for any g≥ 2. Let π=(π_0,π_1) be a permutation of the alphabet 𝒜 of d elements from the corresponding extended Rauzy class satisfying π_0^-1(1)=π_1^-1(d) and π_0^-1(d)=π_1^-1(1). This permutation exists due to Theorem <ref> and by the choice of C, it is not symmetric. Let Ω_π be the translation matrix corresponding to π. By Corollary <ref> there exist symbols a_1,a_2∈𝒜 such that(Ω_π)_a_1a_2=(Ω_π)_a_2a_1=0 and for any rationally independent vector τ∈^𝒜 the numbers (Ω_πτ)_a_2-(Ω_πτ)_a_1and (Ω_πτ)_a_1-((Ω_πτ)_π_0^-1(1)+(Ω_πτ)_π_0^-1(d)) are rationally independent. The proof of the following lemma goes along the same lines as the proof of Lemma 14 in <cit.>. It is mainly based on the recurrence of polygonal Rauzy-Veech induction. The set C_*:={M(π,,τ)∈ C; (π,,τ)∈Θ_π, _a=0 for a∈𝒜∖{π^-1_0(1),π^-1_0(d),a_1,a_2}} is dense in C. Before heading to the proof of the main result we give the proof of Proposition <ref> which treats the density of translation structures on which the vertical flow is reversible. By following again the proof of Lemma 14 in <cit.> we prove that the set C_**:={M(π,,τ)∈ C; (π,,τ)∈Θ_π, _a=0 for a∈𝒜∖{π^-1_0(1),π^-1_0(d)}} is dense in C. The vertical flow on M(π,,τ)∈ C_** is measure-theoretically isomorphic to the vertical flow on a torus given by (_π^-1_0(1),_π^-1_0(1)) and π - the non-trivial permutation of two elements. Since the translation flows on tori are reversible, this concludes the proof. The special representations of the vertical flows associated to the translation structures from C_* as given in Lemma <ref> are special flows over IETs of 3 intervals and under a roof functions which are piecewise constant and have discontinuity points which coincide with the discontinuity points of the IET and one additional discontinuity point inside the middle interval. After one step of either left- or right-hand side Rauzy-Veech induction we get a special representation over rotation and under a piecewise constant roof function with 4 discontinuity points. We now prove some properties of such flows. Let ∈[0,1) be an irrational number, and let T_:/→/ be the rotation by . For any β∈[0,1) let β:=min{{β},{1-β}}. Let {q_n}_n∈ be the sequence of partial denominators associated to . Recall that for every odd n∈, we have a pair of Rokhlin towers {T_^i[-q_n,0)}_i=0,…,q_n-1-1 and {T_^i[0,q_n-1)}_i=0,…,q_n-1 for n≥ 1, which covers /. As a corollary from Theorem 3.9 in <cit.> we get the following result (recall the definitions of f^(r) and Leb^f from subsection <ref>, as well as the definition of μ_t,s given in (<ref>)). Let {(T_^f)_t}_t∈ be the special flow over the rotation by ∈[0,1) under a positive roof function f∈ L^2([0,1),Leb). Suppose that there exists a rigidity sequence {r_n}_n∈ for T_ (which is a subsequence of {q_n}_n∈) such that, setting b_n:=r_n∫_0^1f(y)dy, the sequence {∫_0^1|f^(r_n)(x)-b_n|^2dx}, n∈ is bounded. Then there exists a probability measure P∈𝒫(^2) such that, up to taking a subsequence, (f^(2r_n)-2b_n,f^(r_n)-b_n)_*Leb→ Pweakly. Moreover, along the same subsequence, we have Leb^f_2b_n,b_n→∫_^2Leb^f_-t,-udP(t,u). To prove the next result we need the following remark. Let f:[0,1)→ be a piecewise constant function. Let β_1,…,β_k be the jumps of f and let d_1,…,d_k be their respective values. Then for every x∈[0,1) and every odd n∈, ∑_i=0^q_n-1(f(T_^q_n+i(x))-f(T_^i(x)))=∑_i=0^q_n-1∑_j=1^k -d_jχ_T_^-i[β_j,β_j+q_n)(x). Indeed, the expression f(T_^q_n+i(x))-f(T_^i(x)) takes non-zero value if and only if there is a discontinuity point β_j in the interval (T_^q_n+i(x),T_^i(x)]. However, we have T_^q_n+i(x)=T_^i(x)-q_n. Hence β_j∈ (T_^q_n+i(x),T_^i(x)] if and only if x∈ T_^-i[β_j,β_j+q_n). In other words, if we consider the (not necessarily disjoint) towers U_j:=⋃_i=0^q_n-1T^-i[β_j,β_j+q_n) for j=1,…,k, then ∑_i=0^q_n-1(f(T_^q_n+i(x))-f(T_^i(x)))=∑_j=1^k-d_jχ_U_j(x). In particular, if x∉ U_j for all j=1,…,k, then ∑_i=0^q_n-1(f(T_^q_n+i(x))-f(T_^i(x)))=0. Otherwise, if x∈ U_j for some j=1,…,k, then d_j contributes to the value of the considered expression. We need the following theorem which is a version of Theorem 7.3 in <cit.>. Recall that ξ:^2→ is defined by ξ(t,u):=t-2u. There exists a set Λ⊂/ of full Lebesgue measure such that, for every ∈Λ, there exists a set D_∈(/)× (/) of full Lebesgue measure with the property that, if (β_1,β_2)∈ D_, then * the numbers 0,1-,β_1 and β_2 are distinct; * for every piecewise constant positive function f:/→ with discontinuity points 0,1-,β_1,β_2 and jumps d_β_1 and d_β_2 at β_1 and β_2 respectively, for the special flow {(T^f_)_t}_t∈ there exist probability measures P,Q∈𝒫(^2) such that, up to a subsequence, we have Leb^f_2b_n,b_n→∫_^2Leb^f_-t,-udP(t,u)and Leb^f_-2b_n,-b_n→∫_^2Leb^f_-t,-udQ(t,u)weakly, where b_n:=q_n∫_0^1f(x)dx. Moreover, ξ_*P is atomic with exactly 4 atoms in points 0, -d_β_1, -d_β_2, d_β_1+d_β_2, while ξ_*Q=(-ξ)_*P has exactly 4 atoms in points 0, d_β_1, d_β_2, -(d_β_1+d_β_2). Let Λ⊂/ be the set of irrational ∈/ such that there exists a sequence {k_n}_n∈ of odd numbers such that, for some 1/52≤≤1/25, we have lim_n→∞q_k_nq_k_n=. The set Λ is of full Lebesgue measure. Indeed, the Gauss map G(x)={1/x} is mixing for the absolutely continuous measure with density 1/ln 21/1+x, hence in particular G^2 is ergodic. For any irrational ∈/, let {a_n}_n∈ be the sequence of partial quotients of . Then we have (see <cit.>) 1/21/a_n+1+1<q_nq_n<1/a_n+1. Recall that for any m∈, G^n()∈(1/m+1,1/m] iff a_n=m. Hence by ergodicity of G^2, for almost every ∈[0,1), a_n+1=25 for infinitely many odd numbers n. Thus we obtain the claim. Fix ∈Λ. Recall that for every n∈, / is covered by a pair of towers {T_^i[-q_k_n,0)}_i=0,…,q_k_n-1-1and{T_^i[0,q_k_n-1)}_i=0,…,q_k_n-1. Note that 1-∈ T_^q_k_n-1[0,q_k_n-1). More precisely, 1-=T_^q_k_n-1(0)+q_k_n. Since {q_k_n}_n∈ is a rigidity sequence and f, regardless of the choice of discontinuity points, is of bounded variation, (<ref>) follows from Proposition <ref> and Koksma-Denjoy inequality: |f^(q_k_n)(x)-b_n|≤ Var(f)for every x∈[0,1). In view of Proposition <ref>, taking a subsequence if necessary, we get P=lim_n→∞(f^(2q_k_n)-2b_n,f^(q_k_n)-b_n)_*Leb and Q=lim_n→∞(f^(-2q_k_n)+2b_n,f^(-q_k_n)+b_n)_*Leb. By applying ξ_* to both expressions we obtain ξ_*P=lim_n→∞(∑_i=0^q_k_n-1(f∘ T_^q_k_n+i-f∘ T_^i))_*Leb and ξ_*Q=lim_n→∞(∑_i=1^q_k_n(f∘ T_^-q_k_n-i-f∘ T_^-i))_*Leb. By using the invariance of the Lebesgue measure under T_^2q_k_n, we get ξ_*Q=(-ξ)_*P. Consider the sequence of pairs of disjoint Rokhlin towers V_n:=⋃_i=0^q_k_n-1T^i_[3q_k_n,1/3q_k_n-1)and W_n:=⋃_i=0^q_k_n-1T^i_[2/3q_k_n-1,q_k_n-1-3q_k_n|). We know (see e.g. <cit.>) that 1/2q_k_n<q_k_n-1<1/q_k_n. Hence q_k_n-1/q_k_n<1/q_k_nq_k_n→1/ and q_k_n-1/q_k_n>1/2q_k_nq_k_n→1/2. By using the fact that <1/25, for sufficiently large n we obtain that 12<q_k_n-1/q_k_n<53. In view of (<ref>) we get Leb(W_n)=Leb(V_n)=q_k_n(1/3q_k_n-1-3q_k_n)>q_k_nq_k_n>1/53, for sufficiently large n∈, that is the measures of V_n, W_n are bounded away from 0. By the remark to Lemma 3.4 in <cit.>, this implies that, for almost every β_1∈[0,1), there exists an infinite set N_1 of natural numberssuch that β_1∈ V_n for each n∈ N_1. Once such β_1 and N_1 are fixed, the same argument yields that for almost every β_2∈[0,1) there exists an infinite subset N_2⊂ N_1such that β_2∈ W_n for each n∈ N_2. Using Fubini's theorem, we get that for almost every (β_1,β_2)∈[0,1)×[0,1) there exist infinitely many integers n such that β_1∈ V_n and β_2∈ W_n.Let D_ be the set of such pairs (β_1,β_2). Now we fix (β_1,β_2) in D_. Extracting a subsequence if necessary, we may assume that β_1∈ V_n and β_2∈ W_n for all n. Since V_n and W_n are disjoint and 0,1-∉ W_n∪ V_n for all n∈, the points 0, 1-, β_1 and β_2 are distinct. Note that, by the choice of W_n and V_n the towers ⋃_i=0^q_k_n-1T^-i[β_1,β_1+q_k_n), ⋃_i=0^q_k_n-1T^-i[β_2,β_2+q_k_n) and ⋃_i=0^q_k_nT^-n[0,q_k_n) are pairwise disjoint. Indeed, we have the following inclusions ⋃_i=0^q_k_n-1T^-i[β_1,β_1+q_k_n)⊂⋃_i=0^q_k_n-1T^i_[3q_k_n,1/3q_k_n-1+q_k_n), ⋃_i=0^q_k_n-1T^-i[β_2,β_2+q_k_n)⊂⋃_i=0^q_k_n-1T^i_[2/3q_k_n-1,q_k_n-1-2q_k_n), and ⋃_i=0^q_k_nT^-i[0,q_k_n)⊂⋃_i=0^q_k_n-1T^i_[0,2q_k_n). In view of (<ref>), we get1/3q_k_n-1>4q_k_n for sufficiently large n. In particular 1/3q_k_n-1+q_k_n<2/3q_k_n-1, which shows that the intervals [0,2q_k_n),[3q_k_n,1/3q_k_n-1+q_k_n) and [2/3q_k_n-1,q_k_n-1-2q_k_n)are pairwise disjoint. This implies the desired disjointness of the aforementioned towers. This allow us to control the atoms of limit measures and their respective masses. Suppose now that f has discontinuity points at 0, 1-, β_1 and β_2, where (β_1,β_2)∈ D_ with jumps d_0,d_1-,d_β_1 and d_β_2 respectively. Since k_n is odd, in view of Remark <ref> the expression ∑_i=0^q_k_n-1(f∘ T_^q_k_n+i(x)-f∘ T_^i(x)) may only take non-zero values for x on towers U_1:=⋃_i=0^q_k_n-1T^-i[0,q_k_n), U_2:=⋃_i=0^q_k_n-1T^-i[1-,1-+q_k_n), U_3:=⋃_i=0^q_k_n-1T^-i[β_1,β_1+q_k_n) and U_4:=⋃_i=0^q_k_n-1T^-i[β_2,β_2+q_k_n). We have proved though, that U_3 is disjoint with other towers and the same is true for U_4. Thus we get that ∑_i=0^q_k_n-1(f∘ T_^q_k_n+i(x)-f∘ T_^i(x))=-d_β_1forx∈ U_3, and ∑_i=0^q_k_n-1(f∘ T_^q_k_n+i(x)-f∘ T_^i(x))=-d_β_1forx∈ U_4. On the other hand, U_1∩ U_2=⋃_i=1^q_k_n-1T^-i[0,q_k_n, U_1∖ U_2=[0,q_k_n), and U_2∖ U_1=[q_k_n,2q_k_n). Hence ∑_i=0^q_k_n-1(f∘ T_^q_k_n+i(x)-f∘ T_^i(x))=-d_0-d_1-forx∈ U_1∩ U_2, ∑_i=0^q_k_n-1(f∘ T_^q_k_n+i(x)-f∘ T_^i(x))=-d_0forx∈[0,q_k_n), and ∑_i=0^q_k_n-1(f∘ T_^q_k_n+i(x)-f∘ T_^i(x))=-d_1-forx∈[q_k_n,2q_k_n). Finally ∑_i=0^q_k_n-1(f∘ T_^q_k_n+i(x)-f∘ T_^i(x))=0forx∉ U_1∪ U_2∪ U_3∪ U_4. Note that -d_0-d_1-=d_β_1+d_β_2, since the sum of jumps of f is 0. Moreover Leb([0,q_k_n))=Leb([q_k_n,2q_k_n))=q_k_n→ 0, Leb(U_3)=Leb(U_4)=q_k_nq_k_n→ and Leb(U_1∩ U_2)=q_k_nq_k_n-q_k_n→. Hence we get that ξ_*P=lim_n→∞(∑_i=0^q_k_n-1f∘ T_^q_k_n+i-f∘ T_^i)_*Leb is a measure such that ξ_*P({-d_β_1})=ξ_*P({-d_β_2})=ξ_*P({d_β_1+d_β_2})= and ξ_*P({0})=1-3. Since ξ_*Q=(-ξ)_*P, we obtain the final claim. We now state a reformulation of the above lemma for rotations on arbitrary large circles. If needed, throughout the proof of this lemma, we identify /x with [0,x) for every x∈_>0. There exists a subset Δ_0⊂Δ:={(x,y)∈_>0^2;0< y<x} of full Lebesgue measure in Δ with the property that for every (l,)∈Δ_0 there exists a set D_l,⊂(/l)×(/l) of full Lebesgue measure such that for every (β_1,β_2)∈ D_l, we have * the numbers 0, l-, β_1 and β_2 are distinct in /l and * if T_ is the rotation on /l by ∈/l and h:/l→_>0 is a piecewise constant function withexactly 4 discontinuity points at 0,l-,β_1,β_2 and rationally independent jumps at β_1 and β_2, then the special flow T_^h is weakly mixing and disjoint with its inverse. Let Λ⊂[0,1) and D_⊂[0,1)×[0,1) for ∈Λ be the sets given by Lemma <ref>. For any l∈_>0, let us also denote by l:[0,1)→[0,l) the map given by l(x):=lx. We also consider l as a map between / and /l. For any l∈_>0 let Λ_l:=l(Λ)⊂ [0,l) and set D_l,:=(l× l)(D_l^-1)⊂[0,l)×[0,1) for any ∈[0,l). Define Δ_0:={(x,y); x∈_>0, y∈Λ_x}. Note that Δ_0 is of full Lebesgue measure in Δ and for every l∈_>0 and ∈Λ_l the set D_l, is of full Lebesgue measure in (/l)×(/l). Take (l,)∈Δ_0 and (β_1,β_2)∈ D_l,. By the definition of Λ and D_l^-1, the points 0, l-, β_1 and β_2 are distinct. Let h/l→_>0 be a piecewise constant function which has exactly 4 discontinuity points at 0, l-, β_1 and β_2. Assume that the jumps d_β_1 and d_β_2 at β_1 and β_2 are rationally independent. Consider the special flow T_^h on [0,l)^h. The map (l^-1× Id):[0,l)^h→[0,1)^h∘ l establishesan isomorphism of flows T_^h and T_l^-1^h∘ l. The roof function h∘ l has discontinuities at 0,1-l^-1,l^-1β_1 and l^-1β_2 and has jumps d_β_1 and d_β_2 at l^-1β_1 and l^-1β_2 respectively. Moreover, l^-1∈Λ and (l^-1β_1,l^-1β_2)∈ D_. In view of Lemma <ref>, this gives Leb^h∘ l_2b_n,b_n→∫_^2Leb^h∘ l_-t,-udP(t,u),and Leb^h∘ l_-2b_n,-b_n→∫_^2Leb^h∘ l_-t,-udQ(t,u)weakly, for some increasing to infinity real sequence {b_n}_n∈ and measures P,Q∈𝒫(^2). Furthermore, ξ_*P is atomic and has atoms at 0, -d_β_1, -d_β_2 and d_β_1+d_β_1, while ξ_*Q is also atomic and has atoms at 0, d_β_1, d_β_2 and -(d_β_1+d_β_1). Since d_β_1 and d_β_2 are rationally independent, Proposition <ref> implies that T_l^-1^h∘ l is weakly mixing. Moreover, the rational independence of d_β_1 and d_β_2 also gives ξ_*P≠ξ_*Q which yields P≠ Q. In view of Corollary <ref> this gives that T_l^-1^h∘ l is disjoint with its inverse. Since T_l^-1^h∘ l and T_^h are isomorphic, T_^h is weakly mixing and disjoint with its inverse. We are now ready to give the proof of the main result of this paper.In view of Corollary <ref>, the set of translation structures whose associated vertical flow is weakly mixing and disjoint with its inverse is a G_ set in every connected component of the moduli space. We now show that there is a dense subset of translation structures in each non-hyperelliptic connected component C so that the associated vertical flows are weakly mixing and disjoint with their inverses. Fix a non-hyperelliptic connected component C of the moduli space. Recall that for some d≥ 2 and an alphabet 𝒜 of d elements, there is a permutation π=(π_0,π_1)∈ S_0^𝒜 in the extended Rauzy class associated with C, such that π_1(π_0^-1(1))=dandπ_1(π_0^-1(d))=1. Let Ω:=Ω_π be the translation matrix corresponding to π. Then, in view of Corollary <ref>, there exist symbols a_1,a_2∈𝒜 such that Ω_a_1a_2=Ω_a_2a_1=0 and the numbers (Ωτ)_a_2-(Ωτ)_a_1and (Ωτ)_a_i-((Ωτ)_π_0^-1(1)+(Ωτ)_π_0^-1(d)) are rationally independent for i=1,2 whenever τ is rationally independent.Let Ξ_*:={(π,,τ)∈Θ_π;_a=0 for a∈𝒜∖{π^-1_0(1),π^-1_0(d),a_1,a_2}}Let C_*:={M(π,,τ)∈ C; (π,,τ)∈Ξ_*}. In view of Lemma <ref>, this is a dense subset of C. Hence to prove the density of the desired property in C, it is enough to prove that this property holds for a dense set in C_*. We provethis by finding a dense subset of parameters in Ξ_* such that the associated translation structures have the sought properties.Note that the set Ξ⊂Ξ_* given byΞ:={(π,,τ)∈Ξ_*; T_π, is ergodic; _π_0^-1(1)≠_π_0^-1(d); τ is rationally independent}is dense in Ξ_*. Let ζ=M(π,,τ)∈ C_* with (π,,τ)∈Ξ. Let 𝒯^ζ be the corresponding vertical flow. Recall that it has a special representation T_π,^h over the IET T_π,:[0,||)→[0,||) and under a piecewise constant roof function h[0,||)→_>0 which is constant over exchanged intervals. Moreover, if we consider h={h_a}_a∈𝒜 as a vector of values, where h_a is the value of h over the interval corresponding to a, then h_a=-(Ωτ)_a. However, since (π,,τ)∈Ξ_*, we have that _a=0 for a∈𝒜∖{π^-1_0(1),π^-1_0(d),a_1,a_2}. Thus we can reduce the data describing the above special representation.Let π̂=(π̂_0,π̂_1) bethe permutation on the alphabet 𝒜̂:={π^-1_0(1),π^-1_0(d),a_1,a_2} given byπ̂_0(π^-1_0(1))=1,π̂_0(π^-1_0(d))=4,π̂_0(a_1)=2,π̂_0(a_2)=3andπ̂_1(π^-1_0(1))=4,π̂_1(π^-1_0(d))=1,π̂_1(a_1)=2,π̂_1(a_2)=3.For a∈𝒜̂ let _a:=_a. Moreover, since the intervals corresponding to a∈𝒜∖𝒜̂ are empty, h can be considered as a vector{h_a}_a∈𝒜̂. Then 𝒯^ζ has a special representation T_π̂,^h over the IET T_π̂,:[0,||)→[0,||).Consider the sets Ξ_0,Ξ_1⊂Ξ given byΞ_0:={(π,,τ)∈Ξ; _π^-1_0(1)>_π^-1_0(d)} and Ξ_1:={(π,,τ)∈Ξ; _π^-1_0(1)<_π^-1_0(d)}.We have Ξ_0∪Ξ_1=Ξ. Suppose first that (π,,τ)∈Ξ_0 that is _π^-1_0(1)>_π^-1_0(d). Let ϕ:{(x,y,z,v)∈_>0^4; x>v}→^4_>0 bethe diffeomorphism given byϕ(x,y,z,v):=(x-v,v,y,z).Then after one step of the polygonal right hand side Rauzy-Veech induction on 𝒯_π̂,^h we get a special flow T_^ĥ over the rotation T_:[0,_π^-1_0(1)+_a_1+_a_2)→[0,_π^-1_0(1)+_a_1+_a_2) by =(λ̂):=_a_1+_a_2+_π^-1_0(d) under a piecewise constant function ĥ[0,_π^-1_0(1)+_a_1+_a_2)→_>0 with values h_π^-1_0(1),h_π^-1_0(1)+h_π^-1_0(d),h_a_1,h_a_2 over the consecutive intervals of lengths given by the vector ϕ(_π^-1_0(1),_a_1,_a_2,_π^-1_0(d)). Recall that the flows 𝒯_π̂,^h and T_^ĥ are isomorphic. Let l=l(λ̂):=_π^-1_0(1)+_a_1+_a_2, β_1=β_1(λ̂):=_π^-1_0(1) andβ_2=β_2(λ̂):=_π^-1_0(1)+_a_1. Then ĥ:[0,l)→_>0 has discontinuities at points l-, β_1 and β_2. The jump at the point β_1 is equal to h_a_1-(h_π^-1_0(1)+h_π^-1_0(d)), while at the point β_2 equals h_a_2-h_a_1. Moreover, we haveh_a_1-(h_π^-1_0(1)+h_π^-1_0(d))=-(Ωτ)_a_1+((Ωτ)_π_0^-1(1)+(Ωτ)_π_0^-1(d)),andh_a_2-h_a_1=-(Ωτ)_a_2+(Ωτ)_a_2.Since τ is a rationally independent vector, Corollary <ref> yields the rational independence of the jumps at β_1 and β_2. From now on we treat T_ as a rotation on /l. Furthermore, we also treat ĥ as a piecewise constant function on /l. Then ĥ/l→_>0 gets an additional discontinuity point at 0.Let us considerthe diffeomorphismψ:^4_>0→{(x,y,z,v)∈_>0^4; 0<x-y<z<v<x} given byψ(x,y,z,v):=(x+y+z+v,y+z+v,x+y,x+y+z).Thenψ∘ϕ:{(x,y,z,v)∈^4_>0; x>v}→{(x,y,z,v)∈_>0^4; 0<x-y<z<v<x}is a diffeomorphism andψ∘ϕ ()=(l,,β_1,β_2).Let Δ_0⊂{(x,y)∈_>0^2;y∈(0,x)} and D_l,⊂(/l)×(/l) for (l,)∈Δ_0 be sets given by Lemma <ref>. Then, by Lemma <ref>, for every (l,)∈Δ_0 and (β_1,β_2)∈ D_l,, the special flow T_^ĥ overthe rotation byon /land under a piecewise constant roof function with discontinuity points 0,l-,β_1,β_2 andwith rationally independent jumps at β_1 and β_2 is weakly mixing and disjoint with its inverse. Consider𝒢:={(x,y,z,v)∈_>0^4; (x,y)∈Δ_0, y/x∈∖, (z,v)∈ D_x,y,0<x-y<z<v<x}.In view of Lemma <ref>, Δ_0 is dense in {(x,y)∈_>0; y<x } and D_x,y is dense in (0,x)×(0,x). Therefore 𝒢 is a dense set in {(x,y,z,v)∈_>0^4; 0<x-y<z<v<x}. As ψ∘ϕ is a diffeomorphism, the set (ψ∘ϕ)^-1(𝒢) is dense in {(x,y,z,v)∈_>0^4; x>v}. Hence the set Γ_0:={(π,,τ)∈Ξ_0; ∈(ψ∘ϕ)^-1 (𝒢)and τ is a rationally independent vector} is dense in Ξ_0. By going along the same lines and by using the left-hand side polygonal Rauzy-Veech induction, we find a dense set Γ_1⊂Ξ_1 which has analogous properties. If =(π,,τ)∈Γ_0∪Γ_1 then (l(),())∈Δ_0, (β_1(),β_2())∈ D_l(),() and the vertical flow on M() is isomorphic to a special flow T_()^ĥ on (/l())^ĥ, where ĥ:/l()→_>0 is a piecewise constant roof function with discontinuities at 0, l()-(), β_1(), β_2() and the jumps at β_1() and β_2() are rationally independent.In view of Lemma <ref>, those flows are weakly mixing and disjoint with their inverses.As Γ_0∪Γ_1 is dense in Ξ,it is also dense in Ξ_*.Since M:Θ_π→ C given by (π,,τ)↦ M(π,,τ) is continuous and M(Ξ_*)=C_*, we have that M(Γ_0∪Γ_1) is dense in C_*. Moreover, by Lemma <ref>, C_* is dense in C which yields the result.§ ACKNOWLEDGMENTSThe authors would like to thank M. Lemaczyk for fruitful discussions and for proposing the main ideas used in section 3. We would also like to thank S. Gouezel for pointing out the article <cit.> and for giving some ideas used in section 6. P. Berk and K. Frczek are partially supported by NCN grant nr2014/13/B/ST1/03153.9 AmbrW. Ambrose,Representation of ergodic flows,Ann. of Math. (2) 42 (1941), 723739. AnzH. Anzai,On an example of a measure preserving transformation which is not conjugate to its inverse,Proc. Japan Acad. 27 (1951), 517-522. AF A. Avila, G. Forni,Weak mixing for interval exchange transformations and translation flows,Ann. of Math. (2) 165 (2007), 637664. BerkFrP. Berk, K. Frączek,On special flows that are not isomorphic to their inverses,Discrete Contin. Dyn. Syst. 35 (2015),829855. DaRy A.I. Danilenko, V.V. Ryzhikov,On self-similarities of ergodic flows,Proc. Lond. Math. Soc. 104 (2012), 431-454. dJun A. del Junco,Disjointness of measure-preserving transformations, minimal self-joinings and category,Ergodic theory and dynamical systems, I (College Park, Md., 1979-80), pp. 81-89, Progr. Math., 10, Birkhuser, Boston, Mass., 1981. Fox R.H. Fox, R.B. Kershner,Concerning the transitive properties of geodesics on a rational polyhedron, Duke Math. J. 2 (1936), 147150. FrK. Frączek,Density of mild mixing property for vertical flows of Abelian differentials,Proc. Amer. Math. Soc. 137 (2009), 4129-4142. FrKuLem K. Frączek, J. Kułaga-Przymus, M. Lemańczyk,Non-reversibility and self-joinings of higher orders for ergodic flows,J. Anal. Math. 122 (2014), 163-227. Glas E. Glasner,Ergodic theory via joinings,Mathematical Surveys and Monographs, 101. American Mathematical Society, Providence, RI, 2003. GPC. Goffman, G. Pedrick,A proof of the homeomorphism of Lebesgue-Stieltjes measure with Lebesgue measure,Proc. Amer. Math. Soc. 52 (1975), 196198. Halmos P.R. Halmos,Ergodic Theory,Chelsea, New York, 1956. Ka A.B. Katok,Interval exchange transformations and some special flows are not mixing,Israel J. Math. 35 (1980), 301310. KatokA.B. Katok, A.N. Zemljakov,Topological transitivity of billiards in polygons,Mat. Zametki 18 (1975),291300. KingJ. King,Joining-rank and the structure of finite rank mixing transformations,J. Analyse Math. 51 (1988), 182-227. KhinA.Y. Khinchin,Continued fractions,The University of Chicago Press, Chicago-London, 1964. KonZoM. Kontsevich, A. Zorich,Connected components of the moduli spaces of Abelian differentials with prescribed singularities,Invent. Math. 153 (2003), 631678. MazH. Masur,Interval exchange transformations and measured foliations,Ann. of Math. (2) 115 (1982), 169200. MosJ. Moser,On the volume elements on a manifold,Trans. Amer. Math. Soc. 120 (1965), 286294. Ryz1 V.V. Ryzhikov,Partial multiple mixing on subsequences can distinguish between automorphisms T and T^-1,Math. Notes 74 (2003), 841-847. BorelS.M. Srivastava,A course on Borel sets,Graduate Texts in Mathematics, 180. Springer-Verlag, New York, 1998. VeechW.A. Veech,Gauss measures for transformations on the space of interval exchange maps,Ann. of Math.115 (1982), 201242. YoccozJ.C. Yoccoz,Interval exchange maps and translation surface,Homogeneous flows, moduli spaces and arithmetic, 169, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010. | http://arxiv.org/abs/1703.09111v1 | {
"authors": [
"Przemyslaw Berk",
"Krzysztof Fraczek",
"Thierry de la Rue"
],
"categories": [
"math.DS",
"37A10, 37E35, 37C80"
],
"primary_category": "math.DS",
"published": "20170327143749",
"title": "On typicality of translation flows which are disjoint with their inverse"
} |
./figs/ [email protected] Applications of Artificial IntelligenceTraffic light timing optimization is still an active line of research despite the wealth of scientific literature on the topic, and the problem remains unsolved for any non-toy scenario. One of the key issues with traffic light optimization is the large scale of the input information that is available for the controlling agent, namely all the traffic data that is continually sampled by the traffic detectors that cover the urban network. This issue has in the past forced researchers to focus on agents that work on localized parts of the traffic network, typically on individual intersections, and to coordinate every individual agent in a multi-agent setup. In order to overcome the large scale of the available state information, we propose to rely on the ability of deep Learning approaches to handle large input spaces, in the form of Deep Deterministic Policy Gradient (DDPG) algorithm. We performed several experiments with a range of models, from the very simple one (one intersection) to the more complex one (a big city section). deep learning reinforcement learning traffic light control policy gradient § INTRODUCTION Cities are characterized by the evolution of their transit dynamics. Originally meant solely for pedestrians, urban streets soon shared usage with carriages and then with cars. Traffic organization became soon an issue that led to the introduction of signaling, traffic lights and transit planning. Nowadays, traffic lights either have fixed programs or are actuated. Fixed programs (also referred to as pretimed control) are those where the timings of the traffic lights are fixed, that is, the sequences of red, yellow and green phases have fixed duration. Actuated traffic lights change their phase to green or red depending on traffic detectors that are located near the intersection; this way, actuated traffic light are dynamic and adapt to the traffic conditions to some degree; however, they only take into account the conditions local to the intersection. This also leads to dis-coordination with the traffic light cycles of other nearby intersections and hence are not used in dense urban areas. Neither pretimed or actuated traffic lights take into account the current traffic flow conditions at the city level. Nevertheless, cities have large vehicle detectorinfrastructures that feed traffic volume forecasting tools usedto predict congestion situations. Such information is normally only used to apply classic traffic management actions like sending police officers to divert part of the traffic. This way, traffic light timings could be improved by means of machine learning algorithms that take advantage of the knowledge about traffic conditions by optimizing the flow of vehicles. This has been the subject of several lines of research in the past. For instance, Wiering proposed different variants of reinforcement learning to be appliedto traffic light control <cit.>, and created the Green Light District (GLD) simulator to demonstrate them, which was further used in other works like <cit.>. Several authors explored the feasibility of applying fuzzy logic, like <cit.> and <cit.>. Multi-agent systems where also applied to this problem, like <cit.> and <cit.>.Most of the aforementioned approaches simplify the scenario to a single intersection or a reduced group of them. Other authors propose multi-agent systems where each agent controls a single intersection and where agents maycommunicate with each other to share information to improve coordination(e.g. in a connected vehicle setup <cit.>) or may receive a piece of shared information to be aware of the crossed effects on other agents' performance (<cit.>).However, none of the aforementioned approaches fully profited from the availability of all the vehicle flow information, that is, the decisions taken by those agents were in all cases partially informed. The main justification for the lack of holistic traffic light control algorithms is the poor scalability of most algorithms. In a big city there can be thousands of vehicle detectors and tenths of hundreds of traffic lights. Those numbers amount for huge space and action spaces, which are difficult to handle by classical approaches. This way, the problem addressed in this works is the devisal of an agent that receives traffic data and, based on these, controls the traffic lights in order to improve the flow of traffic, doing it at a large scale.§ TRAFFIC SIMULATIONIn order to evaluate the performance of our work, we make use of a traffic simulation. The base of a traffic simulation is the network, that is, the representation of roads and intersections where the vehicles are to move. Connected to some roads, there are centroids, that act as sources/sinks of vehicles. The amount of vehicles generated/absorbed by centroids is expressed in a traffic demand matrix, or origin-destination (OD) matrix, which contains one cell per each pair of origin and destination centroids. During a simulation, different OD matrices can be applied to different periods of time in order to mimic the dynamics of the real traffic through time. In the roads of the network, there can be traffic detectors, that mimic induction loops beneath the ground that are able to measure traffic data as vehicles go pass through them. Typical measurements that can be taken with traffic detectors include vehicle counts, average speed and percentage of occupancy. There can also be traffic lights. In many cases they are used to regulate the traffic at intersections. In those cases, all the traffic lights in an intersection are coordinated so that when one is red, another one is green, and vice versa (this way, the use of the intersection is regulated so that vehicles don't block the intersection due to their intention to reach an exit of the intersection that is currently in use) . All the traffic lights in the intersection change their state at the same time. This intersection-level configuration of the traffic lights is called a phase, and it is completely defined by the states of each traffic light in the intersection plus its duration. The different phases in an intersection form its control plan. The phases in the control plan are applied cyclically, so the phases are repeated after the cycle duration elapses. Normally, control plans of adjacent intersections are synchronized to maximize the flow of traffic avoiding unnecessary stops. Urban traffic simulation software can keep models at different levels of abstraction. Microscopic simulators simulate vehicles individually computing their positions at every few milliseconds and the the dynamics of the vehicles are governed by a simplified model that drives the behaviour of the driver under different conditions, while macroscopic simulators work in an aggregated way, managing traffic like in a flow network in fluid dynamics. There are different variations between microscopic and macroscopic models, broadly referred to as mesoscopic simulators. To our interests, the proper simulation level would be microscopic, because we need information of individual vehicles and their responses to changes in the traffic lights, mimicking closely real workd dynamics in terms of congestion. As third party simulator we chose Aimsun <cit.>, a commercial microscopic, mesoscopic and macroscopic simulator widely used, both in the private consulting sector and in traffic organization institutions.§ PRELIMINARY ANALYSIS The main factor that has prevented further advance in the traffic light timing control problem is the large scale of any realistic experiment. On the other hand, there is a family of machine learning algorithms whose very strenght is their ability of handle large input spaces, namely deep learning. Recently, deep learning has been successfully applied to reinforcement learning, gainingmuch attention due to the effectiveness of Deep Q-Networks (DQN)at playing Atari games using as input the raw pixels of the game <cit.>. Subsequent successes of a similar approach called Deep Deterministic Policy Gradient (DDPG) were achieved in<cit.>, which will be used in our work as reference articles, given the similarity of the nature of the problems addressed there, namely large continuous state and action spaces. This way, the theme of this work is the application of Deep Reinforcement Learning to the traffic light optimization problem with an holistic approach, by leveraging deep learning to cope with the large state and action spaces. Specifically, the hypothesis that drives this work is that Deep reinforcement learning can be successfully applied to urban traffic light control, having similar or better performance than other approaches. This is hence the main contribution of the present work, along with the different techniques applied to make this application possible and effective. Taking into account the nature of the problem and the abundant literature on the subject, we know thatsome of the challenges of devising a traffic light timing control algorithm that acts at a large scale are:* Define a sensible state space. This includes finding a suitable representation of the traffic information. Deep learning is normally used with input signals over which convolution is easily computable, like images (i.e. pixel matrices) or sounds (i.e. 1-D signals). Traffic information may not be easily represented as a matrix, but as a labelled graph. This is addressed in section <ref>. * Define a proper action space that our agent is able to perform. The naive approach would be to let the controller simply control the traffic light timing directly (i.e. setting the color of each traffic light individually at each simulation step). This, however, may lead to breaking the normal routing rules, as the traffic lights in an intersection have to be synchronized so that the different intersection exit routes do not interfere with each other. Therefore a careful definition of the agent's actions is needed. This is addressed in section <ref>. * Study and ensure the convergence of the approach: despite the successes of Deep Q-Networks and DDPG, granted by their numerous contributions to the stability of reinforcement learning with value function approximation, convergence of such approaches is not guaranteed. Stability of the training is studied and measures for palliating divergence are put in place. This is addressed in section <ref>. * Create a sensible test bed: a proper test bed should simulate relatively realistically the traffic of a big city, including a realistic design of the city itself. This is addressed in section <ref>.§ RELATED WORKIn this section we identify and explore other lines of research that also try to solve the traffic light control problem. §.§ Offline Approaches The most simple traffic light control approaches are those that define fixed timings for the different traffic light phases. These timings are normally defined offline (i.e. not in closed loop). Several differentapproaches have been proposed in the literature for deriving the phase timings, which can be grouped into the following categories [The categorization focuses on both the adaptative nature (or lack thereof) of the approach and the type of algorithms used and their similarity to the approach proposed in this work.]: * Model-based: a mathematical model of the target urban area is prepared and then used to derive an optimal timing, either via derivative calculus, numerical optimization,integer linear programming, or any other method. An exampleof this approach is MAXBAND <cit.>, which defines a model for arterials and optimizes it for maximum bandwidth by means of linear programming. Another example is the TRANSYT system <cit.>, which uses an iterative process to minimize the average journey time in a network of intersections. * Simulation-based: this case is analogous to the model-based, but the core of the validation of the timings is a traffic simulation engine, which is connected to a black box optimization computation that iteratively searches the traffic light timing space to find an optimal control plan. Some examples of this approach are <cit.>, which make use of genetic algorithms together with the CORSIM simulator <cit.>, or <cit.>, which uses particle swarm optimization with the SUMO simulator <cit.>.The usual way of maximizing the success of this kind of methods isto analyze historical traffic data and identify time slots with different traffic volume characteristics; once defined, a different timing strategy is derived for each these time bands. However, not even this partitioning scheme adapts to the dynamism of traffic demand or progressive changes in drivers' behaviour. §.§ Model-based Adaptive Approaches The simplest of these approaches only one intersection into consideration. They define a model (e.g. based on queue theory) that is fed with real detector data (normally from the closest detectors to the intersection). Then, by using algorithmic logic based on thresholds and rules, like <cit.>, or optimization techniques like <cit.>, they try to minimize waiting times. More complex approaches are based on traffic network models of several intersectionsthat are fed with the real time data from multiple traffic detectors. Some of these approaches are heuristically defined algorithms that tune their parameters by performing tests with variations on the aforementioned models. For example, the SCOOT system <cit.> performs small reconfigurations (e.g. individual intersection cycle offsets or cycle splits) on a traffic network model. More recent approaches like <cit.> make use of the information collected by Wireless Sensor Networks (WSN) to pursue the same goal. There are also approaches where more formal optimization methods are employed on the traffic network models fed with read time data, like the case of<cit.>, <cit.>, <cit.> or <cit.>, which compute in real time the switch times of the traffic lights within the next following minutes by solving dynamic optimization problems on realistic models fed with data from real traffic detectors. §.§ Classic Reinforcement Learning Reinforcement Learning has been applied in the past to urban traffic light control. Most of the instances from the literature consist of aclassical algorithm like Q-Learning, SARSA or TD(λ) to control the timing of a single intersection. Rewards are typically based on the reduction of the travel time of the vehicles or the queue lengths at the traffic lights. <cit.> offers a thorough review of the different approaches followed by a subset of articles from the literature that apply reinforcement learning to traffic light timing control. As shown there, many studies use as state space information such as the length of the queues and the travel time delay; such type of measures are rarely available in a real-world setup and can therefore only be obtained in a simulated environment. Most of the approaches use discrete actions (or alternatively, discretize the continuous actions by means of tile coding, and use either ε-greedy selection (choose the action with highest Q value with 1 - ε probability, or random action otherwise) or softmax selection (turn Q values into probabilities by means of the softmax function and then choose stochastically accordingly). In most of the applications of reinforcement learning to traffic control, the validation scenario consists of a single intersection, like in <cit.>. This is due to the scalability problems of classical RL tabular approaches: as the number of controlled intersections increases, so grows the state space, making the learning unfeasible due to the impossibility for the agent to apply every action under every possible state. This led some researchers to study multi-agent approaches, with varying degrees of complexity: some approaches like that from <cit.> train each agent separately, without notion that more agents even exist, despite the coordination problems that this approach poses. Others like <cit.> train each agent separately, but only the intersection with maximum reward executes the action. More elaborated approaches, like in <cit.>, train several agents together modeling their interaction as a competitive stochastic game. Alternatively, some lines of research like <cit.> and <cit.> study cooperative interaction of agents by means of coordination mechanisms, like coordination graphs (<cit.>). As described throughout this section, there are several examples in the literature of the application of classical reinforcement learning to traffic light control. Many of them focus on a single intersection. Others apply multi-agent reinforcement learning techniques to address the problems derived from the high dimensionality of state and action spaces. Two characteristics of most of the explored approaches are that the information used to elaborate the state space is hardly available in a real-world environment and that there are no realistic testing environments used. §.§ Deep Reinforcement Learning There are some recent works that, like ours, study the applicability ofdeep reinforcement learning to traffic light control: Li et al. studied in <cit.> the application of deep learning to traffic light timing in a single intersection. Their testing setup consists of a single cross-shape intersection with two lanes per direction, where no turns are allowed at all (i.e. all traffic either flows North-South (and South-North)or East-West (and West-East), hence the traffic light set only has two phases. This scenario is therefore simpler than our simple network A presented in <ref>. For the traffic simulation, they use the proprietary software PARAllel MICroscopic Simulation (Paramics) <cit.>, which implements the model by Fritzsche <cit.>. Their approach consists of a Deep Q-Network (<cit.>) comprised of a heap of stacked auto-encoders <cit.>, with sigmoid activation functions where the input is the state of the network and the output is the Q function value for each action. The inputs to the deep Q network are the queue lengths of each lane at time t (measured in meters), totalling 8 inputs. The actions generated by the network are 2: remain in the current phase or switch to the other one. The reward is the absolute value of the difference between the maximum North-Source flow and the maximum East-West flow. The stacked autoencoders are pre-trained (i.e. trained using the state of the traffic as both input and output) layer-wise so that an internal representation of the traffic state is learned, which should improve the stability of the learning in further fine tuning to obtain the Q function as output (<cit.>). The authors use an experience-replay memory to improve learning convergence. In order to balance exploration and exploitation, the authors use an ϵ-greedy policy, choosing a random action with a small probability p. For evaluating the performance of the algorithm, the authors compare it with normal Q-learning (<cit.>). For each algorithm, they show the queue lengths over time and perform a linear regression plot on the queue lengths for each direction (in order to check the balance of their queue length). Van der Pol explores in <cit.> the application of deep learning to traffic light coordination, both in a single intersection and in a more complex configuration. Their testing setup consists of a single cross-shaped intersection with one lane per direction, where no turns are allowed. For the simulation software, the author uses SUMO (Simulation of Urban MObility), a popular open-source microscopic traffic simulator. Given that SUMO teleports vehicles that have been stuck for a long time [See <http://sumo.dlr.de/wiki/Simulation/Why_Vehicles_are_teleporting>], the author needs to take this into account in the reward function, in order to penalize traffic light configurations that favour vehicle teleportation. Their approach consists on a Deep Q-Network. The author experiments with two two alternative architectures, taken verbatim respectively from <cit.> and <cit.>. Those convolutional networks were meant to play Atari games and receive as input the pixel matrix with bare preprocessing (downscaling and graying). In order to enable those architectures to be fed with the traffic data as input, an image is created by plotting a point on the location of each vehicle. The action space is comprised of the different legal traffic light configurations (i.e. those that do not lead to flow conflicts), among which the network chooses which to apply. The reward is a weighted sum of several factors: vehicle delay (defined as the road maximum speed minus the vehicle speed, divided by the road maximum speed), vehicle waiting time, the number of times the vehicle stops, the number of times the traffic light switches and the number of teleportations. In order to improve convergence of the algorithm, the authors apply deep reinforcement learning techniques such as prioritized experience replay and keeping a shadow target network, but also experimented with double Q learning <cit.>. They as well tested different optimization algorithms apart from the normal stochastic gradient optimization, such as the ADAM optimizer <cit.>, Adagrad <cit.> or RMSProp <cit.>. The performance of the algorithm is evaluated visually by means of plots of the reward and average travel time during the training phase. The author also explores the behaviour of their algorithm in a scenario with multiple intersections (up to four) by means of a multi-agent approach. This is achieved by training two neighbouring intersections on their mutual influence and then the learned joint Q function is transferred for higher number of intersections. Genders et al. explore in <cit.> thethe application of deep convolutional learning to traffic light timing. Their test setup consists of a single cross-shaped intersection with four lanes in each direction, where the inner lane is meant only for turning left and the outer lane is meant only for turning right. As simulation software, the authors use SUMO, like the work by Van der Pol <cit.> (see previous bullet). However, Genders et al do not address the teleportation problem and do not take into account its effect on the results. Their approach consists of a Deep Convolutional Q-Network. Like in <cit.>, Genders et al. transform the vehicle positions into a matrix so that it becomes a suitable input for the convolutional network. They, however, scale the value of the pixels with the local density of vehicles. The authors refer to this representation as discrete traffic state encoding (DTSE). The actions generated by the Q-Network are the different phase configurations of the traffic light set in the intersection. The reward defined as the variation in cumulative vehicle delay since the last action was applied. The network is fed using experience replay.§ THEORETICAL BACKGROUNDReinforcement Learning (RL) aims at training an agent so that it applies actions optimally to an environment based on its state, with the downside that it is not known which actions are good or bad, but it is possible to evaluate the goodness of their effects after they are applied. Using RL terminology, the goal of the algorithm is to learn an optimal policy for the agent, based on the observable state of the environment and on a reinforcement signal that represents the reward (either positive or negative) obtained when an action has been applied. The underlying problem that reinforcement learning tries to solve is that of the credit assignment. For this, the algorithm normally tries to estimate the expected cumulative future reward to be obtained when applying certain action when in certain state of the environment. RL algorithms act at discrete points in time. At each time step t, the agent tries to maximize the expected total return R_T, that is, the accumulated rewards obtained after each performed action: R_t = r_t+1 + r_t+2 + ⋯ + r_T, where T is the number of time steps ahead until the problem finishes. However, as normally T is dynamic or even infinite (i.e. the problem has no end), instead of the summation of the rewards,the discounted return is used:R_t = r_t+1 + γ r_t+2 + γ^2 r_t+3 + ⋯ = ∑_k=0^∞γ^k r_t+k+1The state of the environment is observable, either totally or partially. The definition of the state is specific to each problem. One example of state of the environment is the position x of a vehicle that moves in one dimension. Note that the state can certainly contain information that condenses pasts states of the environment. For instance, apart from the position x from the previous example, we could also include the speed ẋ and acceleration ẍ in the state vector. Reinforcement Learning problems that depend only on the current state of the environment are said to comply with the Markov property and are referred to as Markov Decision Processes. Their dynamics are therefore defined by the probability of reaching from a state s to a state s' by means of action a:p(s'|s,a) = P(S_t+1=s' | S_t=s, A_t=a)This way, we can define the reward obtained when transitioning from state s to s' by means of action a:r(s,a,s')= 𝔼[ R_t+1 | S_t = s, A_t = a, S_t+1 = s' ]Deep Reinforcement Learning refers to reinforcement learning algorithms that use a deep neural network as value function approximator. The first success of reinforcement learning with neural networks as function approximation was TD-Gammon <cit.>. Despite the initial enthusiasm in the scientific community, the approach did not succeed when applied to other problems, which led to its abandonment (<cit.>). The main reason for its failure was lack of stability derived from: * The neural network was trained with the values that were generated on the go, therefore such values were sequential in nature and thus were auto-correlated (i.e. not independently and identically distributed).* Oscillation of the policy with small changes to Q-values that change the data distribution.* Too large optimization steps upon large rewards.Their recent rise in popularity is due to the success of Deep Q-Networks (DQN)at playing Atari games using as input the raw pixels of the game <cit.>.ℒ(θ) = 𝔼[ (y - Q(s,a ; θ) )^2 ]In DQNs, there is a neural network that receives the environment state as input and generates as output the Q-values for each of the possible actions, using the loss function (<ref>), which implies following the direction of the gradient (<ref>):∇_θℒ(θ)= 𝔼[ ( r + γmax_a' Q(s',a' ; θ) - Q(s,a ; θ) ) ∇_θ Q(s, a; θ) ]In order to mitigate the stability problems inherent to reinforcement learning with value function approximation, in <cit.>, the authors applied the following measures: * Experience replay: keep a memory of past action-rewards and train the neural network with random samples from it instead of using the real time data, therefore eliminating the temporal autocorrelation problem.* Reward clipping: scale and clip the values of the rewards to the range [-1, +1] so that the weights do not boost when backpropagating.* Target network: keep a separate DQN so that one is used to compute the target values and the other one accumulates the weight updates, which are periodically loaded onto the first one. This avoid oscillations in the policy upon small changes to Q-values.However, DQNs are meant for problems with a few possible actions, and are therefore not appropriate for continuous space actions, like in our case. Nevertheless, a recently proposed Deep RL algorithm referred to as Deep Deterministic Policy Gradient or DDPG (<cit.>) naturally accommodates this kind of problems. It combines the actor-critic classical RL approach <cit.> with DeterministicPolicy Gradient <cit.>. The original formulation of the policy gradient algorithm was proposed in <cit.>, whichproved the policy gradient theorem for a stochastic policy π(s,a; θ):(Policy Gradient theorem from <cit.>) For any MDP, if the parameters θ of the policy are updated proportionally to the gradient of its performance ρthen θ can be assured to converge to a locally optimal policy in ρ, being the gradient computed asΔθ≈α∂ρ∂θ = α∑_s d^π(s)∑_a ∂π(s,a)∂θQ^π(s,a) with α being a positive step size and where d^π is defined as the discounted weighting of states encountered starting at s_0 and then following π: d^π(s) = ∑_t=0^∞γ^t P(s_t=s| s_0, π) This theorem was further extended in the same article for the case where an approximation function f is used in place of the policy π. In this conditions the theorem holds valid as long as the weights ofthe approximation tend to zero upon convergence. In our reference articles <cit.> and <cit.>, the authors propose to use a deterministic policy (as opposed to stochastic) approximated by a neural network actor π(s; θ^π) that depends on the state of the environment s and has weights θ^π, and another separate network Q(s,a; θ^Q) implementingthe critic, which is updated by means of the Bellman equation like DQN (<ref>):Q(s_t, a_t) = 𝔼_r_t, s_t+1[ r(s_t, a_t) + γ Q(s_t+1, π(s_t+1)) ]And the actor is updated by applying the chain rule to the loss function (<ref>) and updating the weights θ^π by following the gradient of the loss with respect to them:∇_θ^πℒ ≈𝔼_s [ ∇_θ^π Q(s, π(s|θ^π) | θ^Q) ] = 𝔼_s [ ∇_a Q(s, a | θ^Q) |_a=π(s|θ^π)∇_θ^ππ(s|θ^π) ]In order to introduce exploration behaviour, thanks to the DDPG algorithm being off-policy, we can add random noise 𝒩 to the policy. This enables the algorithm to try unexplored areas from the action space to discover improvement opportunities, much like the role of ε in ε-greedy policies in Q-learning. In order to improve stability, DDPG also can be applied the same measures as DQNs, namely reward clipping, experience replay (by means of a replay buffer referred to as R in algorithm <ref>) and separate target network. In order to implement this last measure for DDPG, two extra target actor and critic networks (referred to as π' and Q' in algorithm <ref>) to compute the target Q values, separated from the normal actor and critic (referred to as π and Q in algorithm <ref>) that are updated at every step and which weights are used to compute small updates to the target networks. The complete DDPG, as proposed in <cit.>, is summarized in algorithm <ref>. § PROPOSED APPROACH In this section we explain the approach we are proposing to address the control of urban traffic lights, along with the rationale that led to it. We begin with section <ref> by defining which information shall be used as input to our algorithm among all the data that is available from our simulation environment. We proceed by choosing a problem representation for such information to be fed into our algorithm in section <ref> for the traffic state and section <ref> for the rewards. §.§ Input InformationThe fact that we are using a simulator to evaluate the performance of our proposed application of deep learning to traffic control, makes the traffic state fully observable to us.However, in order for our system to be applied to the real world, it must be possible for our input information to be derived from data that is available in a typical urban traffic setup. The most remarkable examples of readily available data are the ones sourced by traffic detectors. They are sensors located throughout the traffic network that provide measurements about the traffic passing through them. Although there are different types of traffic detectors, the most usual ones are induction loops placed under the pavement that send real time information about the vehicles going over them. The information that can normally be taken from such type of detectors comprise vehicle count (number of vehicles that went over the detector during the sampling period),vehicle average speed during the sampling period and occupancy (the percentage of time in which there was a vehicle located over the detector). This way, we decide to constrain the information received about the state of the network to vehicle counts, average speed and occupancy of every detector in our traffic networks, along with the description of the network itself, comprising the location of all roads, their connections, etc. §.§ Congestion MeasurementFollowing the self-imposed constraint to use only data that is actually available in a real scenario, we shall elaborate a summary of the state of the traffic based on vehicle counts, average speeds and occupancy. This way, we defined a measured called speed score, that is defined for detector i as: 𝑠𝑝𝑒𝑒𝑑_𝑠𝑐𝑜𝑟𝑒_i =min( 𝑎𝑣𝑔_𝑠𝑝𝑒𝑒𝑑_i𝑚𝑎𝑥_𝑠𝑝𝑒𝑒𝑑_i , 1.0)where 𝑎𝑣𝑔_𝑠𝑝𝑒𝑒𝑑_i refers to the average of the speeds measured by traffic detector i and 𝑚𝑎𝑥_𝑠𝑝𝑒𝑒𝑑_i refers to the maximum speed in the road where detector i is located. Note that the speed score hence ranges in [0, 1]. This measure will be the base to elaborate the representation of both the state of the environment (section <ref>) andthe rewards for our reinforcement learning algorithm (section <ref>). §.§ Data Aggregation Period The microscopic traffic simulator used for our experiments divides the simulation into steps. At each step, a small fixed amount of time is simulated and the state of the vehicles (e.g. position, speed, acceleration) is updated according to the dynamics of the system.This amount of time is configured to be 0.75 seconds by default, and we have kept this parameter. However, such an amount of time is too short to imply a change in the vehicle counts of the detectors. Therefore, it is needed to have a larger period over which the data is aggregated; we refer to this period as episode step, or simply "step" when there is no risk of confusion. This way, the data is collected at each simulation step and then it is aggregated every episode step for the DDPG algorithm to receive it as input. In order to properly combine the speed scores of several simulation steps, we take their weighted average, using the proportion of vehicle counts. In an analogous way, the traffic light timings generated by the DDPG algorithm are used during the following episode step. The duration of the episode step was chosen by means of grid search, determining an optimum value of 120 seconds. §.§ State SpaceIn order to keep a state vector of the environment, we make direct use of the speed score described in section <ref>, as it not only summarizes properly the congestion of the network, but also incorporates the notion of maximum speed of each road. This way, the state vector has one component per detector, each one defined as shown in (<ref>).𝑠𝑡𝑎𝑡𝑒_i = 𝑠𝑝𝑒𝑒𝑑_𝑠𝑐𝑜𝑟𝑒_iThe rationale for choosing the speed score is that,the higher the speed score, the higher the speed of the vehicles relative to the maximum speed of the road, and hence the higher the traffic flow. §.§ Action SpaceIn the real world there are several instruments to dynamically regulate traffic: traffic lights, police agents, traffic information displays, temporal traffic signs (e.g. to block a road where there is an accident), etc. Although it is possible to apply many of these alternatives in traffic simulation software, we opted to keep the problem at a manageable level and constrain the actions to be applied only to traffic lights. The naive approach would be to let our agent simply control the traffic lights directly by setting the color of each traffic light individually at every simulation step, that is, the actions generated by our agent would be a list with the color (red, green or yellow) for each traffic light. However, traffic lights in an intersection are synchronized: when one of the traffic lights of the intersection is green, the traffic in the perpendicular direction is forbidden by setting the traffic lights of such a direction to red. This allows to multiplex the usage of the intersection. Therefore, letting our agent freely control the colors of the traffic lights would probably lead to chaotic situations. In order to avoid that, we should keep the phases of the traffic lights in each intersection. With that premise,we shall only control the phase duration, hence the dynamics are kept the same, only being accelerated or decelerated. This way, if the network has N different phases, the action vector has N components, each of them being a real number that has a scaling effect on the duration of the phase. However, for each intersection, the total duration of the cycle (i.e. the sum of all phases in the intersection) should be kept unchanged. This is important because in most cases, the cycles of nearby intersections are synchronized so that vehicles travelling from one intersection to the other can catch the proper phase, thus improving the traffic flow. In order to ensure that the intersection cycle is kept, the scaling factor of the phases from the same intersection are passed through a softmax function (also known as normalized exponential function). The result is the ratio of the phase duration over the total cycle duration. In order to ensure a minimum phase duration, the scaling factor is only applied to 80% of the duration. §.§ RewardsThe role of the rewards is to provide feedback to the reinforcement learningalgorithm about the performance of the actions taken previously. As commented in previous section, it would be possible for us to define a reward scheme that makes use of information about the travel times of the vehicles. However, as we are self-constraining to the information that is available in real world scenarios, we can not rely on other measures apart from detector data, e.g. vehicle counts, speeds. This way, we shall use the speed score described in section <ref>. But the speed score alone does not tell whether the actions taken by our agent actually improve the situation or make it worse. Therefore, in order to capture such information, we shall introduce the concept of baseline, defined as the speed score for a detector during a hypothetical simulation that is exactly like the one under evaluation but with no intervention by the agent, recorded at the same time step. This way, our reward is the difference between the speed score and the baseline, scaled by the vehicle counts passing through each detector (in order to give more weight to scores where the number of vehicles is higher), and further scaled by a factor α to keep the reward in a narrow range, as shown in (<ref>).𝑟𝑒𝑤𝑎𝑟𝑑_i = α·𝑐𝑜𝑢𝑛𝑡_i ·( 𝑠𝑝𝑒𝑒𝑑_𝑠𝑐𝑜𝑟𝑒_i- 𝑏𝑎𝑠𝑒𝑙𝑖𝑛𝑒_i )Note that we may want to normalize the weights by dividing by the total vehicles traversing all the detectors. This would restrain the rewards in the range [-1, +1]. This, however, would make the rewards obtained in different simulation steps not comparable (i.e. a lower total number of vehicles in the simulation at instant t would lead to higher rewards). The factor α was chosen to be 1/50 empirically, by observing the unscaled values of different networks and choosing a value in an order of magnitude that leaves the scaled value around 1.0. This is important in order to control the scale of the resulting gradients. Another alternative used in <cit.> with this very purpose is reward clipping; this, however, implies losing information about the scale of the rewards. Therefore, we chose to apply a proper scaling instead. There is a reward computed for each detector at each simulation time step. Such rewards are not combined in any way, but are all used for the DDPG optimization, as described in section <ref>. Given the stochastic nature of the microsimulator used, the results obtained depend on the random seed set for the simulation. This way, when computing the reward, the baseline is taken from a simulation with the same seed as the one under evaluation. §.§ Deep Network Architecture Our neural architecture consists in a Deep Deterministic Actor-Critic Policy Gradient approach.It is comprised of two networks: the actor network π and the critic network Q. The actor network receives the current state of the simulation (as described in section <ref>) and outputs the actions, as described in <ref>. As shown in figure <ref>, the network is comprised of several layers. It starts with several fully connected layers (also known as dense layers) with Leaky ReLU activations <cit.>, where the number of units is indicated in brackets, with 𝑛𝑑 being the number of detectors in the traffic network and 𝑛𝑝 is the number of phases of the traffic network. Across those many layers, the width of the network increases and then decreases, up to having as many units as actions, that is, the last mentioned dense layer has as many units as traffic light phases in the network. At that point, we introduce a batch normalization layer and another fully connected layer with ReLU activation. The output of the last mentioned layer are real numbers in the range [0, +∞], so we should apply some kind of transformation that allows us to use them as scaling factors for the phase durations (e.g. clipping to the range [0.2, 3.0]). However, as mentioned in section <ref>, we want to keep the traffic light cycles constant. Therefore, we shall apply an element-wise scaling computed on the summation of the actions of the phases in the same traffic light cycle, that is, for each scaling factory we divide by the sum of all the factors for phases belonging to the same group (hence obtaining the new ratios of each phase over the cycle duration) and then multiply by the original duration of the cycle. In order to keep a minimum duration for each phase, such computation is only applied to the 80% of the duration of the cycle. Such a computation can be pre-calculated into a matrix, which we call the phase adjustment matrix, which is applied in the layer labeled as "Phase adjustment" in figure <ref>, and which finally gives the scaling factors to be applied to phase durations. This careful scaling meant to keep the total cycle duration can be ruined by the exploration component of the algorithm, as described in <ref>, which consists of adding noise to the actions (and therefore likely breaking the total cycle duration), This way, we implement the injection of noise as another layer prior to the phase adjustment. The critic network receives the current state of the simulation plus the action generated by the actor, and outputs the Q-values associated to them. Like the actor, it is comprised of several fully connected layers with leaky ReLU activations, plus a final dense layer with linear activation. §.§ Disaggregated RewardsIn our reference article <cit.>, as well as all landmark ones like <cit.> and <cit.>, the reward is a single scalar value. However, in our case we build a reward value for each detector in the network. One option to use such a vector of rewards could be to scalarize them into a single value. This, however, would imply losing valuable information regarding the location of the effects of the actions taken by the actor. Instead, we will keep then disaggregated, leveraging the structure of the DDPG algorithm, which climbs in the direction of the gradient of the critic. This is partially analogous to a regression problem on the Q-value and hence does not impose constraints on the dimensionality of the rewards. This way, we will have a N-dimensional reward vector, where N is the number of detectors in the network. This extends the policy gradient theorem from <cit.> so that the reward function is no longer defined as r: S × A →ℝ but as r: S × A →ℝ^N. This is analogous to having N agents sharing the same actor and critic networks (i.e. sharing weights θ^π and θ^Q) and being trained simultaneously over N different unidimensional reward functions. This, effectively, implements multiobjective reinforcement learning. To the best of our knowledge, the use of disaggregated rewards has not been used before in the reinforcement learning literature. Despite having proved useful in our experiments, further study is needed in order to fully characterize the effect of disaggregated rewards on benchmark problems. This is one of the future lines of research that can be spawned from this work. Such an approach could be further refined by weighting rewards according to traffic control expert knowledge, which will then be incorporated in the computation of the policy gradients. §.§ ConvergenceThere are different aspects that needed to be properly tuned in order for the learning to achieve convergence:* Weight Initialization has been a key issue in the results cast by deep learning algorithms. The early architectures could only achieve acceptable results if they were pre-trained by means of unsupervised learning so that they could have learned the input data structure <cit.>. The use of sigmoid or hyperbolic tangent activations makes it difficult to optimize neural networksdue to the numerous local minima in the function loss defined over the parameter space.With pre-training, the exploration of the parameter space does not begin in a random point, but in a point that hopefully is not too far from a good local minimum. Pretraining became no longer necessary to achieve convergence thanks to the use of rectified linear activation units (ReLUs) <cit.> and sensible weight initialization strategies. In our case, different random weight initializations (i.e. Glorot's <cit.> and He's <cit.>) gave the best results, finally selecting He's approach. * Updates to the Critic: after our first experiments it became evident the divergence of the learning of the network. Careful inspection of the algorithm byproducts revealed that the cause of the divergence wasthat the critic network Q' predicted higher outcomes at every iteration, as trained according to equation (<ref>) extracted from algorithm <ref>.y_i= r_i + γ Q'(s_i+1,π'(s_i+1|θ^π')|θ^Q')As DDPG learning -like any other reinforcement learning with value function approximation approach- is a closed loop system in which the target value at step t+1 is biased by the training at steps t, driftscan be amplified, thus ruining the learning, as the distance between the desired value for Q and the obtained one differ more and more. In order to mitigate this divergence problem, our proposal consists in reducing the coupling by means of the application of a schedule on the value of the discount factor γ from Bellman's equation, which is shown in figure <ref>.The schedule of γ is applied at the level of the experiment, not within the episode. The oscillation in γ shown in figure <ref> is meant to enable the critic network not to enter in the regime where the feedback leads to divergence. Discount Factor scheduling has been proposed before in <cit.> with positive results, although in that case the schedule consisted in a decaying rate. * Gradient evolution: the convergence of the algorithm can be evaluated thanks to the norm of the gradient used to update the actor network π. If such a norm decreases over time and stagnates around a low value, it is a sign that the algorithm has reached a stable point and that the results might not further improve. This way, in the experiments described in subsequent sections, monitoring of the gradient norm is used to track progress. The gradient norm can also be controlled in order to avoid too large updates that make the algorithm diverge, e.g. <cit.>. This mechanism is called gradient norm clipping and consists of scaling the gradient so that its norm is not over a certain value. Such a value was empirically established as 0.5 in our case. §.§ Summary Our proposal is to apply Deep Deterministic Policy Gradient, as formulated in <cit.>, to the traffic optimization problem by controlling the traffic lights timing. We make use of a multilayer perceptron type of architecture, both for the actor and the critic networks. The actor is designed so that the modifications to the traffic light timings keep the cycle duration. In order to optimize the networks we make use of stochastic gradient descent. In order to improve convergence, we make use of a replay memory, gradient norm clipping and a schedule for the discount rate γ. The input state used to feed the network consists of traffic detector information, namely vehicle counts and average speeds, which are combined in a single speed score. The rewards used as reinforcement signal are the improvements over the measurements without any control action being performed (i.e. baseline). Such rewards are not aggregated but fed directly as expected values ofthe critic network.§ EXPERIMENTSIn this section we describe the experiments conducted in order to evaluate the performance of the proposed approach. In section <ref> we show the different traffic scenarios used while in section <ref> we describe the results obtained in each one, along with lessons learned from the problems found, plus hints for future research. §.§ Design of the ExperimentsIn order to evaluate our deep RL algorithm, we devised increasing complexity traffic networks.For each one, we applied our DDPG algorithm to control the traffic light timing, but also applied multi-agent Q-Learning and random timing in order to have a reference to properly assess the performance of our approach. At each experiment, the DDPG algorithm receives as input theinformation of all detectors in the network, and generates the timings of all traffic light phases. In the multi-agent Q-learning implementation, there is one agent managing each intersection phase. It receives the information from the closest few detectors and generates the timings for the aforementioned phase. Given the tabular nature of Q-learning, both the state space and the action space need to be categorical. For this, tile coding is used. Regarding the state space, the tiles are defined based on the same state space values as DDPG (see section <ref>), clustered in one the following 4 ranges [-1.0, -0.2], [-0.2, -0.001], [-0.001, 0.02], [0.02, 1.0], which were chosen empirically. As one Q-learning agent controls the N_i phases of the traffic lights of an intersection i, the number of states for an agent is 4^N_i. The action space is analogous, being the generated timings one of the values 0.2, 0.5, 1.0, 2.0 or 3.5. The selected ratio (i.e. ratio over the original phase duration) is applied to the duration of the phase controlled by the Q-learning agent. As there is one agent per phase, this is a multi-agent reinforcement learning setup, where agents do not communicate with each other. They do have overlapping inputs, though, as the data from a detector can be fed to the agents of several phases. In order to keep the cycle times constant, we apply the same phase adjustment used for the DDPG agent, described in section <ref>. The random agent generates random timings in the range [0, 1], and then the previously mentioned phase adjustment is applied to keep the cycle durations constant (see section <ref>). Given the stochastic nature of the microscopic traffic simulator used, the results obtained at the experiments depend on the random seed set for the simulation. In order to address the implications of this, we do as follows: * In order for the algorithms not to overfit to the dynamics of a single simulation, we randomize the seed of each simulation. We take into account this also for the computation of the baseline, as described in section <ref>.* We repeat the experiments several times, and present the results over all of them (showing the average, maximum or minimum data depending on the case). §.§ Network AThis network, shown in figure <ref> consists only of an intersection of two 2-lane roads. At the intersection vehicles can either go straight or turn to their right. It is forbidden to turn left, therefore simplifying the traffic dynamics and traffic light phases. There are 8 detectors (in each road there is one detector before the intersection and another one after it). There are two phases in the traffic light group: phase 1 allows horizontal traffic while phase 2 allows vertical circulation. Phase 1 lasts 15 seconds and phase 2 lasts 70 seconds, with a 5-seconds inter-phase. Phases 1 and 2 have unbalanced duration on purpose, to have the horizontal road accumulate vehicles for long time. This gives our algorithm room to easily improve the traffic flow with phase duration changes. The simulation comprises 1 hour and the vehicle demand is constant: for each pair of centroids, there are 150 vehicles.The traffic demand is defined by hand, with the proper order of magnitude to ensure congestion. The definition and duration of the phases were computed by means of the classical cycle length determination and green time allocation formulas from <cit.>. §.§ Network B This network, shown in figure <ref> consists of a grid layout of 3 vertical roads and 2 horizontal ones, crossing in 6 intersections that all have traffic lights.Traffic in an intersection can either go straight, left or right, that is, all turns are allowed, complicating the traffic light phases, which have been generated algorithmically by the software with the optimal timing, totalling 30 phases. There are detectors before and after each intersection, totalling17 detectors. The traffic demand is defined by hand, ensuring congestion. The trafic light phases were defined, like network A, with the classical approach from <cit.>. Four out of six junctions have 5 phases, while the remaining two junctions have 4 and 6 phases each. The traffic demand has been created in a random manner, but ensuring enough vehicles are present and trying to collapse some of the sections of the network. §.§ Network C This network, shown in figure <ref> is a replica of the Sants area in the city of Barcelona (Spain).There are 43 junctions, totalling 102 traffic light phases, and 29 traffic detectors.The locations of the detectors matches the real world. The traffic demand matches that of the peak hour in Barcelona, and it presents high degree of congestion.The number of controlled phases per junction [Note that phases from the network that have a very small duration (i.e. 2 seconds or less) are excluded from the control of the agent] ranges from 1 to 6, having most of them only two phases.§.§ ResultsIn order to evaluate the performance of our DDPG approach compared to both normal Q-learning and random timings on each of our test networks, our main reference measure shall be the episode average reward (note that, as described in section <ref> there is actually a vector of rewards, with one element per detector in the network, that is why we compute the average reward) of the best experiment trial, understanding "best" experiment as the one where the maximum episode average reward was obtained.In figure <ref> we can find the performance comparison for network A. Both the DDPG approach and the classical Q-learning reach the same levels of reward. On the other hand, it is noticeable the differences in the convergence of both approaches: while Q-learning is unstable, DDPG remains remarkably stable once it reached its peak performance.In figure <ref> we can find the performance comparison for network B. While Q-learning maintains the same band of variations along the simulations, DDPG starts to converge. Given the great computational costs of running the full set of simulations for one network, it was not affordable to let it run indefinitely, despite the promising trend.Figure <ref> shows the performance comparison for network C, from which we can appreciate that both DDPG and Q-learning performs at the same level, and that such a level is beneath zero, from which we know that they are actually worse than doing nothing.This way, the performance of DDPG is clearly superior to Q-learning for the simplest scenario (network A), slightly better for scenarios with a few intersections (network B) and at the same level for real world networks. From the evolution of the gradient for medium and large networks, we observed that convergence was not achieved, as it remains always at the maximum value induced by the gradient norm clipping. This suggests that the algorithm needs more training time to converge (probable for network B) or that it diverges (probable for network C). In any case, further study would be needed in order to assess the needed training times and the needed convergence improvement techniques.§ CONCLUSIONSWe studied the application of Deep Deterministic Policy Gradient (DDPG) to increasingly complex scenarios. We obtained good results in network A, which is analogous to most of the scenarios used to test reinforcement learning applied to traffic light control (see section <ref> for details on this); nevertheless, for such a small network, vanilla Q-learning performs on par, but with less stability, though. However, when the complexity of the network increases, Q-learning can no longer scale, while DDPG still can improve consistently the obtained rewards. With a real world scenario, our DDPG approach is not able to properly control the traffic better than doing nothing. The good trend for network B shown in figure <ref>, suggests that longer training time may lead to better results. This might be also true for network C, but the extremely high computational costs could not be handled without large scale hardware infrastructure. Our results show that DDPG is able to better scale to larger networks than classical tabular approaches like Q-learning. Therefore, DDPG is able to address the curse of dimensionality <cit.> regarding the traffic light control domain, at least partially. However, it is not clear that the chosen reward scheme (described in section <ref>) is appropriate. One of its many weaknesses is its fairness for judging the performance of the algorithm based on the individual detector information. In real life traffic optimization it is common to favour some areas so that traffic flow in arterials or large roads is improved, at the cost of worsening side small roads. The same principle could be applied to engineer a more realistic reward function from the point of view of traffic control theory.In order to properly asses the applicability of the proposed approach to real world setups, it would also be needed to provide a wide degree of variations in the conditions of the simulation, from changes in the traffic demand to having road incidents in the simulation. Another aspect that needs further study is the effect of the amount and location of traffic detectors on the performance of the algorithm. In our networks A and B, there were detectors at every section of the network, while in network C their placement was scattered, which is the norm in real world scenarios. We appreciate a loose relation between the degree of observability of the state of the network and the performance of our proposed traffic light timing control algorithm. Further assessment about the influence of observability of the state of the network would help characterize the performance of the DDPG algorithm and even turn it into a means for choosing potential locations for new detector in the real world. Also, the relevance of the provided information is not the same for all detectors; some of them may provide almost irrelevant information while others are key for understanding the traffic state. This is another aspect that should be further studied, along with the effect of the noise present in data delivered by real traffic detectors.An issue regarding the performance of ourapproach is the sudden drops in the rewards obtained through the training process. This suggests that the landscape of the reward function with respect to the actor and critic network parameters is very irregular, which leads the optimization to fall into bad regions when climbing in the direction of the gradient. A possible future line of research that addressed this problem could be applying Trusted Region Policy Optimization <cit.>, that is, leveraging the simulated nature of our setup to explore more efficiently the solution space. This would allow it to be more data efficient, achieving comparable results with less training.We have introduced a concept that, to the best of our knowledge, has not been used before in the deep reinforcement learning literature, namely the use of disaggregated rewards (described in section <ref>). This technique needs to be studied in isolation from other factors on benchmark problems in order to properly assess its effect and contribution to the performance of the algorithms. This is another possible line of research to be spawned from this work. On the other hand, we have failed to profit from the geometric information about the traffic network. This is clearly a possible future line of research, that can leverage recent advances in the application of convolutional networks to arbitrary graphs, similar to <cit.>. Finally, we have verified the applicability of simple deep learning architectures to the problem of traffic flow optimization by traffic light timing control on small and medium-sized traffic networks. However, for larger-sized networks further study is needed, probably in the lines of exploring the results with significantly larger training times, using the geometric information of the network and devising data efficiency improvements.elsarticle-num-names | http://arxiv.org/abs/1703.09035v2 | {
"authors": [
"Noe Casas"
],
"categories": [
"cs.NE"
],
"primary_category": "cs.NE",
"published": "20170327124014",
"title": "Deep Deterministic Policy Gradient for Urban Traffic Light Control"
} |
firstpage–lastpage Mean-Field Theory of Water-Water Correlations in Electrolyte Solutions Michele Ceriotti December 30, 2023. ====================================================================== We have measured a precise optical transmission spectrum for WASP-52b, a highly inflated hot Jupiter with an equilibrium temperature of 1300 K. Two transits of the planet were observed spectroscopically at low resolution with the auxiliary-port camera (ACAM) on the William Herschel Telescope (WHT), covering a wide range of 4000–8750 Å. We use a Gaussian process approach to model the correlated noise in the multi-wavelength light curves, resulting in a high precision relative transmission spectrum with errors on the order of a pressure scale height. We attempted to fit a variety of different representative model atmospheres to the transmission spectrum, but did not find a satisfactory match to the entire spectral range. For the majority of the covered wavelength range (4000–7750 Å) the spectrum is flat, and can be explained by an optically thick and grey cloud layer at 0.1 mbar, but this is inconsistent with a slightly deeper transit at wavelengths > 7750 Å. We were not able to find an obvious systematic source for this feature, so this opacity may be the result of an additional unknown absorber. planets and satellites: individual (WASP 52b)—stars: individual (WASP 52)—techniques: spectroscopic—planets and satellites: atmospheres—atmospheric effects—methods: statistical § INTRODUCTION Exoplanet science is rapidly developing into a mature discipline where detailed characterisation of individual planets is beginning to build a picture of the population of planets as a whole. The characterisation of exoplanet atmospheres, primarily through transmission and emission spectroscopy, has become a key topic. Hot Jupiter atmospheres have been shown to display a wide variety <cit.>, but broadly can be categorised into atmospheres that are “clear", displaying large sodium and potassium features (e.g. HD 209458b), “cloudy", which display muted, or no features at all at optical wavelengths, and “hazy” atmospheres, such as HD 189733b where the dominant feature is a Rayleigh scattering slope caused by sub-micron scale scattering particles high in the atmosphere of the planet. While many high profile discoveries have been space based <cit.>, increasingly, ground based telescopes have also been found to be suitable to the task for the detection and characterization of both narrow <cit.> and broadband features <cit.>. While susceptible to their own systematic issues, progress has been made in identifying and improving the major problems, such as differential slit loss <cit.> and upgrading the Linear Atmospheric Dispersion Corrector (LADC) on the FOcal Reducer and low dispersion Spectrograph (FORS2) <cit.>.Many exoplanets have been shown to present a flat, cloud dominated spectrum <cit.>. Data from HST shows that broad wavelength coverage is required to characterize a hot Jupiter, as most appear flat over ranges of several hundred angstroms without the aid of high resolution measurements <cit.>. So far, there is no obvious correlation between planetary parameters, e.g. equilibrium temperature, surface gravity, and host spectral type, though there are broad predictions about what kinds of condensates are available at different temperature ranges and their impact on observational properties <cit.>. The transmission spectrum of a planet is an emergent property of many branches of atmospheric physics, that likely depends quite sensitively on the specifics of heat transport and vertical mixing in each individual planet. It will probably be necessary to analyse many more systems before the underlying trends begin to emerge.Ground based observations have the advantage of being relatively cheap to perform in large number, compared to space based observations, and they also fill an important role in validating the results from HST and Spitzer, where exoplanet observations are known to suffer from the idiosyncratic systematic errors of these instruments <cit.>. The way these systematics are treated can have a large impact on the resulting transmission spectra, and many early results are controversial <cit.>. Independent detectionswith a wider range of instruments would remove some of the uncertainty stemming from these systematics concerns. Additionally, wavelength dependent trends in the data injected by, for example, stellar activity could be mistaken for real atmospheric features on the planet. The repeatability of a purported signal over several observations is a powerful diagnostic, giving immense value to having multiple nights on target. Ground-based, broad-band transmission spectroscopy therefore remains a vital tool in building an accurate picture of exoplanet atmospheres.WASP-52b is a Saturn-mass planet with an inflated radius that orbits a K type star on a 1.75 day orbit <cit.>. WASP-52b is an excellent target for transmission spectroscopy due to its large scale height, relatively deep transit, and the presence of a nearby bright comparison star. The low density of WASP-52b gives it a scale height of 700 km, which is over three times greater than HD 189733b, despite having a similar equilibrium temperature of 1300 K. Although it orbits a comparatively faint star (V_mag=12), this high scale height means that the Signal-to-Noise achievable on atmospheric features is comparable with HD 189733b, assuming that white noise is the dominant noise component, and potentially better where red noise is a limiting factor. The similarity of the planet's equilibrium temperature and parent star spectral type to HD 189733b make WASP-52b an interesting test of the variations in exoplanet atmospheres.We present two high precision spectroscopic transits of WASP-52b. This paper is organised as follows: a summary of the observations are given in Section <ref>, the reduction of the data is described in Section <ref>, and a short description of the Common mode Gaussian process detrending analysis is presented in Section <ref>. The system parameters are refined in Section <ref> and a low resolution spectrum is presented with a discussion of its features in Section <ref>.§ OBSERVATIONS WASP-52 was observed for 2 full nights on August 22 and 29th 2014 using the auxiliary-port camera (ACAM) on the William Herschel Telescope (WHT). ACAM was used in single slit spectroscopy mode with a custom made extra wide slit of 27". This was to minimise slit losses, which have been found to be a limiting factor in past observations with other instruments <cit.>, though a slit is still of course necessary to avoid contamination from other stars in the field of view and to lower sky background contribution. A slit of 27" arcseconds was chosen after an earlier analysis of the Point Spread Function (PSF) of the instrument in an earlier dataset, with the goal of there being less than 0.1% slit losses. A comparison star with V_mag=13 and a similar spectral type located 3 arcminutes away, well within the unvignetted slit length, was selected to perform differential spectroscopy. The slit was rotated such that both stars were at the center. An exposure time of 100 seconds was used to keep the peak flux well below saturation at all times. The windowed observing mode reduced the readout time to 11 seconds, giving a duty cycle efficiency of 90%. In order to keep the stars centred in the slit, the data were reduced in realtime and the resulting spectra were cross-correlated to produce guiding offsets, using the position of the spectral traces in the X-direction on the chip, and the positions of strong absorption features in the spectra in the Y-direction. Pointing corrections were entered manually at intervals of approximately 20 minutes throughout the observations, with typical amplitudes of 0.1 pixels, these small and frequent corrections prevented the formation of a sawtooth pattern in the trace positions. A small defocus was applied to the telescope (2") to spread the light over a larger number of pixels to reduce the contribution of flat fielding errors, and to create a more stable PSF that is less effected by variable seeing.§ METHOD§.§ ExtractionThe reduction was carried out with custom scripts written in Python. Standard bias corrections were made. However, it was not possible to flat field the data satisfactorily with the sky flats taken due to differential vignetting between the imaging and spectral modes of ACAM. Spectroscopic dome flats did not contain enough blue light to be useful across the whole spectral range. Due to the large number of pixels used during the analysis, and the small spatial scale of flat field variations compared to the variations within the spectra the flat fielding errors are not expected to be an important error source, so the reduction was carried out without flat fielding.The combination of vignetting towards the edge of the chip and the broad wings of the brighter comparison star made it difficult to judge where to place sky background traces. However, the background was found to vary smoothly and predictably over the rest of the chip, so we decided to fit this function directly using a third order iterative sigma clipped polynomial and subtract it from the image on a row by row basis (here rows are perpendicular to the dispersion direction). This approach has the advantage of leveraging information from a larger number of pixels, which increases the accuracy of the background estimate. There is a small curvature in the sky lines in the x-direction due to the alignment of the instrument, but it is significantly less than the width of the lines, so does not effect the background subtraction.We fit the spectral trace for the target and reference star with an iterative cubic spline. Several extraction widths were tested, and it was found that a width of 20 pixels provided the lowest scatter in the final lightcurves on both nights. The adjustments to the guiding kept the brightest region of the spectrum dispersed over the same set of pixels, but the spectral trace was found to rotate slowly throughout the night due to differential refraction, meaning the extreme blue end of the spectrum traversed a slightly larger distance throughout the observation and may suffer an increased noise budget due to the lack of flat-fielding. The bluest end of the usable spectrum at 4000 Å(Pixel row 674) was found to drift by two pixels throughout the night, while the central regions and the red end drift by less than a pixel (0.25").An example spectral extraction is shown in Figure <ref>. The colour scale has been selected to emphasise the target and reference star trails. It is clear that the resulting spectra and their ratios are well behaved with high S/N between pixels 600–2200, which are found to correspond to wavelengths 4000–9000 Å.Persistent artefacts are visible in the images, which we suspect are caused by fringing. These become visible beyond 8750 Å, and make it more difficult to trust the spectra extracted in this region. Bluewards of 4000 Å a combination of low intrinsic flux, detector response, and instrument vignetting drop the count rate below the background, making a total usable range of 4000–8750 Å.An accurate and stable wavelength calibration is important, as errors in wavelength calibration can cause systematic bluewards and redwards slopes in the transmission spectrum that can be mistaken for spectral features in the atmosphere of the planet. Due to the very wide slit, arc calibration frames taken with the same slit produce very few useful features, particularly at the blue ends of the spectrum where the CuAr and CuNe calibration lamps available at the WHT produce very few lines. Using a smaller slit and removing the offset in slit positions is possible, but we found that the wavelength solution drifted noticeably over the 8 hour observation period, which is caused by flexure of the ACAM instrument. Breaking science observations and changing the instrument setup to take regular arcs to recalibrate is undesirable, so we elected instead to self-calibrate the science spectra using telluric and strong stellar features. At the resolution of these observations, the velocity difference between the stellar and telluric features is unimportant. The line positions are measured by fitting a Gaussian profile, which well describes the instrumentally broadened profiles. The central pixel co-ordinates of these features are used to generate a wavelength solution using a 3rd order polynomial fit. Each spectrum is then re-sampled onto a common wavelength grid using pysynphot[Distributed as part of the Space Telescope Science Institute stsci_python package, www.stsci.edu/institute/software_hardware/pyraf/stsci_python], which conserves flux. This produces a stable and consistent wavelength solution for the entire set of observations.Obvious cosmic ray hits are removed from the individual spectra at this stage. A sliding Gaussian weighted average with the FWHM of the instrument is run along the spectra, and points that are 5σ or greater discrepant with this value are replaced by it. Typically only a small number of points per spectrum are affected.To generate a light curve we took the sum of the flux in the desired wavelength range at each time point for both target and comparison, and then took the ratio. To reduce computing time in the systematic noise model analysis (described in section <ref>) we performed a basic fit first to remove extreme outliers. We fit a simple transit model <cit.> with a second order polynomial baseline fit, and removed any points greater than 5σ from the model. Most lightcurves did not require more than one point to be removed. Each lightcurve was then normalised such that the median of the out-of-transit data is 1.The raw white-light (4000-8750 Å) lightcurves for the two nights and their ratio are shown in Figure <ref>. On both nights the dominant trend in the extracted data is the airmass term, which is mostly removed by taking the ratio. Small fluctuations in transparency are likewise shared between target and comparison, so are not present in the ratio.During the first night there is a feature close to the middle of transit, this feature is discussed in more detail in section <ref>. This potential spot-crossing feature is distinct from a bump in the data that is closer to the center of the transit, which is present in both the target and comparison lightcurves and is removed by taking the ratio. The location of this non-astrophysical bump is marked in Figure <ref>. §.§ Fitting procedure In order to generate the final transmission spectrum for each night we follow several steps, each of which will be expanded on in detail. First, the transit and systematics model (described in <ref>) is fit to the whole of the usable spectral range (4000–8750 Å). We then divide the spectrum into 250 Å spectral bins and fit the transit model to each separately, after removing the best-fit systematics model for that night, and fixing the system parameters to the values found for the white light curve. This removes the dominant systematics of the whole spectral range from the individual bins, which significantly lowers the error bars.A Markov Chain Monte Carlo approach is used to maximise model parameter likelihood and estimate their errors. To perform the MCMC integration, we use the emcee code of <cit.>[available at https://github.com/dfm/emcee], which implements the affine-invariant ensemble sampler described in <cit.>.To fit the transit light curve, we use the analytic models described in <cit.>. The eccentricity is fixed to 0, and at each step we draw the period from a distribution with mean and standard deviation values from <cit.>. The inclination (i), system scale (a/R_*), radius ratio (R_p/R_*) and quadratic limb darkening co-efficients (u1,u2) are free parameters, as well as the parameters of the systematics model. LDTK (Limb Darkening Toolkit) <cit.> was used to calculate the priors on the limb darkening parameters for each wavelength range. Each fit has 50,000 steps with 100 walkers, with the first 10,000 steps discarded as burn in. Convergence was checked by visual inspection of the walker chains and the evolution of their mean and standard deviation for each parameter.The light curves for the full spectral range for the two nights are first fit simultaneously in order to calculate the best fitting system parameters, but at all other times the two nights are treated independently. §.§ Gaussian process and white noise model Gaussian processes are a generalized class of functions that can be used to model correlated noise in time series data. They have a wide range of applications, but have increasingly become popular for accurate parameter estimation for exoplanet transits <cit.>. The covariance between data points is modelled with a “kernel", which has a simplified functional form. Including a Gaussian process (GP) noise model allows one to effectively marginalise over the sets of functions that can represent the systematic noise in the data that do not hold any scientific interest. In our case, we use the Gaussian process to model everything in the data that is not explained by the transit model.In this work the Matern 3/2 kernel is used, following <cit.> who found it performed marginally better than other common kernels for transmission spectroscopy. The Matern kernel is similar to the commonly used squared exponential function, but is more sharply peaked, resulting in more flexibility at short timescales:k(t_i,t_j) = α(1 + √(3)(t_i - t_j/τ)) exp(-√(3)(t_i - t_j/τ))where α is the amplitude of the kernel, and τ represents a characteristic time-scale of the correlation. In practice we fit for the logarithm of these values, as they are expected to vary over several orders of magnitude.The python package george[availible at https://github.com/dfm/george] <cit.> is used to efficiently calculate the Gaussian process covariance matrices and the Likelihood at each Markov chain step.There was a slightly higher level of white noise in the data than could be explained by photon noise alone, presumably due to scintillation. To account for this, and other sources of white noise, a Gaussian Process model will often include an additional term, σ_w, to increase the calculated error bars by a fixed amount.For the spectrally resolved fits we found that in the bluest regions the level of additional white noise required was time varying, so we chose to take a similar approach to <cit.> and model the white noise as a smoothly time varying function. This makes sense, since an enhancement to the white noise, most likely due to changing atmospheric conditions, is not necessarily constant. The white noise term in equation <ref> is modelled asσ(t_i) = σ_a exp(-t_i/σ_b) + σ_cwhere σ_a, σ_b and σ_c are additional terms in the model fit.For consistency, we fit all the spectral lightcurves with this white noise model. We did not think it was necessary to re-fit the white light curves with this new model, as their residuals do not vary significantly with time. We find that for most wavelength regions the posteriors using the time varying white noise model are no different to those with the fixed increase, but in the bluest channels, where scintillation is expected to have the greatest effect, the more accurate characterisation of the noise properties slightly narrows the credible intervals of the transit depth, but only marginally changes the mean values. §.§ Application of the model Due to the presence of the spot-crossing event on the first night, we chose to initially fit the two nights separately to ensure that the Gaussian process systematic model was capable of fitting out the event without introducing bias into the other parameter values. Comparison of the parameter values satisfied us that, apart from a small difference in absolute depth which is expected from changes in activity levels, the parameters are consistent on both nights (see Table <ref>). The final parameter values are calculated by fitting the two nights simultaneously.Ultimately, what we are interested in is the relative depth of the wavelength channels, or their conditional probability distribution with respect to each other in order to build a transmission spectrum. Systematic noise sources, such as guiding instability or changes in atmospheric transparency will be the same, or subject to a small scaling factors between bands. It is therefore incorrect to treat a resulting transmission spectrum as a set of independent data-points, and their calculated error bars will be higher than the intrinsic scatter between them. This could also lead to unrealistic model fits, since most goodness of fit statistics, such as χ^2, assume that the datapoints are independent.We use the white light curves for each night to remove the common noise trends. A transit model with our Gaussian process noise model is fit to the normalised white light curve for each night. The best Gaussian process model is then taken and assumed to represent the systematic noise and trends in the data and the spectroscopic lightcurves are divided by this model, thereby removing the systematic components that are common. §.§ Spot crossing event Due to the changing activity levels of the star, it would not a priori be expected that the depth, and hence inferred radius should be consistent between the two nights, as the stellar surface brightness distribution may be inhomogenous due to spots or plages. Indeed, we find that on the night with the apparent spot crossing event the implied transit depth is slightly deeper than the other, although the difference in absolute depths between the two nights is not formally significant, at 0.0102 ± 0.0067. Taking these results at face value, they would imply that the stellar surface was on average 0.4% dimmer on that night. Whilst we unfortunately did not have access to contemporaneous monitoring of WASP-52 during the observation period, <cit.> found that during their second observing season WASP-52 displayed a 16.4 ± 0.04 day period with an amplitude of 0.89%, though there was no significant periodicity found in the first season. The separation between the two transits we observed is 7 days, or roughly half of the rotation period of the star – so it might be expected that the observed spot coverage fraction could be different, despite the observations being close in time.Since <cit.> carried out their analysis, an additional season of SuperWASP-North <cit.> data on WASP-52 has become available. We carried out a harmonic analysis to assess how the activity levels vary between seasons. We used a generalized weighted Lomb Scargle periodogram <cit.> to search for significant periods in the data, which were taken between HJD 2455400 – 2455800. We generate False Alarm Probability (FAP) power levels by using the night shuffling technique presented in <cit.> and <cit.>. Randomly sampling the nights instead of individual datapoints preserves correlated noise in the data, whilst removing any periodicities greater than 1 day, this gives a more realistic and conservative estimate of the FAP. The results are plotted in Figure <ref>, there is a highly significant peak at 17.79 days in the most recent season. To estimate the errors on the period and the amplitude we fit a sine curve to the data and ran an MCMC of 10,000 steps, centred around the peak of the periodogram, with a burn in of 1000 steps. We find that the fractional amplitude is 0.0142^+0.0003_-0.0003 and the period is 17.79^+0.05_-0.05 days. The amplitude is significantly higher than during the first and second seasons observed with WASP, which may be indicative of activity cycles. The period is 1.4 days longer than the period found in the second season, which is likely caused by the active regions occurring at different latitudes and differential rotation.Despite the lack of contemporaneous coverage, the difference in absolute depth between the two nights is therefore consistent with the historical activity levels of the star. Note that the depth on the uncontaminated night is consistent with both the original published value, and the faculae model in <cit.>, but is inconsistent with their spot model. All other parameter values agree to within one sigma, and there is no sign of correlation between the GP hyper-parameters and transit parameters (except depth) in the MCMC corner plotsThe significant difference between the spot model in <cit.> and both the discovery paper and this work strongly supports their conclusion that their data shows the planet crossing faculae rather than spots.Since the spot crossing event appears in our data to be grey to within the level of precision, the approach of removing the common Gaussian process noise model is preferred over attempting to model the spot directly. This would add additional model parameters and degeneracies, and would not improve the level of precision attained in the recovered transmission spectrum. Inspection of the GP model for night 1 (Figure <ref>) reveals at least 2 distinct components which indicate that the idealisation of a single “spot crossing” is unsuitable to this case, and it is more likely to be the crossing of an active region, with multiple spots simultaneously transited. Given the success of the GP model (see section <ref>) we feel it would be unnecessary to attempt to model this explicitly.Modelling the spot crossing with the GP is not strictly correct, as the implicit assumption of the Gaussian process approach is that the noise properties are the same for the whole observation window. In principle it would be possible to specify a compound GP that had an additional kernel active only during the transiting portion of the dataset in order to simulate the additional correlated noise component from inhomogeneities of the stellar surface. However, since the event is not visible in the individual spectral bins after common mode removal (see Figures <ref> and <ref>), we chose not to add additional complexity to the model.An additional possibility is that this feature is not in fact a spot crossing, but an instrumental systematic with no wavelength dependence, features like this are not uncommon in exoplanet lightcurves. Since the systematic appears to be successfully removed by the common mode noise model, and the system parameters for the two independent nights agree well, the source of the feature is not important, as it will not have an effect on our results.§ RESULTS§.§ Updated system parameters We fit the white light curves simultaneously for both nights, using the MCMC GP model, the results are shown in Figure <ref>. We included the transit epoch data from the discovery paper and <cit.> in order to improve the global system parameter estimates. Since the mean stellar surface brightness, and the noise properties are not expected to be the same on both nights, they are allowed to fit separately, but the other system parameters (a/R_*, limb darkening coefficients and inclination) are fit simultaneously. Our final system parameters are listed in Table <ref>, and agree to within 1σ with the values in the discovery paper. This provides the system parameters that are fixed during calculation of the spectroscopic lightcurves. §.§ Transmission spectra For each 250 Å bin, the median value of the transit depth and the 68% credible region are calculated from the MCMC posteriors. We present the resulting transmission spectra for the two nights in Table <ref> and Figure <ref>.Having measurements on two nights fitted independently proved valuable, as we were able to use them as an independent test of the assigned error bars. Assuming that the same transmission spectrum is present on both nights, it is expected that the error-normalised deviation of the points from their average will behave like a set of independent Gaussian measurements, with standard deviation proportional to 1/√(2). Performing this test on the two nights, we find that they do indeed behave as expected for a set of independent measurements of the same distribution, once an absolute depth difference between the two nights has been removed. This indicates that the GP/MCMC method has not underestimated the relative uncertainty in each wavelength bin, and that little common-mode uncertainty remains. It also shows that the spot-crossing event has not introduced any bias to our transmission spectrum.Since the spectra of the two nights do appear to be describing the same underlying spectrum, and the error bars are reasonable, we use the variance weighted average of the two nights as our final result, shown in Figure <ref>It is worth emphasising that since the common-noise has been removed, these error bars are only correct relative to each other, and are not a valid measurement of uncertainty of the absolute depth.As a test for sodium absorption, we repeated the process with increasingly narrow down to 50 Å around the 5900 Å doublet, but found no significant difference to the broader bands. Bands narrower than 50 Åhad significantly larger noise, so could not provide further constraints. Whilst the narrow line core of sodium may be present in the spectrum of WASP-52b it is not possible to detect it at the resolution of the instrument. Higher resolution observations are capable of detecting the line core of sodium even above a cloud/haze deck <cit.>.§ DISCUSSION§.§ Model atmospheres Inspecting the final transmission spectrum in Figure <ref>, the spectrum appears to be largely flat to within a few pressure scale heights, and devoid of any obvious absorption features, though the redmost 4 points are slightly above the median level. Due to the quality and resolution of the available data, a full atmospheric retrieval is not appropriate, especially considering the degeneracies involved in reproducing the flat region of the spectrum (4000–7750Å). Instead we generate a sample of representative atmosphere classes to compare to the data, and qualitatively discuss their applicability to the dataset. A list of all models tested and their goodness of fits are given in Table <ref>.All of the model atmospheres are generated using the NEMESIS radiative transfer code <cit.>, with precalculated correlated k-tables described in <cit.>. Unless otherwise stated, we assume solar abundances of elements. The only major carbon or oxygen bearing molecule with features in our spectral range is water, for which we assume a volume mixing ratio of 3.83 × 10^-4, based on the disequilibrium chemistry model of the terminator of HD 189733b described in <cit.>.We assume simple parametrized Temperature-Pressure profiles generated using the equations presented in <cit.>. For all models except the hot haze model, we assume the zero albedo equilibrium temperature for WASP-52b in the case of uniform redistribution, which is 1300 K (using f=0.25 from <cit.>). We first calculate a Temperature-Pressure profile appropriate for a clear atmosphere with low levels of short-wave scattering k_s=6 × 10^-4 cm^-2g^-1 and γ_0=0.06. A clear atmosphere with solar metalicity without TiO and VO absorption is found to be a very poor due to a clear lack of a broadened sodium feature and Rayleigh slope. These features could be masked by high altitude clouds, or de-emphasised by additional molecular absorption.Several hot Jupiters have been found to have optical spectra that are dominated by Rayleigh slopes, presumably caused by haze. For HD 189733b, which has a similar equilibrium temperature and parent star spectral type to WASP-52b, <cit.> find that the majority of the data taken with HST/ACS are well fit by a Rayleigh slope model with an equlibrium temperature of 1300 K (matching the expected equilibrium temperature of the planet), but bluewards of 6000 Å for data taken with HST/STIS the slope is somewhat steeper, having a best fitting temperature of 2100 ± 500 K <cit.>. <cit.> generate an example analytical Temperature-Pressure profile for HD 189733b using the equations of <cit.> with a strong thermal inversion layer with a thermosphere that reaches 2000 K. This would be capable of reproducing the strong Rayleigh feature seen in the HST/STIS data.We assume the same parameters as for the T-P profile (k_s = 0.0670, γ_0 = 10.0, η = 0.1) which is equivalent to a geometric albedo of 0.5. For WASP-52b, this results in a thermosphere which reaches 2100 K at 0.1 mbar. The steep Rayleigh slope a haze produces in this hot thermosphereis clearly a very poor fit to the data, as seen in Figure <ref>.We also test a more subdued Rayleigh slope, using our original non-inverted T-P profile. The slope can be further de-emphasised by assuming slightly larger particles that exhibit settling and hence have a smaller effective scale height in the atmosphere, we parametrise this by using a fractional scale height for the scattering particles that is 0.3 of the gas pressure scale height. This still produces a poor fit that can be confidently ruled out (see Table <ref>).<cit.> show that many hot Jupiters have their optical and near infrared spectral features partially or fully obscured by grey cloud layers, so it seems likely that our spectrum of WASP-52b could also be flattened by high altitude clouds. <cit.> find a flat spectrum compatible with a cloudy interpretation in their analysis of WASP-52b with high precision 3-band photometry.To model the effect of clouds we insert opaque layers to the radiative transfer code at 0.1, 1 and 10 mbar. All provide significant improvement over the clear spectrum. The flat spectrum provided by the 0.1 mbar cloud layer is the best fit to the data. We are unable to distinguish between clouds at any higher altitude than 0.1 mbar, as the spectrum is already completely flattened. This is consistent with cloud formation theories for HD 189733b, which has the same equilibrium temperature as WASP-52b <cit.>. These clouds would likely be composed of a range of chemical species, with magnesium silicates as important contributors.An alternate explanation could be that unresolved molecular features in the planets atmosphere are de-emphasising the Rayleigh slope and sodium features. As an example, we consider an atmosphere with TiO and VO absorption. In hot Jupiter atmospheres T and V in the gas phase are expected to be practically entirely in the oxide forms from chemical equilibrium, but for cooler planets like WASP-52b with equilibrium temperatures below 1600K condensation and cold trapping remove significant fractions from the atmosphere <cit.>. A full treatment of TiO and VO abundance is beyond the scope of this work, instead, we test the effect of several different un-condensed fractions on the spectrum. We assume solar abundances of T and V <cit.>.A fraction of 1 part per million is indistinguishable from the clear spectrum, but increasing the fraction above this level begins to improve the fit by de-emphasising the Rayleigh slope and sodium feature. We find that an abundance fraction of 0.01 produces a good fit, which is formally indistinguishable from the goodness of fit provided by the cloud dominated model. However, the coolness of the atmosphere of WASP-52b and the lack of firm detections of TiO in other hot Jupiter atmospheres makes this explanation unlikely, so we favour the cloudy model.A higher resolution search for the presence of the sodium feature (which is still present in the TiO case, but mostly obscured in the cloud-dominated case except for a narrow line core) could potentially help break the degeneracy. A search for the water feature in the infra-red would also be a useful diagnostic, as <cit.> shows this is typically obscured in cloudy atmospheres. Our results highlight the need for both low and high resolution observations with a large wavelength coverage to break the degeneracies of exoplanet atmosphere retrieval.Of the models tested, a flat, cloud dominated spectrum provided the best fit to the full dataset - however the residuals are dominated by the reddest 4 points in the spectrum (7750–9000 Å). The significant difference in goodness of fit when including these points may be an unexplained systematic error in the data, or indicate the presence of an unknown absorber. This feature appears to be present on both nights, showing that it is repeatable, but it may be a common artefact of the instrument or the reduction and analysis.The feature may have been introduced as an instrumental or atmospheric effect. There are signs of fringing at the reddest ends of the chip It is not on the region of the chip (y pixel > 2200), but not in the wavelength region corresponding to the feature. The regions of the chip effected by fringing show a dramatically increased high frequency noise, which is distinct to a persistent increase in radius ratio. The error bars in the 7750–8750 Å region are only somewhat larger than the errors elsewhere in the spectrum. OH Meinel bands are present in the telluric spectrum at these wavelengths, but they are removed by our sky background subtraction and differential photometry. An optical blocking filter was not used when taking these data, so redwards of 6000 Å the spectrum is weakly contaminated by the blue end of the second order spectrum. However, this effect is low order and is not able to increase the observed transit depth above the blue end of the spectrum.Incorrect limb darkening parameters can cause errors in radius retrieval <cit.> and could introduce a correlated bias to a transmission spectrum. The limb darkening values and errors were generated with Limb Darkening Tool Kit. LDTK fits a parametrised limb darkening profile to PHOENIX-calculated specific intensity spectra <cit.>. To test that these model profiles were not influencing the shape of the resulting transmission spectrum the fit was repeated without the LDTK derived priors on the limb darkening parameters. To improve convergence and prevent unphysical profiles we followed the formalism of <cit.> and <cit.>. We find that without the additional limb darkening constraints the errors on the resulting spectrum are slightly higher, but the shape of the spectrum is preserved. Therefore it is not likely that limb darkening can account for the 4 outlier points.We can find no obvious systematic reason for this feature in our analysis. While this does not rule additional instrumental effects that we have not considered, it may be a hint of an additional high altitude opacity source. <cit.> find a similar redward feature in their HST/STIS G750L low-resolution transmission spectrum of HAT-P-1b. A similar effect can also be seen on HD 209733b <cit.>, WASP-19b with FORS2 <cit.> and TrES-3b with OSIRIS <cit.>. However, we note that in all these cases, including ours, the unexplained feature is found at the edge of the usable region of the spectra, where the signal to noise is lower and systematics are more likely. This may imply a common instrumental source which we have so far been unable to identify. Alternatively, if this is in fact a real feature, it could be caused by absorption from a molecule not currently included in hot Jupiter atmosphere models. It is also possible that it could be caused by the composition of the particles in the cloud deck, if this layer is vertically extended and has a higher opacity in the red. §.§ Un-occulted spots WASP-52 is known to be an active star, and as discussed in Section <ref>, the star shows periodic modulations in brightness of order 1%, which is likely due to starspots rotating in and out of view. The temperature of the starspots on the very similar K star HD 189733 were found to be 4250 K by <cit.>. Using this temperature, which is 750 K cooler than the photosphere, gives a spot coverage fraction of about 4%. This corresponds well to the lower limit of 4% spot coverage on HD 189733 calculated by <cit.>, and is significantly higher than the 0.3% spot coverage of the Sun at solar maxima <cit.>. The value of 4% spot coverage is an average across an observing season however, and as previously mentioned there was no contemporaneous photometry of WASP-52, so it is possible that the spot coverage was higher than this at the time of these observations.Un-occulted spots can have an effect on a transmission spectrum, mimicking a Rayleigh scattering slope or molecular features such as water <cit.>, so before comparing planetary atmosphere models to the spectra, it is important to quantify what effects un-occulted spots may have.We simulate the effects of spots on the transmission spectrum using stellar spectra synthesised from a grid of atlas9 stellar atmosphere models by <cit.>. We use a log(g) of 4.5 and a temperature of 5000 K for the photosphere, and a range of temperatures for the starspots. The transmission spectrum is generated asδ̃(λ) = δ(λ)1/1 - η(1 - F_λ(spot)/F_λ(phot))where η is the spotted fraction of the visible stellar surface, F_λ(spot) and F_λ(phot) are the spectral radiance of the spots and photosphere, δ(λ) is the `true' transit depth at wavelength λ, here assumed to be a constant value of 0.164, and δ̃(λ) is the observed value.The results are plotted in Figure <ref> for a range of spot temperatures and coverage fractions. The nominal case of 4% coverage with 4250 K spots does not show any significant features within our level of precision. Increasing the spot contrast and the coverage fraction produces a spectrum with a bluewards slope that does not correspond to the observed spectrum.We therefore do not expect that un-occulted spots have an appreciable impact on our transmission spectrum.§.§ Transit timing <cit.> searched for periodic Transit Timing Variations in 10 exoplanet systems, including WASP-52b and found no evidence of periodic variations. We leverage the high precision of the white noise lightcurves and the longer time baseline between the ephemeris reported in the discovery paper <cit.> in order to test for secular drifts in the ephemeris of WASP-52b. The error bars on our ephemerides are probably larger than those reported in <cit.> and <cit.> due to the greater model flexibility allowed by our Gaussian process analysis.We convert the transit dates to BJD in order to calculate the ephemeris using an online tool[http://astroutils.astronomy.ohio-state.edu/time/utc2bjd.html] <cit.>.<cit.> report their central transit time in HJD, which can introduce errors at the .17ex∼1 second level compared to calculating BJD_TBD directly, so we add this uncertainty in quadrature to their reported errors.We calculated the errors using an MCMC with a chain length of 100,000 steps and a burn in chain of 1000 steps. The best fitting ephemeris is shown in Figure <ref>.A linear ephemeris is formally a poor fit, which could be interpreted as evidence of a non-linear ephemeris in this system. However, we believe it is more likely that previous work may have systematically underestimated the size of their errors by not properly accounting for the presence of correlated noise through Gaussian processes or some other technique. Future measurements of the ephemeris of this system may shed more light on the possibility of additional companions.The large time baseline (4 years) between the discovery paper and this work allows a significant improvement in the uncertainty of the period. The aforementioned concern over underestimated errors is not particularly important, as the error budget is dominated by the gradient over time periods this long.§ CONCLUSIONS We have performed transmission spectroscopy on WASP-52b, a highly inflated hot Jupiter orbiting an active K star. Our Gaussian process and common noise model approach to modelling resulted in a precise relative transmission spectrum that was consistent across two nights despite the presence of a potential spot. Our errors are on the order of a single scale height.The possible “spot-crossing” feature on the first night appears to be grey, and is not visible in the spectrally resolved light curves after common-noise term removal. This implies that it is either a wavelength independent systematic, or that the temperature contrast of the spot to the rest of the photosphere is high, which is an expected feature for cooler stars such as WASP-52. We have performed a harmonic analysis of the long term photometry of WASP-52, and find that the activity level may be changing over time, with evidence of differential rotation.We attempted to fit representative model atmospheres to the transmission spectrum, but were unable to find a satisfactory fit to the entire spectral range. For the majority of the spectrum (4000–7750 Å), a cloud layer at 0.1 mbar provides an acceptable fit to the data, but is inconsistent with a slightly deeper transit at wavelengths > 7750 Å. We explore several different systematic reasons for this excess depth, and are unable to find a compelling cause. If it is a real feature of the atmosphere it may be the result of an additional unknown absorber.The cloud layer at 0.1 mbar is consistent with cloud formation theories for HD 189733b, which has the same equilibrium temperature as WASP-52b <cit.>. These clouds would likely be composed of a range of chemical species, with magnesium silicates as important contributors.The transmission spectrum is dramatically different to HD 189733b, which is dominated by a Rayleigh slope caused by high altitude haze. The difference between these transmission spectra provides further evidence of the rich and as yet unexplained diversity within exoplanet atmospheres.§ ACKNOWLEDGEMENTS We thank the anonymous reviewer, whose comments greatly improved this manuscript. Based on observations made with the WHT operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. J.K is supported by an STFC studentship. P.W. and T.L. are supported by STFC consolidated grant (ST/L000733/1).mnras | http://arxiv.org/abs/1703.09285v1 | {
"authors": [
"Tom Louden",
"Peter J. Wheatley",
"Patrick G. J. Irwin",
"James Kirk",
"Ian Skillen"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170327194658",
"title": "A precise optical transmission spectrum of the inflated exoplanet WASP-52b"
} |
TU Wien, Faculty of Mathematics and Geoinformation, Institute of Discrete Mathematics and Geometry, Wiedner Hauptstrasse 8–10, A-1040 Vienna, [email protected] This work was supported by: the Austrian Science Fund (FWF) P23875 (both authors), I1272 (second author) and P30666 (both authors); the first author was supported by the International Academic Mobility Scholarship from Universidad Nacional de Colombia; the second author was supported by the grant no. IN201711, Dirección Operativa de Investigación, Institución Universitaria Pascual Bravo, and by Grant-in-Aid for Early Career Scientists 18K13448, Japan Society for the Promotion of Science.Shizuoka University, Faculty of Sciences. 836 Ohya, Suruga-ku, Shizuoka city, Japan 422-8529 [email protected] http://www.researchgate.com/profile/Diego_Mejia2 [2010]Primary 03E17; Secondary 03E15, 03E35, 03E40Yorioka <cit.> introduced a class of ideals (parametrized by reals) on the Cantor space to prove that the relation between the size of the continuum and the cofinality of the strong measure zero ideal on the real line cannot be decided in ZFC. We construct a matrix iteration of ccc posets to force that, for many ideals in that class, their associated cardinal invariants (i.e. additivity, covering, uniformity and cofinality) are pairwise different. In addition, we show that, consistently, the additivity and cofinality of Yorioka ideals does not coincide with the additivity and cofinality (respectively) of the ideal of Lebesgue measure zero subsets of the real line. On cardinal characteristics of Yorioka ideals Diego A. Mejía December 30, 2023 =============================================§ INTRODUCTION Yorioka <cit.> introduced a characterization of , the σ-ideal of strong measure zero subsets of the Cantor space 2^ω, in terms of σ-ideals _f parametrized by increasing functions f∈ω^ω, which we call Yorioka ideals (see Definition <ref>). Concretely, =⋂{_f:f∈ω^ω increasing} and _f⊆ whereis the σ-ideal of Lebesgue-measure zero subsets of 2^ω. Yorioka used this characterization to show that no inequality between () and :=2^ℵ_0 cannot be decided in ZFC, even more, he proved that ()=_κ (the dominating number on κ^κ) whenever (_f)=(_f)=κ for all increasing f ([Yorioka's original proof assumes (_f)=(_f)==()=κ for all increasing f, butand () can be omitted thanks to the results in Section <ref>.]).Further research on Yorioka ideals has been continued by Kamo and Osuga <cit.>. In <cit.> they proved that, in ZFC, (_f)≤≤≤(_f) for all increasing f and that, for any fixed f, the basic diagram of the cardinal invariants associated to _f (see Figure <ref>) is complete in the sense that no other inequality can be proved in ZFC. On the other hand, in <cit.> they constructed models by FS (finite support) iterations of ccc posets where infinitely many cardinal invariants of the form (_f) are pairwise different. Moreover, if there exists a weakly inaccessible cardinal, then there is a ccc poset forcing that there are continuum many pairwise different cardinals of the form (_f).To continue this line of research, we aim to obtain further consistency results considering several cardinal invariants associated with Yorioka ideals at the same time, that is, to construct models of ZFC where three or more of such cardinal invariants are pairwise different. Given a familyof subsets of a set X, the cardinal invariants associated withare the four cardinals (), (), () and (). The main objective of this paper is to prove the following result.There is a function f_0∈ω^ω and a ccc poset forcing that the four cardinal invariants associated with _f are pairwise different for each increasing f≥^*f_0. Concerning problems of this nature, the consistency of ()<()<()<() with ZFC is a consequence of <cit.>. Quite recently, Goldstern, Kellner and Shelah <cit.> showed the consistency, modulo strongly compact cardinals, of Cichoń's diagram separated into 10 different values, in particular, the four cardinal invariants associated with the meager idealon ℝ are pairwise different is consistent. However, this consistency result alone is unknown without using large cardinals.Though many inequalities between the cardinal invariants associated with Yorioka ideals and the cardinals in Cichoń's diagram are known in ZFC, there are still many open questions. Most of these inequalities had been settled in <cit.>, however, there are a couple of statements from <cit.> whose proofs do not appear anywhere. Namely, ()≤(_g)≤(_f) and (_f)≤(_g)≤() when f,g∈ω^ω are increasing and f(n+1)-f(n)≤ g(n+1)-g(n) for all but finitely many n<ω. We offer our own proofs of these inequalities in Corollaries <ref> and <ref>, even more, in Theorem <ref> we show the stronger fact ([Its proof does not use the fact that _f is an ideal, and it is simpler than Yorioka's argument to prove that _f is a σ-ideal (see <cit.>), though the ideas in both proofs are quite similar.]) that (_g) is above some definable cardinal invariant above () (and dually for (_f)). We use this to prove the following new consistency result.If f∈ω^ω is increasing then there is a ccc poset that forces ()<(_f)<(_f)<(). This definable cardinal we use to prove the theorem above is a cardinal characteristic, denoted by _b,h, that is parametrized by functions b,h∈ω^ω, which form part of what we calllocalization cardinals (Definition <ref>). These cardinals are a generalization of the cardinal invariants Bartoszyński used to characterize () and () in terms of a slalom structure (Theorem <ref>), and they were used by Brendle and the second author in <cit.> to show consistency results about cardinal invariants associated with Rothberger gaps in F_σ ideals on ω. In one of these results, it was constructed a ccc poset that forces infinitely many cardinals of the form _b,h to be pairwise different (even continuum many modulo a weakly inaccessible). An older precedent is known for _b,h, the dual of _b,h: Goldstern and Shelah <cit.> constructed a proper poset (using creatures) to force that ℵ_1-many cardinals of the form _b,h are pairwise different, result that was later improved by Kellner <cit.> for continuum many. In this work, to prove Theorem <ref>, we show in Theorem <ref> a connection between the cardinals of type (_f), (_f) and the localization cardinals.A variation of the localization cardinals that we call anti-localization cardinals, denoted by _b,h and (its dual) _b,h (for b,h∈ω^ω), also play an important role in the study of cardinal invariants associated with Yorioka ideals. The cardinal _b,h is known from Miller's <cit.> characterization ()=min{_b,h:b∈ω^ω} (for any h≥^*1). On the other hand, Kellner and Shelah <cit.> constructed a proper poset that forces that, for the types _b,h and _b,h, continuum many cardinals of each type are pairwise different. In fact, Kamo and Osuga <cit.> discovered a connection between the cardinals of type (_f), (_f) and the anti-localization cardinals (Lemmas <ref> and <ref>), which they use to construct a finite support iteration of ccc posets that forces that, for the types (_f) and _b,1, infinitely many cardinals of each type are pairwise different (even continuum many when a weakly inaccessible cardinal is assumed).We use the connections between the cardinal invariants associated with Yorioka ideals and the localization and anti-localization cardinals to prove Theorems <ref> and <ref>. Even more, in the consistency result of the first theorem we can additionally include that infinitely many cardinals of each type _b,h and _b,1 are pairwise different.There is a function f_0∈ω^ω and a ccc poset forcing that * the four cardinal invariants associated with _f are pairwise different for each increasing f≥^*f_0, * infinitely many cardinals of the form _b,h are pairwise different, and * infinitely many cardinals of the form _b,h and (_f) are pairwise different. Within this result, we merge with Theorem <ref> the consistency result of infinitely many localization cardinals from <cit.>, and the consistency result of infinitely many anti-localization cardinals and (_f) from <cit.>.This result is proved by using a FS iteration, so it forces that ()≤(). Because of this, as ZFC proves that _b,h≤() and ()≤()≤(_f) for any b,h,f, we cannot expect continuum many values in (b) and (c) above. However, Theorem <ref> is stated in such generality that (c) can be obtained for continuum many values when a weakly inaccessible is assumed (of course, in this case (a) can not forced). The same applies for (b) but not simultaneously with (c) because of the limitation in our result that the values for the cardinals in (b) appear below those of (c). A curious fact is that, in our result, any single value from (b) and (c) can be repeated continuum many times for different parameters.The forcing method we use to prove Theorems <ref>, <ref> and <ref> is the technique of matrix iterations to construct two dimensional arrays of posets by FS iterations. This method has been very useful to obtain models where several cardinal characteristics of the continuum are pairwise different. It was introduced for the first time by Blass and Shelah <cit.> to prove the consistency of < with large continuum (i.e. >ℵ_2) whereis the ultrafilter number. Later on, this method was improved by Brendle and Fischer <cit.> when they proved the consistency of =< and ℵ_1<=< with large continuum, the latter assuming the existence of a measurable cardinal in the ground model. The second author <cit.> induced known preservation results in such type of iterations to construct models where several cardinals in Cichoń's diagram are pairwise different. Just a while ago, Dow and Shelah <cit.> used this technique to prove that the splitting numberis consistently singular.To guarantee that the matrix iteration constructed to prove the theorem forces the desired values for the cardinal invariants, we propose a more general version of the classical preservation theory of Judah and Shelah <cit.> and Brendle <cit.>, along with the corresponding version for matrix iterations of the second author <cit.>. This generalization also looks to describe the preservation property that Kamo and Osuga <cit.> proposed to control the covering of Yorioka ideals. Concretely, they proposed a preservation property for cardinal invariants of the form _b,h^𝕀_ω in order to decide values of (_f) thanks to the relations they discovered between both types of cardinals. Although this preservation property is similar to Judah-Shelah's and Brendle's preservation theory, it is not a particular case of it. In view of this, we generalize this classical preservation theory so that the preservation property of Kamo and Osuga becomes a particular case (Example <ref>). Even more, our theory also covers the preservation property defined in <cit.> to decide cardinal invariants related to Rothberger gaps by forcing (Example <ref>). In addition, we add a particular case of our theory to deal directly with the preservation of cardinals of the form _b,h (Example <ref>).This paper is structured as follows. In Section <ref> we review the essential notions of this paper: localization and anti-localization cardinals, Yorioka ideals and related forcing notions and their properties. In Section <ref> we show the connection between (), () and the localization cardinals, and we review some inequalities between the cardinal invariants associated to Yorioka ideals that are known in ZFC. Additionally, we show what happens to the localization and anti-localization cardinals in non-standard cases, that is, in cases where the function b is allowed to take uncountable values. Section <ref> is devoted to our general preservation theory. In Section <ref> we prove the main results of this paper. Finally, we present in Section <ref> discussions and open questions related to this work.§ NOTATION AND PRELIMINARIES Throughout this text, we refer to members of any uncountable Polish space as reals. We write ∃^∞n<ω and ∀^∞n<ω to abbreviate `for infinitely many natural numbers' and `for all but finitely many natural numbers', respectively. For sets A,B, denote by B^A the set of functions from A to B. For functions f,g from ω into the ordinals,f≤ gmeans that f(n) ≤ g(n) for all n <ω. We say that g (eventually) dominates f, denoted by f≤^* g, if ∀^∞_n<ω(f(n)≤ g(n)). Define f<g and f<^*g similarly. F⊆ω^ω is bounded if it is dominated by a single function in ω^ω, i.e, there is a g∈ω^ω such that f≤ ^* g for all f∈ F. A not bounded set is called unbounded. A set D⊆ω^ω is dominating if every f∈ω^ωis dominated by some member of D.We extend some known operations in the natural numbers as operations between functions of natural numbers defined point-wise. For example, for f,g ∈ω^ω, f+g∈ω^ω is defined as (f+g)(n)=f(n)+g(n), likewise for the product and exponentiation. Also, we use natural numbers (and even ordinal numbers) to denote constant functions with domain ω. We denote by f^+ and log f the functions from ω to ω defined by:f^+(n)=∑_j≤ nf(j) and log f (n)=min{ k<ω : f(n)≤ 2^k}.For any set A, 𝕀_A denotes the identity function on A.Given a function b with domain ω such that b(i)≠∅ for all i<ω, h∈ω^ω and n<ω, define _n(b,h)=∏_i<n[b(i)]^≤ h(i), _<ω(b,h)=⋃_n<ω_n(b,h) and (b,h)=∏_n<ω[b(n)]^≤ h(n). Depending on the context, for a set A we denote _<ω(A,h)=_<ω(b,h) where b:ω→{A}. Similarly, we use _n(A,h) and (A,h).Denote ∏ b:=∏_i<ωb(i) and (b):=⋃_n<ω∏_i<nb(i). As a topological space, we endow ∏ b with the product topology where each b(i) has the discrete topology. Note that the sets of the form [s]:=[s]_b:={x∈∏ b: s⊆ x} form a basis of this topology. In particular, if each b(i) is countable then ∏ b is a Polish space, and it is perfect iff ∃^∞ i<ω(|b(i)|≥ 2). §.§ Relational systems and cardinal invariants Many of the classical cardinal invariants can be expressed by relational systems, and inequalities between these cardinals are induced by the Tukey-Galois order between the corresponding relational systems. These notions where defined by Votjas <cit.>.A relational system is a triple =⟨ A_-,A_+,⊏⟩ consisting of two non-empty sets A_+, A_- and a binary relation ⊏⊆ A_-× A_+. If =⟨ A_-,A_+,⊏⟩ is a relational system, define the dual ofas the relational system ^⊥=⟨ A_+,A_-,⊐̸⟩.For x∈ A_- and y∈ A_+, x⊏ y is often read y⊏-dominates x. A family X⊆ A_- is -bounded if there is a member of A_+ that ⊏-dominates every member of X, otherwise we say that the set is -unbounded. On the other hand, Y⊆ A_+ is -dominating if every member of A_- is ⊏-dominated by some member of Y. Define () as the smallest size of an -dominating family, and() as the smallest size of an -unbounded family. Note that ()=(^⊥) and ()=(^⊥). Depending on the relational system, () or () may not exist, case in which the non-existent cardinal is treated as `something' above all the cardinal numbers. Clearly, () does not exist iff ()=1, and () does not exist iff ()=1.Many classical cardinal invariants can be expressed through relational systems. * Letbe a family of subsets of a non-empty set X which is downwards ⊆-closed and ∅∈. Clearly, ,,⊆ and X,,∈ are relational systems, ()=,,⊆, ()=,,⊆, ()= X,,∈, and ()= X,,∈. * :=ω^ω,ω^ω,≤^* is a relational system, =() and =(). * Let b be a function with domain ω such that b(i)≠∅ for all i<ω. Define the relation ≠^* on ∏ b as x≠^* y iff ∀^∞ i<ω(x(i)≠ y(i)). This means that x and y are eventually different. Denote its negation by =^∞. Put (b):=∏ b,∏ b,≠^* ( stands for eventually different). It is clear that (b)^⊥=∏ b,∏ b,=^∞.Functions φ:ω→ [ω]^<ω are often called slaloms.* For two functions x and φ with domain ω, define * x∈^*φ by ∀^∞n<ω(x(n)∈φ(n)), which is read φ localizes x; * x∈^∞φ by ∃^∞n<ω(x(n)∈φ(n)). Denote its negation by φ∌^*x, which is read φ anti-localizes x.* Let b be a function with domain ω such that b(i)≠∅ for all i<ω, and let h∈ω^ω. * Define (b,h):=∏ b,(b,h),∈^* ( stands for localization), which is a relational system. Put _b,h:=((b,h)) and _b,h:=((b,h)), to which we often refer to as localization cardinals.* Define (b,h):=(b,h),∏ b,∌^* ( stands for anti-localization), which is a relational system. Note that (b,h)^⊥=∏ b,(b,h),∈^∞. Define _b,h:=((b,h)) and _b,h:=((b,h)), to which we refer to as anti-localization cardinals. Goldstern and Shelah <cit.>, and Kellner and Shelah <cit.> have studied cardinal coefficients of the form _b,h and _b,h for b∈ω^ω (with a different notation in their work) to provide the first examples of models where continuum many cardinal characteristics of the continuum are pairwise different. On the other hand, Brendle and Mejía <cit.> investigated cardinals of the form _b,h for b∈ω^ω in relation with gap numbers for F_σ-ideals. In this paper we also look at _b,h, the dual cardinal of _b,h.When b is the constant function ω, the localization and anti-localization cardinals provide the following well-known characterizations of classical cardinal invariants. If h∈ω^ω goes to infinity then ()=_ω,h and ()=_ω,h.If h∈ω^ω and h≥^*1 then _ω,h=() and _ω,h=().Fix b and h as in Definition <ref>. The following items justify that the study of localization and anti-localization cardinals can be reduced to the case when 1≤ h and h(i)<|b(i)| for all i<ω. * If |b(i)|≤ h(i) for all but finitely many i<ω, _b,h=1 and _b,h is undefined. On the other hand, if ∃^∞i<ω(h(i)=0) then _b,h=1 and _b,h is undefined. Hereafter, both localization cardinals for b,h are defined only when 1≤^*h and ∃^∞i<ω(h(i)<|b(i)|), even more, if A:={i<ω:1≤ h(i)<|b(i)|} (which is infinite) and ι_A:ω→ A is the increasing enumeration of A, then _b,h=_b∘ι_A,h∘ι_A and _b,h=_b∘ι_A,h∘ι_A (actually, (b,h)(b∘ι_A,h∘ι_A), see Definition <ref>).* If ∃^∞ i<ω(|b(i)|≤ h(i)) then _b,h=1 and _b,h is undefined. On the other hand, if ∀^∞ i<ω(h(i)=0) then _b,h=1 and _b,h is undefined. Hence, both anti-localization cardinals for b,h are defined iff h(i)<|b(i)| for all but finitely many i<ω, and ∃^∞ i<ω(h(i)≥ 1), even more, if A and ι_A are as in (1) then _b,h=_b∘ι_A,h∘ι_A and _b,h=_b∘ι_A,h∘ι_A (actually, (b,h)(b∘ι_A,h∘ι_A), see Definition <ref>). Let =⟨ A_-,A_+,⊏⟩ and =⟨ B_-,B_+,⊏^'⟩ be relational systems. Say thatis Tukey-Galois below , denoted by , if there exist functions φ_-:A_-→ B_- and φ_+:B_+→ A_+ such that, for all x∈ A_- and b∈ B_+, if φ_-(x)⊏^'b then x⊏φ_+(b). Here, we say that the pair (φ_-,φ_+) witnesses . Say thatandare Tukey-Galois equivalent, denoted by, ifand .Assumeand that this is witnessed by (φ_-,φ_+). * If D⊆ B_+ is -dominating then φ_+[D] is -dominating.* If C⊆ A_- is -unbounded then φ_-[D] is -unbounded.In particular, ()≤() and ()≤().* Let ⊆ be two downwards ⊆-closed families of subsets of a non-empty set X such that ∅∈. Clearly, ⟨ X, , ∈⟩⟨ X, , ∈⟩. In particular, ()≤() and ()≤(). * Let b be a sequence of length ω of non-empty sets and h∈ω^ω such that h≥^*1. Define φ_+=𝕀_∏ b and φ_-: ∏ b →(b,h) where φ_-(x):=s_x is defined ass_x(i):={x(i)} if h(i)≠ 0, or s_x(i):=∅ otherwise.Note that, for any x,y∈∏b, s_x∌^*y implies x≠^*y, so (b)(b,h). Even more, (b)(b,1). Hence _b,h≤_b,1=∏ b,∏ b,≠^* and _b,h≥_b,1=∏ b,∏ b,≠^*. In addition, if b∈ω^ω then (b,1)(b,b-1), so _b,1=_b,b-1 and _b,1=_b,b-1. * Let b, b' be sequences of length ω of non-empty sets, and let h,h' ∈ω^ω. If ∀^∞ i<ω(|b(i)|≤|b'(i)|) and h'≤^* h then (b,h)(b',h') and (b',h')(b,h). Hence * _b',h'≤_b,h and _b,h≤_b',h',* _b,h≤_b',h' and _b',h'≤_b,h. By Theorem <ref>, if each b(i) is countable and h≥^*1 then _b,h≤() and ()≤_b,h. In addition, if h goes to infinity then ()≤_b,h and _b,h≤() by Theorem <ref>.* Let b and h be as in (2). If ι:ω→ω is a one-to-one function, then (b∘ι,h∘ι)(b,h) and (b∘ι,h∘ι)(b,h). Hence _b,h≤_b∘ι,h∘ι and _b∘ι,h∘ι≤_b,h, likewise for the anti-localization cardinals. Even more, if ω∖ι is finite then (b∘ι,h∘ι)(b,h) and (b∘ι,h∘ι)(b,h).* If b and h are as in (2), then (b,h)^⊥(b,h), so _b,h≤_b,h and _b,h≤_b,h.Let =⟨ A_-,A_+,⊏⟩ and =⟨ B_-,B_+,⊏^'⟩ be relational systems. Define the following relational systems. * The conjunction ∧=⟨ A_-× B_-,A_+× B_+,⊏_∧⟩ where the binary relation ⊏_∧ is defined as (x,y)⊏_∧ (a,b) iff x⊏ a and y⊏^' b.* The sequential composition (;)=⟨ A_-× B_-^A_+,A_+× B_+,⊏_;⟩ where the binary relation (x,f)⊏_;(a,b) means x⊏ a and f(a)⊏^' b.The following result describes the effect of the conjunction and sequential composition on the corresponding cardinal invariants. Let =⟨ A_-,A_+,⊏⟩ and =⟨ B_-,B_+,⊏^'⟩ be relational systems. Then: * max{(),()}≤(∧)≤()·() and (×)=min{(),()}.* (;)=()·() and (;)=min{(),()}. §.§ Yorioka idealsFor σ∈ (2^<ω)^ω define[σ]_∞:={x ∈ 2^ω:∃^∞n < ω^(σ(n) ⊆ x)}=⋂_n<ω⋃_m ⩾ n[σ(m)]. Also defineh_σ∈ω^ω such that h_σ(i)=|σ(i)| for each i<ω.Define the relation ≪ on ω^ω by f≪ giff ∀k<ω∀^∞n<ω(f(n^k)≤ g(n)).For each f ∈ω^ω define the families𝒥_f:={X⊆ 2^ω:∃σ∈ (2^<ω)^ω(X ⊆ [σ]_∞ and h_σ=f )} and ℐ_f:=⋃_g≫ f𝒥_g.Any family of the form ℐ_f with f increasing is called a Yorioka ideal. It is clear that both _g and _f contain all the finite subsets of 2^ω and that they are downwards ⊆-closed. Note that f≤^* f' implies 𝒥_f'⊆𝒥_f and ℐ_f'⊆ℐ_f, so (ℐ_f)≤(ℐ_f') and (_f)≥(_f'), likewise for _f and _f'. Moreover _f⊆_f and f≪ g implies 𝒥_g⊆ℐ_f, so (𝒥_f)≤cov(ℐ_f)≤cov(𝒥_g) and (𝒥_g)≤non(ℐ_f)≤non(𝒥_f). On the other hand, _f⊆ iff the series ∑_i<ω2^-f(i) converges, and ⊆_f. Hence ⊆_f⊆ when f is increasing, so (𝒩)≤(ℐ_f)≤(𝒮𝒩) and (𝒮𝒩)≤(ℐ_f)≤(𝒩). If f ∈ω^ω is an increasing function then ℐ_f is a σ-ideal. Moreover, 𝒮𝒩=⋂{_f:f increasing}. In contrast, Kamo and Osuga <cit.> proved that _f is not closed under unions when f(i+1)-f(i)≥ 3 for all but finitely many i<ω.Denote ω^↑ω:={d∈ω^ω:d(0)=0 and d is increasing}. For d ∈ω^↑ω and an increasing f∈ω^ω, define g_d^f∈ω^ω by g_d^f(n)=f(n^k+10)whenn ∈ [d(k),d(k+1)). Note that d≤^* e iff g_e^f≤^*g_d^f, and d≤ e iff g_e^f≤ g_d^f. For each d∈ω^↑ω, g^f_d is increasing and g_d^f≫ f for each d ∈ω^↑ω. Conversely, for each g≫ f there exists a d ∈ω^↑ω such that g_d^f≤^* g. In particular, if D⊆ω^↑ω is a dominating family, thenℐ_f=⋃_d ∈ D𝒥_g_d^f.If f∈ω^ω is increasing and c∈ω then f≪ g iff f+c≪ g for all g∈ω^ω. In particular _f=_f+c.Assume f≪ g. Fix a natural number k≥ 1 and choose m>c+2 such that ∀n≥ m(f(n^k+1)≤ g(n)). As f is increasing, for all n≥ m, f(n^k)+c≤ f(n^k+c)≤ f(n^k+n)≤ f(n^k+1)≤ g(n). §.§ ForcingThe basics of forcing can be found in <cit.>, <cit.> and <cit.>. See also <cit.> for further information about Suslin ccc forcing. Unless otherwise stated,we denote the ground model by V. When dealing with an iteration over a model V, we denote by V_α the generic extension at the α-th stage.Recall the following stronger versions of the countable chain condition of a poset. Letbe a forcing notion and κ an infinite cardinal.* For n<ω, B⊆ is n-linked if, for every F⊆ B of size ≤ n, ∃ q∈∀ p∈ F(q≤ p).* C⊆ iscentered if it is n-linked for every n<ω.*is κ-linked if =⋃_α<κP_α where each P_α is 2-linked. When κ=ω, we say thatis σ-linked.*is κ-centered if =⋃_α<κP_α where each P_α is centered. When κ=ω, we say thatis σ-centered.*has κ-cc (the κ-chain condition) if every antichain inhas size <κ.has ccc (the countable chain contidion) if it has ℵ_1-cc.Any κ-centered poset is κ-linked and any κ-linked poset has κ^+-cc.The following generalization of the notion of σ-linkedness is fundamental in this work. Let ρ,π∈ω^ω. A forcing notionis (ρ,π)-linked if there exists a sequence ⟨ Q_n,j:n<ω, j<ρ(n)⟩ of subsets ofsuch that * Q_n,j is π(n)-linked for all n<ω and j<ρ(n), and* ∀ p∈∀^∞n<ω∃j<ρ(n)(p∈ Q_n,j).Here, condition (ii) can be replaced by* ∀ p∈∀^∞n<ω∃j<ρ(n)∃ q≤ p(q∈ Q_n,j)because (i) and (ii') imply that the sequence of Q'_n,j:={q∈:∃ p∈ Q_n,j(q≤ p)} (n<ω and j<ρ(n)) satisfies (i) and (ii).Ifis σ-centered thenis (ρ,π)-linked when ρ:ω→ω goes to +∞. Suppose that =⋃_n<ωP_n where each P_n is centered. For each n∈ω, define Q_n,j=P_j for j<ρ(n). It is clear that⟨ Q_n,j:n<ω, j<h(n)⟩satisfies (i) and (ii) ofDefinition <ref>. Ifis (ρ,π)-linked and π≰^*1 thenis σ-linked. To fix some notation, for each set Ω let _Ω be the finite support product of Cohen forcing :=ω^<ω (ordered by end-extension) indexed by Ω;is Hechler forcing, i.e. the standard σ-centered poset that adds a dominating real; and 1 denotes the trivial poset. These three posets are Suslin ccc forcing notions.The following poset was defined by Kamo and Osuga to increase the cardinal _b,h. Let b,h∈ω^ω such that b≥1 and assumelim_i→+∞h(i)/b(i)=0. Define the (b,h)-eventually different real forcing _b^h as the poset whose conditions are of the form (s,F) and satisfy: * s∈(b),* F⊆(b,h) is finite, and* |F|h(n)< b(n) for each n≥ |s|.It is ordered by (t,F^')≤(s,F) iff s⊆ t, F⊆ F^' and ∀i∈|t|∖|s|∀φ∈ F(t(i)∉φ(i)).If S ⊆(b,h), define ^h_b(S)={ (s,F) ∈^h_b : F ⊆ S } with the same order as _b^h. Denote _b:=^1_b and _b(S):=^1_b(S). If G is _b^h(S)-generic over V and r:=⋃{s:∃F((s,F)∈ G)}, then r∈∏ b and ∀φ∈ S(φ∌^* r). Also, V[G]=V[r]. In particular, ^h_b adds anr∈∏ b such that ∀φ∈(b,h)∩ V(φ∌^* r). Hence, when h≥^*1, r is eventually different from the members of V∩∏ b. The following result is a generalization of <cit.> about the linkedness of ^h_b(S)Let b,h∈ω^ω with b≥ 1. Let π,ρ∈ω^ω and assume that there is a non-decreasing function f∈ω^ω going to infinity and an m^*<ωsuch that, for all but finitely many k<ω, * kπ(k)h(i)<b(i) for all i≥ f(k) and * k∏_i=m^*^f(k)-1((min{k,f(k)}-1)h(i)+1)≤ρ(k).Then, for any S⊆(b,h), ^h_b(S) is (ρ,π)-linked.Fix M>m^* such that, for all k≥ M, (i) and (ii) holds and f(k)>0. Find a non-decreasing function g:ω→ω that goes to infinity such that, for all k≥ M, g(k)<min{k,f(k)} and |∏_i<g(k)b(i)|≤ k. Choose M'≥ M such that g(k)≥ M for all k≥ M'. Put S_k:={s∈∏_i<f(k)b(i):∀ i∈[g(k),f(k))(s(i)≤ g(k)h(i))} when k≥ M', otherwise put S_k:=∅. Note that, for k≥ M', |S_k|≤ k∏_i=g(k)^f(k)-1(g(k)h(i)+1)≤ k∏_i=m^*^f(k)-1((min{k,f(k)}-1)h(i)+1)≤ρ(k), so |S_k|≤ρ(k) for all k<ω. For each s∈ S_k put Q_k,s:={(t,F)∈_b^h(S):t=s,∀ i≥ f(k)(π(k)|F|h(i)<b(i))}. Clearly, Q_k,s is π(k)-linked. It remains to show that Q_k,s:k<ω,s∈ S_k satisfies (ii') of Definition <ref>. If (t,F)∈^h_b(S), choose N such that |t|+|F|+M'≤ g(N). We prove that, for all k≥ N, there is some s∈ S_k such that (s,F)≤(t,F) and (s,F)∈ Q_k,s. Extend t to t' so that |t'|=g(k) and (t',F) is a condition in ^h_b(S) stronger than (t,F). Note that |⋃_φ∈ Fφ(i)|≤ h(i)g(N)≤ h(i)g(k) for each i≥ g(k), thus we can extend t' to an s∈ S_k so that (s,F) is a condition stronger than (t',F). As |F|π(k)h(i)≤ g(N)π(k)h(i)≤ kπ(k)h(i)<b(i) by (i), (s,F)∈ Q_k,s. If S⊆(b,h) and lim_n→+∞h(n)/b(n)=0 then _b^h(S) is σ-linked. In particular, _b^h is a Suslin ccc poset.As lim_n→+∞h(n)/b(n)=0, we can find an increasing function f:ω→ω such that 2kh(i)<b(i) for all i≥ f(k) and k<ω. By Lemma <ref>, ^h_b(S) is (ρ,2)-linked where ρ(k):=k∏_i=1^f(k)-1((min{k,f(k)}-1)h(i)+1). The conclusion is a consequence of Lemmas <ref> and <ref>. [<cit.>] Let b,π,h ∈ω^ω such that π and h are non-decreasing, b≥ 1, both π and h are ≥^*1 and b≥^*hπid_ω+1. If S⊆(b,h) then ^h_b(S) is ((id_ωh)^id_ω,π)-linked. In particular, if h is the constant function 1 then _b(S) is ((id_ω)^id_ω,π)-linked.Use f=𝕀_ω and m^*=1 in Lemma <ref>. To finish this section, we review from <cit.> the poset that increases _b,h.Let b,h∈ω^ω such that b≥1. For R⊆∏ b, define the poset_b^h(R):={(s,F):s∈_<ω(b,h), F⊆ R and ∀ i≥|s|(|F| ≤ h(i)) }ordered by (s',F')≤ (s,F) iff s⊆ s', F⊆ F' and ∀i∈[|s|,|s'|)({x(i):x∈ F}⊆ s'(i)). Put _b^h:=_b^h(∏ b), which is a Suslin ccc poset. Let b,h∈ω^ω such that b≥1 and let R⊆∏ b. If h goes to infinity then _b^h(R) is σ-linked and it generically adds a slalom in (b,h) that localizes all reals in R. In particular, _b^h generically adds a slalom in (b,h) that localizes all the ground model reals in ∏ b.Let b,h,π,ρ∈ω^ω be non-decreasing functions with b≥ 1 and h going to infinity. If {m_k}_k<ω is a non-decreasing sequence of natural numbers that goes to infinity and, for all but finitely many k<ω, k·π(k)≤ h(m_k) and k·|[b(m_k-1)]^≤ k|^m_k≤ρ(k), then _b^h(R) is (ρ,π)-linked for any R⊆∏ b.§ ZFC RESULTS In this section we prove and review some inequalities between the cardinal invariants associated with Yorioka ideals, the cardinals in Cichoń's diagram and localization and anti-localization cardinals. Figure <ref> at the end of this section, which is taken from <cit.>, illustrates some of these inequalities. §.§ Localization and anti-localization cardinals For this subsection, fix a function b with domain ω and h∈ω^ω such that 1≤^* h, b(i)≠∅ for every i<ω and ∀^∞ i<ω(h(i)<|b(i)|). We show that the localization and anti-localization cardinals are characterized by any other known cardinal invariants when b(i) is infinite for infinitely many i<ω. We also show some results for these cardinals when b∈ω^ω, mostly when taking limit values like sup{_b,h:b∈ω^ω}. When h does not go to infinity, the localization cardinals have simple characterizations.If h does not go to infinity then _b,h≥ and _b,h=N+1 where N is the minimum natural number such that A_N:={i<ω:h(i)=N} is infinite. Moreover, _b,h= when ∀^∞ i<ω(|b(i)|≤).Without loss of generality, we assume that each b(i) is a cardinal number. For each l≤ N define x_l∈∏ b(i) by x_l(i):=l when i∈ A_N and h(i)<b(i), or 0 otherwise. Note that no single slalom in (b,h) localizes every x_l for l≤ N, so _b,h≤ N+1. Conversely,by the definition of N, N≤^* h, so N<_b,h. On the other hand, <cit.> states that _b,h= whenever h is constant and b∈ω^ω. So, by Example <ref>(3) and (4), _b,h≥ holds in our case. In addition, if ∀i<ω(|b(i)|≤) then _b,h≤|(b,h)|=|∏ b|=, so the “moreover" part follows by Example <ref>(3). A similar result to the above can be proved for the anti-localization cardinals when the sequence h(i)/|b(i)|:i<ω does not converge to 0 (put h(i)/|b(i)|=0 when b(i) is an infinite set). This proof is supported by the following two facts. The first is based on (2)↔(4) from <cit.>.If κ is an infinite cardinal then (κ,h)(κ,1), so _κ,h=_κ,1 and _κ,h=_κ,1.Wlog assume that h≥ 1. Fix I_n:n<ω the interval partition of ω such that |I_n|=h(n) for all n<ω and put b'(n):=κ^I_n, which has cardinality κ. Hence (b',h)(κ,h). For any S∈(b',h) define y_S∈κ^ω such that, for any n<ω and s∈ S(n) there is some i∈ I_n such that s(i)=y_S(i). Define φ_-:(b',h)→(κ,1) and φ_+:κ^ω→∏ b' by φ_-(S)(i):={y_S(i)} and φ_+(x):= x I_n:n<ω. These maps witness that (b',h)(κ,1). Conversely, by Example <ref>(3), (κ,1)(κ,h). If the sequence h(i)/|b(i)|:i<ω does not converge to 0 then _b,h=N where N is the minimum natural number such that B_N:={i<ω:|b(i)|≤ N· h(i)} is infinite, and _b,h=. For all but finitely many i∈ B_N there is a partition c_i,j:j<N of b(i) such that 0<|c_i,j|≤ h(i) for each j<N. Let ι:ω→ B_N be the increasing enumeration of B_N. Define φ_-:N^ω→(b,h) such that, for each z∈ N^ω, φ_-(z)(ι(i))=c_ι(i),z(i) and φ_-(z)(i)=∅ for all i∉ B_N; and define φ_+:∏ b→ N^ω such that φ_+(x)(i)=j iff x(ι(i))∈ c_ι(i),j. Note that the pair (φ_-,φ_+) witnesses that (N)(b,h), so _b,h≤_N,1 and _N,1≤_b,h. On the other hand, by Example <ref>(2) and Theorem <ref>, _N,1=_N,N-1=N and _N,1=_N,N-1=. Therefore _b,h≤ N and ≤_b,h.The converse inequality for _b,h follows from the fact that (N-1)· h(i)<|b(i)| for all but finitely many i<ω (so N-1<_b,h).Put b'(i)=min{ω,|b(i)|}. As |b'(i)|≤|b(i)| for all i<ω, by Example <ref>(3) _b,h≤_b',h. Note that _b',h≤|∏ b'|≤.Assume that the sequence |b(i)|:i<ω is non-decreasing and |b(0)|≥ℵ_0. Then _b,h≥([κ]^ℵ_0) and _b,h≤([κ]^ℵ_0) where κ=sup_i<ω{|b(i)|}. Recall that ([ω]^ℵ_0)=1 and that ([ω]^ℵ_0) is undefined. If we interpret ([ω]^ℵ_0) as “something" that is above all the ordinals, the inequality _b,h≤([ω]^ℵ_0) makes sense. On the other hand, ([κ]^ℵ_0)=ℵ_1 when κ is uncountable. Wlog assume that each b(i) is an infinite cardinal. Define b'(i):=b(i)^<ω for each i<ω. As |b'(i)|=|b(i)| then (b',h)(b,h), so we can work with b' instead of b. Define φ_-:(b',h)→[κ]^ℵ_0 such that φ_-(S) contains {s(j):j<|s|, s∈ S(i), i<ω}, and define φ_+:[κ]^ℵ_0→∏ b' such that, for any c∈[κ]^ℵ_0, φ_+(c)(i)=c̅ k_c,i where c̅:= c_j:j<ω is some (chosen) enumeration of c and k_c,i is the maximal k≤ i such that c̅ k∈ b'(i). Note that k_c,i:i<ω is a non-decreasing sequence that goes to infinity.It is enough to show that (φ_-,φ_+) witnesses (b,h)[κ]^ℵ_0,[κ]^ℵ_0,⊉, that is, for any S∈(b,h) and c∈[κ]^ℵ_0, if φ_-(S)⊉ c then S∌^*φ_+(c). Choose l∈ c∖φ_-(S). Hence φ_+(c)(i)∉ S(i) for any i≥ N, where N is some natural number such that l=c_j_0 for some j_0< N and {c_j:j≤ j_0}⊆ b'(N).* If b(i) is infinite for all but finitely many i<ω then_b,h=max{([κ]^ℵ_0),()} and _b,h=min{([κ]^ℵ_0),()}where κ=lim inf_i<ω{|b(i)|}. In particular, if κ=ω then _b,h=() and _b,h=(); otherwise, if κ is uncountable then _b,h=ℵ_1.* If b(i) is infinite for infinitely many i<ω then_b,h=min{([λ]^ℵ_0),_ω,h} and _b,h=max{([λ]^ℵ_0),_ω,h}where λ=lim sup_i<ω{|b(i)|}. In particular, if λ=ω then _b,h=_ω,h and _b,h=_ω,h; if λ is uncountable and h goes to infinity then _b,h=ℵ_1.We first show (a). Wlog, we may assume that b(i) is an infinite cardinal for all i<ω. Find a non-decreasing function b_0 from ω into the infinite cardinals such that b_0(i)≤ b(i) for all i<ω, and sup_i<ω{b_0(i)}=κ. By Example <ref>(3) and Theorem <ref>, ()≤_b_0,h≤_b,h and _b,h≤_b_0,h≤(). On the other hand, by Lemma <ref>, ([κ]^ℵ_0)≤_b_0,h≤_b,h and _b,h≤([κ]^ℵ_0), so max{([κ]^ℵ_0),()}≤_b,h and _b,h≤min{([κ]^ℵ_0),()}.Now we show that _b,h≤max{([κ]^ℵ_0),_ω,h}. Let C⊆[κ]^ℵ_0 be a witness of ([κ]^ℵ_0). For each c∈ C choose a witness S_c⊆(b_c,h) of _b_c,h where b_c(i):=b(i)∩ c, and put S:=⋃_c∈ CS_c. It is clear that |S|≤max{([κ]^ℵ_0),_ω,h}, so it is enough to show that, for any x∈∏ b, there is some φ∈ S such that x∈^∞ S. As x is countable, there is some c∈ C such that x⊆ c. Hence x∈∏ b_c and there is some φ∈ S_c such that x∈^∞φ.To see that min{([κ]^ℵ_0),()}≤_b,h, assume that Y⊆∏ b has size less than this minimum and show that there is some φ∈(b,h) such that y∈^∞φ for every y∈ Y. For each y∈ Y choose a c_y∈[κ]^ℵ_0 such that y∈∏ b_c_y. As |Y|<([κ]^ℵ_0), there is some c^*∈[κ]^ℵ_0 such that ⋃_y∈ Yc_y⊆ c^*([For this to happen, it is clear that either κ=ω or Y is countable.]), so Y⊆∏ b_c^*. Hence, as |Y|<()=_b_c^*,h, there is some φ∈(b_c^*,h) as desired.To finish, we show (b). As b≤^*λ, by Example <ref>(3) _λ,h≤_b,h and _b,h≤_λ,h. A similar argument as in (a) guarantees that min{([λ]^ℵ_0),_ω,h}≤_λ,h and _λ,h≤max{([λ]^ℵ_0),_ω,h}.Find an increasing function ι:ω→ω such that |b(ι(i)|:i<ω is a non-decreasing sequence of infinite cardinals and sup_i<ω{|b(ι(i))|}=λ. Put b':=b∘ι and h'=h∘ι. By Example <ref>(4), (5) and Lemma <ref>, _b,h≤_b',h'≤_b',h'≤([λ]^ℵ_0) and ([λ]^ℵ_0)≤_b',h'≤_b',h'≤_b,h, so it remains to show that _b,h≤_ω,h and _ω,h≤_b,h. This is clear by Lemma <ref> when h does not go to infinity; if h goes to infinity then so does h' and, by Theorem <ref> and Example <ref>(3), _b,h≤_b',h'≤_ω,h'=_ω,h and _ω,h=_ω,h'≤_b',h'≤_b,h. In Theorem <ref> the role of h is not too relevant. In (a) we actually have _b,h=_b,1 and _b,h=_b,1. In (b), when h goes to infinity _b,h=min{([λ]^ℵ_0),()}=_b,𝕀_ω and _b,h=max{([λ]^ℵ_0),()}=_b,𝕀_ω. When h does not go to infinity _b,h=_b,1 (see Theorem <ref> and Corollary <ref>(b)) and, although _b,h depends on h, it is a finite number already calculated in Theorem <ref>. The previous result implies that _ω_n,1=max{ℵ_n,()}, _ω_n,𝕀_ω=max{ℵ_n,()}, and _ω_n,1=max{ℵ_n,} for any n<ω. For larger κ, the cardinal ([κ]^ℵ_0) is quite special, e.g., large cardinals are necessary to prove the consistency of ([κ]^ℵ_0)>κ for some κ of uncountable cofinality (for more on this, see e.g. <cit.>).When b is above the continuum, _b,h and _b,h are simpler to calculate. * If ∀^∞ i<ω(|b(i)|≥) then _b,h=(lim inf_i<ω{|b(i)|})^ℵ_0. * If ∃^∞ i<ω(|b(i)|≥) then _b,h=(lim sup_i<ω{|b(i)|})^ℵ_0.It is a direct consequence of Theorem <ref> and the fact that ([κ]^ℵ_0)=κ^ℵ_0 whenever κ≥. This is clear for κ=. If κ>, as |(c)|= for any c∈[κ]^ℵ_0, no cofinal family in [κ]^ℵ_0 can have size <|[κ]^ℵ_0|=κ^ℵ_0. The cases that are not characterized in Theorem <ref> are when b(i) is finite for all (but finitely many) i<ω for the localization cardinals, and when b(i) is finite for infinitely many i<ω for the anti-localization cardinals. Even more, by the following result (for the case κ=ω), the latter case is reduced to the case when b(i) is finite for all i<ω. Let κ be an infinite cardinal. If the set F:={i<ω:|b(i)|<κ} is infinite and ι:ω→ F is the increasing enumeration of F, then _b,h=_b∘ι,h∘ι and _b,h=_b∘ι,h∘ι. If ω∖ F is finite, it is clear that (b,h)(b∘ι,h∘ι). Otherwise, when ω∖ F is infinite, (b,h)(b∘ι,h∘ι)∧(b∘ι',h∘ι') where ι':ω→ω∖ F is the increasing enumeration of ω∖ F. On the other hand,_b∘ι,h∘ι≤_κ,h∘ι=_κ,h∘ι'≤_b∘ι',h∘ι' by Example <ref>(3) and Theorem <ref>. In a similar way, _b∘ι',h∘ι'≤_b∘ι,h∘ι Hence, the result follows by Theorem <ref>.The cardinal _b,h is also simply characterized when |b(i)|≥ for infinitely many i<ω, but in this case it does not matter whether h goes to infinity or not.If ∃^∞ i<ω(|b(i)|≥) then _b,h=min{|∏_i≥ n b(i)|:n<ω}.Put κ:=lim sup_i<ω{|b(i)|}. Without loss of generality we may assume that sup_i<ω{|b(i)|}=κ and that, in case |b(i)|=κ for some i, then ∃^∞ i<ω(|b(i)|=κ) ([By finitely many modifications of b and h, the localization and anti-localization cardinals do not change, see Example <ref>(3).]). This implies that |∏ b|=min{|∏_i≥ n b(i)|:n<ω}. It is clear that _b,h≤|(b,h)|=|∏ b|. To see the converse inequality, split into the following cases. Case I. κ<|∏ b|.Note that, for any φ∈(b,h), there are at most max{,κ}=κ many members of ∏ b that are localized by φ, so no subset of (b,h) of size <|∏ b| is a witness of _b,h. Case II.κ=|∏ b|. This implies that b(i)=κ for some i<ω (otherwise, we would be in Case I by König's theorem). Hence ∃^∞ i<ω(|b(i)|=κ) and κ^ω=κ. If S⊆(b,h) has size <κ, ⋃_φ∈ Sφ(i) has size ≤|S|·ω<κ for every i<ω. Therefore, when |b(i)|=κ, there is some x(i)∈ b(i)∖⋃_φ∈ Sφ(i). If we define x(i)∈ b(i) arbitrarily when |b(i)|<κ, then no slalom from S localizes x.In the context of cardinal invariants of the reals, as ∏ b gives us a Polish space when each b(i) is countable, for our main results we are mainly interested in localization (and anti-localization) cardinals for b:ω→(ω+1)∖{0}. The following result justifies that the study of these cardinals can be reduced to the cases b∈ω^ω or b=ω.Let κ:=lim sup_i<ω{|b(i)|} and assume that it is infinite. If ∃^∞i<ω(|b(i)|=κ) then _b,h=_κ,h and _b,h=_κ,h. Even more, if h goes to infinity then (b,h)(b,𝕀_ω)(κ,𝕀_ω), so in particular _b,h=_b,𝕀_ω=_κ,𝕀_ω and _b,h=_b,𝕀_ω=_κ,𝕀_ω.By Example <ref>(3) and (4) it is clear that (κ,h∘ι)(b,h)(κ,h) when ι:ω→ω is the increasing enumeration of the set {i<ω:b(i)=κ}, so _κ,h≤_b,h≤_κ,h∘ι and _κ,h∘ι≤_b,h≤_κ,h. When h does not go to infinity, _b,h=_κ,h (a finite number that depends only on h) and _b,h=_κ,h=κ^ℵ_0 by Lemmas <ref> and <ref>. To complete the proof, it is enough to show that (κ,h)(κ,𝕀_ω) when h goes to infinity. Choose an increasing function h'∈ω^ω such that h≤ h', and choose another increasing function g:ω→ω such that h(i)≥ h'(n) for all i≥ g(n). Put I_n:=[g(n),g(n+1)) and b'(n):=κ^I_n, which has size κ. The functions φ_-:κ^ω→∏ b' and φ_+:(b',h')→(κ,h), defined by φ_-(x):= x I_n:n<ω and φ_+(S)(i):={t(i):t∈ S(n)} when i∈ I_n, witnesses that (κ,h)(b',h')(κ,h'). On the other hand, (κ,h')(κ,h) by Example <ref>(3), so both relational systems are Tukey-Galöis equivalent. As h'≥𝕀, the same argument above guarantees that (κ,h')(κ,𝕀_ω). Note that if κ:=lim sup_i<ω{|b(i)|} has uncountable cofinality then |b(i)|=κ for infinitely many i<ω, so its localization cardinals are the same as for κ.If κ has uncountable cofinality then _κ,h=sup({κ}∪{_μ,h:μ<κ cardinal}). In particular, _ω_n,h=max{ℵ_n,_ω,h} for any n<ω. By Example <ref>(3), _μ,h≤_κ,h for any μ<κ. On the other hand, κ≤_κ,h (this fact does not depend on the cofinality of κ). Indeed, if S⊆(κ,h) is a family of slaloms of size <κ, for each i<ω there is some x(i)<κ not in ⋃_φ∈ Sφ(i), so x is not localized by the slaloms in S.For each infinite α<κ let S_α⊆(α,h) be a witness of _α,h and put S_κ:=⋃_α<κS_α. It is clear that S_κ has size at most sup({κ}∪{_μ,h:μ<κ cardinal}), so it is enough to show that any z∈κ^ω is localized by some slalom in S_κ. As κ has uncountable cofinality, there is some α<κ such that z∈α^ω, so z is localized by some slalom in S_α.As a consequence, the equality _ω_n,h=max{ℵ_n,_ω,h} can be easily proved by induction on n<ω. By Theorem <ref> and Lemmas <ref> and <ref>, for n<ω, _ω_n,h=max{ℵ_n,()} when h goes to infinity, otherwise _ω_n,h=max{ℵ_n,}. In particular, _ω_1,h=_ω,h.The cardinal _b,h is actually determined when b(i) is uncountable at infinitely many i.If h goes to infinity and infinitely many b(i) are uncountable, then _b,h=ℵ_1.Without loss of generality, assume that each b(i) is a cardinal number. For each α<ω_1 define x_α(i):=α whenever b(i) is uncountable, or 0 otherwise. It is clear that no single slalom from (b,h) localizes every x_α for all α<ω_1. Hence _b,h≤ℵ_1. On the other hand, when F⊆∏ b is countable, by a standard diagonalization argument (as h goes to infinity) it is possible to find a slalom in (b,h) that localizes all the members of F. Now, we look at limits of localization and anti-localization cardinals. Define the following cardinal characteristics. [ := min{_b,𝕀_ω:b∈ω^ω}, := sup{_b,𝕀_ω:b∈ω^ω},; := sup{_b,1:b∈ω^ω}, := min{_b,1:b∈ω^ω}. ] * min{_b,h:b∈ω^ω}= and sup{_b,h:b∈ω^ω}= when h goes to infinity. * sup{_b,h:b∈ω^ω}= and min{_b,h:b∈ω^ω}=.Denote by _h:= min{_b,h:b∈ω^ω}, _h:=sup{_b,h:b∈ω^ω}, _h:= sup{_b,h:b∈ω^ω}, and _h:=min{_b,h:b∈ω^ω}. To see (a) it is enough to show the following. For any b,h'∈ω^ω such that h≤ h' there is some b'∈ω^ω such that (b,h)(b',h') Choose an interval partition I_n:-1≤ n<ω of ω such that h(k)≥ h'(n) for all k∈ I_n (denote h'(-1):=0). Put b'(n):=∏_k∈ I_nb(k). Define φ_-:∏ b→∏ b' by φ_-(x):= x I_n:0≤ n<ω, and define φ_+:(b',h')→(b,h) by φ_+(S)(k)={s(k):s∈ S(n)} whenever k∈ I_n (put S(-1):=∅). It is easy to show that (φ_-,φ_+) is the required Tukey connection.Assume h≤ h'. The claim implies that _h'≤_b',h'≤_b,h and _b,h≤_b',h'≤_h' for any b∈ω. On the other hand, _h≤_h' and _h'≤_h by Example <ref>(3), so equality holds. Therefore, (a) follows by using an h' above both 𝕀_ω and h. Concerning item (b), we haveFor any b,h∈ω^ω with h≥^* 1 there is some b'∈ω^ω such that (b',h)(b,1). Let I_n:n<ω be the interval partition of ω such that |I_n|=h(n) for all n<ω. Put b'(n):=∏_k∈ I_nb(k). Define φ_-:(b',h)→∏ b such that, for any S∈(b',h), φ_-(S) satisfies that ∀ n<ω∀ t∈ S(n)∃ k∈ I_n(φ_-(S)(k)=t(k)) (which is fine because |S(n)|≤|I_n|). On the other hand, define φ_+:∏ b→∏ b' by φ_+(y):= y I_n:n<ω. It is clear that (φ_-,φ_+) is the Tukey connection we want.As in the proof of (a), the claim above can be used to prove (b). Other cardinals like sup{_b,h:b∈ω^ω} are in principle not that interesting, for instance, this supremum above would be _1,h, which is undefined, or at least if b is restricted to be above h, then it would be _h+1,h. Also, when h does not go to infinity, _h and _h are easily characterized by Theorem <ref>.The following is another characterization of () and () in terms of Localization cardinals for b∈ω^ω.()=min{,} and ()=max{,}. In this proof we use the characterization of () and () given in Theorem <ref> Assume that F⊆ω^ω and |F|<min{,}. Therefore, there is some d∈ω^ω such that, for every x∈ F, ∀^∞ i<ω(x(i)<d(i)). On the other hand, as |F|<_d,𝕀_ω, we can find an slalom in (d,𝕀_ω)⊆(ω,𝕀_ω) that localizes all the reals in F (just use a family F'⊆∏ d of the same size as F such that each member of F' is a finite modification of some member of F and viceversa). Therefore, ()≥min{,}).Now we prove ()≤max{,}. Choose a dominating family D⊆ω^ω of sizeand, for each d∈ D, choosea family of slaloms S_d⊆(d,𝕀_ω) that witnesses _d,𝕀_ω. Note that S:=⋃_d∈ DS_d⊆(ω,𝕀_ω) has size ≤max{,} and that every real in ω^ω is localized by some slalom in S, so ()≤|S|.It is easy to see that (ω,𝕀_ω), so ()≤ and ≤(). On the other hand, as ()≤_b,𝕀_ω and _b,𝕀_ω≤() for any b∈ω^ω (see Example <ref>(3)), the converse inequalities follow.The version of this lemma for the anti-localization cardinals is Miller's <cit.> known result ()=min{,} and its dual ()=max{,}, which is proved in Theorem <ref>. Miller also proved that ()=. At the end of Section <ref> we explain why no inequality between each pair of these cardinals can be proved in ZFC . §.§ Additivity and cofinality of Yorioka ideals Though Corollary <ref> and Theorem <ref> are stated in <cit.>, their proofs seem not to appear in any existing reference. We offer original proofs of both results, even more, we provide the following general result that gives us Corollary <ref> as a direct consequence. This is also used in Section <ref> to prove that, consistently, ()<(_f)<(_f)<(). Let f∈ω^ω increasing, b:ω→ω+1∖{0} and h∈ω^ω. If 2^f≪ b and ∃ l^*<ω(h≤^*𝕀_ω^l^*) then min{,_b,h}≤(ℐ_f)and (ℐ_f) ≤max{,_b,h}. Note that ∃ l^*<ω(h≤^*𝕀_ω^l^*) is equivalent to ∃ l^*<ω∀^∞ n(∑_k≤ nh(k)≤ n^l^*). We use the latter assertion in this proof. Fix a bijection r^*:ω→2^<ω such that, for any i,j<ω, i≤ j implies |r^*_i|≤|r^*_j|. By Theorems <ref> and <ref>, it is enough to show that⟨_f, _f, ⊆⟩(⟨ω^↑ω,ω^↑ω,≤^*⟩;(b,h)).Fix X∈_f. There exists a σ∈(2^<ω)^ω with h_σ≫ f such thatX⊆ [σ]_∞. Define h_0∈ω^ω such that h_0(i):=min{⌊log_2b(i)⌋,h_σ(i)}. Note that h_0≫ f+1 because 2^f≪ b. Hence, by Lemma <ref>, there is ad_X∈ω^↑ω such that g^f_d_X+1≤^* h_0. Defineσ_X(i):={[σ(i) g^f_d_X(i) if g^f_d_X(i)+1≤ h_0(i),;otherwise. ].Clearly [σ]_∞⊆ [σ_X]_∞ and, for some N_X<ω, g^f_d_X(i)+1≤⌊log_2b(i)⌋ for all i≥ N_X, so 2^g^f_d_X(i)+1≤ b(i). Therefore 2^≤ g^f_d_X(i)⊆{r^*_j:j<b(i)} for any i≥ N_X. Now, define F_X:ω^↑ω→∏ b byF_X(e)(i):={[j if g^f_e(i) ≤g^f_d_X(i), i≥ N_X, and r^*_j:=σ_X(i) g^f_e(i),;0 otherwise. ].Note that, if e≥^* d_X, then [r^**_F_X(e)]_∞∈_g_e^f⊆_f where r^**_F_X(e):= r^*_F_X(e)(i):i<ω. Besides [σ_X]_∞⊆[r^**_F_X(e)]_∞. Define φ_-:_f→ω^↑ω× (∏ b)^ω^↑ω by φ_-(X):=(d_X,F_X).Fix e∈ω^↑ω and S∈(b,h). Put S_e(n)={j∈ S(n):|r^*_j|=g_e^f(n)} for each n<ω. If ∃^∞n<ω(S_e(n)≠∅)find the interval partition ⟨ I^S_e_n : n<ω⟩ of ω such that |I^S_e_n|=|S_e(n)| and enumerate S_e(n)={j^S_e_i:i ∈ I^S_e_n}. Define τ_e,S∈ (2^<ω)^ω by τ_e,S(i):=r^*_j^S_e_i when i∈ I^S_e_n. We show that [τ_e,S]_∞∈_f, that is, h_τ_e,S≫ f. Choose l^*<ω such that ∀^∞ n(∑_k≤ nh(k)≤ n^l^*) and fix k<ω. Note that ∀ n≥ e(l^*)(f(n^l^*)≤ g_e^f(n)). For i ∈ I^S_e_n with n≥ e(l^*) large enough, i<∑_k≤ n|S_e(k)|≤∑_k≤ nh(k)≤ n^l^*. Hence, for all i∈ I^S_e_n with n≥ e(k· l^*) large enough, h_τ_e,S(i)=|τ_e,S(i)|=g^f_e(n)≥ f(n^k· l^*)≥ f(i^k).Define φ_+:ω^↑ω×∏ b→_f by φ_+(e,S):=[τ_e,S]_∞ if ∃^∞n<ω(S_e(n)≠∅), or φ_+(e,S):=∅ otherwise. It remains to show that, if X∈_f, e∈ω^↑ω, S∈(b,h), d_X≤^*e, and F_X(e)∈^*S, then X⊆ [τ_e,S]_∞. For large enough n<ω, as F_X(e)(n)∈ S(n),F_X(e)(n)=j^S_e_i_n for some i_n∈ I_n^S_e by the definition of S_e(n). This implies that [r^**_F_X(e)]_∞⊆[τ_e,s]_∞. On the other hand, d_X≤^* e implies that X⊆[σ_X]_∞⊆[r^**_F_X(e)]_∞, so X⊆[τ_e,s]_∞. If f∈ω^ω is increasing then ()≤(_f) and (_f)≤().Apply Theorem <ref> with b=ω and h=𝕀_ω. In the previous result we did not use the fact that _f is closed under unions, so it implies that _f is a σ-ideal (see Theorem <ref>). (ℐ_f)≤ and ≤(ℐ_f). Even more, ⟨ω^ω,ω^ω,≤^*⟩≼_T⟨_f,_f,⊆⟩.Let f,f^'∈ω^ω be increasing. If ∀^∞n<ω(f(n+1)-f(n)≤ f^'(n+1)-f^'(n)) then (_f)≥(_f^') and (_f)≤(_f^'). In particular(_f)≤(_id_ω) and (_id_ω)≤(_f). To fix some notation, for each h∈ω^ω define Δ h(0):= h(0) and Δ h(n+1):=h(n+1)-h(n) for all n<ω. By Lemma <ref>, we can assume wlog f(0)≤ f^'(0) and ∀n<ω(f(n+1)-f(n)≤ f'(n+1)-f'(n)), that is, Δ f(n)≤Δ f'(n) for all n<ω. Clearly, m<n implies f(n)-f(m)≤ f'(n)-f'(m) (it can be easily proved by induction on n). As a result f≤ f^', Δ g_d^f(n)≤Δ g_d^f'(n) for all n<ω and g_d^f≤ g_d^f' for any d∈ω^↑ω. By Theorems <ref> and <ref>, it is enough to show _f,_f,⊆(𝐃'; ) where 𝐃':=⟨ω^↑ω,ω^↑ω,≤^*⟩ and :=⟨_f',_f',⊆⟩.Fix X∈_f. There are d_X∈ω^↑ω and σ_X∈(2^<ω)^ω such that X⊆[σ_X]_∞ and h_σ_X=g^f_d_X. For each e∈ω^↑ω and n<ω, define τ_X,e(n)∈2^g_e^f'(n) according to the following cases: if g_e^f(n)≤ d^f_X(n), put ([Intuitively, τ_X,e(n) is the sequence formed by inserting blocks of 0's inside σ_X(n) in the following way: for each k≤ n, insert a block of 0's of length Δ g^f'_e(k)-Δ g^f_e(k) between σ_X(n)(g^f_e(k)-1) and σ_X(n)(g^f_e(k)) (when k=0 and g^f_e(0)=0, just insert the corresponding block behind σ_X(n)(0), and when k=n, just insert the block after σ_X(n)(g^f_e(n)-1)).])τ_X,e(n)(i):={[ σ_X(n)(i-g^f'_e(m-1)+g^f_e(m-1)) if 0≤ i-g^f'_e(m-1)<Δ g^f_e(m);for some m≤ n,;0otherwise ].(consider g^f_e(-1)=g^f'_e(-1):=0 and also note that such m is unique with respect to i because g^f'_e(m-1)≤ i <Δ g^f_e(m)+g^f'_e(m-1)≤ g^f'_e(m)), otherwise τ_X,e(n)(i) is just some fixed member of 2^g_e^f'(n). Note that h_τ_X,e=g_e^f', so [τ_X,e]_∞∈_g_e^f'⊆_f'. Define F_X:ω^↑ω→_f' by F_X(e):=[τ_X,e]_∞ and φ_-:_f→ω^↑ω× (_f')^ω^↑ω by φ_-(X):=(d_X,F_X).Fix e∈ω^↑ω and Y∈_f'. There are e^*≥ e in ω^↑ω and τ'_Y,e∈ (2^<ω)^ω with h_τ'_Y,e=g_e^*^f' such that Y⊆[τ'_Y,e]_∞. Define ρ_Y,e:ω→2^<ω by ρ_Y,e(n)(i)=τ'_Y,e(n)(i+g^f'_e(m-1)-g^f_e(m-1)) when g^f_e(m-1)≤ i<g^f_e(m) ([Intuitively, ρ_Y,e(n) is the sequence that results by cutting from τ'_Y,e(n) the blocks that are indexed by [g^f'_e(k-1)+Δ g^f_e(k),g^f'_e(k)) (for k<ω).]). Note that|ρ_Y,e(n)|=g^f^_e(m) if g^f^'_e^*(n)≥ g^f^'_e(m-1)+Δ g^f^_e(m)g^f_e(m-1)+g^f'_e^*(n)-g^f'_e(m-1)otherwise,where m is the minimum natural number such that g^f'_e^*(n)≤ g^f'_e(m). Clearly, m⩽ n, so |ρ_Y,e(n)|≤ g_e^f(n).[ρ_Y,e]_∞∈_f. It is enough to show that h_ρ_Y,e≫ f.Let k_0=-1 and k_m+1=min{k<ω: ∀ n>k(g^f'_e^*(n)>g^f'_e(m))}. Note that {k_m}_m<ω is a monotone increasing sequence that goes to +∞. Put I_m=(k_m,k_m+1]. Note that, for n∈ I_m, m=min{k<ω:g^f'_e^*(n)≤ g^f'_e(k)} so |ρ_Y,e(n)|≥ g_e^f(m-1).Fix c>0. Find N>e^*(3c) such that, for all m≥ N, k_m>e^*(3c) and (m-1)^2≥ m. If m≥ N^3c and m∈ [e(k),e(k+1)) then k_m+1≤ m^k+10/3c. If n>m^k+10/3c then n^3c>m^k+10, so g^f'_e^*(n)≥ f'(n^3c)> f'(m^k+10)=g^f'_e(m) because e^*(3c)<N≤ N^k+10≤ m^(k+10/3c)< n. Fix n>k_N^3c, so n∈ I_m for some m≥ N^3c. By the subclaim, n ≤ k_m+1≤ m^k+10/3c where k satisfies m∈ [e(k),e(k+1)). Thus h_ρ_Y,e(n)≥ g_e^f(m-1)≥ f((m-1)^k+9)≥ f(n^c) since (m-1)^ k+9≥ m^ k+9/2 and k+9/2≥ k+10/3, so (m-1)^ k+9≥ m^ k+10/3= m^( k+10/3c)c≥ n^c. Define φ_+:ω^↑ω×_f'→_f by φ_+(e,Y)=[ρ_Y,e]_∞.To finish the proof we need to check that, for X∈_f, Y∈_f' and e∈ω^↑ω, if d_X≤^*e and F_X(e)⊆ Y then X⊆[ρ_Y,e]_∞. Fix x∈ X⊆[σ_X]_∞, i.e. ∃^∞n<ω(σ_X(n)⊆ x). Define x'∈2^ω by ([Intuitively, x' results by inserting (infinitely many) blocks of 0's inside x like in the definition of τ_X,e.])x'(i):={[ x(i-g^f'_e(m-1)+g^f_e(m-1)) if 0≤ i-g^f'_e(m-1)<Δ g^f_e(m),; 0otherwise. ]. Clearly, x'∈[τ_X,e]_∞=F_X(e)⊆ Y⊆[τ'_Y,e]_∞. As τ'_Y,e(n)⊆ x' implies ρ_Y,e(n)⊆ x, we conclude that x∈[ρ_Y,e]_∞. We also look at the following cardinal invariants related to Yorioka ideals:[ = min{(ℐ_f):f ∈ω^ω increasing},; = sup{(ℐ_f):f ∈ω^ω increasing},; = min{(ℐ_f):f ∈ω^ω increasing}, and; = sup{(ℐ_f):f ∈ω^ω increasing}. ]It is not necessary to refer to , , supnon andas they are (_𝕀_ω), (_𝕀_ω), (_𝕀_ω) and (_𝕀_ω), respectively. This follows from Theorem <ref> and the fact that 2^ω,_f,∈2^ω,_f',∈ when f≤^* f'. Even more, =⋂{_f:f∈ω^ω increasing} implies ≤(), ≤(), =() and ()≤()^=2^ (see <cit.>). It is already known from <cit.> that ()=, so =() also follows from Theorem <ref>.§.§ Anti-localization cardinals If ∀^∞ i<ω(|b(i)|≥) then _b,h=min_c∈[ω]^ℵ_0{|∏_i∈ cb(i)|}.Consider the following cases. Case I. For any cardinal μ the set {i<ω:|b(i)|=μ} is finite. Hence ∏ b=∏_i∈ cb(i) for any infinite subset c of ω and _b,h≤|∏ b|. Find a monotone increasing function b' from ω into the class of non-zero cardinals such that b'(i)≤|b(i)| for all i<ω and κ:=sup_i<ω{b(i)}=sup_i<ω{|b(i)|}. Hence (b,h)(b',h) and |∏ b'|=|∏ b|=κ^ℵ_0, so it is enough to prove that |∏ b'|≤_b',h. By identifying b'(n) with ∏_i≤ nb'(n), we can find a set E⊆∏ b' of pairwise eventually different functions of size κ^ℵ_0.For any φ∈(b',h) and a∈[ω]^ℵ_0, the set F_φ,a:={y∈ E:∀ i∈ a(y(i)∈φ(i))} has size ≤.Assume that |F_φ,a|≥^+. Choose g:[E]^2→ω such that, for two different y,z∈ E, y(i)≠ z(i) for all i≥ g({y,z}). By Erdős-Rado Theorem there is an uncountable F⊆ F_φ,a and an n<ω such that g({y,z})=n for any different y,z∈ F. But this implies that |F|≤ h(i) for any i∈ a above n, a contradiction. If ⊆(b',h) has size <κ^ℵ_0 then the set of members of E that are not anti-localized by some member ofis equal to ⋃_φ∈⋃_a∈[ω]^ℵ_0F_φ,a. As this union has size ||·<κ^ℵ_0, there is some y∈ E that is anti-localized by every member of . Case II. There is some cardinal κ such that |b(i)|=κ for infinitely many κ. Even more, assume that κ is the minimal such cardinal. Hence, by Lemma <ref> (applied to κ^+), _b,h=_κ,h=_κ,1. Note that min_c∈[ω]^ℵ_0{|∏_i∈ cb(i)|}=κ^ℵ_0, so it is enough to show that _κ,1=κ^ℵ_0. The inequality ≤ is clear. For the converse, when κ<κ^ℵ_0 we can proceed as in Case I using the constant function κ in place of b' (and it is not necessary to use Erdős-Rado Theorem for h=1 because F_φ,a would have size at most 1); when κ^ℵ_0=κ a similar argument to Case II of the proof of Lemma <ref> works. §.§ Covering and uniformity of Yorioka ideals The following results shows a relationship between the cardinals of the relational systems of the form 2^ω,_g,∈ and (b,h). Let b ∈ω^ω with b≥^* 2. If g∈ω^ω and g ≥^* (log b)^+, then _b,1≤(_g) and (_g)≤_b,1. Moreover, (b,1)^⊥⟨ 2^ω, 𝒥_g,∈⟩.Let h,b∈ω^ω and g∈ω^ω monotone increasing. If 1≤^* h≤^* b and b≥^* 2^g∘(h^+-1) then ⟨ 2^ω, 𝒥_g,∈⟩(b,h)^⊥. In particular, (𝒥_g)≤_b,h and _b,h≤(𝒥_g). As a consequence. the cardinalsandcan be characterized as follows. = and =. We first prove that, for all b∈ω^ω, _b,1⩽ and ≤_b,1. Wlog b≥ 2 (By Example <ref>(3)). Put f=(log b)^+. As _f⊇_f, (_f)⩽(_f) and (_f)⩽(_f). By Lemma <ref>, _b,1≤(_f) and (_f)≤_b,1.To prove the converse inequalities, assume that f∈ω^ω is increasing. Choose some g≫ f monotone increasing, so_g⊆_f. Hence (_f)⩽(_g) and (_g)≤(_f). Put b=2^g. By Lemma <ref>, (_g)≤_b,1 and _b,1≤(_g), so (_f)≤_b,1≤ and ≤_b,1≤(_f). On the other hand, by Example <ref>(3) and the previous result, we have≤() and ()≤. In particular (_f)≤() and ()≤(ℐ_f) for every increasing f∈ω^ω. We know that (ℳ)= min{,(ℳ) } and (ℳ)= max{,(ℳ) }, but these equalities can be refined as in the following two results. These yield a version of Lemma <ref> for the anti-localization cardinals.(ℳ)= min{,}.(ℳ)=max{,} The inequality ≥follows from Corollary <ref>.We prove ≤. Let D⊆ω^↑ω be a dominating family of size . By Theorem <ref>, =. For each d ∈ D let E_d⊆∏ d be a witness of _d,1. Then E=⋃_d ∈ DE_d satisfies ∀x∈ω^ω∃ y∈ E(x=^∞y)so, by Theorem <ref>, (ℳ)≤|E|≤·. Hence (ℳ)=max{,(ℳ) }≤max{,}. As a consequence, by Theorems <ref>, <ref> and <ref>, ≤(ℳ) and (ℳ)≤ Figure <ref> summarizes some results of this section. As an application, Yorioka's characterization of () can be reformulated as follows.If ==κ then ()=_κ. In particular, ()=()=κ implies ()=_κ. § PRESERVATION PROPERTIES The preservation properties discussed in this section were developed for FS iterations of ccc posets by Judah-Shelah <cit.> and Brendle <cit.>, later generalized and summarized in <cit.>, <cit.> and <cit.>. We generalize this theory so that preservation properties as in <cit.> (see Example <ref>) become particular cases. Afterwards, we show how to adapt this theory to preserve unbounded reals along FS iterations, which is useful in the context of matrix iterations. §.§ The presevation theory Our notation is closer to <cit.>. The classical preservation theory of Judah-Shelah and Brendle corresponds to the case |Ω|=1 of the definition below. Though the proofs of the facts in this section follow the same ideas as the classical results, the arguments are presented for completeness.Say that =⟨ X,Y,⊏⟩ is a generalized Polish relational system (gPrs) if (I) X is a Perfect Polish space,(II) Y=⋃_e∈ΩY_e where Ω is a non-empty set and, for some Polish space Z, Y_e is non-empty and analytic in Z for all e∈Ω, and(III) ⊏=⋃_n<ω⊏_n where ⟨⊏_n: n<ω⟩is some increasing sequence of closed subsets of X× Z such that, for any n<ω and for any y∈ Y, (⊏_n)^y={x∈ X:x⊏_ny } is closed nowhere dense. If |Ω|=1, we just say thatis a Polish relational system (Prs).For a set A and x∈ X say that x is -unbounded overA if ∀ y ∈ A∩ Y(x⊏̸y). Fix, throughout this section, a gPrs =⟨ X,Y,⊏⟩ as in the previous definition. ⟨ X,(X),∈⟩. In particular, ()≤() and ()≤().Let φ_-=𝕀_X and φ_+:Y→(X) defined by φ_+(y)={x∈ X:x ⊏ y}. Clearly, the pair (φ_-,φ_-) witnesses ⟨ X,(X),∈⟩. Let θ be a cardinal. A family F⊆ X is θ--unbounded if for any E⊆ Y of size <θ there is an x∈ F which is -unbounded over E; say that F is strongly θ--unbounded if |F|≥θ and |{x∈ F:x⊏ y}|<θ for all y∈ Y. For θ≥2, any θ--unbounded family is -unbounded and, for θ regular, if F is a strongly θ--unbounded family then it is |F|--unbounded. In consequence, * If θ≥2 and F⊆ X is θ--unbounded then ()≤|F| and θ≤().* If θ is regular and F⊆ X is strongly θ--unbounded then ()≤|F|≤().The following are useful properties to preserve (strongly) θ--unbounded families in forcing generic extensions. In this context, X,Z and ⊏ are interpreted in transitive models of ZFC as Polish spaces, while Y is interpreted as Y^M=⋃_e∈ΩY^M_e for such a model M containing the information to define Y. As in the case of Polish spaces, we also omit the upper indices M on the interpretation of Y. Letbe aforcing notion and θ a cardinal. *is θ--good if, for any -name ḣ for a member of Y, there existsa non-empty H⊆ Y (in the ground model) of size <θ such that, for any x∈ X, if x is -unbounded overH then ⊩ x̸̸⊏ḣ.*isθ--nice if, for all e∈Ω and for any -name ḣ for a member of Y_e, there exists a non-empty H⊆ Y of size <θ such that, for any x∈ X, if x is -unbounded overH then ⊩ x̸̸⊏ḣ.Say thatis -good (-nice) if it is ℵ_1--good (ℵ_1--nice). Note that θ<θ^' implies that any θ--good poset is θ^'--good. Also, if ⋖ andis θ--good, thenis θ--good. Similar facts hold for niceness. It is clear that anyθ--good forcing notion is θ--nice. The converse holds in some cases as below.Let θ be a regular cardinal. If eitheris θ-cc or |Ω|<θ,thenis θ--nice iff it is θ--good. Assume thateither is θ-cc or |Ω|<θ. Let ḣ an -name for a member of Y. Choose a maximal antichain A inand { e_p :p∈ A}⊆Ω such thatp⊩ḣ∈ Y_e_p for all p∈ A. Put Γ:={e_p:p ∈ A}. By hypothesis, Γ has size <θ. For each β∈Γ define A_β={p∈ A:e_p=β}. As p⊩ḣ∈ Y_β for any p∈ A_β, we can find a -name ẏ_β of a member of Y_β such that p⊩ḣ=ẏ_β for any p∈ A_β.Asis θ--nice, for each β∈Γ there exists a non-empty H_β⊆ Y of size <θ that witnesses niceness for ẏ_β. Put H=⋃_β∈ΓH_β which has size <θ becauseθ is regular. Assume thatx∈ X is -unbounded over H. Given p∈ A, there is a β∈Γ such that p∈ A_β. As x∈ X is -unbounded over H_β, p⊩ x⊏̸ẏ_β and, on the other hand, p⊩ḣ=ẏ_β, so p⊩ x⊏̸ḣ. As A is a maximal antichain, ⊩ x⊏̸ḣ. Let θ be a regular cardinal, λ≥θ a cardinal and letbe a θ--good poset. * If F⊆ X is λ--unbounded, thenforces that it isλ̇'--unbounded where, in the -extension, λ̇' is the smallest cardinal ≥λ.* If (λ)≥θ and F⊆ X is strongly λ--unbounded then ⊩“if λ is a cardinal then F is strongly λ--unbounded".* If ()≥λ thenforces that ()≥λ̇'.* It is enough to consider sets of -names for members of Y of the form {ġ_α}_α<η for some η<λ. For α<η, let H_α⊆ Y of size <θ that witnesses the goodness offor ġ_α. Put H=⋃_α<ηH_α. As|H|<λ, there is some x∈ F -unbounded over H. Thus ⊩ x⊏̸ġ_α for any α<η. * Repeat the argument above with η=1 and find H. Hence ⊩{x∈ F:x⊏ġ_0}⊆⋃_h∈ H{x∈ F:x⊏ h}. That union has size <λ in the ground model. On the other hand, F is forced to have size ≥|λ|. * Consequence of (a) because V“X is λ--unbounded".We now aim to prove that θ--goodness is respected in FS iterations of θ-cc posets. Letbe a forcing notion and let ż be a -name for a real in ω^ω. Apair ( p_n_n<ω,g) is called an interpretation of ż inif g∈ω^ω and, for all n<ω, * p_n∈, p_n+1≤ p_n, and* p_n⊩ż n=g n.Say that this interpretation is below p∈ if, additionally, p_0≤ p.Assume thatis a poset , e∈Ω, f:ω^ω→ Y_e is a continuous function, ż is a -name for a real in ω^ω and ( p_n_n<ω,g) is an interpretation of ż in . If x∈ X, n<ω and x⊏̸_n f(g), then there is a k<ω such that p_k⊩ x ⊏̸_n f(ż). As {y∈ Y_e:x⊏_ny} is closed in Y_e (see Definition <ref>) and f:ω^ω→ Y_e is continuous, f^-1[{y∈ Y_e:x⊏_ny}] is closed in ω^ω. Define C_x:={w∈ω^ω:x⊏_nf(w)} and note that f^-1[{y∈ Y_e:x⊏_ny}]=C_x. If x⊏̸_n f(g) then there is a k<ω such that [g k] ∩ C_x=∅. On the other hand p_k⊩ż k=g k, so p_k⊩ [ż k] ∩ C_x=∅. Hence p_k⊩ż∉ C_x, that is, p_k⊩ x⊏̸_n f(ż). If θ is a cardinal then any poset of size <θ is θ--nice. Moreover, if θ is regular then any such poset is θ--good. In particular,is -good. Put ={p_α:α<μ} where μ:=||<θ. Let e∈Ω and ḣ be a -name for a member of Y_e. Choose a continuous and surjective function f:ω^ω→ Y_e and a -name for a real ż in ω^ω such thatforces that f(ż)=ḣ. For each α<μ, choose an interpretation ( p_α,n_n<ω,z_α) of ż below p_α. We prove that, if x∈ X and ∀α<μ(x⊏̸f(z_α)), then ⊩ x⊏̸ḣ. Fix p∈ and m<ω, so there exists an α<μ such that p=p_α. ByLemma <ref> there exists a k<ω such that p_α,k⊩ x⊏̸_m f(ż). Therefore p_α,k⊩ x⊏̸_mḣ and p_α,k≤ p_α=p.The `moreover' part follows by Lemma <ref>.Let θ be a regular cardinal,a poset anda -name for a poset. Ifis θ-cc, θ--good and it forces thatis θ--good, then ∗ is θ--good Let ḣ be a ∗-name for a member of Y. Wlogforces that ḣ is a -name for a member for Y. Asforces thatis θ--good,forces that there is a nonempty Ḣ⊆ Y of size <θ such that, for any x∈ X, if x -unbounded over Ḣ, then ⊩_ x ⊏̸ḣ. Asis θ-cc, we can find ν<θ in the ground model so that Ḣ={ẏ_α:α<ν}. For each α<ν let B_α be a witness of goodness offor ẏ_α. Put B:=⋃_α<νB_α, which has size <θ. It is easy to see that, if x∈ X is -unbounded over B, then ⊩_∗x⊏̸ḣ.We show that goodness is preserved along direct limits in quite a general way so that the theory of this section can be applied to template iterations as in <cit.>. Say that a partial order I is directed if, for any i,j∈ I, there is a k∈ I such that i,j≤ k. A system⟨_i⟩_i∈ I of posets indexed by a directed partial order I is called a directed system of posets if _i is a complete subposet of _j for all i≤ j in I. For such a directed system, define its direct limit by _i∈ I_i:=⋃_i∈ I_i. Let θ be a regular cardinal, ⟨_i⟩_i∈ I a directed system of posets and :=_i∈ I_i. If |I|<θ and _i is θ--nice for all i∈ I, thenis θ--nice. Let e∈Ω and let ḣ be a -name for member in Y_e. Choose a continuous and surjective function f:ω^ω→ Y_e and a -name for a real ż in ω^ω such thatforces that f(ż)=ḣ. For each i∈ I, find a _i-name for a real ż_i in ω^ω and a sequence ⟨ṗ_i,k_k<ω of _i-names such that _i forces that (ṗ_i,k_k<ω,ż_i) is an interpretation of ż in /_i. Choose H_i⊆ Y of size <θ such that it witnesses goodness of _i for f(ż_i). Put H=⋃_i∈ IH_i, which has size <θ since |I|<θ and θ is regular.We prove that, if x∈ X is -unbounded over H, then ⊩_ x⊏̸ḣ. Assume towards a contradiction that there are p∈ and n<ω such that p⊩_x⊏_nḣ. Choose i∈ I such that p∈_i. Let G be a _i-generic over the ground model V with p∈ G. By the choice of H_i, x⊏̸f(ż_i[G]), in particular, x⊏̸_n f(ż_i[G]). ByLemma <ref>, there is a k<ω such thatṗ_i,k[G]⊩_/_ix⊏̸_nf(ż)=ḣ. On the other hand, by hypothesis, p⊩__i“⊩_/_ix⊏_n ḣ”, a contradiction. We can replace the hypothesis |I|<θ by (I)<θ in the previous result. This is because _i∈ I_i=_i∈ C_i for any cofinal C⊆ I.Let θ be an uncountable regular cardinal and ℙ_δ=⟨ℙ_α,ℚ̇_α⟩_α<δ a FS iteration of θ-cc forcing notions. If, for each α<δ, _α forces that ℚ̇_α is θ--good, then _δ is θ--good.We prove that _α is θ--good by induction on α≤δ. The step α=0 follows by Lemma <ref> and the successor step follows by Lemma <ref>. Assume that α is a limit ordinal. If (α)<θ then _α is θ--good by Theorem <ref> and Lemma <ref>. Assume that cf(α)≥θ. Let ḣ be a _α-name for a member of Y. By θ-cc-ness, there exists a ξ<α such that ḣ is a _ξ-name. As _ξ is θ--good, there is a non-empty H⊆ Y of size <θ that witnesses goodness of _ξ for ḣ. It is clear that H also witnesses goodness of _α for ḣ. Recall that c∈ X is a Cohen real over a model M if c does not belong to any Borel meager set coded in M. It is clear that Cohen forcing adds such a real over the ground model. Indeed, given a metric d on ω such that X, as a complete metric space, is a completion of ω,d, consider ^d:={t∈ω^<ω:∀i<|t|-1(d(t_i,t_i+1)<2^-(i+2))} ordered by end-extension. This is a countable atomless poset (because ω,d is perfect), so it is forcing equivalent to . It is not hard to see that ^d adds a Cauchy-sequence that converges to a Cohen real in X over the ground model.By Definition <ref>(III), any Cohen real in X over the ground model is -unbounded over the ground model. Hence, it is possible to add (strongly) -unbounded families with Cohen reals through FS iterations.If ν is a cardinal with uncountable cofinality and ℙ_ν=⟨ℙ_α,ℚ̇_α⟩_α<ν is a FS iteration of non-trivial (ν)-cc posets, then _ν adds a strongly ν--unbounded family of size ν. The Cohen reals (in X) added at the limit steps of the iteration form a strongly ν--unbounded family of size ν. Let θ be an uncountable regular cardinal, δ≥θ an ordinal, and let ℙ_δ=⟨ℙ_α,ℚ̇_α⟩_α<δ be a FS iteration such that, for each α<δ, _α is a _α-name of anon-trivial θ--good θ-cc poset. Then: * For any cardinal ν∈[θ,δ] with (ν)≥θ, _ν adds a strongly ν--unbounded family of size ν which is still strongly ν--unbounded in the _δ-extension.* For any cardinal λ∈[θ,δ], _λ adds a λ--unbounded family of size λ which is still λ--unbounded in the _δ-extension.* ℙ_δ forces that ()≤θ and|δ|≤(). (a) is a direct consequence of Lemmas <ref>, <ref> and the fact that _δ/_ν, the remaining part of the iteration from stage ν, is θ--good (by Corollary <ref>). On the other hand, by Lemma <ref>, ()≤θ follows from (a) for ν=θ and ()≥|δ| follows from (b) for λ=|δ|. It remains to prove (b) for the case when λ is singular (for λ regular it just follows from (a)). Work in V_λ=V^_λ. Let {δ_ξ:ξ<λ} be the increasing enumeration of 0 and the limit ordinals below λ and, for each ξ<λ, denote by c_ξ the Cohen real in X over V_δ_ξ added by _δ_ξ+1. As explained in the proof of Lemma <ref>, for each ν∈[θ,λ) regular, {c_ξ:ξ<ν}∈ V_ν is (strongly) ν--unbounded in V_ν, and also in V_λ by (a). Thus {c_ξ:ξ<λ} is λ--unbounded. Indeed, if A⊆ Y has size <λ then it has size <ν for some regular ν∈[θ,λ), so there is some ξ<ν such that c_ξ is -unbounded over A. As _δ/_λ is θ--good, then {c_ξ:ξ<λ} is still λ--unbounded in the _δ-extension by Lemma <ref>. §.§ Examples of preservation properties We start with examples of the classical framework, that is, with instances of Polish relational systems.* For every b:ω→ω+1∖{0} with b≰^*1, the relational system (b) (Example <ref>(3)) is a Prs. Indeed, ≠^*=⋃_n<ω≠^*_n where ≠^*_n is defined as x≠^*_n y iff x(i)≠ y(i) for all i≥ n. If b:ω→ω and ν<θ are infinite cardinals, then any ν-centered poset is θ-(b)-good (similar to Lemma <ref> when h=1).* For every b:ω→ω+1∖{0} and h∈ω^ω with h≥^*1, the relational system (b,h) (Definition <ref>) is a Prs. Indeed, ∌^*=⋃_n<ω∌^*_n where ∌^*_n is defined as φ∌^*_n y iff y(i)∉φ(i) for all i≥ n. When b:ω→ω, a similar proof as Lemma <ref> yields that any ν-centered poset is θ-(b,h)-good when ν<θ are infinite cardinals.* The relational system 𝐃=ω^ω,ω^ω,≤^* is a Prs. Clearly, any ω^ω-bounding poset is 𝐃-good. Miller <cit.> proved that , the standard σ-centered poset that adds an eventually different real in ω^ω, is 𝐃-good. A similar proof yields that ^h_b is 𝐃-good for any b:ω→ω∖{0} and h∈ω^ω with lim_n→+∞h(n)/b(n)=0. Likewise, ^h_b is -good for any b:ω→ω∖{0} and h∈ω^ω that goes to infinity.* For ℋ⊆ω^ω denote (ω,ℋ):=⟨ω^ω,(ω,ℋ),∈^*⟩ where(ω,ℋ):=⋃_h∈ℋ(ω,h).If ℋ is countable and non-empty then (ω,ℋ) is a Prs because (ω,ℋ) is F_σ in ([ω]^<ω)^ω. In addition, if ℋ contains a function that goes to infinity then ((ω,ℋ))=() and ((ω,ℋ))=() (as a consequence of Theorem <ref>). If ν<θ are infinite cardinals and θ is regular then any ν-centered poset is θ-(ω,ℋ)-good (<cit.>, see also <cit.>). Moreover, if all the members ofgo to infinity then any Boolean algebra with a strictly positive finitely additive measure is (ω,)-good (<cit.>). In particular, any subalgebra of random forcing is 𝐋𝐜(ω,)-good.Ifis non-empty, then (ω,) is a gPrs where Ω=ℋ, Z=([ω]^<ω)^ω and Y_h=(ω,h) for each h∈ℋ. Like in Example <ref>(4), if ν<θ are infinite cardinals, then any ν-centered poset is θ-(ω,)-nice (see <cit.>) and, by Lemma <ref>, it is θ-(ω,)-good when θ is regular (because any ν-centered poset is θ-cc). For any π,ρ,g_0∈ω^ω with π and g_0 going to +∞, there is a ≤^*-increasing sequence g_n:n<ω such that any (ρ,π)-linked poset is 2-(ω,{g_n:n<ω})-nice (hence (ω,{g_n:n<ω})-good). [Kamo and Osuga <cit.>] Fix a family ℰ⊆ω^ω of size ℵ_1 of non-decreasing functions which satisfies * ∀ e ∈ℰ(e ≤id_ω),* ∀ e ∈ℰ( lim_n→+∞e(n)=+∞ and lim_n→+∞(n-e(n))=+∞),* ∀ e ∈ℰ∃^e^'∈ℰ(e+1≤^* e^') and* ∀ℰ^'∈ [ℰ]^≤ℵ_0∃ e∈ℰ∀ e^'∈ℰ^'(e^'≤^*e).The existence of the family ℰ is a consequence of Lemma <ref> applied to H:=𝕀_ω+1 and g:=𝕀_ω.For b,h ∈ω^ω such that b>0 and h≥^*1, we define (b,h)=_ℰ(b,h) by(b,h):=⋃_e∈ℰ(b,h^e)={φ∈∏_n<ω𝒫(b(n)):∃ e∈ℰ∀ n<ω (|φ(n)|≤ h(n)^e(n))} Let n<ω. For ψ, φ :ω→[ω]^<ω, define the relation ψ▸_nφ iff ∀k≥ n(ψ(k)⊉φ(k)), and define ψ▸φ iff ∀^∞k<ω(ψ(k)⊉φ(k)), i.e., ▸=⋃_n<ω▸_n. Put (b,h):=⟨(b,h^id_ω), (b,h), ▸⟩ which is a gPrs where Ω=ℰ, Z=(b,h^𝕀_ω) and Y_e=(b,h^e) for each e∈ℰ. Note that Y_e is closed in Z.The property θ-(b,h)-good is what Kamo and Osuga <cit.> denote by (∗^<θ_b,h). However, they use (b,h) instead of (b,h^𝕀_ω) for the first coordinate of (b,h) (implicitly), which we think does not work for the second claim in the proof of <cit.>. We believe this can be corrected with the gPrs (b,h) we propose here.When h=1, (b,1)=(b,1),(b,1),▸(b), so (b,1) can be described as a Prs. Even more, ((b,1))=_b,1 and ((b,1))=_b,1. Define ^*(b,h):=∏ b,(b,h),∈^*. When h≥^*1 and b>^* h^e for any e∈ℰ, it is clear that ^*(b,h) is a gPrs.* (b,h)(b,h^𝕀_ω). In particular, _b,h^id_ω≤((b,h)) and ((b,h))≤_b,h^id_ω.* ^*(b,h)(b,h). In particular, _b,h≤(^*(b,h)) and (^*(b,h))≤_b,h. Put φ_-:=𝕀_(b,h^𝕀_ω) and ψ_-=𝕀_∏ b. Define φ_+:∏b→(b,h) by φ_+(y)(i)={y(i)} whenever h(i)>0, and, for some fixed e_0∈, define ψ_+:(b,h)→(b,h) by ψ_+(S)(i)=S(i) whenever e_0(i)>0. It is clear that (φ_-,φ_+) is a Tukey-Galois connection for (a), and (ψ_-,ψ_+) is one for (b). By the previous result, we can use the preservation property of (b,h) to decide the values of _b,h^id_ω and _b,h^id_ω by forcing, likewise for ^*(b,h) and the localization cardinals.Let n<ω and B⊆ be n-linked. If F∈ V has size ≤ n and ȧ is a -name for a member of F, then there exists a c ∈ F such that no p ∈ B forces ȧ≠ c.See e.g. <cit.>.If h,b ∈ω^ω with b>0, b≰^*1 and h≥^*1, then any (h,b^h^id_ω)-linked poset is both 2-(b,h)-good and 2-^*(b,h)-good.This was proved for the gPrs (b,h) in <cit.>. The case of ^*(b,h) follows from the same proof, which we include for completeness. Letbe a poset and let Q_n,j:n<ω, j<h(n) be a sequence that witnesses thatis (h,b^h^id_ω)-linked. Assume that φ̇ is a -name of a member of (b,h). Asis ccc (by Lemma <ref>), we can find an e∈ such thatforces that φ∈(b,h^e). Furthermore, choose e'∈ and N<ω such that h(n)· e(n)>0 and e(n)+1≤ e'(n) for all n≥ N, so we can find a -name φ̇' of a member of (b,h^e) such thatforces φ̇(n)⊆φ̇'(n)≠∅ for all n≥ N. For each n≥ N and j<h(n), as Q_n,j is b(n)^h(n)^n-linked and [b(n)]^≤ h(n)^e(n)∖{∅} has size ≤ b(n)^h(n)^n, by Lemma <ref> there is an a_n,j⊆ b(n) of size ≤ h(n)^e(n) such that p⊮ a_n,j≠φ̇'(n) for all p∈ Q_n,j. Note that ψ(n):=⋃_j<h(n)a_n,j has size ≤ h(n)^e(n)+1≤ h(n)^e'(n), which gives us a ψ∈(b,h^e'). It is clear that * if ϑ∈(b,h^𝕀_ω) and ϑ▸̸ψ then ⊩ϑ▸̸φ̇' (which implies ⊩ϑ▸̸φ̇), and * if x∈∏ b and (x∈^*ψ) then ⊩(x∈^*φ̇') (so ⊩(x∈^*φ̇)).This concludes the proof.If μ<θ are infinite cardinals then any μ-centered poset is both θ-(b,h)-nice and θ-^*(b,h)-nice. In addition, if θ is regular, then any μ-centered poset is both θ-(b,h)-good and θ-^*(b,h)-good. The latter part is a consequence ofLemma <ref>.Letbe a poset such that =⋃_α<μP_α where each P_α is centered. Fix e∈ℰ and a -name φ̇ for a member of (b,h^e). For each α<μ and m∈ω, by Lemma <ref> find a ψ_α(m)∈[b(m)]^≤(h(m)^e(m)) such that no p∈ P_α forcesψ_α(m)≠φ̇(m). Put H:={ψ_α:α<μ}, which is a subset of (b,h^e). Assume that ϑ∈(b,h^id_ω) and ϑ▸̸ψ_α for all α<μ. Fix p∈ and m<ω. Choose α<μ such that p∈ P_α and find a k≥ m such that ϑ(k)⊇ψ_α(k). As p⊮ψ_α(k)≠φ̇(k), there is a q≤ p that forces ϑ(k)⊇ψ_α(k)=φ̇(k).A similar argument yields that, whenever x∈∏ b and (x∈^*ψ_α) for any α<μ,forces that (x∈^*φ̇). The following example abbreviates many facts about the main preservation result in <cit.>. This will not be used in any other part of this text.[Brendle and Mejía <cit.>]Let a̅= a_i:i<ω be a partition of ω into non-empty finite sets and L̅= L_n:n<ω a partition of ω into infinite sets. For each i<ω let φ_i:(a_i)→[0,+∞) be a submeasure. Fix h≥^*1 in ω^ω and let b_a̅(i):=(a_i) for each i<ω. Define P_m(i):={a⊆ a_i:φ_i(a)≤ m} for each i,m<ω. Put Ω:=ω×ω×ℰ (ℰ as in Example <ref>) and, for each (m,n,e)∈Ω, put Y_m,n,e:={n}×(P_m,h^e), which is closed in Z:=ω×(b_a̅,h^𝕀_ω). For each k<ω define the relation ▸'_k⊆([ω]^<ω)^ω× (ω×([ω]^<ω)^ω)) by ϑ▸'_k(n,ψ) iff ϑ(i)⊉ψ(i) for all i∈ L_n∖ k. Put ▸':=⋃_k<ω▸'_k. It is not hard to see that (a̅,φ̅,L̅,h):=(b_a̅,h^𝕀_ω),Y,▸' is a gPrs where Y:=⋃_p∈ΩY_p. The property θ-(a̅,φ̅,L̅,h)-good was studied in <cit.> to preserve Rothberger gaps through FS iterations. Some of its results can be simplified by the theory presented in Subsection <ref>. Note that θ-(a̅,φ̅,L̅,h)-goodness corresponds to <cit.>.§.§ Preservation of -unbounded realsAll the results of this subsection are versions of the contents of <cit.> in the context of gPrs. Though the proofs are similar, we still present them for completeness. Fix, throughout this section, transitive models M and N of (a sufficient large finite fragment of) ZFC with M⊆ N. Given two posets ∈ Mand(not necessarily in M) say thatis a complete suborder ofwith respect to M, denoted by ⋖_M, ifis a suborder ofand every maximal antichain inthat belongs to M is also a maximal antichain in . Clearly, if ⋖_M and G is -generic over N, then G ∩ is -generic over M and M[G ∩]⊆ N[G]. Recall that, ifis a Suslin ccc poset coded in M, then ^M⋖_M^N. Also, if ∈ M is a poset, then ⋖_M.For the following results, fix a gPrs =⟨ X,Y,⊏⟩ coded in M (in the sense that all its components are coded in M).We are interested in preserving -unbounded reals between forcing extensions of M and N.Let be a Suslin ccc poset coded in M. If M“ is -good" then, in N, ^N forces that every c∈ X^N that is -unbounded over M is -unbounded over M^^M. Let Z' be the Polish space whereis defined, and recall the Polish space Z that contains Y (see Definition <ref>). Choose a metric space η,d with η≤ω such that Z, as a complete metric space with metric d^*, is a completion of η,d. Note that any (good) -name of a member of Z can be seen as a name of a Cauchy sequence k̇_m:m<ω in η,d such that d(k̇_m,k̇_m+1)<2^-(m+2) for all m<ω. This can be coded by a member of (Z'×η)^ω×ω. Therefore, “ḣ is a -name of a member of Z" is a conjunction between a ^1_1-statement and a ^1_1-statement in (Z'×η)^ω×ω. Indeed, a -name ḣ of a member of Z is a function ḣ=(p^ḣ,k^ḣ):ω×ω→ Z'×η such that, for each m<ω, * {p^ḣ(m,n):n<ω} is a maximal antichain in(the point is that p^ḣ(m,n) decides that the m-th term of the Cauchy-sequence in η,d converging to ḣ is k^ḣ(m,n)), and* for each n,n'<ω, if p^ḣ(m,n) and p^ḣ(m+1,n') are compatible inthend(k^ḣ(m,n),k^ḣ(m+1,n'))<2^-(m+2)(which means that ḣ is a name of a Cauchy-sequence as described before).The statement in (i) can be expressed as a conjunction between a ^1_1-statement and a ^1_1-statement, while (ii) is a ^1_1-statement. Even more, ifis a Borel subset of Z' then (i) is a ^1_1-statement.[<cit.>] The statement Ψ(ḣ,y̅) that says “ḣ is a -name for a member of Z and, for all x∈ X, if x⊏̸y_n for each n<ω then ⊩_ x⊏̸ḣ" is a conjunction of a ^1_1-statement with a ^1_1-statementin (Z^'×η)^ω×ω× Z^ω. Even more, ifis Borel in Z', then the statement is _1^1. It is just enough to look at the complexity of “⊩_x⊏̸ḣ". This is equivalent to say that “for every p∈ and l<ω there are positive rational numbers r,ε and k,m,n<ω such that p is compatible with p^ḣ(m,n), B(k,r)∩{z∈ Z:x⊏_l z}=∅ and d(k,k^ḣ(m,n))<r-2^-(m+1)-ε" where B(k,r):={z∈ Z:d^*(k,z)<r}. This statement can be written in the form ∀ p∈ Z(p∉ or Θ(p,x,ḣ)) where Θ(p,x,ḣ) is ^1_1 (recall that compatibility inis a ^1_1-relation in Z and that “B(k,r)∩{z∈ Z:x⊏_l z}=∅" is also ^1_1). Hence, as “p∈" is ^1_1, the whole statement is ^1_1. On the other hand, as discussed before the claim, “ḣ is a -name for a member of Z" is a conjunction of a ^1_1-statement with a ^1_1-statement.In M, fix a -name ḣ for a real in Y and a countable H⊆ Y that witnesses the goodness offor ḣ. Enumerate H={y_n:n <ω}. Now, as Ψ(ḣ, y_n:n<ω) is true in M, it is also true in N by Claim <ref> and ^1_1-absoluteness. In N, as c is -unbounded over M, ∀n<ω(c⊏̸y_n), so⊩^N_ c⊏̸ḣ.Assume that ∈ M is a poset. Then, in N, forces that every c∈ X^N that is -unbounded over M is -unbounded over M^. Work within M. Let e∈Ω and ḣ be a -name for a member of Y_e. Fix p∈ and n<ω. Choose a continuous and surjective function f:ω^ω→ Y_e and a -name ż for a real in ω^ω such thatforces that f(ż)=ḣ. Choose an interpretation ( p_k_k<ω,g) of ż below p. In N, as c is -unbounded over M, then c⊏̸f(g), so c⊏̸_n f(g). By Lemma <ref>, there is a k<ω such that p_k⊩^N_ c ⊏̸_n f(ż)=ḣ. Let _0,_1,_0,_1 be partial orders with _0,_1∈ M. Recall that _0,_1,_0,_1 is correct with respect to M if _0 is a complete subposet of _1, _0 is a complete subposet of _1, _i⋖_M_i for each i<2 and, whenever p_0∈_0 is a reduction of p_1∈_1, then p_0 is a reduction of p_1 with respect to _0,_1 (see <cit.>).Let I ∈ M be a directed partial order, ⟨_i⟩_i∈ I∈ M and ⟨_i⟩_i∈ I∈ N directed systems of posets such that, for any i,j∈ I, _i⋖_M_i and _i,_j,_i,_j is correct with respect to M whenever i≤ j. Assume that c∈ X^N is -unbounded over M and that, for each i∈ I, _i forces (in N) that c is -unbounded over M^_i. If =_i∈ I_i and =_i∈ I_i then ⋖_M, _i,,_i, is correct with respect to M for each i∈ I, andforces (in N) that c is -unbounded over M^. By <cit.>, ⋖_M and _i,,_i, is correct with respect to M for each i∈ I.Work within M. Let e∈Ω and ḣ be a -name for a member in Y_e. Choose a continuous and surjective function f:ω^ω→ Y_e and a -name ż for a real in ω^ω such thatforces that f(ż)=ḣ.Work in N. Assume, towards a contradiction, that there are q∈ and n<ω such that q⊩^N_c⊏_nḣ. Choose i∈ I such that q∈_i.Let G be _i-generic over N such that q∈ G. By assumption, ⊩^N[G]_/_ic⊏_n ḣ. In M[G∩_i], find and interpretation ( p_k_k<ω,g) of ż in /_i. In N[G], as c⊏̸f(g), by Lemma <ref> there is a k<ωsuch that p_k⊩^N[G]_/_ic ⊏̸_n f(ż) (by <cit.> /_i⋖_M[G∩_i]/_i, so the previous interpretation of ż in /_i is also an interpretation in /_i). Thus p_k⊩^N[G]_/_ic ⊏̸_nḣ which contradicts ⊩^N[G]_/_ic⊏_nḣ. Let δ be an ordinal, c∈ X^N, ⟨ℙ_α^0,ℚ̇_α^0⟩_α<δ∈ M and ⟨ℙ_α^1,ℚ̇_α^1⟩_α<δ∈ N both FS iterations such that, for any α<δ, if ℙ_α^0⋖_Mℙ_α^1 then, in N, ℙ_α^1 forces that ℚ̇_α^0⋖_M^ℙ_α^0ℚ̇_α^1 and that ℚ̇^1_α forces that c is -unbounded over M^^0_α+1. Then,ℙ_α^0⋖_Mℙ_α^1 for all α≤δ and ^1_δ forces, in N, that c is -unbounded over M^^0_δ. Fix α≤δ. When ^0_α⋖_M^1_α it is not hard to see by induction on β∈[α,δ] that ^0_α,^0_β,^1_α,^1_β is correct with respect to M (the limit step follows by Lemma <ref>, the successor step by <cit.>). The result follows by the case α=0 and by Lemma <ref>.§ CONSISTENCY RESULTS Before we prove the consistency results of this section, we review some facts about matrix iterations, including one result about preservation in the context of general Polish relational systems.A matrix iterationconsists of* two ordinals γ^ and δ^,* for each α≤γ^, a FS iteration ^_α,δ^=^_α,ξ,^_α,ξ:ξ<δ^ such that, for any α≤β≤γ^ and ξ<δ^, if ^_α,ξ⋖^_β,ξ then ^_β,ξ forces ^_α,ξ⋖_V^^_α,ξ^_β,ξ.According to this notation, ^_α,0 is the trivial poset and ^_α,1=^_α,0.By Corollary <ref>, ^_α,ξ is a complete suborder of ^_β,ξ for all α≤β≤γ and ξ≤δ.We drop the upper indexwhen it is clear from the context. If G is ℙ_γ,δ-generic over V we denoteV_α,ξ=V[G∩_α,ξ] for all α≤γ and ξ≤δ . Clearly, V_α,ξ⊆ V_β,η for all α≤β≤γ and ξ≤η≤δ. The idea of such a construction is to obtain a matrix ⟨ V_α,ξ:α≤γ,ξ≤δ⟩ of generic extensions as illustrated in Figure <ref>.The construction of the matrix iterations in our consistency results corresponds to the following particular case.A matrix iterationis standard if* each ^_α,1 (α≤γ^) is ccc, * it consists, additionally, of * a partition S^,T^ of [1,δ^),* a function Δ^𝐦:T^𝐦→{α≤γ^𝐦:α is not limit},* a sequence ⟨_ξ^𝐦:ξ∈ S^𝐦⟩ of Suslin ccc posets coded in the ground model V,* a sequence ^_ξ:ξ∈ T^ such that each ^_ξ is a ^_Δ^(ξ),ξ-name of a poset such that it is forced by ^_γ,ξ to be ccc, and * for each α≤γ and 1≤ξ<δ, ℚ̇_α,ξ^𝐦:=(^_ξ)^V_α,ξ if ξ∈ S^, ^_ξ if ξ∈ T^ and α≥Δ^(ξ), 1 otherwise. In practice, ^_α,ξ:α≤γ is constructed by induction on ξ≤δ (before the ξ+1-th step for ξ∈ T^, _ξ^ is defined). So it is clear that the constructed systemis a matrix iteration. Moreover, each ^_α,ξ is ccc. Again, when there is no place for confusion, we omit the upper index 𝐦.Letbe a standard matrix iteration. Assume that* γ_0≤γ^ has uncountable cofinality and * _γ_0,1=_α<γ_0_α,1Then, for any ξ≤δ^, _γ_0,ξ=_α<γ_0_α,ξ. In particular, for any Polish space X coded in V (by a countable metric space), X^V_γ_0,ξ=⋃_α<γ_0X^V_α,ξ.Letbe a standard matrix iteration and = X,Y,⊏ a gPrs coded in V. Assume that, for each α<γ, * _α+1,1 adds a real ċ_α∈ X that is -unbounded over V_α,1 and * for each ξ∈ S^, _α,ξ forces that _ξ^V_α,ξ is -good.Then _α+1,δ forces that ċ_α is -unbounded over V_α,δ. Even more, if γ has uncountable cofinality and _γ,1=_α<γ_α,1, then ℙ_γ,δ forces ()≤cf(γ)≤(). The first statement is a direct consequence of Lemmas <ref>, <ref> and Corollary <ref>. For the second statement, given an increasing cofinal sequence {α_ζ:ζ<cf(γ)}∈ V in γ, by Lemma <ref> _γ,δ forces that {ċ_α_ζ:ζ<cf(γ)} is strongly cf(γ)--unbounded of size (γ), so ()≤(γ)≤() by Lemma <ref>. For the reader convenience, before we prove our main results we summarize some facts about preservation from the previous sections. We assume that b,h,π,ρ∈ω^ω are increasing. * If, for all but finitelly many k<ω, k·π(k)≤ h(k) and k·|[b(k-1)]^≤ k|^k≤ρ(k) then ^h_b(R) is (ρ,π)-linked for any R⊆∏ b (Lemma <ref>).* If b>^*π𝕀_ω then _b(S) is ((𝕀_ω)^𝕀_ω,π)-linked for any S⊆(b,1) (Lemma <ref>).* Any (h,b^h^𝕀_ω)-linked poset is both 2-^*(b,h)-good and 2-^*(b,h)-good (Lemma <ref>).* If θ is uncountable regular and μ<θ is an infinite cardinal, then any μ-centered poset is both θ-^*(b,h)-good and θ-^*(b,h)-good (Lemma <ref>).* _b,h≤(^*(b,h)) and (^*(b,h))≤_b,h (Lemma <ref>).* _b,h^𝕀_ω≤(^*(b,h)) and (^*(b,h))≤_b,h^𝕀_ω (Lemma <ref>). In relation to (P1) we consider the following structure discussed in <cit.>(see Lemma <ref>(d) below). Let H,g∈ω^ω such that H>𝕀_ω. Define ^g_H:={x∈ω^ω:∀ k<ω(H^(k)∘ x≤^* g)} where H^(0)=𝕀_ω and H^(k+1)=H∘ H^(k), and denote (H,g):=^g_H,^g_H,≤^*. (H,g) whenever ^g_H≠∅. We use the following particular case. Fix the operation σ^*:ω×ω→ω such that σ^*(m,0)=1 and σ^*(m,n+1)=m^σ^*(m,n). Define ρ^*∈ω^ω such that ρ^*(0)=2 and ρ^*(i+1)=σ(ρ^*(i),i+3). Denote ^ρ^*:=^g_H where g(i):=ρ^*(i+1) and H(i):=2^i, and put ^ρ^*:=(H,g). * 𝕀_ω,ρ^*∈^ρ^*.* ^ρ^*.* If x,y∈^ρ^* then x+y, x· y, x^y are in ^ρ^*.* For any x∈^ρ^*, ∀^∞ i<ω(i·|[x(i-1)]^≤ i|^i≤ρ^*(i)).Now, we are ready to prove the main results of this paper. Let μ≤ν≤κ be uncountable regular cardinals and let λ≥κ be a cardinal such that λ^<μ=λ. If * π is an ordinal with |π|≤λ,* θ_ζ:ζ<π is a non-decreasing sequence of regular cardinals in [μ,ν],* λ^<θ_ζ=λ for all ζ<π, and* ζ^*≤π, then there is a ccc poset forcing * MA_<μ,()=(_f')=(_𝕀_ω)=μ and (_𝕀_ω)=(_f')==λ for all increasing f'∈ω^ω, and* there are sets {c_ζ:ζ<π}, {h_ζ:ζ<ζ^*} and {f_ζ:ζ^*≤ζ<π} of increasing functions in ω^ω such that* _c_ζ,h_ζ=θ_ζ for all ζ<ζ^*, * _c_ζ,1=(_f_ζ)=_c_ζ,H=θ_ζ for all ζ∈[ζ^*,π) where H:=(ρ^*)^𝕀_ω, and * there is an increasing function f_π such that (_f)==()=()=ν and =(_f)=κ for any increasing f≥^*f_π. We are allowed to use π=0 in this theorem, in which case we have Theorem <ref> (see the Introduction) and f_π could be found in the ground model. When π>0, we additionally obtain many values for cardinals of the type _b,h and _b,h, even allowing at most λ-many repetitions for each value (as the continuum is forced to be λ, no more that λ-many repeated values are allowed). Fix a bijection g=(g_0,g_1,g_2):λ→κ×(π+1)×λ, and consider t:κν→κ such that t(κδ+α)=α for all δ<ν and α<κ. The ccc poset required is of the form _π+1∗ where _π+1 is the FS iteration of length π+1 of Hechler forcingandis a _π+1-name of a ccc poset constructed by a matrix iteration as defined in step 2 below. First, in the ground model, fix g_-1:=𝕀_ω^𝕀_ω and b_-1:=max{H+1,2^g_-1∘(H^+-1)}. Note that H,g_-1,b_-1∈^ρ^* by Lemma <ref> and, by Lemma <ref>, (_𝕀_ω)≤_b_-1,H and _b_-1,H≤(_𝕀_ω) (this will be used to show (A)).Step 1. For each ζ<ζ^*, let h_ζ∈ℝ^ρ^*∩ V^_ζ+1 be an increasing dominating real over ℝ^ρ^*∩ V^_ζ (recall that ^ρ^*). In V^_ζ+1, choose c_ζ∈^ρ^* such that c_ζ>h_ζ^𝕀_ω (this guarantees that ^*(c_ζ,h_ζ) is a gPrs). For each ζ∈[ζ^*,π], let c_ζ∈ω^ω∩ V^_ζ+1 be an increasing dominating real over V^_ζ, and define f_ζ, g_ζ,b_ζ∈ω^ω∩ V^_ζ+1, all increasing, such that f_ζ≥ (log c_ζ)^+, g_ζ≫ f_ζ and b_ζ≥^* 2^g_ζ∘(H^+-1). Note that, By Lemmas <ref> and <ref>, _c_ζ,1≤(_f_ζ)≤_b_ζ,H and (_f_ζ)≤_c_ζ,1. Step 2. Work in V_0,0:=V^_π+1. According to Definition <ref>, construct a standard matrix iteration 𝐦 that satisfies (i)-(viii) below.* γ^𝐦=κ and δ^𝐦=λκν (as a product of ordinal numbers).* _α,1=_α for each α≤κ.* S=S^𝐦={λρ:0<ρ<κν} and T=T^=[1,δ^)∖ S,* _ξ= for all ξ∈ S.* If ξ=λρ+1 for some ρ<κν, put Δ^(ξ)=t(ρ)+1 and let _ξ be a _t(ρ)+1,ξ-name for (_c_π)^V_t(ρ)+1,ξ=_c_π((c_π,1)∩ V_t(ρ)+1,ξ).For each α<κ and ρ<κν, * let ⟨_α,γ^ρ:γ< λ be an enumeration of all the nice _α,λρ-names for all the posets which underlining set is a subset of μ of size <μ and ⊩__κ,λρ“_α,γ^ρ is ccc" (possible because |_κ,λρ|≤λ=λ^<μ) and * for all ζ<π, let ⟨Ḟ_α,ζ,γ^ρ:γ<λ⟩ be an enumeration of all the nice _α,λρ-names for all subsets of ∏ c_ζof size <θ_ζ.If ξ=λρ+2+ε for some ρ<κν and ε<λ, put Δ^(ξ)=g_0(ε)+1,* whenever g_1(ε)=π put _ξ=_g_0(ε),g_2(ε)^ρ, * whenever g_1(ε)<ζ^* put _ξ=_c_g_1(ε)^h_g_1(ε)(Ḟ_g(ε)^ρ), and * whenever ζ^*≤ g_1(ε)<π put _ξ=_c_g_1(ε)(Ḟ_g(ε)^ρ). Put :=_κ,λκν.We prove that V_κ,λκν satisfies the statements of this theorem.(A) We first show that, for each 0ξ<λκν, _κ,ξ forces that _κ,ξ is μ-(b_-1,ρ^*)-good. The case ξ=0 follows by (P4) (_κ,0=_κ is a FS iteration of countable posets); when ξ=λρ for some ρ<κν, it is clear by (P4); when ξ=λρ+2+ε for some -1≤ε<λ, we split into three subcases: when g_1(ε)=π it is clear by (P4); when g_1(ε)<ζ^*, as h_ζ≥^*𝕀_ω b_-1^H (because h_ζ∈^ρ^* is dominating), it follows by (P1), (P3) and Lemma <ref>; and when either ε=-1 or ζ^*≤ g_1(ε)<π, as c_π,c_ζ>^*𝕀_ω b_-1^H, it follows by (P2) and (P3).Therefore, by Theorem <ref> and (P6),forces _b_-1,H≤((b_-1,𝕀_ω^𝕀_ω))≤μ and λ≤((b_-1,𝕀_ω^𝕀_ω))≤_b_-1,H. As (_𝕀_ω)≤_b_-1,H and _b_-1,H≤(_𝕀_ω),forces ()≤(_f')≤(_𝕀_ω)≤μ and λ≤(_𝕀_ω)≤(_f')≤() for any increasing f'∈ω^ω. On the other hand, since ||=λ,forces =λ.It remains to show that MA_<μ holds in V_κ,λκν(which implies ()≥μ).Letbe a ccc poset of size <μ, wlog its underlining set is a subset of μ, and let 𝒟 be a family of size <μ of dense subsets of . By Lemma <ref>, ,𝒟∈ V_α,λρ for some α<κ and ρ<κν. Asis ccc in V_κ,λρ, there is some γ<λ such that =^ρ_α,γ=𝕋_ξ where ξ=λρ+2+ε and ε=g^-1(α,π,γ). It is clear that, in V_α+1,ξ+1, there is a -generic set over V_α+1,ξ, so this generic set intersects all the members of 𝒟.(B1) Fix ζ<ζ^*. For the inequality _c_ζ,h_ζ^≥θ_ζ: Let F⊆∏ c_ζ∩ V_κ,λκν be a family of size <θ_ζ. By Lemma <ref>, there are α<κ and ρ<κν such that F∈ V_α,λρ, so there is some γ<λ such that F=F_α, ζ, γ^ρ. Hence, the generic slalom added by _c_g_1(ε)^h_g_1(ε)(F_α, ζ, γ^ρ)=𝕋_ξ, where ξ=λρ+2+ε and ε=g^-1(α,ζ,γ), localizes all the reals in F.For the converse, we first show that the iterands in the FS iteration ⟨_κ,ξ,_κ,ξ:ξ<λκν⟩ are θ_ζ-^*(c_ζ,h_ζ)-good posets. Indeed, asis σ-centered, by (P4), _ξ is ^*(c_ζ,h_ζ)-good for each ξ∈ S; as c_π>^* c_ζ^h_ζ^𝕀_ω𝕀_ω (because c_π is dominating over V^_π and c_ζ^h_ζ^𝕀_ω𝕀_ω∈ V^_ζ+1), for each ρ<κν, _κ,λρ+1 forces that _λρ+1 is (h_ζ,c_ζ^h_ζ^𝕀_ω)-linked by (P2) so, by (P3), it is forced to be 2-^*(c_ζ,h_ζ)-good; now we analyse the iterands from (vi)-(viii): for (vi), as _ξ has size <μ≤θ_ζ, _ξ is θ_ζ-^*(c_ζ,h_ζ)-good by (P4); for (vii), when g_1(ε)≤ζ,_ξ is θ_ζ-^*(c_ζ,h_ζ)-good by (P4) because it has size <θ_ζ, else, when ζ<g_1(ε), as h_g_1(ε) is dominating over ^ρ^*∩ V^_ζ+1 and c_ζ^h_ζ^𝕀_ω𝕀_ω∈^ρ^*∩ V^_ζ+1, h_g_1(ε)>^*c_ζ^h_ζ^𝕀_ω𝕀_ω so, by Lemma <ref>, (P1) and (P3),_ξ is 2-^*(c_ζ,h_ζ)-good; finally, for (viii), as g_1(ε)>ζ, c_g_1(ε) is dominating over V^_ζ+1, so c_g_1(ε)>^* c_ζ^h_ζ^𝕀_ω𝕀_ω, and _ζ is 2-^*(c_ζ,h_ζ)-good by (P2) and (P3).Therefore,forces _c_ζ,h_ζ^≤(^*(c_ζ,h_ζ))≤θ_ζ by Theorem <ref> and (P5). (B2) Fix ζ^*≤ζ<π. It is enough to show θ_ζ≤_c_ζ,1 and _c_ζ,H≤θ_ζ. The former inequality is proved by a similar argument as for θ_ζ≤_c_ζ,h_ζ^ in (B1). For the latter, we show that the posets we use in the κ-th FS iteration are θ_ζ-(c_ζ,ρ^*)-good. For each ξ∈ S, _ξ is (c_ζ,ρ^*)-good by (P4) becauseis σ-centered; by (P2), as c_π>^* c_ζ^H𝕀_ω,for each ρ<κν, _κ,λρ+1 forces that _λρ+1 is (ρ^*,c_ζ^H)-linked (note that ρ^*≥^* 𝕀_ω^𝕀_ω) so, by (P3), it is forced to be 2-(c_ζ,ρ^*)-good; for ρ<κν and ε<λ we have following cases: when g_1(ε)=π or g_1(ε)≤ζ,as |_ξ|< θ_ζ, _ξ is θ_ζ-(c_ζ,ρ^*)-good by (P4);when ζ<g_1(ε), c_g_1(ε)>^* c_ζ^H𝕀_ω because c_g_1(ε) is dominating over V^_ζ+1 and c_ζ^H𝕀_ω∈ V^_ζ+1, so, by (P2), _ξ is (ρ^*,c_ζ^H)-linked and, by (P3), it is forced to be 2-(c_ζ,ρ^*)-good.Therefore, by Theorem <ref> and (P6),forces that _c_ζ,H≤((c_ζ,ρ^*))≤θ_ζ. (B3) Note that , as a FS iteration of length λκν, adds a ν-scale and ν-cofinally many Cohen reals. Therefore, it forces ()=()=ν.For ρ<κν denote by r^ρ∈ V_t(ρ)+1,λρ+2∩∏ c_π the generic real added by _t(ρ)+1,λρ+1=(_c_π)^V_t(ρ)+1,λρ+1 over V_t(ρ)+1,λρ+1. This real is eventually different from all the members of V_t(ρ)+1,λρ+1∩∏c_π. Hence ν≤_c_π,1 is a consequence of the following. In V_κ,λκν, for any F⊆∏ c_π of size <ν, there is somer^ρ eventually different from all the members of F. By Lemma <ref> there are α<κ and δ<κν such thatF⊆ V_α,λδ. By the definition of t, find a ρ∈[δ,κν) such that t(ρ)=α. Clearly F⊆ V_α,λρ, so their members are all eventually different from r^ρ. On the other hand, {r_^ρ:ρ<κν} is a family of reals of size ≤κ and, by Claim <ref>, any member of V_κ,λκν∩∏c_π is eventually different from some r_^ρ. Hence _c_π,1≤κ.Fix f∈ω^ω increasing such that f≥ f_π. Then ν≤_c_π,1≤(_f_π)≤(_f) and κ≥_c_π,1≥(_f_π)≥(_f), in fact, _c_π,1=(_f)=ν because (_f)≤≤()=ν. To finish the proof it remains to show that κ≤. For any b'∈ω^ω, asis σ-centered and thus (b',1)-good by (P4), κ≤((b',1))≤_b',1 by Theorem <ref> and (P6). Therefore, by Theorem <ref>, κ≤=min{_b',1:b'∈ω^ω}.If the matrix iteration construction of Theorem <ref> is modified so that S^=∅ (i.e. no use of Hechler forcing) then the cardinal invariants associated with many Yorioka ideals can still be separated. Concretely, the final model still satisfies (A), (B1) and (B2), and also satisfies (_f)=()=ν and ()=(_f)=κ for all increasing f≥^*f_π. However, the values ofandare unknown because it is unclear whether the restricted versions of _c_ζ,h_ζ and _c_ζ (ζ≤π) used in the construction add dominating reals. In the hypothesis of Theorem <ref> assume, additionally, that μ' is a regular cardinal and μ≤μ'≤ν and, instead of κ being regular, just assume that κ^<μ'=κ. As in <cit.>, the forcing construction in Theorem <ref> can be modified so that the matrix iteration allows vertical support restrictions, that is, _A,ξ can be defined for all A⊆κ and ξ≤π. The final model of this construction still satisfies (A), (B1) and (B2), and also satisfies (_f)==μ', ()=()=ν and =(_f)=κ (the latter not necessarily regular) for all increasing f≥^*f_π. Though this is stronger than Theorem <ref>, we do not get to separate more of the cardinal invariants associated with Yorioka ideals. In the context of <cit.>, Theorem <ref> could be modified so that, in (B1), it can be forced that _c_ζ,h_ζ=(_ζ)=θ_ζ for ζ<ζ^* where _ζ:ζ<ζ^* is some sequence of gradually fragmented ideals on ω and (_ζ) denotes the Rothberger number of _ζ. The next result guarantees the consistency of ()<(_f)<(_f)<() for any fixed f. Let μ≤ν≤κ be uncountable regular cardinals and let λ≥κ be a cardinal such that λ^<μ=λ. If f∈ω^ω is increasing then there is a ccc poset forcing that ()=μ, (_f)==()=ν, ()==(_f)=κ and ()==λ. Fix a function b∈ω^ω such that b≫ 2^f, a bijection g=(g_0,g_1):λ→κ×λ, and fix t:κν→κ such that t(κδ+α)=α for δ<ν and α<κ. Put h:=𝕀_ω. According to Defintion <ref>, construct a standard matrix iteration 𝐦 such that:* γ^𝐦=κ and δ^𝐦=λκν, * _α,1=_α for each α≤κ, * If ξ=λρ>0 for some ρ<κν, put Δ^(ξ)=t(ρ)+1 and let _ξ be a _t(ρ)+1,ξ-name for ^V_t(ρ)+1,ξ. * If ξ=λρ+1 for some ρ<κν, put Δ^(ξ)=t(ρ)+1, and let _ξ be a _t(ρ)+1,ξ-name for (_b^h)^V_t(ρ)+1,ξ. For each α<κ and ρ<κν, let ⟨_α,γ^ρ:γ<λ be an enumeration of all the nice _α,λρ-names for all the posets whose underlining set is a subset of μ of size <μ and ⊩__κ,λρ“_α,γ^ρ is ccc" (possible because |_κ,λρ|≤λ=λ^<μ). * If ξ=λρ+2+ε for some ρ<κν and ε<λ, put Δ^(ξ)=g_0(ε)+1 and_ξ=_g(ε)^ρ. By (P1) we can find increasing π, ρ∈ω^ω such that _b^h(F) is (π,ρ)-linked for any F⊆∏ b. Therefore, by Lemma <ref>, there is a ≤^*-increasing sequence 𝒢= g_n:n<ω such that _b^h(F) is (ω, 𝒢)-good for any F⊆∏ b. Also,is (ω, 𝒢)-good (see Example <ref>) and _ξ is μ-(ω, 𝒢)-good when ξ=λρ+2+ε for some ρ<κν and ε<λ. Therefore, in V_κ,λκμ, we have that ()≤μ and λ≤() by Theorem <ref>. The converse inequalities are similar to the proof of (A) of Theorem <ref>.We now show ,_b,h^≤κ and , _b,h^≥ν. For each ρ<κν denote by d^ρ∈ V_t(ρ)+1,λρ+1∩ω^ω the Hechler generic real added by _t(ρ)+1,λρ=_κ,λρ=^V_t(ρ)+1,λρ over V_t(ρ)+1,λρ, and by ψ^ρ∈ V_t(ρ)+1,λρ+2∩(b,h) the generic slalom added by _t(ρ)+1,λρ+1=_κ,λρ+1=(_b^h)^V_t(ρ)+1,λρ+1 over V_t(ρ)+1,λρ+1. In V_κ,λκν, each family of reals of size <ν is dominated by some d^ρ. Let F be such a family. By Lemma <ref> there are α<κ and δ<κν such thatF⊆ V_α,λδ. By the definition of t, find a ρ∈[δ,κν) such that t(ρ)=α. Clearly, F⊆ V_α,ξ where ξ=λρ, so their members are dominated by d^ρ. As a direct consequence, ν≤. On the other hand, {d_^ρ:ρ<κν} is a family of reals of size ≤κ and, by Claim <ref>, any member of V_κ,λκν∩ω^ω is dominated by some d_^ρ. Hence ≤κ. By a similar argument, we can prove: Each family of reals in V_κ,λκν of size <ν is localized by some slalom ψ^ρ. By Theorem <ref>,ν≤min{,_b,𝕀_ω^}≤(_f) and (_f)≤max{,_b,𝕀_ω^}≤κ (since 2^f≪ b), so ν≤(_f) and (_f)≤κ. On the other hand, as _κ,λκν is obtained by a FS iteration of cofinality ν, _κ,λκν adds a ν-(ω)-unbounded family of Cohen reals of size ν by Lemma <ref>, so it forces ()=((ω))≤ν. Also, by Theorem <ref>, ()=((ω))≥κ. Hence, by Corollary <ref>,(_f)≤(_f)≤()≤ν and κ≤()≤(_f)≤(_f). In the previous model, it is clear that ≤ and ≤, so ()= and ()= by Theorem <ref>.To finish this section, we show the consistency of < and <. As the converse strict inequalities hold in the model of Theorem <ref>, each pair of cardinals are independent. In particular, the characterizations ()=min{,} and ()=max{,} (see Theorem <ref>) are optimal. Note that ≤ implies ()==(_𝕀_ω)=, and ≤ implies ()==(_𝕀_ω)=.Let μ≤ν be uncountable regular cardinals and let λ≥ν be a cardinal with λ=λ^<μ. Then, there is a ccc poset that forces ()==μ, ==ν and =. Constructby a FS iteration of length λν of ^𝕀_ω_b for every b∈ω^ω (including those that appear in intermediate extensions), and of all the ccc posets of size <μ with underlying set an ordinal <μ (like in the previous results). We also demand that ^𝕀_ω_b for each b∈ω^ω is used cofinally often. By this construction,is μ--good (by Lemma <ref> and Example <ref>(3)) and it forces MA_<μ, so it forces ()==μ and ==λ by Theorem <ref>. On the other hand, it is clear thatforces _b,𝕀_ω=()=()=_b,id_ω=ν for any b∈ω^ω (also thanks to the Cohen reals added at limit stages), so ==ν. With respect to the pairs , and ,, it is clear that a FS iteration of big size but short cofinality of Hechler posets forces < and < (< can be additionally forced with a matrix iteration as in Theorem <ref> but using only (i)-(iii) for the construction). If after this iteration we force with a large random algebra, then =()=ℵ_1< and <()= is satisfied in the final extension.§ DISCUSSIONS AND OPEN QUESTIONS Though we constructed a model of ZFC where the four cardinal invariants associated with many Yorioka ideals are pairwise different, we still do not know how to construct a model where we can separate those cardinals for all Yorioka ideals. Is there a model of ZFC satisfying (_f)<(_f)<(_f)<(_f) for all increasing f:ω→ω? It is not even known whether there is such a model for just f=id_ω.There are many ways to attack this problem. One would be to find a gPrsthat satisfies (_𝕀_ω)≤() and ()≤(_𝕀_ω) and such that any poset of the form _b is -good. If such a gPrs can be found, then a construction as inTheorem <ref> works. Other way is to adapt the techniques for not adding dominating reals from <cit.> in matrix iterations so that a construction as in Remark <ref> could force ≤μ and λ≤. This is quite possible because _b is 𝐃-good (see Example <ref>(3)), though its restriction to some internal model may add dominating reals (see <cit.>). Success with this method would actually solve the following problem. Is there a model of ZFC satisfying ()<()<()<()? Such a model exists assuming strongly compact cardinals (see <cit.>), but it is still open modulo ZFC alone. Other way to attack Question <ref> is by large products of creature forcing (see <cit.>). Quite recently, A. Fischer, Goldstern, Kellner and Shelah <cit.> used this technique to prove that 5 cardinal invariants on the right of Cichoń's diagram are pairwise different. Such method would also work to solve the dual of Question <ref>, that is, Is there a model of (_f)<(_f)<(_f)<(_f) for all increasing f:ω→ω? There are still many open questions in ZFC about the relations between the cardinal invariants associated with Yorioka ideals and the cardinals in Cichoń's diagram. For instance, it is unknown whether ≤() (or ()≤) is provable in ZFC. In general, very little is known about the additivity and the cofinality of Yorioka ideals. Is it consistent with ZFC that infinitely many cardinal invariants of the form (_f) are pairwise different? Is it true in ZFC that (_f)=(_id_ω) (and (_f)=(_id_ω)) for all (or some) increasing f? Question <ref> is related with finding a gPrs associated to (_f) and (_f).Is it true in ZFC that ()= (and ()=)? Theorem <ref> indicates that the method of Theorem <ref> cannot be used to increase every cardinal of the form (_f) while preserving () small.Concerning the problem of separating the four cardinal invariants associated with an ideal, the consistency of ()<()<()<() is known from <cit.>. Is there a model of ZFC satisfying ()<()<()<()? The same question foris also open.Though in Theorem <ref> we were able to separate (infinitely) many localization and anti-localization cardinals, the -localization cardinals appear below the -anti-localization ones. The reason of this is that the preservation methods we use for the localization cardinals relies on the structure ^ρ^*, so after including one -anti-localization cardinal with functions quite above ρ^* (to preserve the previous localization cardinals), a new forcing to increase some other localization cardinal may not preserve the previous anti-localization ones. In view of this, it would be interesting to improve our preservation methods to allow both type of cardinals appear alternatively.alpha | http://arxiv.org/abs/1703.08634v2 | {
"authors": [
"Miguel A. Cardona",
"Diego A. Mejía"
],
"categories": [
"math.LO",
"03E17, 03E15, 03E35, 03E40"
],
"primary_category": "math.LO",
"published": "20170325010909",
"title": "On cardinal characteristics of Yorioka ideals"
} |
algorithmsection plaintop algorithm[algorithm]singlelinecheck=offequation equation equation* equation* subequations* subequations* subequations subequations align align align* align*equationsection 1]Stella Hadjiantoni 2,3]Erricos Kontoghiorghes [1]University of Kent, UK [2]Cyprus University of Technology [3]Birkbeck University of London, UK A numerical method for the estimation of time-varying parameter models in large dimensions [==========================================================================================[Corresponding author: S. Hadjiantoni, School of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury, Kent CT2 7FS, UK. Email address: [email protected]] A novel numerical method for the estimation of large time-varying parameter (TVP) models is proposed. The updating and smoothing estimates of the TVP model are derived within the context of generalised linear least squares and through numerically stable orthogonal transformations. The method developed is based on computationally efficient strategies. The computational cost is reduced by exploiting the special sparse structure of the TVP model and by utilising previous computations. The proposed method is also extended to the rolling window estimation of the TVP model. Experimental results show the effectiveness of the new updating, window and smoothing strategies in high dimensions when a large number of covariates and regressions are included in the TVP model. time-varying coefficients, recursive estimation, updating, window estimation, matrix algebra § INTRODUCTION The assumption that the coefficients of a linear model are constant over time is often invalid. Recently, models with time-varying structures have been adopted to explain inflation dynamics, to forecast macroeconomic variables under structural change and to model interest rates (Cogley and Sargent, 2005; Primiceri, 2005; Stock and Watson, 2009; Koop and Korobilis, 2013; Zhang and Wu, 2015). A model with time varying coefficients can be given a state space formulation. The most common approach is to use the Kalman filter to provide the updated values of the coefficients, as each new observation is acquired. The Kalman filter is a fast recursive method, especially in small dimensions, but it does not have good numerical properties <cit.>. It is based on matrix inverses, which can be ill-conditioned, and may be the reason for inaccurate results <cit.>. Specifically in a recursive method like the Kalman filter, where at every new data point estimates of the unknown parameters are obtained using previous computations, a numerical error at one iteration of the algorithm can be propagated through to future computations and produce inaccurate results <cit.>. Generalised least squares (GLS) have been applied on a univariate time-varying model in order to derive the Kalman filter and Kalman smoother estimators <cit.>. However, this approach is difficult to implement in practical problems as it requires the inversion of a large variance-covariance matrix which is computationally demanding and numerically unstable. The contribution, herein, is to develop a novel numerical method for the estimation of the multivariate time-varying parameter (TVP) regression model. The proposed method estimates the TVP model by solving a generalised linear least squares problem which yields the best linear unbiased estimator of the model <cit.>. The updating estimates, when new data are acquired, and the smoothing estimates, when existing data are revised, are derived. The method is next extended to the window estimation of the model where data are added and deleted simultaneously. Numerical strategies which update the model to include the effect of new observations and which downdate the model to exclude the effect of old or obsolete observations are employed <cit.>.The novel method is reliable in delivering accurate estimation results, and computationally efficient which makes it feasible to estimate large TVP models.This is achieved in two ways. Firstly, by employing efficiently previous computations when new observations are acquired and by exploiting the sparsity of the multivariate TVP model <cit.>. Secondly, the computational tools which are mainly orthogonal transformations, have the property of being numerically stable and also the capability to limit the computational expense of the estimation procedures <cit.>. Furthermore, the proposed algorithmdoes not require non-singular variance-covariance matrices and also postpones the inversion of matrices up to the last step. The paper is organised as follows. Section UpdateTVPmodel introduces a new numerical method for the estimation of the TVP model based on orthogonal transformations. Section RecursiveTVPSUR considers the multivariate TVP model where the regressions are contemporaneously correlated. The numerical estimation of the model is presented when observations are added to and/or removed from the model. The smoothing estimates of the model are also derived. Section Experiments presents the computational results and finally, Section Conclusions concludes.§ NUMERICAL ESTIMATION OF THE TVP MODEL Consider the univariate time-varying parameter (TVP) model which is given by ψ_t = x_t β_t + ϵ_t, ϵ_t ∼ ( 0, σ^2), t=1,…,Mandβ_t = β_t-1 + η_t, η_t ∼ ( 0, σ^2 Σ_η ), t=1,…,M.Here ψ_t is the observation of the dependent variable y at time t, x_t∈^k is the row vector of explanatory variablesexplanatory variables at time t, β_t ∈^k is the vector of the unknown coefficients which are evolving over time according to the random walk in TVPmodeltB, and ϵ_t and η_t are the error terms with zero mean and variance σ^2 and σ^2 Σ_η, respectively <cit.>. Also 𝔼 (ϵ_t ϵ_t') = 0 if t≠ t' and 𝔼 ( η_t η^T_t' ) = 0 if t ≠ t', t=1,…,M, where M is the sample size. In addition, Σ_η is a known, symmetric and non-negative dispersion matrix. Furthermore, from TVPmodelt, it is easy to derive (see <cit.>) β_t = β_t-1 + η_t =…= β_1 + ∑_s=1^t η_s .Therefore the TVP model TVPmodelt up to time t takes the following form[ ψ_1; ⋮; ψ_t-1; ψ_t ] = [ x_1; ⋮; x_t-1; x_t ]β_t + [ ϵ_1; ⋮; ϵ_t-1; ϵ_t ] - [ x_1 ⋯ x_1; ⋮ ⋱ ⋮; 0 ⋯ x_t-1; 0 ⋯ 0 ][ η_2; ⋮; η_t ]or conformably the compact form y_t = X_t β_t + e_t - A_t u_t, e_t-A_t u_t ∼( 0, σ^2 Ω_t ), where y_t is the vector of observations for the dependent variable up to time t, X_t, e_t, A_t and u_t are analogously defined andΩ_t = I_t + A_t (I_t-1⊗Σ_η ) A^T_t <cit.>. The GLS estimator of the latter model is given by β̂_t = ( X_t^T Ω_t^-1X_t)^-1X_t^T Ω_t^-1y_t.However, the derivation of the GLS estimator is computationally costly and numerically unstable when Ω_t is ill-conditioned <cit.>.An alternative procedure to the GLS methodology is to consider solving a generalised linear least squares problemgeneralised linear least squares problem (GLLSP), that is,β_t, v_targminv_t ^2 subject toy_t = X_t β_t + C_t v_t,where C_t ∈^t × t is upper triangular and non-singular such that Ω_t = C_t C_t^T, v_t is an arbitrary vector, v_t ∼ (0, σ^2 I_t) and · denotes the Euclidean norm. Observe that Ω_t is not formed explicitly but instead its special structure is taken into account. That is, Ω_t= I_t + A_t (I_t-1⊗Σ_η ) A^T_t = [I_t A_t (I_t-1⊗C_η ) ][I_t A_t (I_t-1⊗C_η ) ]^T,where C_η is the Cholesky factor of Σ_η, i.e. Σ_η = C_ηC_η^T. Then the RQ decomposition (RQD) of ( I_t A_t (I_t-1⊗C_η ) ) gives[I_t A_t (I_t-1⊗C_η ) ]P_t,1= [ 0 C_t ],where P_t,1∈^((t-1)k) × ((t-1)k) is orthogonal and C_t ∈^t × t is upper triangular and non-singular. To solve GLLSPt1, the generalised QR decompositiondecomposition!generalised QR decomposition (GQRD) of X_t and C_t is computed, namely, Q^T_t [ X_t y_t ] = [ R_t y_t,A; 0 y_t,B ] k t-kand( Q^T_t C_t ) P_t,2 = U_t = [ U_11,t U_12,t;0 U_22,t ] k t-k , where R_t ∈^k × k, U_t ∈^t × t are upper triangular and non-singular and Q_t, P_t,2 are orthogonal matrices of order t. When Q_t and P_t,2 are applied on GLLSPt1, it givesβ_t, v_targminP_t,2^T v_t ^2subject toQ_t^Ty_t = Q_t^TX_t β_t + Q_t^TC_t P_t,2P_t,2^T v_t.The GLLSP GLLSPt1 now becomesβ_t, v_t,A, v_t,Bargmin [ v_t,A; v_t,B ]^2subject to [ y_t,A; y_t,B ] = [ R_t; 0 ]β_t + [ U_11,t U_12,t;0 U_22,t ][ v_t,A; v_t,B ],where the second part of the restrictions in GLLSPt2 yields v_t,B = U_22,t^-1y_t,B. The GLLSP in GLLSPt2 is then reduced toβ_t, v_t,Aargmin v_t,A^2subject to ỹ_t,A= R_tβ_t + U_11,tv_t,A,whereỹ_t,A = y_t,A - U_12,tv_t,B. The estimator for β_t is derived by setting v_t,A = 0, in order to minimise the argument, and from the solution of the upper triangular system R_t β_t = ỹ_t,A. § MULTIVARIATE TIME-VARYING PARAMETER MODEL A more general case of the TVP model in TVPmodelt is a system of G such regressions which are contemporaneously correlatedcontemporaneous correlation. That is, consider the time-varying parameter seemingly unrelated regressions (TVP-SUR) model[ ψ_1,t; ⋮; ψ_G,t ] = [ x_1,t; ⋱; x_G,t ][ β_1,t; ⋮; β_G,t ] + [ ϵ_1,t; ⋮; ϵ_G,t ], t=1,…,Mand[ β_1,t; ⋮; β_G,t ] = [ β_1,t-1; ⋮; β_G,t-1 ] + [ η_1,t; ⋮; η_G,t ],where x_jt∈^k_j is a row vector of explanatory variables for regression j at time t, ( ϵ_1,tϵ_G,t )^Tis a G× 1 disturbance vector with zero mean and variance covariance matrix Σ = [ σ_ij]_i,j, i,j = 1,…, G. Moreover, η_jt∼ (0, σ_jjΣ_j) and 𝔼 ( η_jtη^T_it) = 0 for i≠ j. As in the constant coefficients seemingly unrelated regressions (SUR) model, when σ_ij≠ 0 for i ≠ j, efficiency will be gained if the estimation of the unknown parameters is executed in a system of the G regressions <cit.>. Furthermore, let K = ∑_i=1^G k_i. Consider the ith regression of the system with all the available observations up to time t, that is, [ ψ_i,1; ⋮; ψ_i,t-1; ψ_i,t ] = [ x_i,1; ⋮; x_i,t-1; x_i,t ]β_i,t + [ ϵ_i,1; ⋮; ϵ_i,t-1; ϵ_i,t ] - [ x_i,1 ⋯ x_i,1; ⋮ ⋱ ⋮; 0 x_i,t-1; 0 ⋯ 0 ][ η_i,2; ⋮; η_i,t ]or in compact form asy_i,t = X_i,tβ_it + e^*_i,t, e^*_i,t∼( 0, σ_ii( I_t + A_i,t ( I_t-1⊗Σ_i ) A^T_i,t) ),where e^*_i,t = e_i,t - A_i,tu_i,t is defined as in TVPmodelt. The TVP-SUR model is then given in matrix form, at time t, by[ y_1,t; y_2,t; ⋮; y_G,t ] = [ X_1,t; X_2,t; ⋱; X_G,t ][ β_1,t; β_2,t; ⋮; β_G,t ] + [ e^*_1,t; e^*_2,t; ⋮; e^*_G,t ],or equivalently byvec( {y_i,t})=( ⊕_i=1^G X_i,t) vec( {β_i,t})+ vec( {e^*_i,t}),where y_i,t∈^t are the response vectors at time t, X_i,t∈^t × k_i are the exogenous matrices at time t with full column rank, β_i,t∈^k_i are the time-varying coefficients at time t and e^*_i,t∈^t are the disturbance terms, i = 1,…,G. Note that {·} denotes a set of vectors and ⊕_i=1^G is the direct sum which for notational convenience will be abbreviated by ⊕_i.The error term in TVPSUR1 has zero mean and variance covariance matrixΩ_t= [ σ_11Ω_1,t σ_12I_t σ_1GI_t; σ_21I_t σ_22Ω_2,t σ_2GI_t; ⋮ ⋮ ⋱ ⋮; σ_G1I_t σ_G2I_t σ_GGΩ_G,t ], = ⊕_i σ_iiA_i,t (I_t-1⊗Σ_i ) A^T_i,t + Σ⊗I_t = ⊕_i C_i,tC_i,t^T + CC^T ⊗I_t = [ ⊕_i C_i,t C⊗I_t ][ ⊕_i C_i,t C⊗I_t ]^T where Ω_i,t = ( I_t + A_i,t (I_t-1⊗Σ_i ) A^T_i,t), C_i,t = √(σ_ii)A_i,t (I_t-1⊗C_i ) and Σ_i = C_i C_i ^T is the Cholesky decomposition of Σ_i. The best linear unbiased estimator of the TVP-SUR model TVPSUR1 is obtained from the solution of the GLLSP [ β_i,t, v_i,targminvec ( {v_i,t} ) ^2subject to; vec ( {y_i,t} ) = (⊕_i X_i,t) vec ( {β_i,t} ) + ( ⊕_i C_i,t C⊗I_t ) vec ( {v_i,t} ), ]where vec ( {v_i,t} ) ∼ ( 0 , I_ (t-1)K +Gt) is such that vec( {e^*_i,t}) =( ⊕_i C_i,tC⊗I_t ) vec ( {v_i,t} ). The solution of SURGLLSP1 is derived by computing the RQD[ ⊕_i C_i,t C⊗I_t ]P̃_t,1 = [0 C̃_t ],and the GQRD Q̃^T_t[ ⊕_i X_i,t vec ( {y_i,t} ) ] = [⊕_i R_i,t vec ( {y_i,tA} );0 vec ( {y_i,tB} ) ], ( Q̃^T_tC_t ) P̃_t,2 = L_t = [ L_11,t L_12,t;0 L_22,t ] K Gt -K,where C̃_t ∈^ Gt × Gt, R_i,t∈^k_i × k_i, i = 1,⋯,G, and L_t∈^ Gt × Gt are upper triangular and non-singular, and P̃_t,1∈^ Gt × ( (t-1)K + Gt ), Q̃^T_t, P̃_t,2, ∈^ Gt × Gt are orthogonal. Note that in the above computations the special structure of the matrices is exploited. Numerically stable and computationally efficient strategies which exploit the special sparse structure of the matrices have been previously developed <cit.>. Using the computations in SURRQD1 and SURGQRD1, the GLLSP in SURGLLSP1 is equivalently given by[ β_i,t, v_i,tA, v_i,tBargmin[ vec ( {v_i,tA} ); vec ( {v_i,tB} ) ]^2subject to;; [ vec ( {y_i,tA} ); vec ( {y_i,tB} ) ] = [ ⊕_i R_i,t; 0 ]vec ( {β_i,t} ) + [ L_11,t L_12,t;0 L_22,t ][ vec ( {v_i,tA} ); vec ( {v_i,tB} ) ], ]where [ vec ( {v_i,tA} ); vec ( {v_i,tB} );] = P̃_t,2^T P̃_t,1 ^T vec ( {v_i,t} ).The solution of the GLLSP in TVPSURGLLSP2 is obtained by solving the triangular systemL_22,tvec ( {v_i,tB} ) = vec ( {y_i,tB} ) for v_i,tB and by setting vec ( {v_i,tA} ) = 0 in order to minimise the argument in TVPSURGLLSP2. The BLUE of β_i,t, i = 1,…,G, is derived from the solution of the triangular system⊕_i R_i,tvec ( {β_i,t} ) = vec ( {ỹ_i,tA} ),where vec ( {ỹ_i,tA} ) = vec ( {y_i,tA} ) - L_12,tvec ( {v_i,tB} ). The steps for the numerical strategy for estimating the TVP-SUR model using orthogonal transformations are summarised in Algorithm AlgTVPSUR.§.§ Updating the TVP-SUR Model with one new Observation Consider now updating each regression in the TVP-SUR model when a new datum is collected. This is defined as the original model TVPSUR1 together with a single new observation in each regression which at time t+1 is given by[ [ ψ_1,t+1; ⋮; ψ_G,t+1 ] = [ x_1,t+1; ⋱; x_G,t+1 ][ β_1,t+1; ⋮; β_G,t+1 ] + [ ϵ_1,t+1; ⋮; ϵ_G,t+1 ],;;[ β_1,t+1; ⋮; β_G,t+1 ] = [ β_1,t; ⋮; β_G,t ] + [ η_1,t+1; ⋮; η_G,t+1 ]. ]The updated TVP-SUR model at time t+1 is written as[ y_1,t; ψ_1,t+1; y_2,t; ψ_2,t+1; ⋮; y_G,t; ψ_G,t+1 ] = [ X_1,t; x_1,t+1; X_2,t; x_2,t+1; ⋱; X_G,t; x_G,t+1 ][ β_1,t+1; β_2,t+1; ⋮; β_G,t+1 ] + [ e^*_1,t; ϵ^*_1,t+1; e^*_2,t; ϵ^*_2,t+1; ⋮; e^*_G,t; ϵ^*_G,t+1 ], where the variance covariance matrix is Ω_t+1 = [ σ _11Ω_1,t+1σ_12I_t+1 σ_1GI_t+1;σ_21I_t+1σ_22Ω_2,t+1 σ_2GI_t+1;⋮⋮⋱⋮;σ_G1I_t+1σ_G2I_t+1 σ_GGΩ_G,t+1 ]and Ω_i,t+1 = ( I_t+1 + A_i,t+1 (I_t⊗Σ_i ) A^T_i,t+1). Notice that the dispersion matrix of each time-varying regression is also updated by X_i,tΣ_i X_i,t^T to encapsulate the new information available, namely,Ω_i,t+1 = [ Ω̃_i,t0;01 ] = [ Ω_i,t + X_i,tΣ_i X_i,t^T0;01 ]. For the recursive estimation of the TVP-SUR model, consider re-arranging the observations of the updated TVP-SUR model MTVPadd1 as follows[ y_1,t; y_2,t; ⋮; y_G,t; ψ_1,t+1; ψ_2,t+1; ⋮; ψ_G,t+1 ] = [ X_1,t; X_2,t; ⋱; X_G,t; x_1,t+1; x_2,t+1; ⋱; x_G,t+1 ][ β_1,t+1; β_2,t+1; ⋮; β_G,t+1 ] + [ ẽ^*_1,t; ẽ^*_2,t; ⋮; ẽ^*_G,t; ϵ^*_1,t+1; ϵ^*_2,t+1; ⋮; ϵ^*_G,t+1 ],which is conformably written as[ vec ( {y_i,t} ); vec ( {ψ_i,t+1} ) ] = [ ⊕_i X_i,t; ⊕_i x_i,t+1 ]vec ( {β_i,t+1} ) + [ vec ( {ẽ_i,t} ); vec ( {ϵ^*_i,t+1} ) ], [ vec ( {ẽ_i,t} ); vec ( {ϵ^*_i,t+1} ) ]∼( 0 ,Ω^*_t+1).Now Ω^*_t+1 is given byΩ^*_t+1= [ Ω̃_t0;0Σ ],where Ω̃_t is the updated variance covariance matrix of the first t observations. That is, TVPSURvarcov is now revised to become Ω̃_t= [ σ _11Ω̃_1,t σ_12I_t σ_1GI_t; σ_21I_tσ_22Ω̃_2,t σ_2GI_t; ⋮ ⋮ ⋱ ⋮; σ_G1I_t σ_G2I_tσ_GGΩ̃_G,t ] = Ω_t + ⊕_i X_i,tΣ_i X_i,t^T= C̃_t C̃_t^T + ⊕_i X_i,tC_iC_i^T X_i,t^T= [ C̃_t ⊕_i X_i,tC_i ][ C̃_t ⊕_i X_i,tC_i ]^T, where C̃_t is from the RQD in SURRQD1 and C_i is the Cholesky factor of Σ_i. Then it follows that Ω^*_t+1 = [ C̃_t ⊕_i X_i,tC_i0;00C ][ C̃_t ⊕_i X_i,tC_i0;00C ]^T.Hence the GLLSP, which yields the BLUE of the updated by one observation TVP-SUR model, is given by[ β̃_i,t+1, v_i,t, v^*_i,t,v_t+1argmin [ vec ( {v_i,t} ); vec ( {v^*_i,t} ); v_t+1 ]^2subject to; ; [ vec ( {y_i,t} ); vec ( {ψ_i,t+1} ) ] = [ ⊕_i X_i,t; ⊕_i x_i,t+1 ]vec ( {β_i,t+1} ) + [ C̃_t ⊕_i X_i,tC_i0;00C ][ vec ( {v_i,t} ); vec ( {v^*_i,t} ); v_t+1 ], ]where previous computations from the solution of the GLLSP SURGLLSP1 can be efficiently utilised to reduce the computational cost. Namely, using the GQRD in SURRQD1 and SURGQRD1 and the solution of TVPSURGLLSP2, the latter GLLSP becomes[ β̃_i,t+1, v_i,tA, v_i,tB, v^*_i,t,v_t+1argmin [vec ( {v_i,tA} );vec ( {v_i,tB} ); vec ( {v^*_i,t} ); v_t+1 ]^2subject to; ; [vec ( {y_i,tA} );vec ( {y_i,tB} ); vec ( {ψ_i,t+1} ) ] = [ ⊕_i R_i,t; 0; ⊕_i x_i,t+1 ]vec ( {β_i,t+1} ) + [ L_11,t L_12,t ⊕_i R_i,tC_i0;0 L_22,t00;000C ][vec ( {v_i,tA} );vec ( {v_i,tB} ); vec ( {v^*_i,t} ); v_t+1 ], ]which reduces to[ β̃, v_i,tA, v^*_i,t, v_t+1argmin [vec ( {v_i,tA} ); vec ( {v^*_i,t} ); v_t+1 ]^2subject to; ; [vec ( {ỹ_i,tA} ); vec ( {ψ_i,t+1} ) ] = [ ⊕_i R_i,t; ⊕_i x_i,t+1 ]vec ( {β_i,t+1} ) + [ L_11,t ⊕_i R_i,tC_i0;00C ][vec ( {v_i,tA} ); vec ( {v^*_i,t} ); v_t+1 ], ]where ỹ_i,tA = y_i,tA - L_12,tv_i,tB. The GLLSP in UTVPSURGLLSP1 is solved in two stages. Firstly, by computing the updating RQD [ L_11,t ⊕_i R_i,tC_i ]P_t+1,1 = [ L̃_11,t 0 ],where L̃_11,t∈^K × K is upper triangular and non-singular, and P_t+1,1∈^ 2K × 2K. Employing TVPSUR.URQD1 in UTVPSURGLLSP1 yields the equivalent GLLSP[β̃, ṽ_i,tA, v_t+1argmin [ vec ( {ṽ_i,tA} );v_t+1 ]^2subject to;; [vec ( {ỹ_i,tA} ); vec ( {ψ_i,t+1} ) ] = [ ⊕_i R_i,t; ⊕_i x_i,t+1 ]vec ( {β_i,t+1} ) + [ L̃_11,t 0; 0 C ][ vec ( {ṽ_i,tA} );v_t+1 ]. ]Secondly, by computing the updating GQRD Q^T_t+1[ ⊕_i R_i,tvec ( {ỹ_i,tA} ); ⊕_i x_i,t+1 vec ( {ψ_i,t+1} ) ] = [⊕_i R_i,t+1 vec ( {y_i,t+1A} );0 vec ( {ψ_i,t+1B} ) ], Q^T_t+1[ L̃_11,t 0; 0 C ]P_t+1,2 = L_t+1 = [ L_11,t+1 L_12,t+1;0 L_22,t+1 ],where R_i,t∈^k_i × k_i, i=1,…,G, L_t+1∈^K × K are upper triangular and non-singular and Q_t+1, P_t+1,2 are orthogonal matrices of order K+G. The GLLSP is now given by[ β̃_i,t+1, v_i,t+1A, v_t+1Bargmin [ vec ( {v_i,t+1A} ); v_i,t+1B ]^2subject to;; [ vec ( {y_i,t+1A} ); vec ( {ψ_i,t+1B} ) ] = [ ⊕_i R_i,t+1; 0 ]vec ( {β_i,t+1} ) + [ L_11,t+1 L_12,t+1;0 L_22,t+1 ][ vec ( {v_i,t+1A} ); v_t+1B ], ]where ( vec ( {v_i,t+1A} )^T v_t+1B^T )^T = P^T_t+1,2 ( (vec ( {v_i,tA} )^T vec ( {v^*_i,t} )^T ) P_t+1,1v_t+1^T )^T. The latter GLLSP is solved in a similar way to TVPSURGLLSP2. The strategy for updating the TVP-SUR model with a single new observation is summarised in Algorithm AlgUTVPSUR.§.§ Multivariate Smoothing Consider now estimating β_i,t based on information up to time M, M>t. That is, at time M the estimates of β_i,t will be re-estimated in order to be revised given the full sample of data. In a way similar to the filtering of the TVP model, it obtains that β_M = β_M-1 + η_M = …= β_t + ∑_s=t+1^M η_s .In order to derive the smoothing estimate of β_i,t, say β_i,t|M, given the full sample, consider the following system of observations for each time-varying regression [ ψ_i,t+1; ψ_i,t+2; ⋮; ψ_i,M ] = [ x_i,t+1; x_i,t+2; ⋮; x_i,M ]β_t|M + [ ϵ_t+1; ϵ_t+2; ⋮; ϵ_M ] + [ x_t+1 0 ⋯ 0; x_t+2 x_t+2 ⋯ 0; ⋮ ⋮ ⋱ ⋮; x_M x_M ⋯ x_M ][ η_t+1; η_t+2; ⋮; η_M ].The latter is given in compact form byy_t+1:M = X_t+1:Mβ_t|M + e_t+1:M + Ã_t+1:Mu_t+1:M,where the error term e_t+1:M + Ã_t+1:Mu_t+1:M has zero mean and variance covariance matrix σ^2 Ω̃_t+1:M = σ^2 ( I_M-t + Ã_t+1:M (I_M-t⊗Σ_η) Ã^T_t+1:M ).The system of regressions in SmoothTVPSUR is used to form the following model [ [ vec ( {y_i,t} ); vec ( {y_i,t+1:M} ) ] = [ ⊕_i X_i,t; ⊕_i X_i,t+1:M ]vec ( {β_i,t|M} ) + [ vec ( {e^*_i,t} ); vec ( {e^*_i,t+1:M} ) ]; ; with[ vec ( {e^*_i,t} ); vec ( {e^*_i,t+1:M} ) ]∼( 0 , [ Ω_t 0; 0 Ω_t|t+1:M ]), ]where the first block of rows in SmoothingTVPSUR is the TVP-SUR model TVPSUR1 used in obtaining the filtering estimates of the model up to time t.Also, the block diagonal elements are given by Ω_i,t+1:M =⊕_i σ_iiÃ_i,t+1:M (I_M-t⊗Σ_i ) Ã^T_i,t+1:M and the off-diagonal elements are σ_ijI_M-t,i,j=1,…,G. Notice that Ã_i,t+1:M, i=1,…,G, are analogous to Ã_t+1:M in TVPmodeltp1M. Also let Ω_t|t+1:M = C̃_t+1:MC̃_t+1:M^T be the Cholesky factorisation of Ω_t|t+1:M. The estimation problem of model SmoothingTVPSUR is now written as the GLLSP[β_i,t|M, v_i,1:t, v_i,t+1:Margmin [ vec ( {v_i,t} ); vec ( {v_i,t+1:M} ) ]^2subject to;; [ vec ( {y_i,t} ); vec ( {y_i,t+1:M} ) ] = [ ⊕_i X_i,t; ⊕_i X_i,t+1:M ]vec ( {β_i,t|M} ) + [ C̃_t0;0 C̃_t+1:M ][ P_1,t^T vec ( {v_i,t} ); vec ( {v_i,t+1:M} ) ]. ]Given SURGQRD1 and TVPSURGLLSP2, it follows that the latter is equivalent to[ β_i,t|M, v_i,tA, v_i,t+1:Margmin [vec ( {v_i,tA} ); vec ( {v_i,t+1:M} ) ]^2subject to;; [vec ( {ỹ_i,tA} ); vec ( {y_i,t+1:M} ) ] = [ ⊕_i R_i,t; ⊕_i X_i,t+1:M ]vec ( {β_i,t|M} ) + [ L_11,t0;0 C̃_t+1:M ][vec ( {v_i,tA} ); vec ( {v_i,t+1:M} ) ]. ]The solution of GLLSPsmothing is analogous to that of the GLLSP in TVPSURGLLSP2 and follows from the updating GQRDQ̃^T_(s)[ ⊕_i R_i,tvec ( {ỹ_i,tA} ); ⊕_i X_i,t+1:M vec ( {y_i,t+1:M} ) ]= [⊕_i R_i,s vec ( {ỹ_i,(s)A} );0 vec ( {y_i,(s)B} ) ]and Q̃^T_(s)[ L_11,t0;0 C̃_t+1:M ]P̃_(s) = [ L̃_11,(s) L̃_12,(s); 0 L̃_22,(s) ].Algorithm AlgSmoothingTVPSUR below summarises the steps for obtaining the smoothing estimates using the proposed method. §.§ Window Estimation of the TVP-SUR ModelWhile a model needs to be updated with the most recent data to keep the estimates up to date, often it is possible that observations will need to be removed from a model so that they no longer affect the estimation results. Observations are excluded from a model because they are old or because they have been detected to be outliers or influential data. Many a time, deleting observations from a model will occur in parallel with adding observations. This is part of the estimation over a rolling window of data and cross validation procedures. Assume that the TVP model TVPSUR1 has been estimated and at time t+1 a rolling window moves forward acquiring one new observation and discarding the oldest one from the model.That is, consider MTVPadd1 and partition y_i,t+1, X_i,t+1 and e^*_i,t+1 as follows y_i,t+1 = [ ψ^(d)_i; y^(r)_i; ψ^(n)_i ] 1 t-1 1 ,X_i,t+1 = [ x^(d)_i; X^(r)_i; x^(n)_i ]ande^*_i,t+1 = [ e^(d)_i; e^(r)_i; e^(n)_i ],where x^(d)_i is the deleted observation from the ith regression of themodel, X^(r)_i are the remaining observations in the model and x^(n)_i is the new observation included in the model. Using the above partitioning, and applying a permutation of the model as in MTVPadd1permuted, [ [ vec ( {y^(r)_i} ); vec ( {ψ^(n)_i} ) ] = [ ⊕_i X^(r)_i; ⊕_i x^(n)_i ]vec ( {β̅_i} ) + [vec ( {e^(r)_i} ); vec ( { e^(n)_i} ) ],; ;[vec ( {e^(r)_i} ); vec ( { e^(n)_i} ) ]∼( 0 , [ Ω̅^(r)0;0Σ ]). ]For the sequential estimation of β̅_i over a window of data, consider the following TVP-SUR model[ [ vec ( {ψ^(d)_i} ); vec ( {y_i,t} ); vec ( {ψ^(n)_i} ) ] = [ ⊕_i x^(d)_i; ⊕_i X_i,t; ⊕_i x^(n)_i ]vec ( {β̅_i} ) + [ vec ( { e^(d)_i} );vec ( {e_i,t} ); vec ( { e^(n)_i} ) ],; ;[ vec ( { e^(d)_i} );vec ( {e_i,t} ); vec ( { e^(n)_i} ) ]∼( 0 , Ω̅). ]The variance covariance matrix in TVPSURwindow2 is given byΩ̅ = [C_d C̃_d,t-d ⊕_i x^(d)_i,tC_i0;0 C̃_t ⊕_i X_i,tC_i0;000C ]Φ[C_d C̃_d,t-d ⊕_i x^(d)_i,tC_i0;0 C̃_t ⊕_i X_i,tC_i0;000C ] ^Tand is such that the effect of the oldest observation is excluded from the current estimate but the new information from the acquired observation will be incorporated. The imaginary unit in TVPSURwindow2 gives the weight needed to downdate the model, that is, to eliminate the affect of the first datum <cit.>.The window estimation problem is then given by[β̃, v^(d)_i,t, v_i,t, v^(n)_iargmin [ vec ( {v^(d)_i,t} ); vec ( {v_i,t} );vec ( { v^(n)_i} ) ]_hsubject to; ; [ vec ( {ψ^(d)_i} ); vec ( {y_i,t} ); vec ( {ψ^(n)_i} ) ] = [ ⊕_i x^(d)_i; ⊕_i X_i,t; ⊕_i x^(n)_i ]vec ( {β̅_i} ) + [C_d C̃_d,t-d ⊕_i x^(d)_i,tC_i0;0 C̃_t ⊕_i X_i,tC_i0;000C ][ vec ( {v^(d)_i,t} ); vec ( {v_i,t} );vec ( { v^(n)_i} ) ], ]where the hyperbolic norm is used together with the imaginary unitto downdate the estimate of the TVP-SUR model <cit.>. Namely, for a complex vectorx, the hyperbolic norm gives x_h = x^H Ψx where Ψ is a signature matrix and ( · )^H denotes the conjugate transpose. Here, ⊕_i x^(d)_i,tC_i and ⊕_i X_i,tC_i is the new information incorporated into the variance covariance matrix of the first t observations due to the inclusion of the new data point. Notice that the information which updates the covariance matrix of the deleted observations, i.e. x^(d)_i,tC_i, is multiplied with the imaginary unit since it has to be excluded from the model. The GLLSP in TVPSURGLLSPwindow is solved by computing the corresponding RQ and generalised QR decompositions in a manner similar to the updating but using hyperbolic transformations when information needs to be removed from the TVP-SUR model TVPSUR1.§ COMPUTATIONAL EXPERIMENTS Experiments have been designed to assess the computational efficiency of the proposed algorithms. Specifically, the strategies presented herein have been compared with existing ones which estimate the model afresh.The computational efficiency of one algorithm compared to another algorithm is defined as the ratio of the computational cost of the two algorithms. Here, the execution time (in seconds) required by each algorithm to compute the desired estimate is presented in order to determine the computational efficiency of the proposed strategies. To analyse the computational performance of the proposed methods and their counterparts, experiments based on synthetic data have been conducted. For the efficient implementation of the new methods, sequential and recursive strategies which exploit the special sparse structure of the matrices are employed<cit.>. Three cases with the corresponding algorithms have been considered.Specifically, it is assumed that the TVP-SUR model has been estimated using the initial dataset and then, new observations enter the dataset and/or old observations are deleted. Firstly, the problem of estimating the TVP model MTVPadd1, which incorporates the effect of a single observation is investigated, by estimating the model afresh (see Algorithm AlgTVPSUR in Section RecursiveTVPSUR)and by implementing the new Updating algorithm (see Algorithm AlgUTVPSUR in Section UpdateTVPSURmodel) which solves the GLLSP UTVPSURGLLSP1. Table TVPtable1 presents the execution times, in seconds, of both algorithms which recursively add the effect of one new observation into the model 100 times. That is, the execution times presented in the third and fourth columns of Table TVPtable1 are the sum of re-estimating the model with one extra observation 100 times. Examples with various numbers of time-varying regressions G and unknown parameters K are shown. The times in Table TVPtable1 confirm that the Updating algorithm outperforms significantly the Afresh algorithm for the estimation of the TVP model. The computational efficiency of the proposed method increases as the dimensions of the models increase.Secondly, consider deriving the smoothing estimates in SmoothingTVPSUR by solving the GLLSP SmoothingTVPSURGLLSP1 afresh or by solving GLLSP GLLSPsmothing which utilises previous computations. The Afresh smoothing algorithm is a variation of Algorithm AlgTVPSUR whereas the new Revising algorithm is Algorithm AlgSmoothingTVPSUR in Section SmoothingTVPSURmodel. Table TVPtable2 compares the two algorithms when each of them, at the end of the period M=60, goes backwards 5 points in time to compute the smoothing estimates.That is, β_M-i|M, i=1,…,5 are estimated. The time presented in each case is the average time required of the corresponding algorithm after 100 iterations. The Revising algorithm is considerably computationally more efficient than the algorithm that solves the GLLSP SmoothingTVPSURGLLSP1 afresh. The computing performance of the Revising algorithm becomes more effective when both the number of regressions G and the number of unknown parameters K increase. Finally, consider estimating the model over a rolling window of data. Namely, let the fixed size estimation window move forward at one point of time to capture the information from the next data point and while excluding the effect of the oldest data point.That is, estimate TVPSURwindow by employing Algorithm AlgTVPSUR or by solving the GLLSP TVPSURGLLSPwindow using the up-downdating algorithm in <cit.>. Table TVPtable3 reports the total time to estimate the model over a window (of fixed size) which rolls ahead one data point 100 times. The ratios of the execution times in Table TVPtable3 confirm that the recursive Up-downdating algorithm performs better than the Afresh algorithm and, similarly to the previous computational results, the efficiency increases when the models' dimensions increase.Overall, the results show that the recursive algorithms which utilise previous computations outperform the algorithms which estimate the model afresh.The computational efficiency becomes notable when both G and K increase. Experiments have been conducted for models of other dimensions; the computational efficiency is similar to Tables TVPtable1, TVPtable2 and TVPtable3.The results show the practical usability of the proposed methods in estimating TVP models of high dimensions. § CONCLUSIONS AND FUTURE WORK The estimation of the multivariate TVP model using alternative numerical methods has been investigated. The TVP model can be written as a general linear model and therefore be estimated with the method of GLS <cit.>. However, using GLS to estimate such a model is computationally expensive and numerically inaccurate due to the computation of large matrix inverses. Therefore, the proposed method considers the equivalent GLLSP to provide the estimates of the model. The GLLSP method has been shown to be computationally faster and numerically more stable than solving the normal equations to obtain the GLS estimator <cit.>. Herein, numerical methods have been investigated for the efficient estimation of the TVP and TVP-SUR models. Various cases have been examined for the efficient estimation of the model when the estimates of the unknown parameters need to be re-computed after changes occur in the dataset. Specifically, the case of updating the model with one new observation and also that of deriving the smoothing estimates of the model are examined. Finally, the simultaneous addition and deletion of observations (up-downdating) within the context of rolling window estimation is explored.The algorithms developed herein, take advantage of the special sparse structure of the models and utilise efficiently previous computations. Experiments have been carried out to analyse the computational performance of the proposed algorithms which update the model with one new observation, compute the smoothing estimates of the model and estimate the model over a rolling window of data. The computational results show that the proposed algorithms are computationally more efficient than their counterparts and that their performance becomes more significant in high dimensions. This demonstrates the usefulness of the proposed methods in practical problems of large-scale TVP models.Future work will consider the estimation of multivariate TVP models using a high-dimensional setting where the number of covariates exceeds the sample size, resulting in a singular variance covariance matrix. Having more parameters to estimate than available observations will affect the estimation of the initial model (see TVPSUR1), but the updating when an extra data point arrives will be straightforward using the novel methods developed herein. Furthermore, models with a more complex time-varying structure should be addressed. For example, to allow for a time-varying variance-covariance matrix. Additionally, the estimation of time-varying parameter vector autoregressive models using the proposed numerical methods and their extension to model selection merit investigation.tocsectionBibliography apalike | http://arxiv.org/abs/1703.09062v2 | {
"authors": [
"Stella Hadjiantoni",
"Erricos J. Kontoghiorghes"
],
"categories": [
"stat.ME",
"stat.CO"
],
"primary_category": "stat.ME",
"published": "20170327133339",
"title": "A numerical method for the estimation of time-varying parameter models in large dimensions"
} |
APS/[email protected] Laboratory of Solid State Microstructures and Department of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, ChinaSpecific heat has been measured in FeSe single crystals down to 0.414 K under magnetic fields up to 16 T. A sharp specific heat anomaly at about 8.2 K is observed and is related to the superconducting transition. Another jump of specific heat is observed at about 1.08 K which may either reflect an antiferromagnetic transition of the system or a superconducting transition arising from Al impurity. We would argue that this anomaly in low temperature region may be the long sought antiferromagnetic transition in FeSe. Global fitting in wide temperature region shows that the models with a single contribution with isotropic s-wave, anisotropic s-wave, and d-wave gap all do not work well, nor the two isotropic s-wave gaps. We then fit the data by a model with two components in which one has the gap function of Δ_0(1+α cos2θ). To have a good global fitting and the entropy conservation for the low temperature transition, we reach a conclusion that the gap minimum should be smaller than 0.15 meV (α = 0.9 to 1), indicating that the superconducting gap(s) are highly anisotropic. Our results are very consistent with the gap structure derived recently from the scanning tunneling spectroscopy measurements and yield specific heat contributions of about 32% weight from the hole pocket and 68% from the electron pockets.Subject AreasCondensed Matter Physics, SuperconductivityHighly anisotropic superconducting gaps and possible evidence of antiferromagnetic order in FeSe single crystals Guan-Yu Chen, Xiyu Zhu, Huan Yang, and Hai-Hu Wen================================================================================================================ The iron-selenium (FeSe) is one of the iron based superconductors with the simplest structure <cit.>. The later effort in enhancing the superconducting transition temperature from about 8.5 K to 37 K by pressure was encouraging <cit.>. However, the interest has been revived recently by exploring the phase diagram under pressure <cit.>. In the normal state under ambient pressure, there is a structural transition from tetragonal to orthorhombic at about 90 K <cit.>. This transition has been proved to be accompanied by the formation of the nematic electronic state <cit.>. In contrast, however, there is no evidence of antiferromagnetic long range order found in the system although the normal state is dominated by very strong spin fluctuations <cit.>. This may help to resolve the disputes about the origin of the nematicity <cit.>. Under pressure, together with the enhancement of superconducting transition temperature, an antiferromagnetic (AF) order appears <cit.>. However, it remains unknown how the AF order extends to the superconducting state in the low pressure region. It might be possible that this AF order is hidden under the superconducting dome and most measurements have been undertaken above the possible AF transition temperature. Angle resolved photoemission (ARPES) <cit.> and scanning tunneling spectroscopy (STS) <cit.> have revealed the existence of both electron and hole pockets with very small area, showing the approximate semi-metal behavior. The closeness of the band edge to the Fermi energy, or the comparable scales of the Fermi energy E_F and the superconducting gap Δ, suggests a possible BCS-BEC crossover <cit.>. The pairing order parameter has been detected by many experimental techniques. The first STS experiment on the sister system FeSe_1-xTe_x has revealed the sign reversal s^± gap. However, in FeSe thick films, a V-shaped spectrum has been observed suggesting the d-wave gap structure <cit.>. Thermal conductivity measurement has been carried out which suggests a nodeless gap but with a very small gap minimum <cit.>. In this paper, we report low temperature specific heat on the FeSe single crystal. We have fitted the experimental data with variable kinds of gap structures. Our results suggest the superconducting order parameter with two components of highly anisotropic gaps. In addition, a clear step like specific heat anomaly at about 1.08 K gives the possible evidence of an AF order in low temperature region. § EXPERIMENTThe FeSe single crystals used for this study were grown by the chemical vapor transport method using Fe and Se powder as the starting materials <cit.>. The mixture of Fe_1.04Se, KCl and AlCl_3 with the ratio of 1:2:4 were put at one end of the quartz tube in a glove box filled with argon. We then sealed the quartz tube and place it into a horizontal tube furnace with tunable temperature gradient. The furnace was heated up to 430 ^∘C and kept for 30 hours, then by adjusting the program the temperature at the end without reactant was tuned to about 370 ^∘C, in order to establish a temperature gradient. The tube was sintered with this temperature gradient for 6 weeks, and finally FeSe single crystals were grown at the cold end of the tube. The magnetization was measured by using a superconducting quantum interference device (SQUID-VSM 7T, Quantum Design). The resistivity was measured with a physical property measurement system (PPMS 16T, Quantum Design) by the standard four probe method. The specific heat was measured with thermal-relaxation method by an option of the PPMS with a He3 insert. This facility allows us to measure specific heat down to 0.414 K. During the specific heat measurement, magnetic fields up to 16 T was applied parallel to the c-axis of the crystal.§ RESULTS AND DISCUSSION§.§ Basic characterization of the sampleThe temperature dependence of resistivity ρ(T) at zero magnetic field is shown in Fig. <ref>. We can see a clear kink at about 88 K, which is related to the structural transition from tetragonal to orthorhombic phase and the nematic transition as well. The onset of the superconducting transition T_c^onset≈ 8.7 K (defined by the crossing point of the extrapolated lines of the normal state and the steep transition part), and the system realizes zero resistivity state at T_c0≈ 8.1 K, which can be seen in the right-bottom inset of Fig. <ref>. The transition width Δ T_c, which is defined asΔ T_c = T_c^onset-T_c0, is 0.6 K. And the residual resistivity ratio (RRR), which is determined by the ratio of ρ(300K)/ρ(T=0K), is about 25.2, where ρ(T=0K) is obtained by linearly extrapolating the normal state resistivity down to zero temperature. The temperature dependence of zero-field-cooled (ZFC) and field-cooled (FC) magnetization at 20 Oe is shown in the upper-left inset of Fig. <ref>. The superconducting volume calculated from the magnetization data is larger than 100% (due to the demagnetization effect), indicates the bulk superconductivity of our sample. The large RRR value and superconducting volume both confirm the high quality of our samples. §.§ Measurement on specific heat and fitting with different gap structuresSpecific heat is sensitive to the quasiparticle density of states (DOS) at the Fermi energy, so it is a useful way to detect superconducting gap structure at low temperatures. In iron based superconductors, specific heat measurements have been done in plenty systems <cit.> showing the multigap feature. In order to study the superconducting gap structures of the FeSe system, we have measured the specific heat of an FeSe single crystal and the temperature dependence of specific heat at zero field is presented in Fig. <ref>(a). A sharp jump of specific heat coefficient can be seen at about 8.2 K, which is corresponding well to the superconducting transition detected by resistivity and magnetization measurements. Another anomaly occurs at about 1.08 K. The second transition at low temperature may reflect the possible antiferromagnetic transition, which have not been observed among previous researches. However, since AlCl_3 have been used during the progress of crystal growth, the possibility that the second jump is arisen from Al impurity (T_c = 1.17 K) could not be excluded. But we will argue that the latter is unlikely. The residual specific heat γ_0=C/T|_T→ 0 determined by extrapolating the data of C/T down to 0 K is negligible, which seems to against the possibility of any nodal gaps in the system, and is consistent with the recent thermal conductivity measurements <cit.>.It is known that the specific heat consists both of the electronic and phonon contributions. Since the phonon contribution will prevail over the electronic part in moderate temperature region, it is very important to extract the superconducting electronic specific heat. Thus the data of the normal state above T_c is fitted by the equation of Debye model which reads as C_n/T =γ_n+β T^2 +η T^4, where γ_n is the normal state electronic specific coefficient, or called as the Sommerfeld coefficient and β T^2 +η T^4 are the phonon contributions according to the Debye model. The fitting curve is shown by the red solid line in Fig. <ref>(a). The fitting function yields γ_n = 6.4 mJ/mol· K^2, β= 0.44 mJ/mol· K^4 and η= -0.00005 mJ/mol· K^6. The Debye temperature estimated here is about 206 K, which can be obtained by using the equation Θ_D = (12π^4 k_B N_A Z/5β)^1/3, where k_B is the Boltzmann constant, N_A is the Avogadro constant and Z is the number of atoms in one unit cell. Our γ_n and Θ_D are close to the values of earlier studies <cit.>. The specific heat jump at T_c, namely Δ C/γ_n T_c with Δ C estimated through entropy conservation near T_c is about 1.6. This is larger than 1.43 predicted by the Bardeen-Cooper-Schriefer (BCS) theory in the weak coupling limit, which may indicate moderate strong coupling in FeSe. The temperature dependence of superconducting electronic specific heat is plotted in Fig. <ref>(b), which is obtained by subtracting the normal state data C_n/T. There is a very small tail above T_c, which may indicate the narrowsuperconducting fluctuation region in FeSe (less than 1 K). The small jump at around 1.08 K can be clearly seen in the insets of Fig. <ref> (a) and (b). A similar anomaly was also seen in earlier report <cit.>, which however exhibits like a shoulder around 2 K there, and the authors attributed this anomaly to a very small second superconducting gap. Since the low temperature anomaly observed here is a clear jumping step, not a knee like, this excludes the possibility that it is a second superconducting gap of the FeSe system, since a multi-band fitting using the same T_c would give rise to a "knee" or "hump" here, not as a sharp step. We would like to attribute this step to either the antiferromagnetic order transition, or the impurity of Al which has a T_c of about 1.17 K. We will address to this issue later.The structure of superconducting gap is a significant issue in determining the pairing mechanism, while, as far as we know, there is no consensus yet concerning whether nodes exist in the superconducting gap of FeSe or not. The STS measurements <cit.> suggest that there might be nodes in this system, while the thermal conductivity measurement <cit.> and the specific heat measurement <cit.> support a nodeless gap. In order to obtain more information about the gap structure of FeSe, we use BCS formula to fit the electronic specific heat in superconducting state, the formula based on BCS theory is shown below, γ_e=4N(E_F)/k_BT^3∫_0^+∞∫_0^2πe^ζ/k_BT/(1+e^ζ/k_BT)^2 (ε^2+Δ^2(θ,T)-T/2dΔ^2(θ,T)/dT) dθ dε, where ζ=√(ε^2+Δ^2(T,θ)), the angle dependence of the gap is entered through Δ (θ) here and the temperature dependence is generated from the BCS gap equation, ε=ħ^2k^2/2m is the kinetic energy of electrons counting from the Fermi energy. For the fitting to one gap, we just use above formula. While for the fitting with two gaps, we use a linear combination of two contributions, each one has a gap. Variable different gap structures have been used to fit our data: single s-wave gap Δ(T, θ) = Δ_0(T), single d-wave gap Δ(T, θ)= Δ_0(T)cos2θ, single extended s-wave gap Δ(T, θ)= Δ_0(T)(1 + α cos2θ), mixture of two isotropic s-wave gapsΔ(T, θ) = Δ_1(T)+Δ_2(T) and mixture of an s-wave gap and an extended s-wave gap. The α in the expression of extended s-wave is the parameter that represents anisotropy. One can see that α = 0 corresponds to the case of an isotropic s-wave gap, while α = 1 corresponds to a zero gap minimum. The optimized fitting parameters of different models are listed in Talbe <ref> and the fitting curves are shown in Fig. <ref>(a)-(c). The insets in Fig. <ref>(a)-(c) provide the enlarged views below 2 K, so we can check the fitting results at low temperatures. Although the origin of the second jump at 1.08 K remains unknown, it will serve as a very nice indicator for checking the appropriate gap structure through the entropy conservation of this transition, especially about the gap minimum. In Fig. <ref>(a), we show the fitting by using three cases of single gap, namely a single isotropic s-wave, single d-wave and singe extended s-wave. The formula used for the extended s-wave naturally has a two-fold symmetry if α is not zero, this form is chosen because it is well-known that the FeSe system has a nematic state. One can see that both of the single isotropic s-wave model and the single extended s-wave model (α = 1) are not possible to describe the experimental data, judging both from the global fitting quality in wide temperature region and the low temperature part. However, it seems a single d-wave fits the data much better. This may suggest that the gap should be highly anisotropic in order to get a global fitting in wide temperature region. A closer scrutiny finds that the d-wave fitting still has some deviation to our data between 2 K and 4 K. Combining with the fact that the residual specific heat coefficient in the zero temperature limit is negligible, we rule out the possibility of d-wave gap(s).These results indicate that a model with only one component/gap is not enough to describe the data. This is reasonable since in the realistic case, there are at least two contributions, one from the hole pocket and one from the electron pocket. The model with two isotropic s-wave gaps is also tried and shown by the red solid line in Fig <ref>(b), which shows a very poor fitting near T_c and the deviation in low temperature is also clear. Therefore isotropic s-wave gap(s) with either single or double contributions cannot fit the data, and anisotropic gaps should be taken into account. Fig. <ref>(b) and (c) show also fitting results by using the gap model with the mixture of an s-wave gap and an extended s-wave gap. Three different values of α have been tried to fit our data. All of these models work well above 2 K when α ranges from 0.9 to 1.05. However, these curves behave quite differently at low temperatures below 2 K. For the case of α = 0.9, as shown by the inset of Fig. <ref>(b), the fitting is relatively better and the low temperature part seems also okay in terms of entropy conservation about the low temperature anomaly. When α = 1, the bending down of the fitting curve at low temperatures seems a little too much to satisfy the entropy conservation. For the case of α = 1.05, the bending down is simply too much without possibility of the conservation of the entropy of the low temperature anomaly. For the case of α = 0.8 or below, we see a flattening of the data starting at higher temperatures, which makes no case for the entropy conservation for the low temperature anomaly. Therefore, in order to have a good global fitting in wide temperature region and the conservation of entropy around the low temperature anomaly, we reach the conclusion that α locates around 0.9. Thus we can conclude that, the gap in FeSe should be highly anisotropic. Taking α = 0.9 to 1, we believe that the minimum of the gap is about 1.5×(1-0.9) = 0.15 meV or smaller. We must mention that, what we used here for the fitting are two components with one isotropic s-wave and one extended s-wave. One can also use a model with two extended s-wave gaps, but that requires more fitting parameters, which would give more uncertainty. About judging to what extent the gap anisotropy is, fitting to specific heat by using one s-wave plus an extended s-wave, or two extended s-wave gaps gives no big difference. §.§ Fitting with the gaps determined by STS experiment Recently, Sprau et al. have done the STS measurements and used Bogoliubov quasiparticle interference imaging (BQPI) to measure the superconducting gap of FeSe <cit.>. They proposed that the two superconducting gaps Δ_α and Δ_ε, which are located at hole pocket and electron pocket respectively, are both extremely anisotropic but nodeless. This highly anisotropic gaps are induced by the orbital selective pairing of the d_yz orbital. In order to check the validity of their model, we use the gap structure determined by them to fit our specific heat data. Since the gaps are not the simple form of a sinusoidal function, therefore we need to fit their data first to an angle dependent function. The angle dependence of Δ_α and Δ_ε from Ref.<cit.> are plotted by the open symbols in Fig. <ref>(a). The definition of the angle θ is illustrated in the inset of Fig. <ref>(a). We try to fit each gap function with a combination of several harmonic of cosine functions, namely Δ=x_1+x_2cos(2θ)+x_3cos(4θ)+x_4cos(6θ)+x_5cos(8θ)+x_6cos(10θ). The fitting results to the gaps are shown as red line and green line in Fig. <ref>(a) and the fitting parameters are listed in Table <ref>. With the expression of the gap structures, we obtain a remarkably good fit to the specific heat data in wide temperature region. In order to check whether the entropy is conserved about the low temperature anomaly, we also try to describe the second jump at 1.08 K by using an isotropic s-wave gap. This treatment is valid for the second anomaly either it is an AF or a superconducting transition. The blue line in Fig. <ref>(b) shows the fitting with the gap structure in Fig. <ref>(a) plus a small s-wave gap with Δ_s = 0.25 meV with T_c = 1.08 K, and the fitting result is perfect. The fitting yields also the contribution of 32% weight from the hole pocket and 68% from the electron pocket. It seems that the gaps determined from the STS measurements can get good support from our specific heat data. The weights determined for different pockets are very important to further investigate the pairing mechanism of FeSe system.§.§ Magnetic field dependence of specific heat in low temperature regionIn order to check whether the gaps have very large anisotropy, we have measured the specific heat of the FeSe single crystal at different magnetic fields. The superconducting electronic specific heat at different fields shown in Fig. <ref> (a) are obtained by deducting the normal state contribution which has been derived above. Here we use the same data for the normal state for all fields supposing only phonon and fermionic contributions in the normal state and both are not dependent on magnetic field. By fitting the data with γ = a + bT^n in the temperature window of about 1.36 K to about 3.8 K, we can get the superconducting electronic specific heat at 0 K at different magnetic fields. Here we must mention that, for the data at about 9 to 12 tesla, the superconductivity related anomaly or enhancement of specific heat has already come to this temperature region, therefore the fitting is done in much narrow temperature region, otherwise the fitting is invalid and would yield unreasonable results about the zero temperature intercept. Above about 12 T, the sample has already come to the normal state above 1.4 K. One can judge this by looking at the data at 14 and 16 tesla, which shows as a plain flat line. Through these treatments and fitting, we can have the magnetic field induced specific heat coefficient defined by Δγ = [C(H) - C(0)]/T|_T→ 0K. The fitting parameters are listed in Table <ref>. The data are shown in Fig. <ref> (b). We then fit the data Δγ (H) =[C(H)-C(0)]/T with the formulaΔγ = p B + q B^1/2 with p:q = 1 : 9 (B in unit of tesla), which seems working quite well for the data. As we know, the value of Δγ depends on the magnetic field induced DOS at the Fermi level. When the superconducting gap is isotropic s-wave, the field dependence of Δγ will follow a linear behavior, because only the vortex core part contributes the quasiparticles in the low temperature limit and the density of vortex linearly goes up with the magnetic field. If the gap structure is nodal like, for example d-wave, a square root relation will be observed due to the Doppler shift effect of the DOS outside the vortex core region <cit.>, which leads to a square-root temperature dependence of magnetic field. The relatively good fit with the formula Δγ = p B + q B^1/2shown by the red solid line in Fig. <ref> (b) does not manifest the d-wave gap, but suggests that the gap should have some deep minimum which leads to clear Doppler shift term. Thus the gaps should have anisotropic feature, and at least one gap should have the minimum which is close to zero. §.§ Concerning the specific heat jump at around 1.08 KNow let's address the possible origin of the specific heat jump at around 1.08 K. In the process of growing the crystal, we have used AlCl_3 as the flux because of its low melting temperature, there is a possibility that this specific anomaly is coming from the impurity of Al. If using the typical value of γ_n and T_c = 1.17 K for Al, we get a composition fraction of about 2% of Al in the sample. Although we could not exclude the possibility of Al as the contribution to this low temperature anomaly, we can however give several arguments to question that this may not be the case. The arguments are as follows: (1) This anomaly was also seen in similar shape in samples without AlCl_3 as flux <cit.>. In that work, a similar anomaly of specific heat appears at around 1.25 K and it strongly depends on the concentration of interstitial Fe. (2) In another earlier study <cit.>, the authors observed a similar specific heat anomaly at around 2-3 K, this is much higher than the critical temperature of Al (T_c = 1.17 K). (3) Most importantly, in our low temperature specific heat measurements, we find that this anomaly appears to a magnetic field at least up to 2000 Oe, but the critical field H_c of Al is about 99 Oe. (4) Finally, from a chemical point of view, it is difficult to understand why AlCl_3 can easily separate into pure Al in the growing process. Therefore we would like to attribute this anomaly as the possible appearance of the long sought antiferromagnetic order. As far as we know, the AF order has never been observed in FeSe at ambient pressure, but it may exist with a very low Neel temperature. Therefore this AF order is very fragile and its appearance will depend on the subtle change of the properties of the sample, for example the concentration of interstitial Fe etc. <cit.>. In this sense, the absence of the AF order in some samples does not imply that it will not happen in other samples, although the superconducting transition temperatures are quite close to each other. As far as we know, in previous neutron diffraction measurements <cit.>, nobody has measured down to the temperature of about 1.08 K. And the usual magnetization measurement to detect the AF order does not work since it is hidden deeply in the superconducting state, and the magnetic screening due to Meissner effect or the vortex state can already cover all the signal from the AF order. Thus we would believe that this low temperature anomaly is most likely to be the AF transition. If this is true, the phase diagram of FeSe needs to be corrected. § SUMMARYSpecific heat down to 0.414 K under magnetic fields up to 16 T has been measured in FeSe single crystals. Variable kinds of gap functions have been tried to fit the data. It is found that a single gap, regardless of the gap functions (isotropic s-wave, anisotropic s-wave, and d-wave gap) cannot be used to fit the data. A combination of two gaps with at least a highly anisotropic gap can fit the data yielding the gap minimum of about 0.15 meV or smaller. We further find that the gaps determined by the recent STS experiment can describe the data perfectly. 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"authors": [
"Guan-Yu Chen",
"Xiyu Zhu",
"Huan Yang",
"Hai-Hu Wen"
],
"categories": [
"cond-mat.supr-con"
],
"primary_category": "cond-mat.supr-con",
"published": "20170325115956",
"title": "Highly anisotropic superconducting gaps and possible evidence of antiferromagnetic order in FeSe single crystals"
} |
[email protected] Quantum Technology Lab, Dipartimento di Fisica, Università degli Studi di Milano, I-20133 Milano, ItalyDipartimento di Fisica e Astronomia, Università di Firenze, I-50019, Sesto Fiorentino (FI), Italy INFN, Sezione di Firenze, I-50019, Sesto Fiorentino (FI), ItalyDipartimento di Fisica e Astronomia, Università di Firenze, I-50019, Sesto Fiorentino (FI), Italy INFN, Sezione di Firenze, I-50019, Sesto Fiorentino (FI), ItalyQuantum Technology Lab, Dipartimento di Fisica, Università degli Studi di Milano, I-20133 Milano, ItalyISC-CNR, UOS Dipartimento di Fisica, Università di Firenze, I-50019, Sesto Fiorentino (FI), Italy Dipartimento di Fisica e Astronomia, Università di Firenze, I-50019, Sesto Fiorentino (FI), Italy INFN, Sezione di Firenze, I-50019, Sesto Fiorentino (FI), ItalyQuantum Technology Lab, Dipartimento di Fisica, Università degli Studi di Milano, I-20133 Milano, Italy INFN, Sezione di Milano, I-20133, Milano, ItalyWe address the dynamics of a bosonic system coupled to either a bosonic or a magnetic environment, and derive a set of sufficient conditions that allow one to describe the dynamics in terms of the effective interaction with a classical fluctuating field. We find that for short interaction times the dynamics of the open system is described by a Gaussian noise map for several different interaction models and independently on the temperature of the environment. In order to go beyond a qualitative understanding of the origin and physical meaning of the above short-time constraint, we take a general viewpoint and, based on an algebraic approach, suggest that any quantum environment can be described by classical fields whenever global symmetries lead to the definition of environmental operators that remain well defined when increasing the size, i.e. the number of dynamical variables, of the environment. In the case of the bosonic environment this statement is exactly demonstrated via a constructive procedure that explicitly shows why a large number of environmental dynamical variables and, necessarily, global symmetries, entail the set of conditions derived in the first part of the work.Effective description of the short-time dynamics in open quantum systems Matteo G. A. Paris December 30, 2023 ======================================================================== § INTRODUCTIONThe modeling of any open quantum system (OQS) inherently implies that of its surroundings. However, knowing the quantum structure of the total Hamiltonian, including the details of the couplings between the principal system A and its environment Ξ, does not usually suffice to develop a simple and meaningful model of the overall system, due to Ξ being made of a very large number N of quantum components, a fact that we will hereafter take as integral to the definition of environment. On the other hand, knowing specific features of Ξ may help selecting a suitable formalism and/or some appropriate approximations, so as to devise the most effective strategies for tackling problems that cannot be otherwise studied. As a matter of fact, the modeling of an effective description of Ξ and of its influence on A usually stems from intuitive and phenomenological arguments <cit.>, or even from an arbitrary choice, rather than a formal derivation. One of the reasons why this is so typical in the study of OQS is that the large-N theories that have been extensively developed and used in quantum-field-theory since the 1970s (comprehensive bibliographies and discussions can be found for instance in Refs. <cit.>) are not trivially applicable when the large-N system is not isolated, but rather coupled with a small, invariably quantum, principal system. Unless one decides that the latter is not “principal” at all, and can be hence neglected, several foundational issues arise in this setting, due to the difficult coexistence of quantum and classical formalisms, possibly made worse by the presence of thermal baths or stochastic agents. Having this issue in mind, here we analyze a specific situation where a principal quantum system A interacts with an equally quantum environment Ξ, which is put into contact with a further external system T. If Ξ is macroscopic and T is a thermal bath at high temperature, it may appear intuitive, and naively understood, that A effectively evolves as if it were under the influence of a classical fluctuating field. This statement, however, has the nature of an ansatz as far as it is not formally inferred, and conditions ensuring its validity are not given. Several OQS have been indeed investigated to assess whether an effective description is viable,where the effects of the environment are described either with a stochastic Schrödinger equation <cit.> or in terms of the coupling with a classical fluctuating field <cit.>. In the latter case, full equivalence has been shown only for single-qubit dephasing dynamics <cit.>, with an explicit construction of the corresponding classical stochastic process. General arguments valid also for bipartite systems have been discussed <cit.> and the effects of the interaction with a classical field have been investigated in detail <cit.>. Parametric representation have also been used to show that classical variables can emerge in quantum Hamiltonians as environmental degrees of freedom <cit.>.In this work we scrutinize the general idea that the dynamics of a quantum system with a macroscopic environment may be effectively described by a non-autonomous, i.e. time-dependent, Hamiltonian acting on the principal system only. In particular, we critically inspect the conditions for the validity of this hypothesis as a tool to understand whether it stems from Ξ being macroscopic, or the temperature being high, or from enforcing some other specific condition. To this aim, in Secs. <ref> and <ref> we study two specific models that go beyond the pure-dephasing, and whose analysis will also serve as explicit guidance for the more abstract approach presented in Sec. <ref>.In particular, in Sec. refs.bosonic we consider the case where A is a bosonic mode coupled with an equally bosonic environment, hereafter called B, which is made of N distinguishable modes that do not interact amongst themselves. The Hamiltonian readsH= ν a^† a +∑_k^N(λ_1k a^†+λ_2ka)b_k+∑_k^N(λ_1k^* a+λ_2k^*a^†)b_k^† + ∑_k^N ω_k b_k^† b_kwhere [a,a^†]=1 and [b_k,b^†_k']=δ_kk', with ν,ω_k∈ℝ and λ_1k,λ_2k∈ℂ, ∀ k. Also, we have set ħ=1, as done throughout this work. Studying the evolution of the reduced density matrix for the principal system, we show that the short-time dynamics defined by Eq. (<ref>) with ω_k ≃ω ∀ k, can be described by an effective Hamiltonian acting on A only, H^ eff_ A(ζ), where the functions ζ embody the remnants of B in the form of classical, possibly fluctuating fields, depending on external parameters such as time and temperature. In what follows we will refer to the condition ω_k ≃ω ∀ k as defining a narrow energy spectrum. The above model has a sibling that describes the case of a spin environment, hereafter called S, made by N distinguishable spin-1/2 particles that do not interact amongst themselves. Its dynamics is studied in Sec. <ref>, as described by the HamiltonianH^=ν a^† a +∑_i^N(g_1i a^†+g_2ia)σ^-_i +∑_i^N(g_1i^* a+ g_2i^*a^†)σ_i^+ + ∑_i^N f_i σ^z_iwhere [σ^+_i,σ^-_i']=2δ_ii'σ_i^z, [σ^z_i,σ^±_i']=±δ_ii'σ^±_i, f_i∈ℝ and g_1i,g_2i∈ℂ, ∀ i. Despite differences with the case of a bosonic environment emerge, essentially due to the specific algebra of the spin operators, the short-time dynamics of this model for f_i∼ f ∀ iis also found to be properly described by an effective Hamiltonian H^ eff_ A(ζ). Upon inspecting the dynamics of both systems in order to retrace the derivation of the short-time dynamics, we notice that no explicit condition on the value of N is involved. This is somehow surprising, given that B and S are named environment insofar as the number N of their quantum components is large, virtually infinite in the case of a macroscopic environment. Therefore, in order to understand whether a relation exists between a large value of N and the assumptions of short-time and narrow energy spectrum ω_k ≃ω used in Secs. <ref>-<ref>, in Sec. <ref> we take on the model (<ref>) from a more abstract viewpoint. More specifically: we generalize the well-established procedures for deriving classical theories as large-N limit of quantum ones <cit.> to the case of composite quantum systems, and find that replacing quantum operatorsby classical fields for N→∞ requires that environmental operators stay well defined in such limit which, in turn, implies the environment to feature some global symmetry. In particular, we show that the renormalization of the couplings, which is necessary for the N→∞ limit to stay physically meaningful, reflects upon the short-time condition previously used. Also, we will discuss how a narrow environmental energy spectrum ω_k ≃ω ∀ k is the key feature that guarantees the existence of a global symmetry in the theory defined by Eq. (<ref>), namely the symmetry under permutation of different modes. Overall, collecting our diverse results, we put forward the conjecture that non-autonomous Hamiltonians for closed quantum systems describe the short-time dynamics of interacting models involving at least one macroscopic subsystem. We also comment upon the symmetry properties allowing this subsystem to emerge as a macroscopic one, and the related features of its energy spectrum. Finally, we discuss the role of such symmetry properties in the design of a general procedure for deriving an effective non-autonomous Hamiltonian from an interacting microscopic model. Sec. <ref> closes the paper with some concluding remarks.§ BOSONIC ENVIRONMENTWe consider the Hamiltonian (<ref>), for either 1) λ_2k=0, with λ_k≡λ_1k finite (linear exchange), or 2) λ_1k=0, with λ_k≡λ_2k finite (parametric hopping), ∀ k, i.e.H_1= ν a^† a + ∑_k ω_k b_k^† b_k + ∑_k ( λ_k^* a b_k^† + λ_k a^† b_k ) ,H_2= ν a^† a + ∑_k ω_k b_k^† b_k + ∑_k ( λ_k^* a^† b_k^† + λ_k a b_k ) . We hereafter use the index j=1,2 to refer to the exchange and hopping case, respectively. The Heisenberg equations of motion (EOM) for the mode operators are Exchange:ȧ = i [H_1,a] = - i ν a - i ∑_k λ_k b_k,ḃ_k = i [H_1,b_k] = - i ω_k b_k - i λ_k^* a, Hopping:ȧ = i [H_2,a] = - i ν a- i ∑_k λ_k^* b_k^† ,ḃ^†_k= i [H_2,b_k^†] =i ω_k b^† _k+ i λ_k a.If the spectrum of the environment is narrow enough to write ω_k≃ω ∀ k, the above EOM can be written asExchange: ȧ = - i ν a - i Λ b , ḃ = - i ω b - i Λ a ,Hopping: ȧ = - i ν a - i Λ b^† , ḃ^† = i ω b^† + i Λ a ,where the bosonic operator b is defined asb ≡1/Λ∑_kλ_kb_k , with Λ^2≡∑_k|λ_k|^2 .The above Eqs. (<ref>-<ref>) are the same EOM that one would obtain starting from the two-mode bosonic HamiltoniansExchange: ν a^† a+ω b^† b+ Λ (a b^†+a^† b) , Hopping: ν a^† a+ω b^† b+ Λ (a^† b^†+a b) ,describing two oscillators, with different frequencies ν and ω, exchanging quanta through a linear interaction. Notice, though, that such direct relation only exists in the case of a narrow spectrum, ω_k∼ω, ∀ k.Both systems of Eqs. (<ref>) and (<ref>) can be solved by Laplace transform, using the rule ȧ̃ (s) = s ã (s) - a(0) to obtain algebraic equations from differential ones. Few calculations lead us, after back-transforming and recalling that the index j=1,2 refers to the Exchange,Hopping respectively, to the solutionsa(t) = e^-i H_j tae^i H_j t = [μ_j(t)a + π_j(t)B_j ]e^- i ω_j t , B_j(t) = e^-i H_j tB_je^i H_j t = [(-)^jπ_j^*(t)a + μ_j^*(t)B_j ]e^- i ω_j t ,where B_1=b, B_2=b^†,μ_j(t)=cos(Δ_j t) -i δ_j/Δ_jsin(Δ_j t) , π_j(t)= -iΛ/Δ_jsin (Δ_j t) , withδ_j=1/2(ν+(-)^jω) ,ω_j=1/2(ν-(-)^jω) ,Δ_j^2 =|δ_j^2-(-)^jΛ^2| , and we have used μ^*_j(t)=μ_j(-t). The overall phase factors in the rightmost terms of Eqs. (<ref>) suggest that a natural interaction picture exists, corresponding to frames rotating at frequency ω_j. We will use these frames in the following, so as to omit those phase factors. Further notice that |μ_j(t)|^2-(-)^j |π_j(t)|^2 =1, ensuring that [a(t), a^† (t)]=[b(t),b^† (t)]=1, ∀ t and also that |μ_j(t)|^2 +(-)^j π^2_j(t) =1, meaning that the evolutions correspond to rotations in the rotating frames.Our goal is now to obtain an effective Hamiltonian H_ A^ eff(ζ), acting on A only, without renouncing to the quantum character of its companion B. This means that we can consider nothing but the time dependence of the reduced density matrix for Aρ_ A(t)= Tr_ B[ e^-i H_j tρ_ A⊗ρ_ Be^i H_j t] ≡ E_j[ρ_ A](t) ,with the notation ρ_ X≡ρ_ X(0) used hereafter. In particular, as already implied by Eq. (<ref>), we want to derive the explicit form of the dynamical map E_j[ρ_ A] upon assuming that at t=0 the system A+B is in a factorized state ρ_ A⊗ρ_ B. Moreover, we specifically take B initially prepared in the state at thermal equilibriumρ_ B= 1/1+n_(n_/1+n_)^b^† b ,where n_=(e^ω/T-1)^-1 is the thermal number of photons, and we have set the Boltzmann constant equal to 1.After this choice, that implicitly means that B further interacts with a third system T, specifically a thermal bath due to the choice of the state in Eq. (<ref>), we can positively move towards the derivation of the field ζ entering H^ eff_ A, and of its possible dependence on some external parameter. To this aim we first write the initial state of A+B using the Glauber formula, ρ_ A⊗ρ_ B = ∬d^2γ'd^2γ”/π^2χ[ρ_ A](γ ') χ[ρ_ B](γ”) D_a^† (γ ')⊗ D_b^† (γ”) ,where χ[ρ](γ)=Tr[ρ D(γ)] is the characteristic function of the state ρ, and D_x(γ) = exp{γ x^† -γ^* x}, with [x,x^†]=1, is the bosonic displacement operator. In order to get the argument of the partial trace in Eq. (<ref>), we use Eqs. (<ref>) to write the evolution of the displacement operators entering Eq. (<ref>),e^-i H_j tD_a^† (γ ') ⊗ D_b^† (γ”)e^i H_j t =D_a^† [μ_j^*(t) γ '+ π_j^*(t) γ” ] ⊗ D_b^† [π_j^*(t) γ '+ μ_j(t) γ” ] .We then perform the partial trace using Tr[D(γ)] = πδ^(2)(γ), so as to getE_j [ρ_ A](t)= ∫d^2γ '/πχ[ρ_ A](γ ') χ[ρ_ B] (-γ 'π^*_j(t)/μ_j(t))D^†(γ '/μ_j(t))= ∫d^2γ/π|μ_j(t)|^2 χ[ρ_ A](γμ_j(t)) χ[ρ_ B](- γπ^*_j(t))D^† (γ) ,where, in the last step, we made the substitution γ'→γμ_j(t).Upon expanding the coefficients (<ref>) for Δ_j t≪ 1, μ_j(t)≃ 1 -i δ_j t + O(t^2) , π_j(t)≃ - i Λ t + O(t^2) ,|μ_j(t)|^2≃ 1 + O(t^2) , and using the explicit form of the characteristic function of the thermal state, χ[ρ_ B](γ) = exp{-|γ|^2 (n_ +1/2)}, we finally writeρ_ A(t) = E_j[ρ_ A](t)= ∫d^2γ/πχ[ρ_ A](γ) e^-|γ|^2σ^2(t)D^† (γ) ,with σ^2(t)=Λ^2t^2 (n_ +12).We now wonder whether the above map is realized by some known unitary evolution involving the interaction with a classical environment only. Indeed, by first noticing that for any state ϱ it isχ[ϱ](γ) e^-|γ|^2σ^2=χ[ϱ__ GN](γ) ,withϱ__ GN≡∫d^2 α/πσ^2 e^-|α|^2/σ^2D(α)ϱD^†(α) ,we recognize in Eq. (<ref>) the Kraus decomposition corresponding to a Gaussian noise (GN) channel, namely a random displacement with Gaussian modulated amplitude [ To prove the equivalence in Eq. (<ref>), start from Tr[ρ_ GND(γ)], insert the definition of ρ_ GN, use the composition rule D^†(α)D(γ)D(α) = D(γ) e^α^* γ - γ^* α and perform the resulting Fourier transform.]. The same map <cit.> describes the evolution of a bosonic system in the presence of a classical fluctuating field, i.e. governed by a non-autonomous Hamiltonian of the formH_ stoc(t) = ν a^† a+ a ζ^*(t) e^i ω_ζ t + a^†ζ(t) e^- i ω_ζ t ,where ζ(t) is a random classical field described by a Gaussian stochastic process ζ(t)=ζ_x(t) + i ζ_y(t) with zero mean [ζ_x(t)]_ζ = [ζ_y(t)]_ζ=0 and diagonal structure of the autocorrelation function[ζ_x (t_1) ζ_x (t_2)]_ζ= [ζ_y (t_1) ζ_y (t_2)]_ζ = K(t_1,t_2) ,[ζ_x (t_1) ζ_y (t_2)]_ζ=[ζ_y (t_1) ζ_x (t_2)]_ζ = 0 . The function σ(t) in Eq. (<ref>) is in this caseσ(t) = ∫_0^t∫_0^tdt_1 dt_2 cos[δ_ζ(t_1-t_2)] K(t_1,t_2) ,where δ_ζ = ω_ζ - ν is the detuning between the natural frequency ν of A and the central frequency ω_ζ of the classical field ζ(t). The map (<ref>) may be obtained, for instance, upon considering the classical environment fluctuating according to a Gaussian Ornstein-Uhlenbeck stochastic process <cit.> characterized by the autocorrelation functionK^_τ(t_1-t_2) = G/2τ e^-1/τ|t_1-t_2| ,where τ is the correlation time, and G is the amplitude of the process. In the short-time limit, one easily finds thatσ(t) =G/2τ t^2 . In conclusion, we have shown that, as far as t≪|Δ_j|^-1, the effective Hamiltonian H_ A^ eff(ζ(t)) equals H_ stoc(t), meaning thatH_ A^ eff(ζ(t))= ν a^† a+ a ζ^*(t) e^i ω_ζ t + a^†ζ(t) e^- i ω_ζ t ,with the field ζ(t) as from Eqs. (<ref>-<ref>), and G = 2τΛ^2(n_ +12).Notice that the dynamical map for A in the short-time limit, Eq. (<ref>), is the same in the exchange and hopping cases. However, due to the j dependence of Δ_j, the condition defining the above short-time limit is different in the two cases. In fact, the difference is removed when the number of environmental modes becomes large, and the effective coupling Λ=√(∑_kλ_k^2) increases accordingly, so thatt≪1/√(|(ν∓ω)^2 ±Λ^2)|large-N ⟶ t≪1/Λ ,which establishes a relation between the short-time constraint and some large-N condition that will be further discussed later on.Overall, we have that the interaction (either exchange or hopping) of an oscillator with a bosonic environment induces a dynamics that is amenable to a description in terms of the interaction with a fluctuating classical field if the following conditions can be, at least approximately, met: *narrow environmental energy spectrum (ω_k ≃ω ∀ k)*short interacting times*environment at thermal equilibrium. It is worth noticing that, if conditions <ref>-<ref> hold, the above description in terms of classical fields is valid at all temperatures.§ MAGNETIC ENVIRONMENT We now consider the situation described by the Hamiltonian (<ref>), i.e. that of a bosonic mode A interacting linearly with a magnetic system S, made of N spin-1/2 particles, each described by its respective Pauli matrices (σ_i ^x, σ_i ^y, σ_i ^z) ≡σ_i. As in Sec. <ref>, we consider both the exchange and the hopping case. Setting 1) g_2i=0, with g_i≡ g_1i finite, and 2) g_1i=0, with g_i≡ g_2i finite, ∀ i, from Eq. (<ref>) we getH_1^= ν a^† a+ ∑_i f_i σ_i^z + ∑_i (g_i^*a σ_i^+ + g_i a^†σ_i^-) , H_2^= ν a^† a+ ∑_i f_i σ_i^z + ∑_i (g_i^*a^†σ_i^+ + g_i a σ_i^-) ,where the superscript S refers to the magnetic nature of the environment. Setting f_i=f, ∀ i, and further choosing f>0, the EOM in the Heisenberg picture are ȧ= i[H_1^, a]=-iν a -i∑_i=1^N g_iσ_i ^- ,σ̇_i^- =i[H_1^, σ_i^-]=-ifσ_i ^-+iag_i^*2σ_i ^z, ȧ= i[H_2 ^, a]=-iν a -i∑_i=1^N g_i^*σ_i ^+ , σ̇_i^+ =i[H_2 ^, σ_i^+]= ifσ_i^+-iag_i2σ_i ^z , where we have related the index of the Hamiltonians H^S_1,2 with the exchange and hopping cases, respectively.Despite Eqs. (<ref>)-(<ref>) have the same form as Eqs. (<ref>)-(<ref>) of the bosonic case, they cannot be solved exactly, due to the different algebra of the spin operators. However, restricting ourselves to physical situations where the operator S^z≡∑_i=1^Nσ_i^z can be replaced by some reasonable expectation value S^z≡N/2σ^z≡-N/2m, (with m>0, due to f being positive) we can rewrite the above EOM in the form ȧ=-iν a -iΛ^S̃^-,Ṡ̃̇ ^- =-ifS̃^- -i Λ^ a ȧ=-iν a -iΛ^S̃^+,Ṡ̃̇^+=ifS̃^+ +iΛ^ a , with g=√(∑_i=1^N|g_i|^2), Λ^=g√(2m), andS̃^+=1/Λ^∑_i=1 g_iσ_i^+ , S̃^-=(S̃^+)^† .In fact, these equations can be derived from the Hamiltoniansν a^† a + f S^z + Λ^ ( a S̃^+ + a^†S̃^-) ,ν a^† a + f S^z + Λ^ (a^†S̃^+ + a S̃^-) ,upon further assuming that the commutation relations[S̃^+, S̃^-]=-1 , [S^z,S̃^+]=S̃^+ , [S^z,S̃^-]=-S̃^- ,hold, meaning that the spin algebra is simplified into a bosonic one.Notice that replacing the total spin operator ∑_i σ_i^z with an expectation value S^z=N/2σ^z we imply that the field f selects the same expectation value σ^z for every spin-1/2, in the spirit of the usual random phase approximation.Once linearized, the EOM (<ref>)-(<ref>) can be solved as in the bosonic case, to get solutions formally analogous to Eqs. (<ref>) for the operators a and S̃_j, with the replacement ℬ_j(t)→S̃_j (t) with S̃_1=S̃^-, S̃_2=S̃^+, and ω→ f in the magnetic expressions corresponding to Eqs. (<ref>).Whatever follows Eq. (<ref>) in Sec. <ref> can be easily retraced until the choice of the initial environmental state ρ_S appears intoρ_A(t) = Tr_S[ e^-i H_j^tρ_A ⊗ρ_Se^i H_j^ t]≡ E^S_j(ρ_A) . Assuming that S is initially prepared in a state at thermal equilibrium, we takeρ_S = 1/1+n_^(n_^/1+n_^)^S̃ ^+ S̃ ^- ,with n_ ^≡N/2(1-m).Despite the formal analogy with Eq. (<ref>), it is important to notice that the temperature-dependence of n_^, and hence that of the dynamical map, is generally different from what we get in the bosonic case, where the thermal number of photons is n_ = (exp{ω/T}-1)^-1. We can, for example, suppose that the magnetic environment thermalizes with the thermal bath so that S^z=-(f)S𝐵_S (x)= -N/2(f) m, where S=N/2 and 𝐵_S (x)=m is the Brillouin function𝐵_S(x)=2S+1/2S(2S+1/2Sx )-1/2S(x/2S) ,with x=S |f|/ T. With this choice, it is n_ ^≡ S (1-𝐵_S (x)) and the dependence on T of the bosonic model is only recovered when T→ 0, being 𝐵_S(x)→ 1-e^-x the low temperature limit of Eq. (<ref>). Notice that, in order for the above representation to stay meaningful in the large-S limit, temperature must scale as T∼ S so as to guarantee a finite x; performing such large-S limit, the Brillouin function turns into the Langevin one, 𝐿(x)=(x)-1/x , which is indeed the classical limit of Eq. (<ref>).We observe that the approximations introduced for the spin system are consistent with our aim of finding an effective classical description for the environment: indeed, once the total spin is guaranteed a constant value S, a classical-like behavior is expected for a spin-system when S≫ 1 <cit.>, and the bosonic expansion given by the Holstein-Primakoff transformation can be safely truncated at its lowest order S^+∼ b^† (if f>0, b^† being a generic bosonic creation operator)<cit.>.We can now write the initial state ρ_ A⊗ρ_S using the Glauber formula as in Eq. (<ref>), with the spin displacement operator defined as D_S̃(γ) = exp{γS̃ ^+ - γ^* S̃ ^-} due to the choice f>0, and hence σ^z<0 (had we taken f<0 it would be D_S̃(γ) = exp{γS̃^- - γ^* S̃ ^+}). Using the solutions of the EOM (<ref>)-(<ref>), one can write the evolution of displacement operators and proceed as done in the previous section up to Eq. (<ref>), thus obtaining that the dynamical map in the magnetic case does also correspond to a Gaussian noise channel. With the additional requirement of a random phase approximation, an effective Hamiltonian of the form of Eq. (<ref>) can hence be written again, allowing us to conclude that the set of conditions sufficient to find an effective classical description is the same as in the bosonic model, the only difference being in the temperature dependence of the standard deviation σ^2, due to the different definition of n_^ in the magnetic case.§ LARGE-N ENVIRONMENT: DERIVING THE CLASSICAL FIELDS In this section we take a more abstract view on the problem of what happens to the principal system A when its environment becomes macroscopic. For the sake of clarity we will specifically refer to the results presented in Secs. <ref>-<ref> and, in particular, to the model (<ref>).Our aim is to understand whether the emergence of an effective Hamiltonian H^ eff_ A(ζ(t)) as in Eq. (<ref>) is a general feature of OQS with macroscopic environments. We also aim at further clarifying the meaning of the conditions <ref>-<ref> given at the end of Sec. <ref>, and the reasons why they seem to be utterly necessary in order to obtain an effective Hamiltonian description. Following suggestions from Refs. <cit.>, the main idea is to show that the emergence of H^ eff_ A(ζ) is related to the crossover from a quantum to a classical environment, possibly observed when the number of components becomes very large. In fact, were the environment described by a classical theory, its effects on the system would naturally be represented by the classical fields ζ. Before introducing the general approach we are going to adopt, let us briefly recall some useful notions.A quantum description of a physical system, or quantum theory Q for short, is based on the introduction of (1) a Hilbert space H, (2) a Lie product [·,·] that defines the commutation rules between the operators on H, and (3) a Hamiltonian H. Traceclass operators on H that represent physical observables usually make up a vector space: this space, together with the above Lie product, is the Lie algebra g of the theory. The expectation values O≡ψ|O|ψ∈ℝ of Hermitian operators, are the (only) physical outputs of the theory, i.e. the experimentally accessible properties of the system.On the other hand, a classical description of a physical system, or classical theory C for short, is defined by (1) a phase space C, (2) a Poisson bracket {·,·}, and (3) a Hamiltonian h(ζ), with ζ representing the set of conjugate variables of the classical phase-space C. Real functions O(ζ) are the (only) physical outputs of the theory, in the same sense as above.The problem of whether or not a system made by quantum particles can be described by a classical theory has been extensively studied in the last decades of the last century. Different approaches (see for instance Sec.VII of Ref. <cit.> for a thorough discussion) all showed that a large number N of quantum constituents is a necessary condition for a system to admit a classical description, but yet it is not a sufficient one, as confirmed by the experimental observation of macroscopic quantum states. In fact, further conditions must be satisfied, that crucially involve symmetry properties of the original quantum theory, and its Lie algebra. Specifically, in Ref. <cit.> it is demonstrated that the N→∞ limit of a quantum theory Q_N, hereafter indicated by Q_N→∞, is a classical theory C if Q_N exhibits a global symmetry. This latter requirement means that it must exist a group of unitary operators, each acting non-trivially on all of the N constituents, that leave the physical observables of the theory invariant (see Ref. [If the theory describes N spin-1/2 particles interacting via an isotropic Heisenberg-like magnetic exchange, one such symmetry can be that defined by operators that rotate the spin of each particle of the same angle. Take instead N particles whose interaction only depends on their distance: the symmetry might be that defined by the same spatial translation of each particle. As for non-interacting, identical, but yet distinguishable particles, a possible global symmetry is that defined by the permutation operators.] for some examples). Indeed, one such symmetry guarantees the existence of a simpler theory Q_k (with k a real parameter defined by N) whose k→ 0 limit, hereafter indicated by Q_k→ 0, is physically equivalent to Q_N→∞, by this meaning that each expectation value that stays finite in the latter limit can be obtained as some expectation value provided by Q_k→ 0. On the other hand, Q_k→ 0, is also a well defined classical theory C, with phase-space Cand classical hamiltonian h(ζ), that therefore provides an effective classical description of the original many-particles quantum system in its macroscopic limit, through the chain Q_N→∞=Q_k→ 0=C (see Fig. 3 of Ref. <cit.> for a graphical depiction of the relation between Q_k, Q_N, and C). Details of the procedure for deriving the above classical theory are given in Appendix, according to the results presented in Ref. <cit.>, and recently used in the framework of OQS <cit.>: suffice it here to say that Q_N and Q_k are related by the fact that their respective Lie algebras, g_N and g_k, are representations of different dimensionality of the same abstract algebra g.Let us now get back to our problem, specifically concentrating upon the model described by the Hamiltonian (<ref>). In order to be used in the framework of OQS dynamics, the results mentioned above and the procedure described in Appendix need being generalized, as we deal with the quantum theory of a bipartite system where just one of the two constituents, namely the environment, is intended to become macroscopic. However, due to the linear structure of the interactions entering Eq. (<ref>), the procedure can still be applied as follows.Keeping in mind that we have to deal with physically meaningful Lie algebras, we first notice that the coupling terms in (<ref>) can be written as a(a^†) tensor-times some sum over k of operators acting on H_ B iff either λ_1k=λ_2k or λ_1(2)k=0, for all k. Taking one or the other of the above conditions true is quite equivalent, as far as the following construction is concerned: for the sake of clarity, and at variance with what done in Secs. <ref>-<ref>, we specifically choose λ_2k=0 and set λ_k≡λ_1k finite, for all k, meaning that we explicitly consider the exchange case only. Further taking ω_k=ω ∀ k, as done in Secs. <ref>-<ref>, we can define the global operatorsE ≡1/N∑_k^N b^†_k b_k and B≡1/√(NΛ^2)∑_k^Nλ_k b_k ,with Λ^2≡∑_k^N|λ_k|^2 as in Eq. (<ref>), and write the Hamiltonian (<ref>) asH=ν a^† a + N[Λ/√(N)(a^† B +aB^†) +ω E ] ;the way N enters Eqs. (<ref>-<ref>) is designed to recognize 1/N as the parameter to quantify quantumness of the environment B, and let all the operators, no matter whether acting on A, B, or A+B, independent on the number of environmental modes.The operators (<ref>), together with the identity, are easily seen to generate a Heisenberg algebra on H_ B, being[B,B^†]=1/N , [B,E]=1/NB , [B^†,E]=-1/NB^† .However, this cannot be regarded as the Lie algebra g_N of some environmental theory, due to the presence of non commuting operators acting on A in Eq. (<ref>), unless the N→∞ limit is taken, as shown below.Explicitly referring to the example given in Sec. IV of Ref. <cit.> and the strategy described in Appendix, we introduce the set of antihermitian operators{L(ϵ,β)≡ iN(ϵ E+β^* B +β B^†)} ,where β∈ℂ, with |β|∝1/√(N), while the coefficients ϵ∈ℝ do not depend on N. In the large-N limit, where terms which are bilinear in β and β^* can be neglected due to their dependence on N, it is [L_1,L_2]= L_3, with L_i≡ L_i(ϵ_i,β_i), β_3=i(ϵ_1β_2-ϵ_2β_1), and ϵ_3=0, meaning that the set (<ref>) is a Lie Algebra. This is indeed the algebra g_N whose recognition represents the first step towards the large-N limit of the quantum theory that describes B. It is easily checked that a possible representation g_k, of the same abstract algebra represented by g_N, is given by the 2× 2 matrices{ℓ(ϵ,β)≡ i(0β^*0ϵ)} ,being [ℓ_1,ℓ_2]= ℓ_3, with ℓ_i≡ℓ_i(ϵ_i,β_i), and β_3,ϵ_3 as above. We underline that the choice of a representation g_k that contains only either β or β^* is also the simplest way to make the presence of non-commuting operators on H_ A in the Hamiltonian (<ref>), harmless as far as the following construction is concerned. The matrices ℓ(ϵ,β) allow us to write[L,[ 1; B ]]≡[ [L,1]; [L,B] ]= [ 0; -i(ϵ B+β^*) ]as[L,[ 1; B ]]= ℓ^†[ 1; B ] ,with ℓ^†≡(ℓ^*)^t, and, quite equivalently,[L,[ 1 B^† ]]= [ 1 B^† ]ℓ .Let us now consider the unitary operatorsU(ϵ,β)≡exp{L(ϵ,β)} :given that, for any pair of operators O and P, it holdse^-POe^P=∑_n(-1)^n/n!n times[P,[P,[...[P,O]...]]] ,from Eqs. (<ref>) and (<ref>) it followsU^-1[ 1; B ]U= u(ϕ,ζ)[ 1; B ]andU^-1[ 1 B^† ]U= [ 1 B^† ]u^†(ϕ,ζ) ,withu(ϕ,ζ)≡( 1 0 ζ ϕ) ,whereϕ=e^iϵ and ζ=β/ϵ( e^iϵ-1 )are obtained by explicitly summing the series in Eq. (<ref>).The fact that the set (<ref>) is a Lie algebra in the large-N limit reflects upon the unitary operators U(ϕ,ζ), in that they form a group in the same limit.In fact, this is just the Lie group corresponding to g_k, sometimes dubbed dynamical <cit.> or coherence <cit.> group, that defines, together with the arbitrary choice of a reference state |0⟩∈H_ B, the Generalized Coherent States (GCS) |u(ϕ,ζ)⟩≡ U(ϕ,ζ)|0⟩ for the theory Q_k. The reason why these states are so relevant, as further commented upon in Appendix, is that the operators B and E are demonstrated <cit.> to transform into B(u)≡u|B|u/N and E(u)≡u|E|u/N, respectively, as N goes to infinity. Therefore, in order to find the large-N limit of the Hamiltonian (<ref>) we now only need to evaluate B(u) and E(u), even without knowing the explicit form of the GCS, to obtain H^ eff_N(ζ) fromHN→∞→ ν a^† a + N[ Λ/√(N)(a^† B(u)+aB^*(u))+ω E(u)] ≡ H^ eff_N(ζ) ,where the relation between |u⟩ and ζ is made explicit below.To proceed accordingly, we choose the reference state for the GCS: |0⟩=Π_k|0⟩_k, with |0⟩_k such that b_k|0⟩_k=0. This implies, given the separable structure of the operators U(ϕ,ζ), that the states |u⟩ are tensor products of single-mode pure states. As a consequence, it is u|BB^†|u =u|∑_k'kb_k'b^†_k|u =u|∑_k b_kb^†_k|u=NE(u), which allows us to determine B(u) and E(u) via⟨u|[ 1; B ]⊗[ 1 B^† ]|u⟩= N ( 1 B^*(u)B(u) E(u) ) ,and finally obtain, by Eqs. (<ref>-<ref>) and again neglecting terms bilinear in β and β^*,⟨0| u(ϕ,ζ)[ 1; B ]⊗[ 1 B^† ]u^†(ϕ,ζ) |0⟩ == ⟨0 | ( 1 ζ^*+ϕ^*B^† ζ+ϕ B ζζ^*+ζϕ^*B^†+ζ^*ϕ B+ϕϕ^*BB^†) | 0|⟩= (1ζ^*ζ1) ,i.e. E(u)=1/N and B(u)=ζ/N.The above result implies that the original Hamiltonian (<ref>) formally transforms, according to Eq. (<ref>), asHN→∞⟶ H^ eff_ A(ζ)= (ν a^† a + ω)+ ζ^* a + ζ a^† ,where we have rescaled ζ→ζΛ /√(N) and (ζ,ζ^*)∈ℝ^2 is generally proved <cit.> to be any point of a classical phase-space M_ B with canonical variables q≡(ζ+ζ^*)/2 and p≡(ζ-ζ^*)/(2i). Notice that |ζ|∝Λ/√(N), which is independent of N by definition.Once Eq. (<ref>) is obtained, we can maintain with confidence that the Hamiltonian (<ref>), originally acting on A+B, formally transforms, as N→∞, into one that exclusively acts on A: however, the presence of the classical field ζ is the remnant of the underlying quantum interaction between A and the huge number of elementary constituents of B, namely the bosonic modes {b_k}_k=1^N. To this respect, notice that the Hilbert space H_ B=⊗_kH_ b_k is replaced by a two-dimensional classical phase-space, M_ B, implying an impressive reduction of dynamical variables. This reduction is the most striking consequence of the global symmetry that the quantum theory for B must exhibit in order to flow into a well defined classical theory when B is macroscopic. In our case, although we did not explicitly used it, the symmetry is that under permutation of the bosonic modes b_k, and that is why we have set ω_k=ω ∀ k. In fact, one can easily check that this is an essential condition for the very same definition of global operators obeying commutation rules of the form (<ref>), which on their turn are necessary to proceed to the definition of the Lie Algebra, and all the rest. At this point, we notice that ω_k=ω ∀ k is just the “narrow environmental energy-spectrum condition” <ref>, discussed at the end of Sec. <ref>. In fact, it immediately strikes that the effective Hamiltonian in Eq. (<ref>) has the same structure of that in Eq. (<ref>), given that the latter refers to an interaction picture that hides the environmental frequency ω. On the other hand, it is somehow puzzling that time does not enter the above construction, which leave us clueless, so far, concerning the relation ζ→ζ(t)e^-iω_ζ t. Looking for the possible origin of a time-dependence in the classical field ζ, we reckon that the results of this section imply the following. Suppose there exists another macroscopic system T that is not coupled with A, and interacts with B in such a way that the above global symmetry is preserved: the presence of T manifests itself in terms of some parameter τ (think about time and/or temperature, for instance) upon which ζ depends, according to the rule ζ=ζ(τ) provided by the classical theory describing B+T. This dependence can be safely imported into the effective cdescription of A via ζ→ζ(τ) in H^ eff_ A(ζ), Eq. (<ref>), as far as the direct interaction between A and T can be neglected, at least on the time scales one is interested in.Finally, we notice that the detuning ν-ω does not play any role in this section, which brings us back to Eq. (<ref>) and the possible relation between the large-N condition here enforced and the short-time approximation previously adopted.§ CONCLUSIONS In this paper, we have addressed the dynamics of a bosonic system coupled to either a bosonic or a magnetic environment. In particular, we have discussed the conditions under which the dynamics of the system may be described in terms of the effective interaction with a classical fluctuating field. Our results show that for both kinds of environments an effective, time-dependent, Hamiltonian description may be obtained for short interaction time and environments with a narrow energy spectrum at thermal equilibrium. The corresponding dynamics is described by a Gaussian noise channel independently of the kind of environment, their magnetic or bosonic nature entering only the form of the noise variance. As far as the energy spectrum is narrow, this effective description is valid at all temperatures and independently on the nature of the interaction between the system and its environment.Moreover, exploiting a general treatment based on the large-N limit of the environment, we have clarified the origin and the meaning of the narrow-environmental-spectrum and short-time conditions. In fact, we find that ω_k ≃ω ∀ k is needed for a global symmetry to emerge and characterize the environment, which is a necessary ingredient for the environment to be described by a small number of macroscopic variables. On the other hand, the large energy scale implied by whatever coupling with a macroscopic environment limits any effective description to short times only. Overall, our results indicate that quantum environments may be described by classical fields whenever global symmetries allows one to define environmental operators that remain well defined when increasing the spatial size of the environment. This is a quite general criterion that may serve as a guideline for further analysis, e.g. for fermionic principal systems and/or hybrid environments. This work has been supported by UniMI through the H2020 Transition Grant 15-6-3008000-625, and by EU through the Collaborative Projects QuProCS (Grant Agreement No. 641277). PV has worked in the framework of the Convenzione Operativa between the Institute for Complex Systems of the Italian National Research Council (CNR) and the Physics and Astronomy Department of the University of Florence. * § Consider a system made by N elements which is described by a quantum theory Q_N that features a global symmetry, as defined in Sec. <ref> (we will equip quantities with the index N to indicate their being relative to this Q_N theory).The procedure described in Ref. <cit.> for deriving the classical theory that formally represents Q_N→∞ can be summarized as follows.The first stepis that of identifying g_N, exploiting the knowledge of the Hilbert space H=⊗_i^N H_i, the Lie product, and the Hamiltonian H_N.As the Hamiltonian H_N represents a physical observable, an effective strategy to identify g_N is that of writing H_N as a linear combination of operators, and see if they belong to some minimal set that generates a representation g_N of some abstract Lie algebra g.The second step of the procedure is that of finding an irreducible representation g_k of g_N, which stands as the Lie algebra for Q_k (notice that this most often implies that an explicit expression for H_k does also become available). Here is where the existence of a global symmetry emerges as a necessary ingredient, as it guarantees that the dimensionality of the representation g_k be significantly smaller than that of g_N. In fact, the way g_k can be most often identified is writing the original Hamiltonian as a linear combination of some global (i.e. acting non trivially upon each subsystem) operators that are invariant under the symmetry-operations, and generate a representation of the same abstract algebra g which is also represented by g_N.In the third step, generalized coherent states (GCS) for Q_k come into play: these are defined, according to the approach either of Gilmore <cit.> or, quite equivalently, of Perelomov <cit.>, starting from the dynamical group of the theory, which is nothing but the Lie group associated to g_k by the usual Lie correspondence <cit.>, and is therefore provided by the above second step. GCS for Q_k, hereafter indicated by |u⟩∈ H_k, enter the procedure due to their being <cit.> in one-to-one correspondence with points u on a manifold M_k, whose cotangent bundle is a classical phase-space C. In other terms, each GCS |u⟩ of the theory Q_k defines a point u∈ M_k and a set of conjugate variables ζ∈ C. In fact, it is demonstrated <cit.> that Q_k→ 0 is a classical theory C, with phase-space the above cotangent bundle C, and hamiltonian h(ζ)=u|H_k|u/N.The last step of the procedure is that of deriving, possibly without knowing the explicit form of the GCS, the exptectation values u|H_k|u, and finally obtain the effective classical hamiltonian describing the original quantum system in the N→∞ limit.The role of the parameters N and k, which has been here understood for the sake of a lighter narration, becomes evident when explicitly employing the procedure, as in Sec. <ref>, where it is k=1/N. | http://arxiv.org/abs/1703.09236v3 | {
"authors": [
"Matteo A. C. Rossi",
"Caterina Foti",
"Alessandro Cuccoli",
"Jacopo Trapani",
"Paola Verrucchi",
"Matteo G. A. Paris"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170327180011",
"title": "Effective description of the short-time dynamics in open quantum systems"
} |
Laboratoire de Physique, Ecole Normale Supérieure de Lyon, 46, allée d'Italie, 69364 Lyon, Cedex 07, France Mathematical Institute, Utrecht University, P.O. Box 80010, 3508 TA Utrecht, The NetherlandsDepartment of Information and Computing Sciences, Utrecht University, P.O. Box 80089, 3508 TB Utrecht, The NetherlandsMathematical Institute, Utrecht University, P.O. Box 80010, 3508 TA Utrecht, The NetherlandsSchool of Mathematics and Statistics, University of Melbourne, Australia Self-avoiding walks on the body-centered-cubic (BCC) and face-centered-cubic (FCC) lattices are enumerated up to lengths 28 and 24, respectively, using the length-doubling method.Analysis of the enumeration results yields values for the exponents γ and ν which are in agreement with, but less accurate than those obtained earlier from enumeration results on the simple cubic lattice. The non-universal growth constant and amplitudes are accurately determined, yielding for the BCC lattice μ=, A=, and D=, and for the FCC lattice μ=, A=, and D=. Exact enumeration of self-avoiding walks on BCC and FCC lattices Nathan Clisby March 27, 2017 ================================================================§ INTRODUCTION The enumeration of self-avoiding walks (SAWs) on regular lattices is a classical combinatorial problem in statistical physics, with a long history, see e.g. <cit.>.Of the three-dimensional lattices, the simple cubic (SC) lattice has drawn the most effort, starting with a paper by Orr <cit.> from 1947, where the number of SAWs Z_N was given for all N up to N_max=6; these results were obtained by hand. In 1959, Fisher and Sykes <cit.> used a computer to enumerate all SAWs up to N_max=9; Sykes and collaborators extended this to 11 terms in 1961 <cit.>, 16 terms in 1963 <cit.>, and 19 terms in 1972 <cit.>.In the following decade Guttmann <cit.> enumerated SAWs up to N_max=20 in 1987, and extended this by one step in 1989 <cit.>. In 1992, MacDonald et al. <cit.> reached N_max=23, and in 2000 MacDonald et al. <cit.> reached N_max=26.In 2007, a combination of the lace expansion and the two-step method allowed for the enumeration of SAWs up to N_max=30 steps <cit.>. Recently, the length-doubling method <cit.> was presented which allowed enumerations to be extended up to N_max=36. To date, this is the record series for the SC lattice. The body-centered-cubic (BCC) and face-centered-cubic (FCC) lattices are in principle equally as physically relevant as the SC lattice, butenumeration is hampered by the larger lattice coordination numbers, which detriments most enumeration methods severely. It is also very slightly more cumbersome to write computer programs to perform enumerations for these lattices. Consequentially, the SC lattice has served as the test-bed problem for new enumeration algorithms, and the literature on enumerations for the BCC and FCC lattices is far more sparse. For the BCC lattice, Z_N was determined up toN_max=15 in 1972 <cit.>,and to N_max=16 in 1989 <cit.>. The current record of N_max=21 was obtained in 1997 by Butera and Comi <cit.> as the N → 0 limit of the high temperature series for the susceptibility of the N-vector model.For the FCC lattice,enumerations up to N_max=12 were performed in 1967 <cit.>, and the record of N_max=14 was achieved way back in 1979 <cit.>.Enumeration results derive their relevance from the ability todetermine critical exponents, which, according to renormalization group theory, are believed to be shared between SAWs on various lattices and real-life polymers in solution <cit.>. Two such exponents are the entropic exponent γ and the size exponent ν. Given the number Z_N of SAWs of all lengths up to N_max and the sum P_N of their squared end-to-end extensions, these two exponents can be extracted using the relationsZ_N= A μ^N N^γ-1(1 + a/N^Δ_1 + O(1/N));P_N/Z_N = σ D N^2ν(1 + b/N^Δ_1 + O(1/N)).In these expressions, the growth constant μ and the amplitudes A and D are non-universal (model-dependent) quantities, while the leading correction-to-scaling exponent is a universal quantity with value Δ_1 = 0.528(8) <cit.>.Sub-leading corrections-to-scaling are absorbed into the O(1/N) term. σ is a lattice specific constant to ensure that our amplitude “D” is the same as in earlier work. σ corrects for the fact that with our definition each step of the walk is of length √(2) for the BCC lattice (leading to σ = 2), and of length √(3) for the FCC lattice (leading to σ = 3). Note that for bipartite lattices, of which the SC and BCC lattices are examples, there is an additional alternating “anti-ferromagnetic” singularity, that is sub-leading but which still must be treated carefully as the odd-even oscillations tend to become amplified by series analysis techniques. Because of universality, the exponents are clearly more interesting from a physics perspective.However, accurate estimates for the growth constant and the amplitudes can also be very helpful for many kinds of computer simulations on lattice polymers.In this paper, we used the length-doubling method <cit.> to measure Z_N and P_N up to N_max=28 and 24, on the BCC and FCC lattices, respectively. These lattices can be easily simulated as subsets of the SC lattice: the collection of sites in which x, y and z are either all even or all odd forms a BCC lattice, and the collection of sites (x,y,z) constrained to even values for x+y+z forms a FCC lattice.We then analysed these results to obtain estimates for the exponents γ and ν, the growth constant μ, and the amplitudes A and D. Our results for the two exponents γ and ν agree with the values reported in literature which are obtained on the SC lattice, reinforcing the credibility of the literature values. Our results for the growth constant μ and the amplitudes A and D for the BCC and FCC lattices are the most accurate ones to date.The manuscript is organized as follows. First, in Sec. <ref> we present a short outline of the length-doubling method, and present the enumeration data. In Sec. <ref> we describe the analysis method we use, before summarising our results and giving a brief conclusion in Sec. <ref>.§ LENGTH-DOUBLING METHODWe first present an intuitive description of the length-doubling method; a more formal description can be found in <cit.>. In the length-doubling method, the number Z_2N of SAWs with a length of 2N steps, with the middle rooted in the origin, is obtained from the walks of length N, with one end rooted in the origin, and the number Z_N(S) of times that a subset S of sites is visited by such a walk of length N.The lowest-order estimate for Z_2N is the number of combinations of two SAWs of length N, i.e. Z_N^2. This estimate is too large since it includes pairs of SAWs which overlap. The first correction to Z_2N is the lowest-order estimate for the number of pairs of overlapping SAWs, which can be obtained from the number Z_N({s}) of SAWs of length N which pass through a single site s. The first correction is then to subtract Z_N({s})^2, summed over all sites s. This first correction is too large, as it includes pairs of SAWs twice, if they intersect twice. The second correction corrects for this over-subtraction, by adding the number Z_N({s,t}) of SAWs that pass through the pair of sites {s,t}.Continuing this process with groups of three sites, etc., the number Z_2N of SAWs of length 2N can then be obtained by the length-doubling formula Z_2N = Z_N^2 + ∑_S ≠∅ (-1)^|S| Z_N^2(S) ,where |S| denotes the number of sites in S.The usefulness of this formula lies in the fact that the numbers Z_N(S) can be obtained relatively efficiently: * Generate each SAW of length N.* Generate for each SAW each of the 2^N subsets S of lattice sites, and increment the counter for each specific subset. Multiple counters for the same subset S must be avoided; this can be achieved by sorting the sites within each subset in an unambiguous way.* Finally, compute the sum of the squares of these counters, with a positive and negative sign for subsets with an even and odd number of sites, respectively, as in Eq. (<ref>). As there are Z_N walks of length N, each visiting 2^N subsets of sites, the computational complexity is 𝒪(2^N Z_N) ∼ (2μ)^N times some polynomial in N which depends on implementation details. This compares favourably to generating all Z_2N∼μ^2N walks of length 2N, provided μ > 2. This is the case on the SC lattice, with μ=4.684, and even more so for the BCC and FCC lattices, as we will show.The length-doubling method can also compute the squared end-to-end distance, summed over all SAW configurations; for details we refer to <cit.>. Details on the efficient implementation of this algorithm are presented in <cit.>.The direct results of the length-doubling method, applied to SAWs on the BCC and FCC lattices, are presented in Tables 1 and 2, respectively. The BCC results for N ≤ 26 and FCC results for N ≤ 22 were obtained and verified by two independent computer programs: SAWdoubler 2.0, available from <http://www.staff.science.uu.nl/ bisse101/SAW/>, and Raoul Schram's program. The BCC results presentedfor the largest problems N=27,28 were obtained by SAWdoubler 2.0 only, and the FCC results for N=23,24 were obtained by Raoul Schram's program only. Thus the largest two problem instances for each lattice were not independently verified since these require a very large amount of computer time and memory. Still, based on our analysis we believe that the given values are correct.§ ANALYSIS We now proceed to analyse our series in order to extract estimates for various parameters. In addition to the expressions for Z_N and P_N / Z_N in Eqs (<ref>) and (<ref>), we also haveP_N= σ A D μ^N N^2ν + γ-1(1 +c/N^Δ_1 + O(1/N)).As discussed earlier, we expect the critical exponentsγ and ν and the leading correction-to-scaling exponent Δ_1 to be the same for self-avoiding walks on the SC, BCC, and FCC lattices.The amplitudes A and D are non-universal quantities, i.e. they are lattice dependent, while σ = 2 for the BCC lattice and σ = 3 for the FCC lattice. In the analysis below, we include a subscript to indicate the appropriate lattice.The BCC lattice is bipartite, which introduces an additionalcompeting correction which has a factor of (-1)^N, so causing odd-even oscillations.We reduce the influence of this additional sub-leading correction by separately treating the sequences for even and odd N. See <cit.> for more detailed discussion on this point for the asymptotic behaviour of Z_N on theSC lattice, which is also bipartite.We now describe the method of analysis we used, which involved two stages: extrapolation of the series via a recently introduced method involving differential approximants <cit.>, and then direct fitting of the extended series with the asymptotic forms in Eqs (<ref>), (<ref>), and (<ref>).We report our final estimates in Table <ref> at the end of the section. §.§ Extrapolation Perhaps the most powerful general-purpose method for the analysis of series arising from lattice models in statistical mechanics is the method of differential approximants, described in <cit.>. The basic idea is to approximate the unknown generating function F by the solution of an ordinary differential equation with polynomial coefficients. In particular if we know r coefficients f_0, f_1, ⋯, f_r-1 of our generating function F, then we can determine polynomials Q_i(z) and P(z) which satisfy the following Kth order differential equation order by order:∑_i=0^K Q_i(z) (z d/dz)^i F(z) = P(z).The function determined by the resulting differential equation is our approximant.The power of the method derives from the fact that such ordinary differential equations accommodate the kinds of critical behaviour that are typically seen for models of interest.Differential approximants are extremely effective at extracting information about critical exponents from the long series that have been obtained for two-dimensional lattice models, such as self-avoiding polygons <cit.> or walks on the square lattice <cit.>. However, differential approximants have been far less successful for the shorter series available for three-dimensional models such asSAWs on the simple cubic lattice <cit.>. For short series, it seems that corrections-to-scaling due to confluent corrections are too strong at the orders that can be reached to be able to reliably determine critical exponents. (In fact, it is extremely easy to be misled by apparent convergence, while in fact estimates have not settled down to their asymptotic values.) The method that has proved most reliable is direct fitting of the asymptotic form <cit.>, which we describe in the next sub-section.However, we can do better than the usual method of performing direct fits of the original series, and adopt a promising new approach recently invented by Guttmann <cit.>,which is a hybrid of the differential approximant and direct fitting techniques. The underlying idea is to exploit the fact that differential approximants can be used to extrapolate series with high accuracy even in circumstances when the resulting estimates for critical exponents are not particularly accurate, or even when the asymptotic behaviour is non-standard such as being of stretched exponential form. The extrapolations can be extremely useful in cases where corrections-to-scaling are large, as the few extra terms they provide may be the only evidence of a clear trend from the direct fits.We have 28 exact terms for the BCC series, and 24 exact terms for the FCC series.We used second order inhomogeneous approximants to extrapolate the series for Z_N, P_N, and P_N/Z_N, where we allowed the multiplying polynomials to differ by degree at most 3. In each case we calculated trimmed mean values, eliminating the outlying top and bottom 10% of estimates, with the standard deviation of the remaining extrapolated coefficients providing a proxy for the confidence interval. Note that this is an assumption, and relies on the extrapolation procedure working well for our problem. In practice, this approach of inferring the confidence interval from the spread of estimates appears to be quite reliable in the cases for which it has been tested.We have also confirmed the reliability of the extrapolations by using the method to “predict” known coefficients from truncated series.We report our extended series in Tables <ref> and <ref>.§.§ Direct fits We then fitted sequences of consecutive terms of the extrapolated series for Z_N and P_N/Z_N to the asymptotic forms given in Eqs (<ref>) and (<ref>), respectively. We found that fits of P_N/Z_N were superior to fits of P_N for estimates of ν and the parameter D, and hence we do not report fits of P_N here.To convert the fitting problem to a linear equation, we took the logarithm of the coefficients, which from Eqs (<ref>) and (<ref>) we expect to have the following asymptotic forms:log Z_N= N logμ+(γ-1) log N + log A + a/N^Δ_1 + O(1/N);logP_N/Z_N = 2νlog N + logσ D +b/N^Δ_1 + O(1/N)We used the linear fitting routine “lm” in the statistical programming language R to perform the fits.In all of the fits, we biased the exponent Δ_1 of the leading correction-to-scaling term, performing the fits for three different choices of Δ_1 = 0.520, 0.528, 0.536 which correspond to the best Monte Carlo estimate of Δ_1 = 0.528(8). We approximated the next-to-leading correction-to-scaling term with a term of order 1/N, which we expect to behave as an effective term which takes into account three competing corrections with exponents -2Δ_1, -1, -Δ_2 ≈ -1. For log Z_N, we fitted log A, logμ, γ, the amplitude a, and the amplitude of the 1/N effective term. For log (P_N/Z_N), we fitted log D, ν, the amplitude b,and theamplitude of the 1/N term. For the BCC lattice, we minimised the impact of the odd-even oscillations by fitting even and odd subsequences separately. We included the extrapolated coefficients in our fits, repeating the calculation for the central estimates and for values which are one standard deviation above and below them.This procedure gave us up to nine estimates for each sequence of coefficients (from the three choices of Δ_1, and the three choices of extrapolated coefficient values). For the central parameter estimates we used the case where Δ_1 = 0.528 (the central value) in combination with the central value of the extrapolated coefficients. We also calculated the maximum and minimum parameter estimates over the remaining 8 cases.For the BCC lattice, we found that 5 of the extrapolated coefficients gave a spread which was only moderately greater than the spread arising from varying Δ_1, effectively extending the series to 33 terms. For the FCC lattice, we found we could use 3 additional coefficients, extending the series to 27 terms.For each of the parameter estimates, we plotted them against the expected relative magnitude of the first neglected correction-to-scaling term. This should result in approximately linear convergence as we approach the N →∞ limit which corresponds to approaching the y-axis from the right in the following figures. In Eqs (<ref>) and (<ref>) we expect that the next term, which is not included in the fits, is of O(N^-1-Δ_1); given that Δ_1 ≈ 0.5, we take the neglected term to be O(N^-3/2). The value of N that is used in the plot is the maximum value of N in the sequence of fitted coefficients, which we denote N_max in the plots.We plot our fitted values in Figures <ref>–<ref>.For ease of interpretation we converted estimates of logμ, log A, and log D to estimates of μ, A, and D. We note that the parameter estimates arising from the odd subsequence of the BCC series for Z_N benefited dramatically from the extrapolated sequence. Examining estimates for γ in Fig. <ref>,in Fig. <ref>, andin Fig. <ref> we see in each case that the trend of the odd subsequence would be dramatically different were it not for the three additional odd terms in the extrapolated sequence. In other cases the additional coefficients are useful, and certainly make the trend for the estimates clearer, but are not as crucial.Our final parameter estimates are plotted on the y-axes.§ SUMMARY AND CONCLUSIONWe give our estimates for γ and ν in Table <ref>, where we also include estimates coming from the literature.We observe that our estimates are consistent with the literature values, but that the recent Monte Carlo estimates of γ and ν, using the pivot algorithm, are far more accurate than the estimates from series.The estimates coming from our enumerations on the BCC and FCC lattices are not quite as precise as the estimates coming from the SC lattice only, but the fact that they are coming from two independent sources, with different systematic errors, makes these new estimates more robust.In addition, our estimates of the non-universal quantities for the BCC lattice are =, =,and =, which should be compared with earlier estimates of 6.5304(13) <cit.> from 1989, and unbiased and biased estimates respectively of 6.53036(9) and 6.53048(12) <cit.> from 1997. Our estimates of the non-universal quantities for the FCC lattice are =, =, and =, which should be compared with earlier estimates of 10.03655 <cit.> from 1979, and 10.0364(6) <cit.> from 1987 (where these estimates come from different analyses of the same N ≤ 14 term series).In conclusion, the length-doubling algorithm has resulted in significant extensions of the BCC and FCC series. The application of a recently invented series analysis technique <cit.>, which combines series extrapolation from differential approximants with direct fitting of the extrapolated series, has given excellent estimates of the various critical parameters. In particular, estimates of the growth constants for the BCC and FCC lattices are far more accurate than the previous literature values.§ ACKNOWLEDGEMENTS This work was sponsored by NWO-Science for the use of supercomputer facilities under the project SH-349-15.Computations were carried out on the Cartesius supercomputer at SURFsara in Amsterdam.N.C. acknowledges support from the Australian Research Council under the Future Fellowship scheme (project number FT130100972) and Discovery scheme (project number DP140101110). 32 urlstyle[Madras and Slade(1993)]madras93 Neal Madras and Gordon Slade. The Self-Avoiding Walk. Probability and its applications. Birkhäuser, Boston, MA, 1993.[Janse van Rensburg(2015)]janse15 E. J. Janse van Rensburg. The Statistical Mechanics of Interacting Walks, Polygons, Animals and Vesicles. Oxford University Press, Oxford, UK, second edition, 2015.[Orr(1947)]orr47 W. J. C Orr. Statistical treatment of polymer solutions at infinite dilution. 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Renormalization group calculation of the universal critical exponents of a polymer molecule. PhD thesis, University of Guelph, 1997.[Li et al.(1995)Li, Madras, and Sokal]Li1995CriticalExponentsHyperscaling Bin Li, Neal Madras, and Alan D. Sokal. Critical exponents, hyperscaling, and universal amplitude ratios for two- and three-dimensional self-avoiding walks. J. Stat. Phys., 80:0 661–754, 1995. | http://arxiv.org/abs/1703.09340v1 | {
"authors": [
"Raoul D. Schram",
"Gerard T. Barkema",
"Rob H. Bisseling",
"Nathan Clisby"
],
"categories": [
"cond-mat.stat-mech",
"math.CO"
],
"primary_category": "cond-mat.stat-mech",
"published": "20170327232414",
"title": "Exact enumeration of self-avoiding walks on BCC and FCC lattices"
} |
aff1,aff2,aff3]Bao-Jun Cai aff1]Bao-An Licor1 [cor1]Corresponding author and speaker: [email protected] aff3]Lie-Wen Chen[aff1]Department of Physics and Astronomy, Texas A& M University-Commerce, Commerce, Texas, 75429, USA [aff2]Department of Physics, Shanghai University, Shanghai 200444, China [aff3]School of Physics and Astronomy and Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong University, Shanghai 200240, China Effects of Neutron-Proton Short-Range Correlation on the Equation of State of Dense Neutron-Rich Nucleonic Matter [ December 30, 2023 =================================================================================================================The strongly isospin-dependent tensor force leads to short-range correlations (SRC) between neutron-proton (deuteron-like) pairs much stronger than those between proton-proton and neutron-neutron pairs. As a result of the short-range correlations, the single-nucleon momentum distribution develops a high-momentum tail above the Fermi surface. Because of the strongly isospin-dependent short-range correlations, in neutron-rich matter a higher fraction of protons will be depleted from its Fermi sea and populate above the Fermi surface compared to neutrons. This isospin- dependent nucleon momentum distribution may have effects on: (1) nucleon spectroscopic factors of rare isotopes, (2) the equation of state especially the density dependence of nuclear symmetry energy, (3) the coexistence of a proton-skin in momentum space and a neutron-skin in coordinate space (i.e., protons move much faster than neutrons near the surface of heavy nuclei). In this talk, we discuss these features and their possible experimental manifestations. As an example, SRC effects on the nuclear symmetry energy are discussed in detail using amodified Gogny-Hartree-Fock (GHF) energy density functional (EDF) encapsulating the SRC-induced high momentum tail (HMT) in the single-nucleon momentum distribution. § SINGLE-NUCLEON MOMENTUM DISTRIBUTION FUNCTION ENCAPSULATING SRC EFFECTS It is well known that the SRC leads to a high (low) momentum tail (depletion) in the single-nucleon momentum distribution function denoted by n_ǩ^J above (below) the nucleon Fermi surface in cold nucleonic matter <cit.>. Significant efforts have been made in recent years both theoretically and experimentally to constrain the isospin-dependent parameters characterizing the SRC-modified n_ǩ^J in neutron-rich nucleonic matter. In particular, it has been found via analyzing electron-nucleus scattering data that the percentage of nucleons in the HMT above the Fermi surface is as high as about 28%±4% in symmetric nuclear matter (SNM) but decreases gradually to about only 1%∼2% in pure neutron matter (PNM). On the other hand, the predicted size of the HMT still depends on the model and interaction used. For instance, the self-consistent Green's function (SCGF) theory using the AV18 interaction predicts a 11%∼13% HMT for SNM at saturation density and a 4%∼5% HMT in PNM <cit.>.For completeness and the ease of the following discussions, we first briefly describe the SRC-modified single-nucleon momentum distribution function encapsulating a HMT constrained by the available SRC data that we shall use in this work. The single-nucleon momentum distribution function in cold asymmetric nuclear matter (ANM) has the following form <cit.>n^J_ǩ(ρ,δ)={[Δ_J,0<|ǩ|<k_F^J,;; C_J(k_F^J/|ǩ|)^4, k_F^J<|ǩ|<ϕ_Jk_F^J. ].Here, k_F^J=k_F(1+τ_3^Jδ)^1/3 is the Fermi momentum where k_F=(3π^2ρ/2)^1/3, τ_3^n=+1 and τ_3^p=-1, respectively, δ=(ρ_n-ρ_p)/(ρ_n+ρ_p) is the isospin asymmetry,and ϕ_J is a high-momentum cut-off parameter.The parameters involved depend on the isospin asymmetry and satisfy the normalization condition <cit.>. The above form of n^J_ǩ(ρ,δ) was found consistent with the well-known predictions of microscopic nuclear many-body theories <cit.> and the recent experimental findings <cit.>. This form of n^J_ǩ(ρ,δ) has been applied to address several issues regarding the HMT effects recently in both nuclear physics and astrophysics.The parameters Δ_J, C_J and ϕ_J are assumed to depend linearly on δ based on predictions of microscopic many-body theoriesY_J=Y_0(1+Y_1τ_3^Jδ) <cit.>. The amplitude C_J and high-momentum cutoff coefficient ϕ_J determine the fraction of nucleons in the HMT via x_J^HMT=3C_J(1-ϕ_J^-1). Moreover, the normalization condition between the density ρ_J and the distribution n_ǩ^J, i.e.,[2/(2π)^3]∫_0^∞n^J_ǩ(ρ,δ)̣̌k=ρ_J=(k_F^J)^3/3π^2 requires that only two of the three parameters, i.e., C_J, ϕ_J and Δ_J, are independent. Here we choose the first two as independent and determine the Δ_J by Δ_J=1-3C_J(1-ϕ_J^-1)=1-x_J^HMT. Meanwhile, the adopted C/|ǩ|^4 shape of the HMT both for SNM and PNM is strongly supported by recent studies both theoretically and experimentally. It is interesting to point out that the |ǩ|^-4 form of the HMT is also found in Bose system theoretically and experimentally, indicating a very general feature of the HMT. For comparisons, we use two HMT parameter sets. The n_ǩ^J adopting a 28% HMT in SNM and a 1.5% HMT in PNM is abbreviated as the HMT-exp set, and that adopting a 12% HMT in SNM and a 4% HMT in PNM <cit.> as the HMT-SCGF set <cit.>. Moreover, the model using a step function for the n_ǩ^J is denoted as the free Fermi gas (FFG) set as a reference. As discussed in more details in ref. <cit.>, the HMT parameters in the HMT-exp (HMT-SCGF) parameter set are ϕ_0≈2.38 (ϕ_0≈1.49), ϕ_1≈-0.56 (ϕ_1≈-0.25), C_0≈0.161 (C_0≈0.121), and C_1≈-0.25 (C_1≈-0.01), respectively. § INCORPORATING SRC EFFECTS IN GOGNY HARTREE-FOCK ENERGY DENSITY FUNCTIONALS In most studies of heavy-ion collisions using transport models, one parameterizes the energy density functionals (EDFs) and determine their parameters by reproducing empirical properties of SNM at the saturation density ρ_0, a selected value of symmetry energy E_sym(ρ_0) and its slope L≡[3ρẸ_sym(ρ)/ρ̣]_ρ_0 as well as main features of nucleon optical potentials extracted from analyzing nucleon-nucleus scatterings, such as the isosclar and isovector nucleon effective masses and their asymptotic values at high momenta at ρ_0, etc., see, e.g., ref. <cit.> for detailed discussions.For example, using a modified Gogny-type momentum-dependent interaction (MDI) <cit.>, a modified GHF-EDF in terms of the average energy per nucleon E(ρ,δ) in ANM at density ρ and isospin asymmetry δ can be written asE(ρ,δ)=∑_J=n,p1/ρ_J∫_0^∞ǩ^2/2Mn_ǩ^J(ρ,δ)̣̌k +A_ℓ(ρ_p^2+ρ_n^2)/2ρρ_0 +A_uρ_pρ_n/ρρ_0 +B/σ+1(ρ/ρ_0)^σ(1-xδ^2) +∑_J,J'C_J,J'/ρρ_0∫̣̌ḳ̌k'f_J(ř,ǩ)f_J'(ř,ǩ')Ω(ǩ,ǩ').The first term is the kinetic energy while the second to fourth terms are the usual zero-range 2-body and effective 3-body contributions characterized by their strength parameters A_ℓ, A_u and B as well as the density dependence σ of the 3-body force <cit.>A_ℓ=A_ℓ^0+2xB/1+σ, A_u=A_u^0-2xB/1+σwhere x controls the competition between the isosinglet and isotriplet 2-body interactions, and it affects only the slope L but not the E_sym(ρ_0) by design <cit.>. The last term in Eq. (<ref>) is the contribution to the Equation of State (EOS) from the finite-range 2-body interactions characterized by the strength parameter C_J,J≡ C_ℓ for like and C_J,J≡ C_u for unlike nucleon paris, respectively, using the notations n=p and p=n. The f_J(ř,ǩ) and n_ǩ^J(ρ,δ) arethe nucleon phase space distribution function and momentum distribution function, respectively. In equilibrated nuclear matter at zero temperature, they are related byf_J(ř,ǩ)=2/h^3n_ǩ^J(ρ,δ)=1/4π^3n_ǩ^J(ρ,δ), ħ=1.For example, in the FFG,n_ǩ^J=Θ(k_F^J-|ǩ|) with Θ the standard step function, then f_J(ř,ǩ)=(1/4π^3)Θ(k_F^J-|ǩ|).The regulating function Ω(ǩ,ǩ') <cit.> originating from the meson exchange theory of nuclear force normally has the form ofΩ(ǩ,ǩ')=[1+(ǩ-ǩ'/Λ)^2]^-1where ǩ and ǩ' are the momenta of two interacting nucleons and Λ is a parameter regulating the momentum dependence of the single-particle potential. For applications to SNM, it is usually determined by fixing the nucleon isoscalar effective mass at the Fermi surface to an empirical value <cit.>. In applying the above formalisms to transport model simulations of nuclear reactions, the f_J(ř,ǩ) and n_ǩ^J(ρ,δ) are calculated self-consistently from solving dynamically the coupled Boltzmann-Uehling-Uhlenbeck (BUU) transport or molecular dynamics equations for quasi-nucleons <cit.>. While in studying thermal properties of hot nuclei or stellar matter in thermal equilibrium, the Fermi-Dirac distributions at finite temperatures are used.Traditionally, one writes the EDF as a sum of kinetic EOS of FFG plus several potential terms. Before making any applications, the model parameters of the EDFs are normally fixed by using step functions for the f_J(ř,ǩ) and n_ǩ^J(ρ,δ) as in a FFG at zero temperature in reproducing properties of nuclei or nuclear matter in their ground states. In reality, however, since all nucleons interact with each other in nuclear medium, they naturally become quasi-nucleons. The normal practice of optimizing the EDFs puts all effects of interactions into the potential part of the EDF thus ignores interaction effects on the kinetic energy of quasi-nucleons. The momentum distribution of these quasi-nucleons in the ground state of the system considered is not simply a step function if SRC effects are considered as we discussed in the previous section.Here, we separate the total EDF into a kinetic energy and several potential parts of quasi-nucleons. The f_J(ř,ǩ) and n_ǩ^J(ρ,δ) with HMTs constrained by the SRC experiments are used in evaluating both the kinetic and the momentum-dependent potential parts of the EDF in ANM at zero temperature. At least for simulating heavy-ion collisions using transport models, how the total EDFs are separated into their kinetic and potential parts are important and have practical consequences in predicting experimental observables. Interestingly, how the SRC may affect the symmetry energy, heavy-ion reactions and properties of neutron stars are among the central issues in our pursuit of understanding the nature of neutron-rich nucleonic matter. Previous attempts to incorporate the experimentally constrained n_ǩ^J(ρ,δ) and f_J(ř,ǩ) with HMT in the non-relativistic EDF and examine their effects on heavy-ion collisions and neutron stars were found very difficult. This is mainly because of the nontrivial momentum dependence of the U_J(ρ,δ,|ǩ|) and the EDF when the SRC-modified n_ǩ^J(ρ,δ) and f_J(ř,ǩ) are used. Since one needs to solve 8-coupled equations simultaneously to obtain self-consistently all model parameters from inverting empirical properties of ANM and nucleon optical potentials at ρ_0, numerical problems associated with the momentum integrals in Eq. (<ref>) using the original Ω(ǩ,ǩ') are very difficult to solve.§ A SURROGATE HIGH-MOMENTUM REGULATING FUNCTION FOR THE MDI ENERGY DENSITY FUNCTIONALTo overcome the numerical problem mentioned above, a surrogate high-momentum regulating function Ω(ǩ,ǩ') that approximates very well the original one while enables all integrals in the EDF and U_J(ρ,δ,|ǩ|)to be analytically expressed was proposed recently in ref. <cit.>. Perturbatively, if Λ is large compared to the momenta scale in the problems under investigation, the Ω(ǩ,ǩ') in Eq. (<ref>) can be expanded as Ω(ǩ,ǩ')≈ 1-ǩ^2/Λ^2-ǩ'^2/Λ^2+2ǩ·ǩ'/Λ^2. Using this as a hint, we parameterize the Ω(ǩ,ǩ') asΩ(ǩ,ǩ')=1+a[(ǩ·ǩ'/Λ^2)^2]^1/4 +b[(ǩ·ǩ'/Λ^2)^2]^1/6,where a and b are two new parameters. It is interesting to note that this Ω(ǩ,ǩ') is invariant under the transformation a→ a/ξ^3/2, b→ξ b and Λ→Λ/ξ^3/2, indicating that we have the freedom to first fix one of them without affecting the physical results. Here we set b=2 and then determine the a and Λ using known constraints as we shall discuss in the following.The advantages of using this new regulating function is twofold: firstly, the basically 1/2 and 1/3 power of ǩ·ǩ'/Λ^2 in the second and third term in (<ref>) is relevant for describing properly the energy dependence of nucleon optical potential <cit.>; secondly, it enables analytical expressions for the EOS and U_J(ρ,δ,|ǩ|) in ANM.We notice that the Ω function is only perturbatively effective at momenta smaller than the momentum scale Λ, indicating that the EDF constructed can only be used to a restricted range of momentum/density. It turns out that the cut-off of the HMT in ANM up to about 3ρ_0 is significantly smaller than the Λ parameter we use here. The above non-relativistic GHF-EDF is denoted as abMDI in the following.We fix all parameters in the model EDF using empirical properties of SNM, ANM and main features of nucleon optical potentials at ρ_0. More specifically, for SNM we adopt E_0(ρ_0)=-16 MeV at the saturation density ρ_0=0.16 fm^-3 with E_0(ρ)=E(ρ,0) the EOS of SNM, its incompressibility K_0≡ [9ρ^2^̣2E_0(ρ)/ρ̣^2]_ρ_0=230 MeV <cit.>, the isoscalar nucleon k-mass, i.e., M_0^∗(ρ)/M=[1+(M/|ǩ|)Ụ_0/|̣ǩ|]^-1_|ǩ|=k_F, is selected as M_0^∗(ρ_0)/M=0.58, and U_0(ρ_0,0)=-100 MeV. For the isospin-dependent part in ANM, we adopt E_sym(ρ_0)=31.6 MeV for the symmetry energy, L≡ L(ρ_0)=58.9 MeV <cit.> for the slope of the symmetry energy and U_sym(ρ_0,1 GeV)=-20 MeV <cit.> for the symmetry potential, respectively. Moreover, the value of Λ is constrained to fall within a reasonable range to guarantee the effect of the high order terms in δ in the EOS of ANM mainly characterized by the fourth order symmetry energy, i.e., E_sym,4(ρ)≡24^-1∂^4E(ρ,δ)/∂δ^4|_δ=0, is smaller than 3 MeV at ρ_0, to be consistent with predictions of microscopic many-body theories. Consequently, 1.40 GeV≲Λ≲1.64 GeV is obtained and the study based onΛ=1.6 GeV is used as the default one. It is worth noting that the single-nucleon potential in SNM thus constructed is consistent with the global relativistic nucleon optical potential extracted from analyzing nucleon-nucleus scattering data <cit.>. Thus, totally five isoscalar parameters, i.e., A_t≡ A_ℓ+A_u,B,C_t≡ C_ℓ+C_u,σ and a for SNM,and three isovector parameters, i.e., A_d≡ A_ℓ-A_u,C_d≡ C_ℓ-C_u and x are all fixed. Details values of these parameters for the three cases using the same set of input physical properties are shown in Tab. <ref> . § SRC EFFECTS ON THE DENSITY DEPENDENCE OF NUCLEAR SYMMETRY ENERGY Now we turn to effects of the SRC on nuclear symmetry energy. Shown in Fig. <ref> are the results obtained using the FFG, HMT-SCGF and HMT-exp parameter sets. By construction, they all have the same E_sym(ρ_0) and L at ρ_0. Also shown are the constraints on the E_sym(ρ) around ρ_0 from analyzing intermediate energy heavy-ion collisions (HIC) <cit.> and the isobaric analog states (IAS) <cit.>. Although the predicted using the three parameter sets can all pass through these constraints, they behave very differently especially at supra-saturation densities. The uncertainty of the due to that of the Λ parameter is also shown in Fig. <ref> for the HMT-exp set with the gray dash-dot lines. It is seen that the uncertainty is much smaller than the SRC effect. For example, the variation of the symmetry energy at 3ρ_0 owing to the uncertainty of Λ is about 2.3 MeV while the SRC effect is about 14.5 MeV.Since the Λ parameter mainly affects the high density/momentum behavior of the EOS, its effects become smaller at lower densities. The reduction of the at both sub-saturation and supra-saturation densities leads to a reduction of the curvature coefficient K_sym≡ 9ρ_0^2^̣2E_sym(ρ)/ρ̣^2|_ρ=ρ_0 of the symmetry energy. More quantitatively, we find that the K_sym changes from -109 MeV in the FFG set to about -121 MeV and -188 MeV in the HMT-SCGF and HMT-exp set, respectively. It is interesting to stress that this SRC reduction of K_sym help reproduce the experimentally measured isospin-dependence of incompressibility K(δ)= K_0+K_τδ^2+𝒪(δ^4) in ANM where K_τ=K_sym-6L-J_0L/K_0. The skewness of SNM J_0≡27ρ_0^3^̣3E_0(ρ)/ρ̣^3|_ρ=ρ_0 is approximately -381, -376 and -329 MeV in the FFG, HMT-SCGF and HMT-exp set, respectively. The resulting K_τ is found to change from -365 MeV in the FFG set to about -378 MeV and -457 MeV in the HMT-SCGF and HMT-exp set, respectively. The latter is in good agreement with the best estimate of K_τ≈-550± 100 MeV from analyzing several different kinds of experimental data currently available <cit.>.It is also interesting to notice that the SRC-induced reduction of within the non-relativistic EDF approach here is qualitatively consistent with the earlier finding within the nonlinear Relativistic Mean-Field (RMF) theory <cit.>. Nevertheless, since there is no explicit momentum dependence in the RMF EDF, the corresponding reduction of is smaller. Obviously, the momentum-dependent interaction makes the softening of the symmetry energy at supra-saturation densities more evident. This naturally leads us to the question why the SRC reduces the at both sub-saturation and supra-saturation densities. The SRC affects the through several terms. First of all, because of the momentum-squared weighting in calculating the average nucleon kinetic energy, the isospin dependence of the HMT makes the kinetic symmetry energy different from the FFG prediction as already pointed out in several earlier studies <cit.>. More specifically, within the parabolic approximation of ANM's EOS the is approximately the energy difference between PNM and SNM. Thus, the larger HMT due to the stronger SRC dominated by the neutron-proton isosinglet interaction increases significantly the average energy per nucleon in SNM but has little effect on that in PNM, leading to a reduction of the kinetic symmetry energy. It is worth emphasizing that we focused on effects of the SRC on the symmetry energy of uniform and cold neutron-rich nucleonic matter within the quasi-nucleon picture. It is known that at verylow densities symmetric nuclear matter is unstable against forming clusters, such as deuterons and alphas.For studies on the symmetry free energy of clustered matter at finite temperature we refer the readers to refs. <cit.>.In summary, within a modified non-relativistic GHF-EDF approach and using a new momentum regulating function, we studied effects of SRC-induced HMT in the single-nucleon momentum distribution on the density dependence of nuclear symmetry energy. After re-optimizing the modified GHF-EDF by reproducing the same empirical properties of ANM, SNM and major features of nucleon optical potential at saturation density, the was found to decrease at both sub-saturation and supra-saturation densities, leading to a reduced curvature K_sym of and subsequently a smaller K_τ for the isospin-dependence of nuclear incompressibility in better agreement with its experimental value.Moreover, the SRC-modified EOS and the single-nucleon potentials in ANM can be used in future transport model simulations of heavy-ion collisions to investigate SRC effects in dense neutron-rich matter in terrestrial laboratories. § ACKNOWLEDGEMENTWe thank X.H. Li, W.J. Guo and X.T. He for helpful discussions in earlier studies on this topic.This work was supported in part by the U.S. Department of Energy, Office of Science, under Award Number DE-SC0013702, the CUSTIPEN (China-U.S. Theory Institute for Physics with Exotic Nuclei) under the US Department of Energy Grant No. DE-SC0009971 and the National Natural Science Foundation of China under Grant No. 11320101004 and 11625521, the Major State Basic Research Development Program (973 Program) in China under Contract No. 2015CB856904, the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education, China, and the Science and Technology Commission of Shanghai Municipality (11DZ2260700). 1Bethe H.A. Bethe, Ann. Rev. Nucl. Part. Sci. 21, 93 (1971). Ant88 A.N. Antonov, P.E. Hodgson, and I.Zh. Petkov, Nucleon Momentum and Density Distribution in Nuclei, Clarendon Press, Oxford, 1988. 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"authors": [
"Bao-Jun Cai",
"Bao-An Li",
"Lie-Wen Chen"
],
"categories": [
"nucl-th",
"astro-ph.HE",
"nucl-ex"
],
"primary_category": "nucl-th",
"published": "20170325212541",
"title": "Effects of Neutron-Proton Short-Range Correlation on the Equation of State of Dense Neutron-Rich Nucleonic Matter"
} |
X[1] > c <[#1]Proof: 𝒱 𝒮 𝒬ω ωtheoremTheorem definitionDefinition corollaryCorollary remarkRemark lemmaLemma assumptionAssumption propositionPropositionDistributed Continuous-Time and Discrete-Time Optimization With Nonuniform Unbounded Convex Constraint Sets and Nonuniform StepsizesPeng Lin, Chunhua Yang and Weihua Gui are with the School of Information Science and Engineering, Central South University, Changsha, China. Wei Ren is with the Department of Electrical and Computer Engineering, University of California, Riverside, USA. E-mail: [email protected], [email protected], [email protected], [email protected]. This work was supported by the National Science Foundation under Grant ECCS-1307678 and ECCS-1611423, the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (61321003), the 111 Project (B17048), the National Natural Science Foundation of China (61573082, 61203080), and the Innovation-driven Plan at Central South University., Wei Ren, Chunhua Yang and Weihua Gui December 30, 2023 ======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== This paper is devoted to distributed continuous-time and discrete-time optimization problems with nonuniform convex constraint sets and nonuniform stepsizes for general differentiable convex objective functions. The communication graphs are not required to be strongly connected at any time, the gradients of the local objective functions are not required to be bounded when their independent variables tend to infinity, and the constraint sets are not required to be bounded. For continuous-time multi-agent systems, adistributed continuous algorithm is first introduced where the stepsizes and the convex constraint sets are both nonuniform. It is shown that all agents reach a consensus while minimizing the team objective function even when the constraint sets are unbounded. After that, the obtained results are extended to discrete-time multi-agent systems and then the case where each agent remains in a corresponding convex constraint set is studied. To ensure all agents to remain in a bounded region, a switching mechanism is introduced in the algorithms. It is shown that the distributed optimization problems can be solved, even though the discretization of the algorithms might deviate the convergence of the agents from the minimum of the objective functions. Finally, numerical examples are included to show the obtained theoretical results.Keywords: Distributed Optimization, Nonuniform Step-Sizes, Nonuniform Convex Constraint Sets § INTRODUCTIONAs an important research direction of control theory, distributed optimization problems for multi-agent systems have attracted more and more attention from the control community <cit.>. The goal of a distributed optimization problem for a multi-agent system is to minimize a desired team objective function cooperatively in a distributed way where each agent can only have access to partial information of the team objective function. During the past few years, several results have been obtained for distributed optimization problems. For example, article <cit.> introduced a discrete-time projection algorithm for multi-agent systems with state constraintsand proved that the optimization problems can be solved when the communication topologies are jointly strongly connected and balanced. Articles <cit.> and <cit.>studieda continuous-time version of the work in <cit.> with convex constraint sets. Article <cit.> gave a distributed continuous-time dynamicalgorithm for distributed optimization, and subsequently, on this basis, articles <cit.> studied the distributed optimization problem for general strongly connected balanced directed graphs and gave the estimate of the convergence rate of the algorithm. Other works about distributed optimization problems could be found in articles <cit.> and the references therein, where new algorithms, e.g., distributed Newton, approximate dual subgradient and zero-gradient-sum algorithms, were given or more complicated cases, e.g., second-order dynamics, time-varying or nonconvex functions, fixed or asynchronous stepsizes and noise, were considered.Though many excellent results have been obtained for the distributed optimization problem, many issues need be further studied, e.g., general convex functions, nonuniform convex constraints and nonuniform stepsizes. For the issues of general convex functions and nonuniform convex constraints, most of the existing results require the gradients or subgradients of the convex functions to be bounded and the convex constraints to be identical and little attention has been paid to general convex functions and nonuniform convex constraints, in particular for multi-agent systems with general directed balanced graphs and unbounded gradients.For example, articles <cit.>studied general convex functions but assumed them to be globally Lipschitz and the graphs are assumed to be strongly connected and balanced. Article <cit.> studied coercive convex functions with unbounded subgradients but the results are limited to the continuous-time multi-agent systems and the convex constraints sets are assumed to be identical for all agents. Article <cit.> studied nonuniform convex constraints but the communication graph is complete and all the edge weights are assumed to be equal. Founded on <cit.>, articles <cit.> gave some results on nonuniform convex constraints but the communication graph is constant and connected, and the objective functions are assumed to be strongly convex or some intermediate variables need be transmitted besides the agent states.Article <cit.> studied a distributed optimization problem with nonuniform convex constraints and gave conditions to guarantee the optimal convergence of the team objective function, but the subgradients and the convex constraint sets are both bounded. Note that <cit.> all adopt a uniform stepsize. For the issue of nonuniform stepsizes, currently, there are few works concerned about this issue. Articles <cit.> studied the distributed optimization problem with nonuniform stepsizes in a stochastic setting, where the communication graphs are required to be undirected and connected. Article <cit.> introduced a kind of nonuniform stepsizes but the discontinuous algorithms were employed to realize the consensus of all agents and the convex constraint sets are assumed to be identical. Article <cit.> also studied the distributed optimization problem with nonuniform stepsizes but some intermediate variables need be transmitted besides the agent states in order to track the average of the gradients.In this paper, we are interested in studying distributed continuous-time and discrete-time optimization problems with nonuniform convex constraint sets and nonuniform stepsizes for general differentiable convex objective functions. The communication graphs might not be strongly connected at any time and it is only required thatthe union of the communication graphs among the time intervals of a certain length be strongly connected. The gradients of the local objective functions considered might not be bounded when their independent variables tend to infinity. First, a distributed continuous-time algorithm is introduced where the stepsizes and the convex constraint sets are both nonuniform. Nonuniform stepsizes mean that the weights of the gradients of the local objective functions in the control input of the agents are nonuniform. That is, the optimal convergence rates of the local objective functions are different, which has great possibility to result in the destruction of the optimal convergence of the team objective function.The existing works (e.g., <cit.>) usually assumed the stepsizes are uniform and took no consideration of nonuniform stepsizes, and hence their approaches are hard to be applied for the case of nonuniform stepsizes.Our approach is to introduce a kind of stepsizes such that the stepsizes of each agent are constructed only based on its own states and the differences between the stepsizes of all agents vanish to zero as time evolves. Though the stepsizes of all agents tend to the same as time evolves, their differences still heavily affect the consensus stability and optimal convergence of the system, especially when the communication graphs are not strongly connected.Moreover, due to the existence of nonuniform convex constraint sets,we need take into account the nonlinearity of the consensus and optimal convergence caused by nonuniform convex constraint sets, which renders the analysis of this case to be very complicated. In particular when the nonuniform constraint sets are unbounded, the gradients of the local objective functions might tend to be unbounded when their independent variables tend to infinity, which makes the existing approaches invalid, e.g., <cit.>, <cit.>, where the nonuniform constraint sets and the subgradients were both bounded. To solve the optimization problem, we perform the analysis in three steps. The first step is to make full use of the convexity of the objective functions and show that our algorithm ensures that all agents remain in a bounded region for all the time. The second step is to analyze the agent dynamics at some key times and show the consensus convergence of all agents. The third step is to estimate the consensus errors and use the estimation of the distance from the agents to the convex constraint sets and the convexity of the objective functions to show the optimal convergence of the optimization problem. After that, we extend the obtained results to discrete-time multi-agent systems, and then study the case where each agent remains in its corresponding convex constraint set. Due to the discretization of the system dynamics, the agents might deviate from the minimum of the objective functions. Such a problem also exists in the centralized optimization system. To deal with it, a switching mechanism is introduced in the algorithms based on each agent's own information under which all agents remain in a bounded region even when the convex constraint sets are unbounded. It is shown that the distributed optimization problems with nonuniform convex constraint sets and nonuniform stepsizes can be solved for the discrete-time multi-agent systems. This paper takes nonuniform unbounded convex constraint sets, nonuniform stepsizes, general differentiable convex objective functions, general switching graphs and the discretization of the algorithms into account simultaneously for the distributed optimization problems. The nonlinearities caused by these factors are different and the coexistence of these nonlinearities would further result in more complicated nonlinearities. Existing works only addressed a fraction of these factors due to the limitations of the algorithms and the analytical approaches. For example, the algorithms in <cit.> cannot be directly applied to the case of convex constraint sets due to the adoption of the integrator operator. The analytical approaches in <cit.> cannot be directly applied in this paper, because the nonsmooth sign functions are used in <cit.> to account for inconsistency in gradients while some intermediate variables need be transimitted besides the agent states in <cit.>. Neither feature is valid in the current paper as continuous functions are used and no intermediate variables are transmitted. Moreover, in <cit.>, the communication graphs are assumed to be strongly connected at all time or the constraint sets are assumed to be identical, which makes the analytical approaches in <cit.> unable to be directly applied in this paper as well. Notation: ℝ^mdenotes the set of all m dimensional real column vectors; ℝ^m× ndenotes the set of all m× n real matrices; ℐ denotes the index set {1,…,n}; 1 represents a column vector of all ones with a compatible dimension; s_i denotes the ith component of the vector s; A_ij denotes the (i,j)th entry of the matrix A; s^T and A^T denote, respectively, the transpose of the vector s and the matrix A; ||s|| denotes the Euclidean norm of the vector s; ∇ f(s) denotes the gradient of the function f(s) at s; diag{A_1,⋯,A_n} denotes a block diagonal matrices with its diagonal blocks equal to the matrices A_i(k);the symbol / denotes the division sign; and P_X(s) denotes the projection of the vector s onto the closed convex set X, i.e., P_X(s)=argmin_s̅∈ Xs-s̅. § PRELIMINARIES AND PROBLEM FORMULATION §.§ PreliminariesLet 𝒢(ℐ,ℰ,𝒜) be a directed communication graph of agents, where ℰ⊆ℐ×ℐ is the set of edges, and 𝒜=[a_ij]∈ℝ^n× n is the weighted adjacency matrix. An edge (i,j) ∈ℰ denotes that agent j can obtain information from agent i. The weighted adjacency matrix 𝒜 is defined as η≤ a_ij≤η̅ for two constants η̅>η>0 if (j,i) ∈ℰ and a_ij=0 otherwise. It is assumed by default that a_ii=0, i.e., (i,i)∉ℰ. The Laplacian of the graph 𝒢, denoted by L, is defined as L_ii=∑_j=1,j≠ i^na_ij and L_ij=-a_ij for all i≠ j. The graph 𝒢 is undirected if a_ij=a_ji for all i,j, and it is balanced if ∑_j=1^na_ij=∑_j=1^na_ji for all i. The set of neighbors of agent i is denoted by N_i={j∈ℐ| (j,i)∈ℰ}. A path is a sequence of edges of the form (i_1,i_2),(i_2,i_3),⋯, where i_j ∈ℐ. The graph 𝒢 is strongly connected, if there is a path from every agent to every other agent, and the graph 𝒢 is connected, if it is undirected and strongly connected <cit.>. <cit.> If the graph 𝒢 is strongly connected, the Laplacian L has one zero eigenvalue associated with eigenvalue vector 1 andall its rest n-1 eigenvalues have positive real parts. Further, if the graph 𝒢 is undirected and connected, all the n-1 nonzero eigenvalues are positive.<cit.> Let f_0(χ): ℝ^r→ℝ be a differentiable convex function. f_0(χ) is minimized if and only if ∇ f_0(χ)=0.<cit.> Suppose that Y≠∅ is a closed convex set in ℝ^r. The following statements hold.(1)For any y∈ℝ^r, y-P_Y(y) is continuous with respect to y and ∇1/2y-P_Y(y)^2=y-P_Y(y);(2)For any y,z∈ℝ^r and all ξ∈ Y, [y-P_Y(y)]^T(y-ξ)≥0, P_Y(y)-ξ^2≤y-ξ^2-P_Y(y)-y^2 and P_Y(y)-P_Y(z)≤y-z. §.§ Problem formulationConsider a multi-agent system consisting of n agents. Each agent is regarded as a node in a directed graph 𝒢[In the following, 𝒢(t), 𝒢(kT), a_ij(t), a_ij(kT), N_i(t) and N_i(kT) will be used to denote the graph, the edge weight and the agent neighbor set at time t or k as defined in Sec. II.A.], and each agent can interact with only its local neighbors in 𝒢. Our objective is to design algorithms using only local interaction and information such that all agents cooperatively find the optimal state that solves the optimization problem[minimize ∑_i=1^nf_i(s); subject to s∈ H=⋂_i=1^nH_i⊆ℝ^r, ]where f_i(s): ℝ^r→ℝ denotes the differentiable convex local objective function of agenti, and H_i denotes the closed convex constraint set of f_i(s). Clearly, ∑_i=1^nf_i(s) is also a differentiable convex function. It is assumed that f_i(s) and H_i are known only toagent i. The problem described above is equivalent to the problem that all agents reach a consensus while minimizing the team objective function ∑_i=1^nf_i(x_i), i.e.,[minimize ∑_i=1^nf_i(x_i); subject to x_i=x_j∈ H=⋂_i=1^nH_i⊆ℝ^r. ]In this paper, our analysis is for the general m case. When no confusion arises, the equations or formula are written in the form of m=1 for notational simplicity.§ DISTRIBUTED CONTINUOUS-TIME OPTIMIZATIONIn this section, we discuss the distributed optimization problem for continuous-time multi-agent systems. The problem has applications in motion coordination of multi-agent systems, where multiple physical vehicles rendezvous or form a formation centered at a team optimal location.Suppose that the agents satisfy the continuous-time dynamicsẋ_i(t)=u_i(t),i∈ℐ,where x_i∈ℝ^r is the state of agent i, and u_i ∈ℝ^r is the control input of agent i. §.§ Assumptions and some necessary lemmas Let 𝒳 denote the optimal set of the constrained optimization problem (<ref>). Before the main assumptions and the necessary lemmas, we need further define the sets X_i≜{x |∇ f_i(x)=0} for all i∈ℐ and X≜{x |∑_i=1^n∇ f_i(x)=0} for later usage. From Lemma <ref>, X_i and X are convex and denote, respectively, the optimal sets of f_i(x) and the team objective function ∑_i=1^nf_i(x) for x∈ℝ^r. Note that in general X is different from 𝒳 but X=𝒳 when H_i =R^r for all i. <cit.> Each set X_i, i∈ℐ, is nonempty and bounded.In Assumption <ref>, we only make an assumption on each f_i(x) rather than the team objective function ∑_i=1^nf_i(x) because ∑_i=1^nf_i(x) is global information for all agents and cannot be used by each agent in a distributed way.H≠∅.In Assumption <ref>, we do not require H_i to be bounded and hence H might be unbounded.<cit.> Under Assumption <ref>,the following two statements hold:(1) lim_y→+∞f_i(y)=+∞ for all i and accordingly lim_y→+∞∑_i=1^nf_i(y)=+∞.(2) All X_i, i∈ℐ, and X are nonempty closed bounded convex sets. <cit.> Under Assumptions <ref> and<ref>, 𝒳 is a nonempty closed bounded convex set.Actually, Lemma <ref> shows the existence and boundedness of the optimal set of the constrained optimization problem (<ref>), 𝒳, under Assumptions <ref> and<ref>. <cit.> The length of the time interval between two contiguous switching times is no smaller than a given constant, denoted by d_w.Under Assumption <ref>, the switching of the graph 𝒢(t) cannot be arbitrarily fast, which prevents the system from exhibiting the Zeno behavior. There exists an infinite sequence of swiching times of the graph 𝒢(t), t_0,t_1,t_2,⋯, such that t_0=0, 0<t_j+1-t_j≤ M and the union of all the graphs during each interval [t_j,t_j+1) is strongly connected for some constant M>d_w and all nonnegative integers j.Assumption <ref> ensures that all agents can communicate with each other persistently.Suppose that the graph 𝒢(t) is balanced for all t. From <cit.>, by rearranging the agent indices, L(t)^T+L(t)/2 can be written as L(t)^T+L(t)/2=diag{L^1(t)^T+L^1(t)/2,⋯,L^h(t)^T+L^h(t)/2}, where each L^i(t)^T+L^i(t)/2 corresponds to a strongly connected component of the agents. From Lemma <ref>, each eigenvalue of L^i(t)^T+L^i(t)/2 is nonnegative and hence all eigenvalues of L(t)^T+L(t)/2 are nonnegative. Before the main results, we first present some necessary lemmas that will be used in the analysis of the main results. Specifically, Lemma <ref> shows aradial growth property of the derivatives of f_i(x),Lemma <ref> shows the consensus convergence property of the stepsizes, Lemma <ref> shows a dependency relationship between the distances from one given point to the convex sets and their intersection, and Lemma <ref> shows the boundedness of the gradients in a bounded region. For clarity, the proofs of Lemmas <ref>-<ref> together with Theorems <ref>-<ref> are provided in the Appendix. <cit.> Let f(s): Ξ↦ℝ be a differentiable convex function and Y be its minimum set in Ξ, where Ξ⊆ℝ^r is a closed convex set. Suppose that Y⊆Ξ is closed and bounded. For any z=λ P_Y(y)+(1-λ)y with λ∈(0,1), 0<∇ f(z)^Ty-P_Y(y)/y-P_Y(y)≤∇ f(y)^Ty-P_Y(y)/y-P_Y(y) for any y∈Ξ-Y.For the system given by q̇_i(t)=arctan (e^x_i(t)) with q_i(0)>0, if lim_t→+∞[x_i(t)-x^*(t)]=0 for all i, lim_t→+∞q_i(t)/q_j(t)=1 for all i,j, where x^*(t)≜1/n∑_i=1^nx_i(t). Let y(t)∈ E for all t, where E is a bounded set. Under Assumption <ref>, if lim_t→+∞y(t)-P_H_i(y(t))=0 for all i, then lim_t→+∞y(t)-P_H(y(t))=0. Let Y be a closed bounded convex set. Then, ∇ f_i(x)<ϱ for all i, all x∈ Y and some constant ϱ>0.§.§ Algorithm and convergence analysis In this subsection, we design a continuous distributed optimization algorithms with nonuniform stepsizes. The algorithm is given by [q̇_i(t)= arctan (e^x_i(t)), q_i(0)>0,; u_i(t)=∑_j∈ N_i(t)a_ij(t)(x_j(t)-x_i(t));- [x_i(t)-P_H_i(x_i(t))]-∇ f_i(x_i(t))/√(q_i(t)) ]for all i. The stepsize of the gradient,1/√(q_i(t)), is used to make the term ∇ f_i(x_i(t))/√(q_i(t)) tend to zero as t→+∞. The role of the term -∇ f_i(x_i(t))/√(q_i(t)) is to make all agents converge to the optimal set of the team objective function and the role of -[x_i(t)-P_H_i(x_i(t))] is to make each agent converge to the convex set H_i.It should be noted that the construction of the stepsize 1/√(q_i(t)) is only based on the ith agent's own states and it does not use the Lipschitz constant or the convexity constant as in the existing works, e.g., <cit.> and <cit.>. The stepsize of each agent 1/√(q_i(t)) is state-dependent and can be nonuniform for all agents. The existing works often assume the stepsizes of all agents to be predesigned and consistent with each other at any time, which exerts a heavy burden on sensing and communication costs of the entire system.In algorithm (<ref>), the role of the inverse tangent functions and the exponential functions is to ensure q̇_i(t) to be upper and lower bounded. In fact, some other more general functions, e.g., saturation function, can be employed to play the same role. Moreover, the stepsizes used in algorithm (<ref>)are in a special form, and it can also be extended to other functions. For easy readability, we do not give the general form of the functions q_i(t) and the stepsizes. Letx(t)=[x_1(t)^T,⋯,x_n^T(t)]^T, Q(t)=diag{1/√(q_1(t)),⋯,1/√(q_n(t))}, ∇ f(x(t))=[∇ f_1(x_1(t))^T,⋯,∇ f_n(x_n(t))^T]^Tand Pz(x(t))=[x_1(t)-P_H_1(x_1(t)),⋯,x_n(t)-P_H_n(x_n(t))]^T. Then the system (<ref>) with (<ref>) can be written as[ ẋ(t)=-L(t)x(t)-Pz(x(t))-Q(t)∇ f(x(t)). ] For convenience of discussion, under Assumption <ref>, let t_j_0<t_j_1<⋯<t_j_m_j with t_j_0=t_j and t_j_m_j=t_j+1 denote all the switching times in the interval [t_j,t_j+1). In the following, we show the effectiveness of this algorithm to solve the optimization problem (<ref>). Suppose that the graph 𝒢(t) is balanced for all t and Assumptions <ref>, <ref>, <ref> and <ref> hold.For arbitrary initial conditions x_i(0)∈ℝ^r, using algorithm (<ref>) for system (<ref>),the following statements hold. (1) All x_i(t) remain in a bounded region for all i and all t.(2) lim_t→+∞x^*(t)-P_H(x^*(t))=lim_t→+∞[x_i(t)-x^*(t)]=0 for all i, where x^*(t) has been defined in Lemma <ref>.Suppose that the graph 𝒢(t) is balanced for all t and Assumptions<ref>, <ref>, <ref> and <ref> hold.For arbitrary initial conditions x_i(0)∈ℝ^r, using algorithm (<ref>) for system (<ref>), all agents reach a consensus, i.e., lim_t→+∞[x_i(t)-x^*(t)]=0 for all i, and minimize theteam objective function (<ref>) as t→+∞.In the existing works, the distributed optimization problems were considered often under the assumption that the stepsizes of the gradients are uniform and explicitly time-dependent for all agents. That is, the stepsizes should be consistent with each other at any time. In algorithm (<ref>), we do not make such an assumption and the stepsizes of the gradients only depend on the agent states. The stepsizes need not have the same value at any time and instead they are usually nonuniform, which greatly relaxes the synchronization requirement on the system.§ DISTRIBUTED DISCRETE-TIME OPTIMIZATIONIn this section, we discuss the distributed optimization problem for discrete-time multi-agent systems. Suppose that the agents satisfy the discrete-time dynamicsx_i((k+1)T)=u_i(kT),i∈ℐ,where x_i∈ℝ^r is the state of agent i, u_i ∈ℝ^r is the control input of agent i, and T>0 is the sample time. In the following, we use “(k)" instead of “(kT)" when no confusion arises. §.§ Assumptions and some necessary lemmasIn Sec. II, when we define the weighted adjacency matrix 𝒜, we assume by default that a_ii=0. In this section, for discussion of discrete-time multi-agent systems, we need to redefine a_ii and make an assumption about the weighted adjacency matrix 𝒜 as shown in the following assumption. <cit.> For all i,j∈ℐ, a_ii(k)≥η, a_ij(k)≥η for some constant 0<η≤1 and each nonzero a_ij(k), ∑_i=1^na_ij(k)=1 and ∑_j=1^n a_ji(k)=1.Under Assumption <ref>, the adjacency matrix of the graph 𝒢(k) is doubly stochastic and its diagonal entries are nonzero. Assumption <ref> is used to generate convex combinations of the agents' states such that the influence of each agent's state is equal in the final consensus value in the distributed optimization algorithms shown later.<cit.> There exists an infinite sequence of switching times of the graph 𝒢(k), k_0,k_1,k_2,⋯, such that k_0=0, 0<k_j+1-k_j≤ M and the union of all the graphs during each interval [k_j,k_j+1) is strongly connected for some positive integer M and all nonnegative integers j.Similar to Assumption <ref>, Assumption <ref> ensures that all agents can communicate with each other persistently.§.§ Distributed optimization with nonuniform stepsizesIn this subsection, we design a discrete-time distributed optimization algorithm with nonuniform stepsizes. The algorithm is given by[ q_i(k+1)=q_i(k)+arctan (e^x_i(k))T, q_i(0)>0,;u_i(k)=w_i(k)(1-γ_i)+P_H_i[w_i(k)]γ_i,; w_i(k)=v_i(k)-gr_i(k)T,;v_i(k)=∑_j∈ N_i(k)∪{i}a_ij(k)x_j(k),;gr_i(k)={[ 0, √(q_i(k))≤∇ f_i(v_i(k))^2,;∇ f_i(v_i(k))/√(q_i(k)), ]. ]where0<γ_i≤1 is a constant for each i.Due to the discretization of the system dynamics, the agents might deviate from the minimum of the team objective function. Such a problem also exists in the centralized optimization system. To deal with it, a switching mechanism is introduced in (<ref>) based on each agent' own information, under which the gradient term would not be too large to result in the divergence of the system. This will be shown in the proof of Theorem <ref>.For Algorithm (<ref>), it can be calculated simply in four steps: (a) v_i(k), q_i(k+1) and ∇ f_i(v_i(k)); (b) gr_i(k) based on the switching mechanism; (c) w_i(k); and (d) P_H_i[w_i(k)] and u_i(k). Though the algorithm computation looks a bit complex due to the existence of the switching mechanism, the algorithm does not require intermediate variables to be transmitted and it is a fully distributed algorithm. Suppose thatAssumptions <ref>, <ref>, <ref> and <ref> hold. For arbitrary initial conditions x_i(0)∈ℝ^r, using algorithm (<ref>) for system (<ref>), if 0<γ_i<1 for all i, all agents reach a consensus, i.e., lim_t→+∞[x_i(k)-x^*(k)]=0 for all i, and minimize theteam objective function (<ref>) as k→+∞. In Theorems <ref> and <ref>, it is not required that each agent remain in its corresponding convex constraint set H_i. In the following theorem, we show that the optimization problem (<ref>) can be solved when all agents remain in their corresponding convex constraint sets.Suppose thatAssumptions <ref>, <ref>, <ref> and <ref> hold. For arbitrary initial conditions x_i(0)∈ H_i, using algorithm (<ref>) for system (<ref>), if γ_i=1 for all i, all agents reach a consensus, i.e., lim_t→+∞[x_i(k)-x^*(k)]=0 for all i, and minimize theteam objective function (<ref>) as k→+∞ while each agent remains in its corresponding constraint sets, i.e., x_i(k)∈ H_i for all i and all k.Since the proposed algorithms are gradient based, the convergence rate of the algorithms is not very fast. This is common for the gradient-based distributed algorithms in the existing literature. In particular, the stepsizes (gradient gains) are nonuniform, which makes the convergence rate slower than that with uniform stepsizes. However, our algorithms are able to deal with the general case of nonuniform stepsizes without intermediate variables being transmitted. In the existing non-gradient-based works, some special assumptions are always made in order to ensure the optimal convergence. For example, in <cit.>, the communication graphs are assumed to be strongly connected and the the convex objective functions are assumed to be strongly convex. In this paper, the communication graphs are only required to be jointly strongly connected, and the convex objective functions are only required to be differentiable (which can be easily extended to the nondifferentiable case by using subgradients). Future work could be directed towards improving the convergence rate of our algorithms. In particular, different dimension might yield different convergence rates. It is worth studying the effects of different dimensions on the convergence rate of the algorithms.§ SIMULATIONSConsider a multi-agent system with 8 agents in ℝ^2.The communication graphs switch among the balanced subgraphs of the graph shown in Fig. <ref>.Each edge weight is 0.5. The sample time is T=0.1 s for the discrete-time algorithm. The local objective functions are adopted as f_1(x_1)=1/2x_11^2+1/2x_12^2, f_2(x_2)=1/2(x_21+1)^2+1/2x_22^2, f_3(x_3)=1/2x_31^2+1/2(x_32+1)^2, f_4(x_4)=1/2(x_41+1)^2+1/2(x_42+1)^2, f_5(x_5)=1/4x_51^4+1/4x_52^4, f_6(x_6)=1/4(x_61+1)^4+1/4x_62^4, f_7(x_7)=1/4x_71^4+1/4(x_72+1)^4, and f_8(x_8)=1/4(x_81+1)^4+1/4(x_82+1)^4.where x_i1 and x_i2 denote the 1st and 2nd components of x_i. The constrained convex sets are adopted as H_1={(x,y)^T∈ℝ^2 |s-[0,0]^T≤ 3} for agents 1 and 5, H_2={(x,y)^T∈ℝ^2 |x≤ 0.5, y≥ 1} for agents 2 and 6, H_3={(x,y)^T∈ℝ^2 |s-[0,3]^T≤ 3} for agents 3 and 7 and H_4={(x,y)^T∈ℝ^2 | x≥ -0.5,y≥ 1} for agents 4 and 8. The team objective function ∑_i=1^8f_i(s) is minimized if and only if s=[-0.5,1]^T. The simulation results for algorithm (<ref>) and algorithm (<ref>) with γ_i=1 for all i are shown in Figs. <ref> and <ref>. It is clear that all agents finally converge to the optimal point. In particular, for algorithm (<ref>), all agents remain in their corresponding constraint sets. All the simulation results areconsistent with our obtained theorems. § CONCLUSIONSIn this paper, distributed continuous-time and discrete-time optimization problems with nonuniform stepsizes and nonuniform possibly unbounded convex constraint sets were studied for general differentiable convex objective functions. One continuous-time algorithm and one discrete-time algorithm were introduced. For these two algorithms, it has been shown that the team objective function is minimized with nonuniform stepsizes and nonuniform convex constraint sets. In particular, for the discrete-time algorithm, it has been shown that the distributed optimization problem can be solved when each agent remain in its corresponding constraint set.xxangelia A. Nedić, A. Ozdaglar, P. A. 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There exists a constant T_0>0 such that t/4<q_i(t)<4t for all t>T_0 and all i. Let Δ_i(t)=q_i(0)-q_1(0)+∫_0^t[arctan(e^x_i(s))-arctan(e^x_1(s))]ds. It is clear that q_i(t)=q_1(t)(1+Δ_i(t)/q_1(t)) for all i. Since lim_t→+∞[x_i(t)-x^*(t)]=0, from the continuity of the function arctan(e^x_i(t)), there exists a constant T_1>T_0 for any ϵ_0>0 such that|arctan(e^x_i(t))-arctan(e^x_1(t))|<ϵ_0 for all t>T_1 and all i. It is clear that lim_t→+∞Δ_i(T_1)/q_1(t)=0 and lim_t→+∞|∫_T_1^t[arctan(e^x_i(s))-arctan(e^x_1(s))]ds/q_1(t)| <4ϵ_0. Since ϵ_0 can be arbitrarily chosen, we havelim_t→+∞Δ_i(t)/q_1(t)=lim_t→+∞[Δ_i(T_1)+ ∫_T_1^t[arctan(e^x_i(s))-arctan(e^x_1(s))]ds]/q_1(t)=0and hence lim_t→+∞q_i(t)/q_1(t)=1 for all i. Therefore, lim_t→+∞q_i(t)/q_j(t)=1 for all i,j.Proof of Lemma <ref>:Under Assumption <ref>, H≠∅. Since E is a bounded set, there exists a closed bounded convex set Y such that E⊂ Y. From Lemma <ref>, y-P_H(y) is a continuous function with respect to y. If this lemma does not hold, there must exist a sequence {t^k} such that lim_k→+∞y(t^k)-P_H(y(t^k))=ψ for some constant 0<ψ≤+∞. Let H_ψ={y|y-P_H(y)=ψ}. Consider the set H_ψ∩ Y. If ∑_i=1^ny-P_H_i(y)=0 for some y, we have y∈ H_i for all i and hence y∈ H, i.e., y-P_H(y)=0.Note thatH_ψ∩ Y is a closed bounded set. The inequality y-P_H(y)=ψ>0 implies thatthe function ∑_i=1^ny-P_H_i(y) on H_ψ∩ Y is lower bounded by a positive constant, denoted by γ. If this is not true, there must exist asequence {ŷ_h∈ H_ψ∩ Y} such that lim_h→+∞∑_i=1^nŷ_h-P_H_i(ŷ_h)=0. Note that the sequence {ŷ_h∈ H_ψ∩ Y} has a subsequence {y̅_h∈ H_ψ∩ Y} such that lim_h→+∞y̅_h=y̅ for a constant vector y̅. Since H_ψ∩ Y is a closed bounded set, y̅∈ H_ψ∩ Y, i.e., ∑_i=1^ny̅-P_H_i(y̅)=0, and hence y̅-P_H(y̅)=0. This contradicts with the definition of H_ψ.Since y(t)∈ Y for all t, lim_k→+∞y(t^k)-P_H(y(t^k))=ψ and lim_t→+∞y(t)-P_H_i(y(t))=0 for all i, there exists a constant N>0 for any ϵ>0 such that ∑_i=1^ny(t^k)-P_H_i(y(t^k))-γ>-ϵ and y(t)-P_H_i(y(t))<ϵ for all t^k>N and t>N. Let ϵ<γ/n+1. It follows that ∑_i=1^ny(t^k)-P_H_i(y(t^k))>γ-ϵ>nϵ>∑_i=1^ny(t)-P_H_i(y(t)) for all t^k>N and t>N, which is a contradiction. Proof of Lemma <ref>: Since each function f_i(x) is differentiable, ∇ f_i(x) exists for any x∈ Y. If ∇ f_i(x) is unbounded for some x∈ Y, there must exist a sequence {y^k∈ Y,k=1,2,⋯} such that lim_k→+∞y^k=y^0 for some y^0∈ Y and lim_k→+∞∇ f_i(y^k)=+∞. Let z^k=y^k+∇ f_i(y^k)/∇ f_i(y^k). It is clear that each f_i(z^k) and each f_i(y^k) are bounded. From the convexity of f_i(y^k), it follows that ∇ f_i(y^k)=∇ f_i(y^k)^T(z^k-y^k)≤ f_i(z^k)-f_i(y^k) and hence lim_k→+∞[f_i(z^k)-f_i(y^k)]≥ +∞ which is a contradiction.Proof of Theorem <ref>:First, we prove that using (<ref>) for (<ref>) all x_i(t) remain in a bounded region and all ∇ f_i(x_i(t)) are bounded for all i and all t.Note that π/4≤arctan(e^x_i(t))≤π/2 for all t and all i. There exists a constant T_0>0 such that 2√(t)>√(q_i(t))>√(t)/2 for all i and all t≥ T_0. Under Assumption <ref>, from Lemma <ref>, X and all X_i are nonempty and bounded. Note from Lemma <ref> that 𝒳 is nonempty and bounded under Assumptions <ref> and <ref>. There is a closed bounded convex set Y={y|y-P_𝒳(y)≤ P_1}⊂ℝ^r for some constant P_1>0 such that x_i(T_0)∈ Y, X⊂ Y, X_i⊂ Y and 𝒳⊂ Y for all i. From the continuity of f_i(x) and Lemma <ref>, let Y be sufficiently large such that f_i(x_i(t))-f_i(z)≥ 4∑_j=1^n[f_j(z)-f_j(z_j)] for all i, all z∈𝒳, all z_j∈ X_j and all x_i(t)∉ Y.Consider the Lyapunov function candidate V_0(t)=1/2x(t)-z 1^2 for some z∈𝒳 and t≥ T_0. Calculating V̇_0(t), we have[V̇_0(t); =(x(t)-z1)^T[-L(t)x(t)-Pz(x(t))-Q(t)∇ f(x(t))] ] for t≥ T_0. Since the graph 𝒢(t) is balanced, then 1^TL(t)=0 and hence z1^TL(t)x(t)=0. Regarding L(t)^T+L(t)/2 as a Laplacian of an undirected graph, we have -x(t)^TL(t)x(t)=-x(t)^TL(t)^T+L(t)/2x(t)≤ 0 from the previous analysis below Assumption <ref>. Since the function 1/2x_i(t)-P_H_i(x_i(t))^2 is convex, it follows that[-(x_i(t)-z)^T(x_i(t)-P_H_i(x_i(t))); ≤1/2z-P_H_i(z)^2-1/2x_i(t)-P_H_i(x_i(t))^2;=-1/2x_i(t)-P_H_i(x_i(t))^2 ]where the last equality holds since z∈𝒳⊂ H⊂ H_i. Thus, we have[ V̇_0(t)≤-1/2∑_i=1^nx_i(t)-P_H_i(x_i(t))^2; -(x(t)-z1)^TQ(t)∇ f(x(t)) ] for t≥ T_0. From the convexity of the function f_i(x_i(t)), we have[ ∇ f_i(x_i(t))^T(z-x_i(t))≤ f_i(z)-f_i(x_i(t))≤ f_i(z)-f_i(z_i). ] It follows that[-(x(t)-z1)^TQ(t)∇ f(x(t)); ≤∑_i=1^n1/√(q_i(t))[f_i(z)-f_i(x_i(t))]. ]Note that f_i(x_i(t))-f_i(z_i)≥0 since z_i∈ X_i for all i. If x_i_0(t)∉ Y for some i_0, we have f_i_0(x_i_0(t))-f_i_0(z)≥ 4∑_j=1,j≠ i_0^n[f_j(z)-f_j(z_j)] for all z_j∈ X_j and hence[ -(x(t)-z1)^TQ(t)∇ f(x(t));≤ -1/√(q_i_0(t))[f_i_0(x_i_0(t))-f_i_0(z)]; +∑_j=1,j≠ i_0^n1/√(q_j(t))[f_j(z)-f_j(z_j)];≤ -1/2√(t)[f_i_0(x_i_0(t))-f_i_0(z)]; +∑_j=1,j≠ i_0^n2/√(t)[f_j(z)-f_j(z_j)] ≤ 0. ] As a result, we have V̇_0(t)≤ 0 for t≥ T_0 if there exists an agent i_0 such that x_i_0(t)∉ Y for some i_0. This implies that all x_i(t) remain in a bounded region for all i and all t. Thus, x_i(t)-x^*(t) is bounded. Note that x_i(t)-P_H(x^*(t))≤x^*(t)-P_H(x^*(t))+x_i(t)-x^*(t)≤x_i(t)-x^*(t)+x^*(t)-s where s∈ H is a constant vector. Hence, x_i(t)-P_H(x^*(t)) is bounded.Further, from Lemma <ref>, it follows that ∇ f_i(x_i(t)) is bounded for all i and all t.Since 2√(t)>√(q_i(t))>√(t)/2 for all i and all t≥ T_0, we have [ x_i(t)-P_H(x^*(t))∇ f_i(x_i(t))/√(q_i(t))<ρ/n√(t) ]for all i, all t≥ T_0 and some constant ρ>0. Moreover, since the graph 𝒢(t) is balanced,[ ẋ^*(t)=-1/n∑_i=1^n[x_i(t)-P_H_i(x_i(t))+∇ f_i(x_i(t))/√(q_i(t))]. ] Consider the Lyapunov function candidateV_1(t)=∑_i=1^nx_i(t)-x^*(t)^2+nx^*(t)-P_H(x^*(t))^2for all t≥ T_0. Calculating V̇_1(t), we have[ V̇_1(t); =2∑_i=1^n[x_i(t)-x^*(t)]^T(ẋ_i(t)-ẋ^*(t)); + 2n[x^*(t)-P_H(x^*(t)]^Tẋ^*(t); =-2x^T(t)L(t)x(t)-2∑_i=1^n[x_i(t)-x^*(t)]^T; × [(x_i(t)-P_H_i(x_i(t))+∇ f_i(x_i(t))/√(q_i(t))]; -2∑_i=1^n[x^*(t)-x_i(t)+x_i(t)-P_H(x^*(t))]^T; ×[x_i(t)-P_H_i(x_i(t))+∇ f_i(x_i(t))/√(q_i(t))]; = -2x^T(t)L(t)x(t)-2∑_i=1^n[x_i(t)-P_H(x^*(t))]^T; ×[x_i(t)-P_H_i(x_i(t))+∇ f_i(x_i(t))/√(q_i(t))]; ≤ -ϕ(t)+2ρ/√(t) ]for all t≥ T_0, where[ ϕ(t)≜2x(t)^TL(t)x(t)+∑_i=1^nx_i(t)-P_H_i(x_i(t))^2, ]∑_i=1^n[x_i(t)-x^*(t)]^Tẋ^*(t)=0, and the last inequality has used the convexity of x_i(t)-P_H_i(x_i(t))^2 and the fact that P_H(x^*(t))∈ H_i and P_H(x^*(t))-P_H_i(P_H(x^*(t)))^2=P_H(x^*(t))-P_H(x^*(t))^2=0. To proceed with our proof, we prove that V_1(t)≤δ for any constant δ>0 when t is sufficiently large. Since all x_i(t) remain in a bounded region for all i and all t, each ẋ_i(t) is bounded for all i,t. There exists a constant h_0>0 such that ϕ̇(t)<h_0 for all t≥ T_0.Consider V_1(t) for t∈ [t_k,t_k+1) and t_k≥ T_0 where t_k has been defined in Assumption <ref>. Suppose that h_1(t_k)=max_s∈ [t_k,t_k+1]ϕ(s) for t_k≥ T_0. Note that it takes at least h_1(t_k)/h_0 time for the value of ϕ(t) to vary from h_1(t_k) to 0 or from 0 to h_1(t_k). By simple calculations, if h_1(t_k)/h_0≤ d_w/2,[ ∫_t_k^t_k+1ϕ(s)ds≥ h_1(t_k)^2/h_0/2 ]and ifh_1(t_k)/h_0≥ d_w/2,[ ∫_t_k^t_k+1ϕ(s)ds≥ (2h_1(t_k)-h_0d_w/2)d_w/4≥ d_w^2h_0/8. ]Let T_0 be sufficiently large for some ϵ<d^2_wh_0/8/(M+1) such that 2ρ/√(t)<ϵand ∇ f_i(x_i(t))/√(q_i(t))<ϵfor all t≥ T_0 and all i. When h_1(t_k)>√(2h_0(M+1)ϵ),[ ∫_t_k^t_k+1ϕ(s)ds≥min{h_1(t_k)^2/h_0/2,d^2_wh_0/8}≥ϵ (M+1). ] From Assumption <ref>, t_k+1-t_k≤ M. It follows that[ V_1(t_k+1)-V_1(t_k)≤∫_t_k^t_k+1[-ϕ(s)+ϵ]d s≤-ϵ. ] Note that[x^T(t)L(t)x(t); =∑_i=1^n∑_j∈ N_i(t)a_ij(t)x_i(t)-x_j(t)^2;≥∑_i=1^n∑_j∈ N_i(t)ηx_i(t)-x_j(t)^2. ]When h_1(t_k)≤√(2h_0(M+1)ϵ), x_i(t)-x_j(t)≤√(2h_0(M+1)ϵ)/√(η) for all i and all j∈ N_i(t) and x_i(t)-P_H_i(x_i(t))≤√(2h_0(M+1)ϵ) for all i. Thus, ẋ_i(t)≤√(2h_0(M+1)ϵ)(1+η̅n/√(η))+ϵ and x_i(t)-x_i(t̅)≤ [√(2h_0(M+1)ϵ)(1+η̅n/√(η))+ϵ](t_k+1-t_k)≤ [√(2h_0(M+1)ϵ)(1+η̅n/√(η))+ϵ]M for all t,t̅∈ [t_k,t_k+1]. Under Assumption <ref>, the union of all the graphs during each interval [t_k,t_k+1), denoted by 𝒢̂(t_k), is strongly connected. In 𝒢̂(t_k), there must exist a directed path between every two agents whose length is no larger than n. Thus,[max_i{x^*(t)-x_i(t)| t∈ [t_k,t_k+1]}; ≤max_i,j{x_i(t)-x_j(t)| t∈ [t_k,t_k+1]};≤ 2n[√(2h_0(M+1)ϵ)(1+η̅n/√(η))+ϵ]M; +n√(2h_0(M+1)ϵ)/√(η). ]From Lemma <ref>, when lim_t→+∞x^*(t)-P_H_i(x^*(t))=0 for all i, lim_t→+∞x^*(t)-P_H(x^*(t))=0. Let [ ϕ_1(t)≜∑_i=1^nx_i(t)-x^*(t)^2; +∑_i=1^nx^*(t)-P_H_i(x^*(t))^2. ]Then there exists a constant δ_1>0 for any ϵ_1>0 such that V_1(t)<ϵ_1 when ϕ_1(t)<δ_1. From Lemma <ref>, we have[ x^*(t)-P_H_i(x^*(t));=x^*(t)-x_i(t)+x_i(t)-P_H_i(x_i(t)); +P_H_i(x_i(t))-P_H_i(x^*(t)); ≤ 2x^*(t)-x_i(t)+x_i(t)-P_H_i(x_i(t)). ]Let ϵ be sufficiently small such that max_i{2x^*(t)-x_i(t)+x_i(t)-P_H_i(x_i(t))| t∈ [t_k,t_k+1]}<√(δ_1/2n) when h_1(t_k)≤√(2h_0(M+1)ϵ). Clearly, x^*(t)-P_H_i(x^*(t))≤√(δ_1/2n) and x^*(t)-x_i(t)≤√(δ_1/2n) for all t∈ [t_k,t_k+1) and hence ϕ_1(t)≤δ_1for all t∈ [t_k,t_k+1) when h_1(t_k)≤√(2h_0(M+1)ϵ). Thus, V_1(t)<ϵ_1for all t∈ [t_k,t_k+1) when h_1(t_k)≤√(2h_0(M+1)ϵ). Clearly, h_1(t_k)>√(2h_0(M+1)ϵ) and hence V_1(t_k+1)-V_1(t_k)≤ -ϵ when V_1(t)≥ϵ_1 for some t∈ [t_k,t_k+1). Thus, there exists a constant T_1>T_0 such that V(t_k)<ϵ_1 for all t_k>T_1.In view of the arbitrariness of ϵ_1, let ϵ_1→0. It follows that lim_k→+∞V_1(t_k)=0.From (<ref>),V̇_1(t)≤2ρ/√(t) for all t≥ T_0. From Assumption <ref>, t_k+1-t_k≤ M. Thus,[ V_1(t)=V_1(t_k)+∫_t_k^tV̇_1(s)ds≤ V_1(t_k)+2Mρ/√(t_k) ]for t∈ [t_k,t_k+1) and hence lim_t→+∞V_1(t)=0.Thus, lim_t→+∞x^*(t)-P_H(x^*(t))=lim_t→+∞[x_i(t)-x^*(t)]=0 for all i. Proof of Theorem <ref>: Note that π/4≤arctan(e^x_i(t))≤π/2 for all t and all i. There exists a constant T_0>0 such that 2√(t)>√(q_i(t))>√(t)/2 for all i and all t≥ T_0. From Theorem <ref>, all x_i(t) remain in a bounded region, and lim_t→+∞[x_i(t)-x^*(t)]=lim_t→+∞x^*(t)-P_H(x^*(t))=0 for all i. It is clear that x^*(t) remains in a bounded region. Moreover, x^*(t)-s is bounded for some s∈ H and hence x^*(t)-P_H(x^*(t)) is bounded for all t. From Lemma <ref>, 𝒳 is a nonempty closed bounded convex set. It follows that f_i(x^*(t)) and f_i(P_𝒳(x^*(t))) are both bounded for all i and all t. Therefore, together with Lemma <ref>, it follows thatthere exists a constant T_1>T_0 for any ϵ_0>0 such that[ x^*(t)-x_i(t)<ϵ_0, x^*(t)-P_H(x^*(t))<ϵ_0, ] [ |1-√(q_i(t)/q_j(t))||f_i(P_H(x^*(t)))|<ϵ_0/2 ]and[ |1-√(q_i(t)/q_j(t))||f_i(P_𝒳(x^*(t)))|<ϵ_0/2 ]for all i,j and all t≥ T_1. Note that all f_i(s) are differentiable functions and lim_t→+∞[x_i(t)-x^*(t)]=lim_t→+∞x^*(t)-P_H(x^*(t))=0. Let T_1 be sufficently large such that |f_i(x^*(t))-f_i(P_H(x^*(t)))|<ϵ_0 and |f_i(x_i(t))-f_i(P_H(x^*(t)))|<ϵ_0for all t≥ T_1. Consider the Lyapunov function candidateV_2(t)=∑_i=1^nx_i(t)-x^*(t)^2+nx^*(t)-P_χ(x^*(t))^2for all t≥ T_1. Similar to the derivation in (<ref>), we have[ V̇_2(t)≤-ϕ(t)-2∑_i=1^n(x_i(t)-P_χ(x^*(t)))^T∇ f_i(x_i(t))/√(q_i(t)). ]Using the convexity of the functions f_i(s), it follows that[V̇_2(t); ≤-2∑_i=1^nf_i(P_H(x^*(t)))-f_i(P_χ(x^*(t)))+f_i(x_i(t))-f_i(P_H(x^*(t)))/√(q_i(t));≤-2∑_i=1^nf_i(P_H(x^*(t)))-f_i(P_χ(x^*(t)))-ϵ_0/√(q_i(t));≤∑_i=1^n4ϵ_0/√(t)-2∑_i=1^nf_i(P_H(x^*(t)))-f_i(P_χ(x^*(t)))/√(q_1(t)); +2∑_i=1^nf_i(P_H(x^*(t)))-f_i(P_χ(x^*(t)))/√(q_1(t))(1-√(q_1(t)/q_i(t)));≤∑_i=1^n8ϵ_0/√(t)-∑_i=1^nf_i(P_H(x^*(t)))-f_i(P_χ(x^*(t)))/√(t);≤8nϵ_0/√(t)-∑_i=1^nf_i(x^*(t))-f_i(P_χ(x^*(t)))+f_i(P_H(x^*(t)))-f_i(x^*(t))/√(t); ≤9nϵ_0/√(t)-∑_i=1^nf_i(x^*(t))-f_i(P_χ(x^*(t)))/√(t);]where the last two inequalities has used the fact that √(t)/2<q_i(t)<2√(t) for all i and t≥ T_0. In the following, we use the properties of the point P_H(x^*(t)) to show the optimal convergence of the system. Define l_0=max_s∈ Hs-P_𝒳(s). Let[ E={s∈ℝ^r |s-P_𝒳(s)≤ l_1} ] for some constant 0<l_1≤ l_0. Suppose that β=min_s∈ H∩∂̅E∑_i=1^n[f_i(s)-f_i(P_𝒳(s))], where ∂̅E={s∈ℝ^r |s-P_𝒳(s)= l_1}.Since P_𝒳(s)∈𝒳, from the definition of 𝒳, we have β>0. From Lemma <ref>,[ ∑_i=1^n[f_i(P_H(x^*(t)))-f_i(P_𝒳(P_H(x^*(t))))]≥β ]for any P_H(x^*(t))∉ E.If x^*(t)-P_𝒳(x^*(t))>l_1+ϵ_0 for t≥ T_1, recalling that x^*(t)-P_H(x^*(t))<ϵ_0 and |f_i(x^*(t))-f_i(P_H(x^*(t)))|<ϵ_0 for t≥ T_1 and all i, it follows that P_H(x^*(t))-P_𝒳(x^*(t))>l_1 and[ ∑_i=1^n[f_i(x^*(t))-f_i(P_𝒳(x^*(t)))]>β-nϵ_0 ]for t≥ T_1.Let ϵ_0 be sufficiently small such that 2ϵ_0+10nϵ_0<β. It follows that for any t≥ T_1 and any x^*(t)-P_𝒳(x^*(t))>l_1+ϵ_0,[ V̇_2(t)≤ -β-nϵ_0/√(t)+9nϵ_0/√(t)<-2ϵ_0/√(t). ]Integrating both sides of this inequality from T_1 to t, we haveV_2(t)-V_2(T_1)≤ -ϵ_0(√(t)-√(T_1)).This implies that there exists a constant T_2>T_1 such that x^*(T_2)-P_𝒳(x^*(T_2))=l_1+ϵ_0. Since x^*(t)-x_i(t)<ϵ_0 for all t≥ T_1, it follows from (<ref>) that [ V_2(T_2)≤ nϵ_0^2+n(l_1+ϵ_0)^2. ] Note that (<ref>) holds for any t≥ T_1 and any x^*(t)-P_𝒳(x^*(t))>l_1+ϵ_0.Thus,[ V_2(t)≤ nϵ_0^2+n(l_1+ϵ_0)^2 ]for all t≥ T_2. In view of the arbitrariness of l_1 and ϵ_0, letting l_1→0 and ϵ_0→0,we have lim_t→+∞x^*(t)-P_𝒳(x^*(t))=0. Since lim_t→+∞[x_i(t)-x^*(t)]=0, we have lim_t→+∞x_i(t)-P_𝒳(x^*(t))=0 for all i. That is, all agents reach a consensus and minimize the team objective function (<ref>) as t→+∞. Proof of Theorem <ref>: Note that π/4≤arctan(e^x_i(k))≤π/2 for all k and all i. There exists a constant T_0>0 such that kT/4<q_i(k)<4kT for all k≥ T_0.Under Assumption <ref>, X and all X_i are nonempty and bounded. Note from Lemma <ref> that 𝒳 is nonempty and bounded under Assumptions <ref> and <ref>. There must exist a closed bounded convex set Y={y|y-z≤ P_2}⊂ℝ^r for some z∈𝒳⊂ H and some constant P_2>0 such that x_i(T_0)∈ Y, X⊂ Y, X_i⊂ Y and 𝒳⊂ Y for all i. Let Y be sufficiently large such that f_i(v_i(k))-f_i(z)≥ 4∑_j=1^n[f_j(z)-f_j(z_j)]T+4nT for all i, all z∈𝒳, all z_j∈ X_j and all v_i(k)∉ Y. Consider the Lyapunov function candidate V_0(k)=max_ix_i(k)-z^2. Calculating x_i(k+1)-z^2, it follows that[x_i(k+1)-z^2; =w_i(k)-z^2+(P_H_i[w_i(k)]-w_i(k))γ_i^2; +2γ_i(w_i(k)-z)^T(P_H_i[w_i(k)]-w_i(k)); ≤w_i(k)-z^2+γ_i^2P_H_i[w_i(k)]-w_i(k)^2; + γ_iz-P_H_i(z)^2-γ_iw_i(k)-P_H_i[w_i(k)]^2; ≤ v_i(k)-z-gr_i(k)T^2 ]for all i, where the first inequality has used the convexity of the function s-P_H_i(s)^2, and the second inequality has used the fact that z∈ H⊂ H_i and z-P_H_i(z)^2=0 and the fact that 0<γ_i<1. When gr_i(k)=0, it follows that[v_i(k)-z-gr_i(k)T^2; ≤∑_j∈ N_i(k)∪{i}a_ij(k)x_j(k)-z^2. ]When gr_i(k)≠0, √(q_i(k))>∇ f_i(v_i(k))^2. It follows that[v_i(k)-z-gr_i(k)T^2;=v_i(k)-z^2+∇ f_i(v_i(k))T/√(q_i(k))^2;-2(v_i(k)-z)^T∇ f_i(v_i(k))T/√(q_i(k));≤ v_i(k)-z^2+T^2/√(q_i(k));- 2[f_i(v_i(k))-f_i(z_i)+f_i(z_i)-f_i(z)]T/√(q_i(k));≤v_i(k)-z^2+2T^2/√(kT);+4[f_i(z)-f_i(z_i)]T/√(kT);-[f_i(v_i(k))-f_i(z_i)]T/√(kT) ]for k≥ T_0, where the first inequality has used the convexity of the function f_i(s) and the second inequality has used the fact that kT/4<q_i(k)<4kT for all k≥ T_0. From the definition of X_i, f_i(z)-f_i(z_i)≥0 and f_i(v_i(k))-f_i(z_i)≥0. If v_i(k)∈Y, then x_i(k)-z≤ P_2/η and it follows from (<ref>) and (<ref>) that[ x_i(k+1)-z^2;≤ P_2^2+2T^2/√(kT)+4[f_i(z)-f_i(z_i)]T/√(kT);≤P_2^2+2T^2/√(T)+4[f_i(z)-f_i(z_i)]√(T). ]If gr_i(k)≠0 and x_i(k)-z> P_2/η, then v_i(k)∉Y and hence[ f_i(v_i(k))-f_i(z)≥ 4∑_j=1^n[f_j(z)-f_j(z_j)]+4nT. ]It follows from (<ref>) and (<ref>) that x_i(k+1)-z^2≤ V_0(k). If gr_i(k)=0 for some i, it follows from (<ref>) and (<ref>) that x_i(k+1)-z^2≤ V_0(k). Summarizing the above analysis, all agents remain in a bounded region. Consequently, all x_i(k) and all v_i(k)are bounded. Since all f_i(s) are differentiable, ∇ f_i(v_i(k)) is bounded for all i. Then there exists a constant T_1>T_0 such that √(q_i(k))>∇ f_i(v_i(k))^2 for all k≥ T_1. Thus,[ gr_i(k)=∇ f_i(v_i(k))/√(q_i(k)) ]for all k≥ T_1.Now, we prove that lim_k→+∞x^*(k)-P_H(x^*(k))=lim_k→+∞[x_i(k)-x^*(k)]=0 for all i. Consider the Lyapunov function candidate[ V_1(k)=V_11(k)+V_12(k) ]for k≥ T_1, where[ V_11(k)=∑_i=1^nx_i(k)-x^*(k)^2 ]and[ V_12(k)=nx^*(k)-P_H(x^*(k))^2 ]for k≥ T_1. Under Assumption <ref>,[ x^*(k+1); =1/n∑_i=1^n[v_i(k)-∇ f_i(v_i(k))T/√(q_i(k))+γ_i(P_H_i(w_i(k))-w_i(k))]; =x^*(k)-1/n∑_i=1^n[∇ f_i(v_i(k))T/√(q_i(k))-γ_i(P_H_i(w_i(k))-w_i(k))] ]for k≥ T_1. It follows that[ V_11(k+1); =∑_i=1^n[w_i(k)+γ_i[P_H_i(w_i(k))-w_i(k)]; -x^*(k+1)^2; ≤∑_i=1^nw_i(k)-x^*(k)^2; -2/n∑_i=1^nγ_i(P_H_i(w_i(k)) -w_i(k))^2; + 1/n∑_i=1^nγ_i(P_H_i(w_i(k))-w_i(k))^2; + ∑_i=1^nγ_i^2P_H_i(w_i(k))-w_i(k)^2+c̃_1/√(k)+c̃_1/k; +2∑_i=1^n(w_i(k)-x^*(k))^Tγ_i(P_H_i(w_i(k))-w_i(k)); - 2∑_i=1^n(w_i(k)-x^*(k))^T; × 1/n∑_i=1^nγ_i(P_H_i(w_i(k))-w_i(k)); ≤∑_i=1^nw_i(k)-x^*(k)^2; -1/n∑_i=1^nγ_i(P_H_i(w_i(k)) -w_i(k))^2; +∑_i=1^nγ_i^2P_H_i(w_i(k))-w_i(k)^2; +2∑_i=1^n(w_i(k)-x^*(k))^Tγ_i(P_H_i(w_i(k))-w_i(k)); +c_1/√(k)+c_1/k; ]for all k≥ T_1 and two constants 0<c̃_1<c_1, where[ |∑_i=1^n(w_i(k)-x^*(k))^T1/n∑_i=1^nγ_i(P_H_i(w_i(k))-w_i(k))|; =|∑_i=1^n(x^*(k)-x^*(k)-1/n∑_i=1^n∇ f_i(v_i(k))T/√(q_i(k)))^T;×1/n∑_i=1^nγ_i(P_H_i(w_i(k))-w_i(k))|≤c_1-c̃_1/2√(k) ]under Assumption <ref> and the terms such as c_1/√(k) and c_1/k can be obtained based on the fact that the variables such as P_H_i(w_i(k)), w_i(k) and ∇ f_i(v_i(k)) are all bounded and the fact that kT/4<q_i(k)<4kT for all k≥ T_0. Also,[V_12(k+1) ≤ nx^*(k+1)-P_H(x^*(k))^2;≤ nx^*(k)-P_H(x^*(k))-1/n∑_i=1^n[∇ f_i(v_i(k))/√(q_i(k));- γ_i(P_H_i(w_i(k))-w_i(k))]^2;≤ nx^*(k)-P_H(x^*(k))^2+2∑_i=1^n[x^*(k)-w_i(k);+w_i(k)-P_H(x^*(k))]^Tγ_i[P_H_i(w_i(k))-w_i(k))];+ c_2/√(k)+c_2/k+1/n∑_i=1^nγ_i(P_H_i(w_i(k))-w_i(k))^2;≤nx^*(k)-P_H(x^*(k))^2;+2∑_i=1^n[x^*(k)-w_i(k)]^T;×γ_i[P_H_i(w_i(k))-w_i(k))]+c_2/√(k)+c_2/k;-∑_i=1^nγ_iP_H_i(w_i(k))-w_i(k))^2;+1/n∑_i=1^nγ_i(P_H_i(w_i(k))-w_i(k))^2 ]for all k≥ T_1 and some constant c_2>0, where the last inequality has used the convexity of 1/2P_H_i(w_i(k))-w_i(k)^2, i.e.,[ [w_i(k)-P_H(x^*(k))]^T[P_H_i(w_i(k))-w_i(k))];≤1/2P_H_i(P_H(x^*(k)))-P_H(x^*(k))^2; -1/2P_H_i(w_i(k))-w_i(k)^2≤ -1/2P_H_i(w_i(k))-w_i(k)^2. ]Under Assumption <ref>, we have 𝒜(k)1=1 and hence 𝒜(k)1x^*(k)=1x^*(k). It follows that[∑_i=1^nv_i(k)-x^*(k)^2; =𝒜(k)x(k)-1x^*(k)^2=𝒜(k)[x(k)-1x^*(k)]^2. ] Note from the form of w_i(k) that ∑_i=1^nw_i(k)-x^*(k)^2≤∑_i=1^nv_i(k)-x^*(k)^2+c_3/√(k)+c_3/kfor some constant c_3>0 and k≥ T_1. Together with (<ref>) and (<ref>), we have that[ V_1(k+1)-V_1(k)≤ -ϕ̅(k) +c_4/√(k)+c_4/k, ]for some constant c_4>0 and k≥ T_1 where[ ϕ̅(k)=[x(k)-1x^*(k)]^T[I-𝒜(k)^T𝒜(k)][x(k)-1x^*(k)];+ ∑_i=1^n(1-γ_i)γ_iP_H_i(w_i(k))-w_i(k))^2. ]By repeated calculations, [ V_1(k_m+1)-V_1(k_m)≤-∑_k=k_m^k_m+1-1ϕ̅(k)+c_5/√(k_m)+c_5/k_m ]for some constant c_5>0.Let T_1 be sufficiently large for any ϵ>0 such that c_5/√(k_m)+c_5/k_m<ϵ for all k≥ T_1 and h_1(k_m)=max_k∈ [k_m,k_m+1)ϕ̅(k) for k_m≥ T_1. When h_1(k_m)>2ϵ for k_m≥ T_1, [ V_1(k_m+1)-V_1(k_m)<-ϵ. ]Now, we need to consider the upper bound of x_i(k)-x^*(k) when h_1(k_m)≤ 2ϵ for k_m≥ T_1. Under Assumption <ref>, the graph 𝒢(k) is balanced for all k. Rearranging the agent indices, it can be obtained that 𝒜(k) can be denoted as 𝒜(k)=diag{𝒜^1(k),⋯,𝒜^h(k)}, where each 𝒜^i(k) corresponds to a strongly connected component of the agents. As a result,[ 𝒜(k)^T𝒜(k)=diag{𝒜^1(k)^T𝒜^1(k),⋯,𝒜^h(k)^T𝒜^h(k)}. ]Note that ∑_i=1^na_ij(k)=∑_j=1^n a_ji(k)=1 under Assumption <ref>. It follows that each row sum of 𝒜^i(k)^T𝒜^i(k) is 1 and hence [I-𝒜(k)^T𝒜(k)]1=0.Note that 𝒜^i(k)^T𝒜^i(k) is symmetric and each of its off-diagonal entries is nonnegative. I-𝒜(k)^T𝒜(k) can be regarded as the Laplacian of a certain undirected graph, denoted by 𝒢̅(k), and each I-𝒜^i(k)^T𝒜^i(k) corresponds to each connected component of 𝒢̅(k), which in turn corresponds to the strongly connected component of 𝒢(k).Under Assumption <ref>, the union of all 𝒢(k) during each interval [k_j,k_j+1) is strongly connected. Hence, the union of all 𝒢̅(k) during each interval [k_j,k_j+1) is connected.Note that each nonzero entry of 𝒜(k) is no smaller than η under Assumption <ref> and hence each nonzero entry of 𝒜(k)^T𝒜(k) is no smaller than η^2. Thus,[x(k)^T([𝒜(k)^T𝒜(k)-I_n]x(k); ≤ -∑_i=1^n∑_j∈ N_i(k)η^2x_i(k)-x_j(k)^2. ]By some calculations similar to (<ref>) and (<ref>), there exists a constant c_6>0 such that x_i(k)-x^*(k)≤ c_6 (√(ϵ)+ϵ) and x^*(k)-P_H_i(x^*(k))≤ c_6 (√(ϵ)+ϵ) for all i and all k_m≤ k<k_m+1-1 when h_1(k_m)≤ 2ϵ for k_m>T_1. Let[ ϕ̅_1(k) ≜∑_i=1^nx_i(k)-x^*(k)^2; +∑_i=1^nx^*(k)-P_H_i(x^*(k))^2. ] From Lemma <ref>, when lim_t→+∞x^*(k)-P_H_i(x^*(k))=0 for all i, lim_t→+∞x^*(k)-P_H(x^*(k))=0. There exists a constant δ_1>0 for any ϵ_1>0 such that V_1(k)<ϵ_1 when ϕ̅_1(k)<δ_1. Let ϵ be sufficiently small such that ϕ̅_1(k)≤δ_1 and hence V_1(k)<ϵ_1 for all k_m≤ k<k_m+1 when h_1(k_m)≤ 2ϵ for k_m>T_1. Note that h_1(k_m)>2ϵ when V_1(k)>ϵ_1 for some T_1<k_m≤ k<k_m+1 and recall that when h_1(k_m)>2ϵ,V_1(k_m+1)-V_1(k_m)<-ϵ. Thus, there exists a constant T_1>T_0 such thatV_1(k_m)≤ϵ_1 for all k_m≥ T_1. In view of the arbitrariness of ϵ_1, let ϵ_1→0. It follows that lim_k_m→+∞V_1(k_m)=0. Since[ V_1(k+1)-V_1(k)≤ -ϕ̅(k)+c_4/√(k)+c_4/k≤c_4/√(k)+c_4/k ]for all k>T_1 and k_m+1-k_m≤ M, it follows that lim_k→+∞V_1(k)=lim_k_m→+∞V_1(k_m)=0.Thus, lim_k→+∞x^*(k)-P_H(x^*(k))=lim_k→+∞[x_i(k)-x^*(k)]=0 for all i. The rest proof is very similar to that of Theorem <ref> and hence omitted. Proof of Theorem <ref>: Since x_i(0)∈ H_i and γ_i=1 for all i, from (<ref>), x_i(k)∈ H_i for all i and all k.By the same approach of the proof of Theorem <ref>, this theorem can be proved. However, it should be noted that x^*(k)-P_H_i(x^*(k))≤x_i(k)-x^*(k)≤ c_6 (√(ϵ)+ϵ) when it has been proved that x_i(k)-x^*(k)≤ c_6 (√(ϵ)+ϵ) for all i and two certain constants c_6>0 and ϵ>0. | http://arxiv.org/abs/1703.08898v5 | {
"authors": [
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"Wei Ren",
"Chunhua Yang",
"Weihua Gui"
],
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"published": "20170327015632",
"title": "Distributed Continuous-Time and Discrete-Time Optimization With Nonuniform Unbounded Convex Constraint Sets and Nonuniform Stepsizes"
} |
scrplain frontmatter/frontmatter scrheadings IDO SIP XDS ODCRO XCS CAS EDC MAS IAI OCS SRD MCS | http://arxiv.org/abs/1703.09144v1 | {
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1Max-Planck-Institut für Astronomie, Königstuhl 17, 69117Heidelberg, Germany 2Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA 3Lawrence Berkeley National Laboratory, CA 94720-8139, USAThe latest measurements of cosmic microwave background electron-scattering optical depth reported by Planck significantly reduces the allowed space ofreionization models, pointing toward a later ending and/or less extended phase transition than previously believed. Reionization impulsively heats the intergalactic medium (IGM) to∼ 10^4 K,and owing to long cooling and dynamical times in the diffuse gas that are comparable to the Hubble time, memory of reionization heating is retained. Therefore, a late-ending reionization has significant implications for the structure of the z∼ 5-6forest.Using state-of-the-art hydrodynamical simulations that allow us to vary the timing of reionization and its associated heat injection, we argue that extant thermal signatures from reionization can be detected via theforest power spectrum at 5 < z < 6. This arises because the small-scale cutoff in the power depends not only on the IGM temperature at these epochs, but is also particularly sensitive to the pressure-smoothing scale set by the IGM full thermal history.Comparing our different reionization models with existing measurements of theforest flux power spectrum at z=5.0-5.4, we find that models satisfying Planck's τ_e constraint favor a moderate amount of heat injection consistent with galaxies driving reionization, but disfavoring quasar driven scenarios. We study the feasibility of measuring the flux power spectrum at z≃ 6 using mock quasar spectra and conclude that a sample of ∼ 10 high-resolution spectra with an attainable signal-to-noise ratio will allow distinguishing between different reionization scenarios.§ INTRODUCTION How and when the first luminous sources reionized diffuse baryons in theintergalactic medium (IGM) is one of the most fundamental open questions incosmology. Recently, the Planck collaboration have released new tighterconstraints on reionization from cosmic microwave background (CMB) observations<cit.>. Using for the first time the low-multipoleEE data from Planck-HFI, the Planck team has significantlyimproved our constraints on thecosmic reionization optical depth, , finding=0.058 ± 0.012 <cit.>.The reionization ofby the UV background from galaxies and/or quasarsresults in the highly ionized IGM probed at z ≲ 6 by observations oftheforest <cit.>. Indeed,observations of complete Gunn-Peterson absorption in the spectra of many of thehighest z∼6 quasars, along with the steep rise of both the optical depth and its sightline-to-sightline scatter with redshift, has led tothe consensus that we are witnessing the end of reionization only at z∼6<cit.>. However,opacity can only set lowerlimits on the redshift of reionization z≳6, because the overlysensitivetransition saturates for volume-averaged neutral fractionsx_≳10^-4, which is far too small to pinpoint the redshift ofreionization. While new constraints have emerged from the possible presence of a IGM damping wing in the highest redshift known quasar at z=7.1<cit.>, and the decreasing strength of emission lines in z∼ 6-7 galaxies<cit.>, the resulting constraints onx_ HI are degenerate with the intrinsic properties of thehigh-z quasars and galaxies that they have been deduced from. We are inneed of another technique to probe when reionization occurred.During reionization, ionization fronts propagate supersonically through the IGM,impulsively heating gas to ∼10^4 K <cit.>.The integrated energy balance of heating and inverse-Compton andadiabatic cooling then gives rise to a power law temperature-density relation, T =T_0 (ρ/ρ̅)^γ-1<cit.>. Another important physical ingredient to describe the thermalstate of the IGM is the gas pressure support that produces an effectivethree-dimensional smoothing of the baryon distribution relative to the darkmatter at a characteristic scale, . In an expanding universe with anevolving thermal state at a given epoch, this scale depends on the entirethermal history of IGM because fluctuations at earlier times expand or fail tocollapse depending on the IGM temperature at that epoch<cit.>. At redshiftz, the level of pressure smoothing depends not on the prevailingpressure/temperature at that epoch, but rather on the temperature of the IGM inthe past. The IGM pressure-smoothing scale, , thus provides an integratedrecord of the thermal history of the IGM, and is sensitive to the timing of andheat injection by reionization events.Measurements of the statistical properties of theforest aresensitive to the thermal state of the IGM through the thermal Doppler broadeningof absorption lines, as well as the pressure-smoothing. The standard approachhas been to compare measurements of different statistics to cosmologicalhydrodynamical simulations<cit.> to deduce the thermalparameters (e.g., T_0, γ or ) that best describe the IGM thermalstate.As a larger number of high-resolution spectra ofquasars have become available at higher redshifts z≳4, the same approachhas been applied to studythe thermal state of the IGM at these redshifts,where theforest is more sensitive to the timing and nature ofhydrogen reionization<cit.>. In the light of the new Planck constraints on reionization, as well as theincreasing number of quasars discovered at high-z<cit.>, it is pertinent torevisit whatobservations with current and upcoming facilities cantell us aboutreionization. This work aims to explore in detail thepossibilities of using theforest 1D flux power spectrum at high-zto constrain HI reionization. For this we have used a new method that has recentlybeen introduced by <cit.>. It builds on the <cit.> model,but enables one to vary the redshifts ofandreionization, as wellas their associated heat injection, allowing one to consistently simulate a more diverserange of reionization histories. This allows for a much comprehensive andconsistent exploration of the space of thermal parameters than previously done.We present here the results of the 1D flux power spectrum at high-z of a newset of hydrodynamical simulations that also improve the resolution used inprevious studies at these redshifts. Additionally, we compare these power spectra with some recent measurements at these redshifts <cit.>.The structure of this paper is as follows. In Section <ref> we describethe characteristics of our hydrodynamical simulations and the different reionization models studied in this work. Section <ref>presents the thermal evolution of the different reionization models obtainedfrom the cosmological hydrodynamical simulations. In Section <ref>we compare the 1D flux power spectrum of each model at 5≤ z≤6 as well aswith the best available observations. We discuss in Section <ref> therelevance of our findings in the context of current observational andtheoretical limitations. We conclude by presenting a summary of our results andan outlook in Section <ref>. In Appendix <ref> we perform a set of convergence test for the optical depth and the 1D flux power spectrum at 5⩽ z⩽6. Throughout this paper we assumed a flat ΛCDM cosmology with the followingfundamental parameters: Ω_ m = 0.3192, Ω_Λ =0.6808, Ω_ b = 0.04964, h = 0.67038, σ_ 8 = 0.826 andn_ s = 0.9655. These values agree within one sigma with the latest cosmological parameter constrains from the CMB <cit.>. The massabundances of hydrogen and helium (X_ p = 0.76 and Y_ p = 0.24) were chosen tobe in agreement with the recent CMB observations and Big Bang nucleosynthesis<cit.>. § SIMULATING REIONIZATION HISTORIES IN LIGHT OF NEW PLANCK CONSTRAINTSThe cosmological hydrodynamical simulations used in this work were performedusing the Nyx code <cit.>.Application of Nyx to studies of the forest, and its convergence and resolution requirements are discussedin <cit.>. We refer to these two works for more details of thenumerical implementation, accuracy, and code performance. To generate theinitial conditions for the simulations, we have used the music code<cit.>, with the transfer function for our cosmological model obtained from camb<cit.>. All simulations discussed in this work used thesame initial conditions and have a box size of length L_ box=20 Mpc h^-1 and1024^3 resolution elements. As is standard in hydrodynamical simulations that model theforest,all cells are assumed to be optically thin to radiation. Thus, radiative feedbackis accounted for via a spatially uniform, but time-varying ultraviolet background (UVB) radiation field, input to the code as a list of photoionization and photoheating rates thatvary with redshift <cit.>. In order to simulate eachreionization model discussed here, we have used the method presented in<cit.>, which allows us to vary the timing and duration ofreionization, and its associated heat injection, enabling us to simulate adiverse range of reionization histories.This method allows us to create the,andphotoionization and photoheating rates, which areinputs to the Nyx code, by volume averaging the photoionization and energyequations. This method requires that each reionization event is defined bythe ionization history with redshift, e.g. x_(z), and its associatedtotal heat injection, Δ T, which depends on the spectral shape andabundance of the ionizing sources, and the opacity of the IGM<cit.>. We direct the reader to<cit.> for the details of this method. In order to determine the reionization histories for our simulations, we explorethe relevant range of reionization models considering the last measurements by Planck and theoptical depth at high redshift, whichset a lower limit forreionization at z=6 <cit.>. In all simulations we also assumed the samereionization model ending atz=3 and which does not produce a significant increase in the IGM temperatureuntil z<5. This model is the same as assumed in the <cit.> UVBmodel, and since we will be comparing with data at z≥5 this assumption will not impact our results <cit.>. In this work weconsider five modelreionization histories constructed using theanalytical formula presented in <cit.> chosen to match the results of radiativetransfer simulations <cit.>,x_=0.5+0.5× g(1/n_1,|z-z_ reion,^0.5|^n_1),z <= z_ reion,^0.50.5-0.5× g(1/n_2,|z-z_reion,^0.5|^n_2), z > z_ reion,^0.5where g is the incomplete gammafunction, n_1=50, n_2=1 and z_ reion,^0.5 is a freeparameter that sets the redshift where x_(z_0)=0.5.We run an early, middle, and latereionization history (EarlyR, MiddleR,LateR), which have a specific reionization redshifts (defined as the redshiftwhere x_=0.999) of z_ reion,= 7.75, 6.55, and 6.0respectively,and are within 1σ of the Planck CMB measurements. We also run two moremodels, one with a very early reionization (VeryEarlyR, z_ reion,=9.70 which is 3σ discrepant with the Planck measurement) and a faster reionization (MiddleR-fast, z_ reion,= 6.55). Asummary of all the relevant parameters used in the runs presented in this workis shown in Table <ref> along with the naming conventions we haveadopted. The full reionization history of each of our models is plotted in the upper leftpanel of Figure <ref>, shown as the redshift evolution of theelectron fraction given by =/=(1+χ)x_ +χx_ where χ=Y_p/(4X_p) and X_p and Y_p are thehydrogen and helium mass abundances, x_(z) is the hydrogen ionizedfraction, and x_(z) is the fraction of helium that is doublyionized[Throughout this paper we made the standard assumption that reionization is perfectly coupled with that of .]. The lower leftpanel of Figure <ref> shows the evolution of thecosmic reionization optical depth, , for each of these models.In the right panel of Figure <ref> we compare the newestPlanck measurements of the CMB polarization EE powerspectrum[Throughout this paper we adopt the conventionD_ℓ=ℓ(ℓ+1)C_ℓ/2π.] low multipoles <cit.>to these reionization models, where we have computed the EE power spectrum usingthe class Boltzmann code <cit.>.For the observedlow multipoles we show the unbiased QML_2 results from <cit.>, illustrating the impressively high precision achieved bythese CMB polarization measurements, which significantly reduces the allowedrange of models.To build the reionization models, we also need to assume the associated totalheat injection, Δ T, duringreionization, which depends on thespectral shape and abundance of the ionizing sources, and the opacity of the IGM<cit.>. To run allthe simulations with the reionization model described above, we assumed ΔT_ = 2 ×10^4 K which is the standard value obtained ingalaxy-drivenreionization models using 1D radiative transfer simulations <cit.>. Quasar-driven scenarios give higher heat injection values, Δ T_∼ 4 ×10^4. Thus, in order to study the effect of different total heat input during reionization, we run three more simulations with the same reionization model as MiddleR, but varying the Δ T_ parameter:MiddleR-cold (Δ T_ = 1 ×10^4 K), MiddleR-warm (ΔT_ = 3 ×10^4 K), and MiddleR-hot (Δ T_ = 4 ×10^4K). § REIONIZATION-DEPENDENT THERMAL HISTORIES Changing the timing and duration of reionization and its associated heatinjection will manifest as changes in the evolution of the parameters governingthe thermal state of the IGM. In Figure <ref> we present theresulting thermal histories for all of these simulations. The upper panel showsthe evolution of temperature T() at the `optimal' overdensity probed by curvature measurements of theforest<cit.>, where we calculate the optimal density ateach redshift using the functional form of (z) given by<cit.>. The evolution of thermal parameters, γ and T_0,governing the density-temperature relation, are shown in the second and thirdpanel from the top, determined by fitting the distribution of densities andtemperatures in the simulation following the linear least-squares methoddescribed in <cit.>[ Changing the thresholds used to do thefit within reasonable IGM densities produces differences just at a few percentlevel <cit.> and in any case it doesnot affect the conclusions presented in this work. We also found no relevanteffects in the main results of this paper when we employed the fittingapproach used in <cit.>.].The evolution of the pressure-smoothing scale, , with redshift is shown inthe bottom panel. To characterize the pressure-smoothing scale in all oursimulations we have followed the approach described by <cit.>.These authors define a pseudo real-spaceflux field, which is thesame as the trueforest flux, but without redshift space effects suchas peculiar velocities and thermal Doppler broadening. This field naturallysuppresses the dense gas that would otherwise dominate the baryon power spectrum,making it robust against the poorly understood physics of galaxy formation and revealing the pressure-smoothing in the diffuse IGM.Inspection of the left panel of Figure <ref> reveals thatsimulations with different reionization histories but the same heat injection during reionization, Δ T, allshare a very similar T_0, T() and γ evolution at z=5-6. This is because once reionization is completed, the IGM thermal state asymptotes to a tight power-law temperature-density relation driven mainly by the photoheating rate and accelerated by Compton and adiabatic cooling. The time to converge to these asymptote valuesis around Δ z∼ 1-2 (a few hundred Myr)once reionization is completed and mainly depends on the amount of heat injected during reionization <cit.>. Since all these models share the same photoionization and photoheating values, once reionization is completed they all converge to the same thermal stateat lower redshift z≲6. However, their pressure-smoothing scale, , remainsvery different at these and lower redshifts. Models in which reionizationhappened at earlier times have a larger pressure-smoothing scale. As discussedabove, this results from the dependence of the IGM pressure-smoothing scale onthe full thermal history <cit.> and notjust on the instantaneous temperature, and it will haveimportant consequences for the statistics of theforest<cit.>. The right panels of Figure <ref> show the thermal historiesfor simulations where we fixed the reionization history, but varied the totalheat injection duringreionization. Models with more heat injection giverise to higher temperatures and slightly lower γ values not only duringreionization, but also at later times while the IGM is stillreaching their asymptote values. Note that similarly as for the different reionization history models, these modelsalso produce a larger pressure-smoothing scale, and these differences persist evenat lower redshifts long after the other thermal parameters, γ and T_0,have relaxed to their asymptotic values. Oncereionization is completed all these models share the same photoionization andphotoheating rates and therefore γ, T_0, and T() thermalparameters asymptote to the same values much faster than the pressure-smoothingscale, which retains memory of the thermal history.The symbols with error bars in Figure <ref> indicate recentobservational constraints on the parameters governing the thermal state of theIGM at high redshift. In particular, the purple square is the <cit.>measurement of T_0 using wavelets at z = 4.20 and black squares are the<cit.> measurements of T() based on the curvaturestatistic. Gray circles and black diamonds represent the joint fits to γand T_0 given by <cit.> and <cit.> respectively,using the 1D flux power spectrum.[These measurements are marginalizedover the mass of a warm dark matter particle. <cit.> and <cit.> used different fitting approaches, but both used thesame grid of hydrodynamical simulations in which the standard reionizationredshift for the runs was z_ reion=12 and the lowest reionizationredshift considered in the grid, by including one simulation, was z_reion=8.] While the thermal parameters measuredby <cit.>, <cit.> and<cit.> appear consistent with themodels discussed here, the <cit.> T_0 measurement suggests asignificantly hotter IGM at z∼4. The origin of this disagreement is unclear, but mayresult from differences in the methods used by these authors and/or thedifferent hydrodynamical simulations compared to the data. Our work aims toshed more light on this issue by comparing the 1D power spectrum measurements at highredshift with an improved set of simulations. § THE 1D FLUX POWER SPECTRUM AT HIGH REDSHIFT Z ∼ 5-6 In order to explore the possibility of discriminating between the reionizationmodels presented here withforest measurements, we calculate the 1Dflux power spectrum, P(k), for each simulation at z=5.0, 5.4, and 6.0.The1D power spectrum of theforest is sensitive to the parametersgoverning the thermal state of the IGM. Pressure smoothing, λ_ P,damps out small-scale fluctuations in the gas, while random thermal motions(sensitive to temperature, or T_0 and γ) Doppler broadens forest lines, further reducing the amount of small-scale structure. Both ofthese effects combine to produce a prominent small-scale (high-k) cutoff in theflux power spectrum P(k) <cit.>.Therefore, by carefully studying thiscutoff, we expect to be able not only to constrain the thermal state of the gas, but also its full thermal history.We have createdforest spectra from the simulation computing the optical depth at a fixed redshift, which can then be easily converted intoa transmitted flux fraction, F_ = e^−τ_. We refer to<cit.> for specific details of these calculations. We computed thepower spectrum, P(k), of the fractional contrast, δ F, at eachredshift defined as δ F=F/F -1. A total of 1024^2 skewers were used at each redshift. We computed the power spectrum of each skewer andthen calculated the average value at each mode, k.The overall level and precise shape of the fractional contrast power is still sensitive to the mean flux because it changes the density-flux mapping. This is to say, a lower mean flux will shiftobservations sensitivity toward lower densities. For this reason, when computing the 1D power spectrumof the fractional contrast from a simulation, it is still very important which mean flux was considered. Following the standard approach, we rescaled the mean flux of each simulation to match a fixed mean flux value. Of course, this rescaling does not affect measurementssince one directly measures a flux contrast field. While the mean flux value has been precisely measured at lower redshift, the measurements at z≳ 5 are more uncertain. At z=5.0 and 5.4, the current best measurements for the mean flux are the binnedvalues computed by <cit.> from the <cit.> high-redshift quasar opacity measurements. These are F=0.14±0.01 forz=5, and F=0.08±0.006 for z=5.4. These values are consistent withthe analytic formula presented by <cit.> derived from their own quasar sample, which areF=0.14603,0.071 respectively. The <cit.> measurements of themean flux using a sample of high-z quasars discovered in the Sloan Digital SkySurvey were F=0.1224±0.03 at z=5.025 andF=0.074^+0.03_-0.06 at z=5.450.These values are also in good agreement with the current best measurements considering their larger errors. In fact, the global fit suggested by <cit.>based on their own measurements gives F=0.1659,0.071, respectively at these redshifts. However, since we will compare our models with the <cit.> observations ofthe 1D flux power spectrum in order to give a better qualitative idea of theresults found in this work, we have considered the mean mean flux values thatgive a better overall normalization to these observations and assumed∼7.5% relative measurement error, which reflects the quoted errors in theresults above: F=0.16±0.01 for z=5, andF=0.055±0.004 for z=5.4. While these values seem to be slightly farfrom the current best observations by <cit.> at z=5.4, they are within the 1σ C.L. found by<cit.> when they performed a marginalized fit of the 1Dflux power spectrum for several parameters that included the mean flux(F=0.148^+0.024_-0.007 at z=5, andF=0.045^+0.02_-0.001 at z=5.4)[This is also in agreementwith the marginalized fit made by <cit.> to thesame dataset. These authors found F=0.142^+0.023_-0.017 at z=5 andF=0.054^+0.014_-0.01 at z=5.4.]. Although the lower mean flux measurements from the 1D flux power spectrum could just indicatesome fluctuation due to the small number of quasars used to compute thepower spectrum at high-z, it definitely highlights the relevance of taking intoaccount the mean flux degeneracy when any astrophysical or cosmologicalinformation from the 1D flux power spectrum is to be extracted. We return to this issue in our discussion of different degeneracies of the 1D flux power spectrum in Section <ref>.The two panels of Figure <ref> show the simulated dimensionless 1D fluxpower spectrum, kP(k)π, computed at z=5.0 and 5.4 for the models where we changed thereionization history butkept the total heat input constant: EarlyR, MiddleR, LateR, MiddleR-fast andVeryEarlyR (see Table <ref>). We first note in the two panels that theoverall power level increases with redshift, which reflects the fact that as theaverage mean flux decreases toward higher z, density fluctuations areexponentially amplified<cit.>. Note also thatat all redshifts the difference between models for the low-k modes is very small andtherefore the use of high-resolution spectra probing to k∼ 0.1 s km^-1 iskey.[Detailed studies of the impact of metalabsorption features in current state-of-the-art high-resolution spectra haveshown that it can increase the 1D flux power spectrum at k>0.1 s km^-1 <cit.>. For this reason, thisis typically the maximum k considered in high-resolution power spectra studies.] The models separate at high-k because their disparate reionizationhistories result in different levels of pressure-smoothing and thermalbroadening (see Fig. <ref>), changing the shape of thesmall-scale (high-k) cutoffs in the power spectra.The color bands for eachmodel show the variation in the 1D flux power spectrum due to the one-sigmauncertainty inthe mean flux value at the corresponding redshift.Note that currently these errorsseems to translate into a ∼ 10% scatter in the 1D flux power spectrum, which is smaller than the current error bars on the z=5.4 measurements <cit.>.The differences between these models at z≳5 are particularly interestingbecause they result primarily from differences in the pressure-smoothing scale,λ_ P, as the other parameters governing the thermal state of theIGM, γ and T_0, are very similar(see left panel of Figure <ref>). These models share exactly the samephotoionization and photoheating rates at the redshifts considered,and differsolely in the timing ofreionization heat injection. These resultshighlight that the 1Dforest power spectrum is sensitive to thedetails ofreionization history even at lower redshifts due to thedifferent pressure-smoothing scale. Any attempts to derive astrophysical orcosmological parameters using high-zforest observations that donot take this issue into account could obtain biased results <cit.>.Figure <ref> show the simulated dimensionless 1D fluxpower spectrum for simulationsin which the timing ofreionization isidentical, but which have different amounts of heat injection, Δ T:Middle-cold (Δ T=1×10^4 K), MiddleR (Δ T=2×10^4 K),MiddleR-warm (Δ T=3×10^4 K), MiddleR-hot (Δ T=4×10^4 K).As expected, the power spectrum shows a larger small-scale cutoff (i.e. toward lower k) forsimulations with a higher heat input during reionization. It is clear that theeffect in the 1D flux power spectrumof a high heat injection during reionization (MiddleR-hot) is degenerate with a reionization modelwith a lower heat input but that completes at higher redshift. Both physicalprocesses produce a higher pressure-smoothing scale at lower redshift. At higher redshifts, z≳ 5 the differences in the power spectrum between simulations shown in Figure <ref> are due not only to the effect of the pressure-smoothing scale, , but alsodue to the differences in the other thermal parameters, γand T_0. This is because at these redshifts the IGM in these models are still reaching the asymptotic temperature-density relationafterreionization (see Figure <ref>). This highlights the other physical process affecting theforest lineswhich is the thermal broadening along the line of sight that also affects thecutoff, in the 1D flux power spectrum. In fact, it is relevant to point out thatthe differences between models with different T() produce larger differencesat k<0.04 s km^-1 in the 1D flux power spectrumthan thermal models that just differ in the pressure-smoothing scale,(seeFigure <ref>). This could open a possibility to distinguish betweenboth physical effects using different k-moderanges of the 1D flux power spectrum, provided thatreionization happens at low enough redshift to still see these effects. §.§ Comparison with Observations <cit.> made measurements of the 1Dforestflux power spectrum at z=5.0 and z=5.4, using a sample of 25 high-resolution quasar spectra. The redshift bins had width dz = 0.4 and contained data from ∼10 quasars per bin.In Figure <ref> and Figure <ref> we also compare themeasurements of <cit.> to our simulation results at the same redshift bins. From these figures it is clear that these measurements already have sufficientprecision to begin distinguishing between different reionization models, oncethe degeneracy due to the mean flux has been taken into account. In what followswe report a first qualitative comparison of this data set with our simulations.A detailed quantitative analysis of these observations using a larger hydrodynamicalgrid of high-resolution large-volume simulations that expands the full parameterspace of the thermal parameters and takes into account relevant degeneracieswill be presented in another paper. We first note from this comparison is that in the context ofour current models and with the caveat that the mean fluxes have been chosen to best fit the power at each redshift, the z=5 and z=5.4 measurements appear to be in agreement withour fiducial model (MiddleR, green line) that uses the Planck τ_e valueand Δ T= 2×10^4 K. This picture is consistent with the conventionalwisdom that galaxies reionized hydrogen <cit.>. Reionization models driven by active galacticnuclei <cit.> have recently gained traction in light of thediscovery of an abundant population of faint AGN at high redshift z∼4-6<cit.>. Such models have higher photoelectric heatingofand would also doubly ionize helium at these high redshifts <cit.> increasing the amount of heat injectionin addition to the oneassociated withreionization and moving the cutoff of the 1D flux powerspectrum to lower k modes.[In the simulations discussed in this work, wedo not consider any high-zreionization model, but this could beeasily added using the same formalism applied toreionization<cit.>.] In any case, current observations ofthe 1D flux power spectrum at z=5 and z=5.4 disfavor high-redshift z≳9 reionization models, far away from Planck constraints on τ_e, even for standard galaxy-driven heat injection. We have shown that this is due tothe dependence of the 1D flux power spectrum cutoff on the timing ofreionization because the pressure-smoothing scale retains memory of the thermalhistory. §.§ Prospects for Measuring the Power Spectrum at z≃ 6Motivated by the results in the previous subsection for the 1D flux powerspectrum at z=5.0,5.4 and the increasing number of high-z quasars that areuncovered by recent surveys<cit.>, we also wish toexplore the 1D flux power spectrum atz=6, and study the feasibility of making a power spectrummeasurement at this redshift with current facilities. Whereas, as discussedpreviously, the power spectrum signal increases toward increasing redshift, themean flux also begins to drop precipitously, lowering the signal-to-noise (S N) ratio level of thequasar spectra, thus increasing the importance of noise in the power spectrummeasurement. At z=6.0 we haveassumed a mean flux value of F=0.011 (τ_ eff=4.5), which is consistent with the latest measurements of the effective optical depth at this redshift <cit.>. We have also assumed that the 1σ error on the mean flux at this redshift will be 10% of its value, which is a reasonable assumption given the large numbers of z∼ 6 quasars recently discovered, ∼ 150. To this end, wecomputed mock observations for the MiddleR simulation assuminghigh-resolution spectra (S N of 35 per resolution element of 8 km s^-1)of 10quasars at z=6.3 employing a path length equivalent of Δ z=0.5 perquasar.For this we calculate the corresponding path length per quasar in cMpcat this redshift. We created random samples for the 10 quasars from thesimulations and added noise realizations to each skewer. We then computed the meanpower spectrum and subtracted off the average noise level.Results of one of these mock observations are shown as black squares in Figure <ref> along side with the error bars computed from a set of 50 mock observations. At the S N considered, the measurement is dominated by cosmic variance. The noise in the quasar spectra isvery significant at this redshift, butthe increase in overall power due to the decrease of the mean flux still allows us to measure of the power using a sample of 10 quasars.The two panels of Figure <ref> show the 1D flux power spectrum forthe same simulations as discussed in Figure <ref> (left panel) andFigure <ref> (right panel), but now at z=6. The color bands foreach model indicate the variation in the 1D flux power spectrum due to 1σ changes in the mean flux. We can see that the overall scale of the flux powerhas increased compared to the values at lower redshift due to the decrease inmean flux. As discussed above, the power increases with redshift as the mean fluxgoes down because this amplifies the fluctuations. Note that the differencesbetween the models have increased relative to z∼5, especially for high-k modes and for simulations where the heat injection is varied. This is because at z=6, we are closer toreionization and therefore not only , but also the other thermalparameters, T_0 and γ (see Figure <ref>), are still affected by the details on how reionization happened <cit.>.Data with a size and S N comparable to our assumed mock are clearly within reach. For example, about 5 such quasar spectra already exist in public telescope archives <cit.>, so a sample of 10 would be accessible with modest allocations of an 8 m class telescope time. We have shown that the differences between models are greater at z=6 than at lower redshift, compensating for the possibly lower S N. In order to understand how reionization heated the IGMand constrain both the reionization history and the heat injected, we therefore need to push as far back into their reionization epoch as possible, where not only , but also γ and T_0 still could have memory of reionization. § DISCUSSION§.§ Convergence of the Results While the resolution and method employed in oursimulations arecurrently state-of-the-art for this type of analysis<cit.>, several aspects of our simulations call forcaution. First, one should consider the level of convergence of the 1D fluxpower spectrum at these high redshifts for our grid of simulations that haveL_ box=20 Mpc h^-1 and N=1024^3. <cit.> reported a carefulconvergence analysis of Nyx simulations of theforest similar to those employed in this work,but they only explored 2≤ z ≤ 4. At z=4, the 1D flux power spectrum oftheir simulations were converged to <5% in terms of spatial resolution(missing power in large k modes) but <8% due to box size effects(overestimated power at large k modes). We show in the Appendix <ref> resolution and box size convergence results atz=5 and z=6 that reach similar conclusions, although approaching ≲10% convergence level at the k∼0.1 s km^-1 modes, as typically usedfor power spectrum measurements, and much better as we move to lowerk modes (larger scales). In any case, although convergence issues have to be takeninto account, they do not seem to change the general conclusions of this work.Similar results at these redshifts but for simulations using the Gadget code canbe found in <cit.> and <cit.>.Another relevant issue is that our simulations do not modelgalaxy formation. Therefore we neglect any impact that the effect ofstellar or black hole feedback could have on the IGM<cit.>. <cit.> showed that at z∼3 this effect could lead to differencesof ∼10% in the 1D flux power spectrum. This is a very relevant issue thatshould be explored in more detail with current state-of-the-art feedback models.However, it is expected that the effects of feedback on the IGM should only decrease at higher redshifts for two reasons: first, theforest istracing lower gas densities at average locations of the universe<cit.>, and the smaller numberof galaxies or black holes at these redshifts make it difficult for any feedbackto alter the thermal state of the IGM.Recent measurements of theoptical depth at high redshift have foundenhanced scatter at z>5.5 that exceeds what can be attributed to densityfluctuations alone <cit.>. It has been argued that theyare driven by fluctuations in the radiation field <cit.>, or thetemperature field <cit.>, both of which may be inevitablebyproducts of a patchy, extended, and late-ending reionization process. Stillothers have interpreted these fluctuations as evidence that reionization wasactually driven by rare AGN <cit.>.In any case, thepossible effect of UVB or temperature fluctuations is currently neglected instandard optically thin simulations like those we used in this work.Previous studies have shown that these effects are manifest on much larger scales,≳10 Mpc h^-1, that is,so much lower k modes than those most sensitive to the thermal state of the IGM<cit.>. For example, <cit.> computed semi-numerical models with and without temperaturefluctuations and showed that they did not generate small-scale power.Considering both temperature and UVB fluctuations is clearly the direction in which the modeling needs to move forward. We plan to study this in detail in the near future with self-consistent hydrodynamical simulations. §.§ Degeneracy with Cosmological Parameters and Warm Dark Matter The 1D flux power spectrum depends not only on the thermal parameters of the IGM,but also on cosmological parameters. Here we have focused our analysis on onespecific cosmological model, but we have checked that when one considers therange of parameters allowed by <cit.> the differences at the kmodes studied in this work are never larger than 8% and the cutoff is unaffected<cit.>. While changes of cosmological parameters within 1σ of the Planck constraints result in minimal changes of the z ∼ 5-6 power spectrumrelative to the parameters governing reionization and its heating effect,this is not true whenone considers dark matter particle properties, such as warm dark matter.Theirfree-streaming horizon leads to a suppression of the small-scale power andtherefore to a degenerate effect on the power spectrum with the thermalparameters <cit.>. This isequivalent to a 3D smoothing at high redshift that continues to decrease asnonlinearities increase. Therefore the small-scale cutoff in warm dark matter models moves from very low k modes (large scales) to higher k modes (smaller scale)at progressively lower redshifts. This happens until the IGM becomes hotter and the IGM temperature (i.e. the T_0 and γ) determines the position of the cutoff. In order to provide reliableWDM constraints, it is therefore essential to marginalize out reionization nuisanceparameters. <cit.> highlighted the degeneracy between differentwarm dark matter masses and the temperature of the IGM. However, these authors didnot discuss the extra degeneracy resulting from the unknown redshift of reionization and its associated heat injection, as we demonstrate in this work. This could be relevant given that the fiducial valuein their models is z_ reion=12 and the lowest reionization redshiftconsidered in their grid, by including one simulation, was z_ reion=8.From their Bayesian analysis, <cit.> and <cit.>obtained very low temperature values at 5<z<5.4 (seeFigure <ref>). Although this is not in complete disagreementwith our qualitative comparison of the measurements to our models,we caution that their simulations use a total of2× 512^3 dark matter and gas particles within a periodic box of L_box=20 cMpc h^-1[Note that <cit.> use 4 N=2× 768^3 and L_box=20 cMpc h^-1 simulations to correct for resolution convergence.]. Based on the resolution convergence tests presented in the <cit.>spectrum using the same code, these simulations could underestimate the powerwith ∼ 20% error at the k modes most relevant to study the cut off of thepower spectrum (0.05 ≲ k ≲ 0.1 s/km). This could produce anartificial increase of the cutoff just due to resolution. §.§ Comparison to Previous Work<cit.> used hydrodynamical simulations with a total of 2×512^3 dark matter and gas particles within a periodic box of L_ box=10cMpc h^-1 to discuss the possibility of constraining the thermal history of the IGMduringreionization by studying the 1D flux power spectrum at z=5 fromcosmological hydrodynamical simulations. To simulate different reionization histories, they adoptedan approach different from ours. Namely they applied a multiplyingfactor to the <cit.> photoheating rates (A×q̇×Δ^B, where Δ is the specific overdensity of that cell). They also usedsimple cutoffs of the <cit.> UVB rates at various redshifts to model different reionization timing. Using these simulations, they showed the effect on the 1D flux power spectrum ofdifferent thermal histories in which they only changed the timing of reionization and tried to study its degeneracy with the temperature of the IGM, T_0.Interestingly, these authors found that this degeneracy between the timing ofreionization and the temperature can in fact be broken using different scales ofthe 1D flux power spectrum. They find that the z=5 1D flux power spectrum ismore sensitive to the timing at 0.03< k <0.13 s km^-1 scales. Our analysis alsoshows that these scales are the most sensitive to the timing of reionization(see Figures <ref>),but it also indicates that information about the lower k modes will be crucial inorder to break the degeneracy between the timing of reionization and otherparameters, the temperature at mean density, T_0 or the mean flux at thatspecific redshift. <cit.> attempt to quantify the effect of a different reionizationtiming using a new parameter, u_0, which is an integral in time of theheating rate per proton mass of the simulation at mean density.[It isdefined as u_0(z)=∫_z^z_ reion∑_in_iq̇_i/ρ̅dz/H(z)(1+z), where ρ̅ is themean background baryon density and n_i and q̇ stand for the densityand photoheating rates of the following species: i= [,,].] With this parameter, the authors try to quantify the pressure-smoothing scale in their models. In orderto facilitate comparison with their work we have computed the value of u_0 atz=4.9 for our simulations and included them in Table <ref>. However,we caution about using this parametrization as u_0 measures the heating just atmean density. Therefore this parametrization will be valid as long as thethe reionization models do not have density-dependent heating. Models with heating rates that depend on the densitybut normalized at mean density <cit.> sharethe same u_0 value, but have a different power spectrum at the k modes that are more sensitive to the thermal history, k>0.03 s km^-1, as they in fact have different pressure-smoothing scales.§ CONCLUSIONSIn this paper we have used state-of-the-art hydrodynamical simulations thatallowed us to self-consistently model different reionization models. We present an ensemble of simulations consistent with the latestmeasurements of the Thompson-scattering optical depth,recently reportedby Planck <cit.>. These models are defined by whenreionization happened, z_ reion, and how much heat was injected into theIGM during reionization, Δ T. Our simulations show that althoughby z∼6 thetemperature of IGM gas has mostly forgotten about reionization heat injection, the pressure-smoothing scale at these redshiftsdepends sensitively on how and when reionization occurred. This is because both thecooling and dynamical times in the rarefied IGM are long, comparable to theHubble time, and therefore memory ofreionization is retained <cit.>.We have found a degeneracy in thepressure-smoothing scale at z<6 between when reionization occurred and theamount of heat injected during reionization. For a fixed reionization history, the pressure-smoothing scale increases as we increase the heat injection.Similarly, the pressure-smoothing scale increases with the redshift of reionization in models with a fixed amount of heat injection.In order to investigate the effects of these different thermal histories on theproperties of theforest, we compute the1D fluxpower spectrum at z∼ 5-6 for our simulation ensemble. Pressure smoothingdamps out small-scale fluctuations in the IGM, while thermal vibrations of IGMgas Doppler broadensforest lines, further reducing the amount ofsmall-scale structure. Both of these effects combine to produce a prominentsmall-scale (high-k) cutoff in the1D flux power spectrum<cit.>. We have found that at these high redshifts, the1D flux power spectrum is especially sensitive to the pressure-smoothingscale of the IGM and not only its temperature. Therefore extant thermalsignatures from reionization can be detected by analyzing theforest power spectrum at these redshifts. We have also conducted a first qualitative comparison of the 1D flux power spectrum measurements at z=5-5.4 made by<cit.> with our simulation ensemble. Taking Planckconstraints on reionization at face value, we have shown that models with a fiducial heat input duringreionization consistent withstandard galaxy-driven reionizationmodels are sufficient to explain the observations. We work on a more complete analysis of this in the near future, with a larger simulation grid, marginalizing out all the different relevant parameters, including the meanflux, and improving upon the reionization modeling.We have also presented a feasibility study of performing a similar measurement at z=6creating mock observations that assumed a realistic sample of quasars at thisredshift both in terms of sample size and S N. We found that combining 10 quasars should be enough to distinguishbetween different thermal histories of the IGM. Our results indicate that quasar spectra at high redshift can not only be useful toconstrain when reionization happenedvia the study ofopacity measurements <cit.>, but also to understand the thermal history of the universe. This is the small-scale structure measured from high-resolution spectra can be used to understand the thermal history of the universe, further constraining the timing and heat injection by reionization.Taking into account thatthere are only a few direct observational probes of reionization currently available, we think thatit is important to push in this direction in the near future.In this regard, pushing these measurements at this and higher redshifts (z∼6) will becrucial to improve the power of theforest to constrain reionization. The new recent fivefold increase in the number of bright quasarsat z>5 is obtained from deep wide-field optical/IR surveys like CFHQS <cit.>,dark energy survey <cit.>, ESO public surveys <cit.>, and Pan-STARRS1 <cit.>. Currently, the total number z > 5.5 quasars available for study is ∼173, so if all could be used to study the cutoff of theforest power spectrum would reduce the errors by a factor of ∼ 4 compared with current measurements. Moreover, large datasets of medium- and high-resolutionspectra are already available at lower redshifts <cit.> that combined with the large redshift data can provide a more comprehensive constraint of reionization. Therefore this requiresstarting to focus now on improving the theory to exploit this increased precision.While modeling the 1D flux power spectrum with hydrodynamical simulations at sufficiently high accuracyis an incredible computational challenge,the advent of high-performance computing power and the high scalability of Nyx has allowed us to significantly improve the accuracy of our predictions in recent years.Finally, given that there are few observables that are sensitive to the thermal state of baryons at the earliest redshifts, the 1D flux power spectrum at z ≳ 5 offers a unique opportunity to explore not onlyreionization, but also constrain other physical scenarios that alter the thermal history of the IGM, for example, models that alter the thermal state of the IGMvia X-ray pre-heating coming fromstarburst galaxies, supernova remnants or miniquasars<cit.>, dark matter annihilationor decay <cit.>, cosmic rays <cit.>, blazar heating <cit.>, broadband intergalactic dust absorption <cit.> or high-z exotic reionization scenarios driven by PopulationIII stars <cit.>.We thank M. White, J. Miralda-Escudé and the members of the ENIGMA group atthe Max Planck Institute for Astronomy (MPIA) for helpful discussions. ZL was partially supported by theScientific Discovery through Advanced Computing (SciDAC) program funded by U.S.Department of Energy Office of Advanced Scientific Computing Research and theOffice of High Energy Physics. Calculations presented in this paper used thehydra cluster of the Max Planck Computing and Data Facility (MPCDF, formerlyknown as RZG) MPCDF is a competence center of the Max Planck Society located inGarching (Germany). We also used resources of the National Energy ResearchScientific Computing Center (NERSC), which is supported by the Office of Scienceof the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.This work madeextensive use of the NASA Astrophysics Data System and of the astro-ph preprintarchive at arXiv.org. This work is based on observations obtained with Planck(http://www.esa.int/Planck), an ESA science mission with instruments andcontributions directly funded by ESA Member States, NASA, and Canada. § NUMERICAL CONVERGENCEIn this section we discuss the convergencetests with spatial resolution and box size for the two fundamental quantities studied in this paper, the mean optical depth and the 1D flux power spectrum. To study the effects of spatial resolution, we have run 4 simulations with the same thermal history (e.g., same UVB; MiddleR) and box size of L_ box=10 Mpc h^-1 but increasing numbers of resolution elements: 128^3(dot-dashed green), 256^3 (dotted blue), 512^3 (dashed red), and 1024^3 (black).To simplify the comparison, simulations performed in the same box size share the same large-scale modes, the only difference being that higher resolution runs have more modes sampled on small scales. These simulations have a cell size of 78, 39, 20, and10 kpc h^-1, respectively, and therefore the 512^3 run has the same spatial resolution as the simulations discussed in this work (L_ box=20 Mpc h^-1 and 1024^3 cells). We also ran one more simulation with the same thermal history(MiddleR), a box size of L_ box=40 Mpc h^-1 and 2048^3 cells in order to study box size effects. We compare this simulation withtwo other runs with the same spatial resolution, but decreasing box size: the L_ box=20 Mpc h^-1 - N_ cell=1024^3 run, which correspond to thesimulations used in this work, and the L_ box=10 Mpc h^-1 - N_ cell=512^3 run also used in the spatial resolution study. The evolution of the mean optical depth τ_ eff for all these simulations is shown in Figure <ref>. We computed its evolution directly from thesimulation mean flux, τ_ eff=-lnF, without any rescaling of the photoionization rate. Thus, all simulations use exactly the same photoionization rates at all redshifts. The left panel shows the convergence of the mean optical depth as we increase the spatial resolution, while the right panel shows the convergence for different box sizes. The simulations discussed in this work (red dashed lines in both panels) show a convergence level below<5%between 4⩽ z⩽6 both in terms of spatial resolution and box size. Figure <ref> shows the convergence tests ofthe 1D flux power spectrum at redshift z=5 (left column) and z=6 (right column) for the same simulations.For this test, we rescaled the mean flux of all the simulations to the same value. We used the fit between mean flux and redshift suggested in <cit.>,obtained using a wide range of data sets between 0<z<6, but the exact values employed do not change our conclusions. For the resolution tests (upper row) and the box size tests (lower row), we find a ≲10% level of convergence for k modes lower than ∼0.04 s km^-1in the simulations with the same resolution and box size used in this work (red dashed lines). The error in these modes is mainly driven by box size effects as the resolution tests show a better convergence. However, for modes more relevant to study the thermal cutoff (0.04< k<0.1 s km^-1) we find a 10% convergence level at z=5 and 15% at z=6 that is mainly driven by spatial resolution effects. Note that this quoted convergence level is the worst level at the highest k mode,0.1 s km^-1, but it decreases as we move to lower k values.It is also very relevant to indicate that spatial resolution effects can only move the cutoff of the 1D flux power spectrum to higher k values as we increase the resolution.Similar results at these redshifts but for simulations using the Gadget code can be found in <cit.> and <cit.>. Convergence results at lower redshifts for the same code used in this paper along with a more detailed discussion can be found in <cit.>. | http://arxiv.org/abs/1703.08633v3 | {
"authors": [
"J. Oñorbe",
"J. F. Hennawi",
"Z. Lukić",
"M. Walther"
],
"categories": [
"astro-ph.CO",
"astro-ph.GA"
],
"primary_category": "astro-ph.CO",
"published": "20170325004828",
"title": "Constraining Reionization with the $z \\sim 5-6$ Lyman-$α$ Forest Power Spectrum: the Outlook after Planck"
} |
B.Verkin Institute for Low Temperature Physics and Engineering, 47, Lenin Ave., 310164 Kharkov, Ukraine Email address: [email protected] http://fntr.ilt.kharkov.ua/fnt/pdf/13/13-8/f13-0873r.pdfFiz. Nizk. Temp., 13, 973 (1987)); (Sov. J. Low Temp. Phys., 13, 501(1987) Measurement of the current-voltage characteristics of point contacts makes it possible to study the properties of individual crystallites in a superconducting ceramic. The critical current density in the superconducting regions of the ceramic La_1.8Sr_0.2CuO_4, with a size of the order of several tens of angstroms, is found to attain values of 10^8 A/cm^2, which are of the same order of magnitude as the pair-breaking current density, as evaluated from the formulas of the standard theory of superconductivity.74.25.Sv; 74,45+c; 74.25.Fy; 74.72.-h; 74.72.DnCritical current density in the superconducting ceramic La_1.8Sr_0.2CuO_4 in S-c-N point contacts I.K. Yanson, L.F. Rybal'chenko, N.L. Bobrov, and V.V. Fisun December 30, 2023 =================================================================================================In superconducting oxides with high critical parameters (T_c, H_c2) the critical current density is usually many orders of magnitude lower than the pair-breaking current density, estimated from the formulas of the standard theory of conductivity <cit.>. This is attributed either to the existence of weak contacts between the superconducting granules in such materials or by the very low vortex-pinning force, in the case of single crystals with a high degree of perfection <cit.>.Measurements of the I-V characteristics of point contacts whose size d is of the order of several tens of angströms makes it possible to study the characteristics of a superconductor under conditions when a vortex structure is not formed and the effect of the nonuniformity of the order parameter at distances greater than d can be disregarded because of the three-dimensional spreading of current from the region of the contact.Because of the low concentration of the superconducting phase in the La_1.8Sr_0.2CuO_4 samples studied, it is very difficult to form an S-c-S point contact between two regions with high critical parameters. Using copper as the counterelectrode, we managed to prepare clamped Cu-La_1.8Sr_0.2CuO_4 point contacts (see the inset to Fig.<ref>), whose characteristics clearly displayed superconductivity at temperatures below T_c^*≈ 27 K, close to the T_c of the bulk sample.The I-V curves shown in Fig.<ref> were taken at different temperatures for one of the point contacts. For low biases and low temperatures the resistance R_0 is due to the copper electrode while for high eV the I-V characteristic is similar to that in the normal state and the differential resistance R_N is determined almost entirely by the resistance to contraction in the ceramic (ρ_ceram≫ρ_Cu). The ratio r=R_N/R_0 varied over a wide range for different contacts and, apparently, is determined by the fraction of superconducting phase in the region near the contact on the side of the ceramic. The maximum values of r (10-20) attained in the best contacts corresponded to a negligible contribution to R_0 from the resistance of the normal phase of the ceramic. At low temperatures the attainment of the critical current I_c is accompanied by a jump in the voltage and hysteresis of the I-V characteristic. The superconductivity in the region of the contact is not destroyed completely, however, as is evidenced by the excess current I_exc, whose value is close to I_c. Apparently, a region of slippage of the order-parameter phase is formed and in this region normal quasiparticles transporting current above I_c are generated. The existence of "critical" and excess currents indicates that the superconductor near the contact is capable of carrying supercurrent with a density j_c∼ I_c/d^2∼ I_exc/d^2.We evaluate the lower limit for the contact diameter, assuming that the electron transit in the copper edge is ballistic; d≳ (16ρ l/3πR_0)^1/2. For R_0 = 83 Ω andρ l = 0.66· 10^-11 Ω· cm^2 we obtain d≳ 37 Å. On the other hand, assuming that the electron motion in the normal ceramic is via diffusion (l< d), we obtain for the upper limit the value d≲ρ_ceram/2R_N≈ 120 Å, where R_N = 750 Ω and ρ_ceram≈ 1.6· 10^-3 Ω· cm is the La_1.8Sr_0.2CuO_4 bulk resistivity. Therefore, 1.0· 10^8 A/cm^2< j_c^exp< 1.1· 10^9 /cm^2. We compare this estimate with the pair-breaking current density j_c =(√(2)/6π√(3))×(cH_cm/λ). Substituting H_cm≈ 4.5· 10^3 Oe and λ≈ 2500 Å into this formula, we have j_c^theor≈ 0.8·10^8A/cm^2. An estimate of the pair-breaking current can also be obtained by using the experimental value of the gap Δ_2, measured on the basis of the position of the dV/dI minimum on the eV axis (see curve a in Fig.<ref>) and the carrier concentration n≈ 10^21 cm^-3, giving p_F≈ 3.2· 10^-20 g· cm/sec. For Δ_2(0)≈ 13 meV we again obtain j_c^theor = en(Δ/p_F)≈ 10^8 A/cm^2. As is known, the gap minima on the dV/dI characteristics of point contacts with direct conduction are due to reflection of quasiparticles from the N-S boundary as a result of the mismatch of the electron parameters at the edges. From Fig.<ref> (curve a) we see that dV/dI has two minima, corresponding to two values of the gap. For the estimate we chose the large gap Δ_2, since the transition to the segment of the I-V curve with the high resistance R_N occurs near it. This transition means that the excess energy of the quasiparticles impinging on the superconductor exceeds the energy gap that determines the current-carrying capacity of the superconductor. We note that, according to our estimates, the contact diameter is comparable with the coherence length ξ≈ 20 Å. Vortices cannot come into being and move, therefore, in the vicinity of the contact.From Fig.<ref> we see that with increasing temperature the energy gaps Δ_1(T) and Δ_2(T) in reduced coordinates decrease more rapidly than I_exc(T) does. These data support our assumption <cit.> that the superconductivity of the ceramic is gapless in an appreciable temperature range below the temperature T_c^*, at which the nonlinearity of the I-V characteristic first appears near V = 0. Material with highly uniform superconducting properties is necessary in order to ascertain whether gapless superconductivity is an intrinsic property of ceramic superconductors or is due to the proximity effect in the presence of neighboring sections of normal phase.We are grateful to B.I. Verkin for his attention and support in the performance of the work. 1V.V. Shmidt, Introduction to the Physic of Superconductors (in Russianl, Nauka, Moscow (1982). 2B. Batlogg, J.P. Remeika, R.C. Dynes, et al.t "Structural instabilities and superconductivity in single crystal Ba(Pb,Bi)O_3 in: Superconductivity in d- and f-band Metals, W. Buckel and W. Weber (eds.), Kernforschungszentrum Karlsruhe GmbH, arisruhef (1982), pp. 401-403. 3I.K. Yanson, L.F. Rybal'chenko, V.V. Fisun, et al., http://fntr.ilt.kharkov.ua/fnt/pdf/13/13-5/f13-0557r.pdfFiz. Nizk. Temp. 13, 557 (1987) [Sov. J. Low Temp. Phys. 13, 315 (1987)]; https://arxiv.org/pdf/1701.01982.pdfarXiv:1701.01982. | http://arxiv.org/abs/1703.08708v2 | {
"authors": [
"I. K. Yanson",
"L. F. Rybal'chenko",
"N. L. Bobrov",
"V. V. Fisun"
],
"categories": [
"cond-mat.supr-con"
],
"primary_category": "cond-mat.supr-con",
"published": "20170325162748",
"title": "Critical current density in the superconducting ceramic $\\rm La_{1.8}Sr_{0.2}CuO_4$ in S-c-N point contacts"
} |
Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria. Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria.Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria. Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria.Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria. Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria.Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, D-85748, Garching, Germany.Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria. Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria.We theoretically show that, despite Earnshaw's theorem, a non-rotating single magnetic domain nanoparticle can be stably levitated in an external static magnetic field. The stabilization relies on the quantum spin origin of magnetization, namely the gyromagnetic effect. We predict the existence of two stable phases related to the Einstein–de Haas effect and the Larmor precession. At a stable point, we derive a quadratic Hamiltonian that describes the quantum fluctuations of the degrees of freedom of the system. We show that in the absence of thermal fluctuations, the quantum state of the nanomagnet at the equilibrium point contains entanglement and squeezing. Quantum Spin Stabilized Magnetic Levitation O. Romero-Isart December 30, 2023 ===========================================According to the Einstein–de Haas and the Barnett effect <cit.>, a change in the magnetization of an object is accompanied by a change in its rotational motion. In particular, if the magnetic moment of a magnet is varied by a single Bohr magneton, it must rotate with an angular frequency ħ/I about the magnetic moment axis to conserve angular momentum. Here I is its moment of inertia about the rotation axis. For a Cobalt sphere of radius R, this corresponds to a frequency ħ/ I≈ 2π× 10^6Hz/(R[nm])^5, where R[nm] is the radius in nanometers.This clear manifestation of the quantum spin origin of magnetization, as prescribed by the gyromagnetic relation, ishence boosted at the nanoscale <cit.>.In this Letter, we explore the role of the quantum spin origin of magnetization in magnetic levitation. Earnshaw's theorem <cit.>, very relevant in this context, prevents magnetic levitation of a non-rotating ferromagnet in a static magnetic field. The theorem can be circumvented by mechanically spinning the magnet, as neatly demonstrated by the Levitron <cit.>, which is a magnetic top of a few centimeters. At the single atom level, magnetic trapping with static fields is also possible by exploiting the fast Larmor precession of its quantum spin <cit.>. In this case, the atom is, from the mechanics point of view, a point particle without rotational degrees of freedom. A magnetic nanoparticle lies in between the Levitron and the atom as both its rotational degrees of freedom and the quantum spin origin of magnetization have to be accounted for. Can a non-rotating magnetic nanoparticle, despite Earnshaw's theorem, be stably levitated with static magnetic fields? We show in this Letter that this is the case. In particular, we predict two stabilization mechanisms that crucially rely on the quantum spin origin of the magnetic moment. At low (large) magnetic fields, the Einstein–de Haas effect (the Larmor precession of its magnetic moment) stabilizes levitation. These results are obtained by deriving a quadratic Hamiltonian which describes the linearized dynamics of the degrees of freedom of the magnet (center-of-mass motion, rotation, and magnetization dynamics) around the equilibrium point. We further show that in the absence of thermal fluctuations, the equilibrium state exhibits both quantum entanglement and squeezing of its degrees of freedom. As discussed in the conclusions, these results could be used to bring and control the rich physics of levitated nanomagnets in the quantum regime. We consider a single magnetic domain nanoparticle (nanomagnet hereafter) in an external static magnetic field (). The nanomagnet is modeled <cit.> as a spherical rigid body of mass M,radius R, and with uniaxial magnetocrystalline anisotropy <cit.>.The dynamics are described in the body frame ,with e_3 aligned to the direction of the anisotropy axis. The body frame is obtained from the laboratory frameby the rotation R(), as defined in <cit.>, where ={,,} are the Euler angles. The Hamiltonian of the system in the body frame is given by <cit.> =^2/2M + ħ^2/2 I+^2-ħ^2 D_3^2 +ħγ_0 ·(,).Here≡ + /(ħ) is the total angular momentum (excluding the center-of-mass angular momentum), ≡ - /(ħ) the spin angular momentum <cit.>,the rigid body angular momentum,the magnetic moment operator, and >0 the gyromagnetic ratio <cit.>. The macrospin approximation is assumed, namely that the total magnetization is constant, which results in ^2=S(S+1) 𝕀 with S ≡ Nf for N identical spin f constituents of the nanomagnet <cit.>. The first (second) term in eq:Ham_Body represents the center-of-mass (rotational) kinetic energy of the nanomagnet.The third term represents the uniaxial magnetocrystalline anisotropy, whose strength is parametrized byD≡ 4π R^3k_a/(3ħ^2 S^2) with k_a being the material-dependent anisotropy constant <cit.>. The last term represents the magnetic dipolar interaction, where (,) ≡ R()().Hereafter, we consider a Ioffe-Pritchard magnetic field, () =(B' x -B” xz /2) - (B' y + B” yz/2) +[B_0 + B”z^2/2 - B”(x^2+y^2)/4 ], where B_0,B' and B”are, respectively, the field bias, gradient and curvature <cit.>.Gravity, which is assumed to be along the z-axis, can be safely neglected since throughout the Letter the condition M g/(μ B”) ≪ (B_0/B”)^1/2, (B'/B”) is fulfilled, where g is the gravitational acceleration, see <cit.> for further details.The rotational angular momentum of the nanomagnet about the anisotropy axis is a constant of motion, [_3,]=0 <cit.>. The degrees of freedom of the system, namely the center-of-mass motion, the rotational motion and the magnetization dynamics, are thus described by 12 independent dynamical variables (see <cit.> for further details). In the following, we address whether the nanomagnet can stably levitate for a given set of physical parameters of our model: mass density ρ_M, magnetization ρ_μ, k_a, R, B_0, B', and B”. Let us first obtain the equilibrium configuration of the system. By writing the Heisenberg equations of motion for the nanomagnet in semiclassical approximation <cit.>, the following relative equilibrium is found (see Fig:Equilibrium): (i) The center of mass is fixed at the center of the trap; (ii) The orientation is given by the body frame aligned to the laboratory frame (e_3 ∥) and rotating about e_3 at the frequency ≡ -ħ_3/I; (iii) The magnetic moment lies along the anisotropy axis e_3 and is anti-aligned to the magnetic field (0)=B_0.The stability of this relative equilibrium is analyzed in the frame co-rotating with the system <cit.>. This is obtained via the unitary transformation Û=exp(-_3 t) which transformsinto +ħ_3.The linear stability of the system is determined by the dynamics of the fluctuations of the degrees of freedom around the equilibrium.The evolution of small fluctuations can be described as a collection of interacting harmonic oscillators through the bosonization procedure presented in <cit.>. In the following, we provide the mapping between the observables and the bosonic operators and refer to<cit.> for its derivation and further details. The fluctuations of the center-of-mass motion are described by three independent harmonic oscillators:≡z_0(+ ),_z ≡ħ/2 z_0(-), for the motion along , and_+ ≡√(2)r_0 (+), _+ ≡ħ/√(2)r_0(-), for the transverse motion, where _± = ± and _± =_x±_y . Here, z_0 ≡√(ħ/(2M)) and r_0 ≡√(ħ/(2M)) are the zero-point motions and≡√(ħ B” S/M) and ≡√(ħ S (B'^2-B_0B”/2)/MB_0) the trap frequencies. We introduced three bosonic modes with [_i,_j]=δ_ij for i,j=z,R,L. The harmonic oscillators describing the fluctuations of the rotational degrees of freedom are obtained as follows. A Holstein-Primakoff mapping generalized to the case where ^2 is not conserved <cit.> leads to = ^†≡_1 -_2 = √(2J), _3=-J -√(2J)(+)/2 -/2+. The k-bosonic mode ([,]=1) describes the fluctuations of _3 around _3= -J ≡ -I/ħ - S. As discussed below, we assume J≫ 1. The j-bosonic mode ([,]=1) describes the fluctuations of ^2 around ^2= J(J+1). Since the components ofin the laboratory frame commute with _3 and _↓↑<cit.>, they need to be bosonized separately as_+=_-^†≡_x + _y = √(2J), _z=-J -√(2J)(+)/2 -/2+. The additional m-bosonic mode ([,]=1) represents the fluctuations of _z around _z=-J. The bosonization of the Euler angle operators is more involved <cit.>.Sinceappears in eq:Ham_Body only inside trigonometric functions, the Euler angle operators can be written as functions of the 9 Wigner D-matrix tensor operators mk1 with m,k=±1,0. The exact bosonization of these operators is given in <cit.> in terms of the j-k-m-bosonic modes. The fluctuations of the magnetic moment under the macrospin approximation (^2=S(S+1) 𝕀 with S ≫ 1) can be described by the standard Holstein-Primakoff approximation, namely <cit.>= ^†≡_1 - _2 = √(2S), _3 = S-. The s-bosonic mode describes fluctuations of _3 around _3=S. In summary, the fluctuations of the degrees of freedom of the nanomagnet around the equilibrium illustrated in Fig:Equilibrium are described by 7 bosonic modes.Let us now derive the Hamiltonian describing the dynamics of these bosonic modes. Note that since _3=_3+_3 is a constant of motion, the bosonic modes are constrained by √(2J) (+) = /2--. Together with the assumption of a slowly rotating nanomagnet, namely | I/ħ|≪ S, which implies J≫1, the bosonization procedure transforms +ħ_3 into a quadratic Hamiltonian that depends on 6 bosonic modes: ,,,,, and . Terms containing the j-mode, as well as non-quadratic terms, are of the order (1/J)^n/2 for n=1,2…, and can thus be safely neglected. The quadratic Hamiltonian reads _q ≡_Z + _T = ħ/2 _Z M_Z_Z + C + C^* .The first term describes the fluctuations of the center-of-mass motion along , where _Z = (,)^T and M_Z=𝕀_2. The second term describes the fluctuations of the other degrees of freedom, where ≡ (,,,,)^T and C ≡[ g- -gg;g _kg ^2 _k ^-1;-g-gg; -g ^2 -g-^2 ;g_k^-1g_μ ]. Here we defined ≡√(S/J), the Larmor frequency ≡ B_0, the anisotropy frequency ≡ħ D S, the Einstein–de Haas frequency ≡ħ S/I, the frequencies _k ≡+ -^2, _μ≡+2-, _±≡(±^2/)/2, and the coupling strength g≡ B'σ_T/B_0. As shown below, the hierarchy between , , andplays a crucial role in the stability of the nanomagnet.The linear stability of the equilibrium can be determined by the Heisenberg equations of motiont _Z=1/ħ[_Z,_Z] ≡K_Z _Z=σ_z M_Z_Z,t =1/ħ [,_T] ≡K_T = G M_T . Here σ_z=diag(1,-1),G ≡𝕀_5⊗σ_z, andM_T=M_T^† is given by rewriting _T as _T = ħ M_T /2, where≡ (,,,,,,,,,)^T. The dynamics of the system are completely characterized by the characteristic polynomials of K_Z and K_T. Linear stability occurs when the roots of the polynomials have zero real part <cit.>.The characteristic polynomial of K_Z reads P_Z(λ) = λ^2 + ^2, which has purely imaginary roots for B”>0.The matrix K_T can be block diagonalized in two 5× 5 complex conjugate blocks whose characteristic polynomial is given by P_T(λ)=a_0 + a_1 λ + a_2λ^2 + a_3 λ^3 + a_4 λ^4 + a_5λ^5, where a_0 ≡ -2^2, a_1 ≡^2(+ ) + ^2, a_2 ≡ -2- 1/22- ^2 - ^2, a_3 ≡ -2++ + 1/2^2, a_4 ≡ 2- - , a_5 ≡ - .The analysis of P_Z(λ) and P_T(λ) as functions of the physical parameters of the problem provides the stability diagram of a magnetically levitated nanomagnet, as discussed below. These polynomials can be alternatively obtained either by using classical equations of motion, or via the linearized Heisenberg equations of motion in semiclassical approximation without performing the bosonization, see <cit.>. Such methods allow to understand the results concerning the stability of a nanomagnet without the need to introduce the bosonization of the quantum mechanical operators. The procedure performed here is richer, since it allows to obtain the quadratic Hamiltonian _q describing the quantum fluctuations around equilibrium as well as the quantum properties of the states in equilibrium.Let us now analyze the linear stability of the system in the non-rotating case, see <cit.> for the ≠ 0 case. The stability phase diagram derived using eq:LinEq_z and eq:LinEq_T, is shown in Fig:Plots. This is a two-dimensional phase diagram with the x-axis given by the bias field B_0 and the y-axis given by the radius of the nanomagnet R. The remaining physical parameters are fixed, see caption of Fig:Plots. Two stable phases are found.The Einstein–de Haas (EdH) phase appears at low magnetic fields and for small radius. In this regime, the dynamics of the magnetization is dominated by the anisotropy interaction (≫). Due to the small moment-of-inertia-to-magnetic-moment ratio (≫), the angular momentum contribution of the macrospin stabilizes the system through the Einstein–de Haas effect <cit.>.That is, the macrospin is locked along the anisotropy direction due to the conservation of energy <cit.>. Even if rotation is absent, the spin-rotation interplay described by the Einstein–de Haas effectstabilizes the non-rotating magnet by keeping the macrospin aligned along the anisotropy direction. The atom (A) phase appears at high magnetic field bias (≫). In this regime, the nanomagnet behaves like a magnetic atom of mass M and spin S <cit.>: the anisotropy interaction can be neglected and the magnetic moment undergoes a free Larmor precession about the local magnetic field direction. The approximated expressions for the borders of the stable phases, given in the caption of Fig:Plots, can be analytically obtained from the discriminant of the characteristic polynomials <cit.>. The stability diagram shows that a non-rotating nanomagnet can be stably levitated in a static field configuration. This opens up the possibility to cool the thermal fluctuations of the degrees of freedom to the quantum regime. The feasibility and analysis of such an experimental proposal will be addressed elsewhere. Let us now analyze the properties of the quantum state at the equilibrium in the absence of thermal fluctuations. This state corresponds to the vacuum state |0⟩ of the normal eigenmodes of the quadratic Hamiltonian eq:Hq: _Z and _i for i=1,…,5 ([_i,_j]=δ_ij).The c-bosonic modes ≡(_1,_1,…,_5,_5)^T are obtained from modesthrough a Bogoliubov transformation =T^-1 such that T^† M_T T = diag(_1,_1,…,_5,_5) and T^† G T = G.At each stable point of the phase diagram, the transformation T exists <cit.> and can be constructed as follows. One firsts obtains the eigenvalues λ_i and eigenvectors _i (for i=1,…,10) of K_T. At a stable point K_T is diagonalizable andλ_i are real and non-degenerate. One then G-orthonormalizes the eigenvectors such that, with an appropriate relabeling, they fulfill _i^† G_j=+δ_ij for i,j=1…5 and _i^† G_j=-δ_ij for i,j=6…10 <cit.>. The Bogoliubov transformation matrix is then given byT=(_1 _1…_5 _5) (the vectors _i and _i are the columns of the matrix), where _i≡(σ_x ⊗1_5)_i^* with σ_x being the non-diagonal-real-valued Pauli matrix.One can now analyze the properties of the vacuum state |0⟩, which is a multi-mode Gaussian state, by the 10× 10 covariancematrix <cit.> Θ_ij ≡1/2⟨0|(_i_j+_j_i)|0⟩=1/2∑_k=1^10 T_ik(T^†)_kj. Note that the _Z-bosonic mode is not included in Θ since it is decoupled from all the other modes. The five 2× 2 diagonal blocks (Σ_a)_ij≡⟨0|(_i_j+_j_i)|0⟩/2 of Θ, with ≡(,) and =,,,,, correspond to the covariance matrices of the a-modes. The off-diagonal blocks describe the correlations between the modes. Entanglement in the pure state |0⟩ can be quantified by 𝒫≡ 5-∑_a 𝒫_a, where 𝒫_a=[2√(det(Σ_a))]^-1 <cit.> is the purity of the a-bosonic mode. This characterizes the bipartite entanglement between one mode and the remaining four. Single-mode squeezing can be quantified via the squeezing parameter ξ≡1/√(2 min_k(θ_k)), where θ_k are the eigenvalues of Θ <cit.>. Fig:Plots shows 𝒫 and ξ as a function of B_0 for a given R in the stable phases. Bipartite entanglement and single-mode squeezing are thus present in the |0⟩ state of a magnetically levitated non-rotating nanomagnet. In conclusion, we showed that the quantum spin origin of the magnetization stabilizes magnetic levitation of a non-rotating nanomagnet, despite the Earnshaw theorem. Such a quantum-spin-stabilized levitation opens the door to experiments aiming not only at demonstrating the predicted phase diagram, but also at bringing a non-rotating nanomagnet to the quantum regime, whose equilibrium states show non-trivial quantum correlations. There are many directions left for further research, some of which we are currently addressing: The experimental proposal and feasibility analysis to prepare the |0⟩ state by sympathetically cooling the degrees of freedom of a levitated nanomagnet near a flux dependent microwave cavity; The analysis and the potential applications ( spin squeezing for magnetic sensing) of the quantum dynamics generated by placing the nanomagnet at the unstable regions of the phase diagram; Levitating, coupling, and cooling several nanomagnets in the quantum regime to study quantum nanomagnetism in a unique substrate-free environment. We hope that such results and their potential applications will trigger further theoretical and experimental research to levitate nanomagnets in the quantum regime. We thank B. Kraus for useful discussions. This work is supported by the European Research Council (ERC-2013-StG 335489 QSuperMag) and the Austrian Federal Ministry of Science, Research, and Economy (BMWFW). 10EdH A. Einstein, and W.J. De Haas, Proc. KNAW, 181, 696 (1915).Barnett S. J. Barnett, Phys. Rev., 6, 239 (1915).Quantum_EdH M. Ganzhorn, S. Klyatskaya, M. Ruben, and W. Wernsdorfer, https://doi.org/10.1038/ncomms114437114432016.Chud_EdH1 E. M. Chudnovsky, http://dx.doi.org/10.1103/PhysRevLett.72.34337234331994.Chud_EinsteindeHaas R. Jaafar, E.M. Chudnovsky, and D.A. Garanin, http://dx.doi.org/10.1103/PhysRevB.79.104410791044102009.bassani2006earnshaw R. Bassani, https://dx.doi.org/10.1007/s11012-005-4503-xMeccanica, 41, 375 (2006).simon1997spin M. D. Simon, L. O. Heflinger, and S. L. Ridgway, http://dx.doi.org/10.1119/1.18488Am. J. Phys., 65, 286 (1997).Dullin H. R. Dullin and R. W. Easton, http://dx.doi.org/10.1016/S0167-2789(98)00251-6Physica D: Nonlinear Phenomena, 126, 1, (1999).Gov1999214 S. Gov, S. Shtrikman, and H. Thomas, http://dx.doi.org/10.1016/S0167-2789(98)00282-6Physica D: Nonlinear Phenomena, 126, 214 (1999).Berry_Levitron M. V. Berry, http://doi.org/10.1098/rspa.1996.0062Proc. R. Soc. Lond. A, 452, 1207 (1996).Sukumar1997 C. V. Sukumar and D. M. Brink, http://dx.doi.org/10.1103/PhysRevA.56.24515624511997.Sukumar2006 D. M. Brink, and C.V. Sukumar, http://dx.doi.org/10.1103/PhysRevA.74.035401740354012006.Note0 The model presented here is based on the following assumptions: (i) single magnetic domain, (ii) macrospin approximation, (iii) rigid body, (iv) sphere, (v) uniaxial anisotropy, (vi) Ioffe-Pritchard magnetic field, (vii) point-dipole approximation, and (viii) dissipation-free dynamics for the system. Assumptions (iv), (v), (vi), could be easily relaxed with the same formalism. Assumptions (ii), (iii) and (vii) could be relaxed by considering internal phonons and other magnonic modes. Relaxing assumptions (i) and (viii) would most like require a different analytical treatment.chikazumi S. Chikazumi and C. D. Graham, Physics of Ferromagnetism (Oxford University Press, Oxford UK, 2009).ORI_Cos_PRB C. C. Rusconi and O. Romero-Isart, https://doi.org/10.1103/PhysRevB.93.054427930544272016.Note3 For convenience, the total spinis defined as the negative of a standard spin operator such that its components satisfy anomalous commutation relations: the body (laboratory) frame components ofsatisfies [_i,_j]=-ijk_k ([_μ,_ν]=-μνλ_λ), for i,j,k=1,2,3 (ν,μ,λ=x,y,z) <cit.>. Note1 We assume that the magnetization arises from the electron spins, for which γ _0 =1.760rad T^-1 s^-1.molecularNM D. Gatteschi, R. Sessoli, and J. Villain, Molecular nanomagnets. Oxford: Oxford University Press, 2011.atomchip J. Reichel and V. Vuletic Atom Chips (Weinheim: Wiley-VCH Verlag, 2011).LinStab_NM C. C. Rusconi, V. Pöchhacker, J. I. Cirac, and O. Romero-Isart, https://doi.org/10.1103/PhysRevB.96.134419961344192017.meiss2007differential J. D. Meiss, http://dx.doi.org/10.1137/1.9780898718232Differential dynamical systems (Siam, US, 2008).Marshalek_RMP A. Klein and E. R. Marshalek, http://dx.doi.org/10.1103/RevModPhys.63.375633751991. Note4 The fact that _3 and _z commute corresponds to the classical result that a rotating top has at most two conserved directions of the angular momentum: the precession about e_z and the mechanical rotation about e_3. See <cit.>for a proof that [_3,_z]=0. HP_Mapping T. Holstein and H. Primakoff, http://dx.doi.org/10.1103/PhysRev.58.1098Phys. Rev. 58, 1098 (1940).Chud_RotStatesNM E. M. Chudnovsky and D. A. Garanin, https://doi.org/10.1103/PhysRevB.81.214423812144232010.O'Keeffe_PRB M. F. O'Keeffe and E. M. Chudnovsky, https://doi.org/10.1103/PhysRevB.83.092402830924022011.Maldonado O. Maldonado, http://dx.doi.org/10.1063/1.5303383450161993.Note2 Since the eigenvalues are all distinct, the eigenvectors 𝐯_i are already G-orthogonal, and need only to be properly normalized.RSimon R. Simon, N. Mukunda, and B. Dutta, https://doi.org/10.1103/PhysRevA.49.15674915671994.ParisPurityGss M. G. A. Paris, F. Illuminati, A. Serafini, and S. De Siena, https://doi.org/10.1103/PhysRevA.68.012314680123142003. | http://arxiv.org/abs/1703.09346v2 | {
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"title": "Quantum Spin Stabilized Magnetic Levitation"
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"published": "20170326160304",
"title": "Application of the boundary control method to partial data Borg-Levinson inverse spectral problem"
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^1 Physikalisches Institut (EP3), Universität Würzburg, Am Hubland, 97074 Würzburg, Germany. ^2 Institute of Physics, PAS, Al. Lotniów 32/46, 02-668 Warsaw, Poland. We present results on the growth of CdTe-HgTe core-shell nanowires, a realization of a quasi one-dimensional heterostructure of the topological insulator HgTe. The growth is a two step process consisting of the growth of single crystalline zinc blende CdTe nanowires with the vapor-liquid-solid method and the overgrowth of these wires with HgTe such that a closed shell is formed around the CdTe core structure. The CdTe wire growth is monitored by RHEED allowing us to infer information on the crystal properties from the electron diffraction pattern. This information is used to find and control the optimal growth temperature. High quality single crystal CdTe nanowires grow with a preferred orientation. For the growth of the conductive HgTe shell structure we find that the supplied Hg:Te ratio is the crucial parameter to facilitate growth on all surface facets. CdTe-HgTe core-shell nanowire growth controlled by RHEED M. Kessel^1, J. Hajer^1, G. Karczewski^2, C. Schumacher^1,C. Brüne^1, H. Buhmann^1 and L. W. Molenkamp^1 December 30, 2023 ==============================================================================================================§ INTRODUCTIONTwo-,<cit.> three- <cit.> and quasi one-dimensional <cit.> topological insulators (TIs) are a relatively new field in condensed matter physics, and have already attracted much attention. They are very interesting due to the topological protected electronic surface states,<cit.> which boast many distinct properties such as spin-momentum coupling.<cit.>An ideal TI hosts conducting surface states, but has an insulating bulk. However, in many material systems, charge transport investigations reveal a significant bulk contribution. Contrary, MBE-grown HgTe layers show high mobility surface states and no bulk conductance both in two- and three-dimensional structures.<cit.> Quasi one-dimensional TI nanowires (NWs) are a novel and interesting topic, as this geometry induces periodic boundary conditions for the surface state along the short perimeter of the NW.<cit.> First investigations on TI NWs were done on Bi_2Se_3 based systems.<cit.> However, crystalline quality and charge carrier mobility are inferior to HgTe and furthermore Bi_2Se_3 has a tendency to be unintentionally doped. More generally, the reduction of size increases the surface to volume ratio. This effect strengthens for a tubular TI, making CdTe NWs shelled with the TI HgTe promising candidates to observe phenomena related to the surface states. HgTe NWs can not be grown with the vapor-liquid-solid (VLS) method due to the high vapor pressure of Hg. Attempts to do so resulted in segmented HgTe wires with inclusions of elemental Te.<cit.> Here we follow an alternative route: We grow CdTe NWs with the VLS method, which are overgrown with HgTe MBE in a second step in order to obtain CdTe-HgTe core-shell NWs.Conventionally, CdTe NWs show a tendency to grow laterally on the substrate. Previous works only reported short vertical CdTe NWs with wurtzite crystal structure.<cit.> Other attempts achieved free-standing zinc blende CdTe NWs, but without specified orientation to the substrate.<cit.> For uniform HgTe shell growth, we need straight, free standing, preferably oriented and single crystalline zinc blende CdTe NWs.Wojtowicz et al. showed, that CdTe NWs can be grown on top of ZnTe NWs.<cit.> Moreover, preferably [111] oriented ZnTe NWs can be grown on GaAs substrates with an Au-based catalyst, thus paving a way for obtaining oriented CdTe wires.<cit.> In this work this approach is refined to achieve the very great uniformity of the NW ensemble. The use of a reflection high energy electron diffraction (RHEED) technique to study the solid-liquid phase transition of the catalyst and the nucleation start has been reported earlier for GaAs NWs with electrons penetrating the nano-crystallites.<cit.> We apply the same technique for our CdTe and HgTe-CdTe structures. In addition to the spotty features reported in <cit.>, we see streaks in the diffraction patterns, when the electron beam is diffracted during reflection at crystalline facets. The narrow temperature limits for uniform growth are controlled by changes in electron diffraction. High crystalline quality and uniformity of the NWs allows to study the atomic periodicity of NW surfaces by RHEED, which are subsequently overgrown with epitaxial HgTe.§ GROWTHStarting point of the sample growth are Si-doped GaAs substrates with (110) and (111)B orientation. Our wires are grown in an UHV cluster using a metallization chamber and three separate MBE chambers optimized for the growth of III-V GaAs compounds, wide gap II-VI materials and Hg containing layers, respectively.In order to prepare the substrates, we thermally remove the natural oxide layer followed by the deposition of 1 nm Au. The NW growth is seeded by liquid eutectic Au-Ga droplets, formed by heating the substrate to 480 ^∘C inside the MBE chamber. Subsequently, the NWs are grown with beam equivalent pressures in the range of 5·10^-7 to 1·10^-6 mbar. Growth starts with a Zn:Te beam pressure ratio of 0.6:1 at 400 ^∘C. We find that the growing ZnTe structures push the droplets across the substrate, causing several droplets to merge and become trapped by three-dimensional ZnTe structures. The redistribution and localization of the droplets is complete after about two minutes. We find that these localized droplets seed vertical wire growth.After the localization of the droplets is complete, the Zn supply is replaced by a Cd flux with a Cd:Te ratio of 1.2:1. The growth of the actual CdTe NWs is initiated by carefully lowering the substrate temperature. At 400 ^∘C CdTe growth is completely suppressed. We lower the temperature slightly until NW growth starts, while two dimensional CdTe growth remains suppressed. The growth of NWs and the optimal growth temperature can be inferred from the RHEED. The optimal substrate temperature to start wire growth is found to be approx. 395 ^∘C. The growth rate of CdTe NWs is highly temperature dependent. Growth rates of up to 0.2 nm/s are observed for substrate temperatures around 395 ^∘C. We achieve crystalline NWs with a high aspect ratio. Straight and uniform CdTe NWs grow along [111]B in a temperature range of about 392 - 397 ^∘C. When the temperature is too low, the NWs grow randomly along different orientations, leading to pronounced kinks. Upon starting NW growth, the sample's radiative loss of energy initially changes significantly, due to the strong surface increase associated with the presence of the NWs. To compensate this effect we increase the substrate temperature slowly by about two Kelvin per hour for three hours during the growth process. Figure <ref> shows CdTe NWs grown under optimized conditions and the temperature increase is controlled by optimizing the RHEED patterns. For long NWs most of the radiated heat remains within the ensemble. In order to grow NWs with lengths up to 2 µm, a further growth step is necessary where the substrate temperature is held constant at about 400 ^∘C for several hours. The self-organized growth procedure described here results in free-standing CdTe NWs oriented along [111] and a high uniformity, being homogeneous over several cm2. In a final step, the NWs are epitaxially overgrown with HgTe at 185 ^∘C substrate temperature. Hg is supplied with a beam equivalent pressure in the range of 1·10^-4 to 6·10^-4 mbar. The Hg:Te ratio can be adjusted using a homebuilt Hg-cell. We find that with a ratio of 200:1 HgTe grows on all NW surfaces. Detailed results can be found in the later sections. § RESULTS AND DISCUSSION §.§ RHEED studies on CdTe nanowire facets In the following section we will discuss how RHEED can be used during growth to gather information on the crystalline structure of the NWs. The method averages over macroscopic sample areas. If the lattice of most of the NWs is identical and furthermore oriented in the same direction, then the interference pattern can be analyzed like the one of a single crystal. Thus, a high ensemble uniformity is needed to allow for detailed analysis and in this case the distance of periodic features in the interference pattern is proportional to the inverse lattice constant. Electrons have a small penetration depth and diffraction during transmission through the bulk region near the surface causes an interference pattern with broadened spots. The period of the spots is a measure for the bulk lattice constant of a single crystal. This is even true for a collection of wires, provided they are all oriented the same. In addition to bulk diffraction spots, we observe lines in the RHEED when the NWs are grown long enough to develop sufficiently large side facets of a uniform lattice constant. These lines are caused by diffraction during reflection, and appear perpendicular to the NW side facets. The high uniformity of the NW ensemble allows us to analyze the periodicity of the surfaces by RHEED. Different kinds of crystallographic surface orientations are observed for our CdTe NWs. The sample rotates azimuthally inside the growth chamber. Each NW surface is in the total reflection condition for one azimuth. Twelve different side facets can be identified by comparing the RHEED obtained from different substrate orientations. These are the low index surfaces {110} and {112} parallel to the growth axis [111]B. We explain the RHEED pattern of NWs grown vertically on a (111)B substrate. With this geometry one can see two alternating diffraction patterns every 30^∘, which are shown in Fig. <ref>. The interference pattern obtained by an electron beam along [112] as well as its schematic illustration are shown in Fig. <ref> (a). The electron beam pointing along [112] is reflected from (110) facets. Lines perpendicular to this surface emerge in the RHEED. One of these lines is denoted by an arrow in reciprocal space shown left-hand side in Fig. <ref> (a). The lines show the same periodicity parallel to the surface as the transmission spots. Therefore the surface atoms show the same period of 1.1 nm along the [111] direction as in the bulk. Taking into account the atomic positions of the zinc blende unit cell we deduce the cross-sectional view of bulk and (110) surface atoms shown right-hand side in Fig. <ref> (a). The pattern obtained by an electron beam along [110] as well as its schematical illustration are shown in Fig. <ref> (b). Zinc blende crystals have a three fold rotational symmetry with respect to the [111] axis and we would not expect the observed mirror symmetry for a single crystal. We suggest that the symmetry observed experimentally is caused by 60^∘, or equally 180^∘, rotational twinning at the bases of the NWs. Most of the CdTe NWs are twin free, except of twinning at the interface to the substrate, as explained in the next section. As in the upper pattern, the spots are a measure for the bulk lattice constant.The electron beam pointing along [110] is deflected during reflection from (112) side facets, which show a more complex pattern compared to the atomically flat (110) side facets. Instead of continuous lines, small and vertically staggered stripes are visible. One of these stripes is indicated by an arrow in reciprocal space shown left-hand side in Fig. <ref> (b).The stripes are located within bands marked with two thin lines in Fig. <ref> (b) parallel to the [111] direction. Half of the stripes are located between two spots and the position of the remaining stripes is identical to the position of the spots. This interference pattern originates from a regularly stepped surface <cit.> consisting of (111) nanofacets. The step length of 2.2 nm along [111] is twice the bulk lattice constant in that direction. The deduced cross-sectional view of the atomic positions within the NW and on its (112) surface is shown right-hand side in Fig. <ref> (b). §.§ HgTe epitaxy on CdTe nanowiresThis section describes the growth of HgTe shell on the CdTe NWs. HgTe growth occurs at temperatures of about 185 ^∘C. The sticking of Hg depends strongly on the polarity of the substrate.<cit.> We find that Hg flux intensity is a critical parameter due to the multifaceted nature of our NWs. The twelve side facets discussed in the previous section have various polarities: A-polar (group II terminated), non-polar, B-polar (group VI terminated), non-polar etc. Supplying a Hg:Te ratio of 200:1, a continuous HgTe shell grows. The nucleation starts at both polar types of facets and the adjacent non-polar facets are overgrown subsequently. Such CdTe NWs with continuous HgTe shells are shown in Fig. <ref> (a).Different growth on polar surfaces lead to the characteristic shell shape. During HgTe overgrowth the stepped and A-polar nanofacets evolve to larger steps resolved by the SEM. The angle between an (111)A step and the NWs' [111]B growth direction is determined by the crystal structure. For B-polar facets no pronounced steps are observed in the SEM. Instead, the growth rate of HgTe is reduced gradually on B-side, leading to uncovered areas near the NW bases, as highlighted (red) in Fig. <ref> (a). This behavior can be understood from the growth rate dependence on the supplied Hg:Te ratio.We find suppressed growth on the B-polar facets with Hg:Te ratios larger than 250:1, while HgTe still grows stoichiometrically for ratios up to 350:1 on A-polar facets. Suppressing growth on B-polar surfaces can be used to obtain another geometry, shown in Fig. <ref> (b). Using a (110) oriented substrate, overgrowth occurs on one side, since such wires are tilted by 35^∘ with respect to the substrate normal. Due to sample-source geometry, the (112)A surface facing the effusion cells has a much larger growth rate compared to those A-polar facets, which are oriented more towards the substrate.We have seen that the growth rate dependence on the Hg:Te ratio and the sample-source geometry have an influence on the heterostructure geometry. Evaporated materials have to be multiply absorbed and desorbed from the NW ensemble to reach the bases of long NWs. Impinging Te is more likely incorporated and much less reflected by the NWs than Hg. Thus, the Hg:Te ratio is effectively larger for the NW bases compared to the top ends. This explains the changes in growth rate on B-polar facets along the wire in Fig. <ref> (a). The differences in growth on the polar facets also reveal the presence of the aforementioned rotational twins in the CdTe wires. The two wires seen in the front of Fig. <ref>(a) are rotational twins. The azimuthal orientation of polar surfaces is constant along each individual wire, thus no further twinning occurs along the wires. We therefore conclude that the origin of the twinning is at the interface to the substrate. The obtained heterostructures have the characteristic shape of a single crystal with faceted side surfaces. We therefore conclude that most of the individual NW heterostructures are single crystalline with a good crystal quality. §.§ Charge transportIn addition to the crystallographic analysis presented in the previous parts of the paper we also performed basic transport studies on CdTe-HgTe core-shell structures and uncovered CdTe NW cores. The uncovered CdTe wires are not conducting, as we can expect from the 1.4 eV band gap of this material. In contrast, we find a high conductivity in the structures overgrown with HgTe, even at low temperature (T=1.7 K). Since these wires are nominally undoped, it is likely that the conductance occurs in the topological surface states of the HgTe capping layer. Figure <ref> (a) shows a magneto-resistance measurement on a single wire sample, as depicted in the inset. The magnetic field is applied perpendicular to the sample plane and results in a positive and parabolic magnetoresistance consistent with the behavior found in many metallic systems.<cit.> Additional, the resistance shows reproducible fluctuations, which are a manifestation of quantum interference of possible carrier trajectories inside the sample.The conductivity can also be modulated by applying a voltage to the backgate. As seen in Figure <ref> (b), the gate effect leads to a similar reproducible fluctuation of the conductance, pointing to universal conductance fluctuations (UCFs). We furthermore do not observe signs for ballistic transport and therefore conclude, that the mean free path for charge carriers in our quasi one-dimensional HgTe is smaller, but the phase coherence length is larger than the sample size of several hundred nanometers. While HgTe layers show no significant bulk- but surface-transport, the present data can not rule out that this is also true for our NWs. Further experiments, including detailed investigation of gate effects to elucidate these issues are underway.§ SUMMARYWe report on a novel growth method for quasi one-dimensional heterostructures containing the TI material HgTe. The concept of a hollow TI-NW, having a trivial insulator as a core is realized for the first time. The growth of CdTe-HgTe core-shell NWs requires high attention in controlling basic parameters such as substrate temperature and the intensity of supplied material fluxes. We successfully optimized the growth conditions enabling us to grow high quality CdTe NWs, using RHEED for fine tuning of the substrate temperature to achieve straight and long wires. With optimized growth parameters a periodic diffraction pattern allows for the detailed analysis of atomic arrangement on the surfaces and in the bulk. The ability to do so reflects the high crystal quality and ensemble uniformity of our CdTe NWs. The high quality of the cores allows us to grow epitaxial HgTe shells around them. The wire structures are used for further investigations on the properties of quasi one-dimensional HgTe. So far, we are able to contact a single NW. Bare CdTe cores are insulating, but the HgTe shell conducts and shows UCFs. A full study of the transport properties will be published elsewhere.§ ACKNOWLEDGEMENTSThis work was supported by the Deutsche Forschungsgemeinschaft in the priority progam “Topological Insulators: Materials - Fundamental Properties - Devices” (DFG-SPP 1666).We thank N. V. Tarakina for the electron transmission micrograph in the inset of Fig. <ref> (b). | http://arxiv.org/abs/1703.09106v1 | {
"authors": [
"M. Kessel",
"J. Hajer",
"G. Karczewski",
"C. Schumacher",
"C. Brüne",
"H. Buhmann",
"L. W. Molenkamp"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170327143345",
"title": "CdTe-HgTe core-shell nanowire growth controlled by RHEED"
} |
B.Verkin Institute for Low Temperature Physics and Engineering, 47, Lenin Ave., 310164 Kharkov, Ukraine Email address: [email protected] http://fntr.ilt.kharkov.ua/fnt/pdf/13/13-7/f13-0771r.pdfFiz. Nizk. Temp., 13, 771 (1987)); (Sov. J. Low Temp. Phys., 13, 442(1987) The structure and physical properties of superconducting compounds Y(La)-Ba(Sr)-Cu-O are studied, the compounds being prepared by the method of cryogenic dispersion of a charge consisting of premix oxides and carbonates. Electrical conductivity and critical current density of the superconductors are measured over a wide temperature range of 10 mK to 300 K. Degradation of the superconductor critical parameters in time and structural characteristics, magnetic susceptibility, heat capacity and acoustic properties are studied, and current-voltage characteristics are determined.74.25.-Q; 74.25.Bt; 74.25.Fy; 74.25.Ld; 74.25.Sv; 74.62.Bf; 74.25.Kc; 74.45+c; 74.72.Bk; 74.72.Dn; 74.72.-hStructure and physical properties of superconducting compounds Y(La)-Ba(Sr)-Cu-O B.I. Verkin, B.B. Banduryan, A.S. Barylnik, A.G. Batrak, N.L. Bobrov, I.S. Braude, Yu.L. Gal'chinetskaya, A.L. Gaiduk, A.M. Gurevich, V.V. Demirskii, V.I. Dotsenko, V.I. Eropkin, S.V. Zherlitsyn, A.P. Isakina, I.F. Kislyak, V.A. Konovodchenko, F.F. Lavrent'ev, L.S. Litinskaya, V.A. Mikheev, V.I. Momot, V.D. Natsik, I.N. Nechiporenko, A.S. Panfilov, Yu.A. Pokhil, A.I. Prokhvatilov, E.Ya. Rudavskii, L.F. Rybal'chenko, I.V. Svechkarev, A.M. Stepanenko, M.A. Strzhemechnyi, L.I. Fedorchenko,V.D. Fil', V.V. Fisun, V.G. Khomenko, V.K. Chagovets, G.A. Sheshin, I.K. Yanson, and V.V. Sergienko December 30, 2023 ==================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================The purpose or the present article is to present a study of the structure and physical properties of the high- temperature superconductors Y(La)-Ba(Sr)-Cu-O, prepared by known technology <cit.>. What was essentially new in the technological process of obtaining superconducting ceramics was the use of cryogenic dispersion of a charge of premix oxides and carbonates by a method described in Ref.<cit.>. Particles of premix oxides and salts with initial dimension of the order of 10 μ m after dispersion had dimensions of 1 μ m or less. From the cryopowder after the passage of the solid phase reaction, mechanically durable samples of given thickness (from 0.1 to 3 mm) and variable diameter were obtained. During electrical measurements copper contacts (glued on with the help of silver paste) and platinum contacts of diameter 30-70 μ m were used for electrodes. § SURFACE MORPHOLOGY OF THE SAMPLESSample surfaces were studied by means of a scanning electron microscope (model BS-300, accelerating voltage 25 kV, regime of secondary electrons, without dusting of the conducting layer); the range of magnification was 50 to 40,000×.The characteristic elements of the surface of the ceramic appear to be particles of spherical shape and pores of irregular shape. The spherical particles are, as a rule, distributed individually, with aggregates of a few particles appearing occasionally. The diameters of the particles fluctuate within the limits 0.2-2 μ m, while the sizes of the aggregates cover the range 23-10 μ m. Under observation it is possible to divide the pores into groups by size, micropores sized 0.2-3 μ m, and macro-pores having diameters in the range 20-30 μ m. These latter are met rather rarely and are probably caused by insufficient pressure during the pressing of the samples, nonconglomeration of particles or gas emission during conglomeration. Typical photographs of sample surfaces are shown in Figs <ref>a and <ref>b. The surface structure of a sample of YBaCuO is shown in Fig.<ref>c. Results of statistical analysis of the microphotographs for La_1.8Sr_0.2CuO_4 are presented in Fig.<ref>. It can be seen that the distribution function of the micropores as well as that of the particles has a sharply defined maximum at a characteristic dimension 0.6-0.8 μ m, which testifies to a sufficiently homogeneous dispersion of the premix powder. § X-RAY STRUCTURE STUDIES OF SUPERCONDUCTORS IN THE RANGE 4.5-293 K Measurements were made with a DRON-0.5 x-ray diffractometer with K_α-irradiation from a copper anode at temperatures in the range 4.5-293 K. The samples were cooled in a special helium cryostat for x-ray studies. Qualitative photograms and diffractograms were obtained, the analysis of which has allowed us to conclude that the samples are 95-97% in the ground phase. The dimension of the region of coherent scattering in the samples under study stood in the range 10^-4-10^-5 cm. Strains were essentially absent.From indexing the x-ray patterns of the compound La_2-xSr_xCuO_4 it was established that the observed reflections emanated from crystals of tetragonal symmetry with lattice parameters a=3.761_6 Å and c=13.174_5 Å at 293 K, which are somewhat smaller than the values in the literature <cit.>. From the observed fading out of the reflections it was also established that the tetragonal lattice of this compound is volume-centered, which further advances the assumption of isostructurality of the ceramic under study with its analog La_2-xBa_xCuO_4, for which the full symmetry of the lattice has already been determined <cit.>. On the basis of this assumption, we modeled the arrangement of the atoms of two formula units in the unit cell with the help of the program "Polikristall-3" on an ES-1033 computer and calculated the intensities of the x-ray reflections, taking into account all scattering factors. From this it was established that at room temperature the compound La_2-xSr_xCuO_4 has the crystal structure of a layered perovskite of the spatial symmetry group I4/mmm. The structure of La_2-xSr_xCuO_4 is schematically represented in Fig.<ref>, and the experimental and calculated values of the angles and intensities of reflections for part of the diffractogram are given in Table <ref>. The coordinates of the atoms, the symmetry of their pointwise positions and the coefficients B of the isotropic thermal factor are given in Table <ref>.During the cooling of samples of a given composition all the way to helium temperatures, no significant changes in the x-ray pattern were detected. On this basis, it can be concluded that the established symmetry of the tetragonal phase of La_2-xSr_xCuO_4 is maintained in the temperature interval 4.5-293 K within the limits of accuracy of the implemented x-ray technique. However, at low temperatures (T < 200 K), it was noted that the (110) reflection on the diffractograms has an apparent asymmetry, which is not detected at room temperature (Fig.<ref>). Such a change in the diffraction picture can be partly related to the appearance at this angles of a new line.On behalf of this proposition, the temperature dependence of the (110) line width, shown in the insert in Fig.<ref>, also testifies. At low temperatures, the width of this line is on average 20% larger than at 293 K. However, in the temperature region of the transition to the superconducting state there is a tendency towards a decrease in the (110) line width. The additional line, not corresponding to the spatial group I4/mmm was indeed detected in the photo- x-ray patterns, taken in order to increase the resolution of the reflections in a 360 mm diameter cassette over the course of 1.5-2 hours. We focused on the (110) reflection. It was found in the study of such x-ray patterns that an additional weak line can be seen on the side of small angles near the (110) reflection of the volume-centered tetragonal lattice. Reliable separation and determination of its angular position were hampered by its low intensity and its location on the falling wing of the strong (110) line. The appearance at small angles of this line testifies most probably to change in the symmetry of the tetragonal crystals of La_2-xSr_xCuO_4 upon lowering of the temperature. Transition to a different symmetry is a result, apparently, of restructuring in the crystals of the mutual orientation of the CuO_6 octahedra or alternating distortions of their equatorial CuO_4 squares. At room temperature, stretched along their own axes, the CuO_6 octahedra are oriented within the La_2-xSr_xCuO_4 lattice parallel to the fourth order axis (Fig.<ref>). Upon lowering of the temperature, periodic deviations arise of small angle of the orientations of the octahedra from the (100) direction or rhombic distortions of the CuO_4 squares which alternate from node to node of the lattice. As a consequence of this, the natural symmetry of the octahedra varies and a doubling of the volume of the unit cell occurs as a result of increase of the lattice parameter in the base surface: a=a_0√(2) where a_0 is the parameter of the high-temperature phase. The parameter c is the same for both phases throughout. This leads to a transformation of the volume-centered tetragonal cell to base-centered orthorhombic. It should be noted that such a transition in the basic ceramic La_2CuO_4 takes place around 520 K. Transformation of the indices of the planes of the reflections during the transition to the new phase corresponds to the following rule: h' = h + k, k' = h - k, l'=l, h', k' and l' are the indices of the planes in the low-temperature phase).This study of the lattice parameters and estimation of the thermal expansion has shown that there is a substantial anisotropy of the coefficients of thermal expansion for crystals of La_2-xSr_xCuO_4, as for other layered structures. Thus, the average values of the coefficients of linear thermal expansion in the base plane and along the direction perpendicular to it in the temperature interval 4.5-293 K are equal to α̅_a= 0.5· 10^-5K^-1 and α̅_c= 1.7· 10^-5K^-1, respectively. In agreement with preliminary data, in the temperature region 25-40 K peculiarities in the behavior of the lattice parameters and anomalies of the linear and volume coefficients of thermal expansion were detected.§ ELECTRICAL RESISTIVITY OF THE SAMPLESWe studied the temperature dependence of the electrical resistivity of the superconductors La_2-xSr_xCuO_4, Y_2-xBa_xCuO_4, and YBa_2Cu_3O_7-y by the four-probe method. Degradation of electrical resistance of the samples with time (Fig.<ref>) was studied. In the course of 47 days of observation it was established that the superconductivity in a sample of La_2-xSr_xCuO_4 at liquid helium temperature disappeared (residual resistance was observed) twice (on the 11th and 17th days); however, afterwards, the sample's superconducting properties were restored. Electrical resistance measurements at superlow temperatures down to 21 mK were carried out with the goal in mind of investigating the possibility of relapsed superconductivity. The sample was placed in thermal contact with the mixing chamber of a ^3He-^4He dilution refrigerator: with the help of a special lacquer it was glued to the body of the chamber across a thin layer of dielectric. Two resistance thermometers were used to measure temperature: a high-temperature Allen-Bradley thermometer which was placed in thermal contact with the body of the mixing chamber, and a low-temperature SPIR thermometer located within the liquid of the mixing chamber. No appearance of electrical resistance was recorded upon cooling of the sample to 21 mK during which a maximum current density of 2 A/cm^2 did not lead to destruction of the superconducting state. It may be noted that upon reheating of the sample to room temperature its electrical resistance did not return to its original value; however, it stood only 2% below it. In a second cooling of the sample down to 77 K, its electrical resistance increased to its original value. Such a trend in the electrical resistance is indicative of some kind of mechanical strains in the samples due to thermocycling.§ CRITICAL SUPERCONDUCTING CURRENTThe temperature dependence of the critical current density for samples of La_1.8Sr_0.2CuO_4 is shown in Fig.<ref>. It may be noted that degradation of the critical current in the samples is not observed.Measurements of the electrical resistance and critical current of thin samples were carried out via a specially elaborated impulse scheme, which allowed us to keep the influence of thermal effects in the contacts on the superconducting parameters to a minimum. In this technique a current pulse of such a magnitude, that the potential across the sample reaches the threshold value, is passed through the sample. In Fig.<ref>b the temperature dependence of the critical current density of thin samples (with cross section ≲ 10^-2 cm^2), measured by such an impulse technique, is shown. The large value of the quantity |∂ j_c/∂ t| for such superconductors should be noted. The current-voltage characteristic of the superconductors is shown in Fig.<ref>. For less than critical currents no resistivity in the superconducting state was recorded upon passage of a low-frequency (≈ 3 Hz) variable current of high amplitude. § HEAT CAPACITY OF THE SUPERCONDUCTORS Measurement of the heat capacity of the superconductors was carried out by the method of absolute calorimetry over the temperature interval 14-300 K. A TSPN- 2V platinum resistance thermometer was used to measure temperature. For example, a sample of La_1.8Sr_0.2CuO_4 with mass greater than 2 gm was placed in a copper container-calorimeter, the mass of which together with the thermometer was around 3 gm. The calorimeter was filled with helium gas and hermetically sealed. The contribution to the heat capacity of the sample to the total heat capacity of the system sample-calorimeter was 30% at 15 K and 40% at 50 K.Measurements were made stepping by T= 0.5-0.8 K. Results are shown in Fig.<ref>. From the investigated temperature range we can separate out the region 28-47 K, ir which there is an anomaly in the heat capacity in the form of a bell-shaped curve. In the insertion in Fig.<ref> we show the difference between the experimental values of the heat capacity and a regular curve. The regular curve (solid line in Fig.<ref>) was obtained by extrapolating the dependence C_P(T) observed above and below the temperatures of the region under question. The maximum deviation from the regular curve occurs at T=38 K and stands at 10%. The anomaly ends in a short step (around 4%) at a temperature of 42-44 K.§ MAGNETIC SUSCEPTIBILITY OF THE SUPERCONDUCTORSMeasurements of the magnetic susceptibility of the superconductors La_2-xSr_xCuO_4 and YBaCuO were carried out by the Faraday method in fields of 0.05-0.85 T over the temperature interval 4.2-300 K. The investigated sample of the ceramic La_1.8Sr_0.2CuO_4 possesses weak paramagnetism at room temperature (χ=0.26· 10^-6 cm^3/g), which abruptly turns into diamagnetism at a temperature around 35 K. The depth of the diamagnetic collapse grows with decrease of the intensity of the magnetic field H (Fig.<ref>a). No noticeable indications of degradation of the superconducting characteristics of the material were detected via its magnetic properties over a period of seven days. § PECULIARITIES OF SOUND TRANSMISSION IN SUPERCONDUCTORSThe temperature dependence of the speed of a longitudinal sound wave with frequency 54.2 MHz in a sample of La_1.8Sr_0.2CuO_4 was measured in the temperature interval 10-130 K (Fig.<ref>). The value of the absolute speed of sound was found from acoustical delay measurements to be (3.5± 0.15)· 10^3 m/sec. Measurements of the variation of the speed of sound were made by comparing the phases of the last signal through the sample and the comparison signal coherent with it, transmitted to the receiver through an electromagnetic delay line. In order to distinguish the auxiliary signal from the probing signal, we used an acoustic delay line of monocrystalline germanium with a known temperature dependence of its speed of sound. Results of our measurements show a tendency towards a softening of the elasticity modulus in the temperature interval 30-50 K. This is indicative of the fact that the system La_2-xSr_xCuO_4 undergoes a structural transition or something similar to it in this temperature interval. § POINT-CONTACT SPECTROSCOPY OF THE SUPERCONDUCTING CERAMIC LA_1.8SR_0.2CUO_4 The determination of the energy gap and the characteristic frequencies of the lattice vibrations in the material under study was made with the help of the method of point-contact (PC) spectroscopy <cit.>. Clamping ceramic-copper heterocontacts served as the object of study.A typical current-voltage characteristic (IVC) and its first derivative (V_1∼ dV/dI) of one of the contacts on which gap peculiarities have been detected is displayed in Fig.<ref>a (the gap peculiarities in V_1 are marked by arrows). (A sharp difference of the Fermi parameters p_F and v_F at the electrodes forming the point-contact, and in addition the presence in the region under consideration of admixtures and structural defects gives the IVC a tunnellike character, where Δ appears in the form of a maximum on V_1(eV) <cit.>.The relative smallness of the excess current I_exc formed in the processes of Andreev reflection of quasi-particles from the N-S boundary, and also the low intensity of the gap peculiarities indicate that the PC contact region is made up predominantly of the nonsuperconducting phase. It is possible to explain the appearance of wide minima on V_1(eV) within the framework <cit.> of the cluster model of the PC (in Fig.<ref>a, near 40 meV) as a result of superposition of semiconducting and metallic types of conduction (taking into account I_exc), belonging to the N and S phases, respectively. In a similar way we can also explain the appearance of the wide maximum near 100 meV (curve 3 in Fig.<ref>a). This conclusion is confirmed by the circumstance that for a variety of contacts the location of the indicated peculiarities (in distinction with the gap peculiarities) fluctuates over wide limits.Thus, the wide minimum near 40 meV cannot be identified with the gap peculiarity; the following fact is indicative of this, namely, that the degree of influence of a magnetic field on the minimum under discussion is significantly higher than on the gap peculiarity (curve 4 in Fig.<ref>a).In the majority of contacts we have observed two gaps each, the maximum values of which were approximately 5 and 12 meV, from which we obtain for the ratio 2Δ/kT_c the values 3.3 and 8.0, respectively. In those cases in which only one gap is observed, the corresponding magnitudes had intermediate values. We note that the values of Δ and 2Δ/kT_c obtained here are close to the values obtained in a previous work <cit.>, in which a ceramic of similar composition was studied, which had a significantly greater relative volume of superconducting phase.The PC spectra (the second derivative of the CVC: V_2∼ d^2V/dI^2), reflecting the energy distributions of the characteristic quasiparticle excitations in the ceramic under study, were measured in contacts with relatively low I_exc), for which the intensity of the above-mentioned parasitic peculiarities are negligibly small.A typical PC spectrum is displayed in Fig.<ref>b. The peaks located in the energy region 15-80 meV can be related to phonon modes, as follows from the calculation of Weber <cit.>. Of special interest are two maxima which are located at significantly higher energies (close to 120 and 180 meV), extremely far from the calculated values of Weber. It may be assumed that these maxima result from surface plasmons in La_1.8Sr_0.2CuO_4, through which strong Cooper pairs may be produced in this material <cit.>. § CONCLUSIONStructural investigations, the study of electrical resistivity, magnetic susceptibility, heat capacity, acoustical properties, investigation of the point-contact spectra indicate the presence of peculiarities near the superconducting transition temperature. 1J.G. Bednorz and K.A. Müller, https://link.springer.com/article/10.1007/BF01303701Z. Phys.-Cond. Matt., 64, 189 (1986). 2 M.K. Wu, J.R. Ashbum. et al., https://doi.org/10.1103/PhysRevLett.58.908Phys. Rev. Lett., 58, 908 (1987). 3 R.J. Cava, R.B. van Dover, B. Batlogg, and E.A. Rietman, https://doi.org/10.1103/PhysRevLett.58.408Phys. Rev. Lett., 58, 408 (1987). 4 B.I. Verkin (ed.), http://www.twirpx.com/file/416877/Cryogenic Technology, Nauk. Dumka, Kiev (1985). 5D. Joigensen, H.B. Schutter, D.G. Aines, et al., https://doi.org/10.1103/PhysRevLett.58.1024Phys. Rev. Lett., 58, 1024 (1987). 6 I.K. Yanson, http://fntr.ilt.kharkov.ua/fnt/pdf/9/9-7/f09-0676r.pdfFiz. Nizk. Temp. 9, 676 (1983) [Sov. J. Low Temp. Phys. 9, 343 (1983)]. 7 G.E. Blonder, M. Tinkham, and T.M. Klapwijk. https://doi.org/10.1103/PhysRevB.25.4515Phys. Rev. B. 25, 4513 (1982). 8 O.I.Shkyarevskii, I.K. Yanson, and N.N. Gribov, Fiz. Nizk. Temp. 14, 479 (1988). 9I.K. Yanson, L.F. Rybal'chenko, V.V. Fisun, et al., http://fntr.ilt.kharkov.ua/fnt/pdf/13/13-5/f13-0557r.pdfFiz. Nizk. Temp. 13, 557 (1987) [Sov. J. Low Temp. Phys. 13, 315 (1987)]; https://arxiv.org/pdf/1701.01982.pdfarXiv:1701.01982. 10 W.Weber, Preprint from https://doi.org/10.1103/PhysRevLett.58.2154Phys. Rev. Lett, to be published (1987). 11 V. Kresin, http://www.w2agz.com/Library/HTSC%20History/Woodstock%20Items/Kresin.wmvReport at the Session of the Am. Phys. Soc., March 1987, New York Hilton, 1987. | http://arxiv.org/abs/1703.08707v1 | {
"authors": [
"B. I. Verkin",
"B. B. Banduryan",
"A. S. Barylnik",
"A. G. Batrak",
"N. L. Bobrov",
"I. S. Braude",
"Yu. L. Gal'chinetskaya",
"A. L. Gaiduk",
"A. M. Gurevich",
"V. V. Demirskii",
"V. I. Dotsenko",
"V. I. Eropkin",
"S. V. Zherlitsyn",
"A. P. Isakina",
"I. F. Kislyak",
"V. A. Konovodchenko",
"F. F. Lavrent'ev",
"L. S. Litinskaya",
"V. A. Mikheev",
"V. I. Momot",
"V. D. Natsik",
"I. N. Nechiporenko",
"A. S. Panfilov",
"Yu. A. Pokhil",
"A. I. Prokhvatilov",
"E. Ya. Rudavskii",
"L. F. Rybal'chenko",
"I. V. Svechkarev",
"A. M. Stepanenko",
"M. A. Strzhemechnyi",
"L. I. Fedorchenko",
"V. D. Fil'",
"V. V. Fisun",
"V. G. Khomenko",
"V. K. Chagovets",
"G. A. Sheshin",
"I. K. Yanson",
"V. V. Sergienko"
],
"categories": [
"cond-mat.supr-con"
],
"primary_category": "cond-mat.supr-con",
"published": "20170325162125",
"title": "Structure and physical properties of superconducting compounds Y(La)-Ba(Sr)-Cu-O"
} |
papersize=, | http://arxiv.org/abs/1703.09100v2 | {
"authors": [
"Jordi Picó-Cortés",
"Fernando Domínguez",
"Gloria Platero"
],
"categories": [
"cond-mat.mes-hall",
"cond-mat.supr-con"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170327142535",
"title": "Signatures of a 4$π$-periodic supercurrent in the voltage response of capacitively shunted topological Josephson junctions"
} |
Speech Enhancementusing a Deep Mixture of ExpertsShlomo E. Chazan, Jacob Goldberger and Sharon Gannot Senior Member, IEEE, Shlomo E. Chazan, Jacob Goldberger and Sharon Gannot are with the Faculty of Engineering, Bar-Ilan University, Ramat-Gan, 5290002, Israel (e-mail: [email protected]; [email protected]; [email protected]). Faculty of Engineering, Bar-Ilan University, Ramat-Gan, 5290002, Israel December 30, 2023 ================================================================================================================================================================================================================================================================================================================================================================================================== In this study we present a DMoE neural-network architecture for single microphone speech enhancement. By contrast to most speech enhancement algorithms that overlook the speech variability mainly caused byphoneme structure, our framework comprises a set of DNN, each one of which is an `expert' in enhancing a given speech typecorresponding to a phoneme. A gatingDNN determines which expert is assignedto a given speech segment.A SPP is then obtained as a weighted average of the expert SPP decisions, with the weights determined by the gating DNN.A soft spectral attenuation, based on the SPP, is then applied to enhance the noisy speech signal.The experts and the gating components of theDMoE network aretrained jointly.As part of the training,speech clustering into different subsets is performed in an unsupervised manner. Therefore, unlike previous methods,a phoneme-labeled dataset is not required for the training procedure.A series of experiments with different noise types verified the applicability of the new algorithm to the task of speech enhancement. The proposed scheme outperforms other schemes that either do not considerphoneme structure or use a simpler training methodology. speech enhancement, deep neural network,mixture of experts § INTRODUCTION There are many approaches to solve the problem of speech enhancement using a single channel <cit.>. Although microphone array algorithms are widely used, there are still applications in which only a single microphone is available. However, the solutions suggested for this task are not always sufficient. Classical algorithms such as the OMLSA estimator and theIMCRA noise estimator approach to robust speech enhancement were developed to even dealwith nonstationary noise environments <cit.>. Nevertheless, when input with rapid changes in noise statistics is processed, the estimator tends to yield musical noise artifacts at the output of the enhancement algorithm. Over the past few years fully-connected DNN-based algorithms have been developed to enhance noisy speech. DAE weretrained to find a non-linear filter between noisy input to clean speech <cit.>. A set of noisy/clean features constituted the database for the training phase. This approach often suffers from speech distortion when an unfamiliar noise is examined. In response, an IBM was proposed,in which the time-frequency bins wherespeech is active are marked`1', and the other bins are set to`0'. Here, the IBM is estimated from the noisy input.The noisy signal is then multiplied by the IBM to reduce the noisy bins. This hard decision approach is however not satisfactory for speech enhancement. This led to the development of an IRM, which applied soft enhancement and had better results. TheseDNN approaches have several drawbacks. First, their fully-connected architecture has to deal with the massive variability ofspeech in the input. Second, these approaches need to be trained on huge databases with varying noises in order to be able to minimize the unfamiliar noise <cit.>. Finally, even if trained on a large database, there can still be a mismatch between the test phase and the training phase, sincenoise in real-life scenarios is always novel. To overcome these hurdles, a new phoneme-based architecture was introduced with an ASR system <cit.>. In this architecture, a set of DNNs were trained separately, one for each phoneme, with its own database to find the IRM. Given a new noisy input, the ASR system outputs the index of the phoneme associated with the current input, and that phoneme DNN is activated to find the IRM. This approach improved performance in terms of noise reduction and more accurate IRM estimation. However,when the ASR system is incorrect, the wrong DNN is activated. Additionally, the continuity of the speech is disrupted by mistakes in the ASR system. Finally, the ASR was not part of the training phase. Chazan et al.<cit.> presented a similar architecture, but instead of the ASR system, another DNN was used as a phoneme-classifier. This approach produced better performance than the fully-connected approach and the phoneme-based architecture with ASR without joint training. Yet, this algorithm still had major drawbacks. First, a phoneme-labeled database is essential, a requirement which is not always tractable. Second, the phoneme-classifier was not part of the joint training, which might prevent the training from producing even better results. Finally, the MSE loss function,used to train the network is not the natural choice for training binary classification tasks such as finding the SPP. The approach presented in this study can be viewed as an extension of the method in Chazan et el. <cit.>,and our goal is to present an enhancement approach that overcomesthe problems describe above. In the next section we described the previous work<cit.>inmore details. The MoE approach, which was introduced more than twenty years ago <cit.>, is based on the principle of divide and conquer. MoE combines the decisions of several `experts', each of which specializes in a different part of the input space. The modelhas three main components: several experts that are either regression functions or classifiers, a gate that makes soft partitions of the input space and defines those regions where the individual expert opinions are trustworthy, anda weighted sum of experts, where the weights are the input-dependent gates. The MoE model allows the individual experts to specialize on smaller parts of a larger problem, and it uses soft partitions of the data implemented by the gate. Previous work on MoE hasfocused on different facets including using different types of expert models such as SVMs <cit.> andGaussian processes <cit.>.A comprehensivesurvey ofMoE theory and applications can be found in <cit.>.In spite of the huge success of deep learning there are not many studies that have explicitly utilized and analyzed MoE as anarchitecture component of aneural network. Eigen et al. <cit.> suggested theextension of MoE to a deep model by stacking two layers of a mixture of experts (where each expert is a feed forward network) followed by a softmax layer.In this work, we presentDMoE modelingfor speech enhancement. The noisy speech signal contains several different regimes which have different relationships between the input and the output based for instance, on phoneme identity or the coarser distinction between voiced and unvoiced phonemes. The proposed enhancement approach is based on aDMoE network where the experts are DNNs, which are combinedby agating DNN. Each expert is responsible for enhancement in a single speech regime and thegating network finds the suitable regime in each time frame. Each expert estimates an SPP andthelocal SPP decisions are averaged, based on the gating function, into afinal SPP result. In our approach there is no need forphoneme-labeled data, since the gating DNN splitsthe input space into sub-areas in an unsupervised manner. The rest of the paper is organized as follows. In Sec <ref> the problem formulation and previous work are presented. Sec. <ref> introduces the new DMoE model, and Sec. <ref> describes the application of DMoE modeling to speech enhancement. The comprehensive experimental results using speech databases with various noise types are presented in Section <ref>. Sec <ref> discuses the attributes of thealgorithm.Finally, some conclusions are drawn and the paper is summarized in Section <ref>. § PROBLEM FORMULATION AND PRIOR ART §.§ Problem formulation Let s(t) denote a sample of speech signal at time t. Let x(t)=s(t)+n(t) denote the observed noisy signal where additive noise n(t) was added to the clean speech. The STFTwith a frame of length L of x(t) is denoted by X_n(k), where n is the frame index and k=0,1,,L-1 denotes the frequency band index. Similarly, S_n(k) and N_n(k) denote the STFT of the speech and the noise only, respectively. Define the log-spectrum of the noisy signal at a single time frame by , such that the k-th component isx_k=|X(k)| where k=0,…,L/2. Note, that the other frequenciescan be obtained by the symmetry of the DFT.Similarly,anddenote the log-spectrum of the corresponding speech and the noise only signals, respectively. Nádas et al. <cit.>suggesteda maximization approximation modeling, in whicheach time-frequency bin of the noisy signal log-spectrum, x_k, is dominated by the maximum between the log-spectrums of the speech, s_k and the noise, n_k, x_k=max(s_k,n_k),k=0,…,L/2. This approximation was utilized and yielded high performance in speech recognition <cit.>, speech enhancement <cit.> and speech separation <cit.>. Following this approximation, a binary mask can be built for each k-th frequency, b_k= 1s_k>n_k 0s_k≤ n_k. We denote the binary-mask vector of all the frequencies at a given time frame as b=[ b_0,…,b_L/2]. In the enhancement task only the noisy signalis observed, and the goal is to find an estimationof the clean speech . If the true binary mask b had been available, a soft attenuation algorithm could have been used, = b·+(1-b) · (-β), where β is the attenuation level. However, the binary-mask b is not available for the enhancement procedure. Instead, we need to compute theSPP vector=[ ρ_0,…,ρ_L/2],from the noisy signal: ρ_k=p(b_k=1|)=p(s_k>n_k|). The goal of this work is to build a DNN with a task-specific architecturetailored to the problem offinding the SPP .Then together with (<ref>) the speech is enhanced: = ·+(1-) · (-β). Training is based on a labeled training dataset that consists of frames (in the log-spectrum) of noisy signals along with the associated true binary masks. In principle, we can traina DNN with a standard architecturebased on apipeline of fully-connected layers.Several studies <cit.> have shown that this approach is beneficial forspeech enhancement in terms of noise reduction. However, there are some drawbacks to using a fully-connected network architecture. The input to the DNN is highly non-homogeneous. Speech is composed of different phonemes andits structure varies over time. Thus, it is a difficult task totraina single DNN to deal with this variability. It is alsoa very difficult task for a single network to preserve the harmony structureof the speech. A single network also hasto address different noise types. Since there are many types of noise there can bea mismatch between the train and test conditions whenan unfamiliar noise is introduced at test time. This leads to a decline inperformance. One possible strategy to overcome this problem is to use a massive database that consists of many hours of recorded data combined with many noise types <cit.>. This approach, however, leads to a long training phase <cit.>.Therefore, a standard fully-connectedDNNmight not be the best solution for the speech enhancement task. §.§ Prior art based on phoneme-labeled data To overcome the problems described above,the speech frames can be grouped such that the variability within each group is not high. For example, we can split the noisy training data according to the phoneme labels. Then, it might be beneficial to train an SPP estimation network separately on each homogeneous subset of the input. In a previous work<cit.>, a phoneme-based architecture was suggested to find the SPP. In this procedure, separate DNN is allocatedfor each phoneme to find the SPP given the phoneme class information. Additionally, a phoneme-classifierproduce the phonemes' posterior distribution. Finlay, theSPPfound by a weighted average of all the phoneme-based SPP estimations. The network istrained jointly to minimize the loss function which was chosen to be the MSE between the final-SPP,and the binary-mask, b.This training takes place in two steps. First, a pre-training stage done uses phoneme-labeled data. In this step each component of the architecture is separately trained to obtain a parameter initialization for the network.In the second stage all phoneme-based sub-networks arejointly trained.This approach was shown to yield better results compared to the widely used fully connected architecture. Nevertheless, this approach still has some drawbacks. First, a phoneme-labeled database is essential forphoneme-based training. This information, however, is not always available. Second, it is not obvious that dividing the speech signals according to the uttered phonemeis the best strategy. Sometimes it is more worthwhile tolet the network automatically split the data in a way that best suits enhancement. § A DEEP MIXTURE OF EXPERTS In this sectionwe present theDeep Mixture of Experts (DMoE) framework, and in the next section we apply it tospeech enhancement. The Mixture-of-Experts (MoE) model introduced by Jacobs et al. <cit.>,provides important paradigms for learning a classifier from data. The objective of this framework is to describe the behavior of certain phenomena, under the assumption that there are separate processes involved in the generation of the data under analysis. The use of MoE makes it possible to combine many simple models to generate a more powerful one.The main idea is based on the`divide-and-conquer' principle that is often used to attack a complex problem by dividing it into simpler problems whose solutions can be combined to yield a solution to the complex problem. The MoE modelis comprised of a set of classifiers that perform the role of experts, and a set of mixing weights determined by a gating functionthat selects the appropriate expert. The experts are responsible for modeling the generation of outputs, given a certain condition on the input space, and are combined by a set of local mixing weights determined by the gating function, which depends on the input.We can view the MoE model as a two step process that produces a decision y given an input feature set x. We first use the gating function to select an expert and thenapply the expert to determine the output label. The index of the selected expert can be viewed as an intermediatehidden random variable denoted by z.Formally, the MoE conditional distributioncan be written as follows:p(y|x;θ) = ∑_i=1^m p(z=i|x;θ_g) p(y|z=i,x;θ_i)such that x is the feature vector, y is the classification result, z is a hidden random variable that selects the expert that is applied and m is the number of experts. The model parameter-set θ is composed of the parameter-set of the gating functionθ_g andparameter-sets θ_1,…,θ_mforthe m experts. In the speech enhancement task x is the log-spectrum vector of the noisy speech, y is thebinary mask information and z is a hidden speech state; e.g., the phoneme identity or a voiced/unvioiced indication. (Note that in the other sections of this paper we denote the binary mask information as b.) In that case the gating function in the enhancement procedure estimates the phoneme given the noisy speech and each expert is associated with a phoneme and is responsible for enhancement of a noisy phoneme utterance. The DMoE is illustrated in Figure <ref>. We next address the problem of learning the MoE parameters (i.e. the parameters of the experts and the gating function) given a training dataset (x_1,y_1),…,(x_n,y_n). The likelihood function of the MoE model parameters is: L(θ) = L(θ_g, θ_1,...,θ_m) = ∑_tlog p ( y_t|x_t; θ).Since the selected expert used to produce y_t from the feature set x_t (i.e. the value of the r.v. z_t) is hidden, it is natural to apply the EM algorithm to find the maximum-likelihood parameters <cit.>. The EM auxiliary function is:Q(θ,θ^0) = E_p(z|x,y;θ^0)( log p (y,z|x;θ))such that θ^0 is the current parameter estimation. Inthe E-step we apply Bayes' rule to estimate the value of the selected expert based on the current parameter estimation:w_ti= p(z_t=i|x_t,y_t;θ^0) =p(y_t|x_t,z_t=i)p(z_t=i|x_t)/p(y_t|x_t). The M-step decouples the parameter estimation of the different components of the MoE model. We can optimize each of the experts and the gating function separately. The updated parameters of the gating function are obtained bymaximizing the weighted likelihood function: L(θ_g) = ∑_t ∑_i w_tilogp(z_t=i|x_t; θ_g) and the updated parameters of the i-th expert are obtained bymaximizing the function:L(θ_i) = ∑_t w_tilogp(y_t|x_t, z_t=i; θ_i)The general EMtheory guarantees a monotoneconvergence of the model-parameter estimations (to a local maximum). In the case where both the experts and the gating functions are implemented by DNNs, we denote this modal Deep Mixture of Experts (DMoE). The compound DNN model is expressed as follows:p_(y|x;θ) = ∑_i p_(z=i|x;θ_g) p_(y|z=i,x;θ_i).In this model θ_g is the parameter-set of the DNN that implements the gating function andθ_i is the parameter-set of the DNN that implements the i-th expert.We can still apply the EM algorithm described above to train the a DMoE. In this case in the M-step we need to train the experts and gating neural networks using the cross-entropy cost function defined by Eq. (<ref>) and (<ref>). We can thus iterate between EM steps and DNN training.There are, however, several drawbacks to using the EM algorithm to train a DMoE. Ineach M-step iteration we need to train a new DNN. There is no closed-form solution for this non-concavemaximization task. The DNN learning is performed by a stochastic gradient ascent and there is no guarantee for monotone improvement of the likelihood score.The EM algorithm is a greedy optimization procedure that is notorious for getting stuck in local optima. In most EM applications there is a closed-form solution for the optimization performed at the M-step. Here, since we need to retrain the experts and gatingDNNs at each M-step(<ref>) (<ref>), even a monotone improvement of the likelihoodis not guaranteed. The main problem of iterating between EM-steps and neural network training, however, is that it does not scale well.The framework requires training a neural network in each iteration of the EM algorithm. For real-world, large-scale networks, even a single training iteration is a non-trivial challenge. In this study we replace the EM algorithm, whichlearns the expert and gating networks at each step separately, by a neural-network training procedure thatsimultaneouslytrains all the sub-networks by directly maximizing the likelihood functionL(θ)= ∑_t=1^n log(∑_i=1^m p_ NN(z_t=i|x_t;θ_g) · p_ NN(y_t|z_t=i,x_t;θ_i))In this architecture, the expertsand the gating networks arecomponents of a singlenetwork and are simultaneously trainedwith the same objective function. In order to update the network parameters we apply a back-propagation algorithm. It can be easily verified that the back propagation equation for the parameter setof the i-th expert is:∂ L/∂_i = ∑_t w_ti·∂/∂_ilog p_ NN(y_t|z_t=i,x_t;θ_i)such that w_ti is the posterior distribution of the gating random variable:w_ti=p_ NN(z_t=i|x_t,y_t;θ) = p_ NN(y_t|x_t,z_t=i;θ_i)p_ NN(z_t=i|x_t;θ_g)/p_ NN(y_t|x_t;θ).Note that this definition coincides with the E-step of the EM algorithm defined in Eq. (<ref>). In a similar way,theback-propagation equation for the parameter set of the gating DNN is:∂ L/∂_g = ∑_t∑_i w_ti·∂/∂_glogp_ NN(z_t=i|x_t;θ_g).Note, that the back-propagation partial derivatives(<ref>) and (<ref>) are exactly thederivatives of the functions (<ref>)and(<ref>)that are optimized by the M-step of the EMalgorithm. By training all the components of theDMoEsimultaneously we thus replace the two steps of the EM iterations by the single step of a gradient ascent optimization. § DEEP MIXTURE EXPERTS FOR SPEECH ENHANCEMENT In this section we apply the DMoE principle to a speech enhancement task and describe the network specifics and training procedure. §.§ Network descriptionThe goal in a speech enhancement task is to find an accurate SPP,from a given noisy signalusing the DMoE model. All the m experts in the proposed algorithm are implemented byDNN with the same structure. The input to each DNN is the noisy log-spectrum frame together with context frames. The network consists of 3 fully connected hidden layers with500 ReLU neurons each. The output layer that provides the SPP binary decisions is composed ofL/2 sigmoid neurons, one for each frequency band. The SPP decision of the i-th expert on the k-th frequency bin is: ρ_i,k=p(b_k=1|;θ_i).Let h_i() be the value of the final hidden layer of the i-th expert. The SPP prediction can be written as ρ_i,k = σ ( w_ikh_i()+a_ik) such that σ is the sigmoid transfer function andw_ik,a_ik are the parameters of the affine input function to the sigmoid neuron. The fact that all the frequency bands are simultaneouslyestimated from h_i() enables the network to reconstruct the harmonic structure. Although the standardMoE approach uses the same input features for both the experts and the gating networks, here the log-spectrum of the noisy signal, , is utilized as the input for the experts alone, and the gating DNN is fed with the corresponding MFCC features denoted by .MFCC, whichis based on frequency bands,is a more compact representation than a linearly spaced log-spectrumandthis frequency warping is knownfor its better representation of sound classes <cit.>. Wefound that using the MFCC representation for the gating DNN both slightly improves performance and significantly reduces the input size. The architecture of the gating DNN is also composed of 3 fully connected hidden layers with 500 ReLU neurons each.The output layer here is asoftmax function that produces the gating distribution on the m experts.The gating procedure therefore is: p_i=p(z=i|;θ_g).The final SPP is obtained by a weighted averageof the deepexperts' decisions: =∑_i=1^mp_i·_i. Given the SPP vector , the enhanced signal is finally obtained using Eq. (<ref>). The suggested DMoE algorithm for speech enhancement is presented in Algorithm <ref>. The network was implemented in Keras <cit.> on top of Theano backend <cit.> with ADAM optimizer <cit.>. To overcome the mismatch between the training and the test conditions, each utterance was normalized prior to the training of the network, such that the sample-mean and sample-variance of the utterance werezero and one, respectively <cit.>. In order to circumvent over-fitting of the DNNs to the training database, we first applied the CMVN procedure to the input, prior to the training and test phases <cit.>. Additionally, the dropout method <cit.> wasutilized on each layer. Finally, the batch-normalization method was applied to train acceleration on each layer <cit.>. §.§ Training the DMoE for speech enhancementTo train the network we need to collect adataset of noisy speech and the corresponding binaryvectors. Unlike <cit.>, hereneither a phoneme-labeled database nor pre-training are needed. Clean speech signals were contaminated with a single noise type in a pre-defined SNR. The log-spectrum of the speech and the noise are known and a binary mask based on the maximization approximation (<ref>) was then computed.Additionally, the corresponding MFCC features were calculated. Finally, in order toenhance the current frame of the noisy signal, context frame information is known to provide better performance; therefore, each input contained four context frames from the past and four from the future.Assume the training set is(_1,_1),...(_N,_N)such that _t is a log-spectrum ofnoisy speech and _t is the corresponding binary mask vector, and N is the length of the database. The training procedure aims to optimize the following log-likelihood function:. L() = ∑_t=1^N log p ( _t |_t ; θ)such that p ( _t |_t ; θ) = ∑_i=1^mp(z_t=i|_t;θ_g) ∏_kp( b_t(k) |_t, z_t=i ; θ_i)with _t,the MFCC coefficients at time t. The back-propagation partial derivative of the parameters of the i-th expert is: ∂ L/∂_i = ∑_t w_ti·∂/∂_i∑_k log p(b_t(k)|z_t=i,x_t;θ_i)where w_ti (see Eq. (<ref>)) is the posterior probability that the i-thexpert was used to produce the binary mask _t. The partial derivative of the gating parameter appears in (<ref>). Another decision we need to make is choosing the number of experts. We found that moving from a single expert to two experts yielded a significant improvement, but adding more experts had little effect. Hence, utilizingOccam's razor principle, we chose the simpler model and set m=2. In Sec <ref> weshowperformance empirically as a function of the number of experts. We also show that when we train DMoE with two experts,the gating network tends to direct voiced frames to one expertand unvoiced frames to the second expert. The training procedure issummarized in Algorithm <ref>. § EXPERIMENTAL STUDY In this section we present a comparative experimental study. We first describe the experiment setup in Sec. <ref>. Objective quality measure results are then presented in Sec. <ref>. Finally, the algorithm is tested on an untrained database in Sec. <ref>. §.§ Experiment setup To test the proposed DMoE algorithm wecontaminated the speech signals with several types of noise from the NOISEX-92 database <cit.>, namely Speech-like, Babble, Car, Room, AWGN and Factory. The noise was added to the clean signal drawn from the test set of the TIMIT database (24-speaker core test set), with5 levels of SNR at -5 dB, 0 dB, 5 dB, 10 dB and 15 dB chosento represent various real-life scenarios.The algorithm was also tested on theWSJ database <cit.> which was collected with a different recording setup. Wecompared the proposed algorithm to theOMLSA algorithm <cit.> with the IMCRA noise estimator <cit.> which is a state-of-the-art algorithm for single microphone speech enhancement. The default parameters of the OMLSA were set according to <cit.>. Additionally, we compared the proposed DMoE algorithm to two other DNN-based algorithms. The first DNNhas a fully-connected architecture andcan be viewed as a single-expert network. We denote this network the DSE. The second DNN is a supervised phoneme-based DMoE architecture<cit.>. The network has 39experts where each expert is explicitlyassociated with a specific phoneme andtraining usesthe phoneme labeling available in the TIMIT dataset. We denoted this phoneme-based supervised networkby S-DMoE. Each expert component in the DSE and S-DMoE networks has the same network architecture as the expert components of the proposedDMoE model. §.§.§ Training Procedure In order to carry out a fair comparison, all the DNN-based algorithms were trained with the same database. We used the TIMIT database <cit.> train set for the training phase and the test set for the testing. Note, that the train and test sets of TIMIT do not overlap. Clean utterances were contaminated with Speech-like noise with an SNR =10 dB. Note, that unlike most DNN-based algorithms <cit.>, we trained the DMoE network only on a single noise type with a single pre-defined SNR value. The speech diversity modeling provided by the expert-set was found to be rich enough to handle noise types that were not presented in the training phase. In order to evaluate the performance of the proposed speech enhancement algorithm, several objective and subjective measures were used. The standard PESQ measure, which is known to have a high correlation with subjective score <cit.>, was used. Additionally,the composite measure suggested by Hu and Loizou <cit.>, was implemented. The composite measureweightsthe LLR, thePESQand the WSS <cit.> to predict the rating of the Cbak, the Csig and the Covl performance. The rating was based on the 1-5 MOS scale, and the clean speech signal had aMOS value of 4.5. Finally, wealso carried out informal listening tests with approximately thirty listeners.[Audio samples comparing the proposed DMoE algorithm with the OMLSA, the DSEand the S-DMoE can be found in <www.eng.biu.ac.il/gannot/speech-enhancement/mixture-of-deep-experts-speech-enhancement/>.] Table <ref> summarizes the experimental setup. §.§ Objective quality measure results We first evaluated the objective results of the proposed DMoE algorithm and compared it with the results obtained by thestate-of-the-art OMLSA algorithm, the DSE and the S-DMoE phoneme-based algorithm <cit.>. The test set was thecore test-set of the TIMIT database. Fig. <ref> depicts the PESQ results for all algorithms for the Speech-like, Room, Factory and Babble noise types as a function of the input SNR. In Fig. <ref> we show the Covl results for the same noises. It is evident that the proposed DMoE algorithm outperformed the competing algorithms on the two objective measures. In order to gain further insights into thecapabilities of the proposed algorithm wecompared the enhancement performance of the DMoE algorithm with the challenging Factory noise environment. We set the SNR in this experiment to 5 dB. Fig. <ref> depicts theSPP comparison between the DSE, the S-DMoE and the proposed DMoE algorithm performance. Clearly the DSE is very noisy, and the DMoE is smoother than the S-DMoE in both the time and frequency domains. §.§ Performance with a different database Here we have trained the DMoE using the TIMIT database. In order to show that the proposed algorithm is immune from the overfitting phenomena we tested the capabilities of the proposed DMoE algorithm when applied to speech signals from other databases. We applied the algorithm to 30 clean signals drawn from the WSJ database <cit.>. The signals were contaminated by Speech, Room and Factory noises, drawn from the NOISEX-92 database,with several SNR levels. Note, that the algorithm was neither trained on that database nor trained with these noise types. Fig. <ref> depicts the PESQ measure of the DMoE algorithm in comparison to the otheralgorithms. It is evident that the performance of proposed algorithm was maintained even for sentences drawn from a database other than the training database. The results for other noise types,not shown here due to space constraints, were comparable. § DISCUSSION In this section, we first discuss the role of the number of experts in Sec <ref>. The experts' performance is tested in Sec <ref>, and finally, the gating is analyzed in Sec <ref>. §.§ Setting the number of experts Theproposed architecture is based on a mixture of m experts and the gating network directs a given input to one of these m experts. In mostMoE studies, finding the number of experts was done by exhaustive search <cit.>. In our case we divide the (log-spectrum) feature space intom simpler subspaces. On one hand, setting a large value for mdivides the problem into many experts thus giving each expert an easier job of enhancing a distinct speech type. On the other hand, when m is large the model complexity is higher, which makes the training task more difficult. Additionally, when the model size is large, the computational demands can make it difficult to usein real time applications. In order to determine the best value of m based on enhancement performance, we conducted an experiment using DMoE models with1, 2, 10, 20, 30 and 39 experts. We use the notationDMoE-m for a DMoE based on m experts. The 6 architectures were trained separately on the train part of the TIMIT database. Wethen tested the obtained networks using the same procedure described in Sec <ref>. Fig. <ref> shows the Covl results, and Fig. <ref> shows the PESQ results. It is clear that moving from a single expert to two experts significantly improves the results. However, as can be seen fromthe results, furtherincreasing the number of experts did not improve the performance much. §.§ The experts' expertise Toillustrate the role of differentexperts,we added non-stationary Factory noise to clean speech from the TIMIT database with SNR=5 dB. The gating DNN was fed with the MFCC features of the noisy input and produced the probability of eachexpert. Now, instead of introducing the experts to the noisy log-spectrum vector , a vector of all-ones was used asinput. This non-informative vectorcan be viewed as a noise-only signal. This vector propagated through the experts and in the end, the SPPwas estimated. Fig. <ref>depicts the output of the enhancing procedure. Figs. <ref> and <ref> show the clean and noisy speech. Fig. <ref> shows the estimated SPP of the fully-connected (single expert) architecture. It is clear that since the input is the same and the MFCC gating-decision data are irrelevant here, no information is preserved and the results therefore are meaningless. Fig. <ref> shows the estimated SPP of the DMoE-2 architecture. Here, the extra information of the gating helps the experts to follow the structure of the input. Note, that only two different structures are shown here, based onthe two experts. Simple inspection showsthat one SPP structure appears in the voiced frames and the other in the unvoiced frames. Finally, in Fig. <ref> the estimated SPP from DMoE-30 is depicted. It is easy to see that this architecture tracks the clean speech more accurately. Now that more experts are present,more speech structures are estimated. This experiment suggests that each expert is responsible for a specific structure of the speech. Consequentially, the experts preserve the speech structure even if introduced to an unfamiliar noise. This leads tomore robust behavior compared to other DNN-based algorithms. §.§ The gating voice/unvoiced decision Fig.<ref> depicts the case of two experts, oneresponsible forenhancingvoiced frames and the other for the unvoiced frames. The gating network therefore needs to classify the frame asvoiced or unvoiced in order to direct it to the appropriate expert. We next show in a more systematic and quantitative manner that this is indeed what the gate network does. Note, that in the training phase no phoneme labels are provided to the gating DNN. The TIMIT database isphoneme labeled. Hence we can collectstatistics on gate decisions for each phoneme separately. Fig. <ref> depicts the gate decision statistic as a function of the phoneme label, andFig. <ref> presents the average structure of the phonemes in the log-spectrum domain. It appears from the gating decisionsthat the gating DNN tends to direct unvoiced phonemes to one expert and the voiced phonemes to the second. This partition, which is obtained in an unsupervised manner, makes sense since the voice/unvoiced structures are dramatically different:the voiced phonemes are characterized by energy in the low frequencies, whereas the unvoiced phonemes are characterized by energy in the high frequencies. § CONCLUSION This article introduced a DMoE model for speech enhancement. This approach divides thechallenging task of speech enhancement into simpler tasks where each DNN expert is responsible for asimpler one. The gating DNN directs the input features to the correct expert. We showed empirically that in the case of two experts, the gating decision is correlated withthe voice/unvoiced status of the input frame. In a fully connected DNN, the input to a single DNN has to deal with bothvoiced and unvoiced frames, which leads toperformance degradation. On the other hand, inthe DMoE model, the gating splits the problem into simpler problems and each expert preserves the pattern of the the spectral structure of frames that are directed to this expert. This approach makes is possible toovercome the well knownproblem of DNN-based algorithms; namely, themismatch between training phase and test phase. Additionally, the proposed DMoE architecture enables training with a small database of noises. The experiments showed that the proposed algorithm outperforms state of the art algorithms as well as DNN-based approaches on the basis of objective and subjective measurements. IEEEtran | http://arxiv.org/abs/1703.09302v1 | {
"authors": [
"Shlomo E. Chazan",
"Jacob Goldberger",
"Sharon Gannot"
],
"categories": [
"cs.SD"
],
"primary_category": "cs.SD",
"published": "20170327203733",
"title": "Speech Enhancement using a Deep Mixture of Experts"
} |
[email protected] Departamento de Física dos Sólidos, Universidade Federal do Rio de Janeiro, 21941-972 Rio de Janeiro, [email protected] Zürich, Computational Physics for Engineering Materials, Institute for Building Materials Schafmattstrasse 6, HIF, CH-8093 Zürich, Switzerland Departamento de Física dos Sólidos, Universidade Federal do Rio de Janeiro, 21941-972 Rio de Janeiro, BrazilWe propose a general method to construct symmetric tensor polynomialsin the D-dimensional Euclidean space which are orthonormal under a general weight. The D-dimensional Hermite polynomials are a particular case of the presentones for the case of a gaussian weight. Hence we obtaingeneralizations of the Legendre and of the Chebyshev polynomials in D dimensions that reduce to the respective well-known orthonormal polynomials in D=1 dimensions. We also obtain new D-dimensional polynomials orthonormal under other weights, such as the Fermi-Dirac, Bose-Einstein, Graphene equilibrium distribution functions and the Yukawa potential. We calculate the series expansion of an arbitrary function in terms of the new polynomials up to the fourth order and defineorthonormal multipoles. The explicit orthonormalization of the polynomials up to the fifth order (N from 0 to 4) reveals an increasing number of orthonormalization equations that matches exactly the number of polynomial coefficients indication the correctness of the present procedure. 74.20.De, 74.25.Dw,74.25.HaChebyshev, Legendre, Hermite and other orthonormal polynomials in D-dimensionsRodrigo C. V. Coelho December 30, 2023 ===============================================================================§ INTRODUCTIONThe theory of orthonormal polynomials is still an unfoldingbranch of Mathematics and Physics <cit.>. It started in the nineteen century and provided the key ingredients for the following centurydevelopment of Quantum Mechanics. Interestingly it was Charles Hermite who firstly introduced tensorial properties to the D-dimensional orthonormal polynomials. For the case of a gaussian weight he obtained symmetric tensorial polynomials and showed them to reduce to the well-known previously found one-dimensional Hermite polynomials by taking the limit D → 1. Since then the D-dimensional Hermite polynomials have been studied under several aspects, such as the obtainment of recurrence formulas <cit.>. Nevertheless the tensorial orthonormalpolynomials have not been systematically studied for weights other than the gaussian one so far,apart from a few attempts,such as the Laguerre weight <cit.> in D=2 with the intent to apply in quantum optics <cit.>. In this paper we propose a general method to construct D-dimensional tensorial polynomials orthonormal under an arbitrary weight. We apply this methodfor the first five polynomials (N=0 to 4) and determine their coefficients. The D-dimensional Hermite polynomials are retrieved as the particular case of the gaussian weight. We obtain D-dimensional generalizations of the Legendre and Chebyshev of first and second kind polynomials. By taking the limit D → 1 the one-dimensional Legendre and Chebyshev of first and second kind polynomials are retrieved. We choose other weights to construct new D-dimensional tensorial polynomials, such as the Bose-Einstein, the Fermi-Dirac and also the Graphene equilibrium distribution functions. Their interest is in the search for solutions of the Boltzmann equation describing semi-classical fluids <cit.>. In such cases the corresponding D-dimensional Euclidean space is that of the microscopic velocity. As a last example we construct the D-dimensional polynomials for the Yukawa weight, which are useful in position space to derive the concept of anorthonormal multipole series expansion. The D-dimensional Hermite polynomials have many interesting applications in Physics ranging from Quantum Optics <cit.> to Statistical Mechanics. In the latter case they offer fundamental aid to solve the Boltzmann equation <cit.> for classical particles. Indeed it was H. Grad who first used the D-dimensional Hermite polynomialsto describe the microscopic velocity space of the Boltzmann equation <cit.>. The Boltzmann equation aimsa statistical description of an ensemble of particles and so describes the motion of a set of particles at a scale between the microscopic and the macroscopic levels. While the microscopic level has a deterministic description of motion, since Newton's law is applied to the individual particles, at the macroscopic level the only laws available are those of conservation of mass and momentum for many particles. For fluids and gases the macroscopic level corresponds to the continuity and to the Navier-Stokes equation, respectively, and it can be shown that both follow from the Boltzmann equation <cit.>. A few decades ago the study of the Boltzmann equation experimented a revival because of a new method developed to solve it on a lattice version of position space. Because of its simplicity this method revolutionized the way to numerically tackle problems in fluid dynamics. It became known as the Lattice Boltzmann Method <cit.> and uses the D-dimensional Hermite polynomials to span the distribution function, which essentially gives the number of particles in a point in phase space. The Gauss-Hermite quadrature is also used in this method to perform integration in the D-dimensional space.Recently it was found that to render the Lattice Boltzmann applicable to semi-classical fluids the weight that render the D-dimensional polynomials must be the equilibrium distribution function itself. Therefore the gaussian weight is not appropriate because it is associated to the Maxwell-Boltzmann equilibrium distribution function whereas for semi-classical fluids the particles obey the Bose-Einstein or the Fermi-Dirac equilibrium distribution functions<cit.>. Therefore D-dimensional polynomials orthonormal under these weights are need in such cases and so, one must go beyond the D-dimensional Hermite polynomials.The remarkably rich tensorial structure of the D-dimensional space is the key element that allows for the existence of the present symmetric tensor polynomials orthonormal under a general weight. This rich tensorial structure was first observed by Harold Grad <cit.>, but never developed to obtain D-dimensional orthonormal polynomials for a general weight. Here we develop this proposal and obtain the D-dimensional orthonormal polynomials. Notice that this rich tensorial structure is not present for D=1 since the Kronecker's delta function is trivial and equal to one. However in higher dimensionsmany tensors can be built as products and sums of the Kronecker's delta function. Harold Grad was the first to notice this wealth of tensors built from Kronecker's delta function <cit.>.This paper is organized as follows. Insection <ref> we propose the general form of the D-dimensional tensorial polynomials and explicitly write the first five ones. In section <ref> the rich tensorial properties ofD-dimensional space. The explicit construction of the first five (N=0 to 4) orthonormal polynomials is carried in section <ref>, which means that all their coefficients are obtained as functions of some integrals over the weight (I_N). Next we apply this general theory to specific weights in section <ref>. The known D-dimensional Hermite polynomials are derived from the present ones and also newD-dimensional generalizations of the Legendre and Chebyshev polynomials of the first and second kinds are proposed here. The projection of such polynomials to D=1 dimensions does give the well-known Hermite in subsection <ref>, Legendre (subsection <ref>) and Chebyshev (subsections <ref> and <ref>) which are projected to D=1 dimension (subsection <ref>). Next we consider D-dimensional polynomials orthonormal under new weights, such as Fermi-Dirac (subsection <ref>), Bose-Einstein (subsection <ref>), graphene (subsection <ref>) and Yukawa potential (subsection <ref>). Finally weexpand a general function in terms of the D-dimensional polynomials in section <ref>, which leads to the proposal of orthonormal multipoles. We reach conclusions in section <ref>. Some useful tensorial identities are discussed in appendix <ref>.§ GENERAL D-DIMENSIONAL POLYNOMIALS Consider the D-dimensional Euclidean space endowed with a weight function ω(ξ ) where the vectorξ≡ (ξ_1,ξ_2,⋯,ξ_D) is defined. We claim here the existence of a set of orthonormal polynomials 𝒫_i_1⋯ i_N( ξ) in this space. ∫d^D ξ ω( ξ )𝒫_i_1⋯i_N( ξ)𝒫_j_1⋯j_M(ξ)= δ_N Mδ_i_1⋯i_N|j_1⋯j_M.The polynomials 𝒫_i_1 ⋯ i_N(ξ) are symmetric tensors in theD-dimensional Euclidean space, expressed in terms of the vector components ξ_i and of δ_ij. TheN^ order polynomialis symmetrical in the indices i_1 ⋯ i_N, and its parityis (-1)^N.𝒫_i_1 ⋯ i_N(-ξ_i_1,…, -ξ_i_k,…,-ξ_i_N)=(-1)^N𝒫_i_1 ⋯ i_N(ξ_i_1,…, ξ_i_k,…ξ_i_N)The following tensors, defined by Harold Grad <cit.>, are expressed as sums of products of the Kronecker's delta function, (δ_ij=1 for i=j and 0 for i≠ j). δ_i_1⋯i_N|j_1⋯j_N ≡δ_i_1 j_1⋯δ_i_Nj_N + ,and,δ_i_1⋯ i_Nj_1⋯ j_N≡δ_i_1 j_1⋯δ_i_Nj_N + .The knowledge of the number of terms in such tensors is useful anddiscussed in more details in section <ref>. The first five (N=0 to 4) polynomials are given by,𝒫_0(ξ) = c_0, 𝒫_i_1(ξ)=c_1 ξ_i_1, 𝒫_i_1 i_2(ξ)=c_2 ξ_i_1ξ_i_2 +f_2(ξ) δ_i_1 i_2,f_2(ξ) ≡c̅_2 ξ^2+ c^'_2, 𝒫_i_1 i_2 i_3(ξ)=c_3ξ_i_1ξ_i_2ξ_i_3+ f_3(ξ) (ξ_i_1δ_i_2 i_3 + ξ_i_2δ_i_1 i_3 +ξ_i_3δ_i_1 i_2), f_3(ξ) ≡c̅_3 ξ^2+ c^'_3,and,𝒫_i_1 i_2 i_3 i_4(ξ)=c_4ξ_i_1ξ_i_2ξ_i_3ξ_i_4+f_4(ξ) (ξ_i_1ξ_i_2δ_i_3 i_4 +ξ_i_1ξ_i_3δ_i_2 i_4 +ξ_i_1ξ_i_4δ_i_2 i_3 +ξ_i_2ξ_i_3δ_i_1 i_4+ ξ_i_2ξ_i_4δ_i_1 i_3 +ξ_i_3ξ_i_4δ_i_1 i_2 ) + g_4(ξ) δ_i_1 i_2 i_3 i_4, f_4(ξ) ≡( c̅_4 ξ^2+ c^'_4 ),g_4(ξ) ≡( d̅_4 ξ^4+d^'_4 ξ^2+d_4 ).Therefore theN^th order polynomial is the sum of all possible symmetric tensors built from products of ξ_i and of δ_ij times coefficients which are themselves polynomials in ξ^2 to maximum allowed power. This proposal yields a unique expression for the N^th order polynomial.We define integrals I_N which are central to the present study. They are assumed to exist and have well defined properties.I_N δ_i_1⋯ i_N≡∫ d^D ξ ω( ξ )ξ_i_1⋯ξ_i_NHereafter the weight functionis assumed to only depend on the modulus of the vector: ω(ξ )= ω(ξ ), ξ≡|ξ|. By symmetry it holds that I_2N+1=0 since the integral vanishes. Using the spherical integration volume, ∫ d^D ξ ω( ξ )= Dπ^D/2/Γ(D/2+1) ∫ dξ ξ^D-1ω(ξ), the I_2N integrals become,I_2N=π^D/2/2^N-1Γ(N+D/2)∫_0^ξ_maxdξ ω(ξ) ξ^2N+D-1.In case that ξ_max = ∞the weight function must have the property ω(ξ) → 0 for ξ→∞ faster than any power of ξ. Next we shall explicitly prove the orthonormality of the first five polynomials.§ D-DIMENSIONAL TENSORS BASED ON THE KRONECKER'S DELTA The orthonormality condition of Eq.(<ref>) shows a rich tensorial structure in D dimensions revealed by the following two important tensors,δ_i_1⋯ i_N| j_1⋯ j_N and δ_i_1⋯ i_N j_1⋯ j_N, defined in Eqs.(<ref>) and (<ref>), respectively. The former is associated with the orthonormality condition while the latter is the totally symmetric tensor introduced in the definition of the functions I_N given by Eq.(<ref>). Both tensors are expressed as sums over several terms each one expressed as a product of Kronecker's delta functions. We determine the number of terms in these two tensors. The tensor δ_i_1⋯ i_N | j_1⋯ j_N has N! terms since this tensor is a sum over all possible permutations of the i's under a fixed set of j's. The tensor δ_i_1⋯ i_N j_1⋯ j_N has (2N-1)!/2^N-1(N-1)!=(2N-1)(2N-3)(2N-5)...1 terms according to the arguments below. Firstly notice that the tensorδ_i_1⋯ i_N j_1⋯ j_N has more terms than δ_i_1⋯ i_N | j_1⋯ j_N and here we seek to find these remainingtensors.δ_i_1⋯ i_N j_1⋯ j_N = δ_i_1⋯ i_N | j_1⋯ j_N +Next we determine the remaining “other tensors” in case N=0 to 4 and determine the number of components of the above tensor by induction. For this we introduce a short notation that only distinguishesindices i from j indices. In this notation the previous expression becomes equal to,δ_i i⋯ i ⋯ j j ⋯j= δ_i i⋯ i | j j ⋯j+ .For N=1 we have that,δ_i j= δ_i| j,and we express this identity with respect to the number of terms simply as 1 = 1. For N=2 notice that, δ_i_1 i_2 j_1 j_2= δ_i_1 j_1δ_i_2 j_2+δ_i_1 j_2δ_i_2 j_1+ δ_i_1 i_2δ_j_1 j_2. Since δ_i_1 i_2 | j_1 j_2= δ_i_1 j_1δ_i_2 j_2+δ_i_1 j_2δ_i_2 j_1 the above relation becomes, δ_i_1 i_2 j_1 j_2= δ_i_1 i_2 | j_1 j_2 + δ_i_1 i_2δ_j_1 j_2. Therefore it holds that δ_i_1 i_2 j_1 j_2= δ_i_1 i_2 | j_1 j_2 + δ_i_1 i_2δ_j_1 j_2. While δ_i_1 i_2 j_1 j_2 contains 3 components, δ_i_1 i_2 | j_1 j_2 has only 2, such that it holds for this decomposition that 3 = 2 ⊕ 1. In the short notation the above relation becomes,δ_i i | j j= δ_i jδ_i j δ_i i j j= δ_i jδ_i j + δ_i iδ_j j.The tensors δ_i jδ_i j and δ_i iδ_j j contain 2 and 1 components, respectively. Thus the short notation gives that δ_i i⋯ i | j j ⋯j≡δ_i jδ_i j⋯δ_i j where the products of δ_i j takes into account all possible permutations. The tensor δ_i i⋯ ij j ⋯j cannot be expressed similarly because not all combinations of δ_i j's are included. Therefore the need to decompose it into δ_i i⋯ i | j j ⋯ j plus other tensors. For N=3 according to the short tensorial notation,δ_i i i j j j = δ_i jδ_i jδ_i j + δ_i iδ_j jδ_i j.To determine the number of components of this tensor, notice that for δ_i_1 i_2 i_3 j_1 j_2 j_3 once a pair is chosen, say δ_i_1 j_1,the previous N=2 is retrieved concerning the number of components. Since there are 5 ways to construct this first pair, the total number of components is 5 times 3, that is, 15 terms. The tensor δ_i jδ_i jδ_i j has 3! components, thus to know the number of terms in the tensor δ_i iδ_i jδ_j j we use the following argument. There are 3 components in δ_i i, namely, δ_i_1 i_2, δ_i_1 i_3 and δ_i_2 i_3 and similarly, 3 components in δ_j j. Once fixed δ_i i, and δ_j j the tensor δ_i j has only one possible component left. Therefore δ_i iδ_i jδ_j j has a total of 3 times 1 times 3, that is 9 components, andthe tensorial decomposition is expressed as 15 = 6 ⊕ 9.Finally for N=4 the i and j short tensorial notation gives that,δ_i i i i j j j j = δ_i jδ_i jδ_i jδ_i j + δ_i iδ_j jδ_i jδ_i j + δ_i iδ_i iδ_j jδ_j jThe same reasoning of the previous cases is used here, namely,once a pair is fixed, say δ_i_1 j_1, the number of terms of the remaining indices is provided by the previous N=3 case. There are 7 ways to construct this first pair, thus the total number of terms is 7 times 15, that is, 105 terms. The tensor δ_i jδ_i jδ_i jδ_i j has 4! terms, The tensor δ_i iδ_j jδ_i jδ_i j has 6 possible terms for δ_i i, δ_i_1 i_2, δ_i_1 i_3, δ_i_1 i_4, δ_i_2 i_3, δ_i_2 i_4 and δ_i_3 i_4,and the same applies for δ_j j. Thus only 2 choices are left for δ_i jδ_i j, once the indices of δ_i i and δ_j j are fixed. Hence the total number of terms is 6 times 6 times 2, namely, 72 terms. For the tensor δ_i iδ_i iδ_j jδ_j j the pairs δ_i iδ_i i have only three terms each and so is for δ_j jδ_j j, such that the total is 3 times 3, namely, 9 terms. The tensorial decomposition is expressed as 105 = 24 ⊕ 72 ⊕ 9. § ORTHONORMALIZATION OF POLYNOMIALS TO ORDER N=4 The coefficients of the polynomials are determined here in terms of the integrals I_N. This is done for the first five polynomials by explicitly computing their inner products. We refer to Eq.(<ref>) by the short notation,(𝒫_(M),𝒫_(N) )≡∫ d^D ξ ω( ξ )𝒫_(M)(ξ)𝒫_(N)(ξ,used here for M,N=0,⋯ 4. The N^th order polynomial is shortly referred as 𝒫_(N) such that the polynomials of Eqs.(<ref>), (<ref>), (<ref>), (<ref>), (<ref>) are called 𝒫_(0), 𝒫_(1), 𝒫_(2), 𝒫_(3) and 𝒫_(4), respectively. The inner product between polynomials with distinct parity vanishes, (𝒫_(even),𝒫_(odd) )=0. Thus the only relevant orthonormality relations are among polynomials with the same parity (odd with odd and even with even). The number of equations given by orthonormalization conditions must be equal to the number of free coefficients. Indeed this is the case, according to Eqs.(<ref>), (<ref>), (<ref>), (<ref>), (<ref>). The total number of coefficients is 14 (c_K for K=0,1,2,3,4, c^'_K for K=2,3,4, c̅_K for K=2,3,4, d_4, d^'_4 and d̅_4). Remarkably14 equations arise from the 9 orthonormalization conditions (𝒫_(M),𝒫_(N) ), as seen below. ∙ (𝒫_(0),𝒫_(0) ) The normalization of the N=0 polynomial is,c_0^2 ∫ d^Dξω(ξ)= 1,which gives that,c_0 = ±1/√(I_0),where Eq. (<ref>) and the definition of I_0 in Eq.(<ref>) have been used. ∙ (𝒫_(1),𝒫_(1) )c_1^2∫ d^Dξ ω(ξ)ξ_i_1ξ_j_1 = δ_i_1 j_1The definition of I_2 in Eq.(<ref>) is invoked to obtain that,c_1 = ±1/√(I_2).TheN=0 and N=1 polynomials are naturally orthogonal because they have distinct parity and to make them orthonormal is enough to normalize them which has been done above by determining the coefficients c_0 and c_1. The polynomial 𝒫_(2) has three coefficients, according to Eq.(<ref>), and so, three equations are needed to determine them. These equations must arise from the tensorial structure of the D dimensional space. ∙ (𝒫_(0),𝒫_(2) ) This conditions is equivalent to c_0 (1,𝒫_(2) )=0 since 𝒫_(0) is a constant.c_0∫ d^Dξω(ξ)[ c_2 ξ_i_1ξ_i_2 + f_2(ξ) δ_i_1 i_2 ] =0.Using the tensorial formulas of appendix <ref> and Eq.(<ref>), it follows that,I_2c_2 + D I_2 c̅_2 + I_0 c_2' = 0.∙ (𝒫_(2),𝒫_(2) ) Remarkably the normalization of the N=2 polynomial leads to multiple equations, in this case the following two equations.∫ d^Dξω(ξ) [ c_2 ξ_i_1ξ_i_2 + f_2(ξ) δ_i_1 i_2 ][ c_2 ξ_j_1ξ_j_2 + f_2(ξ)δ_j_1 j_2 ]=δ_i_1 j_1δ_i_2 j_2 + δ_i_1 j_2δ_i_2 j_1.This is because while the orthonormalization is associated to the tensor δ_i_1 j_1δ_i_2 j_2 + δ_i_1 j_2δ_i_2 j_1 the integration over ξ_i_1ξ_i_2ξ_j_1ξ_j_2 leads to the tensor δ_i_1 j_1δ_i_2 j_2 + δ_i_1 j_2δ_i_2 j_1+δ_i_1 i_2δ_j_1 j_2. This difference is responsible for the onset of more than one condition. Using the tensorial formulas of appendix <ref>, one obtains that:c_2^2 I_4( δ_i_1j_1δ_i_2j_2 + δ_i_1j_2δ_i_2j_1+δ_i_1i_2δ_j_1j_2 ) +[2(D+2)I_4c_2 c̅_2 + 2I_2 c_2 c_2'+ c̅_2^2 D(D+2)I_4 + 2DI_2c̅_2 c_2' + c_2'^2I_0]δ_i_1i_2δ_j_1j_2= δ_i_1j_1δ_i_2j_2 +δ_i_1j_2δ_i_2j_1.The independence of the two tensors leads to two independent equations, given by,c_2^2 I_4 = 1c^2_2I_4 + 2(D+2)I_4c_2 c̅_2 + 2I_2 c_2 c_2'+ c̅_2^2 D(D+2)I_4+ 2DI_2c̅_2 c_2' + c_2'^2I_0 =0.The three equations are promptly solved and the coefficients c_2, c̅_2 and c^'_2 are determined below.c_2 = ±1/√(I_4).Using Eq.(<ref>) to write c_2', we havec_2' = -I_2/I_0c_2 - DI_2/I_0c̅_2.Substituting c_1' in Eq.(<ref>),c̅_2^2 [ D(D+2) I_4 - D^2 I_2^2/I_0]+ c̅_2c_2[ 2(D+2)I_4 - 2DI_2^2/I_0] + c_2^2( I_4-I_2^2/I_0) = 0,which is a equation for c̅_2. The solutions, are:c̅_2 = c_2/D (-1+Δ_2), Δ _2 ≡±√(2/(D+2)-DJ_2), J_2 ≡I_2^2 /I_0 I_4The coefficient c̅_2 must be a real number, and this is the case provided that(D+2)-DJ_2 ≥ 0. From Eq.(<ref>), we calculate c_2', to obtain that,c_2' = - c_2 I_2/I_0Δ _2.The orthogonalization of the first three polynomials has been concluded here. We proceed to the next order (N=3), and because of the increasing difficulty introduce a short notation for tensors, which is discussed in section <ref>. The N=3 polynomialhas three coefficients, similarly to the N=2 case, and so, three equations are needed to determine them. ∙ (𝒫_(1),𝒫_(3))c_1∫ d^Dξω(ξ) ξ_i_1[ c_3ξ_j_1ξ_j_2ξ_j_3+f_3(ξ)(ξ_j_1δ_j_2 j_3 + ξ_j_2δ_j_1 j_3 +ξ_j_3δ_j_1 j_2)]=0.The integrals are calculated with the help of the tensorial formulas of appendix <ref>, such that the above expression becomes [c_3I_4 + c̅_3 I_4 (D+2) + c^'_3 I_2]δ_i j j j=0. The tensor δ_i j j j is short for δ_i_1 j_1 j_2 j_3, according to the notation of section <ref>. Thus we obtain that,c_3I_4 + c̅_3 I_4 (D+2) + c^'_3 I_2 = 0.∙ (𝒫_(3),𝒫_(3) ) This normalization conditiongives the two other equations necessary to calculate the coefficients.∫ d^Dξω(ξ)[ c_3ξ_i_1ξ_i_2ξ_i_3+f_3(ξ)(ξ_i_1δ_i_2 i_3 + ξ_i_2δ_i_1 i_3 + ξ_i_3δ_i_1 i_2)] [ c_3ξ_j_1ξ_j_2ξ_j_3+f_3(ξ)(ξ_j_1δ_j_2 j_3 + ξ_j_2δ_j_1 j_3+ξ_j_3δ_j_1 j_2) ]=δ_i_1 i_2 i_3 | j_1 j_2 j_3We stress the difference between the tensors δ_i i i j j j andδ_i i i | j j j as the reason for multiple equations from a single normalization condition. Using the short notation of section <ref> the integral over the six vector components becomes, ∫ d^Dξω(ξ)ξ_iξ_iξ_iξ_jξ_jξ_j = I_6 δ_i i i j j j, where all permutations of i's and j's are taken into account. From the other side in this short notation, δ_i i i | j j j = δ_i jδ_i jδ_i j, as explained in section <ref>, and, as shown there, δ_i i i j j j = δ_i jδ_i jδ_i j+δ_i iδ_i jδ_j j. Thus the above integrals are computed with the aid of appendix <ref>. Using this notation, Eq.(<ref>) becomes,c_3^2I_6 (δ_ijδ_ijδ_ij + δ_iiδ_ijδ_jj ) +[2I_6 (D+4)c_3c̅_3 + 2I_4c_3c^'_3 +I_2 c^'_3^2+ 2I_4(D+2)c̅_3 c^'_3+ I_6 (D+2) (D+4)c̅_3^2 ]δ_iiδ_ijδ_jj=δ_ijδ_ijδ_ij,which gives two equations:c_3^2I_6 = 1c_3^2I_6 + 2I_6 (D+4)c_3c̅_3 + 2I_4c_3c^'_3 + I_2 c^'_3^2 + 2I_4(D+2)c̅_3 c^'_3+ I_6 (D+2) (D+4)c̅_3^2=0.The solution of the first equation is,c_3 = ±1/√(I_6).Eq.(<ref>) is used to eliminate c^'_3 from Eq.(<ref>),c̅_3^2 (D+2) [ ( I_6-I_4^2/I_2)(D+2) + 2I_6 ] + 2 [( I_6-I_4^2/I_2)(D+2) + 2I_6 ]c_3c̅_3+ ( I_6-I_4^2/I_2) c_3^2 =0.This equation can be solved for c̅_3.c̅_3 = c_3/D+2(-1+Δ _4),and c^'_3 is calculated by Eq.(<ref>):c^'_3 = - I_4/I_2Δ _4 c_3 Δ_4 = ±√(2/(D+4) - J_4(D+2)), J_4 = I_4^2/I_6 I_2.The N=4 polynomial of Eq.(<ref>) sets a new level of difficulty assix equations must be obtained to calculate the six coefficients. ∙ (𝒫_(0),𝒫_(4)) The orthonormalization with the N=0 polynomial means that c_0(1,𝒫_(4))=0.c_0∫ d^D ξω(ξ) [c_4ξ_j_1ξ_j_2ξ_j_3ξ_j_4+f_4(ξ)(ξ_j_1ξ_j_2δ_j_3 j_4 + ξ_j_1ξ_j_3δ_j_2 j_4+ξ_j_1ξ_j_4δ_j_2 j_3+ξ_j_2ξ_j_3δ_j_1 j_4+ ξ_j_2ξ_j_4δ_j_1 j_3 +ξ_j_3ξ_j_4δ_j_1 j_2 )+g_4(ξ)δ_j_1 j_2 j_3 ij_4 ]=0A single equation results from this integral since it can only be proportional to the tensor δ_i i i i.c_4I_4 + 2 [c_4' I_2 + c̅_4 I_4(D+2) ] + d_4I_0 + d_4' I_2 D + d̅_4 I_4(D+2)D = 0.∙ (𝒫_(2),𝒫_(4)) The integration of the N=2 with the N=4 polynomial gives that,∫ d^D ξω(ξ)[c_2 ξ_i_1ξ_i_2 + f_2(ξ) δ_i_1 i_2 ] [c_4ξ_j_1ξ_j_2ξ_j_3ξ_j_4+f_4(ξ)(ξ_j_1ξ_j_2δ_j_3 j_4 +ξ_j_1ξ_j_3δ_j_2 j_4+ ξ_j_1ξ_j_4δ_j_2 j_3+ξ_j_2ξ_j_3δ_j_1 j_4 + ξ_j_2ξ_j_4δ_j_1 j_3+ξ_j_3ξ_j_4δ_j_1 j_2 )+g_4(ξ) δ_j_1 j_2 j_3 j_4 ]=0.Thesixth order tensorial integral, ∫ d^D ξω(ξ)ξ_iξ_iξ_jξ_jξ_jξ_j = I_6 δ_i i j j j j has 15 terms. This tensor can be decomposed as δ_i i j j j j = δ_i iδ_j j j j+δ_i jδ_i j j j, namely, as a sum of two other tensors, whichhave 3 and 12 terms, respectively. This decomposition can be formally expressed as 15=3⊕ 12, as discussed in section <ref>. We notice the presence of the tensor δ_i_1 i_2 j_1 j_2δ_j_3 j_4+δ_i_1 i_2 j_1 j_3δ_j_2 j_4+δ_i_1 i_2 j_1 j_4δ_j_2 j_3+δ_i_1 i_2 j_2 j_3δ_j_1 j_4+δ_i_1 i_2 j_2 j_4δ_j_1 j_3+δ_i_1 i_2 j_3 j_4δ_j_1 j_2, which has 18 terms, and is equal to 2δ_j j j jδ_i i+δ_i i i jδ_i j (2 times 3 plus 12). Therefore Eq.(<ref>) becomes,c_4c_2I_6(δ_i iδ_j j j j+ δ_i jδ_i j j j ) +c_4[c_2'I_4+ c̅_2 I_6 (D+4) ]δ_i iδ_j j j j + c_2[c_4'I_4 + c̅_4 I_6(D+4) ] (2δ_i iδ_j j j j+ δ_i jδ_i j j j) + 2[c_4' c_2' I_2 + (c̅_4 c_2'+ c_4' c̅_2)I_4 (D+2) + c̅_4 c̅_2 I_6 (D+4)(D+2)] δ_i iδ_j j j j+c_2 [d_4I_2+ d_4' I_4 (D+2) + I_6 d̅_4 (D+4)(D+2) ]δ_i iδ_j j j j+ [d_4 c_2' I_0 +(d_4 c̅_2 + d_4' c_2' )I_2D +(d_4' c̅_2 + d̅_4 c_2')I_4(D+2)D+d̅_4 c̅_2 I_6 (D+4)(D+2)D ] δ_i iδ_j j j j =0.This lead to two equations, one proportional to δ_ijδ_ijjj,c_4I_6 + [c_4'I_4 + c̅_4 I_6 (D+4)]=0,and the other proportional to δ_i iδ_j j j j,c_4 c_2 I_6 + c_4[ c_2' I_4 + c̅_2 I_6 (D+4) ] + 2c_2 [c_4'I_4 + c̅_4 I_6 (D+4)]+2[c_4' c_2' I_2+(c̅_4 c_2' + c_4' c̅_2)I_4 (D+2) +c̅_4 c̅_2 I_6 (D+4)(D+2) ] + c_2 [d_4I_2 + d_4' I_4 (D+2) + d̅_4 I_6 (D+4)(D+2) ]+[d_4c_2'I_0 +(d_4 c̅_2 + d_4' c_2')I_2D+ (d_4' c̅_2 + d̅_4 c_2')I_4 (D+2)D + d̅_4 c̅_2 I_6 (D+4)(D+2)D ] = 0,respectively.∙ (𝒫_(4),𝒫_(4)) The normalization of the N=4 polynomial is given by,∫ d^D ξω(ξ) [ c_4ξ_i_1ξ_i_2ξ_i_3ξ_i_4+f_4(ξ) (ξ_i_1ξ_i_2δ_i_3 i_4 + ξ_i_1ξ_i_3δ_i_2 i_4 +ξ_i_1ξ_i_4δ_i_2 i_3+ξ_i_2ξ_i_3δ_i_1 i_4+ ξ_i_2ξ_i_4δ_i_1 i_3+ξ_i_3ξ_i_4δ_i_1 i_2 )+g_4(ξ) δ_i_1 i_2 i_3 i_4 ] [ c_4ξ_j_1ξ_j_2ξ_j_3ξ_j_4+f_4(ξ)(ξ_j_1ξ_j_2δ_j_3 j_4 +ξ_j_1ξ_j_3δ_j_2 j_4 +ξ_j_1ξ_j_4δ_j_2 j_3+ξ_j_2ξ_j_3δ_j_1 j_4+ ξ_j_2ξ_j_4δ_j_1 j_3 + ξ_j_3ξ_j_4δ_j_1 j_2 )+g_4(ξ) δ_j_1 j_2 j_3 j_4] = δ_i_1 i_2 i_3 i_4 | j_1 j_2 j_3 j_4.Here we extensively use the short notation of section <ref>,where δ_i i i i |j j j j= δ_ijδ_ijδ_ijδ_ij. There is the integral ∫ d^Dξω(ξ)ξ_iξ_iξ_iξ_iξ_jξ_jξ_jξ_j = I_8 δ_i i i i j j j j. It holds that δ_i i i i |j j j j = δ_ijδ_ijδ_ijδ_ij + δ_iiδ_jjδ_ijδ_ij + δ_iiδ_iiδ_jjδ_jj. There are 105 terms inδ_i i i i j j j j, 24 inδ_ijδ_ijδ_ijδ_ij, 72 in δ_iiδ_jjδ_ijδ_ij, and 9 in δ_iiδ_iiδ_jjδ_jj. We computed each of theintegrals individually and identify them by the following notation that uses the product of their coefficients c_4, f_4 and g_4. For instance the first one is∫ d^D ξω(ξ)c_4^2 [⋯] = I_8 c_4^2 (δ_ijδ_ijδ_ijδ_ij +δ_iiδ_jjδ_ijδ_ij + δ_iiδ_iiδ_jjδ_jj)The second term is,∫ d^D ξω(ξ)c_4 f_4 [⋯] = 2 [c'_4I_6+c̅_4 I_8 (D+6) ] c_4( δ_iiδ_jjδ_ijδ_ij + 2 δ_iiδ_iiδ_jjδ_jj),where we have used that δ_i i i i j jδ_j j= δ_iiδ_jjδ_ijδ_ij + 2 δ_iiδ_iiδ_jjδ_jj.∫ d^D ξω(ξ)c_4 g_4 [⋯] = 2 [d_4 I_4 + d^'_4 I_6 (D+4) + d̅_4 I_8 (D+6) (D+4)] c_4( δ_iiδ_iiδ_jjδ_jj), ∫ d^D ξω(ξ)f_4 g_4 [⋯] = 4 [c^'_4 d_4 I_2 + (c^'_4d^'_4+c̅_4 d_4)I_4 (D+2)+(c̅_4 d^'_4+c^'_4 d̅_4)I_6(D+4)(D+2) +c̅_4 d̅_4 I_8 (D+6) (D+4) (D+2)] ( δ_iiδ_iiδ_jjδ_jj) ∫ d^D ξω(ξ)f_4^2 [⋯] = [c^'_4^2 I_4+2c^'_4c̅_4I_6(D+4)+c̅_4^2I_8(D+6)(D+4)] ( 4 δ_iiδ_iiδ_jjδ_jj+ δ_iiδ_jjδ_ijδ_ij) ∫ d^D ξω(ξ)g_4^2 [⋯] = [d_4^2I_0+2d_4d^'_4 I_2 D + (d^'_4^2+2d_4 d̅_4)I_4(D+2)D + 2 d^'_4d̅_4 I_6 (D+4) (D+2) D + d̅_4^2 I_8 (D+6)(D+4) (D+2) D ] ( δ_iiδ_iiδ_jjδ_jj)Next these integrals are introduced into Eq.(<ref>) to obtain that,I_8(δ_ijδ_ijδ_ijδ_ij + δ_iiδ_jjδ_ijδ_ij + δ_iiδ_iiδ_jjδ_jj) c_4^2 + 2[c_4' I_6 + c̅_4 I_8 (D+6)]c_4 (2δ_iiδ_iiδ_jjδ_jj + δ_iiδ_jjδ_ijδ_ij) +2[d_4I_4 + d_4'I_6 (D+4)+d̅_4I_8 (D+6)(D+4)]δ_iiδ_iiδ_jjδ_jj + 4 [c_4' d_4 I_2 + (c_4' d_4' + c̅_4 d_4)I_4 (D+2)+ (c̅_4 d_4' + c_4' d̅_4)I_6 (D+4)(D+2) + c̅_4 d̅_4 I_8 (D+6)(D+4)(D+2)]δ_iiδ_iiδ_jjδ_jj + [c_4'^2I_4 + 2c_4' c̅_4 I_6 (D+4)+ c̅_4^2I_8(D+6)(D+4)] (4δ_iiδ_iiδ_jjδ_jj + δ_iiδ_jjδ_ijδ_ij)+[d_4^2 I_0 + 2d_4 d_4' I_2D + (d_4'^2 + 2d_4 d̅_4)I_4 (D+2)D +2d_4' d̅_4 I_6 (D+4)(D+2)D + d̅_4^2 I_8 (D+6)(D+4)(D+2)D] δ_iiδ_iiδ_jjδ_jj = δ_ijδ_ijδ_ijδ_ij.The three remaining equations are the coefficients of the three independent tensors in the above equation.I_8 c_4^2 = 1, I_8 c_4^2 + 2[c_4'I_6 + c̅_4 I_8(D+6)]c_4 + [c_4'^2 I_4 +2c_4' c̅_4 I_6 (D+4)+ c̅_4^2 I_8 (D+6) (D+4)] = 0, I_8 c_4^2 + 4c_4[c_4' I_6 + c̅_4 I_8(D+6)] + 2c_4[d_4 I_4 + d_4'I_6(D+4)+ d̅_4 I_8 (D+6)(D+4)] + 4[c_4' d_4 I_2 +(c_4'd_4' + c̅_4 d_4)I_4(D+2)+ (c̅_4 d_4' + c_4' d̅_4)I_6 (D+4)(D+2) + c̅_4 d̅_4 I_8 (D+6)(D+4)(D+2)]+ 4[c_4'^2I_4+ 2c_4'c̅_4 I_6 (D+4)+c̅_4^2 I_8 (D+6)(D+4)] + [d_4^2I_0 + 2d_4d_4'I_2D + (d_4'^2+ 2d_4 d̅_4)I_4 (D+2)D+ 2d_4' d̅_4 I_6 (D+4)(D+2)D + d̅_4^2 I_8 (D+6)(D+4)(D+2)D] = 0.The six equations given by the Eqs.(<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) can be solved to obtain the six coefficients. Nevertheless notice that coefficients c_4, c̅_4 and c^'_4 are determined from a sub set of equations, namely, Eqs.(<ref>), (<ref>), and (<ref>). From them it follows that,c_4 = ±1/√(I_8), , c̅_4^2 (D+4) [ I_8 (D+6)-I_6^2/I_4(D+4)] + 2 [ I_8 ((D+6)-I_6^2/I_4(D+4) + ]c_4c̅_4+ ( I_8-I_6^2/I_4) c_4^2 =0.Hence one obtains that,c̅_4 = c_4(-1 + Δ _6)/D+4, c_4' = - I_6/I_4Δ _6 c_4.The remaining coefficients are given by,d̅_4 = δ _2/δ _4D(D+2) d_4 + c_4[D-2(D+2)Δ_6]/D(D+2)(D+4), d_4' = -d_4/D( I_0/I_2 + I_4/I_2δ _2/δ _4) + 2I_6Δ_6 c_4/I_4D,and,d_4 = ±√(8δ _4^2 I_4/δ _2)1/√(δ _2 δ _6(D+4) - δ _4^2D),whereΔ _6 = ±√(2/(D+6)- J_6(D+4)),J_6 = I_6^2/I_8I_4, δ _2 = I_0 I_4(D+2) - I_2^2D, δ _4 = I_2I_6(D+4) - I_4^2(D+2), δ _6 = I_4 I_8 (D+6) - I_6^2 (D+4).Notice that δ_2K =2I_2K+2I_2K-2/Δ_2K^2. Hence we have determined all the coefficients in Eqs.(<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>). Some of the coefficients can be summarized insimple formulas for all polynomials, using a general notation:c_K= 1/√(I_2K), ,c^'_K=-c_K I_2K-2/I_2K-4Δ_2K-2, , c̅_K=c_K(-1+Δ_2K-2)/D+2K-4, ,Δ_2K= √(2/(D+2K)-J_2K(D+2K-2))J_2K= I_2K^2/I_2K+2I_2K-2 Notice that we have chosen the positive solutions for the square roots, but the negative ones would also lead to orthogonal polynomials. We have completed the orthonormalization of the first five polynomials (N=0,1,2,3,4). The procedure can be applied for higher polynomials, although it becomes increasingly laborious.§ D-DIMENSIONAL POLYNOMIALS FOR SPECIFIC WEIGHTSIn this section we obtain the D-dimensional polynomials for some specific weights and from them retrieve some well known D=1 polynomials. We also obtain the I_N functions associated to some new weights. §.§ D-dimensional Hermite polynomials We show that the D-dimensional Hermite polynomials are straightforwardly retrieved from the above polynomials for the gaussian weight,ω(ξ) = 1/(2π)^D/2e^-ξ^2/2,ξ_max=∞.To obtain the integrals I_2N of Eq.(<ref>), we note that,∫_0^∞ dξ ω(ξ) ξ^2N+D-1= 2^N-1/π^D/2Γ(N+D/2).Then it follows from Eq.(<ref>) that,I_2N=1.In this limit c_K=1, c̅_K=0c^'_K=-1,d_4=1, d̅_4=0 and d^'_4=0, and the polynomials of Eqs.(<ref>), (<ref>), (<ref>), (<ref>), (<ref>) become,𝒫_0(ξ) = 1, 𝒫_i_1(ξ)= ξ_i_1, 𝒫_i_1 i_2(ξ)= ξ_i_1ξ_i_2 - δ_i_1 i_2 𝒫_i_1 i_2 i_3(ξ)=ξ_i_1ξ_i_2ξ_i_3- (ξ_i_1δ_i_2 i_3 + ξ_i_2δ_i_1 i_3 +ξ_i_3δ_i_1 i_2)and,𝒫_i_1 i_2 i_3 i_4(ξ)=ξ_i_1ξ_i_2ξ_i_3ξ_i_4- (ξ_i_1ξ_i_2δ_i_3 i_4 +ξ_i_1ξ_i_3δ_i_2 i_4 +ξ_i_1ξ_i_4δ_i_2 i_3 +ξ_i_2ξ_i_3δ_i_1 i_4+ ξ_i_2ξ_i_4δ_i_1 i_3 +ξ_i_3ξ_i_4δ_i_1 i_2 )+ δ_i_1 i_2 i_3 i_4We notice that the tensorial basis that spans the new generalized polynomials contains the basis that spans the Hermite polynomials but not vice-versa. The D-dimensional Hermite polynomials 𝒫_i_1⋯ i_N are symmetric tensors in theindices i_1⋯ i_N spanned over the basis formed by the tensors,T_i_1⋯ i_N≡ξ_i_1·ξ_i_2⋯ξ_i_P·δ_i_P+1,i_P+2·δ_i_P+3,i_P+4⋯δ_i_N-1,i_N.This basis is not large enough to span the new generalized polynomials, 𝒫_i_1⋯ i_N, which demand a larger basis formed by the tensorsT_i_1⋯ i_N≡ F( ξ) ξ_i_1·ξ_i_2⋯ξ_i_P·δ_i_P+1,i_P+2·δ_i_P+3,i_P+4⋯δ_i_N-1,i_N,whose scalar functions F( ξ) are polynomials in powers of the vector modulus, 1, ξ^2, ξ^4, ...,ξ^2k. §.§ D-dimensional Legendre polynomialsWe define D-dimensional Legendre polynomials satisfying the general orthonormal relation of Eq.(<ref>) withξ_max = 1 and the following weight:ω(ξ) = 1, ξ_max = 1.From this it follows that,I_2N = π^D/2/2^N-1Γ(N+D/2)∫_0^1 dξξ^2N + D -1= 2^1-Nπ ^D/2/(D+2N) Γ[D+2N/2],Using the expressionscorresponding coefficients up to fourth orderc_N=1/√(2^-Nπ ^D/2/Γ[1+D/2+N]), c̅_2=-2+ √(2(2+D))/D √(π ^D/2/Γ[3+D/2]),c̅_3=-2√(2)+2√(4+D)/(2+D) √(π ^D/2/Γ[4+D/2]),c̅_4 = -4 + 2√(2(6+D))/(4+D) √(π ^D/2/Γ[5+D/2]), c^'_2=-√(2/(2+D )π ^D/2/Γ[3+D/2]),c^'_3=-√(4/(4+D) π ^D/2/Γ[4+D/2]), c^'_4=-√(8/(6+D) π ^D/2/Γ[5+D/2]), d_4=√(16(6+D)π ^3 D/2/(4+D) Γ[3+D/2] Γ[4+D/2]^2/π ^2 D((2+D) (96+D (8+D) (24+D (8+D)))/(8+D) Γ[3+D/2]^4-2 (4+D) (4+D (8+D))/Γ[2+D/2]^3 Γ[5+D/2])), d^'_4=( 2 D (6+D)^3/2Γ[1+D/2] Γ[3+D/2]^2 - 8√(6+D)Γ[3+D/2]^3)d_4 +( 8 √(2)Γ[3+D/2]^3 /D+2-4 √(2)Γ[2+D/2]^2 Γ[4+D/2] )c_4/D (4+D) √(6+D)(-(6+D) Γ[2+D/2]^3+2 Γ[1+D/2] Γ[3+D/2]^2), d̅_4 = c_4 (D-2√(2(6+D))- D √(2(6+D)))+(4+D)^2 (6+D) d_4/D (2+D) (4+D).These coefficients provides orthogonal polynomials for any dimension and, as we will see in subsection <ref>, they give the well known D=1 Legendre polynomials for the particular case with D=1.§.§ D-dimensional Chebyshev polynomials of first kind Following the same procedure used for the Legendre polynomials, we define the D-dimensional Chebyshev polynomials of first kind with the weight ω(ξ) below and integrated under a sphere of radius ξ_max,ω(ξ) = 1/1-ξ^2, ξ_max = 1,with the corresponding integralI_2N = π^D/2/2^N-1Γ(N+D/2)∫_0^1 dξ1/1-ξ^2ξ^2N + D -1 = 2^-Nπ ^1+D/2/Γ[1/2 (1+D+2N)],and coefficients up to fourth orderc_N=1/√(2^-Nπ ^1+D/2/Γ[1+D/2+N]), c̅_2=2 (-1+√(1+D))/D √(π ^1+D/2/Γ[5+D/2]),c̅_3=2 √(2)(-1+√(3+D))/(2+D) √(π ^1+D/2/Γ[7+D/2]),c̅_4 =4 (-1+√(5+D))/(4+D) √(π ^1+D/2/Γ[9+D/2]), c^'_2=-2/√((1+D)π ^1+D/2/Γ[5+D/2]), c^'_3=-2 √(2)/√((3+D)π ^1+D/2/Γ[7+D/2]),c_4'=-4/√((5+D)π ^1+D/2/Γ[9+D/2]) d_4=8 √(π ^3 (1+D)/2/(3+D) (5+D)^2 Γ[5+D/2]^3)/√(π ^2+2 D(-D (2+D)^2/Γ[5+D/2]^4+(4+D) (7+D) (4+D (7+D) (8+D (7+D)))/4 (3+D) Γ[3+D/2]^2 Γ[9+D/2]^2)), d^'_4= 2 (c_4 √(5+D)-d_4 (3+D) (5+D) ) Γ[3+D/2] Γ[5+D/2]^2/D (4+D) (5+D) Γ[5+D/2]^3-2 D (2+D) Γ[3+D/2] Γ[7+D/2]^2, d̅_4 = 1/4 (2+D)(4 c_4 (D-4 √(5+D)-2 D √(5+D))/D (4+D)+8 d_4 (5+D)Γ[5+D/2]^2/D (4+D) Γ[5+D/2]^2-D (2+D) Γ[3+D/2] Γ[7+D/2]).The polynomials with the coefficients above are orthogonal for any dimensions and, as we will see in subsection <ref> for D=1 they give the well known D=1 Chebyshev polynomials of first kind. §.§ D-dimensional Chebyshev polynomials of second kind For the D-dimensional Chebyshev polynomials of second kind, we define the following weight function and maximum radiusω(ξ) = √(1-ξ^2), ξ_max = 1which give the integral and coefficients belowI_2N = π^D/2/2^N-1Γ(N+D/2)∫_0^1 dξ√(1-ξ^2)ξ^2N + D -1 = 2^-1-Nπ ^1+D/2/Γ[1/2 (3+D+2N)], c_N=1/√(2^-1-Nπ ^1+D/2/Γ[3+D/2+N]), c̅_2=2 √(2)(-3+ √(3(3+D)))/3 D √(π ^1+D/2/Γ[7+D/2]),c̅_3=-4+4 √(5+D)/√(3)/(2+D) √(π ^1+D/2/Γ[9+D/2]),c̅_4 =4 √(2)(-3+ √(3(7+D)))/3 (4+D) √(π ^1+D/2/Γ[11+D/2]), c^'_2=-√(8/3(3+D)π ^1+D/2/Γ[7+D/2]), c^'_3=-√(16/3(5+D)π ^1+D/2/Γ[9+D/2]), c^'_4=-√(32/3(7+D)π ^1+D/2/Γ[11+D/2]) d_4=4 √(3(7+D) π ^3 (1+D)/2/(5+D) Γ[9+D/2]^3)/√(π ^2+2 D(-D (2+D)^2/Γ[7+D/2]^4+(4+D) (9+D) (36+D (1+D) (8+D) (9+D))/4 (5+D) Γ[5+D/2]^2 Γ[11+D/2]^2)) d_4^' = 2 (c_4√(3(7+D))-3 d_4(5+D) (7+D) ) Γ[5+D/2] Γ[7+D/2]^2/D (4+D) (7+D) Γ[7+D/2]^3-2 D (2+D) Γ[5+D/2] Γ[9+D/2]^2 d̅_4 = 1/4 (2+D)(-4 c_4 (4 √(3(7+D))+D (-3+2 √(3(7+D))))/3 D (4+D)+24 d_4 (7+D) Γ[7+D/2]^2/D (4+D) Γ[7+D/2]^2-D (2+D) Γ[5+D/2] Γ[9+D/2])The corresponding polynomials are orthogonal for any dimension and, as we will see in subsection <ref>, give the well known D=1 Chebyshev polynomials of second kind. §.§ Projection of Hermite, Legendre and Chebyshev polynomials intoD=1 dimension To obtain the projection of such polynomials in D=1 dimension, it suffices to drop the index of the vector, ξ_i_1→ξ, and to take that δ_i_1 i_2→ 1 since there is only one index, and so, i_1=i_2=1. Thus the tensors based on the Kronecker's delta function have each of its terms equal to one, for instance, δ_i_1 j_1⋯δ_i_Nj_N=1, and so, δ_i_1⋯ i_N| j_1⋯ j_N=N ! and δ_i_1⋯ i_Nj_1⋯ j_N= (2N-1) !/[2^N-1(N-1) !], according to section <ref>. Hence the orthonormality condition of Eq.(<ref>) becomes,∫d^D ξ ω( ξ )𝒫_N( ξ)𝒫_M(ξ)= N! δ_N M Notice that this is not the standard normalization employed in the definition of most of D=1 orthonormal polynomials. §.§.§ Hermite polynomialsThe Hermite polynomials obtained via dimensional reduction from Eqs.(<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) are given by,𝒫_0(ξ) = 1,𝒫_1(ξ)=ξ, 𝒫_2(ξ)=ξ^2 - 1, 𝒫_3(ξ)=ξ^3-3ξ,,𝒫_4(ξ)= ξ^4-6ξ^2+3.The known D=1 Hermitepolynomials, defined with the normalization of probability theory <cit.>, satisfy the following orthonormality condition,∫ _-∞^∞𝐻𝑒_m(x)𝐻𝑒_n(x) e^-x^2/2 d x=√(2π)n!δ _nm.The first five onesare 𝐻𝑒_0(x)=1, 𝐻𝑒_1(x)=x, 𝐻𝑒_2(x)=x^2-1, 𝐻𝑒_3(x)=x^3-3x, and 𝐻𝑒_4(x)=x^4-6x^2+3 and they coincide exactly with the present polynomials. §.§.§ Legendre polynomials It follows by taking D=1 that,𝒫_0(ξ) = 1/√(2), 𝒫_1(ξ) =3/2 ξ,𝒫_2(ξ) =√(5)/2(3ξ^2 -1), 𝒫_3(ξ) =√(21)/2(5ξ^2 - 3), 𝒫_4(ξ) = 3 √(3)/4 ( 35 ξ^4 - 30 ξ^2 + 3).The Legendrepolynomials <cit.> satisfy the following orthonormality condition,∫ _-1 ^1 𝑃_m(x)𝑃_n(x) d x=2/2n+1δ _nm.The first five onesare 𝑃_0(x)=1, 𝑃_1(x)=x, 𝑃_2(x)=(3x^2-1)/2, 𝑃_3(x)=(5x^3-3x)/2, and 𝑃_4(x)=(35x^4-30x^2+3)/8. Thus apart from the normalization they coincide with the present polynomials. §.§.§ Chebyshev polynomials of the first kind 𝒫_0(ξ) =1/√(π), 𝒫_1(ξ) =√(2/π)ξ, 𝒫_2(ξ) =2/√(π)(2ξ^2 -1), 𝒫_3(ξ) = 2√(3/π) (4ξ^3 - 3ξ),𝒫_4(ξ) = 4 √(3/π) ( 8 ξ^4 - 8 ξ^2 + 1).TheChebyshevpolynomials of the first kind <cit.> satisfy the following orthonormality condition,∫ _-1^1𝑇_n(x)𝑇_m(x) dx/√(1-x^2)=0 n≠ m π n=m=0 π/2 n=m≠ 0.The first five onesare 𝑇_0(x)=1, 𝑇_1(x)=x, 𝑇_2(x)=2x^2-1, 𝑇_3(x)=4x^3-3x, and 𝑇_4(x)=8x^4-8x^2+1. Thus apart from the normalization they coincide with the present polynomials. §.§.§ Chebyshev polynomials of the second kind 𝒫_0(ξ) =√(2/π),𝒫_1(ξ) =2 √(2)π ξ, 𝒫_2(ξ) =2/√(π)(4ξ^2 -1), 𝒫_3(ξ) = 8 √(3/π) (2ξ^3 - ξ), , 𝒫_4(ξ) = 4 √(3/π) (16 ξ^4 -12 ξ^2 + 1).TheChebyshevpolynomials of the second kind <cit.> satisfy the following orthonormality condition,∫ _-1^1𝑈_n(x)𝑈_m(x) dx/√(1-x^2)=0 n≠ m π/2 n=m.The first five onesare 𝑈_0(x)=1, 𝑈_1(x)=2x, 𝑈_2(x)=4x^2-1, 𝑈_3(x)=8x^3-4x, and 𝑈_4(x)=16x^4-12x^2+1. Thus apart from the normalization they coincide with the present polynomials. §.§ D-dimensional Fermi-Dirac polynomialsWe seek the set of polynomials orthonormal under the weight defined by the Fermi-Dirac statistical occupation number for u=0.ω(ξ) = 1/z^-1e^-ξ^2/2θ+1ξ_max=∞.The parameter z≡ e^μ/θ is called the fugacity, where μis the chemical potential and θ is the temperature using the so-called reduced units (m=k_B=c=ħ=1). For the Fermi-Dirac weight it holds that,I_2N=(2π)^D/2θ^νg_ν(z),ν≡ N+D/2,g_ν (z) ≡∫_0^∞dx x^ν-1/z^-1e^x+1.It is interesting to consider special limits where the integral g_ν (z) can be explicitly calculated and the I_2Nobtained. One of such limits is when the quantum Fermi-Dirac statistics becomes the classical Maxwell-Boltzmann statistics. This is the small fugacity limit,g_ν (z)= Γ(ν) (z - z^2/2^ν+ z^3/3^ν +⋯ ).The other interesting limit is the so-called Sommerfeld limit <cit.>, used for the treatment of electrons in metals where the chemical potential (Fermi energy) is much larger than the room temperature. In this limit μ/θ=lnz >>1, such that the terms of high order in (θ/μ)^2k can be disregarded.g_ν (z)= Γ(ν)/Γ(ν+1)(lnz)^ν{ 1+ν (ν-1) π^2/61/(lnz)^2+ν (ν-1)(ν-2)(ν-3)7π^4/3601/(lnz)^4+⋯}§.§ D-dimensional Bose-Einstein polynomialsSimilarly to the previous case, we seek the set of polynomials orthonormal under a weight which is the Bose-Einstein statistical occupation number.ω(ξ) = 1/z^-1e^-ξ^2/2θ-1, ξ_max=∞.It holds that,I_2N=(2π)^D/2θ^νh_ν(z),α≡ N+D/2,h_ν (z) ≡∫_0^∞dx x^ν-1/z^-1e^x-1The are also special limits here, the first being when the quantum Bose-Einsteinstatistics becomes the classical Maxwell-Botlzmann one. This is the small fugacity limit,h_ν (z)= Γ(ν) (z + z^2/2^ν+ z^3/3^ν +⋯ ).Notice that in leading order in z, the Fermi-Dirac and the Bose-Einstein polynomials become identical as they reduce to the Maxwell-Boltzmann polynomials at finite temperature. In fact at this limit these polynomials are just scaled versions of the D-dimensional Hermite polynomials. The other interesting limit is that of negative vanishing fugacity near to the onset of the Bose-Einstein condensate. For this we write α≡ - μ/θ, α→ 0. Thus for z ≡ e^-α, one obtains that,h_ν (z)= Γ(1-ν)/α^1-ν+ ∑_i=0^∞(-1)^i/i !ζ(ν-i)α^i,where ζ(s) is the Riemann zeta function, which is defined for Re(s)>1. Thus for ν integer, the above expression must be replaced by,h_m (z)= (-1)^m-1/(m-1) ! (∑_i=1^m-11/i -lnα )α^m-1+∑_[i=0; im-1 ]^∞(-1)^i/i !ζ(m-i)α^i.§.§ D-dimensional Graphene polynomialsGraphene isa two-dimensional sheet of carbon atoms arranged in an hexagonal lattice where electrons move with a relativistic dispersion relation. To dealwith this situation we seek a set of polynomials orthonormal under a weight which the following Fermi-Dirac statistical occupation number for u=0<cit.>.ω(ξ) = 1/z^-1e^|ξ|/θ+1, ξ_max=∞.It holds that,I_2N= 2^D+Nπ ^1/2 (-1+D)θ ^D+2 NΓ[1+D/2+N] g_D+2N(z)The limit of low doping in graphene corresponds to μ→ 0 or, equivalently, z→ 1. The doping of graphene can be chemically adjusted for instance <cit.>.I_2N=2^-N(-2+2^D+2 n) π ^1/2 (-1+D)θ ^D+2 NΓ[1+D/2+n] ζ[D+2 N] +2^-N(-4+2^D+2 N) π ^1/2 (-1+D)θ ^D+2 NΓ[1+D/2+N] ζ[-1+D+2 N] (z-1)+2^-1-Nπ ^1/2 (-1+D)θ ^D+2 NΓ[1+D/2+N] ((-8+2^D+2 N) ζ[-2+D+2 N].. -(-4+2^D+2N) ζ[-1+D+2 N]) (z-1)^2+O[z-1]^3 §.§ D-dimensional Yukawa polynomials Consider the Yukawa potential of nuclear interactions that contains the parameter μ that renders it short ranged. Assume the Yukawa potential as a weight to obtain orthonormal polynomials.ω(ξ) =e^-μξ/ξξ_max=∞.To determine the coefficients that enter the polynomials, it is enough to have the integrals,I_2N= π^D/2/2^N-1Γ(2N+D-1)/Γ(N+D/2)1/μ^2N+D-3.§ EXPANSION OF A FUNCTION IN GENERAL D-DIMENSIONAL POLYNOMIALSSuppose afunction f(ξ) whose argument is the D-dimensional vector ξ such that a small vector u is subtracted from it. We want to expand it in terms of this small correction using the D-dimensional polynomial basis.f(ξ -u)= f( ξ ) ∑_N=0^∞1/N!𝒜_i_1 i_2 ⋯ i_N(u ) 𝒫_i_1 i_2⋯ i_N( ξ),The coefficients 𝒜_i_1 i_2 ⋯ i_N and the polynomials 𝒫_i_1 i_2⋯ i_N only depend on u and ξ, respectively. To help the understanding of the above expansion, we consider the example of a local distribution of charges, ρ( u), that interact through a two-body potential, f(u -u^'), such as the Yukawa potential. We want to obtain the potential v(ξ) far way from this charge distribution, namely at a distanceξ≫u. As the potential produced by each charge at volume d^D u ρ(u) contributes with potential f(ξ -u) to the total potential, one obtains that,v(ξ) = ∫ d^D u f(ξ -u)ρ(u).Therefore we propose here a novel expansion in terms of the orthonormal multipoles 𝒬_i_1 i_2 ⋯ i_N, called in this way because of theorthonormal basis,v(ξ)= f( ξ ) ∑_N=0^∞1/N!𝒬_i_1 i_2 ⋯ i_N𝒫_i_1 i_2⋯ i_N( ξ),𝒬_i_1 i_2 ⋯ i_N≡∫ d^D u 𝒜_i_1 i_2 ⋯ i_N(u )ρ( u).The key ingredient to obtain these orthonormal multipoles is thatthe potential f( ξ) itself enters in the definition of the polynomials 𝒫_i_1 i_2⋯ i_N( ξ) as the weight function that renders them orthonormal. To understand this we turn to the general properties of the expansion of Eq.(<ref>). Notice that it is of the form f(ξ-u)= f(ξ)· S(ξ,u), whichmeans that the expansion S(ξ,u→ 0)→ 1. Thus the corrections in u are necessarily small. The key ingredient that renders the above expansion advantageous is the assumption that the weight function that defines the set of orthonormal polynomials is the expanded function itself for u=0,ω(ξ)≡ f( ξ ).This means that the function has to fulfill the properties previously required for a weight in order to span a polynomial basis. In this case the coefficients are readily determined using the orthonormality condition of Eq.(<ref>).𝒜_i_1 i_2 ⋯ i_N(u )= ∫ d^D ξf(ξ-u) 𝒫_i_1 i_2⋯ i_N( ξ).Hence the above expression once fed back to Eq.(<ref>) gives thecompleteness relation for the polynomials.f(ξ -u)= f( ξ ) ∑_N=0^∞1/N!∫ d^D ξ^'f(ξ^'-u) 𝒫_i_1 i_2⋯ i_N( ξ^') 𝒫_i_1 i_2⋯ i_N( ξ),The completness relation is given byf( ξ ) ∑_N=0^∞1/N! 𝒫_i_1 i_2⋯ i_N( ξ^') 𝒫_i_1 i_2⋯ i_N( ξ) = δ^D(ξ^'-ξ),since f(ξ-u)= ∫ d^D ξ^' δ^D(ξ^'-ξ)f(ξ^'-u). In order to determine other properties of the coefficients, we defineη≡ξ-u to obtain that,𝒜_i_1 i_2 ⋯ i_N(u )= ∫ d^D η ω(η)𝒫_i_1 i_2⋯ i_N( η+u),At this point we introduce N=0 polynomial, which is constant, into this expression.𝒜_i_1 i_2 ⋯i_N(u )= 1/c_0∫d^D η ω(η) 𝒫_0(η)𝒫_i_1 i_2⋯i_N( η+u),Hence the determination of the coefficients is reduced to the expansion 𝒫_i_1 i_2⋯ i_N( η+u) as a sum over polynomials of equal or lower order (M ≤ N), 𝒫_i_1 i_2⋯ i_M( η).𝒫_i_1i_2… i_N(η+u) =𝒰_0(u)𝒫_i_1i_2… i_N(η) +…+ 1/c_0𝒰_i_1i_2… i_N(u)𝒫_0(η),where 𝒰_i_1i_2… i_M(u) is a polynomial in u that multiplies the polynomial in η of order N-M. The sought coefficient is nothing but the term proportional to 𝒫_0(η) in this expansion. The limit u→ 0 in the above expression shows that 𝒰_0(0)=1 while for all other higher order tensors it holds that 𝒰_i_1i_2… i_M(0)=0. In summary the coefficients become,𝒜_i_1i_2… i_N(u)=1/c_0^2𝒰_i_1i_2… i_N(u).and 𝒜_i_1i_2… i_N(u=0)= 0 for N≥ 1. Next we directly obtain the coefficients by directly expanding 𝒫_i_1i_2… i_N(η+u) up to the order N=4. The expansion of the N=0 polynomial is trivial since 𝒫_0(η +u)=c_0𝒜_0 = 1/c_0Expanding the N=1 polynomial, 𝒫_i_1 (η + u) = c_1(η_i_1+u_i_1), so the coefficient is,𝒜(u) = 1/c_0^2c_1 u_i_1.Expanding the N=2 polynomial, 𝒫_i_1i_2(η+u) = c_2 (η_i_1 + u_i_1)(η_i_2+u_i_2) + [c̅_2 (η+ u)^2 + c^'_2] δ_i_1i_2= 𝒫_i_1i_2(η) + (c_2/c_1)[u_i_1𝒫_i_2(η) + u_i_2𝒫_i_1(η)] + (2c̅_2/c_1) u_i_3𝒫_i_3(η) δ_i_1i_2+ 𝒫_0(η)[c_2 u_i_1u_i_2 +c̅_2 u^2 δ_i_1i_2]/c_0, its N=0 coefficient gives that,𝒜_i_1i_2(u) = (c_2 u_i_1u_i_2 + c̅_2 u^2 δ_i_1i_2)I_0,using that c_0^2=1/I_0. Similarly the expansion of the N=3 polynomial, 𝒫_i_1i_2i_3(η + u), contains the N=0 polynomial plushigher order ones that will be omitted for simplicity.𝒫_i_1i_2i_3(η+u) = 𝒫_0/c_0{[ c_3 c̅_2 D c^'_2/c_2(c_2 + Dc̅_2) -c_3 c^'_2/c_2- c̅_3 D c^'_2/(c_2+ Dc̅_2)+ c^'_3+c̅_3u^2 + 2c̅_3 c̅_2 D c^'_2/c_2(c_2+ Dc̅_2) - 2c̅_3 c^'_2/c_2 ] u_i_4δ_i_1i_2i_3i_4 + c_3u_i_1u_i_2u_i_3} + 𝒪(η)Using the equations for the coefficients of polynomials, we have𝒫_i_1i_2i_3(η+u) = 𝒫_0/c_0{[I_2/I_0[c_3+c̅_3(D+2)] + c̅_3u^2 + c^'_3 ]u_i_4δ_i_1i_2i_3i_4 +c_3u_i_1u_i_2u_i_3} + 𝒪(η).Therefore the third order coefficient of the expansion is𝒜_i_1i_2i_3(u) = I_0{ [I_2/I_0[c_3+c̅_3(D+2)]+ c̅_3u^2 + c^'_3]u_i_4δ_i_1i_2i_3i_4 +c_3u_i_1u_i_2u_i_3},Expanding the N=4 polynomial 𝒫_i_1i_2i_3i_4(η+u) in terms ofη is a laborious task as terms in η must be expressed again as functions of polynomials of order N. Nevertheless we only seek the N=0 term and some considerations can be applied to simplify the task. For instance, the odd terms (η_i_1, η_i_1η^2, η_i_1η_i_2η_i_3) do not contribute to the calculation of 𝒜_i_1i_2i_3i_4 and one can take that η^2 = DI_2/I_0 + 𝒪(η). After some algebra, we have:𝒫_i_1i_2i_3i_4(η+u) = 𝒫_0/c_0{c_4 u_i_1u_i_2u_i_3u_i_4 + [ I_2/I_0[c_4+c̅_4(D+4)]+ c_4' + c̅_4 u^2] (δ_i_1i_2u_i_3u_i_4 + δ_i_1i_3u_i_2u_i_4 + δ_i_1i_4u_i_2u_i_3+ δ_i_2i_3u_i_1u_i_4+δ_i_2i_4u_i_1u_i_3+δ_i_3i_4u_i_1u_i_2) +[ 2c̅_4 I_2/I_0u^2 + d^'_4 u^2+ 2DI_2/I_0d̅_4 u^2 + 4I_2/I_0d̅_4 u^2+ u^4 d̅_4 ]δ_i_1i_2i_3i_4} + 𝒪(η).Finally, the N=4 coefficient is𝒜_i_1i_2i_3i_4(u) = I_0 {c_4 u_i_1u_i_2u_i_3u_i_4 + [I_2/I_0[c_4+c̅_4(D+4)]+ c_4' + c̅_4 u^2 ](δ_i_1i_2u_i_3u_i_4 + δ_i_1i_3u_i_2u_i_4+ δ_i_1i_4u_i_2u_i_3+δ_i_2i_3u_i_1u_i_4+δ_i_2i_4u_i_1u_i_3 +δ_i_3i_4u_i_1u_i_2) + [ 2c̅_4 I_2/I_0u^2 +d^'_4 u^2 +2DI_2/I_0d̅_4 u^2 + 4I_2/I_0d̅_4 u^2 + u^4 d̅_4 ] δ_i_1i_2i_3i_4}We summarize the coefficients below, obtained after some additional algebraic manipulation. Notice that they are functions of the integrals I_2N previously defined.𝒜_0(u) = I_0 c_0,𝒜_i_1(u)=I_0 c_1 u_i_1, 𝒜_i_1 i_2(u)= I_0( c_2 u_i_1 u_i_2 + c̅_2 u^2δ_i_1 i_2 ), 𝒜_i_1 i_2 i_3(u)=I_0{ c_3u_i_1 u_i_2u_i_3+ [ c^'_3 (1-J_2 )+c̅_3u^2] (u_i_1δ_i_2 i_3 + u_i_2δ_i_1 i_3 +u_i_3δ_i_1 i_2 ) }, , 𝒜_i_1 i_2 i_3 i_4(u)=I_0{c_4u_i_1 u_i_2u_i_3 u_i_4+[( 1-J_2J_4)c^'_4+c̅_4 u^2] )(u_i_1u_i_2δ_i_3 i_4 + u_i_1u_i_3δ_i_2 i_4+u_i_1u_i_4δ_i_2 i_3+ u_i_2u_i_3δ_i_1 i_4+ u_i_2u_i_4δ_i_2 i_3+u_i_3u_i_4δ_i_1 i_2 )+ [ (2I_2/I_0( c̅_4+(D+2)d̅_4) +d^'_4 ) u^2+d̅_4 u^4 ] δ_i_1 i_2 i_3 i_4}.The contraction between tensors 𝒜_i_1i_2… i_N and 𝒫_i_1i_2… i_N up to N=4 order are obtained below.𝒜_0 𝒫_0 = 1 𝒜_i_1𝒫_i_1 = I_0/I_2 (ξ·u) 𝒜_i_1i_2𝒫_i_1i_2 = I_0/I_4(ξ·u)^2 - I_0(Δ_2^2-1)/I_4Du^2ξ^2 - I_2/I_4Δ_2^2u^2 𝒜_i_1i_2i_3𝒫_i_1i_2i_3 = I_0 (ξ·u) [3(1-J_2)J_4/I_2(D+2)Δ_4^2 - 3J_4/I_4Δ_4^2 u^2- 3 (1-J_2)J_4/I_4Δ_4^2 ξ^2+ 3Δ_4^2-1/I_6(D+2)ξ^2u^2 +1/I_6(ξ·u)^2] 𝒜_i_1i_2i_3i_4𝒫_i_1i_2i_3i_4 = I_0 { c_4^2 (ξu)^2 + 6c_4(c_4' + c̅_4 ξ^2) u^2(ξu)^2+ 3c_4 u^4 (d_4 + d_4'ξ^2 + d̅_4 ξ^4) + 6[ I_2/I_0(c_4+c̅_4(D+4))+ c_4' + c̅_4 u^2 ][ c_4 ξ^2 (ξ·u)^2 + (c^'_4 +c̅_4 ξ^2) [ξ^2 u^2 + (ξu)^2(D+4)]+ (d_4 +d_4'ξ^2 + d̅_4 ξ^4)u^2(D+2)] +3[ u^2 I_2/I_0 2(c̅_4 + D d̅_4 + 2d̅_4)+ d_4' u^2 + d̅_4 u^4 ] [c_4ξ^4 +2(c_4' + c̅_4 ξ^2) ξ^2 (D+2) + (d_4 + d_4' ξ^2 + d̅_4 ξ^4)D (D+2)] } § CONCLUSIONSWe propose here D-dimensional symmetric tensor polynomials orthonormal under a general weight. We show that the number of coefficients of the first five ones (N=0 to 4) matches exactly the number of equations that stem from their orthonormalization, which allows for their obtainment asfunctions of the integrals I_N's. In Statistical Mechanics it is well-known that the D-dimensional Hermite polynomials are a key element to solve the Boltzmann equation for classical particles, which satisfy the Maxwell-Boltzmann statistics <cit.>. The present generalized polynomials are applicable to semi-classical fluids where the particles obey the Bose-Einstein and Fermi-Dirac statistics <cit.>. The proposed generalized polynomials allows for the definition of orthonormal multipoles as shown in case of the Yukawa potential. We foresee many other applications because the proposed polynomials take into account the expanded function as the weight that render them orthonormal in the D-dimensional space.R. C. V. Coelho thanks to FAPERJ and to the European Research Council (ERC) Advanced Grant 319968-FlowCCS for the financial support and to Hans J. Herrmann for the kind hospitality at ETH Zürich. § TENSORIAL IDENTITIES INVOLVING THE I_N Here we obtain especial formulas derived from the definition of the I_N's given in Eq.(<ref>) and obtained from tensorial contractions of the tensor ξ_i_1⋯ξ_i_N, and, consequently, from the δ_i_1⋯ i_N.∫ d^D ξ ω( ξ ) = I_0. ∫ d^D ξ ω( ξ )ξ_i_1ξ_j_1=I_2 δ_i_1 j_1,∫ d^D ξ ω( ξ )ξ^2 =D I_2. ∫ d^D ξ ω( ξ )ξ_i_1ξ_i_2ξ_j_1ξ_j_2 =I_4 δ_i_1 j_1 i_2 j_2,∫ d^D ξ ω( ξ )ξ_i_1ξ_i_2ξ^2 = (D+2) I_4 δ_i_1 j_1,∫ d^D ξ ω( ξ )ξ^4 = (D+2)D I_4. ∫ d^D ξ ω( ξ )ξ_i_1ξ_i_2ξ_i_3ξ_j_1ξ_j_2ξ_j_3 =I_6 δ_i_1 j_1 i_2 j_2 i_3 j_3,∫ d^D ξ ω( ξ )ξ_i_1ξ_i_2ξ_j_1ξ_j_2ξ^2 =(D+4) I_6 δ_i_1 j_1 i_2 j_2, ∫ d^D ξ ω( ξ )ξ_i_1ξ_i_2ξ^4 = (D+4)(D+2) I_6 δ_i_1 j_1, ∫ d^D ξ ω( ξ )ξ^4 = (D+4)(D+2)D I_6 . ∫ d^D ξ ω( ξ )ξ_i_1ξ_i_2ξ_i_3ξ_i_4ξ_j_1ξ_j_2ξ_j_3ξ_j_4 =I_8 δ_i_1 j_1 i_2 j_2 i_3 j_3 i_4 j_4,∫ d^D ξ ω( ξ )ξ_i_1ξ_i_2ξ_i_3ξ_j_1ξ_j_2ξ_j_3ξ^2 =(D+6)I_8 δ_i_1 j_1 i_2 j_2 i_3 j_3, ∫ d^D ξ ω( ξ )ξ_i_1ξ_i_2ξ_j_1ξ_j_2ξ^4 =(D+6)(D+4)I_8 δ_i_1 j_1 i_2 j_2, ∫ d^D ξ ω( ξ )ξ_i_1ξ_j_1ξ^6 =(D+6)(D+4)(D+2)I_8 δ_i_1 j_1, ∫ d^D ξ ω( ξ )ξ^8 =(D+6)(D+4)(D+2)DI_8. | http://arxiv.org/abs/1703.08670v1 | {
"authors": [
"Mauro M. Doria",
"Rodrigo C. V. Coelho"
],
"categories": [
"math-ph",
"cond-mat.stat-mech",
"math.MP"
],
"primary_category": "math-ph",
"published": "20170325094706",
"title": "Chebyshev, Legendre, Hermite and other orthonormal polynomials in D-dimensions"
} |
theoremTheorem[section] lemma[theorem]Lemma corollary[theorem]Corollary proposition[theorem]Proposition definition definitionDefinition[section] remark[theorem]Remark example[theorem]Example assumption[theorem]Assumptionmytheorem[1]theorem-1 for:=definition,remark,plain @addto@macroth@ @preskip #1 1LS10ptRandom Sampling of Latin squares via binary contingency tables and probabilistic divide-and-conquerStephen DeSalvo[Department of Mathematics, University of California Los Angeles. [email protected]]March 24, 2017.8 Abstract. We demonstrate a novel approach for the random sampling of Latin squares of order n via probabilistic divide-and-conquer.The algorithm divides the entries of the table modulo powers of 2, and samples a corresponding binary contingency table at each level. The sampling distribution is based on the Boltzmann sampling heuristic, along with probabilistic divide-and-conquer. Keywords. Random Sampling, Latin Square, Sudoku, Probabilistic Divide-and-Conquer, Rejection Sampling MSC classes: 60C05, 65C50, 60-04 CCS 2012: Mathematics of computing∼Probabilistic algorithms, Theory of computation∼Generating random combinatorial structures§ INTRODUCTION§.§ Motivation When tasked with random sampling from a complicated state space, a popular choice is to run a rapidly mixing Markov chain.Markov chains have proven to be an incredibly powerful and versatile tool, however, there are several reasons why one would consider alternative methods.In particular, unless the chain starts off in stationarity, or has been coupled from the past <cit.>, the error is seldom zero after a finite number of steps; quite often this is an acceptable tradeoff, and through a more complicated analysis one can bound the sampling error.For sampling problems with a large number of constraints, however, it is often difficult to analyze the resulting Markov chain and prove mixing time bounds.In addition, there is no general scheme to parallelize the sequential steps in a Markov chain, and many naive approaches can lead to unintended sampling bias <cit.>.Another paradigm of random sampling, popular in combinatorics, is the Boltzmann sampler <cit.>, where one samples from some combinatorial structure via independent random variables, determined by the form of the generating function to closely resemble the true distribution of component sizes in the random combinatorial structure.The result is a random combinatorial object with a random set of statistics, referred to as a grand canonical ensemble in the statistical mechanics literature.For a fixed set of conditions, an exact sample can be obtained by sampling repeatedly until all of the conditions are satisfied, throwing away samples which miss the target <cit.>.We have the structure which is typical of Boltzmann sampling, which is a collection of independent random variables subject to a condition.However, unlike the plethora of combinatorial structures for which a randomly generated structure with random statistics contains an acceptable amount of bias, there are good reasons to demand that all statistics are satisfied, see for example <cit.>.Fortunately, owing to the large amount of independence in the formulation, Boltzmann samplers are embarrassingly parallel, offering many different effective means of parallelizing the computation.Unfortunately, as with many multivariate combinatorial structures, the rejection costs associated with waiting until all statistics match exactly are prohibitively large, even with effective parallelizing. The Boltzmann sampler has been described as a generalization of the table methods of Nijenhuis and Wilf, also known as the recursive method, see <cit.>, although from our point of view it has a distinctly different flavor.The recursive method champions creating a table of numerical values which can be used to generate individual components of a sample in its correct proportion in an unbiased sample. This approach, however, demands the computing of a lookup table, which can be both prohibitively large in size and prohibitively long to compute each entry, typically by a recursion. Our approach lies somewhere in between the recursive method and Boltzmann sampling, with a distinctly probabilistic twist, and which avoids Markov chains altogether. Our approach, probabilistic divide-and-conquer (PDC) <cit.>, provides an object which satisfies all constraints, and which targets the marginal distribution of components, in this case entries in a table, according to the Boltzmann sampling principle. Rather than sample all entries of the table at once and apply a rejection function which is either 0 or 1, we instead sample one bit of each entry in the table, one at a time, essentially building the table via its bits, starting with the least significant bit.In the case where rejection sampling probabilities are known or can be computed to arbitrary accuracy, the algorithm is unbiased.Probabilistic divide-and-conquer is similar in many respects to the recursive method, by approaching the problem in pieces and selecting a piece in proportion to already observed pieces.However, whereas the recursive method typically involves a more geometric/spatial decomposition of a finite, discrete sample space, we instead champion a probabilistic decomposition of the sample space, by decomposing the random variables themselves which describe random component-sizes.This approach also generalizes in a straightforward manner to sampling from continuous sample spaces and lower dimensional subspaces, see <cit.>, as long as the random variables can be decomposed effectively. In the remainder of this section, we define (r,c)-contingency tables and Latin squares of order n, and describe some of the standard algorithms to randomly sample them.Section <ref> contains the highest-level explanation of our proposed algorithm for Latin squares of order n, and in Section <ref> we present the probabilistic tools which justify this approach.Section <ref> contains the complete statements of the algorithms, along with a word of caution in the final subsection. §.§ (r,c)-contingency tablesAn (r,c)-contingency table is an m by n table of nonnegative integer values with row sums given by r = (r_1, …, r_m) and column sums given by c = (c_1, …, c_n).A table for which the entries are further assumed to be in the set {0,1} is called a binary (r,c)-contingency table. The exact, uniform sampling from the set of (r,c)-contingency tables is a well-studied problem.Owing to the large parameter space, there are a plethora of results spanning many decades pertaining just to counting the number of such tables, see for example <cit.>.There is also interest in various Markov chain approaches, including coupling from the past, see for example <cit.>. The random sampling of contingency tables is an extensive topic, and we do not attempt to recount all previously known results in this area, and instead refer the interested reader to, e.g., the introduction in <cit.> and the references therein.As mentioned previously, the main sampling approach traditionally championed is to use a Markov chain, e.g., starting with the general framework in <cit.>, to define an appropriate state transition matrix, and then prove that such a chain is rapidly mixing; indeed, this approach has been profoundly fruitful, see for example <cit.>.Explicitly, one state transition championed in <cit.> is to sample two ordered rows and two ordered columns uniformly at random, and, if possible, apply the transformation to their entries ([ a_11 a_12; a_21 a_22 ]) →([ a_11+1 a_12 - 1;a_21 -1 a_22 + 1 ]).If such a transformation would force the table to lie outside the solution set, then a different pair of ordered rows and columns is generated.This chain was shown in <cit.> to be ergodic and converge to the uniform distribution over the set of contingency tables.Mixing times were later proved in <cit.> in various contexts.Other Markov chains have also been proposed, see for example <cit.>.The recent book <cit.> also contains many more examples involving Markov chains and coupling from the past to obtain exact samples. An arguably unique approach to random sampling of contingency tables is contained in <cit.>. The algorithm associates to each state a parallelepiped with respect to some basis, and defines some convex set which contains all of the parallelepipeds.Then, one samples from this convex set, and if the sample generated lies within one of the parallelepipeds, it returns the corresponding state; otherwise, restart.This algorithm was shown to be particularly effectivein <cit.> when the row sums are all Ω(n^3/2 mlog m) and the column sums are all Ω(m^3/2 nlog n). Recently, a self-similar PDC algorithm was championed for the random sampling of contingency tables <cit.>, offering an arguably new approach to random sampling of these intricate structures. In related work, the author demonstrated how to utilize PDC to improve upon existing algorithms for exact random sampling of Latin squares of order n <cit.>.The current work, motivated by improving further still the random sampling of Latin squares of order n, extends the original PDC sampling algorithm for contingency tables to the more general case when certain entries are forced to be 0, see for example <cit.>.The probabilistic analysis is straightforward, although the computational complexity changes drastically.Nevertheless, our numerical experiments demonstrate that this approach is simple, practical and executes fairly rapidly for a large class of tables.Thus, we champion our approach for practitioners looking for simple, alternative methods for sampling quickly from these intricate structures. §.§ Latin squares of order n A Latin square of order n is an n × n table of values such that the numbers from the set {1,2,…, n} appear exactly once in each row and in each column. There are many techniques available for random sampling of Latin squares of order n, see for example <cit.>.A Markov chain approach is contained in <cit.>; in particular, they construct and analyze two different Markov chains.The transition states are described via an explicit algorithm, requiring O(n) time to transition from one state to the next, and guaranteed to satisfy all of the Latin square constraints.However, as far as we are aware, even though the stationary distribution was proved to be uniform over the set of Latin squares of order n, neither of the Markov chains presented have been shown to be rapidly mixing.Another approach is to decompose the entries of a Latin square as sums of mutually disjoint permutation matrices, as in <cit.>.This approach has practical limitations, as the straightforward rejection algorithm is prohibitive, and the more involved approach in <cit.> requires an auxiliary computation which dominates the cost of the algorithm.We suspect that this approach may benefit from a probabilistic divide-and-conquer approach, though we have not carried out the relevant analysis. An informal description of our algorithm for random sampling of Latin squares of order n is as follows; see Algorithm <ref> for the complete description.For simplicity of exposition, we assume n = 2^m for some positive integer m, which has no theoretical significance, but avoids cumbersome notation involving rounding.First, sample an n × n binary contingency table with all row sums and column sums equal to n/2.Then consider the subset of entries in which a 1 appears, of which there are n/2 in each row and column, and sample within that subset of entries an (n/2) × (n/2) binary contingency table with all row sums and column sums equal to n/4; do the same for the subset of entries in which a 0 appears. Repeat this process through all m levels.By interpreting the sequence of m 1s and 0s as the binary expansion of a positive integer, we obtain a Latin square of order n. This idea, that of generating a Latin square by its bits, could be considered a straightforward divide-and-conquer algorithm if one simply desires an object from the solution set.What makes our treatment more intricate is our approximation heuristic, designed to target the uniform distribution over the solution set; see Section <ref>. Of course, in order to avoid bias in the sampling algorithm, we must sample each of the n/2^ℓ× n/2^ℓ binary contingency tables in their correct proportion of partially completed Latin squares of order n; this is not such a trivial task, and requires either counting all possible completions, a computationally extensive task, or a more direct analysis of the resulting multivariate probability governing the acceptance probability, given explicitly in Equation (<ref>).While both approaches present technical difficulties, the probabilistic formulation yields a natural approximation heuristic, which we present in Equation (<ref>), which is computable in polynomial time.It assumes independence of columns, a reasonable approximation heuristic in many parameter spaces of interest, while still enforcing several necessary conditions.A more complete probabilistic description of our heuristic is contained in Section <ref>. The following is an example of how to apply the algorithm described above. [( [ 1 1 0 1 0; 0 1 1 0 1; 1 0 1 0 1; 1 0 1 1 0; 0 1 0 1 1 ]);↙↘; ([ 1 0 ▪ 1 ▪; ▪ 1 0 ▪ 1; 0 ▪ 1 ▪ 1; 1 ▪ 1 0 ▪; ▪ 1 ▪ 1 0 ]) ( [ ▪ ▪ 0 ▪ 1; 1 ▪ ▪ 0 ▪; ▪ 0 ▪ 1 ▪; ▪ 1 ▪ ▪ 0; 0 ▪ 1 ▪ ▪ ]);↙; ( [ 1 ▪ ▪ 0 ▪; ▪ 1 ▪ ▪ 0; ▪ ▪ 0 ▪ 1; 0 ▪ 1 ▪ ▪; ▪ 0 ▪ 1 ▪ ]);]This corresponds to the following order 5 Latin square([ 5 1 2 3 4; 4 5 1 2 3; 1 2 3 4 5; 3 4 5 1 2; 2 3 4 5 1 ]).It is not a priori obvious whether the bits at different levels are independent, and whether certain configurations of binary tables at a given level can potentially be completed by a larger number of Latin squares than other configurations.For n=6, we are able to provide a negative answer by a simple observation.According to the OEIS sequence A058527 <cit.>, the number of 6 × 6 binary contingency tables with row sums and column sums equal to 3 is 297200, which does not divide the number of Latin squares of order 6, which is 812851200, see for example <cit.>.Thus, some of theconfigurations of binary tables must necessarily yield a different number of completable Latin squares. Nevertheless, our approach exploits the principles of Boltzmann sampling, which has been previously utilized in various contexts involving contingency tables, see for example <cit.>.There is an extensive history related to partially completed Latin squares of order n, which are Latin squares of order n for which only some of the n^2 entries have been filled in, and none of those entries a priori violate any of the Latin square conditions.In particular, the treatment in <cit.> related to partially completed Latin squares pertains solely to leaving entire entries filled or unfilled, whereas the algorithm described above is of a distinctly different flavor.For example, <cit.> states that all Latin squares of order n with their first k rows filled in can be extended to a partially completed Latin square with the first k+1 rows filled in, for k=1,2,…,n-1.There are also ways of filling in elements for which no possible Latin square can be realized given those elements; in general, deciding whether a Latin square of order n with an arbitrary set of squares filled in can be completed is an NP-complete problem, see <cit.>.It would be interesting to explore the analogous analysis for our proposed bit-by-bit approach. § PROBABILISTIC APPROACH§.§ Probabilistic divide-and-conquer Probabilistic divide-and-conquer (PDC) is an approach for exact sampling by dividing up the sample space into two pieces, sampling each one separately, and then piecing them together appropriately <cit.>.In order to sample using PDC, one starts with a sample space Ω decomposed into two separate sets Ω = 𝒜×ℬ, and writes the target distribution as Ł(S) = Ł((A, B) | E), where A ∈𝒜 and B ∈ℬ have given distributions and are independent, and E ⊂𝒜×ℬ is some measurable event of the sample space. Whereas rejection sampling can be described as sampling a from Ł(A) and b from Ł(B), and then checking whether (a,b) ∈ E (see Algorithm <ref> below), PDC samples x from Ł(A | E) and y from Ł(B | E, a), and returns (x,y) as an exact sample (see Algorithm <ref> below). PDC Lemma <cit.>. Suppose E is a measurable event of positive probability.Suppose X is a random element ofwith distribution Ł(X) = Ł( A | E ),andY is a random element ofwith conditional distributionŁ(Y | X=a ) = Ł( B | E, A=a ).Then (X,Y) =^d S, i.e., the pair (X,Y) has the same distributionas S, given by (<ref>). Often, however, the conditional distributions in Algorithm <ref> are not practical to sample from, and so the following PDC algorithm which uses rejection sampling has been proven to be effective and practical in many situations, see for example <cit.>. Of course, while we have started with the problem of sampling from one conditional distribution, now we have two! Fortunately, by splitting up the randomness into two pieces, which we have the freedom to choose based on available data, it is possible to more effectively target the smaller conditional distribution.A large class of PDC algorithms, for example, choosesandso that, conditional on observing A = a, {b ∈ : (a,b) ∈ E} has cardinality at most 1, i.e., the second stage of Algorithm <ref> is deterministic, and all of the randomness is contained in the first stage. Further details and examples can be found in <cit.>.The example below shows that this idea, decomposing the sample space into two pieces, one of which is deterministic, is a particularly good strategy which requires very little structural knowledge of the sample space, and provides a guaranteed speedup over traditional Boltzmann samplers.Let us pause at this point to note a particularly elegant divide-and-conquer strategy for the random sampling of combinatorial structures of the form Ł( (X_1, …, X_n) | ∑_i=1^n iX_i = n),where X_1, X_2, …, X_n are independent random variables, typically coming from some family of distributions like Poisson, geometric, or Bernoulli. Such structures are typical in combinatorics, see for example <cit.>, and represent an initial benchmark for improvements to the random sampling of classical combinatorial structures like integer partitions, set partitions, etc.Consider two divisions: the first isA_1 = (X_1, X_2, …, X_n),B_1 = ∅and the second is, for any I ∈{1,2,…,n}, A_2 = (X_1, …, X_I-1, X_I+1, …, X_n),B_2 = (X_I),with E = {∑_i=1^n iX_i = n}.The first division is in fact rejection sampling <cit.>, whereas the second has been dubbed PDC deterministic second half, see <cit.>. In order to sample from Ł(A_2 | E) = Ł((X_1, …,X_I-1,X_I+1,…, X_n) | ∑_i=1^n iX_i = n), we use what we call soft rejection sampling, see <cit.>, see also <cit.>.That is, we sample from Ł(A_2), i.e., the unconditional distribution, and apply a rejection probability to obtain the appropriate conditional distribution.The rejection probability is then proportional to( (X_2, …, X_n) = (x_2,…,x_n) | ∑_i=1^n iX_i = n)/( (X_2, …, X_n) = (x_2,…,x_n) )= (X_1 = n - ∑_i=2^n iX_i | X_2=x_2, …, X_n=x_n)/(∑_i=1^n iX_i = n).= (X_1 = n - ∑_i=2^n ix_i)/(∑_i=1^n iX_i = n).Specifically, we obtain the unbiased sampling algorithm in Algorithm <ref> below, see <cit.>, where U denotes a uniform random variable between 0 and 1, independent of all other random variables.Choose any I ∈{1, 2, …, n}.Algorithm <ref> is faster, in terms of the expected number of random bits required, than an exact Boltzmann sampler by a factor of (max_ℓ(X_I = ℓ))^-1. The acceptance set for Boltzmann sampling can be formulated as {(X_1, …, X_I-1, X_I+1,…, X_n, U) : ∑_i≠ I iX_i ≤ nU < (X_I = n - ∑_i ≠ I iX_i) },and the acceptance set for PDC deterministic second half can be formulated as {(X_1, …, X_I-1, X_I+1,…, X_n, U) : ∑_i≠ I iX_i ≤ nU < (X_I = n - ∑_i ≠ I iX_i)/max_ℓ(X_I = ℓ)}. The PDC deterministic second half approach is not simply an application of soft rejection sampling, see <cit.>.Soft rejection sampling is indeed a great complementary technique for which sampling from conditional distributions can be made more accessible, but it is by no means necessary in order to apply PDC in general, especially if one can sample from the conditional distributions directly. A surprisingly efficient division occurs when the sets A, B, and E are such that Ł( B | E, A=a ) is trivial for each a ∈𝒜.In this case, it was shown in <cit.>, that a nontrivial speedup can be obtained by relatively simple arguments. A more recent application in <cit.> uses the recursive method <cit.> in order to obtain both the rejection probability s(a) along with the conditional distributions Ł(B | (a,B) ∈ E).It was proved to have a smaller rejection rate than exact Boltzmann sampling, and also requires less computational resources than the full recursive method.We now recall a division from <cit.> which is recursive and self-similar, and serves as the inspiration for the algorithm championed in this paper.Let 0<x<1, and let Z_1(x), Z_2(x), …, Z_n(x) denote independent geometric random variables with Z_i ≡ Z_i(x) having distribution which is geometric with parameter 1-x^i, i=1,2,…,n.Conditional on ∑_i=1^n iZ_i(x) = n, the random variables Z_i(x), i=1,2,…, n, represent the number of parts of size i in a uniformly chosen integer partition of size n; see <cit.>.Our problem is to sample from Ł(Z_1(x), …, Z_n(x)| ∑_i=1^n iZ_i(x) = n). The division championed in this case is inherently probabilistic, and relies on a property of geometric random variables.Namely, letting G(q) and G'(q^2) denote independent geometric random variables with parameters 1-q and 1-q^2 respectively, and letting B(p) denote an independent Bernoulli random variable with parameter p, we have the decomposition G(q) = B(q/1+q) + 2G'(q^2),where the equality is in distribution.In other words, this decomposition shows that the bits in a geometric random variable are independent, and specifies their distribution. It also suggests a PDC algorithm; that is, sample the least significant bits first, one at a time, and then the remaining part is twice a copy of the first part with a different parameter.As a bonus, many sampling algorithms have a tilting parameter which can be chosen arbitrarily, as is the case for parameter x for integer partitions; after each stage of the PDC algorithm, we may adjust the tilting parameter to more optimally target the remaining sampling problem. Explicitly,we use the divisionA = (ϵ_1, …, ϵ_n),B = (2Z_1(x^2), 2Z_2(x^4), …, 2 Z_n(x^2n)),where ϵ_i denotes the least significant bit of Z_i, distributed as a Bernoulli random variable with parameter x^i/(1+x^i).Even though we have written the geometric random variables in B in terms of a new parameter x^2, since the distribution we are sampling from is conditionally independent of x, we are free to choose any x' ∈ (0,1) which more optimally targets the sampling distribution, which is realized after the bits in A are sampled. An application of Algorithm <ref> in this setting is the following: Sample from A = (ϵ_1, …, ϵ_n) and let a = ∑_i=1^n i ϵ_i. Then we have s(a) = (∑_i=1^n (2i) Z_2i = n-a)/max_ℓ( ∑_i=1^n (2i)Z_2i = ℓ) =(∑_i=1^(n-a)/2 i Z_i = n-a/2)/max_ℓ( ∑_i=1^ℓ iZ_i = ℓ) = p(n-a/2)x^(n-a)/2∏_i=1^(n-a)/2 (1-x^i)/max_ℓ p(ℓ) x^ℓ∏_i=1^ℓ (1-x^i),where p(n) denotes the number of integer partitions of n, which is an extremely well-studied sequence; see for example <cit.> and the references therein.In addition, we have Ł(B | (a,B) ∈ E) is equivalent to the original problem with n replaced by (n-a)/2, hence the description of this algorithm as self-similar.In terms of random sampling of integer partitions, this is also within a constant factor of the fastest possible algorithm; see <cit.>. More importantly, however, is that this type of divide-and-conquer is inherently probabilistic, table-free, and can be applied to other structures which are modeled by geometric random variables, namely,contingency tables. Previous applications of PDC have considered mutually independent random variables X_1, X_2, …, X_n, with target space given by some measurable event E, with the goal to sample from the distributionŁ((X_1, …, X_n)| E).We assume that the unconditional distributions Ł(X_1), …, Ł(X_n) are known and can be sampled.For general two-dimensional tables, we consider sampling problems of the form: X_1,1, …, X_m,n are mutually independent random variables, E is some measurable event, and the goal is to the sample from the distributionŁ([ X_1,1,… X_1,n,;⋮⋮⋮; X_m,1,…X_m,n ] |E ). The uniform distribution over all (r,c)-contingency tables can be obtained, see for example <cit.>, by taking X_i,j to be geometrically distributed with parameter p_i,j = m/m+c_j, i=1,…,m, j=1,…,n, subject to the measurable event E ≡ E_r,c = { [ ∑_ℓ=1^n X_i,ℓ = r_i, ∀ 1≤ i≤ m;;; ∑_ℓ=1^m X_ℓ,j = c_j, ∀ 1 ≤ j ≤ n ]}.Taking X_i,j to be Bernoulli distributed with parameter p_i,j / (1+p_i,j), we obtain the uniform distribution over the set of (r,c)-binary contingency tables.In the case of Latin squares of order n, we consider the parameterization which consists of n × n tables of values, each row of which is a permutation of elements {1,2,…,n}, and each column is a permutation of elements {1,2,…,n}.Let U_i,j denote a collection of i.i.d. discrete uniform random variables from the set {1,2,…,n}, 1≤ i, j ≤ n, and let S_n denote the set of all permutations of the elements {1,…,n}.The sample space for random sampling of Latin squares is then given by Equation (<ref>) with Ł(X_i,j) = Ł(U_i,j) and measurable event E given by E = { [ (X_i,1,…,X_i,n) ∈ S_n,∀ 1≤ i≤ m;; ; (X_1,j,…,X_m,j) ∈ S_n,∀ 1 ≤ j ≤ n ]}. Once this probabilistic framework has been established, the next task is to devise a PDC division that is both practical and amenable to the resulting analysis.We motivate our PDC division in the next section.§.§ An heuristic for contingency tables We first recall a division from <cit.> which is recursive and self-similar, and serves as the inspiration for the algorithm championed in this paper.Let 0<x<1, and let Z_1(x), Z_2(x), …, Z_n(x) denote independent geometric random variables with Z_i ≡ Z_i(x) having distribution which is geometric with parameter 1-x^i, i=1,2,…,n.Conditional on ∑_i=1^n iZ_i(x) = n, the random variables Z_i(x), i=1,2,…, n, represent the number of parts of size i in a uniformly chosen integer partition of size n; see <cit.>.Suppose, for the purpose of this demonstration, that we wish to sample from Ł( (Z_1(x), …, Z_n(x))| ∑_i=1^n iZ_i(x) = n). The division championed in this case is inherently probabilistic, and relies on a property of geometric random variables.Namely, letting G(q) and G'(q^2) denote independent geometric random variables with parameters 1-q and 1-q^2 respectively, and letting B(p) denote an independent Bernoulli random variable with parameter p, we have the decomposition G(q) D= B(q/1+q) + 2G'(q^2),where the equality is in distribution.In other words, this decomposition shows that the bits in a geometric random variable are independent, and specifies their distribution. It also suggests a PDC algorithm; that is, sample the least significant bits first, one at a time, and then the remaining part has a distribution which is two times a geometric distribution with parameter q^2 in place of q.As a bonus, many sampling algorithms have a tilting parameter which can be chosen arbitrarily, as is the case for parameter x for integer partitions; after each stage of the PDC algorithm, we may adjust the tilting parameter to more optimally target the remaining sampling problem. Explicitly,we use the divisionA = (ϵ_1, …, ϵ_n),B = (2Z_1(x^2), 2Z_2(x^4), …, 2 Z_n(x^2n)),where ϵ_i denotes the least significant bit of Z_i, distributed as a Bernoulli random variable with parameter x^i/(1+x^i).Even though we have written the geometric random variables in B in terms of a new parameter x^2, since the distribution we are sampling from is conditionally independent of x, we are free to choose any x' ∈ (0,1) which more optimally targets the sampling distribution, which is realized after the bits in A are sampled. An application of Algorithm <ref> in this setting is the following: Sample from A = (ϵ_1, …, ϵ_n) and let a = ∑_i=1^n i ϵ_i. Then we have s(a) = (∑_i=1^n (2i) Z_2i = n-a)/max_ℓ( ∑_i=1^n (2i)Z_2i = ℓ) =(∑_i=1^(n-a)/2 i Z_i = n-a/2)/max_ℓ( ∑_i=1^ℓ iZ_i = ℓ) = p(n-a/2)x^(n-a)/2∏_i=1^(n-a)/2 (1-x^i)/max_ℓ p(ℓ) x^ℓ∏_i=1^ℓ (1-x^i),where p(n) denotes the number of integer partitions of n, which is an extremely well-studied sequence; see for example <cit.> and the references therein.In addition, we have Ł(B | (a,B) ∈ E) is equivalent to the original problem with n replaced by (n-a)/2, hence the description of this algorithm as self-similar.In terms of random sampling of integer partitions, this is also within a constant factor of the fastest possible algorithm due to the efficient computation of p(n); see <cit.>. More importantly, however, is that this type of divide-and-conquer is inherently probabilistic, table-free, and can be applied to other structures which are modeled by geometric random variables, namely,contingency tables.Next, we revisit and generalize an approach in <cit.> for random sampling of nonnegative integer-valued (r,c)-contingency tables inspired by the above application for integer partitions.The generalization fixes an arbitrary set of entries to be 0, which doesn't change the approximation heuristic; however, it does a priori drastically change the computational complexity of computing the exact sampling rejection probability.We start with a well-known probabilistic model for the entries in a random contingency table, generalized to include entries which are forced to be 0. In this section, we let W be any given m× n matrix with values in {0,1}.For each j=1,2,…,n, we define J_j := {1,…, n}∖{ i : W(i,j) = 1}.Similarly, for each i=1,2,…,m, we define I_i := {1,…, n}∖{j : W(i,j) = 1}.Also, we let h_j = ∑_i=1^m W(i,j) denote the number of entries forced to be zero in column j, for j=1,2,…,n. Let _W=(X_ij)_1 ≤ i ≤ m, 1 ≤ j ≤ n denote a collection of independent geometric random variables with parameters p_ij, such that W(i,j) = 1 implies X_i,j is conditioned to have value 0.If p_ij has the form p_ij = 1 - α_i β_j, then _W is uniform restricted to (r,c)-contingency tables with zeros in entries indicated by W. Let =(X_ij)_1 ≤ i ≤ m, 1 ≤ j ≤ n denote a collection of independent geometric random variables with parameters p_ij, where p_ij has the form p_ij = 1 - α_i β_j.We have (=ξ) =∏_i,j(X_ij=ξ_ij) =∏_i,j(α_i β_j)^ξ_ij(1-α_i β_j) =∏_iα_i^r_i∏_jβ_j^c_j∏_i,j(1-α_i β_j).Since this probability does not depend on ξ, it follows that the restriction ofto (r,c)-contingency tables is uniform.As the collection of random variables are independent, conditioning on any X_i,j = 0 only changes the constant of proportionality, and does not affect the dependence on the ξ, hence (_W=ξ) =∏_i,j : W(i,j) = 0(X_ij=ξ_ij) =∏_i,j: W(i,j)=0(α_i β_j)^ξ_ij(1-α_i β_j) =∏_iα_i^r_i∏_jβ_j^c_j∏_i,j(1-α_i β_j);i.e., it follows that the restriction of _W to (r,c)-contingency tables with forced zero entries indicated by W is uniform.Suppose _W is a collection of independent geometric random variables, where X_i,j has parameter p_ij = m-h_j/m-h_j+c_j, for all pairs (i,j) such that W(i,j) = 0, and p_i,j = 1 for all pairs (i,j) such that W(i,j) = 1.Then the expected column sums of _W are c_1, c_2, …, c_n, and the expected row sums are ∑_j∈ I_1c_j/m-h_j, …, ∑_j∈ I_mc_j/m-h_j.Note first that X_i,j = p_i,j^-1 - 1.Thenfor any j=1,2,…,n,∑_i=1^m X_i,j = ∑_i ∈ J_jm-h_j+c_j/m-h_j - 1= c_jand similarly for any i=1,2,…, m, ∑_j=1^n X_i,j = ∑_j ∈ I_im-h_j+c_j/m-h_j-1 = ∑_j∈ I_ic_j/m-h_j. Next, we describe the PDC algorithm and how the heuristic is utilized. Suppose we have matrices W and 𝒪 with entries in {0,1}.For any set of row sums and column sums (r,c), let Σ(r,c,𝒪, W) denote the number of nonnegative integer-valued (r,c)-contingency tables with entry (i,j) forced to be even if the (i,j)th entry of 𝒪 is 1, and entry (i,j) forced to be 0 if the (i,j)th entry of W is 1.Let 𝒪_i,j denote the matrix which has entries with value 1 in the first j-1 columns, and entries with value 1 in the first i rows of column j, and entries with value 0 otherwise.The algorithm we champion for contingency tables starts by considering the first entry in the first column not forced to be 0, say at entry (s,1), and denote the least significant bit by ϵ_s,1.As we are in the first column, we have Ł(ϵ_s,1 | E) = Ł((q_1/1+q_1) | E), where E is given in Equation (<ref>).In lieu of sampling from this conditional distribution directly, and a more practical alternative is to sample from Ł((q_1/1+q_1)) and reject the outcome with probability proportional to 1-(E | ϵ_s,1). We define r_i(k) = (r_1, …, r_i - k, … r_m) and c_j(k) = (c_1, …, c_j - k, … c_n) for k ∈{0,1}.By Lemma <ref>, we have (E | ϵ_s,1 = k) = Σ(r_s(k),c_1(k),W,𝒪_i,j) · (q_1^c_1-k(1-q_1^2)) (1-q_1)^m-h_1∏_j=2^n q_j^c_j (1-q_j)^m-h_j.Then, since we reject in proportion to this probability, we normalize by all terms which do not depend on k, which gives(E | ϵ_s,1 = k) ∝Σ(r_s(k),c_1(k),W,𝒪_i,j) ·q_1^-k,where we use the notation a(k) ∝ b(k) to mean that a(k)/b(k) is equal to a positive constant independent of k (but potentially depending on all other parameters).Once we accept a value for ϵ_s,1, we then move to the next bit further down in the column and sample analogously, continuing in this manner column by column, left to right.The following is a probabilistic formulation of the exact rejection probability for an arbitrary entry (i,j), which assumes that all bits above the entry in the current column have already been sampled, as well as all bits in all columns to the left of (i,j); see Section <ref> for the complete description of each random variable: f(i,j,k, r, c,W)∝( [ ∑_ℓ∈ J_1^ℓ < j 2ξ”_1,ℓ(q_ℓ^2, c_ℓ) + η_1,j,i'(q_j, c_j)+∑_ℓ∈ J_1^ℓ > jξ'_1,j(q_ℓ, c_ℓ)= r_1; ∑_ℓ∈ J_2^ℓ < j 2ξ”_2,ℓ(q_ℓ^2, c_ℓ) + η_2,j,i'(q_j, c_j)+∑_ℓ∈ J_2^ℓ > jξ'_2,j(q_ℓ, c_ℓ)= r_2;⋮; ∑_ℓ∈ J_i-1^ℓ < j 2ξ”_i-1,ℓ(q_ℓ^2, c_ℓ) + η_i-1,j,i'(q_j, c_j)+∑_ℓ∈ J_i-1^ℓ > jξ'_i-1,j(q_ℓ, c_ℓ)= r_i-1;∑_ℓ∈ J_i^ℓ < j2ξ”_i,ℓ(q_ℓ^2, c_ℓ) + η_i,j,i”(q_j, c_j)+∑_ℓ∈ J_i^ℓ > jξ'_i,j(q_ℓ, c_ℓ)= r_i-k;∑_ℓ∈ J_i+1^ℓ < j2ξ”_i+1,ℓ(q_ℓ^2, c_ℓ) + η_i+1,j,i”(q_j, c_j)+∑_ℓ∈ J_i+1^ℓ > jξ'_i+1,j(q_ℓ, c_ℓ)= r_i+1;⋮;∑_ℓ∈ J_m^ℓ < j2ξ”_m,ℓ(q_ℓ^2, c_ℓ) + η_m,j,i”(q_j, c_j)+∑_ℓ∈ J_m^ℓ > jξ'_m,j(q_ℓ, c_ℓ)= r_m;]) ×( ∑_ℓ∈ I_i, ℓ≤ i 2 ξ_ℓ,j(q_j^2) + ∑_ℓ∈ I_i, ℓ > iξ_ℓ,j(q_ℓ) = c_j - k ). If we could evaluate the multivariate probability above exactly, or to some arbitrarily defined precision, then the bit-by-bit sampling approach would yield an exact sampling algorithm.However, we champion the following approximation to Equation (<ref>) based on the heuristic that the dependencies between the random variables are strongest along the i-th row and the j-th column:F(i,j,k, r, c,W) := ( ∑_ℓ∈ I_i, ℓ < j 2ξ”_i,ℓ(q_ℓ^2, c_j, ϵ) + 2η”_i,j,i + ∑_ℓ∈ I_i, ℓ > jξ'_i,ℓ(q_ℓ) = r_i - k)×( ∑_ℓ∈ J_j, ℓ≤ i 2ξ_ℓ,j(q_j^2) + ∑_ℓ∈ J_j, ℓ > iξ_ℓ,j(q_j) = c_j - k),k ∈{0,1}. Note that while this approximation does not encode the essential global information, it nonetheless enforces several necessary conditions like parity of the row and column, and is fast to compute directly.§.§ An heuristic for Latin squaresThe following is a variation of the motivating example of the previous section.Let 0<x<1, and let X_1(x), X_2(x), …, X_n(x) denote independent Bernoulli random variables with X_i ≡ X_i(x) having distribution which is Bernoulli with parameter x^i/1+x^i, i=1,2,…,n.Conditional on ∑_i=1^n iX_i(x) = n, the random variables X_i(x), i=1,2,…, n, represent the number of parts of size i in a uniformly chosen integer partition of size n into distinct part sizes; see <cit.>.As before, suppose we wish to sample from Ł( (X_1(x), …, X_n(x))| ∑_i=1^n iX_i(x) = n).We then consider the following “spatial" PDC divisionA = (X_1, X_3, X_5, …),B = (X_2, X_4, X_6, …).The result is again a self-similar, recursive PDC algorithm, with the rejection function given by s(a) = (∑_i=1^n/2 (2i) X_2i = n-a)/max_ℓ( ∑_i=1^n/2 (2i)X_2i = ℓ) =p_d(n-a/2)x^(n-a)/2∏_i=1^(n-a)/2 (1+x^i)^-1/max_ℓ p_d(ℓ) x^ℓ∏_i=1^ℓ (1+x^i)^-1,where p_d(n) is the number of integer partitions into distinct part sizes, which can be computed efficiently via, e.g., <cit.>. Let _n denote the set of Latin squares of order n, and let _n,m denote the set of all n× n binary contingency tables with line sums m. Consider any Latin square of order n, and consider the map φ_n : _n ⟶_n,⌈ n/2⌉ which replaces each entry having value x with x2, i.e., each entry becomes a 1 if it is odd and 0 if it is even.The result is an n× n binary contingency table with row sums and column sums ⌈ n/2 ⌉. Let us assume as before that n is a perfect power of 2 for simplicity of exposition. By symmetry, each entry in a Latin square is equally likely to be even or odd, subject to the condition that there are an equal number of odd and even entries in each row and column.It is thus natural to conjecture that the joint distribution of the parity of entries in a random Latin square of order n is close in some sense to a joint distribution of n× n i.i.d. Bernoulli random variables with parameter 1/2, subject to an equal number of 0s and 1s in each row and column; i.e., a uniform distribution on the corresponding set of binary contingency tables _n,n/2.In fact, the standard approach to sampling from binary contingency tables, given in <cit.>, does in fact model each entry as independent subject to the constraints, with entry (i,j) as having a Bernoulli distribution with parameter c_j/m, where c_j is the column sum and m is the number of rows; taking c_j = n/2 and m=n, we obtain our aforementioned natural heuristic for the parity of entries in a Latin square of order n. We may continue this reasoning for tables with prescribed entries forced to be zero, which is the inspiration for Algorithm <ref>. We similarly have analogous lemmas as in the previous section specifically for binary-valued tables, which are also straightforward generalizations from the work in <cit.>, and so we omit the proof. Let _W=(X_ij)_1 ≤ i ≤ m, 1 ≤ j ≤ n denote a collection of independent Bernoulli random variables with parameters p_ij, such that W(i,j) = 1 implies X_i,j is conditioned to have value 0.If p_ij has the form p_ij = 1 - α_i β_j, then _W is uniform restricted to (r,c)-binary contingency tables with zeros in entries indicated by W.Suppose _W is a collection of independent Bernoulli random variables, where X_i,j has parameter p_ij = c_j/m-h_j, for all pairs (i,j) such that W(i,j) = 0, and p_i,j = 1 for all pairs (i,j) such that W(i,j) = 1.Then the expected column sums of _W are c_1, c_2, …, c_n, and the expected row sums are ∑_j∈ I_1c_j/m-h_j, …, ∑_j∈ I_mc_j/m-h_j.There are thus two distinct places where bias can occur in our approach championed in Section <ref>.The first comes from the locations of the even/odd entries; even if we suppose that we may sample from _n,n/2 uniformly at random, there is no a priori reason to believe that φ_n maps the same number of Latin squares of order n to each element of _n,n/2, and indeed this is easily seen to not be the case for n=5, even though the distribution is close to uniform.However, supposing we did know precisely the multiset of counts for the image of φ_n, a natural sampling approach would be to generate samples from _n,n/2 uniformly and apply an appropriate rejection proportional to this multiset of counts (followed by the analogous procedure for the follow-up tables in the algorithm); however, sampling uniformly from _n,n/2, is not always straightforward or efficient, which motivates the following approximation algorithm. In <cit.>, the algorithm championed for binary contingency tables keeps track of an importance sampling weight, which represents the bias introduced by the sampling algorithm. The algorithm proceeds in the same manner as the one we champion for contingency tables, sampling entries column by column from top to bottom.Our suggested modification for binary contingency tables is to ignore the use of the Poisson-binomial distribution altogether, but still apply a rejection which captures several necessary conditions, along with what we suspect is the dominant source of bias in the sampling algorithm.That is, letting X_i,j(p_i,j) denote a Bernoulli random variable with parameter p_j ≡ p_i,j = c_j/n, we propose a rejection probability of the form B(i,j,k, r, c,W) :=(∑_ℓ∈ I_i, ℓ > j X_i,ℓ(p_ℓ) = r_i - k) ×(∑_ℓ∈ J_j, ℓ > i X_ℓ,j(p_j) = c_j - k), k ∈{0,1}.It is plausible that for the random sampling of Latin squares the two sources of bias at times cancel each other out, or potentially exacerbate the individual biases; it would certainly be interesting to explore this in more detail. One can also ask the following fundamental question: Is the range of the injection φ_n equal to _n,⌈ n/2⌉ for all n? In other words, do there exist binary contingency tables which, if generated, doom our approach from the start?The unfortunate answer is yes, found by exhaustive search for n=7, although in practice we have not found there to be very many.The next natural question, of interest in complexity theory, would be if there are necessary and sufficient conditions on _n,⌈ n/2⌉ which indicate in advance whether such a configuration of even/odd entries cannot be completed.One could also ask similar questions pertaining to the later phases of the algorithm.§ ALGORITHMS §.§ Nonnegative integer-valued contingency tables with prescribed 0 entriesSuppose we have matrices W and 𝒪 with entries in {0,1}.For any set of row sums and column sums (r,c), let Σ(r,c,𝒪, W) denote the number of nonnegative integer-valued (r,c)-contingency tables with entry (i,j) forced to be even if the (i,j)th entry of 𝒪 is 1, and entry (i,j) forced to be 0 if the (i,j)th entry of W is 1.Let 𝒪_i,j denote the matrix which has entries with value 1 in the first j-1 columns, and entries with value 1 in the first i rows of column j, and entries with value 0 otherwise.Let Σ(r, c, 𝒪, W; t_1, …, t_r), where each t_ℓ, ℓ = 1, …, r denotes an entry in the table, say t_ℓ = (i_ℓ, j_ℓ), denote the number of nonnegative integer-valued (r,c)-contingency tables as before with the additional assumption that the entries t_1, … t_r are forced to be even.Let J_j ←{1,…, n}∖{ i : W(i,j) = 1}, and let k_j denote the last entry in J_j, for all j=1,2,…,n.Let I_i ←{1,…, n}∖{j : W(i,j) = 1}, and let ℓ_i denote the last entry in I_i, for all i=1,2,…,m. For each 1 ≤ i ≤ m, 1 ≤ j ≤ n, definef(i,j,k,r,c) :=Σ( [ (…,r_i-k,…),; (…,c_j-k,…),;W ])/Σ( [ (…,r_i-1,…),; (…,c_j-1,…),;W ]) + Σ( [ (…,r_i,…),; (…,c_j,…),;W ])if i ∈ J_j ∖{k_j}, and j ∈ I_i ∖{ℓ_i}. Let g_i := ∑_j=1^n W_i,j denote the number of non-zero elements row i of W, for i=1,…,m, and let h_j := ∑_i=1^m W_i,j denote the number of non-zero elements in column j, j=1,2,…,n. Let q_j := c_j/m-h_j+c_j, andlet y_j := q_j^-1 = 1+m-h_j/c_j. Also, let k_j' denote the penultimate entry in J_j, for all j=1,2,…,n, and let ℓ_i' denote the penultimate entry in I_i, for all i=1,2,…,m. Let b(k) be such that r_i-k-b(k) is even. Then we define for each j =1,2,…,n and i ∈ J_j ∖{k_j}, f(i,ℓ_i',k,r,c) := Σ( [(…,r_i-k-b(k),…),; (…,c_ℓ_i'-k,c_ℓ_i-b(k)),;W ]) ·y_ℓ_i^b(k)/Σ( [(…,r_i-1-b(1),…),; (…,c_ℓ_i'-1,c_ℓ_i-b(1)),;W ]) ·y_ℓ_i^b(1)+Σ( [(…,r_i-b(0),…),; (…,c_ℓ_i',c_ℓ_i-b(0)),;W ]) ·y_ℓ_i^b(0) Let v(k) be such that c_j - k-v(k) is even.Similarly we define for each i=1,2,…,m and j ∈ I_i ∖{ℓ_i},f(k_j',j,k,r,c) := Σ( [ (…,r_k_j'-k,r_k_j-v(k)),;(…,c_j-k-v(k),…),;W ]) ·y_j^v(k)/Σ( [ (…,r_k_j'-1,r_k_j-v(1)),;(…,c_j-1-v(1),…),;W ]) ·y_j^v(1)+Σ( [ (…,r_k_j',r_k_j-v(0)),;(…,c_j-v(0),…),;W ]) ·y_j^v(0). Finally, let v(k) be such that c_n-1 - k-v(k) is even, let γ(k) be such that r_m-1-k-γ(k) is even, and let b(k) be such that c_n-γ-b(k) is even.Then we defineA ≡Σ([ (…,r_m-1-k-γ(k),r_m-v(k)-b(k)),; (…,c_n-1-k-v(k),c_n-γ(k)-b(k)),; 𝒪_m,n ]) ·y_n-1^v(k)y_n^γ(k) + b(k), B ≡Σ([ (…,r_m-1-1-γ(1),r_m-v(1)-b(1)),; (…,c_n-1-k-v(1),c_n-γ(1)-b(1)),; 𝒪_m,n ]) ·y_n-1^v(1)y_n^γ(1) + b(1), C ≡Σ([ (…,r_m-1-γ(0),r_m-v(0)-b(0)),; (…,c_n-1-v(0),c_n-γ(0)-b(0)),; 𝒪_m,n ]) ·y_n-1^v(0)y_n^γ(0) + b(0), f(m-1,n-1,k,r,c) :=A/B+C.and let …. For any 1 ≤ i ≤ m, 1 ≤ j ≤ n, recall the definition of Σ(r, c, 𝒪_i,j, W) from the previous section.For k=0,1, let r_i(k) = (r_1,…,r_i-k,…,r_m), and c_j(k) = (c_1,…, c_j-k, …,c_n).Then defineA_i,j(k) := Σ(r_i(k), c_j(k), 𝒪_i,j, W), g(i,j,k,r,c,W) :=A_i,j(k)/A_i,j(0)+A_i,j(1),k ∈{0,1}.The following notations and definitions from <cit.> are used in the algorithms.(q) Geometric distribution with probability of success 1-q, for 0<q<1, with ((q)=k)=(1-q)q^k, k=0,1,2,…. (m, q) Negative binomial distribution with parameters m and 1-q, given by the sum of m independent (q) random variables, with ((m,q) = k) = m+k-1k(1-q)^m q^k.UUniform distribution on [0,1]. We will also denote random variables from this distribution as U or u; should be considered independent of all other random variables, including other instances of U. (p) Bernoulli distribution with probability of success p. Similarly to U, we will also use it as a random variable. ξ_i,j(q) (q) random variables which are independent for distinct pairs (i,j), 1≤ i≤ m, 1≤ j≤ n. ξ_i,j'(q,c_j,W) Random variables which have distributionŁ(ξ_i,j(q) | ∑_ℓ∈ J_jξ_ℓ,j(q) = c_j),and are independent of all other random variables ξ_i,ℓ(q) for ℓ j. 2ξ_i,j”(q,c_j,W) Random variables which have distributionŁ(2ξ_i,j(q^2) | ∑_ℓ∈ J_j 2ξ_ℓ,j(q^2) = c_j ) and are independent of all other random variables ξ_i,ℓ(q) for ℓ j. η'_i,j,s(q, c_j) Random variables which have distribution Ł( ξ_i,j(q) | ∑_ℓ∈ J_j, ℓ≤ s 2ξ_ℓ,j(q^2) + ∑_ℓ∈ J_j, ℓ > sξ_ℓ,j(q) = c_j),and are independent of all other random variables ξ_i,ℓ(q) for ℓ j. 2η”_i,j,s(q, c_j) Random variables which have distribution Ł( 2ξ_i,j(q^2) | ∑_ℓ∈ J_j, ℓ≤ s 2ξ_ℓ,j(q^2) + ∑_ℓ∈ J_j, ℓ > sξ_ℓ,j(q) = c_j),and are independent of all other random variables ξ_i,ℓ(q) for ℓ j. 𝐪 The vector (q_1, …, q_n), where 0<q_i<1 for all i=1,…,n.Consider next the following subroutines to generate a random bit: The previous two procedures are computationally intensive to evaluate, since they a priori rely on the entire solution set, i.e., the set of all potential values of entries.We suggest the following approximation approach, in which the (i,j)-th entry is decided based solely on the row sum r_i and column sum c_j. For each fixed 1 ≤ i ≤ m and 1 ≤ j ≤ n, let t_1, …, t_ν denote the set of entries and the corresponding value which have a deterministic value (either the entire entry or the least significant bit) determined by the line sum conditions after the sampling of the least significant bit of entry (i,j), and let r_i,j(t_1,…, t_ν) and c_i,j(t_1, …, t_ν) denote the residual row and column sums, respectively, given these entries. For example, if there is only one fillable entry in a given row or column, we may uniquely specify its value;if a row sum or column sum is zero, set all fillable entries in that row or column to zero; if there are two entries left in a given row and the row sum is even (and non-zero), then either both least significant bits are 0 or both are 1. * Fill in a set of values {t_1, …, t_ν} into the table.* Check if a priori the entries of any rows or columns are uniquely determined given these values.There are two cases: if a row sum is zero then all remaining entries must be 0, or if a row sum equals the number of undecided entries then they must all be 1. Similarly with columns.* If any values were filled in from the previous step, call the function recursively; stop when no new elements are filled in, or when W is the matrix of all 1s. The rejection procedure we thus champion for approximate sampling of contingency tables is as follows.For each entry (i,j), we calculate the set of all values which would be determined if we set that particular entry's bit equal to 0 or 1 using Procedure Deterministic_Fill.We then sample the bit in proportion to the probability that all of the determined bits were also generated with their uniquely determined values, and apply a rejection probability similar to Equation (<ref>), except we condition on these determined values. This is summarized in Procedure Approximate_Probabilistic below.We defineF(i,j,k, r, c,W, {t_1,…,t_ν}) := ( ∑_ℓ∈ I_i, ℓ < j 2ξ”_i,ℓ(q_ℓ^2, c_j, ϵ) + 2η”_i,j,i + ∑_ℓ∈ I_i, ℓ > jξ'_i,ℓ(q_ℓ) = r_i - k | {t_1, …, t_ν})×( ∑_ℓ∈ J_j, ℓ≤ i 2ξ_ℓ,j(q_j^2) + ∑_ℓ∈ J_j, ℓ > iξ_ℓ,j(q_j) = c_j - k | {t_1 … t_ν}),k ∈{0,1}. Note that in general we cannot be more specific in our notation, since t_1 might contain a deterministic bit, or an entire entry.In each case it is straightforward to rewrite the explicit probability.Note that the 0 in the term F(i,j,0,r_1,c_1,W_1) above is not a typo, since the row sums and column sums are assumed to be updated in the Deterministic_Fill algorithm.In addition, we did not specify whether the random variables X_i,j refer to the entire entry or the least significant bit, since they reflect the information contained in the t_ℓ, ℓ=1,…,ν.In principle, the X_i,j could refer to any statistic related to entry (i,j), though for us we only consider the value of the entry (i,j) or its least significant bit.The time to evaluate Procedure Deterministic_Fill is O(n^2+m^2).Each time we call the function, it iterates through all elements of the table, say twice, at cost O(m n).At worst, we fill in one entire row or column at each recursive call of the function.Since we thus invoke the recursion at most max(n,m) times, the result follows. We now state the main algorithm championed for contingency tables, which depending on several parameters determines the cost of the algorithm and the bias.This algorithm serves as the main inspiration for Algorithm <ref>, and is interesting in its own right.The function we choose for Bit_Sampler determines the bias as well as the complexity, and we have presented three explicit possibilities: Exact, Exact_Probabilistic, and Approximate_Probabilistic.The rejection function g in Line <ref> of Procedure Exact, given in Equation (<ref>), is stated as an expression involving counting the total number of such tables; this is the direct analog of the recursive method, and typically requires a large table of values computed via a recursion.It is similarly difficult to compute the rejection function f in Line <ref> of Procedure Exact_Probabilistic, i.e., the probabilistic equivalent of g, but which motivates our approximation heuristic. Procedure Approximate_Probabilistic offers an alternative which is both practical and an heuristically reasoned approximation for the exact quantities.The resulting algorithm is a generalization of <cit.>, and in fact, it is equivalent to Algorithm <ref> above, except it is based on rejection sampling, where one instead samples from a related distribution and applies a rejection probability which rejects samples in a proportion which yields the desired distribution.One simply replaces lines 12–16 in Algorithm <ref> above with the following lines: 12: X(i,j) ←(c_j / (m-h_j)) 13: if U >f(i,j,X(i,j),r,c,W_b)/max_ℓ∈{0,1}f(i,j,ℓ,r,c,W_b) then 14: Goto Line 12. 15: endif The rejection function f is given by Equation (<ref>).Again, calculating functions g or f become the bottleneck in any computation. The form of rejection function f, however, suggests an approximation using rejection function F given by Equation (<ref>) in place of f.The first term in Equation (<ref>) is a probability over a sum of independent random variables with explicitly known distributions, see Section <ref>, and as such can be computed using convolutions in space/time O(nM^2) or fast Fourier transforms in space/time O(nM log M).Further speedups are possible since only the first few bits of the function are needed on average.It is also straightforward to adapt this approach to binary tables, where we recall that such tables have values restricted to be in {0,1}.A similar approach was obtained specifically for the case of binary contingency tables without specified zeros in <cit.> using an approach called sequential importance sampling, where at each step one keeps track of weights associated with each placement of a 0 or 1 in each entry, and at the end of the algorithm obtains an importance sampling weight associated with each sample.This algorithm is ideal for the first stage of the Latin square algorithm, since we are able to take advantage of the Gale–Ryser necessary and sufficient condition for binary contingency tables <cit.> after completion of each column; subsequent steps utilize Algorithm <ref>, and to our knowledge there is no analogous Gale–Ryser condition applicable to binary tables with an arbitrary prescribed set of zeros. As an example, an application of <cit.> yielded the following set of random bits in each of the five iterations of the algorithm for the random sampling of a contingency table with row sums (40, 30, 30, 50, 100, 50) and column sums (50, 50, 50, 50, 50, 50).[An implementation in Mathematica took about 1 second.] -.5in ( [ 12 123094;270 1155;29 10045;61 243 142; 182 12 32 16 20; 10 19142 14 ]) =( [ 0 0 1 0 1 0; 0 1 0 1 1 1; 0 1 0 0 0 1; 0 1 0 1 0 0; 0 0 0 0 0 0; 0 1 1 0 0 0 ])+2^1 ( [ 0 0 1 0 0 0; 1 1 0 1 0 0; 1 0 1 0 0 0; 1 0 0 1 1 1; 1 1 0 0 0 0; 1 1 0 0 1 1 ]) + 2^2 ( [ 1 1 0 0 0 1; 0 1 0 0 1 1; 0 0 0 0 1 1; 1 0 0 0 1 0; 0 0 1 0 0 1; 0 0 0 1 0 1 ])+ 2^3 ( [ 1 1 0 0 1 0; 0 0 0 1 0 0; 0 1 1 0 0 0; 0 0 1 0 1 0; 0 0 1 0 0 0; 1 0 0 0 0 1 ]) + 2^4( [ 0 0 0 0 0 0; 0 0 0 0 0 0; 0 0 0 0 0 0; 0 0 1 0 0 0; 1 0 0 0 1 1; 0 1 0 0 0 0 ]) + 2^5 ( [ 0 0 0 0 0 0; 0 0 0 0 0 0; 0 0 0 0 0 0; 0 0 0 0 0 0; 0 0 0 1 0 0; 0 0 0 0 0 0 ]).Consider the following example, a 3 × 4 nonnegative integer-valued (r,c)-contingency table with row sums r = (20, 14, 18, 27) and column sums c = (13, 56, 10). Furthermore, we force entries (1,3), (2,1), (3,2), (3,3), (3,4) to be 0.The first step is Line <ref>, which transforms the initial sampling problem into the following:-.5in [[ [ ▪; ▪; ▪ ▪ ▪; ] [ 10; 56; 13 ];[ 20 14 18 27 ] ]⟶ [ [7 ▪ ;▪18 ; 13▪▪▪;][ 10; 56; 13 ]; [ 20 14 18 27 ] ]⟶ ; ;[ [ ▪ ▪; ▪ ▪; ▪ ▪ ▪ ▪; ] [3; 38;0 ];[0 140 27 ] ]⟶[ [;;] [3; 38 ];[ 14 27 ] ] ]Now since this reduces to a standard 2× 2 contingency table, we can apply a standard algorithm or continue with Algorithm <ref>.We suggest in particular the exact sampling algorithm for 2 × n tables given in <cit.>, which for this particular example would sample a uniform number in the set {0, 1, 2, 3} and then uniquely fill in the remaining entries.§.§ Binary contingency tablesOne can formulate an exact sampling rejection function for binary contingency tables as in the previous section, although the approximation algorithm is again the one we champion due to practical computing reasons.It is straightforward but notationally messy to adapt and write out completely the analogous exact sampling procedures in this setting, and so we focus solely on our suggested approximate sampling approach.Let (r,c,W) denote the number of binary (r,c)-contingency tables with entry (i,j) forced to be 0 if the (i,j)th entry of W is 1.Let (r, c, W; t_1, …, t_r), where each t_ℓ, ℓ = 1, …, r denotes an entry in the table, say t_ℓ = (i_ℓ, j_ℓ), denote the number of binary (r,c)-contingency tables as before with the additional assumption that the entries t_1, … t_r are also forced to be 0. Let h_j := ∑_i=1^m W_i,j denote the number of non-zero elements in column j, j=1,2,…,n. Let q_j := c_j/m-h_j, andlet y_j := q_j^-1. For each 1 ≤ i ≤ m and 1 ≤ j ≤ n, let t_1, …, t_r denote the set of entries which are determined by the line sum conditions after sampling of entry (i,j), and let b_1(k), …, b_r(k) denote the deterministic value of the entry in order to satisfy the row and column constraints given that entry (i,j) has value k, where k ∈{0,1}.Then defineA≡([(…,r_i-k,…, r_i_1 - b_1(k),…, r_i_r - b_r(k)),; (…,c_j-k, …, c_j_1 - b_1(k),…, c_j_r - b_r(k)),;W, 𝒪_i,j;t_1, …, t_r ]) · ∏_ℓ=1^r y_j_ℓ^b_ℓ(k),B≡([(…,r_i,…, r_i_1 - b_1(0),…, r_i_r - b_r(0)),; (…,c_j, …, c_j_1 - b_1(0),…, c_j_r - b_r(0)),;W, 𝒪_i,j;t_1, …, t_r ]) · ∏_ℓ=1^r y_j_ℓ^b_ℓ(0), C≡([(…,r_i-1,…, r_i_1 - b_1(1),…, r_i_r - b_r(1)),; (…,c_j-1, …, c_j_1 - b_1(1),…, c_j_r - b_r(1)),;W, 𝒪_i,j;t_1, …, t_r ]) · ∏_ℓ=1^r y_j_ℓ^b_ℓ(1), h(i,j,k,r,c,W) :=A/B+C.Next we present a more probabilistic approach, which yields a natural approximate sampling algorithm.Let B_j(q), j=1,…,n, denote independent Bernoulli random variables with success probability 1-q. We have b(n, q,k):= ( ∑_j=1^n B_j(q_j) = k)= 1/n+1∑_ℓ = 0^n C^-ℓ k∏_j=1^n (1+(C^ℓ-1)(1-q_j)),where C = exp(2π i/n+1) is a (n+1)th root of unity. Define d_n,k := max(b(n, q,c, k), b(n, q,c, k-1))/b(n-1, q,c, k)),and d := max_n,k d_n,k.The expression in (<ref>) is a numerically stable way to evaluate the convolution of a collection of independent Bernoulli random variables using a fast Fourier transform, see for example <cit.>.Next, we consider an m× n matrix W which contains values in {0,1}.A value 1 in entry (i,j) of W implies that this entry is forced to be zero.Given an index set J ⊂{1,2,…, n}, we define similarlyb(n, q,k,J) := ( ∑_j∈ J B_j(q_j) = k). For each i=1,2,…,m, let I_i ←{1,…, n}∖{j : W(i,j) = 1} denote the set of entries in row i which can be non-zero; and for each j=1,2,…,n, let J_j ←{1,…, n}∖{ i : W(i,j) = 1} denote the set of entries in column j which can be non-zero.Also let q = (c_1/(m-h_1), c_2/(m-h_2),…, c_n/(m-h_n)) and let ν_j = (c_j/(m-h_j), c_j/(m-h_j), …, c_j/(m-h_j)).Then we define for i ∈ I_i and j ∈ J_jH(i,j,k,r,c,W) := b(m,q, r_i - k, I_i) × b(n,ν_j, c_j-k, J_j),and 0 otherwise.We therefore define the following procedure to generates bits, noting that the wording of the Deterministic_Fill function defined in the previous section is equally applicable in this setting.For a given set of row sums r = (r_1, …, r_m) and column sums c = (c_1, …, c_n), let (r,c) denote the number of (r,c)-binary contingency tables.Let 𝒪 denote an m× n matrix with entries in {0,1}. For any set of row sums and column sums (r,c), let (r,c,𝒪) denote the number of (r,c)-contingency tables with entry (i,j) forced to be 0 if the (i,j)th entry of 𝒪 is 1, and no restriction otherwise.Let 𝒪_i,j denote the matrix which has entries with value 1 in the first j-1 columns, and entries with value 1 in the first i rows of column j, and entries with value 0 otherwise. Define for 1 ≤ i ≤ m-2, 1 ≤ j ≤ n-2, f(i,j,k,r,c) :=( [ (…,r_i-k,…),; (…,c_j-k,…),;𝒪_i,j ])/( [ (…,r_i-1,…),; (…,c_j-1,…),;𝒪_i,j ]) + ( [ (…,r_i,…),; (…,c_j,…),;𝒪_i,j ]). Procedure Approximate_Probabilistic_Binary has an arithmetic cost of O(n^2+m^2) By Lemma <ref>, each of the two calls to Procedure Deterministic_Fill take at most O(n^2+m^2).In addition, the cost to evaluate H via Equation (<ref>) and Equation (<ref>) is also at most O(n^2+m^2).We thus define the algorithm for approximate sampling of binary (r,c)-contingency tables as follows.§.§ Latin squares of order n Suppose b = (b_1, …, b_k)∈{0,1}^k is some combination of k 0s and 1s, for any k ≥ 1.Given also positive integer n, let n_b denote a sequence of ceiling and floor functions, each followed by a division by 2, applied to ⌈ n/2 ⌉.For example, n_101 = ⌈⌊⌈⌈ n/2 ⌉ /2 ⌉ / 2⌋ / 2 ⌉. We now define elements t_b inductively. Start with t^(0), an m × n matrix of 0s.Let t_0 denote an m × n matrix, conditional on t^(0), which is 1 wherever t is 0, and 0 otherwise. Similarly, let t_1 denote an m × n matrix, conditional on t, which is 1 whenever t is 1, and 0 otherwise.Let t^(1) denote an m× n matrix with each entry consisting of tuple (g_0, g_1), where g_0 is the corresponding element in t_0, and g_1 is the corresponding element in t_1.Now define t^(b), b of size k, conditional on t^(b), b of size k-1, to be an m× n matrix which is 1 whenever Similarly, let W_b denote a matrix in {0,1}^m× n which is conditional onWe now state a complete version of our main algorithm for random sampling of Latin squares.It bears repeating that while a priori this algorithm appears to be a sum over an exponentially growing number of tables, and indeed for exposition purposes we have chosen this form in order to clearly convey the procedure, each of the 2^k calls to Algorithm <ref> inside the innermost for loop can be performed with one loop over each entry in the table, being careful to apply the correct rejection function for each entry (i,j) based on previously observed bits in entry (i,j).This makes the required time O(n^2) times the cost to evaluate and pass the rejection function threshold in Algorithm <ref>. It is easy to adapt this algorithm for the random sampling of n^2 × n^2 Sudoku matrices, with an extra rejection due to the local block constraints; see for example <cit.> for the precise definition. One could start with Algorithm <ref> and perform a hard rejection at each iteration until the binary contingency table generated also satisfies the block constraints, or one could adapt the rejection probability to more effectively target the block constraints. §.§ Algorithmic costs and a word of caution We are unable at present to formulate precise runtime estimates, only to say we have implemented the approximation algorithms in practice and they exhibit roughly the runtime estimates we show below, with one major caveat.Part of the difficulty in precisely costing the algorithms is that, for the exact sampling algorithms, they rely on computing numerical quantities for which no known polynomial time algorithm is known. For the approximation algorithms, we have not yet undertaken a full analysis of the bias, and in fact, we caution that the algorithm can potentially reach a state for which the algorithm cannot terminate due to unsatisfiable constraints.In those cases it is up to the practitioner whether the algorithm ought to halt altogether and restart, or whether some partially completed table should be salvaged; again, since we have not performed a detailed analysis of the bias in these algorithms, we have not explored this matter further, except to note that the algorithm is surprisingly effective when it avoids such unsalvageable states.In each of the estimates below, we assume m denotes the number of rows and n denotes the number of columns in a contingency table, and that W is some m × n matrix with a 1 in the (i,j)-th entry if the (i,j)-th entry of the contingency table is forced to be 0.We also define R := max(r_1, …, r_m),C := max(c_1, …, c_n),M := max(R,C). For Algorithm <ref>, with Procedure Approximate_Probabilistic used for the Bit_Sampler routine, it is easy to see that barring the encounter of a state which cannot be completed, the expected number of random bits required is O(n m log(M)), one for each visit to each entry per bit-level.For the arithmetic cost, each call to Procedure Approximate_Probabilistic involves O(R log R) arithmetic operations, since the evaluation of the function F can be performed in a numerically stable manner using fast Fourier transforms.In addition, each call to Procedure Deterministic_Fill involves O(m^2+n^2) arithmetic operations by Lemma <ref>.Thus, this corresponds to an informal arithmetic cost which is O((m^2+n^2) mnR log(R) log(M)) in the best case scenario. For Algorithm <ref>, it is easy to see that the expected number of random bits is O(nm), with O((n^2+m^2)m n) arithmetic operations, since we iterate through each of the nm entries at most once, each time calling Procedure Approximate_Probabilistic_Binary. Finally, for Algorithm <ref>, the expected number of random bits required is O(n^2 log(n)) with O(n^4 log n) arithmetic operations.Consider Algorithm <ref> with Procedure Approximate_Probabilistic used for the Bit_Sampler routine.The expected number of random bits required for the algorithm to run to completion is O( nm log(M)), with O((m^2+n^2) mnR log(R) log(M)) arithmetic operations. Each call to Procedure Approximate_Probabilistic involves O(R log R) arithmetic operations, since the evaluation of the function F can be performed in a numerically stable manner using fast Fourier transforms.In addition, each call to Procedure Deterministic_Fill involves O(m^2+n^2) arithmetic operations by Lemma <ref>.Finally, since we call these procedures on each of the mn entries, and iterate at most log(M) times, the result follows.For Algorithm <ref>, the expected number of random bits required is O(nm), with O((n^2+m^2)m n) arithmetic operations. Similarly as in the Proof of Theorem <ref>, we iterate through each of the nm entries at most once, each time calling Procedure Approximate_Probabilistic_Binary.By Lemma <ref>, this has cost O(n^2+m^2), hence the result follows.For Algorithm <ref>, the expected number of random bits required is O(n^2 log(n)) with O(n^4 log n) arithmetic operations.We iterate through each of the n^2 entries at most log(n) times, due to the disjoint nature of the divide-and-conquer strategy on the entries. At each entry we potentially call Procedure Deterministic_Fill, which is O(n^2) by Lemma <ref>, and hence the result follows. §.§ Latin squares of order n Suppose b = (b_1, …, b_k)∈{0,1}^k is some combination of k 0s and 1s, for any k ≥ 1.Given also positive integer n, let n_b denote a sequence of ceiling and floor functions, each followed by a division by 2, applied to ⌈ n/2 ⌉.For example, n_101 = ⌈⌊⌈⌈ n/2 ⌉ /2 ⌉ / 2⌋ / 2 ⌉. We now define elements t_b inductively. Start with t^(0), an m × n matrix of 0s.Let t_0 denote an m × n matrix, conditional on t^(0), which is 1 wherever t is 0, and 0 otherwise. Similarly, let t_1 denote an m × n matrix, conditional on t, which is 1 whenever t is 1, and 0 otherwise.Let t^(1) denote an m× n matrix with each entry consisting of tuple (g_0, g_1), where g_0 is the corresponding element in t_0, and g_1 is the corresponding element in t_1.Now define t^(b), b of size k, conditional on t^(b), b of size k-1, to be an m× n matrix which is 1 whenever Similarly, let W_b denote a matrix in {0,1}^m× n which is conditional onWe now state our main algorithm for random sampling of Latin squares, presented to be as simple and concise to follow as possible.As stated, it is a priori the sum of an exponential number of tables; a fortiori, the non-zero entries of these tables form a partition of all n^2 entries, and so one would implement the algorithm below to switch between the various rejection functions and thus avoid an exponential number of calls to Algorithm <ref>.It bears repeating that while a priori this algorithm appears to be a sum over an exponentially growing number of tables, and indeed for exposition purposes we have chosen this form in order to clearly convey the procedure, each of the 2^k calls to Algorithm <ref> inside the innermost for loop can be performed with one loop over each entry in the table, being careful to apply the correct rejection function for each entry (i,j) based on previously observed bits in entry (i,j).This makes the required time O(n^2) times the cost to evaluate and pass the rejection function threshold in Algorithm <ref>. To make this an exact sampling algorithm one would need to apply an additional rejection after each call to Algorithm <ref>.We have suppressed this rejection, as it requires enumerative estimates which are not easily accessible.§ PROOFS§.§ Proof of Theorem <ref> We also have an analogous lemma for the quantitative bounds, cf. <cit.>. Any algorithm which uses Ł((q_j/1+q_j)), where q_j = c_j/m-h_j + c_j, as the surrogate distribution for Ł(ϵ_i,j|E) in rejection sampling, where ϵ_i,j denote the least significant bit of X_i,j, assuming each outcome in {0,1} has a positive probability of occurring, accepts a bit with an incorrect proportion bounded by at most m+2, where m is the number of rows.Let B be distributed as Ł((q_j/1+q_j)).We haveq_j/1+q_j = c_j/m-h_j+2c_j,and so [1/m-h_j+2< (B = 1) <1/2;1/2< (B=0) < m-h_j+1/m-h_j+2. ]We are able to scale up by the larger point probability, i.e., the event {B=0}, which means we accept a generated bit of 1 with probability 1, for which we would have to wait at most an expected m-h_j +2 ≤ m+2 iterations.Thus, at worst we accept a bit m+2 times more likely in one state than the other.All that remains is the costing estimates.For Algorithm <ref>, there is no rejection, and so the cost in terms of the expected number of random bits required is simply the cost of visiting all s entries at most log M times to assign a bit.For Algorithm <ref>, the rejection adds at most O(m) iterations for each entry, which implies O(sm log M) expected random bits required.To calculate the rejection probabilities is a convolution of at most n independent random variables.Each convolution costs a priori O(M^2) directly, or O(M log M) using an FFT.Repeating this n times gives O(nM log M) for each entry, and so the total arithmetic cost is O(snM log M).§.§ Proof of Theorem <ref> We start with a similar treatment as in the previous section, except instead of taking X_i,j to be geometrically distributed, with parameter p_i,j = 1-α_i β_j, we instead take X_i,j to be Bernoulli with parameter q_i,j/1+q_i,j, where q_i,j = 1-p_i,j.In fact, this is precisely the distribution of the least significant bit utilized in Algorithm <ref>. Let _W=(X_ij)_1 ≤ i ≤ m, 1 ≤ j ≤ n denote a collection of independent Bernoulli random variables with parameters p_ij, such that W(i,j) = 1 implies p_i,j = 0, and otherwise p_ij has the form p_ij = α_i β_j/1+α_i β_j. Then _W is uniform restricted to binary (r,c)-contingency tables with zeros in entries indicated by W. The proof is straightforward, as is the analogous result for Lemma <ref> and Lemma <ref>. The proof of uniformity of Algorithm <ref> follows again almost verbatim from <cit.>, aside from straightforward modifications of boundary conditions, summing over index sets j ∈ I_i rather than j=1,2,…,n, and replacing the role of Σ(r,c,𝒪) with (r,c,W).The costing estimates are also straightforward. For Algorithm <ref>, there is no rejection, and so the cost in terms of the expected number of random bits required is simply the cost of visiting all s entries once.For Algorithm <ref>, the rejection adds at most O(m) iterations for each entry, which implies O(sm) expected random bits required.To calculate the rejection probabilities is again a convolution of at most n independent random variables. Each convolution costs a priori O(M^2) directly, or O(M log M) using an FFT. Repeating this n times gives O(nM log M) for each entry, and so the total arithmetic cost is O(snM log M).Since this is a binary contingency table, we must have M ≤max(m,n), although we prefer to keep this as a separate variable to highlight the case of sparse tables. §.§ Proof of Theorem <ref> The fact that a Latin square can be decomposed into its corresponding bits is straightforward, as is the connection with collections of binary (r,c)-contingency tables with restrictions. It is not a priori obvious whether the bits at different levels are independent, and whether certain configurations of binary tables at a given level can potentially be completed by a larger number of Latin squares than other configurations.For n=6, we are able to provide a negative answer, although it would be interesting to explore how non-uniform the distribution truly is.According to the OEIS sequence A058527 <cit.>, the number of 6 × 6 binary contingency tables with row sums and column sums equal to 3 is 297200, which does not divide the number of Latin squares of order 6, which is 812851200, see for example <cit.>.Thus, some of theconfigurations of binary tables yield a different number of completable Latin squares.If one could bound the range of possible completions given such a binary contingency table, then one could obtain quantitative bounds on the number of Latin squares.Next we consider the arithmetic cost. The algorithm requires visiting each entry at most ⌈log_2(n)) ⌉ times. Let us consider the first time entry (1,1) is visited. The rejection probability given in Equation (<ref>) is the product of two convolutions of at most n elements each, for a total of 2× O(n n/2 logn/2).The second time entry (1,1) is visited, the cost is O(n/2n/4logn/4).Summing over at most ⌈log_2(n)⌉ visits, the total cost associated with entry (1,1) is O(n^2). Consider the entry (i,j).The convolutions in Equation (<ref>) have an initial cost of O( (n-i) n-i/2logn-i/2 + (n-j) n-j/2logn-j/2) = O(n n/2logn/2). Thus, summing over all entries O(log n) times we have at most O(n^4 log(n)) arithmetic operations total. The expected number of random bits comes from the rejection probabilities at each step.Consider the first step, which is the random sampling of binary (⌊ n/2⌋ ,⌊ n/2⌋ )-contingency tables.By the previous section, this costs O(n^3), and we repeat this at most O(log n) times.§ OTHER EXTENSIONS One could also consider, e.g., tables with continuous–valued entries.In this case, the conditioning event is more delicate, as we can condition on events of probability zero with sufficient regularity; see <cit.>.One can reason a priori that if the conditioning event E can be written as a random variable T with a density evaluated at various values k ∈(T), then a similar PDC algorithm can be adapted, see <cit.>. In particular, like the decomposition of geometric random variables into their individual bits, an exponential distribution also has an analogous property.For real x, {x } denotes the fractional part of x, and ⌊ x ⌋ denotes the integer part of x, so that x = ⌊ x ⌋ + {x }.Let Y be an exponentially distributed random variable with parameter λ>0, then: * the integer part, ⌊ Y ⌋, and the fractional part, {Y}, are independent <cit.>;* ⌊ Y⌋ is geometrically distributed with parameter 1-e^-λ, and {Y} has density f_λ(x) = λ e^-λ x / (1-e^-λ), 0≤ x < 1. Using this property, a random sampling algorithm for nonnegative real-valued (r,c)-contingency tables is presented in <cit.>.The algorithm first samples the fractional part of each entry of the table; conditional on this first step, the remaining sampling problem is the usual random sampling of nonnegative integer-valued (r',c')-contingency table, for which Algorithm <ref> is applicable. We are not aware of any non-trivial generalizations of Latin squares to real-valued entries.Here is one which may be of interest.Define a partition J_1, …, J_n of the interval [0,n], and demand that that each row and each column of a matrix M has exactly one entry in each of the sets J_1, …, J_n.One can ask, then, to sample from the uniform measure over the set of all such matrices, which has a density with respect to Lebesgue measure.Another generalization would be to take J_1, …, J_n such that J_1 ∪…∪ J_n = [0,n], without the assumption that the sets form a partition, i.e., allow overlap. § CALCULATIONS FOR SMALL N§.§ n=2 Consider the set of Latin squares of order 2, i.e., ([ 1 2; 2 1 ]) and ([ 2 1; 1 2 ]).These correspond to the bit-wise decomposition([ 1 2; 2 1 ]) = 2^1 ([ 0 1; 1 0 ]) + 2^0 ([ 1 ▪; ▪ 1 ])([ 2 1; 1 2 ]) = 2^1 ([ 1 0; 0 1 ]) + 2^0 ([ ▪ 1; 1 ▪ ]),where we use ▪ to denote a forced 0 entry.An application of Algorithm <ref> starts by sampling from X_11, which to yield the uniform distribution over such tables must be 0 or 1 with equal probability.We have c_1 = c_2 = 1, m=2, q_1 = q_2 = 1/3 and q_1/1+q_1 = q_2/1+q_2 = 1/4.The algorithm generates X_11 as a Bernoulli(1/4) and applies one of the following rejection probabilities depending on whether the outcome is 0 or 1. R_0 = (X_12 = 1) (X_21=1) (X_22 = 0) = 3^2/4^2, R_1 = (X_12 = 0) (X_21=0) (X_22 = 1) = 3/4^2.The scaling factor in this case is given by α = min(R_0, R_1) = 3/4^2.Let R̃_0 = R_0 / α, and R̃_1 = R_1 / α. For this particular example we have R̃_0 = 1/3.The overall probability of accepting 0 for the (1,1) entry is then given by the geometric series (X̃_11=0) = (X_11 = 0) R̃_0 + (X_11=0)^2 (1-R̃_0) R̃_0 + (X_11=0)^3 (1-R̃_0)^2 R̃_0 + ⋯ = (X_11=0)R̃_0/1-(X_11=0)(1-R_0) = 1/2.Thus, each outcome is produced with probability 1/2, corresponding to the uniform distribution.§.§ n=3 Rather than apply Algorithm <ref> directly to the case n=3, we shall approach it in the opposite direction by looking at various 3 × 3 tables. Let us take for our next example the 3× 3 table with row sums and column sums all equal to 1, of which there are 6! tables corresponding to the permutation matrices of 3.This case allows us an easy way to asses potential bias in Algorithm <ref>.In this case we have c_1 = 1, m=3, q_1 = 1/4, and q_1/1+q_1 = 1/5.We have R_0 = (X_12 +X_13 = 1) (X_21+X_31=1) = (2 4/5^2)^2 = 4^3/5^4.R_1 = (X_12 +X_13 = 0) (X_21+X_31=0) = (4^2/5^2)^2 = 4^4/5^4.Hence, R̃_0 = 1/4, and as noted previously we must have R̃_1 = 1.The overall probability of accepting a 0 in entry (1,1) is therefore 1/2, by the same geometric series argument, whereas for the uniform distribution over permutation matrices of size 3 it should be 2/3. We thus tend to under sample permutation matrices with a 1 in entry (1,1), with an error proportion not exceeding |1/2 - 2/3| = 1/6. Next, we consider a 3 × 3 table with row sums and column sums all equal to 1, with the following fixed zeros([ ▪; ▪; ▪ ]).WLOG, this covers the case of all possible 3× 3 tables row sums and column sums all equal to 1 with exactly one entry in each row and column fixed to be 0.We have c_1 = 1, m=3, h_1 = 1, and so q_1 = c_1/c_1+m-h_1§ FURTHER DIRECTIONS It is most common when working with Latin squares to normalize the first row and the first column to be the identity permutation {1,2,…,n}.We have not carried out this in the interest of simplicity, but this would be a natural optimization to the current approach.§ RECYCLEOur results are now summarized below. Let m denote the number of rows and n denote the number of columns in a nonnegative integer-valued (r,c)-contingency table, and let M denote the largest row sum or column sum.Let W denote an m× n matrix with a 1 in the (i,j)th entry if the (i,j)-th entry of the contingency table is forced to be 0, and let s := mn - ∑_i,j W_i,j denote the number of entries not forced to be 0.(1) Algorithm <ref> returns a nonnegative integer-valued (r,c)-contingency table, uniform over all such tables.The expected number of random bits required is O( s log M). (2) Algorithm <ref> returns a nonnegative integer-valued (r,c)-contingency table, which is approximately uniform. The expected number of random bits required is O( sm log M), with O(snM log^2 M) arithmetic operations. Item (1) in Theorem <ref> demonstrates that, should one be able to compute certain numerical quantities efficiently, one would have an optimal sampling method. Item (2) in Theorem <ref> offers an alternative explicit and practical approximation algorithm which does not require any enumeration formulas, and instead uses convolution of independent random variables with explicitly known distribution functions.Analogous results hold for binary (r,c)-contingency tables. In addition, motivated by the initial results concerning Shannon's entropy of the collections of Latin squares and Sudoku matrices in <cit.>, later extended in <cit.>, the goal of sampling uniformly from Latin squares and Sudoku matrices allows us to glean important information about various statistics of interest in information theory, e.g., Shannon's entropy, compared to uniformly random matrices without restrictions.Several PDC algorithms were applied recently for the exact sampling from Latin squares and Sudoku matrices in <cit.>, by taking advantage of certain statistics in, e.g., <cit.>, extending the range of practical exact sampling methods for these structures from those in <cit.>.There are, however, alternatives like importance sampling for Sudoku matrices <cit.>, and also Markov chain techniques which have been specifically applied to Latin squares, see for example <cit.>. Algorithm <ref> produces a valid Latin square of order n, approximately uniform over all such Latin squares, which terminates in finite time a.s.The expected number of random bits required is O(n^3 log(n)), with O(n^4 log n) arithmetic operations.One can also generalize the aforementioned tables to having continuous–valued entries, in which case many of the traditional sampling algorithms either break down or require significant adaptation.Using PDC, such generalities are often easily handled, often by adding in just one extra step, see Section <ref>; see also <cit.> and <cit.>.We start with a well-known probabilistic model for the entries in a random contingency table, generalized to include entries which are forced to be 0. In this section, we let W be any given m× n matrix with values in {0,1}.For each j=1,2,…,n, we define J_j := {1,…, n}∖{ i : W(i,j) = 1}, and let k_j denote the last entry in J_j.Similarly, for each i=1,2,…,m, we define I_i := {1,…, n}∖{j : W(i,j) = 1}, and let ℓ_i denote the last entry in I_i, i=1,2,…,m.Also, we let h_j = ∑_i=1^m W(i,j) denote the number of entries forced to be zero in column j, for j=1,2,…,n. Let _W=(X_ij)_1 ≤ i ≤ m, 1 ≤ j ≤ n denote a collection of independent geometric random variables with parameters p_ij, such that W(i,j) = 1 implies X_i,j is conditioned to have value 0.If p_ij has the form p_ij = 1 - α_i β_j, then _W is uniform restricted to (r,c)-contingency tables with zeros in entries indicated by W. Let =(X_ij)_1 ≤ i ≤ m, 1 ≤ j ≤ n denote a collection of independent geometric random variables with parameters p_ij, where p_ij has the form p_ij = 1 - α_i β_j.We have (=ξ) =∏_i,j(X_ij=ξ_ij) =∏_i,j(α_i β_j)^ξ_ij(1-α_i β_j) =∏_iα_i^r_i∏_jβ_j^c_j∏_i,j(1-α_i β_j).Since this probability does not depend on ξ, it follows that the restriction ofto (r,c)-contingency tables is uniform.As the collection of random variables are independent, conditioning on any X_i,j = 0 only changes the constant of proportionality, and does not affect the dependence on the ξ, hence (_W=ξ) =∏_i,j : W(i,j) = 0(X_ij=ξ_ij) =∏_i,j: W(i,j)=0(α_i β_j)^ξ_ij(1-α_i β_j) =∏_iα_i^r_i∏_jβ_j^c_j∏_i,j(1-α_i β_j);i.e., it follows that the restriction of _W to (r,c)-contingency tables with forced zero entries indicated by W is uniform. Suppose _W is a collection of independent geometric random variables, where X_i,j has parameter p_ij = m-h_j/m-h_j+c_j, for all pairs (i,j) such that W(i,j) = 0, and p_i,j = 1 for all pairs (i,j) such that W(i,j) = 1.Then the expected column sums of _W are c_1, c_2, …, c_n, and the expected row sums are ∑_j∈ I_1c_j/m-h_j, …, ∑_j∈ I_mc_j/m-h_j.Note first that X_i,j = p_i,j^-1 - 1.Thenfor any j=1,2,…,n,∑_i=1^m X_i,j = ∑_i ∈ J_jm-h_j+c_j/m-h_j - 1= c_jand similarly for any i=1,2,…, m, ∑_j=1^n X_i,j = ∑_j ∈ I_im-h_j+c_j/m-h_j-1 = ∑_j∈ I_ic_j/m-h_j. The proof of uniformity of Algorithm <ref> now follows almost verbatim from <cit.>, aside from straightforward modifications of boundary conditions and summing over index sets j ∈ I_i rather than j=1,2,…,n.Start by considering any the first entry in the first column not forced to be 0, say at entry (s,1), and denote the least significant bit by ϵ_s,1.As we are in the first column, we have Ł(ϵ_s,1) = Ł((q_1/1+q_1)), and we reject according to the correct proportion (E | ϵ_s,1).In fact, as we reject in proportion to this probability, we normalize by all terms which do not depend on k, which gives(E | ϵ_s,1 = k) ∝Σ([(…,r_s-k,…, r_i_1 - b_1(k),…, r_i_r - b_r(k)),; (…,c_1-k, …, c_j_1 - b_1(k),…, c_j_r - b_r(k)),;W, 𝒪_i,j;t_1, …, t_r ]) ·q_1^-k (∏_ℓ=1^r q_j_ℓ^-b_ℓ(k)).Since (ϵ_s,1 = k) ∝ q_1^k, the rejection function in Equation (<ref>) follows.The rest of the proof follows by induction in a straightforward manner, cf. <cit.>. The second part of the theorem, with rejection function given in Equation (<ref>), is motivated by the following alternative probabilistic formulation of Equation (<ref>): f(i,j,k, r, c,W)∝( [ ∑_ℓ∈ J_1^ℓ < j 2ξ”_1,ℓ(q_ℓ^2, c_ℓ) + η_1,j,i'(q_j, c_j)+∑_ℓ∈ J_1^ℓ > jξ'_1,j(q_ℓ, c_ℓ)= r_1; ∑_ℓ∈ J_2^ℓ < j 2ξ”_2,ℓ(q_ℓ^2, c_ℓ) + η_2,j,i'(q_j, c_j)+∑_ℓ∈ J_2^ℓ > jξ'_2,j(q_ℓ, c_ℓ)= r_2;⋮; ∑_ℓ∈ J_i-1^ℓ < j 2ξ”_i-1,ℓ(q_ℓ^2, c_ℓ) + η_i-1,j,i'(q_j, c_j)+∑_ℓ∈ J_i-1^ℓ > jξ'_i-1,j(q_ℓ, c_ℓ)= r_i-1;∑_ℓ∈ J_i^ℓ < j2ξ”_i,ℓ(q_ℓ^2, c_ℓ) + η_i,j,i”(q_j, c_j)+∑_ℓ∈ J_i^ℓ > jξ'_i,j(q_ℓ, c_ℓ)= r_i-k;∑_ℓ∈ J_i+1^ℓ < j2ξ”_i+1,ℓ(q_ℓ^2, c_ℓ) + η_i+1,j,i”(q_j, c_j)+∑_ℓ∈ J_i+1^ℓ > jξ'_i+1,j(q_ℓ, c_ℓ)= r_i+1;⋮;∑_ℓ∈ J_m^ℓ < j2ξ”_m,ℓ(q_ℓ^2, c_ℓ) + η_m,j,i”(q_j, c_j)+∑_ℓ∈ J_m^ℓ > jξ'_m,j(q_ℓ, c_ℓ)= r_m;]) ×( ∑_ℓ∈ I_i, ℓ≤ i 2 ξ_ℓ,j(q_j^2) + ∑_ℓ∈ I_i, ℓ > iξ_ℓ,j(q_ℓ) = c_j - k ). If we could evaluate the probability above exactly, or to some arbitrarily defined precision, then we would obtain an exact sampling algorithm.However, we surmise that the dependencies between the random variables are strongest along the i-th row and the j-th column, which is why we champion the rejection function in Equation (<ref>), as it captures what is most likely the dominant source of bias, and enforces the parity condition as well.§ ACKNOWLEDGEMENTSThe author would like to acknowledge helpful discussions with Alejandro Morales, Igor Pak, Richard Arratia, and James Zhao.plain | http://arxiv.org/abs/1703.08627v1 | {
"authors": [
"Stephen DeSalvo"
],
"categories": [
"stat.CO"
],
"primary_category": "stat.CO",
"published": "20170324234933",
"title": "Random sampling of Latin squares via binary contingency tables and probabilistic divide-and-conquer"
} |
Band depths based on multiple time instances Department of Statistics, Universidad Carlos III de Madrid, Av. Universidad 30, E-28911 Leganés (Madrid), SpainDepartment of Mathematical Statistics and Actuarial Science, University of Berne, Sidlerstr. 5, CH-3012 Berne, Switzerland * Ignacio Cascos1Ilya Molchanov2 December 30, 2023 ==================================Bands of vector-valued functions f:T↦ℝ^d are defined by considering convex hulls generated by their values concatenated at m different values of the argument. The obtained m-bands are families of functions, ranging from the conventional band in case the time points are individually considered (for m=1) to the convex hull in the functional space if the number m of simultaneously considered time points becomes large enough to fill the whole time domain. These bands give rise to a depth concept that is new both for real-valued and vector-valued functions.*Bands of vector-valued functions f:T↦ℝ^d are defined by considering convex hulls generated by their values concatenated at m different values of the argument. The obtained m-bands are families of functions, ranging from the conventional band in case the time points are individually considered (for m=1) to the convex hull in the functional space if the number m of simultaneously considered time points becomes large enough to fill the whole time domain. These bands give rise to a depth concept that is new both for real-valued and vector-valued functions. § INTRODUCTIONThe statistical concept of depth is well known for random vectors in the Euclidean space. It describes the relative position of x from ℝ^d with respect to a probability distribution on ℝ^d or with respect to a sample x_1,…,x_n∈ℝ^d from it. Given a centrally symmetric distribution (for an appropriate notion of symmetry), the point of central symmetry is the deepest point (center of the distribution), while the depth of outward points is low.The concept of depth has been used in the context of trimming multivariate data, to derive depth-based estimators (e.g. depth-weighted L-estimators or ranks based on the center-outward ordering induced by the depth), to assess robustness of statistical procedures, and for classification purposes, to name a few areas, see <cit.> for extensive surveys and further references.Often, the relative position of a point x with respect to a sample is defined with respect to the convex hull of the sample or a part of the sample. For instance, the classical concept of the simplicial depth appears as the fraction of (d+1)-tuples of sampled points whose convex hull contains x, see <cit.>. Its population version is given by the probability that x is contained in the convex hull of (d+1) i.i.d. copies of the random vector.In high-dimensional spaces the curse of dimensionality comes into play and the convex hull of a finite set of sampled points forms a rather “thin” set and so it is very unlikely to expect that many points belong to it. Even the convex hull of the whole sample becomes rather small if the space dimension d is much larger than the sample size n. The situation is even worse for infinite-dimensional spaces that are typical in functional data analysis. In view of this, a direct generalisation of the simplicial depth and convex hull depth concepts leads to the situation where most points in the space have depth zero, see also <cit.>, who discuss problems inherent with the half-space depth in infinite-dimensional spaces, most importantly zero depth and the lack of consistency, see also <cit.>.One possible way to overcome such difficulties is to consider the depth for the collection of function values at any given time argument value t and then integrate (maybe weightedly) over the argument space. This idea goes back to <cit.> and has been further studied in <cit.>.Another approach is based on considering the position of a function relative to the band generated by functions from the sample. The band generated by real-valued functions is defined as the interval-valued function determined by the pointwise minimum and maximum of the functions from the sample. The corresponding band depth has been studied in <cit.>. In the multivariate case the band becomes a set-valued function that at each point equals the convex hull of the values of functions from the sample, see <cit.>.Another multivariate generalisation of the band depth in <cit.> is based on taking convex combinations of band depths associated to each component. Yet another multivariate functional depth concept was studied in <cit.> by integrating the half-space depth over the time domain, see also <cit.>. It is argued in <cit.> that the multivariate setting makes it possible to incorporate other functional data parameters, such as derivatives, into the sample. It is also possible to combine a function with its smoothed version, possibly with different bandwidths.In this paper we suggest a new concept of multivariate functional depth based on taking convex hulls of the functions' values at m≥1 time points combined to build a new higher-dimensional vector. In a sense, this concept pulls together values of the function at different points and so naturally incorporates the time dependency effects, and so better reflects the shape of curves.Two examples at which these m-band depths are used are presented.The constructions described in Section <ref> remind very much the conventional simplicial band depth, where the main point is to check if a point belongs to the convex hull of a subsample. The underlying convex hull in the functional space is replaced by the band, as in <cit.>. It is shown that the introduced band depth satisfies the main properties described in <cit.>. The theoretical computation of the m-band depth is usually unfeasible, since it requires computing the probabilities that a point belongs to a convex hull of random points. Still, its empirical variant is consistent and rather easy to compute. § REGIONS FORMED BY SAMPLES IN FUNCTIONAL SPACES§.§.§ m-bands Let 𝔼 be a linear space of functions f:T↦ℝ^d whose argument t belongs to a rather general topological space T. For example, 𝔼 may be the family of continuous functions on an interval T or a collection of d-vectors if T is a finite set.Consider functions f_1,…,f_j∈𝔼. The convex hull conv(f_1,…,f_j) of these functions is the family of functions f∈𝔼 that can be represented asf(t)=∑_i=1^jλ_i f_i(t) ,t∈T ,for some non-negative constants λ_1,…,λ_j that sum up to one.If the coefficients λ_1,…,λ_j are allowed to be arbitrary functions of t, we arrive at the family of functions f∈𝔼 such that, for all t∈ T, the value f(t) belongs to the convex hull of f_1(t),…,f_j(t). Following <cit.> for univariate (resp. multivariate) functions, the set of such functions is called the band generated by f_1,…,f_j and is denoted by band(f_1,…,f_j). It is obvious thatconv(f_1,…,f_j)⊂band(f_1,…,f_j).If d=1 (as in <cit.>), then band(f_1,…,f_j) consists of all functions f such thatmin_i=1,…,j f_i(t)≤ f(t)≤max_i=1,…,j f_i(t) , t∈ T .In order to obtain a set of functions with interior points, one should avoid the case when the convex hull of f_1(t),…,f_j(t) is of a lower dimension than d at some t. In particular, for this j should be greater than d.We define nested families of functions that lie between the band and the convex hull generated by the sample.The m-band, band_m(f_1,…,f_j), generated by f_1,…,f_j∈𝔼 is the family of functions f∈𝔼 such that, for all t_1,…,t_m∈ T, the vector (f(t_1),…,f(t_m)) belongs to the convex hull of {(f_i(t_1),…,f_i(t_m)), i=1,…,j}, i.e.(f(t_1),…,f(t_m)) =∑_i=1^j λ_i (f_i(t_1),…,f_i(t_m))for non-negative real numbers λ_1,…,λ_j that sum up to one and may depend on (t_1,…,t_m). [Special cases]If T={t} is a singleton, the functions become vectors in ℝ^d and the m-band is their convex hull for all m≥1.If T is a finite set of cardinality k and d=1, then the functions f_1,…,f_j of t∈ T can be viewed as vectors x_i=(x_i1,…,x_ik)∈ℝ^k, i=1,…,j. The 1-band is the smallest hyperrectangle that contains x_1,…,x_j, which is given by ×[a_l,b_l] for a_l=min(x_il,i=1,…,j) and b_l=max(x_il,i=1,…,j) for l=1,…,k. The 2-band is obtained as the largest set such that its projections on each 2-dimensional coordinate plane equals the projection of the convex hull of x_1,…,x_k. The k-band coincides with the convex hull of x_1,…,x_j.If m=1 and d=1, then we recover the band introduced in <cit.> and given by (<ref>), so that band(f_1,…,f_j)=band_1(f_1,…,f_j).If f∈band_m(f_1,…,f_j), then each convex combination of the values for f_1,…,f_j and f can be written as a convex combination of the values of f_1,…,f_j and soband_m(f_1,…,f_j)=band_m(f_1,…,f_j,f) . The m-band is additive with respect to the Minkowski (elementwise) addition. In particular,band_m(g+f_1,…,g+f_j)=g+band_m(f_1,…,f_j)for all g∈𝔼.The m-band is equivariant with respect to linear transformations, that is,band_m(Af_1,…,Af_j)={Af: f∈band_m(f_1,…,f_j)}for all A:T↦ℝ^d× d with A(t) nonsingular for all t∈ T. If all functions generating an m-band are affected by the same phase variation, the phase of the m-band is affected as shown below,band_m(f_1∘ h,…,f_j∘ h) ={f∘ h: f∈band_m(f_1,…,f_j)}for any bijection h:T↦ T.If d=1 and 𝔼 consists of continuously differentiable functions on T=ℝ, then f∈band_m(f_1,…,f_j) yields that f' belongs to band_m-1(f'_1,…,f'_j). This can be extended for higher derivatives. It is obvious that band_m(f_1,…,f_j) is a convex subset of 𝔼; since the points t_1,…,t_m in Definition <ref> are not necessarily distinct, it decreases if m grows. The following result shows that the m-band turns into the convex hull for large m. Assume thatall functions from 𝔼 are jointly separable, that is there exists a countable set Q⊂ T such that, for all f∈𝔼 and t∈ T, f(t) is the limit of f(t_n) for t_n∈ Q and t_n→ t. Then, for each f_1,…,f_j∈𝔼,band_m(f_1,…,f_j)↓conv(f_1,…,f_j)asm→∞.Consider an increasing family T_n of finite subsets of T such that T_n↑ Q and a certain function f∈𝔼. If m_n is the cardinality of T_n, and f belongs to the m_n-band of f_1,…,f_j, then the values (f(t),t∈ T_n) equal a convex combination of (f_i(t),t∈ T_n), i=1,…,j, with coefficients λ_ni. By passing to a subsequence, assume that λ_ni→λ_i as n→∞ for all i=1,…,j. Using the nesting property of T_n, we obtain thatf(t)=∑λ_i f_i(t) ,t∈T_n .Now it suffices to let n→∞ and appeal to the separability of f. Moreover, under a rather weak assumption, the m-band coincides with the convex hull for sufficiently large m.A set of points in the d-dimensional Euclidean space is said to be in general position if no (d-1)-dimensional hyperplane contains more than d points. In particular, if the set contains at most d+1 points, they will be in general position if and only if they are all extreme points of their convex hull, equivalently, any point from their convex hull is obtained as their unique convex combination.If j≤ d(m-1)+1 and there exists t_1,…,t_m-1∈ T such that the vectors (f_i(t_1),…,f_i(t_m-1))∈ℝ^d(m-1), i=1,…,j, are in general position, thenband_m(f_1,…,f_j)=conv(f_1,…,f_j) .Let f∈band_m(f_1,…,f_j).In view of (<ref>), (f(t_1),…,f(t_m-1)) equals a convex combination of (f_i(t_1),…,f_i(t_m-1)), i=1,…,j, which is unique by the general position condition. By considering an arbitrary t_m∈ T, we see that f(t_m) is obtained by the same convex combination, so that f is a convex combination of functions f_1,…,f_j. In particular, if d=1, then the 2-band of two functions coincides with their convex hull. It suffices to note that if f_1 and f_2 are not equal, then f_1(t_1) and f_2(t_1) are different for some t_1 and so are in general position. The same holds for any dimension d≥ 2.[Linear and affine functions]Let f_1,…,f_j be constant functions. Then their 1-band is the collection of functions lying between the maximum and minimum values of f_1,…,f_j. The 2-band consists of constant functions only and coincides with the convex hull. Together with (<ref>), this implies that the 2-band generated by functions f_i(t)=a(t)+b_i, i=1,…,j, is the set of functions a(t)+b for b from the convex hull of b_1,…,b_j.If f_i(t)=a_it+b_i, i=1,…,j, are affine functions of t∈ℝ, then their 3-band consists of affine functions only and also equals the convex hull. Indeed,(f(t_1),f(t_2),f(t_3))=∑λ_i (a_i(t_1,t_2,t_3)+b_i(1,1,1))yields thatf(t_3)-f(t_1)/f(t_2)-f(t_1) =t_3-t_1/t_2-t_1 .Therefore each f from band_3(f_1,…,f_j) is an affine function. [Monotone functions] Let d=1 and let f_1,…,f_j be non-decreasing (respectively non-increasing) functions. Then their 2-band is a collection of non-decreasing (resp. non-increasing) functions. If all functions f_1,…,f_j are convex (resp. concave), then their 3-band is a collection of convex (resp. concave) functions. The definition of the m-band can be easily extended for subsets F of a general topological linear space 𝔼. Consider a certain family of continuous linear functionals u_t, t∈ T. An element x∈𝔼 is said to belong to the m-band of F if for each t_1,…,t_m∈ T, the vector (u_t_1(x),…,u_t_m(x)) belongs to the convex hull of {(u_t_1(y),…,u_t_m(y)):y∈ F}. Then Definition <ref> corresponds to the case of 𝔼 being a functional space and u_t(f)=f(t) for t∈ T.While the conventional closed convex hull arises as the intersection of all closed half-spaces that contain a given set, its m-band variant arises from the intersection of half spaces determined by the the chosen functionals u_t for t∈ T.§.§.§ Space reduction and time share The m-band reduces to a 1-band by defining functions on the product space T^m.For each j, the m-band band_m(f_1,…,f_j) coincides with band(f^(m)_1,…,f^(m)_j), where f^(m)_i:T^m↦(ℝ^d)^m is defined asf^(m)(t_1,…,t_m)=(f(t_1),…,f(t_m)) .It suffices to note that f^(m)(t_1,…,t_m) belongs to the convex hull of f^(m)_i(t_1,…,t_m), i=1,…,j, if and only if (f(t_1),…,f(t_m)) belongs to the convex hull of (f_i(t_1),…,f_i(t_m)), i=1,…,j. In the framework of Proposition <ref>, it is possible to introduce further bands (called space-reduced) by restricting the functions f^(m)_i to a subset S of T^m. For instance, the 1-band generated by functions f^(2)_1,…,f^(2)_j for the arguments (t_1,t_2)∈ℝ^2 such that |t_1-t_2|=h describes the joint behaviour of the values of functions separated by the lag h. If m=1, then the space reduction is equivalent to restricting the parameter space, which can be useful, e.g. for discretisation purposes.It is possible to quantify the closedness of f to the band by determining the proportion of the m-tuple of time values from T^m when the values of f belong to the band.Define the m-band time-share asTS_m(f;f_1,…,f_j) ={(t_1,…,t_m)∈ T^m:f^(m)(t_1,…,t_m)∈conv({f_i^(m)(t_1,…,t_m)}_i=1^j) .If the functions take values in ℝ, then TS_1(f;f_1,…,f_j) turns into the modified band depth defined in <cit.>.If f belongs to the m-band of f_1,…,f_j, then TS_m(f;f_1,…,f_j)=T^m, while if f belongs to the 1-band of f_1,…,f_j, then {(t,…,t): t∈ T}⊂TS_m(f;f_1,…,f_j). It is also straightforward to incorporate the space reduction by replacing T^m with a subset S. § SIMPLICIAL-TYPE BAND DEPTHS§.§.§ Band depth In the following,we consider the event that a function f belongs to a band generated by i.i.d. random functions ξ_1,…,ξ_j with the common distribution P. The m-band depth of the function f with respect to P is defined bybd_m^(j)(f;P)=𝐏{f∈band_m(ξ_1,…,ξ_j)}=𝐏{(f(t_1),…,f(t_m))∈conv({(ξ_i(t_1),…,ξ_i(t_m))}_i=1^j) ,∀ t_1,…,t_m∈ T} .If m increases, then the m-band narrows, and so the m-band depth decreases.We recall that when d=1 the 1-band coincides with the band introduced in <cit.>. Nevertheless the band depth defined in <cit.> is the sum of bd_m^(j)(f;P) with j ranging from 2 to a fixed value J. The same construction can be applied to our m-bands.The m-band depth of f is influenced by the choice of j, and it increases with j. Unlike to the finite-dimensional setting, where j is typically chosen as the dimension of the space plus one <cit.>, there is no canonical choice of j for the functional spaces. In order to ensure that the m-band generated by ξ_1,…,ξ_j differs from the convex hull, it is essential to choose j sufficiently large, and in any case at least d(m-1)+2, see Proposition <ref>. Furthermore, we must impose stronger conditions on j to avoid the zero-depth problem. If j≤ dm and the joint distribution of the marginals of P at some fixed m time points is absolutely continuous, thenbd_m^(j)(·;P)=0 .If j≤ dm and {(ξ_i(t_1),…,ξ_i(t_m))}_i=1^j are independent and absolutely continuous in ℝ^dm, the probability that any fixed x∈ℝ^dm lies in their convex hull is zero. A theoretical calculation of the m-band depth given by (<ref>) is not feasible in most cases. In applications, it can be replaced by its empirical variant defined in exactly the same way as in <cit.> for the 1-band case. Let f_1,…,f_n be a sample from P. Fix any j∈{dm+1,…,n} and definebd_m^(j)(f;f_1,…,f_n) =nj^-1∑_1≤i_1<⋯<i_j≤n 1_f∈band_m(f_i_1,…,f_i_j) ,so that bd_m^(j)(f;f_1,…,f_n) is the proportion of j-tuples from f_1,…,f_n such that f lies in the m-band generated by the j-tuple.The choice of j affects the results. It is computationally advantageous to keep j small, while it is also possible to sum up the depths over a range of the values for j, asin <cit.>.§.§.§ Time-share depth Assume now that T is equipped with a probability measure μ, for example, the normalised Lebesgue measure in case T is a bounded subset of the Euclidean space or the normalised counting measure if T is discrete. Extend μ to the product measure μ^(m) on T^m. Define the time-share depth bytd_m^(j)(f;P) =𝐄μ^(m)(TS_m(f;ξ_1,…,ξ_j)) .If T is a subset of the Euclidean space, Fubini's Theorem yields that the time-share depth is the average of the probability that (f(t_1),…,f(t_m)) lies in the convex hull of j points in ℝ^dm,td_m^(j)(f;P)=∫𝐏{(f(t_1),…,f(t_m))∈conv({(ξ_i(t_1),…,ξ_i(t_m))}_i=1^j)}dμ^(m)(t_1,…,t_m) . For any j∈{dm+1,…,n}, the empirical time-share depth is given bytd_m^(j)(f;f_1,…,f_n) =nj^-1∑_1≤i_1<⋯<i_j≤n μ^(m)(TS_m(f;f_i_1,…,f_i_j)) ,[Univariate case] Assume that T is a singleton. Then necessarily m=1, the function f is represented by a point x in ℝ^d, and the band depth of x for j=d+1 coincides with the simplicial depth, see <cit.>.Let ξ(t)=a(t)+X, t∈ T, where X is a random variable. Then band(ξ_1,…,ξ_j) for i.i.d. ξ_i(t)=a(t)+X_i, i=1,…,j, is the set of functions bounded above by a(t)+max X_i and below by a(t)+min X_i. Thenbd_1^(j)(a;P)=1-𝐏{X>0}^j-𝐏{X<0}^j.By Example <ref>, band_2(ξ_1,…,ξ_j) consists of functions a(t)+b for the constant b∈[min X_i,max X_i]. Only such functions may have a positive 2-band depth.Let now ξ(t)=a(t)+X, where a:T→ℝ^d and X is an absolutely continuous random vector in ℝ^d which is angularly symmetric about the origin. Thenbd_1^(j)(a;P)=1-2^1-j∑_i=0^d-1j-1 ibeing the probability that the origin belongs to the convex hull of X_1,…,X_j, see <cit.>.§.§.§ Properties of the band depthsFor any j≥ dm+1 we have: * affine invariance. bd_m^(j)(Af+g;P_A,g)=bd_m^(j)(f;P) and td_m^(j)(Af+g;P_A,g)=td_m^(j)(f;P) for all g∈𝔼 and A:T↦ℝ^d× d with A(t) nonsingular for t∈ T.* phase invariance. bd_m^(j)(f∘ h;P^h)=bd_m^(j)(f;P) for any one-to-one transformation h:T↦ T, where P^h(F)=P(F∘ h^-1) for any measurable subset F of 𝔼 when h^-1 is the inverse mapping of h.* vanishing at infinity.bd_m^(j)(f;P)→0 if the supremum of f over T converges to infinity, and td_m^(j)(f;P)→ 0 if the infimum of f over T converges to infinity.The affine invariance of both depths follows from the affine invariance of the m-bands, see (<ref>), (<ref>), while the phase-invariance of the band depth follows from (<ref>).In practice, the functions are going to be evaluated over a finite set of time points, thus T={t_1,…,t_k} and probability P is a distribution on (ℝ^d)^k. Furthermore, the sample of functions f_1,…,f_n to be used to determine an empirical m-band depth should have size at least n≥ j≥ dm+1 . If P is absolutely continuous, for any n≥ j≥ dm+1 we have:* maximality at the center. if P is angularly symmetric about the point (f(t_1),…,f(t_k)), function f will be the deepest with regard to the time-share depth, and td^(j)_m(f;P)=1-2^1-j∑_i=0^dm-1j-1 i.* consistency. band depth sup_f∈𝔼|bd_m^(j)(f;f_1,…,f_n)-bd_m^(j)(f;P)|→ 0 a.s. and time-share depth sup_f∈𝔼|td_m^(j)(f;f_1,…,f_n)-td_m^(j)(f;P)|→ 0 a.s.The properties of the time-share depth rely on formula (<ref>) that makes is possible to write it as an average of the probability that a point lies in the convex hull of independent copies of a random vector. The maximality at center follows from the main result in <cit.> which determines the probability inside the integral in (<ref>), see (<ref>), while the consistency can be proved in a similar way to <cit.> extending the uniform consistency of the empirical simplicial depth <cit.> to the one of the probability that a point lies in the convex hull of a fixed number of independent copies of a random vector. Such an extension, which relies on probabilities of intersections of open half-spaces, can be adapted to prove the consistency of the empirical m-band depth. § DATA EXAMPLES§.§.§ Simulated data Fig. <ref> shows 17 curves which are evaluated at T={1,2,…,9}. Among the 17 curves, there is a clear shape outlier (marked as ) that lies deep within the bunch of curves.Such an outlier will not be detected by the outliergram from <cit.> due to its high depth value with regard to both of the 1-band depth and half-region depth (see <cit.>). Nevertheless, its anomalous shape is detected by any m-band depth with m≥ 2.It is remarkable that curve , which is the deepest curve with respect to the usual band depth and modified band depth (bd^(4)_1 and td^(4)_1) is among the less deep curves for the 2-band depth (bd^(4)_2) and is only the fourth deepest curve for its time-share depth (td^(4)_2). The reason for this last fact is that if we restrict to either of the sets of time points {1,3,5,7,9} or {2,4,6,8}, curveis not a shape outlier with respect to them.§.§.§ Real data The nominal Gross Domestic Product per capita of the 28 countries of the European Union (2004–2013) was obtained from the EUROSTAT web-site and is represented in Fig. <ref>. The missing observation that corresponds to Greece, 2013 was replaced by the value obtained from the FOCUSECONOMICS web-site.The deepest curve with regard to the band depth (bd^(5)_1) is the one of Cyprus. Interestingly, Cyprus suffered the 2012-13 Cypriot financial crisis at the end of the considered period and its GDP per capita experienced a decay in 2013 in comparison with its 2012 figure much greater than the one of any other of the EU countries. Also the Greek curve is among the five deepest ones for bd^(5)_1 despite being the only country with a constant decrement in the second half of the considered time period. If we consider 2-bands, that take into account the shape of the curves, these two curves are not any more considered representative of the evolution of the GDP per capita in the EU.The authors would like to thank María Ángeles Gil for the opportunity to contribute in this tribute to Pedro Gil, to whom we do sincerely appreciate.Most of this work was carried over while IM was supported by the Chair of Excellence Programme of the University Carlos III and the Santander Bank. At that time both authors benefited from discussions with Professor Juan Romo. IM is grateful to the Department of Statistics of the University Carlos III in Madrid for the hospitality. spbasic 99. arr:rom14 A. Arribas-Gil and J. Romo. Shape outlier detection and visualization for functional data: the outliergram. Biostatistics, 15:603–619, 2014.cas10 I. Cascos. Data depth: multivariate statistics and geometry. In W. S. Kendall and I. Molchanov, editors, New Perspectives in Stochastic Geometry, pages 398–426. Oxford University Press, Oxford, 2010.chak:chaud14 A. Chakraborty and P. Chaudhuri. On data depth in infinite dimensional spaces. Ann. Inst. Statist. Math., 66(2):303–324, 2014.claes:hub:slaet:14 G. Claeskens, M. Hubert, L. Slaets, and K. Vakili. Multivariate functional halfspace depth. J. Amer. Statist. Assoc., 109(505):411–423, 2014.due92 L. Dümbgen. Limit theorems for the simplicial depth. Statist. Probab. Lett., 14:119–128, 1992.fraim:mun01 R. Fraiman and G. Muniz. Trimmed means for functional data. Test, 10:419–414, 2001.iev:pag13 F. Ieva and A. M. Paganoni. Depth measures for multivariate functional data. Comm. Stat. Theory and Methods, 42:1265–1276, 2013.kuel:zin13 J. Kuelbs and J. Zinn. Concerns with functional depth. ALEA Lat. Am. J. Probab. Math. Stat., 10(2):831–855, 2013.liu:par:sin99 R. Liu, J. M. Parelis, and K. Singh. Multivariate analysis by data depth: Descriptive statistics, graphs and inference (with duiscussion). Ann. Statist., 27:783–858, 1999.liu90 R. Y. Liu. On a notion of data depth based on random simplices. Ann. Statist., 18:405–414, 1990.lop:rom09 S. López-Pintado and J. Romo. On the concept of depth for functional data. J. Amer. Statist. Assoc., 104:718–734, 2009.lop:rom11 S. López-Pintado and J. Romo. A half-region depth for functional data. Comp. Statist. Data Anal., 55:1679–1695, 2011.lop:sun:lin:gen14 S. López-Pintado, Y. Sun, J. K. Lin, and M. G. Genton. Simplicial band depth for multivariate functional data. Adv. Data Anal. Classif., 8:321–338, 2014.nag17 S. Nagy. Integrated depth for measurable functions and sets. Statist. Probab. Lett., 123:165–170, 2017.niet:bat16 A. Nieto-Reyes and H. Battey. A topologically valid definition of depth for functional data. Statist. Sci., 31(1):61–79, 2016.wen62 J. G. Wendel. A problem in geometric probability. Math. Scand., 11:109–111, 1962.zuo:ser00a Y. Zuo and R. Serfling. General notions of statistical depth function. Ann. Statist., 28:461–482, 2000. | http://arxiv.org/abs/1703.09269v1 | {
"authors": [
"Ignacio Cascos",
"Ilya Molchanov"
],
"categories": [
"math.ST",
"stat.TH"
],
"primary_category": "math.ST",
"published": "20170327190209",
"title": "Band depths based on multiple time instances"
} |
Optimal Timing of Decisions: A General Theory Based on ContinuationValues[ Financial support from Australian Research Council Discovery Grant DP120100321 is gratefully acknowledged.Email addresses: ,] Qingyin Maa and John Stachurskib a, bResearch School of Economics, Australian National University December 30, 2023 Building on insights of <cit.> and subsequent authors, we develop a comprehensive theory of optimal timing of decisions based around continuation value functions and operators that act on them. Optimality results are provided under general settings, with bounded or unbounded reward functions. This approach has several intrinsic advantages that we exploit in developing the theory. One is that continuation value functions are smoother than value functions, allowing for sharper analysis of optimal policies and more efficient computation. Another is that, for a range of problems, the continuation value function exists in a lower dimensional space than the value function, mitigating the curse of dimensionality. In one typical experiment, this reduces the computation time from over a week to less than three minutes. Keywords: Continuation values, dynamic programming, optimal timing § INTRODUCTIONA large variety of decision making problems involve choosing when to act in the face of risk and uncertainty.Examples include deciding if or when to accept a job offer, exit or enter a market, default on a loan, bring a new product to market, exploit some new technology or business opportunity, or exercise a real or financial option.See, for example, <cit.>, <cit.>, <cit.>,<cit.>, <cit.>, <cit.>,<cit.>, <cit.>, and <cit.>. The most general and robust techniques for solving these kinds of problems revolve around the theory of dynamic programming.The standard machinery centers on the Bellman equation, which identifies current value in terms of a trade off between current rewards and the discounted value of future states. The Bellman equation is traditionally solved by framing the solution as a fixed point of the Bellman operator.Standard references include <cit.> and <cit.>. Applications of these methods to optimal timing include <cit.>, <cit.>, <cit.>, <cit.>, and <cit.>.Interestingly, over the past few decades, economists have initiated development of an alternative method, based around continuation values, that is both essentially parallel to the traditional method described above and yet significantly different in certain asymmetric ways (described in detail below).Perhaps the earliest technically sophisticated analysis based around operations in continuation value function space is <cit.>. Inan incumbent firm's exit decision context, Jovanovic proposes an operatorthat is a contraction mapping on the space of bounded continuous functions, andshows that the unique fixed point of the operator coincides with the value of stayingin the industry for the current period and then behave optimally. Intuitively, thisvalue can be understood as the continuation value of the firm, since the firm givesup the choice to terminate the sequential decision process (exit the industry) in thecurrent period. Other papers in a similar vein include <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, and <cit.>.All of the results found in these papers are tied to particular applications, and many are applied rather than technical in nature.It is not difficult to understand why economists often focus on continuation values as a function of the state rather than traditional value functions. One is economic intuition. In a given context it might be more natural or intuitive to frame a decision problem in terms of the continuation values faced by an agent. For example, in a job search context, one of the key questions is how the reservation wage, the wage at which the agent is indifferent between accepting and rejecting an offer, changes with economic environments. Obviously, the continuation value, the value of rejecting the current offer, has closer connection to the reservation wage than the value function, the maximum value of accepting and rejecting the offer. There are, however, deeper reasons why a focus on continuation values can be highly fruitful.To illustrate, recall that, for a given problem, the value function provides the value of optimally choosing to either act today or wait, given the current environment.The continuation value is the value associated with choosing to wait today and then reoptimize next period, again taking into account the current environment.One key asymmetry arising here is that, if one chooses to wait, then certain aspects of the current environment become irrelevant, and hence need not be considered as arguments to the continuation value function.To give one example, consider a potential entrant to a market who must consider fixed costs of entry, the evolution of prices, their own productivity and so on.In some settings, certain aspects of the environment will be transitory, while others are persistent.(For example, in <cit.>, prices and beliefs are persistent while fixed costsare transitory.) All relevant state components must be included in the value function,whether persistent or transitory, since all affect the choice of whether to enter or wait today.On the other hand, purely transitory components do not affect continuation values, since, in that scenario, the decision to wait has already been made.Such asymmetries place the continuation value function in a lower dimensional space than the value function whenever they exist, thereby mitigating the curse of dimensionality.This matters from both an analytical and a computational perspective.On the analytical side, lower dimensionality can simplify challenging problems associated with, say, unbounded reward functions, continuity and differentiability arguments, parametric monotonicity results, etc.On the computational side, reduction of the state space by even one dimension can radically increase computational speed.For example, while solving a well known version of the job search model in section <ref>, the continuation value based approach takes only 171 seconds to compute the optimal policy to a given level of precision, as opposed to more than 7 days for the traditional value function based approach.One might imagine that this difference in dimensionality between the two approaches could, in some circumstances, work in the other direction, with the value function existing in a strictly lower dimensional space than the continuation value function.In fact this is not possible.As will be clear from the discussion below, for any decision problem in the broad class that we consider, the dimensionality of the value function is always at least as large.Another asymmetry between value functions and continuation value functions is that the latter are typically smoother.For example, in a job search problem, the value function is usually kinked at the reservation wage. However, the continuation value function can be smooth.More generally, continuation value functions are lent smoothness by stochastic transitions, since integration is a smoothing operation.Like lower dimensionality, increased smoothnesshelps on both the analytical and the computational side.On the computational side, smoother functions are easier to approximate.On the analytical side, greater smoothness lends itself to sharper results based on derivatives, as elaborated on below.To summarize the discussion above, economists have pioneered the continuation value function based approach to optimal timing of decisions.This has been driven by researchers correctly surmising that such an approach will yield tighter intuition and sharper analysis than the traditional approach in many modeling problems.However, all of the analysis to date has been in the context of specific, individual applications.This fosters unnecessary replication, inhibits applied researchers seeking off-the-shelf results, and also hides deeper advantages.In this paper we undertake a systematic study of optimal timing of decisions based around continuation value functions and the operators that act on them. The theory we develop accommodates both bounded rewards and the kinds of unbounded rewards routinely encountered in modeling economic decisions.[ For example, many applications include Markov state processes (possibly with unit roots), driving the state space and various common reward functions (e.g., CRRA, CARA and log returns) unbounded (see, e.g, <cit.>, <cit.>, <cit.>). Moreover, many search-theoretic studies model agent's learning behavior (see, e.g., <cit.>, <cit.>, <cit.>, <cit.>, <cit.>). To have favorable prior-posterior structure (e.g., both follow normal distributions), unbounded state spaces and rewards are usually required. We show that most of these problems can be handled without difficulty. ]In fact, within the context of optimal timing, the assumptions placed on the primitives in the theory we develop are weaker than those found in existing work framed in terms of the traditional approach to dynamic programming, asdiscussed below.We also exploit the asymmetries between traditional and continuation value function based approaches to provide a detailed set of continuity, monotonicity and differentiability results.For example, we use the relative smoothness of the continuation value function to state conditions under which so-called “threshold policies” (i.e., policies where action occurs whenever a reservation threshold is crossed) are continuously differentiable with respect to features of the economic environment, as well as to derive expressions for the derivative. Since we explicitly treat unbounded problems, our work also contributes to ongoing research on dynamic programming with unbounded rewards. One general approach tackles unbounded rewards via the weighted supremum norm. The underlying idea is to introduce a weighted norm in a certain space of candidate functions, and then establish the contraction property for the relevant operator. This theory was pioneered by <cit.> and has been used in numerousother studies of unbounded dynamic programming. Examples include<cit.>,<cit.>, <cit.> and <cit.>.Another line of research treats unboundedness via the local contraction approach, which constructs a local contraction based on a suitable sequence of increasing compact subsets.See, e.g., <cit.>, <cit.>, <cit.> and <cit.>.One motivation of this line of work is to deal withdynamic programming problems that are unbounded both above andbelow. So far, existing theories of unbounded dynamic programming have been confined tooptimal growth problems. Rather less attention, however, has been paid to the study of optimal timing of decisions. Indeed, applied studies of unbounded problems inthis field still rely on theorem 9.12 of <cit.> (see, e.g.,<cit.>, <cit.>).Since the assumptionsof this theorem are not based on model primitives, it is hard to verify inapplications. Even if they are applicable tosome specialized setups, the contraction mapping structure is unavailable. A recentstudy of unbounded problem via contraction mapping is <cit.>.However, he focuses on a highly specialized decision problem with linearrewards. Since there is no general unbounded dynamic programming theory in thisfield, we attempt to fill this gap.Notably, the local contraction approach exploits the underlying structure of the technological correspondence related to the state process, which, in optimalgrowth models, provides natural bounds on the growth rate of the state process,thus a suitable selection of a sequence of compact subsets to construct localcontractions. However, such structures are missing in most sequential decision settings we study, making the local contraction approach inapplicable. In response to that, we come back to the idea of weighted supremum norm, whichturns out to interact well with the sequential decision structure we explore. To obtain an appropriate weight function, we introduce an innovative idea centered on dominating the future transitions of the reward functions, which renders the classical weighted supremum norm theory of <cit.> as a special case, and leads to simple sufficient conditions that are straightforward to check in applications. The intuitions of our theory are twofold. First, when the underlying state process is mean-reverting, the effect of initial conditions tends to die out as time iterates forward, making the conditional expectations of the reward functions flatter than the original rewards. Second, in a widerange of applications, a subset of states are conditionally independent of the futurestates, so the conditional expectation of the payoff functions is actuallydefined on a space that is lower dimensional than the state space.[ Technically, this also accounts for the lower dimensionality of the continuation value function than the value function, as documented above. Section <ref> provides a detailed discussion.] In each scenario, finding an appropriate weight function becomes an easier job.The paper is structured as follows. Section <ref> outlines the method and provides the basic optimality results. Section <ref> discusses the properties of the continuation value function, such as monotonicity and differentiability. Section <ref> explores the connections between the continuation value and the optimal policy. Section <ref> provides a list of economic applications and compares thecomputational efficiency of the continuation value approach and traditionalapproach. Section <ref> provides extensions and section <ref>concludes. Proofs are deferred to the appendix. § OPTIMALITY RESULTSThis section studies the optimality results. Prior to this task, we introduce somemathematical techniques used in this paper. §.§ PreliminariesFor real numbers a and b let a ∨ b := max{a, b}.If f and g are functions, then (f ∨ g)(x) := f(x) ∨ g(x). If (, ) is a measurable space, then b is the set of -measurable bounded functions fromto , with norm f:= sup_z ∈ |f(z)|. Given a function κ→ [1, ∞), the κ-weighted supremum norm of f → is f_κ:=f/κ = sup_z ∈|f(z)|/κ(z). If f_κ < ∞, we say that f is κ-bounded. The symbol b_κ will denote the set of all functions fromtothat are both -measurable and κ-bounded.The pair (b_κ, ·_κ) forms a Banach space (see e.g.,<cit.>, page 331).A stochastic kernel P on (, ) is a map P ×→ [0, 1] such that z ↦ P(z, B) is -measurable for each B ∈ and B ↦ P(z, B) is a probability measure for each z ∈.We understand P(z, B) as the probability of a state transition fromz ∈ to B ∈ in one step.Throughout, we let := {1,2, …} and _0 := { 0}∪.For all n ∈, P^n (z, B) := ∫ P(z', B) P^n-1 (z,z') is the probability of a statetransition from z to B ∈ in n steps. Given a -measurable function h: →, let (P^n h)(z) :=: _z h(Z_n) := ∫ h(z') P^n(z,z')n ∈_0, where (P^0 h) (z) :=: _z h (Z_0) := h(z). Whenis a Borel subset of^m, a stochastic density kernel (or density kernel) onisa measurable map f:×→_+ such that∫_ f(z'|z) dz' = 1 for all z ∈. We say that the stochastic kernel Phas a density representation if there exists a density kernel f such that P(z, B) = ∫_B f(z'|z)z' z ∈ B ∈𝒵.§.§ Set Up Let (Z_n)_n ≥ 0 be a time-homogeneous Markov process defined on probability space (Ω, , ) and taking values inmeasurable space (, ).Let P denote the corresponding stochastic kernel.Let {_n}_n ≥ 0 be a filtration contained insuch that (Z_n)_n ≥ 0 is adapted to {_n}_n≥0. Let _z indicate probability conditioned on Z_0 = z, while_z is expectation conditioned on the same event. In proofs we take (Ω, ) to be the canonical sequence space, so that Ω = ×_n = 0^∞ andis the product σ-algebra generated by .[ For the formal construction of _z on (Ω, ) given P and z ∈ see theorem 3.4.1 of <cit.> or section 8.2 of <cit.>.]A random variable τ taking values in _0 is called a (finite) stopping time with respect to the filtration {_n}_n≥0 if {τ < ∞} = 1 and {τ≤ n}∈_n for all n ≥ 0.Below, τ = n has the interpretation of choosing to act at time n. Letdenote the set of all stopping times on Ω with respect to the filtration {_n}_n≥0.Let r→ and c→ be measurable functions, referred to below as the exit payoff and flow continuation payoff, respectively.Consider a decision problem where, at each time t ≥ 0, an agent observes Z_t and chooses between stopping (e.g., accepting a job, exiting a market, exercising an option) and continuing to the next stage. Stopping generates final payoff r(Z_t).Continuing involves a continuation payoff c(Z_t) and transition to the next period, where the agent observes Z_t+1 and the process repeats. Future payoffs are discounted at rate β∈ (0, 1).Let v^* be the value function, which is defined at z ∈ by v^*(z) := sup_τ∈_z {∑_t=0^τ-1β^t c(Z_t) + β^τ r(Z_τ)}. A stopping time τ∈ is called an optimal stopping time if it attains the supremum in (<ref>). A policy is a map σ fromto {0, 1}, with 0 indicating the decision to continue and 1 indicating the decision to stop. A policy σ is called an optimal policy ifτ^* defined by τ^* := inf{t ≥ 0| σ(Z_t) =1} is an optimal stopping time. To guarantee existence of the value function and related properties without insisting that payoff functions are bounded,we adopt the next assumption: There exist a-measurable function g→_+ and constants n ∈_0, m, d ∈_+ such that β m < 1, and, for all z ∈,max{∫ |r(z')| P^n (z,z'), ∫ |c(z')| P^n (z,z') }≤ g(z) and ∫ g(z') P(z,z') ≤ m g(z) + d. Note that by definition, condition (<ref>) reduces to |r| ∨ |c| ≤ gwhen n=0. The interpretation of assumption <ref> is that both_z |r(Z_n)| and _z |c(Z_n)| are small relative to some function g suchthat _z g(Z_t) does not grow too quickly. Slow growth in _z g(Z_t) is imposed by (<ref>), which can be understood as a geometric drift condition (see, e.g., <cit.>, chapter 15). To verify assumption <ref>, it suffices to obtain a -measurable function g→_+, and constants n ∈_0, m, d ∈_+ with β m < 1, and a_1, a_2, a_3, a_4 ∈_+ such that ∫ |r(z')| P^n (z,z') ≤ a_1 g(z) + a_2, ∫ |c(z')| P^n (z,z') ≤ a_3 g(z) + a_4 and (<ref>) holds. We use this fact in the applications below. One can show that if assumption <ref> holds for some n, it must hold for all n' ∈_0 such that n' > n. Hence, to satisfy assumption <ref>, it suffices to find a measurable map g and constants n_1, n_2 ∈_0, m,d ∈_+ with β m<1 such that ∫ |r(z')| P^n_1 (z,z') ≤ g(z), ∫ |c(z')| P^n_2 (z,z') ≤ g(z) and (<ref>) holds. One can combine this result with remark <ref> to obtain more general sufficient conditions. Consider first an example with bounded rewards. Suppose, as in <cit.>, that a worker can either accept a current wage offer w_t and work permanently at that wage, or reject the offer, receive unemployment compensation c_0>0 and reconsider next period. Let the current wage offer be a function w_t = w(Z_t) of some idiosyncratic or aggregate state process (Z_t)_t ≥ 0. The exit payoff is r(z) = u(w(z)) / (1 - β), where u is a utility function and β < 1 is the discount factor.The flow continuation payoff is c ≡ c_0. If u is bounded, then we can setg(z) ≡ r ∨ c_0, and assumption <ref> is satisfied with n:=0, m := 1 and d := 0. Consider now Markov state dynamics in a job search framework (see, e.g., <cit.>, <cit.>,<cit.>, <cit.>,<cit.>, <cit.>, <cit.>, <cit.>,<cit.>, <cit.>). Consider the same setting as example <ref>, with state processZ_t+1 = ρ Z_t + b + ϵ_t+1, (ϵ_t)N(0, σ^2). The state space is :=. We consider a typical unbounded problem andprovide its proof in appendix B. Let w_t = exp(Z_t) and the utility ofthe agent be defined by the CRRA form u(w) = {[ w^1-δ/1- δ,δ≥ 0 δ≠ 1; ln w,δ = 1;]. Case I: δ≥ 0 and δ≠ 1. If ρ∈ (-1,1), thenwe can select an n ∈_0 that satisfies β e^|ρ^n| ξ<1, whereξ := |(1- δ)b| + (1 - δ)^2 σ^2 / 2. In this case, assumption<ref> holds forg(z) := e^ρ^n (1 - δ) z + e^ρ^n (δ - 1) zand m := d:= e^|ρ^n| ξ.Indeed, if β e^ξ < 1,then assumption <ref> holds (with n=0)for all ρ∈ [-1,1].Case II: δ = 1. If β |ρ|<1, thenassumption <ref> holds with n:=0, g(z) := |z|, m := |ρ| and d := σ + |b|. Notably, since |ρ| ≥ 1 is not excluded, wages can be nonstationary provided that they do not grow too fast.Assumption <ref> is weaker than the assumptions of existing theory. Consider the local contraction method of<cit.>. The essence is to find a countable increasing sequenceof compact subsets, denoted by {K_j }, such that= ∪_j=1^∞ K_j.Let Γ: → 2^ be the technological correspondence of thestate process ( Z_t)_t ≥ 0 giving the set of feasible actions. To construct local contractions, one need Γ(K_j) ⊂ K_j or Γ(K_j) ⊂ K_j+1 with probability onefor all j ∈ (see, e.g., theorems 3–4 of <cit.>, or assumptions D1–D2 of <cit.>). This assumption isoften violated when (Z_t)_t ≥ 0 has unbounded supports. In example<ref>, since the AR(1) state process (<ref>) travels intertemporally throughwith positive probability, the local contraction method breaks down. The use of n-step transitions in assumption <ref>-(<ref>) has certain advantages. For example, if (Z_t)_t ≥ 0 is mean-reverting, as time iterates forward, the initial effect tends to die out, making the conditional expectations _z |r(Z_n)| and _z |c(Z_n)| flatter than the original payoffs. As a result, finding an appropriate g-function with geometric drift property is much easier. Typically, in Case I of example <ref>, if ρ∈ (-1,1), without using future transitions (i.e., n=0 is imposed),[ Indeed, our assumption in this case reduces to the standard weighted supnorm assumption. See, e.g., section 4 of <cit.>, or assumptions 1-4 of <cit.>.] one need further assumptions such as β e^ξ < 1 (see appendix B), which puts nontrivial restrictions on the key parameters β and δ.Using n-step transitions, however, such restrictions are completely removed. Consider now agent's learning in a job search framework (see, e.g., <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>). We follow <cit.> (section IV) and explore how the reservation wage changes in response to the agent's expectation of the mean and variance of the (unknown) wage offer distribution. Each period, the agent observes an offer w_t and decides whether to accept it or remain unemployed. The wage process (w_t)_t ≥ 0 follows ln w_t =ξ + ϵ_t, (ϵ_t)_t ≥ 0 N(0,γ_ϵ), where ξ is the mean of the wage process, which is not observed by the worker, who has prior belief ξ∼ N(μ,γ).[ In general, ξ can be a stochastic process, e.g., ξ_t+1 = ρξ_t + ϵ_t+1^ξ, (ϵ_t^ξ )N(0, γ_ξ). We consider such an extension in a firm entry framework in section <ref>. ] The worker's current estimate of the next period wage distribution is f(w'|μ,γ)=LN(μ,γ+γ_ϵ). After observing w', the belief is updated, with posterior ξ|w' ∼ N(μ',γ'), where γ' = ( 1 / γ + 1 / γ_ϵ)^-1μ' = γ' ( μ / γ + ln w' / γ_ϵ). Let the utility of the worker be defined by (<ref>). If he accepts the offer, the search process terminates and a utility u(w) is obtained in each future period. Otherwise, the worker gets compensation c̃_0>0, updates his belief next period, and reconsiders. The state vector is z = (w, μ, γ) ∈_++××_++ =:. For any integrable function h, the stochastic kernel P satisfies ∫ h(z') P(z,z') = ∫ h (w',μ', γ') f(w'|μ, γ) w', where μ' and γ' are defined by (<ref>). The exit payoff is r(w) = u(w) / (1-β), and the flow continuation payoff is c ≡ c_0 := u(c̃_0). If δ≥ 0 and δ≠ 1, assumption <ref> holds by letting n:=1, g(μ, γ) := e^(1 - δ) μ + (1 - δ)^2 γ /2, m := 1 and d := 0. If δ = 1, assumption <ref> holds with n :=1, g(μ, γ) := e^-μ + γ/2 + e^μ + γ / 2, m := 1 and d:=0. See appendix B for a detailed proof. Since in example <ref>, the wage process (w_t)_t ≥ 0 is independent and has unbounded support _+, the local contraction method cannot be applied. From (<ref>) we know that the conditional expectation of the reward functions in example <ref> is defined on a space of lower dimension than the state space. Although there are 3 states, _z |r(Z_1)| is a function of only 2 arguments: μ andγ. Hence, taking conditional expectation makes it easier to find an appropriate g function. Indeed, if the standard weighted supnorm method were applied, one need to find a g̃(w, μ, γ) with geometric drift property that dominates |r| (see, e.g., section 4 of <cit.>, or, assumptions 1–4 of <cit.>), which is more challenging due to the higher state dimension. This type of problem is pervasive in economics. Sections <ref>–<ref> provide a systematic study, along with a list of applications.§.§ The Continuation Value Operator The continuation value function associated with the sequential decision problem (<ref>) is defined at z ∈ byψ^*(z) := c(z) + β∫ v^*(z') P(z,dz'). Under assumption <ref>, the value function is a solution to the Bellman equation, i.e., v^* = r ∨ψ^*. To see this, by theorem 1.11 of <cit.>, it suffices to show that _z ( sup_k≥ 0| ∑_t=0^k-1β^t c(Z_t) + β^k r(Z_k) | ) < ∞ for all z ∈. This obviously holds sincesup_k≥ 0| ∑_t=0^k-1β^t c(Z_t) + β^k r(Z_k) | ≤∑_t ≥ 0β^t [|r(Z_t)| + |c(Z_t)|]with probability one, and by lemma <ref> (see (<ref>)in appendix A), the right hand side is _z-integrable for all z ∈. To obtain some fundamental optimality results concerning the continuation valuefunction, define an operator Q by Q ψ (z) = c(z) + β∫max{ r(z'), ψ(z') } P(z,z'). We call Q the Jovanovic operator or the continuation value operator. As shown below, fixed points of Q are continuation value functions. From themwe can derive value functions, optimal policies and so on. To begin with, recall n, m and d defined in assumption <ref>. Let m', d' > 0 such that m+2m' > 1, β(m+ 2m')<1 and d' ≥ d / (m + 2m' -1). Let the weight function ℓ→ be ℓ(z) :=m' ( ∑_t=1^n-1_z |r(Z_t)| + ∑_t=0^n-1_z |c(Z_t)| ) + g(z) + d'. We have the following optimality result.Under assumption <ref>, the following statements are true:1. Q is a contraction mapping on (b_ℓ, ·_ℓ) of modulus β (m + 2m').2. The unique fixed point of Q in b_ℓ is ψ^*.3. The policy defined by σ^*(z) = {r(z) ≥ψ^*(z) } is an optimal policy. If both r and c are bounded, then ℓ can be chosen as a constant, and Q is a contraction mapping of modulus β on ( b , ·). If assumption <ref> is satisfied for n=0, then the weight function ℓ(z) = g(z) + d'. If assumption <ref> holds for n=1, then ℓ(z) = m' |c(z)| + g(z) + d'. Recall the job search problem of example <ref>. Let g, n, m and d be defined as in that example. Define ℓas in (<ref>). The Jovanovic operator is Q ψ(z) = c_0 + β∫max{u(w(z'))/1-β, ψ (z') } f(z'|z) z'. Since assumption <ref> holds, theorem <ref> implies that Q has a unique fixed point in b_ℓ that coincides withthe continuation value function, which, in this case, can be interpreted as theexpected value of unemployment. Recall the adaptive search model of example <ref>. Let ℓ be defined by (<ref>). The Jovanovic operator is Q ψ (μ, γ) = c_0 + β∫max{u(w')/1 - β, ψ (μ', γ') } f(w' | μ, γ)w', where μ' and γ' are defined by (<ref>). As shown in example <ref>, assumption <ref> holds. By theorem <ref>, Q is a contraction mapping on ( b_ℓ, ·_ℓ ) with unique fixed point ψ^*, the expected value of unemployment. Consider an infinite-horizon American option (see, e.g., <cit.> or <cit.>). Let the state process be as in (<ref>) so that the state space :=. Let p_t = p(Z_t) = exp(Z_t) be the current price of theunderlying asset, and γ >0 be the riskless rate of return (i.e.,β = e^-γ). The exit payoff for a call option with a strike price K is r(z) = (p(z) - K)^+,while the flow continuation payoff is c ≡ 0. The Jovanovicoperator for the option satisfies Q ψ(z) = e^-γ∫max{ (p(z')-K)^+, ψ(z') } f(z'|z) z'. If ρ∈ (-1,1), we can let ξ := |b| + σ^2 / 2 and n ∈_0 such that e^-γ + |ρ^n| ξ < 1, then assumption <ref> holds with g(z) := e^ρ^n z + e^-ρ^n z and m := d:= e^|ρ^n| ξ. Moreover, if e^-γ + ξ <1, then assumption <ref> holds (with n=0) for all ρ∈ [-1,1]. For ℓ as defined by (<ref>), theorem <ref> implies that Q admits a unique fixed point in b_ℓ that coincides with ψ^*, the expected value of retaining the option and exercising at a later stage. The proof is similar to that of example <ref> and thus omitted. Firm's R&D decisions are often modeled as a sequential search process for better technologies (see, e.g., <cit.>, <cit.>,<cit.>). In each period, an idea with value Z_t ∈ := _+ is observed, and the firm decides whether to put this idea into productive use, or develop it further by investing in R&D. The former choice gives a payoff r(Z_t) = Z_t. The latter incurs a fixed cost c_0 >0 so as to create a new technology. Let the R&D process be governed by the exponential law of motion (with rate θ>0), F(z'|z) :=(Z_t+1≤ z' | Z_t = z) = 1 - e^ - θ (z' - z) (z' ≥ z), While the payoff functions are unbounded, assumption <ref> is satisfied with n:=0, g(z) := z, m:=1 and d := 1/ θ. The Jovanovic operator satisfies Q ψ(z) = - c_0 + β∫max{ z',ψ(z') } F(z'|z). With ℓ as in (<ref>), Q is a contraction mapping on b_ℓ with unique fixed point ψ^*, the expected value of investing in R&D. The proof is straightforward and omitted.Consider a firm exit problem (see, e.g., <cit.>,<cit.>, <cit.>,<cit.>, <cit.>, <cit.>, <cit.>).Each period, a productivity shock a_t is observed by an incumbent firm, where a_t = a(Z_t) = e^Z_t, and the state process Z_t ∈ := is defined by(<ref>). The firm then decides whether to exit the marketnext period or not (before observing a'). A fixed cost c_f >0 is paid eachperiod by the incumbent firm. The firm's output is q (a, l) = a l^α, where α∈ (0,1) and l is labor demand. Given output and input pricesp and w, the payoff functions are r(z)=c(z) = G a(z)^1/1 - α - c_f, whereG = (α p / w )^1/1-α (1-α) w / α.The Jovanovic operator satisfies Q ψ(z)= ( G a(z) ^1/1 - α - c_f ) + β∫max{ G a(z') ^1/1 - α - c_f, ψ(z') } f(z'|z) z'. For ρ∈ [0,1), choose n ∈_0 such thatβ e^ρ^n ξ<1, whereξ := b/1-α + σ^2/2 (1-α)^2. Then assumption <ref> holds withg(z) := e^ρ^n z / (1-α) and m := d := e^ρ^n ξ. Moreover, if β e^ξ<1, then assumption <ref> holds (with n=0) for all ρ∈ [0,1].The case ρ∈ [-1, 0] is similar. By theorem <ref>, Q admits a uniquefixed point in b_ℓ that corresponds to ψ^*, the expected value ofstaying in the industry next period.[The proof is similar to that of example <ref>. Here we are considering the case ρ∈ [-1,0] and ρ∈ [0,1] separately. Alternatively, we can treat ρ∈ [-1,1] directly as in examples <ref> and <ref>. As shown in the proof of example <ref>, the former provides a simpler g function when ρ≥ 0. ] Consider agent's learning in a firm exit framework (see, e.g., <cit.>, <cit.>, <cit.>, <cit.>). Let q be firm's output, C(q) a cost function, and C(q) x be the total cost, where the state process (x_t)_t ≥ 0 satisfies ln x_t = ξ + ϵ_t, (ϵ_t)N(0, γ_ϵ) with ξ denoting the firm type. Beginning each period, the firm observes x and decides whether to exit the industry or not. The prior belief is ξ∼ N(μ,γ), so the posterior after observing x' is ξ |x' ∼ N(μ', γ'), where γ' = ( 1 / γ + 1 / γ_ϵ)^-1 and μ' = γ' ( μ / γ + (ln x') / γ_ϵ). Let π(p,x) = qmax [pq - C(q)x] be the maximal profit, and r(p,x) be the profit of other industries, where p is price. Consider, for example, C(q) := q^2, and (p_t)_t ≥ 0 satisfies ln p_t+1 = ρln p_t + b + ϵ^p_t+1, (ϵ^p_t)_t ≥ 0 N(0, γ_p). Let z := (p,x, μ, γ) ∈_+^2 ××_+ =:. Then the Jovanovic operator satisfies Q ψ(z) = π(p, x) + β∫max{r(p',x'), ψ(z') } l(p', x' |p, μ,γ) (p', x'), where l(p', x' |p, μ,γ) := h(p'|p) f(x' | μ,γ) with h(p'|p) := LN(ρln p + b, γ_p) and f(x'| μ, γ) := LN(μ, γ+ γ_ϵ). If ρ∈ (-1,1) and |r(p,x)| ≤ h_1 p^2 / x + h_2 for some constants h_1, h_2 ∈_+, let ξ := 2(|b| + γ_p) and choose n ∈_0 such that β e^|ρ^n| ξ < 1. Define δ such that δ≥ e^|ρ^n| ξ / ( e^|ρ^n| ξ - 1 ).[ Implicitly, we are considering ρ≠ 0. The case ρ = 0 is trivial. ] Then assumption <ref> holds by letting g(p,μ,γ) := ( p^2ρ^n + p^-2ρ^n + δ) e^ -μ + γ / 2, m := e^|ρ^n| ξ and d:=0. Hence, Q admits a unique fixed point in b_ℓ that equals ψ^*, the value of staying in the industry.[ In fact, the same result holds for more general settings, e.g., |r(p,x)| ≤ h_1 p^2 / x + h_2 p^2 + h_3 x^-1 + h_4 x + h_5 for some h_1, ..., h_5 ∈_+.]§ PROPERTIES OF CONTINUATION VALUESIn this section we explore some further properties of the continuation valuefunction. As one of the most significant results, ψ^* is shown to be continuously differentiable undercertain assumptions. §.§ Continuity We first develop a theory for the continuity of the fixed point. The stochastic kernel P satisfies the Feller property, i.e., P maps bounded continuous functions into bounded continuous functions. The functions c, r and z ↦∫ |r(z')| P(z,z') are continuous. The functions ℓ and z ↦∫ℓ(z') P(z,z') are continuous. Under assumptions <ref> and <ref>–<ref>, ψ^* and v^* are continuous. The next result treats the special case when P admits a density representation. The proof is similar to that of proposition <ref>, except that we use lemma <ref> instead of the generalized Fatou's lemma of <cit.>to establish continuity in (<ref>). In this way, notably, the continuity of r is not necessary for the continuity of ψ^*. The proof isomitted. Suppose assumptions <ref> and <ref> hold, P admits a density representation f(z'|z) that is continuous in z, and that z ↦∫ |r(z')| f(z'|z)z' and c are continuous, then ψ^* is continuous. If in addition r is continuous, then v^* is continuous. By proposition <ref>, if the payoffs r and c are bounded, assumption <ref> and the continuity of r and c are sufficient for the continuity of ψ^* and v^*. If in addition P has a density representation f, by corollary <ref>, the continuity of the flow payoff c and z ↦ f(z'|z) (for all z' ∈) is sufficient for ψ^* to be continuous.[ Notice that in these cases, ℓ can be chosen as a constant, so assumption <ref> holds naturally.] Based on these, the continuity of ψ^* and v^* of example <ref> can be established. If assumption <ref> satisfies for n=0 and assumption <ref> holds, then assumptions <ref>–<ref> are equivalent to: r, c, g and z ↦_z g(Z_1) are continuous.[ When n=0, ℓ(z) = g(z) + d', so |r| ≤ G(g + d') for some constant G. Since r,g and z ↦_z g(Z_1) are continuous, <cit.> (theorem 1.1) implies that z ↦_z |r(Z_1)| is continuous. The next claim in this remark can be proved similarly. ] If assumption <ref> holds for n = 1 and assumptions <ref>–<ref> are satisfied, then assumption <ref> holds if and only if g and z ↦_z |c(Z_1)|, _z g(Z_1) are continuous.Recall the job search model of examples <ref> and <ref>.By corollary <ref>, ψ^* and v^* are continuous. Here is the proof. Assumption <ref> holds, as was shown. P has a density representation f(z'|z) = N(ρ z + b, σ^2) that is continuous in z.Moreover, r,c and g are continuous. It remains to verify assumption <ref>.Case I: δ≥ 0 and δ≠ 1. The proof of example <ref> shows thatz ↦_z |r(Z_t)| is continuous for all t ∈, and that z ↦_z g(Z_1) is continuous (recall (<ref>)–(<ref>)in appendix B). By the definition of ℓ in (<ref>), assumption<ref> holds.Case II: δ = 1.Recall that assumption <ref> holds for n=0 and g(z) = |z|.Since z ↦∫ |z'| f(z'|z)z' is continuous by properties of the normaldistribution, z ↦_z g(Z_1), _z |r(Z_1)| are continuous.[ Indeed, ∫ |z'| f(z'|z)z' = √(2 σ^2 / π) e^ -(ρ z + b)^2 / 2 σ^2 + (ρ z + b) [ 1 - 2 Φ( -(ρ z + b) / σ) ], where Φ is the cdf of the standard normal distribution. The continuity can also be proved by lemma <ref>. ]Hence, assumption <ref> holds.Recall the adaptive search model of examples <ref> and<ref>. Assumption <ref> holds for n=1,as already shown. Assumption <ref> follows from (<ref>) and lemma <ref>. Moreover, r,c and g are continuous. In the proof of example<ref>, we have shown that(μ, γ) ↦_μ, γ |r(w')|, _μ, γ g(μ', γ')are continuous (see (<ref>)–(<ref>) in appendix B), where _μ, γ g(μ', γ') is defined by (<ref>) and _μ, γ |r(w')| := ∫ |r(w')| f(w'|μ, γ)w'. Since ℓ = m'|c| + g + d' when n = 1, assumptions<ref>–<ref> hold. By proposition <ref>, ψ^* and v^* are continuous.Recall the option pricing model of example <ref>. By corollary <ref>, we can show that ψ^* and v^* are continuous. The proof is similar to example <ref>, except that we use |r(z)| ≤ e^z + K, the continuity ofz ↦∫ (e^z' + K) f(z'|z)z', and lemma <ref> to show that z ↦_z |r(Z_1)| is continuous. The continuity ofz ↦_z |r(Z_t)| (for all t ∈) then follows from induction. Recall the R&D decision problem of example <ref>. Assumption <ref> holds for n=0. For all bounded continuous function h:→, lemma <ref> shows thatz ↦∫ h(z') P(z,z') is continuous, so assumption <ref> holds. Moreover, r, c and g are continuous, and∫ |z'| P(z,z') = ∫_[z, ∞) z' θ e^- θ (z' - z) z' = z + 1 / θimplies that z ↦_z |r(Z_1)|, _z|g(Z_1)| are continuous. Sinceℓ(z) = g(z) + d' when n=0, assumptions <ref>–<ref> hold. By proposition <ref>, ψ^* and v^* are continuous.Recall the firm exit model of example <ref>. Through similar analysis as in examples <ref> and <ref>, we can show that ψ^* and v^* are continuous.Recall the firm exit model of example <ref>. Assumption <ref> holds, as was shown. The flow continuation payoff π(p,x) = p^2 / (4x) since C(q) = q^2. Recall that z=(p,x,μ, γ), and for all integrable h, we have ∫ h(z') P(z,z') = ∫ h(p',x',μ',γ') l(p',x'|p,μ,γ) (p', x'). Since by definition γ' is continuous in γ and μ' iscontinuous in μ, γ and x', assumption <ref> holds by lemma<ref>. Further, induction shows that for some constant a_t and allt ∈, ∫ |π (p', x')| P^t (z,z') = a_t p^2 ρ^t e^-μ + γ / 2,which is continuous in(p, μ, γ).If r(p,x) is continuous, then, by lemma <ref> and induction (similarly as inexample <ref>),(p, μ, γ) ↦∫ |r(p',x')| P^t (z,z') is continuous for all t ∈. Moreover, g is continuous and ∫ g(p',μ', γ') P(z,z') = ( p^2ρ^n+1 e^2 ρ^n b + 2 ρ^2nγ_p + p^-2 ρ^n+1 e^-2 ρ^n b + 2 ρ^2nγ_p + δ) e^ -μ + γ / 2 , which is continuous in (p, μ,γ). Hence, assumptions<ref>–<ref> hold. Proposition <ref> then implies that ψ^* and v^* are continuous.§.§ MonotonicityWe now study monotonicity under the following assumptions. The flow continuation payoff c is increasing (resp. decreasing). The function z ↦∫max{ r(z'), ψ(z')} P(z,z') is increasing (resp. decreasing) for all increasing (resp. decreasing) function ψ∈ b_ℓ. The exit payoff r is increasing (resp. decreasing).If assumption <ref> holds and P is stochastically increasing in the sense that P(z, ·) (first order) stochastically dominates P(z̃, ·) for all z̃≤ z, then assumption <ref> holds. Under assumptions <ref> and <ref>–<ref>, ψ^* is increasing (resp. decreasing). If in addition assumption <ref> holds, then v^* is increasing (resp. decreasing). Let b_ℓ i (resp. b_ℓ d) be the set of increasing (resp. decreasing) functions in b_ℓ. Then b_ℓ i (resp. b_ℓ d) is a closed subset of b_ℓ.[ Let (ϕ_n) ⊂ b_ℓ i such that ρ_ℓ (ϕ_n, ϕ) → 0, then ϕ_n →ϕ pointwise. Since (b_ℓ, ρ_ℓ) is complete, ϕ∈ b_ℓ. For all x, y ∈ with x < y, ϕ(x) - ϕ (y) = [ϕ(x) - ϕ_n (x)] + [ϕ_n (x) - ϕ_n (y)] + [ϕ_n (y) - ϕ (y)]. The second term on the right side is nonpositive, ∀ n. Taking limit supremum on both sides yields ϕ (x) ≤ϕ(y). Hence, ϕ∈ b_ℓ i and b_ℓ i is a closed subset. The case b_ℓ d is similar.] To show that ψ^* is increasing (resp. decreasing), it suffices to verify that Q(b_ℓ i ) ⊂ b_ℓ i (resp. Q(b_ℓ d ) ⊂ b_ℓ d).[ See, e.g., <cit.>, corollary 1 of theorem 3.2.] The assumptions of the proposition guarantee that this is the case. Since, in addition, r is increasing (resp. decreasing) by assumption and v^*=r ∨ψ^*, v^* is increasing (resp. decreasing).Recall the job search model of examples <ref>, <ref> and<ref>. Assumption <ref>, <ref> and<ref> hold. If ρ≥ 0, the stochastic kernel P is stochasticallyincreasing since the density kernel is f(z'|z)=N(ρ z + b, σ^2), soassumption <ref> holds. By proposition <ref>, ψ^* andv^* are increasing.Similarly, we can show that for the option pricing model of example <ref> and the firm exit model of example <ref>, ψ^* and v^* are increasing if ρ≥ 0. Moreover, ψ^* and v^* are increasing in example <ref>. The details are omitted.Recall the job search model of examples <ref>, <ref> and <ref>. Note that r(w) is increasing, μ' is increasing in μ, andf(w'|μ, γ) = N(μ, γ + γ_ϵ)is stochastically increasing in μ. So _μ, γ (r(w') ∨ψ(μ', γ')) is increasing in μ for all candidate ψ that is increasing in μ. Since c ≡ c_0, by proposition <ref>, ψ^* and v^* are increasing in μ. Since r is increasingin w, v^* = r ∨ψ^* is increasing in w.Recall the firm exit model of examples <ref> and<ref>.The flow continuation payoff π(p,x) = p^2 / (4x) is increasing in p and decreasing in x.Since P(r ∨ψ^*) is not a function of x, ψ^* is decreasing in x.If the exit payoff r(p,x) is decreasing in x, then v^* is decreasing in x.If ρ≥ 0 and r(p,x) is increasing in p, sinceh(p'|p)= LN(ρln p + b, γ_p) is stochastically increasing,P(r ∨ψ) is increasing in p for all candidate ψ that is increasing in p. By proposition <ref>, ψ^* and v^* are increasing in p.Recall that μ' is increasing in μ. Since f(x'|μ, γ) := LN(μ, γ + γ_ϵ) is stochastically increasing in μ, P(r ∨ψ) is decreasing in μ for all candidate ψ that is decreasing in μ. By proposition <ref>, ψ^* and v^* are decreasing in μ.§.§ DifferentiabilitySuppose ⊂^m. For i = 1, ..., m, let ^(i) bethe i-th dimension and ^(-i) the remaining m-1 dimensions of .A typical element z ∈ takes form of z = (z^1, ..., z^m). Let z^-i := (z^1, ..., z^i-1, z^i+1, ..., z^m). Given z_0 ∈ and δ >0, letB_δ(z_0^i) := { z^i ∈^(i): |z^i-z_0^i| < δ}and B̅_δ(z_0^i) be its closure.Given h: →, let D_i h(z) := ∂ h(z) / ∂ z^iand D_i^2 h(z) := ∂^2 h(z) / ∂(z^i)^2.For a density kernel f, let D_i f(z'|z) := ∂ f(z'|z) / ∂ z^i andD_i^2 f(z'|z) := ∂^2 f(z'|z) / ∂ (z^i)^2.Let μ(z) := ∫max{r(z'), ψ^*(z') } f(z'|z) dz', μ_i(z) := ∫max{r(z'), ψ^*(z')} D_i f(z'|z) dz', and denote k_1 (z) := r(z) and k_2 (z) := ℓ(z). D_i c(z) exists for all z ∈ () and i=1,...,m. P has a density representation f, and, for i = 1, ..., m: * D_i^2 f(z'|z) exits for all (z,z')∈() ×; * (z,z') ↦ D_i f(z'|z) is continuous; * There are finite solutions of z^i to D_i^2 f(z'|z)=0 (denoted by z_i^* (z', z^-i)), and, for all z_0 ∈ (), there exist δ>0 and a compact subset A ⊂ such that z' ∉ A implies z_i^* (z', z_0^-i) ∉ B_δ(z_0^i). When the state space is unbounded above and below, for example, a sufficient condition for assumption <ref>-(3) is: there are finite solutionsof z^i to D_i^2 f(z'|z)=0, and, for all z_0 ∈(),z'→∞ implies|z_i^* (z', z_0^-i)| →∞. k_j is continuous, and, () ∋ z ↦∫ |k_j (z') D_i f(z'|z)|z' ∈_+ for i=1,...,m and j = 1, 2.The following provides a general result for the differentiability of ψ^*. Under assumptions <ref> and <ref>–<ref>, ψ^* is differentiable at interior points, with D_i ψ^* (z) = D_i c(z) + μ_i (z) for all z ∈ () and i=1,...,m.Fix z_0 ∈ (). By assumption <ref>-(3), there existδ >0 and a compact subset A ⊂ such that z' ∈ A^c implies z_i^*(z',z_0^-i) ∉ B_δ (z_0^i), hencez^i ∈B̅_δ(z_0^i)sup |D_i f(z'|z)|= |D_i f(z'|z)|_z^i = z_0^i + δ∨ |D_i f(z'|z)|_z^i = z_0^i - δ(given z^-i = z_0^-i). By assumption <ref>-(2),given z^-i = z_0^-i, there existsG ∈_+, such thatz^i ∈B̅_δ(z_0^i) sup |D_i f(z'|z)| ≤ z' ∈ A, z^i ∈B̅_δ(z_0^i) sup |D_i f(z'|z)| · (z' ∈ A) + ( |D_i f(z'|z)|_z^i = z_0^i + δ∨ |D_i f(z'|z)|_z^i = z_0^i - δ) · (z' ∈ A^c)≤ G · (z' ∈ A) + ( |D_i f(z'|z)|_z^i = z_0^i + δ + |D_i f(z'|z)|_z^i = z_0^i - δ) · (z' ∈ A^c).Assumption <ref> then shows that condition (2) of lemma <ref> holds. By assumption <ref> and lemma <ref>,D_i ψ^* (z) = D_i c(z) + μ_i (z), ∀ z ∈(), as was to be shown. §.§ SmoothnessNow we are ready to study smoothness (i.e., continuous differentiability), anessential property for numerical computation and characterizing optimal policies. For i=1,...,m and j=1, 2, the following conditions hold: * z ↦ D_i c(z) is continuous on (); * k_j is continuous, and, z ↦∫ |k_j (z') D_i f(z'|z)|z' is continuous on (). The next result provides sufficient conditions for smoothness. Under assumptions <ref>, <ref> and <ref>, z ↦ D_i ψ^*(z) is continuous on () for i = 1,..., m. Since assumption <ref> implies assumptions <ref> and <ref>, by proposition <ref>, D_i ψ^*(z) = D_i c(z) + μ_i (z) on (). Since D_i c (z) is continuous by assumption <ref>-(1), to show that ψ^* is continuously differentiable, it remains to verify: z ↦μ_i (z) is continuous on (). Since |ψ^*| ≤ G ℓ for some G ∈_+, | max{ r(z'), ψ^*(z') } D_i f(z'|z) | ≤ (|r(z')| + G ℓ(z')) |D_i f(z'|z)|, ∀ z',z ∈. By assumptions <ref> and <ref>-(2), the right side of (<ref>) is continuous in z, and z ↦∫ [|r(z')| + G ℓ(z')]|D_i f(z'|z)| z' is continuous. Lemma <ref> then implies that z ↦μ_i (z) = ∫max{ r(z'), ψ^*(z') } D_i f(z'|z) dz' is continuous, as was to be shown.Recall the job search model of example <ref> (subsequently studied by examples <ref>, <ref> and <ref>). For all a ∈, leth(z) := e^ a (ρ z + b) + a^2 σ^2 / 2 / √(2 πσ^2). We can show that the following statements hold: (a) There are two solutions to ∂^2 f(z'|z)/∂ z^2 = 0 z^*(z' ) = z' - b ±σ/ρ; (b) ∫| ∂ f(z'|z)/∂ z| z' = |ρ|/σ√(2/π); (c) | z' ∂ f(z'|z)/∂ z| ≤1/√(2 πσ^2)exp{ - (z' - ρ z - b)^2/2 σ^2}( |ρ|/σ^2z'^2 + |ρ(ρ z + b)|/σ^2 |z'| ); (d) e^a z'| ∂ f(z'|z)/∂ z| ≤ h (z) exp{ - [z' - (ρ z + b + a σ^2)]^2 /2 σ^2}|ρ z'| + |ρ (ρ z + b)|/σ^2, ∀ a ∈; (e) The four terms on both sides of (c) and (d) are continuous in z; (f) The integrations (w.r.t. z') of the two terms on the right side of (c) and (d) are continuous in z. Remark <ref> and (a) imply that assumption <ref>-(3)holds. If δ=1, assumption <ref>-(2) holds by conditions (b), (c), (e), (f) and lemma <ref>. If δ≥ 0 and δ≠ 1, based on (<ref>) (appendix B), conditions (b) and (d)–(f), and lemma <ref>, we can show that assumption<ref>-(2) holds. The other assumptions of proposition <ref> are easy to verify. Hence, ψ^* is continuously differentiable. Recall the option pricing problem of example <ref> (subsequently studied by example <ref> and remark <ref>). Through similar analysis as in example <ref>, we can show that ψ^* is continuously differentiable.[ This holds even if the exit payoff r(z) = (p(z) - K)^+ has a kink at z = p^-1 (K). Hence, the differentiability of the exit payoff is not necessary for the smoothness of the continuation value. ]Recall the firm exit model of example <ref> (subsequently studied by example <ref> and remark <ref>).Through similar analysis to examples <ref>–<ref>, we can show that ψ^* is continuously differentiable. Figure <ref> illustrates. We set β = 0.95, σ = 1, b = 0, c_f = 5, α=0.5, p=0.15, w=0.15, and consider respectively ρ = 0.7 and ρ = -0.7. While ψ^* is smooth, v^* is kinked at aroundz = 1.5 when ρ = 0.7, and has two kinks when ρ = -0.7.§.§ Parametric Continuity Consider the parameter space Θ⊂^k. Let P_θ, r_θ, c_θ, v^*_θ and ψ_θ^* denote the stochastic kernel, exit and flow continuation payoffs, value and continuation value functions with respect to the parameter θ∈Θ, respectively. Similarly, let n_θ, m_θ, d_θ and g_θ denote the key elements in assumption <ref> with respect to θ.Define n := θ∈Θsup n_θ, m:=θ∈Θsupm_θ andd := θ∈Θsupd_θ. Assumption <ref> holds at all θ∈Θ, with β m < 1 and n, d < ∞. Under this assumption, let m'>0 and d' > 0 such that m + 2m'>1, β (m + 2m')<1 andd' ≥ d / (m + 2m' - 1). Consider ℓ: ×Θ→ defined byℓ(z, θ) := m' ( ∑_t=1^n-1_z^θ |r_θ (Z_t)| + ∑_t=0^n-1_z^θ |c_θ (Z_t)| ) + g_θ(z) + d',where _z^θ denotes the conditional expectation with respect toP_θ (z, ·). We implicitly assume that Θ does not include β. However, by letting β∈ [0, a] and a∈ [0,1), we can incorporate β into Θ. β m < 1 in assumption <ref> is then replaced by am<1. All the parametric continuity results of this paper remain true after this change. P_θ(z, ·) satisfies the Feller property, i.e., (z, θ) ↦∫ h(z') P_θ(z,z') is continuous for all bounded continuous function h: →. (z, θ) ↦ c_θ(z), r_θ(z), ℓ(z, θ), ∫ |r_θ (z')| P_θ (z,z'), ∫ℓ (z', θ) P_θ (z,z') are continuous.The following result is a simple extension of proposition <ref>. We omitits proof. Under assumptions <ref>–<ref>, (z, θ) ↦ψ^*_θ(z), v^*_θ(z) are continuous. Recall the job search model of example <ref> (subsequently studied by examples <ref>, <ref>, <ref> and <ref>). Let the parameter spaceΘ := [-1, 1] × A × B × C,where A, B are bounded subsets of _++,, respectively, and C ⊂. A typical element θ∈Θ is θ = (ρ, σ, b, c_0). Proposition <ref> implies that (θ, z) ↦ψ^*_θ(z) and (θ, z) ↦ v^*_θ(z) are continuous. The proof is similar to example <ref> and omitted.The parametric continuity of all the other examples discussed above can beestablished in a similar manner.§ OPTIMAL POLICIESIn this section, we provide a systematic study of optimal timing of decisions when there are threshold states, and explore the key properties of the optimal policies. §.§ Conditional Independence in TransitionsFor a broad range of problems, the continuation value function exists in a lower dimensional space than the value function. Moreover, the relationship is asymmetric. While each state variable that appears in the continuation value must appear in the value function, the converse is not true. The continuation value function can have strictly fewer arguments than the value function (recall example <ref>).To verify, suppose that the state space ⊂^m and can be written as = ×, whereis a convex subset of ^m_0,is a convex subset of ^m-m_0, and m_0 ∈ such that m_0 < m. The state process (Z_t)_t ≥ 0 is then {(X_t, Y_t)}_t ≥ 0, where (X_t)_t ≥ 0 and (Y_t)_t ≥ 0 are two stochastic processes taking values inand , respectively. In particular, for each t ≥ 0, X_t represents the first m_0 dimensions and Y_t the rest m-m_0 dimensions of the period-t state Z_t. Assume that the stochastic processes (X_t)_t ≥ 0 and (Y_t)_t ≥ 0 areconditionally independent, in the sense that conditional on each Y_t, thenext period states (X_t+1,Y_t+1) and X_t are independent.Let z := (x,y) and z' := (x',y') be the current and next period states,respectively. With conditional independence, the stochastic kernelP(z,z') can be represented by the conditional distribution of (x',y') on y,denoted as 𝔽_y (x',y'), i.e.,P(z, z') = P((x,y), (x',y')) = 𝔽_y (x',y'). Assume further that the flow continuation payoffc is definedon , i.e., c: →.[ Indeed, in many applications, the flow payoff c is a constant, as seen in previous examples.]Under this setup, ψ^* has strictly fewer arguments than v^*. While v^* is a function of both x and y, ψ^* is a function of y only. Hence, the continuation value based method allows us to mitigate one of the primary stumblingblocks for numerical dynamic programming: the so-called curse of dimensionality(see, e.g., <cit.>, <cit.>).§.§ The Threshold State ProblemAmong problems where conditional independence exists, the optimal policy isusually determined by a reservation rule, in the sense that the decision processterminates whenever a specific state variable hits a threshold level. In such cases,the continuation value based method allows for a sharp analysis of the optimalpolicy. This type of problem is pervasive in quantitative and theoretical economicmodeling, as we now formulate.For simplicity, we assume that m_0 = 1, in which caseis a convex subset ofandis a convex subset of ^m-1. For each t ≥ 0, X_t represents the first dimension and Y_t the rest m-1 dimensions of the period-t state Z_t. If, in addition, r is monotone on , we call X_t thethreshold state and Y_t the environment state (orenvironment) of period t, moreover, we callthe thresholdstate space andthe environment space. r is strictly monotone on . Moreover, for all y∈, there exists x∈ such that r(x,y) = c(y) + β∫ v^*(x',y') 𝔽_y(x',y'). Under assumption <ref>, the reservation rule property holds.When the exit payoff r is strictly increasing in x, for instance, this property states that if the agent terminates at x ∈ at a given point of time, then he would have terminated at any higher state at that moment. Specifically, there is a decision threshold x̅:→ such that when x attains this threshold level, i.e., x = x̅(y), the agent is indifferent between stopping and continuing, i.e., r(x̅(y), y) = ψ^*(y) for all y ∈. As shown in theorem <ref>, the optimal policy satisfiesσ^*(z) = {r(z) ≥ψ^*(z)}. For a sequential decisionproblem with threshold state, this policy is fully specified by the decision threshold x̅. In particular, under assumption <ref>, we have σ^*(x,y) = {[ { x ≥x̅(y) },;{ x ≤x̅(y)},; ]. Further, based on properties of the continuation value, properties of the decisionthreshold can be established. The next result provides sufficient conditions forcontinuity. The proof is similar to proposition <ref> below and thusomitted. Suppose either assumptions of proposition <ref> or of corollary <ref> hold, and that assumption <ref> holds. Then x̅ is continuous. The next result discusses monotonicity. The proof is obvious and we omit it. Suppose assumptions of proposition <ref> and assumption <ref> hold, and that r is defined on . If ψ^* is increasing and r is strictly increasing (resp. decreasing), then x̅ is increasing (resp. decreasing). If ψ^* is decreasing and r is strictly increasing (resp. decreasing), then x̅ is decreasing (resp. increasing).A typical element y ∈ is y = ( y^1, ..., y^m-1). For givenfunctions h: → and l: ×→,define D_i h(y) := ∂ h(y) / ∂ y^i,D_i l(x,y) := ∂ l(x,y) / ∂ y^i, andD_x l(x,y) := ∂ l(x,y) / ∂ x. The next result follows immediately from proposition <ref> and the implicit function theorem. Suppose assumptions of proposition <ref> and assumption <ref> hold, and that r is continuously differentiable on (). Then x̅ is continuously differentiable on (). In particular, D_i x̅(y) = - D_i r(x̅(y),y) - D_i ψ^*(y) / D_x r(x̅(y),y) for all y ∈ ().Intuitively, (x,y) ↦ r(x ,y) - ψ^*(y) denotes the premium of terminatingthe decision process. Hence,(x,y) ↦ D_i r(x,y) - D_i ψ^*(y), D_x r(x,y) are the instantaneous rates of change of the terminating premium in response to changes in y^i and x, respectively. Holding aggregate premium null, the premium changes due to changes in x and y cancel out. As a result, the rate of change of x̅(y) with respect to changes in y^i is equivalent to the ratio of the instantaneous rates of change in the premium. The negativity is due to zero terminating premium at the decision threshold.Let x̅_θ be the decision threshold with respect to θ∈Θ. We have the following result for parametric continuity. Suppose assumptions of proposition <ref> and assumption <ref> hold. Then (y,θ) ↦x̅_θ(y) is continuous. Define F: ××Θ→ by F(x,y, θ) := r_θ(x,y) - ψ_θ^*(y). Without loss of generality, assume that (x,y,θ) ↦ r_θ(x,y) is strictly increasing in x, then F is strictly increasing in x and continuous. For all fixed (y_0, θ_0) ∈×Θ and ϵ>0, since F is strictly increasing in x and F(x̅_θ_0(y_0), y_0, θ_0)=0, we have F(x̅_θ_0(y_0) + ϵ, y_0, θ_0) > 0 F(x̅_θ_0(y_0) - ϵ, y_0, θ_0) < 0. Since F is continuous with respect to (y,θ), there exists δ>0 such that for all (y,θ) ∈ B_δ((y_0,θ_0)) := { (y, θ) ∈×Θ: (y,θ) - (y_0,θ_0)< δ}, we have F(x̅_θ_0(y_0) + ϵ, y, θ) > 0 F(x̅_θ_0(y_0) - ϵ, y, θ) < 0. Since F(x̅_θ(y), y, θ)=0 and F is strictly increasing in x, we have x̅_θ(y) ∈( x̅_θ_0(y_0) - ϵ, x̅_θ_0(y_0) + ϵ), |x̅_θ(y)-x̅_θ_0(y_0)|<ϵ. Hence, (y, θ) ↦x̅_θ(y) is continuous, as was to be shown. § APPLICATIONSIn this section we consider several typical applications in economics, and compare the computational efficiency of continuation value and value function based methods. Numerical experiments show that the partial impact of lower dimensionality of the continuation value can be huge, even when the difference between the arguments of this function and the value function is only a single variable.§.§ Job Search IIConsider the adaptive search model of <cit.> (section6.6). The model is as example <ref>, apart fromthe fact that the distribution of the wage process h is unknown. The workerknows that there are two possible densities f and g, and putsprior probability π_t on f being chosen. If the current offer w_t is rejected, a new offer w_t+1 is observed at the beginning of next period, and, by the Bayes' rule, π_t updates viaπ_t+1 = π_t f(w_t+1) / [ π_t f(w_t+1) + (1 - π_t) g(w_t+1) ] =: q (w_t+1, π_t). The state space is := × [0,1], whereis a compact interval of _+.Let u(w) := w. The value function of the unemployed worker satisfies v^*(w, π) = max{w/1 - β, c_0 + β∫ v^*(w', q(w', π) ) h_π (w') w' }, where h_π (w') := πf(w') + (1 - π) g(w'). This is a typical threshold stateproblem, with threshold state x := w ∈ and environmenty := π∈ [0,1] =:. As to be shown, the optimal policy is determined by a reservation wage w̅:[0,1] → such that when w = w̅(π), the worker isindifferent between accepting and rejecting the offer. Consider the candidate space (b[0,1], ·). The Jovanovic operator is Q ψ (π) = c_0 + β∫max{w'/1-β, ψ∘ q(w',π) } h_π (w') w'. Let c_0 ∈. The following statements are true: 1. Q is a contraction on (b[0,1], ·) of modulus β, with unique fixed point ψ^*. 2. The value function v^*(w,π) = w/1-β∨ψ^*(π), reservation wage w̅(π) = (1 - β) ψ^*(π), and optimal policy σ^*(w, π) = { w ≥w̅(π) } for all (w,π) ∈. 3. ψ^*, w̅ and v^* are continuous. Since the computation is 2-dimensional via value function iteration (VFI), and is only1-dimensional via continuation value function iteration (CVI), we expect thecomputation via CVI to be much faster. We run several groups of tests and comparethe time taken by the two methods. All tests are processed in a standard Pythonenvironment on a laptop with a 2.5 GHz Intel Core i5 and 8GB RAM.§.§.§ Group-1 ExperimentsThis group documents the time taken to compute the fixed point across different parameter values and at different precision levels. Table <ref> providesthe list of experiments performed and table <ref> shows the result. As shown in table <ref>, CVI performs much better than VFI. Onaverage, CVI is 141 times faster than VFI. In the best case, CVI is 207 timesfaster (in test 5, VFI takes 275.41 seconds to achieve a level of accuracy10^-3, while CVI takes only 1.33 seconds). In the worst case, CVI is109 times faster (in test 5, CVI takes 2.99 seconds as opposed to 327.41seconds by VFI to attain a precision level 10^-6). §.§.§ Group-2 ExperimentsIn applications, increasing the number of grid points provides more accuratenumerical approximations. This group of tests compares how the twoapproaches perform under different grid sizes. The setup and result are summarizedin table <ref> and table <ref>, respectively. CVI outperforms VFI more obviously as the grid size increases.In table <ref> we see that as we increase the number of gridpoints for w, the speed of CVI is not affected. However, the speed of VFI dropssignificantly. Amongst tests 2, 6 and 7, CVI is 219 times faster than VFI onaverage. In the best case, CVI is 386 times faster (while it takes VFI 355.40seconds to achieve a precision level 10^-3 in test 7, CVI takes only 0.92second). As we increase the grid size of w from 100 to 200, CVI is notaffected, but the time taken for VFI almost doubles. As we increase the grid size of both w and π, there is a slight decrease in the speed of CVI. Nevertheless, the decrease in the speed of VFI is exponential. Amongtests 2 and 8–10, CVI is 223.41 times as fast as VFI on average. In test 10, VFItakes 706.34 seconds to obtain a level of precision 10^-3, instead, CVI takesonly 1.83 seconds, which is 386 times faster. §.§.§ Group-3 ExperimentsSince the total number of grid points increases exponentially with the number ofstates, the speed of computation will drop dramatically with an additional state. Toillustrate, consider a parametric class problem with respect to c_0. We set =[0,2], β = 0.95, f = (1,1) and g = (3, 1.2). Let(w, π, c_0) lie in [0, 2] × [10^-4, 1-10^-4] × [0, 1.5] with 100grid points for each. In this case, VFI is 3-dimensional and suffers the "curse ofdimensionality": the computation takes more than 7 days. However, CVI is only2-dimensional and the computation finishes within 171 seconds (with precision10^-6). In figure <ref>, we see that the reservation wage is increasing in c_0 and decreasing in π. Intuitively, a higher level of compensation hinders the agent's incentive of entering into the labor market. Moreover, since f is a less attractive distribution than g and larger π means more weight on f and less on g, a larger π depresses the worker's assessment of future prospects, and relatively low current offers become more attractive. §.§ Job Search IIIRecall the adaptive search model of example <ref> (subsequently studied by examples <ref>, <ref> and <ref>). The value function satisfies v^*(w,μ,γ) =max{u(w)/1-β, c_0 + β∫ v^*(w',μ',γ') f(w'|μ, γ) w' }. Recall the Jovanovic operator defined by (<ref>). This is a threshold state sequential decision problem, with threshold statex:=w ∈_++ =: and environment y := (μ,γ) ∈×_++ =:.By the intermediate value theorem, assumption <ref> holds. Hence, the optimal policy is determined by areservation wage w̅: → such that whenw=w̅(μ, γ), the worker is indifferent between accepting andrejecting the offer. Since all the assumptions of proposition <ref> hold (see example <ref>), by proposition<ref>, w̅ is continuous. Since ψ^* is increasing in μ(see example <ref>), by proposition <ref>,w̅ is increasing in μ. In simulation, we set β=0.95, γ_ϵ = 1.0, c̃_0 = 0.6, and consider different levels of risk aversion: σ=3,4,5,6. The grid points of (μ,γ) lie in [-10,10] × [10^-4,10], with 200 points for the μ grid and 100 points for the γ grid. We set the threshold function outside the grid to its value at the closest grid. The integration is computed via Monte Carlo with 1000 draws.[ Changing the number of Monte Carlo samples, the grid range and grid density produce almost the same results.]Figure <ref> provides the simulation results. There are several key characteristics, as can be seen.First, in each case, the reservation wage is an increasing function of μ, whichparallels the above analysis. Naturally, a more optimistic agent (higher μ)would expect that higher offers can be obtained, and will not accept the offer untilthe wage is high enough. Second, the reservation wage is increasing in γ for given μ ofrelatively small values, though it is decreasing in γ for given μ ofrelatively large values. Intuitively, although a pessimistic worker (low μ) expects to obtain low wage offers on average, part of the downside risks are chopped off since a compensation c̃_0 is obtained when the offer is turned down. In this case, a higher level of uncertainty (higher γ) provides a better chance to "try the fortune" for a good offer, boosting up the reservation wage. For an optimistic (high μ) but risk-averse worker, the insurance out of compensation loses power. Facing a higher level of uncertainty, the worker has an incentive to enter the labor market at an earlier stage so as to avoid downside risks. As a result,the reservation wage goes down. §.§ Firm EntryConsider a firm entry problem in the style of <cit.>. Each period, an investment cost f_t>0 is observed, where{f_t} h with finite mean. The firm then decides whether to incur this cost and enter the market to win a stochastic dividend x_tvia production, or wait and reconsider next period. The firm aims to find a decision rule that maximizes the net returns. The dividend follows x_t = ξ_t + ϵ_t^x,{ϵ_t^x}N(0,γ_x),where ξ_t and ϵ_t^x are respectively a persistent and a transient component, andξ_t = ρξ_t-1 + ϵ_t^ξ,{ϵ_t^ξ} N(0, γ_ξ). A public signal y_t+1 is released at the end of each period t, wherey_t = ξ_t + ϵ_t^y,{ϵ_t^y} N(0,γ_y).The firm has prior belief ξ∼ N(μ,γ) that is Bayesian updated after observing y', so the posterior satisfies ξ | y' ∼ N(μ',γ'), withγ' =[ 1 / γ + ρ^2 / (γ_ξ + γ_y) ]^-1μ' = γ' [ μ / γ + ρ y' / (γ_ξ + γ_y) ]. The firm has utility u(x)= (1-e^-ax) / a, where a>0 is the coefficient of absolute risk aversion. The value function satisfies v^*(f,μ, γ) = max{_μ,γ [u(x)] - f,β∫ v^*(f',μ', γ') p(f',y'|μ, γ) (f',y') }, where p(f',y'|μ, γ) = h(f') l(y'|μ,γ) withl(y'|μ,γ) = N(ρμ, ρ^2 γ + γ_ξ + γ_y).The exit payoff is r(f,μ,γ) := _μ,γ [u(x)] - f = ( 1 - e^-a μ + a^2 (γ + γ_x) /2) / a - f.This is a threshold state problem, with threshold state x := f ∈_++ =:and environment y := (μ, γ) ∈×_++ =:. The Jovanovic operator is Q ψ(μ,γ) = β∫max{_μ',γ' [u(x')] - f', ψ(μ',γ') } p(f', y'|μ,γ) (f',y'). Let n:=1, g(μ, γ) := e^ -μ + a^2 γ / 2, m:=1 andd:=0. Define ℓ according to (<ref>). We use f̅: → to denote the reservation cost.The following statements are true: 1. Q is a contraction mapping on (b_ℓ, ·_ℓ) with unique fixed point ψ^*. 2. The value function v^* (f, μ, γ) = r(f, μ, γ) ∨ψ^* (μ, γ), reservation cost f̅ (μ, γ) = _μ,γ[u(x)] - ψ^*(μ,γ) and optimal policy σ^*(f, μ,γ) = { f ≤f̅(μ,γ)} for all (f,μ, γ) ∈. 3. ψ^*, v^* and f̅ are continuous functions. 4. v^* is decreasing in f, and, if ρ≥ 0, then ψ^* and v^* are increasing in μ. Notably, the first three claims of proposition <ref> have no restriction on the range of ρ values, the autoregression coefficient of {ξ_t}.In simulation, we set β=0.95, a=0.2, γ_x=0.1,γ_y=0.05, and h=LN(0, 0.01). Consider ρ = 1, γ_ξ = 0, and let the grid points of(μ,γ) lie in [-2,10] × [10^-4,10] with200 points for the μ grid and 100 points for the γ grid. The reservation cost function outside of the grid points is set to its value at the closest grid point. The integration in the operator is computed via Monte Carlo with1000 draws.[ Changing the number of Monte Carlo samples, the grid range and grid density produces almost the same results.]We plot the perceived probability of investment, i.e.,{f ≤f̅(μ, γ) }. As shown in figure <ref>, the perceived probability of investment is increasing in μ and decreasing in γ. This parallels propositions 1 and 2 of <cit.>. Intuitively, for given investment cost f and variance γ, a more optimistic firm (higher μ) is more likely to invest. Furthermore, higher γ implies a higher level of uncertainty, thus a higher risk of low returns. As a result, the risk averse firm prefers to delay investment (gather more information to avoid downside risks), and will not enter the market unless the cost of investment is low enough. §.§ Job Search IVWe consider another extension of <cit.>. The setup is as in example <ref>, except that the state process follows w_t=η_t+θ_tξ_t lnθ_t=ρlnθ_t-1+ln u_t where ρ∈ [-1,1],{ξ_t } h and {η_t } v with finite first moments,and { u_t } LN(0, γ_u). Moreover, {ξ_t }, {η_t }and { u_t } are independent, and {θ_t } is independent of{ξ_t} and {η_t }. Similar settings as(<ref>)–(<ref>) appear in many search-theoretic and real options studies (see e.g., <cit.>, <cit.>,<cit.>, <cit.>, <cit.>). We set h = LN(0, γ_ξ) and v = LN(μ_η, γ_η). In this case, θ_t and ξ_t are persistent and transitory components of income, respectively, and u_t is treated as a shock to the persistent component. η_t can be interpreted as social security, gifts, etc. Recall that the utility of the agent is defined by (<ref>), c̃_0 > 0 is the unemployment compensation andc_0 := u(c̃_0). The value function of the agent satisfies v^*(w,θ) = max{ u(w) / 1-β, c_0 + β∫ v^*(w',θ') f(θ'|θ) h(ξ') v(η') (θ', ξ', η') }, where w'=η' + θ' ξ' andf(θ'|θ) = LN(ρlnθ, γ_u) is the density kernel of{θ_t }.The Jovanovic operator is Qψ(θ) = c_0 + β∫max{ u(w') / 1-β, ψ(θ') } f(θ' |θ) h(ξ') v(η') (θ', ξ', η'). This is another threshold state problem, with threshold statex := w ∈_++ =: and environmenty := θ∈_++ =:. Let w̅ be the reservation wage. Recall the relative risk aversion coefficient δ in (<ref>) and the weight function ℓ defined by (<ref>).§.§.§ Case I: δ≥ 0 and δ≠ 1For ρ∈ (-1,1), choose n ∈_0 such thatβ e^ρ^2nσ < 1, whereσ := (1 - δ)^2 γ_u.Let g(θ) := θ^(1- δ) ρ^n + θ^-(1 - δ) ρ^n andm := d:= e^ρ^2nσ. If ρ∈ (-1,1), then the following statements hold: 1. Q is a contraction mapping on (b_ℓ, ·_ℓ) with unique fixed point ψ^*. 2. The value function v^*(w,θ) =w^1 - δ/(1-β)(1 - δ)∨ψ^*(θ), reservation wage w̅(θ) = [(1-β) (1 - δ) ψ^*(θ)]^1/1 - δ, and optimal policy σ^*(w, θ) = { w ≥w̅(θ) } for all (w, θ) ∈. 3. ψ^* and w̅ are continuously differentiable, and v^* is continuous. 4. v^* is increasing in w, and, if ρ≥ 0, then ψ^*, v^* and w̅ are increasing in θ. If β e^(1-δ)^2 γ_u / 2 < 1, then claims 1–3 of proposition <ref> remain true for |ρ| = 1, and claim 4 remains true for ρ = 1. §.§.§ Case II: δ=1 For ρ∈ (-1,1), choose n ∈_0 such thatβ e^ρ^2nγ_u < 1. Let g(θ) := θ^ρ^n + θ^-ρ^n andm := d:= e^ρ^2nγ_u. If ρ∈ (-1,1), then the following statements hold: 1. Q is a contraction mapping on (b_ℓ, ·_ℓ) with unique fixed point is ψ^*. 2. The value function v^*(w,θ) =ln w/1-β∨ψ^*(θ), reservation wage w̅(θ) = e^ (1-β) ψ^*(θ), and optimal policy σ^*(w, θ) = { w ≥w̅(θ) } for all (w, θ) ∈. 3. ψ^* and w̅ are continuously differentiable, and v^* is continuous. 4. v^* is increasing in w, and, if ρ≥ 0, then ψ^*, v^* and w̅ are increasing in θ. If β e^γ_u / 2 < 1, then claims 1–3 of proposition <ref> remain true for |ρ| = 1, and claim 4 remains true for ρ = 1. We choose β=0.95 and c̃_0=0.6 as in<cit.> (section 6.6). Further, μ_η = 0,γ_η = 10^-6, γ_ξ = 5 × 10^-4,γ_u = 10^-4 and δ = 2.5. We consider parametric classproblems with respect to ρ, where ρ∈ [0,1] and ρ∈ [-1,0] aretreated separately, with 100 grid points in each case. Moreover, the grid pointsof θ lie in [10^-4, 10] with 200 points, and the grid is scaled to be more dense when θ is smaller. The reservation wage outside thegrid points is set to its value at the closest grid, and the integration is computed viaMonte Carlo with 1000 draws.When ρ = 0, the state process {θ_t }_t ≥ 0 is independent and identically distributed, in which case each realized persistent component will be forgotten in future stages. As a result, the continuation value is independent of θ, yielding a reservation wage that is horizontal to the θ-axis, as shown in figure <ref>.When ρ >0, the reservation wage is increasing in θ, whichparallels propositions <ref>–<ref>. Naturally, a higher θ puts the agent in a better situation, raising his desired wage level.Moreover, since ρ measures the degree of income persistence, a higherρ prolongs the recovery from bad states (i.e., θ < 1), and hinders the attenuation of good states (i.e., θ > 1).As a result, the reservation wage tends to be decreasing in ρ when θ<1 and increasing in ρ when θ > 1.When ρ < 0, the agent always has chance to arrive at a good state in future stages. In this case, a very bad or a very good current state is favorable since a very bad state tends to evolve into a very good one next period, anda very good state tends to show up again in two periods. If the current state is at a medium level (e.g., θ is close to 1), however,the agent cannot take advantage of the countercyclical patterns. Hence, the reservation wage is decreasing in θ at the beginning and then starts to be increasing in θ after some point.§ EXTENSIONS §.§ Repeated Sequential Decisions In many economic models, the choice to stop is not permanent.For example, when a worker accepts a job offer, the resulting job might only be temporary (see, e.g., <cit.>, <cit.>, <cit.>,<cit.>, <cit.>, <cit.>, <cit.>). Another example is sovereign default (see, e.g.,<cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>,<cit.>, <cit.>, <cit.>),where default on international debt leads to a period of exclusion from international financial markets.The exclusion is not permanent, however. With positive probability, the country exits autarky and begins borrowing from international markets again.To put this type of problem in a general setting, suppose that, at date t, an agent is either active or passive.When active, the agent observes Z_t and chooses whether to continue or exit.Continuation results in a current payoff c(Z_t) and the agent remains active at t+1.Exit results in a current payoff s(Z_t) and transition to the passive state. From there the agent has no action available, but will return to the active state at t+1 and all subsequent period with probability α. There exist a -measurable function g: →_+ and constants n ∈_0, m,d ∈_+ such that β m <1, and, for all z ∈, * max{∫ |s(z')| P^n (z,z'), ∫ |c(z')| P^n (z,z') }≤ g(z); * ∫ g(z') P(z,z') ≤ m g(z) + d. Let v^*(z) andr^*(z) be the maximal discounted value starting at z ∈ in the active and passive state respectively. One can show that, underassumption <ref>, v^* and r^* satisfy[ A formal proof of this statement is available from the authors upon request.] v^*(z)= max{ r^*(z), c(z) + β∫ v^*(z') P(z,z') } and r^*(z) = s(z) + αβ∫ v^*(z') P(z,z') + (1 - α) β∫ r^*(z') P(z,z'). With ψ^* := c + β P v^* we can write v^* = r^* ∨ψ^*. Using this notation, we can view ψ^* and r^* as solutions to thefunctional equations ψ = c + β P (r ∨ψ) and r = s + αβ P(r ∨ψ) + (1 - α) β P r.Choose m', d' >0 such that m+2m' >1, β (m+2m') <1 andd' ≥d/m + 2m' - 1. Consider the weight function κ: →_+ defined by κ (z) := m' ∑_t=0^n-1_z [ |s(Z_t)| + |c(Z_t)| ] + g(z) + d'and the product space (b_κ× b_κ, ρ_κ), where ρ_κ is a metric on b_κ× b_κ defined byρ_κ ((ψ, r), (ψ', r'))= ψ - ψ' _κ∨ r - r'_κ. With this metric, (b_κ× b_κ, ρ_κ) inherits the completeness of (b_κ, ·_κ). Now define the operator L onb_κ× b_κ by L [ ψ; r ] = [c + β P(r ∨ψ); s + αβ P(r ∨ψ) + (1 - α) β P r ]. Under assumption <ref>, the following statements hold: 1. L is a contraction mapping on (b_κ× b_κ, ρ_κ) with modulus β(m+2m'). 2. The unique fixed point of L in b_κ× b_κ is h^* := (ψ^*, r^*). §.§ Sequential Decision with More ChoicesIn many economic problems, agents face multiple choices in the sequential decision process (see, e.g., <cit.>, <cit.>, <cit.>, <cit.>, <cit.>). A standard example is on-the-job search, where an employee can choose from quitting the job market and taking the unemployment compensation, staying in the current job at a flow wage, or searching for a new job (see, e.g., <cit.>,<cit.>, <cit.>). A common characteristic of this type of problem is that different choices lead to different transition probabilities.To treat this type of problem generally, suppose that in period t, the agent observes Z_t and makes choices among N alternatives. A selection of alternative i results in a current payoff r_i (Z_t) along with a stochastic kernel P_i. We assume the following. There exist a -measurable function g: →_+ and constants m, d ∈_+ such that β m <1, and, for all z ∈ and i, j = 1, ..., N, * ∫ |r_i (z')| P_j (z,z') ≤ g(z), * ∫ g(z') P_i (z,z') ≤ m g(z) + d. Let v^* be the value function and ψ^*_i be the expected value of choosing alternative i. Under assumption <ref>, we can show that v^* and (ψ_i^*)_i=1^N satisfy[ A formal proof of this result is available from the authors upon request.] v^*(z) = max{ψ_1^* (z), ..., ψ_N^* (z)}, where ψ_i^*(z) = r_i (z) + β∫ v^*(z') P_i (z,z'), for i = 1, ..., N. (<ref>)–(<ref>) imply that ψ_i^* can be written as ψ_i^*(z) = r_i (z) + β∫max{ψ_1^*(z'), ..., ψ_N^*(z') } P_i (z,z'). for i = 1, ..., N. Define the continuation value functionψ^*:= (ψ_1^*, ..., ψ_N^*). Choose m', d' ∈_++ such that β(Nm' + m) <1 andd' ≥d/Nm' + m -1. Consider the weight functionk: →_+ defined byk (z) := m' ∑_i=1^N |r_i(z)| + g(z) + d'.One can show that the product space( × _i=1 ^N (b_k ), ρ_k)is a complete metric space, where ρ_k is defined byρ_k (ψ, ψ̃) = ∨_i=1^N ψ_i - ψ̃_i _k for all ψ = (ψ_1, ..., ψ_N),ψ̃ = (ψ̃_1, ..., ψ̃_N) ∈×_i=1^N (b_k ).The Jovanovic operator on( ×_i=1^N (b_k ), ρ_k ) is defined by Q ψ = Q [ ψ_1; ...; ψ_N ] = [ r_1 + β P_1 (ψ_1 ∨ ... ∨ψ_N);... ...; r_N + β P_N (ψ_1 ∨ ... ∨ψ_N) ]. The next result is a simple extension of theorem <ref> and we omit its proof. Under assumption <ref>, the following statements hold: 1. Q is a contraction mapping on ( × _i=1 ^N (b_k ), ρ_k ) of modulus β(Nm' + m). 2. The unique fixed point of Q in ( × _i=1 ^N (b_k ), ρ_k ) is ψ^* = (ψ_1^*, ..., ψ_N^*). Consider the on-the-job search model of <cit.>. Eachperiod, an employee has three choices: quit the job market, stay in thecurrent job, or search for a new job. Let c_0 be the value of leisure and θbe the worker's productivity at a given firm, with(θ_t)_ t≥ 0 G(θ). Let p be the current price. The pricesequence (p_t)_t ≥ 0 is Markov with transition probability F(p'|p) and stationary distribution F^*(p). It is assumed that there isno aggregate shock so that F^* is the distribution of prices over firms. Thecurrent wage of the worker is p θ. The value function satisfies v^* = ψ_1^* ∨ψ_2^* ∨ψ_3^*, whereψ_1^* (p, θ) := c_0 + β∫ v^* (p', θ')F^*(p')G(θ')denotes the expected value of quitting the job,ψ_2^* (p, θ) := p θ + β∫ v^*(p', θ)F(p'|p)is the expected value of staying in the current firm, and, ψ_3^* (p, θ) := p θ + β∫ v^*(p', θ')F^*(p')G(θ') represents the expected value of searching for a new job. <cit.> assumes that there are compact supports [θ , θ̅] and [p, p̅] for the state processes (θ_t)_t ≥ 0 and (p_t)_t ≥ 0, where 0 < θ < θ̅ < ∞ and 0 < p < p̅ < ∞. This assumption can be relaxed based on our theory. Let the state space be := _+^2. Let μ_p := ∫ pF^* (p) and μ_θ := ∫θ G(θ). There exist a Borel measurable map g̃: _+ →_+, and constants m̃, d̃∈_+ such that βm̃ < 1, and, for all p ∈_+, * ∫ p'F(p'|p) ≤g̃(p), * ∫g̃(p')F(p'|p) ≤m̃g̃(p) + d̃, * μ_p, μ_θ < ∞ and μ_g̃ : = ∫g̃(p)F^*(p) < ∞. Let m̃ > 1 and m̃' ≥ d / (m̃-1), then assumption<ref> holds by lettingg(p, θ) := θ (g̃(p) + m̃' ), m := m̃ and d := μ_θ (μ_g̃ + m'). By theorem <ref>, Q is a contraction mapping on( ×_i=1^3 b_ℓ, ρ_ℓ).Obviously, assumption <ref> is weaker than the assumption of compact supports.§ CONCLUSIONA comprehensive theory of optimal timing of decisions was developed here. The theory successfully addresses a wide range of unbounded sequential decision problems that are hard to deal with via existing unbounded dynamic programming theory, including both the traditional weighted supremum norm theory and the local contraction theory. Moreover, this theory characterizes the continuation value function directly, and has obvious advantages over the traditional dynamic programming theory based on the value function and Bellman operator. First, since continuation value functions are typically smoother than value functions, this theory allows for shaper analysis of the optimal policies and more efficient computation. Second, when there is conditional independence along the transition path (e.g., the class of threshold state problems), this theory mitigates the curse of dimensionality, a key stumbling block for numerical dynamic programming.§ APPENDIX A Under assumption <ref>, there exist a_1, a_2 ∈_+ such that for all z ∈, * |v^*(z)| ≤∑_t=0^n-1β^t _z [|r(Z_t)| + |c(Z_t)|] + a_1 g(z) + a_2. * |ψ^*(z)| ≤∑_t=1^n-1β^t _z |r(Z_t)| + ∑_t=0^n-1β^t _z |c(Z_t)| + a_1 g(z) + a_2. Without loss of generality, we assume m ≠ 1. By assumption <ref>, we have _z |r(Z_n)| ≤ g(z), _z |c(Z_n)| ≤ g(z) and _z g(Z_1) ≤ m g(z) + d for all z ∈. For all t ≥ 1, by the Markov property (see, e.g., <cit.>, section 3.4.3), _z g(Z_t) = _z [ _z ( g(Z_t)| _t-1) ] = _z ( _Z_t-1 g(Z_1) ) ≤_z ( m g(Z_t-1 )+ d ) = m _z g(Z_t-1) + d. Induction shows that for all t ≥ 0, _z g(Z_t) ≤ m^t g(z) + 1 - m^t/1 - m d. Moreover, for all t ≥ n, apply the Markov property again shows that _z |r(Z_t)| = _z [ _z ( |r(Z_t)| | _t-n) ] = _z ( _Z_t-n |r(Z_n)| ) ≤_z g(Z_t-n). Based on (<ref>) we know that _z |r(Z_t)| ≤ m^t-n g(z) + 1-m^t-n/1-m d. Similarly, for all t ≥ n, we have _z |c(Z_t)| ≤_z g(Z_t-n) ≤ m^t-n g(z) + 1-m^t-n/1-m d. Based on (<ref>) - (<ref>), we can show that S(z):= ∑_t≥ 1β^t_z [ |r(Z_t)| + |c(Z_t)| ] ≤∑_t=1^n-1β^t _z [|r(Z_t)| + |c(Z_t)|] + 2 β^n/1-β m g(z) + 2 β^n+1 d/(1-β m)(1-β). Since |v^*| ≤ |r| + |c| + S and |ψ^*| ≤ |c| + S, the two claims hold by letting a_1 := 2 β^n/1-β m and a_2 := 2 β^n+1 d/(1-β m)(1-β). This concludes the proof. Denote (X,𝒳) as a measurable space and (Y,𝒴,u) as a measure space. Let p: Y × X → be a measurable map that is continuous in x. If there exists a measurable map q: Y × X →_+ that is continuous in x with q(y,x) ≥ |p(y,x)| for all (y,x) ∈ Y × X, and that x ↦∫ q(y,x) u( y) is continuous, then the mapping x ↦∫ p(y,x) u( y) is continuous. Since q(y,x) ≥ |p(y,x)| for all (y, x) ∈ Y × X, we know that (y,x) ↦ q(y,x) ± p(y,x) are nonnegative measurable functions. Let (x_n) be a sequence of X with x_n → x. By Fatou's lemma, we have ∫lim inf_n →∞ [q(y,x_n) ± p(y,x_n)] u( y) ≤lim inf_n→∞∫ [q(y,x_n) ± p(y,x_n)] u( y). From the given assumptions we know that n→∞lim∫ q(y,x_n) u( y) = q(y, x). Combine this result with the above inequality, we have ±∫ p(y,x) u( y) ≤lim inf_n →∞( ±∫ p(y,x_n)u( y) ), where we have used the fact that for any two given sequences (a_n)_n≥ 0 and (b_n)_n ≥ 0 ofwith n→∞lim a_n exists, we have: n →∞lim inf(a_n + b_n) = n →∞lim inf a_n + n →∞lim inf b_n. So lim sup_n →∞∫ p(y,x_n) u( y)≤∫ p(y,x) u( y) ≤lim inf_n →∞∫ p(y,x_n) u( y). Therefore, the mapping x ↦∫ p(y,x) u( y) is continuous. § APPENDIX B : MAIN PROOFS§.§ Proof of Section <ref> Results.In this section, we prove examples <ref>–<ref>. Note that inexample <ref>, P has a densityrepresentation f(z'|z) = N(ρ z + b, σ^2). Case I: δ≥ 0 and δ≠ 1. In this case, the exit payoff is r(z) := e^(1 - δ) z /((1- β) (1 - δ)). Since ∫ e^(1- δ) z' f(z'|z)z' = a_1 e^ρ (1 - δ) z for some constant a_1 > 0, induction shows that ∫ e^(1- δ)z' P^t (z,z') = a_t e^ρ^t (1 - δ) z≤ a_t ( e^ρ^t (1 - δ) z + e^ρ^t (δ - 1) z) for some constant a_t >0 and all t ∈. Recall the definition of ξ in example <ref>. Let n ∈ such that β e^|ρ^n| ξ < 1, g(z) :=e^ρ^n (1 - δ) z + e^ρ^n (δ - 1) z and m := d:= e^|ρ^n| ξ. By remark <ref>, it remains to show that g satisfies the geometric drift condition (<ref>). Let ξ_1 := (1 - δ) b and ξ_2 := (1-δ)^2 σ^2 /2, then ξ_1 + ξ_2 ≤ξ, and we have[ To obtain the second inequality of (<ref>), note that either ρ^n+1(1-δ) z ≤ 0 or ρ^n+1 (δ - 1) z ≤ 0. Assume without loss of generality that the former holds, then e^ρ^n+1 (1 - δ) z≤ 1 and 0 ≤ρ^n+1 (δ - 1) z ≤ρ^n (δ - 1) z ∨ρ^n (1 - δ) z. The latter implies that e^ρ^n+1 (δ -1) z≤ e^ρ^n (1 - δ) z + e^ρ^n (δ - 1) z. Combine this with e^ρ^n+1 (1 - δ) z≤ 1 yields the second inequality of (<ref>).] ∫ g(z') f(z'|z)z' = e^ρ^n+1 (1 - δ) z e^ρ^n ξ_1 + ρ^2nξ_2 + e^ρ^n+1 (δ - 1)z e^ -ρ^n ξ_1 + ρ^2nξ_2 ≤( e^ρ^n+1 (1 - δ) z + e^ρ^n+1 (δ - 1) z) e^ |ρ^n| ξ≤( e^ρ^n (1 - δ)z + e^ρ^n (δ - 1)z + 1 ) e^|ρ^n| ξ = m g(z) + d. Since β m = β e^|ρ^n| ξ<1, g satisfies the geometric drift property, and assumption <ref> holds. In fact, if ρ∈ [0,1), by (<ref>), we can also let g(z) := e^ρ^n (1 - δ) z, then ∫ g(z') f(z'|z)z' = e^ρ^n+1 (1 - δ) z e^ρ^n ξ_1 + ρ^2nξ_2≤( e^ρ^n (1 - δ) z + 1 ) e^ρ^n (ξ_1 + ξ_2)≤( e^ρ^n (1 - δ) z + 1 ) e^ρ^n ξ = m g(z) + d. In this way, we have a simpler g with geometric drift property. If ρ∈ [-1,1] and β e^ξ < 1, by (<ref>)–(<ref>) one can show that assumption <ref> holds with n := 0, g(z) := e^(1 - δ) z + e^(δ - 1) z and m := d := e^ξ. In fact, if ρ∈ [0,1] and β e^ξ_1 + ξ_2 <1, by (<ref>) one can show that assumption <ref> holds with n := 0, g(z) := e^(1 - δ) z and m := d:= e^ξ_1 + ξ_2. In this way, we can treat nonstationary state process at the cost of some additional restrictions on parameter values. Case II: δ = 1. In this case, the exit payoff is r(z) = z / (1 - β). Let n:=0, g(z) := |z|, m :=|ρ| and d := σ + |b|. Since (ϵ_t)_t ≥ 0 N(0, σ^2), by Jensen's inequality, ∫ g(z') f(z'|z)z' = _z |Z_1| ≤ |ρ| |z| + |b| +|ϵ_1| ≤ |ρ| |z| + |b| + √( (ϵ_1^2)) = |ρ| |z| + |b| + σ = m g(z) + d. Since β m = β |ρ| < 1, assumption <ref> holds. This concludes the proof. Case I: δ≥ 0 and δ≠ 1. Recall the definition of g. We have ∫ w'^1 - δ f(w' | μ, γ) w' = e^ (1 - δ)^2 γ_ϵ / 2 · e^ (1- δ) μ + (1 - δ)^2 γ / 2 = e^ (1 - δ)^2 γ_ϵ / 2 g(μ, γ). By remark <ref>, it remains to verify the geometric drift condition(<ref>). This follows from (<ref>)–(<ref>). Indeed, one can show that _μ, γ g(μ', γ') := ∫ g(μ', γ') f(w'| μ, γ) w' = g(μ, γ) Case II: δ = 1. Since |ln a| ≤ a + 1 / a, ∀ a > 0, we have: |u(w')| ≤ w' + w'^-1, and ∫ |u(w')| f(w'|μ, γ) w' ≤ e^μ + (γ + γ_ϵ) / 2 + e^-μ + (γ + γ_ϵ) / 2 = e^γ_ϵ g(μ, γ). Similarly as case I, one can show that _μ, γ g(μ', γ') = g(μ, γ). Hence, assumption <ref> holds in both cases. This concludes the proof. To prove claim 1, based on the weighted contraction mapping theorem (see, e.g., <cit.>, section 3), it suffices to verify: (a) Q is monotone, i.e., Qψ≤ Qϕ if ψ, ϕ∈ b_ℓ and ψ≤ϕ; (b) Q0 ∈ b_ℓ and Q ψ is -measurable for all ψ∈ b_ℓ; and (c) Q(ψ+aℓ)≤ Qψ+a β (m + 2m') ℓ for all a∈_+ and ψ∈ b_ℓ. Obviously, condition (a) holds. By (<ref>)–(<ref>), we have |(Q0)(z)|/ℓ(z)≤|c(z)|/ℓ(z) + β∫|r(z')|/ℓ(z) P(z,dz') ≤ (1 + β) / m' < ∞ for all z ∈, so Q0 _ℓ < ∞. The measurability of Q ψ follows immediately from our primitive assumptions. Hence, condition (b) holds. By the Markov property (see, e.g., <cit.>, section 3.4.3), we have ∫_z' |r(Z_t)| P(z,z') = _z |r(Z_t+1)| ∫_z' |c(Z_t)| P(z,z') = _z |c(Z_t+1)|. Let h(z) := ∑_t=1^n-1_z |r(Z_t)| + ∑_t=0^n-1_z |c(Z_t)|, then we have ∫ h(z') P(z,z') = ∑_t=2^n_z |r(Z_t)| + ∑_t=1^n_z |c(Z_t)|. By the assumptions on m' and d', we have m + 2m' > 1 and (d + d') / (m + 2m') ≤ d'. Assumption <ref> and (<ref>) then imply that ∫ℓ(z') P(z,dz') = m' ( ∑_t=2^n_z |r(Z_t)| + ∑_t=1^n_z |c(Z_t)| ) + ∫ g(z') P(z,dz') + d' ≤ m' ( ∑_t=2^n-1_z |r(Z_t)| + ∑_t=1^n-1_z |c(Z_t)| ) + (m + 2m') g(z) + d + d' ≤ (m + 2m') ( m'/m+ 2m' h(z) + g(z) + d + d'/m + 2m') ≤ (m + 2m') ℓ(z). Hence, for all ψ∈ b_ℓ, a ∈_+ and z ∈, we have Q(ψ+aℓ)(z) = c(z) + β∫max{ r(z'), ψ(z') + aℓ(z')} P(z,dz')≤ c(z) + β∫max{ r(z'),ψ(z')} P(z,dz') + a β∫ℓ(z') P(z,dz')≤ Qψ(z) + a β (m + 2m') ℓ(z). So condition (c) holds. Claim 1 is verified. Regarding claim 2, substituting v^* = r ∨ψ^* into (<ref>) we get ψ^*(z) = c(z) + β∫max{ r(z'), ψ^*(z') } P(z,z'). This implies that ψ^* is a fixed point of Q. Moreover, from lemma <ref> we know that ψ^* ∈ b_ℓ. Hence, ψ^* must coincide with the unique fixed point of Q under b_ℓ. Finally, by theorem 1.11 of <cit.>, we can show that τ̃:= inf{t ≥ 0: v^*(Z_t) = r(Z_t)} is an optimal stopping time. Claim 3 then follows from the definition of the optimal policy and the fact that v^* = r ∨ψ^*. §.§ Proof of Section <ref> Results. Let b_ℓ c be the set of continuous functions in b_ℓ. Since ℓ is continuous by assumption, b_ℓc is a closed subset of b_ℓ (see e.g., <cit.>, section 3). To show the continuity of ψ^*, it suffices to verify that Q (b_ℓ c ) ⊂ b_ℓ c (see, e.g., <cit.>, corollary 1 of theorem 3.2). For all ψ∈ b_ℓ c, there exists a constant G ∈_+ such that |max{ r(z), ψ(z) }| ≤ |r(z)| + G ℓ(z). In particular, z ↦ |r(z)| + G ℓ(z) ±max{ r(z), ψ(z)} are nonnegative and continuous. Let h(z) := max{ r(z), ψ (z) }. Based on the generalized Fatou's lemma of <cit.> (theorem 1.1), we can show that for all sequence (z_m)_m ≥ 0 ofsuch that z_m → z ∈, we have ∫( |r(z')| + G ℓ(z') ± h(z') ) P(z,z') ≤lim inf_m →∞∫( |r(z')| + G ℓ(z') ± h(z') ) P(z_m,z'). Since assumptions <ref>–<ref> imply that lim_m →∞∫( |r(z')| + G ℓ(z') ) P(z_m,z') = ∫( |r(z')| + G ℓ(z') ) P(z,z'), we have ±∫ h(z') P(z,z') ≤lim inf_m →∞( ±∫ h(z') P(z_m,z') ), where we have used the fact that for given sequences (a_m)_m ≥ 0 and (b_m)_m ≥ 0 ofwith m→∞lim a_m exists, we have: m →∞lim inf (a_m + b_m) = m →∞lim a_m + m →∞lim infb_m. Hence, lim sup_m →∞∫ h(z') P(z_m,z') ≤∫ h(z') P(z,z') ≤lim inf_m →∞∫ h(z') P(z_m,z'), i.e., z ↦∫ h(z') P(z,z') is continuous. Since c is continuous by assumption, Q ψ∈ b_ℓ c. Hence, Q (b_ℓ c ) ⊂ b_ℓ c and ψ^* is continuous. The continuity of v^* follows from the continuity of ψ^* and r and the fact that v^*=r ∨ψ^*.Recall μ and μ_i defined in the beginning of section <ref>. The next lemma holds.Suppose assumption <ref> holds, and,for i = 1, ..., m and j = 1, 2 * P has a density representation f such that D_i f(z'|z) exists, ∀ (z, z') ∈() ×. * For all z_0 ∈(), there exists δ>0, such that ∫ |k_j (z')| z^i ∈B̅_δ(z_0^i)sup| D_i f(z'|z) | z' < ∞(z^-i = z_0^-i).Then: D_i μ (z) = μ_i (z) for all z ∈ () and i=1,...,m. For all z_0 ∈(), let { z_n} be an arbitrary sequence of () such that z_n^i → z_0^i, z_n^i ≠ z_0^i and z_n^-i = z_0^-i for all n ∈. For the δ>0 given by (2), there exists N∈ such that z_n^i ∈B̅_δ(z_0^i) for all n≥ N. Holding z^-i = z_0^-i, by the mean value theorem, there exists ξ^i (z',z_n,z_0) ∈B̅_δ(z_0^i) such that |^i (z',z_n,z_0)| := | f(z'|z_n) - f(z'|z_0)/z_n^i-z_0^i| = | D_i f(z'|z)|_z^i = ξ^i (z',z_n, z_0)| ≤z^i ∈B̅_δ(z_0^i)sup| D_i f(z'|z) | Since in addition |ψ^*| ≤ G ℓ for some G ∈_+, we have: for all n≥ N, (a) | max{ r(z'), ψ^*(z')}^i (z',z_n, z_0) | ≤( |r(z')|+ G ℓ(z') ) z^i ∈B̅_δ(z_0^i)sup| D_i f(z'|z)|, (b) ∫( |r(z')|+ G ℓ(z') ) z^i ∈B̅_δ(z_0^i)sup| D_i f(z'|z)| dz' < ∞, and (c) max{ r(z'), ψ^*(z')}^i (z',z_n,z_0) →max{ r(z'), ψ^*(z')} D_i f(z'|z_0) as n→∞, where (b) follows from condition (2). By the dominated convergence theorem, μ(z_n)-μ(z_0)/z_n^i - z_0^i = ∫max{ r(z'), ψ^*(z')}^i (z',z_n,z_0)z'→∫max{ r(z'), ψ^*(z')} D_i f(z'|z_0)z' = μ_i (z_0). Hence, D_i μ(z_0) = μ_i(z_0), as was to be shown.§.§ Proof of Section <ref> Results Assumption <ref> holds due to bounded payoffs. By theorem <ref>, claim 1 holds. Let = [w_l, w_h] ⊂_+. Since c_0 ∈, v^*(w, π) ∈ [w_l / (1 - β), w_h / (1 - β)], then c_0 + β∫ v^* (w', π') h_π (w')w' ∈ [w_l / (1 - β), w_h / (1 - β)]. By the intermediate value theorem, assumption <ref> holds. By theorem <ref> and (<ref>), claim 2 holds. P satisfies the Feller property by lemma <ref>. Since payoff functions are continuous, the continuity of ψ^* and v^* follows from proposition <ref> (or remark <ref>). The continuity of w̅ follows from proposition <ref>. Claim 3 is verified. The exit payoff satisfies | r(f', μ', γ') | ≤ 1/a + ( e^a^2 γ_x / 2 / a ) · e^-a μ' + a^2 γ' / 2 + f'. Using (<ref>), we can show that ∫ e^-a μ' + a^2 γ' / 2 P(z,z') = ∫ e^-a μ' + a^2 γ' / 2 l(y' | μ, γ) y' = e^-a μ + a^2 γ / 2. Let μ_f denote the mean of { f_t}. Combine (<ref>)–(<ref>), we have ∫| r(f',μ',γ') | P(z,z') ≤(1/a + μ_f ) + ( e^a^2 γ_x / 2 / a ) · g(μ, γ). Notice that (<ref>) is equivalent to ∫ g(μ', γ') P(z,z') = g(μ, γ). Hence, assumption <ref> holds with n :=1, m :=1 and d:=0. The intermediate value theorem shows that assumption <ref> holds. By theorem <ref> and the analysis of section <ref>, claims 1–2 hold. For all bounded continuous function f̃: →, we have ∫f̃(z') P(z,z') = ∫f̃ (f',μ', γ') h(f') l(y' | μ, γ) (f', y'). By (<ref>) and lemma <ref>, this function is bounded and continuous in (μ, γ). Hence, assumption <ref> holds. The exit payoff r is continuous. By (<ref>), both sides of (<ref>) are continuous in (μ, γ). By (<ref>)–(<ref>), the conditional expectation of the right side of (<ref>) is continuous in (μ, γ). Lemma <ref> then implies that (μ, γ) ↦_μ, γ |r(Z_1)| is continuous. Now we have shown that assumption <ref> holds. Assumption <ref> holds since g is continuous and (<ref>) holds. Proposition <ref> then implies that ψ^* and v^* are continuous. By proposition <ref>, f̅ is continuous. Claim 3 is verified. Since r is decreasing in f, v^* = r ∨ψ^* is decreasing in f. If ρ≥ 0, then l is stochastically increasing in μ. By (<ref>),P(r ∨ψ) is increasing in μ for all ψ∈ b_ℓ that is increasing in μ, i.e., assumption <ref> holds. Since r is increasing in μ, by proposition <ref>, ψ^* and v^* are increasing in μ. Hence, claim 4 holds. Proof of claim 1. Since w'^1 - δ = ( η' + θ' ξ' )^1 - δ≤ 2 ( η'^1 - δ + θ'^1 - δξ'^1 - δ), we have ∫ w'^1 - δ P(z,z') ≤ 2 ∫η'^1 - δ v(η') η' + 2 ∫ξ'^1 - δ h(ξ') ξ' ·∫θ'^1 - δ f(θ' | θ) θ' = 2 e^ (1 - δ) μ_η + (1 - δ)^2 γ_η / 2 + 2 e^ (1 - δ)^2 (γ_ξ + γ_u) / 2 ·θ^(1 - δ) ρ. Induction shows that ∫ w'^1 - δ P^t (z,z') ≤ a_1^(t) + a_2^(t)θ^(1 - δ) ρ^t≤ a_1^(t) + a_2^(t)( θ^(1 - δ) ρ^t+ θ^-(1 - δ) ρ^t) for some a_1^(t), a_2^(t) > 0 and all t ∈. Define g as in the assumption, then ∫ g(θ') f(θ' | θ) θ' = ( e^ (1 - δ) ρ^n+1lnθ + e^ -(1 - δ) ρ^n+1lnθ) e^ (1 - δ)^2 ρ^2nγ_u / 2 ≤( e^(1 - δ) ρ^n lnθ + e^ -(1 - δ) ρ^n lnθ + 1 ) e^ (1 - δ)^2 ρ^2nγ_u / 2 ≤( g(θ) + 1 ) e^ρ^2nσ = m g(θ) + d. Hence, assumption <ref> holds. Claim 1 then follows from theorem <ref>. Proof of claim 2. Assumption <ref> holds by the intermediate value theorem. Claim 2 then follows from theorem <ref>, assumption <ref> and (<ref>). Proof of claim 3. Note that the stochastic kernel P has a density representation in the sense that for all z ∈ and B ∈, P(z, B) = ∫{ (η' + ξ' θ', θ') ∈ B } v (η') h(ξ') f(θ'|θ) (η', ξ', θ'). Moreover, it is straightforward (though tedious) to show that θ↦ f(θ' | θ) is twice differentiable for all θ', that (θ, θ') ↦∂ f(θ'|θ) / ∂θ is continuous, and that ∂^2 f(θ'|θ) / ∂θ^2 = 0 θ = θ^*(θ') = ã_i e^lnθ' / ρ, i = 1, 2 where ã_1, ã_2 = exp[ γ_u/ρ( -1/2 ρ±√(1/4 ρ^2 + 1/γ_u)) ]. If ρ>0, then θ^*(θ') →∞ as θ' →∞ and θ^*(θ') → 0 as θ' → 0. If ρ < 0, then θ^*(θ') → 0 as θ' →∞ and θ^*(θ') →∞ as θ' → 0. Hence, assumption <ref> holds. Based on (<ref>)–(<ref>) and lemma <ref>, we can show that assumption <ref> holds. By proposition <ref>, ψ^* is continuously differentiable. Since assumption <ref> holds and r is continuously differentiable, by proposition <ref>, w̅ is continuously differentiable. v^* is continuous since v^* = r ∨ψ^*. Proof of claim 4. Assumption <ref> holds since c ≡ c_0. Note that r(w) = r(η + ξθ) = (η + ξθ)^1 - δ / [(1 - β) (1 - δ)] is increasing in θ, and, when ρ >0, f(θ' | θ) is stochastically increasing in θ. Hence, assumption <ref> holds. By propositions <ref> and <ref>, ψ^* and w̅ are increasing in θ. Moreover, r is a function of w, ψ^* is a function of θ, both functions are increasing, and v^* = r ∨ψ^*. Hence, v^* is increasing in w and θ. Since |ln a| ≤ 1 / a + a for all a>0, we have |u(w')| = |ln w' | = |ln (η' + θ' ξ')| ≤ 1/ η' + η' + θ' ξ' . Hence, ∫ |u(w')| P(z,z') ≤∫ (1 / η' + η') v(η') η' + ∫ξ' h(ξ') ξ' ·∫θ' f(θ'| θ) θ' = ( e^-μ_η + γ_η / 2 + e^μ_η + γ_η / 2) + e^ (γ_ξ + γ_u) / 2·θ^ρ. Induction shows that ∫ |u(w')| P^t (z,z') ≤ a_1^(t) + a_2^(t) θ^ρ^t≤ a_1^(t) + a_2^(t)( θ^ρ^t + θ^- ρ^t) for some a_1^(t), a_2^(t) > 0 and all t ∈. Hence, we can define g as in the assumption. Similarly as in the proof of proposition <ref>, we can show that ∫ g(θ') f(θ' | θ) θ' = ( e^ρ^n+1lnθ + e^ -ρ^n+1lnθ) e^ρ^2nγ_u / 2 ≤( e^ρ^n lnθ + e^ -ρ^n lnθ + 1 ) e^ρ^2nγ_u / 2= ( g(θ) + 1 ) e^ρ^2nγ_u / 2 ≤ m g(θ) + d. Hence, assumption <ref> holds. Claim 1 then follows from theorem <ref>. The remaining proof is similar to proposition <ref>.§.§ Proof of Section <ref> Results Regarding claim 1, similar to the proof of theorem <ref>, we can show that ∫κ(z') P(z,z') ≤ (m + 2 m') κ (z) for all z ∈. We next show thatL(b_κ× b_κ, ρ_κ) → (b_κ× b_κ, ρ_κ). For all h := (ψ, r) ∈ b_κ× b_κ, define thefunctions p(z) := c(z) + β∫max{ r(z'), ψ(z')} P(z,z') andq(z) := s(z) + αβ∫max{ r(z'), ψ(z') } P(z,z') + (1 - α) β∫ r(z') P(z,z'). Then there exists G ∈_+ such that for all z ∈, |p(z)|/κ(z)≤|c(z)|/κ(z) + β G ∫κ(z') P(z,z')/κ(z)≤1/m' + β (m + 2m') G < ∞ and|q(z)|/κ(z)≤|s(z)|/κ(z) + β G ∫κ(z') P(z,z')/κ(z)≤1/m' + β (m + 2m') G < ∞. This implies that p ∈ b_κ and q ∈ b_κ. Hence, L h ∈ b_κ× b_κ. Next, we show that L is indeed a contraction mapping on (b_κ× b_κ, ρ_κ). For all fixed h_1 := (ψ_1, r_1) and h_2 := (ψ_2, r_2) inb_κ× b_κ, we haveρ_κ(Lh_1, Lh_2) = I ∨ J, where I := β P (r_1 ∨ψ_1) - β P(r_2 ∨ψ_2) _κ and J := αβ [P(r_1 ∨ψ_1) - P(r_2 ∨ψ_2)] +(1 - α) β (P r_1 - Pr_2) _κ. For all z ∈, we have | ∫ ( r_1 ∨ψ_1 )(z') P(z,z') - ∫ ( r_2 ∨ψ_2 )(z') P(z,z') | ≤∫| r_1 ∨ψ_1 - r_2 ∨ψ_2 | (z') P(z,z') ≤∫ ( |ψ_1 - ψ_2| ∨ |r_1 - r_2| ) (z') P(z,z') ≤ ( ψ_1 - ψ_2 _κ∨ r_1 - r_2 _κ ) ∫κ(z') P(z,z') ≤ρ_κ (h_1, h_2) (m + 2m') κ(z),where the second inequality is due to the elementary fact|a ∨ b - a' ∨ b'| ≤ |a-a'| ∨ |b-b'|. This implies thatI ≤β (m+2m') ρ_κ(h_1, h_2).Regarding J, similar arguments yield J ≤β (m + 2m') ρ_κ(h_1, h_2). In conclusion, we haveρ_κ(L h_1, L h_2) = I ∨ J ≤β (m + 2m') ρ_κ (h_1, h_2). Hence, L is a contraction mapping on(b_κ× b_κ, ρ_κ)with modulus β (m + 2m'), as was to be shown. Claim 1 is verified.Since v^* and r^* satisfy (<ref>)–(<ref>), by (<ref>),h^* := (ψ^*, r^*) is indeed a fixed point of L. To prove that claim 2 holds, it remains to show that h^* ∈ b_κ× b_κ. Since max{ |r^*(z)|, | ψ^*(z)| }≤∑_t=0^∞β^t _z [ |s(Z_t)| + g(Z_t) ], this can be proved in a similar way as lemma <ref>. 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empirical work, author=Ericson, Richard and Pakes, Ariel, journal=The Review of Economic Studies, volume=62, number=1, pages=53–82, year=1995, publisher=Oxford University Press | http://arxiv.org/abs/1703.09832v1 | {
"authors": [
"Qingyin Ma",
"John Stachurski"
],
"categories": [
"math.OC"
],
"primary_category": "math.OC",
"published": "20170327130012",
"title": "Optimal Timing of Decisions: A General Theory Based on Continuation Values"
} |
Speech Enhancementusing a Deep Mixture of ExpertsShlomo E. Chazan, Jacob Goldberger and Sharon Gannot Senior Member, IEEE, Shlomo E. Chazan, Jacob Goldberger and Sharon Gannot are with the Faculty of Engineering, Bar-Ilan University, Ramat-Gan, 5290002, Israel (e-mail: [email protected]; [email protected]; [email protected]). Faculty of Engineering, Bar-Ilan University, Ramat-Gan, 5290002, Israel December 30, 2023 ==================================================================================================================================================================================================================================================================================================================================================================================================0.18in The present paper deals with a prey-predator model with prey refuge proportion to both species and independent harvesting of each species. Our study shows that using refuge as control, it can break the limit circle of the system and reach the required state of equilibrium level. It is established the optimal harvesting policy. The boundedness, feasibility of interior equilibria, bionomic equilibrium have been determined. The main observation is that coefficient of refuge plays an important role in regulating the dynamics of the present system. Moreover the variation of the coefficient of refuge change the system from stable to unstable and vice-versa. Some numerical illustration are given in order to support of our analytical and theoretical findings. Mathematics Subject Classification: 92D25, 92D30, 92D40.Keywords: Harvesting, Refuge, Bionomic equilibrium, Limit cycle, Hopf bifurcation, Numerical simulations§ INTRODUCTIONThe growing needs for more foodand more energy have led to increase exploitation of natural resources. The problem related to multi-species fisheries have been drawingattention of researcher in the recent years (cf. Clark <cit.>, Mesterton-Gibbons <cit.>). The economic progress and ecological balance always have a conflicting interests. Therefore, concerning the conservation for the long term benefits of humanity, there is a wide range of interest in the use of bionomic modeling to gain greater insight in the scientific management of renewable resources like fisheries and forestry. An excellent introduction of optimal management of renewable resources has been presented by (cf. Clark <cit.>). The techniques and issues associatedwith the dynamic economic models of natural resources exploitation are further developed by Clark (cf. Clark <cit.>). Harvesting is one of the most important issue to control extinction and minimization of exploitation of renewable resources. Renewable harvesting policy is indisputably one of the major and interesting problems from ecological and economical point of view. There is also a realistic phenomena for surviving himself is refuge. The use of spatial refuges by the prey is one of the more relevant behavioral traits that affect the dynamics of predator prey systems. By using refuges some fraction of prey population are partially protected against predators. The existence of refuges has significant influence on the coexistence of the predator-prey systems. It has a strong impact for stabilizing and destabilizing the dynamical nature of the system. The study of consequence of prey refuge on the dynamic of prey-predator interaction can be recognized as a major effect in applied mathematics and theoretical ecology (cf. Hassell and May <cit.>, Hassell <cit.>, Holling <cit.>, Hoy <cit.>, Smith <cit.>). Some of the empirical and theoretical works have investigated the effect of refuge and gave a decision that the refuge used by prey have a stabilizing effect on prey predator interaction and prey species can be prevented from extinction by using this policy (cf. Collings <cit.>, Freedman <cit.>, Gonzalez-Olivares and Ramos-Jiliberto <cit.>, Hochberg and Holt <cit.>, Huang et al. <cit.>, Krivan <cit.>, McNair <cit.>, Ruxton <cit.>, Sih <cit.>, Taylor <cit.> ).Also, Brauer and Soudak (cf. Brauer and Soudak <cit.>) studied a class of predator prey models under constant rate of harvesting and under constant percentage of harvesting of both species simultaneously. They have shown that how to classify the possibilities of the quantitative behavior of solution to locate the set of initial values in which the trajectories of the solution approaches to either an asymptotic stable equilibrium or an asymptotic stable limit cycle. Dai and Tang <cit.> studied the following predator-prey model in which two ecological interacting species are harvested independently with constant rates of harvesting: dx/dt =rx(1-x/k)-aϕ(x)y-μ, dy/dt =y(-d+caϕ(x))-h,where x(t), y(t) represents prey and predator species respectively. r, k, d, a, c, ϕ(x) are the intrinsic growth rate, carrying capacity of prey, death rate of predator, maximum per capita consumption rate of predator, conversion rate and general predator response function on prey respectively; μ, h are constant harvesting rates. The most crucial element in an interacting population model is the “functional response"�the expression that describes the rate at which the number of prey is consumed by a predator. They have shown the complicated dynamics of the system. Kar (cf. Kar <cit.>) studied the prey-predator model with prey refuge and individuals are subject to proportional harvesting rates. As far as our knowledge goes there is no work in predator-prey model system with independent harvesting and prey refuge proportional to both the species have done even consideration of prey refuge proportional to both species is one step closure to reality. Keeping this in mind an attempt is made in the present investigation to study the effect of refuge as well harvesting on a Holling Type-II prey-predator model. In this paper we take the following model: dx/dt =rx(1-x/k)-p(1-my)xy/1 +ax(1-my)-q_1E_1x=F_1(x, y), dy/dt = ep(1-my)xy/1 +ax(1-my)-dy-q_2E_2y=F_2(x, y),where x(t), y(t) represents the prey and predator populations respectively at any time t. r, k, p, m, d, q_1, q_2, are all positive constants and have their biological meanings, accordingly r, represent the intrinsic growth rate of prey species, k is the carrying capacity of the prey in absence of predator and harvesting, p>0 is the maximum per capita consumption rate of predator, e (0<e<1) is the efficiency by which predator converts consumed prey into new predator, d>0 is the death rate of predator. E_1≥ 0, E_2≥ 0, denotes the harvesting effort for the prey and predator respectively. q_1E_1x and q_2E_2y, represent the catch of the respective species, where q_1, q_2, are catchability coefficients of the prey and predator species respectively. The present model incorporates a refuge proportional to both the prey-predator determines. i.e., mxy from the predator species, where m∈ [0,1] is a constant. Incorporation of prey refuge leaves(1-my)x of the prey available to be hunted by the predator.§ SOME PRELIMINARIES §.§ Existence and positive invarianceLetting X = (x,y)^t, F : ℝ^2 →ℝ^2, F = (F_1,F_2)^t, the system (<ref>) can be rewritten as dX/dt=F(X). Here F_i ∈ C^∞(ℝ),i=1,2. Since the vector function F is a smooth function of the variables (x,y) in the positive quadrant Ω={(x,y);x>0, y>0}⊂ℝ^2_+, local existence and uniqueness of the solution set hold.§.§ BoundednessAll the solutions of the model (<ref>) are bounded uniformly. Proof. Let us consider a function ξ=x+y/e. For anyζ=d+q_2E_2 >0, dξ/dt+ζξ =rx(1-x/k)-q_1E_1x-1/e(dy+q_2E_2y)+ζ( x+y/e) ≤ k/4r(r+ζ-q_1E_1)^2=κ >0. By applying the theory of differential inequality (cf. Brickhoff and Rota <cit.>), we have the following inequality 0<ξ(x,y)<κ/ζ(1-e^-ζ)+ξ(0)e^-ζ t<max{κ/ζ,ξ(0)}. Therefore,t→ +∞lim sup ξ(t)≤κ/ζ with last bound independent of initial conditions. Hence, all the solutions of the system (<ref>) starting from ℝ^2_+ evolve with respect to time and remain in the compact region ℝ_xy={(x,y)∈ℝ^2_+: ξ(x,y)≤κ/ζ+ϵ}for any ϵ>0. §.§ PersistencePersistence of a predator-prey model system plays an important role in mathematical ecology since the criteria of persistence for ecological systems is a condition that ensuring the long-term survival of all the species. Here we have shown the persistence by using average Lyapunov function (cf. Gard and Hallam <cit.>), together with its boundedness.Considering the average Lyapunov function P(x, y)=x^ρ_1 y^ρ_2,where ρ_1 and ρ_2 are undetermined positive constant. Let us define the function ϕ as follows:Φ(x,y) = Ṗ(x,y)/P(x,y)=ρ_1ẋ/x+ρ_2ẏ/y= ρ_1(r(1-x/k)-p(1-my)y/1 +ax(1-my)-q_1E_1) +ρ_2(ep(1-my)x/1 +ax(1-my)-d-q_2E_2).Now, Φ(0,0)=ρ_1 r-ρ_1 q_1E_1-ρ_2d-ρ_2q_2E_2 > 0, if ρ_1 r>ρ_1 q_1E_1+ρ_2d+ρ_2q_2E_2. Φ(x_1,0) = ρ_1(r(1-x_1/k)-q_1E_1)+ρ_2(epx_1/1 +ax_1-d-q_2E_2)= ρ_2(epx_1/1 +ax_1-d-q_2E_2)>0,if p> d+q_2E_2/e(a+1/k(1-q_1E_1/r)).Hence, the solution of the system (<ref>) is permanent, if the conditions ρ_1 r>ρ_1 q_1E_1+ρ_2d+ρ_2q_2E_2 andp> d+q_2E_2/e(a+1/k(1-q_1E_1/r)) are satisfied. § ANALYSIS OF EQUILIBRIAThe equilibria of the system (<ref>) are (i) the trivial equilibrium E^0(0, 0);(ii) the axial equilibrium E^1(x_1,0)=(k(1-q_1E_1/r),0 ), and the interior equilibrium point (iii) E^*(x^*,y^*). We are interested on the interior equilibrium point E^*, where x^* and y^* are satisfying the following system of equations: r(1-x^*/k)-p (1-my^*)y^*/1 +ax^*(1-my^*)-q_1E_1=0,ep (1-my^*)/1 +ax^*(1-my^*)x^*-d-q_2E_2=0. One can easily found that the interior equilibrium E^* is feasible if the conditionsE_1<r/q_1(1-x^*/k) andE_2<1/aq_2(ep-ad) are satisfied. § LOCAL STABILITY We now investigate the local asymptotically stability (LAS) of (<ref>) around the feasible equilibrium points.The Jacobian matrix at an arbitrary point (x,y) isJ=[[ r-2 rx/k-q_1E_1-py(1-my)/(1+ax(1-my))^2-px(ax(1-my)^2+1-2my/(1+ax(1-my))^2);epy(1-my)/(1+ax(1-my))^2 -d-q_2E_2+e px(ax(1-my)^2+1-2my/(1+ax(1-my))^2) ]].Thus, the Jacobian matrix of the system (<ref>) around the trivial equilibrium point E^0(0,0) isJ_0=[[ r- q_1E_1 0; 0 -d-q_2E_2 ]].The eigenvalues for the steady state E^0(0,0) are (r- q_1E_1) and -(d+q_2E_2). The eigenvalue r- q_1E_1 is positive or negative according to E_1<r/q_1 orE_1>r/q_1, i.e., the equilibrium point E^0(0,0) is locally asymptotically stable (cf. Fig 1(a)) if the BTP (Biotechnical Productivity ) of the prey species (r/q_1) is less than the effort (E_1) for the prey species. If BTP> E_1, then equilibrium point E^0 (0,0) will be a saddle for the system (<ref>).The Jacobian matrix at the point E^1(x_1,0) of the system (<ref>) isJ_1=[[r-2 rx_1/k-q_1E_1 -px_1/(1+ax_1);0 -d-q_2E_2+epx_1/(1+ax_1) ]].The eigenvalues of J_1 are λ_1=r-2 rx_1/k-q_1E_1 and λ_2=-d-q_2E_2+epx_1/(1+ax_1). The eigenvalues are negative if the condition (1-d+q_2E_2/epk-ak(d+q_2E_2))<q_1 E_1/r<1 is satisfied and hence E^1(x_1,0) is locally asymptotically stable (cf. Fig 1(b)). The Jacobian matrix around the interior equilibrium point E^*(x^*,y^*) of the system (<ref>) isJ_*=[[ r-2 rx^*/k-q_1E_1-py^*(1-my^*)/(1+ax^*(1-my^*))^2-px^*(ax^*(1-my^*)^2+1-2my^*/(1+ax^*(1-my^*))^2);epy^*(1-my^*)/(1+ax^*(1-my^*))^2 -d-q_2E_2+e px^*(ax^*(1-my^*)^2+1-2my^*/(1+ax^*(1-my^*))^2) ]].Both the eigenvalues of J_* will be negative if the following conditions hold:eax^*(1-my^*)^2(p+y^*-px^*)+ey^* > p y^*(my^*-p-mex^*) +rx^*(1+ax^*(1-my^*))^2,py^*(1-my^*) > (1+ax^*(1-my^*))^2.Hence, under this parametric conditions (<ref>)-(<ref>) the interior equilibrium point E^*(x^*,y^*) will be locally asymptotically stable. § GLOBAL STABILITY AROUND THE INTERIOR EQUILIBRIUM E^*The interior equilibrium E^* is globally asymptotically stable if the condition 4rme(1+ak)(1+ax^*)>p m^2 y^*^2 is satisfied.Proof. Let us consider the suitable Lyapunov function as followers:V=(x-x^*-x^* lnx/x^*)+(y-y^*-y^* lny/y^*). Clearly V is positive definite for all (x,y)∈𝐑^2_+∖ (x^*, y^*). Taking the time derivative along solution of the system (<ref>), we havedV/dt = (x-x^*)ẋ/x+(y-y^*)ẏ/y= -[α(x-x^*)^2+β(y-y^*)^2+γ(x-x^*)(y-y^*)],where α=r/k+pm(y+y^*)/A-py(1+myy^*)/A,β=mxep/A, γ=px(1+myy^*)+p(1-m(y+y^*))-ep-epy^*-p(y+y^*)/A, A=(1+ax(1-my))(1+ax^*(1-my^*)).Now, 4αβ-γ^2>0, if4rme(1+ak)(1+ax^*)>p m^2 y^*^2. Thus, thequadratic form (<ref>) is positive definite hence dV/dt<0 along all the trajectories in the first quadrant except (x^*, y^*). Also dV/dt|_E^*=0. The proof follows from the suitable Lyapunov function V and Lyapunov-LaSalle's invariant principle (cf. Hale <cit.>). Hence the equilibrium point is globally asymptotically stable if the condition (<ref>) is satisfied.§ BIONOMIC EQUILIBRIUMLet c_1 and c_2 are the fishing cost of prey and predator species per unit effort; p_1and p_2 are the prices of prey and predator per unit biomass. Therefore, the economic rent (net revenue) at any time can be taken as π = (p_1 q_1 x-c_1)E_1 +(p_2 q_2 x-c_2)E_2 = π_x +π_y , whereπ _x=(p_1 q_1 x-c_1)E_1 and π_y=(p_2 q_2 y-c_2)E_2 are the net revenue for the prey and predator species respectively. The bionomic equilibrium is defined as a point where the biological and the economical equilibrium takes places.The bionomic equilibrium (x_∞,y_∞, E_1^∞, E_2^∞) is given by the solutions of following simultaneous equations: r(1-x/k)-p(1-my)y/1 +ax(1-my)-q_1E_1=0,ep(1-my)x/1 +ax(1-my)-d-q_2E_2 =0,(p_1 q_1 x-c_1)E_1 +(p_2 q_2 y-c_2)E_2 =0. The bionomic equilibria are determinedin different case as follows: Case I: If c_2 > p_2 q_2 y, i.e., the fishing cost is greater than the revenue for the predator species, then the predator fishing will be stopped (i.e., E_2=0). Only the prey fishing will be in operational (i.e., E_1< p_1q_1 x ). We then have x_∞=c_1/p_1q_2. Therefore, putting this value in (<ref>) and (<ref>) we have the bionomic equilibrium point (y_∞, E_1^∞) in the y-E_1 plane if the parametric conditionE_1^∞ <r/q_1 (1-c_1/p_1q_1 k) holds.Case II: If c_1> p_1 q_1 x, i.e., thefishing cost is greater than the revenue in the prey fishing then the prey fishery will be closed (i.e., E_1=0) . Only the predator fishing will be in operational (i.e., c_2< p_2q_2 y).In this case the bionomic equilibrium (x_∞, E_2^∞) will be in the first quadrant of x-E_2 plane ifE_2^∞≥1/q_2 (d-p_2 q_2 e r k/4 c_2).Case III: If c_1> p_1q_1 x and c_2> p_2 q_2 y, then the fishing cost is greater than the revenue for both the species and the whole fishery will be closed.Case IV: If c_1< p_1q_1 x and c_2< p_2 q_2 y holds simultaneously, then the revenue for both the speciesbeing positive then the whole fisherywill be in operational, in this case x_∞ =c_1/p_1 q_1, y_∞ =c_2/p_2 q_2 , therefore using this values in (<ref>) and (<ref>) we have the non trivial bionomic equilibrium point (x_∞,y_∞, E_1^∞, E_2^∞) exists if the parametric conditions (i) r/q_1 (1-c_1/p_1 q_1 k)> p(p_2 q_2-m c_2)c_2/p_2 q_2 (p_1 q_1 p_2 q_2 +a c_1(p_2 q_2 -m c_2))and (ii) d< ep (p_2 q_2 -m c_2)c_1/p_1 q_1 p_2 q_2 +a c_1(p_2 q_2- mc_2) are satisfied. § OPTIMAL HARVESTING POLICY The fundamental problem in commercial exploitation of renewable resources is to determine the optimal trade-off between current and future harvests. As observed by Clark (cf. Clark <cit.>). This problem which is very assured of resources conservation is an exceedingly different one not from the Mathematical view point perhaps, but certainly from a political and philosophical view point. However, we look at the problem from the economic view point only and we have to use the time discounting policy to handle the question of inter temporal benefits. This discounting is a normal practice in business management (cf. Solow <cit.>). For determination of an optimal harvesting policy, we consider the present value J^1 of continuous stream of revenue as follows:J^1=∫^∞_0 e^-δ t[( p_1q_1 x-c_1)E_1(t) + (p_2q_2 y-c_2)E_2(t)]dt,where δ denote the instantaneous annual rate of discount. We are to optimize the equation (<ref>) subject to the state equation (<ref>) by using Pontryagian's maximum principle (cf. Pontryagin <cit.>).Let us consider the Hamiltonian function H as follows: H=e^-δ t((p_1 q_1 x-c_1)E_1+(p_2 q_2 y-c_2)E_2)+λ_1(rx(1-x/k)-p(1-my)xy/1 +ax(1-my)-q_1E_1x) +λ_2(ep(1-my)xy/1 +ax(1-my)-dy-q_2E_2y),where λ_1(t) and λ_2(t) are adjoint variables. E_1(t), E_2(t) are the control variables subject to the constraint 0≤ E_i(t)≤ (E_i)_ max, (i=1,2). The control variable E_1(t), E_2(t)appear linearly in the Hamiltonian function H. Assuming that the control constraint are not binding, i.e., optimal solution does not occur at (E_i)_ max or (E_i)_ min, we have singular control (cf. Clark <cit.>) given by ∂ H/∂ E_i=0.Therefore,∂ H/∂ E_1=0, λ_1(t)= e^-δ t(p_1-c_1/q_1 x);∂ H/∂ E_2=0,λ_2(t)= e^-δ t(p_2-c_2/q_2 y). Thus, the shadow prices e^δ tλ_i(t), (i=1,2) do not vary with time in optimal equilibrium. Hence they satisfy the transversally condition at t→ +∞, i.e., they remain bounded as t →∞. Again, ∂ H/∂ E_1=0 λ_1 q_1 x e^δ t=∂π_x/∂ E_1;∂ H/∂ E_2=0 λ_2 q_2 y e^δ t=∂π_y/∂ E_2.From these relations one can say that for each species the user cost of harvesting per unit effort must be equal to the discounted value of the future marginal profit of effort at the steady state level. Now we are to find out the optimal solution of the problem as follows:d λ_1/dt = -∂ H/∂ x= -[e^-δ tp_1 q_1 x E_1+λ_1{r(1-2x/k)-py(1-my)/(1+ax(1-my))^2-q_1E_1} +λ_2 {epy(1-my)/(1+ax(1-my))^2}]Substituting the values of λ_1 and λ_2 from (<ref>), we have the relation between x and y as follows:δ(p_1-c_1/q_1 x) = p_1(r(1-x/k)-py(1-my)/1+ax(1-my))+(p_2-c_2/q_2 y)epy(1-my)/(1+ax(1-my))^2 +(p_1-c_1/q_1 x)(-rx/k+paxy(1-my)^2/(1+ax(1-my))^2) .Againd λ_2/dt = -∂ H/∂ y=-(e^-δ t p_2 q_2 E_2+ λ_1 px(ax(1-my)^2+1-2my)/(1+ax(1-my))^2) +λ_2(-d-q_2 E_2+p e x (ax(1-my)^2+1-2my)/(1+ax(1-my))^2).Substituting the values of λ_1 and λ_2 in (<ref>) we have the following relation in x and yδ (p_2-c_2/q_2 y)=p_2(epx(1-my)/1+ax(1-my)-d)+px(p_1-c_1/q_1 x)(ax(1-my)^2+1-2my)/(1+ax(1-my))^2 -(p_2-c_2/q_2 y)epmxy/(1+ax(1-my))^2.Solving equation (<ref>) and (<ref>) for x and y we have the optimal equilibrium (x^*, y^*) and the optimal harvesting efforts E_1^* and E_2^* can be determined by the following equations:E_1^* = 1/q_1(r(1-x^*/k)-(1-my^*)y^*/1+ax^*(1-my^*)),E_2^* = 1/q_2(-d+ep(1-my^*)x^*/1+ax^*(1-my^*)).§ BIFURCATION ANALYSIS The subject of bifurcation is the study of structurally unstable systems.This topic is a branch of mathematics concerned with dynamical systems whichsuffer sudden qualitative changes in parameters. A small change in parametercauses a topological change. The important question in the field of local bifurcationtheory is that a system depends on a control parameters, as parameter changes, whathappens to the non hyperbolic equilibria. Structurally unstable dynamic systems can beclassified according to the number of parameters that appears in the differentialequations describing the dynamics of the system. For good introduction on the basics of local bifurcation analysis the interested readers are referred to check out the books byGuckenheimer <cit.>, Wiggins <cit.>, and Kuznetsov <cit.>.In our study of bifurcation the parameter r has been chosen for the transcritical bifurcation and m has been taken as bifurcation parameter for the analysis of Hopf- bifurcation for the present system (<ref>).§.§ Existence of transcritical bifurcation around E_1The the system (<ref>) experiences a transcritical bifurcation around the axial equilibrium E_1 as the parameter r crosses its critical value r=q_1 E_1=r_tc. One of the eigenvalues of J_1 will be zero iff Det(J_1)=0, which implies eitherr-E_1q_1=0, or -d-q_2 E_2+epx_1/1+ax_1=0. Let v, w are the eigenvector corresponding to the eigenvalue λ=0 for the matrices J_1 and J_1^T. The eigenvectors v, w are found as v=(v_1, v_2)^T=(1, 0) and w=(w_1, w_2)^T=(1,px_1/epx_1-(1+ax_1)(d+E_2q_2))^T. With these eigenvectors, it is found that (i) w^TF_r(E_1, r_tc)=0, (ii) w^T[DF_r(E_1, r_tc)v]=(1-2x_1/k)≠ 0, (iii) w^T[D^2F(E_1, r_tc)(v, v)]=-2r_tc/k≠ 0. Hence, due to Satomayor theorem (cf. Sotomayar <cit.>), the system experience transcritical bifurcation around the axial equilibrium E^1 at r=r_tc. §.§ Hopf bifurcation It can be easily conclude that the equilibrium point E^* may loss its stability through Hopf bifurcation under certain parametric condition. Considering m as a bifurcation parameter one can detect the threshold value of m = m_h, which satisfy Tr(J_E^*)|_m=m_h=0 and Det(J_E^*)|_m=m_h>0.The transversality condition for the Hopf bifurcation (cf. Carr <cit.>, Hassard and Kazarinoff <cit.>, Perko <cit.>) is d/dm(Tr(J_E^*))|_m=m_h=py_m(2aex_m^2-amx_m y_m^2+ax_m y_m+2ex_m-y_m)/(1+ax_m (-my_m+1))^3|_m=m_h≠0, where x_m and y_m indicate their functionality with respect to the parameter m. The interior equilibrium point E^* loss its stability through the non-degenerate Hopf-bifurcation when the parametric restriction Tr(J_E^*)|_m=m_h=0 and the transversality conditions are satisfied simultaneously.Now we calculate the Lyapunov number to determine the nature of Hopf-bifurcating periodic solutions. Introducing perturbations x=x_1+x_m|_m=m_h, y=y_1+y_m|_m=m_h in (<ref>) andthen expanding in Taylor series, we haveẋ_̇1̇ = a_10 x_1+a_01y_1+a_20x_1^2+a_11x_1 y_1+a_30x_1^3+a_21x_1^2y_1+⋯, ẏ_̇1̇ = b_10 x_1+b_01y_1+b_20x_1^2+b_11x_1 y_1+b_30x_1^3+b_21x_1^2y_1+⋯,where a_10, a_01, b_10, b_01 are the elements of the Jacobian matrix evaluated at the equilibrium point E^* with m=m_h, hence a_10+b_01=0 and Δ=a_10b_01-a_01b_10>0.The expression of the coefficients a_ij and b_ij are given bellow: a_10=∂ F_1/∂ x|_(E^*,m_h),a_01=∂ F_1/∂ y|_(E^*,m_h), a_12=1/2∂^3 F_1/∂ x∂ y^2|_(E^*,m_h),a_21=1/2∂^3 F_1/∂ x^2∂ y|_(E^*,m_h), a_20=1/2∂^2 F_1/∂ x^2|_(E^*,m_h),a_11=∂^2 F_1/∂ x∂ y|_(E^*,m_h), a_30=1/6∂^3 F_1/∂ x^3|_(E^*,m_h);b_10=∂ F_2/∂ x|_(E^*,m_h),b_01=∂ F_2/∂ y|_(E^*,m_h), b_12=1/2∂^3 F_2/∂ x∂ y^2|_(E^*,m_h),b_21=1/2∂^3 F_3/∂ x^2∂ y|_(E^*,m_h), b_20=1/2∂^2 F_2/∂ x^2|_(E^*,m_h),b_11=∂^2 F_2/∂ x ∂ y|_(E^*,m_h), b_30=1/6∂^3 F_2/∂ x^3|_(E^*,m_h). The value of first Lyapunov number (cf. Perko <cit.>), which helps to determine the nature of the stability of limit cycle arising through Hopf-bifurcation is given byσ = -3π/2 a_10Δ^3/2[(a_10b_01a_11^2+a_10a_01(b_11^2+a_20b_11)-2a_10a_01a_20^2-a_01^2(2a_20b_20+b_11b_20) -a_11a_20 (a_01b_10-2a_10^2))-(a_10^2+a_01b_10)(-3a_01a_30+2a_10(a_21+b_12)+(b_10a_12 -a_01b_21))],where the values of a_ij and b_ij, i, j = 0,1,2,3 are included in Appendix A. If the σ<0, the equilibrium point E^* destabilized through a supercritical Hopf-bifurcation, and if σ>0 then the Hopf bifurcation is subcritical.§ NUMERICAL SIMULATIONSIn this section we perform numerical simulation to validate our analytical findings of the previous Section by making uses of the computing Software Packages MATLAB-R2015a and Maple-18. The analytical findings of the present study are summarized and presented Schematically in Tables 1, Table 2 and Table 3. It is very difficult to validate the model results with realistic data so far proportional refuse and harvesting are considered in natural field. These results are verified by means of numerical illustrations of which some chosen ones are shown in figures. We took a hypothetical set of parameter values to illustrate our results. In Section (7)optimal harvesting policy and corresponding effort are determined. For numerical justification we have taken a set of parameter values: r=3, a=0.008, d=0.04, m=0.02, p=0.2, q_1=0.2,q_2=0.6, k=500, e=0.15, p_1=2, p_2=3, c_1=1, c_2=2, δ=.004 for this set of parameter values the optimal equilibrium is found at (188.5858, 30.6567) and the corresponding harvesting efforts are E_1^*=1.8534 and E_2^*=5.8875. In Section (8), it is found that the present system (<ref>) experiences Hopf-bifurcation for the bifurcation parameter m and we find out the threshold value of m and Lyapunov number σ to determine the nature of Hopf-bifurcation (cf. Sen et al. <cit.>). For numerical validation we take the fixed set of parameter values: r=3, a=0.008, d=0.04,p=0.2, q_1=0.2, q_2=0.6,E_1=2, E_1=2, k=500, e=0.15. It is found that the threshold value of the bifurcation parameter m=m_h=0.010695 at which the system (<ref>) experiences Hopf bifurcation around the interior equilibrium E^*. For this choice of parameter values the first Lypunov number is σ=-000143<0. Hence the Hopf-bifurcation is supercritical. It is observed that the trivial and axial equilibria are locally asymptotically stable (LAS) by starting the solution plots from different initial conditions in the neighborhood of E^0 and E^1, all the solution plots eventually converge to the equilibria respectively (cf Fig. 1: (a)-(b)).Fig. 2: (a)-(b) shows the limit cycle behavior of the system (<ref>) around the equilibrium point E^*=(67.86,18.00) for the set of parameter values: r=3, a=0.008, d=0.04, m=0.005, p=0.2 q_1=0.2, q_2=0.6,E_1=2, E_1=2,k=500, e=0.15. In Fig. 3 shows that the system experiences Hopf-bifurcation for bifurcation parameter m=0.01 and the other parameters are same as Fig. 2.Fig. 4: (a)-(b)shows that the system (<ref>) is globally asymptotically stableand conversing to the point E^*(94.99, 23.33) for m=0.015 and the other parameters are same as Fig. 2. Fig. 5 Shows that gradual increase of the coefficient of refuge gives more protection to the prey species and reduce the rate of predation of predator, as a result the volume of prey species became larger and larger and the predator became smaller and smaller until it goes to extinct. This fact is presented in tabular form (cf. Table 3). § CONCLUSIONIn this paper we consider a prey-predator harvesting model with Holling type -II response function incorporating prey refuge proportional to both the species. The novelty of our work lies in taking such kind of refuge function, which is more realistic phenomenon in ecosystem. We emphasize, on refuge coefficient that how the refuge function changes the system dynamics. We also observed from the both mathematical and empirical point of view that traits of behavioral policy of prey refuge has a stabilizing effect on a predator prey dynamics and this policy can help the prey species from extinction (cf. Anderson <cit.>, Cressman and Garay<cit.>, Magalhaes et al. <cit.>, Rudolf et al. <cit.>, Sarwardi et at. <cit.>, Sarwardi et al. <cit.>, Mukherjee <cit.>, Ma et al. <cit.>). The key question is now the refuge alters the evolutionary dynamics of the system. In this paper we have shown that the dynamics of our system depends upon the functionality of refuge construction. In the present paper, it is assumed that functionality of prey refuge depends not only on prey size but on both the species. From the analytical view point the study of influence of prey refuge on the dynamics of interacting population is at present recognized as a significant and challenging issues (cf. Collings <cit.>, Huang <cit.>). We have shown that depending upon the bifurcation parameter m the system exhibits stability as well as bifurcation around some of the equilibrium points. Also we have found that the optimal harvesting policy and the corresponding optimal effort using E_1and E_2 as control parameter.Before ending our conclusion we must say that, there are still some options to improve our model system to have much richer dynamics that what we have found in the present study. Here are given some rooms for our future studies. Firstly, it would be more logical and proven reality that the term representing time delay used in digestion or gestation period for the predator species to produce new born have not been taken into account. Secondly, the harvesting efforts E_1and E_2 can be taken as time dependent functions. Thirdly, the harvesting effort can be taken as non-linear functions of both species. Fourthly, on the basis of the fact that more species could give more stable ecosystem, one can consider one more prey or predator or a pair of predator and prey into the exited system to have more stable system from the biological point of view.The incorporation of all such relaxations existed in the present system in the future updated model would certainly be of some help to empirical researcher to predict their findings one step closer to the real situation from the ecological point of view. In our next paper we will study the dynamics of the system taking non-linear harvesting effort incorporating the taxation to sustain the renewable resources.0.5in Acknowledgements: Authors are thankful to the Department of Mathematics, Aliah University for providing opportunities to perform the present work. The corresponding author Dr. S. Sarwardi is thankful to his Ph.D. supervisor Prof. Prashanta Kumar Mandal, Department of Mathematics, Visva-Bharati (a Central University) for his generous help and continuous encouragement while preparing this manuscript. 0.5in§ APPENDIX The expressions a_ij and b_ij, appearing in (<ref>) are defined bya_ij=1/i!j!∂ ^i+j(F_1)/∂ x^i y^j|_(E^*,m_h), b_ij=1/i!j!∂ ^i+j(F_2)/∂ x^i y^j|_(E^*,m_h) and their explicit values are as follows: a_10=r-q_1E_1 -2x/k- p(1-my) y/(1+ax(1-my))^2|_E^*,a_01=-px(ax(1-my)^2+1-2my)/(1+ax(-my+1))^2|_E^*,b_10=ep(1-my)y/(1+ax(1-my))^2|_E^*, b_01=-d-q_2E_2+epx(ax(1-my)^2+1-2my)/(1+ax(1-my))^2|_E^*,a_11=-p(amxy-ax+2my-1)/(1+ax(-my+1))^3|_E^*,a_20=-r/k+p(my-1)^2ya/(1+ax(1-my))^3|_E^*,a_02=-2pmx(ax+1)/(1+ax(1-my))^3|_E^*,a_30=p(my-1)^3ya^2/(1+ax(1-my))^4|_E^*, b_20=ep(my-1)^2ya/(1+ax(1-my))^3|_E^*, a_21=pa(my-1)(amxy-ax+3my-1)/(1+ax(1-my))^4|_E^*,a_12= pm(a^2mx^2y-a^2x^2+2amxy+1)/(1+ax(1-my))^4|_E^*,b_30=-ep(my-1)^3ya^2/(1+ax(1-my))^4|_E^*, b_11=ep(amxy-ax+2my-1)/(1+ax(1-my))^3|_E^*, b_21=-ep(my-1)a(amxy-ax+3my-1)/(1+ax(1-my))^4|_E^*,b_12=-epm(a^2mx^2y-a^2x^2+2amxy+1)/(1+ax(1-my))^4|_E^*. .2in Clark1 Clark, C. 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Population dynamics of thrips prey and their mite predators in a refuge. Oecologia, Vol. 150, 557-568. Rudolf Rudolf, V. H. W., Armstrong, J., 2008. Emergent impacts of cannibalism and size refuges in prey on intraguild predation systems. Oecologia, Vol. 157, 675-686. Sarwardi1 Sarwardi, S., Mandal, P. K., Ray, S., 2012. Analysis of a competitive prey-predator system with a prey refuge. Biosystems, Vol. 110, 133-148. Sarwardi2 Sarwardi, S., Mandal, P. K., Ray, S., 2013. Dynamical behaviour of a two-predator model with prey refuge. J. Biol. Phys, Vol. 39, 701-722. Mukherjee Mukherjee, D., 2016. The effect of refuge and immigration in a predator-prey system in the presence of a competitor for the prey. Nonlinear Anal.: Real World Appl, Vol. 31, 277-287. Ma Ma, Z., Li, W., Zhao, Y., Wang, W., Zhang, H., Li, Z., 2009. Effects of prey refuges on a predator-prey model with a class of functional responses: the role of refuges. Math Biosci, Vol. 218, 73-79.Huang Huang, Y., Chen, F., Zhong, L., 2006. Stability analysis of prey�predator model with Holling type-III response function incorporating a prey refuge. Appl. Math. Comput, Vol. 182, 672-683. | http://arxiv.org/abs/1703.09275v1 | {
"authors": [
"Md. Manarul Haque",
"Sahabuddin Sarwardi"
],
"categories": [
"math.DS",
"q-bio.PE",
"92D25, 92D30, 92D40"
],
"primary_category": "math.DS",
"published": "20170327193201",
"title": "Dynamics of a harvested prey-predator model with prey refuge depended on both species"
} |
Department of Physics and Physical Oceanography, Memorial University, St. John's, NL, Canada A1B 3X7Department of Physics and Astronomy, University of Manitoba, Winnipeg, MB, Canada R3T 2N2Department of Computer Science, Memorial University, St. John's, NL, Canada, A1B 3X5 Department of Physics and Physical Oceanography, Memorial University, St. John's, NL, Canada A1B 3X7Department of Physics and Astronomy, University of Manitoba, Winnipeg, MB, Canada R3T 2N2 Department of Physics and Astronomy, University of Manitoba, Winnipeg, MB, Canada R3T 2N2 A series of atomistic finite temperature simulations on a model of an FCC lattice of maghemite nanoparticles using the stochastic Landau-Lifshitz-Gilbert (sLLG) equation are presented. The model exhibits a ferromagnetic transition that is in good agreement with theoretical expectations. The simulations also reveal an orientational disorder in the orientational order parameter for T < 0.5 T_cdue to pinning of the surface domain walls of the nanoparticles by surface vacancies. The extent of the competition between surface pinning and dipolar interactions provides support for the conjecture that recent measurements on systems of FCC superlattices of iron-oxide nanoparticles provide evidence for dipolar ferromagnetism is discussed. Dipolar ferromagnetism in three-dimensional superlattices of nanoparticlesJ. van Lierop December 30, 2023 ========================================================================== § INTRODUCTION Magnetism has provided a fertile field in understanding emergent behaviour resulting from strong correlation effects via the signatures of symmetry breaking. While much of the current research in this area is focused on exchange-type driven interactions that arise from the strong correlation effects amongst electrons (e.g. topological insulators), the dipolar interaction, the poor cousin of exchange, also provides examples of novel forms of emergent behaviour (e.g. stripe phases<cit.> and spin ices<cit.>). From a theoretical perspective, the dipolar interaction is especially appealing as it involves no “adjustable parameters", it has no intrinsic length scale and, depending on the context, can be either ferromagnetic or antiferromagnetic. In the seminal works of Luttinger and Tisza<cit.> and Kittel<cit.> it was shown that while the dipolar interaction can, in and of itself, give rise to magnetic order in a three dimensional lattice of point dipoles, it is ferromagnetic only in FCC and BCC structures (with magnetic dipole domains and domain walls) and antiferromagnetic in all others. Despite the obvious significance of magnetic order emerging from dipolar interactions, the experimental verification of this result has been elusive as there is a paucity of real systems in which the dipolar interaction dominates exchange interactions at the atomic or molecular level. A notable exception to this is the arrays of synthetically produced single domain magnetic nanoparticles.While there is now a substantial body of work on well-defined magnetic nanoparticles and their intrinsic nanomagnetism, and it has been shown definitively that their large (effective) dipole moments result in strong interparticle interactions yielding novel physics in disordered systems at high densities<cit.>, experiments on superlattice crystals of nanoparticles (e.g. in FCC arrangements over micron length scales<cit.>) have revealed what can be considered as only tantalizing hints of dipolar ferromagnetism – the collective, cooperative behaviour is still poorly understood. This is due in large part to the subtle interplay between intra-(atomic spin) and interparticle magnetism<cit.>. In this paper we present results from a series of atomistic finite temperature simulations on a model of an FCC lattice of maghemite nanoparticles using the stochastic Landau-Lifshitz-Gilbert (sLLG) equation. These simulations, when taken together with those presented in our earlier studies on ensembles of non-interacting maghemitenanoparticles<cit.>, provide an interesting complement to recent experiments on magnetoferritin (Fe_3O_4/γ-Fe_2O_3) particles<cit.> in which the particles can be self-assembled to form a FCC superlattice with typical length of the order of 1.5-2.0 μm but which can be disassembled following the application of an optical stimulus. Comparing the results from before and after the disassembly clearly show the effects of the FCC ordering on the magnetic properties of the particles. The extent to which these simulation studies support the conjecture that the observed differences serve as a signature of dipolar ferromagnetism, and how further study might help resolve this question, is discussed.§ FCC POINT DIPOLE LATTICE Theoretical studies of ordered arrays of point dipoleswith uniform magnetizations interacting only through dipole interactions have a long history, with particular interest surrounding the prediction of long-range ferromagnetic order inthe case of particles in an FCC lattice configuration<cit.>. The energy of a point dipole lattice can be written asE_d= g'∑_⟨ ij⟩( σ̂_i·σ̂_j/r_i j^3 -3(σ̂_i·r⃗_i j) (σ̂_j·r⃗_i j)/r_i j^5)where σ̂_i are unit vectors defined at each site i, r⃗_ij=R⃗_ij/a is the relative position of two dipoles in units of the nearest neighbour separation a and the sum is over all pairs of atoms ⟨ ij ⟩with i j. The coupling g=μ_0 m^2/4π a^3 k_B Kelvin, where m denotes the magnitude of the dipole moment on each site.As part of this study, simulations on a FCC lattice of point dipoles with periodic boundary conditions were also performed using sLLG and finite size scaling applied to various thermodynamic quantities.A comparison between our later simulation results for the FCC nanoparticle array and this equivalent point dipole FCC lattice will provide corroboration of the simulation results for the nanoparticle array magnetism as we would expect them to agree, qualitatively at least, to leading order. It will also serve to distinguish those properties that may be attributed to the subtle interplay between the internal magnetic structure of the nanoparticles and the FCC lattice internal dipole field.Of particular relevance to the current work is the accurate determination of the Curie temperature T_c of an FCClatticeof point dipoles. Fig. <ref> shows the Binder ratio ⟨ M^4 ⟩ / ⟨ M^2 ⟩^2 of the calculatedoverall magnetization for linear sizes L=4, 8, 16 and 32.We find a ground state energy of E_0/g= -2.962= -2π√(2)/3 and a transition temperature of T_c=T/g=0.625 with the magnetization oriented along the [111] axis, in good agreement withprevious results<cit.>.It is well established that theground state energy of a saturated classical face-centered cubic dipole lattice is independent of theorientation of the magnetization. This degeneracy can be lifted by fluctuations at finite temperature through the mechanism of order by disorder<cit.> and the magnetization axis is determined by an effective induced anisotropy. However, while previous simulation studies by Bouchaud and Zerah<cit.> reported the existence of a reorientation transitionfrom the [111] axis to the [100] axis at approximately T∼ T_c/2, we found no indication of such a transition in any ofour simulations. The absence of such a transition has been confirmed by other independent simulation studies<cit.>. § MULTISCALE MODEL The simulations for the nanoparticle superlattice were performed on a model consisting of 512 (8× 8× 8) spherical maghemite (γ-Fe_2O_3) nanoparticles on an FCC lattice with periodic boundary conditions. The nanoparticles are represented by an atomistic core-shell model consisting of approximately 8200 Fe^3+ spins in which the core has bulk-like exchange and the shell has weak exchange and radial anisotropy as described in Ref. . The model also includes the dipolar interactions between the nanoparticles calculated self consistently using a multi-scale approach that is naturally parallelizable. This represents a more fundamental approach than the more phenomenological models composed of a system of coupled superspins<cit.>. The calculations were all performed using the micromagnetics scripting language MagLua<cit.> that has been successfully applied to a number of atomistic and micromagnetic simulation studies in nanomagnetism<cit.>.Results are presented for nanoparticles of diameter D=7.5 nm with core diameters D_c=6.3 nm and 6.75 nm, which we refer to as the FCd63 and FCd675 superlattices respectively. The nanoparticles are single crystals with a total of 382 spinel unit cells. The numbers of core cells are 226 and 278 and the number of surface cells are 156 and 104for FCd63 and FCd675 respectively. All of the surface cells are incomplete since they are cut by the particle radius whereas the core cells are all complete. Both the FCd63 and FCd675 superlattices show an order/disorder transition from a superparamagnetic configuration to a true ferromagnetic state (i.e. not superferromagnetism<cit.> where exchange interactions amongst nanocrystallites dominate instead of the much weaker dipolar interactions) at a temperature that is consistent with theoretical expectations<cit.>. To simulate a model comprising N=512 (8×8×8 lattice) nanoparticles each consisting of approximately 8200 Fe^3+ ions at an atomistic level that includes the full dipolar interaction using sLLG is, currently, simply not feasible.Instead, we employ a multiscale model in which we assume that the magnetic intraparticle interactions are dominated by exchange and a single-site anisotropy while the interparticle interactions consist solely of the dipolar interaction. Thedipolar interaction energy, Eqn. <ref>, in this multiscale approximation then simplifies toE_eff =g '∑_⟨ kl⟩( ∑_i∈ kσ̂_ki) ·Γ^kl·(∑_j∈ lσ̂_lj)where the subscripts {ki} denote the i^th spin in the k^th nanoparticle and Γ^kl is the interaction tensor between point dipoles located at the centre of each nanoparticle located on a FCC lattice and calculated assuming periodic boundary conditions using the Ewald summation technique. This multiscale approach considerably reduces the computational effort required while retaining the complexity of the spin structure of the individual nanoparticles in combination with the long range dipolar interactions between the nanoparticles.Each of the nanoparticle's Fe^3+ atoms has a moment of m= 5μ_B. The spinel lattice structure of the γ-Fe_2O_3 nanoparticles has tetrahedral (A) and octahedral (B) sites which order ferrimagnetically with a net moment of 1.25 μ_B per atom in the bulk. In order to maintain charge neutrality, 1/6 of the octahedral sites are occupied by vacancies[While in the simulation results reported in this study, the spinel structures of the individual nanoparticles were aligned along a common axis, additional simulations runs in which the orientations were randomly assigned showed no detectable differences for the systems of interest in the present work.]. It is assumed that the surface spins, defined as those spins located in the region D_c/2 < r < D/2, experience a radial single site anisotropy due to the broken translational symmetry. The proportion of surface spins is around 41% and 27% for the FCd63 and the FCd675 superlattices, respectively. The exchange coefficients are those used in Ref. with the surface anisotropy constant K_s/k_B = 10 K.Periodic boundary conditions were applied to the 8×8×8 array. The simulations were parallelized such each of the individual nanoparticles on the FCC array was assigned to a single processor core. The dipole fields at each site on the FCC lattice were calculated and communicated using the Message Passing Interface protocol (MPI).The sLLG time steps were Δ t = 2 × 10^-4 t_u, where the time unit t_u=1 Tesla/γ=5.68× 10^-12 sec, and the damping factor α = 0.5. Since the dipole field changes were very small in a single sLLG time step, tests showed that the dipole field only needed to be updated every 100Δ t with no measurable loss in accuracy. We represent the i^th spin in the k^thnanoparticle by a unit spin σ̂_ki. We calculate the average magnitude of the normalized magnetization associated with the individual nanoparticles (M_n) and the average magnitude of the normalized magnetization of the entire nanoparticle lattice (M_nl) using the definitionsM_n(T)= 4/N∑_k=1^N| ∑_i=1^q_kσ̂_ki/q_k| M_nl(T)= 4/N| ∑_k=1^N∑_i=1^q_kσ̂_ki/q_k|where N is the number of nanoparticles in the lattice and q_k is the number of spins in the k^th nanoparticle.These moments are normalized to the bulk ferrimagnetic magnetization of each nanoparticle, and are equal to one whenthe {σ̂_ki} are aligned along a common axis with the A-site spins anti-parallel to the B-site spins.The factor of 4 is needed to normalize the moments since, as mentioned above, the maximum magnetic moment per site for a ferrimagnetic nanoparticle is 1/4 that of the moment on each Fe atom. We also define the equivalent point dipole FCC lattice consisting of N sites at which is located a temperature dependent dipole moment m⃗_k = m_n(T) σ̂_k (1 ≤ k ≤ N) where m_n(T)=1.25 μ_B M_n(T)⟨ q_k ⟩ is the average magnetic moment of the nanoparticles and σ̂_k is a unit vector defining the orientation of the dipole moment m⃗_k.The energy of the equivalent dipole lattice is therefore given byEqn. <ref> with g→g̃(T) = μ_0 m_n^2(T)/4π a^3 k_B. The magnitude of the normalized magnetization of the equivalent dipole lattice is defined asM_dl=(M_n(T)/N) |∑_k=1^Nσ̂_k |. In what follows we have assumed that the ratio of the nanoparticle diameter to the FCC lattice spacing r = D/a is equal to unity. This is an idealization of the experimental situation<cit.> which consists of synthesized magnetoferritin nanoparticles with a iron-oxide diameter of ∼7 nm encapsulated in a 1 nm protein layer; hence a = 8.5 nm. Our choice of D = 7.5 nmwith r=1 captures the interplay between the long range character of the dipolar interaction between the nanoparticles and their internal spin structure that is the focus of the current work. The presence of the magnetoferritin coating however is important in that it justifies the absence of any effective exchange coupling between the nanoparticles in our model.In order to provide some insight into the dipole ordering of a single nanoparticle, consider the situation where maghemite has no exchange and is a pure dipolar system. The parameter g in Eqn. <ref> would have the value g∼ 0.08 Kelvin. For an FCC lattice structure this would give a dipole ordering temperature of T_c∼ 0.05 Kelvin. By contrast,the large net moment of the maghemite NPs yields a value of g̃≈ 83 K and 58 K for the FCd675 and FCd63 systems respectively<cit.>. This corresponds to T_c ≈ 52 K (FCd675) and36.5 K (FCd63). § MULTISCALE SIMULATION RESULTS Our simulation results are presented in Fig. <ref> for both arrays. The average magnitude of the nanoparticles' magnetization, M_n, exhibits a smooth variation with temperature up to approximately 900 K (not shown), above which it is effectively zero. The increase in |dM_n(T)/dT| for both samples below ∼30 K is due to the partial ordering of the surface moments<cit.>. The frustration of the surface spins due to the radial anisotropy prevents the complete saturation of the nanoparticles' magnetization. The open squares indicate the magnetization, M_dl, ofthe equivalent dipole lattice for which each site has a moment of magnitude M_n at each temperature. The solid dots represent the magnitude of the lattice magnetization, M_nl, obtained using our multiscale approach. The left axis indicates the normalized values of the moments and the right axis indicates the values of the moments in μ_B. The dipole interactions included in the nanoparticle arrays show negligible effect on M_n.The data show that the nanoparticle superlattices and the equivalent point dipole system begin to order ferromagnetically along the [111] axis at T_c≈55 K and 40 K for the FCd675 and the FCd63 superlattices, respectively. The magnetization for both the FCd675 and the FCd63 superlattices, (M_nl) and the equivalent dipole lattice (M_dl) track each other until ≈20 K, below which the surface spins start to order and M_nl drops below M_dl. The increased disorder at low temperatures is more obvious if we eliminate the effect of the temperature dependence of the magnitude of the magnetization of the nanoparticles by plotting the order parameter defined by σ_nl=M_nl/M_n as a function of the reduced temperature T̃ =T/g̃(T), as shown in Fig. <ref>. Note that in the case of the FCd63 superlattice (smaller core nanoparticles) the order parameter, σ_nl, actually decreases with decreasing temperature for T̃ < 0.4. This discrepancy between the equivalent dipole lattice and the nanoparticle superlattices indicates that there is some phenomenon that decreases the orientational order between the nanoparticles.The origin of this additional disordering implied by the reduced order parameter σ_nl observed in Fig. <ref> may be understoodfrom ourprevious simulation studies on ensembles of non-interacting maghemite nanoparticles<cit.>, in which it was shown that the magnetic moment of the nanoparticles were pinned by the surface vacancies in the octahedral B-sites by the surface magnetization. This pinning effect is a result of the frustration that arises as consequence of the competition between the exchange and the surface anisotropy. Because of this competition it was shown in Ref. [rapidComm2016] that for each nanoparticle there exists a Néel-like domain wall in the surface magnetization at the equatorial plane separating the north and south magnetic poles in which the spins, located at the sites within this domain wall, are highly frustrated. As a consequence the presence of a vacancy located at a site within the domain wall will generally result in a lower energy than if the same vacancy were located at a site close to one of the poles. To understand the role of these surface vacancies it is important to keep in mind that, as indicated in Ref. [rapidComm2016], while the distribution of the vacancies among the B-sites is statistically uniform, statistical variance and the crystallographic structure of the nanoparticle will give rise to a spatial clustering of the vacancies. Because of this clustering the number of surface vacancies contained within the domain wall will depend on the orientation of the equatorial plane. While the locations of the vacancies are fixed the location of the equatorial plane, oriented perpendicular to the magnetic moment, is not and hence the number of vacancies located within the domain wall will depend on the direction of the magnetic moment of the nanoparticle. This is illustrated schematically in Fig. <ref>in which 500 randomly distributed points located on the surface of the unit sphere representing the surface vacancies of a magnetic NP are plotted (Fig. <ref>(a)). The points were generated using the Marsaglia algorithm<cit.> . Also shown is a schematic representation of the magnetic moment vector of a magnetic nanoparticle aligned along the z-axis (Fig. <ref>(b)) with the shaded region, defined by -w/2 < z <w/2 with w= 0.2,representing that portion of the NP surface occupied by the domain wall. In Fig. <ref>(c) the vacancy distribution in (a) is shown superposed on the magnetic moment in (b). The number of surface vacancies contained within the domain wall for this particular distribution is calculated to be N_v=43. In Fig. <ref>(d) we show the same distribution of surface vacancies as in (b) but in this case superposed on schematic representation for the magnetic moment vector of the NP rotated by 60about the x-axis. Counting the number of vacancies within in the rotated domain wall we obtain N_v = 33. While the surface vacancy distribution in this illustrative example is much simpler than that of the model NP used in our simulations, it nevertheless serves to highlight two key points. Firstly, the fact that, even although the distribution of surface vacancies is statistically uniform, the effects of clustering due to statistical invariance can nevertheless be significant. The effects of the crystallographic structure of the NP and the finite thickness of the surface, ignored in this simple model, will only serve to enhance the clustering. Secondly, the model explicitly demonstrates the fact that while the locations of the vacancies are fixed, the polar distribution of the vacancies, measured with respect to the axis aligned parallel to the NP magnetic moment vector, will depend on the orientation of the NP magnetic moment. The combination of the arguments that (a) the number of vacancies contained within the vicinity of the magnetic equator depends on the orientation of the NP magnetic moment and (b) that surface vacancies located in the vicinity of the equator will, due to the effects of frustration, result in a lower energy than those located close to one of the poles gives rise to a non-trivial dependence of the nanoparticle energy on the orientation. These arguments lead the conclusion, first presented in Ref. [rapidComm2016], that the nanoparticle energy will have a minimum when the magnetic moment is oriented so that the number of vacancies in the domain wall region is a maximum, a mechanism that corresponds to the pinning of the nanoparticle magnetic moment, (or perhaps more precisely the pinning of the surface domain wall) by the surface vacancies. The precise dependence of this energy on the orientation of the magnetic moment of the nanoparticle will depend not only on the spatial distribution of the surface vacancies also on the degree of frustration associated with the surface spins located within the domain wall. As such the overall magnitude of the energy variation and the functional form of its relationship to orientation of the magnetic moment will be both temperature and field dependent.Evidence for this pinning in the case of the FCd63 and FCd675 superlattices is shown in Fig. <ref> in whichhistograms plotting the average number of surface vacancies in the range z_n→ z_n+Δ z are presented for several temperatures. The data are obtained by averaging the number of surface vacancies in each bin over all the nanoparticles in the ensemble, with the z-axis (with z=cosθ) defined sothat it is aligned parallel to the magnetic moment vector of each individual nanoparticle and passing through its centre as shown schematically (for example) by the arrows in Figs. <ref>(a) and (b)[The difference in the distribution of vacancies in the north (z>0) and south (z<0) poles was observed to be statistically insignificant. To reduce the degree of scatter in Fig. <ref> the data were also averaged over ± z, which accounts for their apparent symmetry with respect to the z-axis.]. For comparison, results for the equivalent non-interacting ensembles as well the for the case of a uniform density are also shown. The peak in the histogram at z= 0 indicating an increasing concentration of vacancies at the equator does not, as one might naively suppose, result from motion of either the Fe atoms or the vacancies. The locations of the Fe atoms and therefore the vacancies are assumed to be fixed. Instead it results from changes in the spin configuration of the individual nanoparticles in response to the magnetic forces acting on the individual atoms and the fluctuations induced by the stochastic field, and is a consequence of the highly frustrated nature of the surface spins located in the vicinity of the equatorial domain wall combined with local inhomogeneities within the distribution of the surface vacancies. All the results show an increasing concentration of vacancies at the equator as the temperature falls below 20 K,at which the surface spins order<cit.>, a clear signature that the pinning energy increases with increasing surface magnetization. The fact that the maximum in the average surface vacancy distributions for both the FCd63 and FCd675 superlattices is slightly lower than for their respective non-interacting ensembles is due to the competition between the dipolar field (absent in the noninteracting ensembles) and the surface vacancies. In addition, the concentration of vacancies at the equator is more pronounced for the FCd63 superlattice than those in the FCd675 superlattice due to the greater fraction of surface spins and the smaller core that enhances the pinning effects of the vacancies at the equator on the overall magnetization alignment, and reduces the magnitude of the nanoparticles' dipole moments.In addition the smaller core results in a reduced value of M_n(T) and g̃(T).This results in a T_c for the FCd63 superlattice that issignificantly lower than for the FCd675 superlattice, hence much closer to the surface ordering temperature. As a result, the FCd675 superlattice is more ordered than the FCd63 superlattice when the surface spins begin to order and the pinning effect of the surface vacancies activates.§ INSIGHTS INTO EXPERIMENTS ON FCC SUPERLATTICLES OF NANOPARTICLESWhile the ferromagnetic order observed in these simulations is consistent with theoretical expectations, experimental evidence for such a transition is elusive. Recent experiments<cit.> directly comparing systems of magnetoferritin (Fe_3O_4/γ-Fe_2O_3) nanoparticles that self assemble to form an FCC superlattice with those obtained after their disassembly following the application of an optical stimuli show significant differences in their magnetic properties that result from the dipolar interaction. However, while it is tempting to assert that these differences may be attributed to the appearance of dipolar ferromagnetism it is by no means obvious that such an assertion is justified. In this section we demonstrate that the picture that emerges from these simulations and those presented in Ref. are at least qualitatively consistent with the experimental data of Ref. , and, more importantly perhaps, how simulations studies similar to those presented in the current work and Ref. could be extended to determine more conclusively whether the existence of dipolar ferromagnetism in FCC nanoparticle superlattices can be inferred from existing experimental studies. Figure <ref> shows the heating and cooling of a system of zero field cooled magnetoferritin nanoparticles in the presence of a 10 mT field. These particles can be self-assembled to form a FCC superlattice with typical length of the order of 1.5-2.0 μm, but which can be disassembled following the application of an optical stimulus, resulting in the typical disordered ensemble of nanoparticles.Data is shown for both the disordered (disassembled) system and the FCC superlattice. The data for both systems exhibit non-ergodic behaviour over the temperature range 2 to 20 K for the FCC superlattice and 2 to 25 K for the disordered system. In the bottom graph the difference between the magnetization on cooling and heating is also shown.The differences in the data clearly show the effects of the dipolar interaction in the case of the FCC superlattice. Of particular interest in the context of the present work is the fact while that the slope |dΔ M_FC - ZFC(T)/dT| in the limitT→ 0 tends to zero, in the case of the disordered system, it remains finite in the case of the FCC superlattice. This indicates that while the magnetization is close to saturation in the case of the disordered system, in the case of the FCC superlattice the magnetization is not saturated along the direction of the applied field.Figure <ref> presentsthe normalized magnetization as a function of the applied field at 2 and10 K for both the FCC superlattice and the disordered system. The principle difference is in the reduced remanent magnetization and the coercivity in the case of the FCC superlattice. This is indicative that the FCC superlattice lattice is composed of system of randomly oriented crystallites.This feature is consistent with the expectation that in the case the disordered system the moments of the individual nanoparticles will align parallel to the field on cooling and, while the pinning effect will result in some measure of disorder at low temperature, the magnetization will nevertheless be close to saturation in the limit T→ 0. In the case of the FCC lattice, on the other hand, the dipole field will dominate and we would expect that, as the system cools, the magnetization would align along the [111] axis that lies closest to the direction of the applied field.Assuming that the crystallographic axes of the individual magnetoferritin superlattice crystallites are randomly oriented, the net magnetization along the direction of the applied field will not saturate in the limit T→ 0. A similar reasoning also provides a plausible explanation for the fact that the normalized remanent magnetization (M_R = lim_H→ 0 M(H)) obtained from the M vs μ_0 H loops presented in the upper panel of Fig. <ref>is greater in the case of the disordered system than that observed for FCC superlattice.Again this may be attributed to the alignment of the magnetization along the randomly oriented [111] axis of the FCC crystallites while in the case of the disordered system the magnetization will be in the direction of the applied field. The qualitative nature of the above discussion is an unfortunate consequence of the fact that current computational capabilities limit the time scales that can be accessed by the atomistic theories methods used in this work to the order of ms. This precludes a more a quantitative interpretation of the intrinsically non-equilibrium/non-ergodic behaviour observed in Fig. <ref> and <ref> based solely on atomistic sLLG simulations. However, as we have shown, such atomistic studies allow us to identify the relaxation processes that dominate at experimental field and temperature sweep rates. In addition, our earlier work<cit.> presented a simple mean field model of the pinning energy calculated as a function of orientation of the magnetization for a given distribution surface vacancies. Based on this model it is possible to determine the orientation of the metastable spin configurations of a nanoparticle, as well as estimates of the activation energies and attempt frequencies separating them. Such information can provide for a quantitive description of the magnetic properties of magnetoferritin nanoparticles involving experimentally relevant time scales.§ CONCLUSIONSOur multiscale simulations reveal that, while the nanoparticle superlatttice orders ferromagnetically in accordance with theoretical expectations, it nevertheless exhibits a degree of disorder at low temperatures. It was shown that this disorder is due the pinning effect that arises as a consequence of a subtle interplay between the single-site anisotropy and the vacancies in the region close to the surface, an effect discussed in an earlier paper on ensembles of non-interacting nanoparticles. In addition, we describe how the results from these simulations, combined with those from earlier studies on non-interacting systems of nanoparticles, support the assertion that certain key differences in the magnetic properties of magnetoferritin arrays before and after disassembly from a FCC superlattice may be attributed to dipolar ferromagnetism.To provide a more definitive case regarding the experimental verification of emergence of dipolar driven ferromagnetism in FCC nanoparticle superlattices than the above qualitative argument, more detailed simulation studies of non-equilibrium properties(ie. heating/cooling and M vs μ_0 H loops) at experimental sweep ratesare required. Such simulation studies, not currently feasible using standard sLLG due to the time scales involved, can play a vital role in this process. We are currently exploring the application of other simulation methods such as Kinetic Monte Carlo<cit.> and Forward Flux Sampling<cit.> that are potentially applicable to experimentally relevant sweep rates.This work was supported by the Natural Sciences and Engineering Research Council of Canada,Compute Canada, ACEnet andWestGrid. | http://arxiv.org/abs/1703.09290v1 | {
"authors": [
"B. Alkadour",
"J. I. Mercer",
"J. P. Whitehead",
"B. W. Southern",
"J. van Lierop"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170327200514",
"title": "Dipolar ferromagnetism in three-dimensional superlattices of nanoparticles"
} |
sgnεℓ_PlTrtr⟨⟩dexp/{}SUSOSpinSL𝔰𝔲𝔰𝔬𝔰𝔩^(γ)1 #11ex∼#1Ursfs Ursfsmn<5> rsfs5 <6><7> rsfs7 <8><9><10><10.95><12><14.4><17.28><20.74><24.88> rsfs10 Ursfsmn Centre de Physique Théorique, Aix Marseille Universit, Universit de Toulon, CNRS, UMR 7332, 13288 Marseille, France.This is a review of the results on black hole physics in the framework of loop quantum gravity. The key feature underlying the results is the discreteness ofgeometric quantities at the Planck scale predicted by this approach to quantum gravity. Quantum discreteness follows directly from the canonical quantization prescription when applied to theaction of general relativity that is suitable for the coupling of gravity with gaugefields and specially with fermions. Planckian discreteness and causal considerations provide the basic structurefor the understanding of the thermal properties of black holes close to equilibrium.Discreteness also provides a fresh new look at more (at the moment) speculative issues such as those concerning the fateof information in black hole evaporation. The hypothesis of discreteness leads also to interesting phenomenology with possible observational consequences. The theory of loop quantum gravity is a developingprogram. This review reports its achievements and open questions in a pedagogical manner with an emphasis on quantum aspects of black hole physics.Black Holes in Loop Quantum Gravity Alejandro Perez===================================§ INTRODUCTIONLoop quantum gravity (LQG) is an approach to a background independent quantization of the gravitational interaction based on the non-perturbative canonical quantization of general relativity. In this framework space-time geometry itself is a dynamical variable that has to be suitably quantized and described in the absence of any background reference geometry. The proposal is still in progress and many important questions remain open. All the same, there are results providing a solid picture of what the quantum nature of space-time at the fundamental scale could be like. One of these key results is that space-time geometric operators acquire discrete spectra: states of the gravitational degrees of freedom can be spanned in terms of spin-network stateseach of which admits the interpretation of an eigenstate of geometry which is discrete and atomistic at the fundamental level <cit.>. Quantum space is made of polymer-like excitations of quantum geometry where one dimensional fluxes of quantised area connect at nodes carrying quantum numbers of volume. The dynamical rules of evolution for these states are also discrete <cit.>. Locality and topology are replaced by the relational notions of connectivity of the underlying network of space quanta.In this framework, the continuum spacetime formulation of general relativity and quantum field theory is seen as the low energy limit of a fundamentally discrete and combinatorial entity. A large body of results in the concrete physical situation defined by quantum aspects of black hole physics have been produced in recent years.This article aims at presenting these recent developments in an organic and pedagogical way. The theory of black holes that follows from the LQG underlying model is still incomplete, partly due to the technical difficulties in defining the notion of black holes in the quantum realm, partly because of the intrinsic difficulties associated with the definition of the dynamics and the low energy limit of the fundamental theory.Nevertheless, the theory indicates a solid conceptual perspective that produces promising insights into the nature of quantum gravity in general. This is an account of a research program in progress.The approach that we will describe can seem quite peculiar from (what sometimes can appear as) the main stream of thought in the holographic tumult of the high energy community. However, we will see, the perspective that arises from our analysis is actually quite conservative and presents many analogies with the behaviour of standard physical systems.The tension with other more popular approaches resides in the complete lack of compliance with any fundamentalnotion of holographic principle <cit.>.Despite this, it can be shown that the theory of black holes stemming from LQG is indeed consistent with the black hole phenomenology derived from semiclassical analysis (we could call this an emergent weak holography). Thus, thepicture of LQG is very different from the bulk-boundary-duality type of quantum gravity scenarioproposed bythe ADS-CFT correspondence <cit.>.We will see that the alternative offered by the LQG treatment may present important advantages in avoiding certain inconsistencies in the description of gravitational collapse and subsequent black hole evaporation. The article will also review the theoretical basis leading to the prediction of discreteness of quantum geometry by LQG. In Section <ref> we will briefly review the construction of the phase space of general relativity starting form an action and variables that satisfy a criterion of naturality once some general principles are stressed. We will see that the roots of discreteness of quantum geometry are found in Heisenberg's uncertaintyrelations for geometric quantities.The inclusion of black holes in terms of boundaries satisfying suitable boundary conditions will be described in Section <ref>. The quantisation of the volume and area operators will be sketched in Section <ref>. In Section <ref>we will apply the formalism to the problem of computing black hole entropy. In Section <ref> we will discuss the problem of the fate of information in black hole evaporation, and some phenomenological ideas with possible observational consequences that are motivated by the discussion of information loss. Throughout this paper there might be sections that seem too technical for a general reader not necessarilyinterested in all the mathematical details.Equations are written to guide the argumentation and, for general readers, are important only in this sense.Once equations are written they call for technical precision (important for those that might be interested in detailed derivations); however, in spite of their apparent complexity due to the presence of indices and other tensorial operations that are often necessary in the presentation of field theoretical notions in the context of general relativity, their message should be transparent when ignoring these details. The reader more interested in the conceptual lineshould read these equations without paying too much of attention to the details of the index structure and concentrate rather on their algebraic form. This is specially so for the construction of the phase space of general relativity; Section <ref> (very important for us as it implies the Poisson non-commutativity of geometry behind quantum discreteness). Classical mechanics is briefly described in its symplectic formulation at the beginning so that all the equations that follow, and are important for gravity, can be interpreted by analogy with these initial equations.Geometric units (G_N=c=1) are used in discussions so that energy, mass, and time are all measured in the same units as legth.§.§ Black hole thermodynamics: an invitation to quantum gravity Black holes are remarkable solutions of general relativity describing the classicalaspects of the late stages of gravitational collapse. Their existence in our nearby universe is by now supported by a great amount of observational evidence <cit.>. When isolated, these systems become very simple as seen by late and distant observers. Once the initial very dynamical phase of collapse has passed (according to physical expectation and the validity of the `no-hair theorem'[The no-hair theorem is a collection of results by Hawking, Israel, Carter and others implying that a stationary (axisymmetric) black hole solution of Einstein's equations coupled with Maxwell fields must be Kerr-Newman <cit.>. Some aspects of this result remain without complete proof and some authors refer to is at the no-hair conjecture (for more details see <cit.> and references therein). The physical relevance of Einstein-Maxwell resides in the fact that gravity and electromagnetism are the only long range interactions. Other forces might be relevant for the description of the matter dynamics during collapse but play no role in describing the final result where matter has already crossed the BH horizon. ]) the system settles down to a stationary situation completely described by a member of the Kerr-Newman family. These are solutions of Einstein's equations coupled with electromagnetism representing a stationary and axisymetric black hole characterised by three parameters only:its mass M, its the angular momentum J, and its electromagnetic charge Q.The fact that the final state of gravitational collapse is described by only a few macroscopic parameters, independently of the details of the initial conditions leading to the collapse, is perhaps the first reminiscence of their thermodynamical nature of black holes. As we will review here, there is a vast degeneracy of configurations (microstates) that can lead to a same final stationary macroscopic state, and the nature of these microstates becomes manifest only when quantum gravity effects are considered.Another classical indication of the thermodynamical nature of black holes (BHs) emerged from the limitations on amount of energy that could be gained from interactions with BHs in thought experiments such as the Penrose mechanism <cit.> and the phenomenon of BH superradiance <cit.>; its field theoretical analog. Later it became clear that such limitations where special instances of the very general Hawking's area theorem <cit.> stating that for natural energy conditions (satisfied by classical matter fields) the area a of a black hole horizon can only increase in any physical process. This is the so-called second law of black hole mechanics which reads: δa≥ 0.This brings in the irreversibility proper of thermodynamical systems to the context of black hole physics and motivated Bekenstein <cit.> to associate to BHs a notion of entropy proportional so their area. Classically, black holes also satisfy the so-called first law of BH mechanics <cit.> which is an energy balance equation relating different nearby stationary BH spacetimes according to δ M=κ/8πδ a _heat?+Ωδ J+Φδ Q,where Ω is the angular velocity of the horizon, Φ is the horizon electric potential, and κ is the surface gravity which plays the roleof a temperature in the analogy with thermodynamics. The surface gravity, defined only in equilibrium,can be related to an intrinsic local geometric quantity associated with the BH horizon; it takes a constant value on the horizondepending only on the macroscopic parameters M, Q and J (for the simplest non-rotating and uncharged BH κ=1/(4M)). The homogeneity of κ on the horizon is called the zeroth law of BH mechanics. The other intensive parameters Ω and Φ are also functions of M, Q and J only (their explicit expression can be found for instance in <cit.>).With the exception of the horizon area a, all the quantities appearing in the first law have an unambiguousphysical meaningfor asymptotic inertial observers at rest at infinity: M is the total mass defined in terms of the Hamiltonian generating time translation for these observers, J is the generator of rotations around the BH symmetry axis, etc. The quantity Φ is the electrostatic potential difference between the horizon and infinity, Ω is the angular velocity of the horizon as seen from infinity, and κ (if extrapolated from the non-rotating case) is the acceleration of the stationary observers as they approach the horizon as seen from infinity <cit.>. It is possible however to translate the first law in terms of physical quantities measures by quasi-local observers close to the BH horizon <cit.>.Thisclarifies the role of the horizon and its near spacetime geometry as the genuine thermodynamical system. The realization that black holes can indeed be considered (in the semiclassical regime) as thermodynamical systems came with the discovery of black hole radiation. In the mid 70's Hawking considered the scattering of a quantum test field on a space time background geometry representing gravitational collapse of a compact source <cit.>. Assuming that very early observers far away from the source prepare the field in the vacuum state, he showed that—after the very dynamical phase of gravitational collapse has ended and the space time settles down to a geometry well described by that of a stationary black hole—late and faraway observers in the future (see Figure <ref>) measure an afterglow of particles of the test field coming from the horizon with a temperatureT =κħ/2π.For the case SchwarzschildBH (Q=J=0), the radiation temperature is T=ħ/(8π M). As black holes radiate the immediate conclusion is that they must evaporate through the (quantum phenomenon of) emission of Hawking radiation. This expectation is confirmed by the study of the the expectation value of the energy-momentum tensor in the corresponding quantum state that shows that there is a net flux of energy out of the BH horizon (see for instance <cit.> and references therein). The quantum energy-momentum tensor violates the energy conditions assumed in Hawkings area theorem and allows the violations of (<ref>): the horizon can shrink. The calculation of Hawking assumes the field to be a test field and thus neglects by construction the back reaction of such radiation. However, it provides a good approximation for the description of black holes that are sufficiently large in order forthe radiated power to be arbitrarily small. This result together with the validity of the first and second laws imply that semiclassical black holes should be associated anentropy (here referred to as Bekenstein-Hawking entropy) given by S_H=a/4ℓ_p^2+S_0where ℓ_p=√(ħ G_N/c^3) is the Planck scale, S_0 is an integration constant that cannot be fixed by the sole use of the first law.In fact, as in any thermodynamical system, entropy cannot be determined by sole thermodynamical considerations.Entropy can either be measured in an experimental setup (this was the initial way in which the concept was introduced) or calculated from basic degrees of freedom using statistical mechanical methods once a model for these fundamental buildingblocks of the system is available. Remarkably, the functional dependence of the entropy of a BH on the area was arguedby Bekenstein first on statistical mechanical terms <cit.>.In this way thermodynamics shows once more its profound insights into the physics of more fundamental degrees of freedom behind macroscopic variables. In this case by shedding light on the nature of the quantum gravitational building blocks of spacetime geometry. As in standard systems, the first law for BHs implies that—when considering energy changes due to the action of macroscopic variables (e.g. work done by changing the volume)—there is a part of the energy that goes into the microscopic molecular chaos (i.e. heat). Thus the first law describing the physics of steam machines and internal combustion engines of the nineteenth century reveals the existence of the microscopic physics of molecules and atoms. Moreover, it is by trying to construct a consistent description of thethermodynamics of photons that Planck made the founding postulate of quantum mechanics <cit.> (more explicit in Einstein <cit.>) that radiation too is made of fundamental building blocks called photons.Similarly, in the present context, equation (<ref>) is a clear physical indication that the smooth spacetime geometry description of the gravitational field must be replaced by some more fundamental atomistic picture.In this way, black holes offer a privileged window for learning about quantum gravity. Mathematically, even though the thermodynamical nature of semiclassical black holes is a robust prediction of the combination of general relativity and quantum field theory as a first approximation of quantum gravity,the precise expression for the entropy of black holes is a question that can only be answered within the framework of quantum gravity. This is a central question for anyproposal of a quantum gravity theory.§.§ Weak Holography A surprising property of the Bekenstein-Hawking entropy of a black hole is that it is proportional to the area a of the event horizon instead of scaling linearly with some three-dimensional volumetric measure of the systems size. The fact that black hole entropy scales as in a lower dimensional system together with the discovery of bounds on the entropy of compact objects (conjectured via the analysis of thought experiments involving black holes and conventional objects; see <cit.>) has led an important part of the quantum gravity community to believe in the so-calledholographic principle<cit.>. In its crudest form the principle states that the classicalphysical world should admit a fundamental description in terms of a hologram on a lower dimensional screen. This is a view that theADS-CFT formulation of string theory incarnates <cit.>.In LQG we do not see any convincing evidence for the need for such a radical principle, and subscribe to some weakernotion that has been described asweak holography<cit.>.The reason for this view is that all the apparently puzzling properties of black holes and their interactions with external agents appear to be completely consistent oncethe following two ingredient are combined: discreteness at Planckian scales, and compatibility with the causal structure predicted by general relativity in the continuum limit. Both ingredients are expected features in LQG.The holographic principle plays no role in the construction of the theory.Causality is one of the keys for understanding the system at hand. This can be clearly illustrated in an intuitive manner with the help of the spacetime representation of gravitational collapse shown in Figure <ref>. Concretely,consider a BH of mass M and an external observer that becomes a stationary observer[A stationary observer is an observer at constant r,θ, and ϕ in a Kerr-Newman spacetime in Boyer-Lindquist coordinates <cit.>. More generally, is an observer whose 4-velocity is parallel to a timelike Killing field for a stationary spacetime. ] in the asymptotic future when its proper time becomes large, τ≫ M, when measured starting at some arbitrary instant around the moment of collapse. In Figure <ref> this is defined by a Cauchy surface Σ placed around the region where the horizon settles down to a stationary one. Due to the presence of the event horizon (itself an outgoing light-like surface trapped with zero radial expansion in the stationary region), an outgoing light wave-front leaving the collapsing matter, outside but close to the event horizon, remains close to the event horizon for a long “time" (more precisely for long values of an affine parameter along the outward-pointing null geodesics) until it finally escapes the strong gravitational region towards infinity (see Figure <ref>). This implies that, in order for the wave front to reach the observer at late times τ≫ M, it must have left the collapsing body from a proper distance ℓ to the horizon at instant Σ that scales as ℓ≈ M exp(-τ/(8M)) (as a simple calculation in the spherically symmetric case would show). In other words, the portion of the collapsing body “seen” by a late observer—for whom the spacetime looks stationary and hence the laws of BH mechanics apply—corresponds to an exponentially-thin hyper annulus given by the region contained between the surface at constant proper distance ℓ from the horizon and the horizon itself.The previous exponential relationship implies that ℓ becomes quickly smaller than ℓ_p for late proper time τ of the external observer. However,we cannot trust the classical expression for ℓ all the way down to transplanckian scales. If we think of these fluctuations as affecting the position of the outgoing wave-front from the boundary of the hyper annulus, then uncertainty in its position sets a natural lower bound for ℓ of the order of Planck's length ℓ≈ℓ_p. Thus the volume outside the BH thatthe very late outside observer can actually see is given byv=aℓ_p.We know the system radiates and it is in a close-to-thermal equilibrium state (at least for large BH masses M in Planck units). Statistical mechanical arguments based on equipartition of probability for volumetric fundamental bits imply that the systems entropy should scale linearly with V in Planck units from which we get that S≈v/ℓ_p^3=a/ℓ_p^2,which is in agreement the Bekenstein-Hawking area entropy law and based on a completely standard statistical mechanicalrationale with no need to invoking an hypothetical holographic principle.One could objet to the above argument that standard statistical mechanical reasoning also suggests that the entropy should grow linearly with the energy of the system. Remarkably, it turns out that energy and area are proportional to each other when one considers the system described above. Useful notions of energy are scarce in general relativity due to its necessary link to a time translational symmetry that is not always available in arbitrary gravitational configurations <cit.>. When the black hole spacetime is asymptotically flat then there are standard definitions of its total energy content such as the ADM mass <cit.>, or the Bondi mass <cit.>, when the spacetime is stationary there is also the Komar mass <cit.>. For the Kerr-Newman BHs (the most general stationary BH solutions classically expected to represent the end result of gravitational collapse) the previous three notions coincide and correspond to the quantity called M in (<ref>). However, non of these energy notions are appropriate for describing the system at hand as they are global notions referring to the energy content of the entire spacetime.For stationary black holes one canshow <cit.>, using perturbation theory and Einstein's equations, that exchanges of energy (as defined by local stationary observers) with the system, defined as theannulus around the horizon mentioned above, are directly related to changes of its area according to the simple law δ E=δ a/8πℓ,where δ E is the standard notion of energy content of the matter falling into BH for local stationary observers, i.e., the one that a calorimeter held stationary close to the horizon would register if captured by the devise. This implies that the natural measure of the internal energy E of the system of interest behaves linearly with the area E∝ a. Once the appropiate local notion of energy is invoked the apparent tension between the area scaling of the entropy and standard thermodynamicsdisappears. In addition to the area-scaling of BH entropy, the holographic hypothesis is said to be supported by entropy bounds for weakly gravitating systems. These bounds where originally proposed by Bekenstein <cit.> who studied suitable thought experiments designed to test the validity of the so-calledgeneralized second law<cit.> of thermodynamics in situations where black holes would be fed with regular matter.Covariant versions of these bounds were constructed by Bousso <cit.>. However, recent results <cit.> strongly suggest that these bounds, when definedin a precise manner, turn out to be valid in the context of standard quantum field theory semiclassically coupled to gravity. Thus their validity in the setting of a theory that is by no means holographic (in the sense of the holographic principle <cit.>) confirms that these bounds cannot be used as physical evidence for the alluded fundamental principle of quantum gravity. Holographic-like behaviour is simply there in standard physics when the situation is befitting.Finally, the generalized second law (GSL) states that the total entropy defined by the Bekenstein-Hawking BH entropy plus the entropy of the external matter can only increase in any physical process. As the BH entropy is expected to arise fromstandard statistical mechanical considerations, which are not different at the fundamental level from those leading to the definition of the entropy of the rest of matter fields, it is widely accepted that the GSL must hold. As in standard statistical mechanics, the second law is hard to prove rigorously (mainly due to the difficulty in defining entropy of matter fields precisely). Nevertheless, versions of the GSL constructed in terms of geometric notions of matter entropy (e.g. entanglement entropy, mutual information, etc.) exist <cit.> and capture a physical meaning that is closely related to the GSL formulated in terms of the standard coarse graining definition of entropy. As in the case of entropy bounds, these proofs rely only on the validity of general relativity, quantum field theory, and the semiclassical formulation where the gravitational field couples to the expectation value of the stress-energy-momentum tensor. Once more no holographic principle needs to be invoked, the GSL (used to motivate holography) is just valid for standard 3+1 dimensional theories carrying genuine bulk degrees of freedom.In conclusion, the black hole system is effectively a 2+1 dimensional system when analyzed by external stationary (and therefore late) observers. The dimension transversal to the horizon is exponentially squeezed by the redshift effect near the horizon and the system becomes effectively 2-dimensional. Consequently, according to a view that enjoys some consensus in the LQG community, there is no need forfundamental screens andfundamental holographic ideas when considering the statistical mechanical origin of the Bekenstein-Hawking area entropy law or any of the black hole phenomenology associated with thought experiments involving interactions with matter and fields in the semiclassical regime. Black holes are special and their thermal properties are encoded in a lower dimensional system: their horizon. Holography, in this weaker sense, is not afundamental property of quantum gravity but simply a property of BHs (and suitable null surfaces); simply a special behaviour of a very special situation. § THE CLASSICAL BASIS OF LOOP QUANTUM GRAVITY In this section we briefly review the main features of the classical theory and the parametrisation of its phase space that defines the starting point for the quantisation program of LQG. The main message of this section is that the action of general relativity when formulated in terms of first order variables (which are suitable for the implementation of the non perturbative quantisation program of LQG) imposes non trivial canonical commutation relations forgeometric quantities. The consequence of this is that suitable geometric observables have discrete spectra in the quantum theory.§.§ Where to start? The choice of the basic fields and action principle The starting point is the choice of the fundamental field variables in terms of which one describes the dynamics of gravity. In the original formulation one uses the metric tensor g_ab (encoding the spacetime geometry) and its dynamics is described by the Einstein-Hilbert action <cit.> S[g_ab]=1/2κ∫√(|g|) R(g_ab) dx^4,where κ=8π G c^-4, g is the determinant of the metric (dv=√(|g|) dx^4 is simply the spacetime volume element), and R(g_ab) is the Ricci scalar: the `trace' R=g^μρg^νσR_μνρσ of the Riemann curvaturetensor R_abcd where g^ab is the inverse metric (g^μνg_μσ=δ^ν_σ). The vacuum Einstein's equations areR_ac=0, where R_ac=g^μνR_aμ cν is the Ricci tensor (we are using here abstract index notation, latin indices denote abstract spacetime indices, greek letters coordinate indices; see <cit.>).Even though the Ricci scalar (or scalar curvature) has a very simple geometric meaning, its dependence on the dynamical field g_ab is quite complicated, namelyR(g_ab)=g^μρ[∂_νΓ^ν_μρ-∂_μΓ^ν_νρ+Γ^α_μρΓ^ν_να-Γ^α_νρΓ^ν_μα], where Γ^ρ_μν=1/2g^ρσ[∂_μ g_νσ +∂_ν g_μσ -∂_σ g_μν] are the Christoffel symbols. Thus, despite the simple geometric meaning of the Einstein-Hilbert action, the Lagrangian of general relativity is quite complicated in terms of metric variables. The algebraic structure of the action can be simplified (in the so-called Palatini formulation) bydeclaring the Christoffel symbols as independent variables. Such modification goes in a good direction; however, there is another, more important, disadvantage of the present choice of variables in the Eintein-Hilbert action or the Palatini modification: one cannot couple fermion fields to gravity described in this form (we come back to this point below).Another disadvantage of the choice of the metric g_ab as a basic variable is the huge (naively infinite) dimensionality of the space of actions that are related to the Einstein-Hilbert action via the renormalization group flow. According to the Wilsonian perspective <cit.> there is an intrinsic uncertainty in theselection of an action principle due to the flow in the space of action principles induced by the integration of quantum fluctuations at scales that are not relevant for the physics of interest. In this sense there is an ambiguity in naming the action principle of a theory: the set of suitable action principles is only limited by the field and symmetry content of the theory. In the case of general relativity in metric variables this corresponds to all possible general covariant functionals of g_ab. This set is characterized by infinitely many coupling constants, concretelyS[g_ab]=1/2 κ∫√(|g|)(R+Λ+α_1 R^2+α_2 R^3+⋯+β_1 R_μνασ R^μνασ⋯) dx^4, whereonly some representative terms have been written with couplingsα_1, α_2,⋯, β_1, β_2, ⋯, etc.If all the infinite dimensional set of couplings defining the above family of metric variable actions would be relevant then it would be impossible to decide what the correct starting point for canonical quantization would be and quantum gravity predictability would be compromised. It is possible, however, that the renormalization group flow selects a final dimensionalspace in this infinite dimensional world of metric gravity actions <cit.>. Such possibility, known as theasymptotic safety scenario,is under present active exploration <cit.>. The necessity of having an action principle that is suitable for the coupling with fermions leads to the type of variables that define the starting point for quantization in LQG. As we will see below the new variables allow for the introduction of natural extended observables which transform covariantly under diffeomorphism, and lead to algebraically simpler action principles (simpler field equations) in a space of actions whose dimensionality is drastically reduced: for pure gravity the space of actions is finite dimensional.§.§.§ The first order formalism In order to couple fermions to general relativity one needs variables where a local action of the rotation group (and more generally Lorentz transformations) is defined. This is naturally achieved by describing the spacetime geometry in terms of an orthonormal frame instead of a metric. Local Lorentz transformations are realized as the set of transformations relating different orthonormal frames. This subsection might seem a bit technical for those that are not familiar with the formalism. Those readers should go through the equations without paying too much of attention to the index structure. The intended message of this part is the algebraic simplicity of the new formulation in comparison the previous one.Concretely one can introduce an orthonormal frame field defined by four co-vectors e_a^I (with the index I=0,⋯, 3; a and other latin indices denote spacetime indices) and writethe spacetime metricas a composite objectg_ab = -e^0_ae^0_b+e^1_ae^1_b+e^2_ae^2_b+e^3_ae^3_b= e^I_ae^J_bη_IJ,where in the second line the internal Minkowski metric η_IJ= diag(-1,1,1,1) is explicitly written. In the familiar three dimensional space there are infinitely many frame-fields related by local rotations; in the present four dimensional Lorentzian setting the choice of an orthonormal frame is also ambiguous. Indeedthe previous (defining) equation is invariant underLorentz transformations: both e and ẽ are solutions with e_a^I→ẽ^I_a=Λ^I_J e^J_a which we will write in matrix notation ase_a →ẽ_a=Λ e_a, where Λ^I_Jsatisfies η_KM=η_IJΛ^I_KΛ^J_M. The physics cannot fix such freedom in the choice of a tetrad; this new symmetry is anadditional gauge symmetry of general relativity when formulated in these variables. As in any gauge theory, derivatives of covariant fields require the introduction of the notion of a connection ω^IJ=-ω^JI (a one-form called the Lorentz connection in this case) defining the covariant derivative. More precisely, if λ^I is an object with internal index transforming covariantly λ→λ̃=Λλ under a Lorentz transformation Λ^I_J then its covariant (exterior) derivative, defined by d_ωλ^I=dλ^I+ω^IJ∧λ_J,also transforms covariantly because ω_a^AB transforms inhomogeneusly under internal Lorentz transformations (<ref>), namely ω→ω̃=ΛωΛ^-1 + Λ dΛ^-1. Thus, the Lorentz connection ω^IJ is an additional field that is necessary in the tetrad formulation to define derivatives in a context where frames can be locally changed by a local Lorentz transformation. In a suitable sense the Lorentz connection plays a role that is similar to that of the Christoffel symbol of the metric formulation. When the gravity field equations are satisfied, this connection is fixed in terms of derivatives of the tetrad field by equations that resemble equation (<ref>). In terms ofe^I and ω^IJ the action principle of gravity drastically simplifies becomingS[e_a^A,ω_a^AB]=1/2 κ∫ϵ_IJKL e^I∧ e^J∧ F^KL(ω), whereF^AB_ab the curvature of the connection ω_a^AB; a two-form valued in the Lie algebra of the Lorentz group with a simple dependence on the connection given byF^AB=dω^AB+ω^AM∧ω_M^ B.The curvature transforms covariantly under a local Lorentz transformation F→Λ F Λ^-1. The internal Levi-Civita symbol ϵ_ABCD—a totally antisymmetric internal tensor such that ϵ_0123=1—is invariant under the simultanneus action of the Lorentz group on its four entries. The action is in this way invariant under the Lorentz gauge transformations (<ref>) and (<ref>). Equations (<ref>) and (<ref>) define the (internal) Lorentz gauge transformations of the basic fields entering the action. Nevertheless, thegauge transformations (<ref>) and (<ref>) need not be listed in addition to (<ref>); the very field equations stemming from the action know about these symmetries. This is specially explicit in the Hamiltonian formulation where gauge symmetries are in direct correspondence with constraints (restrictions among the phase space fields) which in turn are the canonical generators of gauge transformations. These constraints (generators of gauge transformations) are part of the field equations <cit.> (see also <cit.>). We will write them explicitly in Section <ref>.In addition to internal Lorentz transformations the action (<ref>) is invariant under diffeomorphisms(general covariance). At the technical level this comes from the fact that the action (<ref>) is the integral of a 4-form (a completely antisymmetric tensor with 4 contravariant indices): under coordinate transformation x^μ→ y^μ(x) fields transform as tensorse^J_μ dx^μ =e^J_μ∂ x^μ/∂ y^α dy^α ω^JK_μ dx^μ = ω^JK_μ∂ x^μ/∂ y^α dy^α, while the integral remains unchanged as the 4-formtransforms precisely by multiplication by the Jacobian | ∂ x^μ/∂ y^α|. Once more such symmetry will be dictated to us by the equations of motion coming from the action if not explicitly taken into account.This is in fact how Einstein himself was confronted with general covariance: his equations would seem to violate determinism as certain field components would not be entirely determined by the evolution equations. After some struggling with (what became to be know as) the hole argument he realized that the action (<ref>) implied that coordinates have no physical meaning and that only coordinate independentstatements (diffeomorphism invariant in modern jargon) contain physical information (see <cit.> for a modern account). In the present case, these are functions of the basic fields e and ω invariant under the transformations (<ref>) in addition to(<ref>) and (<ref>).The equations of motion coming from (<ref>) follow from δ_e S=0 and δ_ω S=0 respectivelyϵ_IJKL e^J ∧ F(ω)^KL=0 d_ω(e^I∧ e^J)=0.Notice their algebraic simplicity. If the tetrad field is invertible (which basically means that a non degenerate metric can be constructed from it according to(<ref>)) then the previous equations are equivalent to Einstein's equation (<ref>).However, the field equations, as well as the action (<ref>) continue to make sense for degenerate tetrads. For example theno-geometry statee=0—diffeomorphism invariant vacuum—solves the equations and makes perfect sense in terms of the new variables.In this way, guided by the necessity of coupling gravity with fermions, the first order variables and the action (<ref>) introduce a paradigm shift that will be crucial in the quantum theory:the space of solutions (elements of the phase space of the theory (<ref>)) contain degenerate configurations. These configurations arepregeometric in the sense of Wheeler <cit.> and will play a central role in the state space of LQG. Even when these are not important for the description of classical gravitational phenomena they are expected to dominate the physics at the deep Planckian regime. We will see in what follows that these pre-geometric configurations (in the form of quantum excitations) are responsible for the quantum gravitational phenomena associated to black holes (BHs); ranging from their thermal behavior, the relationship of their entropy with their area, to a possible natural explanation the information loss paradox.Another striking property of the tetrad formulation is the radical reduction of the space of actions (formally[The renormalization group flow in first order variables cannot be defined in terms of the usual background field perturbation techniques. The problem is that no well-defined gauge fixing for diffeomorphisms is know around the natural degeneratebackground e=0. If instead a non degenerate background is used then arbitrary terms can be generated by the symmetry breaking that it introduces (see <cit.> for an example in Yang-Mills context, and <cit.> for a discussion in the gravitational case).] expected to be probedby the renormalization group flow. Concretely, if one restricts to the pure gravitational sector the most general action that is compatible with the field content of (<ref>) and its symmetries has only 6 different terms. Indeed all possible gauge invariant 4-forms that can be constructed out of the tetrad e^I and the Lorentz connection ω^IJ areS[e_a^A,ω_a^AB] = 1/2 κ∫ϵ_IJKL e^I∧ e^J∧ F^KL(ω)^ Einstein+Λ ϵ_IJKL e^I∧ e^J∧ e^K∧ e^L ^ CosmologicalConstant+α_1e_I∧ e_J∧ F^IJ(ω)^ Holst+ α_2(d_ω e^I ∧ d_ω e_I-e_I∧ e_J∧ F^IJ(ω))_ Nieh-Yan+ α_3F(ω)_IJ∧ F^IJ(ω)_ Pontrjagin+α_4ϵ_IJKL F(ω)^ IJ∧ F^KL(ω)_ Euler,where d_ω e^I is the covariant exterior derivative of e^I and α_1⋯α_4 are coupling constants.For non-degenerate tetrads Einstein's field equations follow from the previous action independently of the values of the α's: the additional terms are called topological invariants describing global properties of the field configurations in spacetime. The α_1-term is called the Holst term <cit.>, the α_2-term is the Nieh-Yan invariant, the α_3-term is the Pontryagin invariant, and the α_4-term is the Euler invariant.Inspite of not changing the equation of motion these terms can actually be interpreted as producingcanonical transformations in the phase space of gravity [In the presence of Fermions γ controls the strength of an emergent four-fermion interaction <cit.>.]. In such a context the so-called Immirzi parameter <cit.> corresponds to the combination <cit.>γ≡1/(α_1+2α_2).The parameter γwill be particularly important in what follows.§.§.§ Extended variables General covariance is the distinctive feature of general relativity and we have recalled how this is explicitly encoded in the action principles for gravity. The central difficulty of quantum gravity is how generalize what we have learnt about quantum field theory (in the description of other interactions) in order to understand the generally covariant physics of gravity. In general relativity, measurable quantities cannot be defined with the help of coordinates or any non dynamical backgroundas both concepts stop carrying any physical meaning. Localisation of spacetime events is possible only in a relational manner where some degrees of freedom are related to others to produce a generally covariant observable: one that is well defined independently of the coordinates we choose to label events. In the classical theory these observables are always non-local. Localisation in general relativity is always done in a relational fashion using the notion of test observers. Test observers are key in the spacetime interpretation of general relativity; the observables that follow from them are always non local in spacetime. An illustrating example is the case of two free test observers with world lines—geodesics in the spacetime—that meet at some event A, then separate and meet again at an event B. The proper time τ_AB measured by one of the observers between these two eventsis a genuine coordinate independent quantity but is non-local. Another example is the definition of a black hole event horizon which separates those observers that can in principle escape out to infinity from those that cannot: test photons are used to define the horizon in a coordinate independent fashion. All observables are non-local in general relativity.These thoughts led to the idea that extended variables might be best suited for the definition of aquantum theory of gravity. Even when the motivations are sometimes different non local objects are also central in other approaches such as strings, branes <cit.>, twistor theory <cit.>, or causal sets <cit.>. An advantage of the new variables in (<ref>) over the metric variables in (<ref>) is that they allow for the introduction of natural quantities associated to extended subsets (submanifolds) of the spacetime. These quantities are the fluxes of e∧ e and the holonomies of the Lorentz connection ω. More precisely the fluxes are E(α,S)≡∫_Sα_IJ e^I ∧ e^J,where α_IJ is a smearing field andS is a two-dimensional surface. The holonomy assigns an element Λ(ℓ, ω) of the Lorentz group to any one dimensional path in spacetime, by the rule Λ(ℓ, ω)≡ Pexp-∫_ℓω,where Pexp denotes the path ordered exponential. None of these extended variables are diffeomorphism invariant; however, they transform in a very simple way under coordinate transformations: the action of a diffeomorphism on them amounts to the deformation of the surface S and the path ℓ by the action of the diffeomorphism on spacetime points. This behaviour makes these extended variables suitable for the construction of covariant non local operators for the quantum theory.These extended variables are represented in Figure <ref>.The above non-local variables are the basic building blocks in the attempts of giving a meaning to the path integral definition of quantum gravity based on action (<ref>). Such research direction is known as the spin foam approach <cit.>. Even though some applications of spin foams to black holes are available; most of the developments have been achieved in the canonical(or Hamiltonian) formulation. We will see in what follows that the above type of extended variables are also available in the Hamilatonian formulation, but for that we have to briefly describe the phase space structure of general relativity when written in first order variables. §.§ First step towards the quantum theory: the Hamiltonian formulation We need to study the Hamiltonian formulation of gravity formulated in terms of (<ref>). In particular we are interested in obtaining the Poisson brackets between suitable basic variables in terms of which we shall parametrize the phase space of the theory. These Poisson brackets will become the canonical commutation relations in the quantum theory that are responsible for the discreteness of geometric quantities in LQG. In this way, the origin of the Planckian discreteness of geometry is easily seen from the Hamiltonian analysis. We only need to recall a shortcut for the construction of the canonical variables in mechanics, due to the simplicity of the action of gravity in the first order formalism we will be able to derive, via simple algebraic steps, the from of the Poisson brackets for gravity and foresee the seeds of discreteness.§.§.§ The covariant phase space formulation in a nut-shell There is a direct way for obtaining the phase space structure of a field theory from the action principle. The method is easily illustrated by a simple mechanical system with a single degree of freedom andLagrangian L(q,q̇). Under general variations the action changes according to δ S=∫_1^2 [∂ L/∂ q-d/dt(∂ L/∂q̇)] _ e.o.m.δ qdt +.∂ L/∂q̇δ q |_1^2_p δ q,where the boundary term comes from the integration by parts that is necessary to arrive at the equations of motion in the first term. The previous equation contains important information encoded inthe type of variations δ q(t) and its boundary conditions (Figure <ref>). If δ q(t) is arbitrary for intermediate times but it vanishes at the boundary instants 1 and 2, then δ S=0 for those variations gives the equations of motion. If instead δ q(t) are variations defined by infinitesimal differences between solutions of the equations of motion—not necessarily vanishing at the boundary times—, then the first term in (<ref>) vanishes and δ S= pδ q|_1^2. These boundary contributions to the on-shell variation of the action tell uswhat the phase space structure of the system is, i.e., what the momentum p conjugate to q is. In this simple example such method for obtaining the momentum conjugate to q might seem excessive as in this case we already know the recipe p=∂ L/∂q̇; however, it often shows to be the simplest and most direct method when dealing with generally covariant field theories such as the one defined by our action (<ref>). We will use this method to directly access the Poisson commutation relations of geometric variables in gravity. On a slightly more technical level, the boundary term Θ(δ) ≡ pδ q is called the symplectic potential and is a function of δ in the sense that it depends on the specific form of the on shell variation at the boundary—where δ denotes the infinitesimal difference between two solutions, it can be seen as a vector with components δ≡(δ q, δ p). From the symplectic potential Θ(δ) one can obtain the symplectic form Ω(δ, δ') by an additional independent variation δ^' according toΩ(δ, δ')≡δΘ(δ')-δ' Θ(δ)=δ p δ'q-δ'p δ q, i.e., the on-shell antisymetrized variation (exterior field derivative) of the symplectic potential gives the symplectic form.In one simple step, the on-shell antisymetrized variation of the action leads (from (<ref>)) to the conservation of the symplectic form0=(δδ'-δ'δ)S=Ω(δ, δ')|_2-Ω(δ, δ')|_1,and its corollary: Liouville's theorem on the conservation of phase space volume [In the case of N degrees of freedom the volume form in phase space is vol≡ -1/2^N∧^N Ω .]. All these, standard properties of the phase space of a dynamical system with finitely many degrees of freedomcarry over to the field theories with several mathematical subtleties that are not important here. This is the great power of the covariant phase space formalism (see <cit.> for further reading). The symplectic form carries the information about the phase space structure of the system: it defines the dynamically invariant phase space volume measure (Liouville's theorem) and the Poisson brackets of observables (the starting point for quantization). The previous relation between the symplectic form and the symplectic potential also says that Θ(δ)→Θ(δ)+δμ for some function μ does not change the symplectic structure as δδ'μ-δ'δ μ=0. The possibility of changing the symplectic potential by the addition of the variation of a function μ can be shown to encode the notion of canonical transformations. §.§.§ Implementation in gravityNow we are ready to apply the previous techniques to the case of interest. In order to simplify the following analysis we set Λ, α_2, α_3 and α_4 to zero in (<ref>) and get the simpler (Holst) action S=1/2 κ∫ (ϵ_IJKL+1/γη_IKη_JL)( e^I∧ e^J∧ F^KL(ω)),which defines our starting point.The result is not affected if we drop this assumption but the proof becomes more technical <cit.>. Following our recipe, in analogy with (<ref>), we simply need to consider the most general variation of (<ref>) in order to obtain the phase space structure of general relativity in first order variables. As discussed before it will be important to express α_1=1/γ as γ—the Barbero-Immirzi parameter—will play a central role in what follows. Replacing in (<ref>) and varying we obtain δ S=1/2 κ∫ (ϵ_IJKL+1/γη_IKη_JL)(2δ e^I∧ e^J∧ F^KL(ω)+ e^I∧ e^J∧δ F^KL(ω)).The first term does not involve variations of derivatives of the fundamental fields, while the second term does. In fact a well know property of the field strength of a gauge theory is that δ F^KL(ω)=d_ω (δω^IJ) which directly follows from (<ref>).Using this and defining p_IJKL≡ (ϵ_IJKL+1/γη_IKη_JL), we get to the resultbyintegrations by parts as explicitly shown in the following three lines: δ S=1/2 κ∫_M 2 p_IJKLδ e^I∧ e^J∧ F^KL(ω)+p_IJKLe^I∧ e^J∧ d_ω(δω^KL)=1/2 κ∫_M 2p_IJKLδ e^I∧ e^J∧ F^KL(ω)-p_IJKLd_ω(e^I∧ e^J)∧δω^KL+d([p_IJKLe^I∧ e^J]∧δω^KL)=1/2 κ∫_M 2p_IJKLδ e^I∧ e^J∧ F^KL(ω)-p_IJKLd_ω(e^I∧ e^J)∧δω^KL_ e.o.m.+∫_∂ M1/2 κ[p_IJKLe^I∧ e^J]∧δω^KL_pδ q,where in the first line we substituted δ F^KL(ω)=d_ω (δω^IJ) in the second term and then integrated by parts. In the first term (the bulk integral) of the last line we recognise the field equations (<ref>) while the second term (the boundary integral)tells us that P_KL≡ -2 κ^-1p_IJKL e^I∧ e^J is the momentum density conjugate to the Lorentz connection ω^KL. In the Language of the symplectic potential we have Θ(δ)=∫_Σ1/2 κ[p_IJKLe^I∧ e^J]∧δω^KL,where Σ is a spacelike hypersurface (one of the two components of the boundary ∂ M in Figure <ref>) representing the analog of aninstant.However, there is a problem: there are 18 independent components in the ω_a^IJ (6 independent internal configurations of the antisymetric IJ-indices times 3 values of the a-index for the three space coordinates of the spacial boundary Σ) while, naively, the same amount of component are present in the P_ab^IJ only 12 are independent as they are all function of e_a^I! This can be stated by saying that the P_ab^IJ's must satisfy constraints. These constraints (which in the literature a known as thesimplicity constraints) complicate the identification of the genuine phase space variables and must be taken care of. There are two prescriptions for doing this, one is to solve them in some way before going on, the other is the Dirac modification of the Poisson brackets <cit.>. In the present case it will be the easiest to simply solve these constraints by introducing a gauge fixing of the gauge freedom(<ref>).The idea is to reduce the Lorentz symmetry in (<ref>) by demandingthe co-vector e^0 (which defines the time axis of the frame field; the only timeline member of the tetrad) to be perpendicular to the time slices Σ, or equivalently to be aligned with the unit normal n to Σ, namely e^0_a=n_a.This reduces the Lorentz gauge freedom to the rotation sub-group of the Lorentz group that leave invariant the normal to Σ; we denote this SU(2)⊂ SL(2,)[At this stage the rationale would imply that the original gauge group is SO^+(3,1), the proper orthochronous Lorentz group withSO(3) the subgroup obtained via the time gauge. However, for applications including fermions and other features that become clear in the quantum theory it is more convenient to work with the universal coverings SL(2,) and SU(2). ]. This partial gauge fixing is known as thetime-gauge, see Figure <ref>. Such choice is very natural in the Hamiltonian formulation of gravity where the slicing of spacetime in terms of space-like hypersurfaces is already available. The time-gauge amounts to adjusting the time axis in our frame field to the one that is singled out by the foliation. The previous gauge fixing solves the problem of the mismatch of the number of independent components in the momenta as defined in (<ref>). If we explicitly separate the 0 from the i=1,2,3 internal indices then the symplectic potential (<ref>) becomes Θ(δ) = 1/κ∫_Σ( ϵ_0jkle^0∧ e^j∧δω^kl+ 1/γ e^0∧ e^i∧δω_0i)-1/κ∫_Σ( ϵ_0jkle^j∧ e^k ∧δω^l0 +1/γ e^i∧ e^j∧δω_ij) = -1/γκ∫_Σ [ϵ_jkle^j∧ e^k]∧δ(γω^l0+ϵ^lmnω_mn)_Ashtekar-Barbero connection= -1/γκ∫_Σ [ϵ_jkle^j∧ e^k]∧δ A^l, where the first term in the first line vanishes because e^0 is normal to the space slice Σ (it has no space components due to (<ref>) or more precisely its pull back to Σ vanishes). In the second line we used thatϵ_0ijk=ϵ_ijk, simple algebraic properties of ϵ_ijk, and we have factored out γ^-1. In the third line we have defined a new configuration variable A^i≡γω^i0+ϵ^ijkω_jk_Holst, which transforms as a gauge connection under the SU(2) gauge symmetry that remains after the imposition of the time-gauge and is called the Ashtekar-Barbero connection. Now we have 9 A^i_a configuration variables for the 9 conjugate momenta ϵ_jkle^j∧ e^k depending of the 9 components of e_a^i. The strategy of the gauge fixing has worked as there are no additional constraints on momentum variables. Recall our previous discussion on how important it was for the framework to have a connection formulation. For that the factor in front of the second term in the definition of A^i must be precisely 1; this is why one obtains a factor γ^-1 in front of the symplectic potential.From now on we adopt the more compact notationE^i=ϵ^i_jk e^j∧ e^k,and write (<ref>) as Θ(δ) = -1/γκ∫_Σ E_i ∧δ A^i.Notice that the term that makes the connection A^i transform as a connection is the second term in (<ref>) (the first transforms as a vector under an SU(2) rotation) which actually comes directly form the contribution of the Holst terms in (<ref>) to the symplectic potential (as mentioned above there is also a contribution to this term coming from the Nieh-Yan invariant in (<ref>)). Further analysis shows that ω^ik is not free; indeed part of the field equations—equation (<ref>)—imply Cartan's first structure equationd e^i+ω^ik∧ e_k=0, whose solution is a unique function of the triad e^i and we denote ω^ij=ω(e)^ij. The symplectic structure that follows from the recipe (<ref>) and the symplectic potential (<ref>)Ω(δ, δ')= 1/2κγ∫_Σδ A^i∧δ' E^_i-δ' A^i∧δ E^_i,The associated Poisson brackets relations are {E^i(x),E^j(y)}=0{A^i(x),A^j(y)}=0{E^i(x),A^j(y),}=κγ ϵ^ (3)δ^ijδ^ (3)(x,y),where ϵ^ (3)δ^ (3)(x,y) is the Dirac delta distribution with the usual properties that are familiar in the non gravitational context when integrated against text functions with the additional feature of being defined on arbitrary coordinates [More precicely, ϵ^ (3)_abc is a three form Levi-Civita density. Its tensor structure matches the one of the left hand side where we have the (one-form) connection A^i_a times the (two form) E_ab^j. We have dropped tensor indices to improve readability.]. The phase space structure of gravity in connection variables is exactly that of a non-Abelian SU(2) Yang-Mills theory. This combined with background independence will lead to the discreteness of geometric observables in the quantum theory as we will soon show [If one drops the Einstein term in (<ref>) the theory becomes topological (with no local propagating degrees of freedom), but still admits the (E,A) phase space parametrization <cit.>. ].§.§.§ An alternative derivation: the importance of being in 3+1 dimensions But before let us make a little detour that emphasises the importance of the dimensionality of spacetime in the present treatment. There is a peculiar feature of three dimensional space playing a central role in the existence of the E and A canonical variables. If we had started from the simplest action (<ref>), namely S=1/2 κ∫ϵ_IJKL( e^I∧ e^J∧ F^KL(ω))then the symplectic potential would have resulted in Θ(δ)=∫_Σ1/2 κ[p_IJKLe^I∧ e^J]∧δω^KL,which after the time-gauge fixing would have become Θ(δ) =-1/κ∫_Σϵ_jkle^j∧ e^k∧δω^i0 which tells us that E^i=ϵ^i_ jk e^j∧ e^k and ω^i0 are canonical pairs. However, non of these variables transforms as a connection under the remaining SU(2) gauge symmetry. The SU(2) connection formulation found in the previous subsection can be recovered via a canonical transformationthanks to a remarkable property of the geometry of frame fields in three dimensions. Given a frame field (which in 3d corresponds to our field e_a^i) there is a unique solution of the Cartan first structure equation (<ref>) (recall that this equation comes from the field equation (<ref>)) that we call the spin connection ω^ij(e).This is true in any dimension d with the range of index i,j=1,⋯, d. The antisymmetry of the connection in ij implies that there are d_c=d(d-1)/2 independent internal components.The case d=3 is special because only in this case one has d_c=d: the connection has exactly the right amount of components to be added to an object that transforms as a vector under the action of the frame rotation group (SU(2) in this case). Indeed for d=3 we can express this algebraic property by encoding the components of the connection ω^ij in terms of an object with only one internal index (like a vector) using the Levi-Civita internal tensor, namely ω^i(e)≡ϵ^ijkω_jk(e);which can be inverted to give ω^ij(e)≡ϵ^ijkω_k(e). Now in three dimensions only, and interms of this definition, we can write the Cartan equation (<ref>) asd e^i+ϵ^ijkω_j(e)∧ e_k=0.and, most importantly for what follows, taking the variation of Cartan equation and then computing the wedge product with e_i on getsd (δ e^i)∧ e_i+ϵ^ijkδω_j(e)∧ e_k∧ e_i+ϵ^ijkω_j(e)∧δ e_k∧ e_i=0.using (<ref>) to rewrite the third term, and renaming dummy indices, one gets a key result for the foundations of LQG, namely thatϵ^ijke_i∧ e_j ∧δω_k(e)=d(δ e^i∧ e_i).We can use the previous identity to now manipulate (<ref>) and get Θ(δ) =-1/κ∫_Σϵ_jkle^j∧ e^k∧δω^i0 = -1/γκ∫_Σϵ_jkle^j∧ e^k∧δ(γω^i0+ω^i(e))+1/γκ∫_Σϵ_jkle^j∧ e^k∧δω^i(e)_∫_∂Σδ e^i∧ e_i,where we have introduce the Immirzi parameter by adding and substracting a term proportional to the left hand side of(<ref>). Assuming that Σ is compact the last term in the previous expression vanishes due to (<ref>) and we getΘ(δ) =-1/κ∫_Σ E_i∧δA^iin agreement with the previous derivation (<ref>).If ∂Σ≠0 then the last term contributes to the symplectic structure with a boundary term; this will be important in the presence of a BH in Section <ref>. A canonical transformation available in 3+1 dimensions is the wayby which we find the Yang-Mills like parametrisation of the phase space of gravity (the Immirzi parameter labels a one parameter family of these). Some general comments: as it becomes clear from the previous discussion the construction of the phase space of connection variables presented here works naturally only in 3+1 dimensions. There is another possible canonical transformation which leads to the analog of the θ parameter in QCD; its effects on the phase space structure and black holes is studied in <cit.>. Connection variables are also natural in 2+1 dimension where the absence of the simplicity constraints implies that one does not need to introduce the time gauge and can keep manifest Lorentz invariance. It is possible to avoid the time gauge and keep Lorentz invariance in the 3+1 dimensional setting at the price of having non commutative connections due to the contributions of the simplicity constraints to the Dirac brackets <cit.>. Because of this, the quantisation program has not been rigorously realised in this case (see <cit.> for an heuristic approach). The connection parametrisation of higher dimensional gravity is possible but more complicated (due to the presence of simplicity constraints) as has been shown in <cit.>.For a discussion of its quantisation see <cit.>. The formalism has been generalised in order to include supergravity in <cit.>. The calculation of BH entropy in higher dimensions has been studied in <cit.>. §.§ Constraints: the Hamiltonian form of Einstein's equations We have seen how the covariant phase space formulation offers a direct road to obtainingthe phase space structure of general relativity. The Poisson brackets we have obtained in (<ref>) are key in understanding the prediction of Planckian discreteness of geometry (we postpone this discussion to Section <ref>). However, for simplicity we have not discussed in any details the dynamical equation of gravity in the Hamiltonian framework. It is possible to show (for more details see for instance <cit.>) that Einsteins equations split intothe following three constraints on the initial field configuration (E, A) given on a sliceΣ, and the Hamiltonian evolution equation for these data. The constraints are G^i(E,A)=d_AE^i= 0, V_d (E,A)=ϵ^abcE_ab· F_cd= 0S(E,A)= (E_ab× E_de)/√( det(E))· F_cf ϵ^abcϵ^def+⋯= 0,which are called the Gauss constraint, the diffeomorphism constraint, and the scalar constraint (there is an additional term in the last expression that we have omitted for simplicity, the full expression can be found in <cit.>). The · and × denotes the scalar and the exterior product in the internal space.For any quantity O(A,E) (this includes in particular the phase space variables A and E) its evolution is given by the canonical equation Ȯ={O, H[ α, N⃗,M ]} with the HamiltonianH[ α, N⃗,M ]≡∫_Σα_i G^i+N^a V_a+M S,where the fields α, N⃗, and M are completely arbitrary in their space and time dependence; the freedom is associated with coordinate invariance of gravity (encoded in the four free fields N⃗, and M) and the additional SU(2) internal gauge symmetry of the first order formulation in the time gauge. Time evolution is therefore not uniquely defined but all different spacetimes and fields reproduced from one particular initial data satisfying the constraints via (<ref>) can be shown to solve Einsteins equations and be related to each other via a diffeomorphims (coordinate transformation) andgauge transformations. A moment of reflection shows that an initial data (A,E) (i.e., solving the constraints) and (A+δ A, E+δ E) such that δ A={A, H[ α, N⃗,M ]} and δ E={E, H[ α, N⃗,M ]} lead to solutions which are related by a diffeomorphism and gauge transformations. The interpretation of this fact is that (A,E) and (A+δ A, E+δ E) are the very same data in different gauges. Thus, the Hamiltonian generates both time evolution and gauge transformations in agenerally covariant theory <cit.>.Notice also that the Hamiltonian vanishes identically on solutions. The Gauss constraint (<ref>) is specially important. In our gravity context it follows from the covariant field equation (<ref>) but on a more general basis it has a completely geometric origin: it arises from the presence of the underlying SU(2) gauge symmetry (what remains of the original Lorentz symmetry after the time gauge-fixing (<ref>)).Equation (<ref>) is the strict analog of the Gauss law of electromagnetism and Yang-Mills theory.It can be shownthat the smeared versionG[α]≡∫_Σα_i d_A E^i, generates (via Hamilton's equations) SU(2) gauge transformations of the basic variables and hence of any phase space quantity. Explicitly: δ_α A={A,G[α]}=-d_Aα δ_α E={E,G[α]}=[α,E]. These transformations are the generalization of the gauge transformations of electromagnetism to the non-Abelian case. The Gauss law and the transformations it generatesare strictly the same as those of an SU(2) Yang-Mills theory. The phase space parametrization in terms of a non-Abelian electric field E (with a geometric interpretation in this case) and its conjugate SU(2) connection also mimics the natural phase space parametrization of an SU(2) Yang-Mills phase space. This are key features of the variables that follow naturally form (<ref>); they have central importance in the construction of the non-perturbative techniques used to define LQG.In the presence of boundaries some subtleties arise when considering the gaugetransformations generated by (<ref>). We will return to this important point in Section <ref>. § QUANTUM GEOMETRY: THE HORIZON AND THE OUTSIDE In this Section we analyse further the commutation relations we found in (<ref>). On the one hand we will see what kind of commutation relations they imply for geometric observables on the boundary (that can represent a BH horizon), on the other hand we will also derive commutation relations for geometric observables inside (in the bulk). We sketch the quantisation of the observables and present the basics of the theory quantum geometry on which LQG is based.We also review the basic facts about the isolated horizon boundary condition <cit.> representing black holes in the formalism. §.§ Modeling a black hole horizon in equilibrium: the Phase space of an Isolated Horizon As discussed in Section <ref>, the physics of BHs in equilibrium as seen by external, late stationary observers is the physics of an infinitesimal hyper annulus around the horizon with a width that tends exponentially to zero with the propertime of the external observers and becomes quickly shorter than the Planck length. This suggests that the entityencoding the relevant degrees of freedom for the description of the statistical mechanical nature of BH entropy is the2+1 dimensional null hyper-surface defining the BH horizon. With the prospect of quantizing the later degrees of freedom in the canonical framework where LQG techniques have been developed, it is necessary to rethink the Hamiltonian formulation of general relativity in the presence of a null boundarywith suitable boundary conditions incorporating the physical notion of a BH horizon in equilibrium.This has led to the development ofthe so-calledisolated horizon (IH) formalism which we briefly describe below (see <cit.> for a specific review).The definition of thephase space of gravity (as for any field theory) needs special care when boundaries are present. For illustration consider the two situations depicted in figure <ref>: (assuming that the spacetimes of interest are asymptotically flat) on the left panel there is only a boundary at infinity, while on the left panelthe presence of the black hole is modelled by an additionalinternal null boundary Δ. In the presence of such null or time-like boundaries the local conservation of the symplectic structure (encoding the basic Poisson brackets structure of the field theory) does not suffice to grant the conservation of the symplectic structure from one initial Cauchy surface Σ_1 to a later one Σ_2.This is due to the fact that non trivial degrees of freedom excited when specified on Σ_1 can in turn excite those on the timelike or null boundaries of spacetime without being registered on Σ_2. Physically, energy can be carried in or out of the system via the boundaries. This implies that generically there is non trivial symplectic flux leaking across the timelike of null components of the boundary of the spacetime region of interest and hence a lack of conservation of the symplectic form defined on spacelike sections like Σ. The field theory defined on a bounded region is naturally that of an open system.With suitable restrictions of the behaviour of fields on the boundaries one can recover a closed-system field theoretical model (i.e. with a conserved symplectic structure) whichcan represent physically interesting situations. When these boundaries are at infinity the conservation of the symplectic structure follows from the vanishing of the sympletic flux across the boundary at infinity due to suitable fall off conditionson solutions of the field equations (defining asymptotically flat spacetimes). In the presence of the internal boundary Δ (see figure <ref>)boundary conditions must be specified in order to: on the one hand capture the geometry and other degrees of freedom on the boundary in the desired regime (stationary black holes in our case), and, on the other hand, grant the vanishing of the symplectic flux across Δ. If these two conditions are satisfied then one can view the system as a close hamiltonian system with a conserved symplectic structure, and thus contemplate its possible quantization.In order to achieve this goal the definition of isolated horizons declares the boundary Δ to be a null surface with the same topology of that corresponding to a stationary BH horizon, namely Δ=S^2× R, and then (appart from some technical subtleties) fixes the part of the geometry and matter fields that can be freely specified on such a characteristic surface to match those of the corresponding stationary BH horizon, i.e., those of the Kerr-Newman family. Among otherproperties, isolated horizons admit a preferred slicing in terms of spheres S^2 with an intrinsic geometry that is `time' independent in the sense that it does not depend on the S^2 spheres corresponding to the intersection of the spacelike hypersurfaces Σ with Δ.This implies that the area and shape of the horizon are time independent and forbids,by consistency with Einstein's equation, the flux of matter fields and gravitational radiation across the boundary. The definition of isolated horizons is independent of coordinates preserving the slicing, and admits a formulation which does not break SU(2) gauge transformations <cit.>. For that reason fields on the boundary Δ are only fixed up to these two gauge symmetries, hence the boundary condition allows for field variations which are only pure gauge on Δ (a combination of tangent diffeomorphisms and SU(2) gauge transformations). The definition of isolated horizons admits null horizons with local distortion <cit.>. The number of degrees of freedom specifying this distortion is infinite and can be encoded in multipole moments <cit.>. These cases are thought to represent BH horizons in equilibrium with exterior matter distributions causing the distortion due to tidal effects.The local nature of the boundary condition implies a certain ambiguity in the characterisation of time evolution right at the boundary. For stationary black holes the normal is uniquely fixed by requiring it to correspond to the Killing fields whose normalization are fixed by demanding that they generate the symmetries of inertial observers at infinity. Such relationship between symmetries at infinity (which are generated from the Hamiltonian perspective by mass M and angular momentum J) and the symmetry of the horizon along the null normal ℓ is the central ingredient in the validity of the first law (<ref>). In the case of isolated horizons such link is lost due to the very local nature of the definition.A direct technical consequence of that is that one can no longer fix the null normal to the horizon ℓ_ IH by simple global symmetry requirements and sothenull generator ℓ_ IH is defined only up to multiplication by a constant.This ambiguity has important consequences for the first law of BH mechanics which we discussed below. §.§ The laws of isolated horizons The restrictions on the boundary condition that are mentioned above capture the essential features of BH horizons in geometric terms. Despite the limitations described at the end of the previous section it has been shown that isolated horizons satisfy similar mechanical laws as BH horizons <cit.>. The first step is to define the notion of surface gravity κ_ IH which is achieved by the conditions that define a weakly isolated horizon.A slight strengthening of these conditions leads to the notion ofisolated horizons for which the only freedom that remains resides in the value of a constant rescaling of the null normal ℓ_ IH mentioned before. Surface gravity is defined via the intrinsic equation ℓ^a_ IH∇_a ℓ_IH^b=κ_IHℓ_ IH^b. Thus, the scaling ambiguity inℓ^a_IH implies that the surface gravity κ_ IH is up defined up multiplication by a constant as well. Nevertheless, one can show (as for stationary BHs) that κ_ IH is indeed a constant on the horizon even when these are not necessarily spherically symmetric. This is known as thezeroth law of black hole mechanics in analogy with the zeroth law of thermodynamics stating that temperature is uniform in a body at thermal equilibrium. Despite the fact that the definition of isolated horizons does not allow for the flow of matter across the null surface representing the BH horizon, it is possible to prove the validity of thefirst law for isolated horizon (i.e. a balance law analogous to (<ref>)).The way around is the integrability conditions that follow from requiring the existence of a consistent time evolution.Time evolution in general relativity is defined by a timelike vector field (a time flow). When considering time evolution in the canonical framework the flow is generated via Hamilton's equations by a suitable Hamiltonian. In general relativity this Hamiltonian has non trivial contributions (i.e. not vanishing when Einsteins equations are satisfied) only coming from the boundaries [The vanishing of the Hamiltonian density in the bulk—a direct consequence of Einstein's equations—is rooted in general covariance. Due to the absence of any preferred time notion evolution is pure gauge in a generally covariant field theory. Boundary conditions break general covariance by introducing additional structure such as the inertial time flow at infinity or the particular choice of the null normal at the isolated horizon. A consequence of this is the appearance of boundary contributions to the Hamiltonian (see for instance <cit.>; appendix E).]. Consistency of the time evolution demands the boundary contribution to the Hamiltonian coming from the BH horizon to depend on other boundary charges—the area of the horizon a, angular momentum J (which exists when the isolated horizon is axisymmetric <cit.>), and possibly other matter charges such as the electromagnetic charge Q—in a way that is encoded in a differential relation that amounts for the first law. Explicitly,dE_ IH=κ_ IH/8π da+Ω_ IH dJ+ Φ_ IH dQ,where Ω_ IH and Φ_ IH are the angular velocity and electromagnetic potential of the isolated horizon respectively. All intensive quantities are defined up to a constant rescaling inheriting the freedom of the choice of the null normal ℓ_ IH. This freedom precludes the integration of the previous relation to get a unique `state function' E_ IH(a,J,Q). As a consequence there is an infinite family of energy notions for an isolated horizons; one need extra structure in order to extract physics form the previous form of the first law.We will get back to this important point in Section <ref>. For an extensive review on properties of isolated and dynamical horizons see <cit.>. Isolated horizons have been defined in 2+1<cit.> and higher dimensions <cit.>. §.§ Pre-quantum geometry I: Poisson brackets of geometric quantities on the Horizon It turns out that under the restrictions imposed by the isolated horizon boundary conditionthe symplectic flux across the boundary defining the black hole horizon Δ is not zero but is given by the integral of a total differential; hence, it can be written as integrals on the boundary ∂Δ. These boundary fluxes can be absorbed in the definition ofthe symplectic structure of a closed system (i.e. conserved). Concretely, the symplectic structure (<ref>) acquires a boundary term encoding the presence of the internal boundary with its isolated horizon degrees of freedomΩ(δ,δ') =1/2κγ∫_Σδ A^i∧δ' E^_i - δ' A^i∧δ E^_i, + 1/κγ∫_H⊂∂Σδ e_i∧δ' e^i,where H=Δ∪Σ is a cross section of the isolated horizon (see right panel in Figure <ref>).This boundary term comes from the last term in (<ref>). Thus, in addition to the Poisson brackets (<ref>) for the bulk basic variables, the boundary term in the symplectic structure implies the following boundary fields commutation relations: {e^i_a(x), e^j_b(y)}= κγ δ^ijϵ^ (2)_abδ^ (2)(x, y) swhere ϵ^ (2)_ab is the 2d Levi-Civita density. Recall that the e-fields encode the metric information, so the previous equation (which we have shown to come directly from the gravity action (<ref>)) anticipates the non-commutativity of the boundary horizon geometry in the quantum theory.More explicitly, if we consider the two dimensional induced metric tensor g^ (2)_ab≡ e_a^ie^j_bδ_ij on ∂Σ the commutation relations that follow from (<ref>) are {g^ (2)_ab(x), g^ (2)_cd(y)}= κγ(g^ (2)_ac(x)ϵ^ (2)_bd+g^ (2)_bc(x)ϵ^ (2)_ad+g^ (2)_ad(x)ϵ^ (2)_bc+g^ (2)_bc(x)ϵ^ (2)_ad)δ^ (2)(x,y).The previous equation predicts quantum fuzziness of the geometry of the BH horizon (not all the components of the metric can be determined simultaneously due to Heisenberg uncertainty principle). For later application it is instructive to consider the bi-vectors E_B^i=ϵ^i_jk e^j∧ e^k,where E_B^i carries a subindex that stands forboundary to distinguish it from the analogous looking object (<ref>) defined in terms of bulk fields instead (recall discussion on extended variables in Section <ref>).A straightforward calculation shows that (<ref>) implies {E_B^i(x), E^j_B(y)}= κγ ϵ^ij_k E_B^k(x) δ^ (2)(x,y),which is remarkably simple: it corresponds to the algebra of angular momentum generators in standard mechanics. By introducing the smeared version E_B(α)≡∫_Hα_iE_B^i,for an arbitrary smearing field α_i then we can express the previous commutation relations as {E_B(α),E_B(β)}=κγ E_B([α,β]),where [α,β]_k≡ϵ^ij_kα_iβ_j. In the quantum theory this will lead to the discreteness of the BH horizon area spectrum[Similar commutation relations have been proposed via an independent argument in <cit.>.].It is possible to generalise the construction in order to describe higher dimensional BHs <cit.>.§.§.§ The gauge theory way: Chern-Simons formulation Non rotating BHs in equilibrium can be modelled by the isolated horizon boundary condition. When one assumes thehorizon to be spherically symmetric one finds that the curvature of the Ashtekar-Barbero connection in the bulk is related to the `electric' fields on the horizon as a_H/2πF^i(A)= E_B^i,where a_H is the area of the horizon (taken as non dynamical parameter characterizing the isolated horizon), and we assumeK^i=0 which corresponds to the time-symmetric slicing where the analysis of Section <ref> holds [The quasilocal treatment leading to the area Hamiltonian was not available when the first derivation of the CS formulation was proposed <cit.>. A different slicing was then used which lead a factor a_H/(π (1-γ^2)) on thel.h.s of (<ref>).]. As shown in <cit.>, the constraint (<ref>) amounts for the imposition of both diffeomorphism and the Gauss constraint at the boundary, i.e. it is in a precise sense the analog of (<ref>) and (<ref>) together.Because of the boundary condition, no scalar constraint (<ref>) needs to be imposed. All the boundary dynamics is coded by (<ref>). Using the previous equation,and under the assumption that fields satisfy the IH boundary conditions conditions described in Section <ref>, the boundary contribution to symplectic structure (<ref>) can be rewritten in terms of the Ashtekar-Barbero connection on the boundary as -a_H/2πκγ∫_∂Σδ A_i∧δ' A^i.This implies that (on the boundary) the Ashtekar-Barbero connection does not commute with itself, namely {A^i_a(x), A^j_b(y)}= 2πκγ/a_H δ^ijϵ^ (2)δ^ (2)(x, y) . The previous Poisson structure corresponds to that of an SU(2) Chern-Simons theorywith level k=a_H/(2πγ).It can be shownthat in this framework equation (<ref>) is a constraint that completes the Gauss constraint (<ref>) in the presence of a boundary: via Hamiltons equationit is the generator of boundary gauge transformations <cit.> (the relation between E_B and the generators of internal gauge transformations will be clarified further in Section <ref>). Thus, the classical degrees of freedom on an isolated horizon can be described dynamically by a Chern-Simons theory. Historically, the Chern-Simons formulation of isolated horizons was first found in its U(1) gauge fixed form in <cit.>. However, the U(1) gauge fixed theory in the quantum theory when one tries to imposethe gauge fixed version of (<ref>).Theobstacle is that the classical gauge fixing becomes incompatible with Heisenberg's uncertainty principle due to the commutation relations (<ref>). This difficulty is circumvented in the SU(2) formulation which was put forward later <cit.>.The formulation was extended to static BHs with distorsion in <cit.>. There are other parametrizations of the phase space of isolated horizons in the literature establishing a link with BF theories <cit.>; see for instance <cit.>.Rotating black holes do not satisfy the boundary condition (<ref>)<cit.>. Technical difficulties related to the action of diffeomorphisms also arise. For a discussion of these issues and a proposed model <cit.>.Isolated horizons which are not spherically symmetric and not rotating can be mapped to new variables so that the analog of (<ref>) (in the U(1) gauge) is satisfied <cit.>. For simplicity we will concentrate on spherically symmetric black holes in this article.§.§ Pre-quantum geometry II: Poisson brackets of geometric quantities in the bulkHere we show how the Poisson non-commutativity of the geometric variables on a boundary is not a peculiar feature of boundary variables but a generic property of metric observables which remains valid in the bulk. This leads to the non commutativity of the associated quantum operators in LQG and to its main prediction: the fundamental discreteness of the eigenvalues of geometry. This prediction is central for the description of the quantum properties of black holes in this approach to quantum gravity. §.§.§ Fluxes: the building block of quantum geometryGiven an arbitrary surface S in space Σone can define the following classical object which we call theflux of (geometry) E—in analogy with the equivalent quantity in electromagnetism or Yang-Mills theory—by the following expressionE(S,α)≡∫_S α_iE^i,where the smearing field α^i is assumed to have compact support in Σ. This quantity is central in the construction of quantum operators capturing geometric notions in LQG. It is an extended variable (as discussed in Section <ref>) which, through its non locality, allows for the necessary point-splitting regularization of non linear observables in the quantum theory [When applying the canonical quantization recipe, the basic variables E and A mut be promoted to suitable operators acting in a Hilbert space. Because of the distributional nature of the Poisson brackets (<ref>), these operators make sense as distributions as well. Products of these operators at a same point are mathematically ill-defined and lead to the UV divergencies that plague quantum field theories. The extended variables used in LQG are natural regulating structures that resolve this mathematical problem in the definition of (non-linear) geometric observables.].Among the simplest geometric observables one has thearea of a surface S, which can be shown to be given by a(S)=∫_S √(E_xy· E_xy) dx dy, where · denotes the contraction of internal indices (inner product in the internal space) of the E's and x,y are local coordinates on S. The fact that area is given by the previous expression is a simple consequence of the definition (<ref>) and the relationship of the triad e with the metric.Similarly, one can define thevolume of a region R ∈Σ as v(R)=∫_R √(E_xy·(E_yz× E_zx)) dxdydz. Both of which are potentially UV-divergent in the quantum theory due to the fact that they involve the multiplication of operator-valued distributions at the same space point. The statement, that we give here without a proof, is that the quantum operators a(S) and v(R) for arbitrary surfaces S and arbitrary regions R can be defined on the Hilbert space of LQG as functionals of the fluxes (<ref>) for families of regulating surfaces which are removed via a suitable limiting procedure (for details see <cit.>). In this way the fluxes (<ref>)—which arise naturally in the context of the boundary geometry—are also very important when defined in the bulkin terms ofan arbitrary 2-surface S⊂Σ. We will see in what follows that the bulk fluxes also satisfycommutation relations of the type (<ref>). §.§.§Non-commutativity of fluxes; the heart of Planckian discretenessHere we show that the Poisson brackets among fluxes (<ref>) reproduce the algebra of angular momentum generators at every single point on the surface.Here we also show how the appearance of the rotation algebra is related to the SU(2) gauge transformations generated by the Gauss law. Such non commutativity might seem at first paradoxical from the fact that the E^i Poisson commute according to (<ref>). The apparent tension is resolved when one appropriately takes into account the Gauss law (<ref>) and studies carefully the mathematical subtlelties associated with computing the Poisson bracket of an observable smeared on a 2-dimensional surface surface—as (<ref>)—in the context of the field theory on 3+1 dimensions. This subtlety has been dealt with in at least two related ways some time ago <cit.>. Here we follow a simpler and more geometric account recently introduced in<cit.>.We present it in what follows for the interested reader.Without loss of generality we assume S to be a close surface—if the 2-surface S does not close we can extend it to a new surface S' in some arbitrary way in the region outside the support of α to have it closed so that E(S,α)=E(S',α).Using Stokes theorem we can write (<ref>) as a 3-dimensional integral in the interior of S E(S,α) = ∫_ int[S] d(α_iE^i)=∫_ int[S] (d_Aα_i) E^i+α_i (d_AE^i)≈ ∫_ int[S] (d_Aα_i)∧ E^i,where in the second line the symbol ≈ reminds us that we have used the Gauss law (<ref>). More precisely, this implies that the Poisson bracket of any gauge invariant observable[A quantity is gauge invariant if O(E,A)=O(E+δ E, A+δ A) with δ's given by (<ref>) which is equivalent to saying that O(E,A) Poisson commutes with the Gauss generator (<ref>). ] and E(S,α),and Poisson bracket of the same observable and the expression of the right hand side of ≈ coincide. In other words, when considering gauge invariant quantities ≈ amounts to an = sign.It is only at this point—after writing the fluxes in terms of a 3 dimensional smearing of local fields—that we can use the Poisson brackets (<ref>) (whose meaning is a distribution in three dimensions as the Dirac delta functions in (<ref>) explicitly show).But now the new expression of the fluxes(<ref>) explicitly depends on the connection A^i via the covariant derivative d_A. This is the reason at the origin of the non trivial Poisson bracket between fluxes. Direct evaluation of the Poisson bracketsusing (<ref>) yields {E(S,α),E(S,β)}≈∫∫ dx^3 dy^3 {dα_i∧ E^i+ϵ_ijk A^j∧α^k∧ E^i,dβ_l∧ E^l+ϵ_lmn A^m∧β^n∧ E^l} ≈∫∫dx^3 dy^3 {dα_i∧ E^i,ϵ_lmn A^m∧β^n∧ E^l} +{ϵ_ijk A^j∧α^k∧ E^i,dβ_l∧ E^l} +{ϵ_ijk A^j∧α^k∧ E^i,ϵ_lmn A^m∧β^n∧ E^l} ≈κγ∫dx^3ϵ_ijk dα^i∧β^j∧ E^k +ϵ_ijkα^i ∧ dβ^j ∧ E^k+⋯ ≈κγ∫dx^3d_A([α,β])_k∧ E^k ≈κγ E[[α,β], S], where [α,β]_k≡ϵ_kijα^iβ^k, and in the third line we have omitted the explicit computation of the third term of the second line as this one can be guessed from the fact that the result must be gauge invariant. This leads to the sought result: the non commutativity of the fluxes that is at the heart of the discreteness of geometric kinematical observables in LQG. Namely: {E(S,α),E(S,β)}≈κγ E[[α,β], S].We recover in this way in the bulk for the smeared fluxes the same result found in (<ref>) for the boundary. The observables e^i and the Poisson brackets (<ref>) and (<ref>) are not known to be available in the bulk of space Σ. However, recent results <cit.> indicate that there might be a way to extending their to the interior of the space. This could have very important consequences as it would allow for the definition of a new set of observables that could, one the one hand, lead to a natural geometrization of matter degrees of freedom, and, on the other hand, reduce some quantization ambiguities in the definition of the dynamics of LQG. We will comment on these developments in Section <ref>. §.§ Quantum geometryWe are now ready to sketch the construction of the quantum theory. LQG was born from the convergence of two main set of ideas: the old ideas about background independence formulated by Dirac, Wheeler, DeWitt and Misner in the context of Hamiltonian general relativity, and the observation by Wilson, Migdal, among others, that Wilson loops are natural variables in the non perturbative formulation of gauge theories. The relevance of these two ideas is manifest if one formulates classical gravity in terms of the variables that we introduced in Section <ref>that render the some of the equations of general relativity similar to those of standard electromagnetism or Yang-Mills theory (Section <ref>). What is the physical meaning of thenew variables? The triplet of vector potentials A^i have an interpretation that is similar to that of A⃗ in electromagnetism: they encode the `Aharonov-Bohm phase' acquired by matter when parallel transported along a path γ in space—affecting all forms of matter due to the universality of gravity. Unlike in electromagnetism, here the `phase' is replaced by an element of SU(2) associated to the action of a real rotation in space on the displaced spinor. This is mathematically encoded in the Wilson loop (related to the circulation of the magnetic fields B_i) along the loop γ according to W_γ[A]= Pexp∫_γτ_iA^i_a dx^a/ds ds∈ SU(2),where P denotes the path-ordered-exponential[The path ordered exponential is necessary do to the non commutativity of the matrices τ^i.], τ^i are the generators of SU(2), and s is an arbitrary parameter along along γ.This expression is the SU(2) counterpart of the natural extended variables(<ref>) mentioned in Section <ref> before the introduction of the time gauge.In this analogy, the electric fields E^i have a novel physical interpretation: they encode (as reviewed in the previous sections) the geometry of 3-dimensional space, and define in particularthe area surfaces and volume of regions according to (<ref>) and (<ref>).The quantization is performed following the canonical approach, i.e., promoting the phase space variables to self adjoint operators in a Hilbert spacesatisfying the canonical commutation relations according to the rule { , }→ -i/ħ [ , ].As there is no background structure the notion ofparticle, as basic excitations of avacuum representing a state of minimal energy, does not exist. However, there is a naturalvacuum|0⟩_ associated to the state of no geometry or vanishing electric field, i.e. E^i|0⟩_=0. This state represents a very degenerate quantum geometry where the area of any surface and the volume of any region vanishes.Distances are not naturally defined (some ambiguities affect its definition <cit.>); however, all definitions coincide in the statement that the distance between any pair of points is zero in the state |0⟩_. The operator W_γ[A], the quantum version of (<ref>),acts on the vacuum by creating a one-dimensional flux tube of electric field along γ. These fundamentalFaraday lines represent the building blocks of a notion ofquantum geometry.Only those excitations given by closed Wilson lines of quantized electric field are allowed by quantum Einstein's equations, i.e.,loop states. This is due to theGauss constraint (<ref>) ( divergence of the electric field must vanish)that follows directly the equations of motion coming from (<ref>); Section <ref>.Therefore, Faraday lines must always close and form loops. The construction of the Hilbert space of quantum gravity is thus started by considering the set of arbitrary multiple-loop states, which can be used to represent (as emphasized by Wilson in the context of standard gauge theories) the set of gauge invariant functionals of A^i. Multiple-loop states can be combined to form an orthonormal basis of the Hilbert space of gravity. The elements of this basis are labelled by: a closed graph in space, a collection of spins—unitary irreducible representation of SU(2)—assigned to its edges, and a collection of discrete quantum numbers assigned to intersections. As a consequence of (<ref>)the rules of addition of angular momentum must be satisfied at intersections: the total flux of electric field at a node is zero. They are calledspin-network states. Spin network states are eigenstates of geometry as it follows from the rigorous quantization of the notion of area and volume (given by equations (<ref>—<ref>)).In LQG the area of a surface can only take discrete values in units of Planck scale! More precisely, given a surface S and a spin-network state with edges intersecting the surface (at punctures labelled by the dummy index p below) with spins j_1,j_2⋯ then one has a(S) |j_1,j_2⋯⟩=(8πγℓ_p^2∑_p√(j_p (j_p+1)))|j_1,j_2⋯⟩ where we have labelled the spin-network state with the relevant spins only (further details identifying the state, which are not relevant for the area eigenvalue, are not explicitly written for notational simplicity). A particularly important application of this formula is the computation of the eigenvalues of the area of a black hole horizon. A graphical representation of the situation is presented in Figure <ref>: links in the figurecan be interpreted as flux lines of quantum area depending on their colouring by spins. Similarly, the spectrum of the volume operator V(R) isdiscrete and associated to the presence of spin network intersections inside the region R (nodes in Figure <ref> represent volume quanta). More precisely, as mentioned before gauge invariance implies that the quantum numbers of the fluxesassociated to the different spin-network links converging at a node must add up to zero. This condition admits an unambiguous interpretation of nodes as quantum states of a convex polyedron. This interpretation is based on a theorem by Minkowski on discrete Euclidean geometry <cit.> (see Figure <ref>). The properties of the quantum shape of polyedra has been studied numerically via a variational algorithm <cit.>. In figure <ref> we represent a spin-network state including a black hole as a boundary (see below for the description of the boundary (horizon) quantum state). There are three and four valent nodes in the bulk. According to Minkowski theorem four-valent nodes represent quantum tetrahedra where the areas of the four triangles are defined by the area eigenvalues depending on the spins j; three-valent nodes are degenerate (zero-volume and purely quantum) excitations. The discovery of the discrete nature of geometry at the fundamental level has profound physical implications. In fact even before solving the quantum dynamics of the theory one can already answer important physical questions. The most representative example (and early success of LQG) is the computation of black hole entropy from first principles.Another profound potential implication of discreteness concerns the UV divergences that plague standard QFT's. It is well known that in standard QFT the UV problem finds its origin in the difficulties associated to the quantization of product of fields at the same point (representing interactions). A first hint of the regulating role of gravity is provided by the fact that, despite of their non-linearity in E_i, area and volume are quantized without the appearance of any UV divergences. This is expected to provide a universal regulating physical cut-off to the fundamentaldescription of fields in LQG (see important discussion in Section <ref>).Before the imposition of the vector constraint (<ref>) two spin network differing by a tiny modification of their graphs are orthogonal states!—that would seem to make the theory intractable as the Hilbert space would be too large. This is where the crucial role background independence starts becoming apparent as the vector constraint—although is not self-evident—implies that only the information in spin network states up to smooth deformations is physically relevant. Physical states are given by equivalence classes of spin networks under smooth deformations: these states are calledabstract spin networks<cit.>. Abstract spin network states represent a quantum state of the geometry of space in a fully combinatorial manner. They can be viewed as a collection of `atoms' of volume (given by the quanta carried by intersections) interconnected by edges carrying quanta of area of the interface between adjacent atoms. This is the essence of background independence: the spin network states do not live on any pre-established space, they define space themselves. The details of the way we represent them on a three dimensional `drawing board' do not carry physical information. The degrees of freedom of gravity are in the combinatorial information encoded in the collection of quantum numbers of the basic atoms and their connectivity.Finally the full non linearity of the dynamics gravity is encoded in the quantum scalar constraint (<ref>). Quantization of this operator has been shown to be available <cit.>; however, the process suffers from ambiguities <cit.>.These ambiguities are expected to be reduced if strong anomaly freeness conditions are imposed based on therequirement that the constraint algebra is satisfied in a suitable sense (for a modern discussion of this important problem see <cit.> and references therein). §.§ Quantum isolated horizons In the presence of a boundary representing a black hole a given spin-network state intersects the boundary at punctures, Figure <ref>. These puncturesare themselves excitations in the Hilbert space of the boundary Chern-Simons theoryin the isolated horizon model of black holes.The Hilbert space of isolated horizons is the tensor product_H⊗_out where the two factors denote the Hilbert space of the horizon (spanned by puncture-states) and the Hilbert space of the outside bulk (spanned by spin networks) respectively.At punctures the quantum version of (<ref>) must be imposed. This constraint takes the fom (a_H/2πF^i(A)⊗1-1⊗ E^i)|ψ⟩_H |ψ⟩_out =0,where |ψ⟩_H∈_H and |ψ⟩_out∈_out. It can be shown that |ψ⟩_ out breaks individually the SU(2) gauge invariance when the transformation acts non trivially on the horizon <cit.>. It is precisely the addition of the Chern-Simons boundary degrees of freedom and the imposition of (<ref>) which restores the gauge symmetry broken from the point of view of the bulk by the presence of the boundary <cit.>.The Chern-Simons boundary degrees of freedomcan be seen in this sense as would-be-gauge excitations <cit.>. The representation of a generic state is given in Figure <ref>.§.§ The continuum limit A key feature of LQG is the prediction of fundamental discreteness at the Planck scale. States of the gravitational degrees of freedom are spanned in terms of spin-network states (polymer-like excitations of quantum geometry) each of which admits the interpretation of an eigenstate of geometry which is discrete and atomistic at the fundamental level <cit.>. This feature is the trademark of the theory; on a rigorous basis it has been shown that the representation of the basic algebra of geometric observables as operators in a Hilbert space—containing a `vacuum' or `no-geometry' state (<ref>) which is diffeomorphism invariant and hence for which all geometric eigenvalues vanish—is unique <cit.>. In this picture flat Minkowski space-time must be viewed as a highly exited state of the `no-geometry' state (<ref>), where the quantum space-time building blocks are brought together to produce the (locally) flat arena where other particles interact.Thus, there is no a priori notion of space-time unless a particular state is chosen in the Hilbert space.Loop quantum gravity is a concrete implementation of such non-perturbative canonical quantization of gravity <cit.>. Even thoughimportant questions remain open, there are robust results exhibiting features which one might expect to be sufficiently generic to remain in a consistent complete picture.There has been an important activity in trying to construct semiclassical states in the framework. At the canonical level efforts have concentrated in the definition of coherent states of quantum geometry <cit.> representing a given classical configuration. The relationship between the fundamental spin-network state representation of quantum gravity and the Fock state representation of QFTs has been explored in <cit.>. Recently, emphasis has been given to the constructions of states that reproduce the short distance correlations of quantum field theory <cit.>. These results provide useful insights on the nature of the low energy limit of LQG.Clear understanding is still missing mainly because the dynamical aspect of the question—understanding the solution space of (<ref>)—is still under poor control.However, the view that arises from the above studies is that smoothgeometry should emergefrom the underlying discrete fundamental structures via the introduction of coarse observers that are insensitive to the details of the UV underlying structures.One expects that renormalization group techniques should be essential in such context (see <cit.> and references therein). The problem remains a hard one as one needs to recover a continuum limit from the underlying purely combinatorial Wheeler'spre-geometric picture <cit.>. The task is complicated further in that the usual tools—applied to more standard situations where continuity arises from fundamentally discrete basic elements (e.g. condensed matter systems or lattice regularisations of QFTs)—cannot be directly imported to the context where no background geometry is available.Consequently, a given classical space-time cannot directly correspond to a unique quantum state in the fundamental theory: generically, there will be infinitely many different quantum states satisfying the coarse graining criterion defined by a single classical geometry. For instance, there is no state in the LQG Hilbert space that corresponds to Minkowski space-time. Flat spacetime is expected to emerge from the contributions of a large ensemble of states all of which look flat from the coarse grained perspective. Because coarse grained observers are insensitive to Planckian details (quantum pre-geometric defects), flat space-time might be more naturally associated with a density matrix in LQG than to any particular pure state <cit.>.The bulk entropy associated to such mix-state would be non trivialin the sense that it would carry a non trivial entropy density (this will be important in Section <ref>).Unfortunately, due to the difficulties associated with describing the low energy limit of LQG, one cannot give a precise description of the exact nature of the pre-geometric defects that might survive in a state defining a background semiclassical geometry.Nevertheless, there are a variety of structures at the Planck scale that do not seem to play an important role in the construction of the continuum semiclassical space-time, yet they are expected to arise in strong coupling dynamical processes as no selection rule forbids them.For instance one hasclose loops and embeddedknots which are the simplest solutionsto all the quantum constraint-equations including (<ref>) <cit.>. As geometric excitations they are degenerateand carry areabut no volume quantum numbers.More generally the semiclassical weave states in LQG can contain local degenerate contributions such as trivalent spin-network nodes or other configurationswith vanishing volume quanta.On the dynamical side the vertex amplitude ofspin foams<cit.> is known not to impose some of the metricity constraints stronly. This allows for the possibility of pre-geometric structures to survive <cit.> in the semiclassical limit. From the canonicalperspective the quantum Hamiltonian constraint seem to invariably produce such pre-geometric defects at the Planck scale. Finally, there is the q-bit degeneracy of the volume eigenvalues <cit.>: each non-zero eigen-space of the volume of a spin-network node is two dimensional. This local two-fold degeneracy can, by itself, be the source of a non trivial bulk entropy (a recent study of bulk entropy for a volume coarse graining see <cit.>).All this indicates that the emerging picture in an approach as LQG is very different from the bulk-boundary-duality type of quantum gravity scenarios such as the one proposed byADS-CFT correspondence scenarios <cit.>. In LQG the notion of smooth space-time has a capacity for infinite bulk entropy . Such non holographic behaviour at the fundamental level might seem (to some) at odds with the belief in the so-calledholographic principle as a basic pillar of quantum gravity<cit.>.However, further scrutiny shows that, despite its non-holographic nature, the predictions ofLQG are completely consistent with the phenomenology motivating holography. This is basically because, in the physical situations where holographic behaviour arises, bulk entropy only contributes as an irrelevant overall constant.A clear example of this is the validity of covariant entropy bounds satisfied by relative (entanglement) entropy <cit.>. Relative entropy of a state ρ is defined with respect to a reference quantum state (a `vacuum' state) ρ_0 asS_ρ_0(ρ) = - [ρlogρ] + [ρ_0 logρ_0].While giving non trivial information about excitations of the `vacuum' in the mean field approximation (where a space-time background makes sense) such covariant entropy bounds do not constraint the number of fundamental degrees of freedom of quantum geometry. More precisely, the bulk Planckian entropy, being a constant, just cancels out in the subtraction that regularises S_ρ_0(ρ). The point here is that on the basis of the insights of LQG on the nature of thefundamental quantum geometry excitations, the holographic principle is degraded from its status of fundamental principle to a low energy property of quantum fields on curved space-times. This is the perspective that follows from the LQG statistical mechanical account of black hole entropy (see <cit.> for a description in terms of entanglement and a discussion along the lines of the present paper and <cit.> for recent reviews). In this way the framework of LQG, without being fundamentally `holographic', can accommodate the holographic phenomenon when it is valid. The holographic behaviour is a characteristic of suitable systems (black holes, isolated horizons, null surfaces, etc.) but not a fundamental property of the theory.Specialising to ensemble of states that all `look like' Minkowski space-time for suitably defined coarse grained observers we notice that their members must differ by hidden degrees of freedom from the viewpoint of those low energy observers. In particular they would all agree on stating that all the states (even though different at the fundamental scale) have zero ADM (or Bondi) energy. We have seen above that in the particular case of LQG a whole variety of pre-geometric structures, that are well characterized in the strong coupling regime, are potential local defects that—as they do not affect the flatness of the geometry of the semiclassical state—would carry no energy in the usual sense of the concept. These Planckian defects in the fabric of space-time are hidden to the low energy observers but represent genuine degrees of freedom to which other degrees of freedom can correlate to (this plays a key role in a recently introduced perspective for addressing the information puzzle in black hole evaporation, Section <ref>, as well as in natural mechanism for the generation of a cosmological constant, Section <ref>).In this respect one might draw an analogy with frustrated systems in condense matter that do not satisfy the third law as they carry a non trivial residual entropy <cit.> for states describing the continuum limit in LQG. § BLACK HOLE ENTROPY IN LQG We are now ready to present one of the central results about black holes in LQG: the computation of black hole entropy using statistical mechanicalmethods applied to the microscopic states of the horizon as predicted by quantum geometry. As just discussed, a given macroscopic smooth geometry is expected to arise from the coarse graining of the underlying discrete states of LQG. This point of view is the one taken in the computation of BH entropy: for a given (spherically symmetric) isolated horizon one counts the degeneracy of fundamental states satisfying the coarse graining condition that the macroscopic area is a_H.§.§ Direct countingThe discreteness of the area operator (<ref>) predicted by LQG and the previous discussion on the nature of the continuum limit in LQG suggests an obvious definition of the statistical mechanical entropy of a BH in the theory. The microcanonical entropyS≡log()whereis the number of geometry states of the BH horizon such that a-ϵ≤8πγℓ_p^2∑_i=1^N√(j_i(j_i+1))≤a+ϵ,where a is the macroscopic area of the BH and ϵ a macroscopic coarse graining scale that does not enter the leading order contribution to the entropy as in standard statistical systems.Initial estimates of the entropy where based on bounding the number of states <cit.> which immediately led to indications that the entropy would be proportional to the area. The first rigorous countings (valid in the large BH limit) where proposed in <cit.>. An alternative and very simple approach was given in<cit.>. These counting led to an entropy S=γ_0/γa/8πℓ_p^2 where γ_0=0,274⋯[The gauge fixed U(1) Chern-Simons formulation allowed for two different counting prescriptions which led to different values of γ_0; the other possibility gave γ_0=0,237⋯. This ambiguity disappears in the SU(2) (un gauge fixed) formulation. See Section <ref> for references. If for some physical mechanism <cit.> forces punctures to be labelled by the adjoint representation j=1 only, then γ_0=log(3)/(2π√(2)). Such value of γ_0 has been argued to be singled out from a heuristic version of Bohr correspondence and quasinormal modes of spherical BHs <cit.>. This idea (which does not seem to generalized to the rotating BH in any simple way) has not become more solid since its appearance, and is mostly regarded as a strange coincidence.] is a numerical factor determined numerically by the solution of a transcendental equation that we will discuss below (equation (<ref>)). This result was interpreted as a constrain on the value of the Immirzi parameter from semiclassical consistency. More precisely, consistency with the Bekenstein-Hawking value of the entropy of a black hole (which means consistency with classical general relativity coupled semiclassically with quantum fields) requiresγ=γ_0.Sophisticated and very powerful mathematical techniques based on number theory were developed <cit.> for understanding in great detail the combinatorial counting problem at hand. Indeed an exact formula for the entropy can be constructed from this approach (other methods exploiting the connection with conformal field theories were studied <cit.>). These developments were motivated on the one hand by the interest in the computation of logarithmic corrections to the entropy which could allow comparison and contrast with other approaches. On the other hand,numerical investigations of the state counting of small (Planck size) black holes revealed a surprising regular step structure for the entropy<cit.>, i.e. a fine structure of steps around the linear behaviour (<ref>). This was initially thought it could survive the semiclassical limit (large BH limit) and so lead to strong deviations from the standard expectations for Hawking radiation <cit.> (a strong signature of quantum gravity or an inconsistency with the semiclassical picture). The underlying mathematical reasons for this effect was understood thanks to the newly developed number theoretical techniques <cit.>.It became also clear that such fine structure had no physical relevance in the semiclassical regime as it was just a peculiarity of the area density of states for small black holes that does not survive in the thermodynamic limit <cit.>.The problem of computing the entropy of the black hole was greatly simplified by the appeal to a suitable Hamiltonian of the system. Such Hamiltonian is singled out by the notion of time flow defining the stationarity that is inherentto the notion of equilibrium.In what follows we explain how this idea can indeed be implemented. This complements the isolated horizon quasilocal definition of the phase space of idealized black holes (Section <ref>) with the additional structure implying that they are in equilibrium with their immediate vicinity. §.§ The effective area Hamiltonian from local observersThe indeterminacy, mentioned in Section <ref>, of the quantities appearing in the first law for isolated horizon was due to the impossibility of finding a preferred normalization of the null generators at the horizon. Such freedom precludes the definition of a unique time evolution at the horizon and hence of a preferred notion of energy entering the first law. In the case of stationary black holesa preferred time evolution (and energy notion) is provided by the existence of a global time translational symmetry represented by a timelike killing field whose normalization is fixed at infinity by demanding that it coincides inertial observers at rest. This global structure is absent in the isolated horizon formulation and seems at first to compromise the possibility of introducing a physical notion of energy and a standard statistical mechanical account of the BH thermal behaviour.However, we have argued in Section <ref> that only the immediate vicinity of the BH horizon plays an important role in the thermodynamical description of the black hole system. It should be possible to eliminate the above indeterminacy in the first law by assuming that the near horizon geometry only is in equilibriumwithout assuming stationarity ofthe entire spacetime outside the black hole.Indeed, if the local geometry is stationary one can shift the emphasis from observers at infinity toa suitable family of stationary nearby local observers (the local characterization of stationarity in the framework of isolated horizons see <cit.>). As discussed below, these local observers provide the necessary additional structure to recover a preferred energy notion, and a thermodynamical first law. Remarkably, the new (quasi-local) first law and the associated notion of energy are extremely simple and well adapted to the structure of LQG.The key assumption is that the near horizon geometry is isometric to that of aKerr-Newman BHs[Such assumption is physically reasonable due to the implications of the no-hair theorem. At the quantum level one is demanding the semiclassical state of the BH to be picked around this solution.]. A family of stationary observers O located right outside the horizon at a small proper distance ℓ≪√(A) is defined by those following the integral curves of the Killing vector field χ=ξ+Ω ψ=∂_t+Ω ∂_ϕ,where ξand ψ are the Killing fields associated with the stationarity and axisymmetry of Kerr-Newman spacetime respectively, while Ω is the horizon angular velocity (the Killing field χ is timelike outside and close to the horizon).The four-velocity of O is given by u^a=χ^a/χ.It follows from this that O are uniformly accelerated with an acceleration a=ℓ^-1+o(ℓ) in the normal direction.These observers are the unique stationary ones that coincide with thelocally non-rotating observers<cit.> or ZAMOs <cit.> as ℓ→ 0. As a result, their angular momentum is not exactly zero, but o(ℓ).Thus O are at rest with respect to the horizon which makes them the preferred observers for studying thermodynamical issues froma local perspective. It is possible to show that the usual first law (<ref>) translates into a much simpler relation among quasilocal physical quantities associated with O<cit.>.As long as the spacetime geometry is well approximated by the Kerr Newman BH geometry in the local outer region between the BH horizon and the world-sheet of local observers at proper distance ℓ, and, in the leading order approximation for ℓ/√(a)≪ 1, the followinglocalfirst law holdsδ E =κ/8πδ a,whereδ E=∫_W T_μν u^μ dW^ν=χ^-1∫_W T_μνχ^μ dW^ν represents the flow of energy across the world-sheet W defined by the local observers, andκ≡κ/(χ).This is the standard physical energy measured by the local observers; the amount of heat that a calorimeter would register if the falling matter is captured instead of let go into the black hole.The above result follows from the conservation law ∇^a (T_abχ^b)=0 that allows one to write δ E as the flux of T_abχ^b across the horizon. This, in turn, can be related to changes in its areausing Einsteins equations and perturbation theory (more precisely the opticalRaychaudhuri equations) <cit.>.Two important remarks are in order:First, there is no need to normalize the Killing generator χ in any particular way. The calculation leading to (<ref>) is invariant under the rescaling χ→αχ for α a non vanishing constant. This means that the argument is truly localand should be valid for more general black holes with a Killing horizon that are not necessarily asymptotically flat. This rescaling invariance of the Killing generator corresponds precisely to the similar arbitrariness of the generators of IHs as described in Section <ref>.The fact that equation (<ref>) does not depend on this ambiguity implies thatthe local first law makes sense in the context of the IH phase space as long as one applies it to those solutions that are isometric to stationary black hole solutions in the thin layer of width ℓ outside the horizon. The semiclassical input is fully compatible with the notion of IHs. Second, the local surface gravity κ̅ is universal κ̅=ℓ^-1 in its leading order behaviour for ℓ/√(a)≪ 1. This is not surprising and simply reflects the fact that in the limit √(a)→∞, with ℓ held fixed, the near horizon geometry in the thin layer outside the horizon becomes isometric to the corresponding thin slab of Minkowski spacetime outside a Rindler horizon: the quantity κ̅ is the acceleration of the stationary observers in this regime. Therefore, the local surface gravity loses all memory of the macroscopic parameters that define the stationary black hole. This implies that, up to a constant which one sets to zero, equation (<ref>) can be integrated, thus providing an effective notion of horizon energy E=a/8π G_Nℓ, where G_N is Newton's constant.Such energy notion is precisely the oneto be used in statistical mechanical considerations by local observers.Similar energy formulae have been obtained in the Hamiltonian formulation of general relativity with boundary conditions imposing the presence of a stationary bifurcate horizon <cit.>. The area as the macroscopic variable defining the ensemble has been previously <cit.> usedin the context of BH models in loop quantum gravity. The new aspect revealed by the previous equation is its physical interpretation as energy for the local observers.In application of this quasilocal notion of energy to quantum gravity one assumes that the quantum state of the bulk geometry in the local neigborhood of width ℓ outside the (isolated) horizon is well picked around a classical solution whose near horizon geometry is that of a stationary black hole.The thermodynamical properties of quantum IHs satisfying such near horizon condition can be described using standard statistical mechanical methods with the effective Hamiltonian that follows from equation (<ref>) and the LQG area spectrum (<ref>), namely H|j_1,j_2⋯⟩=(γℓ^2_Pl/2 G_Nℓ∑_p√(j_p (j_p+1)))|j_1,j_2⋯⟩ where j_p are positive half-integer spins of the p-th puncture and ℓ_p=√(Għ) is the fundamental Planck length associated with the valuegravitational coupling G in the deep Planckian regime. The analysis that follows can be performed in both the microcanonical ensemble or in the canonical ensemble; ensemble equivalence is granted in this case because the system is simply given by a set of non interacting units with discrete energy levels.§.§ Pure quantum geometry calculationIn this section we compute black hole entropy first in the microcanonical ensemble following a simplified (physicist) version <cit.> Asthe canonical ensemble becomes available with the notion of Hamiltonian (<ref>), we will also derive the results in the canonical ensemble framework.The treatment in terms of the grand canonical ensemble as well as the equivalence of the three ensembles has been shown <cit.>.Denote by s_j the number of punctures of the horizon labelled by the spin j (see Figure <ref>). The number of states associated with a distribution of distinguishable punctures {s_j}_j=1/2^∞ isn({s_j})=∏_j=1/2^∞N!/s_j! (2j+1)^s_j, where N≡∑_j s_j is the total number of punctures. The leading term of the microcanonical entropy can be associated with S=log(n({s̅_j})), where s̅_j are the solutions of the variational conditionδlog(n({s̅_j}))+2πγ_0δ C({s̅_j})=0where γ_0 (the 2π factor is introduced for later convenience) is a Lagrange multipliers for the constraintC({ s_j}) = ∑_j √(j(j+1)) s_j-a/8πγℓ_p^2=0.In words, s̅_j is the configuration maximazing log(n({s_j})) for fixed macroscopic area a. A simple calculation shows that the solution to the variational problem (<ref>) is s̅_j/N=(2j+1)exp(- 2πγ_0 √(j(j+1)) ),from which it follows, by summing over j, that 1=∑_j (2j+1)exp(- 2πγ_0 √(j(j+1))). Numerical evaluation of the previouscondition yields γ_0=0.274⋯. It also follows from (<ref>), and the evaluation of S=log(n({s̅_j})),thatS=γ_0/γa/4ℓ_p^2 as anticipated in (<ref>).In what sense the previousresult constrains the value of the Immirzi parameterγ? One can calculate the temperature of the system usingthe thermodynamical relation1/T=.∂ S/∂ E|_N=G_N γ_0/G_γ 2πℓ, where Eis the energy (<ref>) measured by quasilocal observers. Semiclassical consistency of quantum field theorywith gravity in the near horizon geometry requires the inverse temperature to be given by Unruh's value T^-1=2πℓ. This leads to the following restriction involving γ_0 the Immirzi parameter γ, G, and G_N, namely γ_0=γG/G_N, from which it follows (when replacing it back in (<ref>)) thatS=a/4ħ G_N.Due to quantum effects Newton'sconstant is expected to flow from the IR regime to the deep Planckian one. On the one hand, the UV value of the gravitational coupling is defined in terms of the fundamentalquantum of area predicted by LQG.On the other hand, the low energy value G_N appears in the Bekentein-Hawking entropy formula <cit.>. The semiclassical input that enters the derivation of the entropythrough the assumption of (<ref>) is the ingredient that bridges the two regimes in the present case.§.§.§ Freeing the value of γ by introducing a chemical potential for punctures An interesting possibility is to allow for punctures to have a non trivial chemical potential. Due to the existence of an area gap (depending on the Immirzi parameter) adding or removing a puncture from the state of the horizon is analogous to exchanging a particle carrying non zero energy (<ref>) with the system. At the present stage of development of the theory (with incomplete understanding of the continuum limit) one can investigate the possibility that the number of punctures N of the black hole state represent an additional conserved quantity (a genre ofquantum hair) for semiclassical black holes. The consequence of such generalization is that the Immirzi parameter need non longer be fixed to a particular value: in addition to fixing the value of the area gap, the Immirzi parameter controls the value of the puncture's chemical potential <cit.>.The derivation follows closely the previous one. The difference is that one needs to add a new constraint C^'( s_j})=∑_j s_j-N=0,with an additional Lagrange multiplier that we call σ. The new extremum condition now only fixes a relationship between σ and γ_0 which takes the form σ(γ)= log[∑_j (2j+1) e^-2 πγG/G_N]once (<ref>) is used. The entropy becomesS= a/4ℓ_p^2+σ(γ) N.The first term in the entropy formula is the expected Bekenstein-Hawking entropy while the secondis a new contribution to the entropy which depends on the value of the Immirzi parameter γ. This new contribution comes from the punctures non-trivial chemical potential which is given by μ̅=-T .∂ S/∂ N|_E=-ℓ_p^2/2πℓ σ(γ)where one is again evaluating the equation at the Unruh temperature T=ħ/(2πℓ).The above derivation can be done in the framework of the canonical and grandcanonical ensembles. From the technical perspective it would have been simpler to do it using one of those ensembles. In particular basic formulae allow for the calculation of the energy fluctuationswhich at the Unruh temperature are such that (Δ E)^2/⟨ E⟩^2=(1/N). The specific heat at T_ U is C=Nγ_0^2d^2σ/dγ^2 which ispositive. This implies that, as a thermodynamic system, the IH is locally stable. The specific heat tends to zero in the large γ limit for fixed N and diverges as ħ→ 0. The three ensembles give equivalent results <cit.>.The entropy result (<ref>) might seem at first sight in conflict with, what we could call, thegeometric first law (<ref>) (geometric because it is implied directly by Einsteins equations).However, when translating things back to observers at infinity, the present statistical mechanical treatment implies the following thermodynamical first lawδ M=κħ/2 πδ S+Ωδ J+Φδ Q+ μδ N,where μ=-ℓ_p^2 κσ(γ)/(2π) (the redshifted version of μ̅). It is now immediate to check that the exotic chemical potential term in (<ref>) cancels the term proportional to the number of punctures in the entropy formula (<ref>). Therefore, the above balance equation is just exactly the same as (<ref>).As in the seminal argument by Jacobson<cit.>the validity ofsemiclassical consistency discussed here for generalaccelerated observers in arbitrary local neighbourhoods implies the validity of Einsteins equations <cit.>. §.§ Changing statistics to include matterIn the previous sections only pure geometric excitations have been taken into account. However, from the local observers perspective matter fields are highly exited close to the horizon. More precisely, the quantum state of all non-geometric excitations is seen as a highly excited state at inverse temperature β=2πℓ/ℓ_p^2.This is a necessary condition on the UV structure of the quantum state so that it just looks like the vacuum statefor freely falling observers (at scales smaller than the size of the BH). This is related to the regularity condition of the quantum state granting that the expectation value of the energy momentum tensor is well defined <cit.>: a statement about the two-point correlation function called the Hadamard condition that is intimately related to the UV behaviour of entanglement across the horizon.This has been used to argue <cit.> thatquasilocal stationary observers close to the horizon would find that the number of matter degrees of freedom contributing to the entropy grows exponentially with the horizon area according to D ∝exp(λ a/(ħ G_N)),where λ is an unspecified dimensionless constant that cannot be determined (within the context of quantum field theory) due to two related issues: On the one hand UV divergences of standard QFT introduce regularization ambiguities affecting the value of λ; on the other hand, the value of λ depends on the number of species of fields considered. The degeneracy of states corresponds to the number of matter degrees of freedom that are entangled across the horizon <cit.>. All this implies that matter degrees of freedom might play an important role in the entropy computation, as for each and every state of the quantum geometry considered in the previous section there is a large degeneracy in the matter sector that has been neglected in the counting. Can one take this aspect into account in LQG? At first the question seems a difficult one because of the lack of a complete understanding of the matter sector in the theory. For instance, because λ in (<ref>) depends on the number of species one would seem to need a complete unified understanding of the matter sector to be able to begin answering this question. However, further analysis shows <cit.> that the discrete nature quantum geometry combined with the assumption of the regularity of the quantum state of matter fields across the horizon (embodied in the form of the degeneracy of states (<ref>) for an undetermined λ) plus the additional assumption of indistinguishability of puncture exitationsis sufficient to recover Bekenstein-Hawking entropy.In the treatments mentioned so far punctures were considered distinguishable [In the pure gravity U(1) Chern-Simons formulation the necessity of distinguishability of punctures follows from a technical point in the quantization <cit.>.]. Let us see here what indistinguishability would change. Instead of the microcanonical ensemble, we use now the grand canonical ensemble as this willconsiderably shorten the derivations(keep in mind that all ensembles are equivalent). Thus we start from the canonical partition function which for a systemof non interactive punctures is Q(β,N)=q(β)^N/N! where the N! in the denominator is the Gibbs factor that effectively enforces indistinguishability, and the one-puncture partition function q(β) is given byq(β)=∑_j=1/2^∞ d_j exp(-ħβγ_0/ℓ ),where d_j is the degeneracy of the spin j state (for instance d_j=(2j+1) as inthe SU(2) Chern-Simons treatment) and γ_0 is given by (<ref>). The grand canonical partition function is (β,z)=∑_N=1^∞z^Nq(β)^N/N!=exp(zq(β)).From the equations of state E=-∂_βlog(), and N=z∂_zlog() one gets a/8π G_N ℓ = -z∂_β q(β)N=z q(β)=log().In thermal equilibrium at the Unruh temperature one has β=2πℓħ^-1 and the ℓ dependence disappears from the previous equations. However, for d_j that grow at most polynomially in j, the BH area predicted by the equation is just Planckian and the number of punctures N of order one <cit.>. Therefore, indistinguishability with degeneracies d_j of the kind we find in the pure geometry models of Section <ref> is ruled out because it cannot acommodate BHs that are large in Planck units. If instead we assume that matter degrees of freedom contribute to the degeneracy factor then regularity of the quantum state of matter near the horizon takes the form (<ref>) which in the quantum geometry language translates into D[{s_j}]=∏_j d_j with d_j=exp(λ 8πγ_0). For simplicity lets take ≈ j+1/2<cit.>. We also introducetwo dimensionless variables δ_β and δ_h and write β=β_ U(1+δ_β)—where β_U=2πℓ/ħ—and λ=(1-δ_h)/4. A direct calculation of the geometric series that follows from (<ref>) yieldsq(β)=exp(-πγ_0δ(β))/exp(πγ_0δ(β))-1,where δ(β)=δ_h+δ_β. The equations of state (<ref>) now predict large semiclassicalBHs as follows: for large a/(ħ G_N) and by setting β=2πℓħ^-1 in (<ref>) one can determine δ_h≡δ(2πℓħ^-1) as a function of a and z. The result is δ_h=2√(G_N ħ z/(πγ_0 a))≪ 1. In other words semiclassical consistency implies that the additional degrees of freedom producing the degeneracy (<ref>) must saturate theholographic bound<cit.>, i.e. we getλ=1/4 up to quantum corrections. The entropy is given by the formula S=β E-log(z) N+log() which upon evaluation yieldsS=a/4G_Nħ - 1/2(log(z)-1) (z a/πγℓ_p^2)^1/2 This gives the Bekenstein-Hawking entropy to leading order plusquantum corrections. If one sets the chemical potential of the punctures to zero (as for photons or gravitons) then these corrections remain. One can get rid of the corrections by setting the chemical potentialμ=T_U. Such possibility is intriguing, yet the physical meaning of such a choice is not clear at this stage.The thermal state of the system is dominated by large spins as the mean spin ⟨ j⟩ = a/(Nℓ_p^2) grows like √(a/ℓ_p^2). The conclusions of this subsection hold for arbitrary puncture statistics. This is to be expected because the system behaves as if it were at a very high effective temperature(the Unruh temperature is the precise analog ofthe Hagedorn temperature <cit.> of particle physics). A similar result can be obtained by using Bosonic or Fermionic statistics for the punctures <cit.>. The leading term remains the same, only corrections change. In the case of Bosons the square root correction can be understood as coming from the Hardy-Ramanujan formula giving the asymptotic form of the number of partitions of an integer a in LQG Planck units <cit.>. §.§ Bosonic statistics and the correspondence with the continuum limit The partition function for Bosonic statistics and for z=1 is specially interesting because it produces an expression of the partition function that coincides with the formal continuum path integral partition function <cit.>. Explicitly, from (<ref>) and (<ref>) it follows that (β)=∏_j=1/2^∞∑_s_jexp(2πℓ-β)a_j/8πℓ G_N,where a_j=8πγℓ_p^2 are the area eigenvalues, and we have assumed for simplicity λ=1/4 in (<ref>), namely d_j=exp(a_j/(4G_Nħ)). There is a well known relationship between the statistical mechanical partition function and the Euclidean path integral on a flat background. One has thatZ_sc(β)=∫ Dϕexp{-S[ϕ]} where field configurations are taken to be periodic in Euclidean time with period β. Such expression can be formally extendedto the gravitational context at least in the treatment of stationary black holes. One starts from the formal analog of the previous expression and immediately uses the stationary phase approximation to make sense of it on the background of a stationary black hole.Namely Z_sc(β) = ∫ Dg Dϕexp{-S[g, ϕ]}≈ exp{-S[g_cl, 0]}∫ Dηexp[-∫ dx dy η(x) (δ^2 /δη(x)δη(y)) η(y)]where the first term depends entirely on the classical BH solution g_clwhile the second term represents the path integral over fluctuation fields, both of the metric as well as the matter, that we here schematically denote by η. For local field theories δ_η(x)δ_η(y)=δ(x,y) □_gc where □_gc is a the Laplace like operator(possible gauge symmetries, in particular diffeomorphisms must be gauge fixed to make sense of such formula). In the analytic continuation one sends the Killing parameter t→ -iτ and the space time tube-like region outside the horizon up to the local stationary observers at distance ℓ inherits a positive definite Euclidean metric (for rotating black holes this is true only to first order in ℓ<cit.>). The S^2× representing the black hole horizon shrinks down to a single S^2 in the euclidean and the time translation orbits become compact rotations around the Euclidean horizon with 0≤τ≤β.The tube-like region becomes D× S^2 where D is a disk in the plane orthogonal to and centred at the Euclidean horizon with proper radius 0≤ R≤ℓ (in the Euclidean case the BH horizon shrinks to a point, represented here by the center of D). Recall the Einstein-Hilbert actionS[g_cl, 0]=1/8π G_N ∫_D× S^2√(g)R + boundaryterms On shell the bulk term in the previous integral would seem at first to vanish. However, when β≠2πℓ, the geometry has a conical singularity at the centre of the disk and R contributes with a Dirac delta distribution multiplied by the factor (2πℓ-β). Using the Gibbons-Hawking prescriptionboundary terms <cit.> one can see that they cancel to leading order in ℓ. A direct calculation gives the semi-classical free energy-S[g_cl, 0]=log(Z_cl)=(2πℓ-β) a/8π G_Nℓ.Replacing (<ref>) in (<ref>) and comparing with the form the partition function (<ref>) we concludethatthe inclusion of the holographic degeneracy (<ref>) plus the assumption ofBosonic statistics for punctures makes the results of section <ref> compatible with the continuousformal treatment of the Euclidean path integral. Equation (<ref>) is thuscompatible with the continuum limit.§.§ Logarithmic corrections The equation of state E=-∂_βlog(Z_cl) reproduces the quasilocal energy (<ref>). The entropy is S=β E+log(Z)=A/(4ℓ_p^2) when evaluated at the inverse Unruh temperature β_U=2πℓ. Notice that in the quasi-local framework used here, entropy grows linearly with energy (instead of quadratically as in the usual Hawking treatment). This means that the usual ill behaviour of the canonical ensemble of the standardglobal formulation <cit.> is cured by the quasilocal treatment. Quantum corrections to the entropy come from the fluctuation factorwhich can formally be expressedin terms of the determinant of a second order local (elliptic) differential operator □_g_cl F =∫ Dηexp[-∫ dx η(x) □_g_clη(x)]=[(□_g_cl)]^-1/2.The determinant can be computed from the identity (the heat kernel expansion) log[(□_g_cl)]=∫_ϵ^2^∞ds/s Tr[exp(-s□_g_cl)],where ϵ is a UV cut-off needed to regularize the integral. We will assume here that it is proportional to ℓ_p. In the last equality we have used the heat kernel expansion in d dimensionsTr[exp(-s □_g_cl)]=(4π s)^-d/2∑_n=0^∞ a_n s^n/2,where the coefficients a_n are given by integrals in D× S^2 of local quantities.At first sight the terms a_n with n≤ 2 produce potential important corrections to BH entropy. All of these suffer from regularisation ambiguities with the exception of the term a_2 which leads to logarithmic corrections. Moreover, contributions coming from a_0 and a_1 can be shown to contribute to the renormalization of various couplings in the underlying Lagrangian <cit.>; for instance a_0 contributes to the cosmological constant renormalization.True loop corrections are then encoded in the logarithmic term a_2 and for that reason it has received great attention in the literature (see <cit.> and references therein). Another reason is that its form is regularisation independent.According to <cit.> there are no logarithmic corrections in the SU(2) pure geometricmodel once the appropriate smoothing is used (canonical ensemble). From this we conclude that the only possible source of logarithmic corrections in the SU(2) case must come from the non-geometric degrees of freedom that produce the so called matter degeneracy that plays a central role in Section <ref>. A possible way to compute these corrections is to compute the heat kernel coefficient a_2 for a given matter model. This is the approach taken in reference <cit.>.One can argue <cit.> that logarithmic corrections in the one-loop effective action are directlyreflected as logarithmic corrections in the LQGBH entropy.§.§ Holographic degeneracy from LQG The key assumption that led to the results ofSection <ref>was that matter degeneracy satisfies (<ref>) which was motivated by the regularity Hadamard condition on the vacuum state in the vicinity of the horizon. Can one actually predict such degeneracy directly from the fundamental nature of quantum geometry? Even when a complete model of matter at the Planck scale would seem necessary to answer this question there are indications that the fundamental structure of LQG might indeed allow for such degeneracy when coupling with matter fields. Notice that according to <ref> this question might be directly related to the question of the continuum limit <ref>. The exact holographic behaviour of the degeneracy of the area spectrum has been obtained from the analytic continuation of thedimension of the boundary Chern-Simons theory by sending the spins j_i→ i s-1/2with s∈^+<cit.>.The new continuous labels correspond to SU(1,1) unitary representations that solve the SL(2,) self(antiself)-duality constraintsL^i± K^i=0 (see <cit.>), which in addition comply with the necessary reality condition E· E≥ 0 for the fields E^a_i<cit.>. All this suggests that the holographic behaviour postulated in (<ref>) with λ=1/4 would naturally follow from the definition of LQG in terms of self(antiself)-dual variables, i.e. γ=± i.The same holographic behaviour of the number of degrees of freedom available at the horizon surface is found from a conformal field theoretical perspective for γ=± i<cit.>. A relationship between the termal nature of BH horizons and self dual variables seems also valid according to similar analytic continuation arguments <cit.>. The analytic continuation technique has also been applied in the context of lower dimensional BHs <cit.>.However, these results are at the moment only indications on an interesting behaviour. A clear understanding of the quantum theory in terms of complex Ashtekar variables is desirable on these grounds but unfortunately still missing.Recent investigation of the action of diffeomorphisms on boundaries <cit.> revealed the existence of potentially new degrees of freedom associated to broken residual diffeomorphism around the defects defined by the spin network punctures (as in Figure <ref>). The associated generators are shown to satisfy a Virasoro algebra with central charge c=3. Such CFT degrees of freedom could naturally account for the Bekenstein-Hawking area law and provide a microscopic explanation of (<ref>) . The central feature that makes this possible in principle is the fact that the central charge of the CFT describing boundary degrees of freedom is proportionaltothenumberofpuncturesthatitselfgrows withtheBHarea. Thisisafeaturethatresemblesin spirit previous descriptions <cit.>.However,an important advantage of the present treatment is the precise identification of the underlying microscopic degrees of freedom. Preliminary results (based on the use of the Cardy formula) indicate that the correct value of BH entropycould be obtained without the need of tuning the Immirzi parameter to any special value.It is worth mentioning here the approaches where a holographic degeneracy of the BH density of states arises naturally form symmetry considerations in transverse `r-t plane'of the near horizon geometry <cit.>. Aclear connection or synthesis between these seemingly dual ideas remains open (see <cit.> for a hint of a possible link).Finally, in the relatedgroup field theory approach (GFT) to quantum gravity <cit.> the continuum limit is approached via the notion of condensate states (Bosonic statistics plays here a central role <cit.>). The problem of calculating black hole entropy has been explored in <cit.>. §.§ Entanglement entropy perturbations and black hole entropy Starting from a pure state |0⟩⟨ 0| (“vacuum state") in QFT one can define a reduceddensity matrix ρ= Tr_in(|0⟩⟨ 0|) by taking the trace over the degrees of freedominside the BH horizon. The entanglement entropy is defined as S_ent[ρ]=- Tr(ρlog(ρ)). In four dimensions <cit.>the leading order term of entanglement entropy in standard QFT goes likeS_ent= λa/ϵ^2+correctionswhere ϵ is an UV cut-off, andλ is left undetermined in the standard QFT calculation due to UV divergences and associated ambiguities (recall discussion in Section <ref>).An important one is thatλ is proportional to the number of fields considered; this is known as thespecies problem.These ambiguities seem to disappear if one studies perturbations of (<ref>)when gravitational effects are taken into account <cit.>. The analysis is done in the context of perturbations of the vacuum state in Minkowski spacetime as seen by accelerated Rindler observers. Entanglement entropy is defined by tracing out degrees of freedom outside the Rindler wedge. Such system reflects some of the physics of stationary black holes in the infinite area limit.A key property <cit.> is that,ρ=exp (-2π∫_ΣT̂_μνχ^μ dΣ^ν)/ Tr[exp (-2π∫_ΣT̂_μνχ^μ dΣ^ν)],where Σ is any Cauchy surface of the Rindler wedge. If one considers a perturbation of the vaccum state δρ then the first interesting fact is that the (relative entropy) δ S_ent=S_ent[ρ+δρ]-S_ent[ρ] is UV finite and hence free of regularization ambiguities <cit.>.The secondproperty that follows formally (see below) from (<ref>) is thatδ S_ent=2π Tr(∫_Σδ⟨ T_μν⟩χ^μ dΣ^ν). Now from semiclassical Eintein's equations ∇^μδ⟨ T_μν⟩=0, this (together with the global properties of the Rindler wedge) implies that one can replace the Cauchy surface Σ by the Rindler horizon H in theprevious equation. As in the calculation leading to (<ref>) one can use the Raychaudhuri equation (i.e. semiclassical Eintein's equations)to relate the flux of δ⟨ T_μν⟩ across the Rindler horizon to changes in its area. The result is that δ S_ent=δ A/4G_Nħ independently of the number of species. The argument can be generalized to static black holes <cit.> where a preferred vacuum state exists (the Hartle-Hawking state). In this case the perturbation can send energy out to infinity as well, and the resulting balance equation is δ S_ent=δ a/4G_Nħ+δ S_∞,where δ S_∞=δ E/T_H, and δ E is the energy flow at ^+∪ i^+. Changes of entanglement entropy match changes of Hawking entropy plus an entropy flow to infinity. These results shed light on the way the species problem could be resolved in quantum gravity. The key point being that a is dynamical in gravity and thus grows with the number of gravitating fields. However,as the concept of relative entropy used here is insensitive to the UV degrees of freedom, one key questionis whether the present idea can be extrapolated to the Planck scale (for some results in this direction see <cit.>). Another important limitation of the previous analysis is that property (<ref>) is only valid in a very restrictive sense (see for example <cit.>). Indeed, as shown in <cit.> generic variations involving for instance coherent states will violate (<ref>). Thus this remains a formal remark pointing in an interesting direction that deserves further attention. §.§ Entanglement entropy vs. statistical mechanical entropy One can argue that the perspective that BH entropy should be accounted for in terms of entanglement entropy<cit.>(for a review see <cit.>)and the statistical mechanical derivation presented sofar are indeedequivalent in a suitable sense <cit.>.The basic reason for such equivalence resides in the microscopic structure predicted by LQG <cit.>. In our context, the appearance of the UV divergence in (<ref>) tells us that the leading contribution to S_ent must come from the UV structure of LQG close to the boundary separating the two regions.Consider a basisof the subspace of the horizon Hilbert space characterised by condition (<ref>), and assume the discrete index a labels the elements of its basis.Consider thestate |Ψ⟩=∑_aα_ a|ψ^a_int⟩|ψ^a_ext⟩, where |ψ^a_int⟩ and |ψ^a_ext⟩denote physical states compatible with the IH boundary data a, and describingthe interior and the exterior stateof matter and geometry of the BH respectively. The assumption that such states exist is a basic input of Section <ref>.In the form of theequation above we are assuming that correlationsbetween the outside and the inside at Planckian scalesaremediated by the spin-network links puncturing the separating boundary.This encodes the idea that vacuum correlations are ultra-local at the Planck scale. The proper understanding of the solutions of (<ref>) might reveal a richer entanglement across the horizon (the exploration of this important question is underway <cit.>).This assumption is implicit in the recent treatments <cit.> based on the analysis of a single quantum of area correlation and it is related to the (Planckian) Hadamard condition as defined in <cit.>.We also assume states to be normalized as follows: ⟨ψ_ext^a|ψ_ext^a⟩=1, ⟨ψ_int^a|ψ_int^a⟩=1, and⟨Ψ|Ψ⟩=1. The reduced density matrix obtain from the pure state by tracing over the interior observables isρ_ext = ∑_a p_a |ψ^a_ext⟩⟨ψ^a_ext|, with p_a=|α_a|^2. It follows from this that the entropy S_ext≡ - Tr[ρ_extlog(ρ_ext)] isbounded by micro-canonical entropy of the ensemble as discussed in Section <ref>.If instead one starts from a mixed state encoding an homogeneous statistical mixture of quantum states compatible with(<ref>), then the reduced density matrix leads to an entropy that matches the microcanonical one <cit.>. Such equipartition of probability is a standard assumption in the statistical mechanical description of standard systems in equilibrium. It is interesting to contemplate the possibility of arguing for its validity from a more fundamental level using the ideas of typicality <cit.>.§.§ Hawking radiation The derivation of Hawking radiation from first principles in LQG remains an open problem, this is partly due to the difficulty associated with the definition of semiclassical states approximating space-time backgrounds. Only heuristic models based on simple analogies exist at the moment<cit.>.Without a detailed account of the emission process it is still possible to obtain information from a spectroscopical approach (first applied to BHs in <cit.>) that uses as an input the details of the area spectrum in addition to some semiclassical assumptions <cit.>.The status of the question has improved with the definition and quantisation of spherical symmetric models <cit.>. The approach resembles the hybrid quantisation techniques used in loop quantum cosmology <cit.>. More precisely, the quantum spherical background space-time is defined using LQG techniques, whereas perturbations, accounting for Hawking radiation, are described by a quantum test field (defined by means of a Fock Hilbert space). A fundamental microscopic account of the evaporation in detail would require dynamical considerations where the solutions of (<ref>) describing a semiclassical BH state will have to enter. For an attempt to include dynamics in the present framework see <cit.>; and <cit.> for a related model of evaporation. § INSIGHTS INTO THE HARD PROBLEM: BLACK HOLE QUANTUM DYNAMICS§.§ The information loss problemClassically black holes are causal sinks. According to classical general relativity anything crossing a BH event horizon—Figure <ref>, notice light cone structure—is constrained by the causal structure of spacetime to end up at the singularity, which in the classical theory is regarded as an endpoint of spacetime. When quantum effects are considered the situation changes as black holes evaporate through Hawking radiation. On the one hand, the classical notion of singularity is expected to be described by Planckian scale new physics and so the question of what is the dynamical fate of what falls into this region is expected to have a well defined description. On the other hand, this high curvature region, initially hidden by the BH event horizon, could become eventually visible to outside observersat the end of evaporation or remain forever causally disconnected from the outside region. As these questions concern the physics in the Planckian regime, these questions can only be settled in the framework of a full theory of quantum gravity. Semiclassical considerations meet their end. According to the Hawking effect a black hole in isolation slowly lowers its initial mass Mby the emission of radiation which is very well approximated by thermal radiation (at least while the curvature at the horizon is far from Planckian, or equivalently, while the mass of the BH is much larger than m_p).During this semiclassical-era of evaporation—which lasts an extremely long time τ_evap≈ M^3 in Planck units [In more graphical termsτ_evap≈ 10^54M^3/M^3_⊙τ_univ.where M_⊙ is the solar mass and τ_univ is the age of the universe.]—the black hole also works as an information sink. According to classical gravity and quantum field theory on curved spacetimes, whatever falls into the BH becomes causally disconnected from the outside at least for times τ≤τ_evap. Evolution of the quantum state of matter fields from one instant defined by a Cauchy surface Σ (see Figure <ref>) to another defined by a Cauchy surface Σ^' in its future (embedded in the lower than Planckian curvature region) is unitary. For the spacetime regions which can be well approximated by classical gravity there is part of the Cauchy surfaces Σ^' that is trapped inside the BH and whose future is the classical singularity. As Σ^' is pushed towards the future, the portion inside the BH grows (for instance in terms of its volume) as it approaches more and more the singularity (see dotted portion of Σ^' in Figure <ref>). It is easier to emphasize one aspect of the information lost paradox by concentrating only on the Hawking radiation produced by the BH during its history, and thus neglecting for simplicity of the analysis all the other things that have fell into the BH during its long life (in particular those that led to its formation in the first place). Hawking particles are created by the gravitational tidal interaction of the BH geometry with the vacuum state |0⟩ of matter fields. This state can be expressed as apure state density matrix |0⟩⟨ 0|. It can be precisely shown that when a particle is created by this interaction and send out to the outside, another correlated excitation falls into the singularity <cit.>. It is because these correlated excitations that fell into the BH cannot affect any local experiment outside that an outside observer can trace them out and in this way get a mixed state ρ_T_ BH= Tr_ BH[|0⟩⟨ 0|]. The previous mixed state is thermal state with Hawking temperature T_ BH to an extremely good accuracy while the BH is large in Planck units (the trace is taken at a given instant defined by a Cauchy surface Σ). Yet the overall state of matter, when we do not ignore the fallen correlated excitations, is a pure state! The question of the fate of information in BH physics can then be stated in terms of the question of whether the quantum state of the system after complete evaporation of the BH is again a pure state (the initially lost correlations emerge somehow from the ashes of the end result of evaporation) or the state remains mixed and the information carried by the excitations that fell into the BH are forever lost. If the second scenario is realized then the unitarity of the description of the gravitational collapse and subsequent evaporation would be compromised. This in itself would seem to have very little practical importance for local physics considering the time scales involved for BHs with macroscopic masses. But it could lead to an in principle detectable phenomenology in contexts where small BHs could be created in large numbers classically (such as for primordial BHs in cosmology) or by quantum fluctuations (yet in this case deviations from the semiclassical expectation could become important). Finally, the consideration of such purely theoretical questions can lead to new perspectives possibly useful for understanding the theory we seek.How much information has fallen into the BH at the end of evaporation? The answer really depends on the entire history of the BH. As an extreme example one could think of feeding a BH with matter continuously to compensate for the energy loss via Hawking radiation. In this way a BH can have a lifetime as large as wanted and thus can swallow an unlimited amount of information (independently of its apparent size as seen from the outside). Thus the answer would beinfinite in this case. One could think of the opposite scenario where the BH is created quickly by gravitational collapse and left unperturbed in isolation until it completely evaporates. In this case we would expect that it has absorbed at least all the information that would be necessary to purify the hawking radiation that has been emitted during its evaporation process. Assuming that this process is close to stationary for the most part of the history of the BH and using the generalized second law of thermodynamics one expects the information lost to be of the order of the initial Bekenstein-Hawking entropyS_ BH = a_0/(4ℓ_p^2). The volume inside the BH right before hitting the singularity is huge when calculated at an ultimate instant defined by a constant curvatureslice before the radius of curvature becomes Planckian inside the BH. At that instant the area of the BH as seen from the outside is Planckian. Nevertheless its internal volume defined by this spacelike slice can be as large as the the volume of a ball with a radius R≈ 10^6 × (M/M_⊙) R_ univ where R_ univ denotes the radius of the observable universe (for discussion of the volume inside a BH see <cit.>). Such trapped volume is not bounded in any way by the area of the BH, for instance it can be made as big as desired by feeding the black hole with matter to compensate for its evaporation as in the situation evoked before where an unlimited amount of information would be absorbed by the BH. This is notsurprising as in a curved geometry the volume of a region is not bounded in any way by the apparent size of the region from the outside (the black hole area for instance); in the present case this corresponds to Wheelersbag of gold scenario <cit.>. Hawking 1976 <cit.> formulation of the information paradox can be stated in the questions: is the information falling into the BH region forever lost?or can it be recovered at the end of BH evaporation? It is clear that the answer to these questions is tight to the fate of the causal structure of spacetime across the BH singularity, and is therefore a quantum gravity question.These questions will only be clarified when a solid understanding of the Planckian dynamics becomesavailable. The central interest of Hawking's information paradox is the theoretical challenge it represents; it tells us that one cannot ignore the physics of the singularity.The following scenarios represent some of the main ideas that have been put forward during the last 4 decades:*Black holes are information sinks: A simple possibility is that even when the singularity is replaced by its consistent Planckian description one finds that the excitations that are correlated with the outside can never interact again with it and remain in some quantum gravity sense forever causally disconnected from the outside.There are two possibilities evoked in the literature: The first possibility is that lost information could end entangled in a pre-geometric quantum substrate (where large quantum fluctuations <cit.> prevent any description in terms of geometry); this would be described as a singularity from the point of view of spacetime physics in which case the place where informations ends could be seen as a boundary of spacetime description <cit.>. The second is that to the future of the singularity (a region of large quantum fluctuations at the Planck scale) a new spacetime description becomes available but that the newly born spacetime regions remain causally disconnected from the BH outside: a baby universe <cit.>. *Information is stored in a long-lasting remnant:A concrete proposal consists of assuming that aremnant of a mass of the order of Planck mass at the end of the Hawking evaporation can carry the missing information <cit.>. As the final phase of evaporation lies outside of the regime of applicability of the semiclassical analysis such hypothesis is in principle possible.Notice that this might be indistinguishable from the outside from the baby universe possibility if no information is allowed to come out of this remnant.The Planckian size remnant willlook as a point-like particle to outside observers.In order to purify the state of fields in the future,the remnant must have a huge number of internal degrees of freedom which correlate with those of the radiation emitted during evaporation in addition to those related to the formation history of the BH. If one traces out these degrees of freedom one has a mixed state that represents well the physics of local observers in the future right before the end of evaporation. The entropy of such mixed state is expected to be at least as big as the one of the initial black hole S_BH(M) before Hawking evaporation starts being important. The value of S_BH(M) is a lower bound of the number of such internal states, which, as pointed out above, it can be virtually infinite depending of the past history of the BH. If this particle like object admits a description in terms of an effective field theory (which in itself is not so clear <cit.>) this huge internal degeneracy would lead to an (unobserved) very large pair production rate in standard particle physics situations. One can contemplate the possibility that these remnants could decay via emission of soft photons carrying the missing information back to the outside. However,as the energy available for this is of the order of M_p,remnants would have to be very long lived (with lifetimes of the order of M^4/M_p^4 t_p<cit.>) in order to evacuate all the internal information in electromagnetic, gravitational or any other field-like radiation. Hence, they would basically behave as stable particles and one would run into the previous difficulties with large pair creation rates. The possibility that such remnantscan lead to finite rate production despite of the large degeneracy of their spectrum has been suggested <cit.>. More discussion of remnants and references see <cit.>.Some aspects of the previoustwo scenarios is illustrated in Figure <ref>. *Information is recovered in Hawking radiation:Another proposed scenario for purification of the final state of black hole evaporation consist of postulating that information comes out with the Hawking radiation. This view has been advocated by't Hooft in <cit.> and raised to a postulate by Susskind et al. <cit.> where one declares that “there exists a unitary S-matrix which describes the evolution from infalling matter to outgoing Hawking-like radiation”. See also Page <cit.>. Such view cannot accommodate with the spacetime causal structure representing the BH within the framework of quantum field theory on curved spacetimes (see light cones in Figure <ref>). New physics at low energy scales is invoked to justify that information of the in-falling modes is somehow `registered' at the BH horizon and sent back out to infinity. More precisely, if standard QFT on a curved space-time is assumed to be a valid approximation when the curvature around the black hole horizon is low (for large BHs) then no information on the in-falling modes can leak out the horizon without violating causality. Yet, as argued by Page <cit.>, in order for unitarity to hold such leaking of information must be important when the BH is still large and semiclassical (at Page time corresponding to the time when the BH has evaporated about half of its initial area a). Some peculiar quantum gravity effect must take place at the BH horizon. Further tensions arise when trying to describe the physics from the point of view of freely falling observers who (according to the equivalence principle)should not feel anything special when crossing the BH horizon. In particular, they must find all the information that fell through the horizon right inside. Thus, there is a troublesome doubling of informationin this scenario:a so-called principle of complementarity isevoked in trying to address these issues <cit.>. Hence, the above postulate implies thatquantum gravity effects would be important where they are not expected to be opening the door for paradoxical situations with theoretically convoluted proposed resolutions.The existence of such puzzlinglarge quantum gravity effects in the present scenario was made manifest by the analysis of <cit.> where it is explicitlyshown that (assuming the validity of semiclassical QFT in the vicinity of a large BH) one cannot have information go out of a large BH and across its horizon without a catastrophic violation of the equivalence principle at the BH horizon!A cartoon description of the phenomenon can be given with the help of Fig. <ref> as follows. According to the formalism of QFT on curved space-times the UV structure of the two-point correlation functions is universal for well behaved (Hadamard) states. Physically, this implies that the state of fields looks like`vacuum' to freely falling observers crossing the horizon with detectors sensitive to wavelengths much shorter that the BH size. In the context of the Hawking effect this impliesthat a pair of particles a and b created at the horizon by the interaction of the field with the background geometry must be maximally correlated <cit.>.The statement that the final state of the Hawking radiation is pure (and thus that information has sneaked out of the horizon during the evaporation process) necessitates the existence of non trivial correlations between the early radiation (particle c in Fig. <ref>)and late radiation (particle b) [The analogy with the cooling of a hot body in standard situations is often drawn. An initially hot body can be described by a pure state. While it cools down it emits radiation that looks close to thermal at any particular time. Unitarity implies (among other things) that the early radiation be correlated with the late one to maintain the purity of the state. Such view is misleading when applied to BHs because it disregards the causal structure.]. But correlations between c and b are forbidden by the fact that a and b are maximally correlated.This is due to the so-called monogamy of entanglement in quantum mechanics <cit.>. Relaxing correlations between a and bimplies a deviation from the notion of `vacuum' at the point where a and b are created. This perturbation leads to a divergence of the energy-momentum tensorat the horizon in the past due to its unlimited blue-shift along the horizon towards the past: a `firewall' detectable by freely falling observers. If one is not ready to accept such flagrant violation of the equivalence principle one must admit the inviability of the complementarity scenario.*Information is degraded via decoherence with Planckian degrees of freedom:This is a natural proposal where the fundamental discreteness of quantum geometry at the Planck scales plays a central role in understanding the puzzle of information. The information loss is viewed as a simple phenomenon of decoherence with the quantum gravity substratum reflected in an increase of the von Neumann entropy describing the state of Hawking radiation. This perspective puts in equal footingthe apparent lost of information in the BH context with the degrading of information taking place in the more familiar situations described by standard thermodynamics and captured by the second law.The second law of thermodynamics is not a fundamental principle in physics but rather a statement about the (illusory) apparent asymmetry of time evolution when sufficiently complicated systems are put in special initial conditions and later described statistically in terms of coarse physical variables that are unable to discern all the details of the fundamental system. The idea is easily illustrated in classical mechanics. On the one hand, Liouville's theorem implies that the support of the phase space distribution of the system spans a volume that is time independent; on the other hand, the shape of the support is not restricted by the theorem. An initially simple distribution supported in a ball in Γ will (in suitably complicated systems) evolve into a more and more intricate shape whose apparent phase space volume, when measured with a devise of resolution lower than that of the details of the actual distribution, will grow with time. In a practical sense, the second law implies that information is degraded (yet not lost) in time when encoded incoarse variables. The words in a newspaper are gone when the newspaper is burned but the information they carry continue to be encoded in the correlation among microscopic molecular degrees of freedom that become unavailable in practice. At the quantum level information can be degraded in addition due to decoherence through the entanglement with degrees of freedom that are not accessible to the observer <cit.>. In fact this view leads to a beautiful statement of the foundations of statistical mechanics and sets the fundamental basis for thermodynamics <cit.>. It is the view of the author that this perspective offers the possibility of a simple solution of the information loss paradox in the context of a quantum gravity theory where spacetime geometry is granular or discrete at the fundamental level <cit.>. We have seen that BHs behave like thermodynamical systems. The validity of the laws of BH mechanics and their strict relationship with thermodynamics points to an underlying fundamental description where spacetime is made of discrete granular structures. Without adhering to any particular approach to quantum gravity, the solid theoretical evidence coming from general relativity and quantum field theory on curved spacetimes strongly suggest that BH Horizons are made of Planckian size building blocks: they carry and entropy given by A/4 in Planck units and satisfy the generalized second law (total entropy of matter plus BH entropy can only increase). In the framework of LQG we have seen in Section <ref> that it is precisely the huge multiplicity microscospic quantum states of the BH geometry that can account for its thermal properties. Such microscopic degeneracy of states is also expected in the description of the continuum limit in LQG as argued in Section <ref>.If these expectations are correct then the information puzzle must be understood in terms that are basically equivalent to those valid in familiar situations. Information is not lost in BH evaporation but degraded in correlations with these underlying `atoms' of geometry at Planck scale. In this scenario BH evaporation is represented by Figure <ref> and will be presented in more detail in Section <ref>.§.§ Information loss resolutions suggested by LQG The means for the resolution of the information puzzle, advocated here, can be formulated in the context of the scenario proposed by Ashtekar and Bojowald (AB) in <cit.>.The central idea in the latter paper is that the key to the puzzle of information resides in understanding the fate of the classicalwould-be-singularity in quantum gravity. This view has enjoyed from a steady consensus in the non perturbative quantum gravity literature <cit.>.The scenario was initially motivated by the observed validity of the unitary evolution across the initial big-bang singularity in symmetry reduced models in the context ofloop quantum cosmology<cit.> (see <cit.> and references therein for a modern account). Similar singularity avoidance results due to the underlying discreteness of LQGhave been reportedrecently in the context of spherically symmetric black hole models <cit.> (see also <cit.>). The consistency of the AB paradigm is supported by the analysis of <cit.> in two dimensional CGHS black holes <cit.> where still some assumptions about the validity of quantum dynamics across the singularity are made.Numerical investigations of the CGHS model in the mean field approximation <cit.> strongly suggest the global causal picture proposed in the AB paradigm as well.The spacetime of the AB framework is represented in Figure <ref>. Hypothetical observers falling into the black hole unavoidably meet thewould be singularity. Quantum gravity evolution in the Planckian region takes us across the singularity to the future where the BH has evaporated.In the AB proposal the space-time may rapidly become semiclassical so that our test observer emerges into a flat space-time future above thewould-be-singularity where spacetime is close to a flat spacetime (because all of the mass sourcing gravity has been radiated away to infinity via Hawking radiation).The region where it emerges is in causal contact with the outside. From the view point of an external observer the black hole slowly evaporates until it becomes Planckian. At this final stage the semiclassical approximation fails, curvature at the black hole horizon becomes Planckian andexternal observers become sensitive to the strong quantum gravitational effects which are responsible for the resolutions of the classical would-be-singularity. In practical terms this means that external observers become in causal contact with the strong quantum gravity region and the singularity becomes naked for them.In this framework there is a natural resolution of the question of the fate of information. The full quantum dynamics is unitary when evolving from Σ to Σ^' (see Figure <ref>). The correlations in a Hawking pair a-b created in the vicinity of the BH horizon (Figure <ref>) are maintained by the evolution. The field excitation a falls into the Planckian region where it interacts with the fundamental discrete spacetime foam structure and gets imprinted into the Planckian fabric in what we call a̅. Correlations that make the state pure are not lost, at the end of evaporation the quanta b in the Hawking radiation are entangled with Planckian degrees of freedom a̅ which cannot be encoded in a smooth description of the late physics. These degrees of freedom are associated with a large degeneracy of the flat Minkowski spacetime expects to arise from the continuum limit of LQG via coarse graining: a̅ is a defect in the fundamental structure not detectable for the low energy probes for whom the spacetime seems smooth. The granular structure predicted by LQG can realise the idea of decoherence without dissipation evoked in <cit.>. The account of the fate of information in the context of BHs evaporation would, in this scenario, be very similar to what one believes it happens when burning the newspaper.After burning, the articles in the newspaper remains written in the correlations of the gas molecules diffusing in the atmosphere. After evaporation the information initially available for low energy probes in the initial data that lead to the gravitational collapse is encoded in the correlations with Planckian degrees of freedom which are harder to access. Information gets degraded but not lost: the `fire' of the singularity is a place where the initially low energy smooth physics excitations are forced, by the gravitational collapse, to interact with the Planckian fabric where a new variety of degrees of freedom are exited. The viewpoint developed in considering the question of information in quantum gravity leads to some phenomenological proposals that we briefly describe in what follows. § DISCRETENESS AND LORENTZ INVARIANCE A central assumption behind all the results and perspectives discussed in this article is the compatibility of the prediction of loop quantum gravity of a fundamental discreteness of quantum geometry at the Planck scale with the continuum description of general relativity. As emphasized before the problem of the continuum limit of LQG remains to a large extend openpartlydue to the technical difficulties in reconstructing the continuum from the purelycombinatorial structures of quantum geometry, but also due to the difficulties associated with the description of dynamics in the framework (spacetime is a dynamical question involving the solutions of (<ref>)). A problem that immediately comes to mind is the apparent tension between discreteness and the Lorentz invariance (LI) of the continuum low energy description. Is the notion of a minimum length compatible with Lorentz invariance? The apparent tension was initially taken as an opportunity for quantum gravity phenomenology as such a conflict would immediately lead to observable effects (see <cit.> and references therein). Given the lack of clear understanding of the continuum limit,it was initially assumed that the discreteness of quantum gravity would select a preferred rest framebreaking LI`only' at the deep Planckian regime. However, it was later shown <cit.> that such naive violation of LI would not be compatible with standard QFT at familiar energy scales: violations of LI at Planck scale would generically get amplified via radiative corrections and thus `percolate' from the Planck scale to low energy scales producing effects that would be of the same order of magnitude as the phenomenology predicted by the standard model of particle physics. This is in sharp conflict with observations.These results indicate that discreteness in quantum gravity does not admit a naive interpretation as some granular structure similar to molecules or atoms in a lattice. Compatibility with LI requires a more subtle relation expected to be clarified via the precise understanding of the continuum limit and the solutions of the scalar constraint (<ref>) (the quantum nature of such discreteness is probably one aspect of its elusive nature <cit.>). The key point seems dynamical physical discreteness should be associated to gauge invariant quantities commuting with the scalar constraint. Let us illustrate this with the simpler case of the area operator which is not a gauge invariant observable (a Dirac observable) unless further structure is provided.In the case of the BH models considered here the area of the event horizon is a gauge invariant notion (thanks to the restrictions imposed by the isolated horizon boundary condition) and its discreteness is justified. Notice also that only normal Lorentz transformations preserve the boundary condition and for such the area is an invariant notion. The apparent tension of the discreteness as predicted by our calculations in Section <ref> in view of the expected LI at low energiescan be attenuated with general dynamical considerations as well. Unfortunately, unlike the argument for BHs we just gave, in the general situation the discussion will remain at a more heuristic level until more control on the dynamical question is gained. However, we can be precise if we use the concrete scenario provided by models where time-reparametrization invariance (the gauge symmetry associated to (<ref>)) is eliminated by the use dust or other suitable (massive) matter degrees of freedom as a physical gauge fixing <cit.>.Discreteness of geometry at the Planck scale realizes in relational observables <cit.> like area and volume of regions in the rest frame of matter degrees of freedom. Compatibility, with Lorentz invariance comes from the fact that the discreteness of geometric observables is associated with such preferred `observers' selected by the gauge fixingdegrees of freedom. In these models the Planck length enters in a way that is similar to the mass m of a field in relativistic field theory. The presence of a scale does not break Lorentz symmetry because the meaning of m is that of the rest-mass of the associated particle (it means a definite scale in a special reference frame). Similarly, the discreteness of geometry in the deparametrized context has a meaning in a reference frame determined by the physical degrees of freedom.These models are simplistic in that the matter `rulers' that provide the gauge fixing that eliminates ((<ref>)) are not properly quantized but it illustrates clearly the way in which the apparent tension between discreteness and Lorentz-FitzGerald contraction could be resolved. Waiting for a more detailed understanding we also mention that, in the context of applications to black holes, discreteness of the geometry of null surfaces (themselves a LI object) is the key feature behind the results we have discussed. Another important point concerning BHs is that the results presented here wouldall be preserved (only with a possible modification of the the value of γ_0 in (<ref>)) as long as a non trivial area gap remains (the area spectrum can be continuous as long a there is a minimum non-vanishing area eigenstate <cit.>). In the context of spin foams <cit.> (which provides the framework for understanding the continuum limit dynamically) there are indications that the area gap is a LI feature of quantum geometry [There is for instance the presence of the area gap in the covariantly derived area operator <cit.>, and the persistence of the gap in the LI definition of the area operator in self dual variables <cit.>. See also <cit.> for a general discussion on dynamics versus discreteness.]. Although in such case the physical interpretation in the previous terms seems elusive. §.§ PhenomenologyThe discussion of black hole issues in quantum gravity suggests interesting avenues for phenomenology based on the possible observational implications of Planckian discreteness. Some years ago there was an initial surge of interest in quantum gravitational effects associated with violations of Lorentz invariance mediated by effects associated with a preferred frame where discreteness would realize <cit.>.However, ithas by nowbecomequite clearthat this ideafacessevereproblems. From thedirectobservational side one can concludethat, ifthat kind ofeffects existat all,theymustbe far more suppressed than initiallyexpected <cit.>.From the theoretical side these effects are forbidden by the no-go argument of <cit.>. However,thisresult has created a great puzzle:in what wayspace-time Planckianscalediscreteness (as predicted by LQG) could actually be realized in consistency with the observed Lorentz invariance? Reference <cit.> rules out thedirectandnaive atomistic view of a spacetime made of pieces stuck together in some sort of space-lattice, but it does not offeraclearanswer tothequestion. The answer must come from the dynamical understanding of the theory (the solution of (<ref>)) and the construction of physical observables. A related idea that avoids this no-go argument is that spacetime discreteness can lead to violations of conservation of the energy momentum of matter fields when idealized as propagating in the continuum (no violation of lorentz invariance in the previous naive sense is necessary; see for instance <cit.>). The idea is that discreteness would naturally lead to “energy diffusion” form the low energy field theoretical degrees of freedom to the micro-Planckian structure of spacetime.Such diffusion is generically unavoidable if the decoherence scenario evoked in Section <ref> is realized.Therefore, this phenomenological idea is partly motivated by our considerations of the information puzzle in BH evaporation.Violations of energy momentum conservation are incompatible with Einstein's equations; however, in the context of cosmology, unimodular gravity can be shown to be a good effective description of violations that respect the cosmological principle <cit.>. In that case, the effect of the energy leakage is the appearance of a term in Einstein's equations satisfying the dark energy equation of state with contributions that are of the order of magnitude of the observed cosmological constant.Dimensional analysis, together with the natural hypothesis that Planckian discreteness would primarily manifestin interactions with massive matter (see Section <ref>) in a way that is best captured by the scalar curvature R (vanishing via Eintein's equation for conformally invariant massless matter),lead to the emergence of a cosmological constant in agreement with observations without fine tuning <cit.>.These results are new and poorly understood from the perspective of a fundamental theory of quantum gravity. Nevertheless, they are encouraging and present a fresh view on the dark energy problem that seems promising.Finally, another phenomenological aspect that follows from the discussion of BHs in LQGis the suggestion that quantum effects in gravitational collapse might be stronger than those predicted by the semiclassical framework that leads to Hawking evaporation. These hypothetical strong quantum gravity effects would be important in regions of low curvatures near the event horizon and could actually dominate at some stage of the black hole collapse. The models are motivated by heuristic considerations based on bouncing cosmologies in LQG <cit.> and later refined in <cit.>. The initially proposed spacetimes suffer of certain instabilities <cit.>. In these scenarios black holes would explode in time scales of order M^2 (in Planck units) <cit.>and, as argued, they might lead to precise observational signatures <cit.>. § FUTURE DIRECTIONS AND DISCUSSIONAt present there is no complete understanding of that unified framework of quantum mechanics and gravity that wecall quantum gravity. Several theoretical approaches exist with their advantages and disadvantages depending on the judgement of what physical phenomena are the most relevant guiding principles. Loop quantum gravity is not an exception to such assertion. Important implications of the formalism remain unclear such as the (dynamical) question of the continuum limit or that of the nature of matter at the fundamental scale. In such context, black hole physics is a challenge and an opportunitywhere phenomenology, firmly rooted in predictions of general relativity and quantum field theory on curved spacetimes, guides our steps for the construction of a consistent theory. In this sense Black holes are cosmicmicroscopes of the fundamental structure of space and time. They hide the key for solving the puzzle of quantum gravity.In this article we have reviewed the main achievements of the formalism of loop quantum gravity when applied to black holes. The central featurebehind all these results is the discreteness of geometry at the Planck scale that follows directly (as explained in Section <ref>) from the canonical uncertainty relations of gravity in the first order variables. We argued that there is a finite dimensional ensemble of possible gravitational actions in these variables, Section <ref>, and that the Immirzi parameter arises from the associated coupling constants.In the quantum theory the Poisson non-commutativity of geometric variables implies the discreteness of area and volume whose eigenvalues are modulated by the Immirzi parameter γ (see for instance equation (<ref>)). The parameter γ is thought of as labelling inequivalent quantizations.We have seen that the approach succeeds in explaining the proportionality of black hole entropy with its horizon area without the need of invokingholographic ideas at the fundamental level. Consistency with the low energy semiclassical limit requires a very definite value of the proportionality constant between area and entropy. Two competing perspectives coexist at present. On the one hand, there is the view (motivated by the formalism of quantum isolated horizons) that only geometry degeneracy must contribute to the entropy, Section <ref>. In this case semiclassical consistency is achieved by fixing the value of the Immirzi parameter as in equation (<ref>). On the other hand,if matter degrees of freedom are taken into account and punctures are assumed to be indistinguishable, we have seen, Section <ref>, that it is possible to achieve semiclassical consistency for arbitrary values of the Immirzi parameter (γ only appears in subheading quantum corrections to the entropy). Moreover, if in addition Boson statistics is postulated for punctures then correspondence with the continuum limit holds; Section <ref>. At present there is no consensus on which of the previous alternative views is more appropriate. The second perspective is more challenging as it demands deeper understanding of the of the nature of matter degrees of freedom at the Planck scale. This is a difficult yet potentially promising direction where the properties of black holes can teach us about some aspects of matter coupling of LQG at high energies.In Section <ref> we mention some ideas which can be considered first steps in this direction. We have seen in this article that black holes are modelled in terms of boundaries and the imposition of boundary conditions at the classical level. This approach is natural in the semiclassical context where black holes are large in Planck units and thus radiate so little thatcan be idealised as stationary.Quantum aspects are explored via canonical quantization of the phase space to general relativity restricted by these boundary conditions.In the dynamical regime black holes are more elusive notions. Indeed it is likely that the very notion of black hole (as a trapped region) makes no sense in the full quantum gravity regime (recall discussion in Section <ref>). We have also discussed howsome of the most puzzling issues such as the emergence of the Lorentz invariant continuum, or the fate of information in gravitational collapse requires the full dynamical description of the evaporation process and, what classically would be regarded as, the BH singularity. At present one can argue for possible scenarios on the basis of general features such as discreteness of geometry at the Planck scale. However, the precise treatment ofthese hard questions necessitate full control of the quantum theory and its dynamics at the Planck scale. There is active research on basically two fronts trying to address the dynamical question: the spin foam approach towards the path integral representation of LQG <cit.>, and the canonical Dirac program of regularisation and quantisation of the quantum Einsteins equations <cit.>. 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"authors": [
"Alejandro Perez"
],
"categories": [
"gr-qc",
"hep-ph",
"hep-th"
],
"primary_category": "gr-qc",
"published": "20170327153815",
"title": "Black Holes in Loop Quantum Gravity"
} |
Hadron detection with a dual-readout fiber calorimeter J. Park^m, F. Scuri^b, A. Sill^a and R. Wigmans^a,Corres December 30, 2023 ============================================================Dina Tavares^* ESECS, Polytechnic Institute of Leiria, 2411–901 Leiria, Portugal andCenter for Research and Development in Mathematics and Applications (CIDMA) Department of Mathematics, University of Aveiro, 3810–193 Aveiro, PortugalRicardo Almeida and Delfim F. M. Torres Center for Research and Development in Mathematics and Applications (CIDMA) Department of Mathematics, University of Aveiro, 3810–193 Aveiro, Portugal We study fractional variational problemsof Herglotz type of variable order.Necessary optimality conditions, described by fractional differential equations dependingon a combined Caputo fractional derivativeof variable order, are proved. Two different casesare considered: the fundamental problem,with one independent variable, and the general case,with several independent variables. We end with some illustrative examples of the results of the paper. § INTRODUCTION The theory of fractional calculus is an extension of ordinary calculusthat considers integrals and derivatives of arbitrary realor complex order. Although its birth goes back to Euler,fractional calculus has gained a great importanceonly in recent decades, with the applicabilityof such operators for the efficient dynamic modelingof some real phenomena <cit.>. More recently,a general theory of fractional calculus was presented,where the order of the fractional operators is not constantin time <cit.>. This is a natural extension,since fractional derivatives are nonlocal operatorsand contain memory. Therefore, it is reasonable thatthe order of the derivative may vary along time.The variational problem of Herglotz is a generalizationof the classical variational problem <cit.>.It allows us to describe nonconservative processes,even in the case when the Lagrangian is autonomous(that is, when the Lagrangian does not depend explicitly on time).In contrast to the calculus of variations, where the cost functionalis given by an integral depending only on time, space and velocity,in the Herglotz variational problem the model is given by a differentialequation involving the derivative of the objective functional z and the Lagrange function depends on time, trajectories x and zand on the derivative of x. The problem of Herglotz was posedby Herglotz himself in 1930 <cit.>, but only in 1996,with the works <cit.>, it has gaineda wide attention from the mathematical community. Indeed,since 1996, several papers were devoted to this subject: see <cit.> and references therein.§ PRELIMINARIES In this section we present some needed concepts and results. §.§ The fractional calculus of variable orderWe deal with fractional operators of variable fractional orderon two variables, with range on the open interval (0,1),that is, the order is a function α:[a,b]^2→(0,1). Given a function x:[a,b]→ℝ, we present twodifferent concepts of fractional derivatives of x. First, we recall the definition of fractionalintegral <cit.>. The left Riemann–Liouville fractional integralof orderof x is defined byx(t)=∫_a^t 1/Γ(α(t,))(t-)^α(t,)-1x()dand the right Riemann–Liouville fractional integral of x byx(t)=∫_t^b1/Γ(α(,t))(-t)^α(,t)-1 x()d.For fractional derivatives, we consider two types:Riemann–Liouville and Caputo fractional derivatives.The left Riemann–Liouville fractional derivativeof orderof x is defined byx(t)=d/dt∫_a^t 1/Γ(1-α(t,))(t-)^-α(t,)x()dand the right Riemann–Liouville fractional derivative of x byx(t)=d/dt∫_t^b -1/Γ(1-α(,t))(-t)^-α(,t) x()d.The left Caputo fractional derivative of orderof x is defined byx(t)=∫_a^t 1/Γ(1-α(t,))(t-)^-α(t,)x^(1)()dand the right Caputo fractional derivative of x byx(t)=∫_t^b -1/Γ(1-α(,t))(-t)^-α(,t)x^(1)()d.Motivated by the works <cit.>, we consider here a generalization of previous concepts by introducing a linearcombination of the fractional derivatives of variable fractional order.Let α, β: [a,b]^2→(0,1)be two functions and γ=(γ_1,γ_2) ∈ [0,1]^2 a vector. The combined Riemann–Liouville fractional derivativeof function x is defined byD_γ^,x(t)=γ_1 x(t)+γ_2 x(t).Similarly, the combined Caputo fractional derivativeof function x is defined by^CD_γ^,x(t)=γ_1 x(t)+γ_2 x(t).When dealing with variational problems and necessary optimality conditions, an important ingredient is always an integrationby parts formula. Here, we present two such formulas,involving the Caputo fractional derivative of variable order.If x,y ∈ C^1[a,b], then∫_a^by(t) x(t)dt =∫_a^b x(t)y(t)dt+[x(t) _tI_b^1-y(t) ]_t=a^t=band∫_a^by(t)x(t)dt=∫_a^b x(t)y(t)dt-[x(t)_aI_t^1-y(t)]_t=a^t=b.To end this short introduction to the fractional calculus of variable order, we introduceone more notation. The dual fractional derivativeof (<ref>) is defined byD_γ^,̱=γ_2_aD_t^ +γ_1_tD_T^,where γ=(γ_2,γ_1) and T∈[a,b] is the final time of the problemunder consideration (see (<ref>) below).The dual fractional derivative of(<ref>) is defined similarly. §.§ The fractional calculus of variations Let D denote the linear subspace of C^1([a,b])× [a,b] defined byD={(x,t)∈ C^1([a,b])× [a,b] :x(t)[a,b] }.We endow D with the following norm:(x,t)=max_a≤ t ≤b|x(t)| +max_a≤ t ≤ b|x(t)|+|t|.To fix notation, throughout the text we denote by∂_i ψ the partial derivative of a functionψ:ℝ^n→ℝ with respectto its ith argument, i = 1, …, n. For simplicity of notation,we also introduce the operator [·]_γ^α, β defined by[x]_γ^α, β(t)=(t, x(t),x(t)).Let L be a LagrangianL:C^1([a,b]×ℝ^2 )→ℝ.Consider the following problem of the calculus of variations: minimize functional 𝒥:D→ℝ with𝒥(x,T)=∫_a^TL[x]_γ^α, β(t) dt + ϕ(T,x(T))over all (x,T)∈ D satisfying the initial condition x(a)=x_a, for a given x_a∈ℝ. The terminal time Tand the terminal state x(T) are considered here free.The terminal cost function ϕ:[a,b]×ℝ→ℝ is at least of class C^1.If (x,T) is a minimizer of functional (<ref>) on D,then (x,T) satisfies the fractional differential equations∂_2 L[x]_γ^α, β(t) +D_γ^,̱∂_3 L[x]_γ^α, β(t)=0on [a,T] andγ_2(∂_3 L[x]_γ^α, β(t) - _TD_t^∂_3 L[x]_γ^α, β(t))=0on [T,b]. Moreover, the following transversality conditions hold:L[x]_γ^α, β(T)+∂_1ϕ(T,x(T))+∂_2ϕ(T,x(T))x'(T)=0, [γ_1_tI_T^1-∂_3L[x]_γ^α, β(t) -γ_2_TI_t^1-∂_3 L[x]_γ^α, β(t)]_t=T +∂_2 ϕ(T,x(T))=0, γ_2 [ _TI_t^1-∂_3 L[x]_γ^α, β(t) -_aI_t^1-∂_3L[x]_γ^α, β(t)]_t=b=0.We can rewrite the transversality conditions(<ref>), obtaining the next result.If (x,T) is a minimizer of functional (<ref>) on D,then the fractional Euler–Lagrange equations (<ref>) and (<ref>) are satisfied together with the following transversality conditions:L[x]_γ^α, β(T)+∂_1ϕ(T,x(T))+ x'(T) [ γ_2 _TI_t^1-∂_3L[x]_γ^α, β(t) - γ_1 _tI_T^1-∂_3L[x]_γ^α, β(t)]_t=T =0, [ γ_1 _tI_T^1-∂_3L[x]_γ^α, β(t) - γ_2 _TI_t^1-∂_3L[x]_γ^α, β(t)]_t=T +∂_2 ϕ(T,x(T))=0, γ_2 [ _TI_t^1-∂_3 L[x]_γ^α, β(t) -_aI_t^1-∂_3L[x]_γ^α, β(t)]_t=b=0.§ HERGLOTZ'S VARIATIONAL PRINCIPLEIn this section we present a fractional variational principleof Herglotz depending on Caputo fractional derivatives. Let α,β: [a,b]^2→ (0,1) be two functions.The fractional Herglotz variational problem that we studyconsists in the determination of trajectoriesx ∈ C^1([a,b]), satisfying a given initialcondition x(a)=x_a∈ℝ, and a real T ∈ [a,b]that extremize the value of z(T), where z satisfiesthe following differential equation with dependenceon a combined Caputo fractional derivative operator:ż(t)=L(t,x(t),x(t), z(t) ), t ∈ [a,b],subject to the initial conditionz(a)=z_a,where z_a is a given real number. In the sequel, we use the auxiliary notation[x,z]_γ^α, β(t)=(t,x(t),x(t), z(t) ).The Lagrangian L is assumed to satisfy the following hypothesis: * L ∈ C^1([a,b] ×ℝ^3,ℝ), * t→λ(t)∂_3L[x,z]_γ^α, β(t)is such that _TD_t^( λ(t)∂_3L[x,z]_γ^α, β(t)),( λ(t) ∂_3 L[x,z]_γ^α, β(t) )and D_γ^,̱( λ(t) ∂_3 L[x,z]_γ^α, β(t) )exist and are continuous on [a,b], whereλ(t)=exp(-∫_a^t ∂_4L[x,z ]_γ^α, β(τ)dτ).The following result gives necessary conditionsof Euler–Lagrange type for an admissiblefunction x to be solution of the problem. Let x ∈ C^1([a,b]) be such that z definedby (<ref>) subject to the initial condition (<ref>)has an extremum. Then, (x,z) satisfies the fractional differential equations∂_2 L[x,z]_γ^α, β(t)λ(t) +D_γ^,̱(λ(t) ∂_3 L[x,z]_γ^α, β(t))=0on [a,T] andγ_2((λ(t) ∂_3 L[x,z]_γ^α, β(t)) - _TD_t^(λ (t)∂_3L[x,z]_γ^α, β(t)))=0on [T,b]. Moreover, the following transversality conditions are satisfied:{[ [γ_1 _tI_T^1-(λ (t)∂_3L[x,z]_γ^α, β(t)) -γ_2 _TI_t^1-(λ (t)∂_3L[x,z]_γ^α, β(t))]_t=T=0,;γ_2 [ _TI_t^1-( λ (t)∂_3L[x,z]_γ^α, β(t)) -_aI_t^1-( λ (t)∂_3L[x,z]_γ^α, β(t))]_t=b=0. ].If T<b, then L[x,z]_γ^α, β(T)=0.Let x be a solution to the problem and consider an admissible variationof x, x= x+h, where h∈ C^1([a,b]) is an arbitraryperturbation curve and ∈ℝ represents a small number(→ 0). The constraint x(a)=x_a impliesthat all admissible variations must fulfill the condition h(a)=0. On the other hand, consider an admissible variation of z,z= z+θ, where θ is a perturbationcurve (not arbitrary) such that * θ(a)=0, so that z(a)=z_a, * θ (T)=0, because z(T) is a maximum(z(T)-z(T) ≤ 0) or a minimum(z(T)-z(T)≥0), * θ (t)= dd ε z(x,t)|_ε=0, so that the variation satisfiesequation (<ref>).Differentiating θ with respect to t, we obtain thatddtθ (t) =ddtddεz(x,t) |_ε=0= ddεddtz(x,t) |_ε=0= ddε L(t,x(t)+h(t), x(t)+h(t), z(t) ) |_ε=0and rewriting this relation, we obtainthe following differential equation for θ:θ̇(t) - ∂_4 L[x,z]_γ^α, β(t) θ(t)=∂_2 L[x,z]_γ^α, β(t) h(t)+ ∂_3L[x,z]_γ^α, β(t)h(t).Considering λ(t)=exp(- ∫_a^t∂_4L[x,z]_γ^α, β(τ)dτ),we obtain the solution for the last differential equation:θ(T)λ(T) - θ(a)= ∫_a^T(∂_2 L[x,z]_γ^α, β(t) h(t)+ ∂_3L[x,z]_γ^α, β(t)h(t) ) λ(t) dt.By hypothesis, θ(a)=0. If x is such that z(x,t) definedby (<ref>) attains an extremum at t=T,then θ(T) is identically zero. Hence, we get∫_a^T(∂_2 L[x,z]_γ^α, β(t)h(t)+ ∂_3L[x,z]_γ^α, β(t)h(t) ) λ(t) dt = 0.Considering only the second term in (<ref>),and the definition of combined Caputo derivative,we obtain that∫_a^Tλ(t) ∂_3 L[x,z]_γ^α, β(t) ( γ_1 h(t) + γ_2 h(t) )dt=γ_1∫_a^Tλ(t) ∂_3 L[x,z]_γ^α, β(t)h(t)dt +γ_2[ ∫_a^bλ(t) ∂_3L[x,z]_γ^α, β(t)h(t)dt - ∫_T^bλ(t)∂_3 L[x,z]_γ^α, β(t)h(t)dt ]=⋆.Using Theorem <ref>, and consideringγ =(γ_2, γ_1),we get⋆ = ∫_a^T h(t) D_γ^,̱( λ(t) ∂_3 L[x,z]_γ^α, β(t) ) dt + ∫_T^bγ_2 h(t) [ (λ(t) ∂_3 L[x,z]_γ^α, β(t) ) - _TD_t^( λ(t) ∂_3 L[x,z]_γ^α, β(t) ) ] dt + h(T) [ γ_1_tI_T^1-( λ(t)∂_3 L[x,z]_γ^α, β(t) ) - γ_2_TI_t^1-( λ(t) ∂_3 L[x,z]_γ^α, β(t) ) ]_t=T+ h(b) γ_2[ _TI_t^1-( λ(t)∂_3 L[x,z]_γ^α, β(t) ) - _aI_t^1-( λ(t)∂_3 L[x,z]_γ^α, β(t) ) ]_t=b.Substituting this relation into expression (<ref>), we obtain∫_a^T h(t) [ ∂_2 L[x,z]_γ^α, β(t) λ(t)+ D_γ^,̱( λ(t)∂_3 L[x,z]_γ^α, β(t) )] dt + ∫_T^bγ_2 h(t) [ ( λ(t)∂_3 L[x,z]_γ^α, β(t) )- _TD_t^( λ(t)∂_3 L[x,z]_γ^α, β(t) ) ] dt+ h(T) [ γ_1_tI_T^1-( λ(t)∂_3 L[x,z]_γ^α, β(t) ) - γ_2_TI_t^1-( λ(t) ∂_3 L[x,z]_γ^α, β(t) ) ]_t=T+ h(b) γ_2[ _TI_t^1-( λ(t)∂_3 L[x,z]_γ^α, β(t) ) - _aI_t^1-( λ(t)∂_3 L[x,z]_γ^α, β(t) ) ]_t=b =0.With appropriate choices for the variations h(·),we get the Euler–Lagrange equations (<ref>)–(<ref>)and the transversality conditions (<ref>).Ifand $̱ tend to 1, and if the LagrangianLis of classC^2, then the first Euler–Lagrangeequation (<ref>) becomes∂_2 L[x,z]_γ^α, β(t)λ(t)+(γ_2- γ_1) d/dt[λ(t)∂_3 L[x,z]_γ^α, β(t)]=0.Differentiating and considering the derivativeof the lambda function, we getλ(t) [∂_2 L[x,z]_γ^α, β(t)+(γ_2 - γ_1)[-∂_4 L[x,z]_γ^α, β(t) ∂_3 L[x,z]_γ^α, β(t) +d/dt∂_3 L[x,z]_γ^α, β(t)]]=0.Asλ(t)>0for allt, we deduce that∂_2 L[x,z]_γ^α, β(t) +(γ_2- γ_1)[ d/dt∂_3 L[x,z]_γ^α, β(t) -∂_4 L[x,z]_γ^α, β(t) ∂_3 L[x,z]_γ^α, β(t)]=0.§ THE CASE OF SEVERAL INDEPENDENT VARIABLES We now obtain a generalization of Herglotz's principleof Section <ref> for problems involvingn+1independent variables. DefineΩ=∏_i=1^n [a_i,b_i], withn ∈ℕ,P=[a,b]×Ωand consider the vectors=(s_1, s_2, …, s_n)∈Ω. The new problem consistsin determining the trajectoriesx ∈C^1(P)that give an extremum toz[x,T], where the functionalzsatisfies the differential equationż(t)=∫_ΩL(t,s, x(t,s),x(t,s),. .^CD_γ^1^α_1(·,·),β_1(·,·)x(t,s),…, ^CD_γ^n^α_n(·,·),β_n(·,·)x(t,s), z(t) )d^nssubject to the constraintx(t,s)=g(t,s)(t,s) ∈∂ P,where∂Pis the boundary ofPandgis a given functiong:∂P →ℝ. Assume that* α, α_i, β,β_i: [a,b]^2 → (0,1) with i=1,… n, * γ,γ^1, …, γ^n ∈ [0,1]^2, * d^ns=ds_1… ds_n, * x(t,s), ^CD_γ^1^α_1(·,·),β_1(·,·)x(t,s), …, ^CD_γ^n^α_n(·,·),β_n(·,·)x(t,s)exist and are continuous functions, *the Lagrangian L:P×ℝ^n+3→ℝ is of class C^1. By x(t,s) we mean the Caputo fractional derivativewith respect to the independent variable t and by^CD_γ^i^α_i(·,·),β_i(·,·)x(t,s)we mean the Caputo fractional derivative with respectto the independent variable s_i, i=1,…,n. In the sequel, we use the auxiliary notation[x,z]_n, γ^α, β(t,s) =(t,s,x(t,s),x(t,s),^CD_γ^1^α_1(·,·), β_1(·,·)x(t,s),. . …, ^CD_γ^n^α_n(·,·), β_n(·,·)x(t,s), z(t) ).Consider the functionλ(t)=exp(-∫_a^t∫_Ω∂_2n+4[x,z ]_n, γ^α, β(τ,s) d^ns dτ).If (x,z) is an extremizer to the functional (<ref>),then (x,z) satisfies the fractional differential equation∂_n+2 L[x,z]_n, γ^α, β(t,s)λ(t) +D_γ^,̱(λ(t)∂_n+3L[x,z]_n, γ^α, β(t,s))+ ∑_i=1^n D_γ^i^β_i(·,·), α_i(·,·)(λ(t) ∂_n+3+i L[x,z]_n, γ^α, β(t,s))=0on [a,T] ×Ω andγ_2((λ(t)∂_n+3L[x,z]_n, γ^α, β(t,s)) - _TD_t^(λ (t) ∂_n+3L[x,z]_n, γ^α, β(t,s)))=0on [T,b]×Ω. Moreover, (x,z)satisfies the transversality condition[γ_1 _tI_T^1-(λ (t) ∂_n+3 L[x,z]_n, γ^α, β(t,s)) -γ_2 _TI_t^1-(λ(t)∂_n+3 L[x,z]_n, γ^α, β(t,s))]_t=T=0,s ∈Ω.If T<b, then ∫_Ω L[x,z]_n, γ^α, β(T,s)d^ns=0.Let x be a solution to the problem. Consider an admissible variation of x,x(t,s)= x(t,s)+h(t,s), where h∈ C^1(P) is an arbitraryperturbing curve and ∈ℝ is such that ||≪ 1.Consequently, from the boundary condition (<ref>), h(t,s)=0 for all (t,s)∈∂ P. On the other hand,consider an admissible variation of z, z= z+θ,where θ is a perturbing curve such that θ (a)=0 and θ(t)= dd εz(x,t) |_ε=0.Differentiating θ(t) with respect to t, we obtain thatddtθ (t)=ddtddε z(x,t)|_ε=0= ddεddt z(x,t)|_ε=0= ddε∫_Ω L[x,z]_n, γ^α, β(t,s) d^ns|_ε=0.We conclude thatθ̇(t) =∫_Ω( ∂_n+2 L[x,z]_n, γ^α, β(t) h(t,s)+∂_n+3L[x,z]_n, γ^α, β(t,s)h(t,s).+. ∑_i=1^n∂_n+3+i L[x,z]_n, γ^α, β(t,s) ^CD_γ^i^α_i(·,·), β_i(·,·)h(t,s)+∂_2n+4 L[x,z]_n, γ^α, β(t,s) θ(t)) d^ns.To simplify the notation, define B(t)=∫_Ω∂_2n+4 L[x,z]_n, γ^α, β(t,s)d^ns andA(t) =∫_Ω( ∂_n+2 L[x,z]_n, γ^α, β(t) h(t,s)+∂_n+3 L[x,z]_n, γ^α, β(t,s) h(t,s)+∑_i=1^n∂_n+3+i L[x,z]_n, γ^α, β(t,s)^CD_γ^i^α_i(·,·), β_i(·,·)h(t,s) ) d^ns.Then, we obtain the linear differential equation θ̇(t)-B(t)θ(t)=A(t),whose solution isθ(T)λ(T) - θ(a) = ∫_a^T A(t) λ(t) dt.Since θ(a)=θ(T)=0, we get∫_a^T A(t)λ(t)dt = 0.Considering only the second term in (<ref>), we can write∫_a^T ∫_Ωλ(t) ∂_n+3L[x,z]_n, γ^α, β(t,s)( γ_1 h(t,s) + γ_2 h(t,s) ) d^ns dt=γ_1∫_a^T∫_Ωλ(t) ∂_n+3L[x,z]_n, γ^α, β(t,s)h(t,s)d^ns dt +γ_2[ ∫_a^b∫_Ωλ(t) ∂_n+3L[x,z]_n, γ^α, β(t,s)h(t,s) d^ns dt. - .∫_T^b∫_Ωλ(t)∂_n+3 L[x,z]_n, γ^α, β(t,s)h(t,s) d^nsdt ].Let γ =(γ__2, γ_1).Integrating by parts (cf. Theorem <ref>)and since h(a,s)=0 and h(b,s)=0 for all s ∈Ω,we obtain the following expression:∫_a^T∫_Ω h(t,s) D_γ^,̱( λ(t) ∂_n+3L[x,z]_n, γ^α, β(t,s) ) d^nsdt + γ_2∫_T^b∫_Ω h(t,s) [( λ(t) ∂_n+3L[x,z]_n, γ^α, β(t,s) ).-. _TD_t^( λ(t) ∂_n+3L[x,z]_n, γ^α, β(t,s) ) ] d^nsdt + ∫_Ω h(T,s) [ γ_1_tI_T^1-( λ(t)∂_n+3 L[x,z]_n, γ^α, β(t,s) ).. - γ_2_TI_t^1-( λ(t) ∂_n+3L[x,z]_n, γ^α, β(t,s) ) d^n s]_t=T.Doing similarly for the (i+2)th term of (<ref>),i=1,…, n, letting γ^i =(γ__2^i,γ_1^i), and since h(t,a_i)=h(t,b_i)=0 for all t ∈ [a,b],we obtain∫_a^T∫_Ωλ(t) ∂_n+3+iL[x,z]_n, γ^α, β(t,s) ( γ_1^i_a_i^CD_s_i^α_i(·,·)h(t,s)+ γ_2^i_s_i^CD_b_i^β_i(·,·) h(t,s)) d^ns dt=∫_a^T∫_Ω h(t,s) D_γ^i^ β_i(·,·), α_i(·,·)(λ(t)∂_n+3+iL[x,z]_n, γ^α, β(t,s) ) d^ns dt.Substituting these relations into (<ref>), we deduce that∫_a^T ∫_Ω h(t,s)[ ∂_n+2 L[x,z]_n, γ^α, β(t,s)λ(t)+ D_γ^,̱( λ(t) ∂_n+3L[x,z]_n, γ^α, β(t,s) ) .+∑_i=1^n D_γ^i^ β_i(·,·), α_i(·,·)(λ(t) ∂_n+3+i L[x,z]_n, γ^α, β(t,s) ) d^nsdt + γ_2∫_T^b∫_Ω h(t,s) [( λ(t) ∂_n+3L[x,z]_n, γ^α, β(t,s) ).. - _TD_t^( λ(t) ∂_n+3L[x,z]_n, γ^α, β(t,s) ) ] d^nsdt+ ∫_Ω h(T,s) [ γ_1_tI_T^1-( λ(t) ∂_n+3L[x,z]_n, γ^α, β(t,s) )..-γ_2_TI_t^1-( λ(t)∂_n+3 L[x,z]_n, γ^α, β(t,s))d^ns ]_t=T.We get the Euler–Lagrange equations(<ref>)–(<ref>)and the transversality condition (<ref>) with appropriate choices of h. § ILLUSTRATIVE EXAMPLES We present three examples. Considerż(t)=( x(t))^2+z(t)+t^2-1,t∈ [0,3],x(0)=1,z(0)=0.In this case, λ(t)=exp(-t). The necessary optimality conditions (<ref>)–(<ref>)of Theorem <ref> hold for x(t) ≡ 1.If we replace x by x in (<ref>), we obtainż(t)-z(t)=t^2-1,t∈ [0,3],z(0)=0,whose solution isz(t)=exp(t)-(t+1)^2.The last transversality condition of Theorem <ref> asserts thatL[x,z]_γ^α, β(T)=0 ⇔exp(T)-2T-2=0,whose solution is approximatelyT≈ 1.67835.We remark that z(<ref>) actually attains a minimum valueat this point (see Figure <ref>, (a)):z(1.67835)≈ -1.81685.Consider nowż(t)=(t-1)(x^2(t)+z^2(t)+1),t∈ [0,3],x(0)=0,z(0)=0.Since the first Euler–Lagrange equation (<ref>) reads(t-1)x(t)=0 ∀t∈[0,T],we see that x(t) ≡ 0 is a solution of this equation.The second transversality condition of (<ref>)asserts that, at t=T, we must haveL[x,z]_γ^α, β(t)=0,that is,(t-1)(z^2(t)+1)=0,and so T=1 is a solution of this equation. Substituting x by x in (<ref>), we getż(t)=(t-1)(z^2(t)+1),t∈ [0,3],z(0)=0.The solution to this Cauchy problemis the function z(t)=tan(t^2/2-t)(see Figure <ref>, (b)) and the minimum value isz(1)=tan(-1/2).For our last example, considerż(t)=( x(t) - f(t))^2+t^2-1,t∈ [0,3],x(0)=0,z(0)=0,wheref(t) := t^1-α(t)/2Γ(2-α(t)) -(3-t)^1-β(t)/2Γ(2-β(t)).In this case, λ(t)≡ 1. We intend to find a pair (x,z),satisfying all the conditions in (<ref>),for which z(T) attains a minimum value. It is easy to verify that x(t) = t and T=1 satisfythe necessary conditions given by Theorem <ref>.Replacing x by x in system (<ref>),we get a Cauchy problem of formż(t)=t^2-1,t∈ [0,3],z(0)=0,whose solution isz(t)=t^3/3-t.Observe that this function attains a minimum valueat T=1, which is z(1)=-2/3 (Figure <ref>, (c)). § ACKNOWLEDGEMENTS The authors are grateful to two anonymous refereesfor their comments.99Almeida (MR3274999) [10.3934/dcdsb.2014.19.2367]R. Almeida and A. B. 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Yu,A dynamic-order fractional dynamic system, Chinese Phys. Lett.30 (2013), 046601, 4 pp.Tavares (MR3325357) [10.1080/02331934.2015.1010088]D. Tavares, R. Almeida and D. F. M. Torres,Optimality conditions for fractional variationalproblems with dependence on a combined caputo derivative of variable order, Optim.64 (2015), 1381–1391.arXiv:1501.02082Received July 29, 2016; Revised Feb 03, 2017; Accepted March 27, 2017. | http://arxiv.org/abs/1703.09104v1 | {
"authors": [
"Dina Tavares",
"Ricardo Almeida",
"Delfim F. M. Torres"
],
"categories": [
"math.OC",
"26A33, 34A08, 49K05, 49K10"
],
"primary_category": "math.OC",
"published": "20170327142919",
"title": "Fractional Herglotz variational problems of variable order"
} |
1Department of Astronomy, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91101, USA 2Institute for Astronomy, University of Hawai`i at Mānoa, Hilo, HI 96720-2700, USA 3Department of Physics and Astronomy, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3255, USA 4Inter-University Centre for Astronomy & Astrophysics, Savitribai Phule Pune University Campus, Pune 411 007, IndiaRobo-AO is an autonomous laser guide star adaptive optics system recently commissioned at the Kitt Peak 2.1-m telescope. Now operating every clear night, Robo-AO at the 2.1-m telescope is the first dedicated adaptive optics observatory. This paper presents the imaging performance of the adaptive optics system in its first eighteen months of operations. For a median seeing value of 1.31^'', the average Strehl ratio is 4% in the i^' band and 29% in the J band. After post-processing, the contrast ratio under sub-arcsecond seeing for a 2≤ i^'≤ 16 primary star is five and seven magnitudes at radial offsets of 0.5^'' and 1.0^'', respectively. The data processing and archiving pipelines run automatically at the end of each night. The first stage of the processing pipeline shifts and adds the data using techniques alternately optimized for stars with high and low SNRs. The second “high contrast" stage of the pipeline is eponymously well suited to finding faint stellar companions.§ INTRODUCTION Adaptive optics (AO) systems correct wavefront aberrations introduced by the atmosphere and instrumental optics, restoring the resolution of a telescope to the diffraction limit. Laser guide stars (LGS) were developed in the 1980s to provide AO systems with bright, locatable wavefront reference sources, bringing fainter astrophysical objects into the purview of adaptive optics. Over half of all >8-m aperture telescopes are now equipped with an LGS AO system. The primary application of these AO instruments is for high angular resolution studies of interesting astronomical objects. As such minimizing the overhead has not been a major consideration in the overall design of AO systems on large telescopes.Robo-AO is a robotic LGS AO system designed for maximum target throughput. Unlike LGS systems on large telescopes, it is based on an artificial star produced by Rayleigh scattering of a near UV laser. Robo-AO achieves high target throughput by minimizing overhead times to less than one minute per target. This is accomplished by three key design elements: 1) each step of the observation sequence is automated, allowing tasks that would be performed sequentially by a human operator to be performed in parallel and with minimal delay by the robotic system; 2) the λ=355nm Rayleigh scattering laser guide star is invisible to the human eye. As a result, while coordination with the U.S. Air Force Joint Space Operations Center (JSpOC) is still required to prevent illumination of sensitive space assets, no control measures are required by the Federal Aviation Administration; 3) Robo-AO employs an automated queue scheduler which chooses each new science target based on telescope slew times and approved lasing windows provided in advance by JSpOC. Robo-AO was first commissioned at the Palomar 1.5-m telescope in 2011, where it completed 19 science runs as a PI instrument from May 2012 through June 2015. Full details of the Robo-AO hardware and software can be found in <cit.>, <cit.> and <cit.>. In 2012, the National Optical Astronomy Observatory (NOAO), following the recommendation of the Portfolio Committee which was chartered by thethe National Science Foundation (NSF),decided to divest the Kitt Peak 2.1-m telescope.In 2015, the Robo-AO team made a bid for the telescope and was selected to operate the telescope for three years.Robo-AO was installed at the 2.1-m telescope in November, 2015; since then it has been operating nearly every clear night. As the first dedicated, automated adaptive optics facility, Robo-AO at Kitt Peak is well positioned to support the next generation of large-scale survey programsthat are focused on stellar and exoplanet astronomy (e.g. K2, GAIA, CRTS, PTF, TESS and others), as well as AO follow up of interesting sources. Early science results including Robo-AO KP data can be found in <cit.> and <cit.>.In this paper, we describe the performance of Robo-AO since commissioning. The paper is organized as follows: <ref> introduces the Robo-AO imaging systems; <ref> provides an overview of our automatic data reduction pipelines; <ref> shows the relationships between the weather conditions and the measured seeing; <ref> presents the Strehl ratio and contrast curve statistics as well as the point spread function (PSF) morphology; <ref> describes our automated data archiving system; finally, <ref> describes the newly installed near-IR camera. § SUMMARY OF THE ROBO-AO IMAGING SYSTEMThe Robo-AO imaging system includes two optical relays, each using a pair of off-axis parabolic mirrors. The first relay images the telescope pupil onto a 140-actuator Boston Micromachines micro-electro-mechanical-systems (MEMS) deformable mirror used for wavefront correction. A dichroic then reflects the UV light to an 11×11 Shack-Hartmann wavefront sensor. The second optical relay includes a fast tip-tilt correcting mirror and an atmospheric dispersion corrector (ADC; here, two rotating prisms[From the commissioning of Robo-AO at Kitt Peak in November of 2015 through February of 2017, the right ascension (RA) axis of the 2.1-m telescope suffered from a ∼ 3.7Hz jitter (see <ref> and <ref>) that caused a slight elongation of the stellar PSFs. As a result, the ADCs were not correctly calibrated until an upgrade to the telescope control system removed the jitter.]) located at a reimaged pupil. The output of the second relay is an F/41 beam that is intercepted by a dichroic mirror, which reflects the λ<950nm portion of the converging beam to the visible wavelength filter wheel and EMCCD detector (see Table <ref>). The filter wheel includesg^', r^', i^', and z^' filters, as well as a long-pass “lp600" filter cutting on at 600 nm and extending beyond the red limit of the EMCCD (see Figure 1 in ). The dichroic transmits the longer wavelength light to the near-infrared (NIR) instrument port (see <ref>).Robo-AO was originally designed for simultaneous optical and NIR operations, such that deep science integrations could be obtained in one band while the image displacement could be measured in the other and corrected with the fast tip-tilt mirror. In February of 2017, we achieved first light with a science-grade novel infrared array, a brief summary of which appears in <ref>[A detailed analysis of the operation of this camera, its imaging performance, and its incorporation into an active tip-tilt control loop will be reported elsewhere.]. In this paper, we consider the imaging performance of Robo-AO using the optical imaging camera only. In lieu of an active tip-tilt correction, the EMCCD is run at a framerate of 8.6Hz to allow for post-facto image registration followed by stacking (see <ref>).§ DATA REDUCTION PIPELINES §.§ Overview Image registration and stacking (see <ref>) is accomplished automatically by the “bright star” and “faint star” pipelines, which are optimized for high and low signal-to-noise (SNR) targets, respectively. The data are then processed by the “high contrast” pipeline to maximize the sensitivity to faint companions. These pipelines are described in detail below. §.§ Image Registration Pipelines All observations are initially processed by the “bright star” pipeline. This pipeline generates a windowed datacube centered on an automatically selected guide star. The windowed region is bi-cubically up-sampled and cross correlated with the theoretical point spread function to give the center coordinates of the guide star's PSF in each frame. The full-frame, unprocessed images are then calibrated using the nightly darks and dome flats. Finally, the calibrated full frames are aligned using the center coordinates identified by the up-sampled, windowed frames, and co-added via the Drizzle algorithm <cit.>. These steps are described in detail in <cit.>. After an observation has been processed by the “bright star” pipeline, the core of the brightest star in the frame is fit by a 2D Moffat function. If the full width at half maximum (FWHM) of the function fit to the core is<λ/D, indicating that the stellar centroiding step has failed, the observation is re-processed by the “faint star” pipeline to improve the SNR in the final science image. The individual frames for a given observation are summed to create a master, dark and flat corrected reference image. This frame is then high pass filtered and windowed about the guide star. Each raw short exposure frame is then dark and flat corrected, filtered, and windowed. These individual frames are registered to the master reference frame using the python package written by Adam Ginsburg[<https://github.com/keflavich/image_registration>]. The package finds the offset between the individual and reference frames using DFT up-sampling and registers the images with FFT-based sub-pixel image shifts. Figure <ref> illustrates the strengths and weaknesses of the bright and faint star pipelines. These automatic pipelines have reduced thousands of Robo-AO observations since the instrument was commissioned in November of 2015. Figure <ref> shows a collage of representative observations. §.§ High Contrast Pipeline For science programs that aim to identify point sources at small angular separations from known stars further processing is needed. Our “high contrast imaging” pipeline generates a 3.5^'' frame windowed about the star of interest in the final science frame. A high pass filter is applied to the windowed frame to reduce the contribution of the stellar halo. To whiten correlated speckle noise at small angular separations from the target star we subtract a synthetic PSF generated by Karhunen-Loève Image Processing (KLIP). The KLIP algorithm is based on the method of Principal Component Analysis <cit.>. The PSF diversity needed to create this synthetic image is provided by a reference library of Robo-AO observations – a technique called Reference star Differential Imaging <cit.>. We note that the angular differential imaging approach <cit.> is not possible here because the 2.1-m telescope is an equatorial mount telescope. Our pipeline uses thepackage <cit.>.The full reference PSF library consists of several thousand 3.5^'' square high pass filtered frames that have been visually vetted to reject fields with more than one point source. The PSF library is updated on a nightly basis to ensure that each object's reduction has the opportunity to include frames from the same night. Each frame in the full library is cross correlated with the windowed and filtered science frame of interest. The five frames with the highest cross correlation form the sub-library provided to KLIP. We then adopt only the first principal component (PC) as our synthetic PSF, as including more PCs provides no additional noise reduction on average. A future version of the pipeline will choose the number of PCs automatically for each observation based on SNR maximization. Figure <ref> shows an example of a PSF reduced by the standard data pipeline (panel ), then high pass filtered (panel ), and finally processed with RDI-KLIP (panel ).After a science frame has been fully reduced we use to produce a contrast curve that is properly corrected for small sample statistics and algorithmic throughput losses. The corresponding contrast curves for the three panels are shown in panel .Given that over two hundred new targets are observed during a clear night of Robo-AO observations the reference library is rapidly expanding and increasingly includes PSFs affected by a very wide range of environmental conditions. Hence, speckle noise in a past observation can be further reduced by a fresh RDI-KLIP reduction if the data is more correlated with later PSFs. Clearly a new reduction will benefit from the advantage of the larger reference library.§ SITE PERFORMANCE §.§ Site Geography Kitt Peak is located 56miles southwest of Tucson, Arizona, at an elevation of 6800feet. The 2.1-m telescope is situated 0.4miles to the south of the peak's highest point (the location of the Mayall 4-m telescope). The WIYN 3.5-m and 0.9-m telescopes are respectively 700ft and 400ft to the west of the 2.1-m telescope and at approximately the same elevation. There are no structures at equal or greater elevations to the east of the telescope, and the terrain is relatively flat beyond Kitt Peak in that direction. The 7730ft Baboquivari Peak is 12miles directly south of the telescope.§.§ Seeing MeasurementBefore the start of each science observation, a 10s seeing observation is taken with the AO correction off. During this period the wavefront sensor camera acquires a background image.These seeing observations are dark and flat calibrated and summed without any registration of the individual exposures. The seeing is defined as the FWHM of a two-dimensional Gaussian function fit to this summed frame. Starting in January of 2017, a 90s seeing observation was obtained each hour. Specifically, the Robo-AO queue schedules an observation of a bright (V<8) star within 10^∘ of zenith to refocus the telescope and measure the seeing. As of this writing, there is no significant difference between these “long” and “short” seeing observations. Here on we proceed with the assumption that the 10s seeing measurements are representative of the long-exposure seeing. We display a histogram of these fiducial seeing values in Figure <ref>.Figure <ref> displays the seeing as a function of the seasons. The seeing values measured in a given wavelength are scaled to a fiduciary wavelength of 500 nm by the scaling law seeing_500 nm = seeing_λ× (λ/500 nm)^1/5. §.§ Seeing Contributions We note that our median seeing of 1.31^'' differs from the median seeing of 0.8^'' reported by the adjacent WIYN telescope[< https://www.noao.edu/wiyn/aowiyn/>]. One possible explanation for this discrepancy is thatthe WIYN was built in 1994 with careful attention paid to dome ventilation and telescope thermal inertia. In contrast, the 2.1-m telescope saw first light in 1964 before such considerations were fully appreciated. Figure <ref> demonstrates the challenging thermal conditions at the 2.1-m telescope: during the majority of Robo-AO observations, the mirror is warmer than the ambient dome temperature which in turn is warmer than the outside air. The experience of other observatories indicate that improvements to dome thermalization can significantly improve the measured seeing<cit.>.Another possible cause of the comparatively poor seeing at the 2.1-m telescope is perhaps a more turbulent ground layer. Figure <ref> shows a “wind rose,” or the frequency of wind speeds originating from different directions, for December 2015 through June 2016. We find that during this period the wind most commonly blows from the NNW, or the direction of the higher elevation Mayall 4-m telescope, and rarely from the SE where the terrain is less mountainous. The highest winds (>40mph) come from the north while the south has the largest fraction of low wind speeds (the wind speeds originating from within 20^∘ of due south are under 10mph 73% of the time). Despite these terrain variations, the seeing is not significantly correlated with the wind direction. The wind speed, however, degrades the seeing by several tenths of an arcsecond for winds over 20mph (the dome closes for winds over 40mph). Figure <ref> plots the seeing versus the wind speed, demonstrating that poorer seeing is correlated with higher wind speeds[The mean binned seeing measurements in Figure <ref> are larger than the median of all Robo-AO KP seeing measurements (Figure <ref>) due to binning effects and the difference between the mean and median of the asymmetric distribution of seeing measurements.]. We note that the wind monitor became nonfunctional after June of 2016, and hence further study of the relationship between the seeing and the wind speed will occur after a new wind monitor is in place.§ ADAPTIVE OPTICS PERFORMANCE §.§ Strehl RatioThe goal of an adaptive optics system is to bring the observed PSF closer to its theoretical diffraction-limited shape; hence, an important measure of the AO system's performance is the ratio between the peak intensity of an observed PSF and that of the telescope's theoretical PSF – the Strehl ratio.As the AO performance improves, the Strehl ratio increases.We calculate the Strehl ratio by 1) generating a monochromatic diffraction-limited PSF by Fourier transforming an oversampled image of the pupil, 2) combining several monochromatic PSFs to create a PSF representative of the desired bandpass, 3) re-sampling the polychromatic PSF to match our 0.0175^''/pixel platescale of the up-sampled “drizzled” frames, 4) obtaining the “Strehl factor," or the ratio of the peak intensity to the sum of the intensity in a 3^'' square box, and 5) calculating the Strehl ratio by repeating step 4 for the observed image and dividing by the Strehl factor. These steps are described in detail in <cit.>. Once Robo-AO began regular observationsat the 2.1-m telescope, we noticed that the achieved Strehl ratios were noticeably smaller than those that were achieved (for similar seeing values) at the Palomar 1.5-m telescope.A number of exercises were undertaken to determine possible causes for this degradation. Eventually, we determined that the Telescope Control System (TCS) was the main contributing factor. In Appendix <ref> we discuss the problem in detail. The mitigation consisted of upgrading the TCS (completed February 2017). Below, and for the rest of the paper, we discuss the instrument performance since the TCS upgrade. Figure <ref> plots the Strehl ratio versus the measured seeing for the i^' and lp600 filters. It is clear that the delivered Strehl ratiodrops off quickly as the seeing increases – while Robo-AO achieves >10% Strehl ratio when the seeing is <1.0^'', a 0.25^'' seeing increase halves the Strehl ratio. In Table <ref>we present a detailed error budget under different seeing conditions. This error budget was originally developed by R. Dekany (private communication), and was validated against the on-sky performance of laser AO systems on the Keck telescope, the Hale telescope and the Palomar 1.5-m telescope <cit.>. Since we lacked turbulence profile(s) for the 2.1-m telescope site we adopt a mean C_n^2(h) profile from a MASS-DIMM atmospheric turbulence monitor collected over a year's baseline at Palomar and scaled to the seeing at Kitt Peak.High-order errors are added in quadrature and are dominated by Focal Anisoplanatism (which is an error arising from the finite altitude of the Rayleigh laser guide star resulting in imperfect atmospheric sampling). We estimate one-axis tip-tilt errors as being dominated by bandwidth error for magnitudes greater than 13. As noted in <ref> we did not use the built-in tip-tilt facility but instead resorted to shift and add. We approximate the error resulting from this approach as follows.We assume a standard -3db rejection frequency matching the full-frame rate of the science camera to approximate bandwidth error. The tip-tilt errors are then converted to an equivalent wavefront error and summed in quadrature with the high-order errors. Other high-order and tip-tilt errors include chromatic, scintillation, aliasing, calibration and digitization errors. Strehl ratios are calculated using the Maréchal approximation. The full-widths at half-maximum (FWHM) are calculated from PSF models assuming the residual diffraction-limited, concentrated light, residual seeing, and scattered light halos are proportional to the phase variance of the residual errors. These models have shown accuracy of a few percent for Strehl ratios as low as 4% <cit.>. Figure <ref> demonstrates Robo-AO's ability to approach the predicted Strehl ratio of 14% in sub-arcsecond seeing conditions. §.§ PSF Morphology Figure <ref> shows a representative Robo-AO point spread function (PSF) corresponding to the V=10 star HIP56051. The observation was taken in the i^' band with a total exposure time of 90s. The seeing at the time of the observation was 0.94^'', and the Strehl ratio of the final PSF is 10.17%. The effect of the AO system is to re-arrange the starlight from the equivalent area seeing-limited PSF (dashed curve) to the sharper, observed PSF plotted by the black points. The AO-corrected PSF includes two components: a sharp core and a broader halo, each separately fit by Moffat functions (the light and dark gray curves, respectively). The full width at half maximum (FWHM) of the Moffat function fit to the core is 0.1^''± 0.01^''. This value is consistent with the diffraction limit of 1.028 λ/D = 0.08^''. §.§ Contrast Curves Section <ref> described the “high contrast pipeline,” which produces 5σ contrast curves from the high pass filtered, RDI-PCA reduced science frames. Figure <ref> plots the median and best 10% contrast curves for i^' and lp600 filter science frames. Under sub-arcsecond seeing (the best 10% of cases), the contrast ratio for a 2≤ i^'≤ 16 primary star is five and seven magnitudes at 0.5^'' and 1.0^'', respectively. § DATA ARCHIVE We have developed a fully automated data processing and archiving system[<https://github.com/dmitryduev/roboao-archive>]. The data reduction chain for an observing night proceeds as follows. At the end of each night, the visual camera data are compressed and transferred to the network storage. Next, the darks and dome flats taken at the beginning of each night are combined into master calibration files and applied to the observations. The bright star pipeline is then run on each observation followed by the computation of the Strehl ratio of the resulting image. The high contrast pipeline also produces high pass filtered, PSF-subtracted images and contrast curves for each of these processed images (see Section <ref>). If the “drizzled” image produced by the bright star pipeline does not pass a quality check (i.e. if a 2-component Moffat fit to the PSF has an anomalously narrow core or wide halo) then the faint star pipeline re-reduces the rapid read-out data. Additionally, the “archiver” processes the nightly seeing data, and generates summary plots of the seeing measurements, Strehl ratios, and contrast curves. Completing the full reduction chain for a typical night's worth of data takes a few hours.The “house-keeping” system uses a[An efficient in-memory key-value database]-basedpython package[<https://github.com/coleifer/huey>] to manage the processing queue, which distributes the jobs to utilize all available computational resources. The processing results together with ancillary information on individual observations and system performance are stored in a [<https://www.mongodb.com>] NoSQL database. For interactive data access, we developed a web-based interface powered by the [<https://github.com/pallets/flask>] back-end. It allows the user to access previews of the processing results together with auxiliary data (e.g. external VO images of a field), nightly summary and system performance information. The web application serves as the general interface to the database providing a sophisticated query interface and also has a number of analysis tools. § NEAR-INFRARED CAMERAIn November 2016, we installed a NIR camera for use with Robo-AO. While similar to the camera deployed in engineering tests at the Palomar 1.5-m telescope in 2014 <cit.>, the new camera uses a science-grade detector and faster readout electronics. The detector is a Mark 13 Selex ES Advanced Photodiode for High-speed Infrared Array (SAPHIRA) with an ME-911 Readout Integrated Circuit <cit.>. It has sub-electron readnoise and320×256-pixel array with λ=2.5 μm cutoff. The single-board PB1 `PizzaBox' readout electronics were developed at the Institute for Astronomy and we use 32 readout channels, each capable of a 2 Mpixel/sec sampling rate, for a maximum full-frame read rate of ∼800Hz.The NIR camera attaches to the Robo-AO f/41 infrared camera port that accesses λ > 950nm after transmission through a dichroic. The camera has an internal cold λ < 1.85 μm short-pass filter and an external warm filter wheel with J, H, clear, and blocking filters. The camera has a plate scale of 0.064^'' per pixel and field-of-view of 16.5^''× 20.6^''.We achieved first light on sky in February 2017 during final testing of the upgraded TCS. Initially we used a 1 Mpixel/sec sampling rate (a full frame read rate of 390 Hz) with detector resets every 300 reads. To create a reduced image, we first assembled difference frames between 39 consecutive reads, totaling ∼ 0.1s of integration time, short enough to effectively freeze stellar image displacement. We subtracted a frame median to approximate removing the background. We then synthesized a long exposure image by registering each corrected frame on the brightest target in the field. Figure<ref> shows an example image of a binary star observed in H-band. For the moment, data acquisition and reduction is performed manually. In the coming months, we will optimize the detector readout routines for maximum sensitivity to faint objects (including dithering for background removal), integrate the operation of the camera into the robotic queue and modify our existing data reduction pipeline to handle the NIR data. We will also investigate automating active tip-tilt correction by using either the visible or infrared camera as a tip-tilt camera, as previously demonstrated at Palomar. § CONCLUSIONRobo-AO at the Kitt Peak 2.1-m telescope is the first dedicated adaptive optics observatory. Observing every clear night, Robo-AO has the capacity to undertake LGS AO surveys of large samples. For instance,a 1000-star survey with exposure times of 60s per target can be completed on the timescale of a week. Science programs designed to exploit Robo-AO's unique capabilities are underway. These programs include stellar multiplicity in open clusters, minor planet binarity, major planet weather variability, extragalactic object morphology, sub-stellar companions to nearby young stars, M-star multiplicity, and the influence of stellar companions on asteroseismology. By the summer of 2017, Robo-AO will become the first LGS AO system to operate entirely autonomously, as on-going upgrades to the 2.1-m telescope will remove the need for a human observer.The Robo-AO team thanks NSF and NOAO for making the Kitt Peak 2.1-m telescope available. We thank the observatory staff at Kitt Peak for their efforts to assist Robo-AO KP operations.Robo-AO KP is a partnership between the California Institute of Technology, the University of Hawai`i, the University of North Carolina at Chapel Hill, the Inter-University Centre for Astronomy and Astrophysics (IUCAA) at Pune, India, and the National Central University, Taiwan. The Murty family feels very happy to have added a small value to this important project. Robo-AO KP is also supported by grants from the John Templeton Foundation and the Mt. Cuba Astronomical Foundation. The Robo-AO instrument was developed with support from the National Science Foundation under grants AST-0906060, AST-0960343, and AST-1207891, IUCAA, the Mt. Cuba Astronomical Foundation, and by a gift from Samuel Oschin. These data are based on observations at Kitt Peak National Observatory, National Optical Astronomy Observatory (NOAO Prop. ID: 15B-3001), which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation. C.B. acknowledges support from the Alfred P. Sloan Foundation.KPNO:2.1m (Robo-AO)yahapj § TELESCOPE JITTER After moving Robo-AO from the Palomar 1.5-m telescope to the Kitt Peak 2.1-m telescope, the median Strehl ratio across all wavelengths was initially reduced from 5.8% to 3.2%. The source of this degradation was a ∼3.7Hz vibration in the RA axis. Because Robo-AO mitigates tip/tilt by post facto shift and add rather than a real-time loop, and because its framerate is typically only 8.6Hz, the targets were smeared in the RA direction. Figure <ref> a and b show the power spectral densities of the mean subtracted RA centroid positions of targets observed at Kitt Peak and Palomar, respectively. The peak at ∼3.7Hz is clear in the Kitt Peak data, but is not present at Palomar. The RA-axis smearing for a single test observation is demonstrated in Figure <ref>. The jitter was mitigated by two changes to the system. First, the KPNO staff noticed a ticking sound corresponding to each rotation of the telescope drive worm gear, which was solved by lubrication. This step reduced the height of, but did not eliminate, the PSD peak. Second, we took a test observation in which only sidereal tracking was enabled, and all fine computer guiding was turned off. The peak was absent in this test observation, leading us to conclude that the telescope control system (TCS) was giving erroneous commands that “kicked” the telescope's position. The TCS was replaced in the winter of 2017 to allow the Robo-AO robotic system to fully control the telescope's motion, eliminating the remaining RA jitter (Figure <ref>). Figure <ref> shows a comparison of the Strehl ratios versus the seeing before and after the TCS upgrade. | http://arxiv.org/abs/1703.08867v1 | {
"authors": [
"Rebecca Jensen-Clem",
"Dmitry A. Duev",
"Reed Riddle",
"Maïssa Salama",
"Christoph Baranec",
"Nicholas M. Law",
"S. R. Kulkarni",
"A. N. Ramprakash"
],
"categories": [
"astro-ph.IM"
],
"primary_category": "astro-ph.IM",
"published": "20170326203512",
"title": "The Performance of the Robo-AO Laser Guide Star Adaptive Optics System at the Kitt Peak 2.1-m Telescope"
} |
compute,cere]A. P. Engsig-Karup [email protected], Office phone: +45 45 25 30 73compute]C. Monteserin[compute]Department of Applied Mathematics and Computer Science Technical University of Denmark, 2800 Kgs. Lyngby, Denmark. [cere]Center for Energy Resources Engineering (CERE) Technical University of Denmark, 2800 Kgs. Lyngby, Denmark. cles]C. Eskilsson [cles]Department of Shipping and Marine Technology Chalmers University of Technology, SE-412 96 Gothenburg, Sweden. We present a new stabilised and efficient high-order nodal spectral element method based on the Mixed Eulerian Lagrangian (MEL) method for general-purpose simulation of fully nonlinear water waves and wave-body interactions. In this MEL formulation a standard Laplace formulation is used to handle arbitrary body shapes using unstructured – possibly hybrid – meshes consisting of high-order curvilinear iso-parametric quadrilateral/triangular elements to represent the body surfaces and for the evolving free surface. Importantly, our numerical analysis highlights that a single top layer of quadrilaterals elements resolves temporal instabilities in the numerical MEL scheme that are known to be associated with mesh topology containing asymmetric element orderings. The 'surface variable only' free surface formulation based on introducing a particle-following (Lagrangian) reference frame contains quartic nonlinear terms that require proper treatment by numerical discretisation due to the possibility of strong aliasing effects. We demonstrate how to stabilise this nonlinear MEL scheme using an efficient combination of (i) global L^2 projection without quadrature errors, (ii) mild nonlinear spectral filtering and (iii) re-meshing techniques.Numerical experiments revisiting known benchmarks are presented, and highlights that modelling using a high-order spectral element method provides excellent accuracy in prediction of nonlinear and dispersive wave propagation, and of nonlinear wave-induced loads on fixed submerged and surface-piercing bodies.Nonlinear and dispersive free surface waves Wave-body interaction Marine Hydrodynamics Offshore Engineering Mixed Eulerian Lagrangian High-Order Spectral Element Method Unstructured mesh High-order discretisation. § INTRODUCTION In the last decades, numerical simulation of hydrodynamics for free surface flows and the design of marine structures has become an indispensable tool in engineering analysis. The numerical tools need to accurately and efficiently account for nonlinear wave-wave and wave-body interactions in large and representative marine regions in order to give reliable estimates of environmental loads on offshore structures such as floating production systems, ship-wave hydrodynamics, offshore wind turbine installations and wave energy converters.The development of numerical models for time-domain simulation of fully nonlinear and dispersive water wave propagation have been a research topic since the 1960s and with real engineering applications since the 1970s <cit.>. Models based on fully nonlinear potential flow (FNPF) have been considered relative mature for about two decades, see the review papers <cit.> and the references therein. More recent research focus on improved modelling by incorporating more physics and finally enabling large-scale simulation of nonlinear waves <cit.>. In many applications, computational speed is far more important than the cost of hardware and therefore algorithms that provide speed and scalability of work effort is of key interest.The Boundary Element Method (BEM) is a widely used method in applications for free surface flow <cit.> and flow around complex bodies due to the ease in handling complex geometry <cit.>. However, scalability has been shown to favour Finite Element Methods <cit.>. The use of Finite Element Methods (FEM) for fully nonlinear potential flow has also received significant attention starting with the original work of Wu & Eatock Taylor (1994) <cit.>. Studies of free surface solvers based on second-order FEM are, e.g. <cit.>. For wave-wave interaction and wave propagation with no bodies, the σ-transformed methods can be used to map the physical domain into a fixed computational domain <cit.> is maybe the most relevant approach due to the numerical efficiency. However, most numerical models for free surface flows use the Mixed Eulerian Lagrangian method (MEL) <cit.> for updating the free surface variables, especially for solving wave-body interaction problems. The MEL approach requires re-meshing of and development of techniques that improves the efficiency, e.g. QALE-FEM <cit.>, have been put forward. It is well-known that high-order discretisation methods can give significant reduction in computational effort compared to use of conventional low-order methods, especially for long-time simulations <cit.>. Higher-order schemes that has been proposed for FNPF models include high-order Finite Difference Methods (FDM) <cit.>, high-order BEM <cit.>, the High-Order Spectral (HOS) Method <cit.>and other pseudo-spectral methods <cit.>based on global basis functions in a single element (domain). Indeed, it has been shown that efficient algorithms <cit.> together with other means of acceleration (such as multi-domain approaches and massively parallel computing <cit.> via software implementations on modern many-core hardware <cit.>) render FNPF models practically feasible for analysis of wave propagation on standard work stations - even with a real-time perspective within reach for applications with appreciable domain sizes as discussed in <cit.>. Even though high-order FDM have shown to be very efficient, second-order FEM models remain popular because of the geometrical flexibility and the sparse matrix patterns of the discretisation. In contrast, a high-order finite element method, such as the Spectral Element Method (SEM) due to Patera (1984) <cit.>, has historically received the least attention for FNPF equations. However, SEM is thought to be highly attractive as it combines the high-order accuracy of spectral methods for problems with sufficiently smooth solutions with the geometric flexibility via adaptive meshing capability of finite element methods. A previous study employing SEM for the FNPF model include <cit.> where an Arbitrary Lagrangian Eulerian (ALE) technique was used to track the free-surface motion identified that numerical instabilities were caused by asymmetry in the mesh. In order to stabilise the model, they added a diffusive term proportional to the mesh skewness to the kinematic free surface condition. Recently, a SEM model allowing for the computation of very steep waves has been proposed <cit.>. The use of σ-transformed domains partitioned into a single vertical layer of elements is shown to avoid the fundamental problem of instabilities caused by mesh asymmetry. It was also found that the quartic nonlinear terms present in the Zakharov form <cit.> of the free surface conditions could cause severe aliasing problems and consequently numerical instability for marginally resolved or very steep nonlinear waves. This problem was mitigated through over-integration of the free surface equations and application of a gentle spectral filtering using a cap of 1% of the highest modal coefficient. Finally, in the context of SEM for FNPF models we also mention the HOSE method <cit.>, however, it is only the boundary domains that are discretised with SEM, not the interior domain. Thus, the challenge we seek to address in this work is the development of a robust SEM based FNPF solver with support for geometric flexibility for handling wave-body interaction problems. This is achieved by extending the results of <cit.>, keeping the same de-aliasing techniques. In order to include arbitrary shaped bodies, we discard the σ-transform in favour for the MEL method. To avoid the temporal stability issues associated with asymmetric mesh configurations near the free surface boundary <cit.> we propose to use hybrid meshes consisting of a layer of quadrilaterals at the surface and unstructured triangles below. Eigenvalue analysis of the semi-discrete formulation for small-amplitude waves is used to illustrate that this mesh configuration is linearly stable and during the simulation the vertical interfaces of the quadrilaterals are kept vertical. Global re-meshing is employed where the vertices of the mesh topology are repositioned similar to the technique used in the QALE-FEM method <cit.> to improve general stability of the model for nonlinear waves. This is combined with local re-meshing of the node distributions of the free surface elements to keep the discrete operators well-conditioned for all times, and is shown to increase the temporal stability significantly. In the following, several test cases are used to show the robustness and accuracy of the proposed SEM model based on MEL. §.§ On high-order methods for fully nonlinear potential flow models High-order methods allow for convergence rates faster than quadratic with mesh refinement, and is attractive for improving efficiency of numerical methods - especially for long-time integration <cit.>. Indeed, high-order discretisation methods are needed for realistic large-scale applications. To understand why, we need to first to understand that the key criterion for deciding between practical numerical tools is based on identifying the tool which is the most numerically efficient one, e.g. measured in terms of CPU time. In general, we can express the minimum work effort for a numerical scheme in terms of the discretization parameter asW ≤ c_1 h^-dwhere d is the spatial dimension of the problem and h a characteristic mesh size. At the same time, for a numerical method the spatial error is bound by ||e||≤ c_2 h^qwhere q characterise the order of accuracy (rate of convergence). For SEM the optimal order is q=p+1 with p the order of the local polynomial expansions. So, to be efficient, we desire that the error can be reduced much faster than the work increases, i.e. that ||e||→0 faster than W→∞ when resolution is increased (h→0). Thus,lim_h→0 W||e|| = c_3h^q-dindicating that for larger dimensions, it is necessary that a higher order spatial discretisation is used or that the work is balanced or minimised, e.g. through adapting the mesh resolution. So in three spatial dimensions (d=3) it is not possible to be efficient for large-scale applications with low-order methods since the convergence rate is q≤ 2 for a low-order method. Another key requirement is the development of methods which leads to scalable work effort, i.e. linear scaling with the problem size when subject to mesh refinement pursuing better accuracy. We see that this is only possible if the order of accuracy at least matches the dimension of the problem, i.e. q>d. §.§ Paper contribution The key challenge we seek to address in this work is the development of a FNPF solver with high-order convergence rate that also has support for geometric flexibility for handling arbitrary body shapes and other structures. To this end we exploit and extend the results of the recent work of Engsig-Karup, Eskilsson & Bigoni (2016) <cit.> on a stabilised σ-transformed spectral element method for efficient and accurate marine hydrodynamics applications. We consider in this work a Mixed Eulerian Lagrangian (MEL) method <cit.> based on explicit time-stepping, we discard the σ-transformation to be able to include arbitrary shaped bodies in the fluid domain, and propose a remedy to the temporal stability issues associated with the use of asymmetric mesh configurations near the free surface boundary as identified <cit.> for a classical low-order Galerkin finite element method and <cit.> using a high-order spectral element method for a fully nonlinear potential flow solver. These new developments constitutes a new robust SEM methodology for nonlinear wave-body interactions.§ MIXED EULERIAN-LAGRANGIAN (MEL) FORMULATION The governing equations for Fully Nonlinear Potential Flow (FNPF) is given in the following. Let the fluid domain Ω⊂ℝ^d (d=2) be a bounded, connected domain with piece-wise smooth boundary Γ and introduce restrictions to the free surface Γ^FS⊂ℝ^d-1 and the bathymetry Γ^b⊂ℝ^d-1. Let T:t≥ 0 be the time domain. We seek a scalar velocity potential functionϕ(x,z,t):Ω× T →ℝ satisfying the Laplace problem ϕ=ϕ̃,z = η onΓ^FS∇^2 ϕ =0,-h(x)<z<η inΩ ∂ϕ/∂ z+ ∂ h/∂ x∂ϕ/∂ x = 0,z=-h(x)onΓ^b where h(x):Γ^FS↦ℝ describes variation in the still water depth. The evolution of the free surface boundary is described by η(x,t):Γ^FS× T →ℝ. The notations are illustrated in Figure <ref>.The MEL time marching technique assumes a particle-following (Lagrangian) reference frame for the free surface particles with changes in position in time given byD x/Dt=∇ϕ,x∈Ωwhere ∇=(∂∂ x,∂∂ z) having assumed only two space dimensions.Thus, at the free surface boundary this implies kinematic conditions of the formDx/Dt = ∂ϕ/∂ x, Dη/Dt = ∂ϕ/∂ z = w,z=η,that must be satisfied together with the dynamic boundary condition stated in terms of Bernoulli's equationDϕ/Dt = 1/2∇ϕ·∇ϕ - gz-p,z=η, where the material derivative connects the Eulerian and Lagrangian reference frame through the relationD/Dt≡∂/∂ t + V·∇,where V is a velocity vector for the moving frame of reference that can be chosen arbitrarily. If V=0 we obtain an Eulerian frame of reference and if V=∇ϕ we have a Lagrangian frame of reference. Spatial and temporal differentiation of free surface variables are given by the chain rules ∂ϕ̃/∂ x = ∂ϕ/∂ x|_z=η + ∂ϕ/∂ z|_z=η∂η/∂ x, ∂ϕ̃/∂ t = ∂ϕ/∂ t|_z=η + ∂ϕ/∂ z|_z=η∂η/∂ t, and is useful for expressing the free surface equations valid at z=η(x,t) in terms of free surface variables only, cf. <cit.>, in the form Dx/Dt = ∂ϕ/∂ x = ∂ϕ̃/∂ x - w̃∂η/∂ x,Dη/Dt = w̃,Dϕ̃/Dt = 1/2((∂ϕ̃/∂ x)^2 - 2 w̃∂η/∂ x∂ϕ̃/∂ x + w̃^2(∂η/∂ x)^2 + w̃^2 ) - gη,having assumed a zero reference pressure at the free surface. The tilde '∼' is used to denote that a variable is evaluated at the free surface, e.g.w̃ = ∂ϕ∂ z = ∂ϕ∂ z |_z=η. Note, this formulation contains quartic nonlinear terms as in the Zakharov form <cit.> for the Eulerian formulation of FNPF equations. These nonlinear terms needs proper treatment to deal with aliasing effects.§.§ Boundary conditions For the solution of the Laplace problem, the following free surface boundary condition is specifiedΦ=ϕ̃onΓ^FS,while at fixed vertical boundaries, impermeable wall boundary conditions are assumedn· u =0 onΓ\ (Γ^FS∪Γ^b),where n=(n_x,n_z) denotes an outward pointing unit normal vector to Γ. At rigid vertical wall boundaries the domain boundary conditions at free surface variables imposed are∂_n η = 0, ∂_n ϕ =0 onΓ∩Γ^FS. Wave generation and absorption zones are included using the embedded penalty forcing technique described in <cit.>. § NUMERICAL DISCRETISATION Following <cit.>, we present the discretization of the governing equations in a general computational framework based on the method of lines, where first a semi-discrete system of ordinary differential equations is formed by spatial discretisation in two space dimensions using a nodal SEM. The semi-discrete system is subject to temporal integration performed using an explicit fourth-order Runge-Kutta method. §.§ Weak Galerkin formulation and discretisationWe form a partition of the domain Γ_h^FS⊆Γ^FS to obtain a tessellation 𝒯_h^FS of Γ^FS consisting of N_el non-overlapping shape-regular elements 𝒯_h^FS,k such that ∪_k=1^N_el𝒯_h^FS,k=𝒯_h^FS with k denoting the k'th element. We introduce for any tessellation 𝒯_h the spectral element approximation space of continuous, piece-wise polynomial functions of degree at most P, V={ v_h∈ C^0(𝒯_h); ∀ k ∈{ 1,...,N_el}, v_h|𝒯_h^k∈ℙ^P }.which is used to form finite-dimensional nodal spectral element approximationsf_h(x,t) = ∑_i=1^N^FS f_i(t) N_i(x),where {N_i}_i=1^N^FS∈ V is the N^FS global finite element basis functions with cardinal property N_i(x_j)=δ_ij at mesh nodes with δ_ij the Kronecker Symbol. §.§.§ Unsteady free surface equationsThe weak formulation of the free surface equations (<ref>) is derived in the following form. Find f∈ V where f∈{x,η, ϕ̃} such that∫_𝒯_h^FSDx/Dtv dx= ∫_𝒯_h^FS[ ∂ϕ̃/∂ x - w̃∂η/∂ x] v dx,∫_𝒯_h^FSDη/Dt v dx= ∫_𝒯_h^FS[w̃]vdx,∫_𝒯_h^FSDϕ̃/Dt v dx= ∫_𝒯_h^FS[ 1/2((∂ϕ̃/∂ x)^2 - 2 w̃∂η/∂ x∂ϕ̃/∂ x + w̃^2(∂η/∂ x)^2 + w̃^2 ) - gη]vdx,for all v∈ V. Substitute the expressions in (<ref>) into (<ref>) and choose v(x)∈ V. The discretisation in one spatial dimension becomes ℳDx_h/Dt = 𝒜_x ϕ̃_h - 𝒜_x^w̃_hη_h,Dη_h/Dt =w̃_h, ℳDϕ̃_h/Dt = -ℳ^g η_h + 1/2[ 𝒜_x^(∂ϕ̃/∂ x)_hϕ̃_h +ℳ^w̃_hw̃_h + 𝒜_x^w̃_h^2(∂η/∂ x)_hη_h ] -𝒜_x^w̃_h(∂η/∂ x)_hϕ̃_h, where the following global matrices have been introducedℳ_ij≡∫_𝒯_h^FSN_iN_j dx, ℳ^b_ij≡∫_𝒯_h^FSb(x) N_iN_j dx,(𝒜_x^b)_ij≡∫_𝒯_h^FSb(x) N_i dN_j /d xdx.where f_h∈ℝ^N^FS is a vector containing the set of discrete nodal values.Following <cit.>, the gradients of the free surface state variables are recovered via a global gradient recovery technique based on global Galerkin L^2(Γ^FS_h) projections that work for arbitrary unstructured meshes in the SEM framework as described in <cit.>. Aliasing effects are effectively handled using exact quadrature for nonlinear terms combined with a mild spectral filtering technique <cit.> that gently removes high-frequency noise that may arise as a result of marginal resolution.Remark: The free surface node positions are changing in time, and this implies that the mesh must change accordingly. Thus, the scheme needs to recompute the global spectral element matrices (<ref>) at every time step which impacts the computational efficiency of the scheme.§.§.§ Curvilinear iso-parametric elements To handle arbitrary body shapes we partition the fluid domain Ω_h to obtain another tessellation ℋ_h consisting of N_el^2D=N_el^Q+N_el^T non-overlapping shape-regular elements such that the tessellation can be formed by combining N_el^Q quadrilateral and N_el^T triangular curvilinear elements into an unstructured hybrid mesh such that ℋ_h= 𝒬_h∪𝒯_h = (∪_k=1^N_el^Q𝒬_k)∪(∪_k=1^N_el^T𝒯_k). In two space dimensions, the nodal spectral element approximations takes the formf_h(t, x) = ∑_i=1^n f_i(t) N_i( x).where n is the total degrees of freedom in the discretisation.The curvilinear elements makes it possible to treat the deformations in the free surface and the body surfaces as illustrated in Figure <ref> with two different hybrid unstructured meshes. In both cases a single quadrilateral layer is used just below the free surface.Consider the k'th element ℋ_h^k⊂ℋ_h. On this element, we form a local polynomial expansion expressed as f_h^k( x,t) = ∑_j=1^N_Pf̂_j^k(t)ϕ_j(Ψ_k^-1( x)) = ∑_j=1^N_P f_j^k(t)L_j(Ψ_k^-1( x)),where both a modal and a nodal expansion in the reference element is given in terms of N_p nodes/modes. We have introduced a map to take nodes from the physical element to a reference element, Ψ_k:ℋ_h^k→ℋ_r where ℋ_r is a single computational reference element. This dual representation can be exploited to form the curvilinear representations, since the coefficient vectors are related throughf_h = 𝒱f̂ , 𝒱_ij = ϕ_j( r_i),where ϕ_j, j=1,2...,N_P is the set of orthonormal basis functions and with nodes r={r_i}_i=1^N_p in the reference element that defines the Lagrangian basis. For quadrilaterals a tensor product grid formed by Legendre-Gauss-Lobatto nodes in 1D is used. For triangles, the node distribution is determined using the explicit warp & blend procedure <cit.>. The one-to-one mapping from a general curvilinear element to the reference element is highlighted in Figure <ref>. The edges of the physical quadrilateral elements are defined by the functions Γ_j, j=1,2,3,4, and by introducing iso-parametric polynomial interpolants <cit.> of the same order as the spectral approximations of the formℐ_NΓ_j(s) = ∑_n=0^N Γ(s_n)h_n(s),j=1,...,4,it is possible to represent curved boundaries.Using transfinite interpolation with linear blending <cit.> the affine transformation from the square reference quadrilateral to the physical domain is defined in terms of these edge curves in the formΨ_k(r,s) =12[ (1-r)Γ_4(s)+(1+r)Γ_2(s)+(1-s)Γ_1(r)+(1+s)Γ_3(r) ] -14[ (1-r){ (1-s)Γ_1(-1) + (1+s)Γ_3(-1)}. +. (1-r){ (1-s)Γ_1(1) + (1+s)Γ_3(1)}]. Similarly, using transfinite interpolation with linear blending, the transformation for triangles is given as Ψ_k(r,s) =12[ (1-s)Γ_1(r)-(1+r)Γ_1(-s)+(1-r)Γ_2(s)-(1+s)Γ_2(-r) ] +(1+(r+s)2)Γ_3(r) - (1+r)2Γ_2(-(1+r+s)) + 1+r2Γ_1(1) + r+s2Γ_1(-1).The use of curvilinear elements, implies that the Jacobian of the mapping is no longer constant as for straight-sided triangular elements. Thus, to avoid quadrature errors higher order quadratures are employed in the discrete Galerkin projections and this increases the cost of the scheme proportionally for the elements in question. Note, the introduction of curvilinear elements, can be setup in a pre-processing step and therefore does not add any significant additional complexity to the numerical scheme.§.§.§ Spatial discretisation of the Laplace problem Consider the discretisation of the governing equations for the Laplace problem (<ref>). We seek to construct a linear system of the formℒΦ_h =b, ℒ∈ℝ^n× n, Φ_h,b∈ℝ^n.The starting point is a weak Galerkin formulation that can be expressed as: find Φ∈ V such that ∫_ℋ_h∇·( ∇Φ)v dx = ∮_∂ℋ_h vn· (∇Φ) dx- ∫_ℋ_h( ∇Φ)· (∇ v) dx = 0,for all v∈ V where the boundary integrals vanish at domain boundaries where impermeable walls are assumed.The discrete system operator is defined asℒ_ij≡ -∫_ℋ_h (∇ N_j)· (∇ N_i) dx= -∑_k=1^N_el^2D∫_ℋ_h^k (∇ N_j)· (∇ N_i) dx .The elemental integrals are approximated through change of variables as∫_ℋ_h^k (∇ N_j)· (∇ N_i) dx= ∫_ℋ_r |𝒥^k| (∇ N_j ) · (∇ N_i) dr,where 𝒥^k is the Jacobian of the affine mapping χ^k:ℋ_h^k→ℋ_r. The global assembly of this operator preserves the symmetry, and the resulting linear system is modified to impose the Dirichlet boundary conditions (<ref>) at the free surface. The vertical free surface velocity w̃_h is recovered from the potential Φ_h via a global Galerkin L^2(ℋ_h) projection that involves a global matrix for the vertical derivative.§ NUMERICAL PROPERTIESWe start out by considering the numerical properties of the model related to the temporal stability and convergence of the numerical MEL scheme. Results of comparison with the stabilised Eulerian formulation <cit.> is included since the two schemes are complementary. §.§ Temporal linear stability analysis of semi-discrete system We revisit the analysis of temporal instability following the works of <cit.> by considering the semi-discrete free surface formulation that arise under the assumption of small-amplitude waves Dx/Dt = ∂ϕ̃/∂ x,Dη/Dt = w̃,Dϕ̃/Dt = - gη,The discretization of this semidiscrete systems leads to D/Dt[ [x;η; ϕ̃ ]] = 𝒥[ [x;η; ϕ̃ ]], 𝒥 = [ [ 0 0 ℳ^-1𝒜_x; 0 0𝒥_23; 0-g ℐ 0 ]] , where ℐ_ij=δ_ij and𝒥_23 = [(𝒟_z)_biϕ_i + (𝒟_z)_bbϕ_b] = [ (𝒟_z)_bb -(𝒟_z)_biℒ_ii^-1ℒ_ib]ϕ_b,where 𝒟_z = ℳ̂^-1𝒜̂_z,(𝒜̂_z)_ij≡∬_𝒯_h N_i dN_j /d zdxdz, M̂_ij≡∬_𝒯_hN_iN_j dxdz.and having introduced matrix decompositions of the global matrices of the formℒ = [ [ ℒ_bb ℒ_bi; ℒ_ib ℒ_ii ]],where the subscript indices 'b' refers to the free surface nodes and the 'i' refers to all interior nodes. The eigenspectrum of λ(𝒥) determines the temporal stability of this system. In the context of SEM, we seek to first confirm the results given in <cit.>. By doing a similar eigenanalysis using a triangulated asymmetric mesh, we confirm that we have temporal instability, e.g. see representative results in Figure <ref>. To fix this problem, we need to avoid asymmetric meshes near the free surface layer. Following <cit.> we can also consider a triangulated symmetric mesh in Figure <ref> which turns out to be stable for all polynomial orders except 3 and 4 used in the analysis. This is in line with the results and conclusions presented by Robertson & Sherwin (1999), but reveals that numerical issues may arise for such triangulated meshes in specific configurations. Furthermore, since our objective is to introduce arbitrary shaped bodies inside the fluid domain, in the next experiment we use a hybrid mesh. This mesh is consisting of a layer of quadrilateral below the free surface of the fluid similar to the meshes used in <cit.> and combine this layer with an additional triangulated layer used to represent complex geometries such as a submerged cylinder as illustrated in Figure <ref>. The eigenanalysis shows that we then reach temporal stability for arbitrary polynomial expansion orders. These results are in line with other experiments we have carried out that are not presented here, confirming that by introducing a quadrilateral layer we can fix the temporal stability problem to obtain purely imaginary eigenspectra (to machine precision). Using instead a similar mesh but with slightly skewed quadrilaterals, reveals again that the mesh asymmetry leads to temporal instability as shown in Figure <ref>. So, the temporal instability is associated with the accuracy of the vertical gradient approximation that is used to compute w̃ at the free surface and that determines the dispersive properties of the model. So poor accuracy in w̃ destroys the general applicability of the model since the wave propagation cannot be resolved accurately. This makes it clear that a quadrilateral layer with vertical alignment of nodes close to the free surface provides accurate recovery of the vertical free surface velocities and fixes the temporal instability problem. Remark:This new insight paves the road towards also stabilising nonlinear simulations using a high-order accurate unstructured method such as SEM following the recipe for stabilisations recently laid out in <cit.> for an Eulerian scheme based on quadrilaterals only. In the following, we consider the strenuous benchmark of nonlinear wave propagation of steep nonlinear stream function waves by setting up a mesh with periodicity conditions imposed to connect the eastern and western boundaries. Then, we examine the conservation of energy and accuracy to assess the stability of the model. This analysis shows that the exact integration used in the discrete global projections has a big effect on stabilising the solution. Following <cit.>, a test is performed for different amounts of filtering in the last mode and for different maximum steepness ((H/L)_max). For example, the analysis in Figure <ref> demonstrates that a 1% filter works well for the MEL Formulation. To maintain temporal stability when using explicit time-stepping methods, it is necessary to employ re-meshing<cit.>. The objective is to either prevent strict CFL restrictions although this is not severe for FNPF methods where numerical stability is influenced only weakly by spatial resolution in the horizontal <cit.>, and to retain a proper mesh quality to avoid temporal instabilities induced by the mesh or ill-conditioning in the global operators (discussed in <cit.>). Steep nonlinear waves are associated with larger node movements producing deformations in intra-node positions in the free surface elements. These deformations changes the conditioning of Vandermonde matrices and may impact the accuracy of the numerical scheme. This problem can be prevented by including a local re-mesh operation which changes the interior node distribution while not changing the initial mesh topology. For example, whenever an element goes below 75% or above 125% of its original size and with the initial free surface elements assumed of uniform size. The idea is to stop tracking the original material nodes at the free surface, and reposition the intra-element nodes through a local operation via interpolation to the original node distribution used in each element based on Legendre-Gauss-Lobatto nodes. A simple local re-mesh operation of this kind helps to stabilise the solver by improving accuracy in the numerical solutions and is a technique used in similar FEM solvers, e.g. see <cit.>. The effects of local re-meshing on temporal stability for a steep nonlinear stream function wave is presented in Figure <ref>, and shows that local re-meshing is essential to maintain temporal stability for longer integration times for steep nonlinear waves. A snapshot of a stream function wave after propagating for 50 wave periods the wave is still represented very accurately with essentially no amplitude or dispersion errors, cf. Figure <ref> (a). Furthermore, both mass and energy is conserved with high accuracy, cf. Figure <ref> (b). Also, we compare the MEL method and the Eulerian method due to <cit.> in terms of different stabilisation techniques in Figure <ref>. We find that with our current strategies the stabilised MEL method is more robust than the stabilised Eulerian scheme.§.§ Convergence testsTo validate the high-order spectral element method, we demonstrate for the MEL scheme that the high-order convergence rate is 𝒪(h^P) in line with <cit.> and the accuracy of the method using convergence tests as depicted in Figure <ref>. These tests, demonstrate that we can exert control over approximations errors in the scheme by adjusting the resolution in terms of choosing the points per wave length. For the most nonlinear waves of 90% of maximum steepness, the curves find a plateau at the size of truncation error before convergence to machine precision due to insufficient accuracy in our numerical solution of stream function waves used.§ NUMERICAL EXPERIMENTSWe examine different strenuous test cases that serve as validation of the numerical spectral element model proposed. §.§ Reflection of high-amplitude solitary waves from a vertical wallWe setup a solitary wave as initial condition using the high-order accurate spectral numerical scheme due to <cit.> and consider the propagation of solitary waves of different amplitudes above a flat bed that are reflected by a vertical solid wall. In this experiment, a solitary wave approaches the wall with constant speed and starts to accelerate forward when the crest is at a distance of approximately 2h from the wall. The water level at the wall position grows leading to the formation of a thin jet shooting up along the wall surface. When the maximum of free surface elevation is reached at the wall position it is said that the wave is attached to the wall, at this moment t=t_a and the height of the crest is η_a. Thereafter, the jet forms and reaches its maximum run up η_0 at time t_0. After this event, the jet collapses slightly faster than it developed. The detachment time t_d corresponds to the wave crest leaving the wall. The height of the wave at that instant is η_d, always smaller than the attachment height (i.e. η_d < η_a < η_0). Then a reflected wave propagates in the opposite direction with the characteristics of a solitary wave of reduced amplitude. This adjustment in the height produces a dispersive trail behind the wave. The depression becomes more abrupt for increasing wave steepness. This test case requires the fully nonlinear free surface boundary conditions to capture the steepest solitary waves and their nonlinear interaction with the wall. The case was first studied using a numerical method by Cooker et al. <cit.>, where a Boundary-Integral Method (BIM) was used to solve the Euler equations with fully nonlinear boundary conditions. The results obtained using BIM showed to be in excellent agreement with the experimental data given in <cit.>, however, showed difficulties in the down run phase for the steepest wave of a/h=0.7 indicating that for the most nonlinear cases, numerical modelling is difficult. Accurate simulation of the steepest wavesrequires sufficient spatial resolution and high accuracy in the kinematics to capture the finest details of the changing kinematics of the fluid near the wall. This problem was recently been addressed using other high-order numerical models, e.g. a high-order Boussinesq model with same fully nonlinear free surface boundary conditions based on a high-order FDM model <cit.> and a nodal discontinuous Galerkin spectral element method <cit.> and both studies showed excellent results for the wave run-up and depth-integrated force histories up to a/h=0.5. The numerical experiments are carried out using a domain size x∈[-22.5, 22.5] m, with the initial position of the center of the solitary wave at x_0=0 m. We employa structured mesh consisting of quadrilaterals with N_x elements in a single layer, where N_x∈[40,120] is varied proportional to the wave height to resolve the waves in the range a/h∈[0.2,0.6]. Each element is based on polynomial expansion orders (P_x,P_z)=(6,7).The time step size is chosen in the interval Δ t∈[0.01, 0.02] s. Stabilisation of the numerical scheme is achieved using exact integration and mild spectral filtering using a 1% top mode spectral filter every time step using the MEL scheme. The attachment analysis of <cit.> is reproduced using SEM with the initial profile of the solitary wave <cit.> up to a/h = 0.6 with excellent agreement, cf. Figures <ref> and <ref>. This time history of the wave propagation is illustrated using the MEL scheme in Figure <ref> for a high-amplitude wave with relative amplitude a/h=0.5. The forces are computed numerically as a post-processing step by integration of the local pressure obtained from Bernoulli's equationp = - ρ g z - ρ∂ϕ/∂ t - ρ1/2∇ϕ·∇ϕ,along the wet part of the wall. Approximation of the time derivative ∂ϕ/∂ t is approximated by using the acceleration potential method due to <cit.>. This method estimates ϕ_t by solving an additional boundary value problem based on a Laplace problem defined in terms of ϕ_t in the form∇^2ϕ_t = 0, inΩsubject to the boundary conditions ϕ_t =-gη-1/2|∇ϕ|^2, onΓ^FS ∂ϕ_t/∂ n = 0, onΓ^bThe wall force vector is determined numerically from pressure usingF = ∫_∂Ω pn dS.§.§ Rectangular obstacle piercing the free surface In Lin (2006) <cit.> a numerical method for Navier-Stokes equations is proposed based on transformation of the fluid domain using a multiple-layer σ-coordinate model. Wave-structure interactions of a solitary wave of amplitude a/h=0.1 (mildly nonlinear) with a rectangular obstacle fixed at different positions (seated, mid-submergence, floating) are investigated in a two dimensional wave flume of depth h=1 m. Free surface elevation histories at three different gauge locations are found in excellent agreement with computed VOF solutions. We consider here the experiment corresponding to the obstacle piercing the free surface (width: 5 m, height: 0.6 m, draught: 0.4 m) for validation of the present methodology dealing with semi-submerged bodies.A wave flume of 150 m is sketched in Figure <ref> with the cylinder located at its center (x=0 m). In this scenario, positions of gauges 1 and 3 in <cit.> correspond to x_G1=-31 m and x_G3=26.5 m, respectively. This domain is discretised using hybrid meshes with a top layer of curvilinear quadrilaterals. Below, unstructured grid of triangles is employed in the central part (x ∈ [-4.5; 4.5] m), fitting the body contour (Triangles size is chosen proportional to their distance to the lower corners of the body). The zones away from the structure x ∈ [-75.0; -4.5] m ∪ [4.5 ; 75.0] m are meshed with regular quadrilaterals of width 1 m, small enough to guarantee accurate propagation of this wave according to our tests in Section <ref>. A detail of this mesh corresponding to x ∈ [-10.0 ; 10.0] m is shown in Figure <ref>.Similar to previous experiments with solitary waves, the initial condition (η_0,ϕ̃_0) is generated using the accurate numerical solution due to <cit.> corresponding to a wave peaked at x = -55 m. Using the MEL scheme, the time step size is Δ t=0.02 s. The evolution of the free surface elevations at gauge positions are compared with the Lin (2006) results in Figure <ref>, and excellent agreement is observed.§.§ Solitary Wave Propagation Over a Submerged Semi-circular CylinderWe consider the numerical experiment described in <cit.> on the interaction between a solitary wave of amplitude a/h=0.2 (mildly nonlinear) and a submerged semi-circular cylinder with radius R/h = 0.3. This experiment serves to validate our force estimation, where the acceleration method due to Tanizawa (1995) is used. The initial mesh is illustrated in Figure <ref> and consists of 60 elements in the central part and only 164 elements in total. The computed time evolution of the free surface is given in Figure <ref> (a). In Figure <ref> (d) it is seen how the solitary wave propagates undisturbed from the initial condition until it starts the interaction with the semi-circular cylinder resulting in variation in the horizontal dynamic load. During the interaction, a dispersive trail develops after which the solitary wave restore to it's original form. The results are in good agreement with results of Wang et al. The differences in the force curves given in Figure <ref> (c) is understood to be related to numerical dispersion in the Desingularized Boundary Integral Equation Method (DBIEM) and approximation errors associated with the low-order accurate initial solitary wave. The force curve produced using the SEM solver is for a well-resolved flow and therefore is considered converged to high accuracy. In same figure, the force curve for Morison Equation that takes into account also viscous effects is included and is computed following <cit.>. In figure (b) it is confirmed that the SEM conserves with high accuracy mass and energy during all of the simulation.§.§ Solitary wave propagation over a fixed submerged cylinder Several studies with both experimental and numerical results can be found regarding the interactions of solitary waves with a submerged circular cylinder <cit.>. In the following, we consider two previously reported experiments.First, we consider the introduction of a fixed submerged cylinder inside the fluid domain with a flat bed following the experiments due to Clement & Mas (1995) <cit.>. Analysis of the hydrodynamic horizontal forces exerted on the cylinder and comparison with experimental results <cit.> serve as validation in the experiments reported here. Figure <ref> shows a typical initial mesh for the experiments. The mesh consist of a small quadrilateral layer near the free surface and contains zones of quadrilateral elements only in first and last part of domain. These zones flank a central part, where a hybrid meshing strategy is employed to adapt using an unstructured triangulation to the submerged cylinder surface in the interior. The initial conditions η_0, ϕ_0 for the solitary waves is produced using the method described in <cit.>. In our first experiment, the solitary wave height is a/h=0.286 and corresponds to zone 1 described in <cit.> where interaction is said weak. The cylinder has dimensionless radius D/(2h) = 0.155 and is positioned with submergence z_0/h = -0.29. Figure <ref> shows the resulting time series of the free surface, where we observe a small deformation of the wave during passing the cylinder. This deformation becomes wider at the end and more abrupt in positions just above the cylinder (centered at x=0). The computed mass and energy conservation measures given in Figure <ref> confirms stability and accuracy of the simulation. The dimensionless horizontal component is compared in Figure <ref> with experimental results of <cit.> given in <cit.>. The results are in good agreement for times before the collision (when horizontal force is zero) and significant discrepancies afterwards where viscous effects are important due to vortex shedding in the experiments <cit.>. These effects cannot be captured by potential flow models like the present one.In our second experiment, the wave height is a/h=0.5 and the cylinder has radius D/(2h)=0.25 and is positioned at a submergence z_0/h =-0.5. Following <cit.> this experiment is located in the crest-crest exchange zone 2. In this zone incident and transmitted waves can be clearly distinguished. The fluid motion in this experiment is studied when the wave passes the cylinder. In Figure <ref> and Figure <ref> the horizontal and vertical velocity components in a portion of the fluid domain (-5≤ x ≤ 5) are illustrated at simultaneous times highlighting the changes in the sub-surface kinematics during the solitary wave interaction with the cylinder. While approaching the position of the cylinder the wave looses height and slows down, cf. Figure <ref> and Figure <ref>. Then the birth of a transmitted wave can be observed, cf. Figure <ref>. The transmitted wave is not shifted and leaves behind a dispersive trail that becomes even more abrupt than observed in compared to wave interaction in our first experiment, cf. Figure <ref>. § CONCLUSIONS We have presented a new stabilised nodal spectral element model for simulation of fully nonlinear water wave propagation based on a Mixed Eulerian Lagrangian (MEL) formulation. The stability issues associated with mesh asymmetry as reported in <cit.> is resolved by using a hybrid mesh with a quadrilateral layer with interfaces aligned with the vertical direction to resolve the free surface and layer just below this level. Our linear stability analysis confirms that the temporal instability associated with triangulated meshes can be fixed, paving the way for considering remaining nonlinear stability issues. By combining a hybrid mesh strategy with the ideas described in <cit.> on stabilisation of free surface formulation with quartic nonlinear terms, the model is stabilised by using exact quadratures to effectively reduce aliasing errors and mild spectral filtering to add some artificial viscosity to secure robustness for marginally resolved flows. This is combined with a re-meshing strategy to counter element deformations that may lead to numerical ill-conditioning and in the worst cases breakdown if not used. Our numerical analysis confirm this strategy to work well for the steepest nonlinear water waves when using this stabilised nonlinear MEL formulation. In the MEL formulation we avoid the use of a σ-transform of the vertical coordinate, to introduce bodies of arbitrary geometry that is resolved using high-order curvilinear elements of an unstructured hybrid mesh. By using these elements, only few elements are needed to resolve the kinematics in the water column both with and without complex body surfaces. With this new methodology, we validate the model by revisiting known strenuous benchmarks for fully nonlinear wave models, e.g. solitary wave propagation and reflection, wave-body interaction with a submerged fixed cylinder and wave-body interaction with a fixed surface-piercing structure in the form of a pontoon. The numerical results obtained are excellent compared with other published results and demonstrate the high accuracy that can be achieved with the high-order spectral element method. In ongoing work, we are extending the new stabilised spectral element solvers towards advanced and realistic nonlinear hydrodynamics applications in three space dimensions (cf. the model based on an Eulerian formulation in three space dimensions<cit.>) by extending the current computational framework to also handle moving and floating objects of arbitrary body shape. In ongoing work, we consider freely moving bodies and surface-piercing structures with non-vertical boundary at the body-surface intersections and hybrid modelling approaches <cit.>.§ REFERENCESplain | http://arxiv.org/abs/1703.09697v1 | {
"authors": [
"A. P. Engsig-Karup",
"C. Monteserin",
"C. Eskilsson"
],
"categories": [
"physics.comp-ph",
"math.DS",
"G.1"
],
"primary_category": "physics.comp-ph",
"published": "20170327204140",
"title": "A Stabilised Nodal Spectral Element Method for Fully Nonlinear Water Waves, Part 2: Wave-body interaction"
} |
Hadron detection with a dual-readout fiber calorimeter J. Park^m, F. Scuri^b, A. Sill^a and R. Wigmans^a,Corres December 30, 2023 ============================================================ Let BS(1,n)= ⟨ a,b : a b a ^-1 = b ^n⟩ be the solvable Baumslag-Solitar group for n ≥2. We study representations of BS(1, n) on the plane by orientation preserving homeomorphisms, assuming that a acts as a linear map and b as a map with bounded displacement. We find that the possibilities for a faithful action depend greatly on the Jordan canonical form of the map h defined by the action of a. In case h is diagonalizable over , we shall give examples or prove rigidity theorems depending on the eigenvalues. We also show some rigidity in the cases where h is elliptic or parabolic. Then we give applications to the actions of BS(1, n) by homeomorphisms of the torus. IntroductionFor n, m∈\{0}, the Baumslag-Solitar group BS(m,n) is defined by the presentation BS(m,n)= ⟨ a,b : a b^m a ^-1 = b ^n⟩. These groups were introduced by Baumslag and Solitar in <cit.> to provide the first examples of two generator non-Hopfian groups with a single defining relation. The groups BS(1,n) for n ≥2 are the simplest examples of infinite non-abelian solvable groups, and also provide examples of distorted elements, which are related to Zimmer's conjecture <cit.> and play an important role in surface dynamics <cit.>.Recently there has been an interest on understanding the possible dynamics of Baumslag-Solitar group actions on surfaces, the results usually showing restrictions to these types of actions. J. Franks and M. Handel <cit.> proved that on a surface S of genus greater than one, any distortion element in the group ^1_0(S,area)is a torsion element,and therefore there are no faithful representations of BS(1,n) in ^1_0(S,area).N. Guelman and I. Liousse <cit.> constructed a smooth BS(1, n) action without finite orbits on ^2 that is not locally rigid, but on the other hand, they proved that there are no minimal faithful actions of BS(1, n) on ^2. J. Alonso, N. Guelman and J. Xavier <cit.> proved that there are no faithful representations of BS(1, n) on surfaces where a acts by a (pseudo-)Anosov homeomorphism with stretch factor λ > n. In the case of the torus, they showed that there are no faithful actions where a acts by an Anosov map and b by an area-preserving map. This last result also holds without the area preserving hypothesis if the action is C^1, as shown by Guelman and Liousse in <cit.>. They also show that any faithful action of BS(1, n) on a closed surface where the action of b is C^1 and a acts by a (pseudo-)Anosov map h has a finite orbit contained in the set of singularities of h. In the general scope, these works lie in the context of studying the obstructions to the existence of faithful group actions on certain spaces (a survey of these ideas can be found in<cit.>). Another result in this direction is that of S. Hurtado and J. Xue in <cit.>, concerning Abelian-by-cyclic groups, which generalize the groups BS(1,n). They found that if such a group acts on the torus ^2 by C^r-diffeomorphisms, with r>2, non-finite image and containing an Anosov map, then the action is topologically conjugated to a linear action. In this paper we mainly consider actions of BS(1,n), for any n≥ 2,on the plane by orientation preserving homeomorphisms. To give such an action is the same as giving f,h:^ 2 →^2 orientation preserving homeomorphisms satisfying hfh^-1 = f ^nnamely, f and h are the respective images of b and a under the representation. We compensate for the lack of compactness by supposing that the distance from f to the identity map is uniformly bounded, that is,that there exists K>0 such that|f(x)-x|≤ K for all x∈^ 2We say that f has bounded displacement if this condition holds. Note that this is the case when f is the lift of a map of the torus that is isotopic to the identity. The main motivation for this hypothesis is to consider actions on the plane that are lifted from certain actions on the torus, as we shall see in Section <ref>. In this work we will be interested in the case when h is a linear map. Observe that if we conjugate the action by A∈ GL(2,) we get another action in our hypotheses, since AfA^-1 has bounded displacement with constant ||A||K. Thus we may assume that the associated matrix of h is a canonical form over . First we focus on the case when h is diagonalizable over , thus assuming that its associated matrix is D=([ λ 0; 0 μ ]) λ,μ∈where λμ>0 since h is an orientation preserving homeomorphism. Excluding the cases when either |λ| or |μ| equals 1, our results can be summarized as follows, where actions are assumed to be faithful: λμ 0<|μ|<1 1<|μ|≤ n n<|μ| 0<|λ|<1 No action: Theorem <ref>Product actionNo action: Lemma <ref>1<|λ| ≤ n Product action Strange Rotation action and Product action Preserves horizontal foliation: Corollary <ref>. Product action. n<|λ| No action: Lemma <ref> Preserves vertical foliation: Corollary <ref>. Product action. No action: Lemma <ref> That is, we give examples or prove rigidity theorems depending on the eigenvalues of h. The rigidity theorems are either non-existence of faithful actions, or preservation of a foliation of lines, as indicated in the table. The examples show when such rigidity statements do not hold, namely, the product action (Section <ref>) is a faithful action that preserves a foliation by straight lines, while the strange rotation action (Section <ref>) is a faithful action that does not preserve such a foliation. If one of the eigenvalues is ± 1, say |λ|=1, then|μ|≠ 1 for h to have infinite order. In this case, the construction of the product action for 1<|μ|≤ n still holds, and we give an example of a faithful action for |λ|=1 and |μ|<1 in Section <ref>.For |λ|=1 and |μ|>n there are no faithful actions, by Lemma <ref>.For h∈ SL(2,) we study the other possible canonical forms. In the elliptic case we show non-existence of faithful actions, in Corollary <ref>. For the parabolic case we show that f fixes every point in the eigenspace of h (Corollary <ref>). In particular, if the eigenvalue of h is 1, then its eigenspace is (h), and hence it is the set of global fixed points of the action. This will allow us to prove that there are no faithful actions of BS(1,n) on the torus where a acts as a Dehn twist. Preliminaries We shall use some results from surface homeomorphism theory that we state in this section. For an homeomorphism f we denote by (f) its set of fixed points, and by Ω(f) its non-wandering set, which, we recall, contains the recurrent points of f and in particular its periodic points. The following is the classic Brouwer's Theorem on planar homeomorphisms: Let f:^2→^2 bean orientation preserving homeomorphism with Ω(f)≠∅. Then f has a fixed point.Let f be a surface homeomorphism. A periodic disk chain for fis a finite set U_0, U_1, … U_n-1 of pairwise disjoint topological disks such that for all i=0, …, n-1 we have f(U_i)∩ U_i = ∅, and there exists m_i such that f^m_i (U_i)∩ U_i+1≠∅, where indexes are taken modulo n. Below is Franks's <cit.> adaptation of Brouwer's theorem to periodic disk chains:Let f:^2→^2 bean orientation preserving homeomorphism which possesses a periodic disk chain.Then f has a fixed point. Let f: A → A be an orientation preserving homeomorphism of the open annulusA= S^1× (0,1), and f̃ a lift of f to the covering space à = × (0,1). We will say that there is a positively returning disk for f̃ if there is an open topological disk U⊂Ãsuch that f̃ (U)∩ U= ∅ and f̃ ^n (U) ∩ (U+k)≠∅ for some n, k > 0, where U + k denotes the set {(x+k, t ): (x, t )∈ U}.A negatively returning disk is defined similarly but with k < 0. The following result is Theorem (2.1) in <cit.>: Suppose f: A→ Ais an orientation preserving homeomor- phism of the open annulus which is homotopic to the identity, and satisfies the following conditions:* every point of A is non-wandering, * f has at most finitely many fixed points, and * there is a lift of f to its universal covering space f̃ : Ã→à which possesses both a positively and a negative returning disk, both of which are lifts of some disks in A.Then f has a fixed point. We say that a homeomorphism is recurrent if there exists a sequence n_k, k→∞, such that d(f^n_k, 𝕀)→ 0 uniformly. Next we state a result on sphere homeomorphisms due to Kolev and Pérouème <cit.>: Let f:S^2→ S^2 be a non trivial recurrent orientation-preserving homemorphism.Then f has exactly two fixed points. Basic computations Throughout most of the paper we consider a planar action of BS(1,n)for some n≥ 2, given byorientation preserving homeomorphisms f,h:^2→^2 satisfying the group relation hfh^-1=f^n. The following is an easy consequence of the group relation. Let f and h be bijections such that hfh ^-1 = f ^n.Then:* h^m f h ^-m = f ^n^m for all m≥ 1; * h^-1 ( (f))⊂ (f).Fix an isotopy (f_t)_t∈ [0,1] such that f_0 = 𝕀 and f_1 = f.For all x∈^ 2 we denote by γ _x the arc t↦ f_t (x), t∈ [0,1].More generally, for x∈^ 2 and m≥ 1 define γ ^m_x = ∏ _i= 0 ^m-1γ _f^i (x), where the product is concatenation of arcs. The next lemmas relate to transverse measured foliations that have some compatibility with the action. Let ν be a transverse (signed) measure to an oriented foliation of the plane ^ 2, which can be extended to measure any curve with the following properties: * ν (βγ) = ν (β) + ν (γ), and* ν(β) = ν (γ) if β and γ have the same endpoints.We assume ν satisfies the following further conditions, that relate to the action:* ν (h(γ))= a ν (γ), for some a∈, a≠ 0, * the map x →ν (γ _x) is bounded on ^ 2. Note that the first item implies that h preserves the foliation, i.e. takes leaves to leaves. There are general consequences in the cases when |a|>n and when |a|<1, which shall lead to our rigidity results. If |a|>n, then ν (γ _x )≡ 0. Note that the arcs h^ m γ _h^ -m(x) and γ ^ n^m_x have the same endpoints by item 1 in Lemma <ref>.Therefore,|a|^ m |ν (γ _h^ -m(x))| = |ν (h^ m γ _h^ -m(x))| = |ν (γ ^ n^m_x)| = |∑ _i=0^n^m-1ν (γ _f^i(x))|≤ Cn^ m ,where C is the bound for the map x→ν (γ_x). So,|ν (γ _h^ -m(x))|≤(n/|a|)^ m C,for all m≥ 1 and all x∈^ 2. Since the right term of the last equation is independent of x, if |a|>n we have that ν (γ _x )≡ 0. If F is the foliation of ^2 that has ν as a transverse measure, and |a|>n, then each leaf of F is f-invariant.For all m≥ 1, we have |ν (γ ^ n^m_x)|≤ C |a|^ m.In particular, if |a|<1, then lim_m→∞ν (γ_x^ n^m) = 0 uniformly on x.As before,|ν (γ ^ n^m_x)| = |a|^ m |ν (γ _h^ -m(x))| ≤ C |a|^ m. Rigidity in the diagonal case Let f,h:^2→^2 be orientation preserving homeomorphisms with hfh^-1=f^n, and assume that f has bounded displacement and h is the linear map whose associated matrix isD=([ λ 0; 0 μ ])i.e. h(x,y)=(λ x,μ y). We devote this section to proving the rigidity results announced in the table shown in the Introduction. In the next section we construct examples showing that the hypotheses of these results cannot be relaxed. Observe that dx is a transverse measure to the foliation of ^2 by vertical lines, and that it satisfies the conditions given in Section <ref> with a=λ. The same is true for dy and the horizontal foliation, with a=μ. Then Corollary <ref> gives us that * If |λ|>n, the vertical foliation is preserved by ⟨ f,h⟩.* If |μ|>n, the horizontal foliation is preserved by ⟨ f,h⟩.Next we turn to non-existence of faithful actions under certain conditions on the eigenvalues, where we actually show something stronger, namely that f must be the identity map.If|λ|>n and |μ| > n, then f=𝕀.Using Corollary <ref> with a =λ and ν=dx, one obtains that each vertical lineis preserved by f. So each f-orbit is contained in a vertical line, and using Corollary <ref> with a =μ and ν=dy, one obtains that each horizontal line is also preserved by f. The result follows.If|λ|>n and |μ| <1, then f=𝕀. As in the previous lemma, each f-orbit is contained in a vertical line. Thus for any x∈^2, applying Lemma <ref> with ν=dy shows that x is a recurrent point for the restriction of f to the vertical line containing x. So x must be fixed by f, since line homeomorphisms do not have recurrent points that are not fixed. If|λ|=1 and |μ| >n, then f=𝕀.First we assume that λ=1, μ>n. Using Corollary <ref> with a =μ and ν=dy, we get that f preserves each horizontal line. Moreover, since it has bounded displacement, f preserves the orientation of each horizontal line. Let L=×{0} be the horizontal axis, and note that h|_L=𝕀_L since λ=1. Then we deduce that f|_L^n-1=𝕀_L, and since f|_L is an orientation preserving line homeomorphism, we must have f|_L=𝕀_L.Suppose f≠𝕀, and let p=(x_0,y_0) be a point with f(p)≠ p. We write the restriction of f to the horizontal by p asf(x,y_0)=(ϕ(x),y_0)where ϕ:→ is an orientation preserving homeomorphism. Using that f=h^-kf^n^kh^k for k≥ 0 we see thatf(x,μ^-ky_0)=(ϕ^n^k(x),μ^-ky_0)and since (x_0,μ^-ky_0)→_k(x_0,0) and f fixes (x_0,0),we get by continuity that ϕ^n^k(x_0)→_k x_0. This is absurd, as an orientation preserving line homeomorphism cannot have a recurrent point that is not fixed.For the case λ=-1, μ<-n, we consider the action ⟨ f,h^2 ⟩ of BS(1,n^2). Observing that h^2 has eigenvalues 1 and μ^2>n^2, we get that f=𝕀 from the previous case. If |λ|<1 and |μ| <1, then f= 𝕀. We shall prove this theorem through a series of lemmas. We provide two proofs, one reliant on the work of Oversteegen and Tymchatyn <cit.> classifying recurrent homeomorphisms of the plane, and the other on the theorems of Franks that are stated in Section <ref>.f is recurrent. Moreover, we have d(f^ n^m, 𝕀)→_m 0 uniformly.By Lemma <ref>, dx (γ ^ n^m_x) → _m 0,anddy (γ ^ n^m_x) → _m 0uniformly.The result follows.It follows from <cit.> that f is periodic. This, combined with the fact that nontrivial periodic homeomorphisms have unbounded displacement gives a proof of Theorem <ref>. We include an alternative proof below, which is a nice application of some theorems of Franks relying on Brouwer theory.If f≠𝕀 then (f) =(h) = {(0,0)}. By Theorem <ref>, if f≠𝕀, then f has exactly one fixed point x (we are extending f to S^2 fixing ∞).As the set (f) is h^-1- invariant (by Lemma<ref> item 2.), x must be fixed also by h, which implies that x=(0,0).By Lemma <ref> both h and f map the annulus A= ^2∖{(0,0)} onto itself, thus defining an action of BS(1,n) on A. Moreover, the restriction f|_A:A→ A is isotopic to the identity, since it preserves orientation and both ends of the annulus. So we can assume that for x∈ A the curves γ _x are contained in A. We will need the following general statement for annulus maps, which is a consequence of Theorem <ref>:Let g be a homeomorphism of the open annulus S^ 1× (0,1) that is isotopic to the identity and such that (g) = ∅. Suppose that x∈ S^ 1× (0,1) is a recurrent point for g, and let G be a lift of g to the universal cover× (0,1).If (n_m)_m∈ is any sequence such that g^ n_m(x)→ x, then for any lift x̃ of x we have that(G^n_m(x̃))_1→ + ∞ (G^n_m(x̃))_1→ - ∞where the subindex 1 stands for the projection onto the first coordinate of ×(0,1).Let p:× (0,1)→ S^1× (0,1) be the covering projection. Consider U⊂ S^1× (0,1) a neighbourhood of x such that p^ -1(U)= ∪_α U_α, each U_α projecting homeomorphically onto U. Taking a smaller neighbourhood if necessary, we may assume that g(U)∩ U =∅, since g has no fixed points. Let Ũ be the connected component of p^-1(U) that contains x̃.Since g^ n_m(x)→ x, for any sufficiently large m there exists k_m ∈ such that G^ n_m (x̃)∈Ũ + k_m, in the notation of Section <ref>. Note that it suffices to prove that either k_m→_m ∞ or k_m→_m -∞.Since g has no fixed points, the same is true for G, and by Theorem <ref> we see that G has no periodic disk chains, which implies that k_m≠ 0 for all m.Moreover, the sign of k_m is constant for all m, for otherwise g would have a fixed point by Theorem <ref>.Applying Theorem <ref> again, this time for the returning disks Ũ+k_m, we see that either k_m+1-k_m > 1 for all m, or k_m+1-k_m < -1 for all m. The result follows. Next we show that every f-orbit must turn infinitely many times around this annulus. To give a precise statement, we consider dθ where θ is the angular coordinate on A= ^2∖{(0,0)}, which is well defined on A, and is a transverse measure to the foliation of A by radial lines. Then we show the following.Assume that f≠𝕀. Then for all x∈ A, we have that either dθ (γ_x^ n^ m)→_m +∞ or dθ (γ_x^ n^ m)→_m -∞.Note that f|_A does not have fixed points, by Lemma <ref>. Consider the universal cover p:Ã→ A, and take any lift F:Ã→à of f|_A. We identify A with S^1× (0,1) via homeomorphism, so à is identified with × (0,1). Note that proving the lemma is equivalent to show that for all x̃∈à we have that either (F^ n^m (x̃))_1 →_m ∞or (F^ n^m (x̃))_1 →_m -∞. This is obtained from Lemma <ref>, since for x=p(x̃) we have f^ n^m(x)→_m x by Lemma <ref>. The next result will allow us to lift the action on A to the universal cover Ã. We state it in general, relaxing the conditions on h we assumed for this section. Let ⟨ f,h⟩ be a BS(1,n)-action on ^2 where h is linear and f has bounded displacement and fixes (0,0). Then there exist F,H: Ã→Ã, lifts of f|_A and h|_A respectively, such that HFH^-1= F^n.We consider D the compactification of the plane with the circle of directions.Since h is linear, it extends to D. (For instance, if h(x,y)=(λ x,λ y), then h extends as the identity on ∂ D if λ>0, or as the antipodal map if λ<0.) On the other hand, f also extends to D acting as the identity on the boundary, on account of the bounded displacement hypothesis. We restrict f and h to D∖{(0,0)}, which is homeomorphic to the annulus with one boundary component S^1×(0,1]. So we may take lifts F, H: × (0,1]→× (0,1] of these maps, and we can choose F to be the identity on ×{1}.Now HFH^-1 and F^n are both lifts of the same map f^n and therefore they must be equal, as they coincide (with the identity) on ×{1}.Next we seek to reduce Theorem <ref> to the case with λ = μ. To do this we shall take a suitable conjugation of the action, by the map described in our next result. Suppose that |μ|<|λ|.Let β= log |μ|/log|λ| and consider the map Φ(x,y) = {[(x,y^β)y≥0; (x,-|y|^β)y<0 ].Then f_1= Φ^-1 f Φ has bounded displacement. Notice that since |μ|<|λ|<1 we get that0<1/β<1. From this we can obtain that Φ^-1 is quasi-Lipschitz, namely that |Φ^-1(p) - Φ^-1(q)|≤ |p-q|+C. Thus we have |f_1(p)-p| = |Φ^-1 f Φ(p) - Φ^-1Φ(p)|<|fΦ(p)-Φ(p)| + C ≤ K+Csince |f(q)-q|≤ K for all q.If Theorem <ref> holds for λ=μ, then it also holds forλ≠μ.We suppose without loss of generality that|μ|<|λ|. Then we consider the map Φ defined in Lemma <ref>, which is an homeomorphism, and a direct computation shows thatΦ^-1 h Φ(x,y)=(λ x,λ y)Also, Φ^-1 f Φ has bounded displacement by Lemma <ref>. So by hypothesis we get that Φ^-1 f Φ = 𝕀, which implies that f= 𝕀. The assumption that λ=μ yields that dθ(hγ)=dθ(γ) for any curve γ in A.We will also need to control the map x→ dθ(γ_x) for x∈ A. Note, however, that bounded displacement for f does not imply that this map is bounded, because of the singularity of dθ at the origin. For our proof of Theorem <ref> it will suffice to bound it far from the origin.For every r>0 there is C_r≥ 0 such that |dθ(γ_x)|≤ C_r for all x∈^2∖ B((0,0),r).Let D be the compactification of ^2 used in the proof of Lemma <ref>, and recall that f extends to D as the identity on ∂ D. Thus the map x→ dθ(γ_x) extends continuously to D∖{(0,0)} by setting it to 0 on ∂ D. The result then follows from the compactness of D∖ B((0,0),r). We are now ready to give our second proof of Theorem <ref>: By Corollary <ref>, we assume that λ=μ.Applying Lemma <ref> (item 1) to the lifted action given by Lemma <ref>, we deduce that for all m≥ 1 and x∈ A the curves γ^n^m_x and h^ mγ _h^-m(x) are homotopic in A with fixed endpoints. Then we havedθ (γ^n^m_x) = dθ(h^ mγ _h^-m(x)) = dθ (γ _h^-m(x))recalling that dθ is h-invariant, because we assumed λ=μ. Since |λ|<1, we see that for m large enough h^-m(x) is outside some fixed ball centered at (0,0). So by Lemma <ref> we have that |dθ (γ^n^m_x)|=|dθ (γ _h^-mx)|≤ C_r for some C_r≥ 0 and all m≥ 1. This contradicts Lemma <ref> unless f=𝕀. Examples for the diagonal caseAs we have seen in the previous section, if neither |μ| nor |λ| belong to the interval [1,n], then there is no faithful action of BS(1,n) by planar orientation preserving homeomorphisms with a acting by h(x,y)=(μ x, λ y). In this section we construct examples of such actions when 1<|λ| ≤ n (the casewhere 1<|μ| ≤ n is symmetric). The first family of examples, the product actions, work for any μ≠ 0 and preserve the horizontal foliation. The second construction needs both |λ| and |μ| in (1,n], and gives examples that do not preserve a foliation by lines. We also produce an example with |λ|=1 and |μ|<1.§.§ Product actionsRecall that BS(1,n)=⟨ a,b| aba^-1=b^n⟩. In general, if φ:BS(1,n)→ Homeo_+() is a faithful action on the real line, then we can obtain a faithful action of BS(1,n) on the plane by defining, for g∈ BS(1,n),ψ(g)(x,y)=(φ'(g)(x),φ(g)(y)),where φ' is any (not necessarily faithful) BS(1,n)-action on the line. This is the product action of φ and φ'. Clearly, ψ(b) above has bounded displacement if and only if φ(b) and φ'(b) have bounded displacement. In order to obtain an action on the plane in which ψ(a) is linear, we need to restrict our attention to actions on the line where φ(a) and φ'(a) are linear maps. That is, φ(a)(x)=λ x and φ'(a)(x)=μ x, for some λ,μ∈.When λ=n, then the affine action of BS(1,n) on the line is a faithful action in which the map φ(b)(x)=x+1 has bounded displacement. Hence, the maps h(x,y) = (μ x, ny) and f(x,y) = (x, y+1) provide an example of a a faithful planar BS(1,n)-action. The same holds for f(x,y)=(x,y+c(x)) for any continuous and bounded function c that is not identically zero. If 1<λ<n, one may try to conjugate the affine action above by a homeomorphism of the line of the form c: x↦ x^α, so that c∘φ(a)∘ c^-1(x)=n x. The problem is that then the corresponding φ(b)does not have bounded displacement (this is easy, and we leave it to the reader). So in this case we need to consider a different action on the line. According to <cit.>, up to semi-conjugacy, there is only one other candidate, which we now describe.Let φ(a)(x)=λ x, initially assuming only that λ>0. For k∈ define the fundamental domains D_k={x∈|λ^k≤ x<λ^k+1}, and notice that ^+=(0,+∞)=⋃_k D_k. Let ϕ:→ Homeo_+(D_0) be a continuous homomorphic embedding, i.e. a continuous flow on D_0. For x∈ D_0 we define φ(b)(x)=ϕ(1)(x), and for x∈ D_k we define φ(b)(x)=φ(a^k)∘ϕ(1/n^k)∘φ(a^-k)(x). This ensures that φ extends to a representation of BS(1,n) into Homeo_+(^+), that can be further extended to the whole line by reflection, with 0 as a global fixed point. This action is in fact faithful, see for instance <cit.>. The construction also works for λ<0 with minor changes: using |λ| in the definition of D_0, and defining D_k=φ(a^k)D_0. (In this case ⋃_kD_k is a disconnected union of intervals, instead of ^+.)Now we assume that 1<|λ|≤ n, and we want to see when φ(b) has bounded displacement. Assume for a while that ϕ satisfies(†) There is K>0 such thatm| ϕ(1/m)(x)-x|≤ Kfor all x∈ D_0 and all m∈.Under this assumption, we show that φ(b) has bounded displacement. Indeed, take x∈ D_k with k≥ 0, noting that the case for k< 0 is obtained by compactness, since D_k⊂ [-1,1], and the case for -x∈ D_k is analogous. Defining x_0=φ(a^-k)(x)=λ^-kx∈ D_0, we have|φ(b)(x)-x|= |φ(a^k)∘ϕ(1/n^k)∘φ(a^-k)(x)-x|=|φ(a^k)∘ϕ(1/n^k) (x_0)-φ(a^k)(x_0)|=|λ|^k|ϕ(1/n^k)(x_0)-x_0|≤n^k|ϕ(1/n^k)(x_0)-x_0|≤ K,Then, to produce a planar BS(1,n)-action in our hypotheses, it is enough to consider the mapsh(x,y)=(μ x, λ y),f(x,y)=(x,φ(b)(y)). To finish the construction, we only need to exhibit an -action on D_0 satisfying (†).This can be done by taking ϕ as the flow of a C^1 vector field X on D̅_0 = [1,λ] with X(1)=X(λ)=0. Let K=max |X|. Then m|ϕ(1/m)(x)-x|=m |∫_0^1/m X(ϕ(s)(x)) ds| ≤m ∫_0^1/m |X(ϕ(s)(x))|ds ≤ mK1/m = K. Notice that we can choose X to be non-zero on the interior of D_0, so that the flow ϕ is conjugate to a translation on D_0. §.§ Strange rotationsIn this section we assume |λ|, |μ|∈(1,n], and by symmetry we may take |μ|≤λ|. First we consider the case where λ=μ>0, so we have h(x,y) = (λ x,λ y). Letα = log_λn = log n/logλ, and define f by the following formula in polar coordinates: f(r,θ)= (r,θ + r^-α). Observe that this map is an orientation preserving homeomorphism, though it clearly is not differentiable at the origin. To show it has bounded displacement, recall that in polar coordinates we have ((r,θ_1),(r,θ_2))≤ r |θ_1-θ_2|, wherestands for the Euclidean distance. Then we get|f(p)-p| ≤ r· r^-α = r^1-αwhere p is the point with polar coordinates (r,θ). Since 1<λ≤ n we have α≥ 1, so Equation (<ref>) implies that when r≥ 1 we have |f(p)-p|≤ 1. When r≤ 1 we have |f(p)-p|<2, for f preserves the unit disk.It is also easy to verify the Baumslag-Solitar relation in polar coordinates, noting that h(r,θ)=(λ r,θ):hfh^-1(r,θ)=hf(λ^-1r,θ) =h(λ^-1r,θ+(λ^-1r)^-α)= (r,θ+λ^αr^-α)=(r,θ+nr^-α)=f^n(r,θ)recalling that λ^α=n by definition of α. This example follows the spirit of the construction in <ref>, where the fundamental domains would be D_k:={(x,y)∈^2|λ^k≤ |(x,y)|<λ^k+1}. Still, in this case we obtain a genuinely two dimensional action. It is faithful by the same argument used in <cit.> for the action on the line, noting that the action of f on D_0 has free orbits. (r^-α is irrational for many r∈[1,λ).)The same map f works also for λ=μ<0, taking α=log n/log |λ|. This time we have h(r,θ)=(|λ|r,θ+π), and the same computation follows. Now we adjust this example to the case when |μ|<|λ|, both in the interval (1,n]. Let ⟨ f,h ⟩ be the action just constructed, i.e. h(x,y)=(λ x,λ y) and f as above, and let h_1(x,y) = (λ x,μ y). Consider β=log_|λ| |μ| = log |μ|/log|λ|, and the map Φ(x,y) = (x,(y)|y|^β) which was introduced in Lemma <ref>. It is straightforward to compute that h_1 = Φ h Φ^-1, so we will just conjugate our previous example by Φ. It remains to check that f_1= Φ f Φ^-1 has bounded displacement. Notice that 0<β<1, since 1<|μ|<|λ|, thus the same argument used for Lemma <ref> gives us the desired result. (This is the reverse implication to the one in Lemma <ref>, where we had |μ|<|λ|<1 and thus β>1.)§.§ Examples with eigenvalue ± 1Let us assume that λ=1 and 0<μ<1, so we have h(x,y)=(x,μ y). Fix some K>0 and consider the setsD_k = [-K,K]× [μ^k+1,μ^k)for k∈. Note that D_k=h^kD_0, that they are disjoint and that ⋃_k D_k = [-K,K]×^+, i.e. they work as the fundamental domains used for the previous examples. This construction will follow the same ideas. Let ϕ be a continuous flow supported on the interior of D_0 that preserves each horizontal line, e.g. the flow of an horizontal vector field supported on D_0. We can assume that ϕ(s)→𝕀 uniformly as s→ 0, by the same argument used for (†) in <ref>. Then we define f(x)=ϕ(1)(x) for x∈ D_0, and f(x)=h^k∘ϕ(1/n^k)∘ h^-k(x) for x∈ D_k, which defines an action of BS(1,n) by homeomorphisms of [-K,K]×^+. This action is faithful by the argument in <cit.>. We extend f to the plane by setting f(x)=x for x∉ [-K,K]×^+. Continuity at the points in {± K}×^+ is clear, so we must check it at the points in [-K,K]×{0}, where the condition that μ<1 comes in. Indeed, since μ<1 we see that a small neighbourhood of [-K,K]×{0} only meets D_k for k>0 large enough. On the other hand we note that sup{|f(x)-x|:x∈ D_k} = sup{ |ϕ(1/n^k)(y)-y|:y∈ D_0 }since ϕ is horizontal and h preserves the first coordinate. Hencesup{|f(x)-x|:x∈ D_k}→_k 0as k→+∞, showing continuity. Note that f is clearly bijective, and the same argument above proves continuity for f^-1. Equation (<ref>) also shows that f has bounded displacement, in fact, this construction gives |f(x)-x|<2K for all x∈^2. A similar construction can be done for λ=-1 and -1<μ<0.Elliptic caseThroughout this section we let f,h:^2→^2 be orientation preserving homeomorphisms with hfh^-1=f^n, and assume that f has bounded displacement and h is the linear map whose associated matrix isR=([cos(θ)sin(θ); -sin(θ)cos(θ) ])where θ∈ [0,2π]. We assume that θ/2π is irrational, so that h has infinite order. In this section we prove: In the hypotheses above, we have f=𝕀.Which implies, by conjugation by a linear map, our rigidity result in the elliptic case:There is no faithful BS(1,n)-action by planar orientation preserving homeomorphisms where the action of b has bounded displacement and a acts by an elliptic linear map. Next we develop the lemmas needed to prove Theorem <ref>. For r>0, we let S_r denote the circle centered at the origin with radius r. If x∈(f), then S_|x|⊂ (f).If x∈(f), we have by Lemma <ref> (item 2), that h^-m(x)⊂ (f) for all m≥ 0.Since h is an irrational rotation, the h - backward orbit of x is denseinS_|x|, which implies that S_|x|⊂ (f). (f)≠∅.Note thatf^n^m(0,0) = h^m (f (0,0)) for all m≥ 1, by Lemma <ref> (item 1). If (0,0) is fixed by f we already have the result. If f(0,0) is different from (0,0), we note that it is arecurrent point for h, since h is an irrational rotation. It then follows from Equation (<ref>)that f(0,0) is a recurrent point for f, and by Brouwer's Theorem <ref> there exists x∈(f).It is worth noting that Lemmas <ref> and <ref> do not need the bounded displacement condition on f. Neither does the following. (0,0)∈(f).Let r_0 = inf{|x|: f(x)=x}, noting that r_0<∞ by Lemma <ref>, and that it is actually the minimum of this set, by continuity of f.We shall show that r_0 = 0, for then we would have f(0,0) = (0,0), as desired. Suppose then that r_0>0. By Lemma <ref> we get that S_r_0⊂(f), hence f preserves D_r_0, the open disk of radius r_0, which is also h-invariant, as h is a rotation. We can then do the same argument of Lemma <ref> on the disk D_r_0, observing that Brouwer's theorem holds since the open disk is homeomorphic to the plane, and we obtain a fixed point of f in D_r_0. This contradicts the definition of r_0.By Lemma <ref>, both h and f can be restricted to the annulus A= ^2∖{(0,0)}, and they define an action of BS(1,n) on A. Just as in Section <ref>, we see that f|_A is isotopic to the identity on A, as it preserves orientation and both ends of the annulus, so we can assume that γ_x lies in A for each x∈ A. We also consider dθ as in Section <ref>, and note that since h is a rotation, we havedθ(hγ)=dθ(γ) for any curve γ on A.(f|_A)≠∅.Assume that (f|_A) = ∅, and choose a sequence k_i→_i +∞ such that h^k_i→𝕀 uniformly in compact sets.Then f^n^k_i=h^k_ifh^-k_i→ f uniformly in compact sets, so for any x∈ A we can apply Lemma <ref> to deduce that|dθ(γ_x^n^k_i)|→_m +∞ by the same argument used for Lemma <ref>, since we assumed that f has no fixed point in A. On the other hand, as we did when proving Theorem <ref> at the end of Section <ref>, we use Lemma <ref> to lift the action on A to the universal cover Ã, and then we apply Lemma <ref> (item 1) to this lifted action to deduce that γ_x^n^m and h^mγ_h^-m(x) are homotopic in A for all m≥ 1 and x∈ A. This gives us thatdθ (γ^n^m_x) = dθ(h^ mγ _h^-m(x)) = dθ (γ _h^-m(x))and since h is a rotation, we have h^-m(x)∈ S_|x| for all m. Hence, by compactness, we get that |dθ (γ^n^m_x)|≤ C_x where C_x>0 is a bound that does not depend onm. This contradicts Equation (<ref>), proving the lemma. For 0≤ r<s≤+∞ we consider the annulusA_r,s= {x∈ ^2: r<|x|<s}which is clearly h-invariant. If A_r,s is also f-invariant, the same proof of Lemma <ref> applies to A_r,s instead of A, hence we have the following:If A_r,s is f-invariant, then (f|_A_r,s)≠∅.We are now ready to prove Theorem <ref>: Suppose f≠𝕀, so we have ^2∖ (f)≠∅. Then by Lemma <ref> we can write ^2∖ (f) = ⋃ _α A_α, where A_α=A_r(α),s(α) for some 0≤ r(α)<s(α)≤+∞. Note that each A_α is f-invariant, but contains no fixed points of f by definition, thus contradicting Remark <ref>.Parabolic case §.§ Rigidity on the eigenspaceIn this section we assume that f,h:^2→^2 are orientation preserving homeomorphisms with hfh^-1=f^n, such that f has bounded displacement, and that h is the linear map whose associated matrix isP=([ 1 0; 1 1 ]),i.e. h(x,y)=(x,x+y). Denote the vertical axis by L={(0,y):y∈}, noting that L=(h). Then we shall show: In the conditions above, we have L⊂(f). We show a non-trivial example of such an action in <ref>. Assuming Theorem <ref> we can show the following:Let φ be a BS(1,n)-action by planar orientation preserving homeomorphisms, where φ(b) has bounded displacement and φ(a) is a parabolic linear map. Then the eigenspace of φ(a) is contained in (φ(b)).Recall that the linear map φ(a) is parabolic when its associated matrix is conjugated in GL(2,) to either P orP'=([ -10;1 -1 ]),and in any case the eigenspace is a line, and can be written as (φ(a^2)).Note that the associated matrix of φ(a^2) is conjugated to P in both cases, and that ⟨ b,a^2 ⟩≅ BS(1,n^2). Thus conjugating the action of ⟨ b,a^2 ⟩ and applying Theorem <ref> yields the result.In the rest of this section we prove Theorem <ref>, through a series of lemmas.If the f-orbit of x∈ L is bounded, then f^j(x)∈ L for all j.Note that the h-orbit of any y∉ L is unbounded. Then, if x∈ L and f^j(x)∉ L for some j, we see thatf^jn^k(x)=h^kf^jh^-k(x)=h^kf^j(x)is unbounded as k→∞.In particular, Lemma <ref> applies to points in L that are periodic for f.If x∈ L and f^j(x)∈ L for some j≠ 0, then f(x)∈ L.Observe that if x and f^j(x) belong to L, then f^nj(x)=hf^jh^-1(x)=f^j(x).So f^j(n-1)(x) = x, and by Lemma <ref> we deduce thatf(x)∈ L.For x∈ L there are two possibilities:* If f(x)∈ L, then f^n-1(x) = x and f^j(x)∈ L for all j.* If f(x)∉ L, then f^j (x)∉ L for all j≠ 0. For the first item, note that if x and f(x) are in L, then Equation (<ref>) holds with j=1, giving f^n-1(x) = x, and then apply Lemma <ref>. The second item follows directly from Lemma <ref>.From Lemma <ref> one inmediately deduces the following: (f^n-1)∩ L = {x∈ L: f(x)∈ L}. Let us introduce some notation. PutH^+={(x,y):x>0} H^-={(x,y):x<0}which are respectively the right and left components of ^2∖ L. These are the right and left sides of L, and moreover, we can speak about right and left sides of any oriented and properly embedded topological line on the plane, in particular of f^j(L) for j∈, which are f^j(H^+) and f^j(H^-) respectively. Note that since f^j preserves orientation and has bounded displacement, the right side of f^j(L) meets the right side of L, and the same for the left sides. Letα = sup{ |(f(x))_1|:x∈ L} B=[-α, α]×where the subindex 1 stands for the first coordinate. By definition we have f(L)⊂ B, and note that α≤ K, where K is the bound on the displacement of f. Finally, letC=(f^n-1)∩ L = {x∈ L: f(x)∈ L}i.e. the set in Corollary <ref>, and observe that it is closed. It is also f-invariant: If x∈ C then f^j(x)∈ L for all j by Lemma <ref>, and from f(x),f^2(x)∈ L we deduce that f(x)∈ C, while from f^-1(x),x∈ L we get f^-1(x)∈ C. We shall focus on this set, with the objective of proving that it agrees with L and is contained in (f), thus proving Theorem <ref>.If C=∅, then f^j(L)⊂ B for all j.By definition, if C=∅ then f(L)∩ L =∅. Thus f(L) is contained in either H^+ or H^-. Assume that f(L)⊂ H^+, as the other case is symmetrical. Since f preserves orientation and has bounded displacement, we see that f(H^+)⊂ H^+, and deduce recursively that f^j+1(L) is in the right side of f^j(L) for all j. On the other hand, the fact that f(L)⊂ B implies that for k≥ 1 we havef^ n^k(L)= h^kf(L)⊂ h^k (B) = B.Then for j≥ 1 we pick k with j≤ n^k and note that f^j(L) lies at the right side of L and the left side of f^n^k(L), which intersect in a region contained in B, since f^n^k(L)⊂ B. For j≤ -1 we argue similarly.In fact, the situation described in Lemma <ref> does not happen, as we see next: C is non-empty.Take p∈ L and write f(p)=(a,b). Then we havef^n^k(p) = h^kf h^-k(p) = h^kf(p) = (a,b+ka).On the other hand:f^n^k(f(p)) = h^kf h^-k(a,b) = h^kf(a,b-ka).We define (a_k,b_k) = f(a,b-ka), and so we can writef(f^n^k(p)) = f^n^k(f(p)) = (a_k,b_k+ka_k) Since the displacement of f is bounded by K, when we compare the second coordinates we must have|b_k - b +k(a_k-a)|≤ K,and on the other hand, since (a_k,b_k) = f(a,b-ka), we get|b_k - b + ka|≤ K.From the equations above and the triangular inequality, we deduce that |k(a_k-2a)|≤ 2K. Hence a_k → 2a as k→ +∞. Now we assume that C=∅, aiming to reach a contradiction. Note that in this case α>0, and thus we may choose p∈ L so that |a|>α/2.Since a_k →_k 2a, for this p we have |a_k|>α when k is large enough. On the other hand, Lemma <ref> implies that f^n^k+1(p)∈ B for all k, and then Equation (<ref>) gives that |a_k|≤α for all k. So we get a contradiction. C is connected. Suppose that C is not connected and let I be a bounded complementary interval of C in L. By Lemma <ref> (second point) we see that f^j(I) is disjoint from L for all j≠ 0, and hence the arcs f^j(I) for j∈ are pairwise disjoint. On the other hand we have C=Fix(f^n-1)∩ L, in particular the endpoints of I are fixed by f^n-1, so all the closed arcs f^j(n-1)(I) for j∈ have these same endpoints. Then we take k>0 and note thatf^n^k(n-1)(I) = h^kf^n-1h^-k(I) = h^kf^n-1(I)which cannot be disjoint from f^n-1(I) by a standard Jordan curve argument (see Figure <ref>), that we sketch as follows:We show it for k large, which is sufficient. Divide the arc f^n-1(I) as concatenation of γ and δ, where q=γ(1)=δ(0) has maximum horizontal coordinate, i.e. |q_1| is maximum among the points inf^n-1(I). For the sake of notation assume that f^n-1(I) is at the right side of L, the other case is symmetric. Then h^kf^n-1(I) is contained in the band B'=[0,q_1]× for all k≥ 0. We see that δ splits B' into at least two components, and γ is contained (except for the endpoint q) in the lower unbounded component of B'∖δ. If k is large enough then h^k(q) is in the upper unbounded component of B'∖δ, so h^kγ must intersect δ, as it joins h^kγ(0)=γ(0) with h^k(q) inside B'. This means that f^n^k(n-1)(I)=h^kf^n-1(I) interescts f^n-1(I), thus reaching a contradiction and concluding the proof. Lemma <ref> allows us to writeL=I∪ C∪ Jwhere, for some -∞≤ a ≤ b ≤ +∞, we have I={0}× (-∞,a) and J={0}× (b,+∞). We want to show that I and J are empty. It shall be useful to definef̂=f^n-1K̂=(n-1)K.Note that ⟨f̂,h⟩ is also an action of BS(1,n) by orientation preserving homeomorphisms, where f̂ has bounded displacement with constant K̂, and we have C=(f̂)∩ L. If J≠∅ then f̂(J) meets J exactly at (0,b), so J∪f̂(J) is a line, which is contained in the K̂-neighbourhood of J by the bounded displacement condition. Then there is exactly one component of ^2∖ (J∪f̂(J)) that is contained in the K̂-neighbourhood of J, and we denote it by D_J. Note that D_J can also be defined as the component of ^2∖ (J∪f̂(J)) that is contained in one side of L, namely the side that contains f̂(J). If I≠∅ we can define D_I analogously. These sets are sketched in Figure <ref>.Assume that J≠∅. Then: * The sets f̂^j(D_J) are pairwise disjoint for j∈. * f̂^j(D_J) is in the same side of L as f̂(J) for j≥ 0, and on the opposite side of L for j≤-1.The corresponding statements hold also for I, in case it is non-empty.We need to recall some facts about orientation in the plane. If γ and δ are oriented, properly embedded rays starting at a point p∈^2, the sectors defined by them are the components of ^2∖(γ∪δ). If R is one of those sectors, we can say whether R is at the right or the left of γ, according to whether R lies at the right or left side of any (oriented, properly embedded) linethat extends γ and does not meet R. Note that this is well defined, and that one sector is at the right of γ and left of δ, while the other is at the left of γ and right of δ. If g is an orientation preserving homeomorphism that fixes p, theng takes the sectors defined by γ and δ to the sectors defined by gγ and gδ, preserving the left and right sectors of each ray. Give J the orientation of L (i.e. upwards), so we can see it as a ray starting at (0,b). Recalling that (0,b)∈ C⊂(f̂), we can observe that for j∈ the sets f̂^j(J) are all oriented, properly embedded rays starting at (0,b). The set D_J is one of the sectors defined by J and f̂(J).For simplicity of notation we assume that f̂(J)⊂ H^-, i.e. is at the left side of L, but the argument will work for the other case as well. Under this assumption we have D_J⊂ H^- by definition, and it is the sector at the left of J and the right of f̂(J), according to the definition above. First we show that f̂^-1(J)⊂ H^+: Since f̂^-1 preserves orientation and fixes (0,b), we have that f̂^-1(D_J) is the sector at the right of J and left of f̂^-1(J). Hence if f̂^-1(J) was in the left side of L, it would have to contain H^+, but this cannot happen: Recalling that D_J is in the K̂-neighbourhood of J by definition, we see that f̂^-1(D_J) must be in a 2K̂-neighbourhood of J by bounded displacement, but on the other hand H^+ is not contained in any proper neighbourhood of J. So we have f̂^-1(J)⊂ H^+, and moreover f̂^-1(D_J)⊂ H^+, in particular it is disjoint from D_J.We will show the lemma by induction on |j|, adapting the argument we just made. The base case j=0 is trivial by the definition of D_J. So we take j>0 and assume that for |i|<j the sets f̂^i(D_J) are pairwise disjoint, and contained in H^- for 0≤ i <j and in H^+ for -j<i<0. We need to show that f̂^j(D_J)⊂ H^-, that f̂^-j(D_J)⊂ H^+, and that these sets are disjoint from f̂^i(D_J) for |i|<j (and from each other, but that is given by the previous statements, as H^+ and H^- are disjoint). We give the argument for f̂^j(D_J), the one for f̂^-j(D_J) is analogous.Since f̂^j-1(D_J)⊂ H^- by induction hypothesis, we see thatf̂^j(J) ⊂ H^-, as this curve is in the boundary of f̂^j-1(D_J) and cannot meet L. We consider the line L_j=I∪ C∪f̂^j(J) (which is oriented and properly embedded), and note that H^+ and f̂^i(D_J) for |i|<j are in the right side of L_j. So we need to show that f̂^j(D_J) is in the left side of L_j. For this we proceed as before: We use that f̂ preserves orientation and fixes (0,b) to deduce that f̂^j(D_J) is the sector at the left of f̂^j(J) and the right of f̂^j+1(J). Thus if this sector was not contained in the left side of L_j, it would have to contain I∪ C. On the other hand, f̂^j(D_J) is contained in the (j+1)K̂-neighbourhood of J, which does not contain I∪ C. Thus we have finished the induction step, observing that a similar argument works for f̂^-j(D_J), adjusting the notation as necessary.Note that the argument just presented works for the case whenf̂(J)⊂ H^+ changing notation appropriately. Alternatively, since we did not use the map h, it is valid to replace f̂ by f̂^-1. The result for I, instead of J, is also analogous.Note that Lemma <ref> (item 1) implies that D_J cannot meet f̂^j(J) for any j. * If J≠∅, then f̂(J) is in the left side of L.* If I≠∅, then f̂(I) is in the right side of L. The key fact will be that h translates upwards in H^+ and downwards in H^-. Assume that J≠∅ and f̂(J)⊂ H^+, i.e. is in the right side of L. We will show that h^k f̂(J) intersects D_J for k large enough, which contradicts Lemma <ref> because h^k f̂(J)=f̂^n^k(J), as we remarked above. Consider a vertical band of the formB'=[0,ϵ]×for ϵ>0 small enough so that f̂(J) meets L'={ϵ}×. Note that B'∖f̂(J) has a unique lower unbounded component, and clearly D_J∩ B' is a different component of B'∖f̂(J). Let γ be a small arc with γ(0)∈ J and γ((0,1]) contained in D_J∩ B'. Then h^-kγ is contained in B' for all k, with h^-kγ(0)=γ(0), and since γ(1)∈ H^+ we have that h^-kγ(1) must belong to the lower unbounded component of B'∖f̂(J) for k large enough. Hence h^-kγ must intersect f̂(J), since it joins different components of B'∖f̂(J) inside B'. Therefore h^kf̂(J) meets γ((0,1]), which is in D_J.The second point is analogous, noting that D_I is unbounded in the downwards direction, in which h moves the points in H^-. We are ready to finish the proof of Theorem <ref>: First we show that C=L, i.e. that I and J are empty.Combining both items of Lemma <ref> we see that, assuming J≠∅, we have D_J=H^-∩f̂(H^+), which is to say that f̂^-1(D_J) is the set of the points of H^+ that are mapped to H^- by f̂. Take any p∈f̂^-1(D_J), and note that for k large enough we have h^-k(p)∉f̂^-1(D_J), since p∈ H^+ and f̂^-1(D_J) is in the K̂-neighbourhood of J. It follows that f̂ h^-k(p)∈ H^+, and so h^kf̂ h^-k(p)∈ H^+. On the other hand, applying Lemma <ref> we see that the pointsp∈f̂^-1(D_J) have f̂^j(p)∈ H^- for all j>0.This is a contradiction, as we just showed that f̂^n^k(p)=h^kf̂ h^-k(p) was in H^+. This shows that J=∅, and the same argument works for proving that I=∅.So we have C=L, and are ready to show that L⊂(f). Recall that C is f-invariant, by Lemma <ref> and the definition of C. So we can see f|_L as a line homeomorphism, that preserves the orientation of L since f has bounded displacement. Then f|_L^n-1=f̂|_L = 𝕀_L, and this implies f|_L=𝕀_L by the general theory of orientation preserving line homeomorphisms, thus finishing the proof. §.§ ExampleHere we give an example of a faithful action ⟨ f,h⟩ of BS(1,n) on the plane, with h(x,y)=(x,x+y) as in <ref> and f an orientation preserving homeomorphism with bounded displacement. We follow the same idea we used in the constructions of <ref>, settingD_0={(x,y):0≤ y < |x| }and D_k=h^kD_0 for all k∈. Note that these sets are pairwise disjoint and ⋃_k D_k =^2∖ L, where L is the vertical axis. As in <ref>, given a continuous flow ϕ:→ Homeo_+(D_0) we can define an action of BS(1,n) on ⋃_k D_k by the formula f(x)=h^k∘ϕ(1/n^k)∘ h^-k(x) for x∈ D_k. This action is faithful if ϕ(1) has a free orbit, as shown in <cit.>.We extend f to the plane by setting f(x)=x for x∈ L. Then we need to provide a flow ϕ on D_0 that has a free orbit, and so that f is an homeomorphism with bounded displacement. We take ϕ:→ Homeo_+(D_0) so that: * ϕ(s) preserves each vertical line for all s∈, and* |ϕ(s)(x)-x|<K for all s∈ and x∈ D_0, for some constant K>0.There are many examples of such flows that have free orbits, e.g. the flow defined by a non-zero vertical vector field supported in D_0, that vanishes on the horizontal lines L_j = ×{ y_0+jK} for all j∈ and some fixed y_0.Note that h is a translation on each vertical line, so the conditions on ϕ imply that f has bounded displacement with constant K. To show that f is an homeomorphism, observe that since ϕ is supported on D_0 we have|ϕ(s)(x)-x|<|x_1|x∈ D_0, s∈where x_1 is the first coordinate of x. Since h is a translation on each vertical line, this implies|f(x)-x|<|x_1|x∈^2∖ L.This shows that f and f^-1 are continuous at the points in L, which are the only points where continuity was not immediate. Thus we have built the desired example.A similar construction can be done for h_1(x,y)=(-x,x-y), the other canonical form for a parabolic map, takingD_0={(x,y):x>0, 0≤ y <x}∪{(x,y):x<0, x< y ≤ 0}which is a fundamental domain for h_1 acting on ^2∖ L.Applications to actions on the torus In this section we consider actions of BS(1,n) by homeomorphisms of the torus ^2. As before, such an action is given by two homeomorphisms f,h:^2→^2 satisfying hfh^-1=f^n. In order to apply our results on planar actions, we would like to lift the action on ^2 to the universal cover ^2→^2. So we cite the relevant results in this direction, which are based on Theorem 3 in <cit.>: Let ⟨ f, h ⟩ be a faithful action of BS(1, n) by homeomorphisms of ^2. Then there exists a positive integer k such that f^k is isotopic to the identity and has a lift to the universal cover whose rotation set is the single point {(0,0)}. Moreover, the set of f^k-fixed points is non-empty. Recall that ⟨ f^k, h ⟩ is also an action of BS(1,n), so Theorem <ref> allows us to restrict our study to actions where f is isotopic to the identity. Moreover, we can assume that (f)≠∅ and that f has a lift f̃ to the universal cover with rotation set {(0,0)}.We say that f̃ is the irrotational lift of f. Under these assumptions we can lift the action, by the following result that was proved in <cit.>. Let f̃:^ 2 →^2 be the irrotational lift of f, and h̃: ^ 2 → ^ 2 be any lift of h.Then we have h̃f̃h̃ ^-1 = f̃ ^n.As an application of rigidity in the diagonalizable case, we get a new proof of a result already found in <cit.>: There are no faithful actions ⟨ f, h ⟩ of BS(1,n) by toral homeomorphisms so that h is an Anosov map with stretch factor λ>n.Replacing f by some power of it if necessary, we can lift the action to the plane using Lemma <ref>. Since f is isotopic to the identity, its irrotational lift f̃ has bounded displacement, as it is obtained by litfing the homotopy from 𝕀_^2, beginning at 𝕀_^2. On the other hand, since h is an Anosov map, we can choose h̃ as the lift of h that is a linear map, and its canonical formis D=([λ0;0 λ^-1 ])where λ is the stretch factor of h. As we assumed that λ>n, and clearly λ^-1<1, we can apply Lemma <ref> (up to conjugation), to deduce that f̃=𝕀. Thus the result follows.From our work on the parabolic case, namely Theorem <ref>, we can get the analog of Theorem <ref> for Dehn twist maps:There is no faithful action φ of BS(1,n) by toral homeomorphisms so that φ(a) is a Dehn twist map.We consider f,h:^2→ ^2 a lift of the action φ given by Lemma <ref>, possibly after replacing φ(b) by some power of it. Since φ(b) is isotopic to the identity, the map f commutes with translations by vectors in ^2. By definition of Dehn twist map, we can take h as a linear map which, after conjugation in SL(2,), has an associated matrix of the form P^N=([ 1 0; N 1 ])for N∈, N≠ 0. Note that conjugation by an element of SL(2,) induces a conjugation of φ in Homeo(^2), thus we may assume thath(x,y)=(x,y+Nx)while preserving the ^2-equivariance of f. First we note that conjugating by D=1/N 𝕀 we get in the hypotheses ofTheorem <ref>, and applying it directly we would deduce that φ(b) fixes the points in the meridian circle S_0={0}×/. Our strategy will be to show the same for all the rational meridian circles S_m/l={m/l}×/ for m,l∈, thus obtaining that φ(b)=𝕀 by continuity. Let m,l∈ and consider the vertical line L_m/l={m/l}×. As in <ref>, we let L=L_0 be the vertical axis. Note thath^l(m/l,y)=(m/l,y+mN)=(m/l,y)+(0,mN)thus the mapg(x,y)=h^l(x,y)-(0,mN) fixes all the points in L_m/l. We show that ⟨ f,g ⟩ is an action of BS(1,n^l):gfg^-1(x,y)=gf(h^-l(x,y)+(0,mN)) =h^l(f h^-l(x,y)+(0,mN))-(0,mN) = h^lf h^-l(x,y)+h^l(0,mN)-(0,mN)=f^n^l(x,y)since f commutes with integer translations and h is linear with (h)=L. On the other hand let ϕ(x,y)=(x+m/l,y), i.e. the translation by (m/l,0). Then we clearly have ϕ(L)=L_m/l andϕ h^l ϕ^-1(x,y)= ϕ h^l(x-m/l,y)= ϕ(x-m/l,y+lNx-mN) = (x,y+lNx-mN) = h^l(x,y)-(0,mN) =g(x,y)so setting f_1=ϕ^-1fϕ we get that ⟨ f_1, h^l ⟩ is the action of BS(1,n^l) that we get by conjugating ⟨ f,g⟩ by ϕ^-1. Then we apply Theorem <ref> to ⟨ f_1, h^l ⟩, (after conjugation by 1/lN 𝕀), to get that L⊂(f_1). We deduce that f=ϕ f_1 ϕ^-1 fixes all points in ϕ(L)=L_m/l as desired, thus finishing the proof. 99[AGX]agx J. Alonso, N. Guelman, J. Xavier. Actions of solvable Baumslag-Solitar groups on surfaces with (pseudo-)Anosov elements. Disc. Cont. Dyn. Sys., 35(5), (2015).[BS]bsp G. Baumslag, D. Solitar.Some two generator one-relator non-Hopfian groups. Bull. Amer. Math. Soc., 689 (1962) 199-201.[Brou]brouL. E. J. Brouwer. Beweis des ebenen Translationssatzes. Math. Ann. 72(1912), 37-54.[F]fisher D. Fisher.Groups acting on manifolds: Around the Zimmer program. Geometry, Rigidity, and Group Actions (2011), 72-157.[F]franks J. Franks. Generalizations of the Poincarè-Birkhoff Theorem. Annals of Mathematics, 128 (1), (1988), 139-151.[FH]fh J. Franks, M. Handel. Distortion elements in group actions on surfaces. Duke Math. J. Volume 131, Number 3 (2006), 397-591.[GL1]glp N. Guelman, I. Liousse. Actions of Baumslag-Solitar groups on surfaces. Disc. Cont. Dyn. Sys. Vol. 33, No 5 (2013), 1945-1964.[GL2]gl2 N. Guelman, I. Liousse. Any Baumslag-Solitar action on surfaces with a pseudo-Anosov element has a finite orbit. Preprint, arXiv:1703.09037 (2017)[HX]hurtadoS. Hurtado, J. Xue. Global Rigidity of some Abelian-by-Cyclic group actions on ^2. Geometry and Topology Volume 25, issue 6 (2021), 3133-3178. [KP]kp B. Kolev, M. Peorueme. Recurrent surface homeomorphisms. Math. Proc. Cambridge Philos. Soc., 124(1), (1998), 161-168.[OT]ot L. Oversteegen, E. Tymchatyn. Recurrent homeomorphisms on ^2 are periodic. Proc. Amer. Math. Soc., 110(4), (1990), 1083-1088.[R]rivas jgt C. Rivas. On spaces of Conradian group orderings. J. of Group Theory, 13 (3), (2010), 337-353. [Z]zimmer R. Zimmer Actions of semisimple groups and discrete subgroups. Proc. Internat. Congr. Math. (Berkeley 1986), Vol 2. | http://arxiv.org/abs/1703.09102v2 | {
"authors": [
"Juan Alonso",
"Nancy Guelman",
"Cristobal Rivas",
"Juliana Xavier"
],
"categories": [
"math.DS"
],
"primary_category": "math.DS",
"published": "20170327142904",
"title": "One some planar Baumslag-Solitar actions"
} |
firstpage–lastpage 2002 The Conformal Field Theory on the Horizon of BTZ Black Hole Chao-Guang Huang Revised 02/2017 ============================================================ We investigate the activity induced signals related to rotation in late type stars (FGKM). We analyse the Ca II H&K, the H_α and the radial velocity time-series of 55 stars using the spectra from the HARPS public database and the light-curves provided by the ASAS survey. We search for short term periodic signals in the time-series of activity indicators as well as in the photometric light-curves. Radial velocity data sets are then analysed to determine the presence of activity induced signals. We measure a radial velocity signal induced by rotational modulation of stellar surface features in 37 stars, from late F-type to mid M-type stars. We report an empirical relationship, with some degree of spectral type dependency, between the mean level of chromospheric emission measured by the log_10(R'_HK) and the measured radial velocity semi amplitude.We also report a relationship betweeen the semi amplitude of the chromospheric measured signal and the semi amplitude of the radial velocity induced signal, which strongly depends on the spectral type. We find that for a given strength of chromospheric activity (i.e. a given rotation period) M-type stars tend to induce larger rotation related radial velocity signals than G and K-type stars.Stars: activity — Stars: chromospheres — Stars: rotation — starspots — planetary systems § INTRODUCTION High precision radial velocity (RV) measurements give astronomers the possibility of detecting small exoplanets, down to the mass of the Earth. Unfortunately, intrinsic variationsof the magnetic regions on the stellar surface induce radial velocity variations which, if stable over a few rotation periods, can mimic a planetary signal. Recognising and characterising them is key to disentangle the true planet induced signals <cit.>. Activity induced signals depend mainly on the activity level and spectral type of a star. Highly active stars usually rotate faster, inducing shorter period signals with larger amplitudes. These amplitudes also depend on the strength of the magnetic field, which is a function of the spectral type as it depends strongly on the depth of the convective zone. These signals are produced by a variety of physical phenomena in the stellar surface. Among them, spots are particularly relevant for rapidly rotating late-type dwarfs <cit.>. For slowly rotating stars plages become another important, but not so well understood, source of variability in the timescales of rotation. RV signals induced by plages and spots are not expected to be well correlated <cit.>.The amplitude of the so-called radial velocity jitter has been subject of study for many years <cit.>. Taking advantage of the precision achieved by HARPS (better than 1 m s^-1) and the sampling rate of planet hunting surveys, we present here a detailed analysis of this jitter measuring its period and amplitude in RV time-series data of a sample of late-type dwarfs. We study these rotationally-induded periodic signals, with particular attention at the harmonics of the stellar rotation period <cit.>, and to the relationships between the measured amplitudes and the level of chromospheric emission (log_10(R'_HK)). We also investigate the relation between the amplitude of the induced RV signals and the amplitude of the modulation of the chromospheric emission measured as the Mount Wilson S Index <cit.>.§ STELLAR SAMPLE AND DATA We selected a sample of bright southern stars with HARPS spectra available in the ESO public database. Our initial sample consists of 55 low activity (log_10(R'_HK) -4.4) main-sequence stars covering from late-F to mid-M type. The selected stars are part of the planet-hunting programs using HARPS <cit.> and their rotation periods and magnetic cycles have been investigated using spectroscopic and photometric time-series <cit.>. All of them had been observed more than 20 individual nights as for May 2016, giving a total number of more than 9000 spectra. Figure <ref> shows the distribution of spectral types for the sample of stars. Table <ref> shows relevant data forstars where a reliable detection of an activity induced radial velocity signal was obtained in this work§.§ Spectroscopic Data HARPS is a fibre-fed high-resolution echelle spectrograph installed at the 3.6 m ESO telescope in La Silla Observatory (Chile). The instrument has a resolving power R∼ 115 000 over a spectral range from 378 to 691 nm. It has been designed to attain extreme long-term RV accuracy. It is contained inside a vacuum vessel to avoid spectral drifts due to changes in temperature and air pressure. HARPS comes with its own pipeline providing extracted and calibrated spectra, as well as RV measurements and other data products such as cross-correlation functions and their bisector profiles.For the analysis of the spectral indicators we use the extracted order-by-order wavelength-calibrated spectra produced by the HARPS pipeline. In order to minimize atmospheric effects we create a spectral template for each star to correct the order-by-order flux ratios for the individual spectra. We correct each spectrum for the Earth's barycentric radial velocity and the radial velocity of the star using the measurements given by the pipeline. We finally re-bin them into a constant wavelength step.§.§ Photometry We also use the light curves provided by the All Sky Automated Survey (ASAS) public database. ASAS <cit.> is an all sky survey in the V and I bands running since 1998 at Las Campanas Observatory, Chile.Best photometric results are achieved for stars with V ∼8-14, but this range can be extendedimplementingsome quality control on the data. ASAS has produced light-curves for around 10^7 stars at δ < 28^∘. The ASAS catalogue supplies ready-to-use light curves with flags indicating the quality of the data. For this analysis we reliedonly ongood quality data (grade "A" and "B" in the internal flags). Even after this quality control,there are still some high dispersion measurementswhich cannotbe explained by a regular stellar behaviour. We reject those measurements by de-trending the series and eliminating points deviating more than three times the standard deviation from the median value. § STELLAR ACTIVITY INDICATORS AND RADIAL VELOCITIES We compute the Ca II H&K index, the log_10R'_HK and the Hα index following <cit.> and <cit.>. We use the indexes to re-measure the rotation period of the selected stars and to be able to compare the measured radial velocity semi-amplitudes with the activity indicators.§.§ Radial velocities Radial velocities are taken directly from the measurements of the standard HARPS pipeline, except for M-type stars. The radial velocity measurements in the HARPS standard pipeline is determined by a Gaussian fit of the cross correlation function (CCF) of the spectrum with a digital mask <cit.>. In the case of M-dwarfs, due to the huge number of line blends, the cross correlation function is not Gaussian resulting in a less precise RV measurement and in a loss of sensitivity to the changes in the full width half maximum (FWHM) of the CCF.For this type of stars we opted to do a different modelling of the cross correlation function, using the combination of a second order polynomial with a Gaussian function over a 15km s^-1 window centred at the minimum of the CCF following <cit.>. The center of the Gaussian function is taken as our radial velocity measurement. This allows us to improve the stability of the measured radial velocities as well as to improve the sensitivity of the FWHM measurements. For high signal-to-noise measurements the extracted value is virtually the same. §.§ Quality control of the data As the sampling rate of our data is notwell suited for modelling fast events, such as flares, and their effect in the radial velocity is not well understood, we identify and reject points likely affected by flares by searching for abnormally high measurements of the activity indicators and clear distortions in the Balmer series <cit.>.§ ANALYSIS§.§ Radial velocity signals induced by stellar rotation In order to identify the rotation induced radial velocity signals we first analyse the time-series of the Ca II H&K and Hα activity indicators, the time-series of the variations of the FWHM of the CCF, and the light curve (when available) to determine the rotation period of the star. Some of the rotation periods were previously measured in <cit.>. In these cases we measure it again including the last available data. To search for periodic signals in the series we compute the power spectrum using a generalised Lomb-Scargle periodogram <cit.> and if there is any significant periodicity we fit the detected signal using the RVLIN package <cit.>.To evaluate the false alarm probability of any peak in the periodogram we follow <cit.> modification over the work by <cit.> to obtain the spectral density thresholds for a desired false alarm level. This means our false alarm probability is defined as FAP = 1 - [1 - P (z > z_0]^M where P (z > z_0) = exp (-z_0) is theprobability of z being greater than z_0, with z the target spectral density, z_0 the measured spectral density and M the number of independent frequencies. We search for the power values corresponding to 10 %, 1 % and 0.1% false alarm probability. The mean rotation period of the star is estimated by calculating the weighted average of the detections obtained with different indicators. Errors are the peak-to-peak variation divided by the square root of the number of detections and the global false alarm probability is the combined false alarm probability given by a Fisher's combined probability test<cit.>.Different activity proxies provide information about different parts of the stellar atmosphere. Photometry gives information mainly about photospheric variability, while the S_MW and H_α indexes reflect the variability at different heights of the stellar chromosphere and sometimes even at different latitudes. Thus differential rotation may lead to differences in the rotation period estimates from these three variability indicators which could be larger than the error bars of individual measurements.The period of the induced RV signal should be similar but not necessarily coincident with the average period determined from activity proxies.Once we have a measurement of the rotation period of the star we search for periodic signals in the radial velocity time-series following the same procedure. To do that we first subtract long term trends, if present, using linear fits or incomplete orbits fitting sinusoidal models when the change of slope is apparent. Then we compute the power spectrum using a generalised Lomb-Scargle periodogram <cit.> and iteratively fit all the detected signals with keplerian models using the RVLIN package <cit.> until there are no significant remaining signals. During the process we were able to recover all the published known planetary signals for the analysed stars (using exoplanet.eu as source). The analysis of their signals is beyond the scope of this paper. After cleaning the series from planetary signals we search for periodic signals compatible with the rotation period previously measured. Figure <ref> shows the periodograms for the _MW index,H_α index and radial velocity time-series for the G-type star HD 41248. The three periodograms have a common periodicity which we interpret as the rotation signature of the star in the spectroscopic data. We detect a mean rotation of 23.8 days and a rotation induced radial velocity signal of 25.6 days with a semi amplitude of 2.43 m s^-1. For the K-type star HD 125595 Figure <ref> shows a very similar picture. There is a common periodicity across the different indicators which marks the rotational modulation. We measure a mean rotation period of 38.7 days and a rotation induced radial velocity signal of 36.7 days with a semi amplitude of 2.11 m s^-1. Figure <ref> shows the same scenario for the case of the M-dwarf star GJ 514. There is a common periodicity in the periodograms for the S_MW index,H_α index and radial velocity time-series. Once again we interpret this periodicity as the rotation signal of the star. We measure a mean rotation period of 30.0 days and a rotation induced RV signal of 30.8 days with a semi amplitude of 2.06 m s^-1. Figure <ref> shows the phase folded fits of the three time-series of the three stars using the detected periodicities.By following this procedure for each star we are able to find rotation induced RV signals in 37 stars of our sample. Table <ref> and figure <ref> show the results. There could be multiple reasons why we cannot findthe rotation induced RV signals in the remaining 17 stars of the sample. In some situations – low activity stars, specially in the case of faint stars – the RV induced signal might be below the noise level. In some others, the geometric pattern of activity features might cause signals which are not at the rotation period of the star, but at one of its harmonics. The inclination of the stellar axis related to our line of sight might also play a role in the relative proportion between harmonics and the shape of the RV induced signal by altering the apparent geometric pattern. Even minimizing the amplitude to an undetectable degree in the case of stars aligned pole-on. Short-lived activity regions could also undermine the detectability of the signals. In order to measure a coherent signal it is important for the active regions to last at least a few rotational periods. It has been seen that, in the case of some M-type stars, active regions are stable over long time spans <cit.>, but it does not have to be true for all stars. §.§ Radial velocity signals induced at the harmonics of the rotation period When measuring rotation induced RV signals it is expected to find also modulation with periods around the first harmonics of the rotation period. The relative strength of each of the present signals depends on the surface configuration and the inclination of the rotation axis related to our line of sight. For some particular configurations the signals at the harmonics of the rotation are dominant <cit.>. We performed an analysis equivalent to the previous searching for radial velocity signals at the harmonics of the rotation period. A signal at one of these harmonics would be considered an activity induced signal if there is an equivalent signal in any of the available activity indicators or a strong correlation between the radial velocity signal and at least one activity indicator. We found activity induced signals associated to harmonics of the rotation period in 9 stars, 8 of them with FAPs smaller than 1%. Table <ref> shows the periods and semi-amplitudes of the detected signals. Only one of the detections corresponds to a star not included in Table <ref>. The star GJ 581 shows an induced signal at half the rotation period but not at the rotation period. Of the 8 stars where we detect a signal both at the rotation period and at one of its harmonics, only one star shows the signal larger for the harmonic. The star HD1461 shows a 0.49 ms^-1 signal at the rotation period and0.72 ms^-1 at the harmonic. In the 7 remaining stars the RV signal at the harmonic is always smaller. Figure <ref> shows the combined periodogram for the radial velocities of all the stars from Table <ref>, after removing the planetary signals, with the time axis normalized at the rotation period of each individual star. The rotation signal is the dominant feature in this combined periodogram, being the only clear signal. There is a marginal structure at half the rotation period, telling us once more that some of the stars have signals at the first harmonic of the rotation. No other prominent features are present in our combined periodogram. §.§ Implications for planet-hosting candidates In the process of our search for rotation-induced radial velocity signals we recover all the previously published radial velocity signals attributed to extrasolar planets. For most of them we agree on the planetary origin of the signal, but there are four caseswhere we think a rotation induced signal is more likely to be causing the radial velocity modulation: The cases of HD 40307e,GJ 676 A e, GJ 667C d and GJ 832 c. HD 40307was initially claimed to be the host of 3 super-Earths at 4.3, 9.6 and 20.4 d <cit.>. Then the system was expanded to six planets<cit.>, planet e being a small super-Earth at ∼34.6 day orbital period. This period coincides within the uncertainties with our stellar rotation detection (36.5 d) and therefore we ascribe to rotation the radial velocity signal, as we previously pointed in <cit.>.GJ 676 A has 4 detected planet candidates. Planet b detected at ∼1050 d<cit.> and planets c at ∼4400 d, d at ∼3.6 d and e at ∼35.4 d<cit.>. We recover the signals attributed to planets b, d and e and the trend attributed to planet c. As suggested in <cit.>, the RV signal of planet e is most likely caused by stellar rotation. GJ 667 C is a star that hosts 6 planet candidates. Planets b at ∼7.2 d and c at ∼28.1 d <cit.>, and planets d at ∼91.6 d, e at ∼62.2 d, f at ∼39.0 d and g at ∼256 d <cit.>. <cit.> and <cit.> claimed that planets d and e are artefacts induced by stellar rotation, while<cit.> had already identified the 91 d signal as an alias of an∼106 d activity signal.We support the idea of the signal at ∼91 d being an artefact caused by the stellar rotation. We agree on the ∼106 d rotation period and we also found a power excess at ∼91 d in the S_ MW periodogram <cit.>, supporting the idea that the signal might be the mark of rotation measurements at different latitudes.GJ 832 hosts 2 potential planets. Planet b at ∼3416 d <cit.> and planet c at ∼35.7 d <cit.>, the latter with an orbital period very close to our rotation period measurement. As in <cit.>, we interpret the signal of planet c as the rotation-induced RV signal <cit.>.§ DISCUSSION The previous analysis of radial velocity signals induced by stellar rotation provided a collection of 35 stars where we were able to measure the rotation RV signal and/or one of its harmonics, for stars going from late F to mid M-type. For our sample the distribution of radial velocity induced signals peaks at 1-2 m s^-1. Figure <ref> shows the distribution of the semi-amplitudes of the periodic RV signal induced by the rotation of the stars for the different spectral types. We found that there is a clear relationship between the activity level of the star and the amplitude of the rotation induced RV signal. Figure <ref> shows two linear relationships between the activity level of the star and the logarithm of the amplitude of its induced signal. This relationship is different for M-dwarfs and for FGK stars. We added a few extra points from the literature (Table <ref>) in order to populate a bit more the plot and to increase the activity coverage. Both groups of stars are fit independently:log_10(K)= a ·log_10(R'_ HK) + b,Table <ref> shows the parameters for the different fits for the two groups of stars. We see some discrepancy between our results in Corot-7 and GJ 674 and the results of <cit.> and <cit.>. Our measurement of the amplitudes for the activity induced RV signals is much smaller. In both cases we are using much longer observation baselines, averaging the amplitude over longer periods of time. In the case of Corot-7 <cit.> performed the analysis during a time when the activity level of the star was higher than the average level of our full dataset, and therefore a larger RV amplitude for the activity induced signal is expected. For the case of GJ 674 the sampling rate of the modern data is much better at the scale pertinent for rotation period measurements than in the original dataset. The amplitude of the rotation induced RV signal might have been overestimated. For HD 41248, α Cen B and GJ 176 our measurements are compatible with those previously published. We included the cases of HD 166435, the Sun, GJ 3998 and GJ 15 A as a comparison. The RV semi amplitude of their rotation signals match quite well the expected value according to their mean activity levels. See Table <ref>.Even in the case of slow rotators the radial velocity signals induced by stellar rotation have semi-amplitudes larger than those of terrestrial planets in the habitable zones. In the case of FGK stars these signals can drop below the 1 ms^-1 threshold for stars more quiet than the Sun, reaching sometimes the HARPS precision limit. For M-dwarfs even in the case of very slow rotators these signals are larger than 1 ms^-1. We have not detected any rotation induced RV signal smaler than 1 ms^-1 in an M-dwarf star. Even in the case of the most quiet stars the radial velocity signal induced by the rotation is still larger than those of terrestrial planets. Modelling and removing these signals becomes necessary. In addition, we have studied the relationship between the semi-amplitude of the radial velocity signals and the semi-amplitude of their related activity signals. We found that the amplitude in radial velocity correlates with the amplitude in the S_MW index signals. Figure <ref> shows how the semi-amplitude of the radial velocity signals compare with the semi-amplitude of the S_MW index signals. Even with our limited sample two distinct populations seem quite clear. F to early K stars and mid-K stars to mid-M dwarfs. The jump happens around spectral type K2.5, which corresponds to an effective temperature of 4800 K. This is the surface temperature when the relative abundance of Ca II drops dramatically until being marginal at ∼4000 K. FG and early K in one side, and late-K and M-type stars on the other are fit independently using the following equation:K_V_r= a · K_S_MW + b Table <ref> shows the measured values for the parameters in equation <ref>. Finally, we study the phase difference between the induced RV periodic signals and the periodic modulation in the activity indicators. <cit.> and <cit.> found that these signals are not necessarily in phase. In Figure <ref> we show the phase difference between the RV signal and the S_MW signal against the color of the stars and the mean activity level. We restrict this analysis to those stars where the period estimates of different indicator were consistent within error bars. We see an apparent evolution of the phase shift with the radial velocity signal lagging behind the S_MW signal towards redder stars. Starting in G-type stars, which show small dispersion close to shift zero, and gradually increasing for K-type which reach a phase shift close to 360 degrees (zero again) for mid/late K-type stars, a trend which is seems to be continued for M-type stars. We do not find any correlation between the phase shift and the chromospheric activity level of the stars. Further investigation on a larger sample will be needed to confirm this behaviour.Then we study the phase shift between the RV signal and the H_α index signal finding no clear relationship.We also study the phase shift between the H_α index and the S_MW index and find a group of the stars (HD 30495, HD 59468, Corot-7, HD 209100, GJ 676A, GJ 536, GJ 667C, GJ 674 and GJ 526) show a difference in phase close to phase 180 degrees, while the rest of the stars show a difference in phase close to zero. We do not see a clear correlation between the phase shifts and the average level of activity of each of these two groups.The distinct behaviour of the different spectral types seen in Fig. <ref> and Fig. <ref>, along with phase shift between the RV signal and the S_MW signal (Figure <ref>), give a clue about the nature of the dominant surface features causing the induced RV variations. The relationship between the Ca II H&K emission and the effective change in radial velocity varies for the different spectral types. Ca II H&K emission is generated mainly in the stellar plages <cit.>, which are the main source of stellar induced RV variations in slowly rotating solar type stars <cit.>. In this scenario, and assuming a high convection level - as in the case of the Sun - it is expected the RV signal and the activity signal to be more or less in phase, as we are seeing in our G-type stars. In cooler stars it seems that spots become more and more important. A radial velocity variation cased solely by spots would be expected to show a 90 degrees shift between the RV signal and the activity index signal. The gradual change in the phase might tell us that the equilibrium between the contributions of both types of activity regions gradually changes towards smaller stars, with more complex contributions causing a wide variety of phase shifts between the two signals.§ CONCLUSIONS Using data for 55 late-type stars (from late-F to mid-M) we have analysed the radial velocity time-series searching for periodic signals that match the stellar rotation periods. For 37 stars we have clearly found the induced RV signal by the stellar rotation or one of its harmonics, and measured its semi-amplitude.Our study supports previously reported doubts on the keplerian origin of the periodic RV signals attributed to planets HD 40307 e, GJ 676 A e, GJ 667C d and GJ 832 c, which can also be explained as rotation-induced activity signals.We have investigated the correlation between the level of chromospheric emission, represented by the log_10(R'_HK) index, and the measured semi-amplitude, and obtained a specific linear relationship between this index and the logarithm of the amplitude for each spectral type. We have also investigated the correlation between the amplitude of the radial velocity signals and the amplitude in the activity signals, and we find two different correlations for G to mid-K stars and late-K to mid M dwarfs.We have studied the phase shift between the period radial velocity induced signal caused by stellar rotation and the period signal from the S_MW index. We find an apparent evolution of the phase shift and the color B-V. We do not find a correlation between the phase shift and the activity level.The systematic measurement and characterisation of stellar activity induced RV signals is a necessary step for a reliable identification of the RV signals produced by terrestrial planets at short orbits. For the same activity level, M-type stars show larger activity-induced RV signals than G and K-type stars. Because of their lower stellar mass, both very low activity M dwarfs and late K dwarfs offer a very good opportunity for the detection of terrestrial planets. § ACKNOWLEDGEMENTS This work has been financed by the Spanish Ministry project MINECO AYA2014-56359-P. J.I.G.H. acknowledges financial support from the Spanish MINECO under the 2013 Ramón y Cajal program MINECO RYC-2013-14875. This work is based on data obtained from the HARPS public database at the European Southern Observatory (ESO). This research has made extensive use of the SIMBAD database, operated at CDS, Strasbourg, France and NASAs Astrophysics Data System. We are grateful to all the observers of the following ESO projects, whose data we are using 60.A-9036, 072.C-0096, 073.C-0784, 073.D-0038, 073.D-0578, 074.C-0012, 074.C-0364, 074.D-0131, 075.D-0194, 076.C-0878, 076.D-0130, 076.C-0155, 077.C-0364, 077.C-0530, 078.C-0044, 078.C-0833, 078.D-0071, 079.C-0681, 079.C-0927, 079.D-0075, 080.D-0086, 081.C-0148, 081.D-0065, 082.C-0212, 082.C-0308, 082.C-0315, 082.C-0718, 083.C-1001, 083.D-0040, 084.C-0229, 085.C-0063, 085.C-0019, 085.C-0318, 086.C-0230, 086.C-0284, 087.C-0368, 087.C-0831, 087.C-0990, 088.C-0011, 088.C-0323, 088.C-0353, 088.C-0662, 089.C-0050, 089.C-0006, 090.C-0421, 089.C-0497, 089.C-0732, 090.C-0849, 091.C-0034, 091.C-0866, 091.C-0936,091.D-0469, 180.C-0886, 183.C-0437, 183.C-0972, 188.C-0265, 190.C-0027, 191.C-0505, 191.C-0873, 282.C-5036. | http://arxiv.org/abs/1703.08884v2 | {
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"A. Suárez Mascareño",
"R. Rebolo",
"J. I. González Hernández",
"M. Esposito"
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"published": "20170326234017",
"title": "Characterisation of the radial velocity signal induced by rotation in late-type dwarfs"
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-0.6cm 5.mm 23cm 16cm -0.6cm -0.6cm 0.24cm newthmTheorem theoremTheorem[section] lemma[theorem]Lemma proposition[theorem]Proposition corollary[theorem]Corollary notation[theorem]Notation defthm[theorem]Definition et þdefinition[theorem]Definitionremark example[theorem]Example examples[theorem]Examples exo[theorem]Exercise rema[theorem]Remarkplaintheo[theorem]Theorem definitions[theorem]Definition prop[theorem]Proposition coro[theorem]Corollary conj[theorem]Conjectureremark[theorem]RemarkProof..55em.55em equationsectiongeometrically finiteA B C DE F C G H I J K O L ŵ M N P Q R S T U V W Y Z α ρ θ γ Γ σ Σ φ ε λ Λ Δ δ d-d+1 (p+1)-p (p+1)p ∖ ∖𝔻 ℚP( ) ℙ^1łλ pha ε δ̣ := rad ε dz q.e.d. | http://arxiv.org/abs/1703.08703v2 | {
"authors": [
"Yan Gao",
"Giulio Tiozzo"
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"published": "20170325152136",
"title": "The core entropy for polynomials of higher degree"
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Dynamical alignment of visible and dark sector gauge groups Rainer Dick December 30, 2023 ===========================================================Quantum theories of gravity help us to improve our insight into the gravitational interactions. Motivated by the interesting effect of gravity on the photon trajectory, we treat a quantum recipe concluding a classical interaction of light and a massive object such as the sun. We use the linear quantum gravity to compute the classical potential of a photon interacting with a massive scalar. The leading terms have a traditional 1/r subordinate and demonstrate a polarization-dependent behavior. This result challenges the equivalence principle; attractive and/or repulsive interactions are admissible. §INTRODUCTION Regarding the direct detection of gravitational waves by the Advanced Laser Interferometer Gravitational-wave Observatory (LIGO), one finds that the Newtonian law is not suitable for the high gravity regimes such as black hole systems.<cit.> In addition, one of the challenging subjects in our solar system is the investigation of Newtonian deviation of gravitational potential. In this regard two interesting theories were presented, the so-called MOND and Einstein general relativity.<cit.>. These theories predict deviations of 1/r law and deflection of light by the sun as well. Quantum gravity theories can also be used to obtain quantum corrections for Newtonian potential. Investigating the various interactions between photons and other fields, the gravitational potential may be derived using the field theory approach. These studies provide some modifications to the potential. In this paper, we investigate the effect of light polarization on the interaction of photons with a massive body such as the Sun. Surprisingly, we can obtain a new additional term which is originated from the light polarization. It is notable that such additional polarization term can be expressed based on the quantum prescription. A complete theory of the quantum gravity is not performed yet, however, endeavors are heavily pursued toward this end. Most candidate theories for the quantum gravity, just predict an inappreciable gravitational effect on small extents, where the standard model works very well. So, one may regard that quantum gravity is a mere theoretical scramble? Graviton—a massless boson of spin 2—is the common point of all quantum gravity models, that mediates the gravitational interactions.<cit.> Between the different theories of quantum gravity, the effective field theory is capable of renormalization,<cit.> and could be realized through experimental means.The problem of divergences in quantum gravity, specially the infrared divergences, has been discussed by many authors in several methods.<cit.> In the case of massless particles coupled to gravity, infrared divergences—soft-infrared and collinear singularities—arise, originating from long-range low-energy degrees of freedom. Weinberg showed that the scattering of spin-zero fields is infrared finite, even the fields are massless.<cit.> Moreover, he showed that the collinear divergences vanish after the sum over all diagrams with Eikonal approximation, and the remaining soft infrared divergences are removable as well. References wein2 and NS2011 have emphasized that the infrared divergences in all levels originate from one-loop divergences, and these divergences at one-loop level can be eliminated.<cit.> ٍEvidence shows the gravitational effective theory is well-behaved in the infrared, and there is no reason to be worried about the divergences, especially at tree-level. An effective quantum field theory of gravity is written in the weak gravitational field, where the gravity is supposed to be a linear theory. Based on the effective quantum field theory, one may drive the appropriate vertices and propagators to obtain the scattering amplitudes by using the Feynman rules. The scattering amplitudes contain two types of components, analytic and non-analytic terms. The analytic terms correspond to the local short-distance interactions, while the non-analytic terms are responsible for non-local long rang interactions described by the non-relativistic potential.<cit.> In Ref. bdh02, up to one loop calculation and considering given non-analytic terms, the quantum corrections to the Newtonian potential between two massive scalar object are obtained. For two spin-1/2 charged massive fermions,<cit.> and two massive charged scalars,<cit.> scattering amplitudes are calculated for both Newtonian and Coulomb potentials. The final results of all papers show that they are in agreement upon Newtonian and post-Newtonian terms as predicted by general relativity. In Refs. hr08_1 and hr08_2, authors have calculated the gravitational and electromagnetic-gravitational effects of long distance potential for two massive bodies of different spins, wherein they have recognized spin-independent, spin-orbit, and spin-spin dependent contributions to potential.In addition, authors of Ref. pdv14 have been used some modern techniques employing spinor-helicity variables and on-shell unitarity method at the one-loop level for computing post-Newtonian and quantum corrections of the gravity. Later in Refs. bdhp15 and bdhp16, they applied this method to study the scattering of light and light-like particles of different spins (spin-0, spin-1/2, spin-1), with a heavy scalar (spin-0) particle such as the sun, calculating the bending angle of the particle grazing the sun. Regardless of spins, computations are in agreement with the general relativity up to the post-Newtonian corrections for all the cases. The quantum corrections come afterwards, where they depend on the spin of particles. Although the cited papers above (and many others) have done the calculations up to the one loop, in this paper we calculate the scattering amplitude only in tree level. In fact, our emphasized conclusion is resulted from tree level. One may use the loop diagrams and the required vertices to compute the loop corrections. For the sake of simplicity, we do not carry out this job here.This paper is organized as follow: in section <ref>, we introduce the effective theory of quantum gravity. Section <ref> is devoted to explaining the rule of extracting the classical potential of gravity from scattering amplitude. In section <ref>, we calculate the scattering amplitude of a photon and a massive scalar thoroughly. We obtain the potential in the classical limit and introduce the leading terms for the interaction of light and a massive body in section <ref>. At last, we finish the paper with some conclusions. § EFFECTIVE FIELD THEORY OF GRAVITYWe consider the quantum electrodynamics and the massive scalar field Φ are coupled to General Relativity via the Lagrangianℒ = √(-g)[ 2/κ^2ℛ - 1/4F_μνF_ρσg^μρg^νσ + ( 1/2g^μν∂_μΦ∂_νΦ-1/2M^2 Φ^2)] + ⋯where g is the determinant of the metric tensor g^μν, ℛ is the Ricci scalar, the electromagnetic field is F_μν=∂_μ A_ν - ∂_ν A_μ and the constant κ is related to the gravitational constant asκ^2=32π G. To linearize ℒ, we suppose g_μν=g̅_μν+κ h_μν, where g̅_μν is a background metric, and h_μν is the fluctuation in the gravitational field. In fact, some more extra terms (gauge-fixing and ghost field Lagrangians) are needed to quantize the gravitational field.<cit.> In this paper, we worked in the Natural Units, so we choose ħ=c=1, and the Minkowski metric is η_μν=diag(+1,-1,-1,-1). So one can expand the right-hand side of Eq. (<ref>) with respect to h_μν (or κ), and keep the required leading order terms for the linear theory. Then, it is straightforward to write down the Feynman rules,<cit.> in which we will address them in the appendix, briefly.§ SCATTERING AMPLITUDE AND GRAVITATIONAL POTENTIALOne can use the Feynman rules to calculate the scattering amplitude ℳ. As stated in Ref. don94, scattering amplitude in momentum space is expressed in powers of q^2. The terms proportional to positive powers of q^2, are so-called analytical terms and correspond to local interactions. These terms are related to large momentum transfer, which are not in our attention in this work. The other terms in ℳ, are so-called non-analytical terms which are corresponding to non-local long range interactions. In the classical limit, the potential with negative powers of r comes from these terms, which are our interest in this work. The classical potential can be constructed from quantum gravity by various methods. In Ref. mon95 author has used the technique of Wilson loop to find an expression for the potential energy in terms of vacuum expectation value of a quantized gravitational field. In addition, there is a simple and useful prescription to define the potential energy from scattering amplitudes. Moreover, in Ref. iwa71 Iwasaki has used the Born approximation to conclude the potential via the inverse Fourier transform of scattering amplitudeV(r)=N∫d^3q/(2π)^3 e^-iq·rℳ(q),where N is a normalization factor which should be fixed.§ GRAVITATIONAL SCATTERING OF A PHOTON AND A MASSIVE SCALAR FIELDNow we want to study the scattering of a photon by a massive scalar. In this regard, we use the Feynman diagram in tree level (Fig. <ref>) and use the Feynman rules and propagators and vertices in the appendix. The scattering amplitude is written simply as below iℳ(q) = ϵ^γ_i τ_μνγδ(p_3,p_4) ϵ^*δ_f[ i𝒫^μναβ/q^2] τ_αβ(p_1,p_2, M),where p_1 and p_2 are initial and final 4-momenta of the massive scalar field with mass M, and p_3 and p_4 are initial and final 4-momenta of photon(p_1 p_1) = (p_2 p_2)=M^2,(p_3 p_3) = (p_4 p_4)=0,and ϵ_i and ϵ_f are photon initial and final polarization vectors, where satisfy the relations (ϵ_i p_3)=(ϵ^*_f p_4)=0,and the transferred momentum is q=p_1-p_2=p_4-p_3.The graviton propagator and scalar-scalar-graviton and photon-photon-graviton vertices are introduced in the appendix.Replacing from (<ref>)–(<ref>) in Eq. (<ref>), after contracting indices, we receive toiℳ(q) = iκ^2/2q^2{(p_3p_4)[(ϵ_ip_1)(ϵ^*_fp_2) +(ϵ_ip_2)(ϵ^*_fp_1)]. +(ϵ_iϵ^*_f)[(p_1p_3)(p_2p_4)+(p_1p_4)(p_2p_3)-(p_1p_2)(p_3p_4)] +(p_1p_2)(ϵ_ip_4)(ϵ^*_fp_3)-(p_1p_3)(ϵ^*_fp_2)(ϵ_ip_4)-(p_1p_4)(ϵ_ip_2)(ϵ^*_fp_3) .-(p_2p_3)(ϵ^*_fp_1)(ϵ_ip_4)-(p_2p_4)(ϵ_ip_1)(ϵ^*_fp_3)}.It is interesting to see the Eq. (<ref>) in a more symmetrical formiℳ(q) = iκ^2/2q^2{(ϵ_iϵ^*_f)[(p_1p_3)(p_2p_4)+(p_1p_4)(p_2p_3)-(p_1p_2)(p_3p_4)].+(p_1p_2)[(ϵ_ip_4)(ϵ^*_fp_3)+(ϵ_ip_3)(ϵ^*_fp_4)] +(p_3p_4)[(ϵ_ip_1)(ϵ^*_fp_2)+(ϵ_ip_2)(ϵ^*_fp_1)] -(p_1p_3)[(ϵ^*_fp_2)(ϵ_ip_4) + (ϵ^*_fp_4)(ϵ_ip_2)]-(p_1p_4)[(ϵ_ip_2)(ϵ^*_fp_3) + (ϵ_ip_3)(ϵ^*_fp_2)]-(p_2p_3)[(ϵ^*_fp_1)(ϵ_ip_4) + (ϵ^*_fp_4)(ϵ_ip_1)].-(p_2p_4)[(ϵ_ip_1)(ϵ^*_fp_3) + (ϵ_ip_3)(ϵ^*_fp_1)] }.Using relations (<ref>) and (<ref>), we substitute from (ϵ_i p_2) = (ϵ_i p_1)-(ϵ_i q), (ϵ_i p_4) = (ϵ_i q), (ϵ^*_f p_2) = (ϵ^*_f p_1)-(ϵ^*_f q), (ϵ^*_f p_3) = -(ϵ^*_f q),in (<ref>), and use 2(p_1q)=q^2, and (p_1p_4)+(p_2p_4)=(p_1p_3)+(p_2p_3), (after a bit rearrangement) to getiℳ(q) = iκ^2/2q^2{2(ϵ_ip_1)(ϵ^*_fp_1)(p_3 p_4) -[(ϵ_ip_1)(ϵ^*_fq)+(ϵ^*_f p_1)(ϵ_i q)](p_3 p_4) . +[(ϵ_ip_1)(ϵ^*_fq)-(ϵ^*_f p_1)(ϵ_i q)][(p_1 p_3)+(p_2 p_3)]+(ϵ_i ϵ^*_f)[ (p_1p_3)(p_2p_4)+(p_1p_4)(p_2p_3)-(p_1p_2)(p_3p_4)] . -(ϵ_iq)(ϵ^*_fq)[ (p_1p_2)-(p_1p_3)+(p_1p_4)] }.We can verify the following identityϵ^*_fα(ϵ_iq)-ϵ_iα(ϵ^*_fq) =2(p_3+p_4)_α/q^2(ϵ_iq)(ϵ^*_fq)-4iE/q^2 ϵ_αβγδ p^β_3q^γ S^δ,where S_α is defined as belowS_α=i/2E ϵ_αβγδ ϵ^*β_f ϵ^γ_i(p_3+p_4)^δ.Replacing from the identity (<ref>) in (<ref>) we find iℳ(q) = iκ^2/2q^2{ 2(ϵ_ip_1)(ϵ^*_fp_1)(p_3 p_4) -[(ϵ_ip_1)(ϵ^*_fq)+(ϵ^*_f p_1)(ϵ_i q)](p_3 p_4) 1^2/2^2. +[-2(p_1 p_3)+(p_1 p_4)/q^2(ϵ_iq)(ϵ^*_fq) +4iE/q^2ϵ_αβγδp_1^α p_3^β q^γ S^δ]×[(p_1 p_3)+(p_2 p_3)]-(ϵ_iq)(ϵ^*_fq)[ (p_1p_2)-(p_1p_3)+(p_1p_4)] . 1^2/2^2 + (ϵ_i ϵ^*_f)[ (p_1p_3)(p_2p_4)+(p_1p_4)(p_2p_3) -(p_1p_2)(p_3p_4)]}.Using Eqs. (<ref>) and (<ref>), and definition s=(p_1+p_3)^2, we can simply derive the following relations(p_1q)= -(p_3q)=1/2q^2, (p_3p_4) = -1/2q^2, (p_1p_4) = 1/2(s-M^2+q^2), (p_2p_4) = 1/2(s-M^2), (p_2p_3) = 1/2(s-M^2+q^2), (p_1p_3) = 1/2(s-M^2), (p_1p_2) = M^2-1/2q^2.Replacing from Eqs. (<ref>) in Eq. (<ref>), after some simplifications, we achieve the following relation for the scattering amplitude ℳ(q) = 8π G{ -2(ϵ_ip_1)(ϵ^*_fp_1) +[(ϵ_ip_1)(ϵ^*_fq)+(ϵ_f^* p_1)(ϵ_i q)]1/2.-4(s-M^2+1/2q^2)^2/q^4(ϵ_i q)(ϵ^*_f q) + 8iE(s-M^2+1/2q^2)/q^4ϵ_μβγδp_1^μ p_3^β q^γ S^δ-(ϵ_iq)(ϵ^*_fq)2M^2/q^2.+(ϵ_i ϵ^*_f)[ (s-M^2)^2+sq^2/q^2]},where we have replaced κ^2=32π G. From Eq. (<ref>) we can deduce the non-analytic terms of scattering amplitude ℳ(q)≃-8π G{ -(ϵ^*_f ϵ_i)(s-M^2)^2/q^2 + 4(ϵ^*_fq)(ϵ_iq)(s-M^2)^2/q^4(s-M^2+1/2q^2)/q^4. . + 2(ϵ^*_fq)(ϵ_iq)(2s-M^2)/q^2-8iE (s-M^2+1/2q^2)/q^4 ϵ_μβγδ p_1^μ p_3^β q^γ S^δ}. In the following, we consider some simplifications to reach the classical limit. We assume the massive scalar is too heavy and at rest, and the collision is elastic, hence the initial and final energies of the photon remain unchanged. Under these circumstances we have𝐩_1 ≃ 0 ≃𝐩_2,p_1^0 ≃ p_2^0 ≃ M, p_3^0 ≃ p_4^0=E, 𝐩_3·𝐩_3 ≃𝐩_4·𝐩_4 = E^2,𝐪=𝐩_4-𝐩_3,q^0=p_1^0 - p_2^0=p_4^0 - p_3^0 ≃ 0.We suppose two transverse linear polarization for photon, so the relations (<ref>) read asϵ̂_i·𝐩_3 = ϵ̂_f ·𝐩_4 = 0.Replacing from Eqs. (<ref>) and (<ref>) in (<ref>), we are led toℳ(q)≃-8π G{-ϵ̂_f·ϵ̂_̂î(s-M^2)^2/𝐪^2+ 4(ϵ̂_f·𝐪)(ϵ̂_i·𝐪)(s-M^2)^2/𝐪^4(s-M^2+1/2q^2)/q^4.. - 2(ϵ̂_f·𝐪)(ϵ̂_i·𝐪)(2s-M^2)/𝐪^2 -8iE (s-M^2 - 1/2𝐪^2)/𝐪^4 ϵ_μβγδ p_1^μ p_3^β q^γ S^δ}.From Eqs. (<ref>), (<ref>) and (<ref>), we can investigate the following approximationss ≃ M^2 + 2ME,S_0 ≃i/2E(𝐩_3 + 𝐩_4)·(ϵ̂_i×ϵ̂_f), 𝐒≃ iϵ̂_i×ϵ̂_f,ϵ_μβγδ p_1^μ p_3^β q^γ S^δ≃ -Mϵ_ijkp_3^i q^j S^k = M 𝐒·𝐪×𝐩_3. So Eq. (<ref>) is simplified as belowℳ(q)≃-8π G{-ϵ̂_f·ϵ̂_̂î4M^2 E^2/𝐪^2.+ 4(ϵ̂_f·𝐪)(ϵ̂_i·𝐪)4M^2 E^2/𝐪^4-2(ϵ̂_f·𝐪)(ϵ̂_i·𝐪)(M^2 + 4ME)/𝐪^2. - 4iEM (4ME-𝐪^2)/𝐪^4 𝐪×𝐩_3 ·𝐒}.Taking into account Eq. <ref>, it is notable that the first term is the Newtonian potential term and the second and fourth terms come from identity (<ref>) and former has a part like the Newtonian term, while the third and fourth terms related to the angular momentum. § POTENTIAL Now we are in a position to calculate the classical gravitational potential. Inserting Eq. (<ref>) into Eq. (<ref>) and using the Fourier transformation (see Eqs. (<ref>)–(<ref>) in the appendix for more details), we obtain V(r)≃-8π G N{-ϵ̂_f·ϵ̂_̂î4M^2 E^2/4π r. + 4(ϵ̂_f·ϵ̂_i)4M^2 E^2/8π r - 4(ϵ̂_f·𝐫)(ϵ̂_i·𝐫)4M^2 E^2/8π r^3- 21/4π(ϵ̂_f·ϵ̂_i)(M^2 + 4ME)/r^3 + 23/4π(ϵ̂_f·𝐫)(ϵ̂_i·𝐫)(M^2 + 4ME)/r^5-4iEM -i/8π4ME/r 𝐫×𝐩_3 ·𝐒. +4iEM -i/4π1/r^3 𝐫×𝐩_3 ·𝐒}.In our case study, the normalization factor for the massive scalar and massless photon is chosen as N=1/√(2E_12E_2 2E_3 2E_4)=1/4ME. Simplifying Eq. (<ref>), we findV(r)≃ ϵ̂_f·ϵ̂_̂î-2G M E/r + (ϵ̂_f·𝐫)(ϵ̂_i·𝐫)4G M E/r^3+ (ϵ̂_f·ϵ̂_i)G(M/E + 4)/r^3 - (ϵ̂_f·𝐫)(ϵ̂_i·𝐫)3G(M/E + 4)/r^5 + 4GM E/r 𝐫×𝐩_3 ·𝐒 - 2G/r^3 𝐫×𝐩_3 ·𝐒. Eq. (<ref>) is supposed to be valid in the classical limit, since we have used the linear approximation of gravity. These expressions show that, interestingly, the polarization has considerable effects on the gravitational potential. Comparing our result with Ref. hr08_1 (for a massive spin-1 particle in the field of a massive scalar particle), we find that the first term in Eq. (<ref>) is in agreement with the general relativity. The second term is the main interesting result of this paper. This is a 1/r term, that will not be affected by a further consideration on calculating the loop corrections. Against the first term, this is a repulsive term of the same order. In Ref. hr08_1, within the non-relativistic limits, the transferred momentum is neglected in comparison with the masses in theory, so the results lack such a term. In Refs. bdhp15 and bdhp16 (see Ref. bai17 as well), for massless particles interacting massive particles, the computations are done independently of spin and polarization orientations. For the bending angle of the light grazing the sun, the leading and even the next to the leading terms are in full agreement with the general relativity.§ CONCLUSIONSIn quantum field theory, the interaction of figure <ref> is computed for a time duration of Δ t =ħ/E. So, in the classical limit, we can accept that the polarization of the photon does not change within such a Δ t ∼ 0, i.e. we can choose ϵ̂_i=ϵ̂_f≡ϵ̂. In this limit 𝐒 (Eqs. (<ref>)) vanishes, therefore the last two terms in Eq. (<ref>) are annihilated. For the large r, the first two terms in Eq. (<ref>) are the leading terms in negative powers of r,V(r) ≃ -2GME/r + (ϵ̂·𝐫)^24GME/r^3.That is a distinguished result. We emphasize that Eq. (<ref>) is derived from tree-level amplitude, and loop corrections do not change the result. The first term in Eq. (<ref>) is the usual inverse distance law of gravitational potential for a photon in the field of a mass M predicted by general relativity. The second term in Eq. (<ref>) is a polarization dependent expression of 1/r dependence. The first thing that comes into sight is the difference between the signs of two terms; while the first term is attractive, the second term is repulsive. Meanwhile, the coefficient of the new term is twice as large as the first. Eq. (<ref>) can be rewritten in the form V(r) ≃ -2GME/r(1 - 2cos^2γ) = 2GME/rcos(2γ),where γ∈[0,π] is the angle between 𝐫, the position vector of the photon (light), and its polarization. Depending on the orientation of the polarization vector with the line connecting the photon and the scalar particle, the net force on the photon can be either attractive or repulsive. According to the potential (<ref>), we may assume a plane of motion for a photon around the “sun”. With the polarization of photon perpendicular to this plane at far past, it remains perpendicular to the plane at a later time. In this case, we dealt with the usual attractive gravitational force. Accordingly, the bending of light around the sun is calculated via the first term in (<ref>), which is predicted by the general relativity a century ago.If the polarization vector of light stays in the plane of motion, the second term plays an important role. To illustrate the problem, suppose a beam of light is coming far from the sun, where the position vector of light and its momentum vector are parallel. So the polarization vector of light is perpendicular to the position vector, and the light feels only an attractive force from the sun. With light becoming closer, the polarization vector makes an angle θ with the position vector 𝐫, so a contradictory force presents itself, playing a role in deflection. When the light has the smallest distance from the sun, i.e. at perihelion that 𝐫 is perpendicular to the light path, the net force is repulsive, and its magnitude is equal to the traditional attractive force. Potential (<ref>), makes different patterns for deflection of light. These new patterns can be tested in a solar eclipse. Potential (<ref>) affects the gravitational lensing as well. We will discuss these items in a next paper. § PROPAGATORS, VERTICES AND USEFUL FOURIER TRANSFORMATIONS §.§ Graviton propagator60pt < g r a p h i c s >= i𝒫^μναβ/q^2, 𝒫^μναβ ≡ 1/2(η^μαη^νβ+η^μβη^να-η^μνη^αβ).§.§ Scalar-scalar-graviton vertex60pt < g r a p h i c s >=τ_μν(p_1,p_2,M)= -iκ/2[ p_1μp_2ν+p_1νp_2μ-η_μν(p_1· p_2-M^2)].§.§ Photon-photon-graviton vertex60pt < g r a p h i c s >=τ_μνγδ(p_1,p_2)=iκ{𝒫_μνγδ(p_1· p_2)+1/2[η_μνp_1δp_2γ+η_γδ(p_1μp_2ν+p_1νp_2μ).. ..-(η_μδp_2γp_1ν+η_νδp_2γp_1μ+η_νγp_2μp_1δ+η_μγp_2νp_1δ)]1/2}.§.§ Useful Fourier Transformations∫d^3q/(2π)^3 e^-i𝐪·𝐫(1/𝐪^2) = 1/4π r,∫d^3q/(2π)^3 e^-i𝐪·𝐫(q_i/𝐪^2) = -ir_i/4π r^3,∫d^3q/(2π)^3 e^-i𝐪·𝐫(q_iq_j/𝐪^2) =-1/4π(3r_ir_j/r^5-δ_ij/r^3),∫d^3q/(2π)^3 e^-i𝐪·𝐫(q_iq_j/𝐪^4) = 1/8π(δ_ij/r-r_ir_j/r^3),∫d^3q/(2π)^3 e^-i𝐪·𝐫(q_i/𝐪^4) = -ir_i/8π r. § ACKNOWLEDGMENTS The authors would like to thank John F. Donoghue and Barry R. Holstein for kindly answering our questions.0 LIGO1 LIGO Scientific and Virgo Collab. (B. P. Abbott et al.), Phys. Rev. Lett. 116, 061102 (2016). LIGO2 LIGO Scientific and Virgo Collab. (B. P. Abbott et al.), Phys. Rev. Lett. 116, 241103 (2016). Milg M. Milgrom, Astrophys. J. 270, 365 (1983). FaMc B. Famaey and S. McGaugh, Living Rev. Relativity 15, 10 (2012). Eins1916 A. Einstein, Sitzungsber. K. Preuss. Akad. Wiss. 1, 688 (1916). wien65 S. Weinberg, Phys. Rev. 138, B988 (1965). don94 J. F. Donoghue, Phys. Rev. D 50, 3874 (1994), . don95 J. F. Donoghue, UMHEP-424, C95-06-25 (Talk given at Conference: C95-06-25), . wein2 S. Weinberg, Phys. Rev. 140, B516 (1965). NS2011 S. G. Naculich and H. J. Schnitzer, JHEP 1105, 087 (2011), . Don-Tor J. F. Donoghue and T. Torma, Phys. Rev. D 60, 024003 (1999), . white C. D. White, JHEP 1105, 060 (2011); . akhoury R. Akhoury, R. Saotome and George Sterman, Phys. Rev. D 84, 104040 (2011), [arXiv:1109.0270]. Don-Hol_00 J. F. 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Huang, Phys. Rev. D 95, 064045 (2017), . bohr2002 N. E. J. Bjerrum-Bohr, Phys. Rev. D 66, 084023 (2002), . mon95 G. Modanese, Nucl. Phys. B 434, 697 (1995), . iwa71 Y. Iwasaki, Prog. Theor. Phys. 46, 1587 (1971). | http://arxiv.org/abs/1703.09293v3 | {
"authors": [
"Azizollah Azizi",
"Soudabe Nasirimoghadam"
],
"categories": [
"hep-th",
"gr-qc"
],
"primary_category": "hep-th",
"published": "20170327201214",
"title": "Behavior of light polarization in photon-scalar interaction"
} |
acmcopyrighthttp://dx.doi.org/10.1145/3056540.3064942978-1-4503-5227-7/17/06PETRA '17June 21-23, 2017,Island of Rhodes,Greece$15.006Improving the Accuracy of the CogniLearn System for Cognitive Behavior Assessment Amir Ghaderi University of Texas at Arlington Texas, USA [email protected] GattupalliUniversity of Texas at Arlington Texas, USA srujana.gattupalli@ mavs.uta.eduDylan Ebert University of Texas at Arlington Texas, USA [email protected] Ali SharifaraUniversity of Texas at Arlington Texas, USA [email protected] Athitsos University of Texas at Arlington Texas, USA [email protected] MakedonUniversity of Texas at Arlington Texas, USA [email protected] 30, 2023 ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================= HTKS <cit.> is a game-like cognitive assessment method, designed for children between four and eight years of age. During the HTKS assessment, a child responds to a sequence of requests, such as “touch your head” or “touch your toes”. The cognitive challenge stems from the fact that the children are instructed to interpret these requests not literally, but by touching a different body part than the one stated. In prior work, we have developed the CogniLearn system, that captures data from subjects performing the HTKS game, and analyzes the motion of the subjects. In this paper we propose some specific improvements that make the motion analysis module more accurate. As a result of these improvements, the accuracy in recognizing cases where subjects touch their toes has gone from 76.46% in our previous work to 97.19% in this paper.<ccs2012> <concept> <concept_id>10003120.10003121</concept_id> <concept_desc>Human-centered computing Human computer interaction (HCI)</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010147.10010178</concept_id> <concept_desc>Computing methodologies Artificial intelligence</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010147.10010178.10010224</concept_id> <concept_desc>Computing methodologies Computer vision</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010147.10010257</concept_id> <concept_desc>Computing methodologies Machine learning</concept_desc> <concept_significance>500</concept_significance> </concept> </ccs2012> [500]Human-centered computing Human computer interaction (HCI) [500]Computing methodologies Artificial intelligence [500]Computing methodologies Computer vision [500]Computing methodologies Machine learning § INTRODUCTIONHTKS <cit.> is a game-like cognitive assessment method, designed for children between four and eight years of age. During the HTKS assessment, a child responds to a sequence of requests, such as “touch your head” or “touch your toes”. The cognitive challenge stems from the fact that the children are instructed to interpret these requests not literally, but by touching a different body part than the one stated. For example, a child may be instructed to touch her toes in response to the request “touch your head”. HTKS has been shown to be related to measures of cognitive flexibility, working memory, and inhibitory control. At the same time, HTKS has also been shown to be useful in predicting academic achievement growth for prekindergarten and kindergarten children.In our prior work, we have developed the CogniLearn system <cit.>, that can be used to record the motion of human subjects as they play the HTKS game, and that also analyzes that motion so as to assess how accurately the subjects executed the requested tasks. In CogniLearn, a Microsoft Kinect V2 camera is used for recording human motion. Then, we use the DeeperCut method <cit.> to perform body pose estimation in each frame. Finally, using the body pose estimates from DeeperCut we use a classification module that determines whether the subject touched his or her head, shoulders, knees, or toes. The CogniLearn system compares the part that was touched with the part that should have been touched based on the rules of the game, and assesses the overall accuracy score of the person playing the game. The rest of the paper is organized as follows: In Section <ref> we discuss related work in this area. In Section <ref> we describe the proposed improvements to the prior version of the CogniLearn system. A quantitative evaluation of these improvements is offered in the experiments (Section <ref>). § RELATED WORKSeveral deep-learning methods have been proposed in recent years for video analysis and activity recognition <cit.>, offering significantly improved accuracy compared to previous approaches<cit.>. Deep learning methods have also been used in supervised or unsupervised manner in different tasks in computer vision <cit.>, oftentimes producing state-of-the-art results.In <cit.> we have introduced the CogniLearn system, which is used for automated video capture andperformance assessment during the HTKS assessment. CogniLearn is designed to provide meaningful data andmeasures that can benefit therapists and cognitive experts. More specifically, the motion analysis and evaluation module provides systematic feedback regarding the performance of the HTKS tasks to the human experts. In this paper, we build upon the CogniLearn system, and we suggest some specific improvements in the motion analysis module, that lead to higher recognition accuracy. § OUR METHOD We use DeeperCut <cit.> to estimate the location of human body parts in each color frame of the video. Figure <ref> shows a video frame where we have superimposed the body part locations estimated by DeeperCut. Each color frame of a test video sequence is provided as input to the DeeperCut method. The output of the algorithm is the image location of 12 body parts: head, shoulder(right and left), elbow(right and left), wrist(right and left), hip, knee(right and left), ankle(right and left).After we obtain the body part locations from DeeperCut, we perform an additional step, in order to estimate whether the human, at that frame, is touching his or her head, shoulders, knees, or toes. As a first step, we define a distance D between hands and head, hands and shoulder, hands and knees, and hands and ankles. Using · to denote Euclidean norms, this distance is defined as follows: D(head)= lh - head + rh - head/2D(shoulders)= lh - ls + rh - rs/2D(knees)= lh - lk + rh - rk/2D(ankles)= lh - la + rh - ra/2 In the above definitions, head stands for the (x,y) pixel location of the center of the head in the color frame, as estimated by DeeperCut. Similarly, lh and rh stand for the locations the left and right hand, ls and rs stand for the locations of the left and right shoulder, lk and rk stand for the locations of the left and right knee, and la and ra stand for the locations of the left and right ankle.For example, lh - head denotes the Euclidean distance between the left hand and the center of the head.Based on these D values, one approach for estimating the body part that is being touched is to simply select the body part for which the D score is the smallest. This was the approach used in <cit.>. However, when the person touches the toes or knees, this approach does not work well. When a person bends down to touch the knees or toes with the hands, the head inevitably also gets near to the knees or toes. In that case, two issues may arise. The first one is that the accuracy of the body joint estimator is decreased. The second issue is that thedetected location for the head is near the detected locations for the knees or toes. As a result, for example, when the hands are touching the toes, it frequently happens that the distance of hands to the head is estimated to be smaller than distance of the hands to the toes. These two issues can lead to inaccuracies. As we see in Table <ref>, in the original CogniLearn results of <cit.>, 9.33% of toe frames are classified as head frames, and 14.00% of toe frames are classified as knee frames.In this paper, we propose two heuristic rules to improve the classification accuracy of toe frames:Rule 1: If the distance between the head and the hip is less than a predefined threshold, we can immediately conclude that the hands are touching the toes. Rule 2: Sometimes, when the hands are touching the head, the distance between the hands and the head is estimated to be longer than the distance between the hand and the shoulders. To address this issue, we add a constant bias value to the distance between hands and shoulders, before comparing it with the distance between the hands and the head.In the experiments, we demonstrate that these two rules significantly improve the classification accuracy on toe and head frames, while only minimally affecting the classification accuracy on frames where the hands touch the shoulders or knees. § EXPERIMENTS For our experiments, we use the same dataset that was used in the original CogniLearn paper<cit.>. The dataset includes color videos from 15 participants, whose ages are between 18 and 30 years (while the HTKS assessment has been designed for children between the ages of 4 and 8, at this time we still do not have recorded data available from children of that age). In total, the dataset contains over 60,000 video frames. Figure <ref> shows examples of test frames correctly recognized by our algorithm. The green letter in top left of the images shows the classification output of our system (“T” stands for “toes”, “K” stands for “knees”). Our method is applied on each color frame separately. The goal of our method is to classify each frame into one of four classes, corresponding respectively to whether the human is touching his or her head, shoulders, knees, or toes. Ground truth information is provided for 4,443 video frames, and we use those frames as our test set. The ground truthspecifies, for each frame, which of the four classes belongs to. Accuracy is simply measured as the percentage of test frames for which the output of our system matched the ground truth.We should emphasize that the results that we present are user-independent. None of the 15 subjects appearing in the test set is used to train any part of our models. The only module that uses training is DeeperCut, and we use the pretrained model that has been made available by the authors of <cit.>. §.§ ResultsTable <ref> shows the confusion matrix reported in the original CogniLearn paper <cit.>. As we can see in that table, shoulder and knee frames are recognized at rather high accuracies of 99.63% and98.17% respectively. However, head and toes frames are recognized with lower accuracies, 94.47% and 76.46% respectively. This paper was primarily motivated by the need to improve the accuracy for those two cases.In Table <ref> we report the results from the method proposed in this paper (i.e, when we apply both Rule 1 and Rule 2 from Section <ref>. As we can see, the accuracy for all four categories is more than 94.7%. The accuracy for head frames is marginally improved compared to <cit.>. The accuracy for shoulder and knee frames is slightly worse compared to <cit.>. At the same time, the accuracy for toe frames is now 97.19%, significantly higher than the accuracy of 76.46% reported in <cit.>. Finally, in Table <ref> we show results using a partial implementation of our method, applying only Rule 1, and not Rule 2. We note that the overall accuracy is mostly similar to what we get when we combine Rules 1 and 2. Overall, Rule 1 is by far the biggest contributor to the improvements we obtain over the original results of <cit.>. At the same time, the accuracy for head frames improves from 93.21% to 94.78% when we use Rules 1 and 2, compared to using only Rule 1. Rule 2 was explicitly designed to reduce the percentage of head frames that were classified as shoulder frames. Indeed, using Rule 2 (together with Rule 1) reduces that percentage from 4.96% (obtained using only Rule 1) to 3.39%. Table <ref> shows the overall classification accuracy.In that table, the overall accuracy is defined as the average of the accuracies over the four different classes. The overall accuracy improves from the 92.18% rate of <cit.> to 96.75% when we add Rule 1, and to 97.11% when we also add Rule 2.Figure <ref> shows some sample test frames. More specifically, from each of the four classes we show an example that was classified correctly, and an example that was classified incorrectly. We note that separating the head from the shoulder class can be quite challenging at times, because the distribution of hand positions does not vary much between the two classes. Separating knees and toes can also be difficult, because in frames belonging to both classes the knees are typically occluded, and there is significant overlap between the arms and the legs. This leads to errors in the estimated positions of the hands and the knees.§ CONCLUSIONS AND FUTURE WORK We have propose a method for improving the accuracy of the original CogniLearn<cit.> system in recognizing, for each video frame, whether the human is touching the head, shoulders, knees, or toes in that frame. The experiments have shown that our improvements lead to significantly better accuracy, especially for frames where the human touches the toes. In those cases, the accuracy increased from the 76.46% rate in <cit.> to 97.19%.Our project of automatically capturing and analyzing performance in the HTKS test is still in its initial stages. A high priority for us is to obtain data from children between the ages of 4 and 8, as that is the target age group for the HTKS test. Also, we plan to explore using the depth modality of the Kinect camera in addition to the color modality that we have used in <cit.> and in this paper. Finally, we should note that the HTKS assessment includes a “self-correction” category, in which the subject has started doing an incorrect motion and then self-corrected <cit.>. In the near future we plan to work on developing methods for identifying such self-correction cases, so that our assessment fully matches the formal HTKS description. § ACKNOWLEDGMENTSThis work was partially supported by National Science Foundation grants IIS-1055062, CNS-1338118, CNS-1405985, and IIS-1565328. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors, and do not necessarily reflect the views of the National Science Foundation.abbrv § HEADINGS IN APPENDICESThe rules about hierarchical headings discussed above for the body of the article are different in the appendices. In the appendix environment, the command section is used to indicate the start of each Appendix, with alphabetic order designation (i.e. the first is A, the second B, etc.) and a title (if you include one).So, if you need hierarchical structure within an Appendix, start with subsection as the highest level. Here is an outline of the body of this document in Appendix-appropriate form:§.§ Introduction §.§ The Body of the Paper§.§.§ Type Changes andSpecial Characters§.§.§ Math Equations Inline (In-text) EquationsDisplay Equations §.§.§ Citations§.§.§ Tables§.§.§ Figures§.§.§ Theorem-like Constructs§.§.§ A Caveat for theExpert §.§ Conclusions §.§ Acknowledgments §.§ Additional AuthorsThis section is inserted by ; you do not insert it. You just add the names and information in thecommand at the start of the document.§.§ ReferencesGenerated by bibtex from your .bib file.Run latex, then bibtex, then latex twice (to resolve references) to create the .bbl file.Insert that .bbl file into the .tex source file and comment out the command . § MORE HELP FOR THE HARDYThe sig-alternate.cls file itself is chock-full of succinct and helpful comments.If you consider yourself a moderately experienced to expert user of , you may find reading it useful but please remember not to change it. | http://arxiv.org/abs/1703.08697v1 | {
"authors": [
"Amir Ghaderi",
"Srujana Gattupalli",
"Dylan Ebert",
"Ali Sharifara",
"Vassilis Athitsos",
"Fillia Makedon"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170325143612",
"title": "Improving the Accuracy of the CogniLearn System for Cognitive Behavior Assessment"
} |
Kavli Institute for Theoretical Physics, University of California, Santa Barbara CA 93106, USAKavli Institute for Theoretical Physics, University of California, Santa Barbara CA 93106, USA We theoretically study the interplay between bulk Weyl electrons and magnetictopological defects, including magnetic domains, domain walls, and ℤ_6 vortex lines,in theantiferromagnetic Weyl semimetalsand Mn_3Ge with negative vector chirality.We argue that these materials possess a hierarchy of energies scales which allows a description of the spin structure and spin dynamics using a XY model with ℤ_6 anisotropy.We propose a dynamical equation of motion for the XY order parameter, which impliesthe presence of ℤ_6 vortex lines, the double-domain pattern in the presence of magnetic fields, and the ability to control domains with current.We also introduce a minimal electronic model which allows efficient calculation of the electronic structure in the antiferromagnetic configuration, unveiling Fermi arcs at domain walls, and sharp quasi-bound states at ℤ_6 vortices.Moreover, we have shown how these materials may allow electronic-based imaging of antiferromagnetic microstructure, and propose a possible device based on domain-dependent anomalous Hall effect.Anomalous Hall effect and topological defects in antiferromagnetic Weyl semimetals: Mn_3Sn/Ge Leon Balents December 30, 2023 ============================================================================================= true 2.7in 0.5in comm comm[comm][comm]#1LB: #1 #1JPL: #1 #1#1 Temporary newpage #1[FOOTNOTE: #1]#1 #1 #1#1[#1]The Hall effect has long been a nucleation center for geometry and topology in the physics of solids.In the 1950s, prescient work of Karplus and Luttinger identified Berry curvature of electron wavefunctions as the heart of the anomalous Hall effect (AHE) in ferromagnets<cit.>.In the 1980s, topology entered with the discovery of the quantum Hall effect.These ideas came together in the mid-2000s to unveil broad applications to electronic systems in the form of topological insulators, superconductors <cit.> and semimetals with topological Weyl (and other) fermion excitations <cit.>. The AHE re-appears as one of the key emergent properties of topological semimetals, and coming full circle, mostferromagnets are now believed to host Weyl fermions to which their AHE is at least in part attributed.The dissipationless nature of the Hall effect also makes it interesting for applications.Uses based on ferromagnets may, however, be limited by the difficulty of miniaturization posed by large fields generated by the magnetization.For this reason, antiferromagnetic realizations of AHE may be of practical interest, but the microstructure, dynamics, and AHE of antiferromagnets are relatively uninvestigated.Here we attack these issues in the family of noncollinear antiferromagnets includingMn_3Sn and Mn_3Ge, for which a strong AHE was predicted and then experimentally verified to exist<cit.>.First principles calculations further indicate that in Mn_3Sn and Mn_3Ge there are Weyl nodes around the Fermi level <cit.>.We argue that these materials possess a hierarchy of energies scales which permits a description of the microstructure and spin dynamics as an XY model with ℤ_6 anisotropy.We propose a dynamical equation of motion for the XY order parameter, which implies a rich domain structure, the presence of ℤ_6 vortex lines, and the ability to control domains with current.We further introduce a minimal electronic model which allows efficient calculation of the electronic structure in a textured antiferromagnetic configuration, unveiling Fermi arcs at domain walls, and sharp quasi-bound states at ℤ_6 vortices.We show how these materials may allow electronic-based imaging of antiferromagnetic microstructure (difficult to observe magnetically due to the lattice-scale variations) and propose a possible device based on domain-dependent AHE.Symmetry, order parameter, and implications:The -class material crystallizes in hexagonal lattice structure withspace group P6_3/mmc as shown in Fig. <ref>(a)-(b). Takingas an example, each Mn^4+ ion has a large spin ∼2-3 <cit.> forming a layered Kagome lattice. [Effects of quantum fluctuations may be neglected for such large spins,and hereafter we will treat these spins as classical objects.].The system orders antiferromagnetically in a 120^∘ noncollinear structure as shown in Fig. <ref>(c), with the Neel temperature T_N≈420 K <cit.>. This may be understood from the hierarchy of interactions typical for 3d transition metal ions: Heisenberg exchange J_ij S_i· S_j is largest, followed by Dzyaloshinskii-Moriya (DM) interaction D_ij· S_i× S_j, with single-ion anisotropy (SIA) -K(n̂_i· S_i)^2 the weakest effect.The former two terms select an approximately 120^∘ pattern of spins with negative vector chirality which leaves a U(1) degeneracy: any rotation of spins within the a-b plane leaves the energy unchanged, when the SIA is neglected. Consequently, we can associate with these states an XY order parameter ψ=m_s e^-iθ, where m_s is the magnitude of the local spin moment, and θ is (minus) the angle of some specific spin in the plane.We focus on the ordered phase, in which m_s is uniform, and the free energy may be written in terms of θ alone. Symmetry dictates the form F_s=∫ d^3𝐫( ρ/2|∇θ(𝐫)|^2+ρ_1|𝐊̂(θ)·∇θ|^2-λcos6θ(𝐫) -γ𝐁·𝐊̂ ) . Here ρ and ρ_1 are isotropic and anisotropic stiffnesses, λ is a ℤ_6 anisotropy. We also introduced the XY unit vector 𝐊̂=(cosθ,sinθ,0), which describes coupling γ to a uniform magnetic field 𝐁 (which occurs due to small in-plane canting of the moments <cit.>).Eq. (<ref>) is derived from the microscopicspin Hamiltonian (see Eq. (<ref>) in Method section), which allows us to estimate these parameters.We estimate ρ≈0.568 meV/, ρ_1≈0.011 meV/, and λ≈1.159×10^-7 meV/^3 attemperature 50 K.The ℤ_6 structure of the free energy implies the existence of six minimum energy domains in which θ maximizes λcos 6θ.We take λ>0, for which this is θ = 2π n/6, with n=0,…,5, and the corresponding spin configurations are shown in Fig. <ref>(c).It is convenient to label them as α^+,-, β^+,-, and γ^+,- as shown in Fig. <ref>(c)., the ± superscript denotingdomains which are time-reversal conjugates (θ→θ+π under time-reversal).The long-time dynamics follows from the free energy and the Langevin equation ∂θ(𝐫,t)/∂ t=-μδ F_s/δθ(𝐫,t)+ μη(𝐫,t) + f( j), where η(𝐫,t) represents a random thermal fluctuation at temperature T obeying the Gaussian distribution of zero mean: ⟨η(𝐫,t)⟩=0, and ⟨η(𝐫, t)η(𝐫', t')⟩=2k_BT δ(𝐫-𝐫')δ(t-t') (k_B is the Boltzmann constant.). μ is the damping factor, and hereafter is set to 1. The final term f( j) represents non-equilibrium forces to be discussed later.We note that the overdamped Langevin description with a single time derivative is valid at long times: this is sufficient for most purposes. Neglecting ρ_1 and for 𝐁=0, Eq. (<ref>) becomes the famous (overdamped) sine-Gordon equation.Itsstationary solutions include not only domains but domain walls, which are solitons with a width π/6√(ρ/λ)∼ 110 nm using our estimates.Significantly, the elementary domain walls connect states which differ by Δθ=π/3, which are not time-reversal conjugates.The ρ_1 term leads to orientation-dependence of the domain wall energy, and e.g. faceting of domain boundaries. Six of these minimal domain walls meet at curves in three dimensions which define ℤ_6 vortex lines – see Fig. <ref>(d), around which θ winds by ± 2π.To observe the microstructure predicted by the Langevin model, we carried out a numerical simulation of a thin slab, assuming homogeneity in the z direction and discretizing the 2D continuum model with an effective lattice constant of a_cg=600 – see Method section for details. Figure <ref>(a) shows the spin configuration resulting from a quench from a random initial state of a 576 μm^2 sample in zero applied field at an intermediate stage of evolution. Clearly there are six types of domains in the figure, marked by α^±, β^±, and γ^±.These sixfold domains merge at the vortices and antivortices marked by white and black dots respectively. In Fig. <ref>(b), we show the spin configuration resulting from the same preparation but with an applied magnetic field of B=0.005 T along the [120] axis (y axis).As is clearly shown in the figure, the field preferentially selects just two degenerate β^+ (cosθ=-1/2) and γ^- (cosθ=1/2) domains. The orientation of the domain wall, which tends to be normal to the [100] direction, is fixed by the anisotropic stiffness term.We will show that the double-domain pattern leads to a variety of new physics including domain-wall bound states, novel transport behavior, and domain-wall dynamics.Minimal electronic model and electronic structure:While the ab initio electronic structure ofand Mn_3Ge have been studied extensively, to study electronic properties of magnetic textures with large-scale spatial variations and/or surface/domain wall states is impractical with density functional theory.Therefore we introduce a minimalfour-band tight-binding (TB) model with a single spinor (p_z) orbitals at each Sn.As indicated by the thick dashed lines in Fig. <ref>(a)-(b), we consider the following four hopping processes: t_intra(𝐫_nm)= t_0 𝕀_2× 2+t_J σ·𝐒_nm+(-1)^ξ_mniλ_z σ_z , t_inter(𝐫_nm)=t_1 𝕀_2× 2 , t'_inter(𝐫_nm)= iλ_R 𝐞_soc^𝐫_nm·σ , t”_inter(𝐫_nm)=t_2 𝕀_2× 2 , where the hopping from orbital m centered at _m to orbtial n centered at _n is expressed as a 2× 2 matrix due to the spin degrees freedom of each orbital, and _nm=_n-_m.The model includes three spin-independent hopping terms (t_0 in-layer and t_1 and t_2 inter-layer), a spin-dependent hopping t_J reflecting exchange coupling to the Mn moment S in the middle of the bond across which the electrons hop, and two spin-orbit coupling (SOC) terms λ_z and λ_R, which are important due to the heavy nature of the Sn ions.Details on the ξ_mn and 𝐞_soc^𝐫_nm parameters which define the SOC are given in the Supp. Info.Hereafter we fix the parameters of the model as: t_0=1, t_1=0.5, t_J=-0.5, λ_z=0.5, t_2=-1, and λ_R=0.2.We arrange S_nm spins to reflect the spin order under consideration. In the ordered state we take the spin canting angle ∼ 1.7^∘, corresponding to a net moment ∼ 5% of each Mn spin for each Kagome cell. The bulk bandstructure of the TB model introduced above in the α^+ domain is shown in Fig. <ref>(a).We find that in the α^+ domain (see Fig. <ref>(c)), there are four Weyl nodes at (± 0.3522,0,0) and (∓ 0.3522,± 0.3522,0)at energy E_W1=-2.395t_0, which are denoted by solid blue dots in the inset of Fig. <ref>(a), with the sign corresponding to the chiralities of the Weyl nodes. There are two additional band touching points with quadratic dispersions along the k_z directionat (0,± 0.3564,0) at energy E_W2=-2.480t_0. Since the dispersion is quadratic along k_z, these two additional nodes carry zero Berry flux, and do not make significant contributions to the transport properties. The positions of theWeyl nodes in the other five domains can be obtained by applying C_3z and/or 𝒯 operations to those of the α^+ domain. From magnetic structure to electronic properties: The most interesting feature ofand its relatives is the strong influence of the magnetism on the electronic structure, and the ability to control the latter by modifying the former.The most basic electronic property is the conductivity.In thefamily, a symmetry analysis using crystal symmetries and Onsager relations tightly constrains the conductivity tensor (see Sec. <ref>).In general the antisymmetric part of the Hall conductivity is expressed in terms of a “Hall vector” Q, with 1/2(σ_μν-σ_νμ) = e^2/2π hϵ_μνλ Q_λ.We evaluate Q in a series up to third order in the order parameter ψ, and express the result in terms of K̂, which yieldsQ=q |ψ| K̂ + q̃ |ψ|^3Im[ (K̂_x + i K̂_y)^3] ẑ. where q |ψ| and q̃|ψ|^3 are parameters arising from microscopic modeling (see below). Since we expect the O(|ψ|^3) terms to be small, we observe that the Hall vector is directed along K̂, and lies in the xy plane. Note that, in a Weyl semimetal with all Weyl nodes at the Fermi level, Qis given by the fictitious dipole moment in momentum space of the Weyl points.Whileis a metal and this relation is not quantitively accurate, comparison of the Weyl nodes in the inset of Fig. <ref>(a) shows that it is qualitatively correct. We remark that due to the proportionality between the Hall vector and magnetization, K̂ can be replaced with M in Eq. (<ref>), with a suitable redefinition of q.To verify these symmetry considerations, we carried out a direct calculation of the full bulk conductivity tensor of the microscopic model using the Kubo formula (see Supplementary Information).We show the calculated anomalous Hall conductivity σ_zx in the α^+ domain in Fig. <ref>(b).The result is generically non-zero, but highly dependent upon the Fermi energy (the horizontal axis). Electronic transport, electronic structure, and bound states: The direct connection of the conductivity to the order parameter suggests that transport can be a fruitful probe of magnetic microstructures.When the electronic mean free path is shorter than the length scales of magnetic textures, a local conductivity approximation is adequate: J( r)=σ[ K( r)]E( r). From this relation and Eq. (<ref>), the electrostatic potential can be determined for an arbitrary texture θ(𝐫) (see Supp. Info.), and through inversion, it should be possible to image the magnetic domain structure purely through a spatially-resolved electrostatic measurement.In the full quantum treatment, the electronic structure is non-trivially modified by magnetic textures.The new feature here is the appearance of Fermi arcs at domain walls.This is because a domain wall acts as a sort of internal surface, at which Fermi arc states carry chiral currents, similar to ordinary surfaces.Without loss of generality consider a minimal energy domain wall between the β^+ and γ^- domains, which have K at ± 30^∘ from the y axis.The domains have Weyl points in the k_z=0 plane, with chiralities that differ in the two domains. Distinct electronic properties thus occur when this domain wall is in an xy, xz or yz plane of the crystal. Fig. <ref>(a) shows the surface spectral functions of the β^+ domain for a [100] surface. There are three Fermi arcs connecting the two projected Weyl nodes which are closer to the origin. Fig. <ref>(c) shows the spectral function at the interface of theβ^+ and γ^- domains with the same orientation.It showsdouble the Fermi arcs found at the interface, i.e. 6 instead of 3! Note that some of the projected Weyl nodes are buried in the bulk continuum due to the presence of additional Fermi surfacesaround the Weyl nodes, which causes some of the Fermi arcs to merge into the bulk states before connected to the Weyl nodes. Similarly, there are also Fermi-arc states bound to the domain walls in the xy and zx planes (see Supp. Info.).Short of a challenging measurement of the momentum-resolved density of states at a domain wall, how might one detect the presence of these Fermi arcs and associated bound states? We make two proposals.First, the in-plane transport within a domain wall may exhibit its own anomalous Hall effect.We checked that this indeed occurs for aβ^+-γ^- wall with zx-orientation, by calculating σ_zx for a supercell with two domain walls spread over 30 primitive cells.We find σ_zx=0.044 for the supercell, about two times larger than the bulk value of 0.023 found for the same cell with a uniform β^+ and γ^- state and no domain walls.This enhancement is expected whenever K̂ is normal to the wall in its interior.Second, domain wall bound states can manifest as an intrinsic resistance across the wall, since theytake away from the weight of continuum states which are strongly transmitted and hence contribute to conductance.We verified such a decreased conductance normal to the wall for all domain wall orientations in numerical studies (see Supp. Info.)While we focused on the domain walls, it is worth noting that the ℤ_6 vortex lines may have their own electronic states.Calculations in the Supp. Info. show that these vortex lines show a pronounced 6-fold pattern in their local density states, making them detectable by scanning tunnelling microscopy <cit.>. Current-driven domain-wall dynamics Let us now consider the feedback of the conduction electrons on the spin texture.This is important to control of the magnetic microstructure electronically.In ferromagnets, current-induced forces on domains and domain walls have been extensively studied, through the mechanism of spin-transfer torque<cit.>. Given that the primary order parameter of the antiferromagnet is not the magnetization, it is unclear how consideration of torque, i.e. conservation of angular momentum, applies here.Instead, we take a symmetry-based approach and ask how the current j may appear as a force in the equation of motion for theeasy spin angle θ, Eq. (<ref>).The result (see Supplementary Information) is that the force takes the form f( j) = -∑_a ( p_a j_a ∂_a θ + q_1j·∂_z K̂ + q_2 j_z ∇·K̂). Here p_x=p_y, p_z, q_1 and q_2 are constants.Various arguments (see Supp. Info) suggest that q_1 and q_2, which tend to drive the domain wall along the direction perpendicular to the current flow, are much smaller than p_a, so we henceforth neglect them.Despite the intrinsic antiferromagnetic nature of the system, the p_μ terms appear formally very similar to spin-transfer torques.They could be understood in a hydrodynamic fashion as describing “convection” of the spin texture with or against the current flow: indeed added to Eq. (<ref>) , these terms are equivalent to a Galilean boost and consequently velocity v_a = μ p_a j_a. This leads to concrete experimental proposals. Specifically, in the geometry of Fig. <ref>(b), a current applied along the x direction controls the position of the wall. The non-dissipative Hall voltage measured between two contacts across the y direction at fixed x can thereby be switched by purely electrical means, as the domain wall moves to the left or right of the contacts.The results of this paper provide the framework to design and model the spin dynamics and topologically-influenced electrical transport in the negative vector chirality antiferromagnetsand Mn_3Ge, and the methodology may be applied more broadly to XY-like antiferromagnetic systems. Weyl nodes in the electronic structure induce Fermi arc bound states that influence transport in the presence of domain walls.In addition to advancing the fundamental physics of Weyl fermions in noncollinear antiferromagnets, these results mark the -class of materials as promising candidates for novel magnetic storage devices.§ METHODS §.§ Derivation of the sine-Gordon modelIn this section, we present a derivation of the continuum sine-Gordon energy from a microscopic spin Hamiltonian.We consider the following spin interactionsH_s= J_1 ∑_⟨ ij⟩_xy𝐒_i·𝐒_j + J_2 ∑_⟨ij⟩_z𝐒_i·𝐒_j + ∑_⟨ ij⟩_xy𝐃_ij·𝐒_i×𝐒_j -∑_i K ( 𝐧̂_i ·𝐒_i)^2.Here we indicated a sum over nearest-neighbors in the xy plane by ⟨ ij⟩_xy and similarly nearest-neighbors in successive xy planes by ⟨ ij⟩_z.The spin 𝐒_i is considered as a classical vector with fixed length m_s. The positive constants J_1,J_2 are isotropic Heisenberg interactions on these bonds.We include a Dzyaloshinskii-Moriya interaction specified by the D-vector 𝐃_ij=-𝐃_ji which takes the form allowedby the symmetry of the kagomé lattice. Specifically, if we choose i and j for a given bond so that i→ j proceeds counter-clockwise on the triangle to which the bond belongs, then we have𝐃_ij = D 𝐳̂ + D' 𝐳̂×𝐞̂_ij,where 𝐞̂_ij is the unit vector oriented from site i to site j.It worth to note that prior modeling of the spin interactions in Mn_3Sn have included the D term but not the D' one.The K term is a local easy-axis anisotropy, which is determined by the unit vector 𝐧̂_ioriented along the direction between the spin i and either of its nearest Sb ions as indicated by the gray dashed line in Fig. <ref>(a). We first assume a uniform spin configuration, which is sufficient to describe the ground state, and it determines the Z_6 anisotropy λ. There are six spins per unit cell, which form two triangles, one in each of the two distinct layers.By inspection, we find that there is inversion symmetryin the magnetic ground state and we only have to consider the spins in one triangle. The Heisenberg term in Eq. (<ref>) is minimized by requiring the three spins to lie in a plane at 120 degree angles to one another. The plane of the spins is undetermined by the Heisenberg term, but fixed by the DM interaction. To leading order in the DMterms, the ground state is of the form𝐒^(0)_a = m_s[ cos (-θ - 2π a3);sin(-θ - 2π a3);0 ].We also include small “canting” of the spins away from the rigid configuration.Formally, we do this by writing K→η K, D→η D, D'→η D', and carrying out perturbation theory in η.To do so, we let𝐒_a = m_s[ √(1-u_a^2)cos (-θ - 2π a3+ϕ_a); √(1-u_a^2)sin(-θ - 2π a3+ϕ_a );u_a ], where a=1,2,3 denote the sublattice indices of the kagomeĺattice. We set ϕ_3 = -ϕ_1-ϕ_2 to keep ϕ_1,ϕ_2 linearly independent of θ.We also write u_1,2,3 and ϕ_1,2 in a series in η, u_a = ∑_n=1^∞ u_a,nη^n,ϕ_a =∑_n=1^∞ϕ_a,nη^n. Inserting Eq. (<ref>) into the spin Hamiltonian , we then obtain a formal expansion of the energy order by order in η. Keeping the expansion to the third order in η, then minimizing with respect to u_a and ϕ_a,we obtain the optimal spin configuration to first order in the canting angles, and the ground state energy to third order in η:E_gs = ℰ_0 - K^3/12(J_1+J_2)^2 m_s^2 cos 6θ,where ℰ_0 is a θ-independent constant. The coefficient of cos 6θ in Eq. (<ref>) allows us to determine λ in the sine-Gordon model.Further results are obtained by adding the effect of a Zeeman magnetic field to the energy. We repeat the previous analysis, taking the magnetic field 𝐁→η𝐁 as well.This corresponds to considering the Zeeman energy much small than J_1+J_2, an excellent approximation.It turns out thatthe leading term in the in-plane magnetizationis𝐌_xy = K g m_s/J_1 + J_2[ cosθ; sinθ;0 ]≡K g m_s/J_1 + J_2𝐊̂,One may note that the angle of the net magnetization θ is minus the U(1) rotation angle , which is due to the antichiral spin texture on kagomeĺattice. We refer the readers to Supp. Info. for more details. The out-of-plane magnetization turns out to be parametrically smaller by a factor of D'/J:M_z = -D' K g m_s/√(3)(J_1+J_2)^2sin 3θ.The above equation shows that the magnetization does not stay entirely within the xy plane.For λ>0, where the minimum energy values of θ are multiples of 2π/6, then sin 3θ = 0 and the bulk z-axis magnetization within a uniform domain vanishes. This corresponds to the case in which one of the three spins on each triangle orients along its easy axis, directly toward a neighboring Sn.One can verify that this situation preserves a mirror plane which enforces M^z=0.For λ<0, however, sin 3θ = ± 1 at the minimum values of θ, and so the domains are expected to have a small bulk magnetization, reduced by a factor of D'/√(3)(J_1+J_2) relative to the in-plane magnetization.Since such a z-axis magnetization seems not to have been detected in , we take this as evidence in favor of the λ>0 state. Even for this state, however, we see that the out of plane magnetization M^z becomes non-zero within domain walls.We remark in passing that experiments show that in Mn_3Ge the anomalous Hall conductivity within the xy plane is small but nonvanishing<cit.>, suggesting that the λ>0 state is realized in Mn_3Ge. We continue to study the magnetic susceptibilities in the high-field regime, i.e., when the spontaneous magnetization is much smaller than the field-induced one. When the field is within the xy plane,𝐁 = B(cosα,sinα,0), the in-plane susceptibility is expressed as χ_⊥ c=χ_⊥ c, 0+χ_⊥ c, 1cos6α,whereχ_⊥ c, 0=g^2/J_1+J_2( 1 - √(3)D/J_1+J_2 ) ,χ_⊥ c,1=Kg^2/6(J_1+J_2)^2.It follows that the in-plane magnetization is linear in field with an offset M_xy (see Eq. (<ref>)), and a six-fold modulation.Measurement of the six-fold modulation provides a way to determine K/(J_1+J_2). On the other hand, when the magnetic field is along the z direction, the out-of-plane susceptibility is expressed as χ_∥ c=g^2/J_1+J_2( 1 - √(3)D/3(J_1+J_2) ) . The exchange J_1+J_2 and DM parameter D can be determined by susceptibility measurements using Eq. (<ref>) and (<ref>). To obtain the full continuum theory, we need to allow slow spatial variations of θ.To do so, we introduce the parametrization similar to Eq. (<ref>) but with no assumptions about uniformity or symmetry:𝐒_a,s(𝐫) = m_s[ √(1-u_a,s^2(𝐫))cos (-θ(𝐫) - 2π a3+ϕ_a,s(𝐫)); √(1-u_a,s^2(𝐫))sin(-θ(𝐫) - 2π a3+ϕ_a,s(𝐫) ); u_a,s(𝐫). ]The idea now is to insert the ansatz in Eq. (<ref>) into the spin Hamiltonian, and expand both in powers of ϕ_a,s and u_a,s and in gradients. The leading stiffness terms can be obtained by minimizing the spin Hamiltonian with respect to ϕ_a,s and u_a,s at fixed θ.The result isH = ∑_𝐫 [ 9 m_s^2 a_0^2 J_1 (3J_1+2J_2)/2(3J_1+J_2)( |∂_x θ|^2 + |∂_yθ|^2) + 12 d^2 m_s^2 J_2 |∂_z θ|^2 ].From this the stiffnesses can be read off.The only remaining term in the continuum energy to be discussed is anisotropic gradient one.For simplicity, we neglect the possible effect of the DM interactions on this term, and set D=D'=0. Weanticipate that the anisotropic stiffness appears with a coefficient of order K.We treat the gradients, small canting angle, and K, all of the same order, and expand the energy up to O(ϕ^3).Due to the lack of mixing between z and xy components when D'=0, the out of plane canting components u_a,s have no effect and we can set them to zero.Then one may minimize the energy with respect to ϕ_a,and select the terms second order in gradients.After carrying through this algebra, we obtainH^(3) = ρ_1 ∑_𝐫[ (𝐊̂·∇θ)^2 -12 |∇_⊥θ|^2],with 𝐊̂ = (cosθ,sinθ,0) andρ_1 = 15 m_s^2 a_0^2 K J_1 (9J_1^2 + 3J_1 J_2 + 2J_2^2)/2(J_1+J_2)(3J_1+J_2)^2 .We refer the readers to Supp. Info. for more details about the derivation of the continuum theory.§.§ Evaluations of the sine-Gordon parametersAs discussed above, the exchange interaction J_1+J_2 and DM interaction D can be determined by the susceptibility measurements using Eq. (<ref>) and (<ref>). We have used the data measured at 300 K as reported in Ref. mn3sn-exp, and find that J_1+J_2=5.606 meV, and D=0.635meV.The ratio K/(J_1+J_2) can be determined by measuring the six-fold modulations of the in-plane magnetizations[We thank the groups of Professors Satoru Nakatsuji and Yoshichika Otani for sharing their unpublished data.].It turns out that K=0.187 meV. The finite-temperature effect is taken into account by letting m_s→ m_s(T), where m_s(T) is the mean-field expectation value of a spin 1 at temperature T. In particular, at T=50 K, m_s(T)=0.92. Given the specific values of J_1+J_2, D, K and m_s(T), we evaluate the ℤ_6 anisotropy λ=1.159× 10^-7 meV/^-3, the isotropic stiffness ρ=0.568 meV/, and the anisotropic stiffness ρ_1=0.011 meV/. We may also obtain the canting moment from Eq. (<ref>), which turns out to be 0.061 μ_B per unit cell at 50 K, from which we obtain the Zeeman energy density for in-plane magnetic field B as h=M_xyB=2.814× 10^-3B meV^-3T^-1, where B is the magnitude of the magnetic field in units of Tesla. Note that the estimated canting moment is about 5 times larger than the experimental measurements, and we suspect that themeasured value has underestimated the canting moment due to the cancellation from different domains. §.§ Domain-wall bound statesThe surface spectral functions as shown in Fig. <ref>(a) are calculated using the method proposed in Ref. surface-gf. In order to calculate the domain-wall spectral functions, we include the domain-wall layers coupled to the two semi-infinite domains, and the thickness of the domain wall is N_dw (in units of lattice constants). The spins vary smoothly from one domain to the other across the domain wall.The domain-wall spectral function can be solved using the Dyson equation, G_dw=G_dw^0+ G_dw^0(𝐤,ω)Σ_dw G_dw , where G_dw represents the retarded Green's function of the domain wall including the effects due to the couplings to two domains, while G_dw^0 isthe “bare" Green's function excluding the coupling between the domain wall and the domains, and Σ_dw is the self energy from the coupling. In the above equation the dependence on the 2D wavevector 𝐤 and the frequency ω is implicit. More specifically, G_dw^0=[ G_β^+^s 0 0; 0G_00 0; 0 0 G_γ^-^s ] ,where G_00 is the Green's function of the isolated domain-wall layers, and G_β^+^s and G_γ^-^s denote the surface Green's functions of the β^+ and γ^- domains calculated using the iterative scheme proposed in Ref. surface-gf. The self energy Σ_dw is simply the coupling between the domain wall and the domains, Σ_dw=[0 H_β^+,dw0; H_β^+,dw^†0 H_dw,γ^-;0 H^†_dw,γ^-0 . ] §.§ Symmetry analysis on the conductivity tensor In this section we derive the symmetry-allowed expressions of the bulk conductivity tensor.We consider four generators of the symmetry operations of space group P6_3/mmc: 120^∘ rotation about z axis C_3z, 180^∘ rotation about an in-plane axis which is parallel to [100] and half-way between the z=0 and z=c/2 plane C_2x, a 180^∘ screw rotation about z axis C_2z^s, and finally inversion 𝒫. The full conductivity tensor σ can be expressed as:σ= σ^0+∑_μ=x,yA^μ|ψ|K̂_μ +∑_μ,νB^μν|ψ|^2K̂_μK̂_ν+𝒪(|ψ|^3) , where K_μ is the μth component of the order parameter 𝐊̂=(cosθ,sinθ,0). σ^0 is the term which is independent of magnetic state,while A^μ and B^μν couples to𝐊̂ to the linear and quadratic orders respectively. Due to Onsager reciprocal relation, the terms which are odd (even) in K̂_μ have to be antisymmetric (symmetric). Thus A^μ is antisymmetric, while σ^0 and B^μν are symmetric. The conductivity tensor should be invariant under a symmetry operation g, which means𝒪_g σ^0 𝒪_g^T=σ^0𝒪_g A^μ 𝒪_g^T=∑_μ=x,yΓ_g,μν A^ν𝒪_g B^μν 𝒪_g^T= ∑_μ',ν'Γ_g,μμ'Γ_g,νν' B^μ'ν' ,where 𝒪_g is a 3 × 3 matrix representing the symmetry operation g on a 3D real vector,and Γ_g is a 2× 2 matrixrepresenting the symmetry operation g acting on the xy component of 𝐊̂. The symmetry representations 𝒪_g and Γ_g are tabulated in Table <ref>. After solvingEq. (<ref>), we obtain the symmetry-allowed conductivity tensor: σ_μν= σ_∥δ_μν +(σ_⊥ - σ_∥) δ_μ zδ_ν z + q |ψ|ϵ_μνλK̂_λ + b_1|ψ|^2K̂_μK̂_ν , where σ_⊥ denotes the out-of-plane diagonal conductivity, σ_∥ denotes the isotropic part of the in-plane diagonal conductivity, b_1 denotes the anisotropic part of the in-plane conductivity, and finally q term denotes theanomalous Hall conductivity. We refer the readers to Supp. Info. for more details about the numerical calculations of the conductivities.§.§ Symmetry analysis on the spin-transfer torquesIn this section we provide a derivation of Eq. (<ref>). The general expression for the current-inducedspin-transfer torque is: f(𝐣)=-∑_a,b(B_0^ab j_a ∂_bθ + B_1^ab j_a ∂_bθcosθ+B_2^ab j_a ∂_bθsinθ) , where j_a denotes the a component of the electric current with a,b=x,y,z.The dependence of θ on position and time is implicit.It is convenient to decompose Eq. (<ref>) as f(𝐣)=f_p(𝐣)+f_q(𝐣), where the leading term f_p(𝐣)=-∑_a,b B_0^abj_a∂_bθ, and the subleading term f_q(𝐣)=-∑_a,b(B_1^ab j_a ∂_bθcosθ+B_2^ab j_a ∂_bθsinθ).Let us first consider the f_p(𝐣) term. The U(1) rotation θ is transformed to θ_g after a symmetry operation g. More specifically, C_3z: θ→θ_g=θ+2π/3 , C_2x:θ→θ_g=-θ , C_2z^s: θ→θ_g=θ+π , 𝒫: θ→θ_g=θ . With a symmetry operation g, the spin-transfer torque f_p(𝐣)→f_p,g(𝐣), where f_p,g(𝐣) =-∑_a,a',b,b' B_0^ab 𝒪_g,a'a^T 𝒪_g,bb' j_a' ∂_b' θ_g=-∑_a,bB_0^ab j_a ∂_b θ_g . It follows that ∑_a'b' 𝒪_g,aa'^T B_0^a'b' 𝒪_g,b'b=B_0^ab , where 𝒪_g is tabulated in Table <ref>. After solving Eq. (<ref>), one obtains the p_a j_a∂θ term in Eq. (<ref>). The f_q(𝐣) term is more complicated.Under a symmetry transformation g,f_q(𝐣)→ f_q,g(𝐣), wheref_q,g(𝐣) = -∑_aa'bb'∑_μ=x,y( B_1^ab 𝒪^T_g,a'a 𝒪_g,bb' j_a' ∂_b'θ_g Γ_g,xμ K̂_μ +B_2^ab 𝒪^T_g,a'a 𝒪_g,bb' j_a' ∂_b'θ_g Γ_g,yμ K̂_μ )=-∑_ab( B_1^ab j_a ∂_bθ_g K̂_x+B_2^ab j_a ∂_bθ_g K̂_y ) ,where the matrix Γ_g has been tabulated in Table <ref>, K̂_x=cosθ, and K̂_y=sinθ. From the above equation it follows that ∑_a'b'𝒪_g,aa'^T ( B_1^a'b' Γ_g,xx+B_2^a'b' Γ_g,yx ) 𝒪_g,b'b =B_1^ab ,∑_a'b'𝒪_g,aa'^T ( B_2^a'b' Γ_g,yy+ B_1^a'b' Γ_g,xy ) 𝒪_g,b'b =B_2^ab. One would obtain the q_1, q_2 terms in Eq. (<ref>) after solving the above equations. Acknowledgements:We thank the groups of Professors Satoru Nakatsuji and Yoshichika Otani for introducing us to these materials and sharing their data.This research was supported by the National Science Foundation under grant number DMR1506119. | http://arxiv.org/abs/1703.08910v1 | {
"authors": [
"Jianpeng Liu",
"Leon Balents"
],
"categories": [
"cond-mat.mes-hall",
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170327025601",
"title": "Anomalous Hall effect and topological defects in antiferromagnetic Weyl semimetals: Mn$_3$Sn/Ge"
} |
Hadron detection with a dual-readout fiber calorimeter J. Park^m, F. Scuri^b, A. Sill^a and R. Wigmans^a,Corres December 30, 2023 ============================================================[2]Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Building 3, 143026 Moscow, Russia (mailto:[email protected]@gmail.com) [3]Institute of Numerical Mathematics, Gubkina St. 8, 119333 Moscow, Russia (mailto:[email protected]@skoltech.ru) In this work we generalize the Jacobi-Davidson method to the case when eigenvector can be reshaped into a low-rank matrix. In this setting the proposed method inherits advantages of the original Jacobi-Davidson method, has lower complexity and requires less storage. We also introduce low-rank version of the Rayleigh quotient iteration which naturally arises in the Jacobi-Davidson method. § INTRODUCTION This paper considers generalization of the Jacobi-Davidson (JD) method <cit.> for finding target eigenvalue λ (extreme or closest to a given number) and the corresponding eigenvector x of N× N matrix A:Ax = λ x.We treat the specific case when N=nm and the eigenvector x reshaped intomatrix is exactly or approximately of small rank r. For example, consider a Laplacian operator discretized on tensor product grid; its reshaped eigenvectors are of rank 1.For r≪ n,m our assumption allows to significantly reduce storage of the final solution, at the same time leading to algorithmic complications that we address in this paper. Similarly to the original JD method, we derive the low-rank Jacobi correction equation and propose low-rank version of subspace acceleration.The proposed approach takes the advantage of the original JD method. Compared with the Rayleigh quotient iteration and the Davidson approach, the method is efficient for the cases when arising linear systems are solved both accurately and inexactly.The JD method is known to be a Riemannian Newton method on a unit sphere {x:x= 1} with additional subspace acceleration <cit.>. We utilize this interpretation and derive a new method as an inexact Riemannian Newton method on the intersection ofthe sphere and the fixed-rank manifold.In derivation we assume that the matrix A is real and symmetric, however we test our approach on non-symmetric matrices as well. Complexity of the proposed algorithm scales as 𝒪((n+m) r (R+r)) if A can be approximated asA ≈∑_α=1^R F_α⊗ G_α,where F_α and G_α allowfast matrix-vector multiplication, e.g. they are sparse. Our main contributions are: * We generalize the Jacobi correction equation (Sec. <ref>) and the subspace acceleration (Sec. <ref>) to the case of fixed-rank matrix manifolds.* We introduce low-rank version of the Rayleigh quotient iteration (Sec. <ref>) which naturally arises in the JD method.§ RAYLEIGH QUOTIENT MINIMIZATION ON SPHEREThe first ingredient of the JD method is the Jacobi correction equation.The Jacobi correction equation can be derived as a Riemannian Newton method on the unit sphere <cit.>, which will be useful for our purposes. In this section we provide the derivation, and in Sec. <ref> it will be generalized to the low-rank case.Given a symmetric matrix A∈^n× n the goal is to optimize(x) = x^⊤ A x,subject to x∈ S^n-1, where S^n-1 is a unit sphere considered as an embedded submanifold of ℝ^n with the pullback metric g_x(ξ, η) = ξ^⊤η.The Riemannian optimization approach implies that we optimize (x) on S^n-1, i.e. constraints are already accounted for in the search space. One of the key concepts in the Riemannian optimization is a tangent space which is in fact a linearization of the manifold at a given point.The orthogonal projection of ξ on the tangent space T_x S^n-1 of S^n-1 at x can be written as <cit.>_T_x S^n-1ξ = (I - x x^⊤) ξ.The Riemannian gradient of (<ref>) is(x) = _T_x S^n-1∇ (x) = (I - x x^⊤) (2Ax),where ∇ denotes the Euclidean gradient. The Hessian _x: T_x S^n-1→ T_x S^n-1 can be obtained as <cit.>_x(x) [ξ] = _T_x S^n-1( D ((x)) [ξ] ) =2_T_x S^n-1( D( _T_x S^n-1 Ax) [ξ] ) = 2 _T_x S^n-1 (A ξ + _T_x S^n-1 A x), ξ∈ T_x S^n-1,where D denotes the differential map (directional derivative) and_T_x S^n-1 A x ≡D(_T_x S^n-1)[ξ] A x= - (x^⊤ Ax) ξ - (ξ^⊤ A x) x.Since _T_x S^n-1 x = 0 and _T_x S^n-1ξ = ξ we arrive at_x(x) [ξ] = 2 _T_x S^n-1(A - (x^⊤ A x) I ) _T_x S^n-1ξ. The k-th step of the Riemannian Newton methods looks as_x_k(x_k) [ξ_k] =-(x_k), ξ_k ∈ T_x_k S^n-1,with the retractionx_k+1 = x_k + ξ_k/x_k + ξ_k,which returns x_k + ξ_k back to the manifold S^n-1. Using (<ref>), (<ref>) and (<ref>) we can rewrite (<ref>) as(I - x_k x_k^⊤) (A - (x_k) I )(I - x_k x_k^⊤) ξ_k = - r_k,x_k^⊤ξ_k = 0,where(x_k) = x_k^⊤ A x_k, r_k = (I - x_k x_k^⊤) Ax_k = A x_k - (x_k) x_k.Equation (<ref>) is called the Jacobi correction equation <cit.>. Note that without the projection (I - x_k x_k^⊤) we obtain the Davidson equation(A - (x_k) I ) ξ_k = - r_k,which has solution ξ_k = -x_k collinear to the current approximation x_k. This is the reason for the Davidson equation to be solved inexactly. The original Davidson algorithm <cit.> replaces A by its diagonal part diag(A). By contrast, even if the Jacobi correction equation (<ref>) is solved inexactly using Krylov iterative methods, its solution ξ_k will be automatically orthogonal to x_k which is beneficial for the computational stability. Moreover, since the JD method has the Newton interpretation it boasts local superlinear convergence. The goal of this paper is to extend the Jacobi correction equation (<ref>) and the second ingredient of the JD method — subspace acceleration — to the case of low-rank manifolds. § JACOBI CORRECTION EQUATION ON FIXED-RANK MANIFOLDSLet x∈^nm be an eigenvector of A and X∈^n× m be its matricization: x =(X), wheredenotes columnwise reshape of n× m matrix into nm vector. In this paper we make an assumption that matricisized eigenvector X is approximately of rank r. Therefore, for example, to approximate the smallest eigenvalue we solve the following optimization problemminimize (x) = x^⊤ A xs.t.x ∈ S^nm-1∩_r,where_r = {vec(X),X∈ℝ^n× m: rank(X) = r },which forms a smooth embedded submanifold of ^nm of dimension (m+n)r - r^2 <cit.>. By analogy with the derivation of the Jacobi equation we additionally intersected the manifold_r with the sphere S^nm-1. As we will see from the following proposition S^nm-1∩_r forms a smooth embedded submanifold of ^nm. Hence, optimization problem (<ref>) can be solved using Riemannian optimization techniques.Let =S^nm-1∩_r, then *forms smooth embedded submanifold of ^nm of dimension (n+m)r-r^2 - 1 * The tangent space ofat (X) ∈ with X given by SVD: X= USV^⊤, U^⊤ U = I, V^⊤ V = I, S = diag(σ_1,…,σ_r), σ_1≥…≥σ_r >0 can be parametrized as T_X= { (U_ξ V^⊤ + U V^⊤_ξ + U S_ξ V^⊤): U_ξ⊥ U, V_ξ⊥ V,(S_ξ) ⊥ (S) }. * The orthogonal projection _T_X onto T_X can be written as _T_X=_T_X _r_T_X S^nm-1 = _T_X S^nm-1_T_X _r=_T_X _r- vec(X)vec^⊤(X), where _T_X _r is the orthogonal projection onto the tangent space of _r: _T_X _r = VV^⊤⊗ UU^⊤ + VV^⊤⊗ (I_n - UU^⊤) +(I_m - VV^⊤) ⊗ UU^⊤.The first property follows from the fact that _r and S^nm-1 are transversal embedded submanifolds of ^nm. Indeed, one can easily verify thatT_X _r + T_X S^nm-1 =^nm. Hence, by the transversality property <cit.>forms a smooth embedded submanifold of ^nm of dimension (_r) + (S^nm-1) - (^nm) = (n+m-r)r -1 Let us prove the second property of the proposition. Vector ξ∈ T_X _r can be parametrized <cit.> asξ =(U_ξ V^⊤ + U V^⊤_ξ + U S_ξ V^⊤)with the gauge conditionsU_ξ⊥ U,V_ξ⊥ V.To obtain the parametrization of ξ∈ T_X S^nm-1∩ T_X _r we need to take into account that ξ∈ T_X S^nm-1 and, hence, ξ^⊤ x =0 yielding the additional gauge conditionvec (S_ξ) ⊥vec (S). Let us prove the third property by showing that operators _T_X _r and _T_X S^nm-1 commute and, hence,_T_X=_T_X _r_T_X S^nm-1 = _T_X S^nm-1_T_X _r is an orthogonal projection on the intersection of T_X _r and T_X S^nm-1. Indeed, since (X)(X)^⊤ = (V⊗ U) (S) ((S))^⊤ (V^⊤⊗ U^⊤),andUU^⊤ ( I -UU^⊤) = 0,VV^⊤ ( I -VV^⊤) = 0,we get_T_X _r(X)((X))^⊤ = (X)((X))^⊤ =(X)((X))^⊤_T_X _r.Finally, since _T_X S^nm-1 = I - (X) ((X))^⊤_T_X _T_X=_T_X _r_T_X S^nm-1 = _T_X _r ( I - (X)((X))^⊤) =_T_X _r- (X)((X))^⊤ =_T_X S^nm-1_T_X _r,which completes the proof.§.§ Derivation of Jacobi correction equation on Let us derive the generalization of the original Jacobi correction equation, which is the Riemannian Newton method on .Using (<ref>) and notation x =(X) we obtain(x) =_T_X ∇ (x) = _T_X _r(I - xx^⊤) ∇ (x) = 2_T_X _r(I - xx^⊤) Ax.Similarly to (<ref>) using (<ref>) we get_X(x) [ξ] =2 _T_X(A ξ + _T_XA x) =2 _T_X (A ξ - x ξ^⊤ A x - ξ x^⊤ A x + _T_X _r Ax), ξ∈ T_X.According to (<ref>) _T_Xx = _T_X _r_T_X S^nm-1 x = 0, thus_X(x) [ξ] =2_T_X(A - (x^⊤ A x) I) ξ + _T_X _T_X _r Ax = 2_T_X _r (I - x x^⊤) (A - (x^⊤ A x) I) ξ + _T_X _T_X _r Ax,where the part _T_X _T_X _r Ax corresponds to the curvature of the low-rank manifold. This term contains inverses of singular values. Singular values can be small if the rank is overestimated. This, therefore, leads to difficulties in numerical implementation. Similarly to <cit.> we omit this part and obtain an inexact Newton method, which can be viewed as a constrained Gauss-Newton method. Omitting _T_X _T_X _r Ax we get_X(x) [ξ] ≈ 2_T_X _r (I - x x^⊤) (A -(x) I) ξ,or in the symmetric form _X(x) [ξ] ≈ 2_T_X _r (I - x x^⊤) (A -(x) I) (I - x x^⊤)_T_X _rξ.Using (<ref>) and (<ref>) we can write the linear system arising in the inexact Newton method as(I - x x^⊤) [ _T_X _r(A -(x) I)_T_X _r](I - x x^⊤) ξ = - _T_X _r(I - x x^⊤) Ax,ξ^⊤ x = 0,ξ∈T_X _r .which has the form similar to the original Jacobi correction equation (<ref>) with (A -(x) I) projected on T_X _r.Equation (<ref>) is a linear system of size nm× nm, but the number of unknown elements is equal to dimension of the tangent space (n+m) r - r^2 - 1. Hence, the next step is to derive a local linear system that is of smaller size and is useful for the numerical implementation. The following proposition holds.The solution of (<ref>) written asξ =(U_ξ V^⊤ + U V^⊤_ξ + U S_ξ V^⊤), can be found from the local system(I - BB^⊤) (A-(x) I)_loc (I - BB^⊤) τ_ξ = - (I - BB^⊤) g,B^⊤τ_ξ = 0,where[For an nm× nm matrix C we introduced notation C_v,v = (V_k^⊤⊗ I_n) C( V_k ⊗ I_n) ∈^nr × nr,C_v,u = (V_k^⊤⊗ I_n) C(I_m⊗ U_k) ∈^nr × mr,C_v,vu= (V_k^⊤⊗ I_n) C ( V_k ⊗ U_k) ∈^nr × r^2.Matrices C_u,v,C_u,u, C_u,vu and C_vu,v,C_vu,u,C_vu,vu are defined likewise.]τ_ξ = [ (U_ξ); (V^⊤_ξ); (S_ξ) ], g = [ A_v,v(U S); A_u,u(S V^⊤); A_vu,vu(S) ], B = [ I_r ⊗ U 0 0; 0 V ⊗ I_r 0; 0 0 (S); ], (A - (x) I)_loc =[ (A -(x) I)_v,v (A -(x) I)_v,u(A -(x) I)_v,uv; (A -(x) I)_u,v (A -(x) I)_u,u(A -(x) I)_u,vu;(A -(x) I)_vu,v(A -(x) I)_vu,u (A -(x) I)_vu,vu ], Notice that _T_X _r is a sum of three orthogonal projections _T_X _r = _1 + _2 + _3,_1 =VV^⊤⊗ (I_n - UU^⊤),_2= (I_m - VV^⊤) ⊗ UU^⊤, _3 = VV^⊤⊗ UU^⊤Since _i _j = O,i≠j and _i^2 = _i we obtain[ _1; _2; _3 ] (I - x x^⊤) (A -(x) I) (I - x x^⊤) [ _1 _2 _3 ][ _1ξ; _2ξ; _3ξ ] =[ _1; _2; _3 ](I - x x^⊤) Ax.It is easy to verify that _1 (I - xx^⊤) = _1 = (V⊗ I_n) (V^⊤⊗ (I_n - UU^⊤)),_2 (I - xx^⊤) = _2 = (I_m ⊗ U) ((I_m - VV^⊤) ⊗ U^⊤),_3(I - xx^⊤) = (VV^⊤⊗ UU^⊤) (I - (V⊗ U) (S) ((S))^⊤ (V^⊤⊗ U^⊤)) = (V⊗ U) (I_r^2 -(S) ((S))^⊤ ) (V^⊤⊗ U^⊤).Then from (<ref>)_1 ξ =V ⊗ (I_n - UU^⊤)(U_ξ), _2 ξ = (I_m - VV^⊤) ⊗ U(V_ξ^⊤), _3 ξ =V ⊗ U(S_ξ),Thus, the first block row in (<ref>) can be written asV⊗ (I_n - UU^⊤)( (V^⊤⊗ I)(A -(x) I)(V ⊗ I_n)_(A -(x) I)_v,v(I_r ⊗ (I_n - UU^⊤)) (U_ξ) + (V^⊤⊗ I)(A -(x) I)(I_m ⊗ U)_(A -(x) I)_v,u((I_m - VV^⊤) ⊗ I_r)(V^⊤_ξ)+ (V^⊤⊗ I)(A -(x) I)(V ⊗ U))_(A -(x) I)_v,uv(I_r^2 -(S) ((S))^⊤ ) (S_ξ) = V⊗ (I_n - UU^⊤) (V^⊤⊗ I) A (V ⊗ I_n)_A_v,v (US).Since V has full column rank we obtain exactly the first block row in (<ref>). Other block rows can be obtained in a similar way.§.§ RetractionSimilarly to (<ref>) after we obtained the solution ξ from(<ref>) we need to map the vector x+ξ from the tangent space back to the manifold. The following proposition gives an explicit representation for the retraction on . Let R_r be a retraction from the tangent bundle T _r onto _r, then R(X, Ẋ) = R_r(X, Ẋ)/R_r(X, Ẋ), is a retraction onto . To verify that R is a retraction we need to check the following properties <cit.> * Smoothness on a neighborhood of the zero element in T; * R(X,0) = X for all X∈; * .d/dtR(X, tẊ)|_t=0 = Ẋ for all X∈ and Ẋ∈ T_X. The first property follows from the smoothness of R_r. The second property holds since R_r(X,0) = X andX=1 for X∈. Let us verify the third property: .d/dtR(X, tẊ)|_t=0 = .d/dt(R_r(X, tẊ)/R_r(X, tẊ))|_t=0 = .d/dtR_r(X, tẊ)|_t=0.R_r(X, tẊ)|_t=0 - d/dt.R_r(X, tẊ)|_t=0.R_r(X, tẊ)|_t=0/.R_r(X, tẊ)|_t=0^2. Since (X, Ẋ) = 0 for X∈, we get .d/dtR_r(X, tẊ)|_t=0= .(d/dt R_r(X, tẊ), R_r(X, tẊ))|_t=0 + .(R_r(X, tẊ), d/dt R_r(X, tẊ))|_t=0/2.R_r(X, tẊ)|_t=0= (Ẋ, X) + (X, Ẋ)/2X = 0,X ∈, Ẋ∈ T_X . Substituting the latter expression into (<ref>) and accounting for .R_r(X, tẊ)|_t=0= R_r(X, 0) =X=1we obtain .d/dtR(X, tẊ)|_t=0 = Ẋ which completes the proof. Retraction (<ref>) is a composition of two retractions: first on the low-rank manifold _r and then on the sphere S^n-1. Note that the composition in the reversed order is not a retraction as it does not map to the manifold . A standard choice of retraction on _r is <cit.> R_r(x,ξ) ≡ R_r(x+ξ) = __r (x + ξ), where __r (x+ξ) ≡_y∈_ry - (x+ξ).For small enough correction ξ retraction can be calculated using the SVD procedure <cit.> as follows. First,x + ξ = vec(USV^⊤ + U_ξ V^⊤ + U V^⊤_ξ + U S_ξ V^⊤) =([ U U_ξ ][ S + S_ξ I; I O ][ V V_ξ ]^⊤) .Then we calculate QR decompositions Q_U R_U = [ U U_ξ ], Q_V R_V = [ V V_ξ ].and the truncated SVD with truncation rank r ofR_U [ S + S_ξ I; I O ] R_V^⊤,with r leading singular vectors U_r∈^2r× r, V_r∈^2r× r and the matrix of leading r singular values S_r∈^r× r. Thus, the resulting retraction can be written asR_r(x + ξ) = (Q_U U_r) S_r (Q_V V_r)^⊤.and from (<ref>) the retraction has the formR(x,ξ) ≡ R(x + ξ) = (Q_U U_r) S_r/S_r(Q_V V_r)^⊤. §.§ Properties of the local systemLet us mention several important properties of the matrix (A- (x) I)_loc. Assume that we are looking for the smallest eigenvalue λ_1 and (x) is closer to λ_1 than to the next eigenvalue λ_2, i.e. the matrix (A- (x) I) is nonnegative definite.First, the matrix (A- (x) I)_loc is singular. Indeed, a nonzero vector [(U); -(V^⊤);0 ] is in the nullspace of (A- (x) I)_loc. This is the result of nonuniqueness of the representation of a tangent vector without gauge conditions. However,(A- (x) I)_loc is positive definite on the subspace B^⊤τ_z=0,τ_z =[(U_z); -(V_z^⊤);(S_z) ] where B is defined in (<ref>).Indeed, min_B^⊤τ_z = 0,τ_z≠0(τ_z,(A- (x) I)_loc τ_z ) = min_B^⊤τ_z = 0,τ_z≠0 ((U_z V^⊤ + U V_z^⊤ + US_z V^⊤), (A- (x) I)(U_z V^⊤ + U V_z^⊤ + US_z V^⊤))=min_z ∈ T_x, z≠0(z,(A- (x) I)z) ≥min_z ⊥ x, z≠0(z,(A- (x) I)z) ≥λ_1 + λ_2 - 2(x).The latter inequality follows from <cit.>.Hence, if (x) is closer to λ_1 than to λ_2, the matrix is positive definite.Let us show that the condition number of (I - BB^⊤)(A- (x) I)_loc(I - BB^⊤)does not deteriorate as (x) converges to the exact eigenvalue. The condition number is given asκ = max_τ_z: B^⊤τ_z = 0,τ_z≠0q(τ_z)/min_τ_z: B^⊤τ_z = 0,τ_z≠0 q(τ_z), q(τ_z) = (τ_z, (A- (x) I)_locτ_z)/(τ_z, τ_z).Similarly to (<ref>) one can show thatκ≤max_z: z⊥ x, z≠0 q(z)/min_z: z⊥ x, z≠0 q(z).This expression is a bound for the original Jacobi correction equation and according to <cit.> its condition number does not grow as (x) approaches the exact eigenvalue λ_1.§ SUBSPACE ACCELERATION Since the considered Newton method is inexact or linear systems are solved approximately, we can additionally do the line searchx_new = R(x + α_optξ),where α_opt = _α(R(x + αξ)),which can be found from the Armijo backtracking rule <cit.> or simply approximated without retraction asα_opt≈_α(x + αξ),which can be solved exactly. To accelerate the convergence one can utilize vectors obtained on previous iterations in the Jacobi-Davidson manner. However, to avoid instability and reduce the computational cost we use the vector transport <cit.>. At each iteration we project the basis obtained from previous iterations on the tangent space of the current approximation to the solution.Let us consider this approach in more details.After k iterations we have the basis 𝒱_b-1 = [v_1,…,v_b-1], b≤ k and project it on T_X_k_r:𝒱_b-1 = [_T_X_k_r v_1,…,_T_X_k_rv_b-1].If needed we can carry out additional orthogonalization of 𝒱_b-1 vectors. Note that orthogonalization onto the tangent space is an inexpensive operation since linear combinations of any number of vectors from the tangent space can be at most of rank 2r. Given the solution ξ_k of (<ref>) next step is to expand 𝒱_b-1 with v_b obtained from the orthogonalization of ξ_k with respect to 𝒱_b-1:𝒱_b = [𝒱_b-1, v_b]A new approximation to x is calculated using the Rayleigh-Ritz procedure. Namely, we calculate 𝒱_b^⊤ A 𝒱_band then find the eigenpair (θ, c): 𝒱_b^⊤ A 𝒱_b c = θ c,corresponding to the desired eigenvalue. Finally, the Ritz vector c gives us a new approximation to x: x_k+1 =𝒱_bc.We emphasize that the columns of 𝒱_b are from T_X_k_r, therefore there is no problem with the rank growth. If one wants to maintain fixed rank r it is required to optimize the coefficients c:x_k+1 =R(𝒱_b c_opt),c_opt =_c_1,…,c_b(R(𝒱_b c)).Optimization can be done, e.g. by using the line search over each of c_i sequentially, starting from the initial guess found from (<ref>). However, to reduce complexity one can optimize only over the coefficient in front of v_b, or simply use c instead of c_opt.§ CONNECTION WITH RAYLEIGH QUOTIENT ITERATIONIf the linear system in (<ref>) is solved exactly, JD method without the subspace acceleration is known <cit.> to be equivalent to the Rayleigh quotient iteration:(A - (x_k)I)x̃ = x_k, x_k+1 = x̃/x̃. Let us find how the method will look like when we solve (<ref>) exactly. On the k-th iteration equation (<ref>) looks as(I - x_k x_k^⊤) _T_X_k_r (A -(x_k) I)_T_X_k_rξ_k=-_T_X_k_r (I - x_k x_k^⊤)Ax_k,_T_X_k_rξ_k= ξ_k,x_k^⊤ξ_k =0.Therefore,_T_X_k_r (A -(x_k) I)_T_X_k_r ξ_k - α x_k = -_T_X_k_r (A -(x_k) I) x_k,whereα = x_k^⊤ [_T_X_k_r (A -(x_k) I)_T_X_k_r]ξ_k.Denoting x̃ = x_k + ξ_k, we obtain[_T_X_k_r (A -(x_k) I)_T_X_k_r] x̃ =x_k, _T_X_k_rx̃ = x̃, x_k+1 = R(x̃).where the parameter α was omitted thanks to R(αx̃) = R(x̃).Thus, (<ref>) represents the extension of the Rayleigh quotient (RQ) iteration (<ref>) to the low-rank case and can be interpreted as a Gauss-Newton method. One can expect that the JD method converges faster than the RQ iteration (<ref>) when systems are solved inexactly. As we have shown in Sec. <ref> the condition number of local systems in the proposed JD method does not deteriorate when (x_k) approaches the exact eigenvalue. This property positively influences the convergence, as was investigated for the original JD <cit.>. We will illustrate it in the numerical experiments in Sec. <ref>.§ COMPLEXITYLet us discuss how to solve the Jacobi correction equation numerically for the matrix A given asA = ∑_α=1^R F_α⊗ G_α,where matrices F_α and G_α are of sizes n× n and m× m correspondingly. In complexity estimates we additionally assume that F_α and G_α can be multiplied by a vector using 𝒪(n) and 𝒪(m) operations respectively, e.g. they are sparse.As an example, A can be the Laplacian-type operator with low-rank potential.Even if the initial operator A is sparse, the projected local system A_loc is usually dense. Fortunately, a fast matrix-vector multiplication by A_loc can be done. Let us consider the multiplication by the first block row of A_loc:u= A_v,vvec(U) + A_v,uvec(V^⊤) + A_vu,vuvec(S)=(V_k^⊤⊗ I_n) A (vec (UV_k^⊤ + U_k V^⊤ + U_k S V_k^⊤)=(V_k^⊤⊗ I_n) A (vec (UV_k^⊤ + U_k (V^⊤ +S V_k^⊤)),where we took into account that the vector from the tangent space UV_k^⊤ + U_k V^⊤ + U_k S V_k is of rank 2r instead of 3r as in the case when summing 3 arbitrary rank-r matrices. This slightly decreases the cost of matrix-vector multiplication.Finally substituting (<ref>) into (<ref>)u = (V_k^⊤⊗ I_n)( ∑_α=1^R F_α⊗ G_α) ( ( V_k ⊗ I_n)vec(U) + (I_n⊗ U_k) vec (V^⊤ +S V_k^⊤) ) =( ∑_α=1^R (V_k^⊤F_αV_k) ⊗ G_α)vec(U) + ( ∑_α=1^R (V_k^⊤F_α) ⊗ (G_α U_k) )vec(V^⊤ +S V_k^⊤).Calculation of an r× r matrix V_k^⊤F_αV_k requires 𝒪(nr^2 + n r) operations. Multiplication of V_k^⊤F_αV_k ⊗ G_α by a vector costs 𝒪(mr^2+ m r).Calculation of V_k^⊤F_α and G_α U_k costs 𝒪(n^2r) and 𝒪(m^2r) respectively.As a result, matrix-vector multiplication costs 𝒪((n+m)r^2) operations. Given fast matrix-vector multiplication we can solve (<ref>) by the appropriate Krylov type iterative method. In the next section we discuss how to construct a preconditioner for this system.In subspace acceleration we project vectors of V_b (<ref>) onto the tangent space. Projection of each vector costs 𝒪((m+n)r^2). Thus, assuming that r≪ n,m the complexity of the whole algorithm is 𝒪((n+m) r(R+r)). § BLOCK JACOBI PRECONDITIONING OF THE LOCAL SYSTEMIn the work <cit.> the preconditioner of the typeM_d = ( I - x x^⊤) M ( I - x x^⊤)was proposed, where M is an approximation to A - (x) I. If a system with M can be easily solved, then to solveM_dy = z,one can use the explicit formulay = - λ M^-1 x - M^-1 z, λ = - x^⊤ M^-1 z/x^⊤ M^-1 x.Following this concept we consider a preconditioner of a typeM_d = (I - B B^⊤) M_loc (I - B B^⊤) ,where M_loc is an approximation to (A - (x) I)_loc. Even ifM is easily inverted, this might not be the case for the projected matrix M_loc. Hence, we use a block Jacobi preconditioner M_d = (I - B B^⊤) [ A_v,v -(x) I0;0 A_u,u -(x) I0;00 A_vu,vu -(x) I ] (I - B B^⊤)= [ P^⊥_U (A_v,v -(x) I) P^⊥_U0;0P^⊥_V(A_u,u-(x) I)P^⊥_V0;00P^⊥_S(A_vu,vu-(x) I)P^⊥_S ],where the projection matrices P^⊥_U, P^⊥_V and P^⊥_S are defined asP^⊥_U = I_r ⊗ (I_n - U U^⊤), P^⊥_V=(I_n -V V^⊤ ) ⊗ I_r,P^⊥_S =I_r^2-vec (S)((S))^⊤.Let us note that the system with the matrix A_vu,vu-(x) I can be solved directly since it is of small size r^2 × r^2. Thus, to solve P^⊥_S(A_vu,vu-(x) I)P^⊥_S y = P^⊥_S z, y^⊤(S) = 0, a direct formula can be used (it follows directly from (<ref>))y = (A_vu,vu-(x) I)^-1 P^⊥_Sz - λ_S (A_vu,vu-(x) I)^-1vec(S),whereλ_S =((S))^⊤(A_vu,vu-(x) I)^-1 P^⊥_Sz/((S))^⊤(A_vu,vu-(x) I)^-1vec(S).Let us derive formulas for solvingP^⊥_U (A_v,v -(x) I) P^⊥_U y = z,P^⊥_U y = yor equivalently(I_r ⊗ (I_n - U U^⊤))(A_v,v -(x) I)(I_r ⊗ (I_n - U U^⊤))y = z,(I_r ⊗U^⊤) y = 0,then(A_v,v -(x) I) y - (I_r ⊗ U) Λ = z,where the matrix Λ is chosen to satisfy (I_r ⊗U^⊤) y = 0. For a suitable preconditioner M_vv which approximates (A_v,v -(x) I) we havey - M_vv^-1 (I_r ⊗ U) Λ = M_vv^-1 z,Multiplying the latter equation by (I_r ⊗ U^⊤) we obtainΛ = - [(I_r ⊗ U^⊤)M_vv^-1 (I_r ⊗ U)]^-1 M_vv^-1 z,andy = M_vv^-1 (I_r ⊗ U) Λ + M_vv^-1 z.Similarly for P^⊥_V (A_u,u -(x) I) P^⊥_V y = z,P^⊥_V y = ywe obtain formulasy = M_uu^-1 (V ⊗ I_r) Λ + M_uu^-1 z, Λ = - [(V^⊤⊗ I_r)M_uu^-1 (V ⊗ I_r)]^-1 M_uu^-1 z.Matrices [(V^⊤⊗ I_r)M_uu^-1 (V ⊗ I_r)] and [(I_r ⊗ U^⊤)M_vv^-1 (I_r ⊗ U)] are of size r× r and can be inverted explicitly. The main difficulty is to find M_uu^-1 and M_vv^-1. Their inversion depends on the particular application.For instance, if M = I⊗ F + G ⊗ I, then the inverse can be approximated explicitly as <cit.>M^-1≈∑_k=1^K c_k e^-t_k F⊗ e^-t_k G,which we use later in numerical experiments.Alternatively, one can use inner iterations to solve a system with diagonal blocks. Note that similar to the original JD method, our method is not a preconditioned eigensolver. We use the preconditioner only to solve auxiliary linear systems.§ NUMERICAL EXPERIMENTSIn numerical experiments we find approximation to the smallest eigenvalue of the convection-diffusion operator𝒜 u ≡ - ∂^2 u/∂ x^2- ∂^2 u/∂ y^2+ ∂ u/∂ x + ∂ u/∂ y + V u,(x,y)∈Ω. u|_∂Ω = 0,where Ω = (-1/2, 1/2)^2, and potential V is chosen such that solution is of low rank: V≡ V(x,y) = e^-√(x^2 + y^2)/10. We use a standard second-order finite difference discretization on a n× n tensor product uniform grid to discretize second derivatives and backward difference to approximate first derivatives. The potential V on the grid is approximated by the SVD decomposition with relative accuracy 10^-10 and, hence, represented as a diagonal sum-of-product operator. The discretized operator A is represented in the form (<ref>) with sparse matrices F_α, G_α and R=14. Low-rank version and original JD Let us compare the behaviour of the original JD method and the proposed low-rank version. Figure <ref> shows the residual plot with respect to the number of outer iterations. We set the rank r=5, grid size n=150. One can observe that the low-rank version stagnates near the accuracy of the best rank 5 approximation to the exact eigenvector. We note that the cost of each inner iteration is different: 𝒪(nrR) for the proposed version and 𝒪(n^2) for the original version, so the proposed version is more efficient for large n. Nevertheless, Figure <ref> shows that our method requires fewer number of less expensive iterations to achieve a given accuracy (before stagnation). The less accurately we solve the system, the more gain we observe. Such speed-up may happen due to the usage of additional information about the solution, namely that it is of low rank. Comparison with the low-rank Davidson approach and the Rayleigh quotient iterationIn this experiment we compare performance of the proposed fixed-rank Jacobi-Davidson approach and the proposed Rayleigh quotient inverse iteration (<ref>). We also compare them with the “Davidson” approach when no projectionI - x_k x_k^⊤ is done:[_T_X_k_r (A -(x_k) I)_T_X_k_r]ξ_k = - _T_X_k_r r_k, _T_X_k_rξ_k= ξ_k.Figure <ref> illustrates the results of the comparison. As anticipated, when local systems are solved accurately the Davidson approach stagnates since the exact solution of (<ref>) is -x_k. So, no additional information is added to the previous approximation x_k. This problem does not occur if local systems are solved inexactly. For the Rayleigh quotient iteration we observe opposite behaviour due to the deterioration of condition number of local systems. The Jacobi-Davidson approach yields good convergence in both cases. Comparison with the ALS methodAlternating linear scheme (ALS) method is the standard approach for low-rank optimization. The idea is following: given X = UV^⊤ we minimize Rayleigh quotient (x)≡(U, V) successively over U and V. Minimization over U results in the eigenvalue problem with matrix A_v,v, while minimization over V results in the eigenvalue problem with matrix A_u,u. Note that in the proposed JD method we need to solve local systems, while in the ALS approach we solve local eigenvalue problems. To make comparison fair we ran original JD method to solve local problems in ALS. We choose the fixed number of iterations as choosing fixed accuracy to solve eigenvalue problems in ALS leads to stagnation of the method. Since the inner JD solver has two types of iterations: iterations to solve local problem and outer iterations, we need to tune these parameters to get fair comparison. We tuned them such that each ALS iteration runs approximately the same amount of time as the outer iteration of the proposed JD and gives the best possible convergence. Results are presented on Figure <ref>. On both subfigures the proposed JD method yields the fastest convergence. Subspace acceleration In this part we investigate the behaviour of the subspace accelerated version proposed in Sec. <ref>. First, on Figure <ref> we compare he original subspace acceleration and the version with vector transport when subspace is projected onto the tangent space of the current approximation. No restarts are used. As anticipated the projected version stagnates when accuracy of approximation equals error of low-rank approximation. Apart from that, the convergence behaviour of the methods is comparable, but the projected version is more suitable forlow-rank calculations. To illustrate this point we provide Figure <ref>, where the projected version is compared with the version with no projection. The latter one is implemented with hard rank thresholding of linear combination (<ref>). No additional optimization over coefficients besides Rayleigh-Ritz procedure is done. As we observe from the figure, the projected version outperforms the version without projection. The point is that we exactly optimize coefficients on the tangent space since no rank thresholding in this case is required. If vectors do not belong to the tangent space, rank rapidly grows with the subspace size and rank thresholding can introduce significant error. § RELATED WORKEigenvalue problems with low-rank constraint are usually considered in literature in the context of more general low-rank decompositions of multidimensional arrays, e.g. the tensor train decomposition <cit.>. Two-dimensional case naturally follows from the multidimensional generalization.There are two standard ways to solve eigenvalue problems in low-rank format: optimization of Rayleigh quotient based on alternating minimization, which accounts for multilinear structure of the decomposition, and iterative methods with rank truncation. The first approach has been developed for a long time in the matrix product state community <cit.>. We also should mention altenating minimization algorithms that were recently proposed in the mathematical community. They are based either on the alternating linear scheme (ALS) procedure <cit.> or on basis enrichment using alternating minimal energy method (AMEn) <cit.>. Rank truncated iterative methods include power method <cit.>, inverse iteration <cit.>, locally optimal block preconditioned conjugate gradient method <cit.>. For more information about eigensolvers in low-rank formats see <cit.>. To our knowledge no generalization of the Jacobi-Davidson method was considered. In <cit.> authors consider inexact Riemannian Newton method for solving linear systems with a low-rank solution. They also omit the curvature part in the Hessian and utilize specific structure of the operator to construct a preconditioner. In <cit.> authors proposed a version of inverse iteration based on the alternating linear scheme ALS procedure, which is similar to (<ref>). By contrast, the present work considers inverse iteration on the whole tangent space. We also provide an interpretation of the method as an inexact Newton method.We note that the proposed approach is considered on the fixed rank manifolds. Recently desingularization technique was applied to non-smooth variety of bounded-rank matrices _≤ r <cit.>.§ CONCLUSIONS AND FUTURE WORK The natural next step is to consider generalization to the multidimensional case. Most of the results can be directly generalized to the tensor train decomposition, e.g. (<ref>), (<ref>) and (<ref>). However, to avoid cumbersome formulas and present the method in the most comprehensible way we restricted the paper to the treatment of the two-dimensional case. Moreover, the correct choice of parametrization of the tangent space and efficient practical implementation worth individual consideration. We plan to address them in a separate work and test the method on real-world applications.§ ACKNOWLEDGEMENTSThe authors would like to thank Valentin Khrulkov for helpful discussions, and Marina Munkhoeva for the careful reading of an early draft of the paper.plain | http://arxiv.org/abs/1703.09096v1 | {
"authors": [
"Maxim Rakhuba",
"Ivan Oseledets"
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"categories": [
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"primary_category": "math.NA",
"published": "20170327142015",
"title": "Jacobi-Davidson method on low-rank matrix manifolds"
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#1#2#1#22mu#1#2 | http://arxiv.org/abs/1703.09301v3 | {
"authors": [
"Sergii Babkin",
"Jonathan Stewart",
"Xiaochen Long",
"Michael Schweinberger"
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"categories": [
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"stat.ME"
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"primary_category": "stat.CO",
"published": "20170327203704",
"title": "Large-scale estimation of random graph models with local dependence"
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=0cm | http://arxiv.org/abs/1703.09277v1 | {
"authors": [
"Evgeny Andriyash",
"Mohammad H. Amin"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170327193522",
"title": "Can quantum Monte Carlo simulate quantum annealing?"
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Department of Physics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411008, IndiaSaha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata, India Department of Condensed Matter Physics and Material Science, Tata Institute of Fundamental Research, Dr. Homi Bhabha Road, Mumbai 400 005, India Rutgers Center for Emergent Materials and Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USADepartment of Physics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411008, India Center for Energy Science, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411008, India[][email protected] explore remanent magnetization (μ) as a function of time and temperature, in a variety of rhombohedral antiferromagnets (AFM) which are also weak ferromagnets (WFM) and piezomagnets (PzM). These measurements, across samples with length scales ranging from nano to bulk, firmly establish the presence of a remanence that is quasi static in nature and exhibits a counter-intuitive magnetic field dependence. These observations unravel an ultra-slow magnetization relaxation phenomenon related to this quasi static remanence. This feature is also observed in a defect free single crystal of α-Fe_2O_3, which is a canonical WFM and PzM. Notably, α-Fe_2O_3 is not a typical geometrically frustrated AFM and in single crystal form, it is also devoid of any size or interface effects, which are the usual suspects for a slow magnetization relaxation phenomenon. The underlying pinning mechanism appears exclusive to those AFM which are either symmetry allowed WFM, driven by Dzyaloshinskii-Moriya Interaction (DMI) or can generate this trait by tuning of size and interface. The qualitative features of the quasi static remanence indicate that such WFM are potential piezomagnets, in which magnetization can be tuned by stress alone. Valid PACS appear here Quasi Static Remanence inDzyaloshinskii-Moriya Interaction driven Weak Ferromagnets and Piezomagnets Ashna Bajpai December 30, 2023 ======================================================================================================§ INTRODUCTION Phenomenon of weak ferromagnetism in certain antiferromagnets, including the classic case of α-Fe_2O_3, is associated with the experimental observation of a ferromagnetic (FM) like, spontaneous moment. This feature was initially attributed to a FM impurity phase in an otherwise AFM lattice; such as Fe_3O_4 impurity in α-Fe_2O_3 <cit.>. This controversy was firmly resolved by Dzyaloshiskii in 1958 <cit.>, who proposed a spin canting mechanism that leads to a weak FM like state and Moriya <cit.> who discovered the microscopic origin of this spin canting and its connection with spin orbit coupling (SOC). This is the celebratedDzyaloshinskii-Moriya Interaction (DMI), of the typeD.(S_i X S_j) which is now central to both fundamental and application based trends in contemporary condensed matter physics. Apart from exotic inhomogeneous spin textures and non collinear spin systems such as skyrmions, topological insulators and superconductors, DMI/SOC also brings into fore the role of antiferromagnetic insulators in spintronics <cit.>.In many of the symmetry allowed weak ferromagnets, which include rhombohedral AFMs like α-Fe_2O_3, MnCO_3 and rutile AFMs like NiF_2 or CoF_2, the phenomenon of stress induced moments or piezomagnetism, of the type (M_i = P_iσ_jσ_kσσσ_jσ_k) where σ is stress, was also predicted by Dzyaloshinskii <cit.>. Experimental observations of such stress induced moments were made by Borovik-Romanov in a variety of WFM/PzM single crystals in the seminal work spanning from 1960s to 70s <cit.>. On the similar lines of magnetoelectricity, wherein a magnetic moment can be created by electric field alone - for which Cr_2O_3 is a prototype <cit.>- magnetic moment from stress alone can occur in PzM, for which α-Fe_2O_3 is a prototype<cit.>. It is also interesting that both Cr_2O_3 and α-Fe_2O_3 are isostructural AFM but the piezomagnetic moments are observed in α-Fe_2O_3, not in bulk Cr_2O_3. A picture also emerged with a plausible explanation on the microscopic mechanism of PzM in these systems<cit.>.In some of these WFM/PzM compounds or in their doped versions <cit.>, an unusually slow magnetization relaxation was tracked through the measurement of remanence. This was further seen in ultra-thin films of Cr_2O_3 <cit.>, in FM/AFM core shell systems where Cr_2O_3 appeared as an ultra-thin surface layer <cit.> and also when Cr_2O_3 is encapsulated inside carbon nanotubes <cit.> (CNT). These reports pointed towards some features in remanence which appear to be common, especially for AFM with the possibility of WFM/PzM. Most intriguing among these is ultra slow magnetization relaxation phenomenon, resulting in the observation of a quasi static remanence with a counter-intuitivemagnetic field dependence <cit.>.Interestingly, Cr_2O_3 is not a symmetry allowed WFM/PzM but exhibits quasi static remanence only when it is in an ultra-thin form. It is therefore important to systematically explore whether these features intrinsically exist in symmetry allowed WFM and to investigate the circumstances in which this can appear in systems with altered symmetry conditions, especially due to size/interface effects.In addition, what still remains an open question is whether piezomagnetism will always co-exist in all WFM and if so, what are the foot prints of this phenomenon? It is also important to explore possible means to isolate this subtle effect from routine magnetization measurements, wherein all other field dependent processes contribute for any AFM (canted or otherwise) under magnetic field.In this work we explore remanence in two rhomohedral AFM that are symmetry allowed WFM and PzM. This includes α-Fe_2O_3 with Neel transition temperature (T_N)950 K and MnCO_3 with T_Nσ30 K.Here α-Fe_2O_3 is known to be a pure AFM upto 260 K and a WFM in the temperature range of 260-950 K <cit.>. The temperature at which α-Fe_2O_3 becomes WFM/PzM is also known as Morin Transition, T_M ( 260 K). It is advantageous to have a WFM near the room temperature for practical applications. However the effect is known to be much weaker than MnCO_3<cit.>. We also investigate isostructural compound FeCO_3 with T_Nσ50 K, for which there are conflicting reports in literature about the existence of WFM and PzM <cit.>.For such cases, size effects may play a prominent role as DMI can be dominant and enhanced at surfaces and interfaces <cit.>.We study all three samples in the form of nano and mesoscopic crystals / particles and show a correlation between the structural parameters and the magnitude of pinned moment related to the quasi static remanence.In case of α-Fe_2O_3, which is also a prototypical PzM near the room temperature, we confirm the ultra slow magnetization relaxation in its single crystal form, thus bringing out that the quasi static remanence is intrinsic.We also show that this feature can be substantially tuned by size effects, bycomparing the magnitude of quasi static remanence in the single crystal and nano cubes ofα-Fe_2O_3. § EXPERIMENTAL TECHNIQUES Micro-cubes of MnCO_3 (length2-4 μm), nano-cubes ofα-Fe_2O_3 (length200 nm) and polycrystalline spheres of FeCO_3 (grain size5-10 nm) have been synthesized following the precipitation and hydrothermal routes <cit.> Fig.<ref>a-<ref>c. The single crystal of α-Fe_2O_3 has been grown using Floating Zone technique.Scanning Electron Microscopy images are recorded using ZEISS ULTRA plus field-emission SEM. All the samples have been characterized using X-ray powder diffraction (XRD) using Bruker D8 Advance with Cu Kα radiation (λ = 1.54056 Å). (Supp. Info : Fig. S1-S3).Temperature variation of synchrotron XRD from 20 K-300 K has been conducted in BL-18 beam line, Photon Factory, Japan. The synchrotron XRD data has been fitted using Rietveld Profile Refinement. All three samples stabilize in rhombohedral structure and fitting has been done in hex setting. The XRD data along with the Rietveld fittings at few selected temperatures for each of the sample is shown in Fig.<ref>d-<ref>f. The refined lattice parameters a and c at room temperature for all three samplesare given in table 1.The Temperature variation of refined lattice parameters a and c for the samples are shown in the respective insets in Fig.<ref>d-<ref>f. Here both a and c are normalized with their respective room temperature value.The magnetization measurements have been carried out by using a superconducting quantum interference device (SQUID) magnetometer, Quantum Design MPMS-XL. § RESULTS AND DISCUSSIONS Magnetizationas a function of Temperature ( M_FC vs T)recordedwhile cooling in presence of magnetic field H = 1 kOe is presented Fig.<ref>g-<ref>i for all three samples. This is the routinely known Field Cooled (FC) cycle. The Neel transition temperature for MnCO_3 and FeCO_3as shown in Figure 1, match well with the respective literature values. For both these samples, the His applied in the paramagnetic region, prior to the FC cycle. However, forα-Fe_2O_3, the T_N is 950 K and it is marked schematically in the <ref>h. This is to indicate that in the case of α-Fe_2O_3, the magnetization data is recorded while cooling from 300 K, which is above its Morin Transition temperature (T_M) but below its Neel temperature (T_N).For single crystal ofα-Fe_2O_3, the magnetization (M_ZFC) is also recorded in Zero Field Cooled (ZFC) state as would be shown in the latter part of the text. These factors have important implications while preparing a remanent state for all these samples considered here. §.§Preparation of the Remanent State : FC/ZFC protocolOur primary tool here is DC magnetization in remanent state <cit.>. This enables us to track the magnetization relaxation phenomenon and hence pinning potential landscape in all these WFM.This remanent state is prepared in two experimental protocols, the FC and ZFC. In FC protocol, the sample is cooled in a specified magnetic field,H, whichis appliedmuch above theT_N (or T_M) and the M_FC is recorded while cooling.The H is switched off at 5K,andthereafter the remanent magnetization (or remanence) is experimentally measured in H = 0 state. Thisremanence , prepared after a typical FC cycle, isreferred to as _FC.This can be measured either (i) as a function of increasing temperaturefrom 5K to 300 K or (ii) as a function of time at 5K. In the ZFC protocol, employed only for the single crystal ofα-Fe_2O_3,the H is applied from below the T_M and M_ZFC is measured in warming cycle , right upto 300 K. Thereafter H is switched off andthe corresponding remanence, referred to as _ZFC, is measured as a function of time at 300 K.We emphasize that in all the subsequent data involvingpresented in this work,the magnitude of H indicated in the plots refers to themagnetic field applied duringeither cooling or warming cycle, so as to prepare a remanent state.This remanence (_FC or _ZFC ), the origin of which is the subject matter of investigation here, isexperimentally measuredonly after switching OFF the H. §.§ Temperature Variation of Remanence in MnCO_3Fig.<ref>a shows M_FC vs T, (measured while cooling)in presence of H100 Oe (black dots). The magnitude of M_FCat 5K0.75 emu/g. After removal of H at 5K, a part of the magnetization decays instantaneously. However, a significant part of magnetization remains pinned, resulting in the observation of remanence.This remanence (_FC) shows almost no further decay as a function of time, as long as the temperature is held constant at 5K. As evident from Fig.<ref>a, the magnitude of the _FC at 5K is 0.7 emu/g for this run.On increasing the temperature, _FC vs T (measured while warming) shows a variation which is qualitatively similar to M_FC vs T right up to the T_Nas shown in Fig.<ref>a (blue dots). In the paramagnetic tail, the _FC vanishes, as is expected. Fig.<ref>b shows the same for H30 kOe, for which M_FC12 emu/g whereas the _FC10^-5 emu/g at 5 K. Thus the _FC is vanishingly small for 30 kOe run. We would consider theof this magnitude to be roughly arising from the quenched field of SQUID superconducting magnet, which may be5-10 Oe and can vary from run to run[In SQUID magnetometers, even in H = 0, therecan be some residual magnetic field arising from the superconducting coil. The magnitude of this residual field can be 5 to 10 Oe and its sign can be arbitrary. The vanishingly small value of remanence at H = 30 kOealso sets the base line for any artifacts arising from such residual fields.]. The data contained in Fig.<ref> clearly indicates that the magnitude ofis almost equivalent to that of M_FC for lower (cooling) H whereas it is negligible for very high H. For all the intermediate magnetic fields, the M_FC vs T data are plotted in <ref>c and their corresponding _FC vs T are plotted in Fig.<ref>d. As is evident from these data, the magnetizationincreases with increasing H, consistent with a regular AFM behaviour.However, the corresponding remanence varies with the strength of the magnetic field in an unexpected way. Here the remanence is first seen to rise with increasing H, upto a critical field. Thereafter it decreases with increase in field and eventually vanishes beyond another critical field.To clearly bring out the unusual (cooling) field dependence of the _FC, we compare the magnitude of both M and at 5K. These data points are extracted from different M_FC vs Tand their corresponding _FC vs T runs Fig.<ref>e. Here M_FC is seen to increases with increasing H, as is expected for a regular AFM, whereas the _FC initially rises with increasing H, followed by a sharp drop. The _FC completely vanishes at very high field.The type of field dependence of is not expected for either a regular FM or AFM<cit.>. Thus the H dependence of the remanence (blue dots)brings forward a unique functional form, which is not observed in the routine M vs H isotherm (black dots) . §.§ Remanence in MnCO_3: Variation with Time To check the stability of the remanence as a function of time, we also performed relaxation rate measurements. After a typical M_FC vs T and subsequent removal of H, weobtained _FC vs time, while the temperature is held constant at 5K (Fig.<ref>).These remanence data, obtained for three different cooling fields, again brings forward two distinct magnetization relaxation rate, one of which is ultra-slow.We observe that for measurement times of about two hours, the _FC shows no appreciable decay and this type ofremanence can be termed as quasi static in nature. Consistent with the data presented in Fig.<ref>d, magnitude of the _FC is seen to vary with cooling field H in a way, which is not obvious from the routine temperature M-H isotherms. For the chosen cooling fields of 100 Oe, 1 kOe and 5 kOe, the _FCvalues are93%, 70% and 3% of their corresponding M_FC values. These data also indicate that finding an optimum value of the (cooling) magnetic field enables almost all the in-field magnetization to be retained. For instance,the remanence corresponding to 100 Oe run is 93%of its M_FC value. However, even for the run corresponding to 5 kOe, for which the magnitude of remanence is about3% of itsM_FC value, the relaxation rate is still ultra slow.Thus the data contained in Fig.<ref> confirms presence of the remanence that is quasi static in nature with ultra-slow magnetization dynamics, and exhibits a counter intuitive H dependence (Fig.<ref>e). §.§ Remanence and structural parameters in α-Fe_2O_3,FeCO_3 and MnCO_3Similar measurements were also conducted for α-Fe_2O_3 and FeCO_3 samples.Fig.<ref>a and Fig.<ref>b displays _FC vs H data at 5K (extracted from various _FC vs T runs) for both the samples. These datareveal that the _FC vsH for each of the sample is strikingly different fromcorresponding M_FC vs H shown on the right axis in each plot.Both the samplesexhibit a sharp rise in _FC as a function of (cooling) H and the peak value ofis obtained at different critical H for each sample. This rise is qualitatively similar towhat is seen for MnCO_3 (Fig.<ref>e), thoughthe fall, after the peak isnot as rapid.Overall, the field dependence of remanence is counter-intuitive in all three samples. In addition, all three samples exhibit two distinct time scales for magnetization decay, one of which is ultra slow and can be termed as quasi static. This slow magnetizationrelaxation is evident in _FC vs time measurements as shown in Fig.<ref>c. Forsake of comparison,for each sample the remanent state is prepared in cooling magnetic field of 1 kOe.The magnitude of the remanence is atleast an order of magnitude higher for MnCO_3 as compared to α-Fe_2O_3.This is also consistent with the earlier observations which indicate that the net FM momentdue to spin canting is about an order of magnitude larger larger in MnCO_3 <cit.>. To correlate the observed features inwith structural parameters, the temperature variation of a and c lattice parameters is studied.As can be seen from the inset of Fig.<ref>d, for MnCO_3, both a and c decrease with reducing temperature monotonically till about the T_N, however an expansion in both the lattice parameters is observed just below its AFM transition temperature. In addition, for MnCO_3 the lattice parameter c is seen to fall much rapidly with reducing temperature as compared to a (inset of Fig.<ref>d). On the contrary, for α-Fe_2O_3, the pattern of temperature variation for c and a are quite similar in nature and a slight trend of expansion in both lattice parameters is observed around its WFM region (Fig.<ref>e). For all three samples, both lattice parameters exhibit a slight anomaly below T_N (or around WFM in case of α-Fe_2O_3), however the effect is more pronounced for the MnCO_3. Fig.<ref>d compares normalized c/a ratio as a function of temperature for all three samples. This normalization is w.r.t c/a ratio at 300 K for each sample. We find that the c/a ratio shows a more rapid decline with reducing temperatureand a clear anomalyis observed in the WFM region for MnCO_3. This trend also correlates with the stability and magnitude of the , both of which are relatively higher for MnCO_3 as compared to α-Fe_2O_3. In case of FeCO_3, though the qualitative features in remanenceare similar,butthe morphology of the sample makes its difficult to conclude whether these features are intrinsic or arising due to nano scaling.In this case, the grain size is of the order of 2-5 nm,Fig.<ref>c. This situation is similar to what is observed for Cr_2O_3 which is also isostructural with α-Fe_2O_3 but it is not symmetry allowed WFM in bulk. However, it exhibits slow relaxation and the unusual cooling field dependence of remanence only in ultra thin form <cit.>.Microscopic measurements are needed to confirm the presence of WFM in such cases, including ultra small FeCO_3 grains used in this study. For physical mechanism related to the remanencethat results in ultra-slow magnetization relaxation, a number of phenomena such as glassy phase, superparamagnetism, defect pinning in a regular FM or AFM, exchange bias at FM/AFM interface etc. can be considered. Such phenomena are known to result in slow relaxation with a variety of temporal functional forms<cit.>.However the mechanism behind thequasi static remanence and its unusual(cooling) magnetic field dependencein these samples appears to be different from above mentioned phenomena. For instance, considering size effects,MnCO_3 shows most robust magnetization pinning atlower fields, as shown in Fig.<ref>. However, the sample used for magnetization measurements consists of fairly big crystallites ( 2-4 μm) therefore it is less likely that the slow relaxation is arising from size reduction or nano scaling. It is neither a glassy system, nor a nano scale FM which can exhibit superparamagnetic traits. Crystallites are also regular shaped with well-formed facets therefore the phenomenon ofdefect pinningleading toultra-slow magnetization relaxation is ruled out.Also, for a regular AFM/FM, theshould have shown saturation <cit.> with H, rather than the sharp drop such as seen in Fig.<ref>e.To understand the nature of remanence in AFM withWFM traits and to confirm if this effect is intrinsic, we also explored it in a single crystal (SC). For this purpose, we chose a SC of α-Fe_2O_3 as this sample is well known to exhibit a spin reorientation transition from pure AFM to WFM phase<cit.>.§.§Pure AFM and WFM Phase: Symmetry Considerations Among the samples considered here, α-Fe_2O_3 is known to be both pure AFM (upto 260 K) and WFM (260 K - 950 K) <cit.>.Here pure AFM phase implies that the DMI driven spin canting is not symmetry allowed. As mentioned before, isostructural compound Cr_2O_3which does not exhibit spin canting,in this context, is a pure AFM phase<cit.>. For the sake of clarity, the spin configurations inpure-AFM and WFM state are comparedin Fig.<ref>a. In pure AFM phase, the spins within unit cell are arrangedalong c axis as shown in Fig.<ref>a, configuration (i). Here the red star is theinversion center and the spin configuration can be S1 = -S2 = S3 = -S4 as shown in (i). In WFM state, the spins re-orient to the basal plane, arranged in a specific sequence, in which S1=-S2=-S3=S4. It is important to note that the unit cell is still AFM, but the spin configuration shown in (ii) is essential for DMI driven spin canting. ThisD.(S_i X S_j) type of interactionis possible betweensub-lattices associated with antiferromagnetically coupled spins, with the sign of D consistent with the symmetry considerations discussed in references 1-4.The direction of the net FM moment due to thespin cantingis towards the c direction as is shown schematicallyin Fig.<ref>a(iii). This net FM moment in otherwise AFM isresponsible for weak ferromagnetism. Thespin configurations shown in Fig.<ref>a(ii) is valid for all the rhombohedral AFM discussed here,which are symmetry allowed WFM. For α-Fe_2O_3,the spin reorientation transition from pure AFM (spins along c axis)to WFM state (spins along a axis)occurs at T_M, the Morin transition temperature<cit.>. Thus α-Fe_2O_3 provides a unique opportunity to probe both AFM and WFM phase in the same sample, which individually exist in a wide temperature range. In the following, we present results of remanence measurements in the single crystal of α-Fe_2O_3 in both the regions. §.§ Remanence in a Single Crystal of α-Fe_2O_3 : Variation with TemperatureMain panel of Fig.<ref>b shows M_FC Vs T for a SC of α-Fe_2O_3 sample along a axis. The Morin transition at260 K demarcates the two regions, pure AFM and WFM for this sample. Fromthis, we note that magnitude of M _FC is roughly0.35 emu/g in WFM region and0.015 emu/g in the pure AFM region.After switching off the field at 5K, corresponding _FC vs T in warming cycle is shown in the main panel of Fig.<ref>c.The _FC is found to be negligibly small (10^-5 emu/g) in the pure AFM region and substantially large in WFM region (-0.2 emu/g).Here, the sign of the _FC is found to be negative w.r.t the direction of applied H.From a number of such _FC vs T data along a axis, we find that the sign of _FC at 300 K remains primarily negativeand its magnitude shows a slight decrease with increasing magnetic fields (inset in Fig.<ref>c). It is to be noted thatfor obtaining this data, the H during FC cycle is applied at 300 K, when the sample is in WFM region. This is unlike the case of MnCO_3, where the H can be applied in the paramagnetic region. For obtaining the (cooling) field dependence of remanence unambiguously, such as shown in Fig.<ref>e for MnCO_3, it is preferable to apply the H in the paramagnetic region for preparing individual remanent states.However, in case of α-Fe_2O_3, it is not practically possible to heat the sample above 950K, after each run. Though the sign of the _FC along a axis is not commensurate with the direction applied H while cooling, its magnitude is substantial only in WFM region. To check the stability of this remanence as a function of time, we conducted relaxation measurements both along the c as well as a axis.Since the direction of net FM moment is likely to be towards the c axis of the crystal, we particularly checkedstability of_ZFC as well as_FC along c axis as a function of time. §.§ Remanence in a Single Crystal of α-Fe_2O_3 : Variation with Time In this section we present relaxation rate of remanence in pure AFM and WFM phase of α-Fe_2O_3,measured following the FC and ZFC protocol respectively. For remanence in the pure AFM region, theH is applied from 300 K and M_FC vs T is recorded while cooling (not shown here). The H is switched off at 5K and the _FC is measured as a function of time. This isshown in the inset of Fig.<ref>a.These data further confirm that the remanence is negligiblein pure AFM region10^-5 emu/g (inset of Fig.<ref>a).For preparing the remanent state in WFM region,theH is applied from below the T_M and M_ZFC vs Tis recorded while warming , right upto 300 K (not shown here). At 300 K, the H is switched off and _ZFC is measured as a function of time (main panel, Fig.<ref>a).Here, the remanence is positive and is commensurate with the direction of H applied during ZFC cycle. Thus the remanence is substantial in magnitude in WFM region andit is also fairly stable in time.However, from a number of _ZFC vs time cycles in positive H, we observe that the magnitude of _ZFC in WFM region varies from 0.05-0.2 emu/g but its sign primarily remains negative. This anomaly appears only in the remanence measurements but not in the regular in-field measurements such as shown in Fig.<ref>b. However, such ambiguity with sign has also been observed in the sign of stress induced moments in some WFM/PzM on repeated cooling <cit.>. The reason for such ambiguity in case of remanence, (which does not appear in regular in-field magnetization) is also discussed in the latter part of the text.We also note a slight variation ( 5%) in the magnitude of , from run to run, for the same(cooling) magnetic field. These anomalies are also seen to appear only in the WFM region.Interestingly, wealso observe a slight trend of rise (a few% of total remanence) in _ZFC vs time data,as shown in Fig.<ref>a.The over all relaxation data appears to be a sum of bothtime-decay as well as time-rise of the remanence. Thisindicates that on application of H (while preparing the remanent state) the moments continue to reorient slowly in presence of H and on the removal of H, the time decay is ultra slow as well. This also indicates that the total time spanin which the H is ON forpreparing a particularremanent state is also an important parameter.This couldalso be responsible for variations in the magnitude of the remanence, as observed here.This result prompted us to perform waiting time dependence, usually employed for glassy systems<cit.>.For waiting time runs,tworemanent states are preparedusing the same (cooling) magnetic field. In first case, the H =1 kOe is appliedin ZFC protocol, from below theT_M and the sample is heated right up to 300 K. At 300 K the magnetic field was kept ON for waiting-time of 1 minute, prior to finally switching it OFF for the remanence measurements. The second remanent state is prepared following exactly the same protocol, however this time theH = 1 kOeis kept ON for waiting time of 100 minutes, prior to switching it OFF.These _ZFC vs timedata parallel to c axisare presentedin the main panel of Fig.<ref>b,for 1 min (dots) or 100 min (stars) waiting-time respectively. These dataclearly indicate that the magnitude of the remanencealsochanges with the total time span of the H applied for preparing a particular remanent state. This alsoexplains the the slight differences in the magnitude of remanence from run to run. The inset shows the same for _ZFC parallel to a axis after 100 min of waiting time. Along the a axis, the magnetizationrelaxation is ultra slow and occasionally discrete jumps inremanence are observed, though the change is less than a percent. However, the remanencecontinues toexhibit quasi staticnature. These anomalieswhich exist in the remanent stateare not observed in routine M vs T measurements.α-Fe_2O_3 is not a frustrated AFMand in the single crystal form, size/interface related phenomena cannot account for the waiting time effects and ultra-slow magnetization dynamics. From the observation of quasi static remanence in single crystal, together with similar features observed in MnCO_3, we conclude that the ultra-slow magnetization dynamics can be taken as indicative of the presence of WFM. This ultra-slow dynamics also appears to be associated with the microscopic details of the AFM domain which turn WFM due to spin canting. §.§ Quasi static Remanence and DMI driven Spin CantingConsidering the microscopic reason forquasi static remanence(that leads to the ultra-slow magnetization dynamics as observed here) in these systems, we recall the details of magnetic structure in all these compounds. The spin arrangement shown in Fig.<ref>a(ii) is essential for the observation of WFM. This should also limit the possible ways in which an AFM domain can exist in the WFM region. For a regular AFM, on the application of the H, the induced magnetization is driven by the Zeeman energy and the magnetocrystalline anisotropy. However, the additional factor in WFM will include response from spontaneously canted spins, related to the DMI as well. On removal of H, the reversal of the WFM domain will have to be accompanied by the reversal of the AFM moment which is energetically unfavorable <cit.>. Once a AFM domain withspin-canting is formed, guided by a cooling H appliedfrom above the AFM to PM transition, it is energetically unfavorable for these domains to relax, when the H is removed. This feature is only observed upto a critical value of H which can vary depending on the sample, as is observed here (Fig.<ref> and Fig.<ref>a ). Beyond a critical H, the magnetization dynamics is driven by Zeeman and magnetocrystalline anisotropy. The magnetization relaxation in this case is much faster, similar to what is observed for a normal AFM.However, below this critical field strength, the WFM domain configuration is guided by the sign of H field, when, it is applied from T >> T_N. When the H is applied in WFM region, the spins are already spontaneously canted.This also explains the ambiguity with sign, as observed in case of α-Fe_2O_3.For further confirming that the ambiguity with sign is related to spontaneous spin canting related with DMI and not arising due to measurement related artifacts,werevert back to MnCO_3 which has a T_N30 K and H can be applied in the paramagnetic region. Fig.<ref>shows M_FC vs T data recorded while cooling from above T_N, down to 5 K,in presence of H = + 100 Oe (blue dots) . At 5K the His switched off and the quasi static remanence is observed, which is positive in magnitude as the WFM domain configuration is already guided by the H = + 100 Oe.Temperature still held at 5K, weagain applyH = -100 Oe and subsequent to this, the M vs T is measured in warming cycle (FH cycle) in presence of H = -100 Oe.As is evident from the data shown in Fig.<ref>, once pinned in WFM state from above T_N by a positive H, the negative field cannot change the sign of pinned moment and therefore the sign of remanence. The measured magnetization in presence of H = -100 Oe while warming(black dots) is still positive and clearly a magnetic field applied in WFM region does not make any difference.Thus the observed magnetizationis basically due to the presence of positive remanence, stabilized duringprevious (H = + 100 Oe) FC cycle.This data explains the ambiguity related with the sign of remanence, especially when the H field is applied in WFM region.Over all, these data confirm thatthe quasi static remanenceis observed below a critical value of H in WFM and related to anisotropic exchange. At higher H, the interplay is between Zeeman and exchange energy, as is usually observed for a regular AFM. The ambiguity related with the sign ofin single crystal of α-Fe_2O_3 is related with configuration of AFM domainsin which the spins are spontaneously canted due to DMI, even in the absence of H. On cooling or heating in presence of H leads to stabilization of these canted AFM domains in different configurations, compatible with the interplay of various energy scales involved. This feature again indicates that the net moment related to quasi staticis associated with net FM moment arising due to spontaneous spin canting in otherwise AFM.§.§ Quasi Static Remanence and PiezomagnetismA general consensus in the literature is PzM is connected with the transition from pure AFM to WFM state in an otherwise AFM and one of the mechanism that leads to the WFM state is associated with DMI <cit.>. As mentioned before, the stress induced moments have already been experimentally measured in such WFM systems <cit.>.More importantly, the direction of net FM moment in the WFM phase is seen to coincide with the direction of PzM <cit.>. It is also to be recalled that waiting-timeeffects and ambiguity with sign ( similar to what is observe in remanence data for α-Fe_2O_3 w.r.t the sign of the applied H) have also been observed in the sign of stress induced moments inWFM/PzM on repeated cooling <cit.>. The data presented in Fig.<ref> explains the ambiguity with the sign and the robustness of the pinned moments in WFM region.The presence of quasi static remanencealso shows that once the WFM domains have been formed, guided by the magnetic field from above the magnetic transition temperature, removalof H (or reversingits sign) does not make any difference.The net FM moment arise due to DMI driven canting, their direction can be manipulated only when the H is applied from above T_N.It is also well known that magnetization reversal in piezomoments would require the reversal of WFM sublatticewhich is energetically unfavorable <cit.>.In remanence measurements, this phenomenon is manifested in the form ofultra slow magnetization relaxation (and consequently the quasi static remanence)as observed here.These data presented in Fig.<ref> to Fig.<ref>connect WFM andquasi static remanence. These data also further confirm that WFM phase is intimately related with the onset of transverse PzM in rhombohedral AFM. We emphasizethat the remanence data shown here not only bears a strikingsimilarity with experimentally measured stress induced momentsbut also reveals featureswhich are not obvious inroutine in-field magnetization data.Thus it appears that the remanence measurementscapture the essential physics ofDMI driven WFMbetter than routine M vs T or M vs H and the onset of quasi static remanence can be taken as footprints ofWFM and PzM.From present data it also appears that ultra-slow magnetization dynamics and its unusual magnetic field dependencearises from the WFM and such systems are potential PzM.The magnitude of the WFM/PzM is further related to lattice parameters, especially c/a ratio in all these rhombohedral systems. A systematic study of such canonical WFM/PzM such as presented here, points towards the footprints of this phenomenon by simple magnetization measurements. It is to be emphasized that the system considered here are AFM with WFM trait. These are not frustrated AFM or a disordered glassy system / spin glass in conventional sense, which can exhibit slow relaxation for various other reasons. Therefore it is very interesting to observe ultra-slow relaxation in a completely ordered system in which these features are correlated with DMI/SOC. From our data, it can be concluded that for micro-cubes of MnCO_3 and nano-cubes and single crystalof α-Fe_2O_3,the presence of ultra-slow magnetization dynamics is associated with intrinsic WFM. The temperature variation of remanence data on nano-cubes (Fig.<ref>a)and single crystal of α-Fe_2O_3,(Fig.<ref>c) especially bring out that the magnitude of quasi static remanence can be significantly tunes by nanoscaling, as also has been observed earlier <cit.>. For FeCO_3, data is not sufficient to conclude whether effect is intrinsic or it is arising from the size reduction, as the sample comprises of 5-10 nm particles of FeCO_3. In such cases, the strain in lattice parameters can also stabilize the WFM phase <cit.>, however microscopic measurements are needed to confirm the presence of DMI driven canting.It is to be noted that is relatively hard to stabilize FeCO_3 in the form of macroscopic crystallites for ruling out size effects. However, we are in the process of exploring systematic size effects in FeCO_3. We also assert that for systems which are isostructural AFM with α-Fe_2O_3, such as Cr_2O_3(which is definitely not a symmetry allowed PzM) and FeCO_3 (for which there are conflicting reports in the literature) the strain in the lattice parameter arising from size effects is likely to stabilize the WFM/PzM phase <cit.>.§ CONCLUSIONIn conclusion, we explore two rhombohedral antiferromagnets that are weak ferromagnets and observe an ultra-slow magnetization dynamics and associated with this, a very robust magnetization pinning with unusual magnetic field dependence.These features are intimately related to the weak ferromagnetism arising from spin canting.This spin canting is associated with DMI for the rhombohedral antiferromagnets discussed here. Whether qualitatively similar feature can be observed in other WFM , in which spins are canted but the origin in not DMI driven, is yet to be explored.From present set of data, it is confirmed that the quasi static remanence and its uniquemagneticfield dependence can be taken as foot prints of WFM/PZM sysyems.This feature is intrinsic in nature and the slow relaxation observed here does not relate with magnetization pinning arising from the glassy phase, magnetocrystalline anisotropy or routine exchange bias. The DMI in WFM phase is clearly connected with the possibility of stress induced moments or piezomagnetism. Finally, piezomagnetism, though not as widely explored or utilized, say as piezoelectricity, can have a variety of applications including those related to FM/AFM interfaces, in which the FM moment can be pinned by a PzM, and the effect should be tunable by stress alone.§ ACKNOWLEDGMENTS Authors thank Sunil Nair (IISER Pune) for SQUID magnetization measurements. AB acknowledges Department of Science and Technology (DST), India for funding support through a Ramanujan Grant and the DST Nanomission Thematic Unit Program. SWC is funded by the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant GBMF4413 to the Rutgers Center for Emergent Materials. Authors thank the DST and Saha Institute of Nuclear Physics, India for facilitating the experiments at the Indian Beamline, Photon Factory, KEK, Japan. | http://arxiv.org/abs/1703.09075v2 | {
"authors": [
"Namrata Pattnayak",
"Arpan Bhattacharyya",
"A. K. Nigam",
"Sang-Wook Cheong",
"Ashna Bajpai"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170327134406",
"title": "Quasi Static Remanence in Dzyaloshinskii-Moriya Interaction driven Weak Ferromagnets and Piezomagnets"
} |
Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada Canadian Institute for Advanced Research/Quantum Materials Program, Toronto, Ontario M5G 1Z8, Canada School of Physics, Korea Institute for Advanced Study, Seoul 130-722, KoreaRecently, thermal Hall effect has been observed in the paramagnetic state of Volborthite, which consists of distorted Kagome layers with S=1/2 local moments. Despite the appearance of a magnetic order below 1K, the response to external magnetic field and unusual properties of the paramagnetic state above 1K suggest possible realization of exotic quantum phases. Motivated by these discoveries, we investigate possible spin liquid phases with fermionic spinon excitations in a non-symmorphic version of the Kagome lattice, which belongs to the two-dimensional crystallographic group p2gg. This non-symmorphic structure is consistent with the spin model obtained in the density functional theory (DFT) calculation. Using projective symmetry group (PSG) analysis and fermionic parton mean field theory, we identify twelve distinct ℤ_2 spin liquid states, four of which are found to have correspondence in the eight Schwinger boson spin liquid states we classified earlier. We focus on the four fermionic states with bosonic counterpart and find that the spectrum of their corresponding root U(1) states feature spinon Fermi surfaces. The existence of spinon Fermi surface in candidate spin liquid states may offer a possible explanation of the finite thermal Hall conductivity observed in Volborthite. Fermionic Spin Liquid Analysis of the Paramagnetic State in Volborthite Yong Baek Kim Received ; accepted======================================================================= § INTRODUCTIONThere has been tremendous effort to understand and detect signatures of quantum spin liquid phases in frustrated magnets. The most studied example is the Heisenberg model on the isotropic Kagome lattice, which may be approximately realized in Herbertsmithite, Zn Cu_3 (OH)_6 Cl_2, with small additional exchange interactions present. While a number of experiments on the Herbertsmithite show possible signatures of a spin liquid ground state,<cit.> definite confirmation still remains elusive. It is also important to demonstrate that the appearance of a spin liquid state is generic in this class of frustrated magnets and not confined to the perfectly isotropic Kagome lattice, as small distortions of the lattice or deviation from the Heisenberg model would naturally occur in many related materials. Hence it is useful to examine different materials with related lattice and magnetic structures.Volborthite, Cu_3 V_2 O_7 (OH)_2 · 2 H_2 O, is a prominent example that hosts a distorted version of the Kagome lattice.<cit.> Magnetic order arises below 1K, which is highly suppressed compared to the Curie-Weiss temperature of 140K. The phase diagram in an external magnetic field is highly complex, and the nature of the magnetic order below 1K has not yet been fully understood.<cit.> Recently, the thermal Hall conductivity in Volborthite has been measured and shown to be finite in the paramagnetic state just above 1K.<cit.> This is highly unusual as Volborthite is an insulator and any heat-carrying object should be a charge-neutral excitation, which would not experience the Lorentz force via external magnetic field. Such an anomalous response may be explained if the underlying ground state is a quantum spin liquid that supports fermionic spinons with a Fermi surface. It was proposed earlier that an emergent gauge field in such spin liquid phases inducesa fictitious magnetic field that couples to the charge-neutral spinons, which will eventually lead to a finite thermal Hall conductivity.<cit.>In this work, motivated by the appearance of the finite thermal Hall conductivity, we investigate possible spin liquid phases with fermionic spinons in a distorted Kagome lattice appropriate for Volborthite. A microscopic spin model of Volborthite was obtained earlier from density functional theory (DFT) calculation,<cit.> which suggests that the Kagome layer can be thought of as frustrated J_1 - J_2 spin chains running parallel to each other and interacting through two different interchain couplings J and J'. Analyzing the spatial symmetries of the distorted Kagome lattice described by the J_1 - J_2 - J - J' model further reveals that it belongs to the non-symmorphic two-dimensional crystallographic group p2gg. Non-symmorphic symmetries are interesting from the theoretical perspective because they can prevent the formation of trivial band insulator even though there is an even number of electrons per unit cell. The electron fillings at which a trivial band insulator is possible for each of the 230 space groups are tabulated in Ref. PhysRevLett.117.096404. This is 4n for p2gg, but there are 6 electrons per unit cell in the Volborthite Kagome layer, which is not a multiple of 4. Therefore, the ground state of Volborthite should support either gapless excitations or topological order, by the Hastings-Oshikawa-Lieb-Schultz-Mattis (HOLSM) theorem.<cit.> This suggests that the paramagnetic state of Volborthite cannot be a trivial state, and quantum spin liquid states with a finite excitation gap (topological order) or gapless spinon excitations are possible candidate ground states.Previously, we studied possible quantum spin liquid phases with bosonic spinons<cit.> and the magnetic orders related to them in Ref. 1702.04360. Energetic consideration suggests that a (q,0) spiral order or a (π,π) spin density wave is likely to be the magnetic order observed below 1K and can be obtained via the condensation of bosonic spinons in the spin liquid phases. In the current work, we use a different approach, fermionic parton mean field theory,<cit.> to explore both gapped and gapless spin liquid states, in contrast to the bosonic theory, where only gapped spin liquids are stable. Here, we classify the fermionic spin liquid states in the non-symmorphic Kagome lattice through projective symmetry group (PSG) analysis.<cit.> We are particularly interested in finding out which fermionic states are connected to the bosonic states that give rise to the magnetic orders mentioned above.We identify twelve distinct ℤ_2 spin liquid states that respect the space group of non-symmorphic Kagome lattice and time reversal symmetry, which we label by a number 1,…,6 followed by a letter a or b. In addition, due to the presence of non-symmorphic symmetry, it is shown that only translationally invariant mean field states are allowed. We then explore the relation <cit.> between the twelve fermionic states and the eight bosonic states classified in Ref. 1702.04360 using the idea of symmetry fractionalization <cit.> and fusion rules. Solving the vison PSG,<cit.> we find that the four fermionic states 3a, 3b, 4a and 4b have bosonic correspondence, i.e. each of them can be connected to one of the eight bosonic states. In particular, one of them (4b in our notation) is connected to the most energetically favorable bosonic spin liquid state which undergoes a phase transition to (q,0) spiral order upon spinon condensation. None of the fermionic states is connected to another highly competing bosonic state, from which the (π,π) spin density wave arises.We narrow down our investigation to the four fermionic states with bosonic counterparts and study their generic spectrum. We find that the spinon spectrum in the ℤ_2 spin liquid of 3a, 4a and 4b have Dirac point(s) while 3b is gapped. Given the discovery of thermal Hall effect in Volborthite, we also examine the corresponding root U(1) state of 3a, 3b, 4a and 4b (where the pairing amplitudes in the ℤ_2 spin liquid state are turned off). This is because thermal Hall effect can naturally occur in U(1) spin liquids with spinon Fermi surfaces, where the external magnetic field couples to the U(1) gauge field, which in turn exerts a Lorentz force on the spinons and produces a finite transverse thermal conductivity.<cit.> We find that the root U(1) spectrum of all four states features a spinon Fermi surface, which is consistent with the interpretation of the thermal conductivity experiment. Taken together, these root U(1) spin liquid states, especially the 4b state related to the (q,0) spiral order, may be promising candidates for the paramagnetic phase of Volborthite above 1K. The rest of this paper is organized as follows. In Section <ref>, we introduce the microscopic spin model and the lattice structure of Volborthite. In Section <ref>, we outline the fermionic parton mean field theory. In Section <ref>, we review the essentials of PSG analysis and use it to search for possible fermionic spin liquid states and construct the mean field ansatzes. In Section <ref>, we establish the mapping between bosonic and fermionic spin liquid states by solving the vison PSG. In Section <ref>, we analyze the generic mean field spectrum of the fermionic states that have bosonic counterparts. In Section <ref>, we discuss the relevance of our results to the thermal conductivity experiment on Volborthite.§LATTICE AND HAMILTONIANThe crystal structure of Volborthite can be described by two distorted Kagome layers consisting of CuO_6 octahedra, separated by non-magnetic V_2 O_7 pillars and water molecules.<cit.> We thus analyze a single Kagome net, which has a S=1/2 local moment of Cu^2+ ion at each site. A density functional theory (DFT) calculation <cit.> suggests that the Volborthite Kagome lattice can be viewed as frustrated J_1 - J_2 spin chains with two different interchain couplings J and J', as shown in FIG. <ref>. The relative magnitude of the exchange interactions is given by J:J':J_1:J_2=1:-0.2:-0.5:0.2, where negative (positive) sign indicates ferromagnetic (antiferromagnetic) interaction. The Heisenberg interaction between S=1/2 local moments is given byH = ∑_ij J_ij𝐒_i ·𝐒_jThis microscopic spin model, which is dubbed the J_1 - J_2 - J - J' model, defines a rectangular unit cell with six sublattices. The coordinate of any site can be written as (x,y,s), where x,y ∈ℤ label the unit cell and s=1,…,6 indexes the sublattice. The chain sites have s=1,2,4,5, while the interstitial sites have s=3,6. Let us now inspect the spatial symmetries of the system. The most obvious ones are lattice translation in two independent directions, T_x and T_y. In addition, a rotation by π, or C_2, also leaves the system invariant. Unlike the isotropic Kagome lattice, reflection symmetry is absent. However, a careful inspection reveals that there is a non-symmorphic glide symmetry h, which is a combination of reflection and half lattice translation. We henceforth refer to the Volborthite Kagome layer as non-symmorphic Kagome lattice, since it belongs to the non-symmorphic plane crystallographic group p2gg.There exist (infinitely many) different choices of rotation center and glide axis. We fix the rotation center at the center of the (0,0) unit cell. On the other hand, we choose as the glide axis the horizontal line connecting the sites (x,0,4) and (x,0,5) for all x. The entire space group of the non-symmorphic Kagome lattice can be generated by rotation C_2 and glide h, as the lattice translations T_x ≡ h^2 and T_y ≡ (C_2 h)^2 are not independent. We nevertheless consider four spatial symmetries T_x,T_y,C_2,h, which are shown in FIG. <ref>, for convenience of the subsequent analysis. We also show how a generic site (x,y,s) changes under these symmetry transformations in Appendix <ref>.§FERMIONIC PARTON MEAN FIELD THEORYIn this section, we outline the fermionic parton mean field approach <cit.> for the generic Heisenberg Hamiltonian (<ref>). We define creation (annihilation) operators f_iα^† (f_i α) that create (annihilate) fermionic spinons with spin α∈{↑ , ↓} at site i. Then, we write the spin operator at site i as𝐒_i = 1/2∑_αβ f_i α^†σ_αβ f_i β ,where σ=(σ^x,σ^y,σ^z) is the vector of Pauli matrices. The fermionic operators obey the anticommutation relations{ f_i α, f_j β^†} = δ_ijδ_αβ ,{ f_i α, f_j β} = 0 = { f_i α^†, f_j β^†} .Representing the spin operator as (<ref>) in (<ref>) enlarges the original Hilbert space.<cit.> To obtain the physical spin state, we have to impose the following single occupancy constraint (i.e. one fermion per site), ∑_α f_i α^† f_i α = 1 ,∑_αβ f_i αϵ_αβ f_i β = 0 , where ϵ_αβ is the antisymmetric tensor. At the mean field level, the constraint is replaced by its ground state expectation value <cit.> ∑_α⟨ f_i α^† f_i α⟩ = 1 ,∑_αβ⟨ f_i αϵ_αβ f_i β⟩ = 0 .Next, we define the bond operators χ̂_ij = ∑_α f_i α^† f_j α ,Δ̂_ij = ∑_αβ f_i α [i σ^y]_αβ f_j β ,Ê^a_ij = ∑_αβ f_i α^† [i σ^a]_αβ f_j β ,D̂^a_ij = ∑_αβ f_i α [i σ^y σ^a]_αβ f_j β , where a=x,y,z. χ̂_ij (Δ̂_ij) is known as singlet hopping (pairing) channel, while 𝐄̂_ij (𝐃̂_ij) is known as triplet hopping (pairing) channel. The spin product can be expressed solely in terms of singlet or triplet channel as <cit.> 𝐒_i ·𝐒_j= -1/4( χ̂_ij^†χ̂_ij + Δ̂_ij^†Δ̂_ij - 1 ) = 1/4( 𝐄̂_ij^†·𝐄̂_ij + 𝐃̂_ij^†·𝐃̂_ij - 3 ) . As argued in Ref. PhysRevB.80.064410, due to the sign of ferromagnetic (antiferromagnetic) interaction, decomposition of spin product into singlet (triplet) terms is unstable within an auxiliary field decoupling scheme. Therefore, for ferromagnetic interaction J_ij<0 (antiferromagnetic interaction J_ij>0), we write the spin product as (<ref>) ((<ref>)), and then perform a mean field decoupling to obtain a Hamiltonian quadratic in f-operators.<cit.> With the constraint (<ref>) and (<ref>) enforced by introducing chemical potentials a^l_i ∈ℝ, l=1,2,3, and some constant terms dropped, the mean field Hamiltonian reads[b] H_MF =- 1/4∑_J_ij>0 J_ij[ ( χ_ij^* χ̂_ij + h.c. - |χ_ij|^2 ) .+ . ( Δ_ij^* Δ̂_ij + h.c. - |Δ_ij|^2 ) ]+ 1/4∑_J_ij<0 J_ij[ ( 𝐄_ij^* ·𝐄̂_ij + h.c. - |𝐄_ij|^2 ) .+ . ( 𝐃_ij^* ·𝐃̂_ij + h.c. - |𝐃_ij|^2 ) ]+ ∑_i a^3_i ( ∑_α f_i α^† f_i α - 1 )+ ( ∑_i (a^1_i+ia^2_i) f_i ↓ f_i ↑ + h.c.) .Extremizing the mean field energy with respect to the variational parameters yields the following self consistent equations∂⟨ H_MF⟩/∂𝒪_ij = 0 𝒪_ij = ⟨𝒪̂_ij⟩ ,𝒪_ij = χ_ij, Δ_ij, E^a_ij, D^a_ij ,while the chemical potentials a^l_i are solved such that the constraints (<ref>) and (<ref>) are satisfied. In practice, the self consistent equations (<ref>) are solved iteratively in momentum space.§PROJECTIVE SYMMETRY GROUP ANALYSISWe are interested in the symmetric spin liquid states in non-symmorphic Kagome lattice. Using the method of projective symmetry group (PSG),<cit.> we can identify all possible spin liquid ansatzes that respect the relevant symmetries. The main idea is that the mean field ansatzes for distinct spin liquid states are invariant under symmetry transformations followed by different gauge transformations. Therefore, spin liquid states can be distinguished by these different gauge transformations. <cit.>In order to treat the spin-singlet and triplet terms on equal footing, we closely follow Ref. 1610.06191 and introduceΨ_i = [f_i ↑f_i ↓^†;f_i ↓ -f_i ↑^† ].We can then express the mean field Hamiltonian (<ref>) in the following form <cit.>H_MF = H^0 + H^x + H^y + H^z + C ,whereH^a = ±1/4∑_ij J_ij[ Tr(σ^a Ψ_i U_ij^a Ψ_j^†) + 1/2Tr( U_ij^a † U_ij^a ) ]with + sign for a=0 and - sign for a=x,y,z. σ^0 is the 2 × 2 identity matrix and σ^x,y,z are the Pauli matrices, while the ansatzes are given by U_ij^0= [ χ_ij - Δ_ij^*; - Δ_ij - χ_ij^* ], J_ij>0 , U_ij^a= [ E_ij^aD_ij^a*; - D_ij^aE_ij^a* ], J_ij<0, a=x,y,z . The second term in the square brackets in (<ref>) is irrelevant for the PSG analysis so we can simply ignore it for the moment. Moreover, we introduce u_ij^a ≡± J_ij U_ij^a /4 to simplify the notation.We also write the constraint C explicitly in the Hamiltonian (<ref>), which is often referred to as on-site term. It has the form[b] C= ∑_i Tr(Ψ_i [ a^3_i a^1_i - i a^2_i; a^1_i + i a^2_i-a^3_i ]Ψ_i^†) = ∑_i Tr(Ψ_i ∑_l=1^3 a^l_i τ^l Ψ_i^†)up to some multiplicative constant, which can be absorbed into the definition of chemical potentials a^l_i.An SU(2) spin rotation R=exp(iσ·𝐧̂θ/2) acts on Ψ_i from the left by Ψ_i ⟶ R^†Ψ_i.<cit.> We see from (<ref>) that, while H^0 preserves global spin rotation symmetry of the original Heisenberg Hamiltonian (<ref>), H^x,y,z generically breaks it. We also observe that the mean field Hamiltonian (<ref>) has the following SU(2) gauge redundancy G: Ψ_i ⟶Ψ_i G(i) ,u_ij^a ⟶ G(i)^† u_ij^a G(j) . In (<ref>), we say that a gauge transformation G(i) ∈ SU(2) acts on Ψ_i from the right. The matrix Ψ̃_i ≡Ψ_i G(i) contains the transformed fermionic operators f̃_i α, which leave the representation (<ref>) of spin invariant, satisfying the same anticommutation relations (<ref>) and constraints (<ref>) and (<ref>) as f_i α, so it describes the same fermionic object. Therefore, ansatzes which differ by a gauge transformation give rise to the same physical state.The presence of triplet terms in the mean field Hamiltonian breaks the global spin rotation symmetry. The remaining symmetries are the space group of non-symmorphic Kagome lattice and time reversal symmetry. Now, we are going to use the PSG to identify all possible spin liquid ansatzes that respect these symmetries.Let X be a space group element. In general, there will be a site independent spin rotation R_X ∈ SU(2) associated with X, such that X acts on Ψ_i by <cit.>X: Ψ_i ⟶ R^†_X Ψ_X(i) ,which changes the mean field Hamiltonian byH^a X⟶∑_ijTr(R_X σ^a R^†_X Ψ_X(i) u_ij^a Ψ_X(j)^†) .Let us first inspect the simplest case a=0, in which R_X σ^0 R^†_X=1. We want the Hamiltonian to be invariant under the action of X. By the SU(2) gauge redundancy, this requires u^0_ij to be equal to u^0_X(i)X(j) up to a gauge transformation G_X(𝐫) ∈ SU(2),u^0_X(i)X(j) = G_X(X(i)) u^0_ij G_X^†(X(j)) .This suggests that we can view the action of X and G_X on u^0_ij as <cit.> X: u^0_ij⟶ u^0_X^-1(i)X^-1(j) , G_X: u^0_ij⟶ G_X(X(i)) u^0_ij G_X^†(X(j)) , while the compound operator G_X X acts trivially on u^0_ij. The set of all G_X X that leaves the ansatz invariant is defined as projective symmetry group (PSG). This definition includes X=𝒯, the time reversal operator, whose action will be discussed later.When a=x,y,z in (<ref>), we have to take into account the effect of spin rotation associated with the space group element. We can map the SU(2) rotation to an SO(3) rotation on the Pauli matrices,<cit.>R_X σ^a R^†_X = ∑_b=x,y,z O_X^abσ^b ,such that the triplet Hamiltonian changes by∑_a=x,y,z H^a X⟶∑_ij∑_ab O_X^abTr(σ^b Ψ_X(i) u_ij^a Ψ_X(j)^†) .By SU(2) gauge redundancy, and the fact that O^ab = (O^T)^ba = (O^-1)^ba for any O ∈ SO(3), we must haveu_X(i)X(j)^b = ∑_a (O^-1_X)^ba G_X(X(i)) u_ij^a G_X^†(X(j)) ,or∑_b O_X^ab u_X(i)X(j)^b = G_X(X(i)) u_ij^a G_X^†(X(j)), a=x,y,z.Notice that reflection is not a proper rotation and cannot be described by an SO(3) matrix. We can however generalize SO(3) to O(3) to include improper rotations. In particular, we will see that the action of glide, which is the combination of reflection and half translation, in spin space is described by an O(3) matrix with = -1.Consider now the antiunitary time reversal operator,𝒯 = i σ^y Kwhere i σ^y operates on Ψ_i from the left and K complex conjugates any number on the right. It acts on the Hamiltonian as <cit.>[b] 𝒯 H^a 𝒯^-1 = ∑_ijTr(σ^a* iσ^y Ψ_X(i) K u_ij^a Ψ_X(j)^† K (-iσ^y) ) = Tr(σ^y σ^a*σ^y Ψ_X(i) u_ij^a*Ψ_X(j)^†) ⟶Tr(σ^y σ^a*σ^y Ψ_X(i) (i σ^y) u_ij^a* (-iσ^y) Ψ_X(j)^†) = Tr(σ^aΨ_X(i) (-u_ij^a) Ψ_X(j)^†) .where in the third line we have introduced a gauge transformation i τ^2,<cit.> and in the last line we have used σ^y σ^0*σ^y = σ^0, σ^y u_ij^0*σ^y = -u_ij^0, σ^y σ^a*σ^y = -σ^a, σ^y u_ij^a*σ^y = u_ij^a for a=x,y,z. <cit.> The Hamiltonian must be invariant under the action of 𝒯 if the system has time reversal symmetry, which requires- u_ij^a = G_𝒯(i) u_ij^a G_𝒯^†(j),a=0,x,y,z,for some G_𝒯(𝐫) ∈ SU(2). Note that both σ^a and τ^a are identity (a=0) or Pauli matrices (a=x,y,z or 1,2,3), but σ^a denotes spin rotation while τ^a denotes gauge transformation.Suppose that G_X X ∈PSG. Applying a gauge transformation G(i) ∈ SU(2) on the ansatz, G_X changes as G_X(i) ⟶ G(i) G_X(i) G^†(X^-1(i)) such that G_X X is still a PSG element.<cit.> A subgroup of PSG known as invariant gauge group (IGG) consists of elements of the form G_I I where I is the identity element. It is the group of pure gauge transformation that leave the ansatz invariant.<cit.> Typically, when both hopping and pairing terms are present in the mean field Hamiltonian, the IGG is just ℤ_2 = { - 1, + 1 }. §.§Algebraic PSGThe algebraic relations among the symmetry operations (space group and time reversal) constrain the possible forms of the gauge transformations G_X. In particular, the algebraic identities (<ref>) - (<ref>) impose a set of consistent conditions (<ref>) - (<ref>) on G_X. These consistent conditions (or algebraic constraints) are the same for the singlet and triplet ansatzes, since there is no net spin rotation for symmetry operations that amount to identity. We do have to be careful not to neglect the effect of spin rotation when relating the triplet ansatzes by symmetry, which we will soon discuss. The PSG in which gauge transformation G_X associated with symmetry X satisfies the algebraic constraints is known as algebraic PSG.The final result of algebraic PSG is G_T_x(x,y,s)= τ^0 , G_T_y(x,y,s)= τ^0 , G_C_2(x,y,s)= η_C_2T_y^x+y g_C_2(s) , G_h(x,y,s)= η_C_2T_y^y g_h(s) , G_𝒯(x,y,s)= i τ^2 , with g_X(s) ≡ G_X(0,0,s) for X=C_2,h given in TABLE <ref>. Consistent combinations of the ℤ_2 variables η_C_2, η_C_2T_y, η_C_2𝒯 and η_h𝒯, which arise from the algebraic constraints (<ref>) - (<ref>), result in various forms of g_C_2(s) and g_h(s), which in turn give rise to twelve distinct ℤ_2 spin liquid states that respect the space group of non-symmorphic Kagome lattice and time reversal symmetry. Detailed derivation of the algebraic PSG can be found in Appendix <ref>. §.§Mean Field Ansatz There are four different exchange interactions in the spin model of Volborthite, as discussed in Section <ref>. For ℤ_2 spin liquid states, both hopping and pairing channel are present for every exchange interaction. Each singlet channel contributes one mean field parameter, while each triplet channel contributes three mean field parameters, which correspond to its x,y,z components. Therefore, we have 16 independent mean field parameters, which we denote by χ, Δ, 𝐄', 𝐃', 𝐄_1, 𝐃_1, χ_2 and Δ_2, depending on which exchange couplings they are associated with. We also consider (for reasons that will be explained in Section <ref>) the root U(1) states, which are described by the Hamiltonian (<ref>) without all the pairing terms. Therefore, the root U(1) states have 8 independent mean field parameters.All these mean field parameters are complex-valued in general. However, through (<ref>) and (<ref>), time reversal symmetry restricts the singlet parameters χ_ij and Δ_ij to be real, and the triplet parameters 𝐄_ij and 𝐃_ij to be imaginary, for u_ij^a to be nonzero.Symmetry-related singlet ansatzes u^0_ij are generated by (<ref>), while triplet ansatzes u^a_ij by (<ref>). The matrix O_X ∈ SO(3) in (<ref>) that describes the spin rotation associated with space group element X (<ref>) is trivial for translations X=T_x,T_y. For π-rotation and glide, we have O_C_2 = [ -100;0 -10;001 ] , O_h= [100;0 -10;001 ] . While O_C_2∈ SO(3), we see that O_h having = -1 is an element in O(3) ∖ SO(3).As in the bosonic PSG,<cit.> the gauge matrix (<ref>) and (<ref>) of lattice translations in fermionic PSG are trivial. To construct H_MF, it is therefore sufficient to relate the mean field parameters by (<ref>) or (<ref>) in the (0,0) unit cell, because the same set of relations hold in all other unit cells. In other words, the mean field ansatz does not go beyond the physical unit cell. We explained in Ref. 1702.04360 that this is a consequence of non-symmorphic symmetry. The proof, which involves only few steps of simple algebraic manipulation, will however be repeated here for completeness.Consider the algebraic constraints (<ref>) and (<ref>), which originate from the spatial identities (<ref>) and (<ref>) respectively. Abbreviating G_X X as X̃, we have[b] T̃_x^-1T̃_y^-1T̃_x T̃_y= (η_hh̃^2)^-1T̃_y^-1η_hh̃^2 T̃_y = h̃^-1(h̃^-1T̃_y^-1h̃T̃_y^-1) T̃_y h̃T̃_y = η_hT_yη_hT_y= +1 .(<ref>) rules that the mean field ansatz allowed by PSG can never go beyond the physical unit cell, as it would require T̃_x^-1T̃_y^-1T̃_x T̃_y = -1 should the contrary be true. The same argument holds for three other non-symmorphic plane crystallographic groups pg, p2mg and p4gm. In contrast, the algebraic PSG of isotropic Kagome lattice, where non-symmorphic symmetry is absent, allows certain spin liquid states to have mean field ansatz that enlarges the physical unit cell.<cit.>PSG analysis also requires that the on-site chemical potential ∑_l a^l_i τ^l ≡ u_ii^0 in (<ref>) satisfies u^0_X(i)X(i) = G_X(X(i)) u^0_ii G_X^†(X(i)), X ∈{ T_x,T_y,C_2,h } , - u^0_ii = G_𝒯(i) u^0_ii G_𝒯^†(i) , for consistency.<cit.> By translational invariance, the chemical potential can only have sublattice dependence, so we write u_ii^0 ≡Λ_s, s=1,…,6. Through (<ref>) and (<ref>), time reversal symmetry restricts the chemical potential to beΛ_s = μ_s τ^3 + ν_s τ^1 ,for some μ_s, ν_s ∈ℝ. The remaining symmetries, C_2 and h, further imply that there can only be at most two independent μ_s and ν_s, so in general we have four chemical potentials.We will be interested in the fermionic states 3a, 3b, 4a and 4b, which can be connected to Schwinger boson spin liquid states, as discussed in Section <ref>. In these states, we find only one independent μ_s and ν_s, so the number of chemical potentials is reduced to two, which we argue as follows. Since G_𝒯(x,y,s)= iτ^2, by (<ref>) we have-Λ_s = τ^2 Λ_s τ^2for all s. Furthermore, g_C_2(s) = ± i τ^2 (see TABLE <ref>) for 3a, 3b, 4a and 4b, by (<ref>) we haveΛ_1= τ^2 Λ_5 τ^2 = -Λ_5 ,Λ_2= τ^2 Λ_4 τ^2 = -Λ_4 ,Λ_3= τ^2 Λ_3 τ^2 = -Λ_3 ,Λ_6= τ^2 Λ_6 τ^2 = -Λ_6 ,which implies Λ_3=Λ_6=0. We can further use h to relate the chemical potentials at s=1,2,4,5, which for brevity we show the final result directly,μ_1 = μ_2 = - μ_4 = - μ_5,ν_1 = ν_2 = - ν_4 = - ν_5,for 3a and 3b; μ_1 = μ_2 = - μ_4 = - μ_5,ν_1 = -ν_2 = ν_4 = - ν_5,for 4a and 4b.Therefore, there is only one independent μ_s and ν_s for the fermionic states 3a, 3b, 4a and 4b, which simplifies the calculation.§MAPPING BETWEEN BOSONIC AND FERMIONIC SPIN LIQUID STATES Lu et al. developed a framework to connect bosonic and fermionic spin liquid states on the isotropic Kagome lattice through vison PSG in Ref. 1403.0575. Similar analysis has been carried out in square lattice,<cit.> rectangular lattice,<cit.> and breathing Kagome lattice.<cit.> We would like to do the same for non-symmorphic Kagome lattice, to figure out the correspondence between the eight bosonic states and the twelve fermionic states resulting from PSG analysis in Ref. 1702.04360 and Section <ref>, respectively. The bosonic states (p_2,p_3,p_13) are labeled by three ℤ_2 variables. In particular, we would like to know the fermionic states that are connected to the most energetically favorable bosonic state (1,0,0) and the highly competing (1,1,0), from which a (q,0) spiral order and a (π,π) spin density wave develop, respectively, upon spinon condensation.The mapping is based on the idea of symmetry fractionalization and fusion rule, which we briefly explain below following Ref. 1403.0575. Spinons and visons are collectively known as anyons. Any local excitation in a ℤ_2 spin liquid, which must be a bound state of two anyons of the same type, can only gain a trivial phase factor (+1) under symmetry operations that amount to identity. Therefore, the phase factor picked up by one anyon under these symmetry operations is quantized to be ± 1. This is known as symmetry fractionalization, and the phase factor ± 1 is known as symmetry quantum number. The three types of anyon, bosonic spinon b, fermionic spinon f and vison v, obey the following abelian fusion rule,b × f = v, f × v = b, v × b = f,b × b = f × f = v × v = 1,where 1 represents local excitations carrying integer spin. The fusion rule tells us that vison is a bound state of bosonic spinon and fermionic spinon, etc. Let X be a string of symmetry operations that amount to identity. Suppose that under the action of X, the phase gain of bosonic spinon, fermionic spinon and vison are ϕ_b, ϕ_f and ϕ_v respectively. Then, by the fusion rule, they satisfye^iϕ_f = e^iϕ_t e^iϕ_b e^iϕ_v ,where we have introduced the twist factor e^ϕ_t=± 1. When e^ϕ_t=+1 (e^ϕ_t=-1), the fusion rule is said to be trivial (nontrivial). Nontrivial fusion rule arises due to mutual semion satistics. In short, e^iϕ_t=-1 when anyon of one type, say b, encircles anyon of another type, say v, under the action of X.The symmetry quantum numbers of bosonic and fermionic spinons are known from solving their respective algebraic PSG (see Ref. 1702.04360 and Appendix <ref>). Therefore, the remaining ingredients required to complete the mapping between bosonic and fermionic spin liquid states are the vison PSG and the various twist factors, which we provide in Appendix <ref> and <ref>. The correspondence between bosonic spinon, fermionic spinon and vison PSG on the non-symmorphic Kagome lattice is shown in TABLE <ref>, from which we can easily identify (-1)^p_2 = - η_C_2 , (-1)^p_3 = - η_C_2 T_y , (-1)^p_2+p_3 = - η_C_2 𝒯 , (-1)^p_13 = - η_h 𝒯 . This maps the bosonic state (p_2,p_3,p_13) = (1,0,0), which gives rise to (q,0) spiral order, to the fermionic state 4b, (0,1,0) to 4a, (1,0,1) to 3b, and (0,1,1) to 3a. All other bosonic (fermionic) states do not have fermionic (bosonic) counterpart. For a given lattice, the number of fermion spin liquid states is usually greater than bosonic spin liquid states under PSG classification, which is true in both the isotropic Kagome lattice <cit.> and the non-symmorphic Kagome lattice. However, all the bosonic states in isotropic Kagome lattice have fermionic correspondence,<cit.> unlike the non-symmorphic Kagome lattice, where half of the bosonic states have no fermionic counterpart.§SPECTRUM OF ℤ_2 AND U(1) FERMIONIC STATES WITH BOSONIC CORRESPONDENCE In Section <ref>, we showed that only four out of twelve fermionic spin liquid states have bosonic correspondence. They are 3a, 3b, 4a and 4b, which are connected to the bosonic spin liquid states (0,1,1), (0,1,0), (1,0,1) and (1,0,0) respectively. We narrow down our investigation to these four fermionic states and study their generic mean field spectrum. We consider the ℤ_2 states as well as the corresponding root U(1) states, because thermal Hall effect, which is observed in Volborthite,<cit.> would arise in U(1) spin liquid with spinon Fermi surface. As argued in Ref. PhysRevLett.104.066403,RevModPhys.78.17, when the mean field spin liquid state has U(1) gauge symmetry, the spinons are coupled to an emergent U(1) gauge field. When an external magnetic field is applied, a fictitious magnetic field is induced, which in turn leads to an effective Lorentz force for the spinons and the finite thermal Hall conductivity arises when the spinons form a Fermi surface.Our result is summarized in TABLE <ref>. We find that the root U(1) states of 3a, 3b, 4a and 4b all feature Fermi surfaces. Hence these U(1) spin liquid states may be possible candidates for the paramagnetic phase of Volborthite and provide an explanation of the finite thermal Hall conductivity seen above 1K. We find that the spinon energy bands of these U(1) states are doubly degenerate. For the ℤ_2 spin liquids, we find that 3a, 4a and 4b have Dirac point(s) while 3b has a gapped spectrum. Here, the spinon energy bands of 3a and 4a are doubly degenerate, while those of 3b and 4b are non-degenerate. We plot the dispersion immediately above and below the Fermi level in the root U(1) and ℤ_2 states in FIG. <ref> and <ref>, respectively.§DISCUSSION In this work, we focus on the paramagnetic state of Volborthite above the magnetic ordering temperature T ≈ 1K, where a recent experiment <cit.> reveals finite thermal Hall conductivity. Given that the magnetic ordering temperature is much smaller than the Curie-Weiss temperature of 140K,<cit.> we take the point of view that the paramagnetic state right above 1K may be better considered as a finite temperature version of a proximate paramagnetic quantum state, which may be obtained by suppressing the magnetic order at zero temperature. Hence we are interested in possible quantum (paramagnetic) ground states that are closely related to the magnetic order below 1K and exhibit finite thermal Hall conductivity.We notice that the microscopic spin model derived from DFT <cit.> suggests non-symmorphic symmetry of the underlying lattice structure, which is described by the plane group p2gg and can be regarded as a distorted version of the Kagome lattice. As explained in the main text, this non-symmorphic Kagome lattice has six sites per unit cell with one electron per site in the insulating phase. According to a recent work <cit.> on the extension of the HOLSM theorem, a trivial band insulator is possible in this lattice only when the electron filling is 4n, where n ∈N. Hence any state without broken symmetry in this system must be either topologically ordered (with gapped excitations) or gapless.<cit.> Since Volborthite is an insulator, this means any (non-symmetry-breaking) paramagnetic quantum ground state must be nontrivial.In this context, we investigate possible quantum spin liquid states in the non-symmorphic Kagome lattice. These states are nontrivial paramagnetic quantum ground states potentially relevant to Volborthite. In our previous study,<cit.> we identified a (q,0) spiral order and a (π,π) spin density wave as the most promising magnetically ordered states for the magnetic order seen below 1K in Volborthite. These magnetically ordered states can be obtained from ℤ_2 spin liquid phases with bosonic spinons, via condensation of the spinons. We showed that there exist eight possible bosonic spin liquid states and two of them are related to the (q,0) spiral order and the (π,π) spin density wave respectively.While the bosonic spin liquid states mentioned above are closely related to the magnetic orders, these states are necessarily gapped (topologically ordered) and cannot naturally explain the finite thermal Hall conductivity. Hence, in the current work, we study quantum spin liquid states with fermionic spinons, which allow both gapped and gapless excitation spectra. We find that there are twelve distinct ℤ_2 fermionic spin liquid states. Only four of the twelve ℤ_2 fermionic states are connected to four of the eight ℤ_2 bosonic spin liquid states. Interestingly, the bosonic spin liquid state (which is called (1,0,0) in the main text) closely related to the (q,0) spiral order corresponds to a fermionic spin liquid state (which is called 4b in the main text), while the bosonic state closely related to the (π,π) spin density wave order does not have a fermionic counterpart. Moreover, all of the root U(1) spin liquid states (where the pairing amplitudes vanish) obtained from the four ℤ_2 fermionic spin liquid states mentioned above possess Fermi surfaces.Combining all the information and considering the consistency between different descriptions, we may conclude that the magnetic order below 1K in Volborthite is most likely to be the (q,0) spiral order and the most promising spin liquid state that explains the thermal Hall effect above 1K is the U(1) fermionic spin liquid state with a Fermi surface, which is related to the (q,0) spiral order via the mapping between fermionic and bosonic spin liquid states. The direct confirmation of this conclusion would come from future neutron scattering studies of the excitation spectrum below and above 1K.In the current work, we do not consider the relative energetics of candidate fermionic spin liquid phases, which would require careful examination of projected variational wavefunctions. This would be an excellent topic of future study.We thank Kyusung Hwang for illuminating discussions. This work was supported by the NSERC of Canada and the Center for Quantum Materials at the University of Toronto. Some of the computations were performed on the GPC supercomputer at the SciNet HPC Consortium.<cit.> SciNet is funded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto. §SPACE GROUP AND ALGEBRAIC IDENTITIES OF NON-SYMMORPHIC KAGOME LATTICEWe list below the action of spatial symmetries, T_x, T_y, C_2 and h, on a site (x,y,s) in the non-symmorphic Kagome lattice.T_x: (x,y,s) ⟶(x+1,y,s) .T_y: (x,y,s) ⟶(x,y+1,s) .C_2:(x,y,1) ⟶ (-x,-y,5) ,(x,y,2) ⟶ (-x,-y,4) ,(x,y,4) ⟶ (-x,-y,2) ,(x,y,5) ⟶ (-x,-y,1) ,(x,y,3) ⟶ (-x+1,-y,3) ,(x,y,6) ⟶ (-x,-y+1,6) .h:(x,y,1) ⟶ (x,-y-1,2) ,(x,y,2) ⟶ (x+1,-y-1,1) ,(x,y,3) ⟶ (x,-y,6) ,(x,y,6) ⟶ (x+1,-y,3) ,(x,y,4) ⟶ (x,-y,5) ,(x,y,5) ⟶ (x+1,-y,4) .Inspecting how T_x, T_y, C_2 and h commute with each other leads to the algebraic identitiesT_x^-1 T_y^-1 T_x T_y= I , C_2^2= I , C_2 T_x C_2^-1 T_x= I , C_2 T_y C_2^-1 T_y= I , T_x^-1 h^2= I , h^-1 T_x^-1 h T_x= I , h^-1 T_y h T_y= I , T_x T_y h^-1 C_2 h C_2= I .In addition, time reversal commutes with all the space group elements,𝒯^2= I , X^-1𝒯^-1 X 𝒯 = I,X ∈{ T_x, T_y, C_2, h } .(<ref>) - (<ref>) constrain the possible form of gauge matrices G_X, X=T_x,T_y,C_2,h,𝒯 such that G_XX ∈PSG.§SOLUTION TO FERMIONIC PSGThe algebraic identities (<ref>) - (<ref>) lead to a set of consistent conditions among the gauge matrices,G_T_x^†(T_y^-1(i)) G_T_y^†(i) G_T_x(i) G_T_y(T_x^-1(i)) = η_12τ^0 ,G_C_2(C_2(i)) G_C_2(i) = η_C_2τ^0 ,G_C_2^†(T_xC_2(i)) G_T_x(T_xC_2(i))G_C_2(C_2(i)) G_T_x(i) = η_C_2T_xτ^0 ,G_C_2^†(T_yC_2(i)) G_T_y(T_yC_2(i))G_C_2(C_2(i)) G_T_y(i) = η_C_2T_yτ^0 ,G_T_x^†(i) G_h(i) G_h(h^-1(i)) = η_h τ^0 ,G_h^†(T_x^-1(i)) G_T_x^†(i) G_h(i) G_T_x(h^-1(i)) = η_hT_xτ^0 ,G_h^†(T_yh(i)) G_T_y(T_yh(i)) G_h(h(i)) G_T_y(i) = η_hT_yτ^0 ,G_T_x(T_xT_yh^-1C_2h(i)) G_T_y(T_yh^-1C_2h(i)) G_h^†(C_2h(i))G_C_2(C_2h(i)) G_h(h(i)) G_C_2(i) = η_hC_2τ^0 , [G_𝒯(i)]^2 = η_𝒯τ^0 ,G^†_T_x(i) G^†_𝒯(i) G_T_x(i) G_𝒯(T_x^-1(i)) = η_T_x𝒯τ^0 ,G^†_T_y(i) G^†_𝒯(i) G_T_y(i) G_𝒯(T_y^-1(i)) = η_T_y𝒯τ^0 ,G^†_C_2(i) G^†_𝒯(i) G_C_2(i) G_𝒯(C_2^-1(i)) = η_C_2𝒯τ^0 ,G^†_h(i) G^†_𝒯(i) G_h(i) G_𝒯(h^-1(i)) = η_h𝒯τ^0 ,where the various η_X=± 1 are ℤ_2 variables. Using the standard arguments,<cit.> we can fix G_T_x(x,y,s)=η_12^y τ^0 and G_T_y(x,y,s)=τ^0. Furthermore, we can exploit the global ℤ_2 gauge such that G_T_x⟶η_h G_T_x andG_T_y⟶η_hT_y G_T_y to fix η_h = η_hC_2 = +1.(<ref>) and (<ref>) leads toG_C_2(x,y,s) = η_C_2T_x^x η_C_2T_y^y η_12^δ_s,6 x g_C_2(s) ,where g_X(s) ≡ G_X(0,0,s). Plugging (<ref>) into (<ref>), we findg_C_2 (C_2(s)) g_C_2(s)= η_C_2τ^0,s=1,2,4,5,[g_C_2(3)]^2= η_C_2η_C_2T_xτ^0 ,[g_C_2(6)]^2= η_C_2η_C_2T_yτ^0 ,(<ref>) and (<ref>) leads toG_h(x,y,s) = η_hT_x^x η_hT_y^y η_12^(δ_s,1 +δ_s,2)x g_h(s) ,Applying (<ref>) to i=(x,y,4) and (x,y,5) yieldsη_12^y η_hT_x g_h(4) g_h(5)= τ^0 ,η_12^y g_h(4) g_h(5)= τ^0 ,which implies η_hT_x=+1. Moreover, we must have η_12=+1, since R.H.S. of (<ref>) or (<ref>) is coordinate independent. Applying (<ref>) to sites with different sublattice index, we findg_h(1) g_h(2)= η_hT_yτ^0 , g_h(3) g_h(6)= τ^0 , g_h(4) g_h(5)= τ^0 .Applying (<ref>) to i=(x,y,s) for all s, using (<ref>) to eliminate g_C_2(1) = η_C_2 g_C_2^†(5) and g_C_2(2) = η_C_2 g_C_2^†(4), using (<ref>), (<ref>) and (<ref>) to eliminate g_h(1)=η_hT_yg_h^†(2), g_h(3)=g_h^†(6) and g_h(5)=g_h^†(4), we find six relationsg_h^†(4) g_C_2(4) g_h(2) g_C_2^†(5)= η_C_2η_C_2T_yτ^0 , g_h(4) g_C_2(5) g_h^†(2) g_C_2^†(4)= η_C_2η_C_2T_xη_C_2T_yη_hT_yτ^0 , g_h^†(6) g_C_2(6) g_h(6) g_C_2(3)= η_C_2T_yη_hT_yτ^0 , g_h(2) g_C_2^†(5) g_h^†(4) g_C_2(4)= η_C_2η_hT_yτ^0 , g_h^†(2) g_C_2^†(4) g_h(4) g_C_2(5)= η_C_2η_C_2T_xτ^0 , g_h(6) g_C_2(3) g_h^†(6) g_C_2(6)= τ^0 .Rearranging terms in (<ref>) and (<ref>), we find that they are equal, which implies η_hT_y = η_C_2T_y. Similarly, (<ref>) and (<ref>) are related by hermitian conjugation, which implies η_C_2T_x = η_C_2T_y. (<ref>) and (<ref>) then provide no new information. In summary, there are only four relevant conditions without involving time reversal symmetryg_h(4) g_C_2(5) g_h^†(2) g_C_2^†(4)= η_C_2η_C_2T_yτ^0 , g_h(6) g_C_2(3) g_h^†(6) g_C_2^†(6)= τ^0 ,[g_C_2(3)]^2= η_C_2η_C_2T_yτ^0 ,[g_C_2(6)]^2= η_C_2η_C_2T_yτ^0 . Next, we consider time reversal symmetry. (<ref>) and (<ref>) leads toG_𝒯(x,y,s) = η_T_xT^x η_T_yT^y g_𝒯(s)With (<ref>), applying (<ref>) on i=(x,y,s) for s=3 and 6 yieldsg_h^†(3) g_𝒯^†(3) g_h(3) g_𝒯(6)= η_T_x𝒯η_h𝒯τ^0 , g_h^†(6) g_𝒯^†(6) g_h(6) g_𝒯(3)= η_h𝒯τ^0 .Through (<ref>), we see that (<ref>) and (<ref>) are related by hermitian conjugation, which implies η_T_x𝒯 = +1.Qualitatively different solutions exist for η_𝒯 = ± 1. For η_𝒯 = +1, G_𝒯(i) = ±τ^0 by (<ref>). From (<ref>), the mean field ansatzes satisfy- u_ij^a = G_𝒯(i) u_ij^a G_𝒯^†(j),a=0,x,y,z.For any three sites i,j,k which form a triangle on the non-symmorphic Kagome lattice, any choice of G_𝒯(i),G_𝒯(j),G_𝒯(k) that satisfies G_𝒯(𝐫) = ±τ^0 will render at least one side of the triangle having u_ij^a=0. This is not a relevant physical solution since it changes the lattice structure, and we will simply ignore it and specialize to the case η_𝒯 = -1. (<ref>) then implies G_𝒯(i) = i 𝐚_i ·τ, where 𝐚_i is a real three-component vector of unit length and τ is the vector of Pauli matrices. We can perform a sublattice dependent gauge transformation W_s ∈ SU(2) such that g_𝒯(s) ⟶ W_s g_𝒯(s) W_s^† = i τ^2, without affecting previous gauge fixing.We can further show that η_T_y𝒯=+1. Applying (<ref>) to i=(x,y,s) for s=3 and 6 leads tog^†_C_2(3) g^†_𝒯(3) g_C_2(3) g_𝒯(3)= η_C_2𝒯τ^0g^†_C_2(6) g^†_𝒯(6) g_C_2(6) g_𝒯(6)= η_T_y𝒯η_C_2𝒯τ^0If η_T_y𝒯=-1, then g_C_2(3) and g_C_2(6) must have the forms e^i θτ^2 and i e^i ϕτ^2τ^3, such that (<ref>) cannot be satisfied. Therefore, we must have η_T_y𝒯=+1, which implies G_𝒯(i) = iτ^2 everywhere.Now we proceed to solve the equations (<ref>) - (<ref>), plusg_C_2^†(s) τ^2 g_C_2(s) τ^2= η_C_2 𝒯τ^0 , g_h^†(s) τ^2 g_h(s) τ^2= η_h 𝒯τ^0 ,on a case by case basis. First, note that we have the freedom to perform a sublattice dependent gauge transformation of the form W_s = e^i θ_s τ^2, without affecting previous gauge fixing. The gauge matrices transform as W_s: g_X(s) ⟶ W_s g_X(s) W_X^-1(s) where X=C_2,h.A. η_C_2η_C_2T_y = +1. (<ref>), (<ref>) and (<ref>) give g_C_2(3) = g_C_2(6) = ±τ^0, which further implies η_C_2 𝒯=+1 by (<ref>). Therefore, g_C_2(5)=e^i θ_5 τ^2 and g_C_2(4)=e^i θ_4 τ^2. We perform gauge transformations W_5 = ± e^-i θ_5 τ^2 and W_4 = ± e^-i θ_4 τ^2 to fix g_C_2(s) = ±τ^0 for s=3,4,5,6, which can be further fixed to τ^0 by a global ℤ_2 gauge. Moreover, (<ref>) gives g_h(4) = g_h(2).i. η_h 𝒯=+1. (<ref>) gives g_h(4) = g_h(2) = e^i ϕ_2 τ^2 and g_h(6) = e^i ϕ_6 τ^2. Gauge transformations W_4 = W_2 = e^-i ϕ_2 τ^2 and W_6 = e^-i ϕ_6 τ^2 fix g_h(s) = τ^0 for s=2,4,6. (1)ii. η_h 𝒯=-1. (<ref>) gives g_h(4) = g_h(2) = i e^i ϕ_2 τ^2τ^3 and g_h(6) = i e^i ϕ_6 τ^2τ^3. Gauge transformations W_4 = W_2 = e^-i ϕ_2 τ^2 and W_6 = e^-i ϕ_6 τ^2 fix g_h(s)=i τ^3 for s=2,4,6. (2)B. η_C_2η_C_2T_y = -1.a. η_C_2𝒯 = +1. (<ref>), (<ref>) and (<ref>) require that g_C_2(3) = ± i τ^2 and g_C_2(6) = ± i τ^2. (<ref>) gives g_C_2(5)=e^i θ_5 τ^2 and g_C_2(4)=e^i θ_4 τ^2. Say g_C_2(3) = ± i τ^2, gauge transformations W_5 = ± e^-i θ_5 τ^2 and W_4 = ± e^-i θ_4 τ^2 fix g_C_2(5) = g_C_2(4) = ±τ^0.i. η_h𝒯 = +1. (<ref>) and (<ref>) give g_h(4) = -g_h(2) = e^i ϕ_2 τ^2 and g_h(6) = e^i ϕ_6 τ^2. Gauge transformations W_4 = W_2 = e^-i ϕ_2 τ^2 and W_6 = e^-i ϕ_6 τ^2 fix g_h(4) = - g_h(2) = g_h(6) = τ^0. (<ref>) then forces g_C_2(3) = - g_C_2(6). We fix g_C_2(3) = - g_C_2(6)= i τ^2 and g_C_2(5) = g_C_2(4) = τ^0 by and a global ℤ_2 gauge. (3)ii. η_h𝒯 = -1. (<ref>) and (<ref>) give g_h(4) = -g_h(2) = i e^i ϕ_2 τ^2τ^3 and g_h(6) = i e^i ϕ_6 τ^2τ^3. Gauge transformations W_4 = W_2 = e^-i ϕ_2 τ^2 and W_6 = e^-i ϕ_6 τ^2 fix g_h(4) = - g_h(2) = g_h(6) = i τ^3. (<ref>) then forces g_C_2(3) = g_C_2(6). We fix g_C_2(3) = g_C_2(6) = i τ^2 and g_C_2(5) = g_C_2(4) = τ^0 by a global ℤ_2 gauge. (4)b. η_C_2 𝒯 = -1. (<ref>) requires that g_C_2(s) = i e^i ϕ_s τ^2τ^3. Gauge transformations W_3 = e^-i ϕ_3 τ^2/2, W_6 = e^-i ϕ_6 τ^2/2, W_5 = e^-i ϕ_5 τ^2 and W_4 = e^-i ϕ_4 τ^2 fix g_C_2(s) = i τ^3 for s=3,4,5,6.i. η_h𝒯 = +1. (<ref>), (<ref>) and (<ref>) give g_h(4) = - g_h^†(2) = e^i ϕ_2 τ^2 and g_h(6) = ± i τ^2. Gauge transformation W_4 = W_2^† = ± e^-i ϕ_2 τ^2 fixes g_h(4) = -g_h(2) = ±τ^0. We can further fix g_h(4) = -g_h(2) = τ^0 and g_h(6) = i τ^2 by a global ℤ_2 gauge. (5)ii. η_h𝒯 = -1. (<ref>), (<ref>) and (<ref>) give g_h(4) = i e^i ϕ_2 τ^2τ_3, g_h(2) = -i e^-i ϕ_2 τ^2τ_3 and g_h(6) = ± i τ^1. Gauge transformation W_4 = W_2^† = ± e^-i ϕ_2 τ^2 fixes g_h(4) = -g_h(2) = ± i τ^3. We can further fix g_h(4) = -g_h(2) = i τ^3 and g_h(6) = i τ^1 by a global ℤ_2 gauge. (6)We would like to comment on these solutions. First, only g_C_2(s) for s=3,4,5,6 and g_h(s) for s=2,4,6 are explicitly shown here. The remaining gauge matrices can be related through (<ref>) and (<ref>) - (<ref>). Second, we only consider the overall sign of the product η_C_2η_C_2T_y but not the individual components. One of them, say η_C_2T_y, can be ± 1, which doubles the number of solutions to 12. η_C_2T_y determines the sign of g_h(1), g_C_2(1) and g_C_2(2). Finally, we can perform a gauge transformation on the solutions (3), (4), (5) and (6), such that the gauge matrices g_C_2(s) and g_h(s) appear more symmetric. For (3) and (4), we apply the gauge transformations W_4 = W_5 = i τ^2. For (5) and (6), we apply the gauge transformation W_6 = -i τ^2. The final result is shown in TABLE <ref>.§SOLUTION TO VISON PSGTo see how vison PSG arises, we start from quantum dimer model (QDM), which depicts spin liquid in a generic lattice as linear combination of singlet product states. QDM can be described by an effective ℤ_2 gauge theory, which is further mapped to fully frustrated Ising model (FFIM) on the dual lattice, where the notion of vison creation operator becomes apparent. We then construct the dual lattice of non-symmorphic Kagome lattice, which is dubbed the non-symmorphic dice lattice, and solve the vison PSG. §.§From Quantum Dimer Model to Fully Frustrated Ising ModelQuantum dimer model (QDM) provides a simple picture of spin liquid in terms of spin singlets (or dimers) for a given lattice. The Hilbert space is spanned by different configurations of dimers, each of which is formed by localized S=1/2 moment on two distinct sites of the lattice, such that every site is covered by exactly one dimer (known as hardcore dimer constraint).<cit.> In other words, each state in QDM is essentially a linear combination of independent singlet product states on the lattice. The Hamiltonian of QDM has a kinetic term, which changes the configuration of dimers, and a potential term, which counts the interaction between dimers.<cit.> QDM can be effectively described by a ℤ_2 gauge theory through the introduction of Pauli matrices τ_l defined on each link l connecting two sites on the lattice, such that τ_l^x=-1 (τ_l^x=+1) when l is occupied (unoccupied) by a dimer, while τ_l^z changes the state of l. This leads to the Hamiltonian <cit.>H = - ∑_l J_l τ_l^x - ∑_𝒫Γ_𝒫∏_l ∈𝒫τ_l^z ,where 𝒫 denotes an elementary plaquette, which is either a triangle or a hexagon on Kagome lattice (FIG. <ref>). The first term in (<ref>) corresponds to the potential term while the second term corresponds to the kinetic term. In addition, the hardcore dimer constraint requires <cit.>∏_l ∈𝒮τ_l^x = -1 ,where 𝒮 denotes a `star', the collection of links attached to a given site on the lattice (FIG. <ref>). Note that the Hamiltonian (<ref>) respects the hardcore dimer constraint <cit.>[ H, ∏_l ∈𝒮τ_l^x ] = 0 . ℤ_2 gauge theory on the direct lattice as defined above can be further mapped to fully frustrated Ising model (FFIM) on the dual lattice, which is dice lattice in the case of Kagome lattice. This is done by introducing the operators <cit.>σ_i^x= ∏_l ∈𝒫(i)τ_l^z ,σ_i^z= ∏_l > iτ_l^x ,where i label the sites on dual lattice, which are the center of plaquettes 𝒫 on direct lattice. σ obeys the same Pauli matrices algebra as τ.<cit.> σ_i^z, which involves the product of all links appearing to the right of i (FIG. <ref>), is nonlocal in the direct lattice but local in the dual lattice. In ℤ_2 gauge theory, the flux of an elementary plaquette on the direct lattice is given by σ_i^x. If i is the center of 𝒫, then σ_i^z changes the flux of 𝒫 by ± 1, and we say that a vison is created. Therefore, σ_i^z is understood as the vison creation operator. The definition (<ref>) and the constraint (<ref>) imply <cit.>σ_i^z σ_j^z = λ_ijτ_l^xfor two neighboring sites i and j separated by link l, where λ_ij=± 1 is chosen such that the Gauss law constraint, the equivalent of hardcore dimer constraint in FFIM, is satisfied. Gauss law constraint states that the product of λ_ij around an elementary plaquette on the dual lattice has to be -1 (to see this, write τ_l^x= λ_ijσ_i^z σ_j^z and apply (<ref>)). (<ref>) can then be written asH = - ∑_⟨ ij ⟩λ_ij J_ijσ_i^z σ_j^z - ∑_i Γ_i σ_i^x ,which is the Hamiltonian of FFIM on the dual lattice. Following Ref. PhysRevB.84.094419, we solve (<ref>) for the vison dispersion with soft spin approximation, in which σ_i^z = ± 1 is replaced by a continuous variable ϕ_i ∈ℝ. We also neglect the second term in (<ref>) as our symmetry consideration are restricted to time independent and static configurations. Therefore, (<ref>) becomesH = - ∑_⟨ ij ⟩λ_ij J_ijϕ_i ϕ_j .§.§Non-Symmorphic Dice LatticeNon-symmorphic Kagome lattice can be constructed from isotropic Kagome lattice by replacing the nearest neighbour bonds of the latter with three different bonds. Since the mapping from QDM to FFIM need not preserve the microscopic interactions, we can forget about the spin model suggested by DFT calculation for the moment, and view the non-symmorphic Kagome lattice as consisting of three generically inequivalent links. Its dual lattice, the non-symmorphic dice lattice, is constructed by connecting the center of triangular and hexagonal plaquettes across these links, as shown in FIG. <ref>a. As a result, non-symmorphic dice lattice also has three generically inequivalent links, six sites per unit cell (FIG. <ref>b), and the same space group as non-symmorphic Kagome lattice.Elementary plaquettes on the non-symmorphic dice lattice are the different rhombi (FIG. <ref>b). We must now choose a combination of λ_ij=± 1 such that Gauss law constraint is satisfied. It turns out that the gauge introduced in Ref. PhysRevB.84.094419 is a convenient choice, which we adapt to our system as depicted in FIG. <ref>a. This particular gauge choice enlarges the unit cell such that the original dice lattice effectively becomes a honeycomb lattice with twelve sites per unit cell. Primitive vectors 𝐮 and 𝐯 duplicate the unit cell in two independent directions (FIG. <ref>b), so that the coordinates of any unit cell is 𝐑 = m𝐮 + n𝐯 where m,n ∈ℤ.Consider the action of symmetry transformations of non-symmorphic dice lattice on the effective honeycomb lattice. FIG. <ref>a and b shows how our original gauge choice is modified by translations and glide respectively, while it is invariant under π-rotation and time reversal. To restore the original gauge choice, we have to multiply certain vison field components ϕ_i by the factor -1 as indicated in FIG. <ref>a and b, which corresponds to a ℤ_2 gauge transformation. Such a combination of symmetry and gauge transformation that leaves the Hamiltonian (<ref>) invariant defines the vison PSG.Next, we extract the matrix representation of vison PSG in the order parameter space that describes vison condensation.<cit.> The Hamiltonian (<ref>) in momentum space is obtained through Fourier transform,H= - ∑_⟨ ij ⟩λ_ij J_ijϕ_i ϕ_j = - ∑_i∑_𝐤𝐤'∑_(a,b,δ_I,δ_II,λ,T)λ J_Tϕ_𝐤^ae^i𝐤·(𝐑_i+δ_I)/√(N)ϕ_𝐤'^be^i𝐤' ·(𝐑_i+δ_II)/√(N)= - ∑_k∑_(a,b,𝐝,λ,T)λ J_Tϕ_𝐤^aϕ_-𝐤^b e^i𝐤·𝐝≡∑_k∑_abϕ_𝐤^aD_𝐤^abϕ_-𝐤^b ,where a,b=1,...,12 are sublattice indices, δ_I and δ_II are displacements with respect to coordinate 𝐑_i of unit cell i, 𝐝≡δ_I-δ_II, λ=± 1 reflects the gauge choice in FIG. <ref>a, and T=I,II,III labels the three inequivalent couplings. Diagonalization of D_𝐤 yields the vison dispersion. The order parameters ψ_n are defined through the expansion of vison field in terms of the critical modes,<cit.>ϕ(𝐑) = ∑_nψ_n 𝐯_n e^i 𝐪_n ·𝐑 ,where we label the wave vectors at which minimum occurs by 𝐪_n, and the corresponding eigenvector by 𝐯_n. These eigenvectors are chosen such that they form a complete orthonormal set, ∑_n 𝐯_n 𝐯_n^† = 1. The vison PSG can be viewed as matrix operation on the order parameters as follows,<cit.>[b] G_X X : ϕ^a (𝐑) ⟶ ∑_nψ_n v_n^a' e^i 𝐪_n ·𝐑' G_X (𝐑',a') =∑_n( ∑_m O_nmψ_m ) v_n^a e^i 𝐪_n ·𝐑 ,where (𝐑',a')= X (𝐑,a). O_nm is the matrix representation of vison PSG that we want to determine. With the couplings J_I, J_II, J_III chosen arbitrarily, we find only two critical modes at 𝐪_1 = 𝐪_2 = 0. The PSG matrices are then constructed from the corresponding eigenvectors 𝐯_1 and 𝐯_2,T_x= [0 -1;10 ] , T_y= [0 -1;10 ] , C_2= [ 1 0; 0 1 ] , h= 1/√(2)[ -1 -1;1 -1 ] ,𝒯 = [ 1 0; 0 1 ] ,which result in the symmetry fractionalizationsT_x^-1 T_y^-1 T_x T_y= 1 , C_2^2= 1 C_2 T_x C_2^-1 T_x= -1 , C_2 T_y C_2^-1 T_y= -1 , T_x^-1 h^2= -1 , h^-1 T_x^-1 h T_x= 1 , h^-1 T_y h T_y= -1 , T_x T_y h^-1 C_2 h C_2= -1 ,𝒯^2= 1 , T_x^-1𝒯^-1 T_x 𝒯 = 1 , T_y^-1 T^-1 T_y 𝒯 = 1 , C_2^-1𝒯^-1 C_2 𝒯 = 1 , h^-1𝒯^-1 h 𝒯 = 1 .Furthermore, we find that while different sets of couplings { J_I, J_II, J_III} may change the matrix representations (<ref>) - (<ref>) of vison PSG, they lead to the same symmetry fractionalizations (<ref>) - (<ref>).§TRIVIAL AND NONTRIVIAL FUSION RULEWe check the fusion rule of the algebraic identities listed in Appendix <ref>. Let us consider the motion of a fermionic spinon f, which is the bound state of a bosonic spinon b and vison v as discussed in Section <ref>, under these algebraic identities. We are going to focus on the relative motion between b and v rather than the graphical derivation introduced in Ref. 1603.03041,1605.05322, since the conclusion of the latter seems to depend on how b is placed relative to v initially.First, notice that there is no relative motion between b and v under translations. Therefore, (<ref>) has trivial fusion rule. Next, consider a unitary operator X that squares to identity, for instance C_2. We argue that X^2=I must have nontrivial fusion rule as follows.<cit.> Consider a state |Ψ⟩ = f_i^† f_X(i)^†| 0 ⟩ with | 0 ⟩ being the ground state. Under the action of X,X: f_i^†⟶ X f_i^† X^-1≡ f_X(i)^†and[b] X: |Ψ⟩ ⟶(X f_i^† X^-1) (X f_X(i)^† X^-1) | 0 ⟩= f_X(i) X^2 f_i X^-2| 0 ⟩= e^iϕ_f f_X(i) f_i| 0 ⟩= - e^iϕ_f|Ψ⟩ ,where the minus sign in the last line arises from exchanging two fermions. If we write f_i^† = b_i^† v_i^†, then we haveX: |Ψ⟩⟶ e^iϕ_b e^iϕ_v|Ψ⟩ ,since bosonic operators commute. This shows that X^2=I must have nontrivial fusion rule - e^iϕ_f = e^iϕ_b e^iϕ_v for a unitary operator X. Therefore, (<ref>) have nontrivial fusion rule. An equivalent loop traced by (<ref>) is C_2^-1 T_x C_2 T_x = (C_2^-1)^2 (C_2 T_x)^2, where the operator in each bracket squares to identity. Therefore, (<ref>) has trivial fusion rule. The same argument applies to (<ref>).h is reflection followed by half lattice translation. We conjecture that, since reflection alone squares to identity, and all sorts of translation do not cause any relative motion between b and v, (<ref>) is nontrivial because it involves reflection twice. It also follows that (<ref>) and (<ref>) are trivial, because reflection is first applied and its inverse subsequently. (<ref>) is nontrivial because it involves rotation twice, and reflection and its inverse once.Since 𝒯 is antiunitary, we cannot conclude from previous argument that (<ref>) has nontrivial fusion rule. In fact, it has trivial fusion rule, since 𝒯^2=-1 for half-integer spins (e.g. b and f), while 𝒯^2=+1 for integer spin (e.g. v).<cit.> For X=T_x,T_y, (<ref>) has trivial fusion rule. For X=C_2,h, (<ref>) has nontrivial fusion rule, because, considering the equivalent loop X 𝒯 X^-1𝒯^-1, the effect of 𝒯 X^-1𝒯^-1 should be the same as X, so the twist factor is equal to that under X^2. | http://arxiv.org/abs/1703.09220v1 | {
"authors": [
"Li Ern Chern",
"Robert Schaffer",
"Sopheak Sorn",
"Yong Baek Kim"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170327180000",
"title": "Fermionic spin liquid analysis of the paramagnetic state in Volborthite"
} |
DEDefinition[section] LE[DE]Lemma RE[DE]Remark CO[DE]Corollary THM[DE]Theorem PR[DE]Proposition CN[DE]Conjecture PP[DE]Property CL[DE]Claim pf Proof: defeng[DE]Definition theorem[DE]Theorem lemma[DE]Lemma conjecture[DE]Conjecture corollary[DE]Corollary remark[DE]Remark breakalgorithm[DE]Algorithm breakproblem[DE]Problem claim amsplainThe (theta, wheel)-free graphsPart II: structure theorem Marko RadovanovićUniversity of Belgrade, Faculty of Mathematics, Belgrade, Serbia. Partially supported by Serbian Ministry of Education, Science and Technological Development project 174033. E-mail: [email protected] , Nicolas TrotignonCNRS, LIP, ENS de Lyon. Partially supported by ANR project Stint under reference ANR-13-BS02-0007 and by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program ‘‘Investissements d'Avenir’’ (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).Also Université Lyon 1, université de Lyon. E-mail: [email protected] , Kristina VuškovićSchool of Computing, University of Leeds, and Faculty of Computer Science (RAF), Union University, Belgrade, Serbia.Partially supported by EPSRC grants EP/K016423/1 and EP/N0196660/1, and Serbian Ministry of Education and Science projects 174033 and III44006. E-mail: [email protected]=================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three paths between the samepair of distinct vertices so that the union of any two of the paths induces a hole. A wheel is a graph formed by a holeand a node that has at least 3 neighbors in the hole. In this paper we obtain a decomposition theorem for the classof graphs that do not contain an induced subgraph isomorphic to a theta or a wheel, i.e. the class of (theta, wheel)-freegraphs. The decomposition theorem uses clique cutsets and 2-joins.Clique cutsets are vertex cutsets that work really well in decomposition based algorithms, but are unfortunately not general enough to decompose more complex hereditary graphclasses.A 2-join is an edge cutset that appeared indecomposition theorems of several complex classes, such as perfect graphs, even-hole-free graphs and others.In these decomposition theorems 2-joins are used together with vertex cutsets that are more general than clique cutsets, such as star cutsets and their generalizations(which are much harder to use in algorithms). This is a first example of a decomposition theorem that uses just the combination ofclique cutsets and 2-joins. This has several consequences. First, we can easily transform our decomposition theorem into a complete structuretheorem for (theta, wheel)-free graphs, i.e. we show how every (theta, wheel)-free graph can be built starting from basic graphs that can be explicitlyconstructed, andgluing them together by prescribed composition operations; and all graphs built this way are (theta, wheel)-free.Such structure theorems are very rare for hereditary graph classes, only a few examples are known.Secondly, we obtain an 𝒪 (n^4m)-time decomposition basedrecognition algorithm for (theta, wheel)-free graphs.Finally, in Parts III and IV of this series, we give further applicationsof our decomposition theorem.§ INTRODUCTIONIn this article, all graphs are finite and simple.A prism is a graph made of three node-disjoint chordless paths P_1 = a_1 … b_1, P_2 = a_2 … b_2, P_3 = a_3 … b_3 of length at least 1, such that a_1a_2a_3 and b_1b_2b_3 are triangles and no edges exist between the paths except those of the two triangles.Such a prism is also referred to as a 3PC(a_1a_2a_3,b_1b_2b_3) or a 3PC(Δ ,Δ ) (3PC stands for 3-path-configuration). A pyramid is a graph made of three chordless paths P_1 = a … b_1, P_2 = a … b_2, P_3 = a … b_3 of length at least 1, two of which have length at least 2, node-disjoint except at a, and such that b_1b_2b_3 is a triangle and no edges exist between the paths except those of the triangle and the three edges incident to a.Such a pyramid is also referred to as a 3PC(b_1b_2b_3,a) or a 3PC(Δ ,·).A theta is a graph made of three internally node-disjoint chordless paths P_1 = a … b, P_2 = a … b, P_3 = a … b of length at least 2 and such that no edges exist between the paths except the three edges incident to a and the three edges incident to b.Such a theta is also referred to as a 3PC(a, b) or a 3PC(· ,·). A hole in a graph is a chordless cycle of length at least 4.A wheel W= (H, c) is a graph formed by a hole H (called the rim) together with a node c (called the center) that has at least three neighbors in the hole. A 3-path-configuration is a graph isomorphic to a prism, a pyramid or a theta.Observe that the lengths of the paths in the definitions of 3-path-configurations are designed so that the union of any two of the paths induce a hole. A Truemper configuration is a graph isomorphic to a prism, a pyramid, a theta or a wheel (see Figure <ref>).Observe that every Truemper configuration contains a hole. If G and H are graphs, we say that G contains H when H is isomorphic to an induced subgraph of G.We say that G is H-free if it does not contain H.We extend this to classes of graphs with the obvious meaning (for instance, a graph is (theta, wheel)-free if it does not contain a theta and does not contain a wheel). In this paper we prove a decomposition theorem for (theta, wheel)-free graphs, from which we obtain a full structure theorem and a polynomial time recognition algorithm.This is part of a series of papers that systematically study the structure of graphs where some Truemper configurations are excluded. This project is motivated and explained in more details in the first paper of the series <cit.>. In Parts III and IV of the series (see <cit.>) we give several applications of the structure theorem.§.§ The main result and the outline of the paperA graph is chordless if all its cycles are chordless. By the following decomposition theorem proved in <cit.>, to prove a decomposition theorem for (theta, wheel)-free graphs, it suffices to focus on graphs that contain a pyramid. If G is (theta, wheel, pyramid)-free, then G is a line graph of a triangle-free chordless graph or it has a clique cutset.In Section <ref>, we define a generalization of pyramids that we call P-graphs.The full definition is complex, but essentially, a P-graph is a graph that can be vertexwise partitioned into the line graph of a triangle-free chordless graph and a clique. Clearly, if a(theta, wheel)-free graph contains a pyramid, then it contains a P-graph.We consider such a maximal P-graph and prove that the rest of the graph attaches to it in a special way that entails a decomposition.The decompositions that we use are the clique cutset and the 2-join (to be defined soon). Our main theorem is the following. If G is (theta, wheel)-free, then G is a line graph of a triangle-free chordless graph or a P-graph, or G has a clique cutset or a 2-join. Clique cutsets are vertex cutsets that work really well in decomposition based algorithms, but are unfortunately not general enough to decompose more complex hereditary graphclasses.A 2-join is an edge cutset that appeared indecomposition theorems of several complex classes, such as perfect graphs <cit.>, even-hole-free graphs <cit.> and others.In these decomposition theorems 2-joins are used together with vertex cutsets that are more general than clique cutsets, such as star cutsets and their generalizations(which are much harder to use in algorithms). This is the first example of a decomposition theorem that uses just the combination ofclique cutsets and 2-joins. This has several consequences. First, we can easily transform our decomposition theorem into a complete structuretheorem for (theta, wheel)-free graphs, i.e. we show how every (theta, wheel)-free graph can be built starting from basic graphs that can be explicitlyconstructed, andgluing them together by prescribed composition operations; and all graphs built this way are (theta, wheel)-free.Such structure theorems are very rare for hereditary graph classes, only a few examples are known, such aschordal graphs <cit.>, universally-signable graphs <cit.>, graphs that do not contain a cycle with a unique chord <cit.>,claw-free graphs <cit.> and bull-free graphs <cit.>(for a survey see <cit.>). The second consequence is the following theorem, and the remaining consequences are given in <cit.>. There exists an O(n^4m)-time algorithm that decides whether an input graph G is (theta, wheel)-free. In Section <ref>, we give all the definitions needed in the statement of Theorem <ref>.In particular, we define P-graphs and 2-joins. In Section <ref>, we study skeletons (the skeleton is the root-graph of the line graph part of a P-graph). In Section <ref>, we study the properties of P-graphs. In Section <ref>, we study attachments to P-graphs in (theta, wheel)-free graphs. In Section <ref>, we prove Theorem <ref>. In Section <ref>, we prove Theorem <ref> and describe how a structure theorem is derived from our decomposition theorem.§.§ Terminology and notations A clique in a graph is a (possibly empty) set of pairwise adjacent vertices. We say that a clique is big if it is of size at least 3. A clique of size 3 is also referred to as a triangle, and is denoted by Δ.A diamond is a graph obtained from a clique of size 4 by deleting an edge. A claw is a graph induced by nodes u,v_1,v_2,v_3 and edges uv_1,uv_2,uv_3.A path P is a sequence of distinct vertices p_1p_2… p_k, k≥ 1, such that p_ip_i+1 is an edge for all 1≤ i <k.Edges p_ip_i+1, for 1≤ i <k, are called the edges of P.Vertices p_1 and p_k are the ends of P.A cycle C is a sequence of vertices p_1p_2… p_kp_1, k ≥ 3, such that p_1… p_k is a path and p_1p_k is an edge.Edges p_ip_i+1, for 1≤ i <k, and edge p_1p_k are called the edges of C.Let Q be a path or a cycle.The vertex set of Q is denoted by V(Q).The length of Q is the number of its edges.An edge e=uv is a chord of Q if u,v∈ V(Q), but uv is not an edge of Q. A path or a cycle Q in a graph G is chordless if no edge of G is a chord of Q.Let A and B be two disjoint node sets such that no node of A is adjacent to a node of B. A path P=p_1 … p_k connects A and B if either k=1 and p_1 has neighbors in both A and B, or k>1 and one of the two endnodes of P is adjacent to at least one node in A and the other endnode is adjacent to at least one node in B. The path P is a direct connection between A and B if in G[V(P) ∪ A ∪ B] no path connecting A and B is shorter than P.The direct connection P is said to be from A to B if p_1 is adjacent to a node of A and p_k is adjacent to a node of B.Let G be a graph.For x∈ V(G), N(x) is the set of all neighbors of x in G, and N[x]=N(x) ∪{ x}.Let H and C be vertex-disjoint induced subgraphs of G.The attachment of C over H, denoted by N_H(C), is the set of all vertices of H that have at least one neighbor in C.When C consists of a single vertex x, we denote the attachment of C over H by N_H(x), and we say that it is an attachment of x over H. Note that N_H(x)=N(x)∩ V(H). For S⊆ V(G), G[S] denotes the subgraph of G induced by S. When clear from the context, we will sometimes write G instead of V(G). § STATEMENT OF THE DECOMPOSITION THEOREMWe start by defining the cutsets used in the decomposition theorem. In a graph G, a subset S of nodes and edges is a cutset if its removal yields a disconnected graph. A node cutset S is a clique cutset if S is a clique. Note that every disconnected graph has a clique cutset: the empty set.For a graph G and disjoint sets A,B⊆ V(G), we say that a node cutset S of G separates A and B if S⊆ V(G)∖(A∪ B) and no vertex of A is in the same connected component of G∖ S as some vertex of B.An almost 2-join in a graph G is a pair (X_1,X_2) that is a partition of V(G), and such that:* For i=1,2, X_i contains disjoint nonempty sets A_i and B_i, such that every node of A_1 is adjacent to every node of A_2, every node of B_1 is adjacent to every node of B_2, and there are no other adjacencies between X_1 and X_2.* For i=1,2, |X_i|≥ 3.An almost 2-join (X_1, X_2) is a 2-join when for i∈{1,2}, X_i contains at least one path from A_i to B_i, and if |A_i|=|B_i|=1 then G[X_i] is not a chordless path.We say that (X_1,X_2,A_1,A_2,B_1,B_2) is a split of this 2-join, and the sets A_1,A_2,B_1,B_2 are the special sets of this 2-join. A star cutset in a graph is a node cutset S that contains a node (called a center) adjacent to all other nodes of S. Note that a nonempty clique cutset is a star cutset.If G is a(theta, wheel)-free graph that has a star cutset, then G has a clique cutset.We now define the basic graphs.A graph G is chordless if no cycle of G has a chord, and it is sparse if for every edge e=uv, at least one of u or v has degree at most 2.Clearly all sparse graphs are chordless.An edge of a graph is pendant if at least one of its endnodes has degree 1.A branch vertex in a graph is a vertex of degree at least 3.A branch in a graph G is a path of length at least 1 whose internal vertices are of degree 2 in G and whose endnodes are both branch vertices.A limb in a graph G is a path of length at least 1 whose internal vertices are of degree 2 in G and whose one endnode has degree at least 3 and the other one has degree 1. Two distinct branches are parallel if they have the same endnodes.Two distinct limbs are parallel if they share the same vertex of degree at least 3. Cut vertices of a graph R that are also branch vertices are called the attaching vertices of R.Let x be an attaching vertex of a graph R, and let C_1, … ,C_t be the connected components of R∖ x that together with x are not limbs of R (possibly, t=0, when all connected components of R∖ x together with x are limbs).If x is the end of at least two parallel limbs of R, let C_t+1 be the subgraph of R formed by all the limbs of R with endnode x.The graphs R[V(C_i)∪{ x} ] (for i=1, …, t, if t≠ 0) and the graph C_t+1 (if it exists) are the x-petals of R. For any integer k≥ 1, a k-skeletonis a graph R such that (see Figures <ref>, <ref> and <ref> for examples of k-skeletons for k=1,2,5):*R is connected, triangle-free, chordless and contains at least three pendant edges (in particular, R is not a path).*R has no parallel branches (but it may containsparallel limbs). *For every cut vertex u of R, every component of R∖ u has a vertex of degree 1 in R. *For every vertex cutset S={a, b} of R and for every component C of R∖ S, either R[C∪ S] is a chordless path from a to b, or C contains at least one vertex of degree 1 in R. *For every edge e of a cycle of R, at least one of the endnodes of e is of degree 2 in R. *Each pendant edge of R is given one label, that is an integer from {1, …, k}. *Each label from { 1, … ,k} is given at least once (as a label), and some label is used at least twice. *If some pendant edge whose one endnode is of degree at least 3 receives label i, then no other pendant edge receives label i. *If R has no branches then k=1, and otherwise if two limbs of R are parallel, then their pendant edges receive different labels and at least one of these labels is used more than once. *If k>1 then for every attaching vertex x and for every x-petal H of R, there are at least two distinct labels that are used in H.Moreover, if H' is a union of at least one but not allx-petals, then there is a label i such that both H' and (R∖H')∪{x} have pendant edges with label i. *If k=2, then both labels are used at least twice.Note that ifR is a k-skeleton, then it edgewise partitions into its branches and its limbs. To prove this, let e be an edge of R and P=u… v, where (u)≥(v), the maximal path of R that contains e and whose internal vertices are of degree 2. If (u)=(v)=1, then R is the chordless path induced by V(P), which contradicts <ref>. If (v)=2, then, by the maximality of P, uv is an edge of R. Now, if (u)=2, then R is the chordless cycle induced by V(P), which contradicts <ref>; if (u)≥ 3, thenu is a cut vertex of R that contradicts <ref>. So, (u),(v)≥ 3 and P is a branch of R, or (u)≥ 2 and (v)=1 in which case (u)≥ 3 (by the maximality of P) and P is a limb of R.Also, there is a trivial one-to-one correspondence between the pendant edges of R and the limbs of R: any pendant edge belongs to a unique limb, and conversely any limb contains a unique pendant edge. If R is a graph, then the line graph of R, denoted by L(R), is the graph whose nodes are the edges of R and such that two nodes of L(R) are adjacent in L(R) if and only if the corresponding edges are adjacent in R.A P-graphis any graph B that can be constructed as follows (see Figures <ref>, <ref> and <ref> for examples of P-graphs):* Pick an integer k≥ 1 and a k-skeleton R.* Build L(R), the line graph of R. The vertices of L(R) that correspond to pendant edges of R are called pendant vertices of L(R), and they receive the same label as their corresponding pendant edges in R.* Build a clique K with vertex set { v_1, … ,v_k}, disjoint from L(R).* B is now constructed from L(R) and K by adding edges between v_i and all pendant vertices of L(R) that have label i, for i=1, … ,k. We say that K is the special clique of B and R is the skeleton of B.The next lemma, that is proved in Part I, allows us to focus on (theta, wheel, diamond)-free graphs in the remainder of the paper. If G is a wheel-free graph that contains a diamond, then G has a clique cutset.Observe that P-graphs are generalizations of pyramids (this is why we call them P-graphs).Let us explain this. A pyramid is long if all of its paths are of length greater than 1.Note that in a wheel-free graph all pyramids are long.Every long pyramid Π =3PC(x_1x_2x_3, y) is a P-graph, where K={ y} and R is a tree that is obtained from a claw by subdividing each edge at least once and giving all pendant edges label 1 (see Figure <ref>).It can be checked that a pyramid whose one path is of length 1 (and that is therefore a wheel) is not a P-graph. This is a consequence of Lemma <ref> to be proved soon, but let us sketch a direct proof: the apex of the pyramid is the center of a claw, so it must be in the special clique, which therefore has size 1 or 2.It follows that the skeleton must contain two pendant edges with the same label, and one of them contains a vertex of degree 3, a contradiction to condition <ref>. A long pyramid is a P-graph.In fact, every P-graph contains a long pyramid.Formally we do not need this simple fact, we therefore just sketch the proof: consider three pendant edges of the skeleton for which at most two labels are used (this exists by (i) and <ref>).Consider a minimal connected subgraph T of R that contains these three edges.It is easy to check that T is a tree with three pendant edges and a unique vertex v of degree 3, and that adding to its line graph the vertices of K corresponding toat most two labels yields a long pyramid. To check that the pyramid is long condition <ref> is used, to check that two paths of T linking v to pendant edges with the same label have length at least 2.§ CONNECTIVITY OF SKELETONSIn the following theorem we state versions of Menger's theorem that we use in this paper. Let G be a graph. (i) Let u and v be non-adjacent vertices of G. Then the maximum number of internally vertex-disjoint paths from u to v is equal to the minimum size of a cutset S of G that separates {u} and {v}.(ii) Let A and B be disjoint subsets of V(G). Then the maximum number of vertex-disjoint paths with one endnode in A and the other in B is equal to the minimum size of a cutset S of G that separates A and B.(iii) Letu∈ V(G) and B⊆ V(G)∖{u}. Then the maximum number of paths from u toB that are vertex-disjoint except at u is equal to the minimum size of a cutset S of G that separates {u} and B.Additionally, we will often use the following variant of Menger's theorem, which is due to Perfect <cit.>.Let G be a graph, x∈ V(G) and Y⊆ V(G)∖{x}. A set of k paths P_1,P_2,…,P_k of G is a k-fan from x to Y if V(P_i)∩ V(P_j)={x}, for 1≤ i<j≤ k, and |V(P_i)∩ Y|=1, for 1≤ i≤ k. A fan from x to Y is a |Y|-fan from x to Y. Let G be a graph, x∈ V(G) and Y,Z⊆ V(G)∖{x} such that |Y|<|Z|. If there are fans from x to Y and from x to Z, then there is a fan from x to Y∪{z}, for some z∈ Z∖ Y. For distinct verticesv_1,v_2,…,v_k of G, and pairwise disjoint and non-empty subsets W_1,W_2,…,W_k of V(G)∖{v_1,v_2,…,v_k}, we say that k vertex-disjointpaths P_1,P_2,…,P_k are from {v_1,v_2,…,v_k} to {W_1,W_2,…,W_k} if for some permutation σ∈𝕊_k,P_i∩{v_1,v_2,…,v_k}={v_i} and P_i∩ (W_1∪ W_2∪…∪ W_k) is a vertex of W_σ(i), for 1≤ i≤ k. Let G be a connected graph, v_1,v_2,…,v_k distinct vertices of G and W_1,W_2,…,W_k pairwise disjoint and non-empty subsets of V(G)∖{v_1,v_2,…,v_k}, such that all vertices of W_1 are of degree 1. The following holds: (1) if k=2, and all vertices of W_2 are of degree 1 or W_2={w_2}, then there exist 2 vertex-disjointpaths from {v_1,v_2} to {W_1,W_2}, or a vertex u that separates {v_1,v_2} from W_1∪ W_2;(2) if k=3, W_2={w_2}, W_3={w_3} and there exist 2 vertex-disjointpaths from {v_2,v_3} to {w_2,w_3}, then there exist 3 vertex-disjointpaths from {v_1,v_2,v_3} to {W_1,{w_2},{w_3}}, or there exist vertices u_1 and u_2 such that {u_1,u_2} separates {v_1,v_2,v_3} from W_1∪{w_2,w_3}. Let G' be the graph obtained from G by adding a vertex v (v∉V(G)) and edges vv_i, for 1≤ i≤ k. (1) By Menger's theorem, there is a vertex u that separates {v_1,v_2} from W_1∪ W_2, or two vertex-disjoint paths from {v_1,v_2} to W_1∪ W_2. If the first outcome holds, then we are done, so we may assume that there are vertex-disjoint paths P_1 and P_2 from {v_1,v_2} to W_1∪ W_2. If both W_1 and W_2 contain an endnode of P_1 and P_2, then we are again done.So we assume that both P_1 and P_2 have an endnode in w.l.o.g. W_1, and let these endnodes be v_1' and v_2'. This means that in G' there is a fan from v to {v_1',v_2'}. Since G' is connected, there is a fan from v to some v”∈ W_2, and therefore, by Lemma <ref>, there is a fan from v to {v',v”}, for some v'∈{v_1',v_2'}. This completes the proof of (1).(2) By Menger's theorem, there are vertices u_1 and u_2 that separate {v_1,v_2,v_3} from W_1∪{w_2,w_3}, or three vertex-disjointpaths such that each of them has one endnode in {v_1,v_2,v_3} and the other in W_1∪{w_2,w_3}. If the first outcome holds, then we are done, so we may assume that there are vertex-disjointpaths P_1,P_2,P_3 such that each of them has one endnode in {v_1,v_2,v_3} and the other in W_1∪{w_2,w_3}. Let the endnodes of paths P_1,P_2,P_3 that are in W_1∪{w_2,w_3} be v_1',v_2',v_3'. This means that in G' there is a fan from v to {v_1',v_2',v_3'}. By the conditions of the lemma, there is also a fan from v to {w_2,w_3}, and therefore, by Lemma <ref>, there is a fan from v to {w,w_2,w_3}, for some w∈{v_1',v_2',v_3'}∖{w_2,w_3}. Since {v_1',v_2',v_3'}∖{w_2,w_3} is a subset of W_1, this completes our proof. Recall a standard notion: a block of a graph is an induced subgraph that is connected,has no cut vertices and is maximal with respect to these properties.Recall that every block of a graph is either 2-connected, or is a single edge.Recall that cut vertices of a graph R that are of degree at least 3 are called the attaching vertices of R.Let R be a k-skeleton.If C is a 2-connected block of R, then no two vertices of C that are of degree at least 3 in R are adjacent.In particular, every 2-connected block of R is sparse, no two adjacent vertices of every cycle of R have degree at least 3, and if an edge of R is between two vertices of degree at least 3, then it is a cutedge of R. This is equivalent to condition <ref> in the definition of a P-graph, since an edge of R belongs to a cycle if and only if it belongs to a 2-connected block of R.Let R be a k-skeleton.If e_1 and e_2 are edges of R, then there exists a cycle of R that goes through e_1 and e_2, or there exists a path in R whose endnodes are of degree 1 (in R) and that goes through e_1 and e_2.We set e_1 = u_1v_1 and e_2 = u_2v_2.We apply Menger's theorem to {u_1, v_1} and {u_2, v_2} (or their one-element subsets if these sets are not disjoint).If the outcome is a pair of vertex-disjointpaths, then we obtain the cycle whose existence is claimed.We may therefore assume that the outcome is a cut vertex x that separates e_1 from e_2.Hence, R is vertex-wise partitioned into X_1, {x} and X_2, in such a way that {u_1, v_1}⊆ X_1 ∪{x} and {u_2, v_2}⊆ X_2 ∪{x} and there are no edges between X_1 and X_2. We now show that R[X_1∪{ x} ] contains a path from a vertex of degree 1 in R to x that contains e_1. Since R is connected this is clearly true if an endnode of e_1 has degree 1 in R. So we may assume that both endnodes of e_1 are of degree greater than 1 in R. Let Y_1 be the set of all vertices in X_1 that have degree 1 in R. Note that Y_1≠∅ by <ref> of the definition of the skeleton. Suppose u_1=x. By <ref> of the definition of the skeleton, there exists a path in R[X_1] from a vertex of degree 1 to v_1, and this path can be extended to a desired path by adding the edge v_1u_1. Therefore, by symmetry, we may assume that x∉{ u_1,v_1}. In R[X_1 ∪{x}], we apply Lemma <ref> to {u_1, v_1} and {Y_1,x}. If we obtain a cut vertexy that separates {u_1, v_1} from {x}∪ Y_1, then y is cut vertex of R (separating e_1 from Y_1∪ X_2) and the component of R y that contains e_1 contradicts <ref>.Hence, we obtain two vertex-disjointpaths, whose union yields a path P_1 that contains e_1 from a vertex of degree 1 (in R) to x.A similar path P_2 exists in R[X_2 ∪{x}]. The union of P_1 and P_2 yields the path whose existence is claimed. Let R be a k-skeleton.Every 2-connected induced subgraph D of R has at least 3 distinct vertices that have neighbors outside D. In particular, every 2-connected block of R has at least 3 attaching vertices.Let D be a 2-connected induced subgraph of R. Let u_1 be a degree 1 vertex of R (it exists by (i)). Since R is connected, there is a path P_1=u_1… v_1, where v_1 is the unique vertex of P_1 in D. In particular, v_1 is a vertex of D with a neighbor outside of D.If v_1 is not a cut vertex of R that separates P_1 ∖ v_1 from D∖ v_1, then there is a path P_2=u_1… v_2, where v_2 is the unique vertex of P_2 in D. Otherwise, by <ref>, the component C of R∖ v_1 that contains D∖ v_1 has a vertex u_2 of degree 1 in R, and a path P_2=u_2… v_2, where v_2 is the unique vertex of P_2 from D. So in both cases we get a vertex v_2 distinct from v_1 such that both v_1 and v_2 have neighbors outside D.Since D is 2-connected, v_1 and v_2 are contained in a cycle of D, so by <ref>, v_1v_2 is not an edge of R.Suppose that { v_1,v_2} is not a cutset of R that separates (P_1∪ P_2)∖{ v_1,v_2} from a vertex of D. Then there is a path P_3=u_3… v_3 in R∖{ v_1,v_2}, where u_3 is a vertex of (P_1∪ P_2)∖{ v_1,v_2} and v_3 is the unique vertex of D in P_3, and hence v_1,v_2,v_3 are the desired three vertices.So we may assume that { v_1,v_2} is a cutset of R that separates (P_1∪ P_2)∖{ v_1,v_2} from a vertex of D. By <ref> there is a component C' of R∖{ v_1,v_2} such that C'∩ D≠∅ and R[C'∪{ v_1,v_2}] is not a chordless path. By <ref>, C' contains a vertex u_3 of degree 1 in R, and a path P_3=u_3… v_3, where v_3 is the unique vertex of P_3 in D. Hence v_1,v_2,v_3 are the desired three vertices.Finally, observe that if D is a block then each of v_1,v_2,v_3 is a cut vertex of R, and hence D has at least three attaching vertices.Let R be a k-skeleton.Let x_1 and x_2 bebranch vertices of R (not necessarily distinct).Then, there are two paths P_1 = x_1 … y_1 and P_2 = x_2 … y_2, vertex-disjoint (except at x_1 if x_1=x_2) such that y_1 and y_2 both have degree 1 and are incident with edges with the same label.First suppose that there exists a label i that is used at least twice in R, and such that there does not exist a vertex x and two sets X, Y ⊂ V(R) such that X, Y, {x} form a partition of V(R), x_1, x_2 ∈ X∪{x}, all degree 1 vertices from edges with label i are in Y, and there are no edges between X and Y. Then, by Menger's theorem there exist two vertex-disjointpaths (except at x_1 if x_1=x_2) between {x_1,x_2} and the set of all degree 1 vertices from edges with label i.So, suppose that in R, for every label i that is used at least twice in R, there exists a vertex x and two sets X, Y ⊆ V(R) such that X, Y, {x} form a partition of V(R), x_1, x_2 ∈ X∪{x}, all degree 1 vertices from edges with label i are in Y, and there are no edges between X and Y. We then choose i, x, X and Y subject to the minimality of X. We claim that x is an attaching vertex of R.If x∈{x_1, x_2}, it is true by assumption.Otherwise, if x has a unique neighbor x' in X, then x' is a cut vertex that contradicts the minimality of X (it separatesX{x'} from Y∪{x}).Hence, x has at least two neighbors in X, and at least one in Y, so it is indeed an attaching vertex.Suppose that X∪{x} contains a limb of R ending at x. This limb cannot have x_1 or x_2 as its internal vertex, so we can move it to Y which contradicts the minimality of X.It follows that X∪{x} is an x-petal, or is the union of two x-petals X_1 (that contains x_1) and X_2 (that contains x_2).In this last case, by <ref>, there exists a label j that is used in both X_1 and R∖ X_1. So, there exists a path from x_1 to an edge with label j in X_1∖{x}and a path in R∖ X_1 from x_2 to an edge with label j, and the conclusion follows. When X ∪{x} is an x-petal, we note that there exists another x-petal included in Y∪{x}, because Y∪{x} cannot be a single limb since a label is used twice in Y.Hence, by <ref>, there exists a label j that is used in both X and Y.Let Z be the set ofdegree 1 vertices from X which are the degree 1 ends of edges with label j.First suppose that x=x_1. Since R[Y∪{ x}] is connected, it contains a path from x to a vertex incident to an edge labeled j. If x_2=x then similarly R[X∪{ x}] contains a path from x to a vertex in Z, and the result holds. So we may assume that x_2∈ X. By connectivity of X there exists a path in R[X] from x_2 to a vertex of Z, and the result holds. Therefore, by symmetry, we may assume that x∉{ x_1,x_2}. Now suppose that x_1=x_2. If there are two paths from x_1 to Z∪{ x}, then the result holds (by possibly extending one of the paths from x, through Y, to a vertex incident to an edge labeled j). Otherwise, by Menger's theorem there is a cut vertex that contradicts the minimality of X. Therefore we may assume that x_1≠ x_2.We now apply Lemma <ref> to {x_1, x_2} and {Z,x}.If the conclusion is two disjoint paths, we are done (by extending the path ending in x to an edge with label j in Y).And if the outcome is a cut vertexx' that separates {x_1, x_2} from {x}∪ Z, then we define X' as the union of the components of R{x'} that contain x_1 and x_2.This contradicts the minimality of X.Let R be a k-skeleton. Let P = x_1 … x_2 be a branch of Rand x'_1 a neighbor of x_1 not in P. Then there are three paths P_1 = x_1 … y_1, P'_1 = x_1 x'_1 … y'_1 and P_2 = x_2… y_2, vertex-disjoint except P_1 and P_1' sharingx_1, and such that y_1, y'_1 and y_2 are degree 1 vertices incident with edges with at most two different labels. By Lemma <ref>, there are vertices y_1 and y_2 of degree 1 incident with edges with the same label, such that there exist vertex-disjointpaths from {x_1,x_2} to {y_1,y_2}.We define X as the set of all vertices of degree 1 in R, except y_1 and y_2.Note that X≠∅ by <ref>.We apply Lemma <ref> to {x_1', x_1, x_2} and {X,y_1, y_2}.If the output is three vertex-disjointpaths, then the conclusion of the lemma holds (x_1 needs to be added to the path that starts at x'_1). Otherwise, there exists a cutset {a, b} that separates {x_1', x_1, x_2} from {y_1, y_2}∪ X.This contradicts <ref>.§ PROPERTIES OF P-GRAPHSFor a P-graph B with special clique K and skeleton R, we use the following additional terminology.The cliques of L(R) of size at least 3 are called the big cliques of L(R). Note that they correspond to sets of edges in R that are incident to a vertex of degree at least 3.We denote by K the set that consists of K and all big cliques of L(R).Remove from B the edges of cliques in K. What remains are vertex-disjointpaths, except possibly those that meet at a vertex of K. These paths are segments of B; moreover, a segment is an internal segment if its endnodes belong to big cliques of L(R), and otherwise it is a leaf segment. If S is a leaf segment and u∈ K is an endnode of S, we say that S is a claw segment if S is not the only segment with endnode u; otherwise we say that S is a clique segment. Observe that it is possible that a segment is of length 0, but then it must be an internal segment.Two segments S_1=s_1… t_1 and S_2=s_2… t_2 are parallel if s_1,t_1,s_2,t_2 are all distinct nodes and for some K_1∈ K∖{K}, s_1,s_2∈ K_1 and t_1,t_2∈ K. Note also that every two cliques of B meet in at most one vertex (since R is triangle-free).Let B be a graph that satisfies all the conditions of being a P-graph except that its skeleton fails to satisfies <ref> or <ref>. Then B contains a wheel.Let R be the skeleton of B, and K its special clique.Case 1: when R fails to satisfy <ref>.Supposethat in R there exists an edge e=xy contained in a cycle C such that x and y are both of degree at least 3.If in R e, there are two internally vertex-disjoint paths from x to y, then R contains a cycle with a chord (namely e).So in L(R), e is a vertex that is the center of a wheel.Hence, by Menger's theorem, we may assume that in R e, there is a cut vertex u that separates x from y(note that u is on C).Let X (resp. Y) be the connected component of R{e, u} that contains x (resp. y).We claim that in R[X∪{ u}] there exists a path P_x = x”… x … u such that x” has degree 1 in R.If x is a cut vertexof R, P_x can be constructed as the union of a path from x to u going through the component of Rx that contains u, and a path from a vertex x” of degree 1 (that exists by <ref>) to x going through another component.So, we assume that x is not a cut vertexof R.Hence, from here on, we assume that R x is connected.We observe that {x, u} is a cutset of R, which separates y from each neighbor of x distinct from y.We define u' as the vertex of C e closest to x along C e such that {x, u'} is a cutset of R that separates y from each neighbor of x distinct from y.Let x' be a neighbor of x not in C (this exists since x has degree at least 3 by assumption). Since x is not a cut vertex of R, u'≠ x.Let X' be the connected component of R{x, u'} that contains x'. Suppose xu'∈ E(R). Since R∖ x is connected there is a path P from x' to u' in X'∪{u'}. Together with C this provides a cycle with a chord (namely xu'), which yields a wheel in B. So, xu'∉E(R). Let X_c be the connected component of R{x, u'} that contains the vertices from C e that are between x and u' (possibly, X' = X_c). Note that the vertices of C∖ e that are between u' and uare in the same connected component of R{x, u'} as y, so none of them is in X'∪ X_c. In R[X' ∪ X_c ∪{x, u'}] there are two internally vertex-disjoint paths Q_1 and Q_2 from x to u', for otherwise, by Menger's theorem, a vertex u” from C separates them, and {x, u”} is a cutset that contradicts u' being closest to x ({x,u”} also separates y from each neighbor of x distinct from y).Note that X'∪{u', x} or X_c ∪{u', x} is not a chordless path, since otherwise they induce parallel branches contradicting <ref>. Therefore, by <ref> one of them contains a vertex x” of degree 1.So,there exists a cycle C' (made of Q_1 and Q_2) in R[X' ∪ X_c ∪{x, u'}], and a minimal path in R[X'∪ X_c] from x” to a vertex in C'. This proves that a path visiting in order x”, x and u' exists. We build P_x by extending this path to u along C∖ e.We can build a similar path P_y.In B, the paths P_x and P_y can becompleted to a wheel via K (e is the center of this wheel).Case 2: when R fails to satisfy <ref>.Suppose for a contradiction that some edge xx' of R has label 1, where x has degree at least 3 and x' degree 1.Suppose moreover that another edge of R, say yy' where y' has degree 1, also receives label 1.Let Z be set of all degree 1 vertices of R, except x' and y'.We claim that in R, there exist two vertex-disjoint paths P_y = x”… y' and P_z = x”' … z, where z∈ Z and x”,x”' are some neighbors of x different from x'.For otherwise, by Lemma <ref>, there exists a cut vertex u in R that separates {x”,x”'} from Z ∪{y'}. Then { u,x'} is a cutset of R such that the connected component C of R∖{ u,x'} that contains x fails to satisfy <ref>. Additionally, we may assume that P_y (resp. P_z) does not contain x, since otherwise instead of P_y (resp. P_z) we can take the subpath of P_y (resp. P_z) from a neighbor of x (on this path) to y' (resp. z). Now, in B, the two paths P_y and P_z together with x and vertices from K yield a hole, that is the rim of a wheel centered at the vertex xx' of L(R). Every P-graph is (theta, wheel, diamond)-free.Let B be a P-graph with skeleton R and special clique K. By construction of B, none of the vertices of L(R) can be centres of claws in B. So all centres of claws of B are contained in K and are therefore pairwise adjacent. It follows that B is theta-free.Since R is triangle-free and pendant vertices of L(R) have unique neighbors in K, and by <ref>, B is diamond-free.Suppose that B contains a wheel (H,x).If x∈ K then some neighbor x_1 of x in H does not belong to K, and hence is a pendant vertex of L(R).It follows that the neighborhood of x_1 in L(R) is a clique and that x_1 has a unique neighbor in K. But this contradicts the assumption that x_1 belongs to the hole H of B∖ x. Therefore, x∉K.Since x is a vertex of L(R), it cannot be a center of a claw in B. Since B is diamond-free, x has neighbors x_1,x_2,x_3 in H, where x_2x_3 is an edge and x_1x_2 and x_1x_3 are not. Let x_1' and x_1” be the neighbors of x_1 in H. Note that x has no neighbor in H∖{ x_1,x_1',x_1”,x_2,x_3} and it is adjacent to at most one vertex of { x_1',x_1”}.Suppose x_1∈ K. Then w.l.o.g. x_1'∉K. But then x_1' and x are pendant vertices of L(R) that have the same labels. Since { x,x_2,x_3} induce a triangle in L(R), x corresponds to a pendant edge of R whose one endnode is of degree at least 3, contradicting <ref>. Therefore x_1∉K, and hence it cannot be a center of a claw. Without loss of generality it follows that the neighbors of x in H are x_1',x_1,x_2,x_3 and none of them is in K. In particular, x is not a pendant vertex of L(R).Let e_x be the edge of R that corresponds to vertex x of L(R). Note that the endnodes of e_x are of degree at least 3 in R. So by <ref>, e_x cannot be contained in a 2-connected block of R. It follows that x is a cut vertex of L(R). Let C_1 and C_2 be connected components of L(R)∖ x. Then w.l.o.g. x_1',x_1∈ C_1 and x_2,x_3∈ C_2, and every path in B∖ x from { x_1',x_1} to { x_2,x_3} must go through K. It follows that H must have a chord, a contradiction. If B is a P-graph with special clique K = {v_1, …, v_k} and v a vertex of an internal segment of B, then there exists a hole H in B that contains v, some vertex v_i∈ K and two neighbors of v_i in B K.We view v as an edge of the skeleton R of B.The edge v belongs to a branch of R with ends x_1 and x_2.Let P_1 = x_1… y_1 and P_2 = x_2 … y_2 be the two paths whose existence is proved in Lemma <ref> applied to x_1 and x_2.Let i be the label of edges incident to y_1 and y_2.The hole whose existence is claimed is induced by v_i and the line graph of the union ofP_1, P_2, and the branch of R from x_1 to x_2.Let B be a P-graph with special clique K = {v_1, …, v_k}. Let K_1, K_2, K_3 ∈ K{ K} be three distinct big cliques. Then there exist three paths P_1 = v … u_1, P_2 = v … u_2 and P_3 = v … u_3, vertex-disjoint except at v, with no edges between them (except at v), such that v∈ K and for i∈{ 1,2,3}, K_i∩ P_i={ u_i}. Each of the cliques K_1, K_2 and K_3 is a set of edges from R that share a common vertex.This defines three branch vertices x_1, x_2 and x_3 in R. By Lemma <ref> there are vertex-disjointpaths from {x_1,x_2} to {y_1,y_2}, where y_1 and y_2 are two vertices of R incident with edges that have the same label say 1. We denote by X the set of all the vertices of degree 1 from R different from y_1 and y_2 (X is not empty by (i)).We now apply Lemma <ref> to {x_1, x_2, x_3} and {X,y_1, y_2}.If three vertex-disjoint paths exist (up to a permutation, say Q_1= x_1 … y_1, Q_2= x_2 … y_2 and Q_3 = x_3 … y_3, where y_3 ∈ X and w.l.o.g. y_3 has label 1 or 2), then we are done.Indeed, in L(R), this yields three chordless paths with no edges between them, ending at three vertices with labels 1, 1, 1 or 1, 1, 2. By adding v_1 or v_1, v_2, we obtain the three paths whose existence is claimed.We may therefore assume that the outcome of Lemma <ref> is a set C of at most two vertices that separates {x_1, x_2, x_3} and X∪{y_1, y_2}. This contradicts <ref> or <ref>.Let B be a P-graph with special clique K = {v_1, …, v_k}.Let S be a leaf segment of B, whose ends are in K and in K_2 ∈ K{ K}. Let K_1 ≠ K_2 be a clique in K{ K}.Then there exist three paths P_1 = v … u_1, P_2 = v … u_2 and P_S = v … u_S, vertex-disjoint except at v, with no edges between them (except at v and for one edge in K_2), such that v∈ K, u_S is the endnode of S in K_2, and for i∈{ 1,2}, K_i ∩ P_i={ u_i}. Moreover, P_S =S or P_S v = S.In skeleton R of B, the segment S corresponds to limb with a pendant edge e_S.Each of the cliques K_1 and K_2 is a set of edges from R that share a common vertex.This defines two vertices x_1 and x_2 in R.We suppose first that e_S has a label that is used only once in the skeleton R. We apply Lemma <ref> to x_1 and x_2. This yields paths P_1 and P_2 that have pendant edges with the same label, say 1. Then S, line graphs of P_1 and P_2 and vertex v_1, give the desired three paths.We now suppose that the label of e_s, say 1, is used for anotherpendant edge with a vertex y of degree 1.We denote by X the set of all degree 1 vertices of R, except y and the end of e_s.We apply Lemma <ref> to {x_1, x_2} and {X,y}.If two paths are obtained, note that they do not intersect S (because S is a limb), so by adding S to corresponding paths in B, we obtain the paths that we need. Otherwise, we obtain a cut vertex, that together with any vertex of S yields a cutset of size 2 that contradicts <ref>.Let B be a P-graph withspecial clique K = {v_1, …, v_k} such that k≥ 2.Let S_1 and S_2 be leaf segments of B that have a common endnode v_i in K, and let their other endnodes be in K_1 and K_2, respectively (K_1≠ K_2). Then there existpaths P_1=u'… u_1 and P_2=u”… u_2, vertex-disjointexcept maybe at a vertex of K (when u'=u”) and with no edges between them (except for one edge of K if u'≠ u”, or for edges incident to u' when u'=u”), such that for i∈{ 1,2}, K_i∩ P_i={ u_i}, u',u”∈ K∖{v_i} and v_i∉P_1∪ P_2.In skeleton R of B, the segments S_1 and S_2 correspond to limbs with pendant edges e_1 and e_2, respectively.Each of the cliques K_1 and K_2 is a set of edges from R that share a common vertex.This defines two vertices x_1 and x_2 in R.The label of e_1 and e_2 is i.We denote by X the set of all degree 1 vertices of R that are incident with an edge not labeled with i.We apply Menger's theorem to {x_1, x_2} and X (by <ref> and <ref> we have |X|≥ 2).If two paths are obtained, then we are done. Otherwise, we obtain a cut vertex x, that separates {x_1,x_2} from X. Since x_1 and x_2 are of degree 3 we may assume that x is an attaching vertex, which contradicts <ref>.Let B be a P-graph withspecial clique K = {v_1, …, v_k} such that k≥ 2. Let S_1 be a leaf segment with endnode v_i∈ K, and an endnode in K_1∈𝒦∖{ K}, and let K_2∈𝒦∖{ K,K_1}. Then there exist paths P_1=u_1… u' and P_2=u_2… u” vertex-disjoint except maybe at a vertex of K (when u'=u”) and with no edges between them (except for one edge of K if u'≠ u”, and for edges incident to u' when u'=u”), such that u',u”∈ K∖{v_i},v_i∉P_1∪ P_2,P_1∩ K_1={ u_1} and P_2∩ K_2={ u_2}.In skeleton R of B, the segment S_1 corresponds to a limb with pendant edge e_1. Each of the cliques K_1 and K_2 is a set of edges from R that share a common vertex.This defines two vertices x_1 and x_2 in R.The label of e_1 is i.We denote by X the set of all degree 1 vertices of R that are incident with an edge not labeled with i.We apply Menger's theorem to {x_1, x_2} and X (by <ref> and (xi) we have |X|≥ 2).If two paths are obtained, then we are done. Otherwise, we obtain a cut vertex x, that separates {x_1,x_2} from X. Since x_1 and x_2 are of degree 3 we may assume that x is an attaching vertex, which contradicts <ref>.Let B be a P-graph with special clique K={v_1}. If S is a leaf segment of B and S'an internal segment of B, with an endnode in K'∈𝒦 such that S∩ K'=∅, then there exists a pyramid Π contained in B, such that S and S' are contained in different paths of Π and |Π∩ K'|=2.Let R be the skeleton of B. Let P_S (resp. P_S') be the limb (resp. branch) of R that corresponds to S (resp. S'). Let x be the degree 1 vertex of P_S,let x_1 be the other endnode of P_S, and let y_1 and y_2 be the endnodes of P_S', such that edges incident to y_1 correspond to nodes of K'. Then x_1≠ y_1. Furthermore, let X be the set of all degree 1 vertices of R different from x.If in R there exists a vertex z that separates {y_1,y_2} from X, then for any internal vertex z' of P_S (it exists by <ref> and <ref>), the set {z,z'} is a cutset of R that contradicts (iv). So, by Menger's theorem there are vertex-disjoint paths P'=y_1… x' and P”=y_2… x”, where x',x”∈ X. Suppose that in R∖ y_1 there exists a path from x to (P'∪ P”)∖{y_1}, and let P”' be chosen such that it has the minimum length. Then L(P'∪ P”∪ P”'∪ P_S')∪{v_1} induces the desired pyramid.So, we may assume that y_1 is a cut vertex of R, such that x and (P'∪ P”)∖{y_1} are contained in different connected components of R∖ y_1. Let C_x be the connected component of R∖{y_1} that contains x, let e_x be the edge incident to x and let e_y be an edge of P_S'. By Lemma <ref> there exists a path P in R that contains edges e_x and e_y whose endnodesare of degree 1 in R.Note that P contains P_S'. Let x_1' be a node adjacent to x_1 that does not belong to P. Since x_1≠ y_1, we have {x_1,x_1'}⊆ C_x. Let us apply Lemma <ref> in graph C_x to {x_1,x_1'} and {X_1,x}, where X_1 is the set of all degree 1 (in R) nodes of C_x different from x (X_1 is non-empty, since otherwise for any internal vertex z' of P_S the set {z',y_1} is a cutset of R that contradicts<ref>). If vertex-disjointpaths P_1=P_S and P_1' are obtained, then L(P∪ P_1') and v_1 induce a desired pyramid Π. Otherwise, let z be a vertex of C_x that separates {x_1,x_1'} from X_1∪{x}. But then {z,y_1} is a cutset of R that contradicts <ref>. Let B be a P-graph with special clique K={v_1}. If S_1 and S_2 are leaf segments of B, then there exists a pyramid Π contained in B, such that S_1 and S_2 are contained in different paths of Π.Let x_1 (resp. x_2) be degree 1 vertex of skeleton R of B incident to pendant edge that corresponds to a vertex of S_1 (resp. S_2). Furthermore, letX be the set of all degree 1 vertices of R different from x_1 and x_2. Note that by <ref>, X≠∅. Let P' be a direct connection from {x_1,x_2} to X in R, and w.l.o.g. let x_1 be the neighbor of one endnode of P'. Let P” be a direct connection from x_2 to P'. Then L(P'∪ P”)∪{v_1} induces the desired pyramid.Let B be a P-graph with special clique K={v_1, … ,v_k}. Let v be the vertex of an internal segment of length 0, let K_1∈ K∖{ K} be such that v∈ K_1 and let u∈ K_1∖{ v }. Then B contains a pyramid Π =3PC(uvx,y) such that x∈ K_1 and y∈ K.Let R be the skeleton of B, and let e=x_1x_2 be an edge of R that corresponds to vertex v. Let x_1' be the neighbor of x_1 in R such that x_1x_1' corresponds to vertex u. Let P_1=x_1… y_1, P_1'=x_1x_1'… y_1' and P_2=x_2… y_2 be the three paths obtained by applying Lemma <ref> to x_1,x_1' and x_2. Then y_1,y_1' and y_2 are vertices of degree 1 in R incident with edges with at most two different labels, say i and j. It follows that L({x_1,x_2}∪ P_1∪ P_1'∪ P_2) and { v_i,v_j} induce the desired pyramid in B.§ ATTACHMENTS TO A P-GRAPHLet G be a (theta, wheel)-free graph. If H is a hole of G and v a node of G∖ H, then the attachment of v over H is a clique of size at most 2. In a P-graph B every pair of segments is contained in a hole. Also, every pair of vertices of B is contained in a hole.Follows directly from Lemma <ref> (note that every vertex of B is contained in a segment of B, and every segment contains a vertex that corresponds to an edge of skeleton R of B). Let G be a (theta, wheel, diamond)-free graph and B a P-graph contained in G. If v ∈ G∖ B, then either |N_B (v)|≤ 1 or N_B(v) is a maximal clique of B.Since G is diamond-free, it suffices to show that N_B(v) is a clique.Assume not and let v_1 and v_2 be non adjacent neighbors of v in B.By Lemma <ref>, v_1 and v_2 are contained in a hole H of B. But then H and v contradict Lemma <ref>. Let G be a (theta, wheel, diamond)-free graph and Π =3PC(x_1x_2x_3,y) be a pyramid contained in G. Then Π is a long pyramid and by Lemma <ref> it is a P-graph with special clique { y}.For i=1, 2, 3, we denote by S_i the branch of Π from y to x_i and we denote by y_i the neighbor of y on this path.By Lemma <ref> it follows that the attachment of a node v∈ G∖Π over Π is a clique of size at most 3. For i=1,2,3, we shall say that v is of Type i w.r.t. Π if |N_Π(v)|=i.We now define several kinds of paths that interact with Π. * A crossing of Π is a chordless path P=p_1… p_k in G∖Π of length at least 1, such that p_1 and p_k are of Type 1 or 2 w.r.t. Π, for some i,j∈{1,2,3}, i≠ j, N_Π(p_1)⊆ S_i, N_Π(p_k)⊆ S_j, p_1 has a neighbor in S_i∖{y}, p_k has a neighbor in S_j∖{y}, at least one of p_1,p_k has a neighbor in (S_i∪ S_j) ∖{ x_i,x_j} and no node of P∖{p_1,p_k} has a neighbor in Π. * Let P=p_1… p_k be a crossing of Π such that for some i,j∈{1,2,3}, i≠ j, N_Π(p_1)={ y_i} or { y_i,y}, p_k is of Type 2 w.r.t. Π and N_Π(p_k)⊆ S_j∖{y, y_j}.Moreover, if N_Π(p_1) ={ y_i} then S_i has length at least 3.Then we say that P is a crosspath of Π (from y_i to S_j).We also say that P is a y_i-crosspath of Π. * If P=p_1… p_k is a crossing of Π such that p_1 and p_k are of Type 2 w.r.t. Π and neither is adjacent to { y, y_1, y_2, y_3, x_1, x_2, x_3}, then P is a loose crossing of Π.A long pyramid with a loose crossing is a P-graph. To see this, consider a 1-skeleton made of a chordless cycle C together with three chordless paths P_1,P_2,P_3, all of length at least 2, such that for i∈{ 1,2,3}, P_i∩ C={ v_i}, and v_1,v_2,v_3 are pairwise distinct and nonadjacent. The three pendant edges of the paths receive label 1, and the special clique has size 1.A long pyramid with a crosspath is also a P-graph.The special clique K is {y_i, y} (when N_Π(p_1) ={ y_i}) or {y_i, y, p_1} (when N_Π(p_1) ={ y_i, y}), so it has size 2 or 3.It is easy to check that removing K yields the line graph of a tree that has two vertices of degree 3 and four pendant edges that receive labels 1, 1, 2, 2 when |K|=2 and 1, 1, 2, 3, when |K|=3. Let G be a (theta, wheel, diamond)-free graph.If P=p_1… p_k is a crossing of a Π =3PC(x_1x_2x_3,y) contained in G, then P is a crosspath or a loose crossing of Π. Assume w.l.o.g. that p_1 has a neighbor in S_1∖{ y}, and p_k in S_2∖{ y}.Not both p_1 and p_k can be adjacent to y, since otherwise N_Π(p_1)={ y_1,y} and N_Π(p_k) ={ y_2,y}, and hence S_1∪ S_2∪ P induces a wheel with center y.Suppose that both p_1 and p_k are of Type 2 w.r.t. Π. If p_1 is adjacent to y, then P is a crosspath, since otherwise p_k is adjacent to y_2 and not to y, and hence S_2∪ S_3∪ P induces a wheel with center y_2. So we may assume that neither p_1 nor p_k is adjacent to y. If p_1 is adjacent to y_1, then G[(Π∖{x_2})∪ P] contains a wheel with center y_1. So p_1 is not adjacent to y_1, and by symmetry p_k is not adjacent to y_2. If p_1 is adjacent to x_1, then G[(Π∖{y_2})∪ P] contains a wheel with center x_1. So p_1 is not adjacent to x_1, and by symmetry p_k is not adjacent to x_2. It follows that P is a loose crossing.Without loss of generality we may now assume that p_1 is of Type 1 w.r.t. Π. If p_k is also of Type 1, thenS_1∪ S_2∪ P induces a theta. So p_k is of Type 2. If p_1 is not adjacent to y_1, then G[(Π∖{ x_2} )∪ P] contains a 3PC(y,y), where y is the only neighbor of p_1 on Π. So p_1 is adjacent to y_1. Since S_1∪ S_2∪ P cannot induce a wheel with center y, p_k is not adjacent to y. Since S_2∪ S_3∪ P∪{y_1} cannot induce a wheel with center y_2, N_Π (p_k)⊆ S_2∖{ y,y_2}. If S_1 is of length 2, then G[(Π∖{ y_2} )∪ P] contains a wheel with center x_1. Therefore S_1 is of length at least 3, and hence P is a crosspath. Let G be a (theta, wheel, diamond)-free graph.If G contains a pyramid Π with a crossing P, then G[Π∪ P] is a P-graph.Follows from Lemma <ref> and the fact already mentioned that a pyramid together with a loose crossing or a crosspath is a P-graph. Let S be a segment of a P-graph B such that its endnodes are in K_1 and K_2. Then we say that S∪ K_1∪ K_2 is an extended segment of B. Let B be a P-graph withspecial clique K which is contained in a (theta,wheel,diamond)-free graph G. Let P=u… v be a path in G∖ B whose interior nodes have no neighbors in B and one of the following holds:(1) N_B(u) and N_B(v) are cliques of size at least 2 in B∖ K which are not contained in the same extended segment of B. (2) N_B(u)=K, where |K|≥ 2, and N_B(v) is a clique of size at least 2 which is in B∖ K, but not in an extended clique segment of B. (3) N_B(u)={w}⊆ K, and N_B(v) is a clique of size at least 2 in B∖ K which is not in a extended segment of B incident with w.Then G[B∪ P] is a P-graph contained in G.Let K={v_1,…,v_k} and let R be the skeleton of B. In all three cases neighbors of v in B are in fact in L(R), and they correspond to some edges of R all incident to a single vertexk_2∈ R. By Lemma <ref>, v is adjacent to all vertices that correspond to edges incident to k_2. We now consider each of the cases. (1) Let k_1 be the vertex of R whose incident edges correspond to vertices of the clique N_B(u) in L(R). Note that by Lemma <ref>, u is adjacent to all vertices that correspond to edges incident to k_1. Construct graph R' from R by adding a branch P_R between k_1 and k_2, of length one more than the length of P. We prove that R' is a k-skeleton.By Lemma <ref>, it suffices to check that all conditions other than <ref> and <ref> are met.Since P is of length at least 1, P_R is of length at least 2, and thus (i) holds.Since N_B(u) and N_B(v) are not contained in the same extended segment of B,no branch of R contains both k_1 and k_2, and hence <ref> holds.Note that R and R' have the same degree 1 vertices and the same limbs.It follows that <ref>, <ref>, <ref> and <ref> hold for R'.Let x be a cut vertex of R'.Since R is connected, x is not an internal vertex of P_R.Hence, x is also a cut vertex of R and every component of R' x contains a union of components of R x.It follows that <ref> holds.Also, every x-petal of R' is a union of some x-petals of R and some vertices of P_R, and therefore <ref> holds.To prove <ref> let {a, b} be a cutset of R'.If a and b are in the interior of P_R, one component of R'{a, b} is a chordless path from a to b, and the other contains all the vertices of R' of degree 1, so <ref> holds.If one of a or b, say a, is in the interior of P_R, and the other (so, b) is not, then b is acut vertex of R. Also, every component of R'{a, b} contains a component of R b.Hence <ref> holds because <ref> holds for R.Finally, if none of a and b is in the interior of P_R, then {a, b} is also a cutset of R, and every components of R'{a, b} contains a component of R{a, b}.Therefore, <ref> holds for R' because it holds for R. Thus <ref> holds, and our claim is proven. (2) Construct graph R' from R by adding a chordless path P_R of the same length as P, whose one endnode is k_2 and the remaining nodes are new.Note that pendant edges of R are also pendant edges of R', and R' has one new pendant edge (the one incident to the vertex of degree 1 in R' that is in P_R). Let us assign label k+1 to the new pendant edge. We claim that R' is a skeleton.By Lemma <ref>, there is no need to check <ref> and <ref>.Since P_R is a limb, (i), <ref>, <ref>, <ref> and <ref> hold for R' because they hold for R and since in this case k≥ 2.Let us show that <ref> holds.It could be that the limb that we add to R to build R' is in fact parallel to a limb Q of R, that corresponds to a clique segment S' of B. If the label of pendant edge of Q is used only once, then N_B(v) is contained in an extendedclique segment of B (namely extended segment of S'), a contradiction. So <ref> holds.The conditions <ref>, <ref> and <ref> hold for R' because they hold for R.Indeed, in R', we added a limb, this only possibly adds a vertex of degree 1 to a component,making the condition easier to satisfy. (3) Let w=v_i.We build a path P_R of the same length as P and we consider the graph R' obtained from R by attaching P_R at k_2.Hence, in P_R there is a pendant edge, and we give it label i.We claim that R' is a skeleton.By Lemma <ref>, there is no need to check <ref> and <ref>.Since P_R is a limb, (i), <ref>, <ref>, <ref> and <ref> hold for R' because they hold for R.Condition <ref> also holds, since the limb that we add to build R' has pendant edge with label i that is now used at least twice, and it is not parallel to some other limb with pendant edge i by the condition of the lemma.The conditions <ref>, <ref> and <ref> hold for R' because they hold for R.Indeed, in R' we added a limb, which only possibly adds a vertex of degree 1 to a component,making the condition easier to satisfy.Let G be a (theta, wheel, diamond)-free graph, and let B be the P-graph contained in G with special clique K={v_1,…,v_k} and skeleton R, such that k is maximum, and among all P-graphs contained in G and with special clique of size k, B has the maximum number of segments.Let P=u… v be a chordless path in G∖ B such that u and v both have neighbors in B and no interior node of P has a neighbor in B.Then one of the following holds: (1) N_B(P)⊆ K', where K'∈ K.(2) There exists a segment S of B, of length at least 1, whose endnodes are in K_1∪ K_2 where K_1,K_2∈ K, such that N_B(P)⊆ K_1∪ K_2∪ S. Moreover, if u (resp. v) has a neighbor in K_i ∖ S, for some i∈{ 1,2}, then u (resp. v) is complete to K_i. Before proving the theorem, note that in the proof, conclusion (2) can be replaced by a weaker conclusion :(2') There exists a segment S of B, of length at least 1, whose endnodes are in K_1∪ K_2 where K_1,K_2∈ K, such that N_B(P)⊆ K_1∪ K_2∪ S. Indeed, if (2') is satisfied, then (1) or (2) issatisfied.Let us prove this.Suppose that (2') holds, but neither (1) nor(2) does.Up to symmetry, and by Lemma <ref>, this means that N_B(u) is asingle vertex u' of K_1 ∖ S.If N_B(v) is also asingle vertex v', then by Lemma <ref>, P togetherwith a hole that goes through u' and v' forms a theta (note that since (1) does not hold, v'∈ (S∪ K_2)∖ K_1and hence since R has no parallel branches by <ref>, u'v' is not an edge).By Lemma <ref>, we may therefore assume that N_B(v)=K_2 or N_B(v) is a clique of size 2 in S. We first suppose that K∉{ K_1,K_2}.In R, N_B(u) is an edge y_1y_1', where y_1' is a branch vertex and S corresponds to a branchP'=y_1'… y_2'. We apply Lemma <ref> to y_1,y_1' and y_2'.Let P_1,P_2 and P_3 be the three paths obtained and suppose that label i is used on pendant edges of twoof these paths. Then the graph induced by L(P_1∪ P_2∪ P_3) together with S∖ K_1, P and Kcontains a 3PC(u',v_i) (note that by <ref>, u'v_i is not an edge). Next suppose that K_1=K and let u'=v_i.First observe that if N_B(v)=K_2 and there exists a segment S' of B with endnode v_i and an endnode in K_2,then P satisfies (2) w.r.t. S'. So this cannot happen. It follows that if N_B(v)∩ K=∅ then by part (3)of Lemma <ref>, the maximality of B is contradicted.So let v_j∈ N_B(v)∩ K, where v_j≠ v_i, and let S' be a segment of B with endnode v_i.Let Q be a direct connection from S' to S in B∖ K. Then G[S∪ S'∪ P∪ Q] is a wheel with center v_j,a contradiction. Therefore K_2=K.First suppose that u' is a vertex of an internal segment of B. Then by Lemma <ref>, there exists a hole Hthat contains u' and a vertex v_j∈ K such that neighbors of v_j in H are in B∖ K.If S is not contained in H, then G[H∪ P∪ (S∖ K_1)] contains a 3PC(u',v_j) (note that since u'belongs to an internal segment of B, u'v_j is not an edge).So S is contained in H, and hence v_j is an endnode of S. If N_B(v)=K then G[H∪ P] is a theta.So N_B(v)≠ K.In R, u' is an edge y_1y_1', where y_1' is a branch vertex, and S corresponds to a limb P'=y_1'… y_2'.Let X be the set of all degree 1 vertices of R incident with pendant edges labeled with j not including y_2'(note that X is nonempty) and Y the set of all other degree 1 vertices of R not including y_2'.If in (R∖ P') ∪{ y_1'} there are vertex-disjoint paths P_1 and P_2 from { y_1',y_1} to { X,Y},then G[L(P_1∪ P_2)∪ P∪ (S∖ K_1)∪{ u'}] contains a 3PC(u',v_j).So, by Lemma <ref>, there is a vertex x in R that separates { y_1',y_1} from X∪ Y in (R∖ P')∪{ y_1'},and therefore { y_2',x} is a cutset of R that contradicts <ref>. It follows that u' is an endnode of a leaf segment S' of B.Since (2) does not hold for P and S', N_B(v)≠ K and hence N_B(v) isa clique of size 2 in S.Let v_i (resp. v_j) be the endnode of S (resp. S') in K. Suppose v_i=v_j.Then by <ref>, R has no branches, so by (i), G[(B∖ (S∩ K_1)) ∪ P] contains a3PC(u',v_i) (note that u'v_i is not an edge by <ref>).So v_i≠ v_j. By <ref> there is a segmentS”∉{ S,S'} with an endnode in { v_i,v_j}.Note that S” does not have an endnode in K_1. Let Q be a direct connection from S” to K_1in B∖ K. Then G[(S∖ K_1)∪ S'∪ S”∪ P∪ Q] either contains a 3PC(u',v_i) (if S” has endnode v_i) or 3PC(u',v_j) (if S” has endnode v_j, note that in this case by <ref>, u'v_j is not an edge). Therefore, if (2') holds then (1) or (2) holds. We are now back to the main proof.Suppose the conclusion of thetheorem fails to be true.ByLemma <ref>, it suffices to consider the followingcases.Case 1: For some K_1, K_2∈ K∖{ K},N_B(u)=K_1 and N_B(v)=K_2.Since (1) does not hold, K_1≠ K_2. Let us first prove that no segment of B has endnodes in K_1∪ K_2. Suppose to the contrary that some segment S of B has endnodesin K_1∪ K_2.Since (2') does not hold, S is of length 0,say S=x. So S is an internal segment of B.Let e_x be theedge of R that corresponds to x.By Lemma <ref>, e_x isa cut edge of R, and hence x is a cut vertex of L(R).Fori=1,2, let C_i be the connected component of L(R)∖ xthat contains K_i∖ x. Note that the endnodes of e_x inR are cut vertices of R, and hence by <ref>, C_i has apendant vertex, for i=1,2.It follows that B contains achordless wz-path Q, where w∈ K_1∖ x,z∈ K_2∖ x and no interior node of Q has a neighbor inK_1∪ K_2.But then P∪ Q∪{ x} induces a wheel withcenter x. Therefore, no segment of B has an endnode in K_1∪ K_2.Now, by part (1) of Lemma <ref>, this contradicts the maximality of B. Case 2: For some K_1∈ K∖{ K}, N_B(u)=K_1 and N_B(v)=K. Since (2') does not hold, there is no (leaf) segment with endnodes in K_1 and K, and so by parts (2) and (3) of Lemma <ref> and maximality of B, this case is impossible. Case 3: For some segment S of B, N_B(u)=K and N_B(v)⊆ S. Since (2') does not hold, S is an internal segment of B.Let v' be a neighbor of v in S.Apply Lemma <ref> to B and v'.This provides a hole H in B that contains v' and a single node of K. Note that H contains S because S is a segment.If v' is the only neighbor of v in S, then H and P form a theta, a contradiction.So, by Lemma <ref>, for some vertex v” of S adjacent to v', N_B(v)={ v',v”}. By parts (2) and (3) of Lemma <ref> this contradicts the maximality of B. Case 4: For some K_1∈ K∖{ K} and some internal segment S of B, N_B(u)=K_1 and N_B(v)⊆ S. Let K_2 and K_3 be the end cliques of S. Since (1) and (2') do not hold, K_1∉{ K_2,K_3}. We apply Lemma <ref> to K_1, K_2 and K_3.This provides three paths P_1, P_2 and P_3. If N_B(v)={ v'} then P_1,P_2,P_3,S and Pinduce a theta. So by Lemma <ref> N_B(v)={ v',v”} where v' and v” are two adjacent vertices of S. By part (1) of Lemma <ref> this contradicts the maximality of B. Case 5: For some K_1∈ K∖{ K} and some leaf segment S of B, N_B(u)=K_1 and N_B(v)⊆ S. Let the endnodes of S be in cliques K and K_2∈ K∖{ K}.Since S is a leaf segment of B, it is of length at least 1. Since (2') does not hold, K_1≠ K_2.Let v' be a neighbor of v in S, and let P_1=w_1… w, P_2=w_2… w and P_S=w_s… w be paths obtained when Lemma <ref> is applied to segment S and clique K_1.First, let us assume that N_B(v)={v'}. If v'≠ w and v' is not adjacent to w, then G[P_1∪ P_2∪ P_S∪ P] induces a 3PC(v',w), a contradiction. So, v'=w or v'w is an edge. If v'∈ K, then by part (3) of Lemma <ref> and maximality of B, there is a segment S' with one endnode in K_1 and the other v'. But then P and S' satisfy condition (2'). So, v'∉K, and hence v'w is an edge. Suppose k=1. Let z be a node of K_1 that belongs to an internal segment of B (note that since K_1≠ K_2, and since R is connected by <ref>, it follows that R has a branch and z exists by <ref>). By Lemma <ref> there exists a pyramid Π contained in B such that S and z belong to different paths of Π and |Π∩ K_1|=2. So, N_Π(u) is an edge of a path of Π that contains z. Note that since G is wheel-free, Π is a long pyramid and by Lemma <ref> P is a crosspath of Π. But then G[Π∪ P] is a P-graph with special clique of size greater than 1, contradicting our choice of B (since k=1). Therefore, k>1. Let Q_1 and Q_2 be paths obtained when Lemma <ref> is applied to S and K_1. Then G[Q_1∪ Q_2∪ S∪ P] induces a theta or a wheel, a contradiction. So, by Lemma <ref>, N_B(v) is a clique of size 2.If N_B(v)∩ K=∅, then, by (1) of Lemma <ref>, we have a contradiction to the maximality of B. So, N_B(v)={v',v”}, where v”∈ K. If v”≠ w, then G[P_1∪ P_2∪ P_S∪ P] induces a wheel, a contradiction. So, v”=w. If k=1, then by Lemma <ref> there exists a pyramid Π, contained in B, such that S and z are in different paths of Π, where z is a node of K_1 that belongs to an internal segment of B (it exists by the same argument as in the previous paragraph). Note that w is the center of the claw of Π. But then G[Π∪ P] is a P-graph whose special clique is of size 3, contradicting our choice of B. So k>1. Let Q_1 and Q_2 be paths obtained when Lemma <ref> is applied to S and a node z∈ K_1 that is on an internal segment of B. Then G[Q_1∪ Q_2∪ S∪ P] induces a wheel, a contradiction. Case 6: For some distinct segments S and S' of B, N_B(u)⊆ S and N_B(v)⊆ S'. Let K_1 and K_2 (resp. K_3 and K_4) be the end cliques of S (resp. S'). We divide this case in several subcases.Case 6.1: K∉{K_1,K_2,K_3,K_4}.We may assume that K_3∉{K_1,K_2}. Let P_1, P_2 and P_3 be the 3 paths obtained by applying Lemma <ref> to K_1, K_2 and K_3. Suppose that N_B(u) is a single vertex u'. Since (2') does not hold, v has a neighbor in S'∖ (K_1∪ K_2). But then G[P_1∪ P_2∪ P_3∪ P∪ S∪ (S'∖(K_1∪ K_2))] contains a theta. So, by Lemma <ref>, N_B(u) is a clique of size 2 in S, and similarly N_B(v) is a clique of size 2 in S'. By (1) of Lemma <ref>, this contradicts the maximality of B.Case 6.2: K_4=K and K∉{K_1,K_2}.Case 6.2.1: k=1.By Lemma <ref>, B containsa pyramid Π =3PC(x_1x_2x_3,v_1) such that S and S' are contained in different paths of Π. By part (1) of Lemma <ref>, P cannot be a loose crossing of Π. So by Lemma <ref>, P is a crosspath of Π. But this contradicts our choice of B since k=1. Case 6.2.2: k≥ 2.For i∈{ 1,2}, let x_i be the endnode of S that is in K_i, and let v_i and v_S' be the endnodes of S'. First suppose that K_2=K_3. Let P_1=w… w_1, P_2=w… w_2 and P_S'=w… v_S' be the three paths obtained by applying Lemma <ref> to S' and K_1 (where for i∈{ 1,2}, P_i∩ K_i={ w_i}). Then G[P_1∪ P_2∪ P_S'∪ S] is a pyramid Π =3PC(x_2w_2v_S',w), and S and S' belong to different paths of Π. Suppose v_i=w and N_B(v)=v_i. If u has a unique neighbor u' in S, then G[Π∪ P] contains a 3PC(u',w), and otherwise by part (3) of Lemma <ref> our choice of B is contradicted. So either v_i≠ w or N_B (v)≠{ v_i}. But then by Lemma <ref>, P is a crosspath or a loose crossing of Π, and therefore by Lemma <ref> our choice of B is contradicted. So by symmetry, K_3∉{ K_1,K_2}. Let P_1=w… w_1, P_2=w… w_2 and P_3=w… w_3 be the three paths obtained by applying Lemma <ref> to K_1, K_2 and K_3 (so w∈ K andfor i∈{ 1,2,3}, P_i∩ K_i={ w_i}). Let Q be a direct connection from K_3 to P_1∪ P_2 in B∖ K and H a hole in G[P_1∪ P_2∪ P_3∪ S∪ S'∪ Q] that contains S and S'. Suppose N_B(v)={ w}. If N_B(u)={ u'}, then G[H∪ P] contains a 3PC(u',w). So N_B(u) is a clique of size 2 in S, and hence by Lemma <ref> our choice of B is contradicted. So N_B(v)≠{ w}. Now, let us assume that v_i≠ w and that one of the paths P_1 and P_2 contains a vertex from K∖{w,v_i}. Note that then P_3∩ S'=∅. Let Π' be a pyramid contained in G[P_1∪ P_2∪ P_3∪ S∪ Q] (this pyramid contains S and its claw has center w). Then G[P∪ P_3∪ Q∪ S'] contains a crossing of Π' with an endnode in u, and hence u has two neighbors in S (since u is not adjacent to w). If N_B(v)={v'}≠{v_i}, then G[P_1∪ P_2∪ P_3∪ S∪ S'∪ P] contains a 3PC(u',w), and if N_B(v)={v_i}, then our choice of B is contradicted by Lemma <ref>. So, we may assume that N_B(v)={v',v_i},since otherwiseour choice of B is contradicted by Lemma <ref>. But then G[P_1∪ P_2∪ S∪ S'∪ P] contains a wheel with center v_i, a contradiction.So v_i=w orP_1∩ K,P_2∩ K∈{{w},{w,v_i}}. Then G[P_1∪ P_2∪ P_3∪ S∪ S'∪ Q] contains a pyramid Π (whose claw has center w or v_i), such that S and S' belong to different paths of Π. By our choice of B and Lemma <ref>, P cannot be a loose crossing of Π. So, by Lemma <ref>, P is a crosspath of Π. If the center of the claw of Π is v_i and v_i≠ w, then G[P_1∪ P_2∪ P_3∪ S∪ S'∪ P] contains a theta or a wheel, a contradiction. So, the center of the claw of Π is w. Also w=v_i, since otherwise our choice of B is contradicted by Lemma <ref>. This implies that S' is a claw segment of B. Let Q_1 and Q_2 be the paths obtained when Lemma <ref> is applied to K_1 and S' (we assume that Q_1∩ K_1≠∅). Furthermore, if Q_1 does not contain S, then we can extend Q_1 such that it containsone neighbor of u and such that we do not introduce edges between this new path and Q_2. But then, G[Q_1∪ Q_2∪ S'∪ P] contains a wheel or a theta, a contradiction. Case 6.3: K_2=K_4=K and K_1≠ K_3.Let v_i (resp. v_j) be the endnode of S (resp. S') in K, and let x_S (resp. x_S') be the otherendnode of S (resp. S').Case 6.3.1: v_i=v_j.First, let k=1. By Lemma <ref> B contains a pyramid Π =3PC(x_1x_2x_3,v_i) such that S and S' are contained in different paths of Π. Since P does not satisfy (1) and does not satisfy (2') w.r.t. S nor w.r.t. S', P is a crossing of Π. By the choice of B and since k=1, P cannot be a crosspath of Π. So by Lemma <ref>, P is a loose crossing of Π. But then by part (1) of Lemma <ref>, our choice of B is contradicted.So, let k≥ 2. Let P_1 and P_2 be paths obtained when Lemma <ref> is applied to S and S'. Since P does not satisfy (2'), nodeu (resp. v) has a neighbor in S∖{v_i} (resp. S'∖{v_i}). If u or v is adjacent to v_i, then G[S∪ S'∪ P∪ P_1∪ P_2] contains a wheel with center v_i. Therefore, neither u nor v is adjacent to v_i. Suppose that u has the unique neighbor u' in S. If u'v_i is not an edge, then G[S∪ S'∪ P∪ P_1] contains a 3PC(u',v_i). If u'v_i is an edge, then G[S∪ S'∪ P∪ P_1∪ P_2] contains a wheel with center v_ior a theta. So by Lemma <ref>, N_B(u) is a clique of size 2 that belongs to S∖ v_i, and by symmetry N_B(v) is a clique of size 2 that belongs to S' ∖ v_i. By (1) of Lemma <ref>, this contradicts the maximality of B.Case 6.3.2: v_i≠ v_j.In particular, k≥ 2. First suppose that S and S' are both clique segments of B. Let P_1=w… w_1, P_2=w… w_2 and P_S'=w… w_S' be the three paths obtained by applying Lemma <ref> to S' and K_1. So w∉{v_i,v_j}. Since (2') does not hold u (resp. v) has a neighbor in S∖{v_i} (resp. S'∖{v_j}). Let u' be a neighbor of u in S∖{v_i}. If either N_B (u)={ u'} or u is adjacent to v_i, then G[P_1∪ P_2∪ S∪ (S'∖{ v_j} )] contains a wheel with center v_i or a 3PC(u',w). So by Lemma <ref>, N_B(u) is a clique of size 2 in S∖ K, and by symmetry N_B(v) is a clique of size 2 in S'∖ K. But then by (1) of Lemma <ref> our choice of B is contradicted.So w.l.o.g. we may assume that S is a claw segment. Let Q be a direct connection from K_1 to K_3 in B∖ K. Let S_1 be a segment of B distinct from S that has endnode v_i. Let Q_1 be a direct connection from S_1 to Q in B∖ K. Then G[S∪ S'∪ S_1∪ Q∪ Q_1] is a pyramid Π =3PC(x_1x_2x_3,v_i), in which S and S' are contained in different paths of Π. P cannot be a loose crossing of Π, since otherwise by (1) of Lemma <ref> our choice of B is contradicted. Therefore by Lemma <ref>, P is a v_i'-crosspath of Π, where v_i' is the neighbor of v_i in S (since v has a neighbor in S'∖{v_j}). In particular, uv_i' is an edge and N_B(u)⊆{ v_i',v_i}. By <ref> there exists a leaf segment S_2 of B with endnode v_k such that either v_k∈ K∖{ v_i,v_j}, or v_k=v_j and S_2≠ S'. Let Q_2 be a direct connection from S_2 to Π. Then G[Π∪ S_2∪ Q_2] contains a 3PC(v_i',v_j) (if N_B(u)={v_i'} and j≠ k) or a wheel with center v_i (otherwise). Case 6.4: K_2=K_4=K and K_1=K_3.Let v_i (resp. v_j) be the endnode of S (resp. S') in K, and let x_S (resp. x_S') be the otherendnode of S (resp. S'). Case 6.4.1: k=1.Then by <ref> R has no branches. By (i) B contains a pyramid Π =3PC(x_Sx_S'x,v_1) where S and S' are paths of Π. Since P does not satisfy (1) and does not satisfy (2') w.r.t. S nor w.r.t. S', P is a crossing of Π. By the choice of B and since k=1, P cannot be a crosspath of Π. So by Lemma <ref>, P is a loose crossing of Π. But then by part (1) of Lemma <ref>, our choice of B is contradicted.Case 6.4.2: k≥ 2.Then by <ref>, v_i≠ v_j and w.l.o.g. v_i is an endnode of a leaf segment S_1≠ S. Let Q be a direct connection from S_1 to K_1 in B∖ K. Then S,S',S_1 and Q induce a pyramid Π =3PC(x_Sx_S'x,v_i) (where x is an endnode of Q) such that S and S'v_i are paths of Π. Since P does not satisfy (1) and it does not satisfy (2') w.r.t. S nor w.r.t. S', P is a crossing of Π. By (1) of Lemma <ref> and our choice of B, P cannot be a loose crossing of Π. So by Lemma <ref>, P is a crosspath of Π. If P is a v_j-crosspath of Π then by part (3) of Lemma <ref>, our choice of B is contradicted. So for the neighbor v_i' of v_i in S, P is a v_i'-crosspath of Π. In particular, u is adjacent to v_i' and N_B (u)⊆{ v_i,v_i'}, and N_B (v) is a clique of size 2 of S'∖{v_j}. By <ref> there exists a leaf segment S_2 with endnode v_k∈ K∖{ v_i} such that S_2≠ S'. Let Q_2 be a direct connection from S_2 to Π∖ (S ∪ S' ∪ K). But then G[(Π∖{ x_S'} )∪ S_2∪ Q_2] either contains a 3PC(v_i',v_j) (if k=j and uv_i is not an edge) or a wheel with center v_i (otherwise).Let Π=3PC(x_1x_2x_3,y) be a pyramid contained in a graph G. A hat of Π is a chordless path P=p_1… p_k in G ∖Π such that p_1 and p_k both have a single neighbor in Π and they are adjacent to different nodes of { x_1,x_2,x_3}, and no interior node of P has a neighbor in Π. Let G be a (theta, wheel)-free graph. If G contains a pyramid with a hat, then G has a clique cutset.Let P=p_1… p_k be a hat of Π=3PC(x_1x_2x_3,y) contained in G, with w.l.o.g. N_Π(p_1)={ x_1} and N_Π(p_k)={ x_2}. Assume that G does not have a clique cutset. Then by Lemma <ref>, G is diamond-free. Let S be the set comprised of { x_1,x_2,x_3} and all nodes u∈ G∖Π such that N_Π(u)={ x_1,x_2,x_3}. Since G is diamond-free, S is a clique. Let Q=q_1… q_l be a direct connection from P to Π∖{ x_1,x_2,x_3} in G ∖ S. We may assume w.l.o.g. that a hat P and direct connection Q are chosen so that |V(P)∪ V(Q)| is minimized.By Lemma <ref>, q_l either has a single neighbor in Π or N_Π (q_l) are two adjacent nodes of a path of Π. If a node q_i, i<l, is adjacent to a node of { x_1,x_2,x_3}, then by definition of Q, q_i has a single neighbor in Π. If at least two nodes of { x_1,x_2,x_3} have a neighbor in Q∖ q_l, then a subpath of Q∖ q_l is a hat of Π, contradicting the minimality of P∪ Q. So at most one node of { x_1,x_2,x_3} has a neighbor in Q∖ q_l. Suppose x_i, for some i∈{ 1,2,3}, has a neighbor in Q∖ q_l, and let q_t be such a neighbor with highest index. Then N_Π (q_l)⊆ S_i, since otherwise q_t… q_l is a crossing of Π that contradicts Lemma <ref>. If i=3 then a subpath of (P∖ p_k) ∪ Q or (P∖ p_1) ∪ Q is a hat of Π, contradicting the minimality of P∪ Q. So w.l.o.g. i=1. But then G[(Π∖ y_2)∪ P∪ Q] contains a wheel with center x_1. Therefore, no node of { x_1,x_2,x_3} has a neighbor in Q∖ q_l.Without loss of generality we may assume that P∪{ q_1} contains a chordless path P' from p_1 to q_1 that does not contain p_k. Then N_Π (q_l)⊆ S_1, since otherwise the path induced by P'∪ Q is a crossing of Π that contradicts Lemma <ref>. If N_Π (q_l)={ y} then P'∪ Q∪ S_1∪ S_3 induces a 3PC(x_1,y). So q_l has a neighbor in S_1∖{ x_1,y}. If p_1 is the unique neighbor of q_1 in P, then G[P∪ Q∪ (Π∖ y_2)] contains a wheel with center x_1. So P∪{ q_1} must contain a chordless path P” from p_k to q_1 that does not contain p_1. But then the path induced by P”∪ Q is a crossing of Π that contradicts Lemma <ref>.§ PROOF OF THEOREM <REF> A strip is a triple (H, A, A') that satisfies the following: (i) H is a graph and A and A' are disjoint non-empty cliques of H;(ii) every vertex of H is contained in a chordless path of H whose one endnode is in A, the other is in A', and no interior node is in A∪ A' (such a path is called an AA'-rung). Let B be a P-graph with special clique K, and let V_0 be the set of all vertices of B that are the unique vertex of some segment of length zero. A strip system 𝒮 is any graph obtained from B as follows:* for every segment S = u … v of B of length at least 1, let (H_S, Q_u, S, Q_v, S) be a strip, such that Q_u,S∩ S={u} and Q_v,S∩ S={v}; * V(𝒮) is the union of vertices of H_S, for all segments S of B of length at least 1, and V_0; * if S=u… v, u∈ K, is a claw segment of B, then Q_u,S={u}; * for segments S and S' of length at least 1, if S∩ S'=∅, then V(H_S)∩ V(H_S')=∅; * a clique Q_x, S is complete to a clique Q_x', S' whenever x and x' are in the same clique of K; * a clique Q_x, S is complete to x' whenever x and x' are in the same clique of K and x'∈ V_0;* these are the only edges of the strip system. Furthermore, for a clique K_1∈𝒦, we denote Q_K_1=⋃_u∈ K_1Q_u,S∪ K_1 (where S is a segment of length at least 1 that contains u).Note that any P-graph can be seen as a strip system, where every segment of length at least 1 is replaced by a strip equal to the segment.So, strip system can be seen as a way to thicken a P-graph.In the other direction, consider a graph T induced by V_0 and vertices of one rung from every strip of a strip system S. We say that T is a template of S.Note that in particular B is a strip system with unique template, namely B.Let G be a (theta, wheel)-free graph. Then every template of a strip system of G is a P-graph.We claim that given a P-graph B and a strip system obtained from B (that is contained in G), replacing one segment S=u… v of B by a corresponding rung S'=u'… v' yields another P-graph B'. The lemma then follows from this claim by induction on the number of segments. So let us prove the claim.Let K be the special clique of B and R its skeleton. If u or v, say u, is in K, then let K'={u'}∪ K∖{u}; otherwise let K'=K.By <cit.> a graph is (claw,diamond)-free if and only if it is the line graph of a triangle-free graph. So, B∖ K is (claw,diamond)-free, and hence the same holds for B'∖ K', i.e. B'∖ K' is the line graph of a triangle-free graph R'. Observe that R' can be obtained from R by changing the length of a single branch or limb. Furthermore, in this way no branch of length 1 is obtained since the two cliques of any strip are disjoint. Therefore, R' satisfies all conditions of the definition of a skeleton, except possibly the ones that are concerned with the lengths of the limbs. So, we only need to check that R' satisfies <ref>, which is true by Lemma <ref>.We are ready to prove Theorem <ref>. Proof of Theorem <ref>: Let G be a (theta, wheel)-free graph, and assume that G does not have a clique cutset and that it is not a line graph of triangle-free chordless graph.By Lemma <ref>, G is diamond-free and by Theorem <ref>, G contains a pyramid, and hence a long pyramid (since G is wheel-free). So, by Lemma <ref>, G contains a P-graph. Let B be a P-graph contained in G with maximum size of the special clique K, say |K|=k, and such that out of all P-graphs with special clique of size k it has the maximum number of segments. Let 𝒦 be the set that includes all big cliques of B and K, and let R be the skeleton of B. Furthermore, let 𝒮 be a maximal (w.r.t. inclusion) strip system obtained from B. Claim 1. For every w∈ G∖𝒮 either for someclique K_1∈𝒦,N_𝒮(w)=Q_K_1, or for some segment S of B of length at least 1, N_𝒮(w)⊆ H_S. Proof of Claim 1. Suppose not. Observe that if for some K_1∈𝒦, w has two distinct neighbors in Q_K_1, then since G is diamond-free, w is complete to Q_K_1.First suppose that w is adjacent to a vertex v∈ V_0. By Lemma <ref>, X=(N_G(v)∖ ({ w }∪𝒮)) ∪{ v} is not a star cutset of G, so there exists a chordless path P=w… w' in G∖ (𝒮∪ X) such that w' has a neighbor u' in 𝒮∖{ v } and no interior node of P has a neighbor in 𝒮. By definition of a stripand 𝒮, there is a template of 𝒮 that contains u' and v. By Lemma <ref> we may assume w.l.o.g. that B contains u' and v. By Lemma <ref> applied to P and B, and since w' is not adjacent to v, N_B(P)⊆ K'∈𝒦. In particular, K'∈𝒦∖{ K}, u',v∈ K' and N_B(w')={ u'}. By Lemma <ref>, B contains a pyramid Π=3PC(u'vx,y), with x∈ K' and y∈ K. If N_Π (w)={ v} then P is a hat of Π, contradicting Lemma <ref>. So there exists v'∈ N_Π (w)∖{ v}. By Lemma <ref>, N_Π (w) is a maximal clique of Π. If N_Π (w) ≠{ u' ,v,x} then G[Π∪ P] contains a wheel with center v. So N_Π (w) = { u' ,v,x}, and hence w is complete to Q_K'. It follows that w has a neighbor u” in 𝒮∖ Q_K'. Let B' be a template of 𝒮 that contains v and u”. By Lemma <ref>, B' is a P-graph. By Lemma <ref>, N_B' (w) is a maximal clique of B', and in particular vu” is an edge. It follows that for some K”∈𝒦∖{ K',K}, w is complete to Q_K”. By Lemma <ref> applied to B and v, there exists a hole H in B that contains v, and hence it contains a vertex of K'∖{ v} and a vertex of K”∖{ v}. But then (H,w) is a wheel, a contradiction.Therefore, w is not adjacent to a vertex of V_0. It follows that there exist distinct segments S and S' of B, both of length at least 1, such that w has a neighbor u in H_S,a neighbor v∈ H_S', and there is no clique K_1∈𝒦 such that u and v are both in Q_K_1. Let B' be a template of 𝒮 that contains u and v (it exists by definition of a strip and 𝒮). But then by Lemma <ref>, B' and w contradict Lemma <ref>. This completes the proof of Claim 1. Claim 2. Let S be a segment of B of length at least 1 with endnodes u∈ K_1 andv∈ K_2, where K_1 and K_2 are distinct cliques of 𝒦, K_1≠ K, and let (H_S,Q_u,S,Q_v,S) be the corresponding strip of 𝒮. Then G∖𝒮 cannot contain a chordless path P=w_1… w_2 such that the following hold: * N_𝒮 (w_1)=Q_K_1,* N_𝒮 (w_2)=Q_K_2, or N_𝒮 (w_2) ⊆ H_S and w_2 has a neighbor in H_S∖ Q_K_1, and* no interior node of P has a neighbor in 𝒮∖ Q_u,S. Proof of Claim 2. Assume such a path exists. Let H_S'=H_S∪ P and Q'_u,S=Q_u,S∪{ w_1}. If N_𝒮 (w_2)=Q_K_2 and either K_2≠ K or k>1, then let Q'_v,S=Q_v,S∪{ w_2}, and otherwise let Q'_v,S=Q_v,S. Since w_2 has a neighbor in H_S∖ Q_K_1, H'_S contains a rung with endnode w_1 that contains P, so (H'_S,Q'_u,S,Q'_v,S) is a strip. Since, by maximality of 𝒮, 𝒮'=𝒮∪ P cannot be a strip system, it follows that S is a claw segment (so K_2=K) and N_𝒮 (w_2)=K and k>1. Since S is a claw segment of B, Q_v,S={ v}, and there exists another leaf segment S' of B with endnode v. Suppose that a node u_1 of Q_u,S has a neighbor in interior of P. Let S_1 be a rung of H_S that contains u_1. By Lemma <ref>, B'=(B∖ S)∪ S_1 is a P-graph where S_1 is a claw segment, so by (viii) of the definition of skeleton, u_1v is not an edge. Let H' be a hole of B' that contains S_1 and S'. But then G[H'∪ (P∖ w_1)] contains a 3PC(u_1,v), a contradiction. Therefore, no node of 𝒮 has a neighbor in interior of P. But then by (2) of Lemma <ref>, the choice of B is contradicted. This completes the proof of Claim 2.Claim 3. For a clique K_1∈𝒦∖{ K}, there cannot exist a vertex w of G∖𝒮 such that N_𝒮 (w)=Q_K_1.Proof of Claim 3. Suppose such a vertex exists. Let K' be a maximal clique of G∖{w} that contains Q_K_1. Note that since G is diamond-free, no node of G∖ (K'∪{ w}) is complete to Q_K_1. Since K' cannot be a clique cutset of G, there exists a chordless path P=w… w' in G∖ (𝒮∪ K') such that w' has a neighbor u' in 𝒮∖ Q_K_1, no node of P∖{w} is complete to Q_K_1, and no interior node of P has a neighbor in 𝒮∖ Q_K_1. By Claim 1 one of the following two cases hold. Case 1: For some segment S of B of length at least 1, N_𝒮 (w')⊆ H_S. First suppose that S has an endnode u∈ K_1 and an endnode v∈ K_2, for K_2∈𝒦∖{ K_1}. By Claim 2, a node of Q_K_1∖ Q_u,S must have a neighbor in P∖ w. Let w” be a node of P∖{w} closest to w' that has a neighbor in Q_K_1∖ Q_u,S. So, since G is diamond-free and |K_1|≥ 3, N_𝒮 (w”)={ u”}, where u”∈ Q_K_1∖ Q_u,S. Let B' be a template of 𝒮 that contains u' and u”. By Lemma <ref> B' is a P-graph, and so B' and the w”w'-subpath of Pcontradict Lemma <ref>.So S does not have an endnode in K_1. Let w” be a node of P closest to w' that has a neighbor in Q_K_1. Let u” be a neighbor of w” in Q_K_1, and let B' be a a template of 𝒮 that contains u' and u”. By Lemma <ref> B' is a P-graph, and so B' and the w”w'-subpath of Pcontradict Lemma <ref>. Case 2: For some clique K_2∈𝒦∖{ K_1}, N_𝒮 (w')= Q_K_2. First suppose that there exists a segment S of B of length at least 1 with endnode u∈ K_1 and an endnode v∈ K_2. Then by Claim 2, a node of Q_K_1∖ Q_u,S has a neighbor in P. Let w” be the node of P closest to w' that has a neighbor in Q_K_1∖ Q_u,S. Then N_𝒮 (w”)={ u”}. Let B' be a template of 𝒮 that contains S and u”. By Lemma <ref> B' is a P-graph, and so B' and the w”w'-subpath of P contradict Lemma <ref>.So no segment of B of length at least 1 has an endnode in K_1 and an endnode in K_2. Let w” be the node of P closest to w' that has a neighbor u_1∈ Q_K_1∖ Q_K_2. Let B' be a template of 𝒮 that contains u_1. Then by Lemma <ref>, B' and the w”w'-subpath of P contradict Lemma <ref>.This completes the proof of Claim 3.Claim 4. Let S be a clique segment of B with endnode v∈ K. Then G∖𝒮 cannot contain a chordless path P=w_1… w_2 such that the following hold: * w_1 has a neighbor in H_S∖ Q_K,* N_𝒮 (w_2)=Q_K, and* no interior node of P has a neighbor in 𝒮∖ Q_v,S. Proof of Claim 4. Assume such a path exists. By Lemma <ref>, w.l.o.g. we may assume that w_1 has a neighbor in S∖ K. Let u be an endnode of S different from v, and let K_1∈𝒦∖{ K} such that u∈ K_1. By Claim 3, N_𝒮 (w_1)≠ Q_K_1, and so by Claim 1,N_𝒮 (w_1) ⊆ H_S. Let H'_S=H_S ∪ P and Q'_v,S=Q_v,S∪{ w_2}. Then (H'_S,Q'_v,S,Q_u,S) is a strip and 𝒮'=𝒮∪ P is a strip system that contradicts our choice of 𝒮. This completes the proof of Claim 4. Claim 5. For every connected component C of G∖𝒮, there exists a segment S of B of length at least 1 such that N_𝒮 (C) ⊆ H_S. Proof of Claim 5. Suppose that a connected componentC of G∖𝒮 does not satisfy the stated property. Since Q_K is not a clique cutset, some node of C has a neighbor in 𝒮∖ Q_K. So by Claims 2 and 3 some node w_1 of C has a neighbor in H_S∖ Q_K for some segment S of B of length at least 1. So there exists a chordless path P=w_1… w_2 in C such that w_2 has a neighbor in 𝒮∖ H_S. We choose P to be a minimal such path.First suppose that S is an interior segment of B, and let u∈ K_1 and v∈ K_2 be endnodes of S, where K_1,K_2∈𝒦∖{ K}. By Lemma <ref> w.l.o.g. we may assume that w_1 has a neighbor in S and w_2 has a neighbor in B∖ S. By the choice of P, no interior node of P has a neighbor in B. But then by Lemma <ref>, w_2 is complete to K_1 or K_2, say K_1. By Claim 1 N_𝒮 (w_2)=Q_K_1, contradicting Claim 3. Therefore S is a leaf segment of B. Let u∈ K_1 and v∈ K be the endnodes of S. By the choice of P, no interior node of P has a neighbor in 𝒮∖ Q_v,S. Suppose w_2 has a neighbor in 𝒮∖ Q_K. Then by Lemma <ref>, w.l.o.g. we may assume that w_1 has a neighbor in S∖{ v} and w_2 has a neighbor in B∖ (K∪ S). By Lemma <ref> and Claims 1 and 3, an interior node of P is adjacent to v. Let w_1' be the interior node of P closest to w_2 that is adjacent to v. By Lemma <ref> applied to w_1'w_2-subpath of P,for some leaf segment S' of B with endnode v, w_2 has a neighbor in S'∖ v. Let K_2 be the clique in 𝒦∖{ K} that contains a node of S'. Recall that interior nodes of P do not have neighbors in B∖ v. Also by Claims 1 and 3 and by Lemma <ref>, N_B(w_1) (resp. N_B(w_2)) is either a single node or an edge of S (resp. S'). Suppose k≥ 2. Then by <ref> of the definition of skeleton, K_1≠ K_2. Let P_1 and P_2 be the paths obtained by applying Lemma <ref> to S and S'. Then G[S∪ S' ∪ P_1∪ P_2∪ P] contains a theta or a wheel with center v. So k=1. By Lemma <ref> let Π be a pyramid contained in B such that S and S' are contained in different paths of Π. If w_1 is adjacent to v then G[Π∪ P] contains a wheel with center v. So w_1 is not adjacent to v and by symmetry neither is w_2. If both w_1 and w_2 have unique neighbors in Π, then G[Π∪ P] contains a wheel with center v or a theta. So w.l.o.g. N_B(w_1)={ w_1',w_1”} where w_1'w_1” is an edge of S. Then G[Π∪ P] contains a pyramid Π '=3PC(w_1w_1'w_1”, v). But then, by Lemma <ref>, P∖Π' is a crosspath of Π ' contradicting our choice of B (since k=1). Therefore, N_𝒮 (w_2)⊆ Q_K.Since w_2 has a neighbor outside S, k>1. Let v_2 be a neighbor of w_2 in K∖{ v}. Let w be a node of B∖ K adjacent to v_2, and let Q be a direct connection in B∖ K from w to K_1. By Lemma <ref> w.l.o.g. w_1 has a neighbor in S∖ K. Note that by Claims 1 and 3, N_B(w_1)⊆ S. By Lemma <ref>, N_B(w_1) is a clique of size 1 or 2 in S. If v has a neighbor in P, then G[S∪ P∪ Q] contains a theta or a wheel. So v has no neighbor in P. Then by Lemma <ref>, w_2 is complete to K, and hence by Claim 1, N_𝒮 (w_2)=Q_K. By Claim 4, S is a claw segment of B. So there is a node w' of B∖ (K∪ S) adjacent to v. Let Q' be a direct connection in B∖ K from w' to K_1, and let w_1' be the neighbor of w_1 in S∖ K which is the closest to K_1. If w'_1 is adjacent to v, then G[S∪ P∪ Q] is a wheel with center v. So, w_1' is not adjacent to v, and hence by Lemma <ref>, w_1' is the unique neighbor of w_1 in S. But then G[S∪ P∪ Q'] is a theta. This completes the proof of Claim 5.Suppose G ≠ B. Then by Claim 5, there exists a segment S of B of length at least 1 such that either H_S≠ S or a node of G∖𝒮 has a neighbor in H_S. Let 𝒞 be the union of all connected components C of G∖𝒮 that have a node with a neighbor in H_S. By Claim 5, N_𝒮(𝒞)⊆ H_S. If S is not a claw segment of B, then(H_S∪𝒞,G∖ (H_S∪𝒞)) is a 2-join of G. So we may assume that S is a claw segment of B with an endnode u∈ K.Then, by Claim 5, ((H_S∖{u}) ∪𝒞,(G∖ (H_S)∪𝒞)∪{u}) isa 2-join of G(note that Q_u,S={ u} and by <ref> of the definition of skeleton, every rung of H_S is of length at least 2).§ RECOGNITION ALGORITHMIn this section we give a recognition algorithm and a structure theorem for the class of (theta,wheel)-free graph. For this, most of the necessary work is already done in <cit.> (see Sections 6 and 7, where all important steps in the proof are given for (theta,wheel)-graphs).To obtain a recognition algorithm for (theta,wheel)-free graphs we modify the algorithm given in Theorem 7.6 of <cit.> for only-pyramid graphs. In fact, the only modification that should be made is the change of the subroutine that checks whether a graph is basic. A recognition algorithm for basic (theta,wheel)-free graphs is given in the following lemma.There is an O(n^2m)-time algorithm that decides whether an input graph is the line graph of a triangle-free chordless graph or a P-graph.By Lemma 7.4 from <cit.>, there is an O(n^2m)-time algorithm that decides whether an input graph is the line graph of a triangle-free chordless graph. So, it is enough to give an O(n^2m)-time algorithm that decides whether an input graph is a P-graph.First, in time O(n^2m) we can find the set S of all centers of claws in G. If S=∅, or S does not induce a clique, then G is not a P-graph. So, assume that S induces a non-empty clique. Next, let K be a maximal clique of G that contains S, unless |S|=1, in which case take K=S if the vertex of S is not contained in a clique of size 3, or K is a maximal clique of size at least 3 that contains S otherwise. Now, let G' be the graph obtained from G by removing all vertices of K (and edges incident with them). Using Lemma 7.4 from <cit.> we decide (in time O(n^2m)) whether G' is a line graph of a triangle-free chordless graph, and if it is find R such that G'=L(R) (if G' is not a line graph of a triangle-free chordless graph, then G is not a P-graph). Now, we check whether R is a k-skeleton, where k=|K|. To do this, first we find all pendant edges of R.Wename vertices of K with numbers 1 to k, and give labels to thependant edges of R according to their neighbor in K.We test whether (i), (vi), (vii), (viii) and (xi) in the definition of a k-skeleton are satisfied. Next, we check if (iii) is satisfied (in time O(n(n+m))) and then if (iv) is satisfied (in time O(n^2(n+m))). To check (v), note that an edge e is contained in a cycle of R if and only if R∖ e is connected, that is (v) can be check in time O(m(n+m))=O(n^2m). Branches and limbs of R can be found in time O(n+m) and the number of them is O(m). Hence, (ii) and (ix) can be checked in time O(n+m+m^2)=O(n^2m).Finally, for an attaching vertex x of R all x-petals can be found in time O(n+m), and hence (x) can be be checked in time O(n(n+m)).By the previous lemma, recognition of basic (theta,wheel)-free graphs can be done in the same running time as the recognition of basic only-pyramid graphs (used in <cit.>). Hence, the recognition algorithm for (theta,wheel)-free graphs, that was explained above, has the same running time as the algorithm given in Theorem 7.6 of <cit.>. This proves Theorem <ref>.As in <cit.>, our decomposition theorem for (theta,wheel)-free graphs can be turned into a structure theorem as follows.Let G_1 be a graph that contains a clique K and G_2 a graph that contains the same clique K, and is node disjoint from G_1 apart from the nodes of K.The graph G_1 ∪ G_2 is the graph obtained from G_1 and G_2 by gluing along a clique.Let G_1 be a graph that contains a patha_2 c_2 b_2 such that c_2 has degree 2, and such that (V(G_1){a_2, c_2, b_2}, {a_2, c_2, b_2}) is a consistentalmost 2-join of G_1 (consistent almost 2-join is a special type of almost 2-joins – for the definition see <cit.>). Let G_2, a_1, c_1, b_1 be defined similarly.Let G be the graph built on (V(G_1){a_2, c_2, b_2})∪ (V(G_2){a_1, c_1, b_1}) by keeping all edges inherited from G_1 and G_2, and by adding all edges between N_G_1(a_2) and N_G_2(a_1), and all edges between N_G_1(b_2) and N_G_2(b_1).Graph G is said to be obtained from G_1 and G_2 by consistent 2-join composition. Observe that (V(G_1){a_2, c_2, b_2}, V(G_2){a_1, c_1, b_1}) is a 2-join of G and that G_1 and G_1 are the blocks of decomposition of G with respect to this 2-join.Using the results from <cit.>, it is straightforward to check the following structure theorem. Every (theta,wheel)-free graph can be constructed as follows: * Start with line graphs of triangle-free chordless graphs and P-graphs.* Repeatedly use consistent 2-join compositions from previously constructed graphs.* Gluing along a clique previously constructed graphs.99bondy.murty:book J.A. Bondy and U.S.R. Murty. Graph Theory, volume 244 of Graduate Texts in Mathematics. Springer, 2008.chudnovsky:bull-free M. Chudnovsky. The structure of bull-free graphs II and III - a summary. Journal of Combinatorial Theory B, 102: 252–282, 2012. crst M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas. The strong perfect graph theorem. Annals of Mathematics, 164 (1):51–229, 2006.chudnovsky.seymour:claw-free M. Chudnovsky and P. Seymour. The structure of claw-free graphs. In Surveys in Combinatorics,LMS Lecture Notes Series, 327:153–171, 2005. cckv:universally M. Conforti, G. Cornuéjols, A. Kapoor and K. Vušković. Universally signable graphs. Combinatorica, 17(1):67–77, 1997. cckv:evenhole M. Conforti, G. Cornuéjols, A. Kapoor and K. Vušković. Even-hole-free graphs part I: Decomposition theorem. Journal of Graph Theory, 39(1):6–49, 2002. twf-p1 E. Diot, M. Radovanović, N. Trotignon and K. Vušković. The (theta, wheel)-free graphs, Part I: only-prism and only-pyramid graphs. arXiv:1308.6433 dirac:chordal G.A. Dirac. On rigid circuit graphs. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 25:71–76, 1961.harary.holzmann:lgbip F. Harary and C. Holzmann. Line graphs of bipartite graphs. Revista de la Sociedad Matematica de Chile, 1:19–22, 1974.Perfect H. Perfect. Applications of Menger's graph theorem. J. Math. Anal. Appl., 22:96–111, 1968. twf-p3 M. Radovanović, N. Trotignon and K. Vušković. The (theta, wheel)-free graphs, Part III: cliques, stable sets and coloring. twf-p4 M. Radovanović, N. Trotignon and K. Vušković. The (theta, wheel)-free graphs, Part IV: induced cycles and paths.dsv:ehf M.V.G. da Silva and K. Vušković. Decomposition of even-hole-free graphs with star cutsets and 2-joins. Journal of Combinatorial Theory B 103:144–183, 2013. nicolas.kristina:one N. Trotignon and K. Vušković. A structure theorem for graphs with no cycle with a unique chord and its consequences. Journal of Graph Theory, 63(1):31–67, 2010.kv-surveyK. Vušković,The world of hereditary graph classes viewed through Truemper configurations,Surveys in Combinatorics, LMS Lecture Note Series409:265–325, 2013. | http://arxiv.org/abs/1703.08675v3 | {
"authors": [
"Marko Radovanović",
"Nicolas Trotignon",
"Kristina Vušković"
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"primary_category": "math.CO",
"published": "20170325110725",
"title": "The (theta, wheel)-free graphs Part II: structure theorem"
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Department of Physics and Engineering Physics,University of Saskatchewan, 116 Science Place, Saskatoon,Canada SK S7N [email protected] discuss a dark family of lepton-like particles with their own “private” gauge bosons X_μ and C_μ under a local SU'(2)× U'(1) symmetry.The product of dark and visible gaugegroups SU'(2)× U'(1)× SU_w(2)× U_Y(1) is broken dynamically to the diagonal (vector-like) subgroup SU(2)× U(1) through the coupling oftwo fields M_i to the Higgs field and the dark lepton-like particles. After substituting vacuum expectation values for the fields M_i,the Higgs doublet couples in the standard way to the left-handed SU'(2)doublet Ψ_L and right-handed singlets ψ_1,R, ψ_2,R, but not to the extra gauge bosons. This defines a new Higgs portal, wherethe “dark leptons” can contribute to the dark matter and interact with StandardModel matter through Higgs exchange. It also defines a dark matter model withinternal interactions. At low energies, the Standard Model Higgs boson aligns the two electroweak-type symmetry groups in the visible and dark sectors and generates the masses in both sectors. We also identify charge assignments for SU'(2)× U'(1) in the dark sector which allow for the formation ofdark atoms as bound states of dark lepton-like particles.The simplest single-component dark matter versionof the model predicts a dark matter mass around 96 GeV, but thecorresponding nucleon recoil cross section of1.2× 10^-44 cm^2 is ruled out by the xenon based experiments. However, multi-component modelsor models with a dark SU'(2) doublet mediator instead of the Higgs portal would still be viable. § INTRODUCTION The identification of the dark matter which dominates the large scale structure in the universe remains an open problem. Higgs exchange had been proposed as an option for non-gravitational interactions between dark matter and ordinarymatter <cit.>, and Higgs portal modelswith couplings to the scalar product H^+H of the Higgs doublet H=(H^+,H^0) havebeen discussed extensively for bosonic <cit.>and fermionic <cit.> dark matter, but have meanwhile beenruled out in the lower region of the preferred WIMP mass range between about 100 GeVand about 1 TeV <cit.>. So far the traditional Higgs portal couplings still remain an option for light dark matter, or for WIMPs heavierthan 1 TeV. In the present paper we suggest studies of another kind of Higgs portal which doesnot involve the scalar product H^+H of the Higgs doublet, but may contribute tothe masses of dark fermions, although the Higgs field and the dark fermions transformunder a priori separate SU(2) tranformations at high energies.Mass terms in the dark sector are usually inserted by hand or generated by aseparate symmetry breaking mechanism. However, it is an intriguing questionwhether SU(2) breaking by the Standard Model (SM) Higgs boson could also generatethe dark matter masses without violating the Standard Model gauge symmetries,and yet be safe from constraints arising through couplings to the electroweakgauge bosons. We propose a mechanism to achieve this. The key idea is to havean a priori separate SU'(2)× U'(1) gauge symmetry in the darksector with its own gauge bosons X_μ and C_μ. Dynamical breaking of the dark symmetry, e.g. through scalar fields M_i≡{M_i,ab},which are charged both under the dark and visible gauge groups, then induces standard chiral Yukawa couplings of dark left-handed SU'(2) lepton-like doubletsand right-handed SU'(2) singlets with the Standard Model Higgs doublet,ℒ_H-DM=-√(2)/v_h(m_2 Ψ_L· H·ψ_2,R+ m_1Ψ_L·ϵ·H^*·ψ_1,R)+h.c. These Yukawa couplings in turn break the gauge symmetry of the coupled SM+dark mattermodel to SU_c(3)× SU_w(2)× U_Y(1), because the Higgs couplings alignthe local SU(2)× U(1) gauge transformations in the visible and dark sectors. Irrespective of whether a variant of the model from Sec. 2 turns out to generatethe low energy Higgs alignment (<ref>) of visible and dark sector gauge groups, orwhether some other mechanism is at work, the dark gauge groups should also bebroken up to a possible remnant U(1), in order not to generate too many dark radiation degrees of freedom. This leaves a remnant Yukawa potential interaction in the low energy sector if the remnant dark U(1) is broken at low energy, or a dark photon. Dark photons comply with Planck's CMB constraints on dark radiation if they decouple early enough <cit.>, and additional relativistic degrees of freedom would also favor a highervalue of H_0, thus reducing the tension between Planck and theastronomical measurements of the Hubble constant.The dark leptons, on the other hand, acquire mass terms through their SM-likeYukawa couplings to the Higgs (and possibly also due to internal symmetry breaking in the dark sector) and can constitute the darkmatter, assuming that leptogenesis also had an analog in the dark sector.Neutrality under the internal gauge interactions of the dark sector then implies that the dark matter will consist of dark atoms. The resulting dark matter model is therefore similar to the dark atom models, see <cit.> and references there. The primary new idea is a dynamically generated Higgs alignment betweenvisible and dark gauge groups. Breaking of dark gauge symmetries occurs in particular if the Higgs boson couples both to the visible and dark electroweak type gauge bosons,D_μ H(x) = ∂_μ H(x) -ig_wW_μ(x)·σ/2H(x) -ig_Y/2B_μ(x)H(x)- iq_2X_μ(x)·σ/2H(x) -iq_1/2Y'_h C_μ(x)H(x).Combined with Eq. (<ref>), this defines a class ofrenormalizable Higgs portal models for dark matter withSU_w(2)× U_y(1)× SU'(2)× U'(1) gauge symmetry and spontaneously broken SU(2) factors. This model would not be ruled out (yet) through bounds on the invisible Higgs decay width if all massive particles in the dark sector are heavierthan m_h/2, and if all dark particles with masses below m_h/2 have small masses, m≪ m_h/2.However, as outlined above, in the present investigation we will focus on the case that the Higgs boson does not directly couple to the SU'(2)× U'(1) gauge bosons, but is only charged under the Standard Model gauge group, D_μ H(x)=∂_μ H(x) -ig_wW_μ(x)·σ/2H(x) -ig_Y/2B_μ(x)H(x).Section <ref> provides a dynamical toy model which generates the Higgs portal coupling (<ref>), although the covariant derivative on the Higgs field is only given by (<ref>). Section <ref> then provides a more fulsome discussion ofthe low energy formulation of the theory, and the formation of darkatoms in this theory is discussed in Sec. <ref>.Annihilation cross sections and the constraints from thermal darkmatter creation on the simplest version of dark matter inthis framework are discussed in Section <ref>.Section <ref> summarizes our conclusions.§ A DYNAMICAL MECHANISM FOR ALIGNMENT OF VISIBLEAND DARK GAUGE SYMMETRIESA dynamical mechanism to generate the coupling (<ref>) with a Higgs field which is not charged under the dark gauge groups, cf. (<ref>), can be constructed withscalar fields M_i≡{M_i,ab} which are charged both under the dark gauge group and the Standard Model gauge group. The first index a refers to the fundamental representation of SU'(2) from the left, while the second index b refers to SU_w(2)acting through the adjoint SU_w(2)-matrices from the right. The U'(1)charges are given in terms of the dark fermion chargesby Y'^(M)_i=Y'_L-Y'_i,R, and the U_Y(1) chargesare Y^(M)_i=-Y_h=-1. The covariant derivativesof the M_i-fields are thereforeD_μM_i = ∂_μM_i-iq_2X_μ·σ/2·M_i-iq_1/2(Y'_L-Y'_i,R)C_μM_i+ ig_WM_i·σ/2·W_μ +ig_Y/2Y_hB_μM_i.The following coupling term between the visible and the dark sector is then invariant under the full gauge group SU'(2)× U'(1)× SU_w(2)× U_Y(1),ℒ_MH-DM=-√(2)/v_h( Ψ_L·M_2· H·ψ_2,R + Ψ_L·M_1·ϵ·H^*·ψ_1,R)+h.c. This yields the new Higgs portal coupling (<ref>) in the low energy sector through the potentialV(M_1,M_2)=1/4∑_i=1^2 λ_i[Tr(M_i·M_i^+) -2DetM_i]^2.A 2× 2 matrix M_i satisfies the ground state conditionTr(M_i·M_i^+) =2DetM_iif and only if the matrix is proportional to the unit matrix,M_i=m_i1,up to an additional possible unitary factor V_i. Theparameter m_i in Eq. (<ref>) can be chosen tosatisfy m_i≥ 0. These results can easily be proved using the polardecomposition M_i=H_i·V_i of the matrix M_i into a positive semidefinitehermitian factor H_i and a unitary factor V_i. This simple dynamical model is only a toymodel for the demonstration that the new Higgs portal (<ref>) can arise dynamically, if we are primarily interested in the traditional WIMP mass range,which will be the focus of the remainder of this paper. Any actual phenomenological implementation of the dynamical model (<ref>-<ref>) could only work if dark matter is extremely light.§ DARK MATTER MASSES FROM THE STANDARD MODEL HIGGS BOSONFor all we know, the Higgs field is the only dynamical field (besides gravity) which couples to all massive fields, and indeed generates the masses of those fields through Yukawa couplings. It is therefore compelling to assume that the Higgs field also generates the masses in the dark sector. The Yukawacouplings (<ref>) to fermionic dark matter Ψ_L=(ψ_1,L,ψ_2,L),ψ_1,R and ψ_2,R provide a way to achieve this, and the mechanismoutlined in Sec. <ref> provides a way to generate these couplings in the low energy sector of a renormalizable gauge theory.We assume a single generation of dark lepton-like particles in (<ref>), but the generalization to more generations is straightforward through promotionof the masses m_1 and m_2 to mixing matrices.A priori the dark gauge group SU'(2)× U'(1) acting onthe fields Ψ_L and ψ_i,R and the electroweak gauge group aredifferent symmetries, with SU'(2)× U'(1) acting only in the darksector while the electroweak symmetry only acts in the visible sector. However, the Yukawa couplings (<ref>) of the Higgsdoublet align the transformations in both symmetry groups at low energies,thus breaking the direct product of symmetry groups to its diagonal component,SU'(2)× U'(1)× SU_w(2)× U_Y(1) → SU'(2)× U'(1)=SU_w(2)× U_Y(1).The corresponding dark hypercharge (U'(1)) assignments have to satisfyY'_L-Y'_1,R=-Y_h=-1, Y'_L-Y'_2,R=Y_h=1,where Y_h=1 is the weak hypercharge of the Higgs doublet.The gauge symmetry SU_w(2)× U_Y(1) is therefore implemented forlow energy in the visible and dark sectors through the SU(2)transformations U(x)=exp[iφ(x)·σ/2] and the U(1) transformations exp[iα(x)Y/2], exp[iα(x)Y'/2]. The action of the SM fields is as usual, H'(x)=exp[iα(x)/2]U(x)· H(x), ([ ν'_L(x); e'_L(x); ])=exp[-iα(x)/2]U(x)·([ ν_L(x); e_L(x);]), ν'_R(x)=ν_R(x), e'_R(x)=exp[-iα(x)]e_R(x),… W'_μ(x)·σ =U(x)· (W_μ(x)·σ)· U^+(x) +2i/g_wU(x)·∂_μ U^+(x), B'_μ(x)=B_μ(x)+1/g_Y∂_μα(x),and the corresponding action in the dark sector isΨ'_L(x)≡([ ψ'_1,L(x); ψ'_2,L(x); ]) =exp[i Y'_Lα(x)/2]U(x)·([ ψ_1,L(x); ψ_2,L(x);]), ψ'_1,R(x)=exp[i (Y'_L+1)α(x)/2]ψ_1,R(x), ψ'_2,R(x)=exp[i (Y'_L-1)α(x)/2]ψ_2,R(x), X'_μ(x)·σ =U(x)· (X_μ(x)·σ)· U^+(x) +2i/q_2U(x)·∂_μ U^+(x), C'_μ(x)=C_μ(x)+1/q_1∂_μα(x).Here q_2 and q_1 are the gauge couplings of the dark SU'(2)× U'(1)symmetry group, and the covariant derivatives on the dark matter fields areD_μΨ_L(x)=∂_μΨ_L(x) -iq_2X_μ(x)·σ/2Ψ_L(x) -iq_1/2Y'_L C_μ(x)Ψ_L(x), D_μψ_i,R(x) =(∂_μ-iq_1/2Y'_i,R C_μ(x))ψ_i,R(x), Y'_i,R=Y'_L-(-)^i.The low energy sector of the theory outlined in Sec. <ref> therefore looks like the Standard Model augmented with a fourth lepton-like family with its own “private gauge bosons” X_μ and C_μ. Since we assume m_i>m_h/2, this “fourth family” does not contribute to Z nor Higgs decays, nor does it produce a fourth low mass thermalized particle.The reader will certainly have noticed that the assumed preference forleft-chirality also in the dark sector is not determined by any experimental observation and therefore ambiguous with our current knowledge. Instead, we could just as well use right-handed SU'(2) doublets and left-handed singlets in the dark sector. This would be compelling from an enhanced symmetry point of view. Opposite chirality in the dark sector could restore CP and time-reversal symmetry in particle physics through mappings between visible sector and dark sector fields. CP and time reversal symmetry might then only be hidden from us through the very weakHiggs-coupling between the visible sector and the dark sector. The model would then provide a Higgs portal realization of the old ideaof a CP mirror of the Standard Model <cit.>. However, currently we can constrain the dark sector only through gravitational effects and direct and indirect search limits. We cannot currently distinguish between a V-A or V+A preference in thedark sector, and at this stage a V-A formulation is as good asa V+A formulation.There are obviously infinitely many possible ramifications of this model by also including corresponding families of dark quarks with dark symmetry group SU'_c(N) and corresponding gauge bosons F_μ^a. The Yukawa couplings of the Higgs boson to the dark quark-like multiplets would obviously not align the SU'_c(N) with the Standard Model SU_c(3). In such a model it would be natural to expect the dark matter to consist of darkatoms formed from dark leptons and dark nucleons, with the atoms formed dueto the SU'(2)× U'(1) interactions in the dark sector.However, we will focus on the simplest model of private gauge bosons with one dark family of lepton-like particles and their private SU(2)× U(1)gauge bosons (<ref>,<ref>,<ref>,<ref>). This is practically equivalent to a dark multi-family model with one familybeing lighter than the other families, thus being the relevant family fordark matter. Because we do not need to exclude the presence of heavier lepton-like or quark-like SU'(2)× U'(1) doublets, we also do not have to worry about anomaly cancellation within the lightest lepton-like dark family. However, we will later find that the requirement for darkmatter in the single lepton-like dark family implies Y'_L=0, such thatanomaly cancellation is actually fulfilled already with the single dark doublet Ψ_L. Without dark nucleons from a dark quark sector,Coulomb-type repulsion between the dark leptons could counteract the pull of gravity. Generically this would seem to imply that the SU'(2)× U'(1) couplings q_2 and q_1 should only be ofgravitational strength for not actually preventing dark halo formation.However, we can choose the SU'(2)× U'(1) charges of the dark leptonsin a way which allows for the formation of SU'(2)× U'(1) neutral darkatoms as bound states of the dark leptons, thus alleviating the contraints on the magnitudes of q_1 and q_2.§ DARK ATOMS Recall that the Coulomb potential for electrons and protons in quantum optics (expressed in terms of Schrödinger picture operators)H_C = ∫d^3x∫d^3x' α_S/2|x-x'|[ψ^+_e(x)ψ^+_e(x')ψ_e(x')ψ_e(x) . +.ψ^+_p(x)ψ^+_p(x')ψ_p(x')ψ_p(x) -2ψ^+_e(x)ψ^+_p(x')ψ_p(x')ψ_e(x) ]arises from ∂_μ F^μ0=e(ψ^+_eψ_e-ψ^+_pψ_p)in Coulomb gauge, and substitution of the solution E_(x)=∇∫d^3x' e/4π|x-x'|(ψ^+_e(x')ψ_e(x')-ψ^+_p(x')ψ_p(x'))of Eq. (<ref>) into the contribution from E_ to the electromagnetic field HamiltonianH_EM=1/2∫d^3x(E^2_(x) +E^2_⊥(x)+B^2(x)) =H_C+1/2∫d^3x( E^2_⊥(x)+B^2(x)), E_⊥(x)=-∂A(x,t)/∂ t|_t=0,B(x)=∇×A(x). Substitution of the k-space expansions of the electron andproton operators and normal-ordering then yields the usual electron-electron, electron-(anti-proton), proton-proton,positron-proton etc. repulsion terms, and the electron-positron, electron-proton etc. attraction terms.A corresponding analysis helps to identify any attractive particle-particlecombinations in the dark sector, although for non-abelian dark interactionsthe Coulomb kernel will only yield the perturbative short-distance part of theactual potential if the interaction is asymptotically free, or the Coulombkernel needs to be replaced by a Yukawa kernel for massive gauge fields.In the non-relativistic limit, the equations (with ψ^+_iψ_i≡ψ^+_i,Lψ_i,L+ψ^+_i,Rψ_i,R)∂_μ C^μ 0=-ϱ_C=-q_1/2[Y'_L (ψ^+_1ψ_1+ψ^+_2ψ_2) +ψ^+_1,Rψ_1,R-ψ^+_2,Rψ_2,R], D_μ X_a^μ 0=-ϱ_a=-q_2/2Ψ^+_L·σ_a·Ψ_L,yield the dark sector Coulomb operator H_dC=∫d^3x∫d^3x' ϱ_d^2(x,x') /8π|x-x'|with the charged density-density correlation operatorϱ_d^2(x,x') = ϱ_C^2(x,x') +∑_a=1^3ϱ_a^2(x,x') = q_2^2/4[ ψ^+_1,L(x)ψ^+_1,L(x')ψ_1,L(x')ψ_1,L(x) +ψ^+_2,L(x)ψ^+_2,L(x')ψ_2,L(x')ψ_2,L(x) . -.2ψ^+_1,L(x)ψ^+_2,L(x')ψ_2,L(x')ψ_1,L(x) +4ψ^+_1,L(x)ψ^+_2,L(x')ψ_1,L(x')ψ_2,L(x) ]+ q_1^2/4Y'^2_L [ ψ^+_1,L(x)ψ^+_1,L(x')ψ_1,L(x')ψ_1,L(x) +ψ^+_2,L(x)ψ^+_2,L(x')ψ_2,L(x')ψ_2,L(x) . +.2ψ^+_1,L(x)ψ^+_2,L(x')ψ_2,L(x')ψ_1,L(x) ] + q_1^2/4[ (Y'_L+1)^2ψ^+_1,R(x)ψ^+_1,R(x')ψ_1,R(x')ψ_1,R(x) . + (Y'_L-1)^2ψ^+_2,R(x)ψ^+_2,R(x')ψ_2,R(x')ψ_2,R(x) +. 2(Y'^2_L-1)ψ^+_1,R(x)ψ^+_2,R(x')ψ_2,R(x')ψ_1,R(x) ] + q_1^2/2Y'_L(Y'_L+1) [ ψ^+_1,L(x)ψ^+_1,R(x')ψ_1,R(x')ψ_1,L(x) . +.ψ^+_2,L(x)ψ^+_1,R(x')ψ_1,R(x')ψ_2,L(x) ]+ q_1^2/2Y'_L(Y'_L-1) [ ψ^+_1,L(x)ψ^+_2,R(x')ψ_2,R(x')ψ_1,L(x) . +.ψ^+_2,L(x)ψ^+_2,R(x')ψ_2,R(x')ψ_2,L(x) ]. Most of the terms in Eqs. (<ref>,<ref>) are repulsive between pairs of particles and attractive between particles and anti-particles in the dark sector. The attractive channels between particles and anti-particlesallow for the formation of dark SU'(2) mesons which will decay fast throughthe Higgs portal (<ref>) and accelerate annihilation of any remnant dark anti-leptons. We are therefore interested in attractive terms betweenpairs of particles for the formation of dark atoms.The terms in Eq. (<ref>) with coupling constants q_1^2Y'_L(Y'_L± 1) yield attractive interactions in the two-particlestates ψ^+_1,L(x)ψ^+_1,R(x')|0⟩ and ψ^+_2,L(x)ψ^+_1,R(x')|0⟩ if Y'_L(Y'_L+1)<0, and attractive interactionsin the two-particlestates ψ^+_1,L(x)ψ^+_2,R(x')|0⟩ and ψ^+_2,L(x)ψ^+_2,R(x')|0⟩ if Y'_L(Y'_L-1)<0. However, the resulting bound states would have residual charges q_2/2 under SU'(2) and (2Y'_L± 1)q_1/2 under U'(1), respectively. This would yieldrepulsive Coulomb-type atom-atom interactions and again prevent dark halo collapse unless q_1 and q_2 would be tuned below gravitational strength.However, the Coulomb term for the dark two-leptonstates ψ^+_1,R(x)ψ^+_2,R(x')|0⟩,H_1R,2R=∫d^3x∫d^3x'q_1^2(Y'^2_L-1)/16π|x-x'|ψ^+_1,R(x)ψ^+_2,R(x') ψ_2,R(x')ψ_1,R(x)is attractive if Y'^2_L<1. It has no SU'(2) charge, but aresulting atomic U'(1) charge q_1Y'_L. Setting Y'_L=0therefore yields uncharged dark atoms consisting of the tworight-handed dark leptons. This implies overall charge neutrality of the universe also under the dark gauge groups, in the same way as anomaly cancellation in the Standard Model ensures charge neutrality in the visible sector. At temperatures above 1 TeV there are equal abundances of particle species, and the vanishing sum of visible U_Y(1) hypercharges over SM particle species and dark U'(1) hypercharges over dark sector species ensures overall charge neutralities in both sectors.The potential in the dark sector therefore splits into the left-handed and right-handed parts, H_dC=H_LL+H_RR, with fine structure constants α_L=q_2^2/16π and α_R=q_1^2/16π, and the only attractive particle-particle channel for neutral atoms consisting of the two right-handed dark leptons, which are bound due to the U'(1) interaction.The states of the dark atoms are therefore forseparation r=x_1-x_2 of the two dark leptons and total atomic momentum K given in the standard way by direct transcription of the corresponding results of non-relativistic QFT,|Ψ_n,ℓ,m_ℓ;K(t)⟩ = ∫d^3x_1∫d^3x_2 ψ^+_1,R(x_1)ψ^+_2,R(x_2)|0⟩Ψ_n,ℓ,m_ℓ(x_1-x_2)(2π)^-3/2 ×exp[iK·(m_1x_1+m_2x_2)/M] exp[-iE(K,n)t],with hydrogen type wave functions Ψ_n,ℓ,m_ℓ(r)for coupling constant α_R/4 and reducedmass m_12=m_1 m_2/M, M=m_1+m_2. The energy eigenvalues in the non-relativistic regime areE(K,n)=K^2/2M-1/32n^2α_R^2 m_12.Spin singlet or triplet factors were apparently omitted, since we are not interested in the fine structure of the dark atoms. The chiral projectors in the potential (<ref>) reduce the effective dark sector fine structure constant to α_R/4. This is explained in detail in the Appendix.We also assume α_R<α_S since dark U(1) interactions must be weaker than electromagnetism. The separation of gas and dark matter in bullet-type clusters warrants this conclusion <cit.>.The fact that dark matter halos are much more extended and form lessconcentrated substructures than baryonic matter also tells us that dark matter cannot cool down as efficiently as baryons.§ DARK MATTER ANNIHILATION AND THERMAL CREATION We are interested in the non-relativistic thermal freeze-out of WIMPscale dark leptons. The annihilation of a heavy dark lepton speciesis then dominated by branching ratios into Standard Model particlesthrough the Higgs portal (<ref>). The t and u channel annihilations ψ_iψ_i→ CC and ψ_iψ_i→XX are suppressedwith α_R^2p^2/m_i^2 due to the chirality factors in theamplitudes, which come from the chirality factors (1±γ_5)/2 in thevertices q_1Y'_i,Rψ_iγ^μ C_μ(1+γ_5)ψ_i/4 andq_2ψ_iγ^μX_μ·σ(1-γ_5)ψ_i/4from Eqs. (<ref>,<ref>).The leading order cross sections into the Standard Model states are (with VV=W^+ W^- or VV=ZZ and δ_W=1 for annihilationinto W^+ W^- or δ_W=0 otherwise)σ_ψ_iψ_i→ VV(s) =1+δ_W/64πm_i^2/v_h^4 s√(s-4m_V^2)√(s-4m_i^2)(s-2m_V^2)^2+8m_V^4/(s-m_h^2)^2+m_h^2Γ_h^2, σ_ψ_iψ_i→ ff(s) =N_cm_i^2 m_f^2/16π v_h^4 s(s-4m_f^2)^3/2(s-4m_i^2)^1/2/(s-m_h^2)^2+m_h^2Γ_h^2, σ_ψ_iψ_i→ hh(s) =9m_i^2 m_h^4/64π v_h^4 s√(s-4m_h^2)√(s-4m_i^2)/ (s-m_h^2)^2+m_h^2Γ_h^2. The total annihilation cross section increases with mass m_i for masses above 80 GeV, and therefore the heavier dark lepton specieswill determine both the mass M of the dark atoms and the freezeout temperature. We will assume m_1≲ 0.01 m_2and therefore M≃ m_2.The requirement of thermal freeze out then determines M≃ 96 GeV, see Fig. <ref>, where the thermally averaged <cit.>annihilation cross for a particle with mass m_2≃ M is compared to the required value from thermal dark matter creation. The logarithmically varying required valueof ⟨ vσ⟩ for thermal creationvaries from 3.38× 10^-26 cm^3/s for M=65 GeV to 3.45× 10^-26 cm^3/s for M=100 GeV.A low mass value m_1≲ 1 GeV implies an invisible Higgs decay width which is well below the current limits <cit.>, Γ_h→ψ_1ψ_1≲ 82 keV≃ 0.018Γ_h→SM.This dark matter model is even more predictive than the standard Higgs portal dark matter models because the coupling to the Higgs field is already determined in terms of the mass, g=m_2/v_h≃ M/v_h. The requirement of thermal dark matter creation therefore does not yield a parametrization g(M) of the Higgs portal coupling as a function of the dark matter mass, but determines M. However, the corresponding nucleon recoil cross section σ_DN=g^2/4πm_N^2 M^4/m_h^4 v_h^2(M+m_N)^2is about 1.2× 10^-44 cm^2 for M=96 GeV, m_N=930.6MeV (the weighted average of the nucleon masses in long lived xenon isotopes), and the SVZ value gv_h=210 MeV <cit.> for theeffective Higgs-nucleon coupling. This is in conflict with the exclusion limits from the xenon based directsearch experiments <cit.>. § CONCLUSIONS Alignment of gauge symmetries in the visible and dark sectors through the new Higgs portal is an interesting new tool for dark matter model building. It can arise as a consequence of dynamical symmetry breaking in gauge theories, and it opens a door to fermionic Higgs portalmodels without the need of higher mass-dimension effective vertices.Apparently, the construction presented here opens the Higgs portal into much more complicated and rich dark sectors, even with the possibility of CP and time-reversal reciprocity between the visible and dark sectors, which would be broken through the different mass spectra in the two sectors. Furthermore, the construction also generalizes to alignment of gauge groups through other fields. Every field which transforms in a faithful representation of a symmetry group G can align different copies G_1,G_2,… of the symmetry group, each with their owngauge bosons A^a_i,μ and coupling constants g_i, throughYukawa couplings to fields transforming under the differentsymmetries G_i. In particular, it is conceivable that the dark leptons may not couple to the Standard Model through the Higgsportal, but through a dark scalar SU'(2) doublet H', whichcouples to the dark sector gauge bosons X_μ and C_μ. This would also align the electroweak-type gauge symmetries in the dark and visible sectors through the Yukawa couplings of the scalar SU'(2) doublet. However, it would not be constrained by the current limits from the direct search experiments, since the Yukawa couplings of H'in the visible sector must be weaker than the Higgs couplings,thus also implying a weaker effective H'-nucleon coupling. This work was supported in part by the Natural Sciences andEngineering Research Council of Canada through a subatomicphysics grant. I very much enjoyed the hospitality of the Kavli Institute for Cosmological Physics duringmy sabbatical, and I benefitted greatly from discussions with KICP members and the particle cosmology group, especially Rocky Kolb, Lian-Tao Wang, Andrew Long, Michael Fedderke, and Dan Hooper. I also wish to acknowledge Valeri Galtsev for his excellent computer support at KICP. § APPENDIX: THE SCHRÖDINGER EQUATION IN THE DARK SECTORTo understand the impact of the chiral projectors in the dark sector Coulomb potential (<ref>), we follow the procedure which yields the corresponding Schrödinger equation for the hydrogen atom in the baryonic sector while keeping track of the chiralprojectors.The relevant states for the dark atoms are two-particle states |Ψ(t)⟩=∑_αβ∫d^3x∫d^3x' Ψ_αβ(x,x',t) ψ^+_1α(x)ψ^+_2β(x')|0⟩.The states are written in the Schrödinger picture, and the indicesα,β are Dirac indices.The relevant part of the Hamiltonian of the theory in the sector of Fock space wich is spanned by the states (<ref>) (i.e. suppressing all parts of the Hamiltonian which map into different sectors of Fock space) is with P_R=(1+γ_5)/2,H = ∑_i,α,β∫d^3x ψ_iα(x)( m_iγ_αβ-iγ_αβ·∇) ψ_iβ(x) -∑_α,β,ρ,σP_RαβP_Rρσ∫d^3x∫d^3x' α_R/|x-x'|ψ^+_1α(x)ψ^+_2ρ(x') ψ_2σ(x')ψ_1β(x). The corresponding Hamiltonian without the chiral projectors P_R arises for the electron-proton system from the energy-momentum tensor of QED in Coulomb gauge, see e.g. Sec. 21.4 in Ref. <cit.>.The relativistic Schrödinger equation for the two-particlesystem follows from id|Ψ(t)⟩/dt=H|Ψ(t)⟩ and after decomposition in the basis (<ref>) in the formi∂/∂ tΨ_αβ(x,x',t) =∑_ρσĤ_αβ,ρσΨ_ρσ(x,x',t),with the Hamilton operatorĤ_αβ,ρσ = [m_1γ^0_αρ-i(γ^0·γ)_αρ·∂/∂x]δ_βσ +δ_αρ[m_2γ^0_βσ-i(γ^0·γ)_βσ·∂/∂x']-α_R/|x-x'|P_RαρP_Rβσ. To derive the nonrelativistic limit, we use the Dirac representation of γ matrices and write the 4× 4 matrix Ψ_αβ(x,x',t) in terms of 2× 2matrices in the formΨ(x,x',t) =([ ψ(x,x',t) ϕ(x,x',t); ξ(x,x',t) χ(x,x',t); ]) exp[-i(m_1+m_2)t].Substitution into Eq. (<ref>) then yields in leading order of m_i^-1 and α_R for the “small” components the equationsϕ(x,x',t)=-i/2m_2∂/∂x'ψ(x,x',t) ·σ^T, ξ(x,x',t)=-i/2m_1σ·∂/∂xψ(x,x',t), and χ(x,x',t)=0, and substitution intothe equation following for ψ(x,x',t) from Eq. (<ref>),i∂/∂ tψ= -iσ·∂/∂xξ -i∂/∂x'ϕ·σ^T -α_R/4|x-x'|( ψ+ϕ+ξ+χ),yields the standard two-particle Schrödinger equation up to an extra factor of 1/4 in the Coulomb potential,i∂/∂ tψ(x,x',t)= (-1/2m_1Δ-1/2m_2Δ') ψ(x,x',t) -α_R/4|x-x'|ψ(x,x',t).Separation in center of mass and relative coordinates for the mapping into the effective single-particle equations with masses M=m_1+m_2 and m_12=m_1 m_2/M then proceeds as usual. 88zee Silveira V and Zee A 1985 Phys. Lett. B 161 136JMcD McDonald J 1994 Phys. Rev. D 50 3637bento Bento M C, Bertolami O, Rosenfeld R and Teodoro L 2000 Phys. Rev. D 62 041302(R)cliff Burgess C, Pospelov M and ter Veldhuis T 2001 Nucl. Phys. B 619 709dklm Davoudiasl H, Kitano R, Li T and Murayama H 2005 Phys. Lett. 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J. 648 L109bullet2 Bradač M, Allen S W, Treu T, Ebeling H, Massey R, Morris R G,von der Linden A and Applegate D 2008 Astrophys. J. 687 959; Dawson W A et al. 2012 Astrophys. J. 747 L42 gondolo Gondolo P and Gelmini G 1991 Nucl. Phys. B 360 145atlas ATLAS Collaboration (Aad G et al.) 2015 JHEP 1511 206; 2016 JHEP 1601 172cms CMS Collaboration (Aarrestad T K et al.) 2017 JHEP 1702 135 SVZ Shifman M A, Vainshtein A I and Zakharov V I 1978 Phys. Lett. B 78 443xenon100 XENON Collaboration (Aprile E et al.) 2012 Phys. Rev. Lett. 109 181301lux LUX Collaboration (Akerib D et al.) 2014 Phys. Rev. Lett. 112 091303 rdqm Dick R 2016 Advanced Quantum Mechanics: Materials and Photons 2nd ed (New York: Springer) | http://arxiv.org/abs/1703.09300v2 | {
"authors": [
"Rainer Dick"
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"categories": [
"hep-ph",
"astro-ph.CO"
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"primary_category": "hep-ph",
"published": "20170327203123",
"title": "Dynamical alignment of visible and dark sector gauge groups"
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Huang and HoffmanMonolayer FeSe on SrTiO_3Epitaxial engineering of solid-state heterointerfaces is a leading avenue to realizing enhanced or novel electronic states of matter. As a recent example, bulk FeSe is an unconventional superconductor with a modest transition temperature (T_c) of 9 K. When a single atomic layer of FeSe is grown on SrTiO_3, however, its T_c can skyrocket by an order of magnitude to 65 K or 109 K. Since this discovery in 2012, efforts to reproduce, understand, and extend these findings continue to draw both excitement and scrutiny. In this review, we first present a critical survey of experimental measurements performed using a wide range of techniques. We then turn to the open question of microscopic mechanisms of superconductivity. We examine contrasting indications for both phononic (conventional) and magnetic/orbital (unconventional) means of electron pairing, and speculations about whether they could work cooperatively to boost T_c in a monolayer of FeSe. superconductivity, iron pnictides and chalcogenides, oxides, thin-film interfacesMonolayer FeSe on SrTiO_3 Dennis Huang^1 and Jennifer E. Hoffman^1, 2 ^1Department of Physics, Harvard University, Cambridge, Massachusetts 02138; emails: [email protected], [email protected] ^2Department of Physics & Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada December 30, 2023 ==============================================================================================================================================================================================================================================================================================================§ INTRODUCTION Interface engineering lies at the vanguard of current research in condensed matter physics and novel materials. From a fundamental perspective, quantum-size and electron correlation effects are enhanced in reduced dimensionality, often resulting in unexpected phenomena. From a technological perspective, as the pace of electronics miniaturization fast approches the limit of conventional semiconductors, alternative paradigms, such as atomically-thin materials and interfaces with manifestly quantum behavior, are needed to assume the mantle of next-generation electronics. With improving ability to assemble atomically-sharp interfaces “bottom up” through molecular beam epitaxy (MBE) or mechanical stacking of 2D-layered materials, many possibilites abound.As the central example of this review, interface engineering holds promise in tuning, boosting, or generating superconducting states of matter – low-temperature quantum phases in which electrons form Cooper pairs and charge transport is lossless. Since its discovery in 1911, superconductivity has continued to fascinate and baffle condensed matter physicists, while the goal of realizing room-temperature superconductivity remains elusive. Within the past decade, various examples of interface superconductivity have been observed. When two insulating oxides, LaAlO_3 and SrTiO_3, are put togther, a superconducting electron gas is formed at the interface, albeit with a low transition temperature (T_c) of 200 mK <cit.>. When a bilayer of insulating La_2CuO_4 and metallic La_1.55Sr_0.45CuO_4 is formed, the aggregate system displays a T_c exceeding 50 K <cit.>. And as the latest example, when a single-unit-cell (1UC) layer of FeSe is deposited on SrTiO_3 <cit.>, its T_c skyrockets up to 65 K <cit.> or 109 K <cit.>, an order of magnitude above its bulk T_c of 9 K. In this review, we will cover key experimental and theoretical developments related to 1UC FeSe/SrTiO_3 up to early 2016. We focus on measurements of basic properties and questions of superconducting mechanisms. §.§ Approaching the 2D limit with FeSeFeSe posesses the simplest structure among the iron-based superconductors, consisting of superconducting Se-Fe-Se triple layers stacked by van der Waals forces, with no buffer layers <cit.>. Figure <ref>a shows the structure of one triple layer, which includes Fe atoms arranged in a square lattice and Se atoms staggered above and below the Fe-plane. Due to the staggering, the primitive UC contains two Fe atoms (and two Se atoms). However, since the low-energy bands of FeSe are dominated by Fe 3d orbitals, many theories or spectroscopies reference the 1-Fe UC for convenience.By virtue of its structural simplicity, FeSe should be the prototypical iron-based superconductor to investigate, except it proved difficult to synthesize in high quality at first. Its superconducting polymorph occupies a narrow region in the Fe-Se alloy phase diagram <cit.>, complicating common melt and self-flux growths. In 2011, Song et al. used MBE to grow pristine FeSe films on graphitized SiC <cit.>. Using scanning tunneling microscopy (STM), they resolved clean surfaces with only one atomic defect per 70,000 Se sites. Measurements of tunneling conductance (dI/dV), which is proportional to the local density of states, revealed two signatures of a superconducting state: (1) A V-shaped gap of Δ = 2.2 meV, representing the binding energy of paired electrons, that disappeared above 10 K (Figure <ref>b); (2) vortices in the presence of a perpendicular magnetic field. Although MBE-grown films are not amenable to many bulk and thermodynamic probes, they have other advantages. Both the monolayer limit and interface interactions with different substrates can be readily examined. Song et al. found that the FeSe films interacted weakly with the graphitized SiC substrate (islands could be displaced by an STM tip), and were thereby close to the free-standing limit <cit.>. Upon decreasing film thickness, T_c, as measured by the gap-closing temperature, dropped from 7.8 K (8UC-thick FeSe) to below 2.2 K (1UC-thick FeSe), the base temperature of their experiment (Figure <ref>c). The drop exhibited a 1 - d_c/d dependence, d being the film thickness and d_c being a critical value. This thin-film behavior was explained long ago as resulting from a general, surface boundary condition with the Ginzburg-Landau equation <cit.>. Thus in 2011, it did not appear that the 2D limit of FeSe would be a promising regime to explore, unless new microscopic effects could be introduced. §.§ Monolayer FeSe gets an oxide boost It came as a great surprise a year later that monolayer FeSe could undergo an order-of-magnitude T_c enhancement when grown epitaxially on SrTiO_3(001). The lattice mismatch between bulk FeSe (a = 3.77 Å <cit.>) and SrTiO_3 (a = 3.905 Å <cit.>) is roughly 3%. STM measurements by Wang et al. revealed a topographic period-doubling (Figure <ref>a) and a large U-shaped, double-gap structure (9.0 meV and 20.1 meV) in 1UC FeSe/SrTiO_3 (Figure <ref>b), with closing temperature T_c above their experimental limit of 42.9 K. Intriguingly, this superconductivity boost did not persist or even proximitize low-T_c superconductivity in a second UC of FeSe deposited on the heterostructure. STM dI/dV measurements, whose probing depth is likely limited to the surface FeSe layer, instead showed a semiconducting spectrum on the second FeSe layer (Figure <ref>c). This observation points to an underlying interface effect, one that is atomically localized to the first UC of FeSe on SrTiO_3. Wang et al. speculated that electron-phonon coupling could be enhanced at the interface and boost T_c, based on their previous work with Pb/Si(111) and In/Si(111) films <cit.>.Due to technical challenges, Wang et al. could measure transport only in a Si-capped, 5UC FeSe/SrTiO_3 heterostructure. They measured zero resistance at some temperature lower than 30 K, and extrapolated a resistive onset temperature around 53 K. (As shown by STM spectroscopy in Figure <ref>b,c, the superconducting signal originates from the interface FeSe layer only.) Angle-resolved photoemission spectroscopy (ARPES) measurements in the same year provided initial insights into the role of the interface. ARPES can map filled-state band structure in momentum space. Liu et al. <cit.> found that the Fermi surface (FS) of 1UC FeSe/SrTiO_3 comprises nearly-circular electron pockets at the Brillouin zone (BZ) corner M (Figure <ref>d,e). In contrast to bulk FeSe <cit.>, where additional hole FSs exist at the zone center Γ, here the corresponding hole pocket is sunken 65-80 meV below the Fermi energy (E_F) (Figure <ref>f). Assuming doubly-degenerate electron pockets, a Luttinger count yields 0.10 electrons/Fe atom. Thus, relative to its bulk, 1UC FeSe appears to be electron-doped from the substrate. To provide further support for the superconducting nature of 1UC FeSe/SrTiO_3, Liu et al. resolved nearly-isotropic gaps on the electron pockets at each M point, of values 13±2 meV and 15±2 meV for two samples. They found the gap-closing temperature to be 55±5 K.Before proceeding, we reiterate that monolayer FeSe/SrTiO_3 is not monolayer FeSe. A giant T_c enhancement is present only in the former, due to some effect introduced by the SrTiO_3.§ EXPERIMENTAL CHALLENGES A foremost challenge related to 1UC FeSe/SrTiO_3 has been the characterization of its growth, atomic structure, and superconducting metrics. As a point of emphasis, bulk probes are not effective for this system. Not only is the cross section of a single UC layer miniscule, but also FeSe exhibits extreme air sensitivity, hampering ex-situ measurements. Thus, the basic goal of determining T_c represents a nontrivial endeavor requiring customized and integrated instrumentation in ultra-high-vacuum. Example apparatuses include combined MBE-ARPES-STM systems, double chalcogen-MBE/oxide-MBE chambers, and customized in-situ, four-point probes. In this section, we review various experiments related to film characterization, categorized under three questions: What is T_c? What are the necessary growth conditions? What is the interface structure? We attempt to reflect the sentiments of the scientific community by conveying both the excitement related to the spectacular findings of tour-de-force experiments, and scrutiny related to the challenging nature of these feats and of film quality/homogeneity. §.§ What is T_c? Table <ref> presents a comparison of T_c measurements across different probes, heterostructures, and laboratories. Among various in-situ ARPES measurements <cit.>, there is consensus in a gap-closing temperature T_c ∼65 K. Some variation exists with the degree of post-growth annealing <cit.> (see Subsection <ref> for details). Enhancement of T_c up to 75 K is possible if extra tensile strain is introduced through an additional KTaO_3 substrate <cit.>. A more robust proof of superconductivity would include (1) a zero-resistance state and (2) the Meissner effect (perfect diamagnetism). Due to air sensitivity, ex-situ transport and thermodynamic measurements require film capping, with amorphous Se <cit.>, amorphous Si <cit.>, or epitaxial FeTe <cit.>. In all cases, film characteristics were degraded. Transport measurements of capped heterostructures have found a zero-resistance state below ∼20 K, and a rough onset temperature possibly up to ∼50 K. Similarly, magnetization measurements of capped samples have suffered from weak signals, broadened onset temperatures, or low superconducting volume fractions.Given that many potential applications require some degree of atmosphere exposure, it remains crucial to investigate why capping, particularly epitaxial FeTe, has not worked well. FeTe possesses the same crystal structure as FeSe and its layers interact via van der Waals forces, so naively it should not create a severe disturbance of the FeSe layer below. Several hypotheses have been put forward. Ultrafast spectroscopy revealed an acoustic phonon mode in FeTe that may relax phonon-mediated pairing in FeSe <cit.>. Alternatively, cross-sectional TEM revealed that intermixing with the capping layer can occur, whereby Te atoms substitute Se atoms in the FeSe monolayer <cit.>. As a third possibility, Zhao et al. proposed that FeTe may hole-dope FeSe, reducing T_c <cit.>.In Table <ref>, we distinguish heterostructures that have conducting, Nb-doped SrTiO_3 from those that do not (undoped, bulk-insulating SrTiO_3). In general, transport measurements require an insulating SrTiO_3 substrate, but there are speculations that Nb-doped SrTiO_3 produces higher quality films. Sun et al. <cit.> hinted that “high quality FeSe films are easier to be achieved by MBE growth on conductive STO [SrTiO_3] substrates comparing to insulating STO substrates since the conductive STO substrate shows more flat and homogeneous surface for sample growth."§.§.§ In-situ, micro-four-point measurementsIn late 2014, Ge et al. reported an astonishing new record T_c above 100 K in 1UC FeSe/SrTiO_3 <cit.>. Here, we review their experiment in detail. The authors converted a commercial cryogenic STM into an in-situ, micro-four-point probe by replacing the single STM tip with a set of four Cu/Au wires, separated by 10-100 μm (Figure <ref>a). The four probes were collectively brought towards the sample at a 20^∘ incline using the STM positioning system, until Ohmic contact with the sample was established for each probe. Figure <ref>c shows several four-point I-V curves, which transition from a nonlinear (superconducting, zero resistance) to linear (normal state, Ohmic) line shape as the temperature was raised above T_c.Due to sample inhomogeneity or film damage from probes, linear I-V curves were sometimes observed below T_c. As a result, Ge et al. compiled resistance vs. temperature (R-T) plots in two manners. First, they acquired four-point I-V measurements from separate locations for each temperature (Figure <ref>b). As long as one I-V curve per temperature showed signs of zero resistance, that temperature was deemed to be below T_c. With this method, Ge et al. determined T_c to be 109 K. Alternatively, they were also able to construct R-T plots from measurements at one location, with a sequence of decreasing magnetic fields (Figure <ref>d). With this second method, they demonstrated a similar T_c of 99 K. The magnetoresistance measurements in Figure <ref>e were also acquired at a fixed location.We enumerate questions that have been raised about this experiment, and the authors' responses: * Question: Is the result reproducible on multiple samples? Response: Ten different samples show similar results. Data from four samples are shown in the paper.* Question: Is it possible that the authors simply lost a current lead contact as they cooled, resulting in a sudden drop of the measured V to zero? Response: No, the authors measured full I-V curves at each temperature and magnetic field (B), and extracted a T-dependent and B-dependent critical current (e.g. see Figure <ref>d,e).* Question: Is it possible that the actual T of the sample is lower than the recorded T, giving the appearance of higher T_c? Answer: No, careful calibration measurements show that the temperature of the sample is never more than 2 K less than the recorded temperature.* Question: SrTiO_3 undergoes a structural transition at 105 K. Could this be responsible for the resistive transition observed at 109 K?Response: The authors performed a control experiment on bare, Nb-doped SrTiO_3, and showed that the structural transition produced a negligible signature in the R-T plot [Figure 3b of Ref. <cit.>]. * Question: Don't the measured values of T_c = 109 K and J_c = 1.3 × 10^7 A/cm^2 appear unexpectedly large?Response: The authors performed a control experiment on optimally-doped Bi_2Sr_2CaCu_2O_8+δ, and found T_c = 90 K, J_c ∼ 6000 A/cm^2, in line with expectations. Their J_c value is an order of magnitude higher than that of capped 1UC FeSe/SrTiO_3 films <cit.>, but similar to that of YBa_2Cu_3O_7-x films <cit.>.* Question: Shouldn't there be a Berezinsky-Kosterlitz-Thouless (BKT) effect that broadens the resistive transition for a 2D superconductor? Why is the resistance drop so sharp (Figure <ref>b), such that there are no data points within the transition <cit.>? Response: Below T_c, conduction is 2D and restricted to the superconducting FeSe monolayer. Above T_c, conduction is shorted through the Nb-doped SrTiO_3 substrate, which is 3D and has a much lower resistivity than normal-state FeSe. Thus, the BKT transition is masked by shorting through the metallic substrate. It is also possible that there could be a proximity effect downward into SrTiO_3, such that the total system is not exactly 2D. The authors were able to collect data points within this sharp transition (Figure <ref>d). * Question: In light of the previous question, why not use an insulating SrTiO_3 substrate? Response: The authors cited practical challenges <cit.>: “Further limits exist for detecting films grown on an insulating substrate, as the feedback required to control the contact between the film and the tip is extremely difficult.”* Question: Why doesn't the resistance change when the contact separation is increased tenfold <cit.>? Response: When the probe separation distances are uniform, the resistance should scale with probe separation in both an infinite 2D conductor and a half-infinite 3D conductor. However, when the probe distances are unequal, their relationship to the overall resistance is more complicated [see Supplemental Information of Ref. <cit.>].* Question: How could the resistive transition T_c be higher than the gap-closing temperature T_c measured in-situ by ARPES? Response: ARPES averages signal over a beam spot size, but the in-situ four-point probe may pick up filamentary superconductivity. Indeed, the authors found non-superconducting regions below T_c, but this could be attributed to both intrinsic sample inhomogeneity or film damage from probes. Alternatively, if the out-of-plane coherence length is short, superconductivity might be stronger at the bottom of the FeSe triple layer than at the top. ARPES and STM measure the top, but transport accesses the lowest-resistivity part, which may be located at the buried interface.* Question: Is it possible that the apparent decreasing T_c with increasing B is simply due to gradual sample damage as B is increased? Response: No, the authors showed the same result with increasing B and decreasing B at a fixed location.Despite intense scrutiny, we remain unaware of fatal flaws with the experiment by Ge et al. Nevertheless, there are increasing calls for duplication of this result, as well as complementary in-situ magnetization measurements of the Meissner effect <cit.>. The latter will require specific instrumentation, but will surely fill in an important piece of the puzzle. §.§ What are the necessary growth conditions?An accurate atomic structure is prerequisite to reliable modeling of electronic properties, and yet the former represents another significant experimental challenge for 1UC FeSe/SrTiO_3. Although SrTiO_3 is a workhorse substrate for MBE growth, it is notorious for its numerous nearly-degenerate surface reconstructions that sensitively depend on preparation conditions. With the (001) surface alone, O deficiency can drive the following reconstructions: 2×1, 2×2, c(4×2), c(4×4), 4×4, c(6×2), √(5)×√(5) R26.6^∘, √(13)×√(13) R33.7^∘ <cit.>. Yet some feature of this complex surface interfaced with 1UC FeSe must generate a giant enhancement in T_c. Here, we examine and clarify growth procedures for 1UC FeSe/SrTiO_3. The overall challenge is to identify which steps are necessary and which are supplemental. In the following subsection, we review various measurements of the interface atomic structure.Figure <ref> presents a flowchart with typical growth recipes for 1UC FeSe/SrTiO_3. The recipes can be delineated into a few “primary” steps, which we discuss in turn. §.§.§ SrTiO_3 treatment (in situ and ex situ)Commerically available crystals of SrTiO_3 arrive with contaminated surfaces. In their original report, Wang et al. introduced a novel strategy to clean Nb-doped SrTiO_3: they annealed the substrates in their MBE chamber at 950 ^∘C for 30 minutes, under a Se flux. <cit.>. This treatment produced atomically-flat terraces amenable to STM imaging (albeit lacking atomic resolution). Subsequently, Bang et al. hypothesized that this process created Se substitutions of surface O atoms <cit.>. These Se_O substitutions would then nucleate the growth of the first FeSe monolayer, leaving behind O vacancies that stabilize binding and donate electron carriers.Later films grown on insulating SrTiO_3 involved more conventional and better documented preparation protocols, involving an ex-situ H_2O/acid etch followed by a high-temperature O_2 anneal in a tube furnace <cit.>. The H_2O/acid etch is believed to preferentially remove SrO, which has ionic bonding character, and leave behind a TiO_2-terminated surface <cit.>. It is thus unclear whether the previously-employed Se etch is a necessary procedure for growing epitaxial FeSe on SrTiO_3. Despite the explicit absence of this step here, it is possible that Se_O substitutions are still generated during the deposition of 1UC FeSe.§.§.§ FeSe depositionTo grow stoichiometric FeSe, two conditions are typically employed <cit.>. First, since Se is significantly more volatile than Fe, the substrate temperature is set between the source temperatures: T_Fe > T_substrate > T_Se. At least for growth on “inert,” graphitized SiC, this condition was rationalized as follows: Impinging Fe atoms with temperature ∼ T_Fe will be adsorbed with sticking coefficient close to unity, while impinging Se atoms can stick only if they bind to free Fe on the substrate. Second, to compensate for high Se losses and to mitigate excess Fe clustering, typical molar flux ratios Φ_Se/Φ_Fe range from 5 to 20. We note that with these two conditions (moderate substrate temperature 400-500 ^∘C, excess Se flux), there may still be a sizeable Se chemical potential at the SrTiO_3 surface driving the kinds of Se reactions proposed by Bang et al., but further investigations by STM or other techniques are needed.§.§.§ Post-growth annealingPost growth, the FeSe monolayer on SrTiO_3 becomes superconducting only after an additional vacuum anneal. He et al. used ARPES measurements to show that in this process, the FeSe monolayer is progressively doped with electron carriers <cit.>. The electron doping induces a non-rigid band transformation that eventually leaves the FS with only electron pockets and opens up a gap. The source of electron doping remains an open question. He et al. suggested that the electron doping could arise from O vacancies in SrTiO_3 created during annealing. Berlijn et al. investigated the possibility of Se vacancies, but their calculations revealed Se vacancies to be hole dopants, not electron dopants <cit.>. More recently, cross-sectional TEM imaging by Li et al. suggested the presence of interstital Se atoms trapped at the FeSe/SrTiO_3 interface during growth, which are subsequently released upon annealing. The authors proposed that the removal of these interstitial Se atoms allows O vacancies in SrTiO_3 to effectively donate electron carriers to the FeSe monolayer <cit.>.Overall, some elements of “correct” SrTiO_3 pre-treatment and post-growth annealing appear necessary to produce superconducting 1UC FeSe/SrTiO_3, but many aspects of the growth procedure could be clarified through more systematic investigations. §.§ What is the interface structure? We begin by comparing and contrasting three tools that have been applied to probe the interface atomic structure. * Scanning tunneling microscopy (STM): real-space, atomic-resolution imaging of surface Pro: An in-situ technique commonly integrated with a MBE chamber. Con: An indirect technique that requires additional modeling to make inferences about the buried interface.* Electron diffraction: low-energy (LEED) or reflection high-energy (RHEED): momentum-space information of surface atomic structure Pro: An in-situ technique that can also monitor real-time growth (RHEED). Con: Phase information is unavailable. The interface signal may sometimes be buried after FeSe deposition <cit.>.* Transmission electron microscopy (TEM): real-space, atomic-resolution imaging of exposed cross-section Pro: Direct atomic-resolution imaging of the interface cross-section. Con: An ex-situ technique that requires capping (commonly FeTe). As evinced by Ref. <cit.>, Te atoms from the cap may unintentionally intermix and substitute at least the top-layer Se atoms of 1UC FeSe. The size mismatch between Se and Te can strain the monolayer film, possibly altering its original binding structure to SrTiO_3. Additionally, the technique averages over each column of atoms in the ∼10-100 nm thick section being studied.§.§.§ 2×1 reconstruction The first hint of any interface superstructure was the appearance of dark stripes with 2×1 periodicity in STM topographic images (Figure <ref>a) <cit.>. To explain this structure, Bang et al. proposed an atomic model where half the O atoms on the surface TiO_2 layer are stripped off, and the bottom-layer Se atoms of the FeSe monolayer are laterally registered with the O vacancy sites <cit.>. The authors argued that such arrangement could increase the binding energy, electron-dope the FeSe monolayer, and cause the FeSe monolayer to relax with a 2×1 superstructure. In addition, since there are two equivalent O sites within a TiO_2 UC, the model could naturally explain the observation of half-UC phase shifts that occur either discontinuously at a trench <cit.> (Figure <ref>a), or continuously within a few nanometers of a domain boundary <cit.>. As a word of caution, the 2×1 stripes have not been universally observed. They are absent in AFM topographies <cit.>, which might point to an electronic origin of the stripes, and are also absent in STM topographies of samples prepared in different ways <cit.>. §.§.§ More reconstructions Lee et al. grew 1UC FeSe on SrTiO_3 with neither in-situ Se etching nor ex-situ treatment <cit.>. They simply annealed as-bought substrates up to 830 ^∘C in their MBE chamber, until RHEED measurements detected superstructure spots, typically but not necessarily √(5)×√(5) <cit.> (Figure <ref>b). Subsequent deposition of FeSe and post-growth anneal resulted in superconducting samples with 1×1 diffraction spots. Peng et al. found a qualitatively different behavior in 1UC FeSe/BaTiO_3 <cit.>. After annealing BaTiO_3 at 950 ^∘C under Se flux, their LEED images exhibited 3×3 spots. Curiously, growth of 1UC FeSe produced three distinct domains: one domain commensurate with the BaTiO_3 1×1 UC, with expanded lattice constant 3.99 Å; two domains rotated by ±18.5^∘, commensurate with a BaTiO_3 3×3 supercell, with smaller lattice constant 3.78 Å. Furthermore, ARPES detected superconducting gaps in all three regions, with closing temperature T_c ranging from 70-75 K. More recent experiments have detected superconducting gaps in 1UC FeSe on STO(110) <cit.>, anatase TiO_2(001) <cit.>, and rutile TiO_2 <cit.>, with different lattice constants and surface reconstructions (prior to growth). Taken together (Table <ref>), the variety may imply that neither lattice constant nor the lateral atomic registry between 1UC FeSe and its underlying substrate are critical factors behind the enhanced superconductivity of this heterostructure.§.§.§ Double-TiO_x termination Perhaps what matters is the vertical structure of the interface. Using cross-sectional TEM, Li et al. <cit.> imaged a double-TiO_x termination at the interface of FeTe/1UC FeSe/SrTiO_3 (Figure <ref>c). Zou et al. <cit.> also uncovered a double-TiO_x termination using x-ray diffraction, LEED and RHEED. Although such termination had long been proposed as a candidate model for the 2×1 surface reconstruction <cit.>, it had largely been neglected in atomic models of 1UC FeSe/SrTiO_3 until this point. Roughly speaking, the extra TiO_x termination is half as polar as a bulk TiO_2 layer, and helps SrTiO_3 mitigate a divergence of the electrostatic potential towards its bulk <cit.>. Structural and ferroelectric properties are likely modified near this double-TiO_x termination. Zou et al. argued that the double-TiO_x termination faciliates epitaxial growth of FeSe through stronger binding, and also improves charge transfer from oxygen vacancies <cit.>.Li et al. also used TEM imaging to extract the structural parameters of their FeTe-capped sample. They found the 1UC FeSe to have a 9.5% reduced chalcogen height with 2.5% in-plane lattice tensile strain (compared to bulk values). Furthermore, within a ∼10 nm cross section, the authors imaged a lateral half-UC shift between the bottom Se atoms and topmost Ti atoms. If this feature is characteristic of uncapped 1UC FeSe/SrTiO_3, then it suggests that the heterostructure has local bond disorder due to lattice incommensuration. We note that STM dI/dV measurements do reveal spectral and gap inhomogeneity even in pristine regions of FeSe with no in-plane, atomic-scale defects <cit.>. Further systematic investigations and correlation of disorder with growth procedures is needed.§ ELECTRONIC STRUCTURE AND PAIRINGHaving surveyed a range of experiments characterizing the basic properties of 1UC FeSe/SrTiO_3, we turn to the question of electronic structure and pairing. Superconductors are typically categorized into one of two paradigms: conventional or unconventional (Table <ref>). In a conventional superconductor, electrons are bound into Cooper pairs by attractive interactions mediated by phonons. The resulting energy gap has s-wave angular symmetry and a uniform sign throughout the BZ. In an unconventional superconductor, many believe that quantum fluctuations from a proximate phase (e.g. magnetism) provide the glue to bind electrons. Since these fluctuations are often repulsive, the resulting gap function harbors sign changes throughout the BZ (to be further discussed in Subsection <ref>). This latter class of superconductors, which includes the cuprates and iron pnictides, has long been associated with higher T_c values. However, the tables have turned with the recent discovery of 203 K conventional superconductivity in pressurized H_3S <cit.>. Within months of the 2008 discovery of iron pnictide superconductors, Mazin et al. <cit.> and Kuroki et al. <cit.> proposed an unconventional mechanism of pairing in these compounds. The basic premise was that first, the electron-phonon coupling constant was too small <cit.>; second, the proximity of the superconductor to an antiferromagnetic metal hinted at the role of spin fluctuations; and third, the multiband FS of these compounds, comprising electron pockets at the zone corner M and hole pockets at the zone center Γ, could be crucial. The authors then argued that repulsive spin fluctuations, with wave vector spanning the separated electron and hole pockets, could pair electrons if the gap function reversed sign across the pockets with an overall “s_+-” symmetry (Figure <ref>). Though not free from controversy <cit.>, this framework prevailed in the early years of iron pnictide superconductors. Fast-forwarding to the present, 1UC FeSe/SrTiO_3 poses several theoretical conundrums. First, its FS includes M electron pockets (Figure <ref>e), but not the Γ hole pockets necessary for s_+- pairing. Second, it appears to exhibit traits of both conventionality and unconventionality. In this section, we review contrasting indications for both phononic (Subsection <ref>) and electronic (spin/orbital, Subsection <ref>) mechanisms of pairing. We then evaluate ARPES and STM measurements of gap symmetry and structure (Subsection <ref>). Finally, we discuss multiband, multiboson scenarios of pairing that enable phonons and spin/orbital fluctuations to operate constructively to enhance T_c (Subsection <ref>). These latter ideas are far from being a fait accompli, but exemplify a potential “best-of-both-worlds” path towards creating higher-T_c superconductors. §.§ Phononic mechanisms In their original report, the discoverers of 1UC FeSe/SrTiO_3 proposed some sort of interface-enhanced electron-phonon coupling as the mechanism for high-T_c superconductivity. Here, we discuss subsequent ARPES experiments by Lee, Schmitt, Moore et al. that lent support to this notion <cit.>.§.§.§ Replica bandsWhat Lee, Schmitt, Moore et al. discovered in their ARPES measurements was that each primary electronic band of 1UC FeSe/SrTiO_3 possessed a fainter replica band offset by 100 meV <cit.>. These faint bands were near-duplicates of their primary counterparts, without being offset in momentum or smeared (Figure <ref>a,b). In addition, the replica bands persisted at least to 120 K, well above the gap-opening temperature (T_c = 58±7 K). Such replicas were absent in FeSe films two UC or thicker (Figure <ref>c,d), pointing to an interfacial origin of these features. Similar phenomenology was observed by Peng et al. in 1UC FeSe/BaTiO_3 <cit.>. In their interpretation of the replica bands, Lee, Schmitt, Moore et al. first excluded the possibility of quantum-well states arising from 2D confinement. There is no reason for such states to have identical dispersions. Furthermore, quantum-well states exhibit a well-behaved dependence on layer, in contrast to the abrupt disappearance of replica bands in 2UC FeSe/SrTiO_3. Instead, the authors attributed the replica bands to bosonic shake-off, in analogy to vibrational shake-off observed in photoemission spectroscopy of H_2 molecules. They identified the boson with an optical O phonon band calculated for bulk SrTiO_3 <cit.>. Subsequent calculations of slab SrTiO_3 pointed to a surface phonon mode involving polar vibrations of vertical Ti-O bonds <cit.>. These theoretical comparisons were later corroborated by ARPES measurements that also found replica bands on bare SrTiO_3 <cit.>. However, for an electron-phonon coupling g(q⃗) to produce nearly-identical bands with no momentum smearing, it must be sharply peaked at q⃗ = 0. This differs from the usual assumption of a constant g(q⃗) in theories of phonon-mediated superconductivity, and some modeling is needed to understand its origin.§.§.§ Model of interface electron-phonon couplingTo explain how the electron-phonon coupling g(q⃗) could become sharply peaked at q⃗ = 0, Lee, Schmitt, Moore et al. presented the following model <cit.>: Assume we have a 2D sheet of FeSe at z = 0, and a layer of dipole moments below at the SrTiO_3 surface, z = -h_0 (Figure <ref>). The dipole moments come from vertical stretching of surface Ti-O bonds and are represented by δ p_z(x, y, -h_0). From an electrostatics calculation, these moments induce a potential at the FeSe layer,δΦ(x, y, 0) = ϵ_∥ h_0/ϵ^3/2_⊥∫ dx' dy' δ p_z(x', y', -h_0)/[ϵ_∥ h_0^2/ϵ_⊥ + (x-x')^2 + (y-y')^2]^3/2,where ϵ_∥, ϵ_⊥ are the in-plane and perpendicular dielectric constants in the interface region. Taking the Fourier transform yieldsδΦ(q⃗, 0) = √(ϵ_∥/ϵ_⊥)2π/√(ϵ_⊥)exp[-|q⃗| h_0 √(ϵ_∥/ϵ_⊥)] δ p_z(q⃗, -h_0).It follows that g(q⃗) ∝δΦ(q⃗, 0) ∝exp (-|q⃗|/q_0), where q^-1_0 = h_0 √(ϵ_∥/ϵ_⊥). Intuitively, the q⃗ ∼ 0 coupling hinges upon (1) the FeSe monolayer being sufficiently removed from the dipole layer (large h_0), and (2) the interface region screening lateral charge imbalance much more effectively than vertical charge imbalance (large ϵ_∥/ϵ_⊥). Calculations by Rademaker et al. showed that a ratio of q_0/k_F ∼ 0.1 was needed for replica bands to duplicate primary band features without significant momentum smearing <cit.>. If we take k_F to be 0.20 Å^-1 <cit.> and h_0 to be 4.9 Å <cit.>, the distance between the surface TiO_2 layer and the Fe-plane, then 1/(h_0 k_F) ∼ 1 and we require ϵ_∥/ϵ_⊥ ∼ 100 in the interface region. While one should be wary of interpreting the model interface too literally, an argument suggests that it should contain contributions from both SrTiO_3 and FeSe, with the former having ϵ^STO_∥ ∼ ϵ^STO_⊥ in its 3D bulk limit, and the latter having ϵ^FeSe_∥ ≫ ϵ^FeSe_⊥ due to its 2D nature <cit.>.Alternative speculations regarding the replica bands include O impurity bands <cit.>, or some form of Raman scattering involving SrTiO_3 phonon modes <cit.>. No model details have been presented for impurity bands, and it is unclear whether sufficient cross section and viable selection rules exist for Raman scattering. Peaks and dips have been detected in STM filled- and empty-state d^2I/dV^2 spectra [Supplemental Material of Ref. <cit.>], but the authors have not confirmed their identity as replica bands.§.§.§ FeSe phonon modes In their initial STM measurements of 1UC FeSe/SrTiO_3, Wang et al. reported two gaps in the dI/dV point spectrum, at 9 meV and 20.1 meV respectively <cit.>. This finding appeared to contradict early ARPES measurements of a single isotropic gap on the zone corner electron pockets, with Δ = 13±2 meV in one film and 15±2 meV in another film <cit.>. Coh et al. offered an alternative explanation for the double-gap signature in terms of two FeSe phonon modes, which they argued could enhance T_c when a monolayer of FeSe is locked to a SrTiO_3 substrate <cit.>. Following this report, Tang et al. examined d^2I/dV^2 point spectra of 1UC FeSe/SrTiO_3 and K-coated 2-4 UC FeSe/SrTiO_3 <cit.>. They identified positive-energy dips around 11 meV and 21 meV as FeSe phonon modes. §.§ Electronic mechanismsA feature of the interface phonon-coupling model is that it does not depend at all on 1UC FeSe, as long as the heterostructure has an interface dielectric constant that is sufficiently anisotropic. On one hand, such generality could be desirable for reproducing this mechanism in other systems. On the other hand, the model leaves open the possibility of preexisting pairing interactions within FeSe that are subsequently strengthened by SrTiO_3. Two foil systems suggest that interface phonon coupling plays a secondary role to a primary pairing mechanism within FeSe that is enhanced by electron doping. The first is an FeSe-intercalate, (Li_1-xFe_x)OHFeSe. Fe_Li antisite substitutions increase electron transfer from buffer LiOH layers to FeSe <cit.>, resulting in 40 K superconductivity <cit.>. Importantly, ARPES and STM measurements resolved low-energy bands that are nearly identical to those of 1UC FeSe/SrTiO_3 and gaps of similar magnitudes <cit.>; however, no replica bands were visible. The second system involves coating FeSe with K adatoms, which inject electrons into the surface FeSe layer <cit.>. The resulting superconducting phase has a gap-closing temperature up to 48 K, close to the 65 K value of 1UC FeSe/SrTiO_3. The electronic transition induced by progressive K deposition is rather rich and provides clues of unconventional mechanisms of pairing.§.§.§ Clues from the electron-doping phase diagramIn the paradigm of unconventional superconductors, pairing is likely mediated by quantum fluctuations from nearby electronic phases. Hence, we glean inspiration from the electron-doping phase diagram of FeSe, keeping in mind that it may not be fully representative of 1UC FeSe/SrTiO_3. The first striking feature in the electron-doping phase diagram of FeSe is that T_c evolves through two domes <cit.>: a low-T_c phase is first suppressed, eventually giving way to a higher-T_c phase (Figure <ref>). In general, domes are hallmarks of unconventional superconductivity, less naturally explained within a purely phononic framework <cit.>. Song et al. also found that the higher-T_c phase is insensitive to the disorder of nonmagnetic K adatoms, a point whose implications we revisit in Subsection <ref>. A second observation is that nematic order is suppressed preceding the high-T_c phase <cit.>, although a smaller overlapping tail of the nematic phase may persist due to remnant uniaxial strain from underlying bulk FeSe <cit.>. Nematic order is generally defined as broken rotational symmetry that preserves the translational symmetry of the crystal. In stoichiometric FeSe, nematic order is manifested as a small orthorhombic distortion <cit.> and a large splitting of the Fe 3d_xz and 3d_yz bands <cit.>, without concomitant magnetic order <cit.>. Given the proximity and possible overlap of the nematic phase, it is tempting to ask whether nematic quantum criticality could be at play. Nematic fluctuations would provide attractive q⃗∼ 0 interactions that help bind electrons <cit.>, much like the aforementioned SrTiO_3 phonons.§.§.§ Nematic fluctuationsSince 1UC FeSe bound to SrTiO_3 is nominally tetragonal, nematic order should be globally suppressed. However, if there truly exists a large underlying nematic susceptibility that produces fluctuations, then nanoscale patches of such fluctuations might be pinned around crystalline imperfections that locally break tetragonal symmetry. Using STM as a nanoscale probe, Huang et al. investigated quasiparticle interference (QPI) patterns generated around anistropic defects in 1UC FeSe/SrTiO_3 <cit.> (Figure <ref>). Since QPI anisotropy can arise from random disorder or experimental artifacts, the authors developed a realistic, T-matrix model to specifically detect orbital anisotropy of Fe 3d_xz and 3d_yz bands. By sampling multiple spatial regions of a film, they excluded xz/yz orbital ordering with domain size larger than δ r^2 = 20 nm × 20 nm, xz/yz Fermi wave vector difference larger than δ k = 0.014 π, and energy splitting larger than δ E = 3.5 meV. The lack of detectable ordering pinned around defects disfavors scenarios of a proximate nematic quantum critical point in 1UC FeSe/SrTiO_3, in contrast to K-coated FeSe (Figure <ref>).§.§.§ Spin fluctuations Given the importance of spin fluctuations in many iron pnictide superconductors, their role in pairing should also be considered in 1UC FeSe/SrTiO_3 <cit.>. Several inelastic neutron scattering measurements found that in stoichiometric FeSe, there are stripe spin fluctuations that are enhanced below the orthorhombic transition temperature <cit.> (Figure <ref>). However, magnetic order is absent, owing to some sort of frustration <cit.> or quadrupolar order <cit.>. The nature of spin excitations in 1UC FeSe/SrTiO_3 remains an important open question, especially since both doping and the Se height may tune exchange interactions. Some experimental ingenuity is required, as ex-situ neutron scattering measurements are likely not feasible on 1UC films. One possible approach is the use of STM to map the magnetic-field dependence of impurity signatures, in comparison with theoretical modeling <cit.>.§.§ Gap symmetry and structureWe shift gears and consider pairing from the viewpoint of gap symmetry and structure. In general, such questions have proven more challenging to address in the iron-based superconductors than in the cuprates. Unlike the cuprates, with a single Cu d band and universal d_x^2-y^2 gap symmetry, the multiband FS the of iron-based superconductors can allow a variety of gap structures across their member compounds. Even within the same compound, such as KFe_2As_2, different pairing symmetries can be tuned by pressure <cit.>. To add to the challenge, many candidate gap structures share the same angular symmetry <cit.>, and thus cannot be differentiated by the corner junction experiments that proved instrumental in revealing the d-wave gap of YBa_2Cu_3O_7-x <cit.>. ARPES can resolve gap magnitudes on each specific band, but not their signs. STM QPI measurements carry phase-sensitive information, but can be challenging to interpret or normalize <cit.>. With these complications in mind, there is less likely to be a clear,“smoking-gun” experiment revealing the gap symmetry of 1UC FeSe/SrTiO_3. A more likely scenario is that through multiple experimental measurements, consensus will begin to converge upon a candidate gap function.§.§.§ The candidates Given the Fermi surface of 1UC FeSe/SrTiO_3, with only electron pockets, the primary gap symmetry candidates are “plain” s, “nodeless” d, “bonding-antibonding” s, and “incipient” s_+- (Figure <ref>). Nodal candidates are inconsistent with the fully-gapped structures detected by STM and ARPES (Figure <ref>b). “Plain” s gap symmetry (Figure <ref>a,b) will be discussed in Subsection <ref>. “Nodeless” d (Figure <ref>c,d), which appears most similar to the gap in cuprates, is strictly defined in a 1-Fe, “pseudocrystal momentum” BZ (k̃⃗̃) that only exists when FeSe has exact glide-plane symmetry. When the gap structure is folded into the proper 2-Fe BZ, it is no longer d-wave with respect to regular crystal momentum (k⃗). In addition, nodes are technically created when opposite-sign gaps meet at the pocket crossings <cit.>. However, based on microscopic details, the nodal quasiparticle weight could be weak and elude spectroscopic detection <cit.>.If the folded pockets in the 2-Fe BZ hybridize and detach from each other, then nodes will certainly be avoided, leading to a “bonding-antibonding” s scenario (Figure <ref>e). Here, the inner and outer pockets host gaps of opposite sign.“Incipient” s_+- posits that an opposite-sign gap develops on a sunken zone center hole pocket (Figure <ref>f,g). In the weak-coupling limit, such a gap can still be sizeable <cit.>, the reasons for which we discuss in Subsection <ref>.§.§.§ ARPES measurements Early ARPES investigations of 1UC FeSe/SrTiO_3 reported isotropic gaps (Δ = 13-15 meV) on nearly-circular electron pockets <cit.>. To resolve finer structure, Peng et al. grew 1UC FeSe on SrTiO_3/KTaO_3, whose expanded lattice constant increased pocket ellipticity <cit.>. Alternatively, Zhang et al. changed photon polarizations to selectively probe bands of different orbital characters <cit.>. In both cases, the authors observed two pockets at each corner (main and folded), with no signs of hybridization (Figure <ref>a-c). Momentum distribution cuts across the intersection of the main and folded pockets revealed a single band, with no detectable splitting. Furthermore, gap measurements on equivalent segments of the main and folded pockets showed identical structure. Such lack of sizeable hybridization remains to be understood, given that both spin-orbit coupling or the SrTiO_3 substrate can break glide-plane symmetry. More importantly, it also disfavors scenarios of “bonding-antibonding” s-wave pairing. The authors in both reports also resolved gap anisotropy, with minima directed along the Fe-Se axes. These measurements will provide useful feedback for theoretical gap function calculations.§.§.§ STM measurements Fan et al. employed a multi-pronged STM approach, involving phase-sensitive QPI and defect imaging, to build support for “plain” s-wave superconductivity in 1UC FeSe/SrTiO_3 <cit.>. In particular, the authors found that magnetic adatoms (Cr, Mn) induced in-gap bound states, while non-magnetic adatoms did not (Zn, Ag, K) (Figure <ref>d-g). This observation is consistent with an underlying gap structure without sign changes, but not a fool-proof guarantee of such. Anderson's theorem states that a superconductor with a sign-preserving gap should be robust against the disorder of nonmagnetic impurities. Taken in its equivalent, contrapositive form, the observation of in-gap states induced by nonmagnetic impurities would thereby signal a sign-changing gap. However, the converse statement (“robustness against nonmagnetic impurities” ⟹ “sign-preserving gap”) is not logically identical to the original theorem, so it lacks a“smoking-gun” nature <cit.>. In the case of 1UC FeSe/SrTiO_3, with the “nodeless” d and “bonding-antibonding” s gap structures, the opposite-sign gaps reside on normal-state Fermi pockets with different orbital characters. Given that the impurities in the Fan et al. experiment outside the Fe-plane, they may have had insufficient interorbital scattering strength to produce a pair-breaking effect. This complication is alleviated in the case of s_+- pairing in the iron pnictides. Since both the electron and hole pockets hosting opposite-sign gaps share the same orbital characters, interband scattering mixing the signs is easier.Despite these caveats, the preponderance of current experiments favor same-sign gaps on all Fermi pockets of 1UC FeSe/SrTiO_3 above the other possibilities.§.§ Multiband, multiboson scenarios of pairing In this final subsection, we examine pairing scenarios in which multiple bosons work cooperatively across multiple bands to boost T_c. More specifically, we consider various ways in which attractive interactions (e.g., mediated by phonons) and repulsive interactions (e.g., mediated by spin fluctuations) can fit under the same roof in 1UC FeSe/SrTiO_3 <cit.>.The basic picture can be explained from the T=0 gap equation of a one-band superconductor in the weak-coupling limit: Δ_k⃗ = -∑_k⃗'V_k⃗, k⃗'Δ_k⃗'/2E_k⃗'⃗.Here, V_k⃗, k⃗' is an effective potential that scatters a Cooper pair from (k⃗↑, -k⃗↓) to (k⃗'↑, -k⃗'↓), E_k⃗'⃗ = √(ξ^2_k⃗'⃗ + |Δ^2_k⃗'⃗|) > 0 is the Bogoliubov quasiparticle energy, ξ_k⃗'⃗ is the normal-state quasiparticle energy, and Δ_k⃗ is the gap function. Since any such Δ_k⃗ must obey Eq. <ref> self-consistently, its form is determined by V_k⃗, k⃗' as follows:* Attractive interactions (V_k⃗, k⃗' < 0) increase the gap amplitude if they connect segments of the FS hosting same-sign gaps (Δ_k⃗ > 0 and Δ_k⃗'⃗ > 0, or Δ_k⃗ < 0 and Δ_k⃗'⃗ < 0).* Repulsive interactions (V_k⃗, k⃗' > 0) increase the gap amplitude if they connect segments of the FS hosting opposite-sign gaps (Δ_k⃗ > 0 and Δ_k⃗'⃗ < 0, or Δ_k⃗ < 0 and Δ_k⃗'⃗ > 0).Attractive and repulsive interactions can therefore simultaneously increase the gap amplitude, if the interactions connect different segments of the FS, with appropriate signs in the gap function. In the limit of forward scattering (k⃗ = k⃗'), attractive interactions have the form V_k⃗, k⃗'∝ -δ_k⃗, k⃗'⃗, and from Eq. <ref>, increase the gap amplitude irrespective of the gap sign or functional form. Figure <ref> illustrates a pairing framework for 1UC FeSe/SrTiO_3 involving multiple bosons. We postulate the existence of a “primary” interband interaction peaked around q̃⃗̃ = (π, π), connecting the disparate electron pockets and dictating the overall gap symmetry (Figure <ref>a,b). This interaction could be a repulsive antiferromagnetic spin fluctuation, stabilizing d-wave pairing, or an attractive antiferroorbital fluctuation, stabilizing s-wave pairing. Then in addition, there may be “enhancer” intraband interactions that are necessarily attractive and peaked around q̃⃗̃ = (0, 0) (Figure <ref>c,d). These interactions universally boost pairing irrespective of the underlying gap symmetry, and may come from SrTiO_3 phonons <cit.>, and/or nematic fluctuations.Although this pairing framework is appealing due to its inclusive nature, we emphasize that other than the SrTiO_3 phonon mode, there have been no experimental indications of the other interactions shown in Figure <ref>. Some suggest that DFT calculations of 1UC FeSe/SrTiO_3 with checkerboard antiferromagnetism [q̃⃗̃ = (π, π)] best resemble experimental data <cit.>, thus motivating the possible existence of related spin fluctuations. Others take the orbitally-ordered state of bulk FeSe as a hint of possible ferroorbital [nematic, q̃⃗̃ = (0, 0)] or antiferroorbital [q̃⃗̃ = (π, π)] fluctuations <cit.>.§.§.§ Pairing involving incipient bandsAlternatively, we recall that inelastic neutron scattering measurements have detected stripe spin fluctuations [q̃⃗̃ = (π, 0)] in bulk FeSe <cit.>, similar to many iron pnictide compounds. At first glance, it is unclear whether such interactions, if they persist in 1UC FeSe/SrTiO_3, would be useful for pairing. The usual hole pocket at the BZ center, located Δk̃⃗̃ = (π, 0) away from the Fermi electron pockets, is sunken 65-80 meV below the Fermi energy (Figure <ref>). However, ARPES measurements have demonstrated that in LiFeAs, a superconducting gap can develop on a sunken hole pocket 10 meV below the Fermi energy <cit.>. Motivated by this observation, Chen et al. proposed a “bootstrap” mechanism of pairing involving incipient bands in 1UC FeSe/SrTiO_3 <cit.>.The basic picture proposed by Chen et al. is illustrated in Figure <ref>a. In the weak-coupling limit, interactions such as q̃⃗̃ = (π, 0) spin fluctuations between a Fermi sheet and an incipient band cannot open up a superconducting gap by themselves. However, if there are preexisting interactions, such as phonons, that stabilize a small gap within the Fermi sheet, then spin fluctuations from incipient bands can come into play and enhance pairing. In 1UC FeSe/SrTiO_3, this mechanism results in “incipient” s_+- symmetry, where the Fermi electron pockets host same-sign gaps and the sunken hole pocket at the zone center hosts an opposite-sign gap. From the author's calculations, incipient bands can boost T_c by an order of magnitude, although numbers are sensitive to estimates of interaction strengths and cutoff.An appealing feature of this model in the context of 1UC FeSe/SrTiO_3 is that the Fermi electron pockets and sunken hole pocket barely or do not overlap in energy (depending on the degree of electron doping). Such a superconducting state would likely be immune to nonmagnetic impurities <cit.>, as elastic scattering would not mix opposite-sign quasiparticles. Indeed, QPI measurements have confirmed that there is no electron-hole pocket scattering near the gap energy <cit.>. Thus, the defect experiments by Fan et al. could also be consistent with “incipient” s_+- gap symmetry.Recently, Huang et al. uncovered a Γ electron pocket 80 meV above E_F (Figure <ref>b), using empty-state STM measurements <cit.>. This pocket may be similar to one discovered in K-coated bulk FeSe_0.55Te_0.45 <cit.> and FeSe <cit.>, or a shallow Z-electron Fermi pocket in 3D (Tl, Rb)_yFe_2-xSe_2 <cit.>. Given that this pocket lies within the SrTiO_3 phonon energy (100 meV), it may be interesting to explore whether it has any positive contribution to T_c in 1UC FeSe/SrTiO_3 <cit.>. § SUMMARY AND OUTLOOK In this review, we have surveyed key experimental and theoretical developments related to 1UC FeSe/SrTiO_3 from its time of discovery, 2012, to early 2016. The major themes we have presented and developed can be captured in the following five statements: * Monolayer FeSe on SrTiO_3 exemplifies a dramatic interface effect, in which a unit-cell layer of free-standing FeSe is non-superconducting down to 2.2 K, but subsequently exhibits T_c ranging from 65 K to 109 K when coupled to SrTiO_3.* Experiments probing an air-sensitive, monolayer film are demanding in nature, but this challenge motivates the development and use of improved in-situ instrumentation, such as four-probe STM, which in turn may lead to discoveries of new systems.* Capping of 1UC FeSe/SrTiO_3 has not yet been optimized. Not only is it crucial for protecting films from atmospheric exposure, it simultaneously provides a second interface that could yet be engineered to enhance electronic properties.* Experiments have uncovered clues of both conventional and unconventional mechanisms of pairing in 1UC FeSe/SrTiO_3. While the ARPES replica bands are best explained by cross-interface coupling to SrTiO_3 phonon modes, more experiments verifying their nature and influence on T_c are desirable <cit.>. And although accumulated experience with iron-based superconductors might suggest the importance of spin (and orbital) fluctuations, direct measurements of magnetic excitations in 1UC FeSe/SrTiO_3 are still needed.* Various pairing scenarios have been proposed in which multiple bosons, such as phonons and spin fluctutations, can work cooperatively in a multiband environment to enhance T_c in 1UC FeSe/SrTiO_3. Although these theories still require experimental confirmation, they represent an appealing “best-of-both-worlds” approach to finding and creating superconductors with even higher T_c. Combined with the layered 2D architecture of 1UC FeSe/SrTiO_3 in which these cooperative effects may be realized and engineered, many possibilies abound down the road. § DISCLOSURE STATEMENTThe authors are not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review.§ ACKNOWLEDGMENTSThe authors would like to thank C.-Z. Chang, S. Coh, S. Fang, J.-F. Ge, P. J. Hirschfeld, E. Kaxiras, S. A. Kivelson, D.-H. Lee, I. I. Mazin, Z.-X. Shen, C.-L. Song, T. A. Webb, and K. Zou for valuable discussions that have informed many of the views and ideas presented in this review. Our work on 1UC FeSe/SrTiO_3 was supported by the National Science Foundation under Grants No. DMR-0847433 and No. PHY-1231319 (STC Center for Integrated Quantum Materials), and the Gordon and Betty Moore Foundation's EPiQS Initiative through Grant. No. GBMF4536. J. E. 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"authors": [
"Dennis Huang",
"Jennifer E. Hoffman"
],
"categories": [
"cond-mat.supr-con",
"cond-mat.str-el"
],
"primary_category": "cond-mat.supr-con",
"published": "20170327205035",
"title": "Monolayer FeSe on SrTiO$_3$"
} |
[email protected] Leibniz Institute for Solid State and Materials Research, 01069 Dresden, GermanyLeibniz Institute for Solid State and Materials Research, 01069 Dresden, Germany Leibniz Institute for Solid State and Materials Research, 01069 Dresden, Germany ITCP and CCQCN, Department of Physics, University of Crete, 71003 Heraklion, Greece Institute for Theoretical Physics, TU Braunschweig, 38106 Braunschweig, Germany Department of Chemistry and Food Chemistry, TU Dresden, 01062 Dresden, GermanyDepartment of Chemistry and Food Chemistry, TU Dresden, 01062 Dresden, GermanyDepartment of Chemistry and Food Chemistry, TU Dresden, 01062 Dresden, Germany Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, TN, USAMaterials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN, USA Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN, USA Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA Department of Physics and Center for Quantum Materials, University of Toronto, 60 St. George St., Toronto, Ontario, Canada M5S 1A7 Department of Physics and Center for Quantum Materials, University of Toronto, 60 St. George St., Toronto, Ontario, Canada M5S 1A7 Leibniz Institute for Solid State and Materials Research, 01069 Dresden, Germany Institute of Solid State Physics, TU Dresden, 01069 Dresden, Germany Center for Transport and Devices, TU Dresden, 01069 Dresden, [email protected] Leibniz Institute for Solid State and Materials Research, 01069 Dresden, Germany Center for Transport and Devices, TU Dresden, 01069 Dresden, GermanyThe honeycomb Kitaev-Heisenberg model is a source of a quantum spin liquid with Majorana fermions and gauge flux excitations as fractional quasiparticles. In the quest of finding a pertinent material, α-RuCl_3 recently emerged as a prime candidate. Here we unveil highly unusual low-temperature heat conductivity κ of α-RuCl_3: beyond a magnetic field of B_c≈7.5 T, κ increases by about one order of magnitude, resulting in a large magnetic field dependent peak at about 7 K, both for in-plane as well as out-of-plane transport. This clarifies the unusual magnetic field dependence unambiguously to be the result of severe scattering of phonons off putative Kitaev-Heisenberg excitations in combination with a drastic field-induced change of the magnetic excitation spectrum. In particular, an unexpectedly large energy gap arises, which increases approximately linearly with the magnetic field and reaches a remarkably large ħω_0/k_B≈ 50 K at 18 T. Large field-induced gap of Kitaev-Heisenberg paramagnons in α-RuCl_3 Christian Hess December 30, 2023 ====================================================================Topological quantum spin liquids (QSL) are characterised by massive quantum entanglement of states and constitute peculiar states of matter where quantum fluctuations are so strong that even in the ground state a magnetic long-range ordering is suppressed. Amazingly, despite the inherent quantum disorder, the QSL are conjectured to possess well-defined quasiparticles. These are highly non-trivial, because unlike classical systems, the QSLs' quasiparticles arise from the fractionalisationin a ground state with topological degeneracy and may have anyonic statistics <cit.>.Since QSL ground states are experimentally elusive, the detection and rationalisation of just these QSL quasiparticles appear as the natural path towards identifying a QSL system.Heat conductivity experiments constitute one of the few probes to study such quasiparticle physics because they provide information on the quasiparticles' specific heat, their velocity, and their scattering <cit.>[Energy density is a local operator and will not probe topological degeneracy, however, fractionalisation will certainly leave characteristic fingerprints in heat transport.].In fact, such experiments have been very revealing in clarifying the unconventional ballistic heat-transport characteristics of one-dimensional spinon excitations, the fractional excitations of the spin-1/2 chain <cit.>, and signatures of unconventional spin heat transport in QSL candidate materials which realise spin-1/2 triangular lattices <cit.> or spin-ice systems <cit.>.Experimental realisations of QSLs generally are rare. In the quest of finding a pertinent material to experimentally investigate their physics, α-RuCl_3 recently emerged as a prime candidate for hosting an approximate Kitaev QSL with Majorana fermions and gauge flux excitations asnew kinds of fractional quasiparticles <cit.>.In this material, strong spin-orbit coupling and an edge-sharing configuration of RuCl_6 octahedra yield a honeycomb lattice ofj_eff=1/2 states with dominant Kitaev interaction <cit.>. Long-range magnetic order at T_N≈7 K occurs in as-grown samples of α-RuCl_3 without stacking faults <cit.>, implying a certain degree of further exchange interaction <cit.>. Remarkably, a moderate in-plane magnetic field of ∼8 T, which is far away from full polarisation<cit.>, is sufficient to completely suppress the long-range magnetic order <cit.>.In order to probe the emergence of unusual quasiparticles in this putative Kitaev-QSL, we have measured the thermal conductivity κ of α-RuCl_3 single crystals in magnetic fields up to 18T.Overall, four samples (labeled I to IV) from different crystal growth laboratories have been scrutinised. All samples were of high crystalline quality,evidenced by the onset of magnetic long range order in the range T_N=7.0K (sample II) to T_N=7.4K (sample I), see Methods for details. For fields applied parallel to the planes of α-RuCl_3, we observe a strong impact of the magnetic field on κ, which upon exceeding B_c≈7.5 T, i.e. in the absence of magnetic order, exhibits a qualitatively new behavior: a low-temperature peak arises in the temperature dependence of κ which growswith magnetic field. The analysis of our data unambiguously implies a radical change in the low-energy spectrum of magnetic excitations, i.e. the opening of an energy gap at B>B_c which increases approximately linearly with the magnetic field. § RESULTS The upper panel of Figure <ref> shows representative data of the in-plane thermal conductivity κ_ab of α-RuCl_3 as a function of temperature T in zero field (see the supplementary information (SI) for κ_ab of other single crystals with essentially the same temperature dependence) and at B=16 T, applied parallel to the ab-planes. In zero magnetic field, upon cooling from 300 K down to the base temperature (5.5 K) of our setup, κ_ab increases steadily up to a distinct maximum at around 40K, and decreases steeply at lower temperature. A kink around 7.5 K coincides roughly with the onset of long-range magnetic order of the system <cit.>.At first glance, these zero magnetic field data of κ_ab(T) at T>T_N with a single peak structure resemble that of a conventional phononic heat conductor <cit.>: In such a case the phononic heat conductivity, which can coarsely be estimated as κ_ph∼ c _Vvl, increases strongly with temperature in the low-T regime, where the phononic velocity v and mean free path l are essentially temperature independent, with the phononic specific heat c_V. Towards higher T, phonon umklapp processes increasingly limit l, resulting in a broad peak in κ_ph followed by a fast decline. For antiferromagnetic insulators it is well known that scattering of phonons off paramagnon fluctuations of the incipient long range order may give rise to a significant suppression of κ_ph above and a recovery below the Néel ordering temperature, respectively <cit.>. The whole κ_ab(T) including the observed kink at T_N seems perfectly in line with such a scenario. Strikingly, the application of a large in-plane magnetic field of B=16 T, at which magnetic order is absent, dramatically changes κ_ab, and thereby challenges such a rather conventional interpretation: κ_ab is drastically enhanced at low temperature - a second large peak emerges at around 7 K which even exceeds the one at higher temperature.The unconventional nature of the field-induced double-peak structure in the temperature dependence of κ_ab is further confirmed by a detailed mapping of κ_ab(T,B) up to an in-plane field of B=18 T, as is shown in Fig. <ref>. Apparently, the impact of the magnetic field on the heat transport is profoundly different for B≲ 7.5 T on the one hand and B≳ 7.5 T on the other hand, which clearly defines two field regimes (labeled I and II, respectively) which are separated by a critical field B_c≈7.5 T. In regime I, as is evident from panel (a) of Fig. <ref>, κ_ab slightly decreases for all temperatures T<40K upon increasing the field from zero to B_c. This suppression is most pronounced at T=6 K where it reflects the suppression of the long range magnetic order. A dramatically different field dependence occurs upon further increasing the field (regime II), where κ_ab strongly increases with increasing field. Remarkably, for T≲ 15 K, this increase is essentially linear in magnetic field up to 18 T.§ MAGNETIC HEAT TRANSPORT AT HIGH MAGNETIC FIELD? Without any doubt, the most prominent feature of the present data is the large field-induced low-temperature peak in κ_ab in regime II, which increasingly grows with field (Fig. <ref>b). Note that the critical field B_c, which marks the onset of this regime, coincides withthe complete field-induced suppression of long-rang magnetic order which governs the lowest temperature physics in regime I but is absent in regime II <cit.>. Thus, the low-temperature peak in regime II must be of a qualitatively different origin in contrast to the low-temperature upturn in regime I below T_N, which is closely related to spin fluctuations in the system.A priori, two very different scenarios can be invoked for explaining the nature of a double-peak structure in κ(T) of an electrical insulator which hosts a fluctuating spin system. On the one hand, this could be the signature of magnetic heat transport that, in turn, leads to a pertinent contribution to the (otherwise conventional phononic) heat conductivity. Such a mechanism is common in low-dimensional systems such as spin chains, ladders, and planes <cit.>. In these cases, phonons and magnetic excitations yield two independent transport channels. On the other hand, a double-peak structure is also known to occur in purely phononic heat transport, resulting from the heat carrying phonons scattering off another degree of freedom, such as a spin excitation with a well defined excitation energy ħω_0 <cit.>. Such scattering affects the phononic heat transport over a large temperature range, but has its strongest impact in the temperature regime where the energy of the majority of heat carrying phonons coincides with ħω_0.An unambiguous signature of low-dimensional magnetic heat transport is its anisotropy: spin-heat is transported only along a certain crystal direction along which a significant energy dispersion of the spin excitations exists, i.e., along the directions without a significant magnetic exchange interaction the magnetic heat transport is absent <cit.>. We therefore investigated the heat conductivity of α-RuCl_3 perpendicular to the planes (κ_c), where the magnetic exchange interaction is negligible (sample II, see Fig. <ref>b). Remarkably, apart from minor differences in details (see Fig. <ref> in the SI), we observe practically the same temperature and magnetic field dependence as for κ_ab. The most important findingis the direct comparability of κ(c) to κ(ab) at B=0 and 16 T, with the out-of-plane thermal transport exhibiting the same low-temperature enhancement in both cases, i.e. the presence of a new low-temperature peak. Hence, we can exclude the scenario that transport by the emergent elementary excitations of the spin system gives rise to the low-temperature peak in regime II. This means unambiguously that the field-induced low-temperature peak of κ_ab is primarily phononic, and its peculiar temperature and magnetic field dependence arises from an unusual, field dependent scattering process of the phonons. § MAGNETIC PHONON SCATTERINGAfter having established this first important result, we now move on to rationalising the field and temperature dependence of the heat conductivity more thoroughly.Without further analysis and by invoking the above-mentioned magnetic scattering scenario of phonons one can conclude from the presence of the double-peak structure in regime II with a clear minimum at T_min,that an energetically well-defined magnetic mode with energy ħω_0 exists which scatters primarily the heat carrying phonons of that energy. This can be understood from the fact that the energy of the phonons which predominantly carry heat, ħω̃, is strongly temperature dependent <cit.>. I.e. the two peaks at lower and higher temperature correspond to ω̃<ω_0 andω̃>ω_0, whereasω̃≈ω_0 at T_min. This immediately suggests that a rough quantitative estimate of the scattering magnetic mode energy ħω_0 can be obtained by reading T_min off the data. We therefore plot the temperature derivative ∂κ_ab/∂ T in false colour representation (Fig. <ref>). At B≳11 T, the minimum position T_min depends about linearly upon B. At smaller fields, however, T_min(B) attains a steeper slope and rapidly moves out of the measured temperature window, suggestive of an approximate extrapolation towards B_c at zero temperature (dotted line Fig. <ref>). Thus, while the dominant magnetic scattering mode energy ħω_0 seems to be very close to zero around B_c and at smaller fields, it rapidly develops a substantial size at higher magnetic fields. One can exploit T_min(B) further and estimate the field dependence of the magnetic mode energy ħω_0(B) quantitatively by considering that for a conventional isotropic phononic system, the majority of heat carrying phonons at a certain temperature T possess an energy of about α k_BT with α≈4 <cit.>.For more anisotropic phononic systems, as one might expect for α-RuCl _3, simple dimensional considerations (see Methods) suggest a somewhat reduced α (in particular, α≈2.6 for a hypothetical purely two-dimensional phononic system).Thus, by translating T_min(B) into the phonon energy which is affected strongest by the magnetic scattering one can directly extract the field dependence of the magnetic mode energy as ħω_0(B)≈α T_min(B), with α roughly in the range of 2.6 to 4. In fact, the low-temperature specific heat of α-RuCl_3 has been reported to follow a T^2 rather than a T^3 dependence <cit.>, which indicates a significant anisotropy of the phonons. Yet, the anisotropy of the heat transport is only moderate (see Fig. <ref>) as compared to prototype quasi two-dimensional phononic systems such as graphite <cit.>, implying a significant interlayer lattice coupling and consequently only a moderate anisotropy for the phononic system, in contrast to the two-dimensional nature of the magnetic system.A further corroboration of the scenario of magnetic phonon scattering can be obtained by analysing κ(T,B) in terms of a phononic model which takes the magnetic scattering into account. We follow the usual approach of Callaway <cit.> and express the heat conductivity in terms of an energy dependent combined relaxation time τ_c which takes various scattering processes into account (see Methods). For conventional phononic systems good descriptions of the temperature dependence of κ can be achieved if thestandard expressions for the phonon relaxation times describing umklapp, point defect, and boundary scattering are comprised in τ_c.As one can already conjecture in view of the unconventional double-peak structure of κ_ab(T,B) at B>B_c, a fit of the Callaway model to our data fails in this standard phononic picture. However, a qualitatively reasonable fit is indeed possible if we adapt the model to the afore sketched magnetic scattering scenario by introducing an additional relaxation time τ_mag which describes such scattering. More specifically, we assume the phonons to scatter within an energetically broad magnetic excitation spectrum (mimicking the theoretically predicted <cit.> and experimentally observed <cit.> character of the excitation spectrum of the Kitaev model and of α-RuCl_3, respectively) from a reservoir which is dominated by an energetically sharp and magnetic field dependent low-energy mode at ħω_0 (see Methods for details). Despite the simplicity of this model, it is indeed possible to simultaneously fit the data in regime II with a field independent parameter set for the usual phonon scattering terms and a field dependent magnetic scattering term τ_mag(B), see Fig. <ref>. The values for ω_0(B) extracted thereby are plotted in Fig. <ref> as square symbols. Obviously, they display a field dependence similar to that of the minimum in the heat conductivity, T_min(B), and are in the same energy range as expected from the above considerations with respect to T_min. We emphasise that a qualitatively similar result is reached upon analysing κ_c with the same procedure, which further corroborates this analysis (see SI). At magnetic fields smaller than B_c, the heat conductivity of α-RuCl_3 is always significantly smaller than in the high-field phase. This straightforwardly implies that in regime I the phonon scattering off the magnetic excitation spectrum is even stronger than in regime II (see Fig. <ref>).In view of the vanishing ω_0 at around B_c, this suggests that in regime I the scattering magnetic modes are at relatively low energy if not gapless. Evidently, thefield induced changes of κ_ab in this phase are relatively small (see Fig. <ref>). This small field dependence then naturally is consistent with field induced changes of the magnetic spectrum, connected with the gradual suppression of long range magnetic order. These are more subtle than those apparently induced by high magnetic fields.§ CONCLUSIONIn conclusion, our data and the subsequent analysis clearly show that the heat transport of α-RuCl_3 is primarily of phononic type. More specifically, the field-induced low-temperature peak in the heat conductivity cannot be explained by the expectedexotic excitations of a putative Kitaev-Heisenberg QSL carrying heat.Nevertheless, the magnetic excitations of α-RuCl_3 dramatically impact the phononic heat transport along all directions through scattering of the phonons off the magnetic excitation spectrum.This scattering is particularly strong in regime I, which at first glance seems to be consistent with the incipient long range magnetic order. However, the magnitude of the low-temperature increase of κ in the ordered phase is relatively small as compared to the dramatic enhancement in regime II. This implies that even in the magnetically ordered phase considerable magnetic degrees of freedom exist which is in line with the significantly reduced magnetic moment observed in inelastic neutron scattering<cit.>. These residual degrees of freedom scatter the phonons and are likely to remain quantum disordered all the way down to zero energy.Since the phonon heat conductivity at low temperature primarily is carried by acoustic phonons with small momenta k∼0, it seems natural to conclude that the low energy paramagnons relevant for the scattering possess small momenta as well. On the other hand, the dramatic enhancement of the heat conductivity at higher fields in regime II (B>B_c) implies that these low-energy excitations are increasingly gapped out, i.e. the strongest field-induced change of the excitation spectrum concerns the excitations close to the Γ-point. The field-induced spin gap is surprisingly large, since at 18 T it is already of the order of the extracted Kitaev interaction of α-RuCl_3 <cit.>. One might thus speculate that the field-induced phase at B>B_c isgoverned by new physics where the emergent quasiparticles are indeed different from those of the Kitaev-Heisenberg-paramagnons at B<B_c.Final note:Upon finalizing this manuscript, we became aware of another study of the field dependence of the heat conductivity by Leahy et al. <cit.> where a similar enhancement of κ_ab in the field regime II is reported and is interpreted as the signature of in-plane heat transport by massless so-called proximate Kitaev spin excitations. We stress that the latter interpretation for the field-induced low-temperature peak can be ruled out since this peak is essentially isotropic in κ_ab and κ_c and therefore clearly of phononic origin, as is explained above in detail. § METHODS§.§ Sample preparationSamples I and III have been grown at the TU Dresden. For the synthesis, pure ruthenium-metal powder (99.98%, Alfa Aesar) was filled into a quartz ampoule under argon atmosphere, together with a sealed silica capillary containing chlorine gas (99.5% Riedel-de Haën). The molar ratio of the starting materials was chosen to ensure in-situ formation of RuCl_3 and its consequent chemical transport according to the reaction: RuCl_3(s) + Cl_2(g)→ RuCl_4(g) <cit.>. The vacuum-sealed reaction ampoule (p ≈0.1Pa) was subsequently shaken in order to break the chlorine-containing capillary and to release the gas. The ampoule was kept in the temperature gradient between T_1 = 750^∘ C (starting material) and T_2 = 650^∘ C for 5 days. The value of T_2 was optimised to rule out transformation of the product into the β-modification. The value of T_1 was chosen such as to avoid decomposition of RuCl_3 into elemental ruthenium and RuCl_4 gas species <cit.>. The crystalline product represented pure α-RuCl_3 (according to powder X-ray diffraction) without inclusions of ruthenium. Single-crystal X-ray diffraction studies (Apex II diffractometer, Bruker-AXS, Mo Kα-radiation) and EDXS (Oxford Silicon drift detector X-MaxN, Hitachi SU 8020 SEM, 20kV) of the crystals have confirmed a monoclinic structure <cit.> and the nominal composition. The crystals are black with shiny surfaces, of millimeter size along the ab-plane, and of a thickness less than 0.1 mm. These crystals have been cut into rectangular shape as to have the optimum experimental geometry.Sample II has been grown at Oak Ridge National Laboratory <cit.>. It is significantly larger (roughly 13x1.8x2mm^3) and thus well suited for studying κ _c. For the measurement the crystal has been mounted as-grown.Sample IV has been grown at the University of Toronto <cit.>. It has similar dimensions to samples I and III, roughly a rectangular shape and therefore has been measured as-grown. The crystal quality of all sample batches has been checked rigorously by means of x-ray diffraction and susceptibility measurements (not shown here). All sample handling has been done with great care to avoid the introduction of crystal defects such as stacking faults, which can reportedly be induced by puttingα-RuCl_3 single crystals under mechanical strain <cit.>. §.§ Determination of the Néel temperatureIn order to confirm the connection of the minimum in κ_ab(T) for B≤7.5 T with the Néel temperature T_N of α-RuCl_3, we have measured the temperature dependence of the magnetic susceptibility and of the specific heat on a sample of the same batch as samples I and III as a function of magnetic field (applied parallel to the ab-planes). The magnetic susceptibility has been obtained for a single crystal (m ≈ SI4mg) of the same batch as samples I and III in external magnetic fields B=μ_0 H = 0T-7T||ab using a Superconducting Quantum Interference Device-Vibrating Sample Magnetometer (SQUID-VSM) from Quantum Design. The extracted values for T_N = d (χ× T)/dT are in good agreement with the specific heat results.The specific heat measurements have been performed on a piece of the same single crystal used for the magnetic susceptibility studies (m≈2.5mg) between 1.9 K and 20 K using a heat-pulse relaxation method in a commercial Physical Properties Measurement System (PPMS) from Quantum Design. The magnetic field μ_0 H = 0T-9T has been applied in the ab direction, for which an additional sapphire block has been used for mounting the sample. The heat capacity of the complete sample holder (addenda) had been determined prior to the measurements for the purpose of separating the heat capacity contribution of the sample from the total heat capacity of the setup. Note that the cutting of the sample induced a small amount of stacking faults in the single crystal, which, however, can be well-separated from the intrinsic signal around 7.4 K due to its shifted T_N to slightly higher temperatures. At an external magnetic field of 7.5 T, no signs of a magnetic transition can be detected in our sample in the full temperature regime. Due to a broadening of the magnetic transition in higher fields, we restrain from an entropy-conservation construction to obtain T_N. Instead, the transition temperature is determined from the maximum position of the peak in the specific heat capacity. The difference between both constructions is less than 0.2 K for T_N in zero field. For clarity, only the results of the specific heat are plotted in Fig.<ref>, which at B=7.5 T confirm the absence of a thermodynamic transition down to 1.9 K. §.§ Heat conductivity measurements Steady state thermal conductivity measurements have been performed in the temperature range 5.5K-300K, utilising a home made vacuum setup optimised for low noise measurements in a standard four point probe geometry <cit.>. The in-plane thermal conductivity κ _ab has been measured for samples I, III and IV, the out-of-plane component κ _c for sample II. Magnetic fields up to 18T have been applied parallel to the honeycomb planes for sample I and II and perpendicular for sample III using a standard ^4He bath cryostat with a superconducting magnet coil. The delicateness of the samples requires special means to obtain high quality data. All samples thus have been cooled/heated at slow rates to minimise thermal stress. Furthermore, data have been recorded during heating as well as cooling the samples to rule out hysteretic effects.All these measures underpin the robustness of our results which is evident by the high reproducibility we observe in our κ _ab-data. Temperature sweeps at constant field as well as magnetic field sweeps at constant temperature have been undertaken, resulting in a large, consistent data set. §.§ Modelling of phonon heat conductivity The phonon heat conductivity κ_ph of a three-dimensional isotropic solid can be described by <cit.>: κ_ph =1/3 (2 π)^3 ∫ c_ k v_ k l_ k dkwhere c_ k=d/dTu_ k, v_ k, and l_ k=v_ kτ_ k denote the contribution to the specific heat, the velocity, and the mean free path of a phononic mode with wave vector k. u_ k and τ_ k are the mode's energy and relaxation time, respectively. The Callaway Model <cit.> refers to the evaluation of the above by applying a Debye ansatz for the phonon heat capacity, assuming v_ k=v_s (the speed of sound) for all phonon branches, and by introducing an energy dependent relaxation time τ_c. This yields the low T approximation κ(T) = k_B/2π^2v_s(k_BT/ħ)^3 ∫_0^Θ_D/Tx^4e^x/(e^x-1)^2τ_c(x) dx, withBoltzmann's constant k_B, Planck's constant ħ, and x = ħω / k_BT.Before further specifying the effective phonon scattering rate τ ^-1 _c, a general statement can be deduced from the analytical form of the first term of the integrand. It has a distinct peak at ħω≈ 4 k_B T, determining roughly the energy of those phonons which contribute most to the heat transport at a given temperature T <cit.>. Thus, in presence of a magnetic scatterer with well defined energy ω _0, such scattering can be expected to have the strongest impact on the heat transport at around T≈ħω_0/(4k_B). The layered crystal structure of α-RuCl_3, with its strong intralayer and weak van der Waals interlayer coupling suggests an anisotropic phononic structure with a significantly lower phonon velocity perpendicular to the planes. For the hypothetical extreme case of just two-dimensional phonon propagation, a modification of the first part of the integrand in Eq.<ref> towards (x^3e^x)/(e^x-1)^2 should be considered. In this case energy of the predominantly heat carrying phonons reduces with respect to the three-dimensional case to ħω≈ 2.6 k_B T. Thus, for a very anisotropic lattice such as α-RuCl_3, the scaling factor α which relates the position of the minimum of κ(T) with the energy of a magnetic scatterer can be expected to roughly be in the range 2.6 to 4 times k_BT_min.For conventional phononic systems (i.e. non-magnetic, electrically insulating crystals), the effective phonon scattering time in Eq.<ref> is composed of conventional scattering mechanisms, for which empirical expressions are well established, viz. phonon-phonon umklapp scattering τ_P^-1= AT ω ^3 e^- Θ _D/b T, phonon-defect scattering τ_D^-1= D ω ^4 and phonon-boundary scattering τ_B^-1=v_s L^-1. Following Matthiessen's rule the combined effective scattering rate yields as τ_c,0^-1=τ_P^-1+τ_D^-1+τ_B^-1. As described in the main text, such a conventional model is unable to reproduce the double peak structure of κ _ab which we observe at high magnetic fields.§.§ Modelling of magnetic phonon scattering In order to incorporate the phenomenologically sketched magnetic scattering into the model, we add a further scattering rateτ_mag^-1 which describes the magnetic scattering of the phonons, i.e.τ_c^-1= τ_c,0^-1+τ_mag^-1 In lack of a suitable microscopic theory for describing the magnetic scattering processes of phonons in α-RuCl_3, we use an empirical approach to analytically describe τ_mag, which we adapt from earlier successful modellings of magnetic resonant scattering of phonons off magnetic triplet excitations which are well defined in energy and momentum <cit.>:τ_mag^-1= Cω ^4/(ω ^2 - ω ^2 _0)^2e^-ħω_0/k_BT/1+3e^-ħω_0/k_BT, with a coupling constant C. The second factor ω ^4/(ω ^2 - ω ^2 _0)^2 represents a resonant scattering cross section and the last factor expressing the thermal population of the triplet mode, where we neglect Zeeman splitting. Indeed, a qualitative fit of the peculiar temperature and field dependence of our κ data based on the Callaway model (Eq. <ref>) using the combined relaxation rate τ_c^-1 as given by Eqs. <ref> and <ref> is possible, see Fig. <ref> in the supplementary information. The use of expression for τ_mag apparently has, however, several caveats. Firstly, the chosen resonant scattering cross section describes phonon scattering off an energetically sharply defined magnetic mode at ω_0. However, in quantum magnets with fractional quasiparticles, one typically expects energy-momentum continua rather than a comparatively well-defined energy-momentum dispersions.In principle, the weight of the continuum and its boundaries could be q-dependent. We discard this and replace the sharp resonant cross section by a broad function which mimics the excitation spectrum of α-RuCl_3 and of the Kitaev-Heisenberg model with a high-energy cut-off <cit.>. We therefore replace the resonant factor by the Heaviside function θ(K-ħω), with K the high-energy cut-off.Secondly, one might argue that in view of the excitation continuuman integration over frequencies with respect to the occupation number, respecting, however, a low-energy cut-off of ω≥ω_0 should be considered. Since the exact nature of the excitation is not known we discard such complications and rely on the fact that even if the mentioned integration was to be performed, the temperature dependence would still be governed to leading order by a Boltzmann weight exp(-ħω_0/k_BT). We thus modify τ_mag^-1 and useτ_mag^-1= C θ(K-ħω)e^-ħω_0/k_BT/1+3e^-ħω_0/k_BTfor fitting the data. As can be seen in Fig. <ref>, the quality of the corresponding fits is slightly improved with respect to the first resonant approach.Finally, since one expects the excitation of a Kitaev system to fractionalise into Majorana fermions and gauge fluxes, one might speculate that the occupation function is governed by the statistics of fermions rather than that of triplets. We therefore have tested also whether a τ_mag^-1 of the form τ_mag^-1= C θ(K-ħω)e^-ħω_0/k_BT/1+e^-ħω_0/k_BTimproves the quality of the fit. However, in this case the fit quality is reduced, see Fig. <ref>. Thus, the further analysis of the data is performed using τ_mag^-1 in the form given in Eq. <ref>. §.§ Fitting procedureUpon fitting the data we assumed that the fitting parameters which describe the dynamics of the pure phononic system, i.e. which are captured in τ_c,0 are independent of magnetic field. On the other hand we allow a field dependencefor C and ω_0. Furthermore, in the case of fitting with Eqs. <ref> and <ref> we letK as a free parameter when fitting κ_ab(T,B). When fitting κ_c(T,B), the previously obtained value for K is used as a fixed parameter. To obtain good values for the field independent parameters describing conventional phonon scattering τ_c,0^-1, multiple κ (T) curves were fit simultaneously.Upon fixing the resulting phonon parameters, C(B) and w_0 (B) have then been optimised in a second step for each field separately. All fit parameters are given in the supplementary information in Tables <ref> and <ref>. A comparison of the fits according to using a τ_mag^-1 as described by Eqs. <ref>, <ref>, and <ref> for both κ_ab and κ_c are shown in Fig. <ref> in the supplementary information. After selecting Eq. <ref> as most successful for fitting the data, C(B) and ω_0(B) have been determined for κ_ab at all measured fields B>12 T, i.e. in steps of 0.5 T (the ω_0(B) are plotted as full squares in Fig. <ref>). At smaller magnetic fields, these two parameters, and in particular ω_0 cannot be determined very well because for those fields the measured κ_ab(T) do not possess a clear enough minimum in the measured temperature range. Therefore,the extracted ω_0(B) for fields B>12 T are linearly extrapolated for estimating the field dependence of ω_0 towards smaller fields. Thereby, an upper limit for ω_0(B) for B≤12 T is be obtained (open squares in Fig. <ref>). For those fields, fits to the data are obtained by fixing ω_0(B) to the extrapolated values and keeping only C(B) as a free parameter separately for each field.Fig. <ref> shows a direct comparison of the experimental data with these fits for selected magnetic fields. Furthermore, this figure contains a hypothetical curve where the magnetic scattering is switched of, i.e. τ_mag^-1=0.For κ_c, the determination of C(B) and ω_0(B) has been performed in steps of 1 T (see Table <ref>).§ ACKNOWLEDGMENTSThis work has been supported by the Deutsche Forschungsgemeinschaft through SFB 1143 and through the projects HE3439/12 and HE3439/13. W.B. acknowledges partial support by QUANOMET, CiNNds, and PSM. A.B. and S.E.N were supported by the Energy (US-DOE), Office of Science, Basic Energy Sciences (BES), Scientific User Facilities Division.D.G.M. and P.L.-K. were supported by the Gordon and Betty Moore Foundation's EPiQS Initiative through Grant GBMF4416. J.S. and Y.-J.K. were supported by NSERC through Discovery Grant and CREATE program. § AUTHOR CONTRIBUTIONSR.H. and A.U.B.W. carried out the heat transport and specific heat experiments, respectively. R.H., X.Z., W.B., and C.H. analysed the heat transport data. D.N., A.I., and T.D. grew and characterised samples I and III, P.L.-K., A.B., D.G.M., and S.E.N. grew and characterised sample II, J.S. and Y.-J.K. grew and characterised sample IV. B.B. and C.H. designed and supervised the project. R.H., W.B., and C.H. prepared the manuscript. All authors have read and approved the final version of the manuscript.§ COMPETING FINANCIAL INTERESTSThe authors declare no competing financial interests. naturemag_noURL § SUPPLEMENTARY INFORMATION §.§ Heat conductivity for different samples §.§ In-plane heat transportFigure <ref> shows, additional to the κ _ab(T) data at zero magnetic field of sample I discussed in the main text, the results of κ _ab(T) measurements in zero field performed on samples III and IV. It is evident that the overall temperature dependence of all three samples exhibit the same features, a broad peak at ∼40K followed by a rapid decrease of κ_ab towards lower temperatures and an eventual recovery below T_N (temperature range not measured for sample IV). The peak position of κ _ab displays only a minimal variation from sample to sample, which indicates that the sample purity with respect to point defect scatterers is practically the same. The nevertheless significant difference in absolute values can solely be explained by the uncertainty of the experimental geometry due to the small thickness and irregular sample shapes. For T>200K a change of slope of κ_ ab (T) becomes apparent which is more pronounced for sample IV than for sample I and III.One might conjecture, that the structural phase transition at 155 K <cit.> where the honeycomb layers reorient with respect to each other plays a role in the change of slope ofκ_ab(T).Furthermore, at such high temperatures, unavoidable radiation losses generally become significant in heat transport experiments. These are sample dependent and thus are an additional possible explanation for the observed differences.We mention that for sample III, we measured κ_ab with a magnetic field parallel to the c-axis of the material. A large magnetic field dependence, as observed for fields parallel to the planes, has not been observed.§.§ Out-of-plane heat transportUpon measuring the out-of-plane heat conductivity on sample II, we observed hysteresis-like changes of κ_c(T) near the structural phase transition at T_S∼155K (not shown). Therefore, the low-temperature measurements were performed after an initial cooling of the sample below T_s without further crossing this transition. Nevertheless, the out-of-plane heat conductivity of sample II displayed some instabilities during the measurements, which is indicative of the fragility of α-RuCl_3's crystal structure reported earlier <cit.>. We attribute this to unavoidable inhomogeneous thermal strain during heating/cooling cycles which might change the initial stacking sequence along the c-direction, indubitably altering the phononic transport along the c-axis. Here, we discuss only the consecutively recorded, consistent part of the data. For T>T_S, we show the data recorded during heating the sample in zero field subsequent to the field dependent measurements.§.§ Field dependence of κ_c Fig. <ref> shows the low temperature heat conductivity measured on sample II, perpendicular to the honeycomb planes κ _c at B=0T and selected B>0. A low-temperature enhancement very similar to that of κ_ab is clearly present. In order to track the magnetic field dependence of the minimum in κ_c, we plot in Fig. <ref> the temperature derivative ∂κ_c(T, B)/∂ T in false-colour representation. In analogy to the data for κ_ab as shown in Fig. <ref>, also here the minimum in κ_c is clearly visible and approximately linear in B for B>B_c.§.§ Fit results Fig. <ref> displays a comparison of the fit results according to the Callaway model (Eqs. <ref>, <ref> and <ref>) under consideration of the different forms of τ_mag^-1 as described by Eqs. <ref>, <ref>, and <ref> for both κ_ab (sample I) and κ_c (sample II). The field independent fit parameters for describingκ_ab and κ_c with the Callaway model and τ_mag^-1 (Eq. <ref>) are given in Table <ref>. The field dependent results for C(B) and ω_0(B) for bothκ_aband κ_c are reproduced in Table <ref>.Fig. <ref> shows, in analogy to Fig. <ref>, κ_c data of sample II (open symbols) and fits to the Callaway model (red solid lines) for selected representative magnetic fields. | http://arxiv.org/abs/1703.08623v1 | {
"authors": [
"Richard Hentrich",
"Anja U. B. Wolter",
"Xenophon Zotos",
"Wolfram Brenig",
"Domenic Nowak",
"Anna Isaeva",
"Thomas Doert",
"Arnab Banerjee",
"Paula Lampen-Kelley",
"David G. Mandrus",
"Stephen E. Nagler",
"Jennifer Sears",
"Young-June Kim",
"Bernd Büchner",
"Christian Hess"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170324232801",
"title": "Large field-induced gap of Kitaev-Heisenberg paramagnons in $α$-RuCl$_{3}$"
} |
Physical Layer Security in Wireless Ad Hoc Networks Under A Hybrid Full-/Half-Duplex Receiver Deployment Strategy Tong-Xing Zheng, Member, IEEE, Hui-Ming Wang, Senior Member, IEEE, Jinhong Yuan, Fellow, IEEE, Zhu Han, Fellow, IEEE, and Moon Ho Lee, Life Senior Member, IEEE T.-X. Zheng and H.-M. Wang are with the School of Electronic and Information Engineering, Xi'an Jiaotong University, Xi'an, 710049, Shaanxi, China. Email: [email protected], [email protected]. J. Yuan is with the School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, Australia. Email: [email protected]. Z. Han is with the Electrical and Computer Engineering Department, University of Houston, Houston, TX, USA. Email: [email protected]. M. H. Lee is with the Division of Electronics Engineering, Chonbuk National University, Jeonju 561-756, Korea. Email: [email protected] 30, 2023 ===========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================This paper studies physical layer security in a wireless ad hoc network with numerous legitimate transmitter-receiver pairs and eavesdroppers. A hybrid full-/half-duplex receiver deployment strategy is proposed to secure legitimate transmissions, by letting a fraction of legitimate receivers work in the full-duplex (FD) mode sending jamming signals to confuse eavesdroppers upon their information receptions, and letting the other receivers work in the half-duplex mode just receiving their desired signals. The objective of this paper is to choose properly the fraction of FD receivers for achieving the optimal network security performance. Both accurate expressions and tractable approximations for the connection outage probability and the secrecy outage probability of an arbitrary legitimate link are derived, based on which the area secure link number, network-wide secrecy throughput and network-wide secrecy energy efficiency are optimized respectively. Various insights into the optimal fraction are further developed and its closed-form expressions are also derived under perfect self-interference cancellation or in a dense network. It is concluded that the fraction of FD receivers triggers a non-trivial trade-off between reliability and secrecy, and the proposed strategy can significantly enhance the network security performance. Physical layer security, ad hoc network, full-duplex receiver, outage, stochastic geometry. § INTRODUCTION The rapid development in wireless communications has brought unprecedented attention to information security. Traditionally, security issues are addressed at the upper layers of communication protocols by using encryption. However, the large-scale and dynamic topologies in emerging wireless networks pose a great challenge in implementing secret key management and distribution, particularly in a decentralized wireless ad hoc network without infrastructure support <cit.>. Fortunately, physical layer security, an information-theoretic approach that attains secure transmissions by exploiting the randomness of wireless channels without necessarily relying on secret keys, is becoming increasingly recognized as a promising alternative to complement the cryptography-based security mechanisms <cit.>-<cit.>. Early studies on physical layer security have mainly focused on point-to-point transmissions, and metrics from user viewpoint such as secrecy capacity <cit.>, ergodic secrecy rate <cit.> and secrecy outage probability <cit.>-<cit.> have been used to evaluate the secrecy level in different scenarios/applications. From a network-wide perspective, physical layer security has also shown its potential<cit.>. Many efforts have already been devoted to improve network security in terms of the area secure link number (ASLN) <cit.> and network-wide secrecy throughput (NST) <cit.>. More recently, energy-efficient green wireless network has attracted considerable interests due to energy scarcity, and some research works have been carried out for enhancing network-wide secrecy energy efficiency (NSEE) <cit.>. §.§ Previous Endeavors and MotivationsTo improve the secrecy of information delivery for an ad hoc network, an efficient approach is to degrade the wiretapping ability of eavesdroppers through emitting jamming signals <cit.>. For example, the authors in <cit.>propose a cooperative jamming strategy with single-antenna legitimate transmitters, and when eavesdroppers access a transmitter's secrecy guard zone <cit.>, this transmitter will act as a friendly jammer to send jamming signals to confuse eavesdroppers. This work is extended by <cit.> to a multi-antenna transmitter scenario, and artificial noise <cit.> with either sectoring or beamformingis exploited to impair eavesdroppers. Although these endeavors are shown to achieve a remarkable secrecy throughput enhancement, friendly jammers or multi-antenna transmitters might not be available in many applications. For instance, constrained by the size and hardware cost, a sensor node is usually equipped with only a single antenna. Furthermore, due to a low-power constraint, a sensor has no extra power to send jamming signals. In such unfavorable situations, information transfer is still vulnerable to eavesdropping.Fortunately, recent advances in developing in-band full-duplex (FD) radios provide a new opportunity to strengthen information security in the aforementioned situations. Effective self-interference cancellation (SIC) techniques enable a transceiver to transmit and receive at the same time on the same frequency band <cit.>. Although the transmitter (sensor) is vulnerable to eavesdropping, we can deploy powerful FD receivers such as data collection stations to radiate jamming signals upon their information receptions. By doing so, additional degrees of freedom can be gained for improving network security. In fact, the idea of using FD receiver jamming to improve physical layer security has already been reported by <cit.>-<cit.> for point-to-point transmission scenarios. Specifically, the authors in<cit.> and <cit.> consider a single-input multi-output (SIMO) channel with the receiver using single- and multi-antenna jamming, respectively. The authors in <cit.> consider a multi-input multi-output (MIMO) channel with both transmitter and receiver generating artificial noise. These works are further extended in two-waytransmissions <cit.>, cooperative communications <cit.>, <cit.>, and cellular networks <cit.>. Recently, we have studied the design of the optimal density of the overlaid FD-mode tier to maximize its NST while guaranteeing a minimum network-wide throughput for the underlaid HD-mode tier <cit.>. Generally, investigating the potential benefits of FD receiver jamming techniques in enhancing information security from a network perspective is an interesting, but much more sophisticated issue, since we should take into account numerous interferers and eavesdroppers that are randomly distributed over the network. In addition, using FD receiver jamming in a network is confronted with two fundamental challenges as follows: * Theoretically, activating too many FD receivers to send jamming signals brings severe self- and mutual-interference to legitimate receivers, thus impairing the reliability of the ongoing information transmission. This will result in few secure links being established and accordingly the poor secrecy throughput. * Practically, employing FD receivers incurs more system cost and overhead. FD transceivers are more expensive than half-duplex (HD) transceivers. From energy efficiency perspective, more circuit power is consumed to enable the FD operation or to mitigate the self-interference caused by FD radios, which leads to low energy efficiency. Motivated by these, a proper way to deploy FD receivers is to make a portion of legitimate receivers work in the FD mode simultaneously sending jamming signals and receiving desired signals, and make the rest work in the HD mode just receiving desired signals. This results in a hybrid full-/half-duplex receiver deployment strategy. Then, a question is naturally raised: What should be the optimal fraction of FD receivers in order to optimize the network security performance? To the best of our knowledge, this question has not been answered byexisting literature. So far, a fundamental analysis on the network security performance, in aspects like ASLN, NST and NSEE, is still lacking for a wireless ad hoc network with hybrid full-/half-duplex receivers. This motivates our work. §.§ Our Work and ContributionsIn this paper, we study physical layer security for a wireless ad hoc network under a stochastic geometry framework <cit.>. Each transmitter in this network is equipped with a single antenna and sends a secret message to an intended single-antenna receiver, in the presence of randomly located multi-antenna eavesdroppers. A hybrid full-/half-duplex receiver deployment strategy is proposed, where a fraction of legitimate receivers work in the FD mode receiving desired signals and radiating jamming signals simultaneously, and the remaining receivers work in the HD mode just receiving desired signals. The main contributions of this paper are summarized as follows: * We investigate the fundamental tradeoff between secrecy and reliability via jammers. We analyze the connection outage probability and the secrecy outage probability of a typical legitimate link, and provide both accurate expressions and tractable approximations for them. * We study three important performance metrics on network security, namely, ASLN, NST and NSEE, respectively. We prove that these metrics are all quasi-concave functions of the fraction of FD receivers, and derive the optimal deployment fractions to maximize them. * We further develop insights into the behavior of the optimal fraction of FD receivers with respect to various network parameters. We also provide closed-form expressions for this optimal fraction in special cases, e.g., under a perfect SIC assumption or in a dense network.§.§ Organization and NotationsThe remainder of this paper is organized as follows. In Section II, we describe the system model. In Section III, we analyze the connection outage and secrecy outage probabilities of an arbitrary legitimate link. In Sections IV, V and VI, we optimize the fraction of FD receivers to maximize ASLN, NST and NSEE, respectively. In Section VII, we conclude our work.Notations: bold uppercase (lowercase) letters denote matrices (vectors). (·)^H, (·)^-1,Pr{·}, and 𝔼_A(·) denote Hermitian transpose, inversion, probability, and theexpectation of A, respectively. CN(μ, ν) denotes the circularly symmetric complex Gaussian distribution with mean μ and variance ν. ln(·) denotes the natural logarithm. f^'(q) and f^”(q) denote the first- and second-order derivatives of f(q) on q, respectively. 1_FD(x) is an indicator function with 1_FD(x)=1 for x∈{FD} and 1_FD(x)=0 for x∉{FD}. L_I(s)= 𝔼_I(e^-sI) is the Laplace transform of I. [x]^+≜max(x,0). § SYSTEM MODELWe consider a wireless ad hoc network composed of numerous single-antenna legitimate transmitter-receiver pairs, coexisting with randomly located N_e-antenna eavesdroppers. Each legitimate transmitter sends a secret message to its paired receiver located a distance r_o away[The assumption of a common legitimate link distance is quite generic in analyzing a wireless ad hoc network<cit.>, which eases the mathematical analysis. Nevertheless, in principle the obtained results can begeneralized to an arbitrary distribution of r_o <cit.>.]. We assume that a fraction q of legitimate receivers work in the FD mode such that each of them simultaneously receives the desired signal and radiates a jamming signal to confuse eavesdroppers, and the others work in the HD mode only receiving desired signals. Legitimate receivers and eavesdroppers are distributed according to independent homogeneous Poisson point processes (PPPs) <cit.> Φ_l with density λ_l and Φ_e with density λ_e, respectively. Using the property of thinning for a PPP, the distributions of HD and FD receivers follow independent PPPs Φ^HD with density λ^HD = (1-q)λ_l and Φ^FD with density λ^FD = qλ_l, respectively. We denote by Φ̃^HD and Φ̃^FD the location sets of the transmitters corresponding to HD and FD receivers, respectively. According to the displacement theorem <cit.>, Φ̃^HD and Φ̃^FD are also independent PPPs with densities λ^HD and λ^FD, respectively. We use r_xy to denote the distance between a node located at x and a node at y. For convenience, we use x̃ to denote the location of a transmitter whose paired receiver is located at x. Wireless channels, including legitimate channels and wiretap channels, are assumed to suffer a large-scale path loss governed by the exponent α>2 together with a quasi-static Rayleigh fading with fading coefficients independent and identically distributed (i.i.d.) obeying CN(0,1). Due to uncoordinated concurrent transmissions, the aggregate interference at a receiver dominates the thermal noise. Thereby, we concentrate on an interference-limited scenario by ignoring thermal noise, given that the inclusion of thermal noise results in a more complicated analysis but provides no significant qualitative difference. Throughout this paper, we denote 𝐒∈{HD,FD} by default. For convenience, the legitimate link with an 𝐒 receiver is called an 𝐒-link. Considering a typical 𝐒 receiver located at the origin o, its signal-to-interference ratio (SIR) is given byγ_o^𝐒 = P_th_õ or_o^-α/I^HD+I^FD+1_FD(𝐒)η P_j,where I^HD≜∑_x̃∈Φ̃^HDP_t h_x̃ o r_x̃ o^-α, I^FD≜∑_x̃∈Φ̃^FD∖õ(P_t h_x̃ o r_x̃ o^-α+P_j h_x or_ xo^-α) and η P_j denote the interferences from HD links, from FD links and from the typical FD receiver itself, respectively; P_t and P_j denote the transmit powers of a legitimate transmitter and of an FD receiver, respectively; h_x y denotes the fading channel gain obeying Exp(1); η is a parameter that reflects the SIC capability, and η=0 refers to a perfect SIC while 0<η≤ 1 corresponds to different levels of SIC. Note that r_x̃ o and r_x o in I^HD and I^FD are correlated, and they satisfyr_x̃ o=√(r_x o^2+r_o^2-2r_x or_ocosθ_x), where the angle θ_x is uniformly distributed in the range [0, 2π].For eavesdroppers, we consider a worst-case wiretap scenario where each eavesdropper has multiuser decoding ability and adopts a successive interference cancellation minimum mean square error (MMSE) receiver. The eavesdropper located at e is able to decode and cancel undesired information signals and uses the MMSE detector w^𝐒_e =( R^𝐒_e)^-1 g_õeto aggregate the desired signal, where R^𝐒_e≜∑_x∈Φ^FD∖ oP_j g_xe g_xe^H r_x e^-α +1_FD(𝐒) R_oe with R_oe≜ P_j g_oe g_oe^H r_o e^-α, and g_x e denotes the N_e× 1 complex fading coefficient vector related to the link from a node at x to an eavesdropper at e. The corresponding SIR of the eavesdropper isγ^𝐒_e = P_t g_õe^H( R^𝐒_e)^-1 g_õer_õ e^-α.§.§ Secrecy Performance MetricsWe assume eavesdroppers do not collude with each other such that each of them individually decodes a secret message. To guarantee secrecy, each legitimate transmitter adopts the Wyner's wiretap encoding scheme <cit.> to encode secret information. Thereby, two types of rates, namely, the rate of transmitted codewords R_t and the rate of embedded information bits R_s, need to be designed to meet requirements in terms of the connection outage and secrecy outage probabilities. * Connection outage probability. If a legitimate 𝐒-link can support rate R_t, the legitimate receiver is able to decode a secret message and perfect connection is assured in this link; otherwise a connection outage occurs. The probability that such a connection outage event takes place is referred to as the connection outage probability, denoted as p_co^𝐒. * Secrecy outage probability. In the Wyner's wiretap encoding scheme, the rate redundancy R_e≜R_t -R_s is exploited to provide secrecy against eavesdropping. If the value of R_e lies above the capacity of the most detrimental eavesdropping link, no information is leaked to eavesdroppers and perfect secrecy is promised in the legitimate link <cit.>; otherwise a secrecy outage occurs. The probability that such a secrecy outage event takes place in an 𝐒-link is referred to as the secrecy outage probability, denoted as p_so^𝐒. In this paper, we concern ourselves with the following three important metrics that measure the network-wide security performance from an outage perspective.1) ASLN. A link in which neither connection outage nor secrecy outage occurs is called a secure link <cit.>. To measure how many secure links can be guaranteed under rates R_t and R_s, we use the metric named ASLN, which is defined as the average number of secure links per unit area. Due to the independence of p_co^S and p_so^S, ASLN, denoted as N, is mathematically given byN≜ qλ_l(1-p_co^FD)(1-p_so^FD) + (1-q)λ_l(1-p_co^HD)(1-p_so^HD). 2) NST. To assess the efficiency of secure transmissions, we use the metric named NST <cit.>, which is defined as the achievable rate of successful information transmission per unit area under the requiredconnection outage and secrecy outageprobabilities. The NST, denoted as Ω, under a connection outage probability p_co^𝐒=σ and a secrecy outage probability p_so^𝐒=ϵ is given byΩ≜ qλ_l(1-σ)R^FD_s + (1-q)λ_l(1-σ)R^HD_s,where R^𝐒_s≜ [R^𝐒_t-R^𝐒_e]^+, with R^𝐒_t and R^𝐒_e the codeword rate and redundant rate that satisfy p_co^𝐒(R^𝐒_t)=σ and p_so^𝐒(R^𝐒_e)=ϵ, respectively. The unit of Ω isnats/s/Hz/m^2. 3) NSEE. To evaluate the energy efficiency of secure transmissions, we use the metric named NSEE, denoted as Ψ, which is defined as the ratio of NST to the power consumed per unit area,Ψ≜Ω/λ_l(P_t+P_c)+qλ_lP_j,where P_c combines the dynamic circuit power consumption of transmit chains and the static power consumption in transmit modes <cit.>. The unit of Ψ is nats/Joule/Hz.We emphasize that, the fraction q of FD receivers triggers a non-trivial trade-off between reliability and secrecy, and plays a key role in improving the metrics given above. Intuitively, under a larger q, more FD jammers are activated against eavesdroppers which benefits the secrecy; whereas the increased jamming signals also interfere with legitimate receivers and thus harm the reliability. The overall balance of such conflicting effects needs to be carefully addressed. In Sections IV, V and VI, we are going to respectively determine the optimal fraction q that * maximizes ASLN N given a pair of wiretap code rates R_t and R_s; * maximizes NST Ω given a pair of outage probabilities σ and ϵ; * maximizes NSEE Ψ with and without considering a minimum required NST. Before proceeding, in the following section we first provide some insights into the behavior of the connection outage probability p_co^𝐒 and the secrecy outage probability p_so^𝐒 with respect to network parameters like q, η, etc., which is very important to subsequent network design.§ OUTAGE PROBABILITY ANALYSISIn this section, we derive the connection outage probability and the secrecy outage probability for an arbitrary legitimate link. For ease of notation, we define δ≜2/α, κ≜πΓ(1+δ)Γ(1-δ) and ρ≜P_j/P_t, which will be used throughout the paper. §.§ Connection Outage ProbabilityThe connection outage probability of a typical 𝐒-link is defined as the probability that the SIRγ_o^𝐒 given in (<ref>) falls below an SIR threshold τ_t≜2^R_t-1, i.e.,p_co^𝐒≜Pr{γ_o^𝐒< τ_t}.The general expression of p^𝐒_co is provided by the following theorem. The interested readers are referred to <cit.> for a detailed proof. The connection outage probability of a typical 𝐒-link is given by p^𝐒_co = 1-e^-1_FD(𝐒)ρη r_o^ατ_t e^-κ(1-q)λ_l r_o^2τ_t^δL_I^FD( r_o^ατ_t/P_t), where L_I^FD(r_o^ατ_t/P_t) =exp(-qλ_l∫_0^∞∫_0^2π(1- 1/1+r_o^ατ_tv^-α1/1+r_o^ατ_t(v^2+r_o^2-2vr_o cosθ)^-α/2)vdθ dv).Theorem <ref> provides an exact connection outage probability withthree parts e^-1_FD(𝐒)ρη r_o^ατ_t, e^-κ(1-q)λ_lr_o^2τ_t^δ and L_I^FD(r_o^ατ_t/P_t), reflecting the impacts of the interferences from the typical receiver itself, from HD links and from FD links, respectively. Although with p^𝐒_co given in (<ref>) we no longer need to execute time-consuming Monte Carlo simulations, the double integral in L_I^FD(r_o^ατ_t/P_t) greatly complicates the further analysis, which motivates a more compact form. In the following theorem, we provide the closed-form upper and lower bounds for p^𝐒_co, and refer the interested readers to <cit.> for a detailed proof. Connection outage probability p^𝐒_co is upper and lower bounded respectively by p^𝐒,U_co = 1-e^-1_FD(𝐒)ρη r_o^ατ_t e^-κ r_o^2τ_t^δλ_l(1+ρ^δ q), p^𝐒,L_co = 1-e^-1_FD(𝐒)ρη r_o^ατ_t e^-κ r_o^2τ_t^δλ_l(1+(1+δ)ρ^δ-(1-δ)/2q). Theorem <ref> shows that both bounds for the connection outage probability increase exponentially in η, q and λ_l, because of the increase of self- and mutual-interference. The relationships between the connection outage probability p^FD_co and parameters q and η are validated in Fig. <ref>, wherethe results labeled by η=0 also refer to an HD counterpart. We observe that, although p^FD_co increases as q increases, the effect is not very remarkable when η is large. This is because, in the large η region the self-interference perceived at an FD receiver dominates the interference (including both undesired and jamming signals) from the other network nodes.§.§ Secrecy Outage ProbabilityThe secrecy outage probability of a typical 𝐒-link is defined as the complement of the probability that any eavesdropper's SIR γ^𝐒_e falls below an SIR threshold τ_e≜2^R_e-1, i.e., p^𝐒_so≜ 1-𝔼_Φ^FD𝔼_Φ_e[∏_e∈Φ_ePr{γ^𝐒_e <τ_e|Φ_e,Φ^FD}]. To calculate exact p^𝐒_so is very difficult. Instead, we give an upper bound for p^𝐒_so in the following theorem. Please refer to <cit.> for a detailed proof. Secrecy outage probability p^𝐒_so of a typical 𝐒-link is upper bounded by p^𝐒,U_so = 1 - exp( -λ_e∑_n=0^N_e-1∑_i=0^min(n,1)(κλ^FDρ^δτ_e^δ) ^n-i/(n-i)!Ξ_n,i^𝐒), where Ξ_n,i^𝐒=∫_0^∞∫_0^2πΛ^𝐒_i,θ,vv^2(n-i) e^-κλ^FDρ^δτ_e^δv^2dθ vdv with Λ^𝐒_i,θ,v=(1_FD(𝐒)ρτ_e(v/ √(v^2+r_o^2-2vr_ocosθ))^α)^i/1+1_FD(𝐒)ρτ_e(v/ √(v^2+r_o^2-2vr_ocosθ))^α. In the following sections, we use upper bound p^𝐒,U_so to replace exact p^𝐒_so, not simply for a tractable analysis but also for the following two reasons: on one hand, p^𝐒,U_so provides a pessimistic evaluation of secrecy performance, which actually benefits a robust design; on the other hand, as <cit.> shows, p^𝐒,U_so will converge to p^𝐒_so at the low secrecy outage probability regime, and a low secrecy outage probability is expected in order to guarantee a high level of secrecy.Clearly, since 1_FD(HD)=0, we have Λ^HD_0,θ,v=1 and Λ^HD_i,θ,v=0 for i>0. Substituting these results into (<ref>) yields a closed-form expression for p^HD_so given below,p^HD_so = 1 - exp( -πλ_e∑_n=0^N_e-1(κλ^FDρ^δτ_e^δ) ^n/n!∫_0^∞v^2n×e^-κλ^FDρ^δτ_e^δv^2dv^2)=1-e^-πλ_eN_e/κ qλ_lρ^δτ_e^δ,where the last equality follows from formula <cit.>. As to p^FD_so, the double integral in (<ref>) makes it difficult to analyze. Given that a single-antenna transmitter in a large-scale ad hoc network usually has low transmit power and very limited coverage, we should set the legitimate link distance r_o sufficiently small (compared with the distance between two nodes that are not in pair) to guarantee both reliability and secrecy. In the following, we resort to an asymptotic analysis by letting r_o→ 0 in (<ref>) in order to develop useful and tractable insights into the behavior of p^FD_so. The following corollary gives a quite simple approximation for p^FD_so. In the small r_o regime, i.e., r_o→ 0,p^FD_so in (<ref>) is approximated by p̃^FD_so = 1-exp(-πλ_eN_e/κ qλ_lρ^δτ_e^δ( 1-ρτ_e/N_e/1+ρτ_e)). Recalling Theorem <ref>, plugging r_o→ 0 into Λ^FD_i,θ,v yields Λ^FD_i,θ,v= (ρτ_e)^i/1+ρτ_e, and thus Ξ_n,i^FD=2π(ρτ_e)^i(n-i)!/(1+ρτ_e)(κλ^FDρ^δτ_e^δ)^i-n-1. Substituting Ξ_n,i^FD into (<ref>) completes the proof. We stress that, although Corollary <ref> is established under the assumption r_o→ 0, it actually applies to more general scenarios. Fig. <ref> shows that p̃^FD_so in (<ref>) approximates to p_so^FD in (<ref>) in quite a wide range of r_o and λ_e particularly when λ_f is small, which demonstrates high accuracy for the approximation. Hereafter, unless specified otherwise, we often use this approximation to deal with the secrecy outage probability. Eqn. (<ref>) and (<ref>) clearly show that secrecy outage probabilities increase exponentially with λ_e and N_e. This is ameliorated by increasing q or ρ. In addition, secrecy outage probabilities increase as α increases. This is because, in a large path-loss exponent environment, jamming signals have undergone a strong attenuation before they arrive ateavesdroppers. § AREA SECURE LINK NUMBER In this section, we maximize ASLN N under a given pair of wiretap code rates R_t and R_s by determining the optimal fraction q of FD receivers.To facilitate a robustdesign, we use the upper bounded connection outage probabilityp_co^𝐒,U given in (<ref>), which actually pessimistically assess the connection performance. We also suppose eavesdroppers use a large number of antennas in order to do better wiretapping, which also gives a pessimistic evaluation of the secrecy performance. Resorting to an asymptotic analysis of p^FD_so by letting N_e≫ 1 in (<ref>), p^FD_so shares the same expression as p^HD_so in (<ref>), i.e.,p_so^𝐒=1-e^-πλ_eN_e/κ qλ_lρ^δτ_e^δ.Substituting (<ref>) and (<ref>) into (<ref>) yieldsN =λ_l (q e^-ρη r_o^ατ_t+1-q) e^-κ r_o^2τ_t^δλ_l(1+ρ^δq)-πλ_eN_e/κ qλ_lρ^δτ_e^δ.Introducing an auxiliary function F(q)=(qA+(1-q))e^-Bq-C/q with A≜ e^-ρη r_o^ατ_t<1, B≜κ r_o^2τ_t^δρ^δλ_l and C≜πλ_eN_e/κλ_lρ^δτ_e^δ, we have N=λ_l e^-κ r_o^2τ_t^δλ_lF(q) such that parameter q only exists in F(q). Hence, maximizing N is equivalent to maximizing F(q), which can be formulated as max_q F(q)=(qA+(1-q))e^-Bq-C/q, s.t. 0<q≤ 1.In the following theorem, we prove the quasi-concavity <cit.> of F(q) in q, and give the optimal solution of problem (<ref>). The optimal fraction of FD receivers that maximizes ASLN N is given by q_sl^* = 1, πλ_eN_e > κλ_lρ^δτ_e^δ(1/A+B-1), q_sl^∘, otherwise, where q_sl^∘ is the unique root q of the following equation (A+q^-1-1)(1+Cq^-1-Bq)-q^-1=0. The left-hand side (LHS) of (<ref>) is initially positive and then negative when C≤ 1/A+B-1; and thus, q_sl^∘ can be efficiently calculated using the bisection method with (<ref>). Please refer to Appendix <ref>. Theorem <ref> indicates that as eavesdropper density λ_e or eavesdropper antenna number N_e is sufficiently large such that πλ_eN_e > κλ_lρ^δτ_e^δ(1/A+B-1), all legitimate receivers should work in the FD mode; otherwise a portion of HD receivers are permitted, just as depicted in Fig. <ref>. Although it is difficult to provide an explicit expression for the optimalq_sl^∘ given in (<ref>), we are still able to develop some insights into the behavior of q_sl^∘ in the following corollary. The optimal q_sl^∘ given in (<ref>) monotonically increases with λ_e and N_e, and monotonically decreases with λ_l, r_o, η, ρ, τ_t and τ_e. In the perfect SIC case, i.e., η=0, a closed-form expression on q_sl^∘ can be further given by q^∘,η=0_sl =√(C/B) = 1/κλ_lρ^δ r_o√(πλ_eN_e/τ_t^δτ_e^δ), where q^∘,η=0_sl decreases linearly in λ_l and r_o, and increases linearly in √(λ_e) and √(N_e). Please refer to Appendix <ref>. Corollary <ref> provides some useful insights into the optimal fraction of FD receivers, which will benefit network design. For example, more FD receivers are needed to cope with more eavesdroppers or more eavesdropping antennas; whereas adding legitimate nodes or increasing jamming power allows a smaller fraction of FD receivers. In addition, we should better activate fewer FD receivers when legitimate link distance r_o increases, since the desired signal suffers a greater attenuation and the negative effect of self-interference increases more significantly. Some of the properties in Corollary <ref> are verified in Fig. <ref>, and the others are relatively intuitive. Having obtained the optimal fraction q^*_sl given in (<ref>), the maximum ASLN N^* can be calculated by plugging q^*_sl into (<ref>). Fig. <ref>depicts ASLN as a function of N_e and clearly demonstrates the superiority of our optimization scheme over those fixed-q schemes. For example, the maximum ASLN obtained at q=q_sl^* is nearly twice as large as that obtained at q=0.5 for a small N_e, and is more than twice as large as that obtained at q=0.1 for a large N_e. We observe that, as ρ increases, the ASLN obtained at q=0.5 becomes smaller in the small N_e region whereas becomes larger in the large N_e region. The underlying reason is, when N_e is small, the negative impact of jamming signals on legitimate links is larger than that on wiretap links such that fewer FD jammers should be activated; conversely, as N_e increases, the negative effect of jamming signals on wiretap links increases obviously. In sharp contrast to this, the maximum N^* always increases in ρ regardless of N_e. This is because the optimal fraction q_sl^* adaptively decreases as ρ increases so as to mitigate the negative effect of jamming signals.§ NETWORK-WIDE SECRECY THROUGHPUTIn this section, we maximize NST Ω under a pair of outage probabilities σ and ϵ by determining the optimal fraction q of FD receivers, which can be formulated asmax_q Ω,s.t. 0<q≤ 1.To proceed, we first derive the SIR thresholds τ^𝐒_t and τ^𝐒_e that satisfy p^𝐒_co(τ^𝐒_t)=σ and p^𝐒_so(τ^𝐒_e)=ϵ, respectively. We can easily calculate τ^HD_t from (<ref>); whereas it is in general difficult to derive analytical expressions for τ_t^FD. However, as reported in <cit.>, self-interference can be efficiently mitigated by exploiting the propagation domain, analog circuit domain and digital circuit domain; particularly in analog and digital signal processing it is now feasible to have up to 110 dB SIC capability <cit.>. Such positive news motivates us to consider a perfect SIC case by ignoring self-interference in order to facilitate the design. Letting η=0 in (<ref>) and (<ref>) yields uniform expressions for τ^𝐒_t and τ^𝐒_e, respectively, given byτ^o_t=(σ_o/κ(λ^HD+[1+ρ^δ]λ^FD) r_o^2)^α/2,τ^o_e=(πλ_eN_e/κρ^δλ^FDϵ_o)^α/2,where σ_o≜ln1/1-σ and ϵ_o≜ln1/1-ϵ. Substituting τ^𝐒_t=τ^o_t and τ^𝐒_e=τ^o_e into (<ref>) yieldsΩ=λ_l(1-σ) [ln1+τ^o_t/1+τ^o_e]^+.Clearly, to achieve a positive Ω, we should ensure τ^o_t>τ^o_e, which is equivalent toq>q_m≜(Δ-1)^-1ρ^-δ,where Δ≜σ_oϵ_o/πλ_eN_er_o^2. This indicates, to meet outage probability constraints, a minimum fraction q_m must be guaranteed. Given that q_m<1, the choice of σ and ϵ should satisfyΔ>1+ρ^-δ,i.e., too small a σ and/or too small an ϵ might not be promised. In the following, we only consider the non-trivial case of a positive Ω, i.e., q>q_m; and thus, maximizing Ω in (<ref>) is equivalent to maximizing ln1+τ^o_t/1+τ^o_e. Recalling (<ref>), we introduce the following auxiliary functionw(q)=lnw_1(q)/w_2(q),where w_1(q)=1+β_1(1+ρ^δq)^-α/2 with β_1 ≜(σ_o/κλ_lr_o^2)^α/2, w_2(q)=1+β_2(ρ^δ q)^-α/2 with β_2 ≜(πλ_eN_e/κλ_lϵ_o)^α/2, and w_1(q)>w_2(q)>1 for q∈(q_m,1].Hence, problem (<ref>) changes tomax_q w(q),s.t. 0<q_m<q≤ 1.Fortunately, we also successfully prove the quasi-concavity of w(q) in q and provide the optimal solution of problem (<ref>) in the following theorem. The optimal fraction of FD receivers that maximizes the NST Ω in (<ref>) is q_st^* = ∅,πλ_eN_eϵ_o^-1∈[ X,∞), 1, πλ_eN_eϵ_o^-1∈[ Y, X), q_st^∘, πλ_eN_eϵ_o^-1∈(0, Y), where q_st^* = ∅ corresponds to an empty feasible region of q, X≜σ_o/(r_o^2(1+ρ^-δ)), Y≜(κ^-α/2λ_l^-α/2ρ^-(1+δ)+(1+ρ^-δ) X^-α/2)^-δ< X [Y< ((1+ρ^-δ) X^-α/2)^-δ=(1+ρ^-δ)^-δX< X.], and q_st^∘ is the unique root q that satisfies 1-1+ρ^δq+β_1^-1(1+ρ^δq)^1+α/2/ρ^δq+β_2^-1(ρ^δq)^1+α/2=0. The LHS of (<ref>) is a monotonically increasing function of q in the range q∈(q_m,1], and is first negative and then positive when πλ_eN_eϵ_o^-1∈(0, Y); and thus, the value of q_st^∘ can be efficiently calculated using the bisection method with (<ref>). Please refer to Appendix <ref>. Theorem <ref> shows that when N_e or λ_e is sufficiently small such that πλ_eN_eϵ_o^-1<X, there exists a unique fraction q that maximizes NST Ω; otherwise no positive Ω can be achieved.In the following corollary, we provide some insights into the optimal q^∘_st given in (<ref>). The optimal q^∘_st given in (<ref>) monotonically increases with λ_e, N_e and r_o, and monotonically decreases with σ, ϵ, ρ and λ_l. Please refer to Appendix <ref>. Corollary <ref> indicates that under a moderate constraint on the connection outage probability (a large σ) or on the secrecy outage probability (a large ϵ), we should reduce the portion of FD receivers. This is because, on one hand, reducing FD receivers decreases interference such that greatly benefits legitimate transmissionsespecially when a large σ is tolerable; on the other hand, if a large ϵ is tolerable, we need fewer FD jammers against eavesdropping. It is worth mentioning that the optimal fraction q^∘_st increases as r_o increases, which is just the opposite of what we have observed in Corollary <ref>. The reason behind is that here we have ignored self-interference and meanwhile eavesdroppers who are close to a legitimate transmitter is less impaired by the paired FD receiver as r_o increases, hence more FD receivers are needed.The aforementioned theoretic results are validated in Fig. <ref>, where we see that the optimal fraction q^∘_st deeply depends on parameters r_o and σ.When r_o is large and meanwhile σ is small (e.g., r_o=2, σ=0.1) such that condition (<ref>) is violated, there exists no positive NST no matter how the network design allocates FD and HD receivers. That means in such a legitimate transmission distance, the connection outage probability requirement is too rigorous to satisfy. In order to achieve a certain level of NST, network design should have to relax the connection outage probability constraint or shorten the legitimate distance.Let us recall (<ref>) in Appendix <ref>, it is not difficult to deduce that q^∘_st is inversely proportional to ρ^δ, since ρ^δ q^∘_st keeps constant in ϕ(ρ^δ q^∘_st)=0 when the other parameters are fixed. As a consequence, q^∘_st→ 0 as ρ→∞. If we consider a dense network by letting λ_l→∞ in (<ref>), we can further obtain a simple expression for q_st^∘ given belowq_st^∘,λ_l→∞=(Δ^1/(1+δ)-1)^-1ρ^-δ,which is independent of λ_l, just as shown in Fig. <ref>. This is different from what we can see in Fig. <ref> where the optimal q_sl^* goes to zero as λ_l goes to infinity. This is because a positive secrecy rate mainly depends on the relative interference strength betweenlegitimate nodes and eavesdroppers as λ_l goes to infinity, and a certain portion of FD receivers must be activated to ensure the superiority of the main channel over the wiretap channel in terms of channel quality. Substituting q^*_st given in (<ref>) into (<ref>), we obtain the maximum NST Ω^*. Fig. <ref> compares this NST Ω^* and those obtained at fixed q's.Obviously, activating a proper fraction of FD receivers significantly improves NST. For example, the optimal fraction q_st^* increases NST by about 28% than the equal proportion case (i.e., q=0.5), and by up to 1400% than the small-q case (e.g., q=0.1). We can observe that, too small aσ might not be satisfied while too large a σ results in a small successful transmission probability and accordingly small NST. Therefore, a moderate constraint on the connection outage probability is desirable for improving NST. Fig. <ref> also illustrates the influence of the path loss exponent α on NST. A general trend is that NST increases as α becomes larger. The reason behind is that the distance between a legitimate transmitter-receiver pair is small such that the signal attenuation in a legitimate link is less significant than it is in the eavesdropper link. This implies short-range secure communications might prefer a large path-loss exponent, especially in a sparse-eavesdropper environment. § NETWORK-WIDE SECRECY ENERGY EFFICIENCYIn this section, we determine the optimal fraction q of FD receivers that maximizes the NSEE Ψ with and without considering a required minimum NST. As presented in previous sections, we consider the scenario where self-interference is efficiently canceled. §.§ Without NST ConstraintIn this subsection, we ignore the requirement of a minimum NST. Substituting (<ref>) into (<ref>), the optimizationproblem of interest can be formulated as max_q Ψ=λ_l(1-σ) w(q) /λ_l(P_t+P_c)+qλ_lP_j, s.t. 0<q_m<q≤ 1, where w(q) and q_m have been defined in (<ref>) and (<ref>), respectively. Introducing ρ_c≜P_j/P_t+P_c and the following auxiliary functionJ(q) = w(q)/1+ρ_c q,the object function Ψ of problem (<ref>) can be rewritten in the form of Ψ=1-σ/P_t+P_cJ(q). Clearly, maximizing Ψ is equivalent to maximizing J(q). In the following theorem, we prove the quasi-concavity of J(q) in q, again, and provide the optimal solution to problem (<ref>). The optimal fraction of FD receivers that maximizes the NSEE Ψ in (<ref>) is q_ee^* = ∅,πλ_eN_eϵ_o^-1≥ X, 1, πλ_eN_eϵ_o^-1<X & W/w(1)>δρ_c/1+ρ_c, q_ee^∘, otherwise, where W≜ 1-w_2^-1(1)-1-w_1^-1(1)/1+ρ^-δ and X has been defined in Theorem <ref>. In (<ref>), q_ee^∘ is the unique root q of the following equation Q(q)=0, where Q(q)=w^'(q)(1+ρ_cq)-ρ_cw(q) is initially positive and then negative as q increases; and thus the value of q_ee^∘ can be efficiently computed using the bisection method with (<ref>). Please refer to Appendix <ref>. In the following corollary, we develop some insights into the behavior of q^∘_ee given in (<ref>). The optimal q^∘_ee given in (<ref>) monotonically increases with λ_e, N_e and r_o, and monotonically decreases with σ, ϵ and ρ. Please refer to Appendix <ref>. The properties of q^∘_ee follow Corollary <ref>. Fig. <ref> depicts the optimal fraction q^*_ee and verifies Corollary <ref> well. We see that, too small σ and ϵ might not be simultaneously satisfied (e.g., σ=0.1, ϵ=0.01). As can be observed, the optimal q^*_ee keeps large in the small ρ region, and dramatically decreases as ρ increases. This is because the increase of jamming power provides a relief to the need of FD jammers. In addition, as σ or ϵ decreases, the feasible region of ρ that produces a positive Ψ reduces. This suggests that to meet more rigorous connection outage and secrecy outage constraints, we should consume more power in sending jamming signals. Fig. <ref> depicts NSEE versus ρ for different values of q. We see that as ρ increases, NSEE first increases and then decreases. The underlying reason is that too small jamming power makes NST small whereas too large jamming power leads to large power consumption; both aspects result in small NSEE. We also find that adaptively adjusting the fraction of FD receivers to jamming powersignificantly improves NSEE compared with fixed-q cases, although the latter can approach the optimal performance in some specific regions, e.g., q=1 in the small ρ region. In the following corollary, we reveal how the legitimate node density λ_l influences the optimal allocation between FD and HD receivers and the corresponding NSEE. In a sparse network, i.e., λ_l→ 0, both the optimal fraction q^*_ee and the maximum NSEE Ψ^* keep constant, which are independent of λ_l. Please refer to Appendix <ref>.§.§ With NST ConstraintFor more practical design, we should also take NST into consideration when maximizing NSEE. In this subsection, we impose a constraint on problem (<ref>) that NST Ω lies above threshold Ω^∘, i.e., Ω>Ω^∘. Since we have already obtained the maximum Ω^* in Sec. VI-A, for convenience, we only consider the case Ω^*> Ω^∘ here. If Ω^*≤Ω^∘, we just set Ψ to zero. The optimal fraction of FD receivers that maximizes the NSEE Ψ in (<ref>) subject to the constraint Ω>Ω^∘ is given as follows q_ee^⋆ = ∅,πλ_eN_eϵ_o^-1≥ X, 1, πλ_eN_eϵ_o^-1<X & W/w(1)>δρ_c/1+ρ_c, max(q_st^(1),q^∘_ee),Y≤πλ_eN_eϵ_o^-1<X & W/w(1)≤δρ_c/1+ρ_c, q_ee^+, otherwise, where q_ee^∘ has bee given in (<ref>) and q_ee^+ is determined as q_ee^+= q_ee^∘, q_st^(1)≤ q_ee^∘<q_st^(2), q_st^(1), q_ee^∘<q_st^(1), q_st^(2), q_ee^∘≥ q_st^(2). Let us denote Ω(q) as a function of q. If there exists only one root q∈(q_m,1] that satisfies Ω(q)=Ω^∘, we denote this root as q_st^(1); if there are two such roots, we denote them as q_st^(1) and q_st^(2) such that q_st^(1)<q_st^(2). Please refer to Appendix <ref>. Fig. <ref> shows the maximum NSEE Ψ^* with q_ee^* in (<ref>) and Ψ^⋆ with q_ee^⋆ in (<ref>). As indicated in Corollary <ref>, Ψ^* keeps constant in the small λ_l region, whereas Ψ^⋆ becomes zero since the constraint Ω>Ω^∘ is not satisfied. When λ_l falls in the medium range, the curve of Ψ^* and its counterpart Ψ^⋆ merge and vary smoothly. As λ_l increases further, both Ψ^* and Ψ^⋆ quickly drop to zero. Therefore, a moderate network density is desirable. Fig. <ref> also indicates that although increasing jamming power helps to suppress eavesdroppers, it is at a cost of energy efficiency.To better guide network designers on how to well design the network, Table <ref> summarizes the relationships between the optimal fraction q of FD receivers and key parameters in different objectives. § CONCLUSION AND FUTURE WORKIn this paper, we study physical layer security in a wireless ad hoc network with a hybrid full-/half-duplex receiver deployment strategy. We provide a comprehensive performance analysis and network design under a stochastic geometry framework. We first analyze connection outage and secrecy outage probabilities for a typical legitimate link, and show that enabling more FD receivers increases the connection outage probability but decreases the secrecy outage probability. Based on the analytical results of the dual probabilities, we prove that ASLN, NST and NSEE are all quasi-concave on the fraction of FD receivers, and maximize each of them by providing the optimal fraction. We further develop various useful properties on this optimal fraction. Numerical results are demonstrated to validate our theoretical findings.This paper opens up several interesting research directions. For example, the proposed framework can be extended to investigate the cooperative or multi-antenna FD receivers, where additional degrees of freedom might be gained not only in alleviating the self-interference but also in designing the jamming signals. The benefit of FD receiver jamming techniques can be further exploited by jointly optimizing the allocation between FD and HD receivers and the jamming transmit power of each FD receiver, given that the latter also strikes a non-trivial tradeoff between reliability and secrecy. Another possible direction for future research is to consider the randomness of self-interference and propose an adaptive and intelligent criterion to select work mode for receivers, e.g., letting those receivers with instantaneous self-interference power lying below a certain value work in the FD mode and the rest work in the HD mode. §.§ Proof of Theorem <ref> We start by taking the first-oder derivative of F(q) on qF^'(q) =K(q)e^-Bq-C/q,whereK(q) = (A+1/q-1)(1+C/q-Bq)-1/q.To determine the sign of F^'(q), we first investigate the behavior of K(q) at the boundaries q→ 0^+ and q=1, respectively. Substituting q→ 0^+ into (<ref>) yields lim_q→ 0^+K(q) =lim_q→ 0^+(C/q^2)>0. Substituting q=1 into (<ref>) yields K(1)=A(1+C-B)-1, the sign of which relies on specific values of A, B and C. Consider the following two cases. 1) K(1)> 0: We have A(1+C-B)>1⇒ C-B>1/A-1>0, which yields C/q-Bq≥ C-B>0. Substituting this inequality along with 1/q>1 into (<ref>), we obtain K(q)>A(1+C-B)-1> 0, i.e., F^'(q)>0. This means F(q) monotonically increases in q within the entire range q∈(0,1], and the optimal q that maximizes F(q) or N is q^*=1.2) K(1)< 0: There at least exists one point q∈(0,1] that satisfies K(q)=0 since K(q) is a continuous function of q and lim_q→ 0^+K(q)>0. Denote an arbitrary zero-crossing point q of K(q) as q_o, i.e., K(q_o)=0. To determine the monotonicity of F(q) in q, we first take the second-order derivative of F(q) at q=q_o from (<ref>)F^”(q_o) =K'(q_o)e^-Bq_o-C/q_o,whereK'(q_o)= (B+Cq_o^-2)A+ 2Cq_o^-3-Cq_o^-2-B.Clearly, the sign of F^”(q_o) follows that of K'(q_o). We resort to the equation K(q_o)=0 in (<ref>), which yields A = 1-1/q_o(1-1/1+C/q_o-Bq_o). Given that 0<A<1, we readily obtain C/q>Bq, substituting which combined with 0<q_o≤ 1 into (<ref>) yields K'(q_o)<0, i.e., F”(q_o)<0. Invoking the definition of single-variable quasi-concave function <cit.>, we conclude that F(q) is a quasi-concave function of q, and there exists a unique q that maximizes F(q). In other words, F(q) initially increases and then decreases in q, and the peak value of F(q) is achieved at the unique root q of the equation K(q)=0. By now, we have completed the proof. §.§ Proof of Corollary <ref>Recall (<ref>), and the optimal q^∘_sl satisfies K(q^∘_sl)=0. We first take the first-order derivative of q^∘_sl on A using the derivative rule for implicit functions with K(q^∘_sl)=0, i.e.,d q^∘_sl/d A =-∂ K(q^∘_sl)/∂ A/∂ K(q^∘_sl) /∂ q^∘_sl.From (<ref>), we have ∂ K(q^∘_sl) /∂ A=1+C/q^∘_sl-Bq^∘_sl>0. From (<ref>), we know that ∂ K(q^∘_sl) /∂ q^∘_sl<0. Thus, we obtain d q^∘_sl/d A>0. In a similar way, we can prove d q^∘_sl/d B<0 and d q^∘_sl/d C>0. Observing the expressions of A, B and C directly yields the relationships between the optimal q_sl^∘ and the relevant parameters. For the perfect SIC case, substituting η=0, or, equivalently, A=1, into (<ref>) directly yields the result given in (<ref>). §.§ Proof of Theorem <ref>Taking the first-order derivative of w(q) in (<ref>) on q, i.e.,w^'(q) =w^'_1(q)/w_1(q)- w_2^'(q)/w_2(q),where w^'_1(q)=-ρ^δ[w_1(q)-1]/δ(1+ρ^δq) and w^'_2(q)= -w_2(q)-1/δq. Directly determining either the sign of w^'(q) or the concavity of w(q) from the second-order derivative w^”(q) is difficult. Instead, we prove the quasi-concavity of w(q) on q by reforming w^'(q) as w^'(q)= w^'_1(q)/w_1(q)ϕ(q) with ϕ(q) given byϕ(q)=1-w_1(q)w_2^'(q)/w^'_1(q)w_2(q)=1-(1+ρ^δq)w_1(q)(w_2(q)-1)/ρ^δq (w_1(q)-1) w_2(q).Apparently, the first term w^'_1(q)/w_1(q) is negative. Next, we determine the sign of ϕ(q). Taking the first-order derivative of ϕ(q) on q, and after some algebraic manipulations, we obtainϕ^'(q)=[1+δ w_2(q)]w_1(q)+ρ^δq[w_1(q)-w_2(q)]/δρ^δ q^2 [w_1(q)-1]w_2^2(q)/[w_2(q)-1].Since w_1(q)>w_2(q)>1, ϕ^'(q)>0 always holds, i.e., ϕ(q) monotonically increases with q. When q=q_m, we have w_1(q_m)=w_2(q_m) and w^'_2(q_m)/w^'_1(q_m)=1+ρ^δq_m/ρ^δq_m, thus ϕ(q_m)=-1/ρ^δq_m<0. When q = 1, ϕ(1)=1-β_2(1+ρ^δ)(β_1+(1+ρ^δ)^α/2)/β_1ρ^δ(β_2+ρ),the sign of which depends on β_1 and β_2. Specifically, if 1+β_1^-1(1+ρ^δ)^1+α/2>β_2^-1ρ^1+δ,we have ϕ(1)<0; otherwise, ϕ(1)≥ 0. In the following, we derive the optimal q that maximizes w(q) by distinguishing two cases.1) If ϕ(1)<0, ϕ(q)<0 holds in the entire range q∈(q_m,1]. Accordingly, we have w^'(q)>0, i.e., w(q) monotonically increases with q. Therefore, the optimal q that maximizes w(q) is q^*=1.2) If ϕ(1)≥ 0, ϕ(q) is initially negative and then positive in the range q∈(q_m,1]; the zero-crossing point q that satisfies ϕ(q)=0 is denoted by q_o. We can also conclude that w^'(q) is initially positive and then negative after q exceeds q_o. In other words, w(q) first increases and then decreases with q, and q_o is the solution that yields the peak value of w(q).By now, we have proved the quasi-concavity of w(q) on q. Combined withq_m<1⇒Δ>1+ρ^-δ and the given results, we complete the proof. §.§ Proof of Corollary <ref>Recall (<ref>), and the optimal q^∘_st satisfies ϕ(q^∘_st)=0. Taking the first-order derivative of q^∘_st on β_1 using the derivative rule for implicit functions with ϕ(q^∘_st)=0 yieldsd q^∘_st/d β_1 =-∂ϕ(q^∘_st)/∂β_1/∂ϕ(q^∘_st) /∂ q^∘_st,where ∂ϕ(q^∘_st)/∂β_1=β_1^-2(1+ρ^δq)^1+α/2/ρ^δq+β_2^-1(ρ^δq)^1+α/2>0 and ∂ϕ(q^∘_st)/∂ q^∘_st>0 (see (<ref>)); and thus, d q^∘_st/d β_1<0. Similarly, we can prove d q^∘_st/d β_2>0. From the expressions of β_1 and β_2 given in (<ref>), we can infer that q^∘_st increases in λ_e, N_e and r_o, while decreases in σ and ϵ. As to dq^∘_st/dρ, we first express ϕ(ρ^δ q) asϕ(ρ^δ q)= 1 - 1+ρ^δ q+β_1^-1(1+ρ^δ q)^1+α/2/ρ^δ q+β_2^-1(ρ^δ q)^1+α/2.Taking the first-order derivative of ϕ(ρ^δ q) on ρ^δ q and invoking the equation ϕ(ρ^δ q^∘_st)=0, we can prove ∂ϕ(q^∘_st)/∂ρ>0. Thereby, we have d q^∘_st/d ρ =-∂ϕ(q^∘_st)/∂ρ/∂ϕ(q^∘_st) /∂ q^∘_st<0. As to dq^∘_st/dλ_l, we letϕ(λ_l)= 1 - 1+ρ^δ q+b_1λ_l^α/2(1+ρ^δ q)^1+α/2/ρ^δ q+b_2λ_l^α/2(ρ^δ q)^1+α/2,where b_1≜(κ r_o^2/σ_o)^α/2 and b_2≜(κϵ_o/πλ_eN_e)^α/2. Taking the first-order derivative of ϕ(λ_l) on λ_l and invoking β_1(1+ρ^δq)^-α/2>β_2(ρ^δ q)^-α/2 in (<ref>), we can prove ∂ϕ(q^∘_st)/∂λ_l>0 and d q^∘_st/d λ_l =-∂ϕ(q^∘_st)/∂λ_l/∂ϕ(q^∘_st) /∂ q^∘_st<0. By now, the proof is complete. §.§ Proof of Theorem <ref>We start by giving the first-order derivative of J(q) on q,J^'(q) = Q(q)(1+ρ_cq)^-2,where Q(q) is given in (<ref>). To proceed, we first give the following lemma which is very important to subsequent proof. If w^'(q)>0 for q∈(q_m,1], Q^'(q)<0 holds. Since Q^'(q)=w”(q)(1+Bq), to prove Q^'(q)>0 we need only to prove w^”(q)>0. Taking the derivative of w^'(q) in (<ref>) on q yields w^”(q) which is given in (<ref>) at the top of the next page, where the last equality holds for w_1^”(q)=((1+δ) [w_1(q)-1]ρ^2δ)/(δ^2(1+ρ^δ q)^2) and w_2^”(q)=((1+δ) [w_2(q)-1])/(δ^2q^2). Invoking (<ref>), we can readily obtain the following relationship w^'(q)>0⇒w_2(q)-1/w_1(q)-w_2(q)>ρ^δ q/w_1(q). Plugging the above inequality into (<ref>) yields w^”(q) < -w_1(q)-w_2(q)/δ^2 w_1(q)w_2^2(q)ρ^-δ q(1+ρ^δ q)^2×(1+ρ^δ q[1-w_2(q)/w_1(q)]+δ(1+ρ^δ q)w_2(q))<0, which completes the proof. Next, we are going to determine the sign of Q(q) or J^'(q) in (<ref>). We first determine the sign of Q(q) at the boundaries q_m and 1. Combined with w_1(q_m)=w_2(q_m), we haveQ(q_m)=w_1(q_m)-1/δ w_1(q_m)(1/q_m-ρ^δ/1+ρ^δ q_m)>0.Substituting q=1 into Q(q) yieldsQ(1) =w^'(1)(1+ρ_c)-ρ_cw(1)=α/2(1+ρ_c)(1-w_2^-1(1)-1-w_1^-1(1) / 1+ρ^-δ)-ρ_cw(1),The sign of Q(1) depends on the values of involved parameters. Let us distinguish two cases.1) If Q(1)> 0, we have w^'(1)>0. Since w(q) is quasi-concave in q (Theorem <ref>), w^'(q)>0 holds in the whole range of q∈(q_m,1], which further yields Q^'(q)<0 according to Lemma <ref>. In other words, Q(q) monotonically decreases in q, and thus, Q(q)>Q(1)>0, or, J^'(q)>0. This means J(q) monotonically increases in q, and the optimal q that maximizes J(q) is q=1.2) If Q(1)≤ 0, combined with Q(q_m)>0 in (<ref>), there at least exists one point q∈(q_m,1] that satisfies Q(q)=0 due to the continuity of Q(q) in q. Denote a zero-crossing point q of Q(q) as q_o such that Q(q_o)=0, or, J^'(q_o)=0. To determine the quasi-concavity of J(q) in q, we first take the second-order derivative of J(q) at q=q_o, which isJ^”(q_o)=Q^'(q_o)(1+ρ_cq_o)^-2.Recalling Q(q_o)=0 yields w^'(q_o)=Bw(q_o)/1+ρ_cq_o>0. From Lemma <ref>, we obtain Q^'(q_o)<0, i.e., J^”(q_o)<0. This means J(q) is quasi-concave on q, and the optimal q that maximizes J(q) is the unique root of equation Q(q)=0, i.e., q=q_o. By now, the proof is complete. §.§ Proof of Corollary <ref>Recall (<ref>), and the optimal q^∘_ee satisfies Q(q^∘_ee)=0. Similar to the proof of Corollary <ref>, we take the first-order derivative of q^∘_ee on w_1(q^∘_ee) and on w_2(q^∘_ee), respectively, using the derivative rule for implicit functions with Q(q^∘_ee)=0, and then prove d q^∘_ee/d w_1(q^∘_ee)<0 and d q^∘_ee/d w_2(q^∘_ee)>0. Through observing the monotonicity of w_1(q) and w_2(q) with respect to the parameters involved in Corollary <ref>, we can complete the proof.§.§ Proof of Corollary <ref>The expressions of w_1(q) and w_2(q) in (<ref>) tell that as λ_l→ 0, we havew_1(q)-1/w_1(q)→ 1, w_2(q)-1/w_2(q)→ 1 and w(q)→α/2lnΔ. Substituting these results into (<ref>), we find Q(q) independent of λ_l and so is the root q of Q(q)=0. Plugging the obtained solution q into (<ref>), we can easily conclude that the resulting Ψ is also independent of λ_l. The proof is complete.§.§ Proof of Corollary <ref> Let us recall (<ref>). Obviously, q_ee^⋆=∅ if q_ee^*=∅; q_ee^⋆=1 if q_ee^*=1 and Ω(1)>Ω^∘ simultaneously hold. When q_ee^*=q_ee^∘, let us distinguish two cases. In the first case, there is only one root q∈(q_m,1], denoted as q_st^(1), that satisfies Ω(q)=Ω^∘. If q_st^(1)<q_ee^∘, we have Ω(q_ee^∘)>Ω^∘ and q_ee^⋆=q_ee^∘; otherwise, q_ee^⋆=q_st^(1). 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Haenggi, Stochastic Geometry for Wireless Networks. Cambridge University Press, 2012. Gradshteyn2007Table I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, D. Zwillinger, and S. Technica, Table of Integrals, Series, and Products, 7th ed. New York: Academic Press, 2007. Boyd2004Convex S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, UK: Cambridge Univ. Press, 2004. | http://arxiv.org/abs/1703.08941v1 | {
"authors": [
"Tong-Xing Zheng",
"Hui-Ming Wang",
"Jinhong Yuan",
"Zhu Han",
"Moon Ho Lee"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170327055228",
"title": "Physical Layer Security in Wireless Ad Hoc Networks Under A Hybrid Full-/Half-Duplex Receiver Deployment Strategy"
} |
[email protected] of Computational Science and Modeling, IMX, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland Laboratory for fundamental BioPhotonics, Institutes of Bioengineering and Materials Science and Engineering, School of Engineering, and Lausanne Centre for Ultrafast Science,École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, SwitzerlandPhysical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QZ, UKLaboratory for fundamental BioPhotonics, Institutes of Bioengineering and Materials Science and Engineering, School of Engineering, and Lausanne Centre for Ultrafast Science,École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, SwitzerlandLaboratory of Computational Science and Modeling, IMX, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland Long-range ion induced water-water correlations were recently observed in femtosecond elastic second harmonic scattering experiments of electrolyte solutions. To further the qualitative understanding of these correlations, we derive an analytical expression that quantifies ion induced dipole-dipole correlations in a non-interacting gas of dipoles. This model is a logical extension of Debye-Hückel theory that can be used to qualitatively understand how the combined electric field of the ions induces correlations in the orientational distributions of the water molecules in an aqueous solution. The model agrees with results from molecular dynamics simulations and provides an important starting point for further theoretical work.Mean-Field Theory of Water-Water Correlations in Electrolyte Solutions Michele Ceriotti December 30, 2023. ====================================================================== The electric field of a solvated ion in water induces orientational ordering in the surrounding solvent molecules.However, the length scale over which this ordering persists has been a topic of significant debate, at least in part because the range at which correlations can be detected depends on the experimental probe.<cit.>The results of neutron diffraction,<cit.> X-ray scattering,<cit.>dielectric relaxation,<cit.> and femtosecond pump-probe experiments,<cit.> as well as atomistic simulations of the reorientation timescales of water molecules<cit.> and of the vibrational spectrum of solutions,<cit.> have suggested that the ordering of the surrounding water molecules by ions extends no further than about 3 solvation shells (around 0.8 nm) for sub-molar concentrations.On the other hand, infrared photodissociation experiments,<cit.> and a study combining terahertz and femtosecond infrared spectroscopies,<cit.> have found evidence for ordering extended to longer ranges. Molecular dynamics simulations looking directly at the orientational correlations between water molecules showed that the presence of ionic solutes have an effect on these correlations at distances of more than 1 nm.<cit.> Femtosecond elastic second harmonic scattering (fs-ESHS)<cit.> measurements have recently been used to probe the orientational order of water molecules in H_2O and D_2O electrolyte solutions,<cit.> revealing intensity changes that are already detectable at micromolar concentrations, and which are identical for more than 20 different electrolytes. The non-specificity of the fs-ESHS response, its magnitude, and its onset at low concentration point to its long-range origin. The isotope exchange experiment, together with the recorded polarization combinations (in conjunction with the selection rules for nonlinear light scattering experiments <cit.>) show that the recorded changes in the fs-ESHS response in the concentration range from 1 μM - 100 mM arise from water-water correlations that are induced by the ions (and not from the ions themselves). This effect shows intriguing correlations withchanges in the surface tension of dilute electrolyte solutions, suggesting that the same microscopic phenomenon underlying the second-harmonic signal can have an impact on macroscopic observables. In this Communication we derive an analytical expression for the correlations induced in a non-interacting gas of dipoles by the electric field of ions. This expression is a natural extension to a simple Debye-Hückel model, which has been shown to qualitatively capture the concentration-dependence of the second-harmonic response,<cit.> and can be used to elucidate the nature, the range and the energetics of the weak ion-induced ordering probed by fs-ESHS.[The model derived in this paper is an extension of that found in D.M.W.'s D.Phil thesis, University of Oxford, 2016.] The expression provides a benchmark for a fundamental understanding of the interplay ofion-dipole and ion-ion interactions. By comparison with classical molecular dynamics simulations of dilute NaCl solutions, we demonstrate that both of these factors are needed tocharacterize the ion-induced solvent correlations. We begin by considering the water molecules in an ionic solution to be point dipoles that interact only with the solute, and have no explicit dipole-dipole interactions. Thus, the orientational ordering of these dipoles is caused only by the electric field due to the ions. Although this might appear to be a harsh assumption – and it certainly implies that the model cannot report on short-ranged hydrogen-bonding and dipole-dipole interactions – the dipolar screening is implicitly included through macroscopic quantities such as the dielectric constant and the local field factor. We will also show later that dipole-dipole interactions can be included in a refined version of the model, and have no impact on the long-range behavior. Figs. <ref>(a) and (b) show how this model is built up: firstly, the ions are taken to be point charges in a dielectric continuum, with an appropriate spatial distribution, after which the system is filled with a uniform gas of independent dipoles,<cit.> which will align with the local electric field.We then define the dipole correlation function for two solvent molecules separated by a distance r (that is, the average inner product of two dipoles as a function of their separation), ⟨cosϕ⟩(r) = 1/V∫_V⟨μ̂(R)·μ̂(R + r)⟩_o+id^3R,where μ̂(R) is the unit vector in the direction of the dipole moment of a molecule at R, V is the volume of the system and “o+i” denotes an average over molecular orientations and ionic positions. Fig. <ref>(c) illustrates how the angle ϕ is defined for two representative water molecules. In the Supplementary Information (SI), we show that by taking a Taylor expansion in the reciprocal temperature β = 1/k_ B T, we can make the approximation, ⟨cosϕ⟩(r) ≃1/V(βμ/3)^2∫_V⟨E(R)·E(R+r)⟩_id^3R, where E(R) is the total electric field at position R due to all of the ions in the solution, and μ is the permanent dipole moment of a water molecule. The subscript “i” indicates that the average is taken over the positions of ions. For simplicity of notation, any angular brackets in the following work without a subscript are taken over ion positions only. Eqn. (<ref>) shows that in our model the correlation between dipoles is proportional to the correlation between electric fields, which are taken to be the only source of ordering for the molecules.The electric field E(R) at a given position is the sum of electric fields due to all of the ions. This allows us to write, E(R) = ∑_m∈ionse Z_m f_0/4πϵ_0ϵ_ rℰ(R-r_m), with Z_m the charge of the m^ th ion in units of the electron charge e and r_m the position of this ion, f_0 the Onsager local field factor,<cit.> ϵ_0 the vacuum permittivity, ϵ_ r the solvent dielectric constant, and ℰ(r) the electric field associated with individual ions (most commonly the Coulomb field, r/r^3). This gives ⟨cosϕ⟩(r) ≃ A/V∑_m,n Z_mZ_n∫_V⟨ℰ(R - r_m)·ℰ(R + r - r_n)⟩d^3R, in which we have defined A = (βμ f_0 e/12 πϵ_0ϵ_ r)^2.In the thermodynamic (V→∞) limit the integral in Eqn. (<ref>) is taken over all space and can be most conveniently expressed in reciprocal space,⟨cosϕ⟩(r) ≃ A/V∫ℰ(K)·ℰ(-K) ⟨∑_m,n Z_m Z_n e^iK·(r_m - r_n)⟩ e^iK·rd^3K/(2π)^3, where ℰ(K) is the Fourier transform of the field function ℰ(r). The term in angular brackets is proportional to the charge-charge structure factor S(K) of the ions.<cit.> This gives the dipole correlation function in terms of the ion number density ρ as, ⟨cosϕ⟩(r) ≃ ρ/(2π)^3(βμ e f_0/12 πϵ_0ϵ_ r)^2∫| ℰ(K)|^2 S(K) e^i K·rd^3K. The most appropriate mean-field model can be obtained by taking the field function (r) to be the Coulomb field r / r^3 (corresponding to (K) = -4 π i K / K^2), andusing the Debye-Hückel (DH) structure factor <cit.> S(K) = 2 K^2/K^2 + κ^2, where κ = (2 ρβ Z^2 e^2/ϵ_0ϵ_ r)^1/2 is the inverse Debye length. This gives ⟨cosϕ⟩_DH(r) = ρ/2π(βμ e f_0/3ϵ_0ϵ_ r)^2e^-κ r/r. The variation of ⟨cosϕ⟩_DH(r) with ion concentration is instructive. As seen in Figure <ref>,for small ρ, an increase in concentration leads to an increase in correlation between solvent dipoles, while for large ρ the e^-κ r factor dominates. Increasing the concentration results in ions being more screened and with a lesser propensity to orient solvent dipoles. It should be also noted that, at all of the concentrations shown in Fig. <ref>, the dipolar correlations at distances above 5 nm are very small. However, because the number of water molecules further than 5 nm away is very large,these correlations can be measured by fs-ESHS experiments, a testimony to the exquisite sensitivity of the probe.Eqn. (<ref>) allows us to investigate the interplay between the ion-ion spatial correlations (encoded in S(K))and the ion-dipole orientational correlations (due to the electric field (r)).By changing the form ofS(K), one can estimate the response to an arbitrary distribution of ions: for instance, one could extend this model to investigate the correlations induced by charges on an interface. A particularly instructive example involves a completely uncorrelated arrangement of ions. This random-ion (RI) model is equivalent to setting S(K) = 2,which leads to dipole-dipole correlations corresponding to Eqn. (<ref>) with κ = 0, while the concentration ρ is kept constant. At all concentrations, this RI model leads to increased dipole-dipole correlations, because of thelack of screening of the Coulombic ion-dipoleinteraction by the correlated cloud of counterions. It is worth stressing that, although it might beappealing to qualitatively discuss the dampening of correlations in terms of the exponentially-screenedDH field of an ion, this is not an appropriate model. Such a screened-field/random ions (SF-RI) modelamounts to setting ℰ(r) =-∇ e^-κ r/r = r(e^-κ r/r^3 + κ e^-κ r/r^2) and S(K) = 2. The resulting functional form of the induced dipole-dipole correlations resembles that of the full DH model at short distances, but then leads to unphysical anticorrelations at large distance (see the SI).Fig. <ref> compares the predicted ⟨cosϕ⟩(r) using the full DH theory, the RI and the SF-RI models, and the correlations computed from a MD simulation using a ∼ 20 nm cubic box with about 264,000 TIP4P/2005 water molecules.<cit.> All curves correspond to a salt concentration of 8 mM and a temperature of 300 K. The other physical constants used are described in the SI. Comparison with MD results in Fig. <ref> shows that only the full DH model captures the correct long-range behavior of the dipole-dipole correlations – although the short range structure is clearly absent. Neglecting ion-ion spatial correlations artificially increases the orientational correlations, since randomly distributed ions cannot efficiently screen the fields of other ions.A picture in which one interprets dipole-dipole correlations in terms of the screened electrostatic field of the ions, while providing a qualitativepicture of the physics, is inconsistent with thelinearized-Boltzmann structure of the mean-field model, and fails to quantitatively reproduce the MD results.This comparison demonstrates that the long-ranged dipole-dipole correlations are most naturally interpreted as being due to the bare electric field of the ions. The correlations are modulated by short range interactions (which are not included in this model), and by the presence of ion-ion spatial correlations, which result in the screening of the Coulomb field. This latter effect leads to decreased dipole-dipole correlations and provides an explanation for the saturation of the fs-ESHS signal at high electrolyte concentrations. We note that the mean-field modelcan be further improved to include more physical effects. ⟨cosϕ⟩_DH(r) diverges in the r→ 0 limit because ofthe singularity in the electric field at the ion positions.It is possible to remove this short-distance divergence by restricting the volume of space in which water molecules can be found; however, the fact that two water molecules have a distance of minimum approach, below which ⟨cosϕ⟩(r) is not meaningful, makes the divergence irrelevant. We can also estimate the impact of neglecting dipole-dipole interactions, by re-introducing them in a perturbative fashion.This can be done by following the procedure used to derive the approximation in Eqn. (<ref>), including also the dipole-dipole interaction. In doing so, we find (as described in the SI) that the lowest-order term in ⟨cosϕ⟩(r) that includes the dipole-dipole forces is proportional to β^4 e^-κ r/r^7. This term decays much more rapidly than does the model of Eqn. (<ref>), and makes essentially no contribution at long enough distances: above 0.33 nm, the magnitude of this correction is less than 1 % of the magnitude of ⟨cosϕ⟩_DH(r), and less than 10^-3 % above 1 nm.The computed residual orientational correlation of dipoles at a distance of several nm is extremely small, but since it involves many dipoles the total change in free energy may be non-negligible. In order to elucidate the free energy scale associated with ion-induced long-range dipole-dipole correlations,we evaluate the total energetic contribution associated with the oriented dipoles at distances larger than a chosen cut-off length r_c, which reads (see the SI), <cit.> U = 4πρ_ Sμ∫_r_ c^∞ r^2 E(r) ℒ(βμ E(r)) d r, where ℒ(x) = (x) - 1/x is the Langevin function and ρ_ S is the solvent density. The mean electric field E(r) around an ion is given by Debye-Hückel theory. The integral can be computed by expanding the integrand as a Taylor series in β.Fig. <ref> shows the total energetic contribution of the dipoles oriented by an ion as a function of the electrolyte concentration and for different cut-off distances.At mM concentrations, dipolar order beyond the Bjerrum length (∼ 0.7 nm in water at 300 K) is associated with an energy scale of about 3 k_B T, and even the tails beyond 4 nm correspond to a significant fraction of k_B T. Due to the large number of dipoles in the far region, the collective effect is significant even though each ion-dipole interaction is very small. Thus, it is plausible that ion-induced dipole-dipole correlations extending well beyond the Bjerrum length could lead to measurable changes in the macroscopic energy (as observed in the surface tension measurements of Ref. Chen2016). As this analysis is performed with a very simplified model, this conclusion is not definitive, and a more quantitative analysis should include changes in the long-range dipole-dipole order in the bulk and in the surface region. These changes could then be connected to changes in the free energy of the surface and the bulk region.In conclusion, we have shown that long-range, non-specific electrolyte-induced correlations in water as recently observed in fs-ESHS experiments can be captured by a simple mean-field model that treats water molecules as non-interacting dipoles oriented by theelectrostatic field of ions, which are themselves correlated following Debye-Hückel theory. Although one can intuitively understand theorientational correlations as arising from the exponentially-screened field of correlated ions, a more accurate picture, leading to quantitative predictions of MD simulations, regardsthem as arising from unscreened ion-dipolecorrelations that combine destructivelywhen the physically relevant ion-ion correlations are included.This model is very useful to pinpoint what we think is the main physical origin of the electrolyte-induced change in the fs-ESHS intensity and to estimate the length and energy scale of the effect.It does not, however, explain the dramatic isotope effects that are seen in experiments,<cit.> or the temperature dependence of the fs-ESHS signal.As such it is clearly only a first step in a complete description of the experimental data, which should also include a re-evaluation of the molecular hyperpolarizability tensor,<cit.> particularly when probed by femtosecond laser pulses.<cit.>§ SUPPLEMENTARY INFORMATION See supplementary information for more detailed derivations of the formulas used in the main text, as well as a list of the numerical values of physical constants used.The authors thank Damien Laage for helpful discussions, and Halil Okur and Yixing Chen for critical reading of the manuscript. 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"authors": [
"David M. Wilkins",
"David E. Manolopoulos",
"Sylvie Roke",
"Michele Ceriotti"
],
"categories": [
"cond-mat.stat-mech",
"cond-mat.soft"
],
"primary_category": "cond-mat.stat-mech",
"published": "20170327195602",
"title": "Mean-Field Theory of Water-Water Correlations in Electrolyte Solutions"
} |
firstpage–lastpage Coupling librational and translational motion of a levitated nanoparticle in an optical cavity Zhang-qi Yin December 30, 2023 ==============================================================================================We report discovery of the lowest mass ratio exoplanet to be found by the microlensing method in the light curve of the event OGLE 2016–BLG–1195. This planet revealed itself as a small deviation from a microlensing single lens profile from an examination of the survey data. The duration of the planetary signal is ∼ 2.5hours. The measured ratio of the planet mass to its host star is q = 4.2± 0.7 ×10^-5. We further estimate that the lens system is likely to comprise a cold ∼3 Earth mass planet in a ∼2 AU wide orbit around a 0.2 Solar mass star at an overall distance of 7.1 kpc.gravitational lensing: micro – planets and satellites: detection – stars: individual: OGLE 2016–BLG–1195 § INTRODUCTION In the technique of gravitational microlensing, planetary systems are utilized as naturally occurring lenses of light from background source stars <cit.>.In this technique, one observes the magnification of the source star as the lens star moves across the line-of-sight from Earth. If the lens star has planets, then additional lensing can occur producing perturbations in the profile one would otherwise expect for a single lens. Interestingly, the planetary signal strength is not necessarily weaker for low mass planets, making the technique of microlensing capable of detecting planets down to Earth mass for ground based projects <cit.>and Mars mass for space based projects <cit.>.Of the approximately 3500 extrasolar planets so far discovered, most have been detected by the radial velocity technique <cit.> or transit technique <cit.>. The radial velocity and transit techniques are most sensitive to warm planets with close-in orbits around the host stars. In contrast, gravitational microlensing is most sensitive to cold planets in wider orbits. In planetary formation, an important delimiter in the protoplanetary disk is the “snowline”, beyond which water remains as ice during the planetary formation process <cit.>. It is important to understand the process of planetary formation beyond the snowline, and microlensing is well suited to probe this important region of parameter space.To date, there have been 51 published discoveries of extrasolar planets by microlensing. Most of these have estimated masses above the 12–15 Earth mass threshold that separates the low mass rocky planets from the gas giants. Statistical measures have been derived from microlensing data for giant planets <cit.> with a recent study showing a break in the power law distribution of the planet:host star mass ratio at around10^-4 <cit.>. It is important to probe the distribution of planets with mass ratios below this value. Here we report a microlensing discovery of a planet with the lowest ratio of its mass to its host star amongst microlensing planets.§ OBSERVATIONS The microlensing event OGLE 2016–BLG–1195 (hereafter ob161195)was discovered by the OGLE-IV survey and was alertedby the Early Warning System <cit.> on June 27.57, 2016 (UT). The equatorial coordinates of the event are: α=17:55:23.50 δ=-30:12:26.1 (J2000.0). OGLE-IV monitoring of the event was conducted with the 1.3-m Warsaw telescope located at the Las Campanas Observatory, Chile. The telescope was equipped with the 32 CCD mosaic camera covering 1.4 square degrees with the resolution of 0.26/pixel <cit.>. Observations were obtained through the standard I-band filter. ob161195 was located in one of the frequently observed fields with the standard cadence once per hour. Unfortunately, it occurred during the microlensing Kepler K2C9 campaign <cit.> and was located outside the superstamp monitored continuously by the satellite. Thus, the OGLE cadence of this field for the time of the K2C9 campaign was reduced to three per night.ob161195 was alerted by the MOA collaboration as MOA 2016–BLG–350 on 2016 July 28 10:55 UT approximately 20 hours after the OGLE alert. Since 2006, the MOA microlensing survey has employed a 1.8 m telescope and 80 megapixel camera <cit.> at the University of Canterbury Mt John Observatory near Lake Tekapo, New Zealand. During a single exposure the MOA camera captures a field of view of 2.4 square degrees, with 23 separate target fields on the sky for microlensing survey observations. MOA employs a high cadence observational strategy that aims to routinely cycle through these target fields as many times per night as possible, with some fields being observed more often than others. MOA routinely surveys the Galactic Bulge with a custom broadband red filter, hereafter denoted , which corresponds approximately to the sum of the standard I and R passbands. Occasional observations (once per night) are made in the visual band filter, hereafter , which approximates the standard V passband. Theobservations are reduced in real-time as part of the analysis pipeline that is designed for detecting microlensing events and other astrophysical transients <cit.>. Theobservations are reduced offline as it is not necessary to do this in real-time in our detection of transient events.Event ob161195 occurred in one of the MOA fields that is sampled every 16 minutes. The light curve was well sampled for several days around the peak of the event with the only interruptions due to daylight. On the second night after the MOA alert, visual inspection of observations revealed a possible microlensing anomaly in progress. The lightcurve profile featured a small perturbation that resembles what one would expect from a low mass planet in the lens system <cit.>. The MOA observational cadence of the corresponding field was increased once this anomaly was noticed. An alert was issued to the microlensing community just after the peak of the perturbation but this feature was over within an hour of this alert. As a result, no effective follow-up observations could be carried out. Preliminary models were circulated that indeed showed the perturbation was likely caused by a planet orbiting the lens star. Because the ob161195 planetary anomaly was very short and occurred during Chilean day time it was therefore not possible to confirm the planetary perturbation in the OGLE data. This event is in the observational footprint of the new Korean Microlensing Telescope Network which operates three microlensing survey telescopes in Australia, South Africa, and Chile <cit.>. Their coverage was were not influenced by the MOA and OGLE alerts. However, their data confirm the existence of this perturbation and, in the interests of a completely independent analysis, their observations are presented separately <cit.>.In this work, we present the analysis of the MOA and OGLE data from the point of view of the discovery observations.§ DATA ANALYSIS§.§ Difference Imaging PhotometryIn order to obtain optimized photometry, we carried out an offline re-reduction ofandimages obtained by extracting sub-images centered on the event from the larger observation images. For this offline analysis we selected observations from mid 2011 to the end of 2016. Difference imaging was used to derive the photometry, with each of the MOA passbands treated separately with their own reference images. For offline analysis, we use our own implementation that incorporates a numerical kernel as described by <cit.> with our own modification to allow for a spatial variation of the kernel across the field-of-view in a similar manner to that given by <cit.>.Microlensing events are observed in crowded fields and their centroids on the images are often blended with neighbouring stars. It is important that these centroids are measured carefully because the nearest resolved star on the image will not necessarily be the source star for the microlensing event. The MOA difference images are measured using a reference image analytical PSF model of the form used in the Dophot photometry code of <cit.>. This PSF is then used to measure a given difference image after convolving with the kernel for that observation image. The model PSF is optimized by finding the centroid and shape parameters that give the best photometry on a set of difference images where the flux of the source star is significantly magnified.The baseline photometry, inandseparately, was examined during the off-event years (ie 2011–2015) for any correlations that may be present due to variations in the seeing and the effects of differential refraction (parameterized by the hour angle). We find some small effects present and so we “detrend” the data by modelling the baseline as a function of seeing and the hour angle. We see an improvement in the standard χ^2 goodness of fit of Δχ^2≈698 inand Δχ^2≈23 forin the baseline. These respectiveandmodels were used as corrections to photometry which were subtracted off all the data including those where the source is magnified. Furthermore, all observations with FWHM worse that 4.5 pixels were rejected. This resulted in 13969band and 253band measurements that are used in this study. The OGLE data completed our photometry set with 5365 measurements.Difference imaging photometry measures flux differences for the observation images with respect to the reference image. It is desirable to place these measurements onto an instrumental magnitude scale. The Dophot photometry software <cit.> was run on the difference imaging reference images for each of theandpassbands. The resulting list of extracted stellar objects were cross referenced with each other to produce a single catalog of field stars where instrumental magnitudes in bothandcould be obtained. We add to this catalog an object corresponding to the optimized centroid location (from above) of the source star for the microlensing event. The fluxes of these catalog stars were then measured on a selection of unsubtractedandobservation images using the same procedure that is used to measure the difference images, but using the PSF parameters derived from running Dophot. Linear regression was used to register these fluxes to those as measured by Dophot. Using regression again and the event photometry from the unsubtracted images as a template, the flux differences resulting from the difference imaging analysis can then be transformed onto the same flux scale as those in the Dophotandcatalog.The MOA instrumental magnitudes were calibrated by cross referencing stars in our Dophot catalog to stars in the OGLE-III catalog which provides measurements in the standard Kron-Cousins I and Johnson V passbands <cit.>. From this we derived the following relation between the MOA instrumental magnitudes and colours and the standard magnitudes and colours.I_OGLE-III - = [28.126±0.003] - [0.218±0.002] C_MOA(V-I)|_OGLE-III= [0.505±0.004] + [1.105±0.003] C_MOAwhere C_MOA =-. With these relations, together with our catalog of Dophot measured stars, we identified the well known “red clump giants” on the I_OGLE-III vs (V-I)|_OGLE-III colour-magnitude diagrams. Using only stars within 2 of the event position, we measured the centroid of the clump to be:I_clump = 16.212 ± 0.018 (V-I)|_clump = 2.468 ± 0.007Adopting the intrinsic red clump colour of (V-I)|_RCG,0=1.06 <cit.> and intrinsic magnitude I_RCG,0=14.45 <cit.>, we derive the following for the extinction and reddening towards the direction of the event:A_I= 1.762 ± 0.018E(V-I)= 1.408 ± 0.007 We will use these values in our subsequent modelling and analysis of the source star properties. §.§ Modelling the EventWe modelled the light curve photometry for this event using the image-centered ray-shootingmethod <cit.> to calculate finite source effects. This method has been tested extensively for mass ratios down to 10^-7 <cit.>. The calculation of finite source effects requires an appropriate limb darkening model for the source star. A model independent measurement of the source star colour can obtained by plotting near simultaneousandmeasurements against each other as shown in Fig. <ref>. The slope of the plot gives the ratio of source star fluxes in the respective passbands, or equivalently, the magnitude difference. We derive a model independent instrumental colour index as -=1.476±0.029. Using our instrumental magnitude calibrations and reddening measurements from the previous section, we obtain a dereddened colour index of V-I=0.728±0.033 for the source star. This corresponds to an effective temperature T_eff∼6000K <cit.>. In our modelling we use a linear limb darkening law with parameters appropriate to this value of T_eff and metallicity log g=4.5.Our magnification profile for a binary lensing model is described by the following parameters: the time of closest approach to the barycentre t_0; the Einstein radius crossing time t_E; the impact parameter, u_0, in units of the Einstein ring radius of the source star trajectory with respect the binary lens barycentre; the ratio q of the secondary lens component to the primary; the source radius crossing time t_*; the separation, s, of the binary lens components projected onto the a plane at the lens system perpendicular to the Earth-source line-of-sight; and the angle, ϕ, the source star trajectory makes with the planet-star separation.In our modelling we sought an optimal set of these parameters that can jointly model our observations in the MOAanddata and the OGLE data. The observational data are fully described by the 7 parameters that describe the magnification profile together with 2 flux scaling parameters for each of the passbands. Our goodness of fit is assessed using the standard χ^2 measure combined from the three passbands we use here. We employed the standard technique of searching the phase space of 7 parameters that describe the magnification profile using the Markov Chain Monte Carlo method to find those parameters that minimize the value of χ^2. The measured binary lens model parameters so derived are listed in Table <ref>. The uncertainties in the parameters correspond to their respective range of values that satisfy the standard criterion of χ^2<χ_min^2+1.We find two possible solutions: a “close” model and a “wide” model where the projected separation onto the lens plane is either inside or outside the Einstein radius. The lightcurvetogether with the best fitting wide model is plotted in Fig <ref>. The wide model is only slightly favoured at an insignificant level of Δχ^2≈1. Formally, we are unable to distinguish between the two models. As expected, we find the observed lightcurve is best reproduced with a binary lens model of extreme mass ratio corresponding to a planetary mass for the secondary. The very small measured mass ratio below 10^-4, for the close and wide models, is striking here. The planetary perturbation is covered only by the MOAdata and is the dominant contributor to measurement of the planetary microlensing parameters. The addition of the OGLE data allows for a tighter constraint on our measurement of t_E. Magnification maps are a useful tool for visualizing the possible magnification profiles for a given source star trajectory in a microlensing event <cit.>. In Fig. <ref>, we show these maps for the close and wide solutions. The high sensitivity in this event to such a low mass ratio planetary system is because the planet is very close to the Einstein ring in both the close and wide cases. This results in a region of enhanced magnification on the map that extends a long way along the line separating the planet and the lens star. The planetary perturbation occurs when the source star crosses this line. For this lens system geometry, a wide range of possible values of u_0 would have resulted in a planetary perturbation if one had been observing at that time. We note that the central time and width of the deviation is consistent with that in a relation given by <cit.> who study this effect for events with high peak magnifications where u_0<0.02.Due to the short timescale of the event, we could not measure the microlensing parallax effect. It is a possible that a short term planetary perturbation could be mimicked by a binary source of extreme flux ratio where the fainter companion gets highly magnified <cit.>. We attempted to model our observations with a static binary source single lens model. We parameterize this model with the Einstein crossing time, t_E of the binary source together with the time of closest approach, t_0, and impact parameter, u_0 of the primary component. We then introduce five additional parameters for the secondary companion of the primary. These are its dimensionless separation, d, from the primary, its position angle, ψ, with respect to the trajectory of the primary, and its flux ratio, α, to the primary, the ratio ρ of the angular size of the companion to that of the Einstein ring, and the coefficient, λ, of a linear limb darkening law. We consider finite source effects for the secondary because, if the binary source model is to account for the perturbation we observe, the secondary is expected to be highly magnified and pass close to the lens.The best fitting parameter values are listed in Table <ref>. The negative value of u_0 together with the values for d, ψ, and ρ mean that the secondary lags behind the primary and passes over the lens over the course of the event. With a flux ratio of ∼0.002, the secondary is significantly fainter than the primary but is more highly magnified as expected. In Fig. <ref> we show a close up view of the observed perturbation together with the best fitting binary source model and the wide planetary microlensing model. Overall, the planetary model does a better job at reproducing the features of this perturbation. In particular, the binary source model does not fit the beginning and end of the perturbation as well as the planetary model. The difference in the goodness-of-fit between the two models is Δχ^2≈120. We can compare this to the similar case of OGLE 2005–BLG–390 where a binary source model was excluded in favour of a planetary model at Δχ^2≈46 <cit.>.The parameters of planetary and binary source models considered here are not nested parameters. Strictly speaking, the difference in χ^2 is not an appropriate measure to compare the two. Following the approach of <cit.>, we compare the models using Akaike's Information Criterion AIC=χ^2+n_param and the Bayesian Information Criterion BIC=χ^2+n_paramln(N_data). These are standard criteria used to select a preferred model and they penalize for the number of parameters used. In our data, the planetary model gives the smaller value for both of these criteria. We find for the difference between the models: ΔAIC≈121 and ΔBIC≈129.Here, we conclude that the binary source model is excluded in favour of the planetary models for ob161195. §.§ Observed Source Star Properties The source star fluxes are measured as scaling parameters when determining the best fitting microlensing magnification profile. Though sparsely sampled, the MOAmeasurements cover parts of the light curve where the source star is magnified allowing a measurement of the source star flux in this passband. For the MOA data, we derive an instrumental source star magnitude of =-8.226±0.001 and an instrumental colour -=1.457±0.018. The MOA instrumental colour is in good agreement with the model independent value presented in the previous section. Using our instrumental calibration from Section <ref>, the apparent source star magnitude and colour in the OGLE-III system isI_src = 19.581±0.001(V-I)|_src = 2.113±0.020From our measurements of the red clump in Section <ref>, the extinction corrected and dereddened source star magnitude and colour isI_src,0 = 17.819±0.018(V-I)|_src,0 = 0.705±0.022 An OGLE independent determination of the dereddened color based on OGLE-IV photometry yields (V-I)_0 = 0.67±0.03. This is consistent with MOA result. In Fig. <ref> we plot a I_OGLE-III vs (V-I)_OGLE-III colour magnitude diagram for MOA and OGLE measurements of resolved stars together with the above source star measurements. The position of the source star magnitude and colour measurements is well below the red clump and sub giant regions. Main sequence stars could not be resolved in our Dophot measurements of the reference images. However, we can compare our (V-I, I)|_src,0 measurement with the colour magnitude diagram of main sequence stars of <cit.> based on HST observations of Baade's window. After allowing for extinction and reddening based on the red clump measurements of the HST data <cit.>, our measurements of the source star magnitude and colour are placed well within the main sequence star distribution. Using Table 1 of <cit.> for log g=4.5, our value for (V-I)|_src,0 implies an effective temperature of 5820 K—not too dissimilar to value of 5770 K for the Sun. We conclude that the source star in this event is a Solar-like star.These measurements can be used to calculate the angular source star radius, θ_src using the follow relation involving the apparent magnitude and colours in the OGLE-III system <cit.>log 2θ_src = 0.5014 + 0.4197 (V-I) - 0.2IThis gives θ_src=0.856±0.019 μas. §.§ Estimation of Lens System Parameters For a number of events, it is possible to detect microlensing parallax as the Earth moves on its orbit, and then derive measurements of the absolute masses of the lens system components together with the distance to the lens. The microlensing Einstein ring crossing timescale of around 10 days is too small to allow a detection of parallax. Following the approach of <cit.> we employ standard Bayesian techniques to estimate the parameters in the lensing system. To derive distributions of lens distances and velocities we use a Galactic model comprising a double-exponential disk <cit.> and a bar model from <cit.>. We constrain the mass of the lens and its distance to one relation by a measurement of the angular radius of the Einstein ring. From the microlensing parameters in Section 2 and the source star angular size measurement from Section <ref>, we haveθ_E = θ_srct_E/t_* = 0.261±0.020 mas We make the assumption that planets are equally likely around stars regardless of the planet mass, and the mass of the host star and its distance. We further assume that theplanetary orbital planes have random and uniform orientations. In Table <ref>, we list our estimations of the planet and host star mass, the 3D orbital separation, and the distance to the planetary system. As the characteristic measurement we take the value that maximizes its respective likelihood function. Also provided are the upper and lower limits at the 68% and 99.7% confidence levels. We see very little difference in the estimated values of the parameters when comparing the close and wide models. Even the values for the orbital separation are effectively in agreement. Qualitatively, we have a planet at around ∼1.8 AU from its host star that could be just within or just outside the Einstein ring. In Table <ref>, we present the averages of their respective close and wide values.§ DISCUSSION The planet ob161195Lb has the lowest mass ratio, measured to date, amongst microlensing planets that orbit a single star without any binary companions. The closest contender is the planet in the binary system OGLE-2013-BLG-0341L <cit.>, assuming that the wide binary model, rather than the circumbinary model, is correct for that event. This planet is the sixth microlens planet with a sub-10^-4 mass ratio—the others being OGLE-2013-BLG-0341LBb, OGLE 2005–BLG–390Lb <cit.>, OGLE 2005–BLG–169Lb <cit.>,OGLE-2007-BLG-368Lb <cit.>, and MOA 2009–BLG-266Lb <cit.>.Because the planetary mass ratio is measured in all planetary microlensing detections, the statistical properties of the planetary systems probed by microlensing are most easily described in terms of the mass ratio. A recent statistical analysis of the planetary signals detected by the MOA survey from 2007 through 2012 <cit.>was able to identify a power law break in the mass ratio function based on 23 planets fromthe MOA survey and 7 additional planets from previous analyses<cit.>. Due to the lack of detected planets with mass ratios lower than q = 3× 10^-5, the precise location of this break is somewhat uncertain, at q_ break = 6.7+9.0 -1.8× 10^-5. Thus, the newly discovered planet, ob161195Lb, is close to the mass ratio function break, and it does not provide a strong constraint on the behavior of the mass ratio function below the break. It is likely to be an example of the most common type of planet that orbits beyond the snow line.Although the MOA observational cadence was increased just after the planetary perturbation was noticed, this planet can be regarded as a near “blind survey only” detection where the planetary signal was noticed after the fact. This contrasts with the classic “followup mode” where a real-time alert of either a high magnification or planetary perturbation in progress results in subsequent followup observations by other telescopes or the survey telescope itself. It is expected that most planetary discoveries in new generation microlensing projects KMTNet <cit.> and WFIRST <cit.> will be of the blind survey type. Previous examples of these types of microlensing planet detections are the giant planets MOA 2011–BLG–322Lb <cit.>, MOA 2015–BLG–353Lb <cit.>, and OGLE 2012–BLG–0950Lb <cit.> and the rocky planet MOA 2007–BLG–192 <cit.>.Our statistical analysis in estimating the absolute parameters of this planetary system, does not rule out the possibility of a super Earth planet orbiting a Solar-like star just within the outer edge of the liquid water habitable zone. For the proposed WFIRST microlensing survey, it is expected that a small, but not insignificant, fraction of the microlensing planet yield will comprise planets in the habitable zone <cit.>. High resolution follow-up observations by Keck reveal a possible supermassive planet within the habitable zone of the low mass star star in MOA 2011–BLG–293 <cit.>.However, it is more likely that ob161195Lb is a cold rocky super Earth orbiting an M star. Qualitatively, there is little difference between the close and wide solutions for the 3D orbital separation. Both place the planet at ∼2 AU which would place the planet beyond the snowline, and habitable zone, of its M-star host. It is worth noting that stellar insolation is not the only source of heating for a planet. Internal heating, tidal friction <cit.>, volcanism <cit.>, and radiogenic heating <cit.> can also contribute to the heat budget of a planet to allow liquid water below the planet's surface. The extended sub-surface habitable zones could be more than ten times larger than the circumstellar habitable zones of stars <cit.>. It is possible that ob161195Lb is a rocky planet in this extended habitable zone.It is therefore important to determine the absolute values for planet mass, host star mass, and orbital separation in ob161195L. Unfortunately, this event lies outside the fields-of-view of the recent K2C9 survey <cit.> so a parallax measurement combining ground based and Kepler observations is not possible here. The 95% upper limit on the mass estimation of the lens star in 1.1M_⊙. This corresponds to an upper limit on the brightness of the lens star of I≳19.5. The best prospects for measuring the physical properties of the planet would be to use follow-up high resolution imaging.§ CONCLUSIONS We have discovered a low mass planet in the microlensing event ob161195. This planet has the lowest mass ratio and lowest mass fraction amongst microlensing planets so far detected. Although the absolute masses of the lens system components could not be measured, the measured mass ratio here is an important additional data point in the so far limited sample of mass ratio measurements below the newly discovered break at q∼10^-4.§ ACKNOWLEDGEMENTS The MOA project is supported by JSPS Kakenhi grants JP24253004, JP26247023, JP16H06287, JP23340064 and JP15H00781 and by the Royal Society of New Zealand Marsden Grant MAU1104. The OGLE project has received funding from the National Science Centre, Poland, grant MAESTRO 2014/14/A/ST9/00121 to AU. NJR is a Royal Society of New Zealand Rutherford Discovery Fellow. AS is a University of Auckland Doctoral Scholar. mnras | http://arxiv.org/abs/1703.08639v2 | {
"authors": [
"I. A. Bond",
"D. P. Bennett",
"T. Sumi",
"A. Udalski",
"D. Suzuki",
"N. J. Rattenbury",
"V. Bozza",
"N. Koshimoto",
"F. Abe",
"Y. Asakura",
"R. K. Barry",
"A. Bhattacharya",
"M. Donachie",
"P. Evans",
"A. Fukui",
"Y. Hirao",
"Y. Itow",
"M. C. A. Li",
"C. H. Ling",
"K. Masuda",
"Y. Matsubara",
"Y. Muraki",
"M. Nagakane",
"K. Ohnishi",
"C. Ranc",
"To. Saito",
"A. Sharan",
"D. J. Sullivan",
"P. J. Tristram",
"T. Yamada",
"T. Yamada",
"A. Yonehara",
"J. Skowron",
"M. K. Szymanski",
"R. Poleski",
"P. Mroz",
"I. Soszynski",
"P. Pietrukowicz",
"S. Kozlowski",
"K. Ulaczyk",
"M. Pawlak"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170325023406",
"title": "The Lowest Mass Ratio Planetary Microlens: OGLE 2016-BLG-1195Lb"
} |
[email protected] Department of Physics, Indian Institute of Technology Guwahati, Assam 781039, [email protected] Physical Research Laboratory, Ahmedabad 380009, India [email protected] Department of Physics, Indian Institute of Technology Guwahati, Assam 781039, India We study an extension of the Inert Higgs Doublet Model (IHDM) by three copies of right handed neutrinos and heavy charged leptons such that both the inert Higgs doublet and the heavy fermions are odd under the Z_2 symmetry of the model. The neutrino masses are generated at one loop in the scotogenic fashion. Assuming the neutral scalar of the inert Higgs to be the dark matter candidate, we particularly look into the region of parameter space where dark matter relic abundance is primarily governed by the inert Higgs coupling with the leptons. This corresponds to tiny Higgs portal coupling of dark matter as well as large mass splitting within different components of the inert Higgs doublet suppressing the coannihilations. Such lepton portal couplings can still produce the correct relic abundance even if the Higgs portal couplings are arbitrarily small. Such tiny Higgs portal couplings may be responsible for suppressed dark matter nucleon cross section as well as tiny invisible branching ratio of the standard model Higgs, to be probed at ongoing and future experiments. We also briefly discuss the collider implications of such a scenario.12.60.Fr,12.60.-i,14.60.Pq,14.60.St Lepton Portal Limit of Inert Higgs Doublet Dark Matter with Radiative Neutrino Mass Shibananda Sahoo December 30, 2023 =====================================================================================§ INTRODUCTIONThe observational evidence suggesting the presence of dark matter (DM) in the Universe are irrefutable, with the latest data from the Planck experiment <cit.> indicating that approximately 27% of the present Universe is composed of dark matter. The observed abundance of DM is usually represented in terms of density parameter Ω as Ω_DM h^2 = 0.1187 ± 0.0017where h = (Hubble Parameter)/100 is a parameter of order unity. In spite of astrophysical and cosmological evidences confirming the presence of DM, the fundamental nature of DM is not yet known. Since none of the particles in the Standard Model (SM) can fulfil the criteria of a DM candidate, several beyond Standard Model (BSM) proposals have been put forward in the last few decades. Among them, the weakly interacting massive particle (WIMP) paradigm is the most popular one. Such WIMP dark matter candidates can interact with the SM particles through weak interactions and hence can be produced at the Large Hadron Collider (LHC) or can scatter off nuclei at dark matter direct detection experiments like the ongoing LUX <cit.> and PandaX-II experiment <cit.>. Among different BSM proposals to incorporate dark matter, the inert Higgs doublet model (IHDM) <cit.> is one of the simplest extensions of the SM with an additional scalar field transforming as doublet under SU(2) and having hypercharge Y=1, odd under an imposed Z_2 discrete symmetry. As shown by the earlier works on IHDM, there are typically two mass ranges of DM mass satisfying the correct relic abundance criteria: one below the W boson mass and the other around 550 GeV or above. Among these, the low mass regime is particularly interesting due to stronger direct detection bounds. For example, the latest data from the LUX experiment rules out DM-nucleon spin independent cross section above around 2.2 × 10^-46 cm^2 for DM mass of around 50 GeV <cit.>. In this mass range, as we discuss in details below, the tree level DM-SM interaction through the SM Higgs (h) portal is interesting as it can simultaneously control the relic abundance as well as the DM-nucleon scattering cross section. In this mass range, only a narrow region near the resonance m_DM≈ m_h/2 is currently allowed by the LUX data. Though future DM direct detection experiments will be able to probe this region further, it could also be true that the DM-Higgs interaction is indeed too tiny to be observed at experiments. Such a tiny Higgs portal interaction will also be insufficient to produce the correct relic abundance of DM in this low mass regime. This almost rules out the low mass regime of DM in IHDMm_DM⪅ 70 GeV.Here we consider a simple extension of IHDM by singlet leptons (both neutral and charged) odd under the Z_2 symmetry such that the inert scalar dark matter can interact with the SM particles through these singlet leptons. This new interaction through lepton portal can revive the low mass regime of inert scalar DM even if future direct detection experiment rules out the Higgs portal interaction completely. The lepton portal interactions can also remain unconstrained from the limits on DM-nucleon interactions. Such a scenario is particularly interesting if LHC finds some signatures corresponding to the low mass regime of inert scalar DM while the direct detection continues to give null results. The dominant lepton portal interactions can explain correct relic abundance, null results at direct detection experiments and also give rise to interesting signatures at colliders. The neutral leptons added to IHDM can also give rise to tiny neutrino masses at one-loop level through scotogenic fashion <cit.>. We discuss the constraints on the model parameters from neutrino mass, DM constraints and also make some estimates of some interesting collider signatures while comparing them with the pure IHDM.This article is organised as follows. In section <ref>, we discuss the IHDM and then consider the lepton portal extension of it in section <ref>. In section <ref>, we discuss the dark matter related studies followed by our collider estimates in section <ref>. We finally conclude in section <ref>.§ INERT HIGGS DOUBLET MODEL The inert Higgs Doublet Model (IHDM) <cit.> is an extension of the Standard Model (SM) by an additional Higgs doublet Φ_2 and a discrete Z_2 symmetry under which all SM fields are even while Φ_2 → -Φ_2. This Z_2 symmetry not only prevents the coupling of SM fermions to Φ_2 at renormalisable level but also forbids those terms in the scalar potential which are linear or trilinear in Φ_2. Therefore, the second Higgs doublet Φ_2 can interact with the SM particles only through its couplings to the SM Higgs doublet and the electroweak gauge bosons. The Z symmetry also prevents the lightest component of Φ_2 from decaying, making it stable on cosmological scale. If one of the neutral components of Φ_2 happen to be the lightest Z_2 odd particle, then it can be a potential dark matter candidate. The scalar potential of the model involving the SM Higgs doublet Φ_1 and the inert doublet Φ_2 can be written asV(Φ_1,Φ_2)=μ_1^2|Φ_1|^2 +μ_2^2|Φ_2|^2+λ_1/2|Φ_1|^4+λ_2/2|Φ_2|^4+λ_3|Φ_1|^2|Φ_2|^2 +λ_4|Φ_1^†Φ_2|^2 + {λ_5/2(Φ_1^†Φ_2)^2 + h.c.},To ensure that none of the neutral components of the inert Higgs doublet acquire a non-zero vacuum expectation value (vev), μ_2^2 >0 is assumed. This also prevents the Z_2 symmetry from being spontaneously broken. The electroweak symmetry breaking (EWSB) occurs due to the non-zero vev acquired by the neutral component of Φ_1. After the EWSB these two scalar doublets can be written in the following form in the unitary gauge.Φ_1=[0; v +h /√(2) ] , Φ_2=[ H^+; H+iA/√(2) ]The masses of the physical scalars at tree level can be written asm_h^2= λ_1 v^2 ,m_H^+^2= μ_2^2 + 1/2λ_3 v^2 , m_H^2= μ_2^2 + 1/2(λ_3+λ_4+λ_5)v^2=m^2_H^±+ 1/2(λ_4+λ_5)v^2, m_A^2= μ_2^2 + 1/2(λ_3+λ_4-λ_5)v^2=m^2_H^±+ 1/2(λ_4-λ_5)v^2.Here m_h is the SM like Higgs boson mass, m_H, m_A are the masses of the CP even and CP odd scalars from the inert doublet. Without loss of generality, we consider λ_5 <0, λ_4+λ_5 <0 so that the CP even scalar is the lightest Z_2 odd particle and hence a stable dark matter candidate.The new scalar fields discussed above can be constrained from the LEP I precision measurement of the Z boson decay width. In order to forbid the decay channel Z → H A, one arrives at the constraint m_H + m_A > m_Z. In addition to this, the LEP II constraints roughly rule out the triangular region <cit.>m_H < 80GeV, m_A < 100 GeV, m_A - m_H > 8 GeVThe LEP collider experiment data restrict the charged scalar mass to m_H^+ > 70-90 GeV <cit.>. The Run 1 ATLAS dilepton limit is discussed in the context of IHDM in Ref.<cit.> taking into consideration of specific masses of charged Higgs. Another important restriction on m_H^+ comes from the electroweak precision data (EWPD). Since the contribution of the additional doublet Φ_2 to electroweak S parameter is always small <cit.>, we only consider the contribution to the electroweak T parameter here. The relevant contribution is given by <cit.>Δ T = 1/16 π^2 α v^2 [F(m_H^+, m_A)+F(m_H^+, m_H) -F(m_A, m_H)]where F(m_1, m_2) = m^2_1+m^2_2/2-m^2_1m^2_2/m^2_1-m^2_2lnm^2_1/m^2_2The EWPD constraint on Δ T is given as <cit.>-0.1 < Δ T + T_h < 0.2where T_h ≈ -3/8 πcos^2θ_Wlnm_h/m_Z is the SM Higgs contribution to the T parameter <cit.>. § LEPTON PORTAL EXTENSIONS OF IHDM As discussed in the introduction, considering lepton portal extensions of IHDM is very well motivated, specially from the origin of neutrino mass, dark matter direct detections and other flavour physics observables in the lepton sector. The inert Higgs doublet of the IHDM can couple to the SM leptons, if the model is suitably extended either by Z_2 odd neutral Majorana fermions or by charged vector like leptons, none of which introduce any chiral anomalies.The addition of three copies of neutral heavy singlet fermions N_i, odd under the Z_2 symmetry leads to the upgradation of the IHDM to the scotogenic model <cit.>. Apart from providing another dark matter candidate in terms of the lightest N_i, the model also can explain tiny neutrino masses at one loop level. The relevant interaction terms of these singlet fermions can be written asL⊃M_N N N + (Y_ij L̅_i Φ̃_2 N_j+ h.c.). The Feynman diagram for such one loop neutrino mass is shown in figure <ref>. Using the expression from <cit.> of one-loop neutrino mass (m_ν)_ij = Y_ikY_jk M_k/16 π^2 ( m^2_R/m^2_R-M^2_klnm^2_R/M^2_k-m^2_I/m^2_I-M^2_klnm^2_I/M^2_k)Here m^2_R,I=m^2_H,A are the masses of scalar and pseudo-scalar part of Φ^0_2 and M_k the mass of singlet fermion N in the internal line. The index i, j = 1,2,3 runs over the three fermion generations as well as three copies of N. For m^2_H+m^2_A≈ M^2_k, the above expression can be simply written as(m_ν)_ij≈λ_5 v^2/32 π^2Y_ikY_jk/M_k =m^2_A-m^2_H/32 π^2Y_ikY_jk/M_kIn this model for the neutrino mass to match with experimentally observed limits (∼ 0.1 eV), very tiny Yukawa couplings are required for the right handed neutrino mass of order of 1 TeV. Taking the mass difference m_A-m_H=m_H^±-m_H = 60 GeV, we show the constraints on neutral singlet fermion mass and corresponding Yukawa coupling from correct neutrino mass requirement in figure <ref>. It can be seen that for low mass regime of DM, the neutrino mass constraints force the Yukawa couplings to be smaller than 10^-4, too small to have any impact on dark matter relic abundance calculation, to be discussed below. These neutral fermions can also contribute to charged lepton flavour violation (LFV) at one loop involving N, Φ^±_2. The LFV processes like μ→ e γ remain suppressed in the SM due to the smallness of neutrino masses.Such LFV decays like μ→ e γ are being searched for at experiments like MEG <cit.>. The latest bound from the MEG collaboration is BR(μ→ e γ) < 4.2 × 10^-13 at 90% confidence level <cit.>. However, due to small Yukawa couplings, as required by tiny neutrino mass constraints discussed above, keeps this new contribution toμ→ e γ way below this latest experimental bound, as discussed in the recent works <cit.>.Similar to neutral singlet fermions, one can also incorporate charged singlet leptons χ_L,R with hypercharge Y=2 and odd under the Z_2 symmetry. The relevantLagrangian is L⊃M_χχ̅_L χ_R + Y_ij L̅_i Φ_2 χ_R+ h.c.These leptons can contribute both to dark matter relic abundance as well as LFV decays mentioned above. Since the corresponding Yukawa couplings are not restricted to be small from neutrino mass constraints, they can be sizeable and hence play a non-trivial role in generating DM relic abundance as we discuss below. Such large Yukawa couplings can however give a large contribution to LFV decays like μ→ e γ, with χ, Φ^0_2 in loop. As shown in a recent work <cit.>, the above MEG bound can constrain the product of two relevant Yukawa couplings to be below 10^-9 for χ mass around 100 GeV-1 TeV, too small to have any impact on DM relic abundance. These strict bounds from MEG can however be evaded by choosing diagonal structure of singlet lepton mass matrix M_χ and relevant Yukawa coupling Y. Such a structure can still have non-trivial impact on DM relic abundance, to be discussed below. § DARK MATTER The relic abundance of a dark matter particle ψ which was in thermal equilibrium at some earlier epoch can be calculated by solving the Boltzmann equationdn_ψ/dt+3Hn_ψ = -⟨σ v ⟩ (n^2_ψ -(n^eqb_ψ)^2)where n_ψ is the number density of the dark matter particle ψ and n^eqb_ψ is the number density when ψ was in thermal equilibrium. H is the Hubble expansion rate of the Universe and ⟨σ v ⟩ is the thermally averaged annihilation cross section of the dark matter particle ψ. In terms of partial wave expansion ⟨σ v ⟩ = a +b v^2. Clearly, in the case of thermal equilibrium n_ψ=n^eqb_ψ, the number density is decreasing only by the expansion rate H of the Universe. The approximate analytical solution of the above Boltzmann equation gives <cit.>Ω_ψ h^2 ≈1.04 × 10^9 x_F/M_Pl√(g_*) (a+3b/x_F)where x_F = m_ψ/T_F, T_F is the freeze-out temperature, g_* is the number of relativistic degrees of freedom at the time of freeze-out and M_Pl≈ 10^19 GeV is the Planck mass. Here, x_F can be calculated from the iterative relation x_F = ln0.038gM_Plm_ψ<σ v>/g_*^1/2x_F^1/2The expression for relic density also has a more simplified form given as <cit.>Ω_ψ h^2 ≈3 × 10^-27cm^3 s^-1/⟨σ v ⟩The thermal averaged annihilation cross section ⟨σ v ⟩ is given by <cit.>⟨σ v ⟩ = 1/8m^4_ψT K^2_2(m_ψ/T)∫^∞_4m^2_ψσ (s-4m^2_ψ)√sK_1(√s/T) dswhere K_i's are modified Bessel functions of order i, m_ψ is the mass of Dark Matter particle and T is the temperature.If we consider the neutral component of the scalar doublet Φ_2 to be the dark matter candidate, the details of relic abundance calculation is similar to the inert doublet model studied extensively in the literature <cit.>. In the low mass regime m_H=m_DM≤ M_W, dark matter annihilation into the SM fermions through s-channel Higgs mediation dominates over other channels. As pointed out by <cit.>, the dark matter annihilations H H → W W^* → W f f̅^̅'̅ can also play a role in the m_DM≤ M_W region. Also, depending on the mass differences m_H^+-m_H, m_A-m_H, the coannihilations of H, H^+ and H, A can also play a role in generating the relic abundance of dark matter. The relic abundance calculation incorporating these effects were studied by several groups in <cit.>. Beyond the W boson mass threshold, the annihilation channel of scalar doublet dark matter into W^+W^- pairs opens up suppressing the relic abundance below what is observed by Planck experiment, unless the dark matter mass is heavier than around 500 GeV, depending on the DM-Higgs coupling. Apart from the usual annihilation channels of inert doublet dark matter, in this model there is another interesting annihilation channel where dark matter annihilates into a pair of neutrinos (charged leptons) through the heavy fermion N_i (χ) in the t-channel. Apart from the relic abundance constraints from Planck experiment, there exists strict bounds on the dark matter nucleon cross section from direct detection experiments like Xenon100 <cit.> and more recently LUX <cit.>. For scalar dark matter considered in this work, the relevant spin independent scattering cross section mediated by SM Higgs is given as <cit.>σ_SI = λ^2_L f^2/4πμ^2 m^2_n/m^4_h m^2_DMwhere μ = m_n m_DM/(m_n+m_DM) is the DM-nucleon reduced mass and λ_L=(λ_3+λ_4+λ_5) is the quartic coupling involved in DM-Higgs interaction. A recent estimate of the Higgs-nucleon coupling f gives f = 0.32 <cit.> although the full range of allowed values is f=0.26-0.63 <cit.>. The latest LUX bound <cit.> on σ_SI constrains the η_L-Higgs coupling λ significantly, if η_L gives rise to most of the dark matter in the Universe. According to this latest bound, at a dark matter mass of 50 GeV, dark matter nucleon scattering cross sections above 1.1 × 10^-46 cm^2 are excluded at 90% confidence level. Similar but slightly weaker bound has been reported by the PandaX-II experiment recently <cit.>. We however include only the LUX bound in our analysis. One can also constrain the DM-Higgs coupling λ from the latest LHC constraint on the invisible decay width of the SM Higgs boson. This constraint is applicable only for dark matter mass m_DM < m_h/2. The invisible decay width is given byΓ (h →Invisible)= λ^2_L v^2 64 π m_h√(1-4 m^2_DM/m^2_h)The latest ATLAS constraint on invisible Higgs decay is <cit.>BR (h →Invisible) = Γ (h →Invisible)/Γ (h →Invisible) + Γ (h →SM) < 22 %As we will discuss below, this bound is weaker than the LUX 2016 bound.It should be noted that, there can be sizeable DM-nucleon scattering cross section at one loop level as well, which does not depend on the Higgs portal coupling discussed above. Even in the minimal IHDM such one loop scattering can occur with charged scalar and electroweak gauge bosons in loop <cit.>. The contributions of such one loop scattering can be kept even below future direct detection experiments like Xenon-1T by choosing large mass differences between the components of the inert scalar doublet <cit.>. Such large mass splittings also minimise the role of coannihilation between different inert scalar components on the DM relic abundance. This is in the spirit of the present work's motivation, as the DM abundance is primarily determined by the lepton portal couplings, rather than gauge and Higgs portal couplings. Another one loop scattering can occur, in principle, due to the exchange of photons or Z boson. This is possible through an effective coupling of the form C ∂^μΦ^0_2 ∂^νΦ^0†_2 F_μν with C being the loop factor <cit.>. However, since we have broken the degeneracy of our complex DM candidate Φ^0_2 and reduced it to one scalar and pseudoscalar, we can avoid such one loop scattering by choosing a mass splitting. In fact, one requires a non-zero mass splitting, at least greater than of the order of 𝒪(100keV), typical kinetic energy of DM particles, in order to avoid tree level inelastic scattering of DM off nuclei mediated by Z boson <cit.>. We implement the model in micrOMEGA 4.3.1 <cit.> to calculate the relic abundance of DM. We first reproduce the known results in IHDM by considering the neutral scalar H to be the DM candidate having mass below the W boson mass threshold. In the left panel of figure <ref>, we first show the parameter space of pure IHDM in λ_L-m_DM plane that satisfies the condition Ω_DM h^2 ≤ 0.1187. We have taken both the mass difference m_A-m_H=m_H^±-m_H=60GeV as a typical benchmark value satisfying all other constraints. Such a large benchmark point reduces the coannihilation effects and show the dependence of relic abundance on Higgs portal coupling λ_L in a visible manner.[We have not considered low mass differences in this work as that will make the coannihilations more efficient reducing the dependence of relic abundance on Higgs or lepton portal couplings and here our main motivation is to show the importance of lepton portal couplings.] The blue region in the left panel of <ref> therefore indicates the parameter space where the DM annihilation is either just enough or more than the required one to produce the correct relic abundance. Therefore, considering the additional lepton portal couplings for such values of λ_L will further suppress the relic abundance. Therefore, we choose benchmark values of λ_L-m_DM for our next analysis, from that region of this plot which overproduces the DM in pure IDM, so that an efficient lepton portal annihilation can bring down the relic abundance to the observed range. In the right panel of figure <ref>, we further impose the relic abundance criteria Ω_DM h^2 ∈ 0.1187 ± 0.0017 which reduces the number of allowed points significantly from the one in the left panel. In both the plots we also show the LUX 2016 exclusion line based on the upper bound on DM nucleon scattering cross section.We also show the LHC limit on Higgs invisible decay width which remains weaker than the LUX 2016 bound. The tiny allowed region near m_DM≈ m_h/2 corresponds to the s-channel resonance mediated by the SM Higgs while the allowed region of m_DM close to W boson mass threshold corresponds to the dominance of DM annihilation into three body SM final states mentioned above.After reproducing the known results of IHDM in the low mass regime for a benchmark value of mass splitting, we calculate the DM relic abundance by incorporating the Z_2 odd heavy leptons. In figure <ref>, we show the effect of vector like neutral heavy leptons on relic abundance. To make DM annihilations through lepton portal more efficient, we choose the Higgs portal coupling to be very small λ_L=0.0001 and also keep both the mass splitting within the components of the inert scalar doublet as 60 GeV like before. In the left panel of figure <ref>, the effect of heavy neutral fermion mass on the relic abundance is shown for a fixed value of Yukawa coupling Y=0.2. In the right panel of figure <ref>, the effect of lepton portal Yukawa couplings on DM relic abundance is shown for fixed value of heavy neutral fermion mass M_N=1000 GeV. From both these panels of figure <ref>, it is clear that the leptonic portal can play a non-trivial role in generating the DM relic abundance. While the benchmark values of Higgs portal coupling and mass splitting chosen above produce correct DM abundance only for two different masses, the introduction of lepton portal can result in new allowed region of DM masses. As expected, the maximum effect of lepton portal on DM relic abundance occurs for smaller values of heavy lepton mass or equivalently large values of Yukawa couplings. Since neutral heavy fermion couplings with SM leptons are required to be tiny from neutrino mass constraints as can be seen from figure <ref>, we consider only the effect of heavy charged leptons on DM relic abundance. The effect of charged lepton portal on DM relic abundance will be similar to that of neutral case discussed above. After showing the effect of lepton portal on DM relic abundance for specific values of Yukawa and heavy neutral fermion masses, we do a general scan of these two parameters from the requirement of generating correct abundance. Since neutral heavy fermion portal is not efficient after neutrino mass constraints are incorporated, we do the general scan only for charged heavy lepton portal here. In figure <ref>, we show the allowed parameter space satisfying relic density in the Y-m_DM plane for a benchmark point of IHDM parameters like before and taking the heavy charged fermion mass to be 100 GeV. The left panel of <ref> considers the lepton portal couplings to be of general non-diagonal type while the right panel considers the couplings to be diagonal. As discussed before, such diagonal couplings will evade the constraints from LFV decay. Since a diagonal structure of Yukawa couplings reduces the total number of annihilation channels, one requires larger values of Yukawa couplings to produce the correct relic abundance, compared to the ones in the non-diagonal case. In figure <ref>, we show the allowed parameter space in Y-M_χ plane for two specific dark matter masses m_DM=55, 65 GeV with general non-diagonal Yukawa couplings. The corresponding result for diagonal Yukawa couplings are shown in figure <ref>. It should be noted that these two benchmark values of DM masses in pure IHDM can not give rise to correct relic abundance for small values of Higgs portal couplings as seen from figure <ref>. However, after allowing the lepton portal couplings, we can generate correct relic abundance for such values of DM masses which remain disallowed in the pure IHDM.§ COLLIDER IMPLICATIONS In pure IHDM, the pseudoscalar A can decay into Z and H whereas H^± can decay to either W^±H or W^± A. When m_H^± is close to m_A, then the first decay mode of H^± almost dominates. Depending upon the decay mode of W^± and Z, we have either pure leptonic plus missing transverse energy (MET) or hadronic plus MET or mixed final states from pair production of the inert scalars. Earlier studies in the IHDM <cit.> focussed on pair production of inert scalars and their decays into leptons and MET. In another recent work <cit.>, the authors studied dijet plus MET final states in the context of IHDM at LHC. The dilepton plus dijet plus MET and trilepton plus MET final states have also been studied in a recent work <cit.>. The 8 TeV constraints and 13 TeV projection from monojet plus MET are discussed in another work <cit.>. In the presence of both Z_2 odd neutral and charged vector like leptons, additional channels open up. For example, now H^± can decay to χ^± ν_i or N_i l^±. Similarly, A can decay into l^± χ^∓ or N_i ν̅_i. Since neutrino mass constraints push the mass of neutral leptons typically to the order of TeV range, both H^± and A will mainly decay through charged vector like leptons (VLL)χ^±. Thenχ^± will further decay into l^± H. One can find earlier studies in the context of vector like leptons in references <cit.>. To highlight the difference in collider signatures with comparison to pure IHDM, we have considered a few benchmark points. We choose the following benchmark points all of which correspond to the fixed values of m_h = 125 GeV, λ_L = 0.0001 , λ_2 = 0.1, M_N=1000 GeV, Y=0.001. BP1: m_H = 55 GeV, m_H^+ = m_A = 115 GeV, M_χ=100 GeV, Y_ii=1.5 BP2: m_H = 65 GeV, m_H^+ = m_A = 125 GeV,M_χ=100 GeV, Y_ii=1.5 BP3: m_H = 65 GeV, m_H^+ = m_A = 200 GeV,M_χ=150 GeV, Y_ii=2.0BP4: m_H = 65 GeV, m_H^+ = m_A = 300 GeV,M_χ=150 GeV, Y_ii=2.0. In table <ref>, we have listed the parton level cross sections for final states that contribute to dilepton+MET final states at detector level in both IHDM and IHDM+VLL models for the above benchmark points. It should be noted that for BP1 and BP2, H^± will go through off-shell decay that is, H^±→ W^*^± H with W^*^± decaying leptonically in pure IHDM case due to limited phase space availability.But for BP3 and BP4, H^± will go through on-shell decay that is, H^±→ W^± H with W^± decaying leptonically in pure IHDM case. In IHDM+VLL model, H^± will decay to χ^± that is, H^±→χ^± ν_l with χ^± further decaying into l^± H. It is clearly evident from this table that we have enhancement of the cross section in IHDM+VLL due to opening of new decay modes of H^±. We must highlight one point that it is very difficult to probe heavier charged Higgs mass (like the ones in BP3 and BP4) in pure IHDM case due to small cross section. But in the IHDM+VLL model discussed here, we have sufficient cross section to probe these heavier masses of charged Higgs. Apart from the channels listed in table <ref>, there is another process which contributes to dilepton plus MET final states that is χ^+χ^- production with χ^± decays to l^±H. So as a whole, the dilepton plus MET final state will be an important collider signature to probe the modified IHDM that we discussed in this article. This inspires us to do a full signal versus background study at detector level which we will come up in a separate work <cit.>. § CONCLUSION We have studied a very specific region of parameter space in IHDM where the Higgs portal coupling of DM is very small, as suggested by null results in dark matter direct detection experiments so far. In the low mass regime of DM that is m_DM<M_W, such small value of Higgs portal coupling λ_L may not be sufficient to produce the correct relic abundance of DM except for a a few specific values of m_DM. We then extend this model by heavy neutral and charged leptons which are also odd under the Z_2 symmetry of the IHDM. These heavy leptons can be motivating from neutrino mass as well as LHC phenomenology point of view, apart from their role in producing the correct DM relic abundance in those region of parameter space which can not produce correct relic in pure IHDM. The neutral heavy fermions can generate tiny neutrino masses at one loop level via scotogenic mechanism, requiring the corresponding Yukawa couplings to be small (<10^-4) for TeV scale heavy neutral fermion masses. This keeps the contribution of neutral heavy leptons to DM abundance suppressed. The heavy charged fermion couplings to DM are however, not constrained to be tiny from neutrino mass point of view and hence can be sizeable enough to play a role in DM abundance. We show that the entire low mass regime of IHDM is allowed from relic abundance criteria if the lepton portal parameters are suitably chosen. This does not affect the DM direct detection scattering rates as there are no tree level or one loop couplings of DM with nuclei through leptons. The heavy leptons can also give rise to observable LFV decay rates like μ→ e γ as well as interesting collider signatures like dilepton plus missing energy. Although for simplicity, we choose particular type of Yukawa structure which does not contribute to LFV decay rates, it is in principle possible to choose some structure of the Yukawa couplings which can simultaneously produce correct DM abundance as well as keep the decay rate of LFV decays like μ→ e γ within experimental reach. We also show how the lepton portal extension of IHDM enhances dilepton plus missing energy signals at the LHC, for chosen benchmark points. There can also be lepton number violating signal like same sign dilepton plus dijet plus missing energy in this model, but remain suppressed for the benchmark values chosen in our analysis. We thank P. Poulose for useful discussions while carrying out this work. SS would like to thank Nirakar Sahoo for his constant help in resolving issues in micrOMEGA. Also SS thanks Biswajit Karmakar, Abhijit Saha for technical help in using mathematica and Ashis Kundu, Sourav Chattopadhyay for help in shell scripting.apsrev 99 natexlab#1#1bibnamefont#1#1bibfnamefont#1#1citenamefont#1#1url<#>1urlprefixURL Planck15 P. A. R. Ade et al., [Planck Collaboration], Astron. Astrophys. 594, A13 (2016).. LUX16 Talk on "Dark-matter results from 332 new live days of LUX data" by A. Manalaysay [LUX Collaboration], IDM, Sheffield, July 2016; D. S. Akerib et al. [LUX Collaboration], arXiv:1608.07648. PandaXII A. Tan et al., [PandaX-II Collaboration], Phys. Rev. Lett. 117, 121303 (2016).Barbieri:2006dqR. Barbieri, L. J. Hall and V. S. Rychkov,Phys. Rev. D 74, 015007 (2006) [hep-ph/0603188]. [Cirelli et al.(2006)Cirelli, Fornengo, and Strumia]Cirelli:2005uq authorM. Cirelli, authorN. Fornengo, and authorA. 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Vicente, JHEP 1405, 092 (2014) [arXiv:1312.5329 [hep-ph]] and references therein. Soumya S. Gopalakrishna, T. S. Mukherjee and S. Sadhukhan,Phys. Rev. D 93, no. 5, 055004 (2016) [arXiv:1504.01074 [hep-ph]]. Angelescu:2016mhl A. Angelescu and G. Arcadi,arXiv:1611.06186 [hep-ph]. 2lmet Debasish Borah, P. Poulose, Soumya Sadhukhan, Shibananada Sahoo, In Preparation. | http://arxiv.org/abs/1703.08674v1 | {
"authors": [
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"Shibananda Sahoo"
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"published": "20170325110046",
"title": "Lepton Portal Limit of Inert Higgs Doublet Dark Matter with Radiative Neutrino Mass"
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A molecular-dynamics approach for studying the non-equilibrium behavior of x-ray-heated solid-density matter Ian D. Morris December 30, 2023 ============================================================================================================ Virtual reality (VR) video provides an immersive 360 viewing experience to a user wearing a head-mounted display: as the user rotates his head, correspondingly different fields-of-view (FoV) of the 360 video are rendered for observation. Transmitting the entire 360 video in high quality over bandwidth-constrained networks from server to client for real-time playback is challenging.In this paper we propose a multi-stream switching framework for VR video streaming: the server pre-encodes a set of VR video streams covering different view ranges that account for server-client round trip time (RTT) delay, and during streaming the server transmits and switches streams according to a user's detected head rotation angle. For a given RTT, we formulate an optimization to seek multiple VR streams of different view ranges and the head-angle-to-stream mapping function simultaneously, in order to minimize the expected distortion subject to bandwidth and storage constraints. We propose an alternating algorithm that, at each iteration, computes the optimal streams while keeping the mapping function fixed and vice versa. Experiments show that for the same bandwidth, our multi-stream switching scheme outperforms a non-switching single-stream approach by up to 2.9dB in PSNR.Video streaming, virtual reality, video coding § INTRODUCTION The advent of technologies for camera rigs, fisheye lenses and image-stitching algorithms <cit.> means that 360 virtual reality (VR) video can now be readily generated. A user equipped with a head-mounted display (HMD) such as Oculus Rift[https://www3.oculus.com/en-us/rift/] or HTC Vive[https://www.vive.com/jp/] can enjoy an immersive 360 viewing experience: as the user rotates his head to the left or right, correspondingly different fields-of-view (FoV) of the 360 VR video are rendered for observation. See Fig. <ref> for an illustration. It has been shown <cit.> that such motion parallax visual effect—changing FoVs according to user's head position and rotation angle—is the strongest cue for human's depth perception in a 3D scene, and VR video enables this effect for any head rotation angle from 0 to 360.However, transmitting the entire 360 VR video in high quality over bandwidth-limited networks from a server to a client for real-time playback is challenging. Leveraging on previous works in interactive multiview video streaming (IMVS) <cit.>, we propose a multi-stream switching framework for 360 VR video streaming. The server pre-encodes a set of VR video streams, each covering a different view range of the original 360 video. During streaming, the server transmits and switches among the pre-encoded streams according to a user's detected head rotation angle.By transmitting one video stream covering a limited view range at a time, the server can encode the stream at a higher quality than a single stream covering all 360 viewing angles for the same bandwidth constraint. However, to minimize the adverse effect of interaction delay in motion parallax—even in the face of non-negligible server-to-client round trip time (RTT) delay—each pre-encoded stream must cover a wide enough view range, so that a user's head with rotation angle starting in the view range center would not drift outside the view range in one RTT. This implies that the coded streams tend to overlap in view ranges, resulting in representation redundancies and high storage cost. Thus, multi-stream switching can enable higher visual quality, at the expense of an increase in storage cost due to streams' view range overlaps.Thus, the technical challenge is, for a given RTT, to design multiple VR streams of different view ranges and the head-angle-to-stream mapping function in order to minimize the expected distortion subject to bandwidth and storage constraints. We mathematically formalize this optimization and propose an alternating algorithm that, at each iteration, computes the optimal VR streams while keeping the mapping function fixed and vice versa.Experimental results show that for the same bandwidth constraint, our proposed multi-stream switching scheme outperforms a single-stream approach by up to 2.9dB in PSNR.§ RELATED WORK Using an array of cameras to capture a 3D scene synchronously from slightly shifted viewpoints, IMVS systems <cit.> study how the captured multi-view videos can be pre-encoded into multiple streams. A receiving user can periodically request switches to neighboring camera views, and the server in response switches video streams with minimum discruption to the user's viewing experience. To facilitate stream-switching, new frames like DSC frame <cit.> and merge frame (M-frame) <cit.> were proposed. Unlike IMVS <cit.>, we optimize the division of 360 VR video into multiple streams covering different view ranges given a constant RTT. To the best of our knowledge, we are the first to study this problem for interactive VR video streaming formally. There are recent studies on VR video streaming. Assuming that the 3D scene can be represented by a 3D mesh, <cit.> proposed to first divide the mesh into 3D sub-meshes (tiles). During streaming, a user communicates the desired tiles to the server using MPEG-DASH-SRD <cit.>, an extension of MPEG-DASH <cit.> to specify spatial relationships in media content. Unlike <cit.>, we assume the input to our optimization is a 360 VR video, not 3D mesh.Further, we take the effect of RTT on interaction delay into account explicitly during optimization (to be detailed in Section <ref>).Assuming a camera rig with multiple cameras capturing a 360 view from different angles, <cit.> described a multiview video scheme that divides and codes captured camera views into two types: i) primary views at lower resolution that cover the entire 360 field-of-view, and ii) auxiliary views for the remaining camera views at high resolution. The two video types are coded using multilayer extensions of HEVC.The receiver then performs image stitching to compose a 360 VR view.Instead, we assume 360 VR video is composed at the sender, and the challenge is to design multiple video streams covering different view ranges for interactive streaming.§ SYSTEM OVERVIEWWe overview the operations of our multi-stream switching framework for a given RTT. Denote by T the RTT between server and client. Denote by Δ the time interval between coded frames; 1/Δ is the number of frames per second (fps). For simplicity, assume for now that all frames are intra-coded, so that streams can be switched at any frame.The server starts transmission of an initial video stream to the user at time t=-T/2, assuming the user begins at an initial head rotation angle θ(0). At time t=0, the stream arrives at the client and playback begins. At time t=Δ, the client transmits the first feedback θ(Δ) of the user's head rotation angle to the server. This feedback θ(Δ) arrives at the server at t=T/2+Δ, and the server decides the new stream to transmit corresponding to θ(Δ) using a mapping function f(θ(Δ)). This new stream arrives at the client at time t=T+Δ, exactly T seconds after feedback θ(Δ) was generated. Hence, the transmitted stream must accommodate the change in head rotation angle from θ(Δ) to θ(T+Δ). See Fig. <ref> for an illustration.Consider now the case when the VR streams are coded in Group-of-Pictures (GOP) of H frames each. This means that the frequency at which the server can switch streams is also every H frames. Compared to the previous case of intra-coded frames, each VR stream must now accommodate the change in head rotation angle in time interval T + H Δ.We next formulate the optimization problem to find the multiple VR streams and the mapping function f( ). § PROBLEM FORMULATION§.§ View Interaction ModelWe first define a view interaction model that models a typical view selection process during 360 VR video observation. Denote by θ[n] the central view angle at which an observer is watching straight ahead at discrete time n.For convenience, we define the duration of a discrete time interval to be Δ (time interval between frames), and RTT in discrete instants to be T_s = T/Δ. We assume that θ[n] ∈{1, …, K} is also discrete, where θ[n] 2 π / K is angle in radians between 0 and 2 π. We assume a one-hop Markov view transition model, where the probability of an observer's angle θ[n+1] = j given θ[n] = i is p_i,j. Finally, we assume that the observer changes views only locally per instant, i.e., p_i,j = 0 |i-j| > v_max.At any instant n, the observer has a FoV of size 1+2a ≪ K that defines the angular span a human observes at a time. Hence at time instant n, given central view angle θ[n], the observer's FoV is 𝐑[n] = [θ[n] - a, θ[n] + a]. It means that an observer will see visual distortion if the current video stream is not coded at high enough video quality in this range 𝐑[n].§.§ Expected DistortionWe define the expected distortion an observer sees in a 360 VR video as he naturally rotates his head. We consider first the simple case when the GOP size is a single frame. First, we compute the steady state probabilities 𝐪∈ℝ^K assuming stationary view transition probabilities p_i,j via the Perron-Frobenius Theorem[https://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem]:𝐪 𝐏= 𝐪where 𝐪 is the left eigenvector (row vector) corresponding to the eigenvalue 1 for matrix 𝐏.Denote by 1_k the canonical row vector of length K with the only non-zero entry at position k equals to 1. T_s instants after an observer starts in central angle k, the angle distribution is 1_k 𝐏^T_s. Because an observer's FoV size is 1+2a, we multiply 1_k by a binary circulant matrix 𝐂_a ∈{0, 1}^K × K to account for FoV. For example, 𝐂_1 for K=5 is:𝐂_1 = [ [ 1 1 0 0 1; 1 1 1 0 0; 0 1 1 1 0; 0 0 1 1 1; 1 0 0 1 1 ]] Suppose now that for central angle k, the server transmits stream f(k) with distortion vector 𝐝_f(k), where d_f(k),l is the distortion of angle l in stream f(k). We can then write the expected distortion for this intra-coded streaming system as:D({𝐝_i}, f( )) = ∑_k=1^K q_k1_k𝐂_a 𝐏^T_s𝐝_f(k)where the expected distortion D depends on both the distortion vectors 𝐝_i of different streams i and the mapping function f( ) from angles to streams.If the 360 VR video streams are coded in GOP of H frames each, then the stream-switching delay becomse T_s + H, and the distortion term for each k needs to be computed for all H frames:D({𝐝_i}, f( )) = ∑_k=1^K q_k∑_h=0^H-11_k𝐂_a 𝐏^T_s+h𝐝_f(k) §.§ Rate Constraints Given distortion vector 𝐝_i of stream i, we define the coding rate as r(𝐝_i) = ∑_k=1^K g(d_i,k), where g(d_i,k) is in turn defined as a clipped Laplacian function with parameter σ:g(d) = U(d_max - d) exp( - |d|/σ^2)where U( ) is a step function; i.e., if d ≥ d_max, then rate g(d) is 0. Parameter σ can be chosen according to the 360 VR video characteristics. Because distortion d is non-negative, we can drop the absolute value operator in practice. Having defined r(𝐝_i), we can define a storage constraint as follows. Denote by 𝒮 the set of pre-encoded video streams, by Q the duration in time for the 360 video, and by B the storage budget in bits. We write the storage constraint as:∑_i ∈𝒮 r(𝐝_i) ≤ B/Q We can similarly define a transmission constraint for a transmission budget C in bps.Assuming a mapping function f( ) from angles to streams, we write:∑_k=1^K q_k r(𝐝_f(k)) ≤ C §.§ Objective FunctionAssuming H=1, collecting derived equations (<ref>), (<ref>) and (<ref>), we write an unconstrained Lagrangian objective as:min_{𝐝_i}, f( )∑_k=1^Kq_k1_k𝐂_a 𝐏^T_s𝐝_f(k) +λ∑_i ∈𝒮 r(𝐝_i) + μ∑_k=1^K q_k r(𝐝_f(k))where λ and μ are chosen parameters so that the storage constraint (<ref>) and transmission constraint (<ref>) are satisfied.§ OPTIMIZATION ALGORITHM We take an alternating optimization approach, where we optimize variables {𝐝_i} and f( ) one at a time while keeping the other fixed. When f( ) is fixed, we take the derivative of the objective with respect to d_i,l and set it to 0:∑_k | f(k) = i q_k [ 1_k𝐂_a 𝐏^T_s]_l + ( λ + μ∑_k | f(k)=i q_k )_γ∂ g(d_i,l)/∂d_i,l = 0- 1/γ∑_k | f(k) = i q_k [ 1_k𝐂_a 𝐏^T_s]_l =∂exp( - d_i,l/σ^2)/∂d_i,l - σ^2 log( σ^2/γ∑_k | f(k) = i q_k [ 1_k𝐂_a 𝐏^T_s]_l ) = d_i,l^*where [ ]_l denotes the l-th entry of a vector.For intuition, we can check the boundary cases of (<ref>) as follows. If angle l of stream i is not observed (summation in the argument of log is 0), then the left side of (<ref>) evaluates to ∞, so we can set d^*_i,k to d_max. On the other hand, if angle l is observed with high probability (summation in the argument of log is upper-bounded by 1), assuming σ^2/γ is also upper-bounded by 1, then d_i,l^* is lower-bounded by 0. When streams {d_i} are fixed, we optimize f( ) simply as follows. For each angle k, we identify a stream i for k with the minimum expected transmission cost in (<ref>).§.§ Initialization For a given number |𝒮| of target streams, we perform initialization as follows.We evenly distribute the central angles of |𝒮| streams in {1, …, K}. For each stream i with central angle k, we set distortion d_i,l to a constant d_1 for angle l where |k-l| < T_s v_max; i.e., angle l is reachable in T_s transitions. Otherwise, d_i,l = d_max. d_1 is then adjusted so that the transmission constraint is met for this stream.The number of streams |𝒮| is varied to find a locally optimal solution. § EXPERIMENTS§.§ Experimental Setup We use two 360 VR sequences captured by Kandao Technology[VR sequences will be made available at time of publication.],and , for our experiments. Each video is 1 hour long at 30 fps. FoV is assumed to be 90^∘, and v_max is 5^∘. Video for one FOV has resolution 512 × 512. Number of discrete view angles K is 60, and RTT T_s is 3.We use a linear function to model angle transition probabilities: p_i,j linearly decreases with |i-j|, and the slope of decrease is steeper at π/2 and 3π/2, resulting in higher steady state probabilities q_k at these two angles.As competitor we choose a non-switching scheme called static, which always sends an encoded video covering the entire 360 angles. For practical implementation, both our proposed scheme (called adaptive) and static use two QPs to encode each VR video stream; the two QPs are selected using Lloyd-Max quantizer <cit.> to approximate the theoretical Laplacian RD curve r(d) shown in Fig. <ref> (a). Test videos are first encoded at different QPs to generate empirical RD points, then the parameters of r(d) are fitted. §.§ Experimental Results We assume two different channel bandwidths are available. We vary the available storage and show the tradeoff against visual quality (PSNR) in Fig. <ref> forand . Weight parameters λ and μ are tuned to satisfy bandwidth and storage constraints at each point. Each data point in Fig. <ref> is marked by a square, circle or triangle to denote the optimal number of streams generated: 1, 2 and 3, respectively. static uses 1 stream (squares), and adaptive uses multiple streams (circles and triangles).We observe that adaptive outperforms static for the two sequences—up to 2.9dB in PSNR at the same bandwidth but using more storage. For given channel bandwidth and storage, adaptive selects the optimal number of streams and view range for each stream via optimization of distortion vectors 𝐝_i. Fig. <ref> (b) (distortion versus viewing angle) shows the optimized distortion vectors 𝐝_i for two streams when the storage is 5Gb and bandwidth is 1Mbps.d_max=46 in this case, and the corresponding angle range is not encoded because there is zero probability of being observed (given our view interaction model). In contrast, viewing angles with high probabilities have low distortion values in 𝐝_i. We observe also that the two steams overlap, as discussed in the Introduction, to guarantee good visual quality when user's head rotates in one RTT. Due to the low observe probability at the stream view range boundaries, the associated distortion values are relatively larger.When storage is small, static and adaptive have the same performance for both ch1 and ch2. As more storage becomes available, relative performance of adaptive becomes better for both ch1 and ch2 when multiple streams are employed. On the other hand, by sending only one stream always, static cannot make use of extra storage to improve quality for a given channel bandwidth.§ CONCLUSION Transmitting 360 VR video in high quality over bandwidth-limited networks is difficult. In this paper, we pre-compute mulitple streams covering different overlapping view ranges at the server, and during streaming a single stream is selected corresponding to the user's tracked head rotation angle that minimizes the adverse effect of interaction delay. We formulate an optimization to find the optimal streams and the head-angle-to-stream mapping function simultaneously, solved via an alternating algorithm. Experimental results show that our multi-stream switching approach outperforms a single-stream approach by up to 2.9dB in PSNR.IEEEbib | http://arxiv.org/abs/1703.09090v1 | {
"authors": [
"Gene Cheung",
"Zhi Liu",
"Zhiyou Ma",
"Jack Z. G. Tan"
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"published": "20170327140912",
"title": "Multi-Stream Switching for Interactive Virtual Reality Video Streaming"
} |
^1School of Nano Science, Institute for Research in Fundamental Sciences (IPM), 19395-5531 Tehran, Iran ^2Department of Physics, Payame Noor University (PNU), P. O. Box 19395-3697 Tehran, IranConventional magnetism occurs in systems which contain transition metals or rare earth ions with partially filled d or f shells. It is theoretically predicted that compounds of groups IA and IIA with IV and V, in some structural phases, are ferromagnetic half-metals which made them new candidates for spintronics applications. Employing density functional theory (DFT) we investigate magnetism in binary compounds CaN and CaAs. Regarding the structure of analogous magnetic materials and experimental results of CaAs synthesis, we have considered two cubic structures: rocksalt (RS) and zincblende (ZB), and four hexagonal structures: NiAs, wurtzite (WZ), anti-NiAs, and NaO. The calculated results show that CaN in cubic, NiAs, and wurtzite structures, and CaAs only in zincblende phase have ferromagnetic ground states with a magnetic moment of 1μ _B. Electronic structure analysis of these materials indicates that magnetism originates from anion p states. Existence of flat p bands and consequently high density of states at the Fermi level of magnetic structures gives rise to Stoner spin splitting and spontaneous ferromagnetism. d^0 half-metallic ferromagnetism in CaN and CaAs pnictides: An ab initio study Omid Khakpour^2 December 30, 2023 ==============================================================================§ INTRODUCTION Half-metallic ferromagnetism, which involves metallic conductivity only in one spin channel, renowned to be a key phenomenon in spintronics.Its first-principles introduction by de Groot et al. <cit.> raised enthusiasm among the community to explore this property within materials.For then, several reports have been published regarding the prediction and/or observation of a half-metallic electronic structure in different materials including metallic oxides <cit.>, full and half Heusler alloys <cit.>,and diluted magnetic semiconductors <cit.>.In spite of several half-metallic compound appeared in the literature, the search for novel ferromagnetic half-metals is an active research area. In this context, a significant attention is attracted to the transition metalpnictides and chalcogenides with zincblende (ZB) structure<cit.>. Although, many of transition metal compounds exhibit half-metallic property in the ZB structure, these materials crystallize in a different crystal structure including hexagonal NiAs or orthorhombic structures; hence realization of their half-metallic behavior requires special requirement for synthesis of metastable ZB structure. It is claimed that the ZB structure MnAs <cit.>, CrAs <cit.>,and CrSb <cit.> have been successfully synthesized in the form of thin films. In 2004 Kusakabe et al. <cit.> showed that calcium pnictidesin the ZB structure, in absence of any transition metal element,exhibit half-metal ferromagnetism. These compounds expose an unconventional type of ferromagnetismwhere the spin polarization is in the anion p electrons without any direct involvement of d orbital, while in the conventional transition metals magnets, d electrons are responsible for magnetic ordering.This p orbital magnetization is intrinsic and is not triggered bycrystal defects of any type. The spirit of the exchange is neither double exchange nor p-d exchange.It is explained that these compounds belong to classes of d^0 ferromagnetswhich have been recently observed experimentally and/or theoretically<cit.>. Ferromagnetism in the compounds with no valance d or f electronhas challenged our understanding of magnetism. Later on, Sieberer et al. <cit.> andVolnianska et al. <cit.> studied ferromagnetism inthe binary compounds of I^A/II^A-V elements in tetrahedrally andoctahedrally coordinated structures.They argued that half-metallic ferromagnetism originates from the flat bandmagnetism of holes and may occurs in more structures including RS and NiAs. It was explained that large cell volume, high ionicity and a slighthybridization of anion p orbitals and induced d states around the Fermi level effectively enhance formation of ferromagnetic (FM) state in the system. Our specific aims in this work is to apply accurate first-principles electronic calculations to investigate magnetic propertiesand structural stabilityof two members of binary p magnetic compounds (CaN and CaAs) in six different structures. In section III we will discuss structural and magnetic properties of CaN and CaAs. Electronic structuresof these two compounds are described in section IV.§ COMPUTATIONAL METHOD We performed first-principles calculations based on Density functional theory (DFT) usingpseudopotential technique and spin-dependent generalized gradient approximation (GGA)in the scheme of Perdew, Burke, and Ernzerhof <cit.> by means of Quantum ESPRESSO package <cit.>.The relativistic effects are taken into account in the scalar limit, neglecting the spin-orbit coupling which is expected to be small in light atoms like Ca, N, and As. Using conventional GGA calculations of electronic and magnetic propertiesof materials which do not consist of atoms with inner d or felectrons are thought to be accurate and give reliable results.We used ultrasoft pseudopotentials<cit.>, a kinetic energy cut off of 35 Rydfor the plane wave expansion of Kohn-Sham orbitals, and a kinetic energy cut off of400 Ry for the Fourier expansion of electron density.The Brillouin-zone integrations were performed using a Monkhorst-Pack mesh of12×12×12 k points for cubic structures and equivalent meshes for hexagonal structures. Atomic relaxation of the structures with internal parameters was accurately performed to achieve residual forces less than 0.01 mRy/Bohr and energyaccuracy of better than 0.001 mRy/fu (formula unit). § STRUCTURAL AND MAGNETIC PROPERTIES In this work, we take into account six different structures for binary CaN and CaAs compounds; ZB, wurtzite (WZ), rock-salt (RS), hexagonal NiAs,hexagonal anti NiAs (AsNi), and NaO structures. The tetracoordinated cubic ZB structure is important because several binary compound exhibit half-metallic ferromagnetism in this structure. The fact that many binary semiconductors crystallize in the ZB structure increases the importance of study of this structure for binary half-metals, because these half-metals are expected to be ideal ferromagnetic sources for spin injection into semiconductors.Also the tetracoordinated WZ structure is the hexagonal analogous of the cubic ZB structureand hence is included in our study.The significant electronegativity differences in Ca-N (2.0 Pauling) and Ca-As (1.0 Pauling) pairs <cit.> shows the significant contribution of ionic bonding in CaN and CaAs and hence these compounds may favor a higher coordination atomic arrangementto reduce their total energy.Therefore, the hexacoordinated cubic RS structure is also considered in this work. Recently Liu et al. have claimed that they have grown a layerof RS-CaN on Cu(0 0 1) by means of a new unclear self-assembly mechanism <cit.>.Since some of magnetic transition metal pnictides (II^TM-V compounds)crystallize in the hexagonal NiAs structure, both NiAs and AsNi (anti-NiAs) structures were considered for CaN and CaAs. The natural crystal structure of CaAs seems to be a NaO type lattice <cit.>, which is derived from the AsNi structure, by a 30^∘ rotation clockwise around the z axis, followed by a shift of [0,0,0.25] in the z direction, and finally an increase of the a and b cell parameters by a factor of √(3). The difference between the minimized total energies of the FM and non-magnetic (NM) states, Δ E^FM-NM, of CaN and CaAs in all considered structures are presented in table <ref>. To find the minimized total energies, we have optimized lattice constantsof all structures and atomic positions of the low symmetry NaO structure. The negative values indicate more stability of the FM phase. It is seen that CaN has a FM ground state and non-vanishing magnetic momentin the RS, ZB, NiAs, and WZ structures while the FM behavior of CaAsis limited to the only ZB structure. Both compounds are NM in the NaO phase. Regardless of the crystal structure, in all magnetic ground states,the total magnetic moment per chemical formula unit is 1 μ_B.In contrast to the transition metal pnictides, the small cation spin moment inthe magnetic structures of CaN and CaAs is parallel to anion moment which may be explained by the model of covalent polarization <cit.>. Furthermore, table <ref> lists the equilibrium lattice parameter andbulk modulus for both compounds. It is generally seen that magnetic structures, compared with the nonmagnetic systems, have higher volume and lower bulk modulus. As it was mentioned in the introduction, a large cell volume enhances spin polarization of binary p magnetic materials.The calculated total energies in different volumes for CaN and CaAs are plottedin figure <ref>. According to the figure, the structures with hexacoordinatedanions are energetically more favorable.These structures have smaller cell volumes and higher bulk moduli, indicating that their bonds are stronger due to the smaller interaction distances.The ZB structure has the highest total energy and the experimentally observed structure for CaAs, NaO, with a difference ofabout 0.1 Ry/fu is the most favorable one for both compounds.The calculated equilibrium lattice constants of NaO structure for CaAs are 14.93 and 11.15 Bohr for a and c, respectively, which is in good agreement withthe experimentally found values of 14.85 and 11.19 Bohr.The comparison of the ZB and WZ structures shows that the hexagonal arrangement gives riseto lower energy for both compounds.§ ELECTRONIC STRUCTURE In this section, the detailed investigation of the electronic structureof all considered crystal structures is presented, and the essence of ferromagnetism in these materials is discussed by analyzing their electronic structure.According to a previous study <cit.>, in agreement with ours, for Ca pnictides (CaN, CaP, CaAs, CaSb) in the ZB phase,the energy of spin polarization is the largest for CaN and decreaseswith the increase in atomic number of the anion, and almost vanishes for CaSb.Therefore, anions should have an important contribution in governingmagnetization in these compounds.For better understanding, we calculated free atom energies in the spin unpolarized andpolarized states and found that the spin polarization energy of a free N atom (-3.32 eV)is significantly higher than a free As atom (-1.58 eV). This observation may explain stability of FM state in more structures for CaN, compared with CaAs.In Fig <ref>, the spin resolved total density of states (DOS)of the systems are presented.It is seen that the crystal field splitting between bonding and antibonding states is more pronounced in the tetra coordinated ZB and WZ structures, compared with other systems. The magnetic structures including NiAs, WZ, ZB, and RS structure of CaN andZB structure of CaAs obviously exhibit half-metallic property, as in their majority spin channel there is an energy gap aroundthe Fermi level while their minority spin bands cut the Fermi level. This is different from the half-metallic transition metal pnictides and chalcogenides,where their majority spin channel is metallic and the half-metallic gapappears in their minority spin channel. The spin flip gap, the distance between the majority valence band maximum and the Fermi level, measures the minimum required energy for creatinga majority spin carrier at the top of the majority valence band. Hence, systems with larger spin flip gap are expected to be better source of fully spin polarized current. Among the studied systems, the largest spin flip gap (0.8 eV) is observed in the ZB-CaN.It has been argued <cit.> that the half-metallicity in the ZB structureis mainly caused by the local bonding environment.These alkaline pnictides involve seven valence electrons per formula unit. Since the anion s states are the lowest energy valence states, they must be occupied by two electrons,and the rest five electron occupy the anion p states which can accommodate six electrons.Because of spin splitting, the majority p states lie lower in energy thanthe minority p states, so the majority spin p bands are fulfilledby three electrons and the minority spin p bands are partially filled. As a result of the hole appears in the minority spin valence bands, a total spin momentof 1 μ_B is left on the anion atom.But in the case of transition metal pnictides, there are enough valence electronsto make the anion p states fulfilled, and the antibonding t_2-pbands and the e bands for majority spin channel are also pulled downto accommodate the additional valence electrons,the Fermi level cut the majority spin bands and a half-metallic gap remains inthe minority spin channel. The magnetic moments in these systems are mainly from the transition metals d electrons.The loss of the half-metallicity can be understood by checkingthe change of the Fermi level position and p-d hybridization with the lattice constants. While thelattice constants decrease, the Fermi level will be push up to higher energy and the spin flip gapincreases, but the p-d hybridization will be strengthened and the valence bands characterized by pstates are expanded which would reduce the spin flip gap. When the lattice constants are compressed tosmaller than the critical value, the expanded majority p-character bands would cut the Fermi level and the half-metallicity is destroyed. The inter-metallic CaN and CaAs compounds are made of a strong cationic metal and weak anionicelements, which are known as the Zintle phase. The Zintle phases stand somewhere in-between metallic and ionic compounds. In Zintle phases anions do not reach the stable octet state as isolated species, and hence they need additional bonds to become more stable. Therefore, in these compounds the anion atoms usually connect covalently together and form dimers. Formation of anion dimers in the Zintle phases, electronically balances the system and leads to closed-shell compounds, i.e. the number of electrons provided by theconstituting elements equals the number of electrons needed for covalent bonding in the structure.The spin-dependent band structures along high symmetry lines inthe Brillouin zone for the best half-metallic structure (ZB) and the most stable structure (NaO) are given in Fig <ref>.We know that the anion p states are localized around the nuclei and hence the valance bands are flat and low dispersed.the anion atoms and according to DOS profiles the hybridization betweenthe anion p states and the Ca d states is weak.reflecting the weak polarized-covalent bonds of the anions p orbitals and Ca d states.§ CONCLUSION Conventional magnetism occurs in systems which contain transition metals or rare earth ions with partially filled d or f shells. Recent experimental and/or theoretical observations of magnetism in systems with no valance d or f electron challenge our classical understanding of magnetism. It is theoretically predicted that compounds of groups IA and IIA with IV and V, in some structural phases are ferromagnetic half-metals which made them new candidates for spintronics applications. Employing density functional theory (DFT) we investigate magnetism in binary compounds CaN and CaAs. Regarding the structure of analogous magnetic materials and experimental results of CaAs synthesis, we have considered two cubic structures: rocksalt (RS) and zinc-blende (ZB), and four hexagonal structures: NiAs, wurtzite (WZ), anti-NiAs, and NaO. The calculated results show that CaN in cubic, NiAs, and wurtzite structures, and CaAs only in zincblende phase have ferromagnetic ground states with a magnetic moment of 1 μ_B . Electronic structure analysis of these materials indicates that magnetism originates from anion p states. Existence of flat p bands and consequently high density of states at the Fermi level of magnetic structures gives rise to Stoner spin splitting and spontaneous ferromagnetism. Larger exchange interaction in 2p orbitals of nitrogen respect to 4p orbitals of arsenic causes that CaN has more structures with ferromagnetic ground state than CaAs. Slight hybridization with calcium d states increase density of p states at the Fermi level and enhance ferromagnetism in the system. | http://arxiv.org/abs/1703.08691v1 | {
"authors": [
"Seyed Mojtaba Rezaei Sani",
"Omid Khakpour"
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"published": "20170325132715",
"title": "$d^0$ half-metallic ferromagnetism in CaN and CaAs pnictides: An ab initio study"
} |
calcplain theoremTheorem[section] prop[theorem]Proposition lemma[theorem]Lemma corollary[theorem]Corollary conj[theorem]Conjecture question[theorem]Questiondefinition definition[theorem]Definition *exampleExample | http://arxiv.org/abs/1703.08742v4 | {
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"05A05 (Primary) 05A15, 05A19, 30B70, 05A18 (Secondary)"
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"published": "20170325210113",
"title": "Continued fractions for permutation statistics"
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[email protected] University of Bucharest, Faculty of Physics, Materials and Devices for Electronics and Optoelectronics Research Center, 077125 Magurele-Ilfov, Romania Horia Hulubei National Institute for Physics and Nuclear Engineering, 077126 Magurele-Ilfov, RomaniaNational Institute of Materials Physics, Magurele 077125, Ilfov, RomaniaNational Institute of Materials Physics, Magurele 077125, Ilfov, RomaniaNational Institute of Materials Physics, Magurele 077125, Ilfov, RomaniaNational Institute of Materials Physics, Magurele 077125, Ilfov, RomaniaSchool of Science and Engineering, Reykjavik University, Menntavegur 1, IS-101 Reykjavik, IcelandSchool of Science and Engineering, Reykjavik University, Menntavegur 1, IS-101 Reykjavik, Iceland Icelandic Heart Association, Holtasmari 1, IS-201 Kopavogur, IcelandSchool of Science and Engineering, Reykjavik University, Menntavegur 1, IS-101 Reykjavik, [email protected] National Institute of Materials Physics, Magurele 077125, Ilfov, RomaniaHysteretic effects are investigated in perovskite solar cells in the standard FTO/TiO_2/CH_3NH_3PbI_3-xCl_x/spiro-OMeTAD/Au configuration.We report normal (NH) and inverted hysteresis (IH) in the J-V characteristics occurring for the same device structure, the behavior strictly depending on the pre-poling bias. NH typically appears at pre-poling biases larger than the open circuit bias, while pronounced IH occurs for negative bias pre-poling. The transition from NH to IH is marked by a intermediate mixed hysteresis behavior characterized by a crossing point in the J-V characteristics. The measured J-V characteristics are explained quantitatively by the dynamic electrical model (DEM). Furthermore, the influence of the bias scan rate on the NH/IH hysteresis is discussed based on the time evolution of the non-linear polarization.Introducing a three step measurement protocol, which includes stabilization, pre-poling and measurement, we put forward the difficulties and possible solutions for a correct PCE evaluation. Normal and inverted hysteresis in perovskite solar cells Ioana Pintilie========================================================§ INTRODUCTIONThe dynamic hysteresis phenomena observed in the J-V characteristics of perovskite solar cells (PSCs) fuels an ongoing debate about its origin, accurate determination of the photoconversion efficiency (PCE), and cell stability <cit.>. Several mechanisms have been proposed to explain the hysteretic behavior: ion migration <cit.>, charge trapping and de-trapping <cit.>, ferroelectric polarization <cit.>, capacitive effects <cit.>, charge accumulation at the interfaces <cit.>, or unbalanced distribution of electrons and holes <cit.>. It is already established that the hysteresis is influenced by measurement settings such as the bias scan rate and range <cit.>, but also by the cell pre-conditioning by voltage poling <cit.> and light soaking <cit.>.In the typical hysteresis the reverse-scan current, i.e. measured by reducing the voltage from open circuit bias (V_oc) to short circuit, is larger than in the forward scan. During the reverse scan the cell behaves like a capacitor, releasing charge in the external circuit, and hence an excess of current is obtained, while during the forward scan the opposite situation occurs.We call this behavior normal hysteresis (NH). Recentlyan inverted hysteresis (IH) was reported <cit.> in mixed-cation mixed-halide PSCs, and attributed to charge extraction barriers. It was further shown that methylammonium-lead-iodide (MAPbI_3) PSCs with TiO_2 layer covered by a thin Al_2O_3 insulating shell also exhibits IH. In other studies the IH was explained in terms of ionic accumulation <cit.>,and observed in aged samples with Mo/Ag counter electrodes <cit.>.In this paper we report for the first time the presence of both NH and IH in the same sample, strictly depending on the applied pre-poling bias (V_pol): NH is obtained for V_pol> V_oc and IH for V_pol<0. For 0<V_pol<V_oc the hysteresis loop has a crossing point where the forward and reverse characteristics meet. The experimental data are well reproduced by the dynamic electrical modelwhich is summarized below. Then, we indicate a three-step J-V measurement protocol which is essential for a meaningful evaluation of the hysteretic effects. We discuss the pre-poling bias conditions for the realization of NH, IH and intermediate mixed hysteresis.In connection with the measurement protocol, by controlling the switching between NH and IH, we provide further insight into PSC operation but also a systematic assessment of the dynamic hysteresis, which is crucial for a proper evaluation of the PCE. § THE DYNAMIC ELECTRICAL MODEL (DEM) Several models have been considered for the dynamic hysteresis: Initially a drift-diffusion approach has been formulated, accounting for ionic migration, electronic charge traps, and recombination centers <cit.>, or coupled charge carriers and defect mediated ion motion including bias scan rate and pre-poling <cit.>. Then, an equivalent circuit model with constant capacitance was proposed <cit.>. The DEM, recently formulated by our group <cit.>, is also based on an equivalent circuit, but includes acapacitor accounting for non-linear polarization effectsas depicted in Fig. <ref>.The key assumption is that the slow process governing the time evolution of the non-linear polarization P_nl can be described by a specific relaxation time τ. Since τ is typically of the order of seconds DEM is compatible with ion migration and accumulation at the interfaces resulting in a surface polarization of Maxwell-Wagner-Sillars type. The time dependent J-V characteristics is determined by the coupleddifferential equations: -R_s C_0 ∂ I/∂ t =I_s ( e^q(V + I R_s)/n k_B T-1) + (R_s/R_sh + 1 ) I + V/R_sh + C_0 ∂ V/∂ t + 𝒜∂ P_nl/∂ t - I_ph ,∂ P_nl/∂ t = P_nl,∞(U_c(t)) - P_nl(t)/τ , with initial conditions I(t=0)=I_0 and P_nl(t=0)=P_0, which determines the NH or IH behavior.The current density J is defined as theratio between the current intensity I and the device active area 𝒜.The elements of the circuit are: the series resistance R_s, the shunt resistance R_sh, the diode ideality factor n, the diode saturation current I_s, the photogenerated current I_ph and the geometric capacitance C_0. In contrast to the standard model with a constant capacitance, the time dependence of the non-linear polarization P_nl obeys Eq. (<ref>), with a constant relaxation time τ, whereas the equilibrium polarization P_nl,∞=(U_c/V_oc) P_∞ depends on the voltage on the non-linear capacitor, U_c=V + I R_s,where V is the external applied bias. The constant polarization P_∞ corresponds to U_c=V_oc in the steady state. The relaxation time τ accounts for the decay of the initial polarization P_0 and also for the evolution of the polarization when the applied bias is changed.In fact, as it will be shown in the following, the current componentI_c^(nl)= 𝒜×∂ P_nl/ ∂ t has a crucial role in producingthe hysteretic effects, representing a significant part of the measured current I:I = I_ph - I_d - I_sh - I_c,where I_d is the diode current, I_sh accounts for recombinations andI_c = I_c^(l) + I_c^(nl), with typically negligible linear component I_c^(l) due to the small geometrical capacitance C_0.However, depending on the pre-poling conditions I_c^(nl) can reach magnitudes similarto the measured current <cit.>.Another version of DEM has been subsequently proposed by Ravishankar et. al.<cit.> in terms of a relaxation kinetic constant determined by ion displacement, by reformulating Eq. (<ref>) for the surface polarization voltage [Eq. (6) therein].Both versions of DEM consistently reproduced experimental NH, including the dependence on the bias scan rate and the current overshoot in the reverse scan observed in several experimental studies <cit.>, reproduced only in part by drift-diffusion based models <cit.>. The overshoot is consistent with a relaxation time of the polarizationor charge accumulated at interfaces <cit.> at the beginning of the reverse scan measurement, indicating arelation between the pre-poling bias and the initial polarization. Furthermore, the dependence of the hysteresis on the bias scan rate is consistently described by the DEM: the hysteresis is diminished at very low and very high bias scan rates, being significant at intermediate ones; importantly, the short circuit current is enhanced with the increase of the bias scan rate.It is worth noting that a standard constant capacitance model cannot explain the current overshoot, nor can account for the diminished hysteresis at very high scan rates, unless a vanishing dielectric constant for rapidly varying fields is assumed.Moreover, it predicts only the normal hysteresis behavior. § MEASUREMENT PROTOCOL As already mentioned in other papers<cit.>, in order toobtain reproducible measurements, it is important to adopt a well definedmeasurement protocol. We consider three steps: 1 - the stabilization of the open circuit bias V_oc,2 - the solar cell pre-conditioning phase by pre-poling under 1 sun illumination,and 3 - the actual J-V measurement consisting of a single reverse-forward bias scan starting from the open circuit bias to short-circuit and back. Before each pre-conditioning phase, the solar cell is kept in open circuit under illumination until the steady state is achieved, corresponding to stabilized V_oc and zero current. In this way we reinitialize as far as possible the current state of the solar cell, which may be affected in successive measurements. We choose the open circuit condition for the stabilization phase, instead of the short-circuit, to avoid as much as possible the solar cell deterioration.Pre-poling with illumination is achieved by applying a constant bias V_polfor a time t_pol. For V_pol>V_oc we call the sample over-polarized,for 0<V_pol<V_oc under-polarized, and for V_pol<0 reversely polarized.Regarding the actual measurement, instead of performing individual forward and reverse bias scans, the single step reverse-forward bias scan ensures a better control over the sample pre-poling, eliminating potential pre-poling differences in the two separate scans. § RESULTS AND DISCUSSION Results and discussion. In the following we discuss the NH and IH induced by different pre-poling regimes, the measurements being consistently reproduced by the DEM calculations. The samples are prepared in the standard FTO/TiO_2/CH_3NH_3PbI_3-xCl_x/spiro-OMeTAD/Au configuration, with x=0.4, <cit.> as detailed in the Supporting Information (SI). More details, including the physical properties and time evolution of similar samples, are described in Ref. <cit.>. First, we calibrate the equivalent circuit elements with a measured J-V characteristics showing NH, performed at a bias scan rate of 20 mV/s, with positive pre-poling bias V_pol=1.2V for t_pol=30 s. With this choice, the current overshoot is present in the reverse characteristics and this marked feature enables a more accurate extraction of the model parameters <cit.>. The simulated J-V characteristics of the over-polarized sample is depicted in Fig. <ref>(a) matching closely the detailed features of the experimental data.The obtained DEM parameters are: the photogenerated current I_ph=1.42 mA, the series resistance R_s=95Ω, the shunt resistance R_sh=3 kΩ, the diode ideality factor n=1.53 and saturation current I_s=0.1 pA. The relative dielectric permittivity ϵ_r=100 yields a quite small geometrical capacitance C_0 and its influence over the J-V characteristics is negligible in comparison with the non-linear polarization component. The steady state polarization at the open circuit is P_∞=18 mC/cm^2, corresponding to a stabilized V_oc=0.9 V. The relaxation time τ=9 s fits well with the preliminary indication about the magnitude of the characteristic times involved.The assumed initial polarization P_0=9P_∞ corresponds to the pre-poling stage performed under illumination at 1.2 V for 30 s. The device active area is 𝒜=0.09 cm^2, which is also the illuminated area.By switching the initial polarization P_0 to negative values, J-V characteristics with mixed or completely inverted hysteresis are observed. It is quite remarkable that changing only one parameter, P_0=-9P_∞, the shape of the inverted hysteresis resembles a measured characteristics with V_pol=-1V for 30 s at the same bias scan rate, apart from the diminished short circuit current obtained in the measurement. Taking into account I_ph=1.11 mA the simulated J-V characteristics overlaps quite well with the experimental data, as indicated in Fig. <ref>(b). Our test case shows a mostly inverted hysteresis, with a crossing point in the forward and reverse characteristics at ∼0.23 V.The decrease of I_sc at negative pre-poling was systematically observed in repeated measurements, starting from lower V_pol values to higher ones and in reverse order, which will be discussed later. Apart from this poling effect, a temporary aging of the cell is observed after repeated bias cycling, manifested by an overall decrease of the current in both forward and reverse characteristics. Typically, the samples recover in one day in dark and inert atmosphere conditions. Both hysteretic effects can be explained by analyzing the time evolution of the non-linear polarization P_nl(t), depicted in Fig. <ref>. In the case of NH, P_nl(t=0)=P_0≫ P_∞ is positive and decays rather fast in a time interval of the order of τ in the beginning of the reverse scan. Subsequently, P_nl follows the bias dependent steady state polarization P_nl,∞(U_c(t)) [Eq. (<ref>)] as the bias is further decreased in reverse and then increased in the forward regime. On the other hand, IH is produced by negative poling, yielding a negative initial polarization, P_0<0. The measurement, however, starts from V_oc and P_nl is driven into positive region on the reverse scan, reaching a maximum, and then decreasing towards the zero-bias state. As τ is here smaller than the total time of the reverse scan, the behavior of P_nl in the forward scan is similar for both IH and NH. The condition P_nl,∞=P_nl is correlated either with a minimum or a maximum for P_nl(t). As already indicated, the J_c^(nl) current component is quantitatively significant and plays a determinant role in the features on the J-V characteristics. Depending on its sign the value of the measured current can be enhanced or diminished according to Eq. (<ref>), influencing the determined PCE value. To further investigate the transition between NH and IH, we measured J-V characteristics with several pre-poling voltages, between -1.5V to 1.2V, as shown in Fig. <ref> together with the calculated J-V characteristics with suitably chosen initial polarizations P_0. A crossing point between the reverse and forward characteristics appears and it moves systematically towards lower biases as P_0 (or V_pol) decreases.This behavior was theoretically predicted before, in Ref.<cit.>, as an under-polarization regime (0<P_0<P_∞). On the other hand, the crossing was observed experimentally <cit.> on aged samples with Ag counter electrodes, using a forward-reverse scan protocol starting from -0.1 V, which may be the effect of a small negative bias pre-poling, although not intentionally performed. The crossing of separately measured forward and reverse characteristics was also observed <cit.>.As already indicated, a particular feature is the decrease of the I_sc found in the measured J-V characteristics, as negative poling increases in absolute value. Based on this experimental observation and the fact that R_sh is relatively large, which implies I_sc≈ I_ph, and assuming a relaxation time independenton the poling bias, we adjust the I_ph current to 1.11 mA (P_0=-9P_∞) and 0.96 mA (P_0=-10P_∞) for V_pol = -1 V and -1.5 V, respectively. This may be explained by iodine migration during the poling time, as negative iodine ions tend to accumulate at the TiO_2 mesoporous layer and the induced electric field reduces electron collection during the subsequentmeasurement <cit.>. By contrast, when positive poling is applied the electric field induced by ions is favorable to electron extraction and I_sc does not change significantly.We focus next on the bias scan rate dependence of the IH. For the NHwe obtained a consistent correspondence between the measurements and DEM <cit.>: the NH is widest at intermediate bias scan rates and I_sc increases at higher rates. This behavior was explained in terms of time evolution of non-linear polarization P_nl and the current component J_c^(nl)=∂ P_nl /∂ t introduced in Eq. (<ref>), as also shown (for comparison to IH) in Fig.<ref> of SI.The simulated and measured J-V characteristics showing IH are presented in Fig. <ref>.In the experiment we considered V_pol=-1.4 V, which corresponds in simulations to P_0=-9.5P_∞. At this stage the sample is slightly aged, as one may notice from the 20 mV/s scan, having a smaller I_sc compared to bias pre-poling measurements. We therefore consider I_ph=0.6 mA and an increased τ=30 s and reproduce the sequence of experimental J-V characteristics with different bias scan rates.As in the case of NH at very fast bias scans, the hysteresis amplitude is diminished. However, the behavior of I_sc is opposite: it becomes smaller as the scan rate increases.At very fast bias scan rates there is little variation in P_nl, as it remains close to P_0, and ∂ P_nl/∂ V becomes small in both forward and reverse scans, reducing thus the hysteresis of both NH and IH. In contrast, at short-circuit, the calculated current J_c^(nl)= ∂ P_nl /∂ t has a larger magnitude for NH at higher bias scan rates, and the opposite occurs for IH (Fig. <ref>). This accounts for both measured and calculated decrease in I_sc in IH conditions(Figs. <ref> and <ref>). We also note that ∂ P_nl /∂ t < 0 for NH and ∂ P_nl /∂ t > 0 for IH, which is the reason for the crossing of the forward and reverse characteristics.We therefore conclude that the pre-poling bias is a determinant switching factor from NH to IH, which become two forms of hysteresis previously considered independent.The dynamic hysteresis creates difficulties for an accurate determination of the solar cell PCE.Figure <ref> shows NH (P_0=9P_∞), IH (P_0=-9P_∞) casesand a minimal polarization case (P_0=P_∞), for a 20 mV/s bias scan rate, and the steady state solution (P_∞=0), in the bias range [0,V_oc].The PCE varies drastically in forward and reverse scans of both NH and IH, with a large deviation in the reverse scans compared to the steady state PCE: 27% overestimation for NH and 48% underestimation for IH.In the forward scan of the NH we find a deviation of only 2.3%, whilefor IH we obtain a 28% smaller PCE. However in the minimal polarization case, where the reverse scan starts at V_oc determined before the actual measurement, only a small hysteresis is found, as P_∞≠0. This case provides the most accurate description of the steady state, with a deviation of less than 3% for both forward and reverse scans.Decreasing the bias scan rate would reduce the hysteresis loop even further. However, in practical measurements, for a reasonable measurement time, it is very important to determine firstly V_oc, in order to avoid over-polarization. Secondly, another recommendation within the proposed measurement protocol is taking a continuous reverse-forward cycle, which is essential for removing an unintended initial polarization, e.g. by applying a reverse scan ending into negative voltage followed by a separate forward scan after an arbitrary time interval. A waiting time between reverse and forward bias scans influence significantly the second (forward) scan, as it is further detailed in Fig. <ref> (SI), for an IH test case.Therefore we emphasize the importance of a careful consideration of the cell dynamic polarization. Our three-step protocol allows a systematic evaluation of such hysteretic phenomena, but also provides a realistic determinationof the PCE as the initial polarization is minimized.§ CONCLUSIONS To conclude, we investigated the dynamic hysteresis of standard perovskite solar cells with controlled pre-conditioning. By varying the pre-poling bias, both normal and inverted hysteresis are found for the same sample and the measured J-V characteristics are closely reproduced. The main findings of the present study are: * The NH and IH can be obtained by pre-poling the solar cell with bias V_pol>V_oc and V_pol<0, respectively, and a mixed hysteresis is typically observed for 0<V_pol<V_oc. * The DEM reproduces the two hysteretic behaviors by changing only the initial polarization parameter, which is related to experimental quantities V_pol and t_pol. * At higher bias scan rates the short circuit current I_sc is enhanced for NH and lowered for IH, consistently explained by DEM. * The observed NH/IH behavior reveals once again the importance of controlling the solar cell pre-conditioning, particularly the bias pre-poling; the state induced by a previous measurement may inadvertently affect the next one; the designed three-step protocol, including stabilization, pre-poling and measurement is afforded for the reliability and reproducibility of the experimental data, being especially important for a correct PCE determination. * The presence of both NH and IH unifies apparently contradictory results reported in recent papers. Our results indicate the presence of both normal and the inverted hysteresis in the same device if the required pre-poling conditions can be achieved. The research leading to these results has received funding from EEA Financial Mechanism 2009-2014 under the project contract no 8SEE/30.06.2014. *§ SUPPORTING INFORMATION 1. Device fabrication and characterization. a) Device fabricationWe fabricated perovskite solar cells by successive spin-coating deposition of TiO2 thin and meso-porous layers, CH_3NH_3PbI_3-xCl_x mixed halide perovskite and spiro-OMeTAD on commercial glass/FTO substrate (Solaronix TCO22-7). The compact TiO_2 blocking layer of 150 nm thickness is obtained from bis(acetylacetonate) solution (Aldrich) and annealing at 450 ^∘C for 30 min. The mesoporous TiO_2 film (mp-TiO_2), composed of 20 nm size particles is deposited using a solution of TiO_2 commercial paste (Solaronix Ti-Nanoxide N/SP) diluted in ethanol (1:3, weight ratio). The mp-TiO_2 structures were first annealed at 150 ^∘C for 5 min and then crystallized at 500 ^∘C for 1 h. The CH_3NH_3PbI_3-xCl_x mixt halide perovskite is fabricated using the principle of one-step method <cit.>: a precursor solution containing 369 mg lead iodide, 56 mg lead chloride, 78 mg methyl sulfoxide, 159 mg methyl ammonium iodide (Dysol) and 600 mg dimethylformamide, homogenized for one hour, was spin coated with 2000 rpm/25 s. After 9 s from the start of the spin cycle 150 μl of diethyl ether was dripped on top of the layer, enabling the solvent extraction process to take place. The final perovskite film is obtained after annealing at 65 ^∘C for 1 min and 100 ^∘C for 2 min. All the above described processes were performed in normal laboratory conditions, at 24 ^∘C and humidity between 30% and 40%. The spiro-OMeTAD was spin-coated at 1500 rpm for 30 s, in N_2 enriched atmosphere, at 24 ^∘C and humidity less than 10%. The solution used for this deposition was obtained by mixing 80 mg spiro-OMeTAD (Borun Chemical), 28 μl 4-tert-butylpyridine and 18 μl of bis(trifluoromethane)sulfonimide lithium salt in acetonitrile solution (520 mg ml^-1). Gold counter electrodes of 0.09 cm^2 area have been deposited by magnetron RF sputtering.b) Electrical characterizationThe Photocurrent–Voltage (J–V) characteristics were measured using an Oriel VeraSol-2 Class AAA LED Solar Simulator having AM 1.5 filters and a Keithley 2400 Source Meter. The irradiation intensity was 100 mW/cm^2, calibrated by a Newport standard silicon solar cell 91150. The 1 Sun illumination has been performed through a rectangular aperture of 3×3 mm^2 size, of a geometry identical with that of Au counter electrodes. 2. Simulations and measurements revealing normal hysteresis (NH) or inverted hysteresis (IH), for different bias scan rates and prepoling biases. a) Simulated NH and IH for different bias scan rates NH IH b) Experimental NH and IH induced by bias pre-poling c) Experimental NH and IH - influence of the bias scan rate 3. Influence of the waiting time at short-circuit, between reverse and forward scan, for the case of inverted hysteresis. | http://arxiv.org/abs/1704.03300v1 | {
"authors": [
"George Alexandru Nemnes",
"Cristina Besleaga",
"Viorica Stancu",
"Lucia Nicoleta Leonat",
"Lucian Pintilie",
"Kristinn Torfason",
"Marjan Ilkov",
"Andrei Manolescu",
"Ioana Pintilie"
],
"categories": [
"cond-mat.mes-hall",
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170325132439",
"title": "Normal and inverted hysteresis in perovskite solar cells"
} |
Thermoacoustic tomography for an integro-differential wave equation modeling attenuation Benjamín Palacios^1 & Sebastián Acosta^2 ^1Department of Mathematics, University of Washington, Seattle, WA, USAEmail address: ^2Department of Pediatrics - Cardiology, Baylor College of Medicine, TX, USA Email address:Abstract. In this article we study the inverse problem of thermoacoustic tomography (TAT) on a medium with attenuation represented by a time-convolution (or memory) term, and whose consideration is motivated by the modeling of ultrasound waves in heterogeneous tissue via fractional derivatives with spatially dependent parameters. Under the assumption of being able to measure data on the whole boundary, we prove uniqueness and stability, and propose a convergent reconstruction method for a class of smooth variable sound speeds. By a suitable modification of the time reversal technique, we obtain a Neumann series reconstruction formula.Key words: Multiwave imaging; wave equation; integro-differential equations; attenuation; memory § INTRODUCTIONIt is well known that for biological tissues the attenuation of acoustic waves is frequency-dependent.One way to model this attenuation is to use fractional time derivatives and consequently the representation of the propagation of ultrasound waves by integro-differential equations. Examples of this modeling are frequency power-law attenuation or fractional Szabo models (see for instance <cit.>) where the traveling wave may be assumed to satisfy an equation of the form γ^-2∂_t^2u - Δ u + β∂^k+α_tu = F(t,x),for someα∈(0,1),k=1,2,and where the fractional derivative term can be written as a convolution in time β(x)∂^k+α_tu = ∫^t_-∞Ψ_α(t-s,x)∂^k+1_su(s,x)ds. Assuming, as in thermoacoustics, that the wave field vanishes for negative times, and provided that the kernel is bounded and regular enough, we can perform integration by parts and write the previous integral as a convolution of u with a different kernel, plus time-derivatives of u up to order two. In the case k=2, the sound speed is perturbed resulting in a different speed c^-2 = γ^-2 + βΨ_α(0), which requires conditions on β and Ψ_α in order to get an effective wave speed c>0. We point out there is a recent definition for derivatives of fractional order which employs such continuous and bounded kernels <cit.>.In the present article, we study the inverse problem of finding the initial source f in an attenuating medium, provided boundary data u|_[0,T]×∂Ω and where the acoustic wave u is assumed to satisfies the system{[ ∂_t^2u - c^2Δ u + a ∂_tu + bu + ∫^t_-∞Φ(t-s,x)u(s,x)ds = δ'(t)f(x),∈×^n; u(t,x)=0,t<0. ].We suppose a,b,c∈ C^∞(^n), Φ∈ C^2(^n+1), a,b≥ 0, c_0^-1≥ c ≥ c_0 > 0, and for a fixed open bounded set Ω⊂^n with smooth boundary, we suppose a= b=c-1=Φ = 0 in ^n\Ω̅. We shall use the following notation throughout the paper: P_Φ := ∂_t^2-c^2Δ + a∂_t + b +Φ*·,Φ*u = ∫^t_0Φ(t-s,x)u(s,x)ds.Then P_Φ = ∂_t^2-c^2Δ outside the domain of interest Ω.The Cauchy problem associated with (<ref>) is{[P_Φ u = 0, (t,x)∈(0,∞)×^n,; u|_t=0=f,;∂_tu|_t=0=-af, ].since any solution of (<ref>) extended by zero to (-∞,0)×^n is a solution of (<ref>). Indeed, given a smooth solution u of (<ref>) we consider H(t)u(x,t) where H(t) is the Heaviside function. Then, we can pull out the Heaviside function from the convolution since it integrates on the interval (0,t), thus we getP_Φ(Hu)= uδ' + 2(∂_tu)δ+ auδ + (P_Φ u)Hwith the last term vanishing since P_Φ u = 0.For an arbitrary test function ϕ∈ C^∞_c(^n+1) we have the following,⟨ P_Φ(Hu) , ϕ⟩ = ∫_^n[-(∂_t u)ϕ - u(∂_tϕ) + 2(∂_tu)ϕ + auϕ]|_t=0dx= -∫_^nu∂_tϕ|_t=0dx=⟨ fδ',ϕ⟩,which is the same as problem (<ref>). The thermoacoustic tomography problem in a medium with convolution-type attenuation can be modeled by the following initial value problem (IVP):{[ P_Φ u(t,x) = 0,(t,x)∈(0,T)×^n; u|_t=0=f,;∂_tu|_t=0=-af, ].where we aim to recover the initial source f from boundary measurements u|_(0,T)×∂Ω, assuming the waves propagate freely in the space, that is, we suppose the boundary of Ω does not interact with the outgoing waves. This last assumption has been considered for instance in <cit.>.The problem of thermoacoustic tomography has been broadly studied by many authors. Several reconstruction methods have been proposed for homogeneous media <cit.>, and also for heterogeneous media <cit.>. See also the reviews <cit.> for additional references. The theoretical analysis of the so-called time reversal method has gained considerable attention in the past few years, mainly due to the work of Stefanov and Uhlmann in <cit.>. In its initial formulation, the time reversal technique gives an approximate solution that converges to the exact one as the observation time increases. The problem of recovering the initial source for optimally short measurement time was solved in <cit.> for variable sound speed employing techniques from microlocal analysis. Recently, the focus of the mathematical analysis has been placed on extensions in the following two areas. First, there is the problem of accounting for attenuating media. Homan in <cit.> gave a first extension of Stefanov and Uhlmann's work in this direction by considering the damped wave equation with sufficiently small damping coefficients for the time reversal method to work. In the complete data case, those results were extended to more general damping coefficients in <cit.>. In <cit.> the authors addressed the TAT problem with thermoelastic attenuation. Second, recent publications have addressed the TAT problem in enclosed domains to model the interaction of acoustic waves with reflectors and sensors. The advantages of working with this setting is that it naturally allows to consider partial data and the inverse problem is closely related with boundary control theory. See for instance <cit.>. This article falls in the first group. As far as the authors know, the TAT inverse problem with attenuation of integral type and variable sound speed has not been fully considered in the literature from an analytical point of view. From a heuristic point of view, some advances have been made. For the case of constant wave speed and constant coefficient of attenuation, Modgil et al. <cit.> designed a method based on relating the unattenuated wave field to the attenuated wave field via an integral operator and its subsequent inversion using a singular value decomposition. Treeby et al. <cit.> proposed a reconstruction based on time reversal and the k-space computational method. Attenuation compensation was achieved by separating the absorbing and dispersion terms in the wave equation, and reversing the sign of the absorbing coefficient during the time reversal. This method was modified in <cit.> where the coefficient of attenuation was allowed to vary within the region of interest, but the exponent of the power-law attenuation was still assumed to be constant. However, in some practical settings such as in the presence of bone and soft–tissue, the domain exhibits regions of varying power-law exponents. An appropriate method needs to be devised to avoid blurring and distortions in the reconstruction. Our work is a step in that direction, where the coefficients a,b,c and the kernel Φ in (<ref>) are allowed to vary, which effectively accounts for power-law attenuation of spatially varying exponent. Considering attenuation terms of integral type brings some difficulties to the analysis on the propagation of waves. In particular, the equation is no longer reversible and local in time and consequently it is not possible to use techniques such as Tataru's unique continuation to get uniqueness, at least not in a direct way. Moreover, the microlocal properties of this type of integro-differential operators are not well understood. Nevertheless, it is possible to exploit the fact that an integral term of the sort considered here only presents a compact perturbation of the differential operator. This article can be view as a first attempt to understand the TAT problem in media with memory/attenuation coefficients that vary in space. A subsequent step would be to tackle viscoelastic models, and singular kernels as in the standard definition of fractional derivatives. The paper is structured as follows. In the next section we set the framework under which our analysis is based, such as the well-posedness of the direct problem, the energy space of initial conditions and the hypothesis on the attenuation parameters, namely the damping coefficient and the attenuation kernel. In Section 3 we prove two uniqueness results. The first one is a sharp result on uniqueness for the thermoacoustic inverse problem assuming the distance function from the boundary allows us to foliate the interior of the domain by strictly convex surfaces. In particular we require ∂Ω to be strictly convex. The second main theorem of this section, which does not require convexity of the boundary, assumes that the sound speed satisfies a frequently used condition related with the convexity of the euclidean spheres in the metric induced by the sound speed. The stability question is addressed in Section 4 and we devote Section 5 to show the existence of a Neumann series reconstruction formula. § PRELIMINARIES§.§ Direct problem Let U⊂^n be an open bounded set with smooth boundary, u_0∈ H^1_0(U), u_1∈ L^2(U) and F∈ L^2([0,T]; L^2(U)). We say u is a generalized solution of P_ϕ u = Fin [0,T]× U, u|_[0,T]×∂ U=0, u(0) = u_0, u_t(0) = u_1,if u∈ L^2([0,T];H^1_0(U)), u_t∈ L^2([0,T];L^2(U)), u_tt∈ L^2([0,T];H^-1(U)) and ⟨ c^-2u_tt,φ⟩ + B(u,φ) = (c^-2f,φ)∀φ∈ C^∞_0(U) and for a.e. t∈[0,T]where ⟨·,·⟩ and (·,·) stand for the duality product of H^-1 and H^1_0, and the L^2 inner product respectively, and B(·,·) is the bilinear form given byB(u,φ) = (∇ u,∇φ) + (ac^-2u_t,φ)+ (bc^-2u,φ) + (c^-2Φ*u,φ). The well-posedness follows from Theorems 2.1 and 2.2 in <cit.>.We refer to the appendix for a complete proof. In our case, by finite speed of propagation we can take U to be a large ball containing Ω to ensure we have null Dirichlet conditions.§.§ Energy space and positive-definite kernels Given a domain U⊆^n and a scalar function u(t,x), we define the local energy of = [u,u_t] at time t asE_U((t)) = ∫_U(|∇_x u|^2 + b|u|^2 + c^-2|u_t|^2)dx.In order to give problem (<ref>) a physical sense we need to assume some conditions on the attenuation terms since such system must satisfies that its energy decreases over time. The previous is achieved for instance if a(x)≥ 0 and the kernel Φ is positive-definite, this is ∫^T_0(Φ*y)ydt≥ 0 for all y∈ C([0,T]). We then assume the following:a(x)≥ 0and (-1)^j∂^j_tΦ (t,x) ≥ 0,∀ t≥0, x∈^n,j = 0,1,2.The previous condition guarantees the positive-definiteness of the kernel as shown in <cit.> and <cit.>. Moreover, if we define Ψ(t,x) := -∫^∞_tΦ(s,x)ds,it turns out that -Ψ is also a positive-definite kernel since it satisfies the same condition as Φ.An example of a kernel satisfying Condition <ref> is Φ(t,x) = q(x)e^-α t, for some positive function q ∈ C(^n) and α > 0. In the recent article <cit.>, the authors introduce a new definition for fractional derivatives whose kernel is of the form e^-α t. As a consequence, the analysis carried out in this paper might be applied to fractional models of wave propagation following this new definition of fractional derivatives. Under Condition <ref> we define the extended energy functional at time τ>0, analogously as in <cit.>, to beℰ_U(u,τ) = E_U((τ)) +2∫_[0,τ]× Uac^-2|u_t|^2dxdt+ 2∫_[0,τ]× Uc^-2(Φ*u_t)u_tdxdt,where the last two terms take into account the portion of the energy that is lost due to the attenuation process. If we set U=^n, or by finite propagation speed we take U equal to any sufficiently large ball, in the interval [0,T] the former energy functional E_U is non-increasing since we getd/dtE_U((t)) = - 2∫_[0,τ]× Uac^-2|u_t|^2dxdt- 2∫_[0,τ]× Uc^-2(Φ*u_t)u_tdxdt≤ 0,and integrating in time we deduce that the extended functional is conserved.We adopt the same functional framework as in previous articles related to thermoacoustic tomography. The energy space ℋ(U) of initial conditions is defined to be the completion of C^∞_0(U)× C^∞_0(U) under the energy norm^2_ℋ(U) = ∫_U(|∇_x f_1|^2 + c^-2|f_2|^2)dx.with = [f_1,f_2]. We also let H_D(U) denote the completion of C^∞_0(U) under the normf^2_H_D(U) = ∫_U|∇_x f|^2dx. Notice that ℋ(U) = H_D(U)⊕ L^2(U;c^-2dx) with the latter space denoting the L^2 functions under the weight c^-2dx. Denoting by Ω the region of interest and Σ = [0,T]×∂Ω, we introduce the measurement operator Λ_Φ:ℋ(Ω)∋↦ u|_Σ∈ H^1(Σ), where u satisfies (<ref>).§ UNIQUENESS The first main result of this section, Theorem <ref>, is a uniqueness theorem for the full data case under a particular foliation condition. We work in this part with the more general hyperbolic operator (<ref>) associated to a Riemannian metric g. We then, assuming g=c^-2dx^2, provide a condition for the sound speed that guaranteed the existence of a particular foliation suitable for uniqueness. This is our second main result, Theorem <ref>. Foliation conditions go back to the work on seismology of Herglotz, Weichert and Zoeppritz at the beginning of the 20th century (see <cit.> and reference therein), and have been reintroduced in the literature very recently in <cit.>. This type of assumptions seem to be the natural conditions under one could expect to propagate information from the exterior toward inside the domain. In particular, it has been applied before in the thermoacoustic setting <cit.> to prove uniqueness for the inverse problem of recovering the sound speed (assuming the initial condition is known).Theorem <ref> is a direct consequence of <cit.>, a unique continuation result for hyperbolic equations with a memory term. For the sake of simplicity, the authors proved such result for the wave equation in an Euclidean metric. Nevertheless, in our work we need the full strength of such unique continuation, thus we have included a brief proof in the general case of waves in a general Riemannian setting. We point out that a similar method to the one used in <cit.> was also applied in <cit.> to obtain uniqueness for very general foliations and partial data. It was of fundamental importance in such proof the possibility of using a partial boundary unique continuation result independent of the foliation (see <cit.>). In contrast, in our context is precisely the unique continuation the result we need to prove, which in this case is deeply linked with the foliation. As a consequence we only treat the full data case. Modifying a bit the method mentioned above we are able to prove a second uniqueness result, Theorem <ref>, in the case the sound speed satisfies an specific condition. Let Ω be a bounded open subset of ^n with ∂Ω smooth and strictly convex for a Riemannian metric g. Let T>0 be such that x^n= dist(x,∂Ω) is a smooth function in Ω with non-zero differential for 0≤ x^n≤ T and its level surfaces {x^n=s}, for 0≤ s≤ T, are strictly convex for the metric g as well. If f∈ H_D(Ω) is such that Λ_Φ=0 with = (f,-af), then f=0 in {x∈Ω:dist(x,Ω)<T}. In particular, if T≥ T_1(Ω):= sup_x∈Ω dist(x,∂Ω), then f≡ 0.Under the above hypothesis, this result presents an improvement in the condition imposed on T for uniqueness in the damped wave equation (T>2T_1(Ω) in <cit.>).Let Q = (0,T)×Ω and x^n = dist(x,∂Ω) the signed distance function defined in a neighborhood of the boundary and such that Ω and ∂Ω are characterized respectively by x^n>0 and x^n=0. We define the following weight functionφ(x,t) = (R-x^n) - α t^2 - r^2,which is invariantly defined for any local coordinates (x^1,...,x^n-1) in ∂Ω. Here α = α(Ω,g)>0 is sufficiently small and R,r>0 will be chosen large and close to each other. For ϵ≥ 0 we also consider the setsQ(ϵ)= {(t,x)∈ Q: φ(x,t)>ϵ}, Ω(ϵ)= {x∈Ω:(R-x^n)^2>r^2+ϵ}.By taking r close to R, the set Q(0) is a small neighborhood of {0}×∂Ω inside Q. We recall that in boundary normal coordinates, a Riemannian metric g takes the formg̃_α,β(x',x^n)dx^αdx^β + (dx^2)^2,for α,β≤ n-1. We denote g̃ = (g̃_αβ(x)). Moreover, the strictly convexity of the level surfaces {x^n=s} translates into Π(v,v) = (-1/2∂g̃_αβ/∂ x^n)v^α v^β≥κ_s|v|^2_g̃,∀ v∈ T{x^n=s},with κ_s>0 the smallest eigenvalue (principal curvature) of the second fundamental form Π in {x^n=s}, where R_s=κ_s^-1 can be think as the largest curvature radius of {x^n=s}. The analogous condition for convectors follows from the natural isomorphism ξ_i = g_ij(x)v^j and readsΠ(ξ,ξ) = (1/2∂g̃^αβ/∂ x^n)ξ_αξ_β≥κ_s|ξ|^2_g̃,∀ξ∈ T^*{x^n=s}. In what follows we consider the more general integro-differential operator𝒫_Φ u = u_tt - ∂_j(g^ij(x)∂_iu) + ⟨ A(x),u'⟩ + b(x)u +Φ*u,where u'=(u_x,u_t), g is a Riemannian metric, and the vector-function A, the scalar-function b and the kernel Φ are continuous functions.The next two lemmas also hold if the coefficients A and b are analytic functions in t.Let Ω and T be as in Theorem <ref>. Let f∈ L^2(Ω) and u∈ H^2(Q) be a solution of{[𝒫_Φ u=0 in(0,T)×Ω,; u|_t=0=0in Ω,;∂_tu|_t=0=fin Ω. ]. Ifu=∂_ν u = 0 on ∂ Q(0)∩∂Ω, then u = 0 in Q(0), and in particularf= 0inΩ(0).Given a point y=(y',0)∈∂Ω, let's consider local coordinates (U,(x^1,...,x^n-1)) in the boundary near y'. For ϵ≥ 0 we define the setsQ_y(ϵ)= {(t,x)∈ Q: φ(x,t)>ϵ,x'∈ U}, Ω_y(ϵ)= {x∈Ω:(R-x^n)^2>r^2+ϵ,x'∈ U}.In what follows we take r=R-δ, for some δ>0 small enough, therefore x^n∈[0,δ) in the set Q(0).Let's first consider an arbitrary function u∈ C^∞(Q_y(0)) such that u=∂_ν u=0 on Q_y(0)∩∂Ω, and let ũ(t,x) = χ(x') u(t,x), with χ∈ C^∞_0(U). The idea is to obtain a well known local Carleman estimate for ũ and later use it, along with a partition of unity, to get an analogous estimate in Q(0). Let's denote 𝒫 = u_tt - ∂_j(g^ij(x)∂_iu), the principal part of 𝒫_Φ. By analyzing the conjugate operator 𝒫_τ = e^τφ𝒫e^-τφ, it is possible to deduce (after long computations) a pointwise estimate for v=e^τφũ of the form: C|𝒫_τ v|^2 ≥τ(|v_t|^2+|v_n|^2) + τ^3|v|^2+div_x(Y) + ∂_tZ +4τ(R-δ)^2(1/2∂_ng̃^klv_kv_l) - 2τγ|v_x|^2_g̃for some constant γ>0 depending on the parameter α which is chosen small enough, and with (Y,Z) a vector-valued function depending on lower order derivatives of v and vanishing in ∂ Q_y(0)\{φ=0}. In fact, the previous follows by decomposing 𝒫_τ v as the sum of two operators,𝒫_+v = v_tt - ∂_j(g^ij∂_iv) + τ^2Φ v,Φ = φ^2_t - |φ_x|^2_gand𝒫_-v = 2τ(⟨φ_x,v_x⟩_g - φ_t v_t) + τΨ v,Ψ = ∂_j(g^ij∂_iφ) -φ_tt,and bounding from below the inequality|𝒫_τ v|^2≥ |𝒫_+v|^2 + 2(𝒫_+v)(𝒫_-v).Here we apply the convexity condition on the level surfaces {x^n=s} in (<ref>). By choosing then R large enough and δ small,we arrive to the estimateτ^3|v|^2 + τ(|v_t|^2 + |v_x|^2_g)≤ C(e^τφ|𝒫ũ|^2 - div_x(Y) - ∂_tZ).Because v=e^τφũ, we can bound from below the left hand side of the previous inequality by similar terms but involving now the function u (and the exponential weight function).Integration over Q_y(0)and the Gauss-Ostrogradskiĭ formula give us thatτ∫_Q_y(0)e^2τφ(τ^2|ũ|^2+ |ũ_t|^2 + |ũ_x|^2_g)dxdt≤ C∫_Q_y(0)e^2τφ|𝒫ũ|^2dxdt+ C∫_Γ_y(0)(⟨ X_1u',u'⟩ + ⟨ X_2,u'⟩ u + X_3|u|^2)dS,where dS denotes the surface measure on Γ_y(0) = Q_y(0)∩{φ=0}, and the matrix-function X_1(x,t), the vector-function X_2(x,t), and the scalar-function X_3(x,t) are some continuous functions depending on φ and Q_y(0). Using the continuity of the coefficients in the lower order terms (l.o.t) of 𝒫_Φ and noticing that |𝒫ũ|^2 ≤ 2|𝒫_Φũ|^2 + 2|(l.o.t of 𝒫_Φ)ũ|^2,we can choose τ_0 larger if necessary and absorb the second summand in the right hand side above with the left hand side of (<ref>). Thenτ∫_Q_y(0)e^2τφ(τ^2|ũ|^2+ |ũ_t|^2 + |ũ_x|^2_g)dxdt≤ C∫_Q_y(0)e^2τφ|𝒫_Φũ|^2dxdt+ C∫_Γ_y(0)(⟨ X_1u',u'⟩ + ⟨ X_2,u'⟩ u + X_3|u|^2)dS, The analogous inequality in the larger set Q(0) is obtained by considering a partition of unity and using the compactness of ∂Ω. More precisely, let now u∈ C^∞(Q(0)) and let {U_i}_i be a finite covering of the boundary such that on each U_i we can define boundary local coordinates, and let {χ_i}_i be a finite smooth partition of unity subordinate to {U_i}_i. We also consider a collection of points y_i∈ U_i. Then, denoting u_i=χ_i^1/2 u, and the measure dσ = dt dVol(x) on Q, from the previous estimates we getτ∫_Q(0)e^2τφ(τ^2|u|^2 + |u_t|^2 + |u_x|^2_g)dσ =τ∑_i∫_Q_y_i(0)e^2τφχ_i(τ^2|u|^2 + |u_t|^2 + |u_x|^2_g)dxdt ≤τ∑_i∫_Q_y_i(0)e^2τφ(τ^2|u_i|^2 + |(u_i)_t|^2 + |(u_i)_x|^2_g)dxdt+ Cτ∫_Q(0)e^2τφ|u|^2dσ≤ C(∫_Q(0)e^2τφ|𝒫_Φ u|^2dσ +∫_Γ(0)(⟨ X_1u',u'⟩ + ⟨ X_2,u'⟩ u + X_3|u|^2)dS..+ ∑_i∫_Q(0)e^2τφ|[𝒫_Φ,χ_i]u|^2dxdt + τ∫_Q(0)e^2τφ|u|^2dσ),where notice [𝒫_Φ,χ_i] are differential operators of order 1. We absorb the interior integrals with lower order derivatives of u using the left hand side and getτ∫_Q(0)e^2τφ(τ^2|u|^2 + |u_t|^2 + |u_x|^2_g)dσ ≤ C∫_Q(0)e^2τφ|𝒫_Φ u|^2dσ+ C∫_Γ(0)(⟨ X_1u',u'⟩ + ⟨ X_2,u'⟩ u + X_3|u|^2)dS.It follows from a density argumentthat the previous estimate also holds for functions in H^2(Q) with null Cauchy data in ∂ Q(0)∩∂Ω.Let u be as in the hypothesis of the lemma. Then, u satisfies an inequality of the form (<ref>), without the interior integral in the right hand side. Noticing that the boundary integral does not depends on τ, we let τ goes to infinity and conclude that u=0 in Q(0). The aim of the second lemma is to extend the time for which u is zero. Based again on Carleman estimates we will be able to succeed until we hit the characteristic surface associated to the principal part of 𝒫_Φ, this is the surface {(t,x): T-t=dist(x,∂Ω)}.Let Ω and T be as in Theorem <ref>. If u∈ H^2(Q) is a solution of (<ref>), then u=0in{(t,x)∈ Q: dist(x,∂Ω)<ϵ, 0<t<T- dist(x,∂Ω)}for some 0<ϵ≤ T. From Lemma <ref>, u=0 in some neighborhood {(t,x)∈ Q: (R-x^n)^2>α t^2 + r^2} for appropriate constants α,R,r. It is clear that for sufficiently small ϵ_1,ϵ_2>0, the previous set contains [0,ϵ_1]×{x∈Ω:dist(x,∂Ω)<ϵ_2}.In a neighborhood of ∂Ω we defineψ(t,x) := (ϵ_2-x^n)(T-t-x^n),and for γ>0 we consider the setsQ_γ^ϵ_2 := {(t,x)∈ Q | ψ(t,x)>γ,x^n<ϵ_2},which exhaust Q^ϵ_2 = {(t,x)∈ Q |x^n<ϵ_2, 0<t< T-x^n}, this is Q^ϵ_2 = ⋃_γ>0Q_γ^ϵ_2. Moreover, there exists γ_0>0 such that ∅≠ Q_γ_0^ϵ_2⊂ [0,ϵ_1]×{x∈Ω : x^n<ϵ_2}. We denote by B(t_0,x_0;r) the ball centered at (t_0,x_0) and radius r for the euclidean metric. Given the following Claim. Suppose that for (t_0,x_0)∈ Q^ϵ_2, u vanishes below the level surface {ψ(x,t) = ψ(t_0,x_0)} near (t_0,x_0), this is in Q_ψ(t_0,x_0)^ϵ_2∩ B(t_0,x_0;r) for some r>0. Then, u=0 in a neighborhood of (x_0,t_0). the proof of the lemma is complete by the next argument. Let's assume that suppu∩ Q^ϵ_2≠∅. We can find 0<γ^*≤γ_0 such thatsuppu∩ Q_γ^ϵ_2 = ∅, ∀γ>γ^*andsuppu∩{(t,x)∈ Q^ϵ_2 : ψ(t,x)=γ^*}≠∅.The application of the claim on every contact point (t^*,x^*)∈suppu∩{(t,x)∈ Q^ϵ_2 : ψ(t,x)=γ^*}, contradicts the choice of γ^*. Consequently, we deduce that u=0 on every Q_γ^ϵ_2, γ>0, and therefore u = 0 in Q^ϵ_2.It only remains to show the previous claim. Here is where Carleman estimates play a fundamental role, and as before we will consider a particular choice of weight function which needs to fulfill a pseudo-convex condition with respect to 𝒫 = ∂^2_t - ∂_x^j(g^ij∂_x^i·), in the set {(0,ξ)∈ T^*_(t_0,x_0)Ω}. Moreover, we will take it to be linear and non-increasing in time. Provided the above, it is possible to apply a pseudo-differential Carleman estimate introduced in <cit.> and conclude that u vanishes near (t_0,x_0).Let's consider local coordinates in ∂Ω near some y∈∂Ω such that in those coordinates y = (x_0',0). For some δ>0 to be appropriately chosen, we define the following weight functionφ(t,x) = ψ(t,x)-ψ(t_0,x_0) - 1/2δ|x-x_0|^2where here |·| stands for the euclidean norm and ψ as in (<ref>). Denoting the principal symbol of 𝒫 by p(t,x;θ,ξ) = -θ^2 + |ξ|^2_g, where |ξ|^2_g = g^ij(x)ξ_iξ_j is the norm on covectors induced by g, the pseudo-convexity condition requires to show that φ satisfies (1) Re{p̅,{p,φ}}(t_0,x_0;0,ξ)>0for all ξ≠ 0such thatp(t_0,x_0;0,ξ) = 0,(2) 1/iτ{p̅_φ,p_φ}(t_0,x_0;0,ξ;τ)>0 for all ξ≠ 0, τ>0such that p_φ(t_0,x_0;0,ξ,τ)=0. Here p_φ(t_0,x_0;0,ξ;τ) = p(t,x; θ + iτφ_t, ξ + iτφ_x) and {·,·} is the Poisson bracket {f,h} = ∑_j=1^n∂ f/∂ξ_j∂ h/∂ x_j - ∂ f/∂ x_j∂ h/∂ξ_j + ∂ f/∂θ∂ h/∂ t - ∂ f/∂ t∂ h/∂θ.Recall that we are working in boundary normal coordinates hence the metric g takes the form (<ref>).The first condition is trivially fulfilled since the principal symbol p is elliptic in the set {θ = 0}. Let's use the following notation: the variable appearing in the subindex means we are differentiating with respect to such variable, for instance φ_x' = ∂_x'φ and φ_tx^n = ∂_t∂_x^nφ. To verify the second condition we notice first that φ_x'(t_0,x_0) = ψ_x'(t_0,x_0) = 0, φ_x^n(t_0,x_0) = ψ_x^n(t_0,x_0)= -α and φ_t(t_0,x_0) = ψ_t(t_0,x_0) = -β where α>β>0. In fact,α = (ϵ_2-x^n_0) + (T-t_0-x^n_0),andβ = ϵ_2-x^n_0.Also, denoting δ_ij the Kronecker delta,φ_tt = 0,φ_tx^i = δ_in,φ_x^ix^j = 2δ_inδ_jn - δ·δ_ij .Secondly, it is easy to check that p_φ(t_0,x_0;0,ξ,τ)=0 is equivalent to ξ_n = 0 and |ξ'|^2_g̃ = τ^2(α^2-β^2). Then, after some tedious computations, in the set of points (t_0,x_0;0,ξ;τ) such that p_φ = 0, we get1/iτ{p̅_φ ,p_φ} = 1/τ{Rep_φ,Imp_φ}=8τ^2(α^2 - αβ)+ 4α(1/2∂_ng̃^ij)ξ'_iξ'_j - δ M,with M = 4τ^2α^2 + 4(g̃^jkξ'_j)(g̃^ikξ'_i) such that, for some C>0,M≤4τ^2(α^2 + C(α^2-β^2)). Let's recall the positive-definiteness of the second fundamental form in (<ref>), and denote κ = min_s∈[0,ϵ_2]κ_s. By choosing δ>0 small enough we obtain that1/iτ{p̅_φ ,p_φ}≥ 8τ^2α(α - β)(1+κ/2(α + β)) - 4δτ^2(α^2 + C(α^2-β^2))>0,therefore φ satisfies the second condition of pseudo-convexity. It follows from <cit.> that there exists η,C,d>0 such that any function v supported inside B(t_0,x_0;η) (we of course choose 0<η<r), for which the RHS of the next inequality is finite, satisfies the pseudo-differential Carleman estimateτ^-1Ee^τφv^2_(2,τ)≤ C(Ee^τφP v^2 + e^-dϵτe^τφP v^2 + e^-dϵτe^τφv^2_(1,τ)),for the weighted normsv^2_(m,τ) := ∑_|α|+j≤ mτ^2(m-|α|-j)D^α D^j_tv^2_L^2(^n+1),τ>0;·:=·_(0,τ),and the pseudo-differential operator E :=e^ϵ/2τ|D_t|^2. This operator can also be considered as the convolution operatorEv(x,t) = (τ/2πϵ)^1/2∫ e^-τ|t-s|^2/2ϵv(x,s)ds. We would like to apply the above Carleman estimate to u and eventually deduce that u vanishes near (t_0,x_0). With that in mind we need first to localize it near (t_0,x_0). As in <cit.>, in (ψ'(t_0,x_0))^⊥ = {(θ,ξ):⟨ψ'(t_0,x_0),(θ,ξ)⟩_e⊗ g = 0} we see that |θ|≤ C_1|ξ|_g, hence⟨ (ψ-φ)”(θ,ξ),(θ,ξ)⟩_g = δ|ξ|^2_g≥ c_2|(θ,ξ)|^2_e⊗ g.Therefore, by choosing l_1<0 small enough in magnitude, the set {φ(t,x)> l_1}∩{ψ(t,x)<ψ(t_0,x_0)} is contained in a sufficiently small vicinity of (t_0,x_0). We then localize u by multiplying it with a function of the form χ(φ(t,x)) with χ∈ C^∞() a nondecreasing function such thatχ(s) = {0 for s< l_1,1 for s>l_2 ,.where l_1<l_2<0 are small enough in magnitude, thensupp[u(t,x)χ(φ(t,x))]⊂ B(t_0,x_0;η). In what follows we write χ meaning the composition χ∘φ. Consequently, v = χ u satisfies the inequality (<ref>). We include the integral term in the estimates by noticing that 𝒫(χ u) = χ𝒫u + [𝒫,χ]u = χ𝒫_Φ u - χΦ*u + 𝒫_1u,where 𝒫_1 is a differential operator of order 1 with coefficients supported in {(t,x)|φ(t,x)<l_2}. Consequentlyτ^-1Ee^τφ(χ u)^2_(2,τ)≤ c(E e^τφ𝒫_1 (χ u)^2 + Ee^τφχ(Φ*u)^2+. e^-dϵτe^τφ𝒫 (χ u)^2 + e^-dϵτe^τφ(χ u)^2_(1,τ)).The idea in what remains of the proof is to estimate Ee^τφ(χ u)_(2,τ) by a term of the form e^lτ, with l<0, and use <cit.> to conclude that χ u=0 in {(t,x)|φ(t,x)>l}. Such estimate is obtained in exactly the same way as in the proof of Lemma 6 in <cit.>, where everything reduces to estimate the term with the convolution since the other terms in the right hand side of the last inequality are easily bounded. For the arguments needed to conclude the claim we refer the reader to <cit.>. Proof of Theorem <ref>.Let u be a solution of P_Φ = 0 with initial conditions [f,-af] and such that Λ_Φ u = 0. Due to our assumption on the coefficients of P_Φ, u solves (∂^2_t-Δ)u=0 in (0,T)×(^n∖Ω) with null initial and Dirichlet boundary data. Then, for any x_0∈^n∖Ω, u vanishes in (0,T)× V for some small neighborhood V of x_0 such that V∩Ω = ∅. The previous is a consequence of a sharp domain of dependence for the wave operator in the exterior problem (see <cit.>). Then u=0 in (0,T)×(^n∖Ω) which implies null Neumann data, ∂ u/∂ν|_(0,T)×∂Ω=0. Let's set u̅(t,x) = ∫^t_0u(s,x)dsandΨ(t,x) = -∫^∞_tΦ(s,x)ds.Note that u̅_t(t,x) = u(t,x) and ∂_tΨ = Φ.Moreover∂_t( ∫^t_0Ψ(t-s,x)u̅_s(s,x)ds) = Ψ(0,x)u̅_t(t,x) + ∫^t_0Φ(t-s,x)u̅_s(s,x)ds,which, since u̅(0,x) = 0 and integration by parts, implies∫^t_0( ∫^τ_0Φ (τ-s,x)u(s,x)ds) =∫^t_0Φ(t-s,x)u̅(s,x)ds.We integrate equation (<ref>) on the interval (0,t) for any t>0. It follows from the previous computations that u̅ solves a system of the form (<ref>)with vanishing Cauchy data. In addition, notice that u̅_tt = u_t∈ L^2(Q), so using equation (<ref>) we get c^2Δu̅∈ L^2(Q), which by elliptic regularity implies u̅∈ H^2(Q).We can now apply Lemma <ref> on u̅ and conclude that u = 0 in a set of the form {(t,x)∈ Q: x^n<ϵ0<t<T-x^n}. This implies we have reduced the problem to the smaller domain [0,T-ϵ]×{x∈Ω: x^n>ϵ}. If ϵ=T we are done, otherwise we can apply again Lemma <ref> in the new domain. Iterating this process we conclude the result. ▪ There is a common condition appearing in the literature of Carleman estimates and inverse problems related to the wave equation with variable sound speed. It assumes the existence of some x_0∈^n for which(x-x_0)·∂_xc(x)<c(x) ∀ x∈^n.In geometric terms, (<ref>) says that the spheres with center at x_0 are strictly convex for the metric c^-2dx^2 <cit.>.Such collection of spheres can then be used to foliate the domain Ω and, as you will see in the next corollary (see also Figure <ref>), it allows us to prove unique continuation and consequently uniqueness for the inverse problem without the assumption of Ω and the level surfaces of the distance function, dist(·,∂Ω), being strictly convex. The price we pay by removing the convexity requirement on Ω is the lost of sharpness in the bound of T that guarantee uniqueness. Let Ω⊂^n be an open and bounded subset with ∂Ω smooth, and T>0. Weassume the sound speed c(x) satisfies condition (<ref>) and assume the constant c_0>0 is a lower bound for the sound speed. Let's denote R_Ω=max_r>0{|x-x_0|:x∈∂Ω}, r_Ω = {[ min_r>0{|x-x_0|:x∈∂Ω}, if x∈^n\Ω̅; 0, otherwise, ].and D_Ω = R_Ω-r_Ω.Assume Ω, T and c are as above, and as in the TAT problem, we assume P_Φ = ∂^2_t - Δ outside Ω. If u∈ H^2(Q) is a solution of (<ref>) such that u=∂_ν u=0 on (0,T)×∂Ω, thenu=0in{(t,x)∈ Q: 0<t<T-c^-1_0(R_Ω-|x-x_0|)}.As a consequence, in the thermoacoustic problem, if f∈ H_D(Ω) is such that Λ_Φ=0, with = [f,-af], then f=0in{x∈Ω: |x-x_0|>R_Ω-c_0T},and in particular, f≡ 0 when T≥ c^-1_0D_Ω.From <cit.>, the uniqueness time defined in Theorem <ref> satisfies T_0< c^-1_0D_Ω.Let's extend u to be zero outside Ω in the interval [0,T]. Due to the null Cauchy data, finite speed of propagation and the well-posedness of the exterior problem, u solves (<ref>) in the whole space. Notice that in particular, u=∂_ν u=0 on the Euclidean sphere {x∈^b:|x-x_0|=R_Ω}, for all t∈[0,T]. ∂_ν stands for a generic exterior normal derivative.We denote Σ_r = {x∈^b:|x-x_0|=r} the sphere of center x_0 and radius r, and we set r_0=max{0,c_0T-D_Ω}. By hypothesis, Σ_r with r∈[r_0,R_Ω] are strictly convex surfaces for the metric c^-2dx^2 that foliate Ω (see Figure <ref>(a)). For a given r∈[r_0,R_Ω], let's assume that u=∂_ν u=0on [0,T-c^-1_0(R_Ω -r)]×Σ_r.Since Σ_r is strictly convex we can apply Lemma <ref> with Ω replaced by B(x_0,r), the Euclidean ball of center x_0 and radius r, and deduce that u=0 in{(t,x)∈ (0,T)× B(x_0,r): dist(x,Σ_r)<ϵ,t<T-c^-1_0(R_Ω-r)-dist(x,Σ_r)},for some ϵ>0. Recalling that c_0 is a lower bound for c, we have that dist(x,Σ_r)<c^-1_0(r-|x-x_0|)∀ x∈ B(x_0,r), therefore we can find r_1∈(0,r) such that u vanishes in the smaller set{(t,x): r_1<|x-x_0|< r,0<t<T-c^-1_0(R_Ω-|x-x_0|)}.Moreover, u has null Cauchy data on Σ_r_1 for all t∈(0,T-c^-1_0(R_Ω-r_1)) (see Figure <ref>(b)).If we denote by s the infimum of the radius r≥ r_0 for which u has vanishing Cauchy data in (0,T-c^-1_0(R_Ω - r))×Σ_r, by the first paragraph and the previous argument we know s< R_Ω (since T>0). Moreover, if s>r_0, it must also satisfies the same property, this is, u=∂_ν u=0 in Σ_s for all t∈(0,T-c^-1_0(R_Ω - s)). Consequently, we can still apply the arguments in the paragraph above which leads us to conclude s=r_0.Let now f∈ℋ_D,a(Ω) be as in the hypothesis, and u solution of (<ref>). Analogously to the proof of Theorem <ref>, the function u̅ defined in (<ref>) satisfies a system of the form (<ref>) with null Cauchy data. The result then follows directly from the previous. § STABILITYThe stability with complete data follows directly from the analogous results for the damped and undamped case. Due to the microlocal nature of this property, the minimum time needed to recover f in a stable way is usually larger than the uniqueness time. Indeed, it's necessary to capture information coming from every singularity of the initial source. In a non-trapping domain, such lower bound is related to the valueT_1(Ω) = sup{|γ|_g: γ⊂Ω̅ geodesic for the metricg = c^-2dx^2},being 1/2T_1 when there is no damping coefficient and exactly T_1 for the damped case. Notice that T_1>2T_0 and in the case c satisfies (<ref>), T_1/2≤ (R_Ω-r_Ω)/(α c_0) with (see <cit.>)α = min_x∈Ω̅(1-c^-1(x-x_0)·∂_xc)>0.Let Ω be strictly convex for the metric g=c^-2dx^2. Assume that Ω and T are as in Theorem <ref>(or as in Theorem <ref>). In addition, assume T_1(Ω)<T<∞ if a≠ 0 and 1/2T_1(Ω)<T<∞ otherwise (resp. 2α^-1c^-1_0D_Ω<T<∞ and α^-1c^-1_0D_Ω<T<∞). Then there exists C>0 such thatf_H_D(Ω)≤ CΛ_Φ f_H^1((0,T)×∂Ω).The idea is to compare the observation operator Λ_Φ with its analogous for the undamped and damped case, Λ_0 and Λ_a respectively. These last two operators are known to be stable maps (see <cit.> and <cit.> respectively) and furthermore, from the results of the previous section, we know Λ_Φ is injective. The proof then reduces to show that the respective error operators are compact. We only show this for the case a≡ 0, the proof when there is a damping coefficient is obtained analogously.From <cit.> follows there is a constant C>0 such thatf_H_D≤ CΛ_0f_H^1≤ CΛ_Φ f_H^1 + C(Λ_Φ - Λ_0)f_H^1.Let's denote R = Λ_Φ - Λ_0 and u the attenuated wave related with Λ_Φ. Then, R maps f∈ H_D(Ω) to the boundary data w|_(0,T)×∂Ω of the system{[ (∂^2_t-c^2Δ+b)w = -Φ*u,(t,x)∈(0,T)×^n; w|_t=0=0,; w_t|_t=0=0. ].By finite propagation speed we can work in a larger domain Ω' such that w =u= 0 on its boundary and outside Ω'. Due to the higher regularity theorem in <cit.>, since F(t,x) = -[Φ*u](t,x) satisfies F,F_t∈ L^2((0,T);L^2(Ω')), we obtain that w∈ C((0,T);H^2(Ω')) and w_t∈ C((0,T);H^1(Ω')), and consequently the trace of w in ∂Ω belongs to H^3/2((0,T)×∂Ω), with the latter space compactly embedded in H^1((0,T)×∂Ω). The stability inequality is obtained by recalling the injectivity of Λ_Φ from Theorem <ref> (respectively Theorem <ref>) and applying the classical result <cit.>.§ RECONSTRUCTIONWe aim to construct a Neumann series that allow us to recover f in (<ref>) from boundary measurements as it has been done in <cit.> for the unattenuated case, and in <cit.> for the damped wave equation. However, due to the convolution term we need to modified the equation satisfied by the time reversed wave. Considering the same equation in the backward direction would imply the knowledge of the future. The strategy then is to solve a time reversal problem in such a way that the initial energy of the error function is bounded by the total energy (kinetic, potential and energy lost by attenuation) of the forward wave, inside the domain and at time T, analogously as the argument presented in <cit.>. Such total energy in the whole space has the attribute of being conserved in time, fact that allows us to reduce the proof to an estimate involving the norm of the initial source and the energy of the forward wave outside Ω (see Proposition <ref>). The estimate says that at time T a significant portion of the energy lies outside the domain. It was first used in <cit.> and subsequently applied in <cit.>.Let's introduce the following convolution-type operator[Φ*̃v](s,x) = ∫^T_sΦ(t-s,x)v(t,x)dt,which is the adjoint operator of Φ*(·) under the L^2 inner product in (0,T), this is, for any L^2-functions u,v, ⟨Φ*u,v⟩_L^2(0,T) = ⟨ u,Φ*̃v⟩_L^2(0,T).Indeed, denoting by χ_I the indicator function in the interval I⊂,∫^T_0[Φ*u](t)v(t)dt= ∫∫χ(t)_[0,T]χ(s)_[0,t]Φ(t-s)u(s)v(t)dsdt= ∫∫χ(s)_[0,T]χ(t)_[s,T]Φ(t-s)u(s)v(t)dsdt = ∫^T_0[Φ*̃v](s)u(s)ds. Following the same approach than the latest results in reconstruction for TAT in the enclosure case as well as in the attenuated case for the damped wave equation, the idea is to consider the right back projection system that will make the error operator to be a contraction.In the same way as in the proof of uniqueness, instead of working with u with setu̅(t,x) = ∫^t_0u(s,x)ds,and Ψ(t,x) as in (<ref>).Then, they satisfy {[ ∂_t^2u̅- c^2Δu̅ +a∂_tu̅ +pu̅+ Ψ*∂_tu̅=0 in(0,T)×^n,; u̅|_t=0=0 in ^n;∂_tu̅|_t=0=f in ^n ].with p(x) = b(x) - Ψ(x,0)≥ 0. Notice we do not use (<ref>) to obtain an equation as in (<ref>) and we keep a derivative inside the convolution. If Λ̅_Ψ:L^2(Ω;c^-2dx)→ H^1((0,T)×∂Ω) denotes the observation operator for this problem, this is Λ̅_Ψ f = u̅|_(0,T)×∂Ω, by well-posedness of the direct problem we have the following relation, Λ̅_Ψ f = ∫^t_0[Λ_Φ f](t)dt,∀ f∈ H_D(Ω).For the data h̅ = Λ̅_Ψ f, we consider the solution v of the system{[ (∂^2_t-c^2Δ -a∂_t+ p - Ψ*̃∂_t)v =0in (0,T)×Ω,;v|_t=T =ϕ,;v_t|_t=T =0,; v|_(0,T)×∂Ω = h̅, ].with ϕ the harmonic extension of h̅(T,·) in Ω. Notice that problem (<ref>) is well-posed. This is due to the convolution term that involves values of v in the interval (t,T), thus by doing the change of variables t→ T-t we get an IBVP of the form (<ref>) which is uniquely solvable. We define the Time Reversal operator by A:H^1_(0)([0,T]×∂Ω)→ L^2(Ω;c^-2dx), Ah = v_t(0,·), and denote by K the error operator defined as follows,K:L^2(Ω;c^-2dx)→ L^2(Ω;c^-2dx), Kf = w_t(0,·),with w = u̅-v, the error function that solves problem (<ref>). In what follows we suppose the domain Ω is non-trapping (i.e. T_0(Ω)<∞). The main result of this section is the nextLet Ω be strictly convex for the metric g=c^-2dx^2. Assume that Ω and T are as in Theorem <ref>(or as in Theorem <ref>). In addition, assume T_1(Ω)<T<∞ if a≠ 0 and 1/2T_1(Ω)<T<∞ otherwise (resp. 2α^-1c^-1_0D_Ω<T<∞ and α^-1c^-1_0D_Ω<T<∞, with α as in (<ref>)). Then AΛ̅_Ψ = Id-K with K_ℒ(L^2(Ω;c^-2dx))<1, and for any initial condition of (<ref>) of the form =(f,-af) with f∈ H_D(Ω), the thermoacoustic inverse problem has a reconstruction formula given byf=∑^∞_m=0K^mAh̅,h̅ = Λ̅_Ψ f.Notice the error function w = u̅-v satisfies the equation{[ (∂_t^2 - c^-2Δ + p) w = -au̅_t - av_t-Ψ*∂_tu̅- Ψ*̃∂_tvin (0,T)×Ω,;w|_t=T = u̅^T - ϕ,;w_t|_t=T = u̅^T_t,;w|_(0,T)×Γ =0. ].with u̅^T = u̅(T,·) and ∂_tu̅^T = ∂_tu̅(T,·). Moreover, we can write Kf = f - Ah̅ = w_t(0),withh̅ = Λ̅_Ψ f.We want to estimate the norm of Kf, hence we need to compute the energy of w. Multiplying (<ref>) by 2c^-2w_t and integrating over (0,T)×Ω we obtainE_Ω(w,0)= E_Ω(w,T) +2∫_[0,T]×Ωac^-2u̅_tw_tdxdt + 2∫_[0,T]×Ωac^-2v_tw_tdxdt+ 2∫_[0,T]×Ωc^-2(Ψ*∂_tu̅)w_tdxdt +2∫_[0,T]×Ωc^-2(Ψ*̃∂_tv)w_tdxdt= E_Ω(w,T) +2∫_[0,T]×Ωac^-2|u̅_t|^2dxdt - 2∫_[0,T]×Ωac^-2|v_t|^2dxdt+ 2∫_[0,T]×Ωc^-2(Ψ*∂_tu̅)∂_tu̅dxdt -2∫_[0,T]×Ωc^-2(Ψ*̃∂_tv)∂_tvdxdt -2∫_[0,T]×Ωc^-2(Ψ*∂_tu̅)∂_tvdxdt + 2∫_[0,T]×Ωc^-2(Ψ*̃∂_tv)∂_tu̅dxdt.Neglecting the integration in the spatial variable in the last two terms for a moment, we can use the identity (<ref>) which makes them cancel each other out. Furthermore, it follows from the same identity and Condition (<ref>) on the kernels (which guarantees positive-definiteness) that∫_[0,T]×Ωc^-2(Ψ*̃∂_tv)∂_tvdxdt = ∫_[0,T]×Ωc^-2(Ψ*∂_tv)∂_tvdxdt≥ 0.In consequence we getE_Ω(w,0) ≤ E_Ω(w,T)+2∫_[0,T]×Ωac^-2|u̅_t|^2dxdt+ 2∫_[0,T]×Ωc^-2(Ψ*∂_tu̅)∂_tu̅dxdt.The choice of the time reversal system (<ref>) helps to minimize the total energy in the dynamic satisfied by the error function w in a similar way as the functions ϕ helps to minimize the energy of w at time T. Indeed, by integration by parts we have that(u̅^T-ϕ,ϕ)_H_D(Ω) = -∫_Ω(u̅^T - ϕ)Δϕ dx + ∫_∂Ω(u̅^T-ϕ)∂_νϕ dS = 0,thereforeE_Ω(w(T)) = u̅^T-ϕ^2_H_D(Ω)+u̅^T_t^2_L^2(Ω) = E_Ω(u̅(T)) - ϕ_H_D(Ω)^2.From the above relations (<ref>) and (<ref>), we deduceKf^2_L^2(Ω;c^-2dx)≤ E_Ω(w,0)≤ℰ_Ω(u̅,T),where recall the term in the right hand side is the extended energy functional associated to (<ref>) and defined in (<ref>). By conservation of the extended energy in ^n, ℰ_Ω(u̅,T) = ℰ_^n(u̅,T) - E_Ω^c(u̅,T) = f^2_L^2(c^-2dx) - E_Ω^c(u̅,T).The conclusion of the theorem follows from the next proposition which is known to hold when there is no integral term. There is C>0 so that for all f∈ L^2(Ω;c^-2dx) and u̅ solutions of (<ref>),f^2_L^2(Ω;c^-2dx)≤ CE_Ω^c(u̅,T). An inequality of this form was first proved in <cit.> (see (5.15) in the same article) for the case of the unattenuated wave equation, and later extended to the damped case in <cit.>, requiring a larger lower bound for the measurement time though. Such estimate is obtained by microlocalizing near the singularities and studying how their energy is transmitted across the boundary provided they hit the boundary in a transversal way. By considering strictly convex domains we can be sure that all singularities meet that requirement. When there is no damping coefficient the analysis of the singularities can be decoupled to those following the positive sound speed and negative sound speed. The time needed then for the estimate to hold equals the time needed to get at least one signal from each singularity of the initial condition, this is T>1/2T_1(Ω). In contrast, the appearance of a damping term makes no longer possible such microlocal decoupling, and therefore it makes necessary to wait until both signals, issued from every singularity of the initial condition, reach the boundary, or in other words T>T_1(Ω).Let's prove the above proposition. Denote by U̅(x,t) the solution of the damped wave equation {[ (∂^2_t+ a∂_t- c^2Δ + b)U̅(t,x) = 0,(t,x)∈(0,T)×^n;U̅|_t=0=0,;U̅_t|_t=0=f. ].Denoting = [0,f]∈ℋ(Ω), from the paragraph above follows there is C>0 so thatf^2_L^2(Ω;c^-2dx) = ^2_ℋ(Ω)≤ CE_Ω^c(U,T).Furthermore, defining W̅ = U̅-u̅ we obtainf^2_L^2(Ω;c^-2dx) ≤ C(E_Ω^c(u̅,T)+E_Ω^c(W̅,T))and letting (t) = [u̅(t),u̅_t(t)], W̅(t) = [W̅(t),W̅_t(t)], the previous inequality impliesf_L^2(Ω;c^-2dx)≤ C(T)_H^1(Ω^c)⊗ L^2(Ω^c) + CW̅(T)_H^1(Ω^c)⊗ L^2(Ω^c),where the error function W̅ satisfies the IVP{[ (∂^2_t+a∂_t-c^2Δ+b)W̅ = Ψ*∂_tu̅, (t,x)∈(0,T)×^n; W̅|_t=0=0,; W̅_t|_t=0=0. ].We claim the bounded map L^2(Ω;c^-2dx)∋ f↦(T)∈ H^1(Ω^c)⊗ L^2(Ω^c) is injective. In fact, it can be decomposed as the composition of two injective bounded maps, the first one being the observation operator Λ̅_Ψ, which is injective since (<ref>) is equivalent (following the computation in (<ref>)) to a system of the form (<ref>) where the method used to prove Theorem <ref> (resp. Theorem <ref>) can be applied, and our choice of T>1/2T_1≥ T_0 (resp. T>α^-1c^-1_0R_Ω≥1/2T_1). The second map is the exterior IBVP map that takes Dirichlet boundary data h̅∈ H^1_(0)([0,T]×∂Ω) to (T)∈ H^1(Ω^c)⊗ L^2(Ω^c), where v̅ solves:{[ (∂^2_t - c^2Δ) v̅(t,x) = 0,(t,x)∈(0,T)×^n∖Ω;v̅|_t=0=0,; ∂_tv̅|_t=0=0,;v̅|_[0,T]×∂Ω = h̅. ].To see the injectivity of the latter map, consider h̅∈ H^1_(0)([0,T]×∂Ω) such that v̅(T) = v̅_t(T) = 0, with v̅ solution of (<ref>). By domain of dependence and reversibility in time of the exterior problem, we have that v̅ vanishes in {(t,x)∈ (0,∞)×^n\Ω̅: dist_e(x,∂Ω)>t} and also in {(t,x)∈ (0,∞)×^n\Ω̅: dist_e(x,∂Ω)>|T-t|}. Therefore v̅=0in{(t,x)∈ (0,3T/2)×^n\Ω̅: dist_e(x,∂Ω)>T/2}. Applying Tataru's unique continuation theorem on any p∈{dist_e(x,∂Ω)>T/2}, we deduce that v̅ = 0 in (^n\Ω̅)∩{(t,x)∈ (0,∞)×^n : |x-p| + |t-3T/4|<3T/4},which implies that h̅ vanishes for t∈(T/2,T) (see Figure <ref>). We can now apply the same argument replacing T by T/2 and get that h̅ is null in the interval (T/4,T/2). Iterating this process we finally conclude that h̅ = 0 for all t∈(0,T). Our second claim is that the map L^2(Ω;c^-2dx)∋ f↦(T)∈ H^1(Ω^c)⊗ L^2(Ω^c) is compact. It is in fact a composition of the bounded maps L^2(Ω;c^-2dx)∋ f↦u̅_t∈ L^2((0,T);L^2(^n)), u̅_t↦(T)∈ H^2(Ω^c)⊗ H^1(Ω^c) and the compact embedding H^2(Ω^c)⊗ H^1(Ω^c)↪ H^1(Ω^c)⊗ L^2(Ω^c). The continuity of the second map for those Sobolev spaces is due to <cit.> since denoting F := Ψ*u̅_t, then F,F_t∈ L^2((0,T);L^2(Ω^c)).It follows from <cit.> that for a different constantf_L^2(Ω;c^-2dx)≤ C(T)_H^1(Ω^c)⊗ L^2(Ω^c).The proposition is then proved by recalling the finite speed of propagation and applying Poincare's inequality on a large ball minus Ω. ▪ We conclude the proof of Theorem <ref> by joining (<ref>), (<ref>) andProposition <ref>, hence for some C>1,Kf^2_L^2(Ω;c^-2dx) ≤f^2_L^2(Ω;c^-2dx) - E_Ω^c(u,T)≤f^2_L^2(Ω;c^-2dx) - C^-2f^2_L^2(Ω;c^-2dx)≤ (1-C^-2)f^2_L^2(Ω;c^-2dx). § APPENDIX§.§ Well-posedness of the direct problemFor the existence of solutions we follows the proof of <cit.>. Let's assume without lost of generality that u_0 = 0. For a fixed t_0∈ (0,T] letℰ_t_0 = {v(t)|v(t)∈ C^∞([0,t_0];H^1_0(U)), v(0) = 0},with two inner product given by(v,w)_1:= ∫^t_0_0{(v_t(t),w_t(t)) ⟩ + ( ∇ v(t),∇ w(t))}dtand (v,w)_2:= (v,w)_1 + t_0 (v_t(0), w_t(0)),and respective norms ·_1 and ·_2. Let F_t_0 be the completion of ℰ_t_0 under the norm ·_1. It can be proved, for instance by Stone-Weierstrass, that u∈ F_t_0 is a generalized solution in the interval [0,t_0] if and only ifℬ(u,v) = 𝒟(f,v) + t_0(c^-2u_1,v_t(0))_L^2(U),∀ v∈ℰ_t_0,whereℬ(u,v)= ∫^t_0_0(t-t_0)[ (c^-2u_t(t),v_tt(t)) - (∇ u(t),∇ v(t))- (c^-2au_t(t),v_t(t))- (c^-2bu(t),v_t(t)) - ∫^t_0(c^-2Φ(t-τ)u_τ(τ),u_t(t))dτ]dt+ ∫^t_0_0(c^-2u_t(t),v_t(t))dt, 𝒟(f,v)= -∫^t_0_0(t-t_0)(c^-2f(t),v_t(t))dt.where (<ref>) is obtained by using the test function (t-t_0)v_t(t) with v∈ℰ_t_0, in (<ref>). Notice that applying integration by parts we get that the bilinear form ℬ satisfies that for all v∈ℰ_t_0 (recall v(0) = 0),ℬ(v,v) = 1/2∫^t_0_0[(c^-2v_t(t),v_t(t)) + (∇ v(t),∇ v(t)) + (c^-2bv(t),v(t))]dt - ∫^t_0_0(t-t_0)[ (c^-2av_t(t),v_t(t)) - (c^-2Φ(0)v(t),v(t))-∫^t_0(c^-2Φ(t-s)v(s),v(t))ds ]dt+ t_0/2(c^-2v_t(0),v_t(0)).Therefore, recalling that 0<c_0≤ c≤ c^-1_0, we bound from bellow and choosing t_0>0 small enough and using Poincare's inequality we getℬ(v,v) ≥ 1/2min{1,c^2_0}v^2_2 - Ct_0([a_∞ + Φ(0)_∞ + t_0Φ_∞)v^2_1≥δv^2_2.for some δ>0. On the other hand, |𝒟(f,v)|≤ t_0c^-2_0f_L^2v_2 |t_0(c^-2u_1,v_t(0))| ≤ t_0^1/2c^-2_0u_1_L^2v_2Then, similarly as in <cit.>, we get the existence of weak solutions on the interval [0,t_0]. Iterating this argument for the intervals [t_0,2t_0], [2t_0,3t_0] etc, we conclude the existence on [0,T]. The uniqueness follows the same ideas as in <cit.>. plain | http://arxiv.org/abs/1703.09271v1 | {
"authors": [
"Sebastian Acosta",
"Benjamin Palacios"
],
"categories": [
"math.AP"
],
"primary_category": "math.AP",
"published": "20170327190358",
"title": "Thermoacoustic tomography for an integro-differential wave equation modeling attenuation"
} |
Fluid Communities: A Competitive, Scalable and Diverse Community Detection Algorithm Fluid Communities Parés F., Garcia-Gasulla D. et al. Ferran ParésBarcelona Supercomputing Center (BSC), Barcelona, Spain, [email protected] Dario Garcia-GasullaBarcelona Supercomputing Center (BSC), Barcelona, Spain,[email protected] Armand Vilalta Barcelona Supercomputing Center (BSC), Barcelona, Spain Jonatan Moreno Barcelona Supercomputing Center (BSC), Barcelona, Spain Eduard Ayguadé Barcelona Supercomputing Center (BSC) & UPC - BarcelonaTECH, Barcelona, Spain Jesús Labarta Barcelona Supercomputing Center (BSC) & UPC - BarcelonaTECH, Barcelona, Spain Ulises Cortés Barcelona Supercomputing Center (BSC) & UPC - BarcelonaTECH, Barcelona, Spain Toyotaro Suzumura Barcelona Supercomputing Center (BSC) & IBM T.J. Watson, New York, USA * Ferran ParésBoth authors contributed equally to this work, Dario Garcia-Gasulla, Armand Vilalta, Jonatan Moreno, Eduard Ayguadé, Jesús Labarta, Ulises Cortés and Toyotaro Suzumura December 30, 2023 =======================================================================================================================================================================================*We introduce a community detection algorithm (Fluid Communities) based on the idea of fluids interacting in an environment, expanding and contracting as a result of that interaction. Fluid Communities is based on the propagation methodology, which represents the state-of-the-art in terms of computational cost and scalability. While being highly efficient, Fluid Communities is able to find communities in synthetic graphs with an accuracy close to the current best alternatives. Additionally, Fluid Communities is the first propagation-based algorithm capable of identifying a variable number of communities in network. To illustrate the relevance of the algorithm, we evaluate the diversity of the communities found by Fluid Communities, and find them to be significantly different from the ones found by alternative methods.We introduce a community detection algorithm (Fluid Communities) based on the idea of fluids interacting in an environment, expanding and contracting as a result of that interaction. Fluid Communities is based on the propagation methodology, which represents the state-of-the-art in terms of computational cost and scalability. While being highly efficient, Fluid Communities is able to find communities in synthetic graphs with an accuracy close to the current best alternatives. Additionally, Fluid Communities is the first propagation-based algorithm capable of identifying a variable number of communities in network. To illustrate the relevance of the algorithm, we evaluate the diversity of the communities found by Fluid Communities, and find them to be significantly different from the ones found by alternative methods. § INTRODUCTIONCommunity detection (CD) extracts structural information of a network unsupervisedly. Communities are typically defined by sets of vertices densely interconnected which are sparsely connected with the rest of the vertices from the graph. Finding communities within a graph helps unveil the internal organization of a graph, and can also be used to characterize the entities that compose it (groups of people with shared interests, products with common properties, ).One of the first CD algorithms proposed in the literature is the Label Propagation Algorithm (LPA) <cit.>. Although other CD algorithms have been shown to outperform it, LPA remains relevant due to its scalability (with linear computational complexity O(E)) and yet competitive results <cit.>. In this paper we propose a novel CD algorithm: the Fluid Communities (FluidC) algorithm, which also implements the efficient propagation methodology. This algorithm mimics the behaviour of several fluids (communities) expanding and pushing one another in a shared, closed and non-homogeneous environment (a graph), until equilibrium is found. By initializing a different number of fluids in the environment, FluidC can find any number of communities in a graph. To the best of our knowledge, FluidC is the first propagation-based algorithm with this property, which allows the algorithm to provide insights into the graph structure at different levels of granularity.§ RELATED WORKThe most recent evaluation and comparison of CD algorithms was made by <cit.>, where the following eight algorithms were compared in terms of Normalized Mutual Information (NMI) and computing time: Edge Betweenness <cit.>, Fast greedy <cit.>, Infomap <cit.>, Label Propagation <cit.>, Leading Eigenvector <cit.>, Multilevel (Louvain) <cit.>, Spinglass <cit.> and Walktrap <cit.>. The performance of these eight algorithms was measured on artificially generated graphs provided by the LFR benchmark <cit.>, which defines a more realistic setting than the alternative GN benchmark <cit.>, including scale-free degree and cluster size distributions. One of the main conclusions of this study is that the Multilevel algorithm is the most competitive overall in terms of CD quality.A similar comparison of CD algorithms was previously reported by <cit.>. In this work twelve algorithms were considered, some of them also present in the study of <cit.> (Edge Betweenness, Fastgreedy, Multilevel and Infomap). In this study, the algorithms were compared under the GN benchmark, the LFR benchmark, and on random graphs. In their summary, authors recommend using various algorithms when studying the community structure of a graph for obtaining algorithm-independent information, and suggest Infomap, Multilevel and the Multiresolution algorithm <cit.> as the best candidates. Results from both <cit.> and <cit.> indicate that the fastest CD algorithm is the well-known LPA algorithm, due to the efficiency and scalability of the propagation methodology.§ FLUID COMMUNITIES ALGORITHM The Fluid Communities (FluidC) algorithm is a CD algorithm based on the idea of introducing a number of fluids (communities) within a non-homogeneous environment (a non-complete graph), where fluids will expand and push each other influenced by the topology of the environment until a stable state is reached.Given a graph G=(V,E) composed by a set of vertices V and a set of edges E, FluidC initializes k fluid communities 𝒞 = {c_1..c_k}, where 0<k≤|V|. Each community c∈𝒞 is initialized in a different and random vertex v ∈ V. Each initialized community has an associated density d within the range (0,1]. The density of a community is the inverse of the number of vertices composing said community: d(c) = 1/ | v ∈ c| Notice that a fluid community composed by a single vertex (every community at initialization) has the maximum possible density (d=1.0). FluidC operates through supersteps. On each superstep, the algorithm iterates over all vertices of V in random order, updating the community each vertex belongs to using an update rule. When the assignment of vertices to communities does not change on two consecutive supersteps, the algorithm has converged and ends.The update rule for a specific vertex v returns the community or communities with maximum aggregated density within the ego network of v. The update rule is formally defined in Equations <ref> and <ref>. 𝒞_v^' = argmax_c ∈𝒞∑_ w ∈{v, Γ(v)} d(c) ×δ(c(w),c)δ(c(w),c) = {[1,if c(w) = c;0,if c(w) ≠ c ]. where v is the vertex being updated, 𝒞^'_v is the set of candidates to be the new community of v, Γ(v) are the neighbours of v, d(c) is the density of community c, c(w) is the community vertex w belongs to and δ(c(w),c) is the Kronecker delta. Notice that 𝒞_v^' could contain several community candidates, each of them having equal maximum sum. If 𝒞_v^' contains the current community of the vertex v, v does not change its community. However, if 𝒞_v^' does not contain the current community of v, the update rule chooses a random community within 𝒞_v^' as the new community of v. This completes the formalization of the update rule: c^'(v) = {[ x ∼𝒰(𝒞^'_v) ,if c(v) ∉𝒞^'_v;c(v) ,if c(v) ∈𝒞^'_v; ].where c^'(v) is the community of vertex v at the next superstep, 𝒞_v^' is the set of candidate communities from equation <ref> and x ∼𝒰(𝒞_v^') is the random sampling from a discrete uniform distribution of the 𝒞_v^' set.Equation <ref> guarantees that no community will ever be eliminated from the graph since, when a community c is compressed into a single vertex v, c has the maximum possible density on the update rule of v (1.0) guaranteeing c∈𝒞_v^', and thus c^'(v)=c. An example of FluidC algorithm behaviour is shown in Figure <ref>.§.§ PropertiesFluidC is asynchronous, where each vertex update is computed using the latest partial state of the graph (some vertices may have updated their label in the current superstep and some may not). A straight-forward synchronous version of FluidC (one where all vertex update rules are computed using the final state of the previous superstep) would not guarantee that densities are consistent with Equation <ref> at all times (a community may lose a vertex but its density may not be increased immediately in accordance). Consequently, a community could lose all its vertices and be removed from the graph.FluidC allows for the definition of the number of communities to be found, simply by initializing a different number of fluids in the graph. This is a desirable property for data analytics, as it enables the study of the graph and its entities at several levels of granularity. Although a few CD algorithms already had this feature (Walktrap, Fastgreedy), none of those were based on the efficient propagation method. Another interesting feature of FluidC is that it avoids the creation of monster communities in a non-parametric manner. Due to the spread of density among the vertices that compose a community, a large community (when compared to the rest of communities in the graph) will only be able to keep its size and expand by having a favourable topology (having lots of intra-community edges which make up for its lower density). Figure <ref> shows two cases of this behaviour, one where a large community is able to defend against external attack, and one where it is not.FluidC is designed for connected, undirected, unweighted graphs, and variants of FluidC for directed and/or weighted graphs remain as future work. However, FluidC can be easily applied to a disconnected graph G^' just by performing an independent execution of FluidC on each connected component of G^' and appending the results.§ EVALUATION To evaluate performance we use the LFR benchmark <cit.>, measuring NMI obtained on a set of graphs with six different graph sizes (|V|= 233, 482, 1000, 3583, 8916 and 22186) and 25 different mixing parameters (μ from 0.03 to 0.75). The mixing parameter is the average fraction of vertex edges which connect to vertices from other communities <cit.>. To guarantee consistency, 20 different graphs were generated for each combination of graph size and mixing parameter. This results in a total of 3,000 different graphs (6 graph sizes × 25 mixing parameter values × 20 graphs), and is the same evaluation strategy used in <cit.>. Besides the graph size and mixing parameter, the LFR benchmark also requires a list of hyperparameters to generate a graph. We use the same ones defined in <cit.>; maximum degree 0.1×|V|, maximum community size 0.1×|V|, average degree 20, degree distribution exponent -2 and community size distribution exponent -1.These experiments were performed on the five most competitive CD algorithms evaluated in <cit.> (Fastgreedy, Infomap, Label Propagation, Multilevel and Walktrap) and FluidC. We report the results in Figure <ref>, where each panel contains two plots. The bottom one shows average performance in NMI on the 20 graphs built for the various values of μ (shown on the horizontal axis), while the top one shows the corresponding standard deviation (Std). Each plot line represents the results of an algorithm on a different graph size (see panel legend). Among all normalization variants of the Mutual Information metric we use the geometric normalization, dividing by square root of both entropies. Before analyzing the results, let us clarify two aspects of the evaluation process. First, the LFR benchmark may generate disjoint graphs. When this is the case, an independent execution of FluidC is computed on each connected subgraph separately, and the communities found on the different subgraphs are appended to measure the overall NMI. And second, FluidC requires to specify the number of communities to be found k, which is an unknown parameter. For comparability reasons, we report the results obtained using the k resulting in highest modularity. This is analogous to what other algorithms which also require k do (Fastgreedy and Walktrap).In our experiments, Multilevel achieved top NMI results on most generated graphs, while Fastgreedy and LPA were clearly inferior to the rest of algorithms. The remaining three algorithms, Walktrap, Infomap and FluidC, were competitive, and had results close to Multilevel. In the context of Walktrap and Infomap, FluidC has a rather particular behaviour. It is better on large graph sizes than Walktrap; for the largest graph computed (|V|=22,186), FluidC outperforms Walktrap for all μ values in the range [0.33,0.66]. FluidC is also more resistant to large mixing parameters than Infomap, which is unable to detect relevant communities for μ>0.55. The performance of FluidC is slightly sub-optimal (NMI between 0.9 and 0.95) on low mixing parameters (μ≤0.4). This is because communities generated in a graph with low mixing parameters are very densely connected, and only have a few edges connecting them with other communities, edges that act as bottlenecks. These bottlenecks can sometimes prevent the proper flow of communities in FluidC, which leads to sub-optimal results. In practice, the sub-optimal performance of FluidC on graphs with very low mixing parameter is a minor inconvenience. Real world graphs are often large and have relatively high mixing parameters, a setting where FluidC is particularly competent. In contrast, the recommended algorithm for processing graphs with low mixing parameters would be LPA, as it finds the optimal result in these cases, and it is faster and scales better than the alternatives. § SCALABILITYThe main purpose of the FluidC algorithm is to provide high quality communities in a scalable manner, so that good quality communities can also be obtained from large scale graphs. In the previous section we saw how the performance of FluidC in terms of NMI is comparable to the best algorithms in the state-of-the-art (Multilevel, Walktrap and Infomap). Next we evaluate FluidC scalability, to show its relevance in the context of large networks.To analyze the computational cost of FluidC we first compare the cost of one full superstep (checking and updating the communities of all vertices in the graph) with that of LPA. LPA is the fastest and more scalable algorithm in the state-of-the-art <cit.>, which is why we use it as baseline for scalability along this section. Figure <ref> shows the average time per iteration, using the same type of plots used in the NMI evaluation. Results indicate that the computing time per iteration of FluidC is virtually identical to that of LPA for all graph sizes and mixing parameters. Significantly, both algorithms are almost unaffected by a varying mixing parameter.Beyond the cost of a single superstep, we also explore the total number of supersteps needed for the algorithm to converge. Figure <ref> shows that information for both FluidC and LPA. For this experiment we set the FluidC parameter k to the ground truth. This comparison is relevant for mixing parameters up to 0.5. Beyond that value, LPA produces a single monster community after three supersteps (NMI = 0.0, see Figure <ref>, Panel e). Nevertheless, the number of supersteps needed by FluidC to converge is never above 13. These results indicate that FluidC and LPA are similar both in time per superstep and number of supersteps, which implies that both algorithms are analogous in terms of computational cost (with linear complexity O(E)) and scalability. To provide further evidence in that regard, and to also evaluate the scalability of the most relevant alternatives, next we consider the evaluation of larger graphs. We generate graphs with 60,000 and 150,000 vertices following the same methodology described in <ref>, and measure the computing times of LPA, FluidC, Multilevel and Walktrap (the three fastest algorithms according to <cit.> plus FluidC). Figure <ref> shows the scalability of each algorithm, where the continuous lines in the background correspond to different mixing parameters (from 0.03 to 0.75), and the big dashed line with markers indicates the computed mean over all the 25 mixing parameters. According to the results shown in Figure <ref>, LPA has the lowest computing time, closely followed by FluidC. However, LPA results are mistakenly optimistic, since the algorithm is particularly fast for large mixing parameters, where it obtains zero NMI after doing only three supersteps (see Panel e of Figure <ref>, and Figure <ref>). If this is taken into account, LPA and FluidC have an analogous scalability.Walktrap is considerably slower than the rest, and results for graphs larger than 3,000 vertices are not shown. Multilevel is 5x slower than LPA/FluidC, and its cost grows faster. While the slope of LPA/FluidC computed through a linear regression is roughly 10^-6, the slope of Multilevel is close to 10^-5. Processing a large scale graph like PageGraph <cit.> (|V|=3,500M vertices) would take approximately 9 hours for FluidC while more than two days (approximately 49 hours) for Multilevel.§ DIVERSITY OF COMMUNITIES Synthetic graphs generated by benchmarks like LFR can be used to evaluate the ability of algorithms at finding the community structures pre-defined by those benchmarks. This kind of results are useful for understanding the strengths and weaknesses of algorithms certain structural properties (μ and graph size). However, graphs obtained from real world data will rarely contain a single community structure, as complex data can be typically sorted in several coherent but unrelated ways (group products by sell volume or by sell dates). For this reason it is recommended to use several CD algorithms when analyzing a given graph, to obtain a variety of algorithm independent information <cit.>. In this context, the relevance of a CD algorithm is also affected by how diverse are the communities it is capable of finding, in comparison with the communities that the rest of the algorithms find.To evaluate the relevance of FluidC in terms of diversity, we execute the six previously evaluated algorithms on a set of graphs with multiple ground truths. If two algorithms consistently find different ground truths from the ones available within multi-ground truth graphs, it can be argued that both may provide different insights into the community structure of a graph. We build a total of 30 multi-ground truth graph, each of them containing two independent ground truths, each ground truth composed by four communities. Our multi-ground truth graphs are obtained by first generating two independent graphs (of size |V|=22,186) with four disconnected communities (μ=0) and then appending the edges of both. The rest of the LFR parameters used are the same ones used for evaluation at <ref>, except for minimum and maximum community size which are 0.2|V| and 0.3|V| respectively.Given a multi-ground truth graph G with two ground truths T_1,T_2, we categorize a CD algorithm 𝒜 in one of three values of θ; the algorithm finds T_1 over T_2 (θ=1), the algorithm finds T_2 over T_1 (θ=-1), the algorithm finds T_1 and T_2 to the same degree (θ=0). 𝒜 finds ground truth T_x over ground truth T_y when NMI(𝒜,T_x) - NMI(𝒜,T_y) > α, where α is a predefined threshold.After one hundred executions, the behaviour of an algorithm a multi-ground truth graph is represented by one hundred θ values. A similarity between the behaviour of two algorithms on the same graph can be computed by applying the Chi-square test on the two corresponding series of θ values, resulting in a probability of both series having the same categorical distribution. Qualitatively, that probability refers to how likely both algorithms are of having the same behaviour. We built 30 similarity matrices, each one corresponding to a multi-ground truth graph. For visualization purposes, we average them in Figure <ref>.The resultant similarity matrix provides qualitative insight on the diversity of the algorithms under consideration. To validate the consistency of the approach the α threshold, we report the similarity matrix corresponding to setting α=0.5 and α=0. Since the NMI score is in the range [0,1], α=0.5 implies that an algorithm finds one ground truth over the alternative at least by doubling its NMI score. On the other hand, the more permissive α=0 considers that an algorithm finds one ground truth over the alternative just by obtaining a larger NMI. In this case, θ=0 values are rare, as that would require both NMI to be identical. Additional values of α within the range [0,0.5] were analyzed, and the corresponding similarity matrices where found to be consistent with the two reported ones.The similarity matrices of Figure <ref> indicate that Multilevel, Walktrap and Fastgreedy have similar behaviours on multi-ground truth graphs, as all three algorithms result in very similar θ values. Significantly, this relation is stronger when α=0, which indicates that all three algorithms consistently prioritize the same ground truth over the alternative. Notice all three algorithms use hierarchical bottom-up agglomeration, while Fastgreedy and Multilevel are modularity-based methods <cit.>.The remaining three algorithms, Infomap, LPA and FluidC, show a much more diverse behaviour. The pattern with which each of these algorithms finds one ground truth over the alternative differs significantly from the other five. These results indicate that the proposed FluidC algorithm provides significantly diverse communities when compared to the algorithmic alternatives. In this context, FluidC may contribute to provide algorithm-independent information by being computed alongside Multilevel and Infomap on the same graph, all of which have a competitive performance according to the LFR benchmark as shown in <ref>.§ REPRODUCIBILITYAll the experiments presented in this paper have been computed on a computer with OpenSUSE Leap 42.2 OS (64-bits), with an Intel(R) Core(TM) i7-5600U CPU @ 2.60GHz and 16GB of DDR3 SDRAM. An open source implementation of the FluidC algorithm has been made publicly available to the community at Github (<github.com/FerranPares/Fluid-Communities>). It has been integrated into the(<github.com/networkx>) and(<github.com/igraph>) graph libraries. For consistency, all scalability experiments were performed using thegraph library.§ CONCLUSIONSIn this paper we propose a novel CD algorithm called Fluid Communities (FluidC). Through the well established LFR benchmark we demonstrate that FluidC identifies high quality communities (measured in NMI, see Figure <ref>), getting close to the current best alternatives in the state-of-the-art (Multilevel, Walktrap and Infomap). In this context, FluidC is particularly competent on large graphs and on graphs with large mixing parameters. The main limitation of FluidC in terms of NMI performance is that it does not fully recover the ground truth communities on graphs with small mixing parameters due to the effect of bottleneck edges. However, at larger mixing parameters (a more realistic environment) FluidC gets competitive to state-of-the-art algorithms. Although FluidC does not clearly outperform the current state-of-the-art in terms of NMI on the LFR benchmark, the importance of the contribution can be summarized both in terms of scalability and diversity.In terms of scalability, FluidC, together with LPA, represents the state-of-the-art in CD algorithms. Both belong to the fastest and most scalable family of algorithms in the literature with a linear computational complexity of O(E). However, while the performance of LPA rapidly degrades for large mixing parameters, FluidC is able to produce relevant communities at all mixing parameters. The next algorithm in terms of scalability is the Multilevel algorithm, which takes roughly 5x more seconds to compute, and which scales slightly worse (see Figure <ref> and the mentioned slopes). Thus, we consider FluidC to be highly recommendable for computing graphs of arbitrary large size.In terms of diversity, FluidC is the first propagation-based algorithm to report competitive results on the LFR benchmark, and also the first propagation-based algorithm which can find a variable number of communities on a given graph. Providing coherent and diverse communities is particularly important for unsupervised learning tasks, such as CD, where typically there is not a single correct answer. To measure the diversity provided by FluidC, we computed its behaviour on multi-ground truth graphs, and compare it with the alternatives. Results indicate that FluidC uncovers sets of communities which may be consistently different than the ones obtained by the other algorithms. Considering both performance on LFR and diversity, we conclude that a thorough community analysis of a given graph would benefit from the inclusion of Multilevel, Infomap and FluidC.§ ACKNOWLEDGEMENTSThis work is partially supported by the Joint Study Agreement no. W156463 under the IBM/BSC Deep Learning Center agreement, by the Spanish Government through Programa Severo Ochoa (SEV-2015-0493), by the Spanish Ministry of Science and Technology through TIN2015-65316-P project and by the Generalitat de Catalunya (contracts 2014-SGR-1051), and by the Japan JST-CREST program.spbasic | http://arxiv.org/abs/1703.09307v3 | {
"authors": [
"Ferran Parés",
"Dario Garcia-Gasulla",
"Armand Vilalta",
"Jonatan Moreno",
"Eduard Ayguadé",
"Jesús Labarta",
"Ulises Cortés",
"Toyotaro Suzumura"
],
"categories": [
"cs.DS",
"cs.SI",
"physics.soc-ph"
],
"primary_category": "cs.DS",
"published": "20170327205229",
"title": "Fluid Communities: A Competitive, Scalable and Diverse Community Detection Algorithm"
} |
In Vivo Evaluation of the Secure Opportunistic Schemes Middleware using a Delay Tolerant Social NetworkCorey E. Baker1, Allen Starke2, Tanisha G. Hill-Jarrett3, Janise McNair21Department of Electrical and Computer Engineering, University of California, San Diego2Department of Electrical and Computer Engineering, University of Florida3Department of Clinical & Health Psychology, University of FloridaEmail: [email protected], [email protected], [email protected], [email protected] December 30, 2023 =======================================================================================================================================================================================================================================================================================================================================================================================================================Over the past decade, online social networks (OSNs) such as Twitter and Facebook have thrived and experienced rapid growth to over 1 billion users. A major evolution would be to leverage the characteristics of OSNs to evaluate the effectiveness of the many routing schemes developed by the research community in real-world scenarios. In this demonstration, we showcase the Secure Opportunistic Schemes (SOS) middleware which allows different routing schemes to be easily implemented relieving the burden of security and connection establishment. The feasibility of creating a delay tolerant social network is demonstrated by using SOS to enable AlleyOop Social, a secure delay tolerant networking research platform that serves as a real-life mobile social networking application for iOS devices. AlleyOop Social allows users to interact, publish messages, and discover others that share common interests in an intermittent network using Bluetooth, peer-to-peer WiFi, and infrastructure WiFi. § INTRODUCTIONOver the past decade, online social networks (OSNs) such as Twitter and Facebookhave thrived and experienced rapid growth to over 1 billion users <cit.>. A major limitation of OSNs is the dependence on Internet which is oftensparse, difficult to maintain, or unavailable in rural areas or developingcommunities. In developed communities with cellular infrastructure, networks can becomeoverwhelmed by too many users, particularly during emergencies.In natural disaster situations, Internet and cellularcommunication infrastructures can be severely disrupted, prohibiting users fromnotifying family, friends, and associates about safety, location, food, water, andother resources. In addition, natural disasters typically damage infrastructure, which increases network traffic demandson any available undamaged infrastructure, causing congestion and delays.Opportunistic communication can seamlessly supplement Internet connectivitywhen needed and keep communication channels open even during high-use andextreme situations. Furthermore, opportunistic communication can also serve as a low-cost solution for smart cities, allowing developing and metropolitan areas to route smart city data through mobileand stationary nodes such as pedestrians, vehicles, street lights, public transportation.DTN routing has the ability to deliver data in an intermittent network,but a major challenge for DTN routing is assessing real-worldperformance <cit.>. To truly understand the reliability of DTNs and their ability to support social networks, it is imperative that DTN routing schemes are evaluatedin vivo with use-cases that are replicable, comparable, and availableto a variety of researchers. In this demonstration, we present the Secure Opportunistic Schemes (SOS) middleware, anovel middleware that facilitates secure message delivery in cases wheremobile connectivity is limited, unavailable, or non-existent. The SOS middleware supports real-life delay tolerant social networks onmobile devices. This allows mobile devices to leverage SOS to dynamically deliver messages tointerested nodes when network infrastructure is not available and improvemessage delivery when infrastructure is available.Additionally, the AlleyOop Social research platform is leveraged,which serves as an overlay application for SOS to create a delay tolerantsocial network for Apple iOS devices <cit.>. AlleyOop Social is named after the basketball play known as an “alley oop”. An “alley oop” occurs when one player throws the ball close to the basket, but itis not able to reach the final destination. While the ball is in flight, a teammate that is closerto the basket catches the ball and scores. In the same regard, AlleyOop Social enables wireless mobile users tocommunicate over longer distances by sending messages that cannot reachthe final destination, but are “caught” by intermediate mobile devices,which continue to catch and pass the messages until they are delivered tothe final destination. § RELATED WORK In recent years, a number of social-aware and social-based routing schemes have leveraged social interactions to deliver data using delay tolerant networks(DTNs) <cit.>. However, related work has primarily evaluated routing protocols in simulationenvironments, which provide valuable analyses, but are based on syntheticmobility patterns to emulate node movement and tend to use abstract models to imitate the radio response of real commodity wireless technologies<cit.>.There are a few studies that have taken on the approach of demonstrating DTNsin realistic environments <cit.>. However, these studies do not consider other significant aspects, such as usersecurity and privacy along with the limitation of operating with only theepidemic routing scheme. Various middlewares <cit.>, testbeds <cit.>,and mobile applications have been developed toaddress providing deployable delay tolerant networking applications which canoperate with minimal infrastructure and effectively evaluate DTN routingprotocols. § SECURE OPPORTUNISTIC SCHEMES (SOS) MIDDLEWARE The SOS middleware is an underlying framework that turns the AlleyOop Socialresearch platform into a delay tolerant mobile social network.The SOS middleware takes a modular approach to abstract away much of thecomplexity involved in implementing opportunistic routing schemes such asdevice dicovery, establishing D2D connections, and handling device security andprivacy.DTNs are intended to provide an overlay architecture above the existingtransport layer and ensure reliable routing during intermittency <cit.>.Building on the knowledge gained from previous middlewares <cit.>, SOS hides the complexity of the network stack (session andpresentation) within the ad hoc manager, message manager, and routing manager, allowing any mobile application to run at the application layer as depicted inFigure <ref>.Different from other middlewares such as the Haggle Project <cit.>, a separate instance of the SOS middleware is intended to run within each mobile applicationas opposed to a daemon which often requires devices to be rooted or jailbroken.Designing SOS in this manner allows for the middleware to be integrated within anymobile application in iOS, enabling them to support opportunistic communicationwithout jailbreaking devices along with being compliant with App Store regulations.§.§ Application Mobile applications serve as an overlay to the SOS middleware. Applications can be of any form such as social networking, medical, or anyother type of application that would like to share data opportunistically. Mobile applications are responsible for providing a user interfaceto users and storing data to local or online storage systems.The SOS Middleware provides a number of API's for sending/receiving data,surrounding user notification, routing protocol selection, and security andprivacy preferences.Existing mobile applications can simply add the SOS middleware as a frameworkand start using the aforementioned API's to send and receive data.Applications are responsible providing the data to be sent as well ashandling data once it has been received and decrypted. §.§ Routing manager The routing manager is responsible for leveraging D2D connections totransform any application into a delay tolerant networking application thatdelivers messages to out of range nodes in the midst of intermittency.Routing in SOS is designed for modularity, permitting additionalDTN routing schemes to be developed on top of the message manager and runseamlessly under the Application layer.Designing SOS in this manner allows for a flexible middleware that enablesapplications to dynamically change based on user preference without theneed of modifying hardware or other layers in the software stack.Currently, the routing manager in SOS has two DTN routing protocolsimplemented: epidemic routing and interest-based routing.Epidemic routing is a simple routing scheme that achieves effectiveness throughgratuitous replication and delivery of messages upon node encounters <cit.>.The IB routing protocoloperates in a similar manner to epidemic routing, except, instead ofpropagating messages to all users, messages are only propagated to interestedusers who are subscribed to the publisher of the original message.Due to the modular nature of the SOS middleware, additional routingprotocols can be added to the routing manager.APIs are available to all protocols in the routing manager tofacilitate communication between the message manager and theapplication layer.Both the IB and Epidemic routing protocols are written in less than 100 linesof Swift code. §.§ Message manager The message manager notifies the respective protocol used in therouting manager whenever a new peer has been discovered or lost. Additionally, the message manager is responsible for taking action whenever aconnection state changes. For example, if the disconnection between two users is lost, the message manager knows what messages were not transferred.Lastly, the message manager translates messages between the routing managerand ad hoc manager in a common format for both layers to interpret.§.§ Ad hoc manager The ad hoc manager manages Apple's multipeer connectivity (MPC) framework,which allows communication between iOS, macOS, and tvOS devices usingpeer-to-peer WiFi, Bluetooth personal area networks, or infrastructure WiFi networks[Apple Inc., Multipeer connectivity framework reference, https://developer.apple.com/library/ios/documentation/MultipeerConnectivity/ Reference/MultipeerConnectivityFramework/].To the best of our knowledge, SOS is the first middleware to leverage MPC to evaluate multiple delay tolerant routing schemes.The ad hoc manager is responsible for viewing discovered peers, establishingD2D connections, encrypting connections, encrypting data from end-to-end,generating keys, validating certificates, as well as signing and verifyingdata sent and received data.Apple's documentation on how to use MPC is detailed, but the company doesnot disclose specific details on how MPC works.For example, specifics about the encryption methods MPC uses are not provided.Details about how the SOS middleware handles security and privacy are elaboratedon in Section <ref>.§ PRIVACY AND SECURITY In regard to network security there is no “one-size-fits-all"approach <cit.>.Security concerns may become exacerbated in delay tolerant and ad hocapplications where nodes are vulnerable to attacks such as eavesdropping,denial of service, and compromised devices.Providing secure communication that prevents an adversary from accessing and/ormodifying data is a fundamental requirement of any DTN application <cit.>. Previous research discusses security in opportunistic applications conceptuallyand makes no claims that the implementations are secure <cit.>.The intent of the section is to provide a novel, but simple concept andimplementation of an initial layer of security for DTN protocols and enablethe overlaying mobile application to detect the identity of its users, send encryptedinformation, verify the originating source of the information being forwarded, and ensure that data have not been modified — all with minimal dependence on centralized infrastructures. Additional security can be added to AlleyOop Social by incorporating mechanismssuch as distributing CA functionality amongst nodes <cit.>,or integrating trust measurements within the routing schemes <cit.>available in the routing manager discussed in Section <ref>. To enable the initial layer of security in the SOS middleware, AlleyOop Socialleverages conventional public-key infrastructure (PKI) techniques to create a one-timePKI requirement that occurs during initial download and user-signup for the application.AlleyOop Social assumes that users will have Internet connectivityduring the initial download and installation of the mobile app.After the one-time infrastructure requirement, Internet connectivity isno longer needed for privacy, security, and message dissemination. The process of generating keys and receiving X.509 certificates in AlleyOop Social'sone-time infrastructure requirement is depicted inFigure <ref>.Using the one-time infrastructure requirement in Figure <ref> is not without limitations.The obvious shortfall is the “one-time” requirement.A fair assumption is that the AlleyOop Social application along with othersusing the SOS middleware will acquire their mobile applications from the AppleApp Store, which currently requires an Internet connection.Assuming users sign up shortly after acquiring the application addresses someconcerns with the “one-time” requirement.Additionally, if a connection between a device and the cloud is somehowcompromised, or a malicious device attempts to provide someone else'sunique user-identifier during user sign-up, a certificate with the wrongcredentials could be generated by the CA.To circumvent this issue, the cloud can ask the CA to compareand validate the unique user-identifier provided in the certificate with theunique user-identifier affiliated with the logged in user.Other limitations are also prevalent with the current security scheme such asan Internet connection is required to revoke specific user certificates, updateCA root certificates, replenish expired certificates, and notify users of knownmalicious devices. § MESSAGE DISSEMINATION After sign-up is complete and a mobile device receives its respective certificate andAlleyOop Social CA root certificate, the user can disseminate messages to other AlleyOop Social users using any DTN routing protocol discussed in Section <ref>.Whenever a user creates a message or performs an action such as follow/unfollowof a user, AlleyOop Social performs the following two operations: 1) saves the action to the local databaseon the mobile device and 2) synchronizes the action with the cloud when the Internetbecomes available.Once an action is saved to the local database of the device it canbe disseminated using a DTN routing protocol to interested AlleyOop Social users without the use of Internet. The following sections expound upon how messages are disseminated after being created by a user in the AlleyOop Social application layer and passedto the routing manager.§.§ Advertisements and node discovery Mobile devices roam freely advertising and browsing for basic information in plain-text toassist other AlleyOop Social enabled devices with making the decision of whether or not to requesta connection.For example, the epidemic and IB routing protocols discussed in Section <ref>advertises a plain-text key/value dictionary consisting of UserID/MessageNumber.The key field in the dictionary is a 10 byte unique user identificationstring.The value field of the dictionary is the latest MessageNumber thatthe advertising device has for the particular UserID.A browsing node is now able to quickly decide whether it is interested in the MessageNumber for the respective UserID string and whether itshould request a connection from the advertising node.Figure <ref> depicts a typical scenario in AlleyOop Social where Bob's device is interested in messages from Alice's device. §.§ Forwarder Selection & Dissemination Depending on the DTN routing protocol being used in the Routing layer of themobile device, prospective nodes can become message forwarders for other users.For example, in epidemic and IB routing, a node becomes a messageforwarder for a particular user-identifier whenever a new message is requestedand received.When a node becomes a message forwarder, it follows a similar process to theone outlined in Figure <ref>, with particulardifferences that are shown in Figures <ref>.Figure <ref> shows the interaction betweenBob and Carol when Carol is interested in Alice's message that Bob is forwarding.The process is similar to message dissemination inFigure <ref>, except Bob sendshis certificate to Carol to establish an encrypted connection and in addition, forwards Alice's certificate. § REAL WORLD EVALUATIONThe AlleyOop Social application was available for beta testing in Apple TestFlight app for 7 days.AlleyOop Social had thirty one (31) testers who downloaded the application aroundthe United States. Due to limited amount of users and a critical mass of users who are sociallyrelated to each other, this section will constrain the results to users whopassed at least one (1) D2D message using the IB routing protocol in Gainesville, FL. Ten devices used AlleyOop Social in a ∼11km x 8km area depicted inFigure <ref>, resulting in users posting 259 unique messages.§.§ Social relationships Many of the students were friends before the field study and typically interactedduring the school week.Individual users were given the freedom to choose other users to subscribe to; therefore, all users did not follow each other. The digraph 𝒢(V,E) formed by the total nodes whoparticipated n = |V(𝒢)| = 10 is depicted in Figure <ref>. A social relationship between a node pair i,j ∈ E is an edge e_i,j, meaning that user i follows user j.The edge e_i,j does not necessarily mean the edge e_j,i exists because some users did not follow each other back as in the case for node 1 and node 3 in Figure <ref>.The density of the social relationships is 0.64, meaning that the majorityof the possible social relationships were formed naturally by the participatingnodes.The compactness of 𝒢 can be determined by calculating the averageshortest path length between all node relationship pairs∑_i≥ jl(i,j) / n(n-1)/2 = 1.3, along with the maximum shortestpath length, otherwise known as the diameter d between any two nodesd(𝒢) = max_i,j∈ V l(i,j) = 2.The compactness of the social relationship graph reveals that even if a userdoes not follow another user directly, there is still an indirect follower that is two degrees away. The center nodes (6 and 7) of the social relationship graph has a radius of1 which reflects the nodes with the smallest eccentricity_i,j =max_i,j ∈ V l(i,j). Additional features can be determined by translating Figure <ref> to a undirected graph.This means that if a two-way relationship did not already exist, it will existin the undirectional graph making e_i,j = e_j,i for all i,j∈ E.Now the network transitivity is computed to be T(𝒢) = 3* number oftriangles / number of connected triads =0.80 which measures the extent that a friend k of a friend j is also a friend of i. §.§ Message dissemination The social relationship graph in Figure <ref> provides anoverall understanding of nodes' interests in messages along with providing insightinto how nodes may cluster due to who they follow.Figure <ref> does not provide any insight on physical nodelocations or mobility during the evaluation.Figure <ref> assists with understanding node mobility by showingwhere users created messages (blue) and passed messages (red) in Gainesville, FL.A total of 967 messages were disseminated from user-to-user using IB routingin AlleyOop Social.The total amount of subscriptions made by the ten (10) active users was 46. Figure <ref> provide the delay results formessages disseminated via “1-hop” and “All” hops. In regard to “All” messages, Figure <ref> shows that0.43 of the messages delivered had a delay of 24 hours or less,while 0.90 of the messages had a delay of 94 hours or less. In regard to “1-hop” delay, that 0.44 of the messages delivered had adelay of 24 hours or less, while 0.92 of the messages had a delay of 94hours or less for “1-hop” messages.In regard to message delivery, Figure <ref> shows that 0.30of the subscriptions had a delivery ratio greater than 0.80 for “All” messages.0.50 of the subscriptions had a delivery ratio greater than 0.70 for allmessages.0.25 of the subscriptions had a delivery ratio of 0.80 for “1-hop” messages.Users delivered 0.826 of the 967 messages via 1-hop.The additional 0.174 were delivered using 2-hops or more and is depicted in “All”. The compactness of the social relationships between the nodesdiscussed in Section <ref> partially explains whythe majority of the messages were delivered within “1-hop”.Note the low density due to real people being able to operate freely ina large city area (88km^2), which resulted 0.93 of the messages being delivered within in 94 hours of creation.DTN simulations typically model 50 to 100 nodes in a constrained simulationspace rangingbetween 0.25km^2 - 4km^2.In addition, node mobility tends to become stationary, for at least 5-8hours a day due to the human requirement to sleep, thus limiting possibleinteractions between nodes.The results at such a low density provide promising insight into delay tolerant social networks and suggest further investigations at higher densities are needed. § DEMONSTRATION During the demonstration attendees will be able to download AlleyOopSocial on their iOS devices via Apple TestFlight.Users can follow friends, post new messages, as well as toggle between DTN routing schemes inside the application.We will demonstrate both the online and offline modes by disconnecting mobiledevices from cellular and WiFi networks. IEEEtran | http://arxiv.org/abs/1703.08947v2 | {
"authors": [
"Corey E. Baker",
"Allen Starke",
"Tanisha G. Hill-Jarrett",
"Janise McNair"
],
"categories": [
"cs.NI",
"cs.SI"
],
"primary_category": "cs.NI",
"published": "20170327064227",
"title": "In Vivo Evaluation of the Secure Opportunistic Schemes Middleware using a Delay Tolerant Social Network"
} |
Greedy walks on two lines Katja Gabrysch December 30, 2023 ========================= Maltese is a morphologically rich language with a hybrid morphological system which features both concatenative and non-concatenative processes. This paper analyses the impact of this hybridity on the performance of machine learning techniques for morphological labelling and clustering. In particular, we analyse a dataset of morphologically related word clusters to evaluate the difference in results for concatenative and non-concatenative clusters. We also describe research carried out in morphological labelling, with a particular focus on the verb category. Two evaluations were carried out, one using an unseen dataset, and another one using a gold standard dataset which was manually labelled. The gold standard dataset was split into concatenative and non-concatenative to analyse the difference in results between the two morphological systems. § INTRODUCTION Maltese, the national language of the Maltese Islands and, since 2004, also an official European language, has a hybrid morphological system that evolved from an Arabic stratum, a Romance (Sicilian/Italian) superstratum and an English adstratum <cit.>.The Semitic influence is evident in the basic syntactic structure, with a highly productive non-Semitic component manifest in its lexis and morphology <cit.>. Semitic morphological processes still account for a sizeable proportion of the lexicon and follow a non-concatenative, root-and-pattern strategy (or templatic morphology) similar to Arabic and Hebrew, with consonantal roots combined with a vowel melody and patterns to derive forms. By contrast, the Romance/English morphological component is concatenative (i.e. exclusively stem-and-affix based). Table <ref> provides an example of these two systems, showing inflection and derivation for the words eżaminato examine taking a stem-based form, and gidebto lie from the root gdb which is based on a templatic system. Table <ref> gives an examply of verbal inflection, which is affix-based, and applies to lexemes arising from both concatenative and non-concatenative systems, the main difference being that the latter evinces frequent stem variation.To date, there still is no complete morphological analyser for Maltese. In a first attempt at a computational treatment of Maltese morphology, Farrugia:08b used a neural network and focused solely on broken plural for nouns <cit.>. The only work treating computational morphology for Maltese in general was by Borg:14, who used unsupervised techniques to group together morphologically related words. A theoretical analysis of the templatic verbs <cit.> was used by Camilleri:13, who created a computational grammar for Maltese for the Resource Grammar Library <cit.>, with a particular focus on inflectional verbal morphology. The grammar produced the full paradigm of a verb on the basis of its root, which can consist of over 1,400 inflective forms per derived verbal form, of which traditional grammars usually list 10. This resource is known as Ġabra and is available online[<http://mlrs.research.um.edu.mt/resources/gabra/>]. Ġabra is, to date, the best computational resource available in terms of morphological information. It is limited in its focus to templatic morphology and restricted to the wordforms available in the database. A further resource is the lexicon and analyser provided as part of the Apertium open-source machine translation toolkit <cit.>. A subset of this lexicon has since been incorporated in the Ġabra database.This paper presents work carried out for Maltese morphology, with a particular emphasis on the problem of hybridity in the morphological system. Morphological analysis is challenging for a language like Maltese due to the mixed morphological processes existing side by side. Although there are similarities between the two systems, as seen in verbal inflections, various differences among the subsystems exist which make a unified treatment challenging, including: (a) stem allomorphy, which occurs far more frequently with Semitic stems; (b) paradigmatic gaps, especially in the derivational system based on semitic roots <cit.>; (c) the fact that morphological analysis for a hybrid system needs to pay attention to both stem-internal (templatic) processes, and phenomena occurring at the stem's edge (by affixation).First, we will analyse the results of the unsupervised clustering technique by Borg:14 applied on Maltese, with a particular focus of distinguishing the performance of the technique on the two different morphological systems. Second, we are interested in labelling words with their morphological properties. We view this as a classification problem, and treat complex morphological properties as separate features which can be classified in an optimal sequence to provide a final complex label.Once again, the focus of the analysis is on the hybridity of the language and whether a single technique is appropriate for a mixed morphology such as that found in Maltese.§ RELATED WORK Computational morphology can be viewed as having three separate subtasks — segmentation, clustering related words, and labelling (see Hammarstrom:11). Various approaches are used for each of the tasks, ranging from rule-based techniques, such as finite state transducers for Arabic morphological analysis <cit.>, to various unsupervised, semi- or fully-supervised techniques which would generally deal with one or two of the subtasks. For most of the techniques described, it is difficult to directly compare results due to difference in the data used and the evaluation setting itself.For instance, the results achieved by segmentation techniques are then evaluated in an information retrieval task.The majority of works dealing with unsupervised morphology focus on English and assume that the morphological processes are concatenative <cit.>. Goldsmith:01 uses the minimum description length algorithm, which aims to represent a language in the most compact way possible by grouping together words that take on the same set of suffixes. In a similar vein, Creutz and Lagus Creutz:05a,Creutz:07 use Maximum a Posteriori approaches to segment words from unannotated texts, and have become part of the baseline and standard evaluation in the Morpho Challenge series of competitions <cit.>. Kohonen:10 extends this work by introducing semi- and supervised approaches to the model learning for segmentation. This is done by introducing a discriminative weighting scheme that gives preference to the segmentations within the labelled data. Transitional probabilities are used to determine potential word boundaries <cit.>. The technique is very intuitive, and posits that the most likely place for a segmentation to take place is at nodes in the trie with a large branching factor. The result is a ranked list of affixes which can then be used to segment words. vandenBosch:99 and Clark Clark:02,Clark:07 apply Memory-based Learning to classify morphological labels. The latter work was tested on Arabic singular and broken plural pairs, with the algorithm learning how to associate an inflected form with its base form. Durrett:13 derives rules on the basis of the orthographic changes that take place in an inflection table (containing a paradigm). A log-linear model is then used to place a conditional distribution over all valid rules. Poon:09 use a log-linear model for unsupervised morphological segmentation, which leverages overlapping features such as morphemes and their context. It incorporates exponential priors as a way of describing a language in an efficient and compact manner. Sirts:13 proposed Adaptor Grammars (AGMorph), a nonparametric Bayesian modelling framework for minimally supervised learning of morphological segmentation. The model learns latent tree structures over the input of a corpus of strings. Narasimhan:15 also use a log-linear model, and morpheme and word-level features to predict morphological chains, improving upon the techniques of Poon:09 and Sirts:13. A morphological chain is seen as a sequence of words that starts from the base word, and at each level through the process of affixation a new word is derived as a morphological variant, with the top 100 chains having an accuracy of 43%. It was also tested on an Arabic dataset, achieving an F-Measure of 0.799. However, the system does not handle stem variation since the pairing of words is done on the basis of the same orthographic stem and therefore the result for Arabic is rather surprising. The technique is also lightly-supervised since it incorporates part-of-speech category to reinforce potential segmentations. Schone and Jurafsky Schone:00,Schone:01 and Baroni:02 use both orthographic and semantic similarity to detect morphologically related word pairs, arguing that neither is sufficient on its own to determine a morphological relation. Yarowsky:00 use a combination of alignment models with the aim of pairing inflected words. However this technique relies on part-of-speech, affix and stem information. Can:12 create a hierarchical clustering of morphologically related words using both affixes and stems to combine words in the same clusters. Ahlberg:14 produce inflection tables by obtaining generalisations over a small number of samples through a semi-supervised approach. The system takes a group of words and assumes that the similar elements that are shared by the different forms can be generalised over and are irrelevant for the inflection process. For Semitic languages, a central issue in computational morphology is disambiguation between multiple possible analyses. Habash:05 learn classifiers to identify different morphological features, used specifically to improve part-of-speech tagging. Snyder:08 tackle morphological segmentation for multiple languages in the Semitic family and English by creating a model that maps frequently occurring morphemes in different languages into a single abstract morpheme. Due to the intrinsic differences in the problem of computational morphology between Semitic and English/Romance languages, it is difficult to directly compare results. Our interest in the present paper is more in the types of approaches taken, and particularly, in seeing morphological labelling as a classification problem. Modelling different classifiers for specific morphological properties can be the appropriate approach for Maltese, since it allows the flexibility to focus on those properties where data is available.§ CLUSTERING WORDS IN A HYBRID MORPHOLOGICAL SYSTEM The Maltese morphology system includes two systems, concatenative and non-concatenative. As seen in the previous section, most computational approaches deal with either Semitic morphology (as one would for Arabic or its varieties), or with a system based on stems and affixes (as in Italian). Therefore, we might expect that certain methods will perform differently depending on which component we look at. Indeed, overall accuracy figures may mask interesting differences among the different components.The main motivation behind this analysis is that Maltese words of Semitic origin tend to have considerable stem variation (non-concatenative), whilst the word formation from Romance/English origin words would generally leave stems whole (concatenative)[Concatenative word formations would always involve a recognisable stem, though in some cases they may undergo minor variations as a result of allomorphy or allophomy.]. Maltese provides an ideal scenario for this type of analysis due to its mixed morphology. Often, clustering techniques would either be sensitive to a particular language, such as catering for weak consonants in Arabic <cit.>, or focus solely on English or Romance languages <cit.> where stem variation is not widespread.The analysis below uses a dataset of clusters produced by Borg:14, who employed an unsupervised technique using several interim steps to cluster words together. First, potential affixes are identified using transitional probabilities in a similar fashion to <cit.>. Words are then clustered on the basis of common stems. Clusters are improved using measures of orthographic and semantic similarity, in a similar vein to <cit.>. Since no gold-standard lexical resource was available for Maltese, the authors evaluated the clusters using a crowd-sourcing strategy of non-expert native speakers and a separate, but smaller, set of clusters were evaluated using an expert group. In the evaluation, participants were presented with a cluster which had to be rated for its quality and corrected by removing any words which do not belong to a cluster. In this analysis, we focus on the experts' cluster dataset which was roughly balanced between non-concatenative (NC) and concatenative (CON) clusters. There are 101 clusters in this dataset, 25 of which were evaluated by all 3 experts, and the remaining by one of the experts. Table <ref> provides an overview of the 101 clusters in terms of their size. Immediately, it is possible to observe that concatenative clusters tend to be larger in size than non-concatenative clusters. This is mainly due to the issue of stem variation in the non-concatenative group, which gives rise to a lot of false negatives. It is also worth noting that part of the difficulty here is that the vowel patterns in the non-concatenative process are unpredictable. For example qsimdivision is formed from qasamto divide qsm, whilst ksurbreakage is formed from kiserto break ksr. Words are constructed around infixation of vowel melodies to form a stem, before inflection adds affixes. In the concatenative system there are some cases of allomorphy, but there will, in general, be an entire stem, or substring thereof, that is recognisable. §.§ Words removed from clusters As an indicator of the quality of a cluster, the analysis looks at the number of words that experts removed from a cluster — indicating that the word does not belong to a cluster. Table <ref> gives the percentage of words removed from clusters, divided according to whether the morphological system involved is concatenative or non-concatenative. The percentage of clusters which were left intact by the experts were higher for the concatenative group (61%) when compared to the non-concatenative group (45%).The gap closes when considering the percentage of clusters which had a third or more of their words removed (non-concatenative at 25% and concatenative at 20%). However, the concatenative group also had clusters which had more than 80% of their words removed. This indicates that, although in general the clustering technique performs better for the concatenative case, there are cases when bad clusters are formed through the techniques used. The reason is usually that stems with overlapping substrings are mistakenly grouped together. One such cluster was that for ittraletter, which also got clustered with ittraduċitranslate and ittratattreated, clearly all morphologically unrelated words. However, these were clustered together because the system incorrectly identified ittra as a potential stem in all these words. §.§ Quality ratings of clusters Experts were asked to rate the quality of a cluster, and although this is a rather subjective opinion, the correlation between this judgement and the number of words removed was calculated using Pearson’s correlation coefficient. The trends are consistent with the analysis in the previous subsection; Table <ref> provides the breakdown of the quality ratings for clusters split between the two processes and the correlation of the quality to the percentage of words removed. The non-concatenative clusters generally have lower quality ratings when compared to the concatenative clusters. But both groups have a strong correlation between the percentage of words removed and the quality rating, clearly indicating that the perception of a cluster's quality is related to the percentage of words removed.§.§ Hybridity in clustering Clearly, there is a notable difference between the clustering of words from concatenative and non-concatenative morphological processes. Both have their strengths and pitfalls, but neither of the two processes excel or stand out over the other. One of the problems with non-concatenative clusters was that of size. The initial clusters were formed on the basis of the stems, and due to stem variation the non-concatenative clusters were rather small. Although the merging process catered for clusters to be put together and form larger clusters, the process was limited to a maximum of two merging operations. This might not have been sufficient for the small-sized non-concatenative clusters. In fact, only 10% of the NC clusters contained 30 or more words when compared to 22% of the concatenative clusters. Limiting merging in this fashion may have resulted in a few missed opportunities. This is because there's likely to be a lot of derived forms which are difficult to cluster initially due to stem allomorphy (arising due to the fact that root-based derivation involves infixation, and in Maltese, vowel melodies are unpredictable). So there are possibly many clusters, all related to the same root. The problem of size with concatenative clusters was on the other side of the scale. Although the majority of clusters were of average size, large clusters tended to include many false positives. In order to explore this problem further, one possibility would be to check whether there is a correlation between the size of a cluster and the percentage of words removed from it. It is possible that the unsupervised technique does not perform well when producing larger clusters, and if such a correlation exists, it would be possible to set an empirically determined threshold for cluster size. Given the results achieved, it is realistic to state that the unsupervised clustering technique could be further improved using the evaluated clusters as a development set to better determine the thresholds in the metrics proposed above. This improvement would impact both concatenative and non-concatenative clusters equally. In general, the clustering technique does work slightly better for the concatenative clusters, and this is surely due to the clustering of words on the basis of their stems. This is reflected by the result that 61% of the clusters had no words removed compared to 45% of the non-concatenative clusters. However, a larger number of concatenative clusters had a large percentage of words removed. Indeed, if the quality ratings were considered as an indicator of how the technique performs on the non-concatenative vs the concatenative clusters, the judgement would be medium to good for the non-concatenative and good for the concatenative clusters. Thus the performance is sufficiently close in terms of quality of the two groups to suggest that a single unsupervised technique can be applied to Maltese, without differentiating between the morphological sub-systems.§ CLASSIFYING MORPHOLOGICAL PROPERTIES In our approach, morphological labelling is viewed as a classification problem with each morphological property seen as a feature which can be classified. Thus, the analysis of a given word can be seen as a sequence of classification problems, each assigning a label to the word which reflects one of its morphological properties. We refer to such a sequence of classifiers as a `cascade'.In this paper, we focus in particular on the verb category, which is morphologically one of the richest categories in Maltese. The main question is to identify whether there is a difference in the performance of the classification system when applied to lexemes formed through concatenative or non-concatenative processes. Our primary focus is on the classification of inflectional verb features. While these are affixed to the stem, the principal issue we are interested in is whether the co-training of the classifier sequence on an undifferentiated training set performs adequately on both lexemes derived via a templatic system and lexemes which have a `whole', continuous stem. §.§ The classification system The classification system was trained and initially evaluated using part of the annotated data from the lexical resource Ġabra. The training data contained over 170,000 wordforms, and the test data, which was completely unseen, contained around 20,000 wordforms.A second dataset was also used which was taken from the Maltese national corpus (mlrs — Malta Language Resource Server[<http://mlrs.research.um.edu.mt/>]). This dataset consisted of 200 randomly selected words which were given morphological labels by two experts. The words were split half and half between Semitic (non-concatenative) and Romance/English (concatenative) origin. The verb category had 94 words, with 76 non-concatenative, and 18 concatenative. This is referred to as the gold standard dataset. A series of classifiers were trained using annotated data from Ġabra, which contains detailed morphological information relevant to each word. These are person, number, gender, direct object, indirect object, tense, aspect, mood and polarity. In the case of tense/aspect and mood, these were joined into one single feature, abbreviated to TAM since they are mutually exclusive. These features are referred to as second-tier features, representing the morphological properties which the system must classify. The classification also relies on a set of basic features which are automatically extracted from a given word. These are stems, prefixes, suffixes and composite suffixes, when available[Composite suffixes occur when more than one suffix is concatenated to the stem, usually with enclitic object and indirect object pronouns, as in qatil-hu-lihe killed him for me.], consonant-vowel patterns and gemination.A separate classifier was trained for each of the second-tier features. In order to arrive at the ideal sequence of classifiers, multiple sequences were tested and the best sequence identified on the basis of performance on held-out data (for more detail see Borg:16). Once the optimal sequence was established, the classification system used these classifiers as a cascade, each producing the appropriate label for a particular morphological property and passing on the information learnt to the following classifier. The verb cascade consisted of the optimal sequence of classifiers in the following sequence: Polarity (Pol), Indirect Object (Ind), Direct Object (Dir), Tense/Aspect/Mood (TAM), Number (Num), Gender (Gen) and Person (Per). The classifiers were trained using decision trees through the weka data mining software <cit.>, available both through a graphical user interface and as an open-source java library. Other techniques, such as Random Forests, SVMs and Bayes, were also tested and produced very similar results. The classifiers were built using the training datasets. The first evaluation followed the traditional evaluation principles of machine learning, using the test dataset which contained unseen wordforms from Ġabra, amounting to just over 10% of the training data. This is referred to as the traditional evaluation. However, there are two main aspects in our scenario that encouraged us to go beyond the traditional evaluation. First, Ġabra is made of automatically generated wordforms, several of which are never attested (though they are possible) in the mlrs corpus. Second, the corpus contains several other words which are not present in Ġabra, especially concatenative word formations. Thus, we decided to carry out a gold standard (GS) evaluation to test the performance of the classification system on actual data from the mlrs corpus. The evaluation in this paper is restricted to the verb category. §.§ Evaluation Results We first compare the performance of the classification system on the test dataset collected from Ġabra to the manually annotated gold standard collated from the mlrs corpus. These results are shown in Figure <ref>. The first three features in the cascade — Polarity, Indirect Object and Direct Object — perform best in both the traditional and gold standard evaluations. In particular, the indirect object has practically the same performance in both evaluations. A closer look at the classification results of the words reveals that most words did not have this morphological property, and therefore no label was required. The classification system correctly classified these words with a null value. The polarity classifier on the other hand, was expected to perform better — in Maltese, negation is indicated with the suffix -x at the end of the word. The main problem here was that the classifier could apply the labels positive, negative or null to a word, resulting in the use of the null label more frequently than the two human experts. The errors in the classification of the morphological property TAM were mainly found in the labelling of the values perfective and imperative, whilst the label imperfective performed slightly better. Similarly, the number and gender classifiers both had labels that performed better than others. Overall, this could indicate that the data representation for these particular labels is not adequate to facilitate the modelling of a classifier.As expected, the performance of the classifiers on the gold standard is lower than that of a traditional evaluation setting. The test dataset used in the traditional evaluation, although completely unseen, was still from the same source as the training data (Ġabra) — the segmentation of words was known, the distribution of instances in the different classes (labels) was similar to that found in the training data. While consistency in training and test data sources clearly make for better results, the outcomes also point to the possibility of overfitting, particularly as Ġabra contains a very high proportion of Semitic, compared to concatenative, stems. Thus, it is possible that the training data for the classifiers did not cover the necessary breadth for the verbs found in the mlrs corpus. To what extent this is impacting the results of the classifiers cannot be known unless the analysis separates the two processes. For this reason, the analysis of the verb category in the gold standard evaluation was separated into two, and the performance of each is compared to the overall gold standard performance. This allows us to identify those morphological properties which will require more representative datasets in order to improve their performance. Figure <ref> shows this comparison.The first three classifiers — polarity, indirect object and direct object — perform as expected, meaning that the concatenative lexemes perform worse than the non-concatenative. This confirms the suspicion that the coverage of Ġabra is not sufficiently representative of the morphological properties in the concatenative class of words. On the other hand, the TAM and Person classifiers perform better on the concatenative words. However, there is no specific distinction in the errors of these two classifiers.One overall possible reason for the discrepancy in the performance between the traditional and gold standard evaluation, and possibly also between the concatenative and non-concatenative words, is how the words are segmented. The test data in the traditional evaluation setting was segmented correctly, using the same technique applied for the training data. The segmentation for the words in the mlrs corpus was performed automatically and heuristically, and the results were not checked for their correctness, so the classification system might have been given an incorrect segmentation of a word. This would impact the results as the classifiers rely upon the identification of prefixes and suffixes to label words.§ CONCLUSIONS AND FUTURE WORK This paper analysed the results of the clustering of morphologically related words and the morphological labelling of words, with a particular emphasis on identifying the difference in performance of the techniques used on words of Semitic origin (non-concatenative) and Romance/English origin (concatenative).The datasets obtained from the clustering technique were split into concatenative and non-concatenative sets, and evaluated in terms of their quality and the number of words removed from each cluster. Although generally, the clustering techniques performed best on the concatenative set, scalability seemed to be an issue, with the bigger clusters performing badly. The non-concatenative set, on the other hand, had smaller clusters but the quality ratings were generally lower than those of the concatenative group. Overall, it seems that the techniques were geared more towards the concatenative set, but performed at an acceptable level for the non-concatenative set. Although the analysis shows that it is difficult to find a one-size-fits-all solution, the resulting clusters could be used as a development set to optimise the clustering process in future. The research carried out in morphological labelling viewed it as a classification problem. Each morphological property is seen as a machine learning feature, and each feature is modelled as a classifier and placed in a cascade so as to provide the complete label to a given word. The research focussed on the verb category and two types of evaluations were carried out to test this classification system. The first was a traditional evaluation using unseen data from the same source as the training set. A second evaluation used randomly selected words from the mlrs corpus which were manually annotated with their morphological labels by two human experts. There is no complete morphological analyser available for Maltese, so this was treated as a gold standard. Since the classifiers were trained using data which is predominantly non-concatenative, the performance of the classification system on the mlrs corpus was, as expected, worse than the traditional evaluation. In comparing the two evaluations, it was possible to assess which morphological properties were not performing adequately. Moreover, the gold standard dataset was split into two, denoting concatenative and non-concatenative words, to further analyse whether a classification system that was trained predominantly on non-concatenative data could then be applied to concatenative data. The results were mixed, according to the different morphological properties, but overall, the evaluation was useful to determine where more representative data is needed. Although the accuracy of the morphological classification system are not exceptionally high for some of the morphological properties, the system performs well overall, and the individual classifiers can be retrained and improved as more representative data becomes available. And although the gold standard data is small in size, it allows us to identify which properties require more data, and of which type. One of the possible routes forward is to extend the grammar used to generate the wordforms in Ġabra and thus obtain more coverage for the concatenative process. However, it is already clear from the analysis carried out that the current approach is viable for both morphological systems and can be well suited for a hybrid system such as Maltese. § ACKNOWLEDGEMENTS The authors acknowledge the insight and expertise of Prof. Ray Fabri. The research work disclosed in this publication is partially funded by the Malta Government Scholarship Scheme grant.eacl2017 | http://arxiv.org/abs/1703.08701v1 | {
"authors": [
"Claudia Borg",
"Albert Gatt"
],
"categories": [
"cs.CL",
"I.2.7"
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"primary_category": "cs.CL",
"published": "20170325145627",
"title": "Morphological Analysis for the Maltese Language: The Challenges of a Hybrid System"
} |
Yoneda Structures and KZ Doctrines Charles Walker December 30, 2023 ==================================empty empty This paper addresses the problem of decentralized abstractions for multiple mobile manipulators with 2nd order dynamics. In particular, we propose decentralized controllers for the navigation of each agent among predefined regions of interest in the workspace, while guaranteeing at the same time inter-agent collision avoidance and connectivity maintenance for a subset of initially connected agents. In that way, the motion of the coupled multi-agent system is abstracted into multiple finite transition systems for each agent, which are then suitable for the application of temporal logic-based high level plans. The proposed methodology is decentralized, since each agent uses local information based on limited sensing capabilities. Finally, simulation studies verify the validity of the approach. § INTRODUCTIONMulti-agent systems have gained a significant amount of attention in the last decades, due to the several advantages they yield with respect to single-agent setups. A recent direction in the multi-agent control and robotics field is the use of temporal logic languages for motion and/or action planning, since they provide a fully-automated correct-by-design controller synthesis approach for autonomous robots. Temporal logics, such as linear temporal logic (LTL), computation tree logic (CTL) or metric-interval temporal logic (MITL), provide formal high-level languages that can describe planning objectives more complex than the usual navigation techniques. The task specification is given as a temporal logic formula with respect to a discretized abstraction of the robot motion modeled as a finite transition system, and then, a high-level discrete plan is found by off-the-shelf model-checking algorithms, given the finite transition system and the task specification <cit.>. There exists a wide variety of works that employ temporal logic languages for multi-agent systems, e.g., <cit.>. The discretization of a multi-agent system to an abstracted finite transition system necessitates the design of appropriate continuous-time controllers for the transition of the agents among the states of the transition system <cit.>. Most works in the related literature, however, including the aforementioned ones, either assume that there exist such continuous controllers or adopt single- and double-integrator models, ignoring the actual dynamics of the agents. Discretized abstractions, including design of the discrete state space and/or continuous-time controllers, have been considered in <cit.> for general systems and <cit.> for multi-agent systems. Another important issue concerning multi-agent abstractions that has not been addressed in the related literature is the collision avoidance between the robotic agents, which, unlike the unrealistic point-mass assumption that is considered in many works, can be more appropriately approximated by unions of rigid bodies. This work addresses the problem of decentralized abstractions for a team of mobile robotic manipulators, represented by a union of 3D ellipsoids, among predefined regions of interest in the workspace. Mobile manipulators consist of a mobile base and a robotic arm, which makes them suitable for performing actions around a workspace (e.g., transportation of objects). In <cit.> the authors consider the navigation of two mobile manipulators grasping an object, based on 3D ellipsoids, whereas <cit.> deals with general-shape multi-agent navigation, both based on point-world transformations. Navigation of ellipsoidal agents while incorporating collision-avoidance properties was also studied in <cit.> for single-integrator dynamics, by transforming the ellipsoids to spheres. In our previous work <cit.>, we addressed a hybrid control framework for the navigation of mobile manipulators and their interaction with objects in a given workspace, proposing, however, a centralized solution.In this work, we design robust continuous-time controllers for the navigation of the agents among the regions of interest. The proposed methodology is decentralized, since each agent uses only local information based on limited sensing capabilities. Moreover, we guarantee (i) inter-agent collision avoidance by introducing a novel transformation-free ellipsoid-based strategy, (ii) connectivity maintenance for a subset of the initially connected agents, which might be important for potential cooperative tasks, and (iii) kinematic singularity avoidance of the robotic agents. The rest of the paper is organized as follows. Section <ref> provides necessary notation and preliminary background and Section <ref> describes the tackled problem. The main results are given in Section <ref> and Section <ref> presents simulation results. Finally, <ref> concludes the paper. § PRELIMINARIES§.§ NotationThe set of positive integers is denoted as ℕ whereas the real and complex n-coordinate spaces, with n∈ℕ, are denoted as ℝ^n and ℂ^n, respectively; ℝ^n_≥ 0, ℝ^n_> 0, ℝ^n_≤ 0 and ℝ^n_< 0 are the sets of real n-vectors with all elements nonnegative, positive, nonpositive, and negative, respectively. The notation x is used for the Euclidean norm of a vector x ∈ℝ^n. Given a a scalar function y:ℝ^n→ℝ and a vector x∈ℝ^n, we use the notation ∇_xy(x) = [∂ y∂ x_1, …, ∂ y∂ x_n]^⊤∈ℝ^n.Define by I_n ∈ℝ^n × n, 0_m × n∈ℝ^m × n, the identity matrix and the m × n matrix with all entries zero, respectively; ℬ_c,r = {x ∈ℝ^3: x-c≤ r} is the 3D sphere of center c∈ℝ^3 and radius r ∈ℝ_≥ 0. The boundary of a set A is denoted as ∂ A and its interior as ∘A = A\∂ A. The vector connecting the origins of coordinate frames {A} and {B} expressed in frame {C} coordinates in 3D space is denoted as p^ C_ B/A∈ℝ^3.For notational brevity, when a coordinate frame corresponds to an inertial frame of reference {I}, we will omit its explicit notation (e.g., p_ B = p^ I_ B/I, ω_ B = ω^ I_ B/I). All vector and matrix differentiations are derived with respect to an inertial frame {I}, unless otherwise stated. §.§ Cubic Equations and Ellipsoid CollisionConsider the cubic equation f(λ) = c_3λ^3+c_2λ^2+c_1λ + c_0 = 0 with c_ℓ∈ℝ,∀ℓ∈{0,…,3} and roots (λ_1,λ_2,λ_3)∈ℂ^3, with f(λ_1)=f(λ_2)=f(λ_3)=0. Then, given its discriminant Δ = (c_3)^4∏_i∈{1,2} j∈{i+1,…,3}(λ_i-λ_j)^2, the following hold:* Δ = 0 ⇔∃ i,j∈{1,2,3}, with i≠ j, such that λ_i=λ_j, i.e., at least two roots are equal, * Δ > 0 ⇔λ_i∈ℝ,∀ i∈{1,2,3}, and λ_i≠λ_j, ∀ i,j∈{1,2,3}, with i≠ j, i.e., all roots are real and distinct.<cit.>Consider two planar ellipsoids 𝒜 = {z∈ℝ^3s.t.z^⊤ A(t) z ≤ 0}, ℬ = {z∈ℝ^3s.t.z^⊤ B(t) z ≤ 0}, with z=[p^⊤ 1]^⊤ being the homogeneous coordinates of p∈ℝ^2, and A, B:ℝ_≥ 0→ℝ^3×3 terms that describe their motion in 2D space. Given their characteristic polynomial f:ℝ→ℝ with f(λ) = (λ A - B), which has degree 3, the following hold: * ∃λ^*∈ℝ_>0 s.t.f(λ^*)=0, i.e, the polynomial f(λ) always has one positive real root,* 𝒜∩ℬ = ∅⇔∃λ^*_1, λ^*_2 ∈ℝ_<0, with λ^*_1≠λ^*_2, and f(λ^*_1)=f(λ^*_2)=0, i.e., 𝒜 and ℬ are disjoint if and only if the characteristic equation f(λ) = 0 has two distinct negative roots.* 𝒜∩ℬ≠∅ and ∘𝒜∩∘ℬ=∅⇔∃λ^*_1, λ^*_2 ∈ℝ_<0, with λ^*_1=λ^*_2, and f(λ^*_1)=f(λ^*_2)=0, i.e., 𝒜 and ℬ touch externally if and only if the characteristic equation f(λ) = 0 has a negative double root.§ PROBLEM FORMULATIONConsider N∈ℕ fully actuated agents with 𝒱{1,…,N}, N≥ 2, composed by a robotic arm mounted on anomnidirectional mobile base, operating in a static workspace 𝒲 that is bounded by a large sphere in 3D space, i.e. 𝒲 = ℬ_p_0,r_0 = {p∈ℝ^3s.t. ‖ p - p_0 ‖ < r_0}, where p_0∈ℝ^3 is the center of 𝒲, and r_0∈ℝ_≥ 0 is its radius. Without loss of generality, we consider that p_0 = 0_3× 1, corresponding to an inertial frame {I}.Within 𝒲 there exist K disjoint spheres around points of interest, which are described by π_k = ℬ_p_k,r_k = {p∈ℝ^3s.t. ‖ p - p_k ‖≤ r_k}, k∈𝒦{1,…,K}, where p_k∈ℝ^3 and r_k∈ℝ_>0 are the center and radius of the kth region, respectively. The regions of interest can be equivalently described by π_k = {z∈ℝ^4s.t.z^⊤ T_π_kz ≤ 0 }, where z=[p^⊤, 1]^⊤ is the vector of homogeneous coordinates of p∈ℝ^3, andT_π_k = [I_3p_k; 0^⊤_3× 1 -r^2_k ], ∀ k∈𝒦.The dynamic model of each agent is given by the second-order Lagrangian dynamics:M_i(q_i)q̈_i + N_i(q_i,q̇_i)q̇_i + g_i(q_i) + f_i(q_i, q̇_i) = τ_i,∀ i∈𝒱, where q_i∈ℝ^n_i is the vector of generalizedcoordinates (e.g., pose of mobile base and joint coordinates of the arms), M_i:ℝ^n_i→ℝ^n_i× n_i is the positive definite inertia matrix,N_i:ℝ^n_i×ℝ^n_i→ℝ^n_i× n_i is the Coriolis matrix, g_i:ℝ^n_i→ℝ^n_i is the gravity vector, f_i:ℝ^n_i×ℝ^n_i→ℝ^n_i is a term representing friction and modeling uncertainties and τ_i∈ℝ^n_i is the vector of joint torques, representing the control inputs. Without loss of generality, we assume that n_i = n∈ℕ,∀ i∈𝒱. In addition, we denote as {B_i} the frame of the mobile base of agent i and p_ B_i:ℝ^n→ℝ^3 its inertial position.Moreover, the matrix Ṁ_i - 2N_i is skew-symmetric <cit.>, and we further make the following assumption: There exist positive constants c_i such that ‖ f_i(q_i,q̇_i)‖≤ c_i‖ q_i ‖‖q̇_i ‖, ∀ (q_i,q̇_i)∈ℝ^n×ℝ^n, i∈𝒱.We consider that each agent is composed by ℓ_i rigid links (see Fig. <ref>) with 𝒬_i = {1,…,ℓ_i} the corresponding index set. Each link of agent i is approximated by the ellipsoid set <cit.> ℰ_i_m(q_i) = {z∈ℝ^4s.t.z^⊤ E_i_m(q_i)z ≤ 0}; z=[p^⊤, 1]^⊤ is the homogeneous coordinates of p∈ℝ^3, and E_i_m:ℝ^n→ℝ^4×4 is defined as E_i_m(q_i) = T^-T_i_m(q_i)Ê_i_mT^-1_i_m(q_i), where Ê_i_m = diag{a^-2_i_m,b^-2_i_m,c^-2_i_m,-1} corresponds to the positive lengths a_i_m,b_i_m,c_i_m of the principal axes of the ellipsoid, and T_i_m:ℝ^n→ℝ^4×4 is the transformation matrix for the coordinate frame {i_m} placed at the center of mass of the m-th link of agent i, aligned with the principal axes of ℰ_i_m:T_i_m(q_i) = [ R_i_m(q_i) p_i_m(q_i); 0^⊤_3× 11 ], with R_i_m:ℝ^n→ℝ^3×3 being the rotation matrix of the center of mass of the link, ∀ m∈𝒬_i,i∈𝒱. For an ellipsoid ℰ_i_m, i∈𝒱,m∈𝒬_i, we denote as ℰ^xy_i_m,ℰ^xz_i_m, ℰ^yz_i_m its projections on the planes x-y, x-z and y-z, respectively, with corresponding matrix terms E^xy_i_m,E^xz_i_m, E^yz_i_m. Note that the following holds for two different ellipsoids ℰ_i_m and ℰ_j_l:ℰ_i_m(q_i)∩ℰ_j_l(q_j) ≠∅∘ℰ_i_m(q_i)∩∘ℰ_j_l(q_j) = ∅⇔ ℰ^s_i_m(q_i)∩ℰ^s_j_l(q_j) ≠∅ ∘ℰ^s_i_m(q_i)∩∘ℰ^s_j_l(q_j) = ∅,∀ s∈{xy,xz,yz}, i.e., in order for ℰ_i_m, ℰ_j_l to collide (touch externally), all their projections on the three planes must also collide. Therefore, a sufficient condition for ℰ_i_m and ℰ_j_l not to collide is ℰ^s_i_m(q_i)∩ℰ^s_j_l(q_j)=∅, for somes∈{xy,xz,yz}. In view of Proposition <ref>, that means that the characteristic equations f^s_i_m,j_l(λ) (λℰ^s_i_m(q_i) - ℰ^s_j_l(q_j))=0 must always have one positive real root and two negative distinct roots for at least one s∈{xy,xz,yz}. Hence, be denoting the discriminant of f^s_i_m,j_l(λ) = 0 as Δ^s_i_m,j_l, Proposition <ref> suggests that Δ^s_i_m,j_l must remain always positive for at least one s∈{xy,xz,yz}, since a collision would imply Δ^s_i_m,j_l=0, ∀ s∈{xy,xz,yz}. Therefore, by defining the function δ:ℝ→ℝ_≥ 0 as: δ(x) = ϕ_δ(x), x > 0, 0, x ≤ 0,where ϕ_δ is an appropriate polynomial that ensures that δ(x) is twice continuously differentiable everywhere (e.g. ϕ_δ(x)=x^3), we can conclude that a sufficient condition for ℰ_i_m and ℰ_j_l not to collide is δ(Δ^xy_i_m,j_l) + δ(Δ^xz_i_m,j_l) + δ(Δ^yz_i_m,j_l) > 0, since a collision would result in Δ^s_i_m,j_l = 0 ⇔δ(Δ^s_i_m,j_l) = 0, ∀ s∈{xy,xz,yz}. Next, we define the constant d̅_ B_i, which is the maximum distance of the base to a point in the agent's volume over all possible configurations, i.e. d̅_ B_i = sup_q_i∈ℝ^n{‖ p_ B_i(q_i) - p_i(q_i) ‖}, p_i∈⋃_m∈𝒬_iℰ_i_m(q_i). We also denote d̅_ B = [d̅_ B_1,…,d̅_ B_N]^⊤∈ℝ_≥ 0^N. Moreover, we consider that each agent has a sensor located at the center of its mobile base p_ B_i with a sensing radius d_con_i≥ 2max_i∈𝒱{d̅_ B_i} + ε_d, where ε_d is an arbitrarily small positive constant. Hence, each agent has the sensing sphere 𝒟_i(q_i)={p∈ℝ^3s.t. ‖ p - p_ B_i(q_i) ‖≤ d_con_i} and its neighborhood set at each time instant is defined as 𝒩_i(q_i) = {j∈𝒱\{i} s.t. ‖ p_ B_i(q_i) - p_ B_j(q_j)‖≤ d_con_i}. As mentioned in Section <ref>, we are interested in defining transition systems for the motion of the agents in the workspace in order to be able to assign complex high level goals through logic formulas. Moreover, since many applications necessitate the cooperation of the agents in order to execute some task (e.g. transport an object), we consider that a nonempty subset 𝒩_i ⊆𝒩_i(q_i(0)), i∈𝒱, of the initial neighbors of the agents must stay connected through their motion in the workspace. In addition, it follows that the transition system of each agent must contain information regarding the current position of its neighbors. The problem in hand is equivalent to designing decentralized control laws τ_i,i∈𝒱, for the appropriate transitions of the agents among the predefined regions of interest in the workspace.Next, we provide the following necessary definitions. An agent i∈𝒱 is in region k ∈𝒦 at a configuration q_i∈ℝ^n, denoted as 𝒜_i(q_i)∈π_k, if and only if ‖ p_i_m(q_i) - p_k ‖≤ r_k - max{α_i_m,β_i_m,c_i_m},∀ m∈𝒬_i ⇒‖ p_ B_i(q_i) - p_k ‖≤ r_k - d̅_ B_i. Agents i,j∈𝒱, with i≠ j, are in collision-free configurations q_i,q_j∈ℝ^n, denoted as 𝒜_i(q_i)≢𝒜_j(q_j), if and only if ℰ_i_m(q_i)∩ℰ_j_l(q_j)=∅, ∀ m∈𝒬_i,l∈𝒬_j. Given the aforementioned discussion, we make the following assumptions regarding the agents and the validity of the workspace:The regions of interest are * large enough such that all the robots can fit, i.e., given a specific k∈𝒦, there exist q_i, i∈𝒱, such that𝒜_i(q_i)∈π_k, ∀ i∈𝒱, with 𝒜_i(q_i)≢𝒜_j(q_j), ∀ i,j∈𝒱, with i≠ j. * sufficiently far from each other and the obstacle workspace, i.e., ‖ p_k - p_k'‖≥max_i∈𝒱{ 2d̅_ B_i} + r_k + r_k' + ε_p,r_0 -p_k ≥max_i∈𝒱{ 2d̅_ B_i},∀ k,k'∈𝒦, k≠ k', where ε_p is an arbitrarily small positive constant.Next, in order to proceed, we need the following definition. Assume that 𝒜_i(q_i(t_0))∈π_k, i∈𝒱, for some t_0∈ℝ_≥ 0,k∈𝒦, with 𝒜_i(q_i(t_0))≢𝒜_j(q_j(t_0)), ∀ j∈𝒱\{i}. There exists a transition for agent i between π_k and π_k', k'∈𝒦, denoted as (π_k,t_0)(π_k',t_f), if and only if there exists a finite time t_f≥ t_0, such that 𝒜_i(q_i(t_f))∈π_k' and 𝒜_i(q_i(t))≢𝒜_j(q_j(t)), ℰ_i_m(q_i(t))∩ℰ_i_ℓ(q_i(t)),ℰ_i_m(q_i(t))∩π_z = ∅,∀ m,ℓ∈𝒬_i, m≠ℓ, j∈𝒱\{i},z∈𝒦\{k,k'}, t∈[t_0,t_f]. Given the aforementioned definitions, the treated problem is the design of decentralized control laws for the transitions of the agents between two regions of interest in the workspace, while preventing collisions of the agents with each other, the workspace boundary, and the remaining regions of interest. More specifically, we aim to design a finite transition system for each agent of the form <cit.>𝒯_i = (Π, Π_i,0, , 𝒜𝒫_i, ℒ_i, ℱ_i),where Π = {π_1,…,π_K} is the set of regions of interest that the agents can be at, according to Def. <ref>, Π_i,0⊆Π is a set of initial regions that each agent can start from, ⊂(Π×ℝ_≥ 0)^2 is the transition relation of Def. <ref>, 𝒜𝒫_i is a set of given atomic propositions, represented as boolean variables, that hold in the regions of interest, ℒ_i:Π→2^𝒜𝒫_i is a labeling function, and ℱ_i:Π→Π^|𝒩_i | is a function that maps the region that agent i occupies to the regions the initial neighbors 𝒩_i of agent i are at. Therefore, the treated problem is the design of bounded controllers τ_i for the establishment of the transitions . Moreover, as discussed before, the control protocol should also guarantee the connectivity maintenance of a subset of the initial neighbors 𝒩_i,∀ i∈𝒱. Another desired property important in applications involving robotic manipulators, is the nonsingularity of the Jacobian matrix J_i:ℝ^n→ℝ^6× n, that transforms the generalized coordinate rates of agent i∈𝒱 to generalized velocities <cit.>. That is, the set 𝕊_i = {q_i∈ℝ^n s.t. (J_i(q_i)[J_i(q_i)]^⊤) = 0 } should be avoided, ∀ i∈𝒱.Formally, we define the problem treated in this paper as follows:Consider N mobile manipulators with dynamics (<ref>) and K regions of interest π_k,k∈𝒦, with q̇_i(t_0) < ∞, A_i(q_i(t_0))∈π_k_i, k_i∈𝒦, ∀ i∈𝒱 and 𝒜_i(q_i(t_0))≢𝒜_j(q_j(t_0)), ℰ_i_m(q_i(t_0))∩ℰ_i_ℓ(q_i(t_0)) =∅, ∀ i,j ∈𝒱,i≠ j, m,ℓ∈𝒬_i,m≠ℓ. Given nonempty subsets of the initial edge sets 𝒩_i⊆𝒩_i(q_i(0))⊆𝒱, ∀ i∈𝒱, the fact that (J_i(q_i(t_0))[J_i(q_i(t_0))]^⊤) ≠ 0, ∀ i∈𝒱, as well as the indices k'_i∈𝒦,i∈𝒱, such that ‖ p_k'_i - p_k'_j‖ + r_k'_i + r_k'_j≤ d_con_i, ∀ j∈𝒩_i,i∈𝒱, design decentralized controllers τ_i such that, for all i∈𝒱:* (π_k_i,t_0)(π_k'_i,t_f_i), for some t_f_i≥ t_0,* r_0 - (‖ p_ B_i(t) ‖+ d̅_ B_i) > 0,∀ t∈ [t_0,t_f_i],* j_i^*∈𝒩_i(q_i(t)), ∀ j_i^*∈𝒩_i, t∈ [t_0,t_f_i],* q_i(t)∈ℝ^n\𝕊_i, ∀ t∈ [t_0,t_f_i]. The aforementioned specifications concern 1) the agent transitions according to Def. <ref>, 2) the confinement of the agents in 𝒲, 3) the connectivity maintenance between a subset of initially connected agents and 4) the agent singularity avoidance. Moreover, the fact that the initial edge sets 𝒩_i are nonempty implies that the sensing radius of each agent i covers the regions π_k_j of the agents in the neighboring set 𝒩_i. Similarly, the condition ‖ p_k'_i - p_k'_j‖ + r_k'_i + r_k'_j≤ d_con_i, ∀ j∈𝒩_i, is a feasibility condition for the goal regions, since otherwise it would be impossible for two initially connected agents to stay connected. Intuitively, the sensing radii d_con_i should be large enough to allow transitions of the multi-agent system to the entire workspace.§ MAIN RESULTS§.§ Continuous Control DesignTo solve Problem <ref>, we denote as φ_i:ℝ^Nn→ℝ_≥ 0 a decentralized potential function, with the following properties: * The function φ_i(q) is not defined, i.e., φ_i(q) = ∞, ∀ i∈𝒱, when a collision or a connectivity break occurs, * The critical points of φ_i where the vector field ∇_q_iφ_i(q) vanishes, i.e., the points where ∇_q_iφ_i(q) = 0, consist of the goal configurations and a set of configurations whose region of attraction (by following the negated vector field curves) is a set of measure zero. * It holds that ∇_q_iφ_i(q) + ∑_j∈𝒩_i(q_i)∇_q_iφ_j(q) = 0 ⇔ ∇_q_iφ_i(q) = 0 and ∑_j∈𝒩_i(q_i)∇_q_iφ_j(q) = 0, ∀ i∈𝒩,q∈ℝ^Nn.More specifically, φ_i(q) is a function of two main terms, agoal function γ_i:ℝ^n→ℝ_≥ 0, that should vanish when 𝒜_i(q_i)∈π_k'_i, and an obstacle function,β_i:ℝ^n→ℝ_≥ 0 is a bounded that encodes inter-agent collisions, collisions between the agents and the obstacle boundary/undesired regions of interest, connectivity losses between initially connected agents and singularities of the Jacobian matrix J_i(q_i);β_i vanishes when one or more of the above situation occurs.Next, we provide an analytic construction of the goal and obstacle terms. However, the construction of the function φ_i is out of the scope of this paper. Examples can be found in <cit.>[In that case, we could choose φ_i = 11-ϕ_i, where ϕ_i is the proposed function of <cit.>] and <cit.>. §.§.§ γ_i - Goal Function Function γ_i encodes the control objective of agent i, i.e., reach the region of interest π_k'_i. Hence, we defineγ_i:ℝ^n→ℝ_≥ 0 asγ_i(q_i) = ‖ q_i - q_k'_i‖^2,where q_k'_i is a configuration such that r_k' - p_ B_i(q_k'_i) - p_k'_i≤d̅_ B_i - ε, for an arbitrarily small positive constant ε, which implies 𝒜_i(q_k'_i)∈π_k'_i, ∀ i∈𝒱. In case that multiple agents have the same target, i.e., there exists at least one j∈𝒱\{i} such that π_k'_j = π_k'_i, then we assume that 𝒜_i(q_k'_i)≢𝒜_j(q_k'_j). γ_i(η_i) = {η_i,η_i > (r_k'_i - d̅_ B_i)^2 ϕ_i(η_i), (r_k'_i - d̅_ B_i)^2 ≥η_i ≥ 0.where η_i:ℝ^n→ℝ_≥ 0, with η_i(q_i) = ‖ p_ B_i - p_k'_i‖^2, denotes the distance of agent i's base to the center of π_k'_i. Note that the condition η_i < (r_k'_i - d̅_ B_i)^2 is equivalent to 𝒜_i(q_i)∈π_k'_i. The term ϕ_i(η_i) is an appropriate polynomial that has its global minimum when η_i = c^2_i and guarantees that γ_i is non-negative and twice continuously differentiable everywhere. The constant c_i is chosen such that (r_k'_i - d̅_ B_i)^2 > c^2_i > d̅^2_ B_i, in order to ensure that, when the global minimum of γ_i is reached (i.e. ‖ p_ B_i - p_k'_i‖ = c_i), the bounding sphere of radius d̅_ B_i of agent i will not contain the center p_k'_i or region π_k'_i, i.e. p_k'_i∉⋃_m∈𝒬_iℰ_i_m(q_i), ∀ q_i∈{q_i∈ℝ^ns.t. γ_i(η_i(q_i))=0 }, where ⋃_m∈𝒬_iℰ_i_m(q_i)⊆{p∈ℝ^3s.t. ‖ p - p_ B_i‖ = d̅_ B_i, ‖ p_ B_i - p_k'_i‖ = c_i } (see Figs. <ref> and <ref>). In that way, by designing γ_i such that it is minimized in a circle around the center of π_k'_i, we facilitate cases where more than one agents have the same goal region. In these cases, the goal functions γ_i of the agents that have the same goal region will be minimized in different positions in the workspace, through the obstacle functions that are introduced next. Note also that Assumption <ref> ensures that a choice of c_i such that (r_k'_i - d̅_ B_i)^2 > c^2_i > d̅^2_ B_i is feasible.§.§.§ β_i - Collision/Connectivity/Singularity FunctionThe function β_i encodes all inter-agent collisions, collisions with the boundary of the workspace and the undesired regions of interest, connectivity between initially connected agents and singularities of the Jacobian matrix J_i(q_i),∀ i∈𝒱.Consider the function Δ_i_m,j_l:ℝ^2n→ℝ_≥ 0, with Δ_i_m,j_l(q_i,q_j) = δ(Δ^xy_i_m,j_l(q_i,q_j)) + δ(Δ^xz_i_m,j_l(q_i,q_j)) + δ(Δ^yz_i_m,j_l(q_i,q_j)), whereΔ^s_i_m,j_l:ℝ^2n→ℝ_≥ 0 is the discriminant of the cubic equation {λ E^s_i_m(q_i) - E^s_j_l(q_j)}=0, ∀ s∈{xy,xz,yz}, for two given ellipsoids ℰ_i_m and ℰ_j_l, m∈𝒬_i,l∈𝒬_j,i,j,∈𝒱, and δ as defined in (<ref>).As discussed in Section <ref>, a sufficient condition for the ellipsoids ℰ_i_m and ℰ_j_l not to collide, is Δ_i_m,j_l(q_i(t),q_j(t)) > 0, ∀ t∈ℝ_≥ 0.Additionally, we define the greatest lower bound of the Δ_i_m,j_l when the point p_j_l is on the boundary of the sensing radius ∂ D_i(q_i) of agent i, as Δ_i_m,j_l = inf_(q_i,q_j)∈ℝ^2n{Δ_i_m,j_l(q_i,q_j)} s.t. ‖ p_ B_i(q_i) - p_j_l(q_j) ‖ = d_con_i, ∀ m∈𝒬_i,l∈𝒬_j,i,j∈𝒱. Since d_con_i > 2max_i∈𝒱{d̅_ B_i} + ε_d, it follows that there exists a positive constant ε_Δ such that Δ_i_m,j_l≥ε_Δ > 0,∀ m∈𝒬_i,l∈𝒬_j,i,j∈𝒱, i≠ j.Moreover, we define the function Δ_i_m,π_k:ℝ^n→ℝ_≥ 0, withΔ_i_m,π_k(q_i) = δ(Δ^xy_i_m,π_k(q_i))+δ(Δ^xz_i_m,π_k(q_i))+δ(Δ^yz_i_m,π_k(q_i)), where Δ^s_i_m,π_k:ℝ^n→ℝ is the discriminant of the cubic equation (λ E^s_i_m(q_i) - T^s_π_k), with T^s_π_k the projected version of T_π_k in (<ref>), s∈{xy,xz,yz}, and δ as given in (<ref>). A sufficient condition for ℰ_i_m and region π_k, k∈𝒦 not to collide is Δ_i_m,π_k(q_i(t))>0, ∀ t∈ℝ_≥ 0, m∈𝒬_i,i∈𝒱.We further define the function η_ij,c:ℝ^n×ℝ^n→ℝ, with η_ij,c(q_i,q_j) = d^2_con_i-‖ p_ B_i(q_i) - p_ B_j(q_j)‖^2, and the distance functions β_i_m,j_l:ℝ_≥ 0→ℝ_≥ 0, β_ij,c:ℝ→ℝ_≥ 0, β_iw:ℝ_≥ 0→ℝ asβ_i_m,j_l(Δ_i_m,j_l)=ϕ_i,a(Δ_i_m,j_l),0 ≤Δ_i_m,j_l < Δ̅_i_m,j_l,Δ̅_i_m,j_l,Δ̅_i_m,j_l≤Δ_i_m,j_l,β_ij,c(η_ij,c)= 0,η_ij,c < 0, ϕ_i,c(η_ij,c), 0 ≤η_ij,c < d^2_con_i, d^2_con_i,d^2_con_i≤η_ij,c, β_iw(‖ p_ B_i‖^2)=(r_w - d̅_ B_i)^2 - ‖ p_ B_i‖^2, where Δ̅_i_m,j_l is a constant satisfying 0 < Δ̅_i_m,j_l≤Δ_i_m,j_l, ∀ m∈𝒬_i,l∈𝒬_j,i,j∈𝒱,i≠ j, and ϕ_i,a, ϕ_i,c are strictly increasing polynomials appropriately selected to guarantee that the functions β_i_m,j_l, and β_ij,c, respectively, are twice continuously differentiable everywhere, with ϕ_i,a(0) = ϕ_i,c(0) = 0, ∀ i∈𝒱.Note that the functions defined above use only local information in the sensing range d_con_i of agent i. The function β_i_m,j_l becomes zero when ellipsoid ℰ_i_m collides with ellipsoid ℰ_j_l, whereas β_ij,c becomes zero when agent i loses connectivity with agent j. Similarly, β_iw encodes the collision of agent i with the workspace boundary. Finally, we choose the function β_i:ℝ^Nn→ℝ_≥ 0 as β_i(q) =((J_i(q_i)[J_i(q_i)]^⊤))^2β_iw(‖ p_ B_i‖^2)∏_j∈𝒩_iβ_ij,c(η_ij,c) ∏_(m,j,l)∈Tβ_i_m,j_l(Δ_i_m,j_l)∏_ (m,k)∈LΔ_i_m,π_k(q_i),∀ i∈𝒱, where T = 𝒬_i×𝒱×𝒬_j, L = 𝒬_i×(𝒦\{k_i,k'_i}), and we have omitted the dependence on q for brevity.Note that we have included the term ((J_iJ^⊤_i))^2 to also account for singularities of J_i, ∀ i∈𝒱 and the term ∏_(m,j,l)∈Tβ_i_m,j_l(Δ_i_m,j_l) takes into account also the collisions between the ellipsoidal rigid bodies of agent i. With the introduced notation, the properties of the functions φ_i are: * β_i(q)→ 0 ⇔ (φ_i(q) →∞), ∀ i∈𝒱, * ∇_q_iφ_i(q)|_q_i=q^⋆_i = 0,∀ q^⋆_i∈ℝ^ns.t. γ_i(q^⋆_i) = 0 and the regions of attraction of the points {q ∈ℝ^Nn: ∇_q_iφ_i(q)|_q_i=q_i = 0, γ_i(q_i) ≠ 0 }, i∈𝒱, are sets of measure zero. By further denoting 𝔻_i = {q∈ℝ^Nn: β_i(q) > 0 }, we are ready to state the main theorem, that summarizes the main results of this work. Under the Assumptions <ref>-<ref>, the decentralized control laws τ_i: 𝔻_i×ℝ^n→ℝ^n, withτ_i(q,q̇_i) = g_i(q_i) - ∇_q_iφ_i(q) - ∑_j∈𝒩_i(q_i)∇_q_iφ_j(q)- ĉ_i(q_i,q̇_i)‖ q_i ‖q̇_i - λ_iq̇_i,∀ i∈𝒱, along with the adaptation laws ĉ̇_i:ℝ^n×ℝ^n→ℝ:ĉ̇_i(q_i,q̇_i) = σ_i‖q̇_i‖^2‖ q_i‖,with ĉ_i(q_i(t_0),q̇_i(t_0)) < ∞,σ_i∈ℝ_≥ 0 , ∀ i∈𝒱, guarantee the transitions (π_k_i,t_0)(π_k'_i,t_f_i) for finite t_f_i,i∈𝒱 for almost all initial conditions, while ensuring β_i > 0,∀ i∈𝒱, as well as the boundedness of all closed loop signals, providing, therefore, a solution to Problem <ref>. The closed loop system of (<ref>) is written as:M_i(q_i)q̈_i + N_i(q_i,q̇_i)q̇_i + f_i(q_i,q̇_i) = -∇_q_iφ_i(q_i) - λ_iq̇_i - ĉ(q_i,q̇_i)‖ q_i ‖q̇_i- ∑_j∈𝒩_i(q_i)∇_q_iφ_j(q),∀ i∈𝒱. Due to Assumption <ref>, the domain where the functions φ_i(q) are well-defined (i.e., where β_i > 0) is connected. Hence, consider the Lyapunov-like function V:ℝ^N×ℝ^Nn×ℝ^N×𝔻_1×…×𝔻_N→ℝ_≥ 0, withV(φ, q̇, c,q) = ∑_i∈𝒱φ_i(q) + 1/2[q̇^⊤_iM_i(q_i)q̇_i + 1/σ_ic_i^2]where φ and c are the stack vectors containing all φ_i and c_i, respectively, i∈𝒱, and c_i:ℝ^n×ℝ^n→ℝ, with c_i(q_i,q̇_i) = ĉ_i(q_i,q̇_i) - c_i, ∀ i∈𝒱. Note that, since there are no collision or singularities at t_0, the functions β_i(q), i∈𝒱, are strictly positive at t_0 which implies the boundedness of V at t_0. Therefore, since q̇_i(t_0)<∞ and ĉ_i(t_0)<∞, ∀ i∈𝒱, there exists a positive and finite constant M<∞ such that V_0 V(φ(q(t_0)), q̇(t_0), c̃(q(t_0),q(t_0)) ≤ M.By differentiating V, substituting the dynamics (<ref>), employing the skew symmetry of Ṁ_i - 2N_i as well as the property ∑_i∈𝒱 ( [∇_q_iφ_i(q)]^⊤q̇_i + ∑_j∈𝒩_i(q_i) [∇_q_jφ_i(q)]^⊤q̇_j ) = ∑_i∈𝒱( [∇_q_iφ_i(q) ]^⊤ +∑_j∈𝒩_i(q_i) [∇_q_iφ_j(q)]^⊤) q̇_i, we obtain V̇ = ∑_i∈𝒱{q̇^⊤_i ( ∇_q_iφ_i(q) + ∑_j∈𝒩_i(q_i)∇_q_iφ_j(q) + τ_i - g_i(q_i))-q̇_i^⊤ f_i(q_i,q̇_i) + 1/σ_ic_iĉ̇_i},which, by substituting the control and adaptation laws (<ref>), (<ref>) becomes:V̇ = ∑_i∈𝒱{ -λ_i‖q̇_i ‖^2 - ĉ_i‖q̇_i ‖^2 ‖ q_i ‖- q̇_i^⊤ f_i(q_i,q̇_i)+ c_i‖q̇_i ‖^2 ‖ q_i ‖ ≤ ∑_i∈𝒱{ -λ_i‖q̇_i ‖^2 - (ĉ_i-c_i-c_i)‖q̇_i ‖^2 ‖ q_i ‖where we have used the property ‖ f_i(q_i,q̇_i) ‖≤ c_i‖ q_i ‖‖q̇_i‖. Since c_i = ĉ_i - c_i, we obtain V̇≤ - ∑_i∈𝒱λ_i‖q̇_i‖^2,which implies that V is non-increasing along the trajectories of the closed loop system. Hence, we conclude that V(t)≤ V_0 ≤ M, as well as the boundedness of c_i, φ_i, q̇_i and hence of ĉ_i, ∀ i∈𝒱, t≥ t_0. Therefore, we conclude that β_i(q(t)) > 0, ∀ t≥ t_0, i∈𝒱. Hence, inter-agent collisions, collision with the undesired regions and the obstacle boundary, connectivity losses between the subsets of the initially connected agents and singularity configurations are avoided.Moreover, by invoking LaSalle's Invariance Principle, the system converges to the largest invariant set contained inS = {(q,q̇)∈𝔻_1×…×𝔻_N×ℝ^Nn s.t. q̇ = 0_Nn× 1}.For S to be invariant, we require that q̈_i = 0_n× 1, ∀ i∈𝒱, and thus we conclude for the closed loop system (<ref>) that ∇_q_iφ_i(q) = 0_n× 1, ∀ i∈𝒱,since f_i(q_i,0_n×1) ≤ 0, ∀ q_i∈ℝ^n, in view of Assumption <ref>. Therefore, by invoking the properties of φ_i(q), each agent i∈𝒱 will converge to a critical point of φ_i, i.e., all the configurations where ∇_q_iφ_i(q) = 0_n×1, ∀ i∈𝒱. However, due to properties of φ_i(q), the initial conditions that lead to configurations q_i such that∇_q_iφ_i(q)|_q_i=q_i = 0_n×1 and γ_i(q_i) ≠ 0 are sets of measure zero in the configuration space <cit.>. Hence, the agents will converge to the configurations where γ_i(q_i) = 0 from almost all initial conditions, i.e., lim_t→∞γ_i(q_i(t)) = 0. Therefore, since r_k' - p_ B_i(q_k'_i) - p_k'_i≤d̅_ B_i - ε, it can be concluded that there exists a finite time instance t_f_i such that 𝒜_i(q_i(t_f_i))∈π_k', ∀ i∈𝒱 and hence, each agent i will be at its goal region p_k'_i at time t_f_i, ∀ i∈𝒱. In addition, the boundedness of q_i,q̇_i implies the boundedness of the adaptation laws ĉ̇_i, ∀ i∈𝒱. Hence, the control laws (<ref>) are also bounded.Note that the design of the obstacle functions (<ref>) renders the control laws (<ref>) decentralized, in the sense that each agent uses only local information with respect to its neighboring agents, according to its limited sensing radius. Each agent can obtain the necessary information to cancel the term ∑_j∈𝒩_i(q_i)∇_q_iφ_j(q) from its neighboring agents.Finally, note that the considered dynamic model (<ref>) applies for more general manipulation robots (e.g. underwater or aerial manipulators), without limiting the proposed methodology to mobile ones.§.§ Hybrid Control Framework Due to the proposed continuous control protocol of Section <ref>, the transitions (π_k_i,t_0)(π_k'_i,t_f_i) of Problem <ref> are well-defined, according to Def. <ref>. Moreover, since all the agents i∈𝒱 remain connected with the subset of their initial neighbors 𝒱_i and there exist finite constants t_f_i, such that 𝒜_i(q_i(t_f_i))∈π_k'_i,∀ i∈𝒱, all the agents are aware of their neighbors state, when a transition is performed. Hence, the transition system (<ref>) is well defined, ∀ i∈𝒱. Consider, therefore, that 𝒜_i(q_i(0))∈π_k_i,0, k_i,0∈𝒦, ∀ i∈𝒱, as well as a given desired path for each agent, that does not violate the connectivity condition of Problem <ref>. Then, the iterative application of the control protocol (<ref>) for each transition of the desired path of agent iguarantees the successful execution of the desired paths, with all the closed loop signals being bounded.Note that, according to the aforementioned analysis, we implicitly assume that the agents start executing their respective transitions at the same time (we do not take into account individual control jumps in the Lyapunov analysis, i.e., it is valid only for one transition). Intuition suggests that if the regions of interest are sufficiently far from each other, then the agents will be able to perform the sequence of their transitions independently.Detailed technical analysis of such cases is part of our future goals.§ SIMULATION RESULTSTo demonstrate the validity of the proposed methodology, we consider the simplified example of three agents in a workspace with r_0 = 12 and three regions of interest, with r_k = 4, ∀ k∈{1,2,3} m. Each agent consists of a mobile base and a rigid link connected with a rotational joint, with d̅_ B_i = 1m, ∀ i∈{1,2,3}. We also choose p_1 = [-5,-5]m, p_2 = [6,-4]m, p_3 = [-3,6]m.The initial base positions are taken as p_ B_1 = [-3,-4]^⊤m, p_ B_2 = [3,-4]^⊤m, p_ B_3 = [-4,-5]^⊤m with d̅_ B_i = 1.25m, ∀ i∈{1,2,3}, which imply that 𝒜_1(q_1(0)),𝒜_3(q_3(0))∈π_1 and 𝒜_2(q_2(0))∈π_2 (see Fig. <ref>). The control inputs for the agents are the 2D force acting on the mobile base, and the joint torque of the link. We also consider a sensing radius of d_con_i = 8m and the subsets of initial neighbors as 𝒩_1 = {2}, 𝒩_2 = {1,3}, and 𝒩_3 = {2}, i.e., agent 1 has to stay connected with agent 2, agent 2 has to stay connected with agents 1 and 3 and agent 3 has to stay connected with agent 2. The agents are required to perform two transitions. Regarding the first transition, we choose π_k'_1 = π_2 for agent 1, π_k'_2 = π_1 for agent 2, and π_k'_3 = π_3, for agent 3. Regarding the second transition, we choose π_k'_1 = π_1, π_k'_2 = π_2, and π_k'_3 = π_2. The control parameters and gains where chosen as k_i = 5, λ_i =10, ρ_i=1, and σ_i = 0.01, ∀ i∈{1,2,3}. We employed the potential field from <cit.>. The simulation results are depicted in Fig. <ref>-<ref>. In particular, Fig. <ref> and <ref> illustrate the two consecutive transitions of the agents. Fig. <ref> depicts the obstacle functions β_i which are strictly positive, ∀ i∈{1,2,3}. Finally, the control inputs are given in Fig. <ref> and the parameter errors c are shown in Fig. <ref>, which indicates their boundedness. As proven in the theoretical analysis, the transitions are successfully performed while satisfying all the desired specifications.§ CONCLUSIONS AND FUTURE WORKS In this paper we designed decentralized abstractions for multiple mobile manipulators by guaranteeing the navigation of the agents among predefined regions of interest, while guaranteeing inter-agent collision avoidance and connectivity maintenance for the initially connected agents. We proposed a novel approach for ellipsoid collision avoidance as well as appropriately chosen potential functions that are free of undesired local minima. Future efforts will be devoted towards addressing abstractions of cooperative tasks between the agents by employing hybrid control techniques as well as abstraction reconfiguration due to potential execution incapability of the transitions. ieeetr | http://arxiv.org/abs/1703.08692v3 | {
"authors": [
"Christos K. Verginis",
"Dimos V. Dimarogonas"
],
"categories": [
"cs.SY"
],
"primary_category": "cs.SY",
"published": "20170325133421",
"title": "Robust Decentralized Abstractions for Multiple Mobile Manipulators"
} |
A. R. Booker]Andrew R. Booker Department of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, United Kingdom, e-mail: [email protected] H. Then]Holger Then Alemannenweg 1, 89537 Giengen, Germany, e-mail: [email protected] Let L be a degree-2 L-function associated to a Maass cusp form. We explore an algorithm that evaluates t values of L on the critical line in time O(t^1+). We use this algorithm to rigorously compute an abundance of consecutive zeros and investigate their distribution. The authors wish to express their thanks to Andreas Strömbergsson and Pankaj Vishe for offering deep insight into their methods.H. T. thanks Brian Conrey, Dennis Hejhal, Jon Keating, Anton Mellit, and Franzesco Mezzadri for inspiring discussions. A. B. and H. T. acknowledge support from EPSRC grant EP/H005188/1. Rapid computation of L-functions attached to Maass forms [ December 30, 2023 ========================================================§ INTRODUCTION In <cit.>, the first author presented an algorithm for the rigorous computation of L-functions associated to automorphic forms. The algorithm is efficient when one desires many values of a single L-function or values of many L-functions with a common Γ-factor. In this paper, we explore the prototypical case of a family of L-functions to which that does not apply, namely Maass cusp forms in the eigenvalue aspect.As described in <cit.>, one of the main challenges when computing L-functions is the evaluation of the inverse Mellin transform of the associated Γ-factor. Rubinstein <cit.> describes an algorithm based on continued fractions that performs well in practice, but for which it seems to be very difficult to obtain rigorous error bounds. On the other hand, the algorithm in <cit.>, following Dokchitser <cit.>, uses a precomputation based on simpler power series expansions that are easy to make rigorous; it works well for motivic L-functions of low weight, but suffers from catastrophic precision loss when the shifts in the Γ-factor grow large, as is the case for Maass forms.A well-known similar problem occurs when one attempts to evaluate an L-function high up in the critical strip. Rubinstein, following an idea of Lagarias and Odlyzko <cit.>, has demonstrated that this can be dealt with effectively by multiplying by an exponential factor to compensate for the decay of the Γ-factor; specifically, for a complete L-function Λ(s) of degree d, one works with e^-iθ sΛ(s) for a suitable θ<π d/4.This idea can be made to work for general L-functions, including those associated to Maass forms (albeit with the problems related to precision loss noted above, if the Γ-factor is not fixed), and Molin <cit.> has worked out rigorous numerical methods in quite wide generality.For the specific case of Maass cusp forms, Vishe <cit.> (see also <cit.>) has shown that the “right” factor to multiply by to account for the variation in both t and the Γ-factor shifts is not the exponential function e^-iθ s, but rather the hypergeometric functions+ϵ+ir/2s+ϵ-ir/21/2+ϵ -tan^2θwhere ϵ denotes the parity of the Maass form, and 1/4+r^2 is its Laplacian eigenvalue. To understand the motivation for this factor, consider first the case of a classical holomorphic cuspform f, for which the L-function is defined via the Mellin transformΛ(s)=∫_0^∞ f(iy)y^sdy/y.Since f is holomorphic and vanishes in the cusp, we can change the contour of integration from (0,∞) to e^iθ(0,∞) for some θ∈(-π/2,π/2); writing y=e^iθu for u∈, we obtaine^-iθ sΛ(s)=∫_0^∞ f(ie^iθu)u^sdu/u.Thus, Rubinstein's exponential factor arises naturally from a contour rotation.For a Maass form f of weight 0 and even parity (say), we similarly have the integral representationΛ(s)=∫_0^∞ f(iy)y^s-1/2dy/y.Since f is no longer holomorphic in this case, we cannot simply rotate the contour in this expression, but we are free to start with the rotated integral ∫_0^∞ f(ie^iθu)u^s-1/2du/u and try to relate it back to Λ(s). As we show in <ref>, this can be done, and the two are related essentially by the factor (<ref>).We analyze this strategy in greater detail in <ref>, but the upshot is that to compute Maass form L-functions for a wide range of values of r and t, it suffices to compute f(ie^iθu) for suitable values of θ and u.In turn, using modularity to move each point to the fundamental domain, the problem reduces to computing the K-Bessel function K_ir(y) for various r and y. Fortunately, that is a problem that underlies all computational aspects of Maass cusp forms and has been well studied; see, for instance, <cit.>. §.§ Numerical results In <ref>, using as input the rigorous numerical Maass form data of <cit.> we compute values of the corresponding Maass form L-functions and use the resulting numerical data to test conjectures about the distribution of zeros of Maass form L-functions in the t- and r-aspects.In particular, we show that the phenomenon of zero repulsion around 1/2± ir that Strömbergsson observed <cit.> disappears in the large eigenvalue limit.We derive rigorous error estimates and use the interval arithmetic package MPFI <cit.> throughout our computations to manage round-off errors. Thus, modulo bugs in the software or hardware, our numerical results are rigorous.§.§ Prior results For a Maass form L-function on Γ_1(N), the zeros obey: * By absolute convergence of the Euler product, there are no zeros for s>1.* There are no zeros on the line s=1.* For s≤0 there are trivial zeros which arise from the functional equation of the completed L-function and the poles of the Γ-factor. The trivial zeros are the only zeros for s≤0, and they are located at s=-2n-ϵ-ir and s=-2n-ϵ+ir, ∀ n∈_≥0.* Inside the critical strip, {s∈:0< s<1}, there are countably many zeros.All the zeros inside the critical strip are non-trivial.* According to the Grand unified Riemann Hypotheses (GRH) all non-trivial zeros of Maass form L-functions on Γ_1(N) are conjectured to be on the critical line, {s∈: s=1/2}.* All non-trivial zeros are conjectured to be of order one.* Assuming GRH, the distribution of non-trivial zeros follows random matrix theory (RMT) predictions <cit.>.* Strömbergsson <cit.> has observed a zero-free region on the critical line for t around r.§ PRELIMINARIES ON MAASS FORMS Let f∈ L^2(Γ_1(N)\)̋ be a cuspidal Maass newform and Hecke eigenform of weight 0 and level N. Then f has a Fourier expansion of the formf(x+iy)=√(y)∑_n=1^∞ a_nK_ir(2π ny)cos^(-ϵ)(2π nx),where a_n is the nth Hecke eigenvalue of f, 1/4+r^2 is its eigenvalue for the hyperbolic Laplacian -y^2(∂^2/∂ x^2 +∂^2/∂ y^2), ϵ∈{0,1} indicates the parity, andcos^(-ϵ)(x):= cosx if ϵ=0, sinx if ϵ=1.Moreover, f is related to its dual f̅ via the Fricke involution, so thatf(z)=wf(-1/Nz),for some complex number w with |w|=1.Associated to f we have the L-function L(s,f), defined for (s)>1 by the seriesL(s,f):=∑_n=1^∞a_n/n^s.It follows from (<ref>) that L(s,f) continues to an entire function and satisfies a functional equation relating its values at s and 1-s̅. To see this, let _2F_1 denote the Gauss hypergeometric functionαβγz:= 1+∑_n=1^∞(α)_n(β)_n z^n/(γ)_n n!,and consider the family of Γ-factorsγ_θ(s,f):= i^-ϵw^-1/2(cosθ/√(N))^1/2-sΓ_(s+ϵ+ir)Γ_(s+ϵ-ir)·s+ϵ+ir/2s+ϵ-ir/21/2+ϵ-tan^2θ,where Γ_(s):=π^-s/2Γ(s/2) and θ∈(-π/2,π/2) is a parameter. By <cit.>, we haveγ_θ(s,f) = 4 i^-ϵ w^-1/2(cosθ/√(N))^1/2-s1/π^s+ϵ∫_0^∞ K_ir(2t) cos^(-ϵ)(2tan(θ)t)/(2tan(θ)t)^ϵ t^s+ϵdt/tfor s>0. (Note that for a Maass form with odd reflection symmetry, viz. ϵ=1, (<ref>) has a removable singularity at θ=0; this is related to the fact that the complete L-function is the Mellin transform of ∂f/∂x rather than f.) Making the substitution t↦π ncos(θ)u, we can express the complete L-function Λ_θ(s,f):=γ_θ(s,f)L(s,f) as the Mellin transform of the Maass form along a ray in the upper half plane:Λ_θ(s,f) = γ_θ(s,f) L(s,f)= c_θ(s,f) ∑_n=1^∞ a_n ∫_0^∞ (cos(θ)u)^1/2 K_ir(2π ncos(θ)u) cos^(-ϵ)(2π nsin(θ)u) u^s-1/2du/u = c_θ(s,f) ∫_0^∞ f(ie^iθu) u^s-1/2du/u, wherec_θ(s,f) := 4 w^-1/2/(2π itanθ)^ϵ N^1/2(s-1/2).Splitting the integral at u=1/√(N) and employing (<ref>) completes the analytic continuation:Λ_θ(s,f) = c_θ(s,f) {∫_1/√(N)^∞ f(ie^iθu) u^s-1/2du/u + ∫_0^1/√(N) wf(-1/Nie^iθu) u^s-1/2du/u}.Using that f(-z) = (-1)^ϵ f(z) and making the substitution u↦1/Nu, we obtain the functional equation:Λ_θ(s,f)=c_θ(s,f) ∫_1/√(N)^∞ f(ie^iθu) u^s-1/2du/u + c_θ(1-s,f)∫_1/√(N)^∞f(ie^iθu) u^1/2-sdu/u= Λ_θ(1-s,f).In particular, Λ_θ(1/2+it,f)∈ for t∈.§ RIGOROUS COMPUTATION OF L-FUNCTIONS We describe an algorithm based on the fast Fourier transform that allows one to evaluate Λ_θ(s,f) quickly, if one is interested in many points.The integral (<ref>) is essentially a Fourier transformation,<ref>a Λ_θ(σ+it,f) = c_θ(σ+it,f) ∫_-∞^∞ f(ie^iθe^u) e^u(σ-1/2) e^iut du.Similarly for the integral (<ref>),<ref>aγ_θ(σ+it,f) = c_θ(σ+it,f) ∫_-∞^∞ (cos(θ)e^u)^1/2 K_ir(2πcos(θ)e^u) cos^(-ϵ)(2πsin(θ)e^u) e^u(σ-1/2) e^iut du. In order to use the fast Fourier transform to computeg(t) = ∫_-∞^∞ĝ(u) e^iut du,we first need to discretize the integral.To that end, let A,B>0 be parameters such that q=AB is an integer.In the Poisson summation formula,∑_k∈ g(m/A+kB) = 2π/B∑_l∈ e(ml/q) ĝ(2π l/B),we solve for g(m/A), which results ing(m/A) = 2π/B∑_l=-C'^C e(ml/q) ĝ(2π l/B) + _g, _g := 2π/B∑_{l∈:l<-C' or l>C} e(ml/q) ĝ(2π l/B) - ∑_k≠0 g(m/A+kB).In <ref> we will derive precise bounds for this error term.§ BOUNDS Let Q(s,f) be the analytic conductor, defined byQ(s,f) := N s+ϵ+ir/2πs+ϵ-ir/2π.Note that γ(s,f) satisfies the recurrence γ(s+2,f) = Q(s,f) γ(s,f). Further, we defineχ(s,f) := γ(1-s,f)/γ(s,f),so that L(s,f) = χ(s,f) L(1-s,f).<cit.> For s in the strip {s∈ : -1/2≤ s≤3/2},L(s,f)^2 ≤χ(s,f)Q(s,f)sup_t∈L(32+it,f)^2.The estimate in Lemma <ref> is not optimal since, for s=1/2+it for large t, the right-hand side grows quadratically in t, whereas the convexity estimate would give O(t^1+ε). Moreover, for ϵ=0 and s=1± ir the bound becomes useless, since L(1± ir,f)<∞, whereas lim_s→1± irχ(s,f)Q(s,f)→∞. Nevertheless, the estimate is clean and uniform in all parameters, and suffices for our purposes if we keep away from s=1± ir.For s in the strip {s∈ : 1/2≤ s<1},L(s,f)≤ 3N^1/2( s+D_s,f)withD_s,f := 3 s -1 + ϵ + r + (2 s-1)^2/1- s+ϵ. Recall that Γ_(s) satisfies the recurrence, reflection, and duplication formulass Γ_(s) = 2πΓ_(2+s), Γ_(s)Γ_(2-s) = 1/sin(π/2s), andΓ_(s)Γ_(1+s) = 2^1-sΓ_(2s).Hence, for ϵ∈{0,1} and t∈,Γ_(2+ϵ-it)/Γ_(1+ϵ+it)^2 = .t^2 cosh(π/2t)/2π t sinh(π/2t) for ϵ=0(1+t^2) sinh(π/2t)/2π t cosh(π/2t) for ϵ=1 }≤(1+ϵ)^2+t^2/π^2,which yieldsχ(s,f)Q(1-s,f)≤ 4N^-1/2Q(s,f)for s=1.On the critical line we haveχ(s,f)Q(1-s,f)/Q(s,f)_ s=1/2 = 1,and by the Phragmén-Lindelöf theorem,sup_1/2≤ s≤1χ(s,f)Q(1-s,f)/Q(s,f)≤max{4N^-1/2,1}≤ 4.Thus, in the strip 1/2≤ s<1 we haveQ(s,f)^2/ Q(1-s,f) ≤ N(s+D_s,f/2π)^2.Using the Kim–Sarnak estimate <cit.> p^-ϑ≤α_p≤ p^ϑ with ϑ=7/64 in the Euler product givessup_ s=3/2L(s,f) = sup_ s=3/2∏_p1/(1-α_pp^-s)(1-α_p^-1p^-s)≤ζ(32+ϑ)ζ(32-ϑ) < 3π.Inserting the last three bounds in Lemma <ref> yields the corollary.For s=σ+it with 0<σ≤1 and 0<θ<δ<π/2,γ_θ(σ+it,f) < E_σ,θ,δ e^-(δ-θ)twithE_σ,θ,δ := c_θ(σ,f)/(cos(δ-θ))^1/2{cosh(1)σ^-1(σ^-1+log(2)+e^-1)/(2π)^σ (cos(δ-θ)cosθ)^σ-1/2 + Γ(σ-1/2, cosδ/cos(δ-θ)cosθ)/ 2(2πcosδ)^σ-1/2}. For γ_θ we have the integral representation (<ref>). Since γ_θ(σ-it,f) = γ_θ(σ+it,f), it is enough to prove the lemma for non-negative t.Making the change of variables u↦ u+i(δ-θ) and moving the contour of integration back to the real line, we getg(t) := γ_θ(σ+it,f) = ∫_ĝ(u) e^iut du = ∫_ĝ(u+i(δ-θ)) e^i(u+i(δ-θ))t du,andg(t)≤ e^-(δ-θ)t{∫_{u∈:2π e^ucos(δ-θ)cosθ < 1}ĝ(u+i(δ-θ)) du + ∫_{u∈:2π e^ucos(δ-θ)cosθ≥ 1}ĝ(u+i(δ-θ)) du}.We bound the first integral using the estimates K_ir(z)≤log(2/ z)+e^-1 <cit.> and cos^(-ϵ)(tan(δ-θ)tan(θ) z)≤cosh(1), and the second integral using K_ir(z)<(π/2 z)^1/2 e^- z and cos^(-ϵ)(z)≤ e^ z.For s=σ+it with 0<σ≤1, t=m/A≤ T≤B/2, and 0<θ<δ<π/2,∑_k≠0γ_θ(σ+i(m/A+kB),f) < E_σ,θ,δ/sinh((δ-θ)B/2).Using Lemma <ref> together with m/A+kB≥(k-1/2)B yields the stated bound.For s=σ+it with 1/2≤σ<1, t=m/A≤ T≤B/2, and 0<θ<δ<π/2,∑_k≠0Λ_θ(σ+i(m/A+kB),f) < 3N^1/2E_σ,θ,δ/sinh((δ-θ)B/2)( B/2+D_σ,f+B/1-e^-(δ-θ)B).By Corollary <ref> and Lemma <ref>,Λ_θ(σ+it,f) < 3N^1/2(t+D_σ,f) E_σ,θ,δe^-(δ-θ)t.Applying the estimate (k-1/2)B≤m/A+kB≤(k+1/2)B and summing up results in the stated bound.For 1/2≤σ≤1, B>0, C≥B/2πlog1+B/(2π)/4πcosθ, C'=C+B/2πlog N, and Λ̂_σ,θ(u,f):=c_θ(σ,f)f(ie^iθe^u) e^u(σ-1/2),2π/B∑_{l∈:l<-C' or l>C} e(ml/q) Λ̂_σ,θ(2π l/B,f) < 2π/B56 N^1/4/(2πtanθ)^ϵ e^πC/B - 2πcos(θ)e^2πC/B.Applying a_n≤2n^1/2, K_ir(y)<(π/2y)^1/2e^-y, and cos^(-ϵ)(x)≤1 in the Fourier expansion of the Maass form givesf(x+iy)<1/e^2π y-1,and by Fricke involutionf(ie^iθe^u)=f(ie^-iθe^-u/N) <1/e^2πcos(θ)max{e^u,e^-u/N}-1. For σ≤1,2π/B∑_l>C e(ml/q) Λ̂_σ,θ(2π l/B,f) < 2π/B∑_l>C4N^1/4/(2πtanθ)^ϵe^πl/B/e^2πcos(θ)e^2π l/B-1.Writing a:=2πcosθ, u:=2πl/B, u_0:=2πC/B, with a>0 and u_0<u, we havee^-a e^u+u/2/1-e^-a e^u < e^-a e^u_0(1+u-u_0)+u/2/1-e^-a e^u_0.Summing the resulting geometric series gives2π/B∑_l>C e(ml/q) Λ̂_σ,θ(2π l/B,f)< 2π/B4N^1/4/(2πtanθ)^ϵe^-a e^u_0+u_0/2/(1-e^-a e^u_0) (1-e^-(a e^u_0-1/2)2π/B) < 2π/B4N^1/4/(2πtanθ)^ϵ 7 e^-a e^u_0+u_0/2for 2π/B(a e^u_0-1/2) ≥1/2, and similarly for the sum over l<-C.For 1/2≤σ≤1,2π/B∑_l<-C' e(ml/q) Λ̂_σ,θ(2π l/B,f) < 2π/B∑_l>C'4N^1/4/(2πtanθ)^ϵ1/e^2πcos(θ)e^2π l/B/N-1.Writing a/N:=2πcosθ/N, u:=2πl/B, u_0':=u_0+log N=2πC'/B, with a>0 and u_0'<u, we havee^-a/Ne^u/1-e^-a/Ne^u < e^-a/Ne^u_0'(1+u-u_0')/1-e^-a/Ne^u_0',and summing up gives2π/B∑_l<-C' e(ml/q) Λ̂_σ,θ(2π l/B,f) < 2π/B4N^1/4/(2πtanθ)^ϵe^-a e^u_0/(1-e^-a e^u_0) (1-e^-a e^u_02π/B). For 1/2≤σ≤1, B>0, C≥B/2πlog1+B/(2π)/4πcosθ, C'≥3B/2π, and γ̂_σ,θ(u,f) := c_θ(σ,f) (cos(θ)e^u)^1/2 K_ir(2πcos(θ)e^u) cos^(-ϵ)(2πsin(θ)e^u) e^u(σ-1/2),2π/B∑_{l∈:l<-C' or l>C} e(ml/q) γ̂_σ,θ(2π l/B,f)< 2π/BN^1/4/(2πtanθ)^ϵ{ 6 e^πC/B-2πcos(θ)e^2πC/B + 23 (cosθ)^1/6(π r/2) e^-π C'/3B}. We have K_ir(y)<(π/2y)^1/2e^-y and cos^(-ϵ)(x)≤1, so that2π/B∑_l>C e(ml/q) γ̂_σ,θ(2π l/B,f) < 2π/B∑_l>C4N^1/4/(2πtanθ)^ϵ 2^-1 e^-2πcos(θ)e^2πl/B e^πl/B.Writing a:=2πcosθ, u:=2πl/B, u_0:=2πC/B, with a>0 and u_0<u, we havee^-a e^u + u/2 < e^-a e^u_0(1+u-u_0) + u/2,and summing the geometric series yields2π/B∑_l>C e(ml/q) γ̂_σ,θ(2π l/B,f)< 2π/B4N^1/4/(2πtanθ)^ϵ 2^-1e^-a e^u_0 + u_0/2/1 - e^-(a e^u - 1/2)2π/B < 2π/B6N^1/4/(2πtanθ)^ϵ e^-a e^u_0 + u_0/2for 2π/B(a e^u_0-1/2)≥1/2.The argument is slightly different for the sum over l<-C. Using that cosh(π r/2)K_ir(y)<4y^-1/3 <cit.> and cos^(-ϵ)(x)≤1, we have2π/B∑_l<-C' e(ml/q) γ̂_σ,θ(2π l/B,f)< 2π/B∑_l>C'4N^1/4/(2πtanθ)^ϵ4(2π)^-1/3/cosh(π r/2) (cos(θ)e^-2πl/B)^1/6 < 2π/B23N^1/4/(2πtanθ)^ϵ (cosθ)^1/6(π r/2) e^-π C'/3Bforπ C'/3B≥1/2. The following two Lemmata show that we can circumvent division by zero in L(s,f)=Λ_θ(s,f)/γ_θ(s,f). For any y∈ with y>5, Γ_(x+iy) is an increasing function of x∈. It suffices to show that ∂/∂ xlogΓ_(x+iy) = ψ_(x+iy) > 0. Suppose first that x≥1.Then, differentiating Binet's formula for logΓ_, we haveψ_(x+iy) = 1/2log(x^2+y^2) - x/x^2+y^2 - 1/2log(4π) - ∫_0^∞( 1 - 1/t + 2/e^2t-1) e^-xtcos(yt) dt ≥1/2log(x^2+y^2) - x/x^2+y^2 + ψ_(1) + 1.For x≥1, this is easily seen to be minimum at x=1, so we obtainψ_(x+iy) ≥1/2log(1+y^2) - 1/1+y^2 + ψ_(1) + 1. For x≤1 we use the reflection formula ψ_(z)=ψ_(2-z)-π(π/2z) to see thatψ_(x+iy) = ψ_(2-(x+iy)) - π(π2(x+iy)),and apply the above to obtain a bound for ψ_(2-(x+iy)). We calculate that(π2(x+iy)) = 2e^π ysin(π x)/1 - 2e^π ycos(π x) + e^2π y,and with a little calculus we see that this is bounded in modulus by 1/sinh(πy).Thus, altogether we haveψ_(x+iy) ≥1/2log(1+y^2) - 1/1+y^2 - π/sinh(πy) + ψ_(1) + 1for all x∈ and y≠0.Note that the right-hand side is strictly increasing for y>0.Using the known value ψ_(1)=-γ-1/2log(16π)=-2.53587…, it is straightforward to see that ψ_(x+iy) is positive for y>5.For r>5, 1/2≤σ≤1, 0<θ_1<θ_2<π/2, cosθ_1≤(4+t^2-r^2)^-1/2, and cosθ_2=e^-π/2rcosθ_1,maxγ_θ_1(s,f),γ_θ_2(s,f)≥2/3 (cosθ_2)^1/2+ϵ(2π)^-ϵ( π/rsinh(π r))^1/2. We follow the proof of <cit.>, generalizing it and making the implied constants explicit.Using <cit.> and writingg_θ(s,f) := Γ_(s+ϵ+ir) Γ_(-2ir)Γ_(1+2ϵ)/Γ_(1-s+ϵ-ir)s+ϵ+ir/21-s+ϵ+ir/2ir+1cos^2θ,we getγ_θ(s,f) = i^-ϵ w^-1/2 N^1/2(s-1/2){ (cosθ)^1/2+ϵ+ir g_θ(s,f) + (cosθ)^1/2+ϵ-irg_θ(s,f)}.By Lemma <ref>, for σ≥1/2 and t+r>5,Γ_(s+ϵ+ir)/Γ_(1-s+ϵ-ir) = Γ_(2σ-1+1-σ+ϵ+i(t+r))/Γ_(1-σ+ϵ-i(t+r))≥ 1.Next,Γ_(-2ir)Γ_(1+2ϵ) = (2π)^-ϵ( π/rsinh(π r))^1/2,and for 1/2≤σ≤1,(s+ϵ+ir/2+n)(1-s+ϵ+ir/2+n)/ (ir+1+n)(1+n) < 1+t^2-r^2/4.Hence for cos^2θ≤(4+t^2-r^2)^-1,s+ϵ+ir/21-s+ϵ+ir/2ir+1cos^2θ = 1 + ∑_n=1^∞Θ(1/4^n) = 1 + Θ( 1/4/1-1/4),where Θ(x) stands for a value of absolute size at most x.r>5 implies that t+r>5 or t-r>5.Thusmaxg_θ(s,f),g_θ(s,f)≥ (2π)^-ϵ( π/rsinh(π r))^1/2 (1-Θ(13)).Adjusting the phase factor (cosθ)^ir in (<ref>) suitably, i.e. taking θ=θ_1 and θ=θ_2 with (cosθ_2)^2ir=e^-iπ(cosθ_1)^2ir, respectively, completes the proof. § INTERPOLATING ZEROS We compute values of L on a grid, but we are ultimately interested in the zeros, which are not regularly spaced. To zoom in on the zeros, we interpolateh(t) := e^-(t-t_0)^2/2b^2 g(t)with g(t)=Λ_θ(σ+it,f) and g(t)=γ_θ(σ+it,f), respectively. The function h has the advantage that it decays rapidly at ∞ and is approximately bandlimited, which allows us to use the Whittaker–Shannon sampling theorem <cit.>h(t) - ∑_m∈ h(m/A) (π A(t-m/A))≤ 4∫_π A^∞ĥ(u) du.Truncating the sum over m and bounding the error of truncation, we get an effective interpolation formula for h. For g(t):=Λ_θ(σ+it,f), G:=max_t≤ Tg(t), 1/A≤ J≤ T-t_0, and b>0,∑_m/A-t_0>J h(m/A) (π A(t-m/A)) < √(2π)bA{ G(J-1/A/√(2)b) .. + 3N^1/2E_σ,θ,δe^-(δ-θ)T{2b/√(2π) e^-(T-t_0-1/A)^2/2b^2 + (t_0+D_σ,f) (T-t_0-1/A/√(2)b) }}. For t≤ T, we have g(t)≤ G, and for t>T, we bound g(t) by (<ref>).Hence,∑_m/A-t_0>J h(m/A) (π A(t-m/A))≤∑_m/A-t_0>Je^-(m/A-t_0)^2/2b^2g(m/A) < ∑_m/A-t_0>Jm/A≤ T e^-(m/A-t_0)^2/2b^2 G + ∑_m/A-t_0>Jm/A>T e^-(m/A-t_0)^2/2b^2 3N^1/2(m/A+D_σ,f) E_σ,θ,δe^-(δ-θ)m/A < G ∫_m/A-t_0>J-1/A e^-(m/A-t_0)^2/2b^2 dm + 3N^1/2 E_σ,θ,δe^-(δ-θ)T∫_m/A>T-1/A e^-(m/A-t_0)^2/2b^2(m/A+D_σ,f) dm. For g(t):=γ_θ(σ+it,f), G:=max_t≤ Tg(t), 1/A≤ J≤ T-t_0, and b>0,∑_m/A-t_0>J h(m/A) (π A(t-m/A)) < √(2π)bA{ G(J-1/A/√(2)b) .. + E_σ,θ,δe^-(δ-θ)T(T-t_0-1/A/√(2)b) }.The proof is similar to that of the previous lemma, except that for t>T we bound g(t) by Lemma <ref>.For g(t):=Λ_θ(σ+it,f), 1/2≤σ≤1, u_0≥0, π A≥ u_0, and b>0,4∫_π A^∞ĥ(u) du < 2c_θ(σ,f)e^u_0(σ-1/2)/1-e^-2πcos(θ)/√(N){√(2π)/2b(b/√(2)(π A-u_0)) .. + 2e^-u_0(σ-1/2)/(2πcosθ)^σ-1/2Γ(σ-12,2πcos(θ)e^u_0). } By Fourier convolution,ĥ(v) = b/√(2π)∫_ e^-b^2/2(v-u)^2-i(v-u)t_0ĝ(u) duwith ĝ(u):=c_θ(σ,f)f(ie^iθe^u)e^u(σ-1/2), as defined in (<ref>a). Using (<ref>) and writing a:=2πcosθ givesĥ(v) < b/√(2π)∫_ e^-b^2/2(v-u)^2c_θ(σ,f)e^u(σ-1/2)/ e^amax{e^u,e^-u/N}-1 du.Changing the order of integration, we have4∫_π A^∞ĥ(v) dv < 4b/√(2π)c_θ(σ,f)∫_du e^u(σ-1/2)/e^amax{e^u,e^-u/N}-1∫_π A^∞ dv e^-b^2/2(v-u)^2 = 2c_θ(σ,f)∫_e^u(σ-1/2)(b/√(2)(π A-u))/e^amax{e^u,e^-u/N}-1 du. Let u_0∈[0,π A] and set x:=b/√(2)(π A-u). For u<u_0, we have x>0 and (x)<e^-x^2, while for u≥ u_0, (x)<2. Moreover, for u<u_0, a>0, σ≥1/2,e^u(σ-1/2)/e^amax{e^u,e^-u/N}-1 < e^u_0(σ-1/2)/1-e^-a/√(N),while for u≥ u_0,1/e^amax{e^u,e^-u/N}-1 < 1/(1-e^-a/√(N))e^a e^u.Thus,4∫_π A^∞ĥ(v) dv < 2c_θ(σ,f)e^u_0(σ-1/2)/1-e^-a/√(N){∫_-∞^u_0 e^-b^2/2(π A-u)^2 du+ 2e^-u_0(σ-1/2)∫_u_0^∞e^u(σ-1/2)/e^a e^u du }.Identifying the integrals with the complementary error function and the incomplete gamma function completes the proof.For g(t):=γ_θ(σ+it,f), 1/2≤σ≤1, u_0≥0, π A≥ u_0, and b>0,4∫_π A^∞ĥ(u) du < 2c_θ(σ,f){2/b e^u_0(σ-1/2) (2πcos(θ)e^u_0)^1/6(π r/2) (b/√(2)(π A-u_0)) .. + (2πcosθ)^1/2-σΓ(σ-12,2πcos(θ)e^u_0) }.By Fourier convolution,ĥ(v) = b/√(2π)∫_ e^-b^2/2(v-u)^2-i(v-u)t_0ĝ(u) du,with ĝ(u) := c_θ(σ,f) (cos(θ)e^u)^1/2 K_ir(2πcos(θ)e^u) cos^(-ϵ)(2πsin(θ)e^u) e^u(σ-1/2), as in (<ref>a). For u<u_0, we have cosh(π r/2)K_ir(y)<4y^-1/3 <cit.>, cos^(-ϵ)(x)≤1, and e^u(σ-1/2)<e^u_0(σ-1/2). Hence, writing a/2π:=cosθ>0, we have∫_-∞^u_0 e^-b^2/2(v-u)^2ĝ(u) du < c_θ(σ,f) e^u_0(σ-1/2)∫_-∞^u_0 e^-b^2/2(v-u)^24(2π)^-1/3/cosh(π r/2)(a/2π e^u)^1/6 du.For u≥ u_0≥ 0, we use K_ir(y)<(π/2y)^1/2e^-y, and cos^(-ϵ)(x)≤1 to obtain∫_u_0^∞ e^-b^2/2(v-u)^2ĝ(u) du < c_θ(σ,f)∫_u_0^∞ e^-b^2/2(v-u)^21/2 e^-a e^u e^u(σ-1/2) du.Therefore, after interchanging the order of integration,4∫_π A^∞h(v) dv < 4b/√(2π)c_θ(σ,f){ e^u_0(σ-1/2)∫_-∞^u_04(2π)^-1/2/cosh(π r/2) (a e^u)^1/6∫_π A^∞ e^-b^2/2(v-u)^2 dv du .. + ∫_u_0^∞1/2 e^-a e^u e^u(σ-1/2)∫_π A^∞ e^-b^2/2(v-u)^2 dv du}< 2c_θ(σ,f){ e^u_0(σ-1/2)4(2π)^-1/2/cosh(π r/2) (a e^u_0)^1/6∫_-∞^u_0 e^-b^2/2(π A-u)^2 du + ∫_u_0^∞ e^-a e^u e^u(σ-1/2) du },where we have employed (x)≤ e^-x^2 for x≥0, and (x)<2 otherwise.Evaluating the integrals completes the proof. § DETECTING ZEROS For each Maass form L-function under consideration, we compute rigorously many values on the critical line.For instance, Figure <ref> shows a graph ofZ(t,f) := L(12+it,f) e^iγ(12+it,f)for the first even Maass form L-function on Γ(1). Let Ω_γ_θ := {t∈:γ_θ(12+it)=0} be the set of zeros on the critical line of the Γ-factor. For values of θ_1 and θ_2 chosen according to Lemma <ref>,Ω_γ_θ_1∩Ω_γ_θ_2 = ∅.This follows immediately from Lemma <ref>. Fixing the value of θ, say θ=θ_1, there is a risk of hitting a zero of γ_θ_1 (to within the internal precision) when evaluating Z_θ_1 for some specific value of t. In all of our computations, we never observed this in practice, i.e. we never had to deal with division by zero. However, computing at finite absolute precision, we sometimes come close to a zero of γ_θ_1 and experience some loss of precision in the division by γ_θ_1. Since we also compute for a second value of θ, θ=θ_2, chosen according to Lemma <ref>, we may always ensure the accuracy of the computed values of Z. For each Maass form L-function under consideration, we rigorously compute all zeros on the critical line up to some height. The search for zeros is faciliated by the following lemma.(a) Let Z∈ C^1() be real valued, and assume it has consecutive simple zeros at t_0, t_1 and t_2, with Z'(t_0)>0. Then ∃ t_a,t_b,t_c,t_d,t_e,t_f such that the following holds:[ t Z'(t)Z(t) quadrant of Z'+iZ; t_a<t<t_0>0<0 4; t_0<t<t_b>0>0 1; t_b≤ t≤ t_c>01or2; t_c<t<t_1<0>0 2; t_1<t<t_d<0<0 3; t_d≤ t≤ t_e<03or4; t_e<t<t_2>0<0 4; t_2<t<t_f>0>0 1; ] (b) Let Z∈ C^n() be real valued, and assume it has a zero of order n at t_0, with (d^ndt^nZ)(t_0)>0.Then ∃ t_a,t_b such that the following holds:[t d^ndt^nZ d^n-1dt^n-1Z d^n-2kdt^n-2kZ d^n-2k-1dt^n-2k-1Z;t_a<t<t_0 >0 <0 >0 <0;t=t_0 >0 =0 =0 =0;t_0<t<t_b >0 >0 >0 >0;]for k∈, but 0<2k<n. The lemma follows from elementary analysis and the intermediate value theorem. If the quadrants of Z'+iZ for consecutive t_j are not ordered as given in Lemma <ref>(a), there is either a zero of order greater than 1 which is to be investigated according to Lemma <ref>(b), or the sequence is not yet fine enough. We expect the sequence to be fine enough if for successive t_j the quadrants of Z'+iZ do change by at most by 1, and when they change, they do so in agreement with Lemma <ref>. There is no proof that the expectation in Remark <ref> holds, and one can construct sequences (t_j) that contradict the expectation. Nevertheless, with some reasonable choices in the construction of the sequence (t_j) and its refinements, the expectation turns out to be reliable in practice.Namely, for every Maass form L-function that we considered, we never overlooked any zero, as proven after the fact using Turing's method.§ TURING'S METHOD Turing's method for verifying the Riemann hypothesis for arbitrary L-functions is described in <cit.>. For t not the ordinate of some zero or pole of Λ, letS(t) := 1/π∫_∞^1/2L'/L(σ+it,f)dσ.By convention, we make S(t) upper semicontinuous, i.e. when t is the ordinate of a zero or pole, we define S(t)=lim_→0^+ S(t+).We select a particular branch of logγ(s) by using the principal branch of logΓ.With this choice, setΦ(t) := 1/πlogγ(1/2+it,f).We further defineN(t) := Φ(t)+S(t),which relates to the number of zeros in the critical strip up to height t.For t_1<t_2 let Ω_L denote the multiset of zeros with imaginary part in (t_1,t_2], and let N(t_1,t_2) denote their number, counting multiplicity,N(t_1,t_2) := #Ω_L(t_1,t_2).Then, we haveN(t_1,t_2) = N(t_2)-N(t_1). <cit.> For ϑ=764 and σ>ϑ+1, definez_ϑ(σ) := ( ζ(2σ+2ϑ)ζ(2σ-2ϑ)/ζ(σ+ϑ)ζ(σ-ϑ))^1/2andZ_ϑ(σ) := ( ζ(σ+ϑ)ζ(σ-ϑ) )^1/2.Suppose t_1 and t_2 satisfy(t_i± r)^2≥(52+ϵ)^2+X^2, i=1,2for some X>5, and setC_ϑ := log Z_ϑ(3/2) + ∫_3/2^∞logZ_ϑ(σ)/z_ϑ(σ) dσ - ∫_3/2^5/2log z_ϑ(σ) dσ + (log 4) z_ϑ'(32)/z_ϑ(32).Thenπ∫_t_1^t_2 S(t) dt ≤1/4logQ(3/2+it_2) + (log2 - 1/2)logQ(3/2+it_1) + 2C_ϑ + 2/√(2)(X-5). For 0≤ t_1<t_2, assume Ω̃_L(t_1,t_2) is a given multiset of zeros with imaginary part in (t_1,t_2], i.e. Ω̃_L(t_1,t_2) ⊆Ω_L(t_1,t_2). LetN_Ω̃_L(t_1,t_2) := #Ω̃_L(t_1,t_2), counting multiplicity,N_Ω̃_L(t) := N_Ω̃_L(t,0)+Φ(0)+S(0),andS_Ω̃_L(t) := N_Ω̃_L(t)-Φ(t).Ifπ∫_t_1^t_2( S_Ω̃_L(t) + 1 ) dtexceeds the right-hand side of the bound in Theorem <ref>, then the set Ω̃_L(0,t_1) contains all zeros with imaginary part in (0,t_1]. Ω̃_L(0,t_1) = Ω_L(0,t_1).If Ω̃_L(0,t_1) were a proper subset of Ω_L(0,t_1), then we would have N_Ω̃_L(t_1)<N(t_1), whence S_Ω̃_L(t)+1≤ S(t) ∀ t∈(t_1,t_2]. But the integral of the latter is bounded by Theorem <ref>. § NUMERICAL RESULTS We consider consecutive Maass cusp forms on (2,)=Γ(1). Booker, Strömbergsson, and Venkatesh <cit.> have rigorously computed the first 10 Maass cusp forms on (2,) to high precision.Bian <cit.> has extended these computations to a larger number of Maass cusp forms. The readily available list of rigorously computed Maass cusp forms is consecutive for the first 2191 Maass cusp forms, which covers all Maass cusp forms whose Laplacian eigenvalue λ=r^2+1/4 falls into the range 0≤ r≤178.Previous numerical computations of some non-trivial zeros for a few even Maass form L-functions were made by Strömbergsson <cit.>. We extend his results by rigorously computing, for each of the first 2191 consecutive Maass form L-functions on (2,), many values of Z, including all non-trivial zeros up to T=30000, at least.At the time of Strömbergsson's work, even the numerical data pertaining to the Maass cusp forms for (2,) was not rigorously proven to be accurate, so he had no reason to carry out his computations of the zeros with more than heuristic estimates for the error. Making use of the rigorous data sets described above, we have rigorously verified the correctness of Strömbergsson's results. In particular his lists of zeros are consecutive and accurate.Moreover, we confirm his observation of a zero-free region on the critical line for t near r, when r is small.We note that some theoretical results, such as Cho's theorem <cit.> on simple zero of Maass form L-functions, assumed the correctness of Strömbergsson's numerical results. With our verification, Cho's theorem becomes unconditional. Our lists of zeros contain more than 60000 consecutive non-trivial zeros per Maass form L-function.All these zeros are simple. The first several zeros of the first five Maass form L-functions are listed in Table <ref>. For f a Maass cusp form on (2,) with spectral parameter 0≤ r≤178, all non-trivial zeros with 0≤ t≤30000 of the corresponding Maass form L-function are simple and on the critical line. For each Maass form L-function we prove, using Corollary <ref>, that the corresponding list of rigorously computed zeros is consecutive for 0≤ t≤30000, and that all the zeros are indeed simple and on the critical line. According to a conjecture of Montgomery <cit.>, the distribution of non-trivial zeros should follow random matrix theory (RMT) predictions.In case of Maass form L-functions, the distribution of non-trivial zeros is expected to conform to that of eigenvalues of large random matrices from the Gaussian unitary ensemble (GUE) <cit.>. This raises the question of how GUE statistics relate to the zero-free region around t=r observed by Strömbergsson <cit.>—are the GUE statistics asymptotically correct in the large t aspect only?We investigate this question by distinguishing between zeros with small and large absolute ordinate, respectively.For a given Maass form L-function there are only a finite number of zeros with small ordinate, and the resulting statistics would be poor. Knowing the zeros for many Maass form L-functions, we can evaluate on a common scale the distribution of zeros for each L-function and collate the statistics of many of them together.Let f be a Maass cusp form with spectral parameter r and parity ϵ.Consider the zeros of the associated Maass form L-function. We unfold the zeros,x_i := Φ(t_i),in order to obtain rescaled zeros x_i with a unit mean density.Then s_i := x_i+i-x_i defines the sequence of nearest-neighbor spacings, which has mean value 1 as i→∞. Now, the distribution of nearest-neighbor spacings is given by∫_0^s P_f(s') ds' := lim_j→∞#{i≤ j:s_i≤ s}/#{i≤ j},where the index f denotes the corresponding Maass cusp form. Distributions of rescaled nearest-neighbor spacings are expected to be independent of the specific parameter values of corresponding Maass cusp forms f and can be collated by writingP(s):=1/#{f}∑_f P_f(s). To distinguish between zeros with small and large absolute ordinate, we define the respective nearest-neighbor spacings distributions,∫_0^s P_f,t<r,-n(s') ds' := lim_j→∞#{i≤ j:s_i≤ s,0<t_i-1, t_i+n-1<r}/#{i≤ j:0<t_i-1, t_i+n-1<r}, ∫_0^s P_f,t=r,+n(s') ds' := lim_j→∞#{i≤ j:s_i≤ s, t_i-1-n<r<t_i+n}/#{i≤ j:t_i-1-n<r<t_i+n}, ∫_0^s P_f,t>r,-n(s') ds' := lim_j→∞#{i≤ j:s_i≤ s, t_i-n>r}/#{i≤ j:t_i-n>r},where n is a non-negative integer, as well as their collated versionsP_t<r,-n(s) := 1/#{f}∑_f P_f,t<r,-n(s),P_t=r,+n(s) := 1/#{f}∑_f P_f,t=r,+n(s),P_t>r,-n(s) := 1/#{f}∑_f P_f,t>r,-n(s).For the first 2191 Maass form L-functions on (2,), the resulting distributions are displayed in Figure <ref>, in comparison with the Wigner surmiseP_GUE(s) := 32/π^2 s^2 e^-4s^2/π.As is visible, the distribution of zeros resembles GUE statistics for both small and large absolute ordinate, and there appears to be no distinction between the statistics of the two cases. Only the distribution of zeros that are in absolute size closest to the value of the spectral parameter might show a stronger level repulsion than the Wigner surmize.However, it is unclear whether this seemingly stronger level repulsion is just an artefact of the limited number of 2191 spacings that contribute to the histogram of P_t=r,+0. If we take three times as many spacings into account, as is the case with P_t=r,+1, we again find a close resemblance to the Poisson distribution. We speculate that the GUE statistics hold for all t, not only in the large t aspect.Since the GUE statistics are based on rescaled zeros, x_i=Φ(t_i), they do not contradict a zero-free region on the critical line.The density of zeros is described by Φ', and according to the Γ-factor, the density of zeros is smaller for t in a neighborhood of r.In particular, for small values of the spectral parameter r, the density Φ' becomes negative for t near r; see Figure <ref>.There are finitely many Maass form L-functions on (2,) that have such a region where Φ' is negative. By inspection, we find that no zero falls into a negative density region.Moreover, the zeros seem to be repelled away from the negative density regions resulting in the zero-free region around r.Finally, we investigate the repulsion from zero of the rescaled first zero x_1 in dependence of the parity ϵ of the Maass form L-function. For this we consider the distributions of the rescaled first zero,∫_0^x P_ϵ=e(x') dx' := lim_r→∞#{j: x_1<x, r_j≤ r,ϵ=e}/#{j:r_j≤ r, ϵ=e},for e∈{0,1}. The resulting distributions are displayed in Figure <ref>. Close to the origin of the plots, they show a stronger level repulsion than the Wigner surmise P_GUE(x).99Bian2017 C. Bian, A. R. Booker, and M. Jacobson, Unconditional computation of the class groups of real quadratic fields, in preparation.Booker2006 A. R. Booker, Artin's conjecture, Turing's method, and the Riemann hypothesis, Experiment. Math. 15 (2006), 385–408.BookerStrombergssonThen2013 A. R. Booker, A. Strömbergsson, and H. Then, Bounds and algorithms for the K-Bessel function of imaginary order, LMS J. Comp. Math. 16 (2013), 78–108.BookerStrombergssonVenkatesh2006 A. R. Booker, A. Strömbergsson, and A. Venkatesh, Effective computation of Maass cusp forms, Int. Math. Res. Notices 2006 (2006), article ID 71281.Brown1967 J. L. Brown Jr., On the error in reconstructing a non-bandlimited function by means of the bandpass sampling theorem, J. Math. Anal. Appl. 18 (1967), 75–84.Cho2013 P. J. Cho, Simple zeros of Maass L-functions, Int. J. Number Theory 9 (2013), 167–178.Dokchitser2004 T. Dokchitser, Computing special values of motivic L-functions, Experiment. Math. 13 (2004), 137–149. Good1983 A. Good, On various means involving the Fourier coefficients of cusp forms, Math. Z. 183 (1983), 95–129.GradshteynRyzhik2007 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, 2007.KeatingSnaith2000 J. P. Keating and N. C. Snaith, Random matrix theory and ζ(1/2+it), Commun. Math. Phys. 214 (2000), 57–89.KimSarnak2003 H. H. Kim and P. Sarnak, Refined estimates towards the Ramanujan and Selberg conjectures, J. Amer. Math. Soc. 16 (2003), 139–183.LagariasOdlyzko1979 J. C. Lagarias and A. M. Odlyzko, On computing Artin L-functions in the critical strip, Math. 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Notices 1999 (1999), 839–851.Vishe2012 P. Vishe, Rapid computation of L-functions for modular forms, Int. Math. Res. Notices (2012), doi:10.1093/imrn/rns112 | http://arxiv.org/abs/1703.08863v1 | {
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"title": "Rapid computation of $L$-functions attached to Maass forms"
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Following our earlier finding based on RHIC data on the dominant jet production from nucleus corona region, we reconsider this effect in nucleus-nucleus collisions at LHC energies. Our hypothesis was based on experimental data, which raisedthe idea of a finite formation time for the produced medium.At RHIC energy and in low density corona region this time reaches about 2 fm/c.Following this hypothesis, the nuclear modification factor R_AA at high p_t should be independent on particle momentum, and the azimuthal anisotropy of high p_t particles, v_2,should be finite. A separate prediction held that,at LHC energy, the formation time in the corona region should be about 1 fm/c.New data at LHC show that R_AA is not flat and is rising with p_t. We add to our original hypothesis an assumption that a fast partontraversing the produced medium loses the fixed portion of its energy. A shift of about 7 GeV from the original power law p^-6 production cross section in ppexplains well all the observedR_AA dependencies. The shift of about 7 GeV is also valid at RHIC energy. We also show that the observed at LHC dependenceof v_2 at high p_t and our previous predictions agree. De-anonymization of Social Networks with Communities: When Quantifications Meet Algorithms Luoyi Fu^* Shanghai Jiao Tong University [email protected] Xinzhe Fu^* Shanghai Jiao Tong University [email protected] Zhongzhao Hu Shanghai Jiao Tong University [email protected] Zhiying Xu Shanghai Jiao Tong University [email protected] Xinbing Wang Shanghai Jiao Tong University [email protected] December 30, 2023 =========================================================================================================================================================================================================================================================================================================================================================================================Over the last 17 years of relativistic nucleus-nucleus collisions at RHIC and LHC, a set of observables was found which confirms the formation of high energy and high density matter. Among these features are the strong jet suppression manifested in particle suppression at high transverse momentum, p_t, and large particle anisotropy. There is also a long list of models and theoretical assumptions to explain these effects. In our view, when one talks aboutjet suppression, a significant effect of particle production from the nucleus corona region is often ignored or underestimated. In a previous publication based purely on experimental data at RHIC, a simple model was proposed <cit.> to explain the angular dependence in the reaction plane of the nuclear modification factor R_AA. The model nicely described the centralityand azimuthaldependence (or factor v_2 for high p_t) of R_AA at RHIC energy. In the model, there is one free parameter of about 2.3 fm/c which wasinterpreted as plasma formation time at the low density corona region. The physical meaning of this parameter is that fast partons have roughly this time to escape from the produced medium and, theafter, they are absorbed by the absolutely opaque central region. This value of T_0=2.3 fm/c is not “crazy large” because the number of nuclear collisions, N_coll, near corona region is rather small, but it should be less than 0.8 fm/c in the core region of the produced matter <cit.>. Time, necessary to form the strongly interacting colored matter, should be proportional to the mean distance between the interaction or collision points with a color exchange.This distance, itself, should be inversely proportional to the square root of the density of such interactions. The picture in some sense is similar to the percolation scenario <cit.>.If the densityof N_coll in x-y plane of colliding nuclei near the corona region is ρ_periph then the formation time T(r) versus density ρ(r) will be: T(r)=T_0·√(ρ_periph/ρ(r)),where r is the distance from the center. In Fig. <ref> we plot the evolution of the formation time versus the distance from the center of the region for the colliding Au+Au nuclei in the 0-5% centrality bin. For the N_coll density distribution of colliding nucleons we used density profiles generated for our first publication <cit.>. To demonstrate how formation time works we show in Fig. <ref> two extreme cases: when the fast parton is produced in the center of the colliding region, arrow 1, and near the surface at a depth of about 2 fm from the Woods-Saxon radius, arrow 2. The first parton moves with the speed of light along its world line 1 only for about 0.8 fm and then is stopped by the produced matter. The second parton will survive. The proposed model in <cit.> works well at RHIC energy.In “The last call for prediction” published prior to the start of LHC we also proposed some features which should be observed at LHC if a similar picture with formation time is valid <cit.> (seepages 119–121 and figures 99–100 in the e-print version). As we already mentioned, the formation time should be proportional to the mean distance between interactions with the color exchange. It means also that only part of the nucleon-nucleon inelastic cross section will contribute to the process:single- and double diffractive and soft process with meson exchange will not be relevant here. If a relativistic rise of the total nucleon-nucleon, NN,cross section comes purely from the contribution from the colored parton hard scatterings, we can estimate the relative value of hard scatterings to the totalnucleon-nucleon cross section. At √(()S_NN)=20 GeV the NN total cross section is at its minimum of30 millibars- there is almost no hard scattering, butmostly soft nucleon-nucleon interactions with meson exchange. At200 GeV the cross section rises by 13 mb, at 5500 GeV – by 49 mb. The formation time of the colored mattershould be proportional to one over the square root of these numbers because the density of N_coll is proportional to the cross section. If we getT=2.3 fm/c at 200 GeV, then from the rise of the NN total cross section, we estimate T=1.2 fm/c near the corona region at around 5 TeV ofLHC energy. In the center of the collision zone it will be about three times shorter. Calculations show that such a value of T should give a constant R_AA=0.1 for high p_t particles in the most central collisions. We have to emphasize that the value ofT around 1.2 fm/c is valid within uncertainty of 5%in the LHC energy range of 2.7–5 TeV. It comesfrom a little change of pp total cross section between 85 mb and 90 mb if one interpolates the existing pp data <cit.>, thus, and relative change of hard scattering contribution is on the level of 5 mb.Predictions made in <cit.> assume that the core of the produced matter is opaque, but experimental data for Pb+Pb collisions obtained byALICE,CMS and ATLAS <cit.> show that R_AA is continuously rising at high p_t. It means that the core of the collision zone becomes more transparent for fast particles. It is natural to assume that the parton loses some portion of its energy. We found that a constant energy loss of 7 GeV describes well the data for R_AA versus p_t. Particle, namely pion,production cross section at LHC energy follows a simple power law p^-6 <cit.> at high p_t. Thus, the energy drop by 7 GeV becomes less significant with increasingparton p_t. In Fig. <ref> we present results for the R_AA versus p_t from CMS data <cit.> and our calculations for most central collisions. There are two contributions: a constant value of 0.1 for a particle from the corona region, as was predicted in ref. <cit.>, and a new momentum dependent component when matter becomes more transparent for fast parton, which loses 7 GeV.This provides excellent agreement with the data. In Fig. <ref> we show a similar plot for mid-central collisions. In this case thecontribution to R_AA from the corona region reaches 0.35 <cit.>, but the penetrating partoncontribution is about the same. Out of curiosity we checked how this 7 GeV energy loss works at RHIC and addedthis component to the previous calculation with the corona region and absolutely black core, Fig. <ref>. The only difference here is that the production cross section at RHIC follows a more steep power law p^-8 <cit.>. Within the error bars our line follows the experimental points. Such a large energy loss (7 GeV) at RHIC explains why the assumption about the complete black core with some corona contribution worked so well – the loss is too big for produced particles at RHIC.Our model worked well at RHIC for the observed large azimuthal of highp_t particles or parameter v_2. Nearly 10 years ago we did a prediction for v_2 at LHC <cit.>. It seems that the prediction is valid. In Fig. <ref> we compare our estimations with CMS results at p_t=15 GeV/c <cit.>. The prediction of a large v_2 even at LHC is confirmed, the sensitivity to the collision geometry persists up to high p_t. There is a deviation at small N_part but this is due to the well known effect of distortion by the initialgeometry fluctuations (see for example, PHOBOS paper <cit.>). We also can explain the observed drop of v_2 with particle or jet momentum above 15 GeV/c. The corona effect for in- and out-of-plane particle production is diluted by penetrating partons with energy loss. For example, looking at Fig. <ref> and Fig. <ref>, one can see that at p_t=40 GeV/c particles from the corona region count for about one half of the total yield at this momentum. Thus,v_2 should drop to about a factor of 2. This what is qualitatively seen by the three experiments <cit.>. In conclusion, we demonstrate that in PbPb collisions at LHC the contribution from the corona region and the assumption of a finite formation time for the colored strongly interacting matter are the reasons for the observed centrality and momentum dependence of particle R_AA. At LHC energies, a fast parton escapes the interaction zone by losing about 7 GeV. Within our model this value does not depend on momentum, centrality, energy density, and, probably, on beam energy. The observed azimuthal angular asymmetry at a high transverse momentum is well described at RHIC and LHC energies.This work was partially supported by the RFBR grant numbers 14-22-03069-ofi-m and14-02-00570-a. We would like to thank Keith Guzik for help with the text. 99jetp_l_2007 V. S. Pantuev,JETP Lett.85, 104 (2007) doi:10.1134/S0021364007020026[arXiv:hep-ph/0506095].density V. S. Pantuev, Proceedings of 22nd Winter Workshop on Nuclear Dynamics (WWND 2006),URL: http://rhic.physics.wayne.edu/ bellwied/sandiego06/, [arXiv: hep-ph/0604268].satz H. Satz,Nucl. Phys. A 642, 130 (1998) doi:10.1016/S0375-9474(98)00508-9 [hep-ph/9805418]. last_call N. Armesto et al.,J. Phys. G 35, 054001 (2008) doi:10.1088/0954-3899/35/5/054001[arXiv:0711.0974 [hep-ph]]. pdg C. Patrignani et al. (Particle Data Group), Chin. Phys. C, 40, 100001 (2016), URL: http://pdg.lbl.gov/. alice_pt K. Aamodt et al. [ALICE Collaboration],Phys. Lett. B 696, 30 (2011) doi:10.1016/j.physletb.2010.12.020 [arXiv:1012.1004 [nucl-ex]].cms_ptS. Chatrchyan et al. [CMS Collaboration],Phys. Lett. B 712, 176 (2012) doi:10.1016/j.physletb.2012.04.058 [arXiv:1202.5022 [nucl-ex]].atlas_ptG. Aad et al. [ATLAS Collaboration],JHEP 1509, 050 (2015) doi:10.1007/JHEP09(2015)050 [arXiv:1504.04337 [hep-ex]]. alice B. B. Abelev et al. [ALICE Collaboration],Eur. Phys. J. C 74, no. 10, 3108 (2014)doi:10.1140/epjc/s10052-014-3108-8[arXiv:1405.3794 [nucl-ex]].cms V. Khachatryan et al. [CMS Collaboration], Submitted to: JHEP,[arXiv:1611.01664 [nucl-ex]].phenixA. Adare et al. [PHENIX Collaboration], Phys. Rev. Lett.101, 232301 (2008) doi:10.1103/PhysRevLett.101.232301[arXiv:0801.4020 [nucl-ex]].cms_v2 S. Chatrchyan et al. [CMS Collaboration],Phys. Rev. Lett.109, 022301 (2012) doi:10.1103/PhysRevLett.109.022301 [arXiv:1204.1850 [nucl-ex]].phobosB. Alver et al. [PHOBOS Collaboration],Phys. Rev. Lett.98, 242302 (2007) doi:10.1103/PhysRevLett.98.242302 [nucl-ex/0610037].ALICE_v2 J. Adam et al. [ALICE Collaboration],Phys. Lett. B 753, 511 (2016) doi:10.1016/j.physletb.2015.12.047 [arXiv:1509.07334 [nucl-ex]].ATLAS_v2 G. Aad et al. [ATLAS Collaboration],Phys. Rev. Lett.111, no. 15, 152301 (2013) doi:10.1103/PhysRevLett.111.152301 [arXiv:1306.6469 [hep-ex]]. | http://arxiv.org/abs/1703.09021v3 | {
"authors": [
"V. S. Pantuev"
],
"categories": [
"hep-ph",
"nucl-ex",
"nucl-th"
],
"primary_category": "hep-ph",
"published": "20170327115333",
"title": "Corona effect in AA collisions at LHC"
} |
IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, Vol. xx, No. x, December 2016 Dai et al.: Reweighted infrared patch-tensor model with both non-local and local priors for single-frame small target detection Reweighted Infrared Patch-Tensor Model With Both Non-Local and Local Priors for Single-Frame Small Target Detection Yimian Dai, Yiquan Wu Y. Dai is with the College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China (e-mail: [email protected]). Y. Wu is with the College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China, and also with Key Laboratory of Spectral Imaging Technology CAS, Xi'an Institute of Optics and Precision Mechanics of CAS, Xi'an 710000, China (e-mail: [email protected]). Manuscript received December 19, 2016; revised March 29, 2017.December 30, 2023 ================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================ Many state-of-the-art methods have been proposed for infrared small target detection. They work well on the images with homogeneous backgrounds and high-contrast targets. However, when facing highly heterogeneous backgrounds, they would not perform very well, mainly due to: 1) the existence of strong edges and other interfering components, 2) not utilizing the priors fully. Inspired by this, we propose a novel method to exploit both local and non-local priors simultaneously. Firstly, we employ a new infrared patch-tensor (IPT) model to represent the image and preserve its spatial correlations. Exploiting the target sparse prior and background non-local self-correlation prior, the target-background separation is modeled as a robust low-rank tensor recovery problem. Moreover, with the help of the structure tensor and reweighted idea, we design an entry-wise local-structure-adaptive and sparsity enhancing weight to replace the globally constant weighting parameter. The decomposition could be achieved via the element-wise reweighted higher-order robust principal component analysis with an additional convergence condition according to the practical situation of target detection. Extensive experiments demonstrate that our model outperforms the other state-of-the-arts, in particular for the images with very dim targets and heavy clutters. infrared small target detection, infrared patch-tensor model, reweighted higher-order robust principal component analysis, non-local self-correlation prior, local structure prior. § INTRODUCTIONInfrared small target detection is a key technique for many applications, including early-warning system, precision guided weapon, missile tracking system, and maritime surveillance system <cit.>.Traditional sequential detection methods, such as 3D matched filter <cit.>, improved 3D filter <cit.>, and multiscan adaptive matched filter <cit.>, are workable in the case of static background, exploiting the target spatial-temporal information.Nevertheless, with the recent fast development of high-speed aircrafts <cit.> like anti-ship missiles, the imaging backgrounds generally change quickly due to rapid relative motion between the imaging sensor and the target.The performance of the spatial-temporal detection method degrades rapidly.Therefore, the research of single-frame infrared small target detection is of great importance and has attracted a lot of attention in recent years.Different from general object or saliency detection tasks, the main challenge of infrared small target detection is lacking enough information. Due to the long imaging distance, the target is always small without any other texture or shape features.As the target type, imaging distance, and neighboring environment differ a lot in real scenes, the target brightness could vary from extremely dim to very bright (see <ref> for example).In the absence of spatial-temporal information and the target features like shape and size, the characteristics of the background <cit.> and the relation between the background and target are very important priors for single-frame infrared small target detection.Thus how to design a model to incorporate and exploit these priors is vital for infrared small target detection in a single image. §.§ Prior work on single-frame infrared small target detectionThe previously proposed single-frame infrared small target detection methods could be roughly classified into two categories.In the first type, a local background consistent prior is exploited, assuming the background is slowly transitional and nearby pixels are highly correlated.As a result, the target is viewed as the one that breaks this local correlation.Under this assumption, the classical methods, such as two-dimensional least mean square (TDLMS) filter <cit.> and Max-Median filter <cit.>, enhance the small target by subtracting the predicted background from the original image.Unfortunately, besides the targets, they enhance the edges of the sky-sea surface or heavy cloud clutter as well, since these structures also break the background consistency as the target does.To differentiate the real target and high-frequency change, some edge analysis approaches <cit.> have been proposed to extend these methods to estimate the edge direction in advance and preserve the edges.Bai et al. <cit.> designed a new Top-Hat transformation using two different but correlated structuring elements.Another class of local prior based methods exploits the local contrast, which is computed by comparing a pixel or a region only with its neighbors.The seminal work of Laplacian of Gaussian (LoG) filter based method <cit.> has motivated a broad range of studies on the Human Visual System (HVS), and has led to a series of HVS based methods, e.g., Difference of Gaussians (DoG) <cit.>, second-order directional derivative (SODD) filter <cit.>, local contrast measure (LCM) <cit.>, improved local contrast measure (ILCM) <cit.>, multiscale patch-based contrast measure (MPCM) <cit.>, multiscale gray difference weighted image entropy <cit.>, improved difference of Gabors (IDoGb) <cit.>, local saliency map (LSM) <cit.>, weighted local difference measure (WLDM) <cit.>, local difference measure (LDM) <cit.>, etc. The second type of single-frame infrared small target detection methods which has not been explored extensively, exploits the non-local self-correlation property of background patches, assuming that all background patches come from a single subspace or a mixture of low-rank subspace clusters.Then, target-background separation can be realized with the low-rank matrix recovery <cit.>.Essentially, this type of methods attempts to model the infrared small target as an outlier in the input data.To this end, Gao et al. <cit.> generalized the traditional infrared image model to a new infrared patch-image model via local patch construction.Then the target-background separation problem is reformulated as a robust principal component analysis (RPCA) <cit.> problem of recovering low-rank and sparse matrices, achieving a state-of-the-art background suppressing performance.To correctly detect the small target located in a highly heterogeneous background, He et al. <cit.> proposed a low-rank and sparse representation model under the multi-subspace-cluster assumption.§.§ MotivationExisting methods detect the infrared small target by either utilizing the local pixel correlation or exploiting the non-local patch self-correlation.From our observation, the unsatisfying performance of local prior methods <cit.> in detecting the dim target under complicated background mainly lies in their imperfect grayscale based center-difference measures. The saliency of a dim but true target would be easily overwhelmed by the measured saliency of some rare structures. We call this phenomenon the rare structure effect. In contrast, the non-local prior methods <cit.> suffer from the salient edge residuals. Its intrinsic reason is because the strong edge is also a sparse component as the same as the target due to lack of sufficient similar samples.Since the target might be dimmer than the edge, they would simultaneously be wiped out by simply increasing the threshold.Our key observation is that the non-local prior and local prior are not equivalent, and in fact they are complementary for the problem of infrared small target detection, as illustrated in <ref>.The often appearing false alarm components in local (non-local) prior methods could be well suppressed by the non-local (local) prior methods.For example, the stubborn strong edges in the non-local prior based methods, can be easily identified by the local edge analysis approaches. Naturally, an intuitive way to solve above-mentioned dilemma is to extract the local structure information and merge it into the non-local prior based detection framework.Therefore, how to simultaneously and appropriately utilize both the local and non-local priors has been an important issue for improving the detection performance under very complex backgrounds.To the best of our knowledge, very few single-frame infrared small target detection methods concern this problem.To address this issue, we propose a single-frame small target detection framework with reweighted infrared patch-tensor model (RIPT). Our main contributions consist of the following three folds: * To dig out more information from the non-local self-correlationship in patch space, we generalize the patch-image model to a novel infrared patch-tensor model (IPT) and formulate the target-background separation task as an optimization problem of recovering low-rank and sparse tensors. * To incorporate the local structure prior into the IPT model, an element-wise weight is designed based on structure tensor, which helps to suppress the remaining edges and preserve the dim target simultaneously. * To reduce the computing time, we adopt a reweighted scheme to enhance the sparsity of the target patch-tensor.Considering the particularity of infrared small target detection, an additional stopping criterion is used to avoid excessive computation. The proposed RIPT model is validated on different real infrared image datasets. Compared with the state-of-the-art methods, our proposed model achieves a better background suppressing and target detection performance.The remaining of this paper is organized as follows. We propose the non-local correlation based IPT model in Section <ref>. The details of the local structure weight construction are described in Section <ref>. In Section <ref>, we further propose the reweighted IPT model and its detailed optimization scheme is also provided. Section <ref> presents detailed experimental results and discussion. Finally we conclude this paper in Section <ref>.§ NON-LOCAL CORRELATION DRIVEN INFRARED PATCH-TENSOR MODEL To dig out more spatial correlationships, we develop a novel target-background separation framework named infrared patch-tensor model in this section.Before describing the details, several mathematical notations are defined first in <ref>. Given an infrared image, it could be modeled as a linear superposition of target image,background image and noise image:f_ = f_ + f_ + f_,where f_, f_, f_, and f_ represent the input image, background image, target image, and noise image, respectively.Via a window sliding from the top left to the bottom right over an image, we stack the patches into a 3D cube (see the construction step in <ref>).Then <ref> is transferred to the patch space with spatial structure preserved:ℱ = ℬ + 𝒯 + 𝒩,where ℱ, ℬ, 𝒯, 𝒩∈ℝ^I× J× P are the input patch-tensor, background patch-tensor, and target patch-tensor, respectively.I and J are the patch height and width, P is the patch number.Background patch-tensor ℬ.Generally, the background is considered as slowly transitional, which implies high correlations among both the local and non-local patches in the image, as illustrated in <ref>(a). Although patches p_1,p_2,p_3 locate in the different region of the image, they are equivalent. Based on this non-local correlationship, the state-of-the-art IPI model imposed the low-rank constraint to background patch-image.As a patch-image is the mode-3 unfolding matrix of a patch-tensor, the patch-image model could be viewed as a special case of the proposed patch-tensor model essentially. Since the main difficulty of detecting the infrared small target in a single image is lacking enough information, only considering the low-rank structure in one unfolding is insufficient to deal with highly difficult scenes.Naturally, it motivates us to think whether we can utilize the other two unfolding modes. Actually, the mode-1 and mode-2 unfolding matrices of the infrared patch-tensor are also low-rank.In <ref>(b) – (d), the singular values of all the unfolding matrices rapidly decrease to zero, which demonstrates that every unfolding mode of the background patch-tensor is intrinsically low-rank. Therefore, we can consider the background patch-tensor ℬ as a low-rank tensor, and their unfolding matrices are also all low-rank defined as: (B_(1)) ≤ r_1, (B_(2)) ≤ r_2, (B_(3)) ≤ r_3,where r_1, r_2, and r_3 all are constants, representing the complexity of the background image.The larger their values are, the more complex the background is. Target patch-tensor 𝒯.Since the small target merely occupies several pixels in the whole image, the target patch-tensor is an extremely sparse tensor in fact. That is:‖𝒯‖_0≤ k,where k is a small integer determined by the number and size of the small target.Noise patch-tensor 𝒩.In this paper, we just assume that the noise is additive white Gaussian noise and ‖𝒩‖_≤δ for some δ > 0. Thus we have ‖ℱ-ℬ-𝒯‖_≤δ.It should be noted that although the values of parameters k, r, δ are different depending on the images, we would not directly use them in the following sections.Ideally, we would like to obtain a low-rank and sparse decomposition and solve the following problem:min_ℬ, 𝒯 (B) + λ‖𝒯‖_0, s.t. ℬ + 𝒯 = ℱ.Unfortunately, the rank computation of a given tensor is a NP-hard problem in general <cit.>.In Ref. <cit.>, Goldfarb and Qin proposed the Higher-order RPCA (robust tensor recovery) through replacing the rank by a convex surrogate Tucker-rank (ℬ), and ‖𝒯‖_0 by ‖𝒯‖_1 to make the above problem tractable.In the singleton model, the tensor rank regularization term is defined as the sum of all the nuclear norms of the mode-i unfoldings, i.e., (ℬ) = ∑_i ‖B_(i)‖_*, i = 1,2,3.With this relaxation, our proposed IPT model with random noise assumption can be solved by minimizing the following convex problem:min_ℬ, 𝒯∑_i=1^3‖B_(i)‖_* + λ‖𝒯‖_1, s.t. ‖ℱ - ℬ - 𝒯‖_F≤δ.λ is a weighting parameter that controls the global trade-off between the background patch-tensor and the target patch-tensor.Larger λ can shrink those non-target but sparse components to zeros in the target patch-tensor.Nevertheless, it will also over-shrink the dim target which should be preserved.On the contrary, a smaller λ does retain the dim target, but it keeps the strong cloud edges as well.Therefore, adopting a global constant weighting parameter λ is not a reasonable scheme for detecting the infrared small target in a complex scene.Naturally, it motivates us to design an entry-wise weighting scheme.§ INCORPORATING LOCAL STRUCTURE PRIORIn this section, we focus our emphasis on combining the local structure prior and non-local correlation prior together into our model.we construct a local structure weight and interpret it as an edge salience measure.For the sake of simplicity, the local structure weight is designed on the basis of the image structure tensor.Structure tensor is widely used in many partial differential equation (PDE) based methods <cit.> to estimate the local structure information in the image, including edge orientation.To integrate the local information, the structure tensor is constructed based on a local regularization of a tensorial product, which is defined asJ_α(∇ u_σ) = G_α * (∇u_σ⊗∇u_σ) = [ J_11 J_12; J_21 J_22;],where u_σ is a Gaussian-smoothed version of a given image u.σ > 0 is the standard deviation of the Gaussian kernel which denotes the noise scale, making the edge detector ignorant of small details.J_α is a symmetric and positive semi-definite matrix, which has two orthonormal eigenvectors denoted as w and v = w^, where w = ( 2 J_12, J_22 - J_11 + √(( J_22 - J_11)^2 + 4 J_12^2 ))^⊤, w = w / |w|.w points to the maximum contrast direction of the geometry structure while v points to the minimum direction <cit.>.Their corresponding eigenvalues λ_1 and λ_2 can be calculated viaλ_1, λ_2 = ( J_11 + J_22) ±√(( J_22 - J_11)^2 + 4 J_12^2 ).These two values can be used as two feature descriptors of the local geometry structure, where at the flat region, λ_1 ≈λ_2 ≈ 0; at the edge region, λ_1 ≫λ_2 ≈ 0; at the corner region, λ_1 ≥λ_2 ≫ 0.For the sake of low computational cost, we take λ_1 - λ_2 as the edge awareness feature since its value of the edge pixel is much larger than that belongs to the flat region and corner. By applying <ref> and <ref> to every pixel in the input image f_F, two matrices L_1 and L_2 can be obtained, which consists of the large and small structure tensor eigenvalues of all the pixels, respectively.Then we can transform L_1 and L_2 to their corresponding patch-tensors ℒ_1 and ℒ_2.Finally, we define the local structure weight patch-tensor as follows𝒲_LS = exp( h ·( ℒ_1 - ℒ_2 ) - d_min/d_max - d_min),where h is a weight stretching parameter, d_max and d_min are the maximum and minimum of ℒ_1 - ℒ_2, respectively.<ref> displays the edge awareness maps of <ref>, which demonstrates that the structure tensor based local structure weight has a good performance in identifying the edges.It should be noticed that for the sake of displaying effect, <ref> is created via a normalized version of -exp( -h ·( ℒ_1 - ℒ_2 ) - d_min/d_max - d_min).In the proposed algorithm, we still calculate the local structure weight via <ref>. With the help of 𝒲_LS, we can rewrite <ref> into a weighted IPT model:min_ℬ, 𝒯∑_i=1^3‖B_(i)‖_* + λ‖𝒲_LS⊙𝒯‖_1, s.t. ‖ℱ - ℬ - 𝒯‖_F≤δ,where ⊙ is the notation of Hadamard product.Thanks to the weighted IPT model (<ref>), the strong edges could be well suppressed in the target image.§ REWEIGHTED INFRARED PATCH-TENSOR MODEL AND ITS SOLUTION §.§ Reweighted infrared patch-tensor modelThe computing time is also a major concern in infrared small target detection. Generally, the stopping criterion of a RPCA algorithm is that the reconstruction error is less than a certain small value. To meet this criterion, WIPT needs dozens of iterations, which is still time-consuming. An interesting phenomenon we find is that the non-zero entry number in the target patch-tensor has ceased to grow before the algorithm converges. In fact, in the target image, true target merely occupies the brightest of the few.In the second half of the iteration, their values barely change. Therefore, considering the particularity of infrared small target, it is reasonable to replace the reconstruction error with the target patch-tensor sparsity in our proposed model.The algorithm stops iteration once the non-zero entry ceases to grow. Then the sparsity of the target patch-tensor becomes critical in reducing the computing time. We hope the non-zero entries keep decreasing as the iteration goes on, leaving the final target image the sparsest. Unfortunately, the real situation is against our expectation in IPT and WIPT, where the target image deteriorates as the algorithm converges.Naturally, it motivates us to take a sparsity enhancing approach to solve this problem. In Ref. <cit.>, Candès proposed a reweightedℓ_1 minimization for enhancing sparsity.By minimizing a sequence of weighted ℓ_1 norm, a significant performance improvement is obtained on sparse recovery.Inspired by it, many improved RPCA models have been proposed <cit.>.Motivated by these state-of-the-art models, we adopt a similar reweighted scheme for the values in the target patch-tensor.The large weights discourage non-zero entries, and the small weights preserve non-zero elements.The sparsity enhancement weight is defined as follows 𝒲_SE^k+1(i,j,p) = 1/|𝒯^k(i,j,p) | + ϵ,where ϵ > 0 is a preset constant to avoid division by zero.Then besides the relative error ‖ℱ - ℬ^k+1 - 𝒯^k+1‖_F / ‖ℱ‖_F < ε, we could add a new end condition that counts the non-zero entry element, namely ‖𝒯^k+1‖_0 = ‖𝒯^k‖_0. With the help of this empirical observation, the computing time could be largely decreased, as illustrated in <ref> and <ref>.Another intrinsic characteristic that both Ref. <cit.> and Ref. <cit.> neglect is the fact that the small target is always brighter than its neighborhood environment in infrared images due to the physical imaging mechanism <cit.>.Therefore, besides the sparsity constraint <cit.> of the target patch-tensor, it is reasonable to assume that all the entries in 𝒯 are non-negative.To this end, we incorporate this target non-negative prior into the reweighted IPT model via rewriting <ref> as follows𝒲_SE^k+1(i,j,p) = δ( 𝒯^k (i,j,p) )= 1/𝒯^k(i,j,p) + ϵ,if 𝒯^k(i,j,p) > 0;∞,if 𝒯^k(i,j,p) ≤ 0,where δ(·) is an indicator function.We combine the local structure weight 𝒲_LS and sparsity enhancing weight 𝒲_SE^k to get the adaptive weight as follows𝒲^k = 𝒲_LS⊙𝒲_SE^k.Finally, we generalize the proposed IPT model and WIPT model to a novel reweighted infrared patch-tensor model (RIPT) as followsmin_ℬ, 𝒯∑_i=1^3‖B_(i)‖_* + λ‖𝒲⊙𝒯‖_1, s.t. ℬ + 𝒯 = ℱ.§.§ Solution of RIPT model In this subsection, we show how to solve the proposed RIPT model as a reweighted robust tensor recovery problem via an Alternating Direction Method of Multipliers (ADMM) <cit.>.The augmented Lagrangian function of <ref> is defined as ℒ =∑_i=1^N‖B_i,(i)‖_* + λ‖𝒲⊙𝒯‖_1 + ∑_i=1^N1/2μ‖ℬ_i + 𝒯 - ℱ‖^2 - ⟨𝒴_i, ℬ_i + 𝒯 - ℱ⟩ ,where 𝒴_i∈ℝ^I× J× P,i = 1, 2, 3 are the Lagrange multiplier tensors, and μ is a positive penalty scalar.ADMM decomposes the minimization of ℒ into two subproblems that minimize ℬ_i and 𝒯, respectively. More specifically, the iterations of ADMM go as follows:Updating ℬ_i with the other terms fixed.ℬ_i^k+1 = min_ℬ_i‖B_i,(i)‖_* + 1/2μ‖ℬ_i - (ℱ + μ𝒴_i^k - 𝒯^k) ‖^2_F Updating 𝒯 with the other terms fixed.𝒯^k+1 =min_𝒯λ‖𝒲^k⊙𝒯‖_1 + ∑_i=1^N1/2 μ‖𝒯 - ( ℱ + μ𝒴_i^k - ℬ_i^k+1) ‖^2_F Updating the multiplier 𝒴_i^k+1 with the other terms fixed.𝒴_i^k+1 = 𝒴_i^k + 1/μ^k( ℱ - ℬ_i^k+1 - 𝒯^k+1) , i = 1, ⋯, N. The subproblems (<ref>) and (<ref>) can be solved via the following two operators, respectively. ℬ_i^k+1= 𝒯_i, μ( ℱ + μ𝒴_i^k - ℰ^k)𝒯^k+1= 𝒮_μλ/N𝒲^k[ 1/N∑_i=1^N( ℱ + μ𝒴_i^k - ℬ_i^k+1) ]From <ref>, it could be observed that the weighting parameter determines the soft-threshold, controlling the trade-off between the target patch-tensor and background patch-tensor.Therefore, our element-wise adaptive weight tensor could simultaneously preserve the small target and suppress the strong edges. Finally, the solution of the proposed model is given in <ref>. §.§ Detection ProcedureIn <ref>, we present the whole procedure of detecting the infrared small target via the model proposed in this paper. The detailed steps are as follows: * Given an infrared image f_, its local structure feature map f_ is calculated via <ref>. * The original patch-tensor ℱ and local structure weight patch-tensor 𝒲_LS are constructed from f_ and f_. * <ref> is performed to decompose the patch-tensor ℱ into the background patch-tensor ℬ and target patch-tensor 𝒯. * The background image f_ and target image f_ are reconstructed from the background patch-tensor ℬ and target patch-tensor 𝒯, respectively. For the sake of implementation convenience, we adopt the uniform average of estimators (UAE) reprojection scheme <cit.>. * The target is segmented out as the same as Ref. <cit.>. The adaptive threshold is determined by: t_ = max(v_min, f̅_ + kσ),where f̅_ and σ are the average and standard deviation of the target image f_. k and v_min are constants determined empirically. § EXPERIMENTAL VALIDATIONTo fully evaluate the proposed algorithm, we conduct a series of experiments using images of various scenarios and include ten state-of-the-art methods for comparison. §.§ Experimental setup Datasets. We test the proposed model on extensive real infrared images to cover different scenarios, as illustrated in <ref>, varying from the flat backgrounds with salient targets to complex backgrounds with heavy clutters and extremely dim targets. All targets are labeled with red boxes.Since some targets are so dim that could hardly be observed by human eyes directly, we enlarge the demarcated area. Taking into account that the biggest difficulty of current infrared small target detection is how to detect those very dim targets with strong clutters, a good detection performance on those extremely complex images is more convincing than the satisfying result on relatively simple images.Therefore, in the following experiments, our main focus is put on the datasets with complex scenes that <ref>(a) – (d) and (l) belong to.The detailed characteristics of these five sequences are presented in <ref>. Baselines and Parameter settings.The proposed algorithm is compared with ten state-of-the-art solutions, including three filtering based methods (Max-Median <cit.>, Top-Hat <cit.>, TDLMS <cit.>), three HVS based methods (PFT <cit.>, MPCM <cit.>, WLDM <cit.>), and four recently developed low-rank methods (IPI <cit.>, PRPCA <cit.>, WIPI <cit.>, NIPPS <cit.>).<ref> summarizes all the methods involved in the experiments and their detailed parameter settings.For all the low-rank methods, i.e. IPI, PRPCA, WIPI, NIPPS, IPT, and RPIT, they are all solved via ADMM.All the algorithms are implemented in MATLAB 2016b on a PC of 3.4 GHz and 4GB RAM. The source code of our method is publicly available at <https://github.com/YimianDai/DENTIST>. Evaluation Metrics.For a comprehensive evaluation, four metrics including the local signal to noise ratio gain (LSNRG), background suppression factor (BSF), signal to clutter ratio gain (SCRG), and receiver operating characteristic (ROC) curve are adopted in comparison of background suppressing performance.LSNRG measures the local signal to noise ratio (LSNR) gain, which is defined as= _/_,where _ and _ are the LSNR values before and after background suppression.LSNR is given as = P_ / P_. P_ and P_ are the maximum grayscales of the target and neighborhood, respectively.BSF measures the background suppression quality using the standard deviation of the neighborhood region. It is defined as:BSF = σ_in/σ_out, where σ_in and σ_out are the standard variances of background neighborhood before and after background suppression.The most widely used SCRG is defined as the ratio of signal-to-clutter ratio (SCR) before and after processing:= SCR_out/SCR_in,where SCR represents the difficulty of detecting the infrared small target, and it is defined by SCR = |μ_t - μ_b| / σ_b.μ_t is the average target grayscale. μ_b and σ_b are the average grayscale and standard deviation of the neighborhood region.For all these three metrics, the higher their values are, the better background suppression performance the detection method has.All three metrics are computed in a local region, as illustrated in <ref>.The target size is a × b, and d is the neighborhood width. we set d = 20 in this paper.Among all the existing metrics, the detection probability P_d and false-alarm rate F_a are the key performance indicators, which are defined as followsP_d= number of true detections/number of actual targets, F_a= number of false detections/number of images.The ROC curve shows the trade-off between the true detections and false detections.§.§ Validation of the proposed methodIn this subsection, we take a closer look at the proposed method by validating their robustness against various scenes and noisy cases.At last, the roles of the patch-tensor, local structure weight, and sparsity enhancement weight are examined in depth to evaluate each prior individually. §.§.§ Robustness to various scenesIn <ref>, we show the separated target images for the images of <ref>.Observing <ref>, it can be clearly seen that the background clutters are completely wiped out, leaving the target the only sole component in the target image.Since <ref> contains a lot of different scenarios, it is fair to say that the proposed method is quite robust to various scenes. §.§.§ Robustness to noiseNoise is another key influence factor.In <ref>, we evaluate the proposed method's performance in the case of noise with different levels.When the noise standard deviation is 10, the proposed method could well enhance the targets and suppress the clutters and noise.As the noise standard deviation increases to 20, RIPT still detects the target in <ref>(m) – (n) and (q) – (r), but fails in <ref>(o) – (p).Nevertheless, this failure is acceptable, since the target is totally overwhelmed by the noise in <ref>(o) – (p) (see the enlarged box). Therefore, the noise influence depends not only on the intensity of the noise itself but also on the original contrast of the target. As long as the polluted target can maintain a weak contrast like <ref>(c) – (d), the proposed method is still able to detect it. §.§.§ Roles of components in the proposed model To further understand the effects of the components in the proposed RIPT model, we evaluate each prior individually with experiment to investigate how these priors influence the final detection performance. The ROC curves of IPI, IPT, IPT with sparsity enhancement weight (SIPT), WIPT, and RIPT for Sequence 1 – 4 are given in <ref>, leading to the following observations. (1) The four patch-tensor based methods outperform the state-of-the-art IPI method, which demonstrates that the patch-tensor model, involving mode-1 and mode-2 unfolding matrices, does contribute to the final detection performance.(2) By comparing WIPT and RIPT with IPT and SIPT, we see that incorporating local structure prior significantly improves the detection probability.(3) Although the sparsity enhancement weight does not contribute to the final detection performance, it significantly reduces the iteration number, as illustrated in <ref>. These observations indicate that the introduced priors are effective, and, when combined together, lead to excellent performance as reported in the next subsections. §.§ Algorithm Complexity and Computational Time The proposed model is solved via ADMM, which has been proved a 𝒪(1/k) convergence <cit.>.Therefore, our solving algorithm is ensured to converge.The algorithm complexity and computational time for <ref>(a) with various methods are given in <ref>.The image size is M× N, and m,n are the rows and columns of the patch-image or mode-3 unfolding. Although the computational complexities of these methods seem the same, their computing time differs a lot. For the filtering and HVS based methods, the difference in computing time lies in whether the code could be vectorized. For the low-rank methods, the dominant factor is the iteration number. It can be seen from the data in <ref> that the low-rank methods are generally slower than the filtering and HVS based methods.Nevertheless, considering low-rank method could handle more difficult scenes, this trade-off is acceptable.Among the low-rank methods, the RIPT costs the least time.The underlying reason is that both the local structure weight and sparsity enhancement weight help to reduce the iteration number.In addition, unlike the weight in WIPI, the time for constructing the weight is neglectable in RIPT. §.§ Parameters analysisFor the proposed model, the related parameters, such as the patch size, sliding step, weight stretching parameter h, weighting parameter λ, and penalty factor μ, are all important factors, which usually affects the model fitness on the real databases.Therefore, a better performance can be obtained by finely tuning these parameters.Nevertheless, the optimal values are always related to the infrared image content. In <ref>, we give the ROC curves for different model parameters on Sequence 1 – 4 to evaluate their influence.These parameters are validated to obtain a local optimal value with other parameters fixed.The stepped shape of our ROC curves might seem a bit odd. It is because we have adopted a much larger weighting parameter λ than normal RPCA-based foreground-background separation tasks in order to better fit the actual situation of single-frame infrared small target detection.§.§.§ Patch size I× JIt not only has a large impact on the separation, but also influences the computational complexity.The matrix size of mode-1 and mode-2 unfoldings of the patch-tensor is I × (J· P); the matrix size of mode-3 unfolding is J × (I· P).Obviously, a smaller patch size will lead to a smaller computational complexity.On one hand, we hope a larger patch size to ensure that the target is sparse enough.On the other hand, a larger patch size reduces the correlationships between the non-local patches, which degrades the separation results.To seek a balance between a low computational burden, target sparsity, and background correlationship, we vary the patch size P from 10 to 60 with ten intervals and provide the ROC curves in the first row of <ref>.By observing the ROC curves, we can have the following conclusions.Firstly, a smaller patch size is easier to raise false alarms while a larger patch size may lead to a relatively lower detection probability, which just demonstrates our above analysis about the patch size.Secondly, the proposed RIPT method is not very sensitive to the patch size.The detection result of the patch size among 20 – 60 is acceptable. Thirdly, 30 seems a good choice for Sequence 1 – 4 since it achieves the best performance in ROC. §.§.§ Sliding stepThe sliding step influences the patch-tensor size as well.To reduce the computational complexity, we prefer a larger sliding step, which means smaller matrices to perform SVD.Nevertheless, a larger sliding step also reduces the redundancy of the original patch-tensor 𝒯 and undermines the final detection results since our proposed model is based on the non-local repentance of correlated patches.To investigate its influence, we vary the sliding step S from 8 to 16 with two intervals.The results are displayed in the second row of <ref>.It could be observed that the ROC curve of small sliding step like 8 tends to have a more sharp shape, but its overall detection probability remains relatively low.The best value for sliding step is among 12 to 14, here we pick 12.In addition, by comparing the first row with the second row of <ref>, we can conclude that the algorithm is quite robust to the variation of step length.§.§.§ Weight stretching parameter hIt controls the local structure weight's suppression degree to the clutter edges.We vary h from 6 to 22 in the experiment and illustrate the ROC curves in the third row of <ref>.Generally, we would like a larger h which suppresses the undesirable non-target components thoroughly.Nevertheless, since the target-clutter distinguishing scheme is not always perfect, an overlarge h would also wipe out some targets.A typical example is the different performance of h = 18 or 22 among four sequences.For Sequence 2 and 3, h = 18 or 22 achieves the best performance. But, they perform the worst for Sequence 1 and 4. It is because the target moves along the cloud edge in many frames of Sequence 1 and 4, and an overlarge h would easily mistake the target as the edge and suppress it completely, resulting in a lower detection probability.On the contrary, a smaller h might preserve the small target, but it also retains some non-target components, making the false-alarm ratio relatively high.For Sequence 2 and 3, when the detection probability is fixed, the false-alarm ratio of h = 6 is the largest.In order to seek a balance, we set the optimal h as 10 in the following experiment.§.§.§ Weighting parameter λDespite the usage of local structure weight, fine tuning of λ is still of great importance.We show the effects of λ in the fourth row of <ref>.Since λ is set as L / √(min(I,J,P)) in our model, instead of directly varying λ, we vary L from 0.3 to 1.5.From the illustration, it can be observed that a large λ does keep the false-alarm ratio being quite low like.For example, the ROC curves of L = 1.0 and L = 1.5 for Sequence 2 are straight line segments.But their detection probabilities are also low because many dim targets are suppressed by the overlarge threshold.On the contrary, when the detection probability is fixed, the false-alarm ratio of L = 0.3 is always higher than the others, suggesting that a too small L is also not a good choice.§.§.§ Penalty factor μIt is precisely the shrinking threshold of <ref>, which influences the low-rank property of the background patch-tensor.With a smaller μ, more details are preserved in the background patch-tensor.Thus fewer non-target components are left in the target patch-tensor.Nevertheless, the small target might be preserved in the background patch-tensor as well, resulting no target in the target image.On the contrary, a larger μ would lead to more non-target components lying in the target patch-tensor.Thus, it is necessary to choose an appropriate value for μ to keep the balance between detection probability and false-alarm ratio.Since we set μ = C_μstd(vec(ℱ)), instead of varying μ directly, we investigate the influence of the penalty factor on Sequence 1 – 4 by varying C_μ from 0.5 to 3.The results are shown in the last row of <ref>, from which we can observe that an overlarge or too small μ is not an optimal choice and the best value for our four sequences is about 0.7. §.§ Comparison with State-of-the-Arts In this subsection, we first compare the proposed model with the other state-of-the-art methods on the ability of clutter suppression.<ref> – <ref> show the separated target images by twelve tested methods for four representative frames of Sequence 1 – 4 in <ref>.It can be seen that the classical Max-Median filter does enhance the tiny targets in <ref>(b) - <ref>(b).Nevertheless, many non-target pixels are also enhanced simultaneously, especially in <ref>(b) and <ref>(b), which would raise many false alarms.In <ref>(a) - <ref>(a) produced by TDLMS, the phenomenon of enhancing non-target isolated points does not exist, but the cloud edges are highlighted, making them much brighter than the small target.Since the given target size matches the real target size exactly, the Top-Hat operator succeeds to enhance the target region.If not match, the Top-Hat operator is likely to lose the target.No matter whether the given and real target sizes match, Top-Hat cannot well suppress the background clutters.Many strong clutters still remain in resulting images, as illustrated in <ref>(c) - <ref>(c).Although PFT can retain the target to a certain extent, the target is not necessarily the brightest and there are also many non-target salient residuals, as shown in <ref>(d) - <ref>(d). MPCM and WLDM failed to achieve good results because they suffered from a phenomenon we named rare structure effect which is caused by the inaccuracy of the local dissimilarity measure and often happens when the target is extremely dim. In next subsection, we will further discuss this phenomenon. In fact, the common and intrinsic reason behind the unsatisfactory performance of all these six methods lies in their pre-set assumption about the target shape, namely a hot spot brighter than its neighborhood.Nevertheless, when the target is too dim to maintain its significant contrast over non-target components, just like <ref>(a) – (d), they might not perform as well as they usually do. The last six tested methods are all low-rank based methods. Comparing with above six methods, their results contain fewer background details. Relatively speaking, the effects of PRPCA and WIPI are not as good as the other four methods.Different from the other low-rank based methods that all build their low-rank assumptions on the data structure composed of patches, PRPCA supposes the individual patch is low-rank.Thus in PRPCA, each patch is applied to an individual RPCA process. Then all the separated target patches are synthesized into a target image.By comparing <ref>(g) and <ref>(g) with <ref>(h) and <ref>(h) , it can be seen that fewer edges were left by IPI than PRPCA.It is because the rare structure in a patch is not necessarily rare in the patch-image due to the redundancy of the whole image.Therefore, the results of IPI and IPT are much better than those of PRPCA.As to WIPI, considering the targets in Sequence 1 – 4 is much dimmer than those in Ref. <cit.>, it is fair to say that the steering kernel based patch-level weight is still not robust enough to handle all of the complex infrared backgrounds.From <ref>(l) and <ref>(l), we can see that with the help of the local structure weight, the non-target components were suppressed completely via our proposed model.For example, the cloud residuals in <ref>(g) by IPI is brighter than its target, while in <ref>(l), it is wiped out thoroughly.Based on above comparisons, it is fair to conclude that the proposed RIPT model achieves the most satisfying performance in infrared background suppression among twelve tested methods. For infrared small target detection, the biggest difficulty is the interference of complex backgrounds.These undesirable background clutters raise the false alarm rates, and might even overwhelm the dim targets.Therefore, the ability of successfully suppressing the background clutters is a major concern in evaluating an infrared small target detection method. Quantitative evaluating indices are also widely used to assess this ability.<ref> shows the experimental data of all twelve tested methods for <ref>(a) – (d).The gray-scale of every separated target image is rescaled to the range 0–255.It could be observed that our proposed method gets the highest scores among all indices and all tested images.Different from the filtering based methods, for the low-rank based methods, Inf, namely infinity, is quite common, which just means that the target neighborhood completely shrinks to zero.In addition, it should be noted that the high scores in these three quantitative indices merely reflect the good suppression performance in a local region, and not necessarily mean a good global suppression ability. To further reveal the advantage of the proposed model, we display the ROC curves of Sequence 1 and Sequence 3 – 5 for comparison in <ref>. The most interesting point is the performances of the state-of-the-art WLDM on Sequence 1, 3, 4 and Sequence 5 are very different.For Sequence 5, WLDM performs very well but fails in Sequence 1 – 4. We believe the reason lies in the rare structure effect which is a born problem for local contrast method. NIPPS's performance is slightly better than the IPI model. Finally, the proposed algorithm achieves the highest detection probability for the same false-alarm ratio, meaning that the proposed RIPT model has a better performance than the other models.§ CONCLUSIONTo further suppress the strong edges while preserving the spatial correlation, a reweighted infrared patch-tensor model for small target detection is developed in this paper, simultaneously combining non-local redundant prior and local structure prior together.A local structure weight is designed based on the structure tensor and served as an edge indicator in the weighted model.In addition, a sparsity enhancement scheme is adopted to avoid the target image being contaminated.Then the target-background separation task is modeled as a reweighted robust tensor recovery problem, which can be efficiently solved via ADMM.Detailed experimental results show that our proposed model is robust to various scenarios and obtainsthe clearest separated target images compared with the state-of-the-art target-background separation methods. § ACKNOWLEDGMENTSThe authors would like to thank the editor and anonymous reviewers for their helpful comments and suggestions.This work was supported in part by the National Natural Science Foundation of China under Grants no. 61573183, and Open Research Fund of Key Laboratory of Spectral Imaging Technology, Chinese Academy of Sciences under Grant no. LSIT201401.[ < g r a p h i c s > ]Yimian Dai received the B.S. degree from Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing, China, in 2013, where he is currently pursuing the Ph.D. degree. His current interests include target detection, image restoration, and machine learning. [ < g r a p h i c s > ]Yiquan Wu received his M.S. and Ph.D. degrees from Nanjing University of Aeronautics and Astronautics in 1987 and 1998, respectively. He is at present a professor and Ph.D. supervisor in the Department of Information and Communication Engineering at the Nanjing University of Aeronautics and Astronautics, where he is involved in teaching and research in the areas of image processing and recognition, target detection and tracking, and intelligent information processing. | http://arxiv.org/abs/1703.09157v1 | {
"authors": [
"Yimian Dai",
"Yiquan Wu"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170327155727",
"title": "Reweighted Infrared Patch-Tensor Model With Both Non-Local and Local Priors for Single-Frame Small Target Detection"
} |
Service Overlay Forest Embedding forSoftware-Defined Cloud Networks Jian-Jhih Kuo1, Shan-Hsiang Shen12, Ming-Hong Yang13, De-Nian Yang1, Ming-Jer Tsai4 and Wen-Tsuen Chen14 1Inst. of Information Science, Academia Sinica, Taipei, Taiwan2Dept. of Computer Science & Information Engineering,National Taiwan University of Science & Technology, Taipei, Taiwan3Dept. of Computer Science & Engineering, University of Minnesota, Minneapolis MN, USA4Dept. of Computer Science, National Tsing Hua University, Hsinchu, TaiwanE-mail: {lajacky,sshen3,curtisyang,dnyang,chenwt}@iis.sinica.edu.tw and [email protected] December 30, 2023 =========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== A simple robust genuinenly multidimensional convective pressure split (CPS) , contact preserving, shock stable Riemann solver (GM-K-CUSP-X) forEuler equations of gasdynamics is developed. The convective and pressure components of the Euler system are seperated following the Toro-Vazquez type PDE flux splitting [Toro et al, 2012]. Upwind discretization of these components are achieved using the framework of Mandal et al [Mandal et al, 2015]. The robustness of the scheme is studiedon a few two dimensional test problems. The results demonstrate the efficacy of the scheme over the corresponding conventional two state versionof the solver. Results from two classic strong shock test cases associated with the infamous Carbuncle phenomenon, indicate that thepresent solver is completely free of any such numerical instabilities albeit possessing contact resolution abilities.Such a findingemphasizes the preexisting notion about the positive effects that multidimensional flow modeling may have towards curing of shock instabilities.Keywords : Genuinely multidimensional, Riemann solver, Contact preserving, Convective Pressure Split, numerical shock instability, Carbuncle phenomenon. § INTRODUCTION Past few decades have witnessed commendable advancement in the computation of high speed flows. Birth of the upwind schemes for finite volume methods is amilestone in this regard. Among these characteristics based schemes, those based on the solution of two state ‘Riemann problems’at cell interfaceare the most popular class of schemes lately. Godunov <cit.> proposed the first Riemann problem based algorithm (also called Riemann solvers) that used the exact solution of a Riemann problem toconstruct a first order accuratescheme. However, this exact solver was critized for being prohibitively expensive because of the iterative technique involved<cit.>.To mitigate this shortcoming, a class ofapproximate Riemann solvers have been proposed and continue to be developed. Details about some famous approximate Riemann solvers can be found in<cit.>. Such algorithms have brought forth an era of much cheaper yet accurate computation of gasdynamic flows.Since most of these approximate schemes are inherently one dimensional in their framework, they enjoy accurate and robust resolution ofboth linear and nonlinear wave fields arising in one dimensional problems. However, accurate resolution of flow featuresin practically relevantmultidimensional problems where important flow features are oblique to the grid still pose a major challenge. Thisis mainly because the wave system in such problems possessinfinitely many propagation direction as compared to the limited ones in a one-dimensional problem. Further vorticitywave enters the formulation and has to be dealt with appropriately <cit.>. Currently, the standard way of extending the one-dimensional schemes to multidimensions is through a dimensionally split operator approach motivated in <cit.> where local one dimensional Riemann problems are solved in a direction normal to each cell interface.Such a strategy has been criticised because of its unnatural selection of the interface normal as the only wave propagation direction even for problems with infinitely many propagation directions. This lacuna is attributed to cause for example, pressure disturbances across a grid oblique shear wave <cit.>.Another disturbing problem encountered by dimensionally split approximate Riemann solvers is the occurrence of various forms of numerical shockinstabilities when simulating strong normal shocks. A catalouge of many such failures can be found in <cit.>. It has been observed that only those schemes that have exact shear wave resolution abilities are known to produce these instabilities. There is increasing evidence <cit.> to consider that such failures occur due to lack of adequate dissipation across cell faces which are normal to the shock wave front.These problems solicit a strong need for a genuinely multidimensional formulation that resolves all characteristic fields while introducingappropriate dissipation along necessary directions.Probably the earliest attempt at creating a multidimensional Riemann solver was by Raithby <cit.> who suggesteddiscretizing the convective terms along flow dominant directions. Davis <cit.> and others <cit.> introduced the idea ofrotated Riemann solvers that involves identifying grid oblique shock orientations and solving Riemann problems across them. Roe and others <cit.> suggested using extrapolated data from the left and right states across a Riemann solverto construct simple wave solutions for a multidimensional Riemann problem. This technique is not generally valid for arbitrarygoverning equations and demands substantial reformulationfor three dimensional problems. Collela <cit.> is credited with proposing the CTU method that uses the characteristics of the system to incorporate diagonal cell contirbutions to interface fluxes in a predictor corrector framework.Leveque <cit.> proposed a multidimensional scheme in which cross derivative terms that constitute the transverse contributions areincluded by interpreting the usual one dimensional Gudonov method in a fluctuation splitting framework. Ren et al <cit.> constructed an operator split predictor corrector scheme based on CTU and Leveque's wave propagation methodfor both Euler and Navier Stokes systems. While the predictor step solves linearized Euler equations in characteristic variables, the corrector step adds viscouscontributions.Wendroff <cit.> introduced a multistate Riemann solver that attempted the extension of one dimentional HLLE scheme <cit.> into several dimensions. Thescheme used an expensive nine point stencil and suffered unacceptable dissipation due to unnatural wavespeed selection <cit.>. Another multistate Riemann solver is from the work of Brio et al <cit.>. Multidimensional effects are incorporated by adding correction terms to the standardface normal fluxes at every computational cell interface. These correction terms are obtained by solving a three state Riemann problem using Roe's FDS at the cornersof the respective cells. In a similar spirit, Balsara <cit.> presented a generic multidimensional HLLE solver (GM-HLLE) with simple closed form expressionsthat can be extended to any hyperbolic system. This method too relies on construction of multidimensional correction terms like <cit.> wherein these terms are obtained using a four state HLLE Riemann solver at interface corners.By building upon the wave model introduced in GM-HLLE, Balsara <cit.> further proposed a multidimensional HLLC solver for Euler and MHD systems. To dealwith contact discontinuities in two dimensions, a set of twelve possible contact orientations on a given cell are included in the wave model. Although posessing closed form expressions in two dimensions, such a wave model would not be easily tractable in three dimensions.Improvements to the model was proposed in <cit.> by reformulating the scheme in terms of characteristic variables but this too remainscomplicated to implement on a computer code.Mandal et al <cit.> proposed a multidimensional convective-pressure split scheme (GM-HLLCPS-Z) based on Zha-Bilgen <cit.> way of splitting Euler fluxes. The scheme consists of a wave speed averaged upwinding for the convective part and GM-HLLE type discretization for the pressure part. This scheme is basically a multidimensional extension of HLL-CPS strategy <cit.> that uses a HLL type discretizationfor distinctly treating convective and pressure flux parts. The splitting of full Euler flux into convective and pressure parts can beachieved by adopting AUSM type or Zha-Bilgen type PDE splitting <cit.>. Diffusion control is achieved by careful tuning of thedissipation vector of the pressure flux. This renders the scheme with stationary and moving contact preserving abilities in addition tocapability of surviving the most stringent test problems. Toro has corroborated such a method by showing that exact contact preserving ability can be incorporated into convective-pressure split framework by discretizing the pressure fluxes using a Riemann solver <cit.>. Surprisingly, although contact preserving, HLL-CPS is found toevade most common forms of numerical shock instabilities, particularly the carbuncle phenomenon. Recently Xie et al <cit.> has proposed a contact capturing convective-pressure split scheme named K-CUSP-X. The schemeuses exactly similar discretization as HLL-CPSfor both convective and pressure fluxes but differs only in the method of splitting the Euler fluxes into these components; instead of using Zha-Bilgen or AUSM way of PDE splitting, it adopts Toro-Vazquez method wherein the pressure terms embedded in the energy is also seperated <cit.>.However unlike HLL-CPS, the present investigations reveal that this scheme is found to suffer from numerical shock instabilites.In this paper, a new genuinely multidimensional scheme based on the conventional K-CUSP-X is proposed.A multidimensional correctionterm similar to <cit.> and <cit.> is constructed by solving appropriate four state Riemann problem at the corners of each interface. The multidimensional terms areincorporated such that the final fluxes can be calculated at interfaces with the familiar ease of pre-existing dimensional split methods. Adhereing to the existing convention, this new scheme may becalled GM-K-CUSP-X. It will be the purpose of this paper to demonstrate that such a construction is not only as accurate as GM-HLL-CPS-Z,but also cures the numerical shock instability that plagued the corresponding two state conventional K-CUSP-X Riemann solver. The positive effect of genuinely multidimensionalflow modeling on K-CUSP-X was first demonstrated in <cit.> wherein authors demonstrated that shock instability associated with standing shock problem <cit.> was completely removed by extending the original two state solver into its genuinely multidimensional variant. Such a finding corroborates the pre existing opinion thatmultidimensional dissipation acts as a reliable cure for numerical shock instability problems and incentivizes the need for extendingthe accurate Riemann solvers in literature into their robust genuinely multidimensional counterparts. This paper is organized as follows. In the next section, the governing equations and the type of PDE flux splitting used is detailed. In Section 3, the second order version of the new formulation is described. In Section 4, results for some complex flow problems like double Mach reflection and multidimensional Riemann problem are discussed. Further, two classic shock instability test problems, the odd-even decoupling problem and standing shock problem are used to study instability behaviour of the newly developed scheme. Section 5 contains some concluding remarks. § PRELIMINARIES §.§ Governing equation Consider the two-dimensional Euler equations in differential form∂𝐔/∂ t + ∂𝐅/∂ x + ∂𝐆/∂ y=0where 𝐔=[ρ ρ u ρ v ρ e]^T is the vector of conserved quantities. 𝐅 and 𝐆 are the flux vectors in the x and y directions respectively given by𝐅=[ρ u;ρ u^2 + p; ρ uv; u(ρ e + p) ]𝐆=[ρ v; ρ uv;ρ v^2 + p; v(ρ e + p) ]In the present work, the above flux vectors are split into corresponding convective and pressure parts followingthe approach of Toro-Vazquez <cit.> . Using ideal gas law, the splitflux vectors can be written as𝐅_1=u[ρ;ρ u;ρ v; 1/2ρ (u^2+v^2) ]𝐅_2=[0;p;0; γ/γ-1 pu ]𝐆_1=v[ρ;ρ u;ρ v; 1/2ρ (u^2+v^2) ]𝐆_2=[0;0;p; γ/γ-1 pv ]where 𝐅_1 and 𝐆_1 are the convective fluxes while 𝐅_2 and 𝐆_2 are the pressure fluxes. γrepresents the ratio of specific heat capacities. As suggested by Toro et al <cit.> the convective fluxes 𝐅_1 and 𝐆_1 can be interpreted assimple advection of mass, momentum and kinetic energy along the x and y directions respectively, while the pressure fluxes 𝐅_2 and 𝐆_2 are interpreted to be sonic impulses that spread in all directionswith reference to these convecting particles.This type of PDE splitting primarily differs from those that already exist in the literature like <cit.> and <cit.>,in terms of thequantity concerning energy that is being advected. Such differences may have strong bearing on the robustness of these schemes.§.§ Finite volume discretization Consider the integral form of equation (<ref>)∂/∂ t∫_Ω𝐔 dΩ + ∮_dΩ(𝐅+𝐆)·𝐧̂ ds=0where 𝐧̂=(n_x,n_y) is the unit vector along the face normal. Consider a Cartesian cell of area Δ x ×Δ y as shown in Figure <ref>. The finite volume discretization for the cell (i,j) can be written as𝐔̅_i,j^n+1=𝐔̅_i,j^n+Δ t/Δ x(𝐅̅^'_i-1/2,j-𝐅̅^'_i+1/2,j)+Δ t/Δ y(𝐆̅^'_i,j-1/2-𝐆̅^'_i,j+1/2)where n is the time level, i and j are cell indices. The (̅.̅)̅ quantities depict the respective averaged quantities. The total fluxes at the cell interfaces (i+1/2,j) and (i-1/2,j) are denoted as 𝐅̅^'_i+1/2,j and𝐅̅^'_i-1/2,j respectively. Similarly 𝐆̅^'_i,j+1/2 and𝐆̅^'_i,j-1/2 are the y directional total fluxes in corresponding y interfaces. § FORMULATION In the proposed scheme the interface flux will comprise of both a two state conventional Riemann flux and a multidimensional flux. These multidimensional fluxes are sought from the solution of four state Riemann problem that occurs at each corner of a Cartesian cell. To see this more clearly, consider a typical cell (i,j) as shown in Figure <ref>. The four constant states that come together at corner c_1 at t=0 and formsa two-dimensional Riemann problem are marked as LU (Left-Upper), LD (Left-Down), RU (Right-Upper) and RD (Right-Down). Although realistically,the waves pertaining to this multistate Riemann problem will travel in infinitely many directions and swap a curvilinear area inspace at any later time t=T,for sake of simplicity, the simple wave model consisting of only four waves as introduced in <cit.> is used in this work.Accordingly the wave propagation at the corners is assumed to span, at any time T, a rectangular region. Figure <ref> represents a three dimensional view of thesewaves as they evolve in time where the shaded region depicts the domain of influence of this multidimensional Riemann problem.In principle, the Equation <ref> can be integrated along the space-time volume of Figure <ref> appropriatelyto obtain the time averaged multidimensional fluxes 𝐅^* in x-direction and 𝐆^* in y-direction. For a typical cell interface (i+1/2,j) the totalflux will be an ensemble of the two multidimensional fluxes from corners c_1 and c_4 denoted as 𝐅^*_i+1/2,j+1/2, 𝐅^*_i+1/2,j-1/2 and thesingle mid-point flux𝐅_i+1/2,j^mid. This is shown in Figure <ref>. A conservative ensemble of the corner and mid point fluxes are achieved by using Simpson's rule of intergration <cit.> along the interface as given by, 𝐅̅^'_i+1/2,j=1/6𝐅^*_i+1/2,j+1/2+4/6𝐅_i+1/2,j^mid+1/6𝐅^*_i+1/2,j-1/2 §.§ Evaluation of flux at the midpoint of the cell interface This section deals with determination of the two state Riemann flux at the mid point of a typical interface (i+1/2,j) denoted as𝐅_i+1/2,j^mid.Following the Convective Pressure Split(CPS) philosophy the total flux at this interface is first split into convective and pressure parts. This work uses the Toro-Vazquez type flux splitting as mentioned in Equation (<ref>),𝐅_i+1/2^mid = 𝐅_1_i+1/2,j^mid + 𝐅_2_i+1/2,j^mid These convective and pressure parts are discretized independently following the original HLL-CPS strategy <cit.>. It may be noted that theinterface (i+1/2,j) admits only the x-directional Riemann flux 𝐅_i+1/2,j^mid while they-directional flux, 𝐆_i+1/2,j^mid is zero on it.§.§.§ Evaluation of convective flux at the midpoint of the cell interface (𝐅_1_i+1/2,j^mid)The upwind discretization of the convective flux denoted by 𝐅_1_i+1/2,j^mid at the mid point of the cell interface (i+1/2,j),following the strategy of HLL-CPS method <cit.>, is given by,𝐅_1_i+1/2,j^mid=M_k[ρ;ρ u;ρ v; 1/2ρ (u^2+v^2) ]_k a_k k= L u̅≥0 R u̅<0 M_k=u̅/u̅-S_L^cu̅≥0u̅/u̅-S_R^cu̅<0 a_k= u_L-S_L^c u̅≥0 u_R-S_R^c u̅<0where, the average local x-directional velocity at the interface is taken asu̅=u_L+u_R/2. Dependingon the sign of the average local x-directional velocity, left (L) or right (R) states are selected for upwinding.S_L^c and S_R^c are carefully selected wave speeds which are discussed in section <ref>.§.§.§ Evaluation of pressure flux at the midpoint of the cell interface (𝐅_2_i+1/2,j^mid) The pressure flux at the midpoint of the cell interface is obtained by applying a HLL type discretization of the pressure flux vector<cit.>.𝐅_2_i+1/2,j^mid=S_R^c/S_R^c-S_L^c𝐅_2_L-S_L^c/S_R^c-S_L^c𝐅_2_R+S_R^cS_L^c/S_R^c-S_L^c(𝐔_R-𝐔_L)The above equation can be rewritten as𝐅_2_i+1/2,j^mid=1/2(𝐅_2L+𝐅_2R) + δ𝐔_2where 𝐅_2L and 𝐅_2R are the left and right side x-directional pressure fluxes normal to the interface (i+1/2,j) and δ𝐔_2 is the numerical diffusion given byδ𝐔_2=S_R^c+S_L^c/2(S_R^c-S_L^c)(𝐅_2_L-𝐅_2_R)-S_L^cS_R^c/S_R^c-S_L^c[ ρ_L-ρ_R; (ρ u)_L-(ρ u)_R; (ρ v)_L-(ρ v)_R; (ρ e)_L-(ρ e)_R ]It is clear from Equation <ref> that the second term will give rise to numerical diffusion across a contact. Thus in order to capture the contact discontinuity accurately,density terms in δ𝐔_2 are replaced by the pressure terms by using isentropic assumption a̅_c^2=δ p/δρas described in the reference <cit.>.δ𝐔_2=S_R^c+S_L^c/2(S_R^c-S_L^c)(𝐅_2L-𝐅_2R)-S_R^cS_L^c/a̅_c^2(S_R^c-S_L^c)[ p_L-p_R; (pu)_L-(pu)_R; (pv)_L-(pv)_R; a̅_c^2/2 (p_L-p_R) + 1/2[(pq^2)_L-(pq^2)_R] ] where a̅_c is the average speed of sound at the interface given by a̅_c=a_L+a_R/2 and q^2=u^2+v^2 is twice thelocal kinetic energy per unit mass. §.§.§ Selection of wave speeds for the 1D Riemann problem at the interface (𝐢+1/2,𝐣) The wave speeds are selected according to conventional HLL-CPS method <cit.>S_L= min (0, u_L-a_L,u^*-c^*) S_R= max (0, u_r+a_r,u^*+c^*)where u^* and c^* are given by <cit.>,u^*= u_L+u_R/2+a_L-a_R/γ-1 c^*= a_L+a_R/2+ γ-1/4(u_L-u_R)Supersonic conditions are taken care by including '0' in the above expressions. For a stationary flow the wave speeds are modified asS_L^c=-a̅_cS_R^c=a̅_cIt is to be noted that midpoint fluxes at other interfaces can be obtained in the similar manner.§.§ Evaluation of flux at the corner of the cell interfaceThis section will detail how to evaluate the fluxes 𝐅^*_i+1/2,j+1/2 and 𝐆^*_i+1/2,j+1/2 that resultsfrom the interaction of four Riemann states at a representative corner c_1. These fluxes forms themultidimensional component of the interface fluxes and are shown in Figure <ref>. Once again, resorting to the (CPS) philosophy, the total flux at this corner is split into convective (𝐅_1^*_i+1/2,j+1/2 and 𝐆_1^*_i+1/2,j+1/2) andpressure fluxes (𝐅_2^*_i+1/2,j+1/2 and 𝐆_2^*_i+1/2,j+1/2) originating at the corneras per Equation <ref>. Analogous to the procedure for evaluation of the two state mid point flux at the interface, the split convective and the pressure parts at thecorners will also be evaluated using different upwind strategies. §.§.§ Evaluation of convective flux at the corner c_1 of the cell interface(𝐅_1^*_i+1/2,j+1/2) Following <cit.>, the x-directional convective flux is evaluated as,𝐅_1^*_i+1/2,j+1/2=u̅(S_U[ρ;ρ u;ρ v; 1/2ρ (u^2+v^2) ]_k_1-S_D[ρ;ρ u;ρ v; 1/2ρ (u^2+v^2) ]_k_2)/S_U-S_Dwhere u̅ is the wave speed averaged x-directional local fluid speed at the corner defined as, u̅=u_LUS_U-u_LDS_D+u_RUS_U-u_RDS_D/2(S_U-S_D)Based on the direction of the average x-directional flow, the upwind states k_1,k_2 are chosen as, If u̅>0, k_1=LU and k_2=LD (i.e upwinding is done from the left states).If u̅<0, k_1=RU and k_2=RD (i.e upwinding is done from the right states).In similar spirit, the y directional convective flux is evaluated as, 𝐆_1^*_i+1/2,j+1/2=v̅(S_R[ρ;ρ u;ρ v; 1/2ρ (u^2+v^2) ]_k_1-S_L[ρ;ρ u;ρ v; 1/2ρ (u^2+v^2) ]_k_2)/S_R-S_Lwhere v̅ is the the wave speed averaged y-directional local fluid speed at the corner defined as,v̅=v_RUS_R-v_LUS_L+v_RDS_R-v_LDS_L/2(S_R-S_L)The states k_1,k_2 are chosen accordingly as, If v̅>0, k_1=RD and k_2=LD (i.e upwinding is done from the down states).If v̅<0, k_1=RU and k_2=LU (i.e upwinding is done from the up states).Since the above strategy is developed for a subsonic case, slight modification u̅ and v̅ is done toextend the above formulation to supersonic cases: 1. If the flow is supersonic in positive x-direction, then at the corner upwinding is done from left states. Therefore for the evaluation of x-directional convective flux u̅ is taken asu̅=u_LUS_U-u_LDS_D/S_U-S_D2. If the flow is supersonic in negative x-direction the upwinding is done from right states. Therefore for the evaluation of x-directionalconvective flux u̅ is taken asu̅=u_RUS_U-u_RDS_D/S_U-S_D3. If the flow is supersonic in positive y-direction the upwinding is done from down states. Therefore for the evaluation of y-directionalconvective flux v̅ is taken asv̅=v_RDS_R-v_LDS_L/S_R-S_L4. If the flow is supersonic in negative y-direction the upwinding is done from upper states. Therefore for the evaluation of y-directionalconvective flux v̅ is taken asv̅=v_RUS_R-v_LUS_L/S_R-S_L§.§.§ Evaluation of pressure flux at the corner c_1 of the cell interface (𝐅_2^*_i+1/2,j+1/2) Following <cit.>, the x-directional convective flux is evaluated as,𝐅_2^*_i+1/2,j+1/2 =𝐅_2_LU S_RS_U+𝐅_2_RDS_LS_D-𝐅_2_LDS_RS_D- 𝐅_2_RUS_LS_U/(S_R-S_L)(S_U-S_D) -2S_RS_L/(S_R-S_L)(S_U-S_D)(𝐆_2_RU-𝐆_2_LU+𝐆_2_LD- 𝐆_2_RD)+S_RS_L/(S_R-S_L)(S_U-S_D)(S_U(𝐔_RU-𝐔_LU)-S_D(𝐔_RD-𝐔_LD))Similarly the y-directional pressure flux is evaluated as, 𝐆_2^*_i+1/2,j+1/2 =𝐆_2_RD S_RS_U+𝐆_2_LUS_LS_D-𝐆_2_RUS_RS_D- 𝐆_2_LDS_LS_U/(S_R-S_L)(S_U-S_D)-2S_US_D/(S_R-S_L)(S_U-S_D)(𝐅_2_RU-𝐅_2_LU+𝐅_2_LD- 𝐅_2_RD)+S_US_D/(S_R-S_L)(S_U-S_D)(S_R(𝐔_RU-𝐔_RD)-S_L(𝐔_LU-𝐔_LD))The above equations can be rewritten as,𝐅_2^*_i+1/2,j+1/2 =1/2(𝐅_2 _L+𝐅_2_R) + δ𝐔_2𝐱-2S_RS_L/(S_R-S_L)(S_U-S_D)(𝐆_2_RU-𝐆_2_LU+𝐆_2_LD- 𝐆_2_RD)𝐆_2^*_i+1/2,j+1/2 =1/2(𝐆_2 _D+𝐆_2_U) + δ𝐔_2𝐲 -2S_US_D/(S_R-S_L)(S_U-S_D)(𝐅_2_RU-𝐅_2_LU+𝐅_2_LD- 𝐅_2_RD)where𝐅_2_L=𝐅_LUS_U-𝐅_LDS_D/S_U-S_D 𝐅_2_R=𝐅_RUS_U-𝐅_RDS_D/S_U-S_D𝐆_2_D=𝐆_RDS_R-𝐆_LDS_L/S_R-S_L𝐆_2_U=𝐆_RUS_R-𝐆_LUS_L/S_R-S_LTheδ𝐔_2𝐱 and δ𝐔_2𝐲 terms in Equations (<ref>,<ref>), are thenumerical diffusion terms in x and y-directions respectively. To remove the numerical dissipation across a contact wave, the dissipation terms in x and y-directions are remodeled using isentropic expression as,δ𝐔_2𝐱 =S_R+S_L/2(S_R-S_L)(𝐅_2 _L-𝐅_2_R)-S_RS_L/(S_R-S_L)(S_U-S_D)(a̅^2)[ S_U(p_LU-p_RU)-S_D(p_LD-p_RD); S_U((pu)_LU-(pu)_RU)-S_D((pu)_LD-(pu)_RD); S_U((pv)_LU-(pv)_RU)-S_D((pv)_LD-(pv)_RD); S_U(e^*_LU-e^*_RU)-S_D(e^*_LD-e^*_RD) ]δ𝐔_2𝐲 =S_U+S_D/2(S_U-S_D)(𝐆_2_D-𝐆_2_U)-S_US_D/(S_R-S_L)(S_U-S_D)(a̅^2)[ S_R(p_RD-p_RU)-S_L(p_LD-p_LU); S_R((pu)_RD-(pu)_RU)-S_L((pu)_LD-(pu)_LU); S_R((pv)_RD-(pv)_RU)-S_L((pv)_LD-(pv)_LU); S_R(e^*_RD-e^*_RU)-S_L(e^*_LD-e^*_LU) ]where e^*_k is given ase^*_k=a̅^2/γ-1p_k+1/2p_k(u^2+v^2)_kand a̅ asa̅=a_LU+a_RU+a_LD+a_RD/4 The flux contributions due to the genuinely multidimensional Riemann problem at the other corner c_4 of the interfacecan be obtained in a similar manner. Most importantly, it must be noted that while 𝐅^*_i+1/2,j ±1/2contributes to the total interface flux at the interface (i+1/2,j), 𝐆^*_i + 1/2,j + 1/2 and𝐆^*_i + 1/2,j - 1/2 contributes to thetotal interface flux at the interfaces (i,j+1/2) and (i,j-1/2) respectively.§.§ Selection of wave speeds for the multidimensional Riemann problem at the corners As previously mentioned, the present work adopts a simple wave model as proposed by <cit.> to represent the waves emerging fromthe four state Riemann problem at the corners of every interface. A top view of the area swept by these four waves S_L,S_R,S_U,S_D is shown in Figure <ref>. The rectangle ABCD in Figure <ref>depicts the domain of influence of the four state Riemann problem at corner c_1 at time T on x-y plane. An estimate for these wavespeeds can be obtained as,S_R =max(0,λ_x^N(𝐔_RU),λ_x^N(𝐔_RD),λ̅_x^N(𝐔_LU,𝐔_RU),λ̃_x^N(𝐔_LD,𝐔_RD))S_L =min(0,λ_x^1(𝐔_LU),λ_x^1(𝐔_LD),λ̅_x^1(𝐔_LU,𝐔_RU),λ̃_x^1(𝐔_LD,𝐔_RD))S_U =max(0,λ_y^N(𝐔_RU),λ_y^N(𝐔_LU),λ̅_y^N(𝐔_RD,𝐔_RU),λ̃_y^N(𝐔_LD,𝐔_LU))S_D =min(0,λ_y^1(𝐔_RD),λ_y^1(𝐔_LD),λ̅_y^1(𝐔_RD,𝐔_RU),λ̃_y^1(𝐔_LD,𝐔_LU))where typically,λ_x^1(𝐔_k) denote smallest x-directional wave speed in the state 𝐔_k,λ_x^N(𝐔_k) denote largest x-directional wave speed in the state 𝐔_k,λ̃_x^1(𝐔_k,𝐔_l) denotes smallest x-directional wave speed from Roe averaged state between 𝐔_k and 𝐔_l,λ̃_x^N(𝐔_k,𝐔_l) denotes largest x-directional wave speed from Roe averaged state between 𝐔_k and 𝐔_l such that k,l ∈{LU,LD,RU,RD} and k≠ l.If u̅=0 and v̅≠0, then x-directional wave speeds are modified as S_L=-a̅ and S_R=a̅.If u̅≠0 and v̅=0, then y-directional wave speeds are modified as S_D=-a̅ and S_U=a̅.If u̅=0 and v̅=0, then wave speeds are modified as S_L=-a̅, S_R=a̅,S_D=-a̅ and S_U=a̅. It should be noted that zero has been added in the above expressions in order to ensure fully one sided flux in supersonic flow.§ RESULTS §.§ Isentropic vortex problemA second order accurate version of the present solver is developed using the SDWLS strategy <cit.> whose details are omitted here for brevity. Formal order of accuracy of the second order version of GM-K-CUSP-X is investigated using the isentropic vortex problem <cit.>. The problem involves an isentropic vortex centered initially at (0,0) and made to traverse the domain diagonally under a periodic boundary condition. A Cartesian domain of size [-5,5] × [-5,5] is used. The initial conditions consists of a unperturbed state given by (ρ,p,u,v)= (1.0,1.0,1.0,1.0). The temperature is defined as T=p/ρ. A perturbation is added to the flow as,(δ u, δ v) =ϵ/2πe^0.5(1-r^2)(-y,x) δ T=-γ-1/8 γπ^2ϵ^2 e^(1-r^2) δρ=δ T^1/γ-1 δ p=δρ^γHere, ϵ which defines the strength of the vortex is taken as 5.0. r denotes the Cartesian distance from vortex center. Theaccuracy of the scheme is measured in L_1 and L_∞ norms of the density variable. The formal order of accuracy O is obtained by the formula, O= log_10(η_2) - log_10(η_1)/log_10(Δ x_2) - log_10(Δ x_1)where, η_1 and η_2 depict consecutive norms (L_1 or L_∞) for progressively refined grids with dimensions Δ x_1 and Δ x_2 respectively. The results are tabulated in table <ref>. It is observed from the analysis that GM-K-CUSP-X scheme is able to achieve second order accuracy on sufficiently refined grids in both L_1 and L_∞ norms.§.§ Two dimensional Riemann problems Two dimensional Riemann problems provides an excellent test case to assess the qualitative improvement of genuinely multidimensional formulation of GM-K-CUSP-X over the corresponding conventional two state K-CUSP-X scheme. Second order versions of both solvers are used for these test cases. The first two dimensional Riemann problem investigated in the present work consists of a double Mach reflection and an oblique shockwave propagating at an angle to the grid. The initial conditions of the 2D Riemann problem are given by <cit.>, A Cartesian grid of size 2000×2000 spanning [-1,1]×[-1,1] is used. The solution is evolved up to a time of 1.05 units with a CFL number of 0.95 as suggested in reference <cit.>.The density contours at the final time are shown in Figure <ref>. Result for K-CUSP-X solver under identical conditions is given for comparison. It is observed that the mushroom cap structure is well resolved by GM-K-CUSP-X as compared to the original K-CUSP-X. Further GM-K-CUSP-X is able to resolve the Kelvin-Helmholtz roll up much better than K-CUSP-X. The second Riemann problem consist evolution of two weak shock waves and two contact waves and can be set using conditions <cit.>, A 2000X2000 Cartesian grid spanning [-1,1]×[-1,1] domain is used. CFL number of 0.95 is used and the solution is evolved for a time of 0.5 units. The results obtained are represented using iso density contours. For comparison, resultsobtained under identical conditions by corresponding two state conventional K-CUSP-X is also provided in Figure <ref>.Although both solvers are able to resolve the resultant contact waves and Mach reflections <cit.>, only GM-K-CUSP-X is able toresolve the Kelvin Helmholtz instability along the Mach stems. §.§ Double Mach reflection problem This problems deals with a Mach 10 shock inclined at 60^o with the x-axis and propagating downstream of a rectangular duct of size[0,4]×[0,1] and interacting with a reflective bottom boundary wall. Detailed initial and boundary conditions can be found in<cit.>. The problem is solved on a Cartesian mesh of size 1920×480 with a CFL of 0.7 and evolved up to time of 0.2 units. The second order accurate results obtained for GM-K-CUSP-X are represented using iso density contours and are compared with thatobtained using K-CUSP-X under same conditions. It is visible from the result that the present solver is able to resolve the complex shock structures crisply and also the slipping contact line emerging from the triple point. § NUMERICAL SHOCK INSTABILITY TESTS One of the objectives of the present work is to demonstrate the effectof genuinely multidimensional formulation on the numerical shock instabilitycharacteristics of a Riemann solver. Two standard test cases namely, the odd-even decoupling problem <cit.> and the standingshock problem <cit.> will be used to carry out the investigations. Since shock instability is most explicitly observed in the first order simulations, first order version of solvers will be used for these tests.§.§ Odd-Even decoupling problemOdd-even decoupling test consists of a M = 6 shock propagating down a computational duct whose centerline grid is slightly perturbedto induce random oscillations into the initial conditions <cit.>. It has been long argued that odd-even decoupling and the Carbuncle have the same origin <cit.>. Much alike the Carbuncle phenomenon, the moving shock profile in the odd-even decoupling problemalso deteriorates over time (producing the recognizable bulge at the centerline) and pollutes the after shock flow field. The originalK-CUSP-X scheme was found to have slight after shock perturbations in this test. In comparison, the behaviorof GM-K-CUSP-X scheme on this problem is shown in <ref> after the solution has evolved for t = 140 units. It is clearly observed that the genuinely multidimensional extension is able to preserve the shock profile without any instabilities. §.§ Standing shock instabilityTo clearly differentiate the behavior of K-CUSP-X and GM-K-CUSP-X schemes, a standing shock instability test case is used. The detailsof the test are available in <cit.>. As seen in the figure <ref>, K-CUSP-X fails miserably in this test case. However, from figure <ref> it is very evident that GM-K-CUSP-X scheme is free of this instability. The reason for the stabilizing nature of GM-K-CUSP-X can be discerned from observing Equations (<ref>), (<ref>),(<ref>), (<ref>). Discretization of the convective terms using Equations (<ref>), (<ref>)employs a wave weighted averaging that strives to accurately model the underlying multidimensional wave evolution phenomenon. Specifically, it is easy to note that Equations (<ref>) and (<ref>) defines wave averaged convection velocities in x and y directions that depends on all the adjoining states at the corners. Such a formulation admits information from transverse cells which would contribute todissipation that would help supressing the instabilities. A similar observation can be found from Equations (<ref>), (<ref>) which describe the discretization of the pressure terms.For example, in Equation<ref> for multidimensional flux 𝐅_2^*_i+1/2,j+1/2, the third term on right hand side consist of terms 𝐆_2_RU, 𝐆_2_LU,𝐆_2_LD and 𝐆_2_RD clearly indicating the influence of y-directional flux terms in the evaluation of x-directional flux. Equation <ref> depicts a vice versa situation for the flux 𝐆_2^*_i+1/2,j+1/2. These multidimensional coupling terms along with the wave averaged convective terms may be providing the additional cross dissipation that damps any unprecedented growth ininstabilities thereby making the present scheme immune to shock instabilities. These equations reveal that the discretization of convective and pressure fluxes in the corner of interfaces using GM-HLLE strategy may have a positive impact in curing such instabilities. § CONCLUSIONSThe present work introduces a new genuinely multidimensional contact preserving Riemann solver GM-K-CUSP-X. This scheme is based uponToro-Vazquez type PDE level flux splitting and the ensuing convective-pressure fluxes are discretized following <cit.>.While the convective fluxes are upwinded based on appropriate wave speeds that emerge from the interacting states, the pressure fluxes are treated in a HLLE framework. Restoration of stationary contact preservation ability is improved by explicitly reducing the numerical dissipation in the pressure flux discretization.The resulting solver is found to produce improved results as compared to the original K-CUSP-X solver on standard test problems.Particularly, interesting flow features like Kelvin-Helmholtz roll up and mushroom cap structure in two dimensional Riemann problem andcomplex shock interactions in double Mach reflection problem are well resolved.Further, the present genuinely multidimensional solver is able to mitigate various forms of shock instabilities that plagued the corresponding conventional two state Riemann solver, K-CUSP-X.Such a finding reassures the pre existing notion that multidimensional dissipation is one of the most promising methodsto cure shock instabilities. Due to the simplicity of the formulation, the present solver can be easily extended to unstructured framework and three dimensional problems in principle.ieeetr20torovaz_2012 E.F. Toro, V. Cendon Flux splitting schemes for the Euler equations, Comput Fluids, 70(2012) 1-12.mandal_2015 J.C. Mandal , V. Sharma A genuinely multidimensional convective pressure flux split Riemann solver for Euler equations, J. Comput. Phys., 297(2015) 669-688.mandal_2008 J. C Mandal, J. Subramanian, On the link between weighted least-squares and limiters used in higher-order reconstructions for finite volume computations of hyperbolic equations, Appl Numer Math, 58(2008) 705-725.godunov_1959 S. K. Godunov, A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sb., 47(1959) 357-393. roe_1981P. L. Roe,Approximate Riemann solvers, parameter vectors, and difference schemes,J. Comput. Phys., 43(1981) 357-372.harten_1983A. Harten, P. D. 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Gressier, A matrix stability analysis of the carbuncle phenomenon, J. Comput. Phys., 197(2004) 647-670.sangeeth_2016S. Simon, Mandal J. C.,Genuinely multidimensional cure for numerical shock instability in a contact preserving Riemann solver, Proceedings of Sixth International Congress on Computational Mechanics and Simulation, Indian Institute of Technology Bombay, Mumbai (2016), 307-310. mandal_2011 J. C Mandal,P. R Sundeep, High resolution finite volume computations on unstructured grids using solution dependent weighted least squares gradients, Comput & fluids, 44(2011) 23-31.Chauvat_2005 Y. Chauvat, J. M. Moschetta, J. Gressier, Shock wave numerical structure and the carbuncle phenomenon, Int. J. Numer. MethodsFluids, 47(2005) 903-909. | http://arxiv.org/abs/1703.09621v1 | {
"authors": [
"S. Sangeeth",
"J. C. Mandal"
],
"categories": [
"math.NA",
"physics.comp-ph"
],
"primary_category": "math.NA",
"published": "20170327112632",
"title": "An accurate and robust genuinely multidimensional Riemann solver for Euler equations based on TV flux splitting"
} |
#1⟨#1 ⟩ #1( #1 ) #1{ #1 } #1[ #1 ]↑↓∇⃗∂N_ch #1 #1#1⟨ #1 ⟩↑̆↓̣"hat"Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden. Department of Applied Physics, School of Engineering Sciences, KTH Royal Institute of Technology, Electrum 229, SE-16440 Kista, Sweden University of Southampton, Southampton, SO17 1BJ, United KingdomEuropean XFEL GmbH, Holzkoppel 4, 22869 Schenefeld, Germany Cambridge, United Kingdom Department of Applied Physics, School of Engineering Sciences, KTH Royal Institute of Technology, Electrum 229, SE-16440 Kista, SwedenDepartment of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden.Swedish e-Science Research Center (SeRC), KTH Royal Institute of Technology, SE-10044 Stockholm, Sweden Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden. We present a simple and fast method to simulate spin-torque driven magnetisation dynamics in nano-pillar spin-valve structures. The approach is based on the coupling between a spin transport code based on random matrix theory and a micromagnetics finite-elements software. In this way the spatial dependence of both spin transport and magnetisation dynamics is properly taken into account. Our results are compared with experiments. The excitation of the spin-wave modes, including the threshold current for steady state magnetisation precession and the nonlinear frequency shift of the modes are reproduced correctly. The giant magneto resistance effect and the magnetisation switching also agree with experiment. The similarities with recently described spin-caloritronics devices are also discussed. Micromagnetic simulations of spin-torque driven magnetisation dynamics with spatially resolved spin transport and magnetisation texture Jonas Fransson December 30, 2023 ======================================================================================================================================= § INTRODUCTION The orientation of the magnetization in a magnetic film can be influenced using a spin-polarized current. Consequently, a direct current can transfer spin angular momentum between magnetic layers, separated by either a normal metal or a thysichin insulating layer. This effect is called spin transfer torque (STT) and was first discussed in the 1970s in the context of moving magnetic domain walls <cit.> and fully understood in the 1990s <cit.>. STT has been of profound importance for the development of spintronic devices such as read-heads based on the giant magnetoresitive (GMR) effect <cit.>, the spin-transfer torque magnetic random-access memory (STT-MRAM) <cit.> and spin-torque nano-oscillators (STNO)<cit.>.Until very recently <cit.> the approaches to theoretically describe the magnetization dynamics induced by a spin torque usually greatly simplified or neglected the description of either the spatial inhomogeneity of the spin torque, or the three-dimensional magnetization texture <cit.>.In the present work, we go beyond such approaches by coupling a finite element micromagnetic method <cit.> to a numerical solver for spin transport, based on continuous random matrix theory (CRMT) <cit.>. In this way the effect of spin torque is described, with the transport and magnetic degrees of freedom treated on an equal footing. The spatial inhomogeneity of both spin transport and magnetization dynamics is thus explicitly included. In our implementation, CRMT is parametrised by the same set of experimentally accessible parameters as in Valet-Fert theory <cit.>, so that our numerical simulations contain no free parameters.We demonstrate the capabilities of our computational method by addressing theoretically the effect of spin torque on the magnetization dynamics in the perpendicularly magnetized circular spin-valve nanopillar experimentally investigated in Ref. [naletov11]. This configuration is obtained by saturating the device with a large applied field perpendicular to the layers. The device setup is kept very simple and as such serves as a prototype for spin-valve structures, which find application also in the emerging field of spin-caloritronics <cit.>. In particular, our configuration corresponds to a circular precession of the magnetisation, allowing for a precise identification of the spin wave (SW) modes.This is crucial if one wants to couple the system to an external rf signal, since only signal with the same symmetries of the SW modes can excite the magnetisation. Breaking the axial symmetryresults in a more complicated configuration where modes with different symmetries mix up <cit.>.Moreover, the system can be described using the language of coupled oscillators <cit.> and in particular of the discrete nonlinear Schrödinger equation <cit.> (DNLS). Indeed, this setup corresponds to the simplest realisation of the DNLS, containing only two elements. Here spin transfer torque physically corresponds to a magnon chemical potential <cit.> that controls the propagation of the energy and magnetisation currents between the two layers. The DNLS appears in many branches of Physics, including Bose-Einstein condensates, photonics waveguides and photosynthetic reactions. Understanding the dynamics in our setup can therefore shed lighton a very general oscillator model. The remainder of this paper is organized as follows: in Sec. II we describe the geometry of the nanopillar and also briefly review the classification of SW modes in a perpendicularly magnetised nanopillar. Those modes are then identified by means of micromagnetic simulations at zero current (thus without spin torque). This study was previously performed in Ref. [naletov11], to which we refer for a thorough discussion.Sec. III reviews key concepts of scattering approach and CRMT to describe transport in magnetic multilayers, and consitutes an extension of the material presented in Refs. [waintal00] and [rychkov09]. In Sec. IV we describe how to couple the CRMT transport code to Nmag <cit.>, in order to simulate the effect of spatially inhomogeneous spin transfer torque and magnetisation dynamics on the same footing. Sec. V contains our micromagnetic simulations of current driven spin dynamics. Here we identify the SW modes excited by spin transfer torque, the critical current for auto oscillations, and the frequency shift beyond the critical threshold <cit.>.Using CRMT we provide a precise characterisation of the magnetoresistance. Our simulations are then compared with the experimental results found in Ref. naletov11.Finally, in the conclusion we summarise the main results of this paper and point out further possible developments.§ PHYSICAL SYSTEM AND MODEL§.§ Spin valve structureThe nanopillar studied here is displayed in Fig.<ref>(a). It consists of a trilayer structure made of two Permalloy (Ni_80Fe_20alloy) disks Py_a and Py_b separated by a 10 nm Cu spacer. The disks have diameter d=200 nm and thicknesses t_a=4 nm (upper disk) and t_b=15 nm (lower disk). The upper disk is connected to a 25 nm Au contact and the lower disk to a 60 nm Cu contact.In the experiment<cit.>, the sample was mounted inside a magnetic resonance force microscope (MRFM) and the whole apparatus was placed inside a vacuum chamber operated at room temperature. The external magnetic field H_ext, was oriented along the pillar axis z, which corresponds to the precession axis of the magnetisation.The MRFM consists of an ultra-soft cantilever with a 100 nm diameter magnetic sphere glued to its tip. The sphere is positioned precisely above the center of the nano-pillar, so as to retain the axial symmetry. The mechanical-FMR spectroscopy consists in recording, by optical means, the vibration amplitude of the cantilever as a function of the bias out-of-plane magnetic field in the presence of a RF field with fixed frequency excitation <cit.>. The dynamics can be excited also by injecting a dc current along z, which excites the dynamics in the thin layer due to spin transfer torque. In the steady state, the combined torques exerted by the RF excitations and spin transfer compensate the damping, so that the local magnetization vector precesses regularly at the Larmor frequency along a circular orbit, see Fig.<ref>a). The instantaneous magnetization M can be decomposed into a large static component M_∥ and a small oscillating component M_⊥, where M_∥ is parallel to the local precession axis (i.e., the direction of the external magnetic field), and M_⊥ is perpendicular to that axis. An essential feature of our system is that different SW modes have different spatial distribution of the phase of M_⊥ inside the magnetic disks. Those modes can be excited only by RF fields with the same rotational symmetries, giving selection rules for the excitation of the SW modes <cit.>.The dipolar force acting on the cantilever is proportional to the spatially averaged value of the longitudinal (static) component of the magnetization inside the whole nanopillar,M_z=1/V∫_VM_z(r)d^3r. The latter is not subject to any selection rule, so that the the mechanical-FMR setup detects all possible SW modes. Experimentally, it has been observed that the presence of the cantilever introduces a shift of +0.57 GHz in the SW spectrum. This has be taken into account in our simulations.The dynamics of the magnetization can be excited also by means of an rf current flowing along the axis of the pillar, which generates an rf orthoradial Oersted field, and by mean of an homogeneous in-plane RF magnetic field.We remark that, although in experiments the dynamics driven by a combination of RF fields and STT, in our simulations the dynamics is excited by STT only.The effect of the RF field is modelled by setting different initial condition for the magnetisation. In our circuit, a positive current corresponds to a flow of electrons from the bottom Py_b thick layer to the top Py_a thin layer,and stabilizes the parallel configuration due to the spin transfer effect.Vice-versa, a negative current stabilizes the thick layer and destabilizes the thin one. At low current, thick and thin layer are thus the fix and free layer correspondingly. However, at high enough current, both layers precess due to the repulsive dipolar interaction, as will be discussed later.§.§ Magnetization dynamics In this section, we briefly review the magnetization dynamicsin our system. For a more comprehensive discussion, see Ref. [naletov11].The local dynamics of the magnetisation M^j, which depends continuously on the position r^j in the layer j=(a,b), is described by the Landau-Lifshitz-Gilbert equation <cit.>:1/γṀ^̇j̇ = M^j ×H_eff^j + αM^j ×Ṁ^j +M^j ×S^j.Here, γ<0 is the gyromagnetic ratio in the magnetic layer.The first term on the right-hand side of Eq.(<ref>) describes the adiabatic torque, that accounts for the precession of the magnetization vector around the local equilibrium direction. This precession axis is defined by the effective magnetic field experienced locally by the magnetization, H^j_eff= -∂ F/∂M^j, which contains all the static contributions to the free energy F of the layers <cit.>. In particular, the effective field contains contributions from applied field, exchange interaction, and dipolar interaction between the layers. We refer to Refs.gurevich96,naletov11,borlenghi15b,borlenghi11 for the explicit expressions.The second terms on the right-hand side of Eq.(<ref>) is the damping torqued^j=α^jṀ^jproportional to the Gilbert damping parameter α^j. We introduce here the notation M^j ≡ M_s^jm^j, with M_s^j the norm of the magnetization (a constant of the motion)and m^j the unit vector along the magnetization direction.The dissipative term d^j is responsible for the finite linewidth (full width at half height)of the resonance peaks, Δ H=2 d. For a normally magnetized nanopillar, with circular precession of the magnetization, the simple relation α=|γ| Δ H/ (2ω) holds <cit.>.If a charge current I_dc is flowing between two layers j and j^', the Slonczewski-Berger <cit.> spin transfer torque readsS^j=I_dc/2πλm^j×m^j'.The latter depends on the relative angle between the magnetization m^j in the layer j and the spin polarization of the current, which coincides with the direction of the magnetization m^j^' of the polarizer (here the thick layer). The termλ^j=2 e M_s^j V/η h has the dimension of a distance. Here, h is the Planck constant, e the absolute value of the electron charge.η is the spin polarisation of the current and V the volume of the thin layer. Since d^j and S^j are collinear, spin transfer torque can compensate the damping torque, as shown in Fig.<ref>.When the dc current through the nano-pillar reaches the threshold current I_th=-2 πλα H_eff, the thin layer starts auto-oscillating. Combining Eqs.(<ref>) and (<ref>), it is possible to define an effective damping for the thin layer, d=α(1-I_dc/I_th),which depends linearly on the spin polarized dc current <cit.>. The critical thresohold corresponds to the value of the current I_dc at which the effective damping vanishes and the system starts auto-oscillating. §.§ Coupled oscillator model and classification of the SW modes Since our layers are thinner than 15 nm, one can assume that the magnetization dynamics is uniform along the thickness. In this approximation, the linearized LLG equation simplifies to two equations describing the circular precession of the transverse magnetization projections M_x^j and M_y^j around the z axis, which depends only on the two spatial variables (x,y) in the layer j<cit.>. The two real equations of each layer can be rewritten as one complex equation for the dimensionless spin-wave amplitudec_j=M^j_x + i M^j_y/√(2M^j_s(M^j_s+M^j_z)),that depends on the polar coordinates (r_j,ϕ_j) of disk j. The dynamics of the two disks, written in terms of the c_js, is described by the equations <cit.>ċ_a=iω_ac_a-[Γ_a--Γ_a+]c_a+ih_abc_b ċ_b=iω_bc_b-[Γ_b--Γ_b+]c_b+ih_bac_a,which are the equations of motion of two coupled nonlinear oscillators with resonance frequency ω_j (p_j) and damping rates Γ_j(p_j).Both depend on the SW power p_j=|c_j|^2, which describes the amplitude of the oscillations in each disk.From hereon, to keep the notation simple we do not write explicitly the dependence on p_j.The frequencies ω_j=γ |H^j_eff| are proportional to the local magnetic field, while the damping rates Γ_j- are proportional to α_jω_j. Both can be therefore controlled by means of the applied field along z. The terms Γ_j+, proportional to I_dc are due to spin transfer torque, which can compensate the damping and lead to auto oscillations of the layers. In the present case, those terms do not have the same sign. At positive current, Γ_a+ is positive, while Γ_b+ is negative, favouring the auto oscillations in the thin layer a, and stabilising the thick layer b. The dipolar coupling strength h_jj^' is an effective term obtained by averaging the dipolar field over the volumes of the samples, see Refs. naletov11 for the explicit expression.Eqs. (<ref>) and (<ref>) describe the dynamics of a nonlinear Schrödinger dimer, the simplest realisation of the discrete nonlinear Schrödinger equation (DNLS) <cit.>. Upon multiplying Eqs.(<ref>) and (<ref>) respectively by c_a^*and c_b^* and summing them with their complex conjugate equations, one has the following continuity equation for the SW powerṗ_a=-2(Γ_a--Γ_a+) p_a+j^p_ab.and a similar equation for p_b. The magnetisation current j^p_ab=2[h_abc_ac^*_b] describe the transfer of M_z between the two layers, and is essentially the SW current written for a discrete systems with only two spins <cit.>. Upon writing c_a=√(p_a(t))e^iϕ_a(t), the current readsj_ab^p=2h_ab√(p_ap_b)sin[ϕ_a(t)-ϕ_b(t)+β]. The quantity β comes from the condition of dissipative coupling between the oscillators <cit.>. When the two oscillators are synchronised, ϕ_a≈ϕ_b and the magnetisation current approaches the constant value j^p_ab∝sinβ. On the other hand, if the oscillators are not synchronised,the magnetisation current oscillates around zero and vanishes in average. Within this DNLS formulation, the spin transfer torque that appears in Eq.(<ref>) plays the role of a magnon chemical potential, that by controlling the lifetime of the excitations, controls also the SW current between them.The diagonalization of the LLG equation in a confined geometry leads to a discrete series of normal modes having each a differenteigen-value, ω/(2π), the so-called Larmor precession frequency. The normal modes of the system are numbered according to the number of half waves in the vibration. In the case of a 2D axially symmetric structure, the normal modes are identified by two integers: ℓ and m, respectively the mode number in the azimuthal and radial directions. The analytical expression of the normal modes of a perpendicularly magnetized disk is found in Refs. [damon61,naletov11] c_ℓ,m(r,ϕ,t)=J_ℓ (k_ℓ,m r) e^+ i ℓϕ e^- i ω_ℓ m t,where J_ℓ are the Bessel functions of the first kind and k_ℓ,m is the modulus of the in-plane SW wave-vector, which depends on the boundary conditions.The above labelling can be extended to the case of two different magnetic disks coupled by dipolar interaction.In the perpendicular geometry, the strength of the dynamical dipolar coupling is attractive (lower in energy) when both layers vibrate in antiphase,because the dynamical dipolar charges in each layer are alternate <cit.>.Thus the binding stateB corresponds to a collective motion where the two layers vibrate anti-symmetrically and the anti-binding state A to a collectivemotion where the two layers vibrate symmetrically. The B modes correspond to a precession amplitude that is larger in the thick layer b, while the A modes correspond to a precession amplitude that is larger in the thin layer a<cit.>, see Fig.<ref> for a cartoon.The dynamical dipolar coupling does not modify the nature of the modes hence, in order to describe the dynamics of the bi-layer system, we shall just add a new index B or A indicating if the precession occurs in antiphase (mostly in the thick layer) or in phase (mostly in the thin layer), respectively. There are thus three indices to label the observed eigen-modes: the usual azimuthal and radial indices for a single disk (ℓ,m), plus an additional index referring to the symmetrical or anti-symmetrical (A or B) coupling between both layers.The identification of the SW modes and their symmetry is essential to couple the oscillator to an external source, since SW modes can couple only to asource with the same symmetry. Here the ℓ index determines the rotational symmetry of the SW mode. The ℓ=0 modes correspond to SW that do not rotate in the x-y plane and can be excited only by a spatially uniform in-plane RF field, while the ℓ=1 modes correspond to SW that rotate around the disk in the same direction as the Larmor precession, and can be excited only by an RF Oersted field with orthoradial symmetry. Thus, exciting the system with these different means gives two different spectra.§ MICROMAGNETIC SIMULATIONS AT ZERO CURRENTIn this section we describe the SW spectra by means of micromagnetic simulations without spin-transfer torque. Those simulations were performed with the NMAG micromagnetic software <cit.>, where the sample is described by finite elementtetrahedral mesh. The latter has a maximum intersite distance of 6 nm,of the order of the Py exchange length. The micromagnetic parameters are the same used in Ref.[naletov11] and are reported in Tab.I for convenience. The dynamics at each of the i=1,...,N nodes of the mesh of disk j=(a,b) is described by the following LLG equation for the unit magnetisation vector, m_i^j=M_i^j/M_s: ṁ_i^j=-γ(m_i×H_effi^j)+α/M_s(m^j_i×ṁ_i^j) The integration of the LLG equation at each mesh site is performed by the Sundials ODE solver <cit.>, which is based on variable steps multistep methods. The field H^j_i at each mesh node has contributions from applied field, first neighbour exchange interaction and long range dipolar interaction, responsible for the coupling between the layers.The quantity of interest is the space-averaged magnetisation m^j(t)=1/V^j∫_V^jm^j(r^j,t)d^3r, which for our finite-elements mesh reduces to m^j=1/N∑_i=1^Nm_i^j. From this quantity, the SW amplitudes c_j(t) are calculated. The power spectrum, shown in Fig. <ref>, is given by the Fourier transform of the time series of the collective SW amplitude averaged over the sample thicknesses:c=(c_at_a+c_bt_b)/(t_a+t_b).The modes with ℓ=0 (displayed in blue tones) are excited starting from an initial condition where the magnetization uniformly tilted 8^∘ in the x direction with respect to the precession axis z. Instead, the the modes with ℓ=+1 (displayed in red tones) are excited by applying to the magnetization aligned with the z axis the orthoradial vector field perturbation θ(r,z)=ϵẑ×ρ̂. Here ϵ=0.01 and ρ̂ is the unit vector in the radial direction. Starting from these conditions, we have computed the time evolution of the system for 120 ns, with an integration time step of 5 ps. From the Fourier transform, the maximum frequency is 100 GHz and the frequency resolution 15 MHz. The frequencies of the SW modes are displayed in Tab.<ref> and <ref> and compared with the experimental values. § CONTINUOUS RANDOM MATRIX THEORY (CRMT) FOR SPIN TRANSPORTThis section contains a thorough review of the CRMT semi-classical theory of spin-dependent transport in magnetic multilayers. We follow closely the material presented in Refs. [waintal00,rychkov09,borlenghi11] and we extend it by providing an explicit formula for spin torque, that will be used inmicromagnetics simulations of next section. §.§ Scattering matrix approachWithin the scattering matrix formulation developed by Landauer and Buttiker <cit.>, a sample is defined by the scattering matrix S which expresses the outgoing propagating modes in term of the incoming ones. The incoming modes are filled according to the Fermi-Dirac distribution of the leads to which they are connected. From the elements of the scattering matrix various physical quantities can be calculated,such as the conductance G, the spin currents J and charge current I.The system contains N_ch≫ 1 propagating modes (or channels) per spin. In particular, one has N_ch≈ A/λ_F^2, where A is the transverse area of theelectrode and λ_F is the Fermi wavelength. The amplitude of the wave-function on the different modes is given by the vector ψ_i± with N_ch elementsψ_i±=([ ψ_i±; ψ_i± ]).The latter contains the amplitudes for the right (+) and left (-) moving electron direction with spinσ=,̆$̣ along thez-axis in regioni=0,2of the multilayer, see Fig.<ref> for a cartoon. TheSmatrix is a4N_ch×4N_chunitary matrix that relates the outgoing modes to the ingoing ones: ([ ψ_0-; ψ_1+ ])=S([ ψ_0+; ψ_1- ]), see Fig.<ref> (a) for a schematic of the system. and consists of2N_ch×2N_chtransmissiont,t'and reflectionr,r'sub-blocks.S=([ r't; t'r ]). Here the(r,t)and(r^',t^')describe reflection and transmission respectively from left to right and from right to left.The transmission and reflection matrices have an internal spin structure: t=([ t_̆̆ t_̣̆; t_̣̆ t_̣̣ ])wheret_σσ'areN_ch× N_chmatrices containing amplitudes for transmission betweenσ'andσspin states, capturing both spin preserving and spin-flip phenomena.The conductance of the system is given by the Landauer formula <cit.> G=e^2/h[t^† t], while the spin current in region 0 reads∂J⃗_0/∂μ=1/4π[tσ⃗t^†],μbeing the difference of chemical potential between the two electrodes <cit.>. §.§ Random Matrix Theory (RMT)The scattering matrix approach is fully quantum and contains interference effects such as weak localization or universal conductance fluctuations <cit.>.The system studied in this paper is a nanopillar of 250 nm of diameter connected to top and bottom electrodes. Those contain≈ 10^4-10^5propagative channels. Here the scattering is not perfectly ballistic (mismatch at the interfaces, surface roughness or impurity scattering) so that channels get mixed up. Random Matrix Theory(RMT) <cit.> assumes that this mixing is ergodic: an electron entering the system in a given mode will leave it in an arbitrary mode, acquiring a random phase in the process.In this case, the transmission and reflection probabilities of an electron are well caracterized by their average over the propagative channels. This average is obtained by taking the trace over theN_chof the original reflection and transmission matrices.For instance, the hat matrixt̂<cit.> is defined as t̂_ση,σ'η'=1/N_ch_N_ch[t_σσ' t^†_ηη'].Explicitely, this reads t̂=1/N_ch_N_ch[ t_↑↑ t_↑↑^†t_↑↑t_↑↓^†t_↑↓t_↑↑^†t_↑↓t_↑↓^†;t_↑↑t_↓↑^†t_↑↑t_↓↓^†t_↑↓t_↓↑^†t_↑↓t_↓↓^†;t_↓↑t_↑↑^†t_↓↑t_↑↓^†t_↓↓t_↑↑^†t_↓↓t_↑↓^†;t_↓↑t_↓↑^†t_↓↑t_↓↓^†t_↓↓t_↓↑^†t_↓↓t_↓↓^† ],with the same structure for the reflection hat matrix.The elements of this matrix correspond to the probability for an electron with a given spin to be transmitted (t̂) or reflected (r̂) by the system. In particular, the terms(1/N_ch)Tr_N_ch( t_t_^†)≡ T_and(1/N_ch)Tr_N_ch( t_t_^†)≡ T_correspond to the probability to transmit an electron with up and down spin correspondingly. The terms at the corners,(1/N_ch)Tr_N_ch( t_t_^†)≡ T_andT_correspond to transmission probabilities with spin flip. The so-called "mixing transmission",(1/N_ch)Tr_N_ch( t_t_^†)=T_mxis a complex number whose amplitude measures how much of a spin transverse to the magnetic layer can be transmitted through the system, while its phase amounts for the corresponding precession.T_mxdecays exponentially with the size of the ferromagnet, and accounts for spin transfer effect <cit.>. The other off diagonal elements in the hat matrix Eq.(<ref>) can be ignored <cit.>. In the basis parallel to the magnetisation, Eq.(<ref>) therefore becomes: t̂=[ T_00 T_;0 T_mx00;00 T_mx^*0; T_00 T_ ], with the same structure for the reflection matrixr̂. The hat-matrixŜhas a form similar to Eq.(<ref>),Ŝ= ([ r̂'t̂; t̂'r̂ ]). In order to describe transport in non-collinear multilayers, where the orientation of the magnetization changes inside the system, one needs to rotate the originalSmatrix asS̃= R_θ,n⃗S R_θ,n⃗^†in the chosen working basis. Here the matrixR_θ,n⃗=exp(-iσ⃗·n⃗ θ/2)is the rotation matrix of angleθaround the unit vectorn⃗that brings the magnetization onto the z-axis of the working basis. In term of hat matrices, this translates directly intoŜ̃̂= R̂_θ,n⃗ŜR̂_θ,n⃗^†, withR̂_ση,σ'η'=R_σσ' R^*_ηη'a unitary matrix.From Eq.(<ref>), the conductance is given by G=1/ℛ_shT_+T_+T_+T_ where the Sharvin resistanceℛ_sh=h/N_che^2, is a material property that can be experimentally measured, related to the numberN_chof transverse propagative channels for the electrons crossing the system.Eq.(<ref>) is analogous to the Landauer formula Eq.(<ref>) and consists of the sum of all the possible transmission processes (spin preserving and spin flipping) for an electron.In analogy with the modesψ_i±σfor the scattering matrix, the4-vectors P_i±are introduced P_i±=[P_i±,; P_i±, mx; P^*_i±, mx;P_i±,;].The components o P_i±,have interpretation in term of probabilities for an electron to propagate in the regioniof the system.The "mixing" components,P_mxare complex numbers which correspond to probability to find the electron with spin along thex(real part) ory(imaginary part) axis. Inside magnetic layers where thezaxis xill correspond to the direction of the magnetisation, they will correspond to the probability for the spin to have a part transverse to the magnetisation.For the following discussion, it is convenient to consider a system made of two conductors connected in series,described by the two hat matricesŜ_aandŜ_b, see Fig. <ref> (b). The space is thus divided into three regions: region 0 and 2 respectively at the leftmost and rightmost part of the system, and region 1 in between the two hat matrices. In analogy with Eq.(<ref>) which expresses the amplitudes of the outgoing modes in term of the incoming ones, in subsystemsaandbone has respectively ([ P_0-; P_1+ ])= Ŝ_a([ P_0+; P_1- ]), ([ P_1-; P_2+ ])= Ŝ_b([ P_1+; P_2- ]),while for the total systema+b([ P_0-; P_2+ ])=Ŝ_a+b([ P_0+; P_2- ]).Here we have used the addition law of hat matrices, a fundamental property that will be useful in the derivation of spin currents and spin torque inside the system. According to this rule, given the hat matrices for separate systemsaandb, the hat matrices of the composed systema+bread <cit.>t̂_a+b = t̂_a1/1̂-r̂^'_br̂_at̂_b r̂_a+b = r̂_b+t̂^'_b1/1̂-r̂_ar̂^'_br̂_at̂_b with1̂the4× 4identity matrix. Similar expression holds forr_a+b^'andt_a+b^'.The main result of Refs.[rychkov09,borlenghi11] is that the spin current in regioniof the system can be expressed in terms of probability vectors as follows: J_i=N_ch/4πσ⃗· (P_i+- P_i-), whereσ⃗=(σ⃗_,σ⃗_,σ⃗_,σ⃗_)is the vector of components of Pauli matrices. To express the spin current in region 1 in between the two conductors as a function of the hat matrices, we use Eqs.(<ref>)-(<ref>) to eliminateP_2+andP_2-and obtain([ P_1+; P_1- ])= ([ 1/1̂-r̂_ar̂_b^'t̂_a^' 1/1̂-r̂_ar̂_b^'r̂_at̂_b; 1/1̂-r̂_b^'r̂_ar̂_b^'t̂_a^' 1/1̂-r̂_b^'r̂_at̂_b ]) ([ P_0+; P_2- ]) The theory is completed imposing boundary conditions on the incoming electrons on both sides of the system. For normal electrodes one has, P_0+=([ μ_0; 0; 0; μ_0 ]),P_2-=([ μ_2; 0; 0; μ_2 ])whereμ_0andμ_2are the respective chemical potentials of the two electrodes. The generalization to magnetic electrodes is done imposing different chemical potentials for majority and minority electrons in the leads. By taking as boundary conditionsμ_0=eU, withUthe potential difference between the two sides andμ_2=0, The spin current finally reads𝒥_1=N_ch/4πJeUwhere the spin current per channel and per unit of potential difference reads J=σ⃗·1-r̂_b^'/1̂-r̂_ar̂_b^'t̂_a.Eq.(<ref>) allows one to compute the spin current in the region between two bulk materials of arbitrary thicknesses. In the multilayer considered here, the currentsJ_a/b±δare calculated at positionsδ=±1nm before and after the two magnetic layers, as shown in Fig.<ref>. Thetorque is the spin current absorbed by each layer, i.e. the quantity τ^j = N_ch/4πf_jeUwithf_j=J_j-δ-J_j+δ,j=a,b. §.§ From scattering matrices to CRMTIn order to calculate the current in different regions of the multilayer, one needs to calculate the matricesŜ(L)as a function of the positionLinside the system.The main result of Refs.[rychkov09,borlenghi11] is that the matrixŜ(L+δ L)for an infinitesimal increment of the positionδ Lis entirely characterised by two matricesΛ^tandΛ^r, defined as t̂(δ L) = 1 - Λ^t δ L, r̂(δ L) = Λ^r δ LOnceŜ(δ L)is known, one can make use of the addition law Eqs.(<ref>) and (<ref>) to obtain a differential equation to computeŜ(L+δ L). By taking the limitδ L→ 0one getsthe two CRMT differential equations that describe hat matrices as a function of the length of the system: ∂r̂/∂ L = Λ^r -Λ^tr̂ -r̂Λ^t+r̂Λ^rr̂ ∂t̂/∂ L =-Λ^tt̂ + r̂Λ^r t A bulk magnetic material is characterised by four independent parametersΓ_↑,Γ_↓Γ_ sfandΓ_ mx, which describe spin preserving and spin flip phenomena.With this parametrisation, one has for the transmissionΛ^t= [ Γ_ +Γ_ sf 0 0-Γ_ sf; 0 Γ_ mx 0 0; 0 0 Γ^*_ mx 0;-Γ_ sf 0 0Γ_+Γ_ sf ],and for the reflectionΛ^r= [ Γ_ -Γ_ sf 0 0 Γ_ sf; 0 0 0 0; 0 0 0 0; Γ_ sf 0 0Γ_-Γ_ sf ], These four parameters correspond in turn to 5 different lengths.The two most important one are the mean free paths for majority (σ=) and minority (σ=) electrons defined asℓ_σ = 1/Γ_σ.Next comes the spin diffusion lengthl_sf = [4Γ_sf (Γ_ + Γ_)]-1/2.Last come the complex numberΓ_mx = 1/l_⊥ + i/l_Lwherel_⊥is the penetration length of transverse spin current inside the magnet whilel_Lis the Larmor precession length. Upon integrating Eq. (<ref>) one obtains for the mixing transmission. T_mx(L)=e^-L/l_⊥ -iL/l_ L.which shows the exponential decay of the transverse spin current, absorbed by the layer, giving the phenomenon of spin transfer torque. The two lengthsl_⊥andl_Lare of the order of the nm <cit.>, since spin transfer torque is a phenomenon that occurs close to the interface. For bulk materials, another important quantity is the current polarisation,P_σ=T_σ-T_σ/T_σ+T_σ. The latter decays exponentially as a function of the distance in the material, with spin-flip length scale of the order of 15 nm for Py, and reaches a constant value. The behaviour of spin and charge currents in a multilayer for different magnetic configurations have been extensively studied inRef.[borlenghi11], to which we refer for a thorough discussion. The CRMT parameters (Γ_$̆, Γ_$̣ andΓ_sf) are in 1-1 correspondence with the Valet-Fert (VF) parameters<cit.> [ρ_$̆, ρ_$̣(resistivities for majority and minority electrons) andl_sf(spin-flip diffusion length)].Using the standard notations for the average resistivityρ^*and polarizationβ[ρ_↑(↓)= 2ρ^* (1∓β)]onehas <cit.>:1/l_ sf =2√(Γ_ sf)√(Γ_↑ + Γ_↓) β = Γ_↓ - Γ_↑/Γ_↑ + Γ_↓ ρ^*/ R_ sh = (Γ_↑+Γ_↓)/4This parametrization does not fix the mixing coefficientsΓ_mx, which only play a role in non collinear configurations <cit.>. In the VF theory, the interfaces are characterised by the average resistancer^b*and polarisationγwithr_, =2r^b*(1 ±γ)]. In the CRMT formalism, interfaces are modelled as a virtual material with their own transmission and reflection hat matrices. Those are related to the VF parameters as follows T_ = (1+e^-δ)/2/1+2(r^b*/ℛ_sh)(1-γ)T_ = (1-e^-δ)/2/1+2(r^b*/ℛ_sh)(1-γ)T_ = (1+e^-δ)/2/1+2(r^b*/ℛ_sh)(1+γ)T_ = (1+e^-δ)/2/1+2(r^b*/ℛ_sh)(1+γ)while for the reflection coefficient one hasR_ =1-1/1+2(r^b*/ℛ_sh)(1-γ)R_ =1-1/1+2(r^b*/ℛ_sh)(1+γ) andR_=R_=0.At this point the parametrisation of CRMT is complete. Eqs.(<ref>) and (<ref>) have been solved numerically to simulate a one dimensional magnetic multilayer. Each material and interface are simulated separately, and the whole system is recovered by applying the addition law Eqs.(<ref>) and (<ref>). In order to calculate magnetoresistance and spin torque as a function of the magnetic configuration, we simulate transport the nano-pillar depicted in Fig.<ref> by keeping the magnetisationM_bfixed and rotating the magnetisationM_aof an angleθin spin space. Such rotation is obtained by applying to the hat matrices of the thin layer the transformation Eq.(<ref>).Fig.<ref>a) and b) shows respectively the angular dependence of spin torque and resistance inside the nanopillar for a single propagative channel and witheU=1.The resistance difference between parallel and antiparallel configuration is 27 mΩ. We remark that this calculation, performed without adjustable parameters, reproducesexperimental data within the10%.We note also that in our system, the current is spin-polarized by the thick layer and impinges the thin layer, exerting a torquethat destabilizes its magnetization. Because of multiple reflections of spin polarized electrons between the two Py layers,STT tends to stabilize the thick layer, increasing its effective damping. Thus, at low current, one can consider the thick layer as "fixed" and the thin layer as "free”. However, at high current both layers undergo a coupled precession (see next section). § SIMULATIONS OF CURRENT DRIVEN DYNAMICSThis section contains the main results of the paper. Here solve simultaneously the transport equations coupled to the LLG equation with spin transfer torque, by coupling CRMT to Nmag. The approach described here is valid for one dimensional systems, where the magnetisation varies only along the direction of propagation of electrons. In the present case the magnetisation is uniform alongzinside the material, but its dynamics is different in each layer. A generalization of CRMT for a fully three dimensional spin transport has been recently developed <cit.>. However, for the case considered here the onedimensional CRMT approach captures the physics well and is extremely fast. The other advantage of CRMT is that it is parametrised by thesame set of experimentally accessible parameters as the VF theory (reported in Tabs.<ref> and <ref>), so that no free parameter is needed to characterise realistic systems and materials.To include the effect of spin transfer torque into Nmag, the LLG equation Eq.(<ref>) at nodeiof diskj=a,bneeds to be modified as follows ṁ^j_i= -γ(m^j_i× H^j_ effi)-α^j/M_s^j(m^j_i×ṁ^j_i)+ gμ_B/ħ C M_sτ^j_iŵ_i^j. Heregis the Landé factor,μ_Bis the Bohr magneton andCis the volume associated to each site of the mesh.τ^j_iis the torque given by Eq.(<ref>) For our relatively homogeneous mesh, we have taken this volume as the total volumeV_jof diskj=a,bdivided by the total number of sitesN_j. For simplicity, we have defined the vectorŵ_i^j=m_i^j× (m_i^j ×m_i^j^').To include STT in our micromagnetic simulations, we adopt the following self-consistent loop:* At time t and site i in disk j, Nmag computes the vector m_i^j * The magnetic configuration is passed to the CRMT solver which computes the torque τ^j_i * Then the quantity m_i^j (t)+τ_i^j(t)Δ t is set as new initial condition for the Nmag solver,which performs the time integration of the LLG equation between times t and t+Δ t. At this point spin torque is re-calculated as a function of the new magnetic configuration and the loop starts again. The integration time stepΔ tis of the order of the ps.Since Nmag uses a finite element mesh while CRMT a finite difference discretisation of space, to implement the coupling,we divide the mesh into columns of sites, each columnkrepresenting a CRMTpillar with cross sectionS_k. The whole system is then considered as an assembly of CRMT pillars connected in parallel, as shown in Fig.<ref>. Each column contains two sites, lying onsurfaces of thin (a) and thick (b) layer facing each other, with magnetizationsm_ak(t), andm_bk(t)correspondingly.From the angleθ_kbetween these two sites, one obtains the torquesτ^j_k(θ)that act on the magnetisation vectorsm^j_k, and the resistanceR_k(θ_k). This procedure takes into account the three dimensional texture of the magnetization, while electronic transport is along zonly, without considering the lateral diffusion of spins.Since the system is an assembly of columns connected in parallel, the currentI_kflowing in each column is given by the total currentI_dcdivided by the number of columnsN_k, which in our system corresponds to the number of sites lying at the surface of each disk. Each column has a cross sectionS_k≈ S/N_k. The currentI_kis related to the potential difference between the conductors via Ohm'slaw:eU=R_k(θ)I_dc/N_k, where the resistanceR_kdepends on the transmission probabilityT_k(θ)for an electron to cross columnk: R_K=ℛ_sh/T_k(θ)S_k.Inserting Eq.(<ref>) into Eq.(<ref>), and recalling thatℛ_ sh=h/(e^2N_ch), the torque finally readsτ_jk=gμ_B/2eV_jN_j/N_kI_dcf_jk(θ)/T_k(θ)ŵ_j.We remark that Eq.(<ref>) is quite general and can be applied to systems with arbitrary geometry, provided that electronic transport is one dimensional only. § RESULTS AND DISCUSSIONWe turn now to the discussion of the coupled Nmag-CRMT simulations. A qualitative description of the different dynamical regimes is given by the time evolution ofM_z, displayed in Fig.<ref>. Panel (a) shows the region beyond critical current, where both the magnetisations are fixedand aligned with thezaxis. Panels (b) shows the region slightly beyond critical current, where the thin layer precesses and the thick one remains fixed. Finally, panel (c) displays the high current regime, where the thin layer is reversed and both layers undergo coupled precession. This behaviour is due to the repulsive character of dipolar interaction: as the magnetisation of the thin layer reverses, it repels the magnetisation of the thick one, causing it to precess.Fig.<ref> shows the resistance as a function of the dc current, for values of the applied field between1T ands1.3T. One can recognise clearly the three dynamical regions discussed above. In particular, the resistance starts increasing around-6mA, indicating the region of critical currents. The resistance increases with the current until it reaches a peak between-13and-16mA, corresponding to the reversal of the magnetisationm^aof the thin layer. Note that at increasing field the resistivity graph moves rightwards towards higher currents. This is due to the fact that the precession frequency increases proportionally to the applied field, so that the dampingΓ_a+=≈ 2α_aω_aalso increases anda higher current is needed to compensate it. The power spectrum of the system, with the modesℓ=0and+1, is shown respectively in Fig.<ref> (a) and (b) in logarithmic scale. The modesA_00andA_10, corresponding to the dynamics of the thin layer, increase with the current and dominate the spectrum aroundI_dc=-5mA. Beyond the critical threshold, their frequency increases with the field. On the other hand, the modesB_00andB_10, which dominate the spectrum at zero currents, decrease until they almost disappear near the critical threshold. Fig.<ref> shows the linewidth and frequencies of the modesA_00andA_10as a function of the dc current. In panel (a), one can see that the linewidths of both modes decrease linearly of an order of magnitude at increasing current, and they vanish at the critical threshold, where the damping is compensated and the system begins to auto oscillate.Note that the two modes have slightly different critical currents, of about5(ℓ=0) and5.5(ℓ=1)mA. The behaviour of the spectrum and the critical currents agree with the experimental result of Refs.[thesis,naletov11]. The frequencies of the excited modes remain constant until the critical threshold, and then starts increasing linearly with a rate around 2 GHz per mA. This behaviour also agrees with experimental data.We conclude this section by noting that the physics discussed here is very similar to that of the spin-caloritronics diode <cit.>.In Fig.<ref>, one can see that increasing the electrical dc current leads to an increase of the magnon current between the two disks. This current describes the transfer of magnetic momentM_zbetween the two disks, and corresponds to the usual SW current written for a system of only two spins<cit.>.The increase of the SW current with the dc current is due to the fact that spin transfer torque excites only one mode, which eventually dominates the spectrum, inducing a phase locking between the two disks. In this kind of discrete systems, the magnon current is indeed a measure of the phasesynchronisation of the system. § CONCLUSIONSBy coupling CRMT to micromagnetic simulations with the Nmag code weallow description, on an equal footing and without free parameters, of both transport and magnetic degrees of freedom. The results of this work have important consequences regarding the characterizationand the optimization of the performance of STNOs.Using the developed method, we have identified the nature of the modes that auto-oscillate when the current exceeds the critical threshold. In particular, we have predicted different critical currents for the modesA_00andA_10,and a nonlinear frequency shift of the order of 2 GHz per mA. The precise determination of the SW mode symmetry of the auto-oscillating mode is important for the phase synchronization of a STNO to an external source. In fact, it will be successful only if the latter can couple efficiently to the spin transfer driven auto-oscillation modes,i.e., if it has the appropriate symmetry.The flexibility of CRMT and of the procedure to couple it with Nmag allows one to use our numerical tool to simulate different geometries and materials. Further development and investigations are possible. A multiscale approach which combines systematically CRMT with a fully quantum approach has been implemented <cit.>. This should allow to compute current driven dynamics in a large variety of systems, including multi-terminal devices and tunnel junctions. We remark that in our simulations we have not taken into account the lateral diffusion of spins, since the system is described usingone dimensional CRMT columns, where electrons propagate only along thezdirection.This 1D model of transport is effective to describe selection rules into a perpendicularly magnetized nanopillar with magneticfield applied along e_z, but more complicated configurations (such as magnetic vortexes and multi-terminal spin valves) may require a fully three dimensional description.Finally, further investigation is necessary to understand the behaviour of the system at high current, where nonlinear effects(such as the dependence of the Gilbert damping on current <cit.>) may play an important role. The present work can be considered as an intermediate step towards a fully 3D description of transport and magnetization dynamics in realistic systems. § ACKNOWLEDGEMENTSWe wish to thank R. Lassalle-Balier and J. Dubois for fruitful discussions, X. Waintal, O. Klein and G. de Loubens for useful comments and assistance in the analysis of the experiments. Financial support from Vetenskapsrådet (VR),Carl Tryggers Stiftelse (CTS), Stiftelsen Olle Engqvist Byggmästare and Swedish Energy Agency (STEM) is gratefully acknowledged.The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the National Supercomputer Center (NSC), Linköping University, the PDC Centre for High Performance Computing (PDC-HPC), KTH, and the High Performance Computing Center North(HPC2N), Umeå University.38 natexlab#1#1 bibnamefont#1#1 bibfnamefont#1#1 citenamefont#1#1 url<#>1 urlprefixURL [Berger(1978)]berger78authorL. Berger, journalJournal of Applied Physicsvolume49, pages2156 (year1978).[Berger(1996)]berger96authorL. Berger, journalPhys. Rev. Bvolume54, pages9353 (year1996).[Slonczewski(1996)]slonczewski96authorJ. 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# Locality from Quantum Gravity: All or NothingNetta Engelhardt^a and Sebastian Fischetti^b [[email protected], [email protected]]^aDepartment of Physics, Princeton UniversityPrinceton, NJ 08544, USA^bTheoretical Physics Group, Blackett Laboratory, Imperial College London SW7 2AZ, UKIn a full theory of quantum gravity, local physics is expected to be approximate rather than innate. It is therefore important to understand how approximate locality emerges in the semiclassical limit.Here we show that any notion of locality emergent from a holographic theory of quantum gravity is “all or nothing”: local data is not obtained gradually from subregions of the boundary, but is rather obtained all at once when enough of the boundary is accessed. Our assumptions are mild and thus this feature is quite general; in the special case of AdS/CFT, a slightly different manifestation follows from well-known and familiar properties.Written for the Gravity Research Foundation 2017 Awards for Essays on Gravitation1cm 1.2 Recent developments suggest that local – that is, semiclassical – gravitational physics is not fundamental, but rather emerges from an appropriate limit of a complete theory of quantum gravity.A mysterious aspect of this emergence is its rarity: generic states of quantum gravity are non-classical. As our own universe is one of the special semiclassical states, it is of vital importance to understand how a theory of quantum gravity gives rise to local physics. What precisely do we mean by local physics? An ordinary quantum field theory is local if the commutator of any field ϕ at two spacelike-separated points x and y vanishes: [ϕ(x),ϕ(y)]=0. This concept is only well-defined in a system that has at least (i) an approximate notion of points and (ii) the information to determine the causal separation between the two points (whether they are spacelike-, timelike-, or null-separated). These two data constitute a semiclassical conformal geometry, a more primitive and coarser construct than a full semiclassical geometry: the latter contains the data required to measure distances between points, which is absent in the former. Understanding the emergence of locality is thus tantamount to understanding the emergence of a semiclassical conformal geometry.Because we are ultimately interested in describing our own universe, we restrict our attention to semiclassical (conformal) spacetimes containing matter.Specifically, we make the following assumption: A1A1 A semiclassical spacetime contains a weakly interacting quantum matter field ϕ(x).While this assumption is a reasonable expectation of any putative theory of quantum gravity, little else is known explicitly about such theories.Fortunately, the holographic principle <cit.> provides valuable insights: any theory of quantum gravity in (d+1) dimensions is believed to be expressible indirectly in terms of a non-gravitational theory in d dimensions; typically these theories are termed the “bulk” and “boundary”, respectively.In this Essay, we draw from our earlier work in <cit.> to show that the holographic principle implies that the emergence of semiclassical locality from a holographic theory of quantum gravity obeys a key property: it is “all or nothing”.To facilitate our arguments, we first make our assumptions about holography explicit: A2A2 The boundary theory lives on a geometry that can be embedded as a timelike or null hypersurface in the bulk; A3A3 For each n, the boundary contains an object O_n(X_1,…,X_n) which in the limit of a semiclassical bulk is related to the n-point correlator of ϕ(x) as lim_x_i →X_i ϕ(x_1) ⋯ϕ(x_n) = O_n(X_1,…,X_n), where x_i and X_i label points in the bulk and boundary, respectively.While the reader may protest that <ref> seems draconianly restrictive, it is in fact rather mild: for timelike separated X_i, the left hand side of (<ref>) is morally the S-matrix of the bulk theory. Any boundary theory should contain an object that encodes bulk scattering data if it is to describe the bulk.The three ingredients <ref>-<ref> can now be combined.First, an immediate consequence of <ref>: if the bulk is semiclassical, the correlator ϕ(x_1) ⋯ϕ(x_n) (taking n ≥ 4) is singular when the x_i are null-separated from a common vertex y, as long as the corresponding position-space Feynman diagram (shown in Figure <ref>) conserves energy-momentum at y <cit.> (see also earlier work by <cit.>). These so-called lightcone singularities – in which the x_i are distinct and some are timelike separated – are therefore sensitive to the causal structure of the bulk.It then follows from <ref> and <ref> that whenever the bulk is well-approximated by a semiclassical conformal geometry, the boundary object O_n(X_i) is singular when the boundary points X_i are null-separated from a bulk point y (at which energy-momentum is conserved)[In the context of AdS/CFT, these were termed “bulk-point singularities” in <cit.>.]; see Figure <ref>. Thus we immediately obtain a necessary condition for the emergence of a semiclassical conformal geometry: the O_n(X_i) must exhibit singularities when the X_i are distinct and at least some are timelike separated. Bulk locality plays an essential role in the above observation, as these singularities in O_n(X_i) may be traced to the shared vertex y, which uniquely identifies a bulk point.It is therefore instructive to ask whether this argument can be reversed: that is, is a semiclassical bulk causal structure encoded in the singularities of O_n(X_i)?As shown in <cit.>, the answer is yes.Because the singularities of O_n(X_i) correspond to null-separation from a bulk point y, they can be used to construct spacelike slices of the boundary corresponding to the intersection of lightcones of bulk points with the boundary; these so-called “lightcone cuts” are shown in Figure <ref>.Two such cuts are tangent if and only if the corresponding bulk points are null-separated; this feature implies that the geometric structure of the cuts represents the bulk conformal geometry.The upshot is that if the singularity structure of O_n(X_i) defines cuts that give rise to a consistent conformal geometry via the procedure of <cit.>, then that is the emergent bulk dual.Manifest in this construction is the emergence of locality: the very notion of a bulk point y is contained in the singularity structure of O_n(X_i) for some boundary points X^(y)_i null-separated from y, and the conformal geometry in a neighborhood of y is contained in the singularity structure of O_n(X_i) in a neighborhood of X^(y)_i.We have thus arrived at a sufficient and necessary condition for the emergence of locality from a state of holographic quantum gravity: the object O_n(X_i) must feature singularities on boundary spatial slices, and the geometry of those spatial slices must give rise via the procedure of <cit.> to a consistent semiclassical conformal geometry. §.§.§ All or Nothing Can we use this precise correspondence to understand properties of emergent bulk locality?The above construction of a bulk point y (and the conformal geometry at y) from singularities of the object O_n(X_i) relies on lightcone singularities; these exist only if (i) the X_i are all null-separated from a common vertex y, and (ii) energy-momentum is conserved at y.This latter condition implies that the construction of y requires access to a “sufficiently spread-out” set of boundary points X_i, as shown in Figure <ref>.A dramatic consequence is immediate: if the X_i are restricted to lie in too small a region , then O_n(X_i) may not be singular even if the X_i are null separated from y; thus it is impossible to identify y from data in .As is enlarged, however, the point y and the conformal geometry at y can be fully constructed as soon as reaches the critical size necessary for points into conserve momentum at y.The essential feature here is that the point y is not obtained gradually and little-by-little as is enlarged; rather, it is obtained completely and suddenly as soon as becomes sufficiently large.This feature follows very generally in any holographic framework obeying properties <ref>-<ref>; this demonstrates our advertised claim that the reconstruction of (approximate) locality in holographic quantum gravity is “all-or-nothing”. A holographic description of the theory of quantum gravity in the bulk cannot partially describe local data: it must describe it fully or not at all.§.§.§ An Illustrative Example: AdS/CFT This “all-or-nothing” emergence is a general feature of holographic frameworks of quantum gravity.We now consider a concrete example of such a framework: the anti-de Sitter/conformal field theory (AdS/CFT) correspondence <cit.>.This correspondence, which provides the most explicit formulation of holography, relates string (or M) theory with asymptotically AdS boundary conditions to a CFT on the asymptotic boundary of AdS.Since the asymptotic boundary is timelike, property <ref> holds.Moreover, there exists a limit in which the bulk is well-described by semiclassical gravity; in this limit, bulk matter fields are weakly coupled, so property <ref> holds as well.Finally, the AdS/CFT dictionary implies that each CFT operator (X) has a dual bulk quantum field ϕ(x) and vice versa, and that the correlators (X_1) ⋯(X_n) are obtained from the correlators ϕ(x_1) ⋯ϕ(x_n) as the x_i are taken to the asymptotic boundary.Thus property <ref> holds, with O_n(X_1,…,X_n) = (X_1) ⋯(X_n).AdS/CFT is therefore a particular manifestation of the more general holographic framework <ref>-<ref>.In fact, access to the full AdS/CFT dictionary, rather than just the limited holographic dictionary of properties <ref> and <ref>, shows that the “all-or-nothing” property of local bulk data is quite robust.To see this, recall that a bulk operator ϕ(y) at the point y can be reconstructed from boundary data in a spatial region if y lies between the so-called Ryu-Takayanagi (RT) surface of and the boundary <cit.>, as shown in Figure <ref>.Thus if is sufficiently small, ϕ(y) cannot be reconstructed from any data in ; as is enlarged, ϕ(y) (and any other fields at y) can be reconstructed as soon as the RT surface of encloses y.This feature was interpreted in terms of quantum error correction in <cit.>, where it was noted that the underlying mechanism is Page's theorem <cit.>[We thank Daniel Harlow for calling our attention to this.].Our arguments indicate that the above feature of the AdS/CFT correspondence is a manifestation of the more general principle identified in this Essay: that quantum gravity in general describes locality in an abrupt, global manner, giving access to full data or none, with no continuum in between. In retrospect, this should perhaps have been anticipated: quantization typically induces discretization. There is no reason to expect that a quantum theory of spacetime should produce spacetime in an altogether continuous way; indeed, it does not. §.§.§ Acknowledgements We thank Daniel Harlow for comments on an earlier version of this essay.The work of NE was supported in part by NSF grant PHY-1620059.SF was supported by STFC grant ST/L00044X/1. jhep | http://arxiv.org/abs/1703.09222v2 | {
"authors": [
"Netta Engelhardt",
"Sebastian Fischetti"
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"hep-th",
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"primary_category": "hep-th",
"published": "20170327180000",
"title": "Locality from Quantum Gravity: All or Nothing"
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