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Visual Translation Embedding Network for Visual Relation Detection Hanwang Zhang^†, Zawlin Kyaw^, Shih-Fu Chang^†, Tat-Seng Chua^ ^†Columbia University, ^National University of Singapore {hanwangzhang, kzl.zawlin}@gmail.com; sfchang@ee.columbia.edu; dcscts@nus.edu.sg December 30, 2023 ==================================================================================================================================================================================================================Visual relations, such as “person ride bike” and “bike next to car”, offer a comprehensive scene understanding of an image, and have already shown their great utility in connecting computer vision and natural language. However, due to the challenging combinatorial complexity of modeling subject-predicate-object relation triplets, very little work has been done to localize and predict visual relations. Inspired by the recent advances in relational representation learning of knowledge bases and convolutional object detection networks, we propose a Visual Translation Embedding network (VTransE) for visual relation detection. VTransE places objects in a low-dimensional relation space where a relation can be modeled as a simple vector translation, i.e., subject + predicate ≈ object. We propose a novel feature extraction layer that enables object-relation knowledge transfer in a fully-convolutional fashion that supports training and inference in a single forward/backward pass. To the best of our knowledge, VTransE is the first end-to-end relation detection network. We demonstrate the effectiveness of VTransE over other state-of-the-art methods on two large-scale datasets: Visual Relationship and Visual Genome. Note that even though VTransE is a purely visual model, it is still competitive to the Lu's multi-modal model with language priors <cit.>.§ INTRODUCTIONWe are witnessing the impressive development in connecting computer vision and natural language, from the arguably mature visual detection <cit.> to the burgeoning visual captioning and question answering <cit.>. However, most existing efforts to the latter vision-language tasks attempt to directly bridge the visual model (, CNN) and the language model (, RNN), but fall short in modeling and understanding the relationships between objects. As a result, poor generalization ability was observed as those models are often optimized on specialized datasets for specific tasks such as image captioning or image QA. <cit.>.As illustrated in Figure <ref>, we take a step forward from the lower-level object detection and a step backward from the higher-level language modeling, focusing on the visual relations between objects in an image. We refer to a visual relation as atriplet[When the context is clear, we always refer to object in normal font as a general object andin teletype to the tail object in a relation.], where the predicate can be verb (--), spatial (--), preposition (--), and comparative (--) <cit.>. Visual relations naturally bridge the vision and language by placing objects in a semantic context of what, where, and how objects are connected with each other. For example, if we can detect -- and -- successfully, the reasoning behind the answer “gray” to the question asked in Figure <ref> will be explicitly interpretable using dataset-independent inference, , QA over knowledge bases <cit.>, and thus permits better generalization or even zero-shot learning <cit.>. In this paper, we present a convolutional localization network for visual relation detection dubbed Visual Translation Embedding network (VTransE). It detects objects and predicts their relations simultaneously from an image in an end-to-end fashion. We highlight two key novelties that make VTransE effective and distinguishable from other visual relation models <cit.>:Translation Embedding. Since relations are compositions of objects and predicates, their distribution is much more long-tailed than objects. For N objects and R predicates, one has to address the fundamental challenge of learning 𝒪(N^2R) relations with few examples <cit.>. A common solution is to learn separate models for objects and predicates, reducing the complexity to 𝒪(N+R). However, the drastic appearance change of predicates makes the learning even more challenging. For example,appearance largely varies from -- to --. To this end, inspired by Translation Embedding (TransE) in representing large-scale knowledge bases <cit.>, we propose to model visual relations by mapping the features of objects and predicates in a low-dimensional space, where the relation triplet can be interpreted as a vector translation, , + ≈ . As shown in Figure <ref>, by avoiding learning the diverse appearances of -- with large variance, we only need to learn thetranslation vector in the relation space, even though the subjects and/or objects can be quite diverse.Knowledge Transfer in Relation. Cognitive evidences show that the recognition of objects and their interactions is reciprocal <cit.>. For example,anddetections serve as the context forprediction, which in turn constrains the articulation of the two objects, and thus benefiting object detection.Inspired by this, we explicitly incorporate knowledge transfer between objects and predicates in VTransE. Specifically, we propose a novel feature extraction layer that extracts three types of object features used in translation embedding: classeme (, class probabilities), locations (, bounding boxes coordinates and scales), and RoI visual features. In particular, we use the bilinear feature interpolation <cit.> instead of RoI pooling <cit.> for differentiable coordinates. Thus, the knowledge between object and relation—confidence, location, and scale—can be transfered by a single forward/backward pass in an end-to-end fashion.We evaluate the proposed VTransE on two recently released relation datasets: Visual Relationship <cit.> with 5,000 images and 6,672 unique relations, and Visual Genome <cit.> with 99,658 images and 19,237 unique relations. We show significant performance improvement over several state-of-the-art visual relation models. In particular, our purely visual VTransE can even outperform the multi-modal method with vision and language priors <cit.> in detection and retrieval, and a bit shy of it in zero-shot learning.In summary, our contributions are as follows: 1) We propose a visual relation detection model dubbed Visual Translation Embedding network (VTransE), which is a convolutional network that detects objects and relations simultaneously. To the best of our knowledge, this is the first end-to-end relation detection network; 2) We propose a novel visual relation learning model for VTransE that incorporates translation embedding and knowledge transfer; 3) VTransE outperforms several strong baselines on visual relation detection by a large performance gain.§ RELATED WORKOur work falls in the recent progress on grounding compositional semantics in an image <cit.>. It has been shown that high-quality groundings provide more comprehensive scene understanding, which underpins many vision-language tasks such as VQA <cit.>, captioning <cit.> and complex query retrieval <cit.>. Visual relation detection not only ground regions with objects, but also describes their interactions. In particular, our VTransE network draws on recent works in relation learning and object detection. Visual Relation Detection. Different from considering relations as hidden variables <cit.>, we relate to explicit relation models which can be divided into two categories: joint model and separate model. For joint models, a relation triplet is considered as a unique class <cit.>. However, the long-tailed distribution is an inherent defect for scalability. Therefore, we follow the separate model that learns subject, object, and predicate individually <cit.>. But, modeling the large visual variance of predicates is challenging. Inspired by TransE that has been successfully used in relation learning in large-scale knowledge base <cit.>, our VTransE extends TransE for modeling visual relations by mapping subjects and objects into a low-dimensional relation space with less variance, and modeling the predicate as a translation vector between the subject and object. Note that there are works <cit.> that exploit language priors to boost relation detection, but we are only interested in visual models.Object Detection. VTransE is based on an object detection module composed of a region proposal network (RPN) and a classification layer. In particular, we use Faster-RCNN <cit.>, which is evolved from its predecessors <cit.> that requires additional input of region proposals. Note that VTransE cannot be simply considered as appending a relation prediction layer to Faster-RCNN. In fact, we propose a novel feature extraction layer that allows knowledge transfer between objects and relations. The layer exploits the bilinear interpolation <cit.> instead of the non-smooth RoI pooling in Faster-RCNN and thus the reciprocal learning of objects and predicates can be achieved in a single forward/backward pass. Note that VTransE can be married to any object detection network that contains an RPN such as the very recent SSD <cit.> and YOLO <cit.>. § OUR APPROACH: VTRANSE NETWORKVTransE is an end-to-end architecture that completes object detection and relation prediction simultaneously. As illustrated in Figure <ref>, it builds upon an object detection module (, Faster-RCNN), and then incorporates the proposed feature extraction layer and the translation embedding for relation prediction. §.§ Visual Translation EmbeddingGiven any valid relation, Translation Embedding (TransE) <cit.> represents -- in low-dimensional vectors s, p, and o, respectively, and the relation is represented as a translation in the embedding space: s + p ≈ o when the relation holds, and s + p ≉ o otherwise. TransE offers a simple yet effective linear model for representing the long-tail relations in large knowledge databases <cit.>. To transfer TransE in the visual domain, we need to map the features of the detected objects into the relation space, which is consistent to recent works in visual-semantic embeddings <cit.>. Suppose 𝐱_s,𝐱_o∈ℝ^M are the M-dimensional features ofand , respectively. Besides learning a relation translation vector 𝐭_p∈ℝ^r (r≪ M) as in TransE[In experiments, we tested r∈{100,200,...,1000} and found that r = 500 is a good default.], VTransE learns two projection matrices 𝐖_s,𝐖_o∈ℝ^r× M from the feature space to the relation space. Thus, a visual relation can be represented as:𝐖_s𝐱_s+𝐭_p≈𝐖_o𝐱_o. As in TransE, we can use a large-margin metric learning loss function for relations:ℒ = ∑_(s,p,o)∈ℛ∑_(s',p,o')∈ℛ' [d(𝐖_s𝐱_s+𝐭_p,𝐖_𝐨𝐱_𝐨)+1 - d(𝐖_s𝐱_s'+𝐭_p,𝐖_𝐨𝐱_𝐨')]_+where d() is a proper distance function, ℛ is the set of valid relations, and ℛ' is the set of invalid relations. However, unlike the relations in a knowledge base that are generally facts, , --, visual relations are volatile to specific visual examples, , the validity of -- depends on the heights of the specific car and person in an image, resulting in problematic negative sampling if the relation annotation is incomplete. Instead, we propose to use a simple yet efficient softmax for prediction loss that only rewards the deterministically accurate predicates[In fact, predicate is multi-labeled, , both -- and -- are correct. However, most relations are single-labeled in the datasets, , 58% in VRD <cit.> and 67% in VG <cit.>.], but not the agnostic object compositions of specific examples:ℒ_rel =∑_(s,p,o)∈ℛ-logsoftmax(𝐭^T_p(𝐖_o𝐱_o-𝐖_s𝐱_s)),where the softmax is computed over p. Although Eq. (<ref>) learns a rotational approximation for the translation model in Eq. (<ref>), we can retain the translational property by proper regularizations such as weight decay <cit.>. In fact, if the annotation for training samples is complete, VTransE works with softmax (Eq. (<ref>)) and negative sampling metric learning (Eq. (<ref>)) interchangeably.The final score for relation detection is the sum of object detection score and predicate prediction score in Eq. (<ref>): S_s,p,o = S_s+S_p+S_o.§.§ Feature ExtractionWe propose a Feature Extraction Layer in VTransE to extract 𝐱_s and 𝐱_o. There are three types of features that characterize the multiple facets of objects in relations:Classeme. It is an (N+1)-d vector of object classification probabilities (, N classes and 1 background) from the object detection network. Classeme is widely used as semantic attributes in various vision tasks <cit.>. For example, in relation detection, classeme is a useful prior for rejecting unlikely relations such as --.Location. It is a 4-d vector (t_x,t_y,t_w,t_h), which is the bounding box parameterization in <cit.>, where(t_x,t_y) specifies a scale-invariant translation and (t_w, t_h) specifies the log-space height/width shift relative to its counterpartor . Takeas an example:t_x = x-x'/w', t_y = y-y'/h',t_w = logw/w', t_h = logh/h'where (x,y,w,h) and (x',y',w',h') are the box coordinates ofand , respectively. Location feature is not only useful for detecting spatial or preposition relation, but also useful for verbs, ,is usually abovewhen the predicate is . Visual Feature. It is a D-d vector transformed from a convolutional feature of the shape X× Y× C. Although it is as the same size as the RoI pooling features used in Faster-RCNN, our features are bilinearly interpolated from the last conv-feature map, so as to achieve end-to-end training that allows knowledge transfer (cf. Section <ref>). The overall feature 𝐱_s or 𝐱_o is a weighted concatenation of the above three features (M = N+D+5), where the weights are learnable scaling layers since the feature contribution dynamically varies from relation to relation. As shown in Figure <ref>, the proposed feature extraction layer couples the Object Detection Module and the Relation Prediction Module. §.§ Architecture DetailsA training image for VTransE is labeled with a list of -- triplets, where every uniqueoris annotated with a bounding box. At testing time, VTransE inputs an image and outputs a set of detected objects and the relation prediction scores for every pair of objects. Object Detection Network. VTransE network starts from the Faster-RCNN <cit.> object detection network with the VGG-16 architecture <cit.>. At training time, we sample a mini-batch cotaining 256 region proposal boxes generated by the RPN of Faster-RCNN, each of which is positive if it has an intersection over union (IoU) of at least 0.7 with some ground truth regions and it is negative if the IoU < 0.3. The positive proposals are fed into the classification layer, where each proposal outputs an (N+1) class probabilities and N bounding box estimations. Then, we perform non-maximum suppression (NMS) for every class with the IoU > 0.4, resulting in 15.6 detected objects on average, each of which has only one bounding box. The reasons of performing NMS for object detection are two folds: 1) we need a specific object class for each region to match with the relation ground truth, and 2) we need to down-sample the objects for a reasonable number of candidate relations. At test time, we sample 300 proposal regions generated by RPN with IoU > 0.7. After the classification layer, we perform NMS with IoU > 0.6 on the 300 proposals, resulting in 15–20 detections per image on average.Bilinear Interpolation. By removing the final pooling layer of VGG-16, we use the last convolutional feature map 𝐅 of the shape W' × H'× C (the pink cube in Figure <ref>), where C =512 is the number of channels, W' = ⌊W/16⌋, and H' = ⌊H/16⌋, where W and H are the width and height of the input image. 𝐅 encodes the visual appearance of the whole image and is used for extracting visual features for the object detection and relation prediction.In order to achieve object-relation knowledge transfer, the relation error should be back-propagated to the object detection network and thus refines the objects. However, the widely-used RoI pooling visual feature in Fast/Faster R-CNN is not a smooth function of coordinates since it requires discrete grid split for the proposal region, resulting in zero coordinate gradients back-propagated from the feature extraction layer.To this end, we replace the RoI pooling layer with bilinear interpolation <cit.>. It is a smooth function of two inputs: the feature map 𝐅 and an object bounding box projected onto 𝐅, and the output is a feature 𝐕 of the size X× Y× C (the orange cube in Figure <ref>). Each entry value in 𝐕 can be efficiently interpolated from 𝐅 in a convolutional way:V_i,j,c = ∑^W'_i'=1∑^H'_j' = 1F_i',j',ck(i'-G_i,j,1)k(j'-G_i,j,2),where 𝐆∈ℝ^X× Y × 2 records the positions of the X× Y grid split in the input bounding box and k(x) = max(0,1-|x|) is the bilinear interpolation kernel. Note that the grid position 𝐆 matrix is a linear function of the input box. Therefore, the gradients from 𝐕 can be back-propagated to the bounding box coordinates.Optimization. We train the VTransE network end-to-end by stochastic gradient descent with momentum <cit.>. We follow the “image-centric” training strategy <cit.>, , the mini-batch arises from a single image that contains many object regions and relations. The loss function is a multi-task loss combining the object detection loss ℒ_obj and the relation detection loss ℒ_rel in Eq. (<ref>), allowing reciprocal learning for objects and relations. In particular, we find that a reasonable loss trade-off is ℒ_obj+0.4ℒ_rel. Since object detection and relation prediction have different sample sizes, we normalize ℒ_obj and ℒ_rel by the mini-batch size. For model initializations, we pre-train Faster-RCNN on the objects in the relation datasets to initialize the object detection network and randomly initialize the VTransE component with Gaussian weights. For end-to-end training, we also replace the RoI pooling layer in the object detection network with bilinear interpolation. For efficiency, we do not fine-tune the VGG-16 CNN. Generally, we need 2 – 3 epochs for the model to converge. For a single image that has been resized to the longer side of 720 pixels, the training runs in 2.0 fps and the testing runs in 6.7 fps on a Titan X GPU using Caffe and Python. Note that we can always plug-in faster object detection networks such as SSD <cit.> and YOLO <cit.> for more efficient training and testing. § EXPERIMENTSWe are going to validate the effectiveness of the proposed VTransE network by answering the following questions. Q1: Is the idea of embedding relations effective in the visual domain? Q2: What are the effects of the features in relation detection and knowledge transfer? Q3: How does the overall VTransE network perform compared to the other state-of-the-art visual relation models?§.§ Datasets and MetricsTo the best of our knowledge, there are only two datasets for visual relation detection at a large scale. We used both: VRD. It is the Visual Relationships dataset <cit.>. It contains 5,000 images with 100 object categories and 70 predicates. In total, VRD contains 37,993 relation annotations with 6,672 unique relations and 24.25 predicates per object category. We followed the same train/test split as in <cit.>, , 4,000 training images and 1,000 test images, where 1,877 relationships are only in the test set for zero-shot evaluations. VG. It is the latest Visual Genome Version 1.2 relation dataset <cit.>. Unlike VRD that is constructed by computer vision experts, VG is annotated by crowd workers and thus the objects and relations are noisy. Therefore, we contact the authors for an official pruning of them. For example, “young woman” and “lady” are merged to the WordNet hypernym “woman”. We filtered out relations with less than 5 samples. In summary, VG contains 99,658 images with 200 object categories and 100 predicates, resulting in 1,174,692 relation annotations with 19,237 unique relations and 57 predicates per object category. We split the data into 73,801 for training and 25,857 for testing.Following <cit.>, we used Recall@50 (R@50) and Recall@100 (R@100) as evaluation metrics for detection. R@K computes the fraction of times a true relation is predicted in the top K confident relation predictions in an image. Note that precision and average precision (AP) are not proper metrics as visual relations are labeled incompletely and they will penalize the detection if we do not have that particular ground truth. For the relation retrieval task (cf. Section <ref>), we adopted the Recall rate@5 (Rr@5), which computes the fraction of times the correct result was found among the top 5, and Median rank (Med r), which is the median rank of the first correctly retrieved image <cit.>. In fact, for datasets with more complete annotations (, VG), even if the recall is low, the actual precision could be high since the number of ground truth in an image is usually larger than 50/100. Therefore, the retrieval task measured by Rr@5 and Med r provides a complementary evaluation. §.§ Evaluations of Translation Embedding (Q1) Setup. Visual relation detection requires both object detection and predicate prediction. To investigate whether VTransE is a good model for relations, we need to isolate it from object detection and perform the task of Predicate Prediction: predicting predicates given the ground-truth objects with bounding boxes.Comparing Methods. We compared 1) JointBox, a softmax classifier that classifies the images of theandjoint bounding boxes into predicates, and 2) VTransE that classifies the predicate of a pair ofandboxes. For fair comparison, we only use the RoI pooling visual features of boxes for the two methods. Note that JointBox represents many visual relation models in predicate prediction <cit.>Results. From Table <ref>, we can see that VTransE formulated in Eq. (<ref>) outperforms conventional visual models like JointBox. This is because the predicate model parameters of VTransE—the translation vectors—are able to capture the essential meanings of relations between two objects mapped into a low-dimensional relation space. Figure <ref> illustrates that VTransE can predict correct predicates with diversity while JointBox is more likely to bias on certain visual patterns. For example, JointBox limitsin cars, but VTransE can generalize to other subjects like plane and bus. Moreover, by inspecting the semantic affinities between the predicate parameter vectors in Figure <ref>, we can speculate that JointBox does not actually model relations but the joint object co-occurrence. For example, in JointBox, the reason whyis close toandis largely due to the co-occurrence of -- and --; however, VTransE is more likely to understand the meaning ofas its neighbors areand , and it is far fromand .§.§ Evaluations of Features (Q2)Setup. We evaluated how the features proposed in Section <ref> affect visual relation detection. We performed Relation Detection <cit.>: the input is an image and the output is a set of relation triplets and localizations of bothandin the image having at least 0.5 overlap with their ground-truth boxes simultaneously.Comparing Methods. We ablated VTransE into four methods in terms of using different features: 1) Classeme, 2) Location, 3) Visual, and 4) All that uses classeme, locations, visual features, and the fusion of the above with a scaling layer (cf. Figure <ref>), respectively. Note that all the above models are trained end-to-end including the object detection module. To further investigate the feature influence on relations, we categorized the predicates into four categories: verb, spatial, preposition and comparative (cf. Supplementary Material for the detailed category list).Results. From Figure <ref>, we can see the details of what features are good at detecting what relations: 1) fusing all the features with a learned scaling layer can achieve the best performance on all types of relations; 2) classeme can generally outperform visual features in various kinds of relations as it characterizes both the high-level visual appearances (, what an object looks like) and composition priors (,is more likely to - than ); 3) for spatial relations, location features are better; however, for preposition relations, all features perform relatively poor. This is because the spatial and visual cues of prepositions are volatile such as -- and --. Table <ref> shows that the end-to-end training of VTransE can improve the object detection. This is mainly due to that the proposed feature extraction layer allows knowledge transfer so that the errors made by relation prediction can be back-propagated to the front object detection module. In fact, the improvement can be expected since we incorporate additional relation labels besides object labels. As shown in Figure <ref>, compared to the pre-trained Faster-RCNN module, the object detection module trained by VTransE can generally improve bounding boxes, such as minor refinement or even recovery from drastic dislocation and corrections for wrong detections. This demonstrates that relations place objects in a contextual scene. For example, relation can recoverfrom the wrong detection , even though the correct detection should be , which is semantically similar to . This correction is likely inferred by the relation --/.§.§ Comparison with State-of-The-Arts (Q3)Setup. As we will introduce later, some joint relation models can only detect a joint bounding box for an entire relation; thus, besides relation detection, we performed Phrase Detection <cit.>: the input is an image and the output is a set of relation triplets and localizations of the entire bounding box for each relation that having at least 0.5 overlap with the ground-truth jointandbox.For more extensive evaluations, we also performed two additional tasks. 1) Relation Retrieval: image search with the query of a relation triplet. We first detect the relation query in gallery (, test) images and then score them according to the average detection scores of the query relation. An image with at least one successful query relation detection is considered as a hit. This task is a representation of the compositional semantic retrieval <cit.>; We selected the top 1,000 frequent relations as queries. 2) Zero-shot Learning <cit.>: individual , , andare seen in both training and test, but some specific triplet compositions are only in the test set. Due to the long-tailed relation distribution, it is a practical setting since it is impossible to collect data for all triplets. Comparing Methods. We compared the VTransE network to four state-of-the-art visual relation detection models. 1) VisualPhrase <cit.>: a joint relation model that considers every unique relation triplet as an relation class. For fair comparison, we replace the original DPM object detection model <cit.> with Faster-RCNN <cit.>; 2) DenseCap <cit.>: it detects sub-image regions and generate their descriptions simultaneously. It is an end-to-end model using bilinear interpolated visual features for region localizations. We replace its LSTM classification layer with softmax for relation prediction. Thus, it can be considered as an joint relation model; 3) Lu's-V (V-only in <cit.>): it is a two-stage separate model that first uses R-CNN <cit.> for object detection and then adopts a large-margin JointBox model for predicate classification; 4) Lu's-VLK (V+L+K in <cit.>): a two-stage separate model that combines Lu's-V and word2vec language priors <cit.>. In addition, we compared VTransE to its two-stage training model VTransE-2stage that apply Faster-RCNN for object detection and then perform predicate predication using translation embedding as in Q1.As we have no training source codes of Lu's methods, we cannot apply them in VG and we quoted the results of VRD reported in their paper <cit.>. Moreover, as the joint relation models such as VisualPhrase and DenseCap can only detect relation triplet as a whole, they are not applicable in zero-shot learning. Therefore, we only report zero-shot results (detection and retrieval) on VRD for the official 1,877 zero-shot relations <cit.>. Results. From the quantitative results in Table <ref> and the qualitative results in Figure <ref>, we have: 1) Separate relation models like VTransE and Lu's-V outperform joint models like VisualPhrase and DenseCap significantly, especially on VRD. This is because the classification space of joint models for all possible relationships is large (, 6,672 and 19,237 training relations in VRDand VG, respectively), leading to insufficient samples for training infrequent relations.2) For separate models, better object detection networks, such as Faster-RCNN v.s. R-CNN used in VTrasnE and Lu's, are beneficial for relation detections. As shown in Figure <ref>, on VRD dataset, Lu's-VLK mistakesasandas . We believe that this is a significant reason why their visual model Lu's-V is considerably worse than VTransE.3) Even though VTransE is a purely visual model, we can still outperform Lu's-VLK which incorporates language priors, , on VRD measured by R@50 and Med r, we are 20%, 2%, and 230% relatively better in phrase detection, relation detection, and relation retrieval, respectively. First, the classeme feature can serve as a similar role as language priors. Second, location feature is indispensable to relations. Take the -- relation query as an example in Figure <ref>, when there are multipledetections in an image, Lu's-VLK usually relatesto the wrong , regardless the fact that the spatial distance is far. Similar examples can be also found in the false detection -- of Lu's-VLK.4) The end-to-end VTransE is better than VTransE-2stage across all the tasks on both datasets. Together with the results in Q2, they demonstrate the effectiveness of reciprocal learning between objects and relations.From the zero-shot quantitative results in Table <ref> and the qualitative results in Figure <ref>, we have: 1) The performances of ours and the compared methods degrade drastically, , for relation detection, VTransE and Lu's-VLK suffer 88% and 79% performance (R@100) drop, respectively. This is the key limitation of VTransE. Perhaps this is because our transformation from feature space to relation space in Eq. (<ref>) is too generic, especially for verbs, and thus fails to capture the relation-specific visual deformations. For example, VTransE cannot discriminate between -- and --. One remedy is to incorporate predicate and object models <cit.>, although it will increase the model complexity from 𝒪(N+R) to 𝒪(NR), where N is the number of objects and R is the number of predicates.2) Both as visual models, our VTransE is significantly better than Lu's-V in zero-shot relation predictions; nevertheless, as a multi-modal model, Lu's-VLK surpasses VTransE by exploiting language priors. But, since visual relations are volatile to specific examples, language priors are not always correct—Lu's-VLK can be misled by frequent language collocations which are invalid in visual examples, , the mismatch ofandin -- and --. § CONCLUSIONSWe focused on the visual relation detection task that is believed to offer a comprehensive scene understanding for connecting computer vision and natural language. Towards this task we introduced the VTransE network for simultaneous object detection and relation prediction. VTransE is an end-to-end and fully-convolutional architecture that consists of an object detection module, a novel differentiable feature extraction layer, and a novel visual translation embedding layer for predicate classification. Moving forward, we are going to 1) model higher-order relations such as ----, 2) tackle the challenge of zero-shot relation learning, and 3) apply VTransE in a VQA system based on relation reasoning. § SUPPLEMENTARY MATERIAL§.§ Predicate Categorization§.§.§ 70 predicates in Visual Relationship DatasetVerb: attach to, carry, contain, cover, drive, drive on, eat, face, feed, fly, follow, hit, hold, kick, lean on, look, lying on, park behind, park next, park on, play with, pull, rest on, ride, sit behind, sit next to, sit on, sit under, skate on, sleep next to, sleep on, stand behind, stand next to, stand on, stand under, talk, touch, use, walk, walk beside, walk next to, walk past, walk to, watch, wear.Spatial: above, adjacent to, behind, below, beneath, beside, in, in the front of, inside, near, next to, on, on the left of, on the right of, on the top of, outside of, over, under.Preposition: across, against, at, by, has, with.Comparative: taller than. §.§.§ 100 predicates in Visual Genome DatasetVerb: adorn, attach to, belong to, build into, carry, cast, catch, connect to, contain, cover, cover in, cover with, cross, cut, drive on, eat, face, fill with, fly, fly in, grow in, grow on, hang in, hang on, hit, hold, hold by, lay in, lay on, lean on, look at, mount on, paint on, park, play, print on, pull, read, reflect in, rest on, ride, say, show, sit at, sit in, sit on, stand behind, stand on, standing by, standing in, standing next to, support, surround, swing, throw, touch, use, walk, walk in, walk on, watch, wear, wear by, write on.Spatial: above, behind, below, beneath, between, in, in front of, in middle of, inside, near, next to, on, on back of, on bottom of, on side of, on top of, outside, over, under, underneath.Preposition: across, against, along, around, at, beside, by, for, from, have, of, part of, to, with.Comparative: small than, tall than. §.§ Incomplete Annotation in VRDMore qualitative relation retrieval results from VRD are shown in Figure <ref>. We can see that even though most of the top 10 results are correct, none of them is labeled as the corresponding query relation. We believe that this incomplete annotation in VRD is the main cause of the low retrieval performance, as reported in Table 3 and Table 4 from the paper.§.§ Demo LinkWe provide a demo running on an Amazon server (US East N. Virginia) ) <cvpr.zl.io>. The interface is illustrated in Figure <ref>. Note that for clear visualizations, only detections scored about top 20% of each predicate type are shown. ieee

http://arxiv.org/abs/1702.08319v1

http://arxiv.org/abs/1702.08085v4

firstpage–lastpage 2017Internal rotation of 13 low-mass low-luminosity red giants in the Kepler field S. A. Triana <ref>,<ref> E. Corsaro <ref>,<ref>,<ref> J. De Ridder <ref> A. Bonanno <ref> F. Pérez Hernández<ref>,<ref> R. A. García <ref>Accepted 2017 May 2. Received 2017 May 2; in original form 2017 February 25 =========================================================================================================================================================================================================== The donor in the microquasar GRS 1915+105 is a low-mass giant. Such a star consists of a degenerate helium core and a hydrogen-rich envelope. Both components are separated by a hydrogen burning shell. The structure of such an object is relatively simple and easy to model. Making use of the observational constraints on the luminosity and the radius of the donor, we constrain the mass of this star with evolutionary models. We find a very good agreement between the constraints from those models and from the observed rotational broadening and the NIR magnitude. Combining the constraints, we find solutions with stripped giants of the mass of ≥0.28 M_ and of the spectral class K5 III, independent of the distance to the system, and a distance-dependent upper limit, ≲1 M_. We also calculate the average mass transfer rate and the duty cycle of the system as a function of the donor mass. This rate is much below the critical rate (at which the system would become persistent), and the duty cycle is less than 20 per cent. binaries: general – stars: evolution – stars: individual: V1487 Aql – X-rays: binaries – X-rays: individual: GRS 1915+105. § INTRODUCTIONis a low mass X-ray binary, which appears to be the most distinct Galactic microquasar <cit.>. Its optical component got a variable-star name V1487 Aql. Its outburst began on 1992 August 15 <cit.>, and it has remained in the outburst state since then. The system contains a black hole and a low mass K–M III giant donor <cit.>, and it has a long period of P=33.85± 0.16 d <cit.>. Its donor fills its Roche lobe and supplies the matter accreted by the black hole. The mass of the black hole, M_1, can be constrained from the radial velocity amplitude, which was measured by VLT as K_2= 126± 1 km s^-1 <cit.>. The value of the mass depends on the inclination, i, as M_1∝ (K_2/sin i)^3. In the case of , the value of i can be determined for the jets only, where it depends on the distance to the source <cit.>. The distance is currently determined as d= 8.6^+2.0_-1.6 kpc from radio parallax measurements <cit.>, which determination is consistent with an independent estimate of d≲ 10 kpc by <cit.> based on considering the jet kinetic power, and which yields M_1=12.4^+2.0_-1.8 <cit.>.While the mass of the black hole may be considered relatively precisely known, the mass of the K–M III donor is more poorly constrained. Its mass can be constrained by rotational broadening, which has been measured by <cit.> as vsin i=21± 4 km s^-1, where the standard deviation v for an individual measurement is ∼2–3 km s^-1, and the total uncertainty includes estimated systematic errors. This value combined with their K_2 measurement yields q≃ 0.042± 0.024. For the distance-dependent black-hole mass estimate of <cit.>, this gives M_2≃ 0.52± 0.31. Thus, the donor has a mass substantially lower than that of an isolated star of the same spectral class <cit.>, i.e., it is a 'stripped giant' (e.g., , hereafter WRS83). We explore here another possibility to constrain the donor mass based on modelling of its internal structure. Since the donor is not a main sequence star but rather an evolved low-mass giant, its structure is relatively simple and easy to model. Making use of the observational constraints on the luminosity and the radius of the donor, we construct an evolutionary model of this star and attempt to constrain its mass. For that, we also need the effective temperature of the donor. The currently most accurate NIR observations of this system are those of <cit.>, who matched their spectra to those of K0, K1, K2, K5 III and M0 III template stars. Thus, we adopt here the possible range of the spectral classes[<cit.> gave the range of the spectral class of the donor as K0–3 III, but that choice was not based on any additional constraints with respect to those of <cit.> (J. McClintock, private communication.) Also, they gave M_2≃ 0.58± 0.33, which appears to be due to a typo.] from K0 III to M0 III. § OBSERVATIONALLY DETERMINED PARAMETERSThe third Kepler law and the relation between the donor radius (which is equal to the radius of its Roche lobe) and the separation, a, for M_2≪ M_1 <cit.> area =[G(M_1+M_2)]^1/3 P^2/3/(2π)^4/3, R_2=2 a/3^4/3(M_2/M_1+M_2)^1/3,respectively. This yields the standard formula,R_2=(2 G M_2)^1/3(P/9π)^2/3≃ 1.945 (M_2/)^1/3 (P/1d)^2/3.This formula makes the radius of the donor filling its Roche lobe one of the most accurately determined parameters of a binary. The orbital period is usually known with high precision and the dependence on M_2 is weak. Moreover, equation (<ref>) does not depend on the distance to the binary system. For discussion of the accuracy of equation (<ref>) see, e.g., <cit.>. Given the small uncertainty on the period of , its contribution to the error is negligible. We note that the Roche-lobe radius for a given donor mass of equation (<ref>), R_2(M_2,P), is identical to that implied by the rotational broadening,R_2=P (vsin i)/2sin i,for M_2=q(vsin i, K_2) M_1(K_2, P, i, q) substituted in the former. Here q follows from the standard rotational-broadening relationship (e.g., ), which solution can be found asq=[r(y)-1]^2/3 r(y), r(y)=2^-1/3[2+27y+3^3/2√(y(4+27y))]^1/3≥ 1,y=(3^4/3 vsin i/2 K_2)^3,while the black-hole mass isM_1=P K_2^3(1+q)^2/2 Gsin^3 i.Equations (<ref>–<ref>) assume corotation, which is very likely in .We can relate the inclination to the distance by assuming the inner jet has the same direction as the binary axis, which yields i=arctan[2μ_ aμ_ r d/ (μ_ a-μ_ r)c], where μ_ a and μ_ r are, respectively, the angular velocities of the approaching and receding jet. <cit.> used the weighted average of the inclination resulting from the observations by <cit.> and <cit.>. On the other hand, <cit.> argued that the jet direction changes between the projected distance from the centre of 0.3” (observed by ) and 1” <cit.>, and thus the results of the former are more relevant for estimating the binary plane orientation, resulting in i=59^+5_-4. <cit.> argued that the alignment is very likely, and this assumption has indeed been universally used in the mass estimates for . On the other hand, an also likely and widely accepted model for the low-frequency QPOs/breaks in the power spectra of black-hole binaries is the Lense-Thirring precession of an inner hot part of the accretion flow (e.g., ). This model requires a misalignment between the black-hole spin and binary axes, though the minimum required misalignment appears not to be specified.does show low-frequency QPOs, which appear very similar to those in other black-hole binaries (e.g., ), and the above model can apply to it. Also, the black-hole binary GRO J1655–40 has an accurate determination of the orbital axis inclination of 68.65± 1.5 <cit.>, which is significantly different from the jet axis inclination of 85± 2 <cit.>. If the axes are different inas well, the alignment-based estimates would become inaccurate, and, e.g., the marked difference between the black-hole mass ofof M_1=12.4^+2.0_-1.8 <cit.> and the average for accreting low-mass black-hole binaries of 7.8± 1.2 <cit.>[We note here that the above average needs to be updated, e.g., taking into account the revision of the black-hole mass in Nova Muscae from 7.2.± 0.7 to 11.0^+2.1_-1.4 <cit.>.] may disappear.Given this uncertainty, we show the constraints on the radius and mass of the donor as functions of the inclination directly in Fig. <ref>, and of the distance (assuming the alignment) in Fig. <ref>.In the former, we show the range of the inclinations obtained by <cit.> assuming the alignment. We can also constrain the size and mass of the donor vs. the distance by using the observed NIR flux of the donor (as first done by ). The unveiled donor K magnitude (at λ=2.2μm) and the extinction towards the system have been estimated as 14.5–15.0 and 2.2± 0.3 by <cit.> and <cit.>, respectively. This gives the extinction-corrected magnitude of K ≃ 12.0–13.1, or the flux per unit wavelength at 2.2μm as (2.2–6.2)× 10^-8 erg s^-1 cm^-3. By approximating the stellar spectrum as a blackbody at the effective temperature, B_λ (T_ eff), i.e., F_λ= B_λ(T_ eff) (R_2/d)^2, we can then obtain the stellar radius as a function of the distance. The results for the temperatures within the range of T_ eff=3690–4660 K <cit.>, corresponding to the adopted range of the spectral classes of K0 III–M0 III, are shown by the dashed red curves in Fig. <ref>. These results are basically equivalent, and almost the same as those obtained using the Barnes-Evans relation <cit.>, as given for the K magnitude by equation (1a) of <cit.>, and for the range of the surface brightness of F_ K=3.81–3.86 (which approximately corresponds to K0–M0 giants, see fig. 2 of ). Combining the above constraints for the allowed range of the distance, we find 12.5≲ R_2/≲ 21 and 0.25≲ M_2/≲ 1. The upper limit is larger than that of <cit.> because they gave that from propagation of errors, while here we give it for the entire allowed range of d. At a given distance, the constraints are more stringent, as shown on Fig. <ref>. We see in Fig. <ref> that the constraints from the NIR flux agree with those from the rotational broadening (assuming the jet-binary alignment), but they impose only minor additional constraints. The current constraints could be improved with future more accurate measurements of the rotational broadening and the donor star's flux in the K band.§ THE MODELS OF V1487 AQL §.§ The core mass–radius planeIn order to calculate evolutionary models of stripped giants, we used the Warsaw stellar-evolution code (described in ). The code was calibrated to reproduce the Sun at the solar age. This calibration resulted in the chemical composition of the H mass fraction of X = 0.74, the metallicity of Z = 0.014, and the mixing length parameter of α = 1.55.To reproduce the present state of the donor, we followed the evolution of a 1 star, which was maintained at a constant mass until hydrogen was nearly exhausted in its centre. Then, the mass removal from the surface started and continued until the donor star reached six different values of the mass, 0.26, 0.28, 0.3, 0.5, 0.8 and 0.9. Until then, the He core had not formed. The further evolution was followed at a constant total mass. The H-burning shell moves outwards, increasing the mass of the He core, M_ c, and decreasing that of the H-rich envelope. Generally, this causes an increase of the radius of the stripped giant. However, when the mass of the remaining envelope gets sufficiently low, the giant starts to shrink, as shown in Fig. <ref>.The particular choice of 1 star for the starting configuration was dictated by the reason of the computational convenience.Our motivation was to obtain thermal-equilibrium stars of a given mass. Since the structure of such stars does not depend on their evolutionary history, the choice of the initial configuration or the prescription for mass removal from the surface was unimportant[<cit.> suggested that this initial configuration could involve a donor star as massive as 5. The large amount of the mass from the donor accreted by black hole would then help to explain its claimed large spin. However, <cit.> assumed fully conservative mass transfer throughout the system history, which included epochs of highly super-Eddington accretion, during which strong outflows most likely took place, questioning that assumption.]. To check that this is indeed the case, we made several experiments. First, during earlier calculations (described in ) we calculated tracks for a 0.2 stripped giant in two cases: one in which the mass removal from the initial 1 star started when it was near the end of the core H burning (but H was not yet completely exhausted), and second in which the mass removal started whenthe initial 1 star developed already a substantial (0.12) He core. We found that the structure and the further evolution of the 0.2 stars was identical in both cases (for the core mass range of 0.12–0.19). Second, we applied two different (arbitrary) rates of the mass removal, 1.3 × 10^-9 and 1.3 × 10^-8/y, to the initial 1 star. Again, the resulting stripped giants were insensitive to these details. Finally, we followed the evolution of 0.5 stripped giant in two cases: one in which the starting configuration was 1 star and another in which it was a 1.4 star. Again, both resulting tracks were identical. Additionally, we followed the evolution of 1.0 and 1.4 stars without any mass removal (un-stripped giants).The results of our calculations are presented in Fig. <ref>, which show the evolutionary tracks in the core mass–radius diagram. The tracks are shown for the stripped giants with the five lowest considered masses. The stars evolve at constant mass and the driving mechanism is the progress of the H-burning shell moving outwards. The radii of the partially stripped giants generally increase with M_ c, except for the shrinking when the masses of their envelopes become very low.Fig. <ref> also shows values of the Roche-lobe radius of the donor calculated with the formula (<ref>) for the considered masses. Possible solutions that we consider as models for the donor are given by the intersections between a horizontal line (for a given mass) and the corresponding evolutionary track.From Fig. <ref>, we immediately see that a stripped giant of the mass 0.26 cannot provide a solution since during its evolution it never attains a sufficiently large radius. We have checked that the same is true for the mass of 0.27. So, we are left with the remnants of the mass ≥0.28, which minimum value and its corresponding radius we show in Figs. <ref> and <ref>. In further discussion, we shall consider remnants of the masses ≳ 0.28. Magnified portions of the relevant tracks and horizontal lines from Fig. <ref>(a) are shown in Fig. <ref>(b). The possible solutions given by the intersections between the horizontal line (for a given mass) and the evolutionary track (for the same mass) are marked with consecutive capital letters, A, B, C and D for M_2=0.28, 0.3, 0.5 and 0.8, respectively, while the primed letters indicate the intersections during the final evolution stages. We point out that the solutions A' and B' are unphysical as they do not assure a continuous mass transfer between the components of the binary system since solutions lie on the declining parts of the evolutionary tracks. During this evolutionary phase, the star shrinks with the growing mass of the core. Therefore, any mass outflow would be quickly stopped. The parameters of the physical solutions are given in Table 1, including their effective temperature and luminosity. We note that the dependence of the stellar radius on the core mass can be considered for three different situations. First, we can consider the evolution of isolated giants at constant mass, as in Fig. <ref>. While this evolution does depend on the stellar mass, this dependence is relatively weak, and we can provide a fitting formula for the main dependence of R_2(M_ c) averaged over M_2, as also done by WRS83. Then, we can consider R_2(M_ c) during the mass transfer via Roche-lobe overflow, i.e., with a simultaneous decrease of the donor mass. This gives usually steeper dependencies. It corresponds to considering also the second term in equation (1) for Ṙ_2 of WRS83. Finally, we can consider R_2(M_ c) for our solutions, imposing the stellar radius equal to that of the Roche lobe of a given mass. This does not correspond to any evolutionary sequence, but just parameterises our results for P=33.85 d. This, in turn, gives a flatter dependence. We compare our radius vs. the core mass dependencies with the results of WRS83 (and of , who retained only the first order in their formulae). They correspond to the first case above, i.e., for evolution of an isolated star, and averaging over the dependence on M_2. Given that we use different chemical composition and more contemporary physics (especially opacities), our results, while qualitatively similar, are quantitatively different. While WRS83 obtained R_2∝ M_ c^5.1, we find on average for M_2≥ 0.5,R_2≃ 14 [M_ c/(0.25)]^4.3,for the evolution of a giant at a constant mass. We do see some dependence on M_2 in Fig. <ref>, especially for low masses (see also ). Also, this dependence excludes ranges of M_ c close to M_2, when the star starts to shrink. The second case above is discussed in Section <ref>. If we consider only the radii equal to the Roche-lobe radii at the period of(the third case), we find R_2∝ M_ c^3.2. §.§ The radius–luminosity planeSo far, we have made use only of the value of the radius of the donor as a function of its mass. We can also use a second parameter, its luminosity, L_2. We present our treatment below, which currently leads to relatively loose constraints, given that L_2 has been so far only roughly estimated. However, a future more accurate measurement will lead to a precise determination of the mass.The luminosity can be estimated from the range of the allowed effective temperatures, L_2=4 R_2^2 σ T_ eff^4, where σ is the Stefan-Boltzmann constant, and we use the values corresponding to giants of the spectral classes K0 III–M0 III, T_ eff=3690–4660 K <cit.>. This uncertainty leads to the corresponding uncertainty of the luminosity estimate. The range of L_2 at a given R_2 corresponding to the adopted range of T_ eff is shown in Fig. <ref>.Our evolutionary model predicts the luminosity at any stage, and we compare the values corresponding to our solutions, see Table 1, with the above constraints in Fig. <ref>. We have found that all our solutions lie in middle of the allowed range, indicating a good agreement of our evolutionary calculations with the standard parameters of giants of <cit.>, in spite of the reduction of the mass due to accretion mass loss[We do not show here the solution A', which also agrees with above luminosity constraints. It was this solution that <cit.> advocated as the best model. Unfortunately, it was not noted at that time that this model did not assure a continuous mass transfer since it corresponded to the radius decreasing with time, and hence was unphysical.]. All of the solutions have the temperatures of ≃ 3990–4160 K, which correspond to the spectral class of K5 III <cit.>. We thus find that the evolutionary considerations do not provide any significant upper limit on the mass and radius. For example, we have also calculated models with the final mass of 0.9 (solution E), 1.0 (solution F), and 1.4 (solution G), with the last one being well above the mass range allowed by the observations, and found that those solutions also lie in the middle of the range of L_2 allowed by the adopted range of T_ eff. We show their parameters in Table 1.One comment that should be made about our models concerns the luminosity of a star losing mass through Roche lobe overflow. If the mass outflow is very rapid, the surface luminosity of the star may become significantly lower than that without outflow (as noted by ). It is known that the black hole in the system is accreting matter roughly at the Eddington rate, ∼ 10^-7/yr, though the rate of the mass loss from the donor is much lower, see Section <ref>. However, we have found that even the Eddington rate is not high enough to decrease substantially the stellar surface luminosity. We superimposed a mass outflow at about Eddington rate on our model A, and found that the internal radiation flux in the outer layers started to decrease toward the surface, indicating that these layers were departing from thermal equilibrium. However, this departure was very small. The surface luminosity of the model with the outflow was smaller by less than 1 per cent compared with the model without outflow.The obtained values of the luminosity show a similar steep dependence (excluding M_2=0.28 and 0.3) on the core mass to that of WRS83 of L_2∝ M_ c^8.1, but our normalization is ∼50 per cent higher,L_2≃ 50 [M_ c/(0.25)]^8.15,with the relative error <3 per cent. Combining it with the dependence of R_2 assuming the Roche-lobe radius for the observed P (the third case at the end of Section <ref>), we find L_2∝ R_2^2.5, which is shown in Fig. <ref>.§ THE MASS OUTFLOW RATE FROM THE DONORHaving constructed the models describing the internal structure ofthe donor, we can calculate the rate of the mass transfer between the components of the binary system implied by our models. To do so, let us locate each of our models in a binary with P=33.85 d and a 12.4 black hole. We assume the conservative mode of the mass transfer (conservation of the total mass and of the total orbital angular momentum). Then, we calculate numerically at which rate of the mass outflow from the star the changes of the stellar radius will follow the changes of the Roche lobe around it. The resulting rates are given in Table 1 and shown in Fig. <ref>, where we see a strong dependence of Ṁ_2 on M_2. In addition to the Roche-lobe overflow, the donor loses mass via stellar wind, whose rate can be estimated using equation (106) of <cit.>. This gives -Ṁ_ 2,wind≃ (3.7–4.5)× 10^-10y^-1 for our considered cases. We see it is much less than the Roche-lobe rate in most cases, and comparable to it only for the lowest allowed mass. Given that only a small fraction of the wind is captured by the black hole, we neglect its effect hereafter.Our numerical results can be compared with the theoretical expression for Ṁ_2 separated into three physical terms,Ṁ_2 =( dR_2/ dM_2)^-1 dR_2/ dM_ cṀ_ c,where dR_2/ dM_2 is the rate of the Roche-lobe changes with the changing mass, given by equation (12) of WRS83, with R_2 M_2^-5/3 at M_2≪ M_1, and dR_2/ dM_ c is the derivative along the solution with the mass transfer, i.e., the second case discussed in Section <ref>. We find R_2∝ M_ c^c_1 along the solutions with the mass transfer, and c_1≃ 4.53–4.88 for M_2=0.5–1.4. The time derivative of the core mass is linked to the luminosity and the efficiency of hydrogen burning (accounting for neutrino losses), ϵ_ H≃ 6.0× 10^18 erg/g (as given by WRS83), -Ṁ_ c= L/(X' ϵ_ H). Here, X' is the H content immediately above the H-burning shell, which, due to the evolutionary H burning, can be substantially lower than the intial X=0.74 (which still approximately equal that at the stellar surface). In our cases with M_2=0.5–1.4, X'≃ 0.62–0.63. We find the values of Ṁ_2 obtained from equation (<ref>) using the obtained values of c_1 and X' to agree very well with our numerical results, shown in Table 1 and Fig. <ref>. We note that Ṁ_2→ 0 at the exact minimum possible mass, because then dR_2/ dM_ c→ 0. Thus, arbitrarily low values of Ṁ_2 can be obtained when the minimum allowed mass is approached, but this requires significant fine-tuning, see Fig. <ref>. We thus see that for the allowed range of M_2 of 0.28–1, -Ṁ_2≲ 2× 10^-8y^-1. We can compare our values with fitting expressions given in literature. The rate given by equation (15) of WRS83 in terms of M_2 and M_ c (and with a very weak dependence on M_1) is equivalent to our equation (<ref>), but we have now different numerical coefficients appearing in that formula. The formula (25a) in WRS83, giving Ṁ_2 in terms of M_2 and P (with the same weak dependence on M_1 as in their eq. 15) is shown by the dashed line in Fig. <ref>. We see it has a different slope from that shown by our values. We have fitted the values of Ṁ_2 for M_2≥ 0.5, and obtained-Ṁ_2≃ 7.4× 10^-10(P/1d)^0.93(M_2/)^1.9y^-1,which is shown in Fig. <ref> by the solid line. The dependence on P has been kept unchanged with respect to WRS83. The disagreement at the lowest masses is a boundary effect, caused by Ṁ_2→ 0 at the exact minimum possible mass (as discussed above). Our results supersede those of <cit.>, who used the fits of WRS83 with a previous estimate of M_2≃ (1.2± 0.2).We compare the obtained mass transfer rates with the average accretion rate required to power the outburst of . We used for that the results of <cit.>. From their fig. 5, we infer the average observed bolometric flux corresponding to ∼3/4 of the Eddington luminosity for M_1=14, pure H and at d=12.5 kpc assumed by them, which is 7.1× 10^-8 erg cm^-2 s^-1. This corresponds to L_1≃ 6.2(d/8.6kpc)^2× 10^38 erg s^-1, while <cit.> estimated L_1≃ 5.7(d/8.6kpc)^2× 10^38 erg s^-1, a very similar value. We thus adopt L_1= 6(d/8.6kpc)^2× 10^38 erg s^-1, which corresponds to the accretion rate of 1.1(d/8.6kpc)^2 (ϵ/0.1)^-1× 10^-7/y, where ϵ is the accretion efficiency. The accreted mass during the 25 y of the outburst is ≃ 2.7(d/8.6kpc)^2 (ϵ/0.1)^-1× 10^-6. See <cit.> for estimates of the disc mass in .At the observational best-fit donor mass value of M_2≃ 0.5 and at our corresponding value of Ṁ_2 (Table 1), the duty cycle (defined as the fraction of the time spent in outbursts) is thus about 5 per cent. At the maximum allowed mass of 1, it is 20 per cent, while it is much less at the lowest allowed donor masses. Our estimated range of the mass transfer rate is much below the critical rate at which the system would become persistent. <cit.> have obtained the critical rate as ≃ 4× 10^15 (P/1h)^1.59g s^-1, which gives -Ṁ_ crit≃ 3× 10^-6y^-1 for , i.e., much more than our estimated rates (and much more than even the accretion rate during the outburst). The transientness parameter, defined by <cit.> as Ṁ_2/Ṁ_ crit, is thus < 7× 10^-3. The comparison with other black-hole transients shown in their fig. 4 suggests then the average recurrence time of at least 30 y, with a large uncertainty, and consistent with that estimated from the duty cycle of ≳130 y.A potentially important effect we have neglected is the illumination of the donor by the X-ray source (pointed out by ). At the radius for M_2=0.5, the donor subtends a solid angle of 0.006× 4 (see equation <ref>), and the ratio of the illuminating luminosity (neglecting shielding and assuming isotropy) to L_2/2 (emitted by the hemisphere facing the black hole) is 33. At the estimated duty cycle, the average illuminating luminosity is larger than the intrinsic one, and illumination will slightly increase the stellar radius, which will in turn increase the mass transfer rate (e.g., ). However, a detailed treatment of donor illumination, including calculations of the albedo and the time dependence, is beyond the scope of this paper. We note only that such an effect appears to be clearly seen in another black-hole binary with a giant donor, GX 339–4, which has P≃ 1.76 d <cit.> and M_2≲ 1 <cit.>. Given its frequent outbursts, its average mass transfer rate required to power the outbursts can be relatively accurately estimated as ∼ 10^-8y^-1 <cit.>. On the other hand, the theoretical mass transfer rate based on the formalism of WRS83 is ≲ 10^-9y^-1 <cit.>, see also equation (<ref>), which discrepancy is likely to be accounted for by illumination.§ CONCLUSIONS We have studied constraints on the present mass and radius of the donor in /V1487 Aql by constructing evolutionary models of the internal structure of this star. We have compared the resulting radii and luminosity of our physical models to observational constraints. We find solutions with stripped giants of the mass of ≥0.28 and of the spectral class K5 III, independent of the distance to the system, and a distance-dependent upper limit, ≲1. We have also obtained the constraints from the observed NIR magnitude of the companion, which are found to be consistent with the other constraints. Overall, we have found very good agreement of our theoretical models with observational constraints.Then, we have estimated the mass transfer rate, which we found to be ≲ 2× 10^-8y^-1. Comparing to the average luminosity of , we obtain the duty cycle of ≲ 20 per cent. Thus, the expected average recurrence time is ≳ 130 y. The mass transfer rate and duty cycle become four times lower at M_2≃ 0.5, which is the observationally-determined best-fit mass. The obtained allowed values of the mass transfer rate are more than two orders of magnitude below the critical rate, at which the system would become persistent. The main quantities that would profit from more accurate measurements are the rotational broadening, which would allow a better constraint on the mass ratio, and the donor luminosity, which would allow a direct comparison with the predictions of our theoretical model. Both quantities can be more precisely measured aftertransits to quiescence. However, the time of this future event remains currently uncertain <cit.>.§ ACKNOWLEDGEMENTS We thank Joanna Mikołajewska and Tomaso Belloni for valuable discussions, and the referee for valuable suggestions. 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http://arxiv.org/abs/1702.07907v2

carlos.benavides-riveros@physik.uni-halle.de Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06120 Halle (Saale), Germany Theoretical andPhysical Chemistry Institute, National Hellenic Research Foundation, GR-11635 Athens, Greece Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06120 Halle (Saale), Germany The concept of correlation is central to all approaches that attempt the description of many-body effects in electronic systems. Multipartite correlation is a quantum information theoretical property that is attributed to quantum states independent of the underlying physics. In quantum chemistry, however, the correlation energy (the energy not seized by the Hartree-Fock ansatz) plays a more prominent role. We show that these two different viewpoints on electron correlation are closely related. The key ingredient turns out to be the energy gap within the symmetry-adapted subspace. We then use a few-site Hubbard model and the stretched H_2 to illustrate this connection and to show how the corresponding measures of correlation compare.31.15.V-, 31.15.xr, 31.70.-fRelating correlation measures: the importance of the energy gap Miguel A. L. Marques December 30, 2023 ===============================================================§ INTRODUCTION Since P.-O. Löwdin in the fifties, one usually defines correlation energy in quantum chemistry by the difference between the exact ground state (GS) energy of the system and its Hartree-Fock (HF) energy <cit.>:E_ corr = E_ GS - E_ HF.Since E_ HF is an upper bound on E_ GS the correlation energy is negative by definition. Beyond HF theory, numerous other methods (such as, e.g., configuration interaction or coupled-cluster theory) aim at reconstructing the part of the energy missing from a single-determinantal description. In fact, one common indicator of the accuracy of a method is the percentage of the correlation energy it is able to recover. Rigorous estimates of the error of the HF energy are already known for Coulomb systems with large atomic numbers <cit.>.In density-functional theory (DFT), nowadays the workhorse theory for both quantum chemistry and solid-state physics, the correlation energy has a slightly different definition. Instead of HF energy, one can use as reference the energy obtained by the (exchange only) optimized effective potential method <cit.> which is slightly higher than the HF energy. Clearly, the choice of the reference energy is arbitrary, as the correlation energy is not a physical observable. It remains, however, a very useful tool in understanding and quantifying the magnitude of many-body effects in given systems.In recent years a considerable effort has been devoted to characterize the correlation of a quantum system from a quantum-information theoretical viewpoint <cit.>. A priori, fermionic correlation is a property of the many-electron wave function. For the ground state |Ψ_ GS⟩, the total correlation can be quantified by the minimal (Hilbert-Schmidt) distance of |Ψ_ GS⟩⟨Ψ_ GS| to a single Slater determinant state <cit.> or just to the HF ground state |Ψ_ HF⟩⟨Ψ_ HF|,D(Ψ_ GS,Ψ_ HF)≡ 1/2 Tr[(|Ψ_ GS⟩⟨Ψ_ GS| -|Ψ_ HF⟩⟨Ψ_ HF|)^2]= 1-|⟨Ψ_ GS|Ψ_ HF⟩|^2 .This is closely related to theL^2-norm Ψ_ GS-Ψ_ HF^2, that however is not a good distance measure since it depends on the global phases of the respective states (which remains a problem even after restricting to real-valued wave functions). The distance (<ref>) is bounded between 0 and 1, reaching the upper value when the overlap between the two wave functions vanishes. Note that maximising this distance for fixed |Ψ_ GS⟩ over all single Slater determinants is not equivalent to the minimisation of the energy that leads to the Hartree-Fock orbitals. In fact, such procedure leads to the so-called Brueckner orbitals <cit.>, which are more “physical” than Hartree-Fock or Kohn-Sham orbitals, as they represent much better single-particle quantities <cit.>. We note in passing that in DFT it is less common to measure correlation from the overlap of the wave functions, as the Kohn-Sham Slater determinant describes a fictitious system and not a real one. Further correlation measures involving directly the N-fermion wave function are the Slater rank for two-electron systems <cit.>, the entanglement classification for the three-fermion case <cit.> or the comparison with uncorrelated states <cit.>.The nonclassical nature of quantum correlations and entanglement has enormous implications for quantum cryptography or quantum computation. Yet, quantifying correlations and entanglement for systems of identical particles is a part of an ongoing debate <cit.>. From a practical viewpoint, measuring correlation is even more challenging for identical particles since typically only one- and possibly two-particle properties are experimentally accessible. As a consequence, also simplified correlation measures involving reduced density operators were developed. These are, e.g., the squared Frobenius norm of the cumulant part of the two-particle reduced density matrix <cit.>, the entanglement spectrum and its gap<cit.>, the von-Neumann entropy S(ρ̂_1) = - [ρ̂_1 logρ̂_1] of the one-particle reduced density operator ρ̂_1 or just the l^1-distance δ(n⃗) of the decreasingly-ordered natural occupation numbers n⃗ (the eigenvalues of ρ̂_1) to the “Hartree-Fock”-point n⃗_ HF = (1_1,…,1_N,0_N+1,…) <cit.>.A first elementary relation between all those correlation measures and the concept of correlation energy is obvious: Each measure attains the minimal value zero whenever the exact ground state is given by a single Slater determinant <cit.>, i.e. the correlation energy vanishes. Furthermore, a monotonous relationship between the von-Neumann entropy of ρ̂_1 and the density functional definition of correlation energy has already been observed for some specific systems <cit.>.In this paper we establish a connection between those two viewpoints on electron correlation by providing a concise universal relation between the distance measure (<ref>) and the correlation energy E_ corr. Furthermore, due to the continuity of the partial trace similar relations between measures involving reduced density operators and E_ corr follow then immediately.The paper is arranged as follows. Section <ref> presents our main results, while section <ref> illustrates them for molecular systems.The last section provides a conclusion. Technical aspectsof our work are presented in the appendix.§ MAIN RESULTSOur starting point is the following theorem. Let Ĥ be a Hamiltonian on the Hilbert space $̋ with a unique ground state|Ψ_GS⟩and an energy gapE_gap = E_ES-E_GSto the first excited state. Then, for any|Ψ⟩ ∈$̋ with energy E = ⟨Ψ|Ĥ|Ψ⟩ we have|⟨Ψ_ GS|Ψ⟩|^2 ≥E_ ES - E/E_ gap. The significance of this theorem concerns the case of energy expectation values E = ⟨Ψ|Ĥ|Ψ⟩ within the energy gap [E_ GS,E_ ES], and relates the energy picture with the structure of the quantum state. A state |Ψ⟩ has a good overlap with the ground state whenever its energy expectation value E is close to the ground state energy, when measured relatively to the energy gap E_ gap.To prove this theorem we use the spectral decomposition of Ĥ = ∑_E E P̂_E, where P̂_E is the orthogonal projection operator onto the eigenspace of energy E. This yieldsE = ⟨Ψ|Ĥ|Ψ⟩ ≥ E_ GS⟨Ψ|P̂_ GS|Ψ⟩ +E_ ES∑_E≥ E_ ES⟨Ψ|P̂_E|Ψ⟩= E_ GS⟨Ψ|P̂_ GS|Ψ⟩ +E_ ES (1-⟨Ψ|P̂_ GS|Ψ⟩),where we used in the last line ∑_E≥ E_ ESP̂_E=1-P̂_ GS. By using ⟨Ψ|P̂_ GS|Ψ⟩=|⟨Ψ_ GS|Ψ⟩|^2 this leads to Eq. (<ref>) which completes the proof.From this result, we can deduce that the distance between the ground state of any Hamiltonian (with a unique ground state) and the corresponding HF ground state is bounded from above by a function depending on the energy gap of the system according toD(Ψ_ GS,Ψ_ HF) ≤|E_ corr|/E_ gap. In practice, the Hamiltonian at hand typically exhibits symmetries. For instance, the electronic Hamiltonian Ĥ of atoms and molecules commutes with the total spin. The ground state inherits this symmetry, i.e. it lies in an eigenspace ℋ_σ = π̂_σℋ of the symmetry operators, where π̂_σ denotes the restriction to that subspace with eigenvalue σ. Numerical methods are usually adapted to the ground state symmetry (if possible). A prime example is the restricted HF, a specific HF ansatz for approximating ground states with the correct spin symmetries. These considerations on symmetries allow for a significant improvement of estimate (<ref>): |Ψ_ GS⟩ and |Ψ_ HF⟩ are not only ground state and HF ground state of Ĥ, respectively, but also of the restricted HamiltonianĤ_σ = π̂_σĤπ̂_σ^†acting on the symmetry-adapted Hilbert space ℋ_σ. Application of the estimate (<ref>) to Ĥ_σ implies an improved upper bound: E_ gap no longer refers to the gap to the first excited state but to the first excited state within the symmetry-adapted space ℋ_σ of the ground state (and may therefore increase considerably). In the following, E_ ES will therefore stand for the energy of the first excited state with the same symmetries as the ground state.The estimate (<ref>) is our most significant result. It establishes a connection between both viewpoints on electron correlation and shows that the dimensionless quantity |E_ corr| / E_gap provides a universal upper bound on correlations described by the wave function. This result also underlines the importance of the energy gap being the natural reference energy scale. Furthermore, it is worth noting that estimate (<ref>) implies a similar estimate for the simplified correlation measure δ(n⃗) = _l^1(n⃗,n⃗_ HF)=∑_i=1^N(1-n_i)+∑_j > Nn_j, since (see Appendix <ref>):δ(n⃗)/2N≤ D(Ψ_ GS,Ψ_ HF).Before we continue a note of caution is in order here. One might be tempted to apply estimate (<ref>) to metals. However, since metals have a vanishing energy gap and also E_ ES = E_ GS < E_HF, i.e. |E_ corr| > E_ gap, our estimate has no relevance for them.To illustrate our results, in the next section we use simple, analytically solvable systems, namely the two- and three-site Hubbard model, which are well known for their capability of exhibiting both, weak and strong (static) correlation. We study also the stretching of H_2, which is considered a paradigm of the difficulties that single-determinant methods have with bond dissociation <cit.>.§ NUMERICAL INVESTIGATIONS §.§ Hubbard model Besides its importance for solid-state physics, the Hubbard model is one of the paradigmatic instances used to simplify the description of strongly correlated quantum many-body systems. The Hamiltonian (in second quantization) of the one-dimensional r-site Hubbard model reads:Ĥ = -t/2∑_i,σ (c^†_iσ c_(i+1)σ + h.c. ) + 2 U ∑_in̂_i↑n̂_i ↓,i ∈{1,2,…,r}, where c^†_iσ and c_iσ are the fermionic creation and annihilation operators for a particle on the site i with spin σ∈{↑, ↓} and n̂_iσ = c^†_iσ c_iσ is the particle-number operator. The first term in Eq. (<ref>) describes the hopping between two neighboring sites while the second represents the on-site interaction. Periodic boundary conditions in the case r > 2 are also assumed. Achieved experimentally very recently with full control over the quantum state <cit.>, this model may be considered as a simplified tight-binding description of the H_r molecule <cit.>.For two fermions on two sites, the eigenstates of Ĥ are described by four quantum numbers |E, s,m, p⟩, E being the energy, (s, m) the spin eigenvalues and p the eigenvalue of the operator swapping both sites. The dimension of the Hilbert space is 6, which splits in two parts according to the total spin: There are three triplet spin states with 0-energy, |0, 1, 1, -1⟩, |0, 1, -1, -1⟩ and |0, 1, 0, -1⟩, and three singlets, one of them |2U, 0, 0, -1⟩. The other two singlets |E_ GS, 0, 0, 1⟩ and |E_ ES, 0, 0, 1⟩ span the spin and translation symmetry-adapted Hilbert space ℋ_0,0,1. A straightforward computation yields for the ground state E_ GS = U - √(U^2 + t^2) and for the excited state E_ ES = U + √(U^2 + t^2). The restricted HF energy, E_ HF = -t + U, is a reasonable approximation to the exact ground state energy only for small values of U/t. The unphysical behaviour observed for larger values can be explained by the contribution of ionic states to the HF wave function <cit.>. The energy gap is given by 2 √(U^2 + t^2). Since the subspace of s=m=0 and p=+1 is two-dimensional and since the restricted HF ground state belongs to it as well, we have that the equality in (<ref>) holds: D(Ψ_ GS, Ψ_ HF)= |E_ corr|/E_ gap.For the ground state |Ψ_ GS⟩, the corresponding natural occupation numbers follow as n_±(U/t) = (1 ± 1/√(1 + U^2/t^2))/2, each one with multiplicity two. Note that by defining the dimensionless energy gap Δ = E_ gap/t we can express the occupation numbers as a function of Δ, leading to n_± (Δ) = 1/2 ± 1/Δ. This result shows that the one-particle correlation measures (von-Neumann entropy and δ-distance) also depend on the energy gap. In particular, the distance of the natural occupation numbers to the HF-point follows as δ(Δ) = 2 - 4/Δ which turns out to saturate the inequality (<ref>).To study the Hubbard model for more than two sites, we first recall that the Hamiltonian (<ref>) commutes with the total spin vector operator, its z-component and the translation operator (from the lattice site i to the next site i+1), with eigenvalues e^i2π p/r with p∈{0,1,…,r-1}. The Hamiltonian is block diagonal with respect to those symmetries (see Appendix <ref>). For the case of three fermions on three sites, the spectrum of the Hubbard model restricted to the subspace that corresponds to s = 1/2, m = 1/2 and p = 2 is given by <cit.>:E_j(U,t) = -2√(Q)cos(θ-2π j/3) + 4U/3,j ∈{0,1,2},where Q = 28 U^2/9 + 3 t^2/4 and cosθ = 8 U^3/(27Q^3/2). The dimensionless energy gap is Δ(U/t) = (E_ ES - E_ GS)/t = -2√(3Q)sin[(θ-π)/3]/t. For positive values of the dimensionless coupling U/t, Δ(U/t) =3/2 + 4(U/t)^2/9 + 𝒪((U/t)^3). For negative values, the energy gap is bounded from above: Δ(U/t) → 1.73205.In Fig. <ref> we plot several correlation measures as a function of U/t for this model. As expected, all curves increase monotonically with the strength of the interaction. For the positive region U/t ≥ 0, the curve for |E_ corr|/E_ gap follows very closely the one for D(Ψ_ GS,Ψ_ HF) confirming the significance of our estimate (<ref>). Both curves converge to the same value (2/3) for U/t →∞ . However, for negative values of U/t the estimate loses its significance. This is based on the fact that a significant part of the weight of the HF ground state lies on higher excited states. In addition, the energy gap is getting of the same order of magnitude as the correlation energy, leading to a rapid growth of our bound. In the strong correlation regime, beyond U/t < -3.375, |E_ corr| > E_ gap and our estimate has no significance. For positive values of U/t the energy gap increases monotonously. Note that the quantity |E_ corr|/E_ gap provides a much better estimate on the quantum state overlap (<ref>) than the von-Neumann entropy or the l^1-distance to the HF-point. The latter ones (the blue curves in Fig. <ref>) saturate very soon in contrast to the red and black ones. This shows the limitation of the one-particle picture to measure total fermion correlation. §.§The stretched H_2 As a second example we look at the archetypal instance of strong (static) correlation, i.e. the stretched dihydrogen H_2 <cit.>, which we analyze numerically using a cc-pVTZ basis set. In its dissociation limit, this system is commonly used as a benchmark to produce exchange-correlation functionals for strong static correlations <cit.>. The HF approach describes well the equilibrium chemical bond, but fails dramatically as the molecule is stretched. It is also known that DFT functionals describe the covalent bond well, but the predicted energy is overestimated in the dissociation limit due to delocalization, static-correlation and self-interaction errors <cit.>. Around the equilibrium separation (0.74 Å), electronic correlation is not particularly large and the HF state therefore approximates significantly well the ground-state wave function. The first excited state of H_2 with the same symmetry of the ground state (s = m = 0) is the second excited state. Around the equilibrium geometry, the energy gap diminishes as the interatomic distance is elongated. As for the two-site Hubbard model, close to equilibrium, the bound |E_ corr|/E_ gap provides a good estimate on the correlation measure D(Ψ_ GS,Ψ_ HF). Remarkably, as shown in Fig. <ref>, beyond the equilibrium bound, where the static correlation effects can be observed, |E_ corr|/E_ gap reproduces the behaviour of the distance D(Ψ_ GS,Ψ_ HF). The same holds for the δ-distance, which is largely due to the fact that for two-fermion models the value of the first occupation number is approximately the square of the projection of the ground state onto the HF configuration. In contrast, the von-Neumann entropy saturates very soon.§ CONCLUSION In conclusion, we have connected both viewpoints on fermion correlation by providing the universal estimate (<ref>). It connects the measure of total fermion correlation (as property that can be attributed to quantum states independent of the underlying physics) and the correlation energy (commonly used in quantum chemistry). The quantity that connects both measures is the energy gap of the corresponding block Hamiltonian with the same symmetry as the ground state. Moreover, due to the continuity of the partial trace, similar estimates follow for several correlation measures resorting to reduced-particle information only. Yet, as it can be inferred from their early saturation shown in Fig. <ref>, the significance of such simplified correlation measures is limited.Since the quantity |E_ corr|/E_ gap provides an estimate on the overlap between the HF and the exact ground state wave function our work may allow one to use the sophisticated concept of multipartite entanglement developed and explored in quantum information theory for a more systematic study of strongly correlated systems. In particular, our work suggests an additional tool for describing the possible failure of DFT in reconstructing specific properties of a given quantum system. This failure can be either attributed to a rather poor reconstruction of the systems ground state energy or to the failure of the effectivemethod (e.g. Kohn-Sham)in reconstructing many-particle properties from one-particle information. The latter case would be reflected by poor saturationof the inequality (<ref>) while the first one corresponds toa large correlation energy (requiring a multi-reference methodinstead <cit.>).§ ACKNOWLEDGEMENTSWe thank D.Gross and M.Springborg for helpful discussions. C.L.B.R. thanks the Clarendon Laboratory at the University of Oxford for the warm hospitality. We acknowledge financial support from the Hellenic Ministry of Education (through ESPA) and from the GSRT through “Advanced Materials and Devices” program (MIS:5002409) (N.N.L.), the Oxford Martin Programme on Bio-Inspired Quantum Technologies, the UK Engineering, Physical Sciences Research Council (Grant EP/P007155/1) (C.S.) and the DFG through projects SFB-762 and MA 6787/1-1 (M.A.L.M.).§ PROOF OF ESTIMATE (<REF>)We consider an N-fermion Hilbert space ℋ_N^(f) where the underlying one-particle Hilbert space ℋ_1^(d) has dimension d ∈ℕ∪{∞}. For |Ψ⟩∈ℋ_N^(f) we can determine its one-particle reduced density operator ρ̂_1 (trace-normalized to N) and the vectorλ⃗ = (λ_i)_i=1^d of decreasingly-ordered eigenvalues of ρ̂_1 (natural occupation numbers). Let {|χ_j⟩}_j=1^d be a Brueckner orthonormal basis for ℋ_1^(d), i.e. the specific Slater determinant |χ⟩=|χ_1,…,χ_N⟩ maximizes the overlap with |Ψ⟩. Furthermore we introduce δ(x⃗)= ∑_i=1^N (1-x_i)+∑_j=N+1^d x_j and n̂_i as the particle number operator for state |χ_i⟩. Obviously, δ(λ⃗) is the l^1-distance of λ⃗ to the “Hartree-Fock”-point (1,…,1,0,…). In the following we prove the estimateδ(λ⃗)/2 (N,d-N)≤ 1-|⟨χ|Ψ⟩|^2 .For this, we introduce the particle number expectation values n_i = ⟨Ψ|n̂_i|Ψ⟩ and δ̂= δ((n̂_i)_i=1^d). Since the spectrum of δ̂ is given by {0,1,…,2M} with M=(N,d-N) we findδ(n⃗)= ⟨Ψ|δ̂|Ψ⟩ =∑_d=1^2M dP̂_d Ψ_L^2^2 ≤ 2M∑_d=1^2MP̂_d Ψ_L^2^2 = 2M (1-P̂_0 Ψ_L^2^2) ,where we have used the spectral decomposition δ̂=⊕_d=0^2M dP̂_d of δ̂. By using P̂_0=|χ⟩⟨χ| and the fact that the vector λ⃗ of decreasingly-ordered eigenvalues of ρ̂_1 majorizes any other vector of occupation numbers (particularly n⃗) we obtainδ(λ⃗) ≤δ(n⃗) ≤ 2M (1-|⟨χ|Ψ⟩|^2) .Since |χ⟩ maximizes the overlap with |Ψ⟩, we eventually find for the Hartree-Fock ground state |Ψ_ HF⟩ (or any other single Slater determinant)δ(λ⃗)/2M≤ 1-|⟨Ψ_ HF|Ψ⟩|^2 ,i.e. estimate (<ref>). § ANALYTIC SOLUTION OF THE HUBBARD MODEL FOR THREE ELECTRONS ON THREE SITESIn this section we recall the analytical solution of the three-site Hubbard model for three electrons which was already presented in Ref. <cit.>. For the Hubbard model, the one-body reduced density matrix is diagonal in the basis of the Bloch orbitals, which satisfy T̂_1 |q⟩ = e^iφ q|q⟩, where φ = 2π/r, T̂_1 is the 1-particle translation operator and T̂ = ⊗^N_i=1T̂_1. The creation operators in the Bloch basis set read: c̃^†_qσ = 1/√(r)∑_k=1^r e^iφ q k c^†_kσ, q∈{0,1,…,r-1}. To block diagonalize this Hamiltonian one can employ the natural-orbital basis set generated by {|q⟩} and then split the total Hilbert space with respect to the spin quantum numbers s and m. For example, the only state with maximal magnetic number m = r/2 is c̃^†_0↑c̃^†_1↑⋯c̃^†_r-1↑| vac⟩, which spans the one-dimensional subspace _̋r/2, r/2,η (η = 0 for r odd or η = r/2 otherwise), as defined by the direct sum of the total Hilbert space:=̋⊕^N/2_s=s_-⊕^s_m=-s⊕^r-1_p=0_̋s,m,p.For the case of three fermions on three sites, the total spin quantum number s can take only two values 32 and 12. For s=12, thanks to the fact that the Hamiltonian is invariant under simultaneous flipping of all spins, the results for m= -12 are identical to the case m= 12. The latter is related to the eight dimensional Hilbert space _̋1/2,1/2,2⊕_̋1/2,1/2,1⊕_̋1/2,1/2,0, where_̋1/2,1/2,2 =span{|0↑0↓2↑⟩, |2↑2↓1↑⟩,|1↑1↓0↑⟩}, _̋1/2,1/2,1 =span{|0↑0↓1↑⟩,|2↑2↓0↑⟩,|1↑1↓2↑⟩}.The translation invariant subspace _̋1/2,1/2,0 is two dimensional and can be built with two spin-compensated linear combinations of the following three configurations: |0↑1↑2↓⟩, |0↑1↓2↑⟩ and |0↓1↑2↑⟩. As an elementary exercise one verifies that the Hamiltonian restricted to each one of the subspaces _̋1/2,1/2,2 and _̋1/2,1/2,1 leads to the same 3× 3 matrix. Indeed,Ĥ|__̋1/2,1/2,1 = Ĥ|__̋1/2,1/2,2.It is worth noting that the same configuration appears in the description of the spin-compensated Lithium isoelectronic series <cit.>. Moreover, since the diagonalization of any of the Hamiltonians (<ref>) can be performedanalytically, an expression for the energy spectrum can be exactly known<cit.>:E_j = -2√(Q)cos(θ-2π j/3) + 4U/3,for j = 0, 1, 2. Here Q = 28 U^2/9 + 3 t^2/4 and cosθ = 8 U^3/(27Q^3/2). See Fig. <ref>.The energy gap E_ gap is then given by E_1 - E_0.

http://arxiv.org/abs/1702.08422v2

http://arxiv.org/abs/1702.08348v2

§ INTRODUCTION The motivation to study neutral kaon mixing comes from the role it plays in indirect CP violation observed in kaon decays. The degree of indirect CP violation is a well known, experimentally measured quantity <cit.>, which can be expressed as a simple product of the kaon mixing bag parameter, Wilson coefficients and a function of CKM matrix elements and other known quantities. Thus precise lattice calculations of kaon mixing help to constrain the CKM matrix and beyond the standard model (BSM) theories (where new CP violating phases can can appear.) In the standard model (SM), kaon mixing is mediated by the W boson and the leading order contributions are given by the box diagrams. An operator product expansion (OPE) separates the short and long distance contributions. Theshort distance part is given by the matrix element ⟨K̅^0|O_1|K^0|$⟩, whereO_1is aΔS = 2effective four-quark operator, given in equation <ref>, and can be found non-perturbatively using lattice simulations.Beyond the standard model, where the mediating particle is not restricted to the W boson, a range of operators become available as more dirac and colour structures are possible. A basis of 5ΔS =2effective BSM operators can be defined <cit.>, O_1=[s̅_a γ_μ (1-γ_5) d_a][s̅_b γ_μ (1-γ_5) d_b]O_2=[s̅_a (1-γ_5) d_a][s̅_b (1-γ_5) d_b]O_3 =[s̅_a (1-γ_5) d_b][s̅_b (1-γ_5) d_a]O_4=[s̅_a (1-γ_5) d_a][s̅_b (1+γ_5) d_b] O_5 =[s̅_a (1-γ_5) d_b][s̅_b (1+γ_5) d_a]where a and b are colour indices, and dirac indices are not shown. This basis is referred to as the SUSY basis, but is model independent. By calculating the matrix elements of these operators we can place boundaries on BSM theories. § SIMULATIONSThe simulations are of domain wall fermion QCD with Iwasaki gauge fields.We present new results from two ensemble sets; the48^3and64^3physical point, details of which can be found in ref. <cit.>. Alongside the physical point ensembles, in our analysis we include results from the ensemble sets24^3and32^3in our analysis. Details of our32^3can be found in ref. <cit.>, however24^3has been recalculated with a physical strange mass. A summary of simulation parameters for the ensembles included is shown in table <ref>. We have also performed calculations on a finer lattice, referred to as48^3fine, but the NPR calculation is still ongoing and so we have not included those results here, as they cannot yet be included in our analysis. This will be included in a later journal publication writing up this work, and hence our present results should be viewed as preliminary.§ MEASUREMENTS The Bag Parameter is defined as the matrix elements normalised by their VSA values, expressed as,0.5 B_1(μ) = ⟨K̅|O_1(μ)|K|⟩/8/3m_K^2 f_K^2 0.5 B_i(μ) = ⟨K̅|O_i(μ)|K|⟩/N_i m_K^2 f_K^2 (m_K/m_u(μ)+m_s(μ))^2for the SM and BSM case respectively, whereN_i=(−5/3, 1/3,2, 2/3)fori≥2.We also measure an alternative quantity, the ratio of the BSM operators to the SM operator, which we define as <cit.>,R_i(μ) =[ f_K^2/m_K^2]_expt[ m_K^2/f_K^2⟨K̅|O_i(μ)|K|⟩/⟨K̅|O_1(μ)|K|⟩]_Lat.At the continuum physical point,R_i(μ)is a simple ratio of the BSM over the SM matrix element. § GLOBAL FIT The physical value ofR_i(μ)andB_i(μ)are recovered by performing a simultaneous chiral and continuum extrapolation. We first must renormalise the bare results. This was done non-perturbatively with non-exceptional kinematics (RI-SMOM) <cit.> for24^3and32^3in ref. <cit.>. In these preliminary results, we have reused the renormalisation constants calculated for the24(32)^3ensembles to renormalise48(64)^3. Next we adjust the valence strange mass to the physical point. Here there are multiple strange masses, we interpolate linearly. If there is only one strange mass for an ensemble, we extrapolate, taking the gradient from an ensemble set with the same lattice spacing and multiple strange masses.We ignore the size of the sea strange mass effects as they are expected to be negligible compared to the final error. These were measeured in <cit.> Thirdly, the simultaneous fit is preformed, as in ref. <cit.>, according to one of two methods:*A fit to a linear ansatz:Y(0,m_π^2/(4π f_π)^2) = Y(a^2,m^2_ll/(4π f_π))[1+c_a a^2 + c_m m_ll^2/(4π f_ll)^2].* A fit according to NLO SU(2) chiral perturbation theory:Y(0,m_π^2/(4π f_π)^2) = Y(a^2,m^2_ll/(4π f_ll))[1+c_a a^2 + m_ll^2/(4π f_ll)^2 ( c_m + C_i log(m_ll^2/Λ^2)) ].ais the lattice spacing inGeV^-1,m_llis the pseudoscalar light-light meson mass.c_aandc_mare free parameters. The fixed coefficientsC_iare given in <cit.>. We then extrapolate to the physical point.§ RESULTS We present here results for the ratios and bag parameters.In table <ref> the bare unrenormalised values gained from the correlator fits are shown. We then present the preliminary continuum, physical point results gained from our fitting procedure in table <ref>.[It should be noted that all the errors presented on the new results here, are statistical errors only, and we are yet to calculate the systematic errors. We also have not shown here the errors arising from the matching to ms but these will be included in the future.] Plots of the fits are shown in figures <ref> and <ref>. We compare these new results to those previously gained by our collaboration. In <cit.> a result for the SM bag parameter including the physical point data was found,B_k^ms(3GeV) = 0.5293 ±0.0017 ±0.0106 .This is consistent with the new result, even though we did not use overweighting of the physical point data and did not include reweighting for the sea strange mass. This agreement validates our statement that we can ignore these effects for the less well determined BSM matrix elements. These new results are (althoughpreliminary, and still to be improved upon) compatible with our collaborations previous BSM results <cit.> (shown in table <ref>) which were obtained with the same fitting approach.§ CONCLUSIONS We present preliminary results for the BSM bag parameters and ratios including physical pion mass data which appear to be consistent with the previous results in ref. <cit.>. We have dramatically reduced the chiral error on our results by including lattices with physical pions and kaons in our fits. We will include data from an ensemble with fine lattice spacing in our analysis for the final results. § ACKNOWLEDGEMENTS The ensemble results were computed on the STFC funded DiRAC BG/Q system in the Advanced Computing Facility at the University of Edinburgh. N.G. is supported by Leverhulme Trust, grant RPG-2014-118. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement 279757. We are grateful to A. Khamseh for her contribution to calculating NPR for the fine ensemble which will be included in the final analysis. 99 Agashe:2014kda K. A. Oliveet al. [Particle Data Group Collaboration],Review of Particle Physics Chin. Phys. C38 (2014) 090001. Gabbiani:1996hiF. Gabbiani, E. Gabrielli, A. Masiero and L. Silvestrini,A Complete analysis of FCNC and CP constraints in general SUSY extensions of the standard model Nucl. Phys. B477, 321 (1996)[hep-ph/9604387]. 1411.7017T. Blumet al. Domain wall QCD with physical quark masses Phys.Rev. D93 (2016) no.7, 074505 [hep-lat/1411.7017]Aoki:2010dyY. Aokiet al. [RBC and UKQCD Collaborations],Continuum Limit Physics from 2+1 Flavor Domain Wall QCD, Phys. Rev. D83, 074508 (2011)[hep-lat/1011.0892].Babich:2006bhR. Babich, N. Garron, C. Hoelbling, J. Howard, L. Lellouch and C. Rebbi,K0 - anti-0 mixing beyond the standard model and CP-violating electroweak penguins in quenched QCD with exact chiral symmetry Phys. Rev. D74, 073009 (2006)[hep-lat/0605016]. Sturm:2009kbC. Sturm, Y. Aoki, N. H. Christ, T. Izubuchi, C. T. C. Sachrajda and A. Soni,Renormalization of quark bilinear operators in a momentum-subtraction scheme with a nonexceptional subtraction point Phys. Rev. D80, 014501 (2009) [hep-ph/0901.2599].Garron:2016mva N. Garron, R. J. Hudspith and A. T. Lytle,Neutral Kaon Mixing Beyond the Standard Model with n_f=2+1 Chiral Fermions Part 1: Bare Matrix Elements and Physical Results [hep-lat/1609.03334]. Becirevic:2004qdD. Becirevic and G. Villadoro,Remarks on the hadronic matrix elements relevant to the SUSY K0 - anti-K0 mixing amplitude Phys. Rev. D70, 094036 (2004)[hep-lat/0408029].

http://arxiv.org/abs/1703.00392v1

The Long-Moody construction and polynomial functors Arthur Souli Accepted ???. Received ???; in original form December 30, 2023 ================================================================== In 1994, Long and Moody gave a construction on representations of braid groups which associates a representation of 𝐁_n with a representation of 𝐁_n+1. In this paper, we prove that this construction is functorial and can be extended: it inspires endofunctors, called Long-Moody functors, between the category of functors from Quillen's bracket construction associated with the braid groupoid to a module category. Then we study the effect of Long-Moody functors on strong polynomial functors: we prove that they increase by one the degree of very strong polynomiality.Published in Annales de l'Institut Fourier, Volume 69 (2019) no. 4 p. 1799-1856. This work was partially supported by the ANR Project ChroK, ANR-16-CE40-0003. 2010 Mathematics Subject Classification: 18D10, 18A25, 20C07, 20C99, 20J99, 20F36, 20F38, 57M07, 57N05. Keywords: braid groups, functor categories, Long-Moody construction, polynomial functors.§ INTRODUCTION Linear representations of the Artin braid group on n strands 𝐁_n is a rich subject which appears in diverse contexts in mathematics (see for example <cit.> or <cit.> for an overview). Even if braid groups are of wild representation type, any new result which allows us to gain a better understanding of them is a useful contribution.In 1994, as a result of a collaboration with Moody in <cit.>, Long gave a method to construct from a linear representation ρ:𝐁_n+1→ GL(V) a new linear representation ℒℳ(ρ):𝐁_n→ GL(V^⊕ n) of 𝐁_n (see <cit.>). Moreover, the construction complicates in a sense the initial representation. For example, applying it to a one dimensional representation of 𝐁_n+1, the construction gives a mild variant of the unreduced Burau representation of 𝐁_n. This method was in fact already implicitly present in two previous articles of Long dated 1989 (see <cit.>). In the article <cit.> dating from 2008, Bigelow and Tian consider the Long-Moody construction from a matricial point of view. They give alternative and purely algebraic proofs of some results of <cit.>, and they slightly extend some of them. In a survey on braid groups (see the Open Problem 7 in <cit.>), Birman and Brendle underline the fact that the Long-Moody construction should be studied in greater detail.Our work focuses on the study of the Long-Moody construction ℒℳ from a functorial point of view. More precisely, we consider the category 𝔘β associated with braid groups. This category is an example of a general construction due to Quillen (see <cit.>) on the braid groupoid β. In particular, the category 𝔘β has natural numbers ℕ as objects. For each natural number n, the automorphism group Aut_𝔘β(n) is the braid group 𝐁_n. Let 𝕂-𝔐𝔬𝔡 be the category of 𝕂-modules, with 𝕂 a commutative ring, and 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) be the category of the functors from 𝔘β to 𝕂-𝔐𝔬𝔡. An object M of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) gives by evaluation a family of representations of braid groups { M_n:𝐁_n→ GL(M(n))} _n∈ℕ, which satisfies some compatibility properties (see Section <ref>). Randal-Williams and Wahl use the category 𝔘β in <cit.> to construct a general framework to prove homological stability for braid groups with twisted coefficients. Namely, they obtain the stability for twisted coefficients given by objects of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).In Proposition <ref>, we prove that a version of the Long-Moody construction is functorial. We fix two families of morphisms { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ and {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ, satisfying some coherence properties (see Section <ref>). Once this framework set, we show: [Theorem A (Proposition <ref>) ] There is a functor 𝐋𝐌_a,ς:𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)→𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡), called the Long-Moody functor with respect to coherent families of morphisms { a_n} _n∈ℕ and {ς_n} _n∈ℕ, which satisfies for σ∈𝐁_n and M∈ Obj(𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡))𝐋𝐌_a,ς(M)(σ)=ℒℳ(M_n)(σ). Among the objects in the category 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) the strong polynomial functors play a key role. This notion extends the classical one of polynomial functors, which were first defined by Eilenberg and Mac Lane in <cit.> for functors on module categories, using cross effects. This definition can also be applied to monoidal categories where the monoidal unit is a null object. Djament and Vespa introduce in <cit.> the definition of strong polynomial functors for symmetric monoidal categories with the monoidal unit being an initial object. Here, the category 𝔘β is neither symmetric, nor braided, but pre-braided in the sense of <cit.>. However, we show that the notion of strong polynomial functor extends to the wider context of pre-braided monoidal categories (see Definition <ref>). We also introduce the notion of very strong polynomial functor (see Definition <ref>). Strong polynomial functors turn out inter alia to be very useful for homological stability problems. For example, in <cit.>, Randal-Williams and Wahl prove their homological stability results for twisted coefficients given by a specific kind of strong polynomial functors, namely coefficient systems of finite degree (see <cit.>).We investigate the effects of Long-Moody functors on very strong polynomial functors. We establish the following theorem, under some mild additional conditions (introduced in Section <ref>) on the families of morphisms { a_n} _n∈ℕ and {ς_n} _n∈ℕ, which are then said to be reliable. [Theorem B (Corollary <ref>) ] Let M be a very strong polynomial functor of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) of degree n and let { a_n} _n∈ℕ and {ς_n} _n∈ℕ be coherent reliable families of morphisms. Then, considering the Long-Moody functor 𝐋𝐌_a,ς with respect to the morphisms { a_n} _n∈ℕ and {ς_n} _n∈ℕ, 𝐋𝐌_a,ς(M) is a very strong polynomial functor of degree n+1.Thus, iterating the Long-Moody functor on a very strong polynomial functor of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) of degree d, we generate polynomial functors of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡), of any degree bigger than d. For instance, Randal-Williams and Wahl define in <cit.> a functor 𝔅𝔲𝔯_t:𝔘β→ℂ[t^±1]-𝔐𝔬𝔡 encoding the unreduced Burau representations. Similarly, we introduce a functor 𝔗𝔜𝔐_t:𝔘β→ℂ[t^±1]-𝔐𝔬𝔡 corresponding to the representations considered by Tong, Yang and Ma in <cit.>. These functors 𝔅𝔲𝔯_t and 𝔗𝔜𝔐_t are very strong polynomial of degree one (see Proposition <ref>), and moreover, we prove that the functor 𝔅𝔲𝔯_t is equivalent to a functor obtained by applying the Long-Moody construction. Thus, the Long-Moody functors will provide new examples of twisted coefficients corresponding to the framework of <cit.>.This construction is extended in the forthcoming work <cit.> for other families of groups, such as automorphism groups of free groups, braid groups of surfaces, mapping class groups of orientable and non-orientable surfaces or mapping class groups of 3-manifolds. The results proved here for (very) strong polynomial functors will also hold in the adapted categorical framework for these different families of groups.The paper is organized as follows. Following <cit.>, Section <ref> introduces the category 𝔘β and gives first examples of objects of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡). Then, in Section <ref>, we introduce the Long-Moody functors, prove Theorem A and give some of their properties. In Section <ref>, we review the notion of strong polynomial functors and extend the framework of <cit.> to pre-braided monoidal categories. Finally, Section <ref> is devoted to the proof of Theorem B and to some other properties of these functors. In particular, we tackle the Open Problem 7 of <cit.>.We will consider a commutative ring 𝕂 throughout this work. We denote by 𝕂-𝔐𝔬𝔡 the category of 𝕂-modules. We denote by 𝔊𝔯 the category of groups. We take the convention that the set of natural numbers ℕ is the set of nonnegative integers { 0,1,2,…}.Let ℭ𝔞𝔱 denote the category of small categories. Let ℭ be an object of ℭ𝔞𝔱. We use the abbreviation Obj(ℭ) to denote the objects of ℭ. For 𝔇 a category, we denote by 𝐅𝐜𝐭(ℭ,𝔇) the category of functors from ℭ to 𝔇. If 0 is initial object in the category ℭ, then we denote by ι_A:0→ A the unique morphism from 0 to A. The maximal subgroupoid 𝒢𝔯(ℭ) is the subcategory of ℭ which has the same objects as ℭ and of which the morphisms are the isomorphisms of ℭ. We denote by 𝒢𝔯:ℭ𝔞𝔱→ℭ𝔞𝔱 the functor which associates to a category its maximal subgroupoid.The author wishes to thank most sincerely his PhD advisor Christine Vespa, and Geoffrey Powell, for their careful reading, corrections, valuable help and expert advice. He would also especially like to thank Aurlien Djament, Nariya Kawazumi, Martin Palmer, Vladimir Verchinine and Nathalie Wahl for the attention they have paid to his work, their comments, suggestions and helpful discussions. Additionally, he would like to thank the anonymous referee for his reading of this paper.§ THE CATEGORY 𝔘Β The aim of this section is to describe the category 𝔘β associated with braid groups that is central to this paper. On the one hand, we recall some notions and properties about Quillen's construction from a monoidal groupoid and pre-braided monoidal categories introduced by Randal-Williams and Wahl in <cit.>. On the other hand, we introduce examples of functors over the category 𝔘β.We recall that the braid group on n≥2 strands denoted by 𝐁_n is the group generated by σ_1, ..., σ_n-1 satisfying the relations: * ∀ i∈{ 1,…,n-2}, σ_iσ_i+1σ_i=σ_i+1σ_iσ_i+1;* ∀ i,j∈{ 1,…,n-1} such that | i-j|≥2, σ_iσ_j=σ_jσ_i.𝐁_0 and 𝐁_1 both are the trivial group. The family of braid groups is associated with the following groupoid.The braid groupoid β is the groupoid with objects the natural numbers n∈ℕ and morphisms (for n,m∈ℕ):Hom_β(n,m)=𝐁_n if∅ if . The composition of morphisms ∘ in the groupoid β corresponds to the group operation of the braid groups. So we will abuse the notation throughout this work, identifying σ∘σ'=σσ' for all elements σ and σ' of 𝐁_n with n∈ℕ (with the convention that we read from the right to the left for the group operation). §.§ Quillen's bracket construction associated with the groupoid β This section focuses on the presentation and the study of Quillen's bracket construction 𝔘β (see <cit.>) on the braid groupoid β. It associates to β a monoidal category whose unit is initial. The category 𝔘β has further properties: Quillen's bracket construction on β is a pre-braided monoidal category (see Section <ref>) and β is its maximal subgroupoid. For an introduction to (braided) strict monoidal categories, we refer to <cit.>.A strict monoidal category will be denoted by (ℭ,♮,0), where ℭ is the category, ♮ is the monoidal product and 0 is the monoidal unit.§.§.§ Generalities In <cit.>, Randal-Williams and Wahl study a construction due to Quillen in <cit.>, for a monoidal category S acting on a category X in the case S=X=𝔊 where 𝔊 is a groupoid. It is called Quillen's bracket construction. Our study here is based on <cit.> taking 𝔊=β.<cit.> A monoidal product ♮:β×β⟶β is defined by the usual addition for the objects and laying two braids side by side for the morphisms. The object 0 is the unit of this monoidal product. The strict monoidal groupoid (β,♮,0) is braided, its braiding is denoted by b_-,-^β. Namely, the braiding is defined for all natural numbers n and m such that n+m≥2 by:b_n,m^β=(σ_m∘⋯∘σ_2∘σ_1)∘⋯∘(σ_n+m-2∘⋯∘σ_n∘σ_n-1)∘(σ_n+m-1∘⋯∘σ_n+1∘σ_n)where {σ_i} _i∈{ 1,…,n+m-1} denote the Artin generators of the braid group 𝐁_n+m. We consider the strict monoidal groupoid (β,♮,0) throughout this section.<cit.> Quillen's bracket construction on the groupoid β, denoted by 𝔘β, is the category defined by:* Objects: Obj(𝔘β)=Obj(β)=ℕ;* Morphisms: for n and n' two objects of β, the morphisms from n to n' in the category 𝔘β are given by:Hom_𝔘β(n,n')=βcolim[Hom_β(-♮ n,n')].In other words, a morphism from n to n' in the category 𝔘β, denoted by [n'-n,f]:n→ n', is an equivalence class of pairs (n'-n,f) where n'-n is an object of β, f:(n'-n)♮ n→ n' is a morphism of β, in other words an element of 𝐁_n'. The equivalence relation ∼ is defined by (n'-n,f)∼(n'-n,f') if and only if there exists an automorphism g∈ Aut_β(n'-n) such that the following diagram commutes.(n'-n)♮ n[d]_g♮ id_n[r]^fn' (n'-n)♮ n[ur]_f' * For all objects n of 𝔘β, the identity morphism in the category 𝔘β is given by [0,id_n]:n→ n.* Let [n'-n,f]:n→ n' and [n”-n',g]:n'→ n” be two morphisms in the category 𝔘β. Then, the composition in the category 𝔘β is defined by:[n”-n',g]∘[n'-n,f]=[n”-n,g∘(id_n'-n♮ f)]. The relationship between the automorphisms of the groupoid β and those of its associated Quillen's construction 𝔘β is actually clear. First, let us recall the following notion.Let (𝔊,♮,0) be a strict monoidal category. It has no zero divisors if for all objects A and B of 𝔊, A♮ B≅0 if and only if A≅ B≅0. The braid groupoid (β,♮,0) has no zero divisors. Moreover, by Definition <ref>, Aut_β(0)={ id_0}. Hence, we deduce the following property from <cit.>.The groupoid β is the maximal subgroupoid of 𝔘β. In addition, 𝔘β has the additional useful property.<cit.> The unit 0 of the monoidal structure of the groupoid (β,♮,0) is an initial object in the category 𝔘β.Let n be a natural number and ϕ∈ Aut_β(n). Then, as an element of Hom_𝔘β(n,n), we will abuse the notation ϕ=[0,ϕ]. This comes from the canonical functor:β → 𝔘β ϕ∈ Aut_β(n)↦ [0,ϕ].Finally, we are interested in a way to extend an object of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡) to an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡). This amounts to studying the image of the restriction 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)→𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡).Let M be an object of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡). Assume that for all n,n',n”∈ℕ such that n”≥ n'≥ n, there exists an assignment M([n'-n,id_n']):M(n)→ M(n') such that:M([n”-n',id_n”])∘ M([n'-n,id_n'])=M([n”-n,id_n”])Then, we define a functor M:𝔘β→𝕂-𝔐𝔬𝔡 (assigning M([n'-n,σ])=M(σ)∘ M([n'-n,id_n']) for all [n'-n,σ]∈ Hom_𝔘β(n,n')) if and only if for all n,n'∈ℕ such that n'≥ n:M([n'-n,id_n'])∘ M(σ)=M(ψ♮σ)∘ M([n'-n,id_n'])for all σ∈𝐁_n and all ψ∈𝐁_n'-n.Note that for n'=n, M([n'-n,id_n'])=Id_M(n). Let us assume that relation (<ref>) is satisfied. We have to show that the assignment on morphisms is well-defined with respect to 𝔘β. First, let us prove that our assignment conforms with the defining equivalence relation of 𝔘β (see Definition <ref>). For n and n' natural numbers such that n'≥ n, let us consider [n'-n,σ] and [n'-n,σ'] in Hom_𝔘β(n,n') such that there exists ψ∈𝐁_n'-n so that σ'∘(ψ♮ id_n)=σ. Since M is a functor over β, M([n'-n,σ])=M(σ')∘(M(ψ♮ id_n)∘ M([n'-n,id_n'])). According to the relation (<ref>) and since M satisfies the identity axiom, we deduce that M([n'-n,σ])=M(σ')∘ M(ψ♮ id_n)∘ M([n'-n,id_n'])=M([n'-n,σ']).Now, we have to check the composition axiom. Let n, n' and n” be natural numbers such that n”≥ n'≥ n, let ([n'-n,σ]) and ([n”-n',σ']) be morphisms respectively in Hom_𝔘β(n,n') and in Hom_𝔘β(n',n”). By our assignment and composition in 𝔘β (see Definition <ref>) we have that:M([n”-n',σ'])∘ M([n'-n,σ])=M(σ')∘(M([n”-n',id_n”])∘ M(σ))∘ M([n'-n,id_n']).According to the relation (<ref>), we deduce that:M([n”-n',σ'])∘ M([n'-n,σ]) = M(σ')∘(M([n”-n',id_n”])∘ M(σ))∘ M([n'-n,id_n']). = M(σ')∘(M(id_n”-n'♮σ)∘ M([n”-n',id_n”]))∘ M([n'-n,id_n']).Hence, it follows from relation (<ref>) that:M([n”-n',σ'])∘ M([n'-n,σ])=M(σ'∘(id_n”-n'♮σ))∘ M([n”-n,id_n])=M([n”-n',σ']∘[n'-n,σ]). Conversely, assume that the functor M:𝔘β→𝕂-𝔐𝔬𝔡 is well-defined. In particular, composition axiom in 𝔘β is satisfied and implies that for all n,n'∈ℕ such that n'≥ n, for all σ∈𝐁_n:M([n'-n,id_n'])∘ M(σ)=M([n'-n,id_n'-n♮σ]).It follows from the defining equivalence relation of 𝔘β (see Definition (<ref>)) that for all ψ∈𝐁_n'-n:M([n'-n,id_n'])∘ M(σ)=M([n'-n,ψ♮σ]).We deduce from the composition axiom that relation (<ref>) is satisfied. Let M and M' be objects of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) and η:M→ M' a natural transformation in the category 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡). Then, η is a natural transformation in the category 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) if and only if for all n,n'∈ℕ such that n'≥ n:η_n'∘ M([n'-n,id_n'])=M'([n'-n,id_n'])∘η_n. The natural transformation η extends to the category 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) if and only if for all n,n'∈ℕ such that n'≥ n, for all [n'-n,σ]∈ Hom_𝔘β(n,n'):M'([n'-n,σ])∘η_n=η_n'∘ M([n'-n,σ]).Since η is a natural transformation in the category 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡), we already have η_n'∘ M(σ)=M'(σ)∘η_n'. Hence, this implies that the necessary and sufficient relation to satisfy is relation (<ref>). §.§.§ Pre-braided monoidal category We present the notion of a pre-braided category, introduced by Randal-Williams and Wahl in <cit.>. This is a generalization of that of a braided monoidal category.<cit.> Let (ℭ,♮,0) be a strict monoidal category such that the unit 0 is initial. We say that the monoidal category (ℭ,♮,0) is pre-braided if:* The maximal subgroupoid 𝒢𝔯(ℭ,♮,0) is a braided monoidal category, where the monoidal structure is induced by that of (ℭ,♮,0).* For all objects A and B of ℭ, the braiding associated with the maximal subgroupoid b_A,B^ℭ:A♮ B⟶ B♮ A satisfies:b_A,B^ℭ∘(id_A♮ι_B)=ι_B♮ id_A:A⟶ B♮ A.Recall that the notation ι_B:0→ B was introduced in Notation <ref>.Since the groupoid (β,♮,0) is braided monoidal and it has no zero divisors, we deduce from <cit.> the following properties.The category 𝔘β is pre-braided monoidal. The monoidal structure (𝔘β,♮,0) is defined on objects as that of (β,♮,0) and defined on morphisms letting for [n'-n,f]∈ Hom_𝔘β(n,n') and [m'-m,g]∈ Hom_𝔘β(m,m'):[m'-m,g]♮[n'-n,f]=[(m'-m)♮(n'-n),(g♮ f)∘(id_m'-m♮(b_m,n'-n^β)^-1♮ id_n)].In particular, the canonical functor β→𝔘β is monoidal.The category (𝔘β,♮,0) is pre-braided monoidal, but not braided. Indeed, as Figure 1 shows, the pre-braiding defined on 𝔘β is not a braiding: Figure 1 shows that b_1,2^β∘(ι_1♮ id_2)≠ id_2♮ι_1 whereas these two morphisms should be equal if b_-,-^β were a braiding. §.§ Examples of functors associated with braid representations Different families of representations of braid groups can be used to form functors over the pre-braided category 𝔘β to the category 𝕂-𝔐𝔬𝔡. Namely, considering { M_n:𝐁_n→𝕂-𝔐𝔬𝔡} _n∈ℕ representations of braid groups, or equivalently an object M of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡), we are interested in the situations where Proposition <ref> applies so as to define an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡). Tong-Yang-Ma resultsIn 1996, in the article <cit.>, Tong, Yang and Ma investigated the representations of 𝐁_n where the i-th generator is sent to a matrix of the form Id_i-1⊕ T⊕ Id_n-i-1, with T a m× m non-singular matrix and m≥2. In particular, for m=2, they prove that there exist up to equivalence only two non trivial representations of this type. We give here their result and an interpretation of their work from a functorial point of view, considering the representations over the ring of Laurent polynomials in one variable ℂ[t^±1].Let 𝔤𝔯 denote the full subcategory of 𝔊𝔯 of finitely generated free groups. The free product *:𝔤𝔯×𝔤𝔯→𝔤𝔯 defines a monoidal structure over 𝔤𝔯, with 0 the unit, denoted by (𝔤𝔯,*,0).Let (ℕ,≤) denote the category of natural numbers (natural means non-negative) considered as a poset. For all natural numbers n, we denote by γ_n the unique element of Hom_(ℕ,≤)(n,n+1). For all natural numbers n and n' such that n'≥ n, we denote by γ_n,n':n→ n' the unique element of Hom_(ℕ,≤)(n,n'), composition of the morphisms γ_n'-1∘γ_n'-2∘⋯∘γ_n+1∘γ_n. The addition defines a strict monoidal structure on (ℕ,≤), denoted by ((ℕ,≤),+,0).Let 𝐁_-:(ℕ,≤)→𝔊𝔯 and GL_-:(ℕ,≤)→𝔊𝔯 be the functors defined by:* Objects: for all natural numbers n, 𝐁_-(n)=𝐁_n the braid group on n strands and GL_-(n)=GL_n(ℂ[t^±1]) the general linear group of degree n.* Morphisms: let n be a natural number. We define 𝐁_-(γ_n)=id_1♮-:𝐁_n↪𝐁_n+1 (where ♮ is the monoidal product introduced in Example <ref>). We define GL_-(γ_n):GL_n(ℂ[t^±1])↪ GL_n+1(ℂ[t^±1]) assigning GL_-(γ_n)(φ)=id_1⊕φ for all φ∈ GL_n(ℂ[t^±1]).For all natural numbers n≥2, for all i∈{ 1,…,n-1}, we denote by incl_i^n:𝐁_2≅ℤ↪𝐁_n the inclusion morphism induced by:incl_i^n(σ_1)=σ_i. <cit.> Let η:𝐁_-⟶ GL_- be a natural transformation. Assume that for all natural numbers n≥2, for all i∈{ 1,…,n-1}, the following diagram is commutative:𝐁_n@->[rr]^η_n GL_n(ℂ[t^±1]) 𝐁_2@->[rr]_η_2@->[u]^incl_i^n GL_2(ℂ[t^±1]).@->[u]_id_i-1⊕-⊕ id_n-i-1Here, two such natural transformations η and η' are said to be equivalent if there exists a natural equivalence μ:GL_-⟶ GL_- such that μ∘η=η' or if η'=η^* where -^* denotes the dual representation. Then, η is equivalent to one of the following natural transformations.* The trivial natural transformation, denoted by 𝔦𝔡: for every generator σ_i of 𝐁_n, 𝔦𝔡_n(σ_i)=Id_GL_n(ℂ[t^±1]).* The unreduced Burau natural transformation, denoted by 𝔟𝔲𝔯: for all generators σ_i of 𝐁_n,𝔟𝔲𝔯_n,t(σ_i)=Id_i-1⊕ B(t)⊕ Id_n-i-1,with B(t)=[[ 0 t; 1 1-t ]].* The natural transformation denoted by 𝔱𝔶𝔪: for every generator σ_i of 𝐁_n if n≥2,𝔱𝔶𝔪_n,t(σ_i)=Id_i-1⊕ TYM(t)⊕ Id_n-i-1,with TYM(t)=[[ 0 t; 1 0 ]]. We call it the Tong-Yang-Ma representation.The unreduced Burau representation (see <cit.> or <cit.> for more details about this family of representations) is reducible but indecomposable, whereas the Tong-Yang-Ma representation is irreducible (see <cit.>). We can also consider a natural transformation using the family of reduced Burau representations (see <cit.> for more details about the associated family of representations): these are irreducible subrepresentations of the unreduced Burau representations.Let :(ℕ,≤)→𝔊𝔯 be the functor defined by:* Objects: for all natural numbers n, (n)=GL_n-1(ℂ[t^±1]) the general linear group of degree n-1.* Morphisms: let n be a natural number. We define (γ_n):GL_n-1(ℂ[t^±1])↪ GL_n(ℂ[t^±1]) assigning GL_-(γ_n)(φ)=id_1⊕φ for all φ∈ GL_n-1(ℂ[t^±1]). The reduced Burau natural transformation, denoted by 𝔟𝔲𝔯:𝐁_-→ is defined by:* For n=2, one assigns 𝔟𝔲𝔯(σ_1) to be the automorphism of ℂ[t^±1] defined by the multiplication by -t.* For all natural numbers n≥3, we define for every Artin generator σ_i of 𝐁_n with i∈{ 2,…,n-2}:𝔟𝔲𝔯_n,t(σ_i)=Id_i-2⊕B(t)⊕ Id_n-i-2withB(t)=[[100;1 -tt;001 ]]and𝔟𝔲𝔯_n,t(σ_1)=[[ -tt;01 ]]⊕ Id_n-3;𝔟𝔲𝔯_n,t(σ_n-1)=Id_n-3⊕[[10;1 -t ]]. Let us use these natural transformations to form functors over the category 𝔘β. Indeed, a natural transformation η:𝐁_-→ GL_- (or ) provides in particular:* a functor 𝔑:β⟶ℂ[t^±1]-𝔐𝔬𝔡;* morphisms 𝔑([n'-n,id_n']):𝔑(n)→𝔑(n') for all natural numbers n'≥ n, such that the relation (<ref>) of Proposition <ref> is satisfied.Therefore, according to Proposition <ref>, it suffices to show that the relation (<ref>) is satisfied to prove that 𝔑 is an object of 𝐅𝐜𝐭(𝔘β,ℂ[t^±1]-𝔐𝔬𝔡).Recall that 0 is a null object in the category of R-modules, and that the notation ι_G:0→ G was introduced in Notation <ref>. Let n∈ℕ. For all natural numbers n and n' such that n'≥ n, we define ι_ℂ[t^±1]^⊕ n'-n⊕ id_ℂ[t^±1]^⊕ n:ℂ[t^±1]^⊕ n↪ℂ[t^±1]^⊕ n' the embedding of ℂ[t^±1]^⊕ n as the submodule of ℂ[t^±1]^⊕ n' given by the n last copies of ℂ[t^±1]. Tong-Yang-Ma functor:This example is based on the family introduced by Tong, Yang and Ma (see Theorem <ref>). Let 𝔗𝔜𝔐_t:β→ℂ[t^±1]-𝔐𝔬𝔡 be the functor defined on objects by 𝔗𝔜𝔐_t(n)=ℂ[t^±1]^⊕ n for all natural numbers n, and for all numbers n≥2, for every Artin generator σ_i of 𝐁_n, by 𝔗𝔜𝔐_t(σ_i)=𝔱𝔶𝔪_n,t(σ_i) for morphisms. For all natural numbers n and n' such that n'≥ n, we assign 𝔗𝔜𝔐_t([n'-n,id_n']):ℂ[t^±1]^⊕ n↪ℂ[t^±1]^⊕ n' to be the embedding ι_ℂ[t^±1]^⊕ n'-n⊕ id_ℂ[t^±1]^⊕ n (where these morphisms are introduced in Notation <ref>). For all natural numbers n”≥ n'≥ n, for all Artin generators σ_i∈𝐁_n and all ψ_j∈𝐁_n'-n, our assignments give:𝔗𝔜𝔐_t(ψ♮σ)∘𝔗𝔜𝔐_t([n'-n,id_n']) =(Id_j-1⊕ TYM(t)⊕ Id_(n'-n)-j-1⊕ Id_n'-n+i-1⊕ TYM(t)⊕ Id_n'-i-1) ∘(ι_ℂ[t^±1]^⊕ n'-n⊕ id_ℂ[t^±1]^⊕ n).Remark that (Id_j-1⊕ TYM(t)⊕ Id_(n'-n)-j-1)∘ι_ℂ[t^±1]^⊕(n'-n)=ι_ℂ[t^±1]^⊕(n'-n). Hence we deduce that𝔗𝔜𝔐_t(ψ♮σ)∘𝔗𝔜𝔐_t([n'-n,id_n'])=𝔗𝔜𝔐_t([n'-n,id_n'])∘𝔗𝔜𝔐_t(σ)for all σ∈𝐁_n and all ψ∈𝐁_n'-n. According to Proposition <ref>, our assignment defines a functor 𝔗𝔜𝔐_t:𝔘β→ℂ[t^±1]-𝔐𝔬𝔡, called the Tong-Yang-Ma functor. Burau functors:Other examples naturally arise from the Burau representations.Let 𝔅𝔲𝔯_t:β⟶ℂ[t^±1]-𝔐𝔬𝔡 be the functor defined on objects by 𝔅𝔲𝔯_t(n)=ℂ[t^±1]^⊕ n for all natural numbers n, and for all numbers n≥2, for every Artin generator σ_i of 𝐁_n, by 𝔅𝔲𝔯_t(σ_i)=𝔟𝔲𝔯_n,t(σ_i) for morphisms. For all natural numbers n and n' such that n'≥ n, we assign 𝔅𝔲𝔯_t([n'-n,id_n']):ℂ[t^±1]^⊕ n↪ℂ[t^±1]^⊕ n' to be the embedding ι_ℂ[t^±1]^⊕ n'-n⊕ id_ℂ[t^±1]^⊕ n (where these morphisms are introduced in Notation <ref>). As for the functor 𝔗𝔜𝔐, the assignment for 𝔅𝔲𝔯 implies that for all natural numbers n”≥ n'≥ n, for all σ∈𝐁_n and all ψ∈𝐁_n'-n, 𝔅𝔲𝔯_t([n'-n,id_n'])∘𝔅𝔲𝔯_t(σ)=𝔅𝔲𝔯_t(ψ♮σ)∘𝔅𝔲𝔯_t([n'-n,id_n']). According to Proposition <ref>, our assignment defines a functor 𝔅𝔲𝔯_t:𝔘β⟶ℂ[t^±1]-𝔐𝔬𝔡, called the unreduced Burau functor. The dual version of the functor 𝔅𝔲𝔯_t was already considered by Randal-Williams and Wahl in <cit.>.Analogously, we can form a functor from the reduced Burau representations. Let 𝔅𝔲𝔯_t:β⟶ℂ[t^±1]-𝔐𝔬𝔡 be the functor defined on objects by 𝔅𝔲𝔯_t(0)=0 and 𝔅𝔲𝔯_t(n)=ℂ[t^±1]^⊕ n-1 for all nonzero natural numbers n, and by 𝔅𝔲𝔯_t(σ_i)=𝔟𝔲𝔯_n,t(σ_i) for morphisms for every Artin generator σ_i of 𝐁_n for all numbers n≥2.For all natural numbers n and n' such that n'≥ n, we assign 𝔅𝔲𝔯_t([n'-n,id_n']):ℂ[t^±1]^⊕ n-1↪ℂ[t^±1]^⊕ n'-1 to be the embedding ι_ℂ[t^±1]^⊕ n'-n⊕ id_ℂ[t^±1]^⊕ n-1 (where these morphisms are introduced in Notation <ref>). Repeating mutadis mutandis the work done for the functor 𝔗𝔜𝔐, the assignment for 𝔅𝔲𝔯_t implies that for all natural numbers n”≥ n'≥ n, for all σ∈𝐁_n and all ψ∈𝐁_n'-n, 𝔅𝔲𝔯_t([n'-n,id_n'])∘𝔅𝔲𝔯_t(σ)=𝔅𝔲𝔯_t(ψ♮σ)∘𝔅𝔲𝔯_t([n'-n,id_n']). According to Proposition <ref>, our assignment defines a functor 𝔅𝔲𝔯_t:𝔘β⟶ℂ[t^±1]-𝔐𝔬𝔡, called the reduced Burau functor. Lawrence-Krammer functor:The family of Lawrence-Krammer representations was notably used to prove that braid groups are linear (see <cit.>). For this paragraph, we assign 𝕂=ℂ[t^±1][q^±1] the ring of Laurent polynomials in two variables and consider the functor GL_- of Definition <ref> with this assignment. Let 𝔏𝔎:𝔘β→ℂ[t^±1][q^±1]-𝔐𝔬𝔡 be the assignment:* Objects: for all natural numbers n≥2, 𝔏𝔎(n)=1≤ j<k≤ n⊕V_j,k, with for all 1≤ j<k≤ n, V_j,k is a free ℂ[t^±1][q^±1]-module of rank one. Hence, 𝔏𝔎(n)≅(ℂ[t^±1][q^±1])^⊕ n(n-1)/2 as ℂ[t^±1][q^±1]-modules. Moreover, one assigns 𝔏𝔎(1)=0 and 𝔏𝔎(0)=0.* Morphisms:* Automorphisms: for all natural numbers n, for every Artin generator σ_i of 𝐁_n (with i∈{ 1,…,n-1}), for all v_j,k∈ V_j,k (with 1≤ j<k≤ n),𝔏𝔎(σ_i)v_j,k= v_j,k if ,tv_i,k+(t^2-t)v_i,i+1+(1-t)v_i+1,k if ,v_i+1,k if ,tv_j,i+(1-t)v_j,i+1-(t^2-t)qv_i,i+1 if ,v_j,i+1 if ,-qt^2v_i,i+1 if . * General morphisms: let n,n'∈ℕ, such that n'≥ n. For all natural numbers j and k such that 1≤ j<k≤ n, we define the embedding 𝔙_j,k^n,n':V_j,k∼⟶V_j+(n'-n),k+(n'-n)↪1≤ j<k≤ n'⊕V_j,k of free ℂ[t^±1][q^±1]-modules. Then we define 𝔏𝔎([n'-n,id_n']):1≤ j<k≤ n⊕V_j,k→1≤ j<k≤ n'⊕V_j,k to be the embedding 1≤ j<k≤ n⊕𝔙_j,k^n,n'. Since we consider a family of representations of 𝐁_n (see <cit.>), the assignment 𝔏𝔎 defines an object of 𝐅𝐜𝐭(β,ℂ[t^±1]-𝔐𝔬𝔡).Let n, n' and n” be natural numbers such that n”≥ n'≥ n. It follows directly from our definitions of 𝔏𝔎([n'-n,id_n']), 𝔏𝔎([n”-n',id_n”]) and 𝔏𝔎([n”-n,id_n”]) that relation (<ref>) of Proposition <ref> is satisfied.According to the definition of 𝔏𝔎(σ_l) with σ_l an Artin generator of 𝐁_n'-n, for all v_j,k∈ V_j,k with 1+(n'-n)≤ j<k≤ n', 𝔏𝔎(σ_l)v_j,k=v_j,k. Hence for all ψ∈𝐁_n'-n:𝔏𝔎(ψ♮ id_n)∘𝔏𝔎([n'-n,id_n'])=𝔏𝔎([n'-n,id_n']).Note also that for all l∈{ 1,…,n-1}, for all v_j,k∈ V_j,k with 1+(n'-n)≤ j<k≤ n', it follows from the assignment of 𝔏𝔎 that:𝔏𝔎(id_n'-n♮σ_l)(v_(n'-n)+j,(n'-n)+k)=𝔏𝔎(σ_n'-n+l)(v_(n'-n)+j,(n'-n)+k)=𝔏𝔎([n'-n,id_n'])(𝔏𝔎(σ_l)(v_j,k)).Therefore, this implies that for all σ∈𝐁_n, 𝔏𝔎([n'-n,id_n'])∘𝔏𝔎(σ)=𝔏𝔎(id_n'-n♮σ)∘𝔏𝔎([n'-n,id_n']). Hence, 𝔏𝔎 satisfies the relation (<ref>) of Proposition <ref>. Hence, the assignment defines a functor 𝔏𝔎:𝔘β→ℂ[t^±1][q^±1]-𝔐𝔬𝔡, called the Lawrence-Krammer functor.§ FUNCTORIALITY OF THE LONG-MOODY CONSTRUCTION The principle of the Long-Moody construction, corresponding to Theorem 2.1 of <cit.>, is to build a linear representation of the braid group 𝐁_n starting from a representation 𝐁_n+1. We develop a functorial version of this construction, which leads to the notion of Long-Moody functors (see Section <ref>). Beforehand, we need to introduce various tools, which are consequences of the relationships between braid groups and free groups (see Section <ref>). Finally, in Section <ref>, we investigate examples of functors which are recovered by Long-Moody functors. §.§ Braid groups and free groups This section recalls some relationships between braid groups and free groups. We also develop tools which will be used throughout our work of Sections <ref> and <ref>.We consider the free group on n generators, which we denote by 𝐅_n=⟨ g_1,…,g_n⟩.We denote by e_𝐅_n the unit element of the free group on n generators 𝐅_n, for all natural numbers n. Recall that the category of finitely generated free groups is monoidal using free product of groups (see Notation <ref>). The object 0 being null in the category 𝔤𝔯, recall that ι_𝐅_n:0→𝐅_n denotes the unique morphism from 0 to 𝐅_n as in Notation <ref>.Let n be a natural number. We consider ι_𝐅_1*id_𝐅_n:𝐅_n↪𝐅_n+1. This corresponds to the identification of 𝐅_n as the subgroup of 𝐅_n+1 generated by the n last copies of 𝐅_1 in 𝐅_n+1. Iterating this morphism, we obtain for all natural numbers n'≥ n the morphism ι_𝐅_n'-n*id_𝐅_n:𝐅_n↪𝐅_n'. Let {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ be a family of group morphisms from the free group 𝐅_n to the braid group 𝐁_n+1, for all natural numbers n. We require these morphisms to satisfy the following crucial property.For all elements g∈𝐅_n, for all natural numbers n'≥ n, the following diagram is commutative in the category 𝔘β: 1♮ n@->[rrr]^ς_n(g)@->[d]_id_1♮[n'-n,id_n']1♮ n@->[d]^id_1♮[n'-n,id_n']1♮ n'@->[rrr]_ς_n'(e_𝐅_n'-n*g)1♮ n'. Condition <ref> will be used to prove that the Long-Moody functor is well defined on morphisms with respect to the tensor product structure in Theorem <ref>. Moreover, it will also be used in the proof of Propositions <ref> and <ref>. Condition <ref> is equivalent to assume that for all natural numbers n, for all elements g∈𝐅_n, the morphisms {ς_n} _n∈ℕ satisfy the following equality in 𝐁_n+2:((b_1,1^β)^-1♮ id_n)∘(id_1♮ς_n(g))=ς_n+1(e_𝐅_1*g)∘((b_1,1^β)^-1♮ id_n). Let n and n' be natural numbers such that n'≥ n. The equality (<ref>) implies that for all 1≤ k≤ n'-n, the following diagram in the category β is commutative :1♮ n'@->[rrrr]^id_n'-(n+k)♮ς_n+k-1(e_𝐅_k-1*g)@->[d]_id_n'-(n+k)♮(b_1,1^β)^-1♮ id_(k-1)+n 1♮ n'@->[d]^id_n'-(n+k)♮(b_1,1^β)^-1♮ id_(k-1)+n1♮ n'@->[rrrr]_id_n'-(n+k)♮ς_n+k(e_𝐅_k*g) 1♮ n'.Hence composing squares, we obtain that the following diagram is commutative in the category β: 1♮⋯♮(1♮1)♮ n@->[rrr]^id_n'-n-1♮(b_1,1^β)^-1♮ id_n@->[d]_id_n'♮ς_n(g)1♮⋯♮1♮(1♮ n)@->[d]_id_n'-1♮ς_n+1(e_𝐅_1*g)@->[rrr]^ id_n'-n-2♮(b_1,1^β)^-1♮ id_1+n ⋯@->[rr]^(b_1,1^β)^-1♮ id_n'-1 1♮ n'@->[d]^ς_n'(e_𝐅_1*g)1♮⋯♮1♮ n@->[rrr]_id_n'-n-1♮(b_1,1^β)^-1♮ id_n1♮⋯♮1♮(1♮ n)@->[rrr]_ id_n'-n-2♮(b_1,1^β)^-1♮ id_1+n ⋯@->[rr]_(b_1,1^β)^-1♮ id_n'-1 1♮ n'.By definition of the braiding (see Definition <ref>), we deduce that the composition of horizontal arrows is the morphism (b_1,n'-n^β)^-1♮ id_n in β. Recall from Proposition <ref> that id_1♮[n'-n,σ]=[n'-n,(id_1♮σ)∘((b_1,n'-n^β)^-1♮ id_n)]. Hence Condition <ref> is satisfied if we assume that the equality (<ref>) is satisfied for all natural numbers n.Conversely, assume that Condition <ref> is satisfied. Condition <ref> with n'=n+1 ensures that:[1,((b_1,1^β)^-1♮ id_n)∘(id_1♮ς_n(g))]=[1,ς_n'(e_𝐅_1*g)∘((b_1,1^β)^-1♮ id_n)].Since Aut_𝔘β(1)=𝐁_1 is the trivial group, we deduce from the defining equivalence relation of 𝔘β (see Definition <ref>) the equality in 𝐁_n+2:((b_1,1^β)^-1♮ id_n)∘(id_1♮ς_n(g))=ς_1+n(e_𝐅_1*g)∘((b_1,1^β)^-1♮ id_n).It follows from Lemma <ref> that, for i≥2, ς_n(g_i) is determined by ς_k(g_1) for k≤ n by the equalities (<ref>).The family ς_n,1, based on what is called the pure braid local system in the literature (see <cit.>), is defined by the following inductive assignment for all natural numbers n≥1.ς_n,1:𝐅_n ⟶ 𝐁_n+1g_i ⟼ σ_1^2 ifσ_1^-1∘σ_2^-1∘⋯∘σ_i-1^-1∘σ_i^2∘σ_i-1∘⋯∘σ_2∘σ_1 ifWe assign ς_0,1 to be the trivial morphism.The family of morphisms {ς_n,1} _n∈ℕ satisfies Condition <ref>.Relation (<ref>) is trivially satisfied for n=0. Let n≥1 be a fixed natural number. By definition <ref>, we have (b_1,1^β)^-1=σ_1^-1. Moreover, for all i∈{ 2,…,n}, we haveς_n+1(e_𝐅_1*g_i-1)=ς_n+1(g_i) andid_1♮ς_n,1(g_i-1)=σ_2^-1∘⋯∘σ_i-1^-1∘σ_i^2∘σ_i-1∘⋯∘σ_2.We deduce that:((b_1,1^β)^-1♮ id_n)∘(id_1♮ς_n,1(g_i-1))∘(b_1,1^β♮ id_n)=ς_n,1(g_i).Hence Relation (<ref>) of Lemma <ref> is satisfied for all natural numbers.Let us consider the trivial morphisms ς_n,*:𝐅_n→0_𝔊𝔯→𝐁_n+1 for all natural numbers n. The relation of Lemma <ref> being easily checked, this family of morphisms {ς_n,*:𝐅_n→𝐁_n+1} _n∈ℕ satisfies Condition <ref>. Action of braid groups on automorphism groups of free groups:There are several ways to consider the group 𝐁_n as a subgroup of Aut(𝐅_n). For instance, the geometric point of view of topology gives us an action of 𝐁_n on the free group 𝐅_n (see for example <cit.> or <cit.>) identifying 𝐁_n as the mapping class group of a n-punctured disc Σ_0,1^n: fixing a point y on the boundary of the disc Σ_0,1^n, each free generator g_i can be taken as a loop of the disc based y turning around punctures. Each element σ of 𝐁_n, as an automorphism up to isotopy of the disc Σ_0,1^n, induces a well-defined action on the fundamental group π_1(Σ_0,1^n)≅𝐅_n called Artin representation (see Example <ref> for more details). In the sequel, we fix a family of group actions of 𝐁_n on the free group 𝐅_n: let { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ be a family of group morphisms from the braid group 𝐁_n to the automorphism group Aut(𝐅_n). For the work of Sections <ref> and <ref>, we need the morphisms a_n:𝐁_n→ Aut(𝐅_n) to satisfy more properties.Let n and n' be natural numbers such that n'≥ n. We require (ι_𝐅_n'-n*id_𝐅_n)∘(a_n(σ))=(a_n'(σ'♮σ))∘(ι_𝐅_n'-n*id_𝐅_n) as morphisms 𝐅_n→𝐅_n' for all elements σ of 𝐁_n and σ' of 𝐁_n'-n, ie the following diagrams are commutative: 𝐅_n@->[rr]^a_n(σ)@->[d]_ι_𝐅_n'-n*id_𝐅_n𝐅_n@->[d]^ι_𝐅_n'-n*id_𝐅_n𝐅_n@->[rr]^ι_𝐅_n'-n*id_𝐅_n@->[dr]_ι_𝐅_n'-n*id_𝐅_n𝐅_n' 𝐅_n'@->[rr]_a_n'(id_n'-n♮σ)𝐅_n' 𝐅_n'.@->[ur]_a_n'(σ'♮ id_n) Condition <ref> will be used to define the Long-Moody functor on morphisms in Theorem <ref>. Moreover, it will also be used for the proof of Propositions <ref> and <ref>. We will also require the families of morphisms {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ to satisfy the following compatibility relations.Let n be a natural number. We assume that the morphism given by the coproduct ς_n*(id_1♮-):𝐅_n*𝐁_n→𝐁_n+1 factors across the canonical surjection to 𝐅_na_n⋊𝐁_n. In other words, the following diagram is commutative:𝐅_n@^(->[r]@->[dr]_ς_n 𝐅_na_n⋊𝐁_n@->[d]𝐁_n@_(->[l]@->[dl]^id_1♮-𝐁_n+1.where the morphism 𝐅_na_n⋊𝐁_n→𝐁_n+1 is induced by the morphism 𝐅_n*𝐁_n→𝐁_n+1 and the group morphism id_1♮-:𝐁_n→𝐁_n+1 is induced by the monoidal structure. This is equivalent to requiring that, for all elements σ∈𝐁_n and g∈𝐅_n, the following equality holds in 𝐁_n+1:(id_1♮σ)∘ς_n(g)=ς_n(a_n(σ)(g))∘(id_1♮σ). Condition <ref> is essential in the definition of the Long-Moody functor on objects in Theorem <ref>. We fix a choice for these families of morphisms {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ.The families {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ are said to be coherent if they satisfy conditions <ref>, <ref> and <ref>.A classical family is provided by the Artin representations (see for example <cit.>). For n∈ℕ, a_n,1:𝐁_n→ Aut(𝐅_n) is defined for all elementary braids σ_i where i∈{ 1,…,n-1} by:a_n,1(σ_i):𝐅_n ⟶ 𝐅_ng_j ⟼g_i+1 if g_i+1^-1g_ig_i+1 if g_j if .It clearly follows from their definitions that the morphisms { a_n,1:𝐁_n→ Aut(𝐅_n)} _n∈ℕ satisfy Condition <ref>.The morphisms { a_n,1:𝐁_n→ Aut(𝐅_n)} _n∈ℕ together with the morphisms {ς_n,1:𝐅_n↪𝐁_n+1} _n∈ℕ of Example <ref> satisfy Condition <ref>.Let i be a fixed natural number in { 1,…,n-1}. We prove that the equality (<ref>) of Condition <ref> is satisfied for all Artin generator σ_i and all generator g_j of the free group (with j∈{ 1,…,n}). First, it follows from the braid relation σ_iσ_i+1σ_i=σ_i+1σ_iσ_i+1 that:σ_1+i^-1∘σ_i^-1∘σ_1+i^-2∘σ_i^2∘σ_1+i^2∘σ_i∘σ_1+i=σ_i^-1∘σ_1+i^2∘σ_i,and we deduce that:σ_1+i^-1∘ς_n,1(a_n,1(σ_i)(g_1+i))∘σ_1+i=ς_n,1(g_1+i).Also, the braid relation σ_i+1∘σ_i∘σ_i+1=σ_i∘σ_i+1∘σ_i implies that σ_i+1^-1∘σ_i^-1∘σ_i+1^2∘σ_i∘σ_i+1=σ_i^2 and a fortiori:σ_1+i^-1∘ς_n,1(a_n,1(σ_i)(g_i))∘σ_1+i=ς_n,1(g_i).Finally, for a fixed , the commutation relation σ_iσ_j=σ_jσ_i and the braid relation σ_iσ_i+1σ_i=σ_i+1σ_iσ_i+1 give directly:ς_n,1(g_j)=σ_1+i^-1∘ς_n,1(a_n,1(σ_i)(g_j))∘σ_1+i.The families of morphisms { a_n,1:𝐁_n→ Aut(𝐅_n)} _n∈ℕ and {ς_n,1:𝐅_n→𝐁_n+1} _n∈ℕ are coherent.Consider the family of morphisms {ς_n,*:𝐅_n→𝐁_n+1} _n∈ℕ of Example <ref> and any family of morphisms { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ. Then Condition <ref> is always satisfied. As a consequence, these families of morphisms {ς_n,*:𝐅_n→𝐁_n+1} _n∈ℕ and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ are coherent if and only if the family of morphisms { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ satisfies Condition <ref>. §.§ The Long-Moody functors In this section, we prove that the Long-Moody construction of <cit.> induces a functor 𝐋𝐌:𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)→𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡). We fix families of morphisms {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ, which are assumed to be coherent (see Definition <ref>).We first need to make some observations and introduce some tools. Let F be an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) and n be a natural number. A fortiori, the 𝕂-module F(n+1) is endowed with a left 𝕂[𝐁_n+1]-module structure. Using the morphism ς_n:𝐅_n→𝐁_n+1, F(n+1) is a 𝕂[𝐅_n]-module by restriction.Let us consider the augmentation ideal of the free group 𝐅_n, denoted by ℐ_𝕂[𝐅_n]. Since it is a (right) 𝕂[𝐅_n]-module, one can form the tensor product ℐ_𝕂[𝐅_n]𝕂[𝐅_n]F(n+1). Also, for all natural numbers n and n' such that n'≥ n, the morphism ι_𝐅_n'-n*id_𝐅_n:𝐅_n↪𝐅_n' canonically induces a morphism ι_ℐ_𝕂[𝐅_n'-n]*id_ℐ_𝕂[𝐅_n]:ℐ_𝕂[𝐅_n]↪ℐ_𝕂[𝐅_n']. In addition, the augmentation ideal ℐ_𝕂[𝐅_n] is a 𝕂[𝐁_n]-module too:The action a_n:𝐁_n→ Aut(𝐅_n) canonically induces an action of 𝐁_n on ℐ_𝕂[𝐅_n] denoted by a_n:𝐁_n→ Aut(ℐ_𝕂[𝐅_n]) (abusing the notation).For any group morphism H→ Aut(G), the group ring 𝕂[G] is canonically an H-module and so is the augmentation ideal ℐ_G, as a submodule of 𝕂[G]. If the family of morphisms { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ is coherent with respect to the family of morphisms {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ, the relation of Condition <ref> remains true mutatis mutandis, for all natural numbers n and n', considering the induced morphisms a_n:𝐁_n→ Aut(ℐ_𝕂[𝐅_n]) and ι_ℐ_𝕂[𝐅_n'-n]*id_ℐ_𝕂[𝐅_n]:ℐ_𝕂[𝐅_n]→ℐ_𝕂[𝐅_n']. In the following theorem, we define an endofunctor of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) corresponding to the Long-Moody construction. It will be called the Long-Moody functor with respect to {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ.Recall that we have fixed coherent families of morphisms {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ. The following assignment defines a functor 𝐋𝐌_a,ς:𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)→𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).* Objects: for F∈ Obj(𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)), 𝐋𝐌_a,ς(F):𝔘β→𝕂-𝔐𝔬𝔡 is defined by:* Objects: ∀ n∈ℕ, 𝐋𝐌_a,ς(F)(n)=ℐ_𝕂[𝐅_n]𝕂[𝐅_n]F(n+1).* Morphisms: for n,n'∈ℕ, such that n'≥ n, and [n'-n,σ]∈ Hom_𝔘β(n,n'), assign:𝐋𝐌_a,ς(F)([n'-n,σ])(i𝕂[𝐅_n]v)=a_n'(σ)(ι_ℐ_𝕂[𝐅_n'-n]*id_ℐ_𝕂[𝐅_n])(i)𝕂[𝐅_n']F(id_1♮[n'-n,σ])(v),for all i∈ℐ_𝕂[𝐅_n] and v∈ F(n+1). * Morphisms: let F and G be two objects of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡), and η:F→ G be a natural transformation. We define 𝐋𝐌_a,ς(η):𝐋𝐌_a,ς(F)→𝐋𝐌_a,ς(G) for all natural numbers n by:(𝐋𝐌_a,ς(η))_n=id_ℐ_𝕂[𝐅_n]𝕂[𝐅_n]η_n+1. In particular, the Long-Moody functor 𝐋𝐌_a,ς induces an endofunctor of the category 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡).When there is no ambiguity, once the morphisms {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ are fixed, we omit them from the notation 𝐋𝐌_a,ς for convenience (especially for proofs).For this proof, n, n' and n” are natural numbers such that n”≥ n'≥ n. * First let us show that the assignment of 𝐋𝐌 defines an endofunctor of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡). The two first points generalize the proof of <cit.>. Let F, G and H be objects of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡).* We first check the compatibility of the assignment 𝐋𝐌(F) with respect to the tensor product. Consider σ∈𝐁_n g∈𝐅_n, i∈ℐ_𝕂[𝐅_n] and v∈ F(n+1). Since (id_1♮σ)∘ς_n(g)=ς_n(a_n(σ)(g))∘(id_1♮σ) by Condition <ref>, we deduce that:𝐋𝐌(F)(σ)(i𝕂[𝐅_n]F(ς_n(g))(v)) = a_n(σ)(i)𝕂[𝐅_n]F(id_1♮σ)(F(ς_n(g))(v)) = a_n(σ)(i)𝕂[𝐅_n](F(ς_n(a_n(σ)(g)))∘ F(id_1♮σ))(v) = a_n(σ)(i· g)𝕂[𝐅_n]F(id_1♮σ)(v) =𝐋𝐌(F)(σ)(i· g𝕂[𝐅_n](v)). * Let us prove that the assignment 𝐋𝐌(F) defines an object of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡). According to our assignment and since a_n and id_1♮- are group morphisms, it follows from the definition that 𝐋𝐌(F)(id_𝐁_n)=id_𝐋𝐌(F)(n). Hence, it remains to prove that the composition axiom is satisfied. Let σ and σ' be two elements of 𝐁_n, i∈ℐ_𝕂[𝐅_n] and v∈ F(n+1). From the functoriality of F over β and the compatibility of the monoidal structure ♮ with composition, we deduce that F(id_1♮(σ'))∘ F(id_1♮(σ))=F(id_1♮(σ'∘σ)). Since a_n is a group morphism, we have:(a_n(σ'∘σ))(i)=a_n(σ')(a_n(σ)(i)). Hence, it follows from the assignment of 𝐋𝐌 that:𝐋𝐌(F)(σ'∘σ)(i𝕂[𝐅_n]v) =(a_n(σ'∘σ))(i)𝕂[𝐅_n]F(id_1♮(σ'∘σ))(v) = a_n(σ')(a_n(σ)(i))𝕂[𝐅_n](F(id_1♮(σ'))∘ F(id_1♮(σ)))(v) =𝐋𝐌(F)(σ')∘𝐋𝐌(F)(σ)(i𝕂[𝐅_n]v). * It remains to check the consistency of our definition of 𝐋𝐌 on morphisms of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡). Let η:F→ G be a natural transformation. Hence, we have that:G(id_1♮σ)∘η_n+1=η_n'+1∘ F(id_1♮σ).Hence, it follows from the assignment of 𝐋𝐌 that:𝐋𝐌(G)(σ)∘𝐋𝐌(η)_n=𝐋𝐌(η)_n'∘𝐋𝐌(F)(σ)Therefore 𝐋𝐌(η) is a morphism in the category 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡). Denoting by id_F:F→ F the identity natural transformation, it is clear that 𝐋𝐌(id_F)=id_𝐋𝐌(F). Finally, let us check the composition axiom. Let η:F→ G and μ:G→ H be natural transformations. Let n be a natural number, i∈ℐ_𝕂[𝐅_n] and v∈ F(n). Now, since μ and η are morphisms in the category 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡):𝐋𝐌(μ∘η)_n(i𝕂[𝐅_n]v) = i𝕂[𝐅_n](μ_n+1∘η_n+1)(v)=𝐋𝐌(μ)_n∘𝐋𝐌(η)_n(i𝕂[𝐅_n]v).* Let us prove that the assignment 𝐋𝐌 lifts to define an endofunctor of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡). Let F, G and H be objects of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).* First, let us check the compatibility of the assignment 𝐋𝐌(F) with respect to the tensor product. In fact, this compatibility being already done for automorphisms (see <ref>), the remaining point to prove is the compatibility of 𝐋𝐌(F)([n'-n,id_n']). Let g∈𝐅_n, i∈ℐ_𝕂[𝐅_n] and v∈ F(n+1). It follows from Condition <ref> that in 𝐁_n+1:id_1♮[n'-n,id_n'-n♮ς_n(g)]=ς_n'(e_𝐅_n'-n*g)∘(id_1♮[n'-n,id_n']).Since (ι_ℐ_𝕂[𝐅_n'-n]*id_ℐ_𝕂[𝐅_n])(i· g)=(e_ℐ_𝕂[𝐅_n'-n]*i)·(e_𝐅_n'-n*g), we deduce that:𝐋𝐌(F)([n'-n,id_n'])(i𝕂[𝐅_n]F(ς_n(g))(v)) =(ι_ℐ_𝕂[𝐅_n'-n]*id_ℐ_𝕂[𝐅_n])(i)𝕂[𝐅_n']F(id_1♮[n'-n,id_n'])(F(ς_n(g))(v)) =(ι_ℐ_𝕂[𝐅_n'-n]*id_ℐ_𝕂[𝐅_n])(i· g)𝕂[𝐅_n']F(id_1♮[n'-n,id_n'])(v) =𝐋𝐌(F)([n'-n,id_n'])(i· g𝕂[𝐅_n]v). * Let us prove that the assignment 𝐋𝐌(F) defines an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) using Proposition <ref>. Recall the compatibility of the monoidal structure ♮ with respect to composition and that F is an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡). Consider [n'-n,σ]∈ Hom_𝔘β(n,n'). It follows from our assignment, that:𝐋𝐌(F)([n'-n,σ])=𝐋𝐌(F)(σ)∘𝐋𝐌(F)([n'-n,id_n']).Moreover, the composition of morphisms introduced in Definition <ref> implies that:𝐋𝐌(F)([n”-n,id_n”]) =𝐋𝐌(F)([n”-n',id_n”])∘𝐋𝐌(F)([n'-n,id_n']).Hence, the relation (<ref>) of Proposition <ref> is satisfied. Let σ∈𝐁_n and ψ∈𝐁_n'-n. Since (ι_n'-n*id_n)∘(a_n(σ))=(a_n'(ψ♮σ))∘(ι_n'-n*id_n) by Condition <ref>, we deduce that:𝐋𝐌(F)(ψ♮σ)∘𝐋𝐌(F)([n'-n,id_n']) =𝐋𝐌(F)([n'-n,id_n'])∘𝐋𝐌(F)(σ).Hence the relation (<ref>) of Proposition <ref> is also satisfied. Therefore, according to Proposition <ref>, since 𝐋𝐌(F) is an object of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡), the assignment 𝐋𝐌(F) defines an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).* Finally, let us check the consistency of our assignment for 𝐋𝐌 on morphisms. Let η:F→ G be a natural transformation. We already proved in <ref> that 𝐋𝐌(η) is a morphism in the category 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡). Since η is a natural transformation between objects of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡), we have that:G(id_1♮[n'-n,id_n'])∘η_n+1=η_n'+1∘ F(id_1♮[n'-n,id_n']).Hence, it follows from the assignment of 𝐋𝐌 that:𝐋𝐌(G)([n'-n,id_n'])∘𝐋𝐌(η)_n=𝐋𝐌(η)_n'∘𝐋𝐌(F)([n'-n,id_n']).Hence the relation (<ref>) of Proposition <ref> is satisfied, and we deduce from this last proposition that 𝐋𝐌(η) is a morphism in the category 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡). The verification of the composition axiom repeats mutatis mutandis the one of <ref>.Recall the following fact on the augmentation ideal of the free group 𝐅_n where n∈ℕ.<cit.> The augmentation ideal ℐ_𝕂[𝐅_n] is a free 𝕂[𝐅_n]-module with basis the set {(g_i-1)| i∈{ 1,…,n}}. This result allows us to prove the following properties.The functor 𝐋𝐌_a,ς:𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)→𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) is reduced and exact. Moreover, it commutes with all colimits and all finite limits.Let 0_𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡):𝔘β→𝕂-𝔐𝔬𝔡 denote the null functor. It follows from the definition of the Long-Moody functor that 𝐋𝐌(0_𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡))=0_𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).Let n be a natural number. Since the augmentation ideal ℐ_𝕂[𝐅_n] is a free 𝕂[𝐅_n]-module (as stated in Proposition <ref>), it is therefore a flat 𝕂[𝐅_n]-module. Then, the result follows from the fact that the functor ℐ_𝕂[𝐅_n]𝕂[𝐅_n]-:𝕂-𝔐𝔬𝔡→𝕂-𝔐𝔬𝔡 is an exact functor, the naturality for morphisms following from the definition of the Long-Moody functor (see Theorem <ref>).Similarly, the fact that the functor 𝐋𝐌_a,ς commutes with all colimits is a formal consequence of the commutation with all colimits of the tensor products ℐ_𝕂[𝐅_n]𝕂[𝐅_n]- for all natural numbers n. The commutation result for finite limits is a property of exact functors (see for example <cit.>).Let F be an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) and n a natural number. For all k∈{ 1,…,n}, we denote F(n+1)_k=𝕂[(g_k-1)]𝕂[𝐅_n]F(n+1) with g_k a generator of 𝐅_n. We define an isomorphism_n,F:ℐ_𝕂[𝐅_n]𝕂[𝐅_n]F(n+1)⟶ [k=1]n⊕F(n+1)_k≅(F(n+1))^⊕ n (g_k-1)𝕂[𝐅_n]v⟼ (0,…,0,k-thv,0,…,0).Thus, for η:F→ G a natural transformation, with :∀ n∈ℕ, _n((𝐋𝐌(η))_n)=η_n+1^⊕ n.Hence, we can have a matricial point of view on this construction (see <cit.>). Similarly, the study of Bigelow and Tian in <cit.> is performed from a purely matricial point of view. Case of trivial ς:Finally, let us consider the family of morphisms {ς_n,*:𝐅_n→𝐁_n+1} _n∈ℕ of Example <ref>.As stated in Example <ref>, we only need to consider a family of morphisms { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ which satisfies Condition <ref> so that the families {ς_n,*:𝐅_n→𝐁_n+1} _n∈ℕ and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ are coherent.We denote by 𝔛:𝔘β→𝕂-𝔐𝔬𝔡 the constant functor such that 𝔛(n)=𝕂 for all natural numbers n. We have the following remarkable property.Let F be an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ a family of morphisms which satisfies Condition <ref>. Then, as objects of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡), 𝐋𝐌_a,ς_*(F)≅𝐋𝐌_a,ς_*(𝔛)𝕂⊗F(1♮-).Remark <ref> shows that there is an isomorphism of 𝕂-modules of the form:𝐋𝐌_a,ς_*(F)(n)@->[rr]^_n,F(F(n+1))^⊕ n@->[rr]^(_n,𝔛𝕂⊗id_F(1♮ n))^-1𝐋𝐌_a,ς_*(𝔛)(n)𝕂⊗F(1♮ n) .It is straightforward to check that this isomorphism is natural if ς is trivial.§.§ Evaluation of the Long-Moody functor A first step to understand the behaviour of a Long-Moody endofunctor is to investigate its effect on the constant functor 𝔛. This is indeed the most basic functor to study. Moreover, as Proposition <ref> shows, the evaluation on this functor is the fundamental information to understand a given Long-Moody endofunctor when we consider the family of morphisms {ς_n,*:𝐅_n→𝐁_n+1} _n∈ℕ of Example <ref>.Fixing coherent families of morphisms {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ, we consider the Long-Moody functor 𝐋𝐌_a,ς:𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡)→𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡).For a fixed natural number n, using the isomorphism _n of Remark <ref>, we observe that 𝐋𝐌_a,ς(𝔛)(n)≅𝕂^⊕ n.Let y be an invertible element of 𝕂. Let y𝔛:β→𝕂-𝔐𝔬𝔡 be the functor defined for all natural numbers n by y𝔛(n)=𝕂 and such that:* if n=0 or n=1, then y𝔛(id)=id_𝕂;* if n≥2, for every Artin generator σ_i of 𝐁_n, (y𝔛)(σ_i):𝕂→𝕂 is the multiplication by y.For an object F of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡), we denote the functor y𝔛𝕂⊗F:β→-𝔐𝔬𝔡 by yF. §.§.§ Computations for 𝐋𝐌_1 Let us assume that 𝕂=ℂ[t^±1]. Let us consider the coherent families of morphisms {ς_n,1:𝐅_n↪𝐁_n+1} _n∈ℕ (introduced in Example <ref>) and { a_n,1:𝐁_n→ Aut(𝐅_n)} _n∈ℕ (introduced in Example <ref>). We denote by 𝐋𝐌_1 the associated Long-Moody functor. We are interested in the behaviour of the functor t^-1𝐋𝐌_1(t𝔛):β⟶ℂ[t^±1]-𝔐𝔬𝔡 on automorphisms of the category 𝔘β. Indeed, adding a parameter t is necessary to recover functors specifically associated with the category 𝔘β, such as 𝔅𝔲𝔯_t (see Section <ref>). Let us fix n a natural number and σ_i an Artin generator of 𝐁_n.Beforehand, let us understand the action a_n,1:𝐁_n⟶ Aut(ℐ_𝕂[𝐅_n]) induced by a_n,1:𝐁_n→ Aut(𝐅_n). We compute:a_n,1(σ_i):ℐ_𝕂[𝐅_n] ⟶ ℐ_𝕂[𝐅_n]g_j-1⟼g_i+1-1if g_i+1^-1g_ig_i+1-1=[g_i-1]g_i+1+[g_i+1-1](1-g_i+1^-1g_ig_i+1)if g_j-1if Hence, we have the following result. As objects of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡), t^-1𝐋𝐌_1(t𝔛)=𝔅𝔲𝔯_t^2.Using the isomorphism _n of Remark <ref>, we obtain that for σ_i an Artin generator of 𝐁_n:t^-1𝐋𝐌_1(t𝔛)(σ_i)=Id_i-1⊕[[ 0 t^2; 1 1-t^2 ]]⊕ Id_n-i-1=𝔅𝔲𝔯_t^2(σ_i). Recovering of the Lawrence-Krammer functor:Let us first introduce the following result due to Long in <cit.>. We assume that 𝕂=ℂ[t^±1][q^±1]. For this paragraph, we assume that 1+qt=0, q has a square root, q^2≠1 and q^3≠1.We denote by 𝔛':β⟶ℂ[t^±1][q^±1]-𝔐𝔬𝔡 the constant functor such that 𝔛'(n)=ℂ[t^±1][q^±1] for all natural numbers n. Generally speaking, for F an object of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡) the representation of 𝐁_n induced by F will be denoted by F_|𝐁_n.<cit.> Let n be a natural number such that n≥4. Then, the Lawrence-Krammer representation 𝔏𝔎_|𝐁_n is a subrepresentation of q^-1(𝐋𝐌_1(q(t^-1𝐋𝐌_1(t𝔛))))_|𝐁_n. We first need to introduce new tools. Let n and m be two natural numbers. Let w_n=(w_1,…,w_n)∈ℂ^n such that w_i≠ w_j if . We consider the configuration space: Y_w_n,m={(z_1,…,z_m)| z_i∈ℂ, z_i≠ w_k for , z_i≠ z_j if } .The two following results due to Long will be crucial to prove Proposition <ref>.<cit.> Let n be a natural number and ρ:𝐁_n+1→ GL(V) be a representation of 𝐁_n with V a ℂ[t^±1][q^±1]-module. Then, the representation defined by Long in <cit.>, which we denote by ℒℳ, is a group morphism:q^-1ℒℳ(qρ):𝐁_n→ GL(H^1(Y_w_n,1,E_ρ))for E_ρ a flat vector bundle associated with ρ (see <cit.>).<cit.> For all natural numbers m, there is an isomorphism of abelian groups:H^m+1(Y_w_n,m+1,E_𝔛_|𝐁_n)≅ H^1(Y_w_n,1,H^m(Y_w_n+1,m,E_𝔛_|𝐁_n)).In particular, for m=1, H^2(Y_w_n,2,E_𝔛_|𝐁_n)≅ H^1(Y_w_n,1,H^1(Y_w_n+1,2,E_𝔛_|𝐁_n)). By Proposition <ref>, we can write as a representation:q^-1ℒℳ(q(t^-1ℒℳ(t𝔛))):𝐁_n→ GL(H^1(Y_w_n,1,E_t^-1ℒℳ(t𝔛))).A fortiori by Lemma <ref>, q^-1ℒℳ(q(t^-1ℒℳ(t𝔛_|𝐁_n))) is an action of 𝐁_n on H^2(Y_w_n,2,E_𝔛_|𝐁_n). In particular, for m=2 and n≥4, according to <cit.>, the representation of 𝐁_n factoring through the Iwahori–Hecke algebra H_n(t) corresponding to the Young diagram (n-2,2) is a subrepresentation of q^-1ℒℳ(q(t^-1ℒℳ(t𝔛_|𝐁_n))). Moreover, this representation is equivalent to the Lawrence-Krammer representation by <cit.>. By the definition of the Long-Moody construction (see <cit.>), q^-1ℒℳ(q(t^-1ℒℳ(t𝔛_|𝐁_n))) is the representation q^-1(τ_1𝐋𝐌_1)(q(t^-1𝐋𝐌_1(t𝔛)))_|𝐁_n.We denote by 𝔏𝔎^≥4:β⟶(ℂ[t^±1])[q^±1]-𝔐𝔬𝔡 the subfunctor of the Lawrence-Krammer defined in Example <ref> which is null on the objects such that n<4. The result of Proposition <ref> implies that:The functor 𝔏𝔎^≥4 is a subfunctor of q^-1(τ_1𝐋𝐌_1)(q(t^-1𝐋𝐌_1(t𝔛)))^≥4.§.§.§ Computations for other cases Let us introduce examples of Long-Moody functors which arise using other actions a_n:𝐁_n→ Aut(𝐅_n). Wada representationsIn 1992, Wada introduced in <cit.> a certain type of family of representations of braid groups. We give here a functorial approach to this work.Let Aut_-:(ℕ,≤)→𝔊𝔯 be the functor defined by:* Objects: for all natural numbers n, Aut_-(n)=Aut(𝐅_n) the automorphism group of the free group on n generators;* Morphisms: let n be a natural number. We define Aut_-(γ_n):Aut(𝐅_n)↪ Aut(𝐅_n+1) assigning Aut_-(γ_n)(φ)=id_1*φ for all φ∈ Aut(𝐅_n), using the monoidal category (𝔤𝔯,*,0) recalled in Notation <ref>. Let us consider two different non-trivial reduced words W(g_1,g_2) and V(g_1,g_2) on 𝐅_2, such that:* the assignments g_1↦ W(g_1,g_2) and g_2↦ V(g_1,g_2) define a automorphism of 𝐅_2;* the assignment (W,V):𝐁_2⟶ Aut(𝐅_2):[(W,V)(σ_1)](g_j)= W(g_1,g_2)if V(g_1,g_2)ifis a morphism.Two morphisms (W,V):𝐁_2⟶ Aut(𝐅_2) and (W',V'):𝐁_2→ Aut(𝐅_2) are said to be swap-dual if W'(g_1,g_2)=V(g_2,g_1) and V'(g_1,g_2)=W(g_2,g_1), backward-dual if W'(g_1,g_2)=(W(g_1^-1,g_2^-1))^-1 and V'(g_1,g_2)=(V(g_1^-1,g_2^-1))^-1, inverse if (W',V')=(W,V)^-1.<cit.> Let W(g_1,g_2) and V(g_1,g_2) be two words on 𝐅_2. A natural transformation 𝒲:𝐁_-→ Aut_- is said to be of Wada-type if for all natural numbers n, for all i∈{ 1,…,n-1}, the following diagram is commutative (we recall that incl_i^n was introduced in Notation <ref> and Aut_-(γ_2,i) in Definition <ref>):𝐁_n@->[rr]^𝒲_n Aut(𝐅_n) 𝐁_2@->[rr]_(W,V)@->[u]^incl_i^n Aut(𝐅_2).@->[u]_Aut_-(γ_2,i)∗ id_𝐅_n-i-1 Note that therefore a Wada-type natural transformation is entirely determined by the choice of (W,V). Wada conjectured a classification of these type of representations. This conjecture was proved by Ito in <cit.>.<cit.> There are seven classes of Wada-type natural transformation 𝒲 up to the swap-dual, backward-dual and inverse equivalences, listed below.* (W,V)_1,m(g_1,g_2)=(g_2,g_2^mg_1g_2^-m) where m∈ℤ;* (W,V)_2(g_1,g_2)=(g_1,g_2);* (W,V)_3(g_1,g_2)=(g_2,g_1^-1);* (W,V)_4(g_1,g_2)=(g_2,g_2^-1g_1^-1g_2);* (W,V)_5(g_1,g_2)=(g_2^-1,g_1^-1);* (W,V)_6(g_1,g_2)=(g_2^-1,g_2g_1g_2);* (W,V)_7(g_1,g_2)=(g_1g_2^-1g_1^-1,g_1g_2^2). Note that the action given by the first Wada representation with m=1 is a generalization of the Artin representation.The actions given by the k-th Wada-type natural transformation will be denoted by a_n,k:𝐁_n↪ Aut(𝐅_n). In particular, for k=1 with m=1, we recover the Artin representation (see Example <ref>). For all 1≤ k≤8, it clearly follows from their definitions that the families of morphisms { a_n,k:𝐁_n→ Aut(𝐅_n)} _n∈ℕ satisfy Condition <ref>. Hence, for 1≤ k≤8, we consider a family of morphisms {ς_n,k:𝐅_n→𝐁_n+1} assumed to be coherent with respect to the morphisms { a_n,k:𝐁_n↪ Aut(𝐅_n)} _n∈ℕ (in the sense of Definition <ref>). Such morphisms ς_n,k always exist because we could at least take the family of morphisms {ς_n,*:𝐅_n→𝐁_n+1} (see Example <ref>). We denote by 𝐋𝐌_k:𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡)→𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡) the corresponding Long-Moody functor defined in Theorem <ref> for k∈{ 1,…,8}. Let us imitate the procedure of Section <ref>. We assume that 𝕂=ℂ[t^±1]. Let n be a fixed natural number. Let us consider the case of k=2. Using the isomorphism _n of Remark <ref>, we obtain the functor 𝐋𝐌_2(𝔛):β→ℂ[t^±1]-𝔐𝔬𝔡, defined for σ_i∈𝐁_n by:𝐋𝐌_2(F)(σ_i)=(F(σ_i))^⊕ n.For k=3, using _n, we compute that the functor t^-1𝐋𝐌_3(t𝔛):β→ℂ[t^±1]-𝔐𝔬𝔡 is defined for σ_i∈𝐁_n by:t^-1𝐋𝐌_3(t𝔛)(σ_i) = Id_i-1⊕[[ 0 -ς_n,3(g_i); 1 0 ]]⊕ Id_n-i-1.Hence, the functor t^-1𝐋𝐌_3(t𝔛) is very similar to the one associated with the Tong-Yang-Ma representations (recall Definition <ref>). We deduce that the identity natural equivalence gives t^-1𝐋𝐌_3(t𝔛)≅𝔗𝔜𝔐_-ς_n,3(g_i) as objects of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡).For the actions given by the Wada-type natural transformation 4, 5, 6 and 7 in Theorem <ref>, the produced functors t^-1𝐋𝐌_i(t𝔛):β⟶ℂ[t^±1]-𝔐𝔬𝔡 are mild variants of what is given by the case i=1.§ STRONG POLYNOMIAL FUNCTORS We deal here with the concept of a strong polynomial functor. This type of functor will be the core of our work in Section <ref>. We review (and actually extend) the definition and properties of a strong polynomial functor due to Djament and Vespa in <cit.> and also a particular case of coefficient systems of finite degree used by Randal-Williams and Wahl in <cit.>.In <cit.>, Djament and Vespa construct a framework to define strong polynomial functors in the category 𝐅𝐜𝐭(𝔐,𝒜), where 𝔐 is a symmetric monoidal category, the unit is an initial object and 𝒜 is an abelian category. Here, we generalize this definition for functors from pre-braided monoidal categories having the same additional property. In particular, the notion of strong polynomial functor will be defined for the category 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡). The keypoint of this section is Proposition <ref>, in so far as it constitutes the crucial property necessary and sufficient to extend the definition of strong polynomial functor to the pre-braided case. §.§ Strong polynomiality We first introduce the translation functor, which plays the central role in the definition of strong polynomiality.Let (𝔐,♮,0) be a strict monoidal small category, let 𝔇 be a category and let x be an object of 𝔐. The monoidal structure defines the endofunctor x♮-:𝔐⟶𝔐. We define the translation by x functor τ_x:𝐅𝐜𝐭(𝔐,𝔇)→𝐅𝐜𝐭(𝔐,𝔇) to be the endofunctor obtained by precomposition by the functor x♮-. The following proposition establishes the commutation of two translation functors associated with two objects of 𝔐. It is the keystone property to define strong polynomial functors.Let (𝔐,♮,0) be a pre-braided strict monoidal small category and 𝔇 be a category. Let x and y be two objects of 𝔐. Then, there exists a natural isomorphism between functors from 𝐅𝐜𝐭(𝔐,𝔇) to 𝐅𝐜𝐭(𝔐,𝔇):τ_x∘τ_y≅τ_y∘τ_x. First, because of the associativity of the monoidal product ♮ and the strictness of 𝔐, we have that τ_x∘τ_y=τ_x♮ y and τ_y∘τ_x=τ_y♮ x. We denote by b_-,-^𝔐 the pre-braiding of 𝔐. The key point is the fact that as b_-,-^𝔐 is a braiding on the maximal subgroupoid of 𝔐 (see Definition <ref>), b_x,y^𝔐:x♮ y≅⟶y♮x defines an isomorphism. Hence, precomposition by b_x,y^𝔐♮ id_𝔐 defines a natural transformation (b_x,y^𝔐♮ id_𝔐)^*:τ_x♮ y→τ_y♮ x. It is an isomorphism since we analogously construct an inverse natural transformation ((b_x,y^𝔐)^-1♮ id_𝔐)^*:τ_y♮ x→τ_x♮ y. In Proposition <ref>, the natural isomorphism is not unique: as the proof shows, we could have used the morphism (b_y,x^𝔐)^-1♮ id_𝔐 instead to define an isomorphism between τ_x♮ y(F) and τ_y♮ x(F). In fact, a category only needs to be equipped with natural (in x and y) isomorphisms x♮ y≅ y♮ x to satisfy the conclusion of Proposition <ref>. Let us move on to the introduction of the evanescence and difference functors, which will characterize the (very) strong polynomiality of a functor in 𝐅𝐜𝐭(𝔐,𝒜). Recall that, if 𝔐 is a small category and 𝒜 is an abelian category, then the functor category 𝐅𝐜𝐭(𝔐,𝒜) is an abelian category (see <cit.>).From now until the end of Section <ref>, we fix (𝔐,♮,0) a pre-braided strict monoidal category such that the monoidal unit 0 is an initial object, 𝒜 an abelian category and x denotes an object of 𝔐.For all objects F of 𝐅𝐜𝐭(𝔐,𝒜), we denote by i_x(F):τ_0(F)→τ_x(F) the natural transformation induced by the unique morphism ι_x:0→ x of 𝔐. This induces i_x:Id_𝐅𝐜𝐭(𝔐,𝒜)→τ_x a natural transformation of 𝐅𝐜𝐭(𝔐,𝒜). Since the category 𝐅𝐜𝐭(𝔐,𝒜) is abelian, the kernel and cokernel of the natural transformation i_x exist. We define the functors κ_x=(i_x) and δ_x=coker(i_x). The endofunctors κ_x and δ_x of 𝐅𝐜𝐭(𝔐,𝒜) are called respectively evanescence and difference functor associated with x.The following proposition presents elementary properties of the translation, evanescence and difference functors. They are either consequences of the definitions, or direct generalizations of the framework considered in <cit.> where 𝔐 is symmetric monoidal.Let y be an object of 𝔐. Then the translation functor τ_x is exact and we have the following exact sequence in the category of endofunctors of 𝐅𝐜𝐭(𝔐,𝒜):0⟶κ_xΩ_x⟶Idi_x⟶τ_x_x⟶δ_x⟶0.Moreover, for a short exact sequence 0⟶ F⟶ G⟶ H⟶0 in the category 𝐅𝐜𝐭(𝔐,𝒜), there is a natural exact sequence in the category 𝐅𝐜𝐭(𝔐,𝒜):0⟶κ_x(F)⟶κ_x(G)⟶κ_x(H)⟶δ_x(F)⟶δ_x(G)⟶δ_x(H)⟶0.In addition:* The translation endofunctor τ_x of 𝐅𝐜𝐭(𝔐,𝒜) commutes with limits and colimits.* The difference endofunctors δ_x and δ_y of 𝐅𝐜𝐭(𝔐,𝒜) commute up to natural isomorphism. They commute with colimits.* The endofunctors κ_x and κ_y of 𝐅𝐜𝐭(𝔐,𝒜) commute up to natural isomorphism. They commute with limits.* The natural inclusion κ_x∘κ_x↪κ_x is an isomorphism.* The translation endofunctor τ_x and the difference endofunctor δ_y commute up to natural isomorphism.* The translation endofunctor τ_x and the endofunctor κ_y commute up to natural isomorphism.* We have the following natural exact sequence in the category of endofunctors of 𝐅𝐜𝐭(𝔐,𝒜):0⟶κ_y⟶κ_x♮ y⟶τ_xκ_y⟶δ_y⟶δ_x♮ y⟶τ_yδ_x⟶0.In the symmetric monoidal case, this is <cit.>: the numbered properties are formal consequences of the commutation property of the translation endofunctors given by Proposition <ref>. Hence, the proofs carry over mutatis mutandis to the pre-braided setting.Using Proposition <ref>, we can define strong polynomial functors.We recursively define on n∈ℕ the category 𝒫ol_n^strong(𝔐,𝒜) of strong polynomial functors of degree less than or equal to n to be the full subcategory of 𝐅𝐜𝐭(𝔐,𝒜) as follows:* If n<0, 𝒫ol_n^strong(𝔐,𝒜)={ 0};* if n≥0, the objects of 𝒫ol_n^strong(𝔐,𝒜) are the functors F such that for all objects x of 𝔐, the functor δ_x(F) is an object of 𝒫ol_n-1^strong(𝔐,𝒜).For an object F of 𝐅𝐜𝐭(𝔐,𝒜) which is strong polynomial of degree less than or equal to n∈ℕ, the smallest d∈ℕ (d≤ n) for which F is an object of 𝒫ol_d^strong(𝔐,𝒜) is called the strong degree of F.By Proposition <ref>, the category (𝔘β,♮,0) is a pre-braided monoidal category such that 0 is initial object. This example is the first one which led us to extend the definition of <cit.>. Thus, we have a well-defined notion of strong polynomial functor for the category 𝔘β. The following three propositions are important properties of the framework in <cit.> adapted to the pre-braided case. Their proofs follow directly from those of their analogues in <cit.>.<cit.> Let 𝔐' be another pre-braided strict monoidal category such that such that its monoidal unit is an initial object and α:𝔐→𝔐' be a strong monoidal functor. Then, the precomposition by α provides a functor𝒫ol_n^strong(𝔐,𝒜)→𝒫ol_n^strong(𝔐',𝒜).<cit.> The category 𝒫ol_n^strong(𝔐,𝒜) is closed under the translation endofunctor τ_x, under quotient, under extension and under colimits. Moreover, assuming that there exists a set 𝔈 of objects of 𝔐 such that:∀ m∈ Obj(𝔐),∃{ e_i} _i∈ I∈ Obj(𝔈) whereis finite,m≅i∈ I♮e_i,then, an object F of 𝐅𝐜𝐭(𝔐,𝒜) belongs to 𝒫ol_n^strong(𝔐,𝒜) if and only if δ_e(F) is an object of 𝒫ol_n-1^strong(𝔐,𝒜) for all objects e of 𝔈. Let n be a natural number. Let F be a strong polynomial functor of degree n in the category 𝐅𝐜𝐭(𝔐,𝒜). Then a direct summand of F is necessarily an object of the category 𝒫ol_n^strong(𝔐,𝒜).According to Proposition <ref>, the category 𝒫ol_n^strong(𝔐,𝒜) is closed under quotients. The category 𝒫ol_n^strong(𝔐,𝒜) is not necessarily closed under subobjects. For example, we will see inSection <ref> that for 𝔐=𝔘β and 𝒜=ℂ[t^±1]-𝔐𝔬𝔡, the functor 𝔅𝔲𝔯_t is a subobject of τ_1𝔅𝔲𝔯_t (see Proposition <ref>), 𝔅𝔲𝔯_t is strong polynomial of degree 2 (see Proposition <ref>) whereas τ_1𝔅𝔲𝔯_t is strong polynomial of degree 1 (see Proposition <ref>). If we assume that the unit 0 is also a terminal object of 𝔐, then κ_x is the null endofunctor, δ_x is exact and commutes with all limits. In this case, the category 𝒫ol_n^strong(𝔐,𝒜) is closed under subobjects.If we consider 𝔐=𝔘β, then each object n (ie a natural number) is clearly 1^♮ n. Hence, because of the last statement of Proposition <ref>, when we will deal with strong polynomiality of objects in 𝐅𝐜𝐭(𝔘β,𝒜), it will suffice to consider τ_1.<cit.> Let F be an object of 𝐅𝐜𝐭(𝔐,𝒜). Then, the functor F is an object of 𝒫ol_0^strong(𝔐,𝒜) if and only if it is the quotient of a constant functor of 𝐅𝐜𝐭(𝔐,𝒜). Finally, let us point out the following property of the strong polynomial degree with respect to the translation functor.Let d and k be natural numbers and F be an object of 𝐅𝐜𝐭(𝔘β,-𝔐𝔬𝔡) such that τ_k(F) is an object of 𝒫ol_d^strong(𝔘β,-𝔐𝔬𝔡). Then, F is an object of 𝒫ol_d+k(𝔘β,-𝔐𝔬𝔡).We proceed by induction on the degree of polynomiality of τ_k(F). First, assuming that τ_k(F) belongs to 𝒫ol_0^strong(𝔘β,-𝔐𝔬𝔡), we deduce from the commutation property 6 of Proposition <ref> that τ_k(δ_1F)=0. It follows from the definition of τ_k(F) (see Definition <ref>) that for all n≥2, δ_1(F)(n)=0. Hencetimesδ_1⋯δ_1δ_1(F)≅0and therefore F is an object of 𝒫ol_k(𝔘β,-𝔐𝔬𝔡). Now, assume that τ_k(F) is a strong polynomial functor of degree d≥0. Since (τ_k∘δ_1)(F)≅(δ_1∘τ_k)(F) by the commutation property 6 of Proposition <ref>, (τ_k∘δ_1)(F) is an object of 𝒫ol_d-1^strong(𝔘β,-𝔐𝔬𝔡). The inductive hypothesis implies that δ_1(F) is an object of 𝒫ol_d+k^strong(𝔘β,-𝔐𝔬𝔡). Let us consider the atomic functor 𝔄_n (with n>0), which is strong polynomial of degree n (see Example <ref>). Then τ_k(𝔄_n)≅𝔄_n-k^⊕ n is strong polynomial of degree n-k, for k a natural number such that k≤ n. This illustrates the fact that d+k is the best boundary for the degree of polynomiality in Lemma <ref>. §.§ Very strong polynomial functors Let us introduce a particular type of strong polynomial functor, related to coefficient systems of finite degree (see Remark <ref> below). We recall that we consider a pre-braided strict monoidal category (𝔐,♮,0) such that the monoidal unit 0 is an initial object and an abelian category 𝒜.We recursively define the category 𝒱𝒫ol_n(𝔐,𝒜) of very strong polynomial functors of degree less than or equal to n to be the full subcategory of 𝒫ol_n^strong(𝔐,𝒜) as follows:* If n<0, 𝒱𝒫ol_n(𝔐,𝒜)={ 0};* if n≥0, a functor F∈𝒫ol_n^strong(𝔐,𝒜) is an object of 𝒱𝒫ol_n(𝔐,𝒜) if for all objects x of 𝔐, κ_x(F)=0 and the functor δ_x(F) is an object of 𝒱𝒫ol_n-1(𝔐,𝒜).For an object F of 𝐅𝐜𝐭(𝔐,𝒜) which is very strong polynomial of degree less than or equal to n∈ℕ, the smallest d∈ℕ (d≤ n) for which F is an object of 𝒱𝒫ol_d(𝔐,𝒜) is called the very strong degree of F.A certain type of functor, called a coefficient system of finite degree, closely related to the strong polynomial one, is used by Randal-Williams and Wahl in <cit.> for their homological stability theorems, generalizing the concept introduced by van der Kallen for general linear groups <cit.>. Using the framework introduced by Randal-Williams and Wahl, a coefficient system in every object x of 𝔐 of degree n at N=0 is a very strong polynomial functor.As we force κ_x to be null for all objects x of 𝔐, the category 𝒱𝒫ol_n(𝔐,𝒜) is closed under kernel functors of the epimorphisms. In particular, this category is closed under direct summands. However, 𝒱𝒫ol_n(𝔐,𝒜) is not necessarily closed under subobjects. For instance, as for Remark <ref>, we have that the functor 𝔅𝔲𝔯_t is strong polynomial of degree 2 (see Proposition <ref>), the functor τ_1𝔅𝔲𝔯_t is very strong polynomial of degree 1 (see Proposition <ref>), but 𝔅𝔲𝔯_t is a subobject of τ_1𝔅𝔲𝔯_t (see Proposition <ref>).The category 𝒱𝒫ol_n(𝔐,𝒜) is closed under the translation endofunctor τ_x, under kernel of epimorphism and under extension. Moreover, assuming that there exists a set 𝔈 of objects of 𝔐 such that:∀ m∈ Obj(𝔐),∃{ e_i} _i∈ I∈ Obj(𝔈) (whereis finite),m≅i∈ I♮e_i,then, an object F of 𝐅𝐜𝐭(𝔐,𝒜) belongs to 𝒱𝒫ol_n(𝔐,𝒜) if and only if κ_e(F)=0 and δ_e(F) is an object of 𝒱𝒫ol_n-1(𝔐,𝒜) for all objects e of 𝔈.The first assertion follows from the fact that for all objects x of 𝔐, the endofunctor τ_x commutes with the endofunctors δ_x and κ_x (see Proposition <ref>). For the second and third assertions, let us consider two short exact sequences of 𝐅𝐜𝐭(𝔐,𝒜): 0⟶ G⟶ F_1⟶ F_2⟶0 and 0⟶ F_3⟶ H⟶ F_4⟶0 with F_i a very strong polynomial functor of degree n for all i. Let x be an object of 𝔐. We use the exact sequence (<ref>) of Proposition <ref> to obtain the two following exact sequences in the category 𝐅𝐜𝐭(𝔐,𝒜):0⟶κ_x(G)⟶0⟶0⟶δ_x(G)⟶δ_x(F_1)⟶δ_x(F_2)⟶0; 0⟶0⟶κ_x(H)⟶0⟶δ_x(F_3)⟶δ_x(H)⟶δ_x(F_4)⟶0.Therefore, κ_x(G)=κ_x(H)=0 and the result follows directly by induction on the degree of polynomiality. For the last point, we consider the long exact sequence (<ref>) of Proposition <ref> applied to an object F of 𝒱𝒫ol_n(𝔐,𝒜) to obtain the following exact sequence in the category 𝐅𝐜𝐭(𝔐,𝒜):0⟶κ_y(F)⟶κ_x♮ y(F)⟶τ_xκ_y(F)⟶δ_y(F)⟶δ_x♮ y(F)⟶τ_yδ_x(F)⟶0.Hence, by induction on the length of objects as monoidal product of { e_i} _i∈ I, we deduce that κ_m(F)=0 for all objects m of 𝔐 if and only if κ_e(F)=0 for all objects e of 𝔈. Moreover, since 𝒱𝒫ol_n(𝔐,𝒜) is closed under extension and by the translation endofunctor τ_y, the result follows by induction on the degree of polynomiality n. Let F be an object of 𝐅𝐜𝐭(𝔐,𝒜). The functor F is an object of 𝒱𝒫ol_0(𝔐,𝒜) if and only if it is isomorphic to τ_kF for all natural numbers k.The result follows using the long exact sequence (<ref>) of Proposition <ref> applied to F.The following example show that there exist strong polynomial functors which are not very strong polynomial in any degree.Let us consider the categories 𝔘β and 𝕂-𝔐𝔬𝔡, and n a natural number. Let 𝕂 be considered as an object of 𝕂-𝔐𝔬𝔡 and 0 be the trivial 𝕂-module. Let 𝔄_n be an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡), defined by:* Objects: ∀ m∈ℕ, 𝔄_n(m)=𝕂 if 0otherwise.* Morphisms: let [j-i,f] with f∈𝐁_n be a morphism from i to j in the category 𝔘β. Then: 𝔄_n(f)= id_𝕂 if 0otherwise. The functor 𝔄_n is called an atomic functor in 𝕂 of degree n. For coherence, we fix 𝔄_-1 to be the null functor of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡). Then, it is clear that i_p(𝔄_n) is the zero natural transformation. On the one hand, we deduce the following natural equivalence κ_1(𝔄_n)≅𝔄_n and a fortiori 𝔄_n is not a very strong polynomial functor. On the other hand, it is worth noting the natural equivalence δ_1(𝔄_n)≅τ_1(𝔄_n) and the fact that τ_1(𝔄_n)≅𝔄_n-1. Therefore, we recursively prove that 𝔄_n is a strong polynomial functor of degree n.Contrary to 𝒫ol_n^strong(𝔐,𝒜), a quotient of an object F of 𝒱𝒫ol_n(𝔐,𝒜) is not necessarily a very strong polynomial functor. For example, for 𝔐=𝔘β and 𝒜=𝕂-𝔐𝔬𝔡, let us consider the functor 𝔄_0 defined in Example <ref>, which we proved to be a strong polynomial functor of degree 0. Let 𝔄 be the constant object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) equal to 𝕂. Then, we define a natural transformation α:𝔄→𝔄_0 assigning:∀ n∈ℕ,α_n= id_𝕂 if t_𝕂 otherwise.Moreover, it is an epimorphism in the category 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) since for all natural numbers n, coker(α_n)=0_𝕂-𝔐𝔬𝔡. We proved in Example <ref> that 𝔄_0 is not a very strong polynomial functor of degree 0 whereas 𝔄 is a very strong polynomial functor of degree 0 by Proposition <ref>. Finally, let us remark the following behaviour of the translation functor with respect to very strong polynomial degree.Let d and k be a natural numbers and F be an object of 𝒱𝒫ol_d(𝔐,-𝔐𝔬𝔡). Then the functor τ_k(F) is very strong polynomial of degree equal to that of F.We proceed by induction on the degree of polynomiality of F. First, if we assume that F belongs to 𝒱𝒫ol_0(𝔐,-𝔐𝔬𝔡), then according to Proposition <ref>, τ_k(F)≅ F is a degree 0 very strong polynomial functor. Now, assume that F is a very strong polynomial functor of degree n≥0. Using the commutation properties 5 and 6 of Proposition <ref>, we deduce that (κ_1∘τ_k)(F)≅(τ_k∘κ_1)(F)=0 and (δ_1∘τ_k)(F)≅(τ_k∘δ_1)(F). Since the functor δ_1(F) is a degree n-1 very strong polynomial functor, the result follows from the inductive hypothesis. The previous proof does not work for strong polynomial functors since the initial step fails. Indeed, considering the atomic functor 𝔄_1, which is strong polynomial of degree 1 (see Example <ref>), then τ_2(𝔄_0)=0. §.§ Examples of polynomial functors over 𝔘β The different functors introduced in Section <ref> are strong polynomial functors. Very strong polynomial functors of degree one:Let us first investigate the polynomiality of the functors 𝔅𝔲𝔯_t and 𝔗𝔜𝔐_t.The functors 𝔅𝔲𝔯_t and 𝔗𝔜𝔐_t are very strong polynomial functors of degree 1.For the functor 𝔅𝔲𝔯_t, the proof is mutatis mutandis the same as the one for the dual version considered in <cit.>. We will thus focus on the case of the functor 𝔗𝔜𝔐_t. Let n be a natural number. By Remark <ref>, it is enough to consider the application i_1𝔗𝔜𝔐_𝔱([0,id_n])=ι_ℂ[t^±1]^⊕ n'-n⊕ id_ℂ[t^±1]^⊕ n. This map is a monomorphism and its cokernel is ℂ[t^±1]. Hence κ_1𝔗𝔜𝔐_t is the null functor of 𝐅𝐜𝐭(𝔘β,ℂ[t^±1]-𝔐𝔬𝔡). Let n' be a natural number such that n'≥ n and let [n'-n,σ]∈ Hom_𝔘β(n,n'). By naturality and the universal property of the cokernel, there exists a unique endomorphism of ℂ[t^±1] such that the following diagram commutes, where the lines are exact. It is exactly the definition of δ_1𝔗𝔜𝔐_t([n'-n,σ]).0@->[r]ℂ[t^±1]^⊕ n@->[rrr]^ι_ℂ[t^±1]⊕ id_ℂ[t^±1]^⊕ n@->[d]_𝔗𝔜𝔐([n'-n,σ]) ℂ[t^±1]^⊕ n+1@->[rr]^π_n+1@->[d]^τ_1(𝔗𝔜𝔐)([n'-n,σ])ℂ[t^±1]@->[r]@.>[d]^∃!00@->[r]ℂ[t^±1]^⊕ n'@->[rrr]_ι_ℂ[t^±1]⊕ id_ℂ[t^±1]^⊕ n' ℂ[t^±1]^⊕ n'+1@->[rr]_π_n'+1ℂ[t^±1]@->[r] 0.For all (a,b)∈ℂ[t^±1]⊕ℂ[t^±1]^⊕ n=ℂ[t^±1]^⊕ n+1, τ_1(𝔗𝔜𝔐_t)([n'-n,σ])(a,b)=(a,𝔗𝔜𝔐_t([n'-n,σ])(b)). Therefore, (π_n'+1∘τ_1(𝔗𝔜𝔐_t)([n'-n,σ]))(a,b)=a=π_n+1(a,b). Hence, id_ℂ[t^±1] also makes the diagram commutative and thus δ_1𝔗𝔜𝔐_t([n'-n,σ])=id_ℂ[t^±1]. Hence, δ_1𝔗𝔜𝔐_t is the constant functor equal to ℂ[t^±1]. A fortiori, because of Proposition <ref>, δ_1𝔗𝔜𝔐_t is a very strong polynomial functor of degree 0.The particular case of 𝔅𝔲𝔯_t:Let 𝒯_1:𝔘β⟶ℂ[t^±1]-𝔐𝔬𝔡 be the subobject of the constant functor 𝔛 (see Notation <ref>) such that 𝒯_1(0)=0 and 𝒯_1(n)=ℂ[t^±1] for all non-zero natural numbers n.It follows from Definition <ref> that δ_1𝒯_1≅𝔄_0 (where 𝔄_0 is introduced in Example <ref>). Therefore, 𝒯_1 is a strong polynomial functor of degree 1, but is not very strong polynomial. Nevertheless, it is worth noting that κ_1𝒯_1=0.The functor 𝔅𝔲𝔯 is a strong polynomial functor of degree 2. This functor is not very strong polynomial. More precisely, we have the following short exact sequence in 𝐅𝐜𝐭(𝔘β,ℂ[t^±1]-𝔐𝔬𝔡):0@->[r]𝔅𝔲𝔯_t@->[r]τ_1𝔅𝔲𝔯_t@->[r]𝒯_1@->[r] 0 . The natural transformation i_1(𝔅𝔲𝔯_t)_n:𝔅𝔲𝔯_t(n)→τ_1𝔅𝔲𝔯_t(n) (introduced in Definition <ref>) is defined to be ι_ℂ[t^±1]^⊕ n'-n⊕ id_ℂ[t^±1]^⊕ n-1. Let n≥2 be a natural number. This map is a monomorphism (so κ_1𝔅𝔲𝔯_t=0) and its cokernel is ℂ[t^±1]. Repeating mutatis mutandis the work done in the proof of Proposition <ref>, we deduce that for all [n'-n,σ]∈ Hom_𝔘β(n,n') (with n'≥ n≥2), δ_1𝔅𝔲𝔯_t([n'-n,σ])=Id_ℂ[t^±1]. In addition, since 𝔅𝔲𝔯_t(1)=0 and τ_1𝔅𝔲𝔯_t(1)=ℂ[t^±1], we deduce that δ_1𝔅𝔲𝔯_t(1)=ℂ[t^±1] and for all n'≥1, for all [n'-1,σ]∈ Hom_𝔘β(1,n'), δ_1𝔅𝔲𝔯_t([n'-1,σ])=Id_ℂ[t^±1]. Hence, we prove that δ_1𝔅𝔲𝔯_t≅𝒯_1 where 𝒯_1 is introduced in Definition <ref>. The results follow from the fact that δ_1𝒯_1≅𝔄_0 by Remark <ref>.For formal reasons (see Proposition <ref>), 𝔅𝔲𝔯_t is a subfunctor of τ_1𝔅𝔲𝔯_t. The following proposition illustrates Remarks <ref> and <ref>.The functor τ_1𝔅𝔲𝔯_t is a very strong polynomial functor of degree 1.Repeating mutatis mutandis the work done in the proof of Proposition <ref>, we prove that δ_1τ_1𝔅𝔲𝔯_t is the constant functor equal to ℂ[t^±1] (denoted by 𝔛 in Notation <ref>). Since 𝔛 is a constant functor, δ_1τ_1𝔅𝔲𝔯_t is by Proposition <ref> a very strong polynomial functor of degree 0.A very strong polynomial functor of degree two:We could have defined the unreduced Burau functor of Example <ref> assigning ((ℂ[t^±1])[q^±1])^⊕ n to each object n∈ℕ.Abusing the notation, (ℂ[t^±1])[q^±1]:𝔘β→(ℂ[t^±1])[q^±1]-𝔐𝔬𝔡 denotes the constant functor at (ℂ[t^±1])[q^±1]. The functor 𝔅𝔲𝔯_tℂ[t^±1]⊗(ℂ[t^±1])[q^±1] is denoted by 𝔅̌𝔲̌𝔯̌_̌ť:𝔘β→(ℂ[t^±1])[q^±1]-𝔐𝔬𝔡.These functors (ℂ[t^±1])[q^±1] and 𝔅̌𝔲̌𝔯̌_̌ť are also very strong polynomial of degree one (the proof is exactly the same as the one for 𝔅𝔲𝔯_t in Proposition <ref>).Considering the modified version of the unreduced Burau functor of Remark <ref>, then δ_1𝔏𝔎 is equivalent to 𝔅̌𝔲̌𝔯̌_̌ť.We consider the application i_1𝔏𝔎([0,id_n]). This map is a monomorphism and its cokernel is 2≤ l≤ n+1⊕V_1,l. Let n and n' be two natural numbers such that n'≥ n. Let [n'-n,σ]∈ Hom_𝔘β(n,n'). By naturality and because of the universal property of the cokernel, there exists a unique endomorphism of (ℂ[t^±1])[q^±1]-modules such that the following diagram commutes, where the lines are exact. It is exactly the definition of δ_1𝔏𝔎([n'-n,σ]).0@->[r]1≤ j<k≤ n⊕V_j,k@->[rrr]^𝔏𝔎([1,id_1+n])@->[d]_𝔏𝔎([n'-n,σ]) 1≤ i<l≤ n+1⊕V_i,l@->[rrr]^oooπ_n@->[d]^τ_1(𝔏𝔎)([n'-n,σ]) 2≤ l≤ n+1⊕V_1,l@->[r]@.>[d]^∃!00@->[r]1≤ j'<k'≤ n'⊕V_j',k'@->[rrr]_𝔏𝔎([1,id_1+n']) 1≤ l'≤ n'+1⊕V_i',l'@->[rrr]_oooπ_n' 2≤ l'≤ n'+1⊕V_1,l'@->[r] 0.Let i∈{ 1,…,n-1}, l∈{ 2,…,n+1} and v_1,l be an element of V_1,l. Then we compute:τ_1𝔏𝔎(σ_i)v_1,l=𝔏𝔎(σ_1+i)(v_1,l)= v_1,l if ,tv_1,i+1+(1-t)v_1,i+2-(t^2-t)qv_i+1,i+2 if ,v_1,i+2 if .We deduce that in the canonical basis {𝐞_1,2,𝐞_1,3,…,𝐞_1,n+1} of 2≤ l≤ n+1⊕V_1,l: δ_1𝔏𝔎(σ_i)=Id_i-1⊕[[ 0 t; 1 1-t ]]⊕ Id_n-i-1=𝔅̌𝔲̌𝔯̌_̌ť(σ_i).So as to identify δ_1𝔏𝔎, it remains to consider the action on morphisms of type [1,id_n+1]. According to the definition of the Lawrence-Krammer functor, we have τ_1(𝔏𝔎)([1,id_n+1])=𝔏𝔎(σ_1^-1)∘𝔏𝔎([1,id_n+2]) and:𝔏𝔎(σ_1)(v_1,k)= v_2,k if ,-qt^2v_1,2 if .It follows that for all v_i,l∈ V_i,l with 1≤ i<l≤ n+1:π_n+1∘τ_1(𝔏𝔎)([1,id_n+1])(v_i,l)= v_1,l+1 ifand ,0otherwise.Hence, we deduce that for all 2≤ l≤ n+1, δ_1𝔏𝔎([1,id_n+1])(v_1,l)=v_1,l+1=𝔅̌𝔲̌𝔯̌_̌ť([1,id_n+1])(v_1,l). The functor 𝔏𝔎 is a very strong polynomial functor of degree 2.Let n be a natural number. By Remark <ref>, we only have to consider the application i_1𝔏𝔎([0,id_n]). Since this map is a monomorphism with cokernel 1≤ i≤ n⊕V_i,n+1, κ_1𝔏𝔎 is the null constant functor. Since the functor 𝔅̌𝔲̌𝔯̌_̌ť is very strong polynomial of degree one (following exactly the same proof as the one of Proposition <ref>), we deduce from Lemma <ref> that 𝔏𝔎 is very strong polynomial of degree two. § THE LONG-MOODY FUNCTOR APPLIED TO POLYNOMIAL FUNCTORS Let us move on to the effect of the Long-Moody functors on (very) strong polynomial functors. For this purpose, it is enough by Remark <ref> to consider the cokernel of the map i_1𝐋𝐌. First, we decompose the functor τ_1∘𝐋𝐌 (see Proposition <ref>) so as to understand the behaviour of the image of i_1𝐋𝐌 through this decomposition. This allows us to prove a splitting decomposition of the difference functor (see Theorem <ref>). This is the key point to prove our main results, namely Corollary <ref> and Theorem <ref>. Finally, we give some additional properties of Long-Moody functors with respect to polynomial functors.Let {ς_n:𝐅_n↪𝐁_n+1} _n∈ℕ and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ be coherent families of morphisms (see Definition <ref>), with associated Long-Moody functor 𝐋𝐌_a,ς (see Theorem <ref>), which we fix for all the work of this section (in particular, we omit the "a,ς" from the notation). §.§ Decomposition of the translation functor We introduce two functors which will play a key role in the main result. First, let us recall the following crucial property of the augmentation ideal of a free product of groups, which follows by combining <cit.> and <cit.>.Let G and H be groups. Then, there is a natural 𝕂[G∗ H]-module isomorphism:ℐ_𝕂[G∗ H]≅(ℐ_𝕂[G]𝕂[G]⊗𝕂[G∗ H])⊕(ℐ_𝕂[H]𝕂[H]⊗𝕂[G∗ H]). In the statement of Proposition <ref>, recall that the augmentation ideal ℐ_𝕂[G] (respectively ℐ_𝕂[H]) is a free right 𝕂[G]-module (respectively 𝕂[H]-module) by Proposition <ref>. Moreover, the group ring 𝕂[G∗ H] is a left 𝕂[G]-module (respectively left 𝕂[H]-module) via the morphism id_G∗ι_H:G→ G*H (respectively ι_G∗ id_H:H→ G*H ).Let n and n' be natural numbers such that n'≥ n. We consider the morphism id_𝐅_n*ι_𝐅_n'-n:𝐅_n↪𝐅_n'. This corresponds to the identification of 𝐅_n as the subgroup of 𝐅_n' generated by the n first copies of 𝐅_1 in 𝐅_n'.In addition, the group morphism id_𝐅_n*ι_𝐅_n'-n:𝐅_n↪𝐅_n' canonically induces a 𝕂-module morphism id_ℐ_𝕂[𝐅_n]*ι_ℐ_𝕂[𝐅_n'-n]:ℐ_𝕂[𝐅_n]↪ℐ_𝕂[𝐅_n']. For F an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡), we consider the functor (τ_1∘𝐋𝐌)(F). For all natural numbers n, by Proposition <ref>, we have a 𝕂[𝐅_1+n]-module isomorphism: ℐ_𝕂[𝐅_1+n]𝕂[𝐅_1+n]F(n+2) ≅((ℐ_𝕂[𝐅_1]𝕂[𝐅_1]𝕂[𝐅_1+n])⊕(ℐ_𝕂[𝐅_n]𝕂[𝐅_n]𝕂[𝐅_1+n]))𝕂[𝐅_1+n]F(n+2).Now, by Remark <ref>, the 𝕂[𝐅_n+1]-module F(n+2) is a 𝕂[𝐅_1]-module via F(ς_1+n(id_𝐅_1∗ι_𝐅_n)):𝐅_1→ Aut_𝕂-𝔐𝔬𝔡(F(n+2)) and 𝕂[𝐅_n]-module via F(ς_1+n(ι_𝐅_1∗ id_𝐅_n)):𝐅_n→ Aut_𝕂-𝔐𝔬𝔡(F(n+2)).Therefore, because of the distributivity of tensor product with respect to the direct sum, we have the following proposition.Let F∈ Obj(𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)) and n be a natural number. Then, we have the following 𝕂-module isomorphism:τ_1𝐋𝐌(F)(n)≅ (ℐ_𝕂[𝐅_1]𝕂[𝐅_1]F(n+2))⊕(ℐ_𝕂[𝐅_n]𝕂[𝐅_n]F(n+2)). For all natural numbers n and F∈ Obj(𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)), we denote by* υ(F)_n the monomorphism of 𝕂-modules (id_ℐ_𝕂[𝐅_1]∗ι_ℐ_𝕂[𝐅_n])𝕂[𝐅_1+n]id_F(n+2):ℐ_𝕂[𝐅_1]𝕂[𝐅_1]F(n+2)↪τ_1𝐋𝐌(F)(n),* ξ(F)_n the monomorphism of 𝕂-modules (ι_ℐ_𝕂[𝐅_1]*id_ℐ_𝕂[𝐅_n])𝕂[𝐅_1+n]id_F(n+2):ℐ_𝕂[𝐅_n]𝕂[𝐅_n]F(n+2)↪τ_1𝐋𝐌(F)(n),associated with the direct sum of Proposition <ref>. The aim of this section is in fact to show that this 𝕂-module decomposition leads to a decomposition of τ_1𝐋𝐌 (see Theorem <ref>) as a functor.§.§.§ Additional conditions We need two additional conditions so as to make the decomposition of Proposition <ref> functorial. First, we require the morphisms { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ to satisfy the following property.Let n and n' be natural numbers such that n'≥ n. We require a_1+n'((b_1,n'-n^β)^-1♮ id_n)∘(ι_𝐅_n'-n*id_𝐅_n+1)∘(id_𝐅_1∗ι_𝐅_n)=id_𝐅_1∗ι_𝐅_n'. In other words, the following diagram is commutative: 𝐅_1@->[d]_id_𝐅_1*ι_𝐅_n@->[rrrr]^id_𝐅_1∗ι_𝐅_n'𝐅_1+n' 𝐅_1+n@->[rrrr]_ι_𝐅_n'-n*id_𝐅_1+n𝐅_n'-n*𝐅_1+n≅𝐅_1+n'.@->[u]_a_1+n'((b_1,n'-n^β)^-1♮ id_n) Condition <ref> will be used to define an intermediary functor (see Proposition <ref>). In addition, we will assume that the morphisms { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ satisfy the following condition.Let n and n' be natural numbers such that n'≥ n. We require a_n'(id_n'-n♮-):𝐁_n→ Aut(𝐅_n') maps to the stabilizer of the homomorphism id_𝐅_n'-n*ι_𝐅_n:𝐅_n'-n⟶𝐅_n', ie for all element σ of 𝐁_n the following diagram is commutative: 𝐅_n'-n@->[rr]^id_𝐅_n'-n*ι_𝐅_n@->[dr]_id_𝐅_n'-n*ι_𝐅_n𝐅_n'𝐅_n'.@->[ur]_a_n'(id_n'-n♮σ) Condition <ref> will be used in the proof of Propositions <ref> and <ref>.The relations of Conditions <ref> and <ref> remain true mutatis mutandis, for all natural numbers n, considering the induced morphisms a_n:𝐁_n→ Aut(ℐ_𝕂[𝐅_n]) and id_ℐ_𝕂[𝐅_n]*ι_ℐ_𝕂[𝐅_n'-n]:ℐ_𝕂[𝐅_n]↪ℐ_𝕂[𝐅_n'].If the morphisms { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ also satisfy conditions <ref> and <ref>, the coherent families of morphisms {ς_n:𝐅_n↪𝐁_n+1} _n∈ℕ and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ are said to be reliable.The coherent families of morphisms { a_n,1:𝐁_n→ Aut(𝐅_n)} _n∈ℕ and {ς_n,1:𝐅_n↪𝐁_n+1} _n∈ℕ of Examples <ref> and <ref> are reliable.Recall from Definition <ref> that (b_1,n'-n^β)^-1=σ_1^-1∘σ_2^-1∘⋯∘σ_n'-n^-1. We consider the element e_𝐅_n'-n*g_1*e_𝐅_n=g_n'-n+1∈𝐅_(n'-n)+1+n. The definition of a_n,1 gives that a_1+n',1(σ_n'-n)(g_n'-n)=g_n'-n+1. Therefore, we have that:a_1+n',1(σ_n'-n^-1)(g_n'-n+1)=g_n'-n.Iterating this observation, we deduce that a_1+n'((b_1,n'-n^β)^-1♮ id_n)(g_n'-n+1)=g_1∈𝐅_1+n'. Hence, the family of morphisms { a_n,1:𝐁_n→ Aut(𝐅_n)} _n∈ℕ satisfies Condition <ref>.Similarly to Example <ref> earlier, for all g∈𝐅_n'-n and each Artin generator σ_i∈𝐁_n, a_n'(id_n'-n♮σ_i)(g*e_𝐅_n)=g*e_𝐅_n. Hence, the family of morphisms { a_n,1:𝐁_n→ Aut(𝐅_n)} _n∈ℕ satisfies Condition <ref>.From now until the end of Section <ref>, we fix coherent reliable families of morphisms {ς_n:𝐅_n↪𝐁_n+1} _n∈ℕ and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ.§.§.§ The intermediary functorsThe functor τ_2:Let us consider the factor ℐ_𝕂[𝐅_1]𝕂[𝐅_1]F(n+2) of τ_1𝐋𝐌(F)(n) in the decomposition of Proposition <ref>.For all objects F of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡), for all natural numbers n, we denote ℐ_𝕂[𝐅_1]𝕂[𝐅_1]F(n+2) by (F)(n). Recall the monomorphisms {υ(F)_n:(F)(n)↪τ_1𝐋𝐌(F)(n)} _n∈ℕ of Definition <ref>.Let F be an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡). For all natural numbers n and n' such that n'≥ n, and for all [n'-n,σ]∈ Hom_𝔘β(n,n'), assign: (F)([n'-n,σ])=id_ℐ_𝕂[𝐅_1]𝕂[𝐅_1]F(id_2♮[n'-n,σ]).This defines a subfunctor (F):𝔘β→𝕂-𝔐𝔬𝔡 of τ_1𝐋𝐌(F), using the monomorphisms {υ(F)_n} _n∈ℕ.Let us check that the assignment (F) is well defined with respect to the tensor product. Let n and n' be natural numbers such that n'≥ n, and [n'-n,σ]∈ Hom_𝔘β(n,n') with σ∈𝐁_n'. Recall from Proposition <ref> that id_2♮[n'-n,σ]=[n'-n,(id_2♮σ)∘((b_2,n'-n^β)^-1♮ id_n)]. On the one hand, by Condition <ref>, we have: (id_2♮σ)∘ς_1+n'(g_1)=ς_1+n'(a_1+n'(id_1♮σ)(g_1))∘(id_2♮σ).Hence, it follows from Condition <ref> that (id_2♮σ)∘ς_1+n'(g_1)=ς_1+n'(g_1)∘(id_2♮σ).On the other hand, Condition <ref> gives that g_1=a_2+n'((b_1,n'-n^β)^-1♮ id_n+1)(g_n'-n+1)and by Condition <ref> we have g_1=a_2+n'(id_1♮(b_1,n'-n^β)^-1♮ id_n)(g_1).By the definition of the braiding b_-,-^β (see Definition <ref>), we deduce that:ς_1+n'(g_1) =ς_1+n'(a_2+n'((b_2,n'-n^β)^-1♮ id_n)(g_n'-n+1)).Then, it follows from the combination of Conditions <ref> and <ref> that as morphisms in 𝔘β:[n'-n,ς_1+n'(g_1)∘((b_2,n'-n^β)^-1♮ id_n)]=[n'-n,((b_2,n'-n^β)^-1♮ id_n)∘(id_n'-n♮ς_1+n(g_1))].Hence, we deduce from the relations (<ref>) and (<ref>) that:[n'-n,((id_2♮σ)∘((b_2,n'-n^β)^-1♮ id_n))∘(id_n'-n♮ς_1+n(g_1))]=[n'-n,ς_1+n'(g_1)∘((id_2♮σ)∘((b_2,n'-n^β)^-1♮ id_n))].A fortiori, F(id_2♮[n'-n,σ])∘ F(ς_1+n(g_1))=F(ς_1+n'(g_1))∘ F(id_2♮[n'-n,σ]). Hence, our assignment is well defined with respect to the tensor product.Let us prove that the subspaces (F)(n) are stable under the action of 𝔘β. Let i∈ℐ_𝕂[𝐅_1] and v∈ F(n+2). We deduce from the definition of the monoidal structure morphisms of 𝔘β (see Proposition <ref>) and from the definition of the Long-Moody functor (see Theorem <ref>) that, for all i∈ℐ_𝕂[𝐅_1] and for all v∈ F(n+2):((τ_1𝐋𝐌(F)([n'-n,σ]))∘υ(F)_n)(i𝕂[𝐅_1]v)= a_1+n'(id_1♮σ)(a_1+n'((b_1,n'-n^β)^-1♮ id_n)(ι_ℐ_𝕂[𝐅_n'-n]*id_ℐ_𝕂[𝐅_1]*ι_ℐ_𝕂[𝐅_n])(i))𝕂[𝐅_n'+1]F(id_1♮ id_1♮[n'-n,σ])(v).It follows from Condition <ref> that:a_1+n'((b_1,n'-n^β)^-1♮ id_n)(ι_ℐ_𝕂[𝐅_n'-n]*id_ℐ_𝕂[𝐅_1]*ι_ℐ_𝕂[𝐅_n])(i)=(id_ℐ_𝕂[𝐅_1]*ι_ℐ_𝕂[𝐅_n'])(i).Since by Condition <ref>, a_1+n'(id_1♮σ)(id_ℐ_𝕂[𝐅_1]*ι_ℐ_𝕂[𝐅_n'])(i)=(id_ℐ_𝕂[𝐅_1]*ι_ℐ_𝕂[𝐅_n'])(i) for all elements σ of 𝐁_n', we deduce that:(τ_1𝐋𝐌(F)([n'-n,σ])∘υ(F)_n)(i𝕂[𝐅_1]v) =(υ(F)_n'∘(F)([n'-n,σ]))(i𝕂[𝐅_m]v).Therefore, the functorial structure of τ_1𝐋𝐌(F) induces by restriction the one of (F).Now, we can lift this link between (F) of τ_1𝐋𝐌(F) to endofunctors of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).Let F and G be two objects of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡), and η:F→ G be a natural transformation. For all natural numbers n, assign :((η))_n=id_ℐ_𝕂[𝐅_1]𝕂[𝐅_1]η_n+2.Then we define a subfunctor :𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)→𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) of τ_1𝐋𝐌 using the monomorphisms {υ(F)_n} _n∈ℕ. The consistency of our definition follows repeating mutatis mutandis point (<ref>) of the proof of Theorem <ref>. It directly follows from the definitions of ((η))_n, υ(G)_n and τ_1∘𝐋𝐌 (see Definition <ref>) that υ(G)_n∘()(η)_n=(τ_1∘𝐋𝐌)(η)_n∘υ(F)_n.In fact, we have an easy description of the functor .There is a natural equivalence ≅τ_2 where τ_2 is the translation functor introduced in Definition <ref>.Let F be an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡). By Proposition <ref>, for all natural numbers n, we have an isomorphism:χ_n,F:ℐ_𝕂[𝐅_1]𝕂[𝐅_1]F(n+2)≅⟶F(n+2). (g_1-1)𝕂[𝐅_n]v⟼vIt follows from Definition <ref> and Proposition <ref> that the isomorphisms {χ_n,F} _n∈ℕ define the desired natural equivalence χ→τ_2.The functor 𝐋𝐌∘τ_1:Now, let us consider the part ℐ_𝕂[𝐅_n]𝕂[𝐅_n]F(n+2) of τ_1∘𝐋𝐌(F)(n) in the decomposition of Proposition <ref>. In fact, we are going to prove that these modules assemble to form a functor which identifies with 𝐋𝐌(τ_1F). We recall from Theorem <ref> and Definition <ref> the following fact.The functor 𝐋𝐌∘τ_1:𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)→𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) is defined by:* for F∈ Obj(𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)), ∀ n∈ℕ, (𝐋𝐌∘τ_1)(F)(n)=ℐ_𝕂[𝐅_n]𝕂[𝐅_n]F(n+2), where F(n+2) is a left 𝕂[𝐅_n]-module using F(id_1♮ς_n(-)):𝐅_n→ Aut_𝕂-𝔐𝔬𝔡(F(n+2)). For n,n'∈ℕ, such that n'≥ n, and [n'-n,σ]∈ Hom_𝔘β(n,n'):(𝐋𝐌∘τ_1)(F)([n'-n,σ])=a_n'(σ)(ι_ℐ_𝕂[𝐅_n'-n]*id_ℐ_𝕂[𝐅_n])𝕂[𝐅_n']F(id_1♮ id_1♮[n'-n,σ]). * Morphisms: let F and G be two objects of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡), and η:F→ G be a natural transformation. The natural transformation (𝐋𝐌∘τ_1)(η):(𝐋𝐌∘τ_1)(F)→(𝐋𝐌∘τ_1)(G) for all natural numbers n is given by:((𝐋𝐌∘τ_1)(η))_n=id_ℐ_𝕂[𝐅_n]𝕂[𝐅_n]η_n+2.For all F∈ Obj(𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)), the monomorphisms {ξ(F)_n} _n∈ℕ (see Definition <ref>) allow to define a natural transformation ξ'(F):(𝐋𝐌∘τ_1)(F)→(τ_1∘𝐋𝐌)(F) where, for all natural numbers n: ξ'(F)_n=(ι_ℐ_𝕂[𝐅_1]*id_ℐ_𝕂[𝐅_n])𝕂[𝐅_1+n]F((b_1,1^β)^-1♮ id_n).This yields a natural transformation ξ':𝐋𝐌∘τ_1→τ_1∘𝐋𝐌.Let n and n' be natural numbers such that n'≥ n, and [n'-n,σ]∈ Hom_𝔘β(n,n') with σ∈𝐁_n'. Let i∈ℐ_𝕂[𝐅_n], v∈ F(n+2) and g∈𝐅_n. By Condition <ref> (using Lemma <ref> with n'=n+1) the following equality holds in 𝐁_n+2:((b_1,1^β)^-1♮ id_n)∘(id_1♮ς_n(g))=ς_1+n(e_𝐅_1*g)∘((b_1,1^β)^-1♮ id_n).Recall that F(n+2) is a 𝕂[𝐅_n]-module via F(ς_1+n∘(ι_𝐅_1∗ id_𝐅_n)) and τ_1F(n+1) is a 𝕂[𝐅_n]-module via F(id_1♮(ς_n∘ id_𝐅_n)). Then it follows that the assignment ξ'(F)_n is well-defined with respect to the tensor product structures of (𝐋𝐌∘τ_1)(F)(n) and (τ_1∘𝐋𝐌)(F)(n). Moreover, we compute that:((τ_1∘𝐋𝐌)(F)([n'-n,σ]))∘(ξ'(F)_n)(i𝕂[𝐅_n]v) = a_1+n'(id_1♮σ)(a_1+n'((b_1,n'-n^β)^-1♮ id_n)(ι_ℐ_𝕂[𝐅_1+n'-n]*id_ℐ_𝕂[𝐅_n])(i)) 𝕂[𝐅_n'+1]F((b_1,1^β)^-1♮[n'-n,σ])(v).It follows from Condition <ref> that:a_1+n'((b_1,n'-n^β)^-1♮ id_n)∘(ι_ℐ_𝕂[𝐅_1+n'-n]*id_ℐ_𝕂[𝐅_n])(i)=(ι_ℐ_𝕂[𝐅_1+n'-n]*id_ℐ_𝕂[𝐅_n])(i).Again by Condition <ref>, we deduce that:a_1+n'(id_1♮σ)∘(ι_ℐ_𝕂[𝐅_1+n'-n]*id_ℐ_𝕂[𝐅_n])(i)=ι_ℐ_𝕂[𝐅_1]*a_n'(σ)(ι_ℐ_𝕂[𝐅_n'-n]*id_ℐ_𝕂[𝐅_n])(i).Hence, we deduce that:((τ_1∘𝐋𝐌)(F)([n'-n,σ]))∘(ξ'(F)_n)=(ξ'(F)_n')∘((𝐋𝐌∘τ_1)(F)([n'-n,σ])). Let η:F→ G be a natural transformation in the category 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) and let n be a natural number. Since η is a natural transformation, we have:G((b_1,1^β)^-1♮ id_n)∘η_n+2=η_n+2∘ F((b_1,1^β)^-1♮ id_n).Hence, we deduce from the definitions of τ_1∘𝐋𝐌 (see Theorem <ref>) and of 𝐋𝐌∘τ_1 (see Remark <ref>) that:ξ'(G)_n∘(𝐋𝐌∘τ_1)(η)_n=(τ_1∘𝐋𝐌)(η)_n∘ξ'(F)_n.§.§.§ Splitting of the translation functor Now, we can establish a decomposition result for the translation functor applied to a Long-Moody functor.There is a natural equivalence of endofunctors of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡):τ_1∘𝐋𝐌≅τ_2⊕(𝐋𝐌∘τ_1). Recall the natural transformations υ:→τ_1∘𝐋𝐌 (introduced in Proposition <ref>) and ξ':𝐋𝐌∘τ_1→τ_1∘𝐋𝐌 (defined in Proposition <ref>). The direct sum in the category 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) (induced by the direct sum in the category 𝕂-𝔐𝔬𝔡) allows us to define a natural transformation:υ⊕ξ':⊕(𝐋𝐌∘τ_1)⟶(τ_1∘𝐋𝐌)(F).This is a natural equivalence since for all natural numbers n, we have an isomorphism of 𝕂-modules according to Proposition <ref>: (F)(n)⊕(𝐋𝐌∘τ_1)(F)(n)≅(τ_1∘𝐋𝐌)(F)(n). We conclude using Proposition <ref>.§.§ Splitting of the difference functor Recall the natural transformation i_1:Id_𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)→τ_1 of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡). Our aim is to study the cokernel of i_1∘𝐋𝐌. We recall that for F an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡), for all natural numbers n, (i_1𝐋𝐌)(F)_n=𝐋𝐌(F)([1,id_1+n]) (see Definition <ref>).Explicitly for all elements i of ℐ_𝕂[𝐅_n], for all elements v of F(n):(i_1𝐋𝐌)(F)_n(i𝕂[𝐅_n]v)=(ι_ℐ_𝕂[𝐅_1]*id_ℐ_𝕂[𝐅_n])(i)𝕂[𝐅_1+n]F(id_1♮ι_1♮ id_n)(v).The natural transformation 𝐋𝐌∘ i_1:Let us consider the exact sequence (<ref>) in the category of endofunctors of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) of Proposition <ref>:0@->[r]κ_1@->[r]^Ω_1Id@->[r]^i_1 τ_1@->[r]^_1 δ_1@->[r] 0 .Since the Long-Moody functor is exact (see Proposition <ref>), we have the following exact sequence:0@->[r]𝐋𝐌∘κ_1@->[rr]^ooo𝐋𝐌(Ω_1)𝐋𝐌@->[rr]^𝐋𝐌(i_1)𝐋𝐌∘τ_1@->[rr]^𝐋𝐌(_1)𝐋𝐌∘δ_1@->[r] 0 .From the definition of 𝐋𝐌 (see Theorem <ref>), we deduce that for F an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡), for all natural numbers n, for all elements i of ℐ_𝕂[𝐅_n], for all elements v of F(n):𝐋𝐌(i_1)(F)_n(i𝕂[𝐅_n]v)=i𝕂[𝐅_n]F(ι_1♮ id_1♮ id_n)(v).Recall the natural transformation ξ':𝐋𝐌∘τ_1→τ_1∘𝐋𝐌 introduced in <ref>.As natural transformations from 𝐋𝐌 to τ_1∘𝐋𝐌, which are endofunctors of the category 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡), the following equality holds:ξ'∘(𝐋𝐌(i_1))=i_1𝐋𝐌. Let F be an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡). Let n be a natural number. Let i be an element of ℐ_𝕂[𝐅_n] and let v be an element of F(n). Since (b_1,1^β)^-1∘(ι_1♮ id_1)=id_1♮ι_1 by Definition <ref>, we deduce from Proposition <ref>, Remark <ref> and Remark <ref>, that:(ξ'∘(𝐋𝐌(i_1)))(F)_n(i𝕂[𝐅_n]v)=(id_1*i)𝕂[𝐅_1+n]F(id_1♮ι_1♮ id_n)(v)=(i_1𝐋𝐌)(F)_n(i𝕂[𝐅_n]v). Decomposition results:Lemma <ref> leads to the following key results.There is a natural equivalence in the category 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡):δ_1∘𝐋𝐌≅τ_2⊕(𝐋𝐌∘δ_1).Moreover, there is a natural isomorphism κ_1∘𝐋𝐌≅𝐋𝐌∘κ_1.It follows from the definition of i_1 (see Proposition <ref>) and from Lemma <ref> that the following diagram is commutative and the row is an exact sequence: 0@->[r]κ_1∘𝐋𝐌@->[rr]^oooΩ_1𝐋𝐌𝐋𝐌@->[rr]^i_1𝐋𝐌@=[d] τ_1∘𝐋𝐌@->[rr]^_1𝐋𝐌δ_1∘𝐋𝐌@->[r] 0𝐋𝐌@->[rr]^𝐋𝐌(i_1)𝐋𝐌∘τ_1.@^(->[u]_by Lemma <ref>^ξ'We denote by i_𝐋𝐌∘τ_1^⊕ the inclusion morphism 𝐋𝐌∘τ_1↪τ_2⊕(𝐋𝐌∘τ_1). The functor 𝐋𝐌∘κ_1 is also the kernel of the natural transformation i_𝐋𝐌∘τ_1^⊕∘(𝐋𝐌∘ i_1), as the inclusion morphism i_𝐋𝐌∘τ_1^⊕:𝐋𝐌∘τ_1↪τ_2⊕(𝐋𝐌∘τ_1) is a monomorphism. Then, recalling the exact sequence (<ref>), we obtain that the following diagram is commutative and that the two rows are exact: 0@->[r]κ_1∘𝐋𝐌@->[rr]^oooΩ_1𝐋𝐌𝐋𝐌@->[rr]^i_1𝐋𝐌@=[d] τ_1∘𝐋𝐌@->[rr]^_1𝐋𝐌δ_1∘𝐋𝐌@->[r] 00@->[r]𝐋𝐌∘κ_1@->[rr]_ooo𝐋𝐌(Ω_1)𝐋𝐌@->[rr]_i_𝐋𝐌∘τ_1^⊕∘(𝐋𝐌(i_1))τ_2⊕(𝐋𝐌∘τ_1)@->[rr]_id_τ_2⊕(𝐋𝐌(_1))@->[u]_υ⊕ξ'^by Proposition <ref>τ_2⊕(𝐋𝐌∘δ_1)@->[r] 0A fortiori, by definition of δ_1 (see Definition <ref>) and the universal property of the cokernel, we deduce that:τ_2⊕(𝐋𝐌∘δ_1)≅δ_1∘𝐋𝐌.Furthermore, by the unicity up to isomorphism of the kernel, we conclude that κ_1∘𝐋𝐌≅𝐋𝐌∘κ_1.§.§ Increase of the polynomial degree The results formulated in Theorem <ref> allow us to understand the effect of the Long-Moody functors on (very) strong polynomial functors.Let F be a non-null object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡). If the functor F is strong polynomial of degree d, then:* the functor τ_2(F) belongs to 𝒫ol_d^strong(𝔘β,-𝔐𝔬𝔡);* the functor 𝐋𝐌(F) belongs to 𝒫ol_d+1^strong(𝔘β,-𝔐𝔬𝔡). We prove these two results by induction on the degree of polynomiality. For the first result, it follows from the commutation property 5 of Proposition <ref> for τ_2. For the second result, let us first consider F a strong polynomial functor of degree 0. By Theorem <ref>, we obtain that δ_1𝐋𝐌(F)≅τ_2(F). Therefore 𝐋𝐌(F) is a strong polynomial functor of degree less than or equal to 1. Now, assume that F is a strong polynomial functor of degree n≥0. By Theorem <ref>: δ_1𝐋𝐌(F)≅𝐋𝐌(δ_1F)⊕τ_2(F). By the inductive hypothesis and the result on τ_2, we deduce that 𝐋𝐌(F) is a strong polynomial functor of degree less than or equal to n+1. For all natural numbers d, the endofunctor 𝐋𝐌 of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) restricts to a functor:𝐋𝐌:𝒫ol_d^strong(𝔘β,-𝔐𝔬𝔡)⟶𝒫ol_d+1^strong(𝔘β,-). Let d be a natural number and F be an object of 𝒫ol_d^strong(𝔘β,-𝔐𝔬𝔡) such that the strong polynomial degree of τ_2(F) is equal to d. Then, the functor 𝐋𝐌(F) is a strong polynomial functor of degree equal to d+1.Let d be a natural number and F be an object of 𝒱𝒫ol_d(𝔘β,-𝔐𝔬𝔡) of degree equal to d. Then, the functor 𝐋𝐌(F) is a very strong polynomial functor of degree equal to d+1.Using Lemma <ref>, it follows from Corollary <ref> that 𝐋𝐌(F) is a strong polynomial functor of degree equal to n+1.Since the functor 𝐋𝐌 commutes with the evanescence functor κ_1 by Theorem <ref>, we deduce that (κ_1∘𝐋𝐌)(F)≅(𝐋𝐌∘κ_1)(F)=0. Moreover, using Theorem <ref>, we have:(κ_1∘(δ_1∘𝐋𝐌))(F)≅(κ_1∘τ_2)(F)⊕(κ_1∘(𝐋𝐌∘δ_1))(F).Therefore, the fact that τ_2 commutes with the evanescence functor κ_1 (see the commutation property 6 of Proposition <ref>) and Theorem <ref> together imply that:(κ_1∘(δ_1∘𝐋𝐌))(F)≅(τ_2∘κ_1)(F)⊕(𝐋𝐌∘(κ_1∘δ_1))(F).The result then follows from the fact that F is an object of 𝒱𝒫ol_n(𝔘β,-𝔐𝔬𝔡) and τ_2 is a reduced endofunctor of the category 𝐅𝐜𝐭(𝔘β,-𝔐𝔬𝔡). By Proposition <ref>, 𝔛 is a very strong polynomial functor of degree 0. Now applying the Long-Moody functor 𝐋𝐌_1, we proved in Proposition <ref> that t^-1𝐋𝐌_1(t𝔛) is naturally equivalent to 𝔅𝔲𝔯_t^2, which is very strong polynomial of degree 1 by Proposition <ref>. §.§ Other properties of the Long-Moody functors We have proven in the previoussection that a Long-Moody functor sends (very) strong polynomial functors to (very) strong polynomial functors. We can also prove that a (very) strong polynomial functor in the essential image of a Long-Moody functor is necessarily the image of another strong polynomial functor.Let d be a natural number. Let F be a strong polynomial functor of degree d in the category 𝐅𝐜𝐭(𝔘β,-𝔐𝔬𝔡). Assume that there exists an object G of the category 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) such that 𝐋𝐌(G)=F. Then, the functor G is a strong polynomial functor of degree less than or equal to d+1 in the category 𝐅𝐜𝐭(𝔘β,-𝔐𝔬𝔡).It follows from Theorem <ref> that:δ_1F≅τ_2(G)⊕(𝐋𝐌∘δ_1)(G).According to Corollary <ref>, the functor τ_2(G) is an object of the category 𝒫ol_d-1^strong(𝔘β,-𝔐𝔬𝔡), and because of Lemma <ref> the functor G is an object of the category 𝒫ol_d+1^strong(𝔘β,-𝔐𝔬𝔡). The Long-Moody functor 𝐋𝐌:𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡)⟶𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡) is not essentially surjective.Let l be a natural number. Let E_l:𝔘β⟶𝕂-𝔐𝔬𝔡 be the functor which factorizes through the category ℕ, such that E_l(n)=𝕂^⊕ n^l for all natural numbers n and for all [n'-n,σ]∈ Hom_𝔘β(n,n') (with n, n' natural numbers such that n'≥ n), E_l([n'-n,σ])=ι_ℂ[t^±1]^⊕ n'^l-n^l⊕ id_ℂ[t^±1]^⊕ n^l. In particular, for all natural numbers n, for every Artin generator σ_i of 𝐁_n, E_l(σ_i)=id_𝕂^⊕ n^l. It inductively follows from this definition and direct computations that E_l is a very strong polynomial functor of degree l.Let us assume that 𝐋𝐌 is essentially surjective. Hence, there exists an object F of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡) such that 𝐋𝐌(F)≅ E_l. Because of the definition of 𝐋𝐌(F) on morphisms (see Theorem <ref>), this implies that for all natural numbers n and for all σ∈𝐁_n, a_n(σ)=id_n. Also, if 𝐋𝐌 is essentially surjective, there exists an object T of the category 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡) such that we can recover the Burau functor from 𝐋𝐌(T), ie something like α𝐋𝐌(T) (see Notation <ref>) with α∈𝕂. We deduce from the definition of 𝐋𝐌(T) on objects and morphisms that for all n≥1, T(n)=𝕂 and for all generator σ_i of 𝐁_n:𝐋𝐌(T)(σ_i)=T(σ_i)· Id_n.Then necessarily, for all i∈{ 1,…,n}, T(σ_i)=δ such that δ^2=t and we consider δ^-1𝐋𝐌(T). We deduce that there exists a natural transformation ω:δ^-1𝐋𝐌(T)≅→𝔅𝔲𝔯_t. This contradicts the fact that for all σ∈𝐁_n, a_n(σ)=id_n. The proof of Proposition <ref> shows in particular that a Long-Moody functor 𝐋𝐌 is not essentially surjective on very strong polynomial functors in any degree. In <cit.>, Birman and Brendle ask “whether all finite dimensional unitary matrix representations of 𝐁_n arise in a manner which is related to the construction” recalled in Theorem <ref>. Since the Tong-Yang-Ma and unreduced Burau representations recalled in Theorem <ref> are unitary representations, the proof of Proposition <ref> shows that any Long-Moody functor (and especially the one based on the version of the construction of Theorem <ref>) cannot provide all the functors encoding unitary representations. Therefore, we refine the problem asking whether all functors encoding families of finite dimensional unitary representations of braid groups lie in the image of a Long-Moody functor.Another question is to ask whether we can directly obtain the reduced Burau functor 𝔅𝔲𝔯_t by a Long-Moody functor. Recall that for all natural numbers n, 𝔅𝔲𝔯_t(n)=ℂ[t^±1]^⊕ n-1 and 𝐋𝐌(F)(n)≅(F(n+1))^⊕ n for any Long-Moody functor 𝐋𝐌 and any object F of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡) (see Remark <ref>). Therefore, for dimensional considerations on the objects, it is clear that we have to consider a modified version of the Long-Moody construction. This modification would be to take the tensor product with ℐ_𝐅_n-1 on 𝐅_n-1, the 𝕂-module F(n+1) being a 𝕂[𝐅_n-1]-module using a morphism 𝐅_n-1→(𝐅_n-1a_n'⋊𝐁_n+1)→𝐁_n+1 for all natural numbers n, where a_n':𝐁_n+1→ Aut(𝐅_n-1) is a group morphism. plainIRMA, Universit de Strasbourg, 7 rue Ren Descartes, 67084 Strasbourg Cedex, FranceE-mail address:

http://arxiv.org/abs/1702.08279v6

Understanding Convolution for Semantic Segmentation [ December 30, 2023 ===================================================Understanding Convolution for Semantic Segmentation [ December 30, 2023 =================================================== Let f:^d-1×^d-1→ be a function of the form f(,') = g(,') for g:[-1,1]→. We give a simple proof that shows that poly-size depth two neural networks with (exponentially) bounded weights cannot approximate f whenever g cannot be approximated by a low degree polynomial. Moreover, for many g's, such as g(x)=sin(π d^3x), the number of neurons must be 2^Ω(dlog(d)). Furthermore, the result holds w.r.t. the uniform distribution on ^d-1×^d-1. As many functions of the above form can be well approximated by poly-size depth three networks with poly-bounded weights, this establishes a separation between depth two and depth three networks w.r.t. the uniform distribution on ^d-1×^d-1.§ INTRODUCTION AND MAIN RESULT Many aspects of the expressive power of neural networks has been studied over the years. In particular, separation for deep networks <cit.>, expressive power of depth two networks <cit.>, and more <cit.>. We focus on the basic setting of depth 2 versus depth 3 networks. We ask what functions are expressible (or well approximated) by poly-sized depth-3 networks, but cannot be approximated by an exponential size depth-2 network. Two recent papers <cit.> addressed this issue. Both papers presented a specific function f:^d→ and a distributionon ^d such that f can be approximated w.r.t.by a (d)-size depth 3 network, but not by a (d)-size depth 2 network. In <cit.> this was shown for f being the inner product mod 2 andbeing the uniform distribution on {0,1}^d×{0,1}^d. In <cit.> it was shown for a different (radial) function and some (unbounded) distribution.We extend the above results and prove a similar result for an explicit and rich family of functions, and w.r.t. the uniform distribution on ^d-1×^d-1. In addition, our lower bound on the number of required neurons is stronger: while previous papers showed that the number of neurons has to be exponential in d, we show exponential dependency on dlog(d). Last, our proof is short, direct and is based only on basic Harmonic analysis over the sphere. In contrast, <cit.>'s proof is rather lengthy and requires advanced technical tools such as tempered distributions, while <cit.> relied on the discrepancy of the inner product function mod 2. On the other hand, <cit.> do not put any restriction on the magnitude of the weights, while we and <cit.> do require a mild (exponential) bound. Let us fix an activation function σ:→. For ∈^n we denote σ() = (σ(x_1),…,σ(x_n)). We say that F:^d-1×^d-1→ can be implemented by a depth-2 σ-network of width r and weights bounded by B ifF(,') = w^T_2σ(W_1 + W'_1' + b_1)+b_2 ,where W_1,W'_1∈ [-B,B]^r× d, w_2∈ [-B,B]^r, b_1∈ [-B,B]^r and b_2∈ [-B,B]. Similarly, F:^d-1×^d-1→ can be implemented by a depth-3 σ-network of width r and weights bounded by B ifF(,') = w^T_3σ(W_2σ(W_1 + W'_1' + b_1)+b_2) + b_3for W_1,W'_1∈ [-B,B]^r× d, W_2∈ [-B,B]^r× r, w_3∈ [-B,B]^r, b_1, b_2∈ [-B,B]^r and b_3∈ [-B,B]. DenoteN_d,n=d+n-1d-1-d+n-3d-1=(2n+d-2)(n+d-3)!/n!(d-2)! .Let μ_d be the probability measure on [-1,1] given by dμ_d(x) = Γ(d/2)/√(π)Γ(d-1/2)(1-x^2)^d-3/2dx and defineA_n,d(f) = min_p is degree n-1 polynomialf-p_L^2(μ_d)Our main theorem shows that if A_n,d(f) is large then (,')↦ f(,') cannot be approximated by a small depth-2 network. Let N:^d-1×^d-1→ be any function implemented by a depth-2 σ-network of width r, with weights bounded by B. Let f:[-1,1]→ and define F:^d-1×^d-1→ by F(,') = f(,'). Then, for all n,N - F_L^2(^d-1×^d-1)≥A_n,d(f)(A_n,d(f) - 2rBmax_|x|≤√(4d)B+B|σ(x)|+2B/√(N_d,n)) Let us consider the case that σ(x)=max(0,x) is the ReLU function, f(x) = sin(π d^3x), n=d^2 and B=2^d. In this case, lemma <ref> implies that A_n,d(f) ≥1/5eπ. Hence, to have 1/50e^2π^2-approximation of F, the number of hidden neuorons has to be at least,√(N_d,d^2)/20eπ 2^2d(1+√(4d)) + 2^d+1 = 2^Ω(dlog(d))On the other hand, corollary <ref> implies that F can be ϵ-approximated by a ReLU network of depth 3, width 16π d^5/ϵ and weights bounded by 2π d^3§ PROOFSThroughout, we fix a dimension d. All functions f:^d-1→ and f:^d-1×^d-1→ will be assumed to be square integrable w.r.t. the uniform measure. Likewise, functions f:[-1,1]→ and f:[-1,1]× [-1,1]→ will be assumed to be square integrable w.r.t. μ_d or μ_d×μ_d. Norms and inner products of such functions are of the corresponding L^2 spaces. We will use the fact that μ_d is the probability measure on [-1,1] that is obtained by pushing forward the uniform measure on ^d-1 via the function ↦ x_1. We denote by _n:L^2(μ_d)→ L^2(μ_d) the projection on the complement of the space of degree ≤ n-1 polynomials. Note that A_n,d(f)=_n,df_L^2(μ_d). §.§ Some Harmonic Analysis on the SphereThe d dimensional Legendre polynomials are the sequence of polynomials over [-1,1] defined by the recursion formulaP_n(x)=2n+d-4/n+d-3xP_n-1(x) - n-1/n+d-3P_n-2(x) P_0≡ 1, P_1(x)=xWe also define h_n:S^d-1× S^d-1→ by h_n(,') = √(N_d,n)P_n(,'), and for ∈ S^d-1 we denote L^_n(') = h_n(,'). We will make use of the following properties of the Legendre polynomials. * For every d≥ 2, the sequence {√(N_d,n)P_n} is orthonormal basis of the Hilbert space L^2(μ_d).* For every n, ||P_n||_∞=1 and P_n(1)=1.* L_i^,L_j^' = P_i(,')δ_ij. §.§ Main ResultWe say that f:^d-1×^d-1→ is an inner product function if it has the form f(,') = ϕ(,') for some function ϕ:[-1,1]→. Let _d⊂ L^2(^d-1×^d-1) be the space of inner product functions. We note thatf^2 = __'ϕ^2(,')=_ϕ^2 = ϕ^2Hence, the correspondence ϕ↔ f defines an isomorphism of Hilbert spaces between L^2(μ_d) and _d. In particular, the orthonormal basis {√(N_d,n)P_n}_n=0^∞ is mapped to {h_n}_n=0^∞. In particular,_n(∑_i=0^∞α_ih_i ) = ∑_i=n^∞α_ih_iLet ,'∈^d-1. We say that f:^d-1×^d-1→ is (,')-separable if it has the form f(,') = ψ(,,',') for some ψ:[-1,1]^2→. We note that each neuron implements a separable function.Let _,'⊂ L^2(^d-1×^d-1) be the space of (,')-separable functions. We note thatf^2 = _,'ψ^2(,, ',')=ψ^2Hence, the correspondence ψ↔ f defines an isomorphism of Hilbert spaces between L^2(μ_d×μ_d) and _,'. In particular, the orthonormal basis {√(N_d,n)P_n ⊗√(N_d,m)P_m}_n,m=0^∞ is mapped to {L_n^⊗ L_n^'}_n,m=0^∞.The following theorem implies theorem <ref>, as under the conditions of theorem <ref>, any hidden neuron implement a separable function with norm at most Bmax_|x|≤√(4d)B+B|σ(x)|, and the bias term is a separable function with norm at most B. Let f:^d-1×^d-1→ be an inner product function and let g_1,…,g_r:^d-1×^d-1→ be separable functions. Thenf-∑_i=1^rg_i^2 ≥_nf(_nf - 2∑_i=1^rg_i/√(N_d,n)) We note that_,'h_n(,')L_i^()L_j^'(')= _L_i^()_'h_n(,')L_j^'(') = _L_i^()_'L^_n(')L_j^'(') = δ_nj_L_i^()P_n(,') = δ_nj/√(N_d,n)_L_i^()L_n^'() = δ_njδ_ni P_n(,') /√(N_d,n)Suppose now that f = ∑_i=n^∞α_ih_i and suppose that g = ∑^r_j=1 g_j where each g_j depends only on _j,,'_j,' for some _j,'_j∈ S^d-1.Write g_j(,')=∑_k,l=0^∞β^j_k,lL^_j_k()L^'_j_k('). By equation (<ref>), L^_j_k()L^'_j_l(') is orthogonal to f whever k l. Hence, if we replace each g_j with ∑_k=0^∞β^j_k,kL^_j_k()L^'_j_k('), the l.h.s. of (<ref>) does not increase. Likewise, the r.h.s. does not decrease. Hence, we can assume w.l.o.g. that each g_j is of the form g_j(,')=∑_i=0^∞β^j_iL^_j_i()L^'_j_i('). Now, using (<ref>) again, we have thatf-g^2 = ∑_i=0^∞α_i h_i - ∑_j=1^r β^j_iL^_j_i ⊗ L^'_j_i^2 ≥ ∑_i=n^∞α_i h_i - ∑_j=1^r β^j_iL^_j_i ⊗ L^'_j_i^2 ≥ ∑_i=n^∞α_i^2- 2∑_i=n^∞∑_j=1^rα_i h_i, β^j_iL^_j_i ⊗ L^'_j_i= _nf^2-2∑_i=n^∞∑_j=1^r β^j_iα_iP_i(_j,'_j)/√(N_d,k)≥ _nf^2-2∑^r_j=1∑_i=n^∞|β^j_i| |α_i|/√(N_d,n)≥ _nf^2-2∑^r_j=11/√(N_d,n)√(∑_i=n^∞ |β^j_i|^2)√(∑_i=n^∞ |α_i|^2)≥ _nf^2 - 2_nf∑_j=1^r g_j/√(N_d,n) §.§ Approximating the cosine function Define g_d,m(x) = sin(π√(d)m x). Then, for any d≥ d_0, for a universal constant d_0>0, and for any degree k polynomial p we have∫_-1^1 (g_d,m(x)-p(x))^2dμ_d(x) ≥m-k/4eπ m We have that (e.g. <cit.>) dμ_d(x) = Γ(d/2)/√(π)Γ(d-1/2)(1-x^2)^d-3/2dx. Likewise, for large enough d and |x|< 1/√(d) we have 1-x^2 ≥ e^-2x^2≥ e^-2/d and hence (1-x^2)^d-3/2≥ e^-d-3/d≥ e^-1. Likewise, since Γ(d/2)/Γ(d-1/2)∼√(d/2), we have that for large enough d and |x|≤1/√(d), dμ_d(x)≥√(d)/2eπ. Hence, for f≥ 0 we have∫_-1^1 f(x)dμ_d(x) ≥∫_-d^-1/2^d^-1/2 f(x)dμ_d(x) ≥√(d)/2eπ∫_-d^-1/2^d^-1/2f(x)dx = 1/2eπ∫_-1^1f(t/√(d))dtApplying this equation for f = g_d,m - p we get that∫_-1^1 (g_d,m(x)-p(x))^2dμ_d(x) ≥1/2eπ∫_-1^1 (sin(π m x) - q(x))^2dxWhere q(x):=p(x/√(d)). Now, in the 2m segments I_i = (-1+i-1/m,-1 + i/m),i∈ [2m] we have at least m - k segments on which x↦sin(π m x) and q do not change signs and have opposite signs. On each of these intervals we have ∫_I (sin(π m x) - q(x))^2dx ≥∫^1/m_0sin^2(π m x)dx =1/2m.Let σ(x)=max(x,0) be the ReLU activation, f:[-R,R]→ an L-Lipschitz function, and ϵ>0. There is a functiong(x) = f(0) + ∑_i=1^m α_iσ(γ_i x-β_i)for which g-f_∞≤ϵ. Furthermore, m≤2RL/ϵ, |β_i|≤ R, |α_i|≤ 2L, γ_i∈{-1,1}, and g is L-Lipschitz on all . Let f:[-1,1]→ [-1,1] be an L-Lipschitz function and let ϵ>0. Define F:^d-1×^d-1→ [-1,1] by F(,') = f(,'). There is a function G:^d-1×^d-1→ [-1,1] that satisfies F-G_∞≤ϵ and furthermore G can be implemented by a depth-3 ReLU network of width 16d^2L/ϵ and weights bounded by max(4,2L)By Lemma <ref> there is a depth-2 network _square that calculates x^2/2 in [-2,2], with an error of ϵ/2dL and has width at most 16dL/ϵ and hidden layer weights bounded by 2, and prediction layer weights bounded by 4. For each i∈ [d] we can compose the linear function (,')↦ x_i+x'_i with _square to get a depth-2 network _i that calculates (x_i+x'_i)^2/2 with an error of ϵ/2dL and has the same width and weight bound as _square. Summing the networks _i and subtracting 1 results with a depth-2 network _inner that calculates ,' with an error of ϵ/2L and has width 16d^2L/ϵ and hidden layer weights bounded by 2, and prediction layer weights bounded by 4.Now, again by lemma <ref> there is a depth-2 network _f that calculates f in [-1,1], with an error of ϵ/2, has width at most 2L/ϵ, hidden layer weights bounded by 1 and prediction layer weights bounded by 2L, and is L-Lipschitz. Finally, consider the depth-3 network _F that is the composition of _inner and _f. _F has width at most 16d^2L/ϵ weight bound of max(4,2L), and it satisfies|_F(,') - F(,')|=|_f(_inner(,')) - f(,')| ≤|_f(_inner(,')) - _f(,')| + |_f(,')- f(,')| ≤L|_inner(,') -,'| + ϵ/2≤Lϵ/2L + ϵ/2 = ϵ

http://arxiv.org/abs/1702.08489v1

On the Molecules of Puiseux Monoids]On the molecules of numerical semigroups, Puiseux monoids, and Puiseux algebras Department of Mathematics UC Berkeley Berkeley, CA 94720 Department of Mathematics Harvard University Cambridge, MA 02138 felixgotti@berkeley.edu felixgotti@harvard.edu Department of Mathematics University of Florida Gainesville, FL 32611 marlycormar@ufl.edu [2010]Primary: 20M13, 20M25; Secondary: 13G05, 20M14 A molecule is a nonzero non-unit element of an integral domain (resp., commutative cancellative monoid) having a unique factorization into irreducibles (resp., atoms). Here we study the molecules of Puiseux monoids as well as the molecules of their corresponding semigroup algebras, which we call Puiseux algebras. We begin by presenting, in the context of numerical semigroups, some results on the possible cardinalities of the sets of molecules and the sets of reducible molecules (i.e., molecules that are not irreducibles/atoms). Then we study the molecules in the more general context of Puiseux monoids. We construct infinitely many non-isomorphic atomic Puiseux monoids all whose molecules are atoms. In addition, we characterize the molecules of Puiseux monoids generated by rationals with prime denominators. Finally, we turn to investigate the molecules of Puiseux algebras. We provide a characterization of the molecules of the Puiseux algebras corresponding to root-closed Puiseux monoids. Then we use such a characterization to find an infinite class of Puiseux algebras with infinitely many non-associated reducible molecules. [ Marly Gotti December 30, 2023 =====================§ INTRODUCTIONLet M be a commutative cancellative monoid. A non-invertible element of M is called an atom if it cannot be expressed as a product of two non-invertible elements.If x ∈ M can be expressed as a formal product of atoms, then such a formal product (up to associate and permutation) is called a factorization of x. If every non-invertible element of M has a factorization, then M is called atomic. Furthermore, the atoms and factorizations of an integral domain are the irreducible elements and the formal products of irreducible elements, respectively. All the undefined or informally-defined terms mentioned in this section will be formally introduced later on.The elements having exactly one factorization are crucial in the study of factorization theory of commutative cancellative monoids and integral domains. Aiming to avoid repeated long descriptions, we call such elements molecules. Molecules were first studied in the context of algebraic number theory by W. Narkiewicz and other authors in the 1960's. For instance, in <cit.> and <cit.> Narkiewicz studied some distributional aspects of the molecules of quadratic number fields. In addition, he gave an asymptotic formula for the number of (non-associated) integer molecules of any algebraic number field <cit.>. In this paper, we study the molecules of submonoids of (_≥ 0,+), including numerical semigroups, and the molecules of their corresponding semigroup algebras.A numerical semigroup is a cofinite submonoid of (_0,+), where _0 = {0,1,2,…}. Every numerical semigroup is finitely generated by its set of atoms and, in particular, atomic. In addition, if N ≠_0 is a numerical semigroup, then it contains only finitely many molecules. Notice, however, that every positive integer is a molecule of (_0,+). Figure <ref> shows the distribution of the sets of molecules of four numerical semigroups. We begin Section <ref> pointing out how the molecules of numerical semigroups are related to the Betti elements. Then we show that each element in the set _≥ 4∪{∞} (and only such elements) can be the number of molecules of a numerical semigroup. We conclude our study of molecules of numerical semigroups exploring the possible cardinalities of the sets of reducible molecules (i.e., molecules that are not atoms).A submonoid of (_≥ 0,+) is called a Puiseux monoid. Puiseux monoids were first studied in <cit.> and have been systematically investigated since then (see <cit.> and references therein). Albeit a natural generalization of the class of numerical semigroups, the class of Puiseux monoids contains members having infinitely many atoms and, consequently, infinitely many molecules. A Puiseux monoid is prime reciprocal if it can be generated by rationals of the form a/p, where p is a prime and a is a positive integer not divisible by p. In Section <ref>, we study the sets of molecules of Puiseux monoids, finding infinitely many non-isomorphic Puiseux monoids all whose molecules are atoms (in contrast to the fact that the set of molecules of a numerical semigroup always differs from its set of atoms). In addition, we construct infinitely many non-isomorphic Puiseux monoids having infinitely many molecules that are not atoms (in contrast to the fact that the set of molecules of a nontrivial numerical semigroup is always finite). We conclude Section <ref> characterizing the sets of molecules of prime reciprocal Puiseux monoids. The final section of this paper is dedicated to the molecules of the semigroup algebras of Puiseux monoids, which we call Puiseux algebras. Puiseux algebras have been recently studied in <cit.>. First, for a fixed field F we establish a bijection between the set molecules of a Puiseux monoid and the set of non-associated monomial molecules of its corresponding Puiseux algebra over F. Then we characterize the molecules of Puiseux algebras of root-closed Puiseux monoids. We conclude this paper using the previous characterization to exhibit a class of Puiseux algebras having infinitely many molecules that are neither monomials nor irreducibles.§ MONOIDS, ATOMS, AND MOLECULES §.§ General NotationIn this section we review the nomenclature and main concepts on commutative monoids and factorization theory we shall be using later. For a self-contained approach to the theory of commutative monoids we suggest <cit.> by P. A. Grillet, and for background on non-unique factorization theory of atomic monoids and integral domains the reader might want to consult <cit.> by A. Geroldinger and F. Halter-Koch.We use the double-struck symbols ℕ and ℕ_0 to denote the set of positive integers and the set of nonnegative integers, respectively, while we letdenote the set of primes. If R ⊆ and r ∈ℝ, then we setR_≥ r := {x ∈ R : x ≥ r}.The notation R_> r is used in a similar way. We let the symbol ∅ denote the empty set. If q ∈_> 0, then the unique a,b ∈ such that q = a/b and (a,b)=1 are denoted by 𝗇(q) and 𝖽(q), respectively. For Q ⊆_>0, we call𝗇(Q) := {𝗇(q) : q ∈ Q}and𝖽(Q) := {𝖽(q) : q ∈ Q}the numerator set and denominator set of Q, respectively. In addition, if S is a set consisting of primes and q ∈_> 0, then we set𝖣_S(q) := { p ∈ S : p |𝖽(q) }and𝖣_S(Q) := ∪_q ∈ Q𝖣_S(q).For p ∈, the p-adic valuation on _≥ 0 is the map defined by (q) = (𝗇(q)) - (𝖽(q)) for q ∈ℚ_> 0 and (0) = ∞, where for n ∈ the value (n) is the exponent of the maximal power of p dividing n. It can be easily seen that the p-adic valuation satisfies that (q_1 + … + q_n) ≥min{(q_1), …, (q_n) } for every n ∈ and q_1, …, q_n ∈_> 0.§.§ MonoidsThroughout this paper, we will tacitly assume that the term monoid by itself always refers to a commutative and cancellative semigroup with identity. In addition, we will use additive notation by default and switch to multiplicative notation only when necessary (in which case, the notation will be clear from the context). For a monoid M, we let M^∙ denote the set M ∖{0}. If a,c ∈ M, then we say that a divides c in M and write a |_M c provided that c = a + b for some b ∈ M. We write M = ⟨ S ⟩ when M is generated by a set S. The monoid M is finitely generated if it can be generated by a finite set; otherwise, M is said to be non-finitely generated. A succinct exposition of finitely generated monoids can be found in <cit.>.§.§ Atoms and MoleculesThe set of invertible elements of M is denoted by M^×, and M is said to be reduced if M^× contains only the identity element. An element a ∈ M ∖ M^× is an atom provided that for all u,v ∈ M the fact that a = u + v implies that either u ∈ M^× or v ∈ M^×. The set of atoms of M is denoted by 𝒜(M), and M is called atomic if M = ⟨𝒜(M) ⟩. Let M be a reduced monoid. Then the factorization monoid 𝖹(M) of M is the free commutative monoid on 𝒜(M). The elements of 𝖹(M), which are formal sums of atoms, are called factorizations. If z = a_1 + … + a_n ∈𝖹(M) for some a_1, …, a_n ∈𝒜(M), then |z| := n is called the length of z. As 𝖹(M) is free on 𝒜(M), there is a unique monoid homomorphism from 𝖹(M) to M determined by the assignment a ↦ a for all a ∈𝒜(M). Such a monoid homomorphism is called the factorization homomorphism of M and is denoted by ϕ_M (or just ϕ when there is no risk of ambiguity involved). For x ∈ M, the sets𝖹(x) := 𝖹_M(x) := ϕ^-1(x) ⊆𝖹(M)and 𝖫(x) := 𝖫_M(x) := {|z| : z ∈𝖹(x)}are called the set of factorizations and the set of lengths of x, respectively. Clearly, M is atomic if and only if 𝖹(x) ≠∅ for all x ∈ M.Let M_red denote the set of classes of M under the equivalence relation x ∼ y if y = x + u for some u ∈ M^×. It turns out that M_red is a monoid with the addition operation inherited from M. The monoid M_red is called the reduced monoid of M (clearly, M_red is reduced). Note that an element a belongs to 𝒜(M) if and only if the class of a belongs to 𝒜(M_red). If M is an atomic monoid (that is not necessarily reduced), then we set 𝖹(M) := 𝖹(M_red) and, for x ∈ M, we define 𝖹(x) and 𝖫(x) in terms of 𝖹(M) as we did for the reduced case.As one of the main purposes of this paper is to study elements with exactly one factorization in Puiseux monoids (in particular, numerical semigroups), we introduce the following definition. Let M be a monoid. We say that an element m ∈ M ∖ M^× is a molecule provided that |𝖹(m)| = 1. The set of all molecules of M is denoted by ℳ(M). It is clear that the set of atoms of any monoid is contained in the set of molecules. However, such an inclusion might be proper (consider, for instance, the additive monoid _0). In addition, for any atomic monoid M the set ℳ(M) is divisor-closed in the sense that if m ∈ℳ(M) and m' |_M m for some m' ∈ M ∖ M^×, then m' ∈ℳ(M). If the condition of atomicity is dropped, then this observation is not necessarily true (see Example <ref>).§ MOLECULES OF NUMERICAL SEMIGROUPSIn this section we study the sets of molecules of numerical semigroups, putting particular emphasis on their possible cardinalities. A numerical semigroup is a cofinite additive submonoid of _0. We let 𝒩 denote the class consisting of all numerical semigroups (up to isomorphism). We say that N ∈𝒩 is nontrivial if _0 ∖ N is not empty, and we let 𝒩^∙ denote the class of all nontrivial numerical semigroups. Every N ∈𝒩 has a unique minimal set of generators A, which is finite. The cardinality of A is called the embedding dimension of N. Suppose that N has embedding dimension n, and let N = ⟨ a_1 , …, a_n ⟩ (we always assume that a_1 < … < a_n). Then (a_1, …, a_n) = 1 and 𝒜(N) = {a_1, …, a_n}. In particular, every numerical semigroup is atomic. When N is nontrivial, the maximum of _0 ∖ N is called the Frobenius number of N. Here we let 𝖥(N) denote the Frobenius number of N. See <cit.> for a friendly introduction to numerical semigroups. For k ≥ 1, consider the numerical semigroup N_1 = ⟨ 2, 21 ⟩, whose molecules are depicted in Figure <ref>. It is not hard to see that x ∈ N_1^∙ is a molecule if and only if every factorization of x contains at most one copy of 21. Therefore ℳ(N_1) = {2m + 21n : 0 ≤ m < 21, n ∈{0,1},and(m, n) ≠ (0, 0)}. In addition, if 2m + 21n = 2m' + 21n' for some m,m' ∈{0,…, 20} and n,n' ∈{0,1}, then one can readily check that m = m' and n = n'. Hence |ℳ(N_1)| = 41. §.§ Betty ElementsLet N = ⟨ a_1, …, a_n ⟩ be a minimally generated numerical semigroup. We always represent an element of 𝖹(N) with an n-tuple z = (c_1, …, c_n) ∈^n_0, where the entry c_i specifies the number of copies of a_i that appear in z. Clearly, |z| = c_1 + … + c_n. Given factorizations z = (c_1, …, c_n) and z' = (c'_1, …, c'_n), we define(z,z') = (min{c_1,c'_1}, …, min{c_n,c'_n}).The factorization graph of x ∈ N, denoted by ∇_x(N) (or just ∇_x when no risk of confusion exists), is the graph with vertices 𝖹(x) and edges between those z, z' ∈𝖹(x) satisfying that (z,z') ≠ 0. The element x is called a Betti element of N provided that ∇_x is disconnected. The set of Betti elements of N is denoted by Betti(N). Take N to be the numerical semigroup ⟨ 14, 16, 18, 21, 45 ⟩. A computation in SAGE using thepackage reveals that N has nine Betti elements. In particular, 90 ∈Betti(N). In Figure <ref> one can see the disconnected factorization graph of the Betti element 90 on the left and the connected factorization graph of the non-Betti element 84 on the right. Observe that 0 ∉Betti(N) since |𝖹(0)| = 1. It is well known that every numerical semigroup has finitely many Betti elements. Betti elements play a fundamental role in the study of uniquely-presented numerical semigroups <cit.> and the study of delta sets of BF-monoids <cit.>. For a more general notion of Betti element, meaning the syzygies of an ^n-graded module, see <cit.>. In a numerical semigroup, Betti elements and molecules are closely related. Let N be a numerical semigroup. An element m ∈ N is a molecule if and only if β∤_N m for any β∈Betti(N). For the direct implication, suppose that m is a molecule of N and take α∈ N such that α|_N m. As the set of molecules is closed under division, |𝖹(α)| = 1. This implies that ∇_α is connected and, therefore, α cannot be a Betti element. The reverse implication is just a rephrasing of <cit.>.§.§ On the Sizes of the Sets of MoleculesObviously, for every n ∈ there exists a numerical semigroup having exactly n atoms. The next proposition answers the same realization question replacing the concept of an atom by that of a molecule. Recall that 𝒩^∙ denotes the class of all nontrivial numerical semigroups. {|ℳ(N)| : N ∈𝒩^∙} = _≥ 4. Let N be a nontrivial numerical semigroup. Then N must contain at least two atoms. Let a and b denote the two smallest atoms of N, and assume that a < b. Note that 2a and a+b are distinct molecules that are not atoms. Hence |ℳ(N)| ≥ 4. As a result, {|ℳ(N)| : N ∈𝒩^∙}⊆_≥ 4∪{∞}. Now take x ∈ with x > 𝖥(N) + a b. Since x' := x - a b > 𝖥(N), we see that x' ∈ N and, therefore, 𝖹(x') contains at least one factorization, namely z. So we can find two distinct factorizations of x by adding to z either a copies of b or b copies of a. Then 𝖥(N) + a b is an upper bound for ℳ(N), which means that |ℳ(N)| ∈_≥ 4. Thus, {|ℳ(N)| : N ∈𝒩^∙}⊆_≥ 4. To argue the reverse inclusion, suppose that n ∈_≥ 4, and let us find N ∈𝒩 with |ℳ(N)| = n. For n = 4, we can take the numerical semigroup ⟨ 2, 3 ⟩ (see Figure <ref>). For n > 4, consider the numerical semigroup N = ⟨ n-2, n-1, …, 2(n-2)-1 ⟩. It follows immediately that 𝒜(N) = {n-2, n-1, …, 2(n-2)-1}. In addition, it is not hard to see that 2(n-2), 2(n-2)+1 ∈ℳ(N) while k ∉ℳ(N) for any k > 2(n-2)+1. Consequently, ℳ(N) = 𝒜(N) ∪{2(n-2), 2(n-2)+1}, which implies that |ℳ(N)| = n. Therefore {|ℳ(N)| : N ∈𝒩}⊇_≥ 4, which completes the proof. The monoid (_0,+) is the only numerical semigroup having infinitely many molecules. In Proposition <ref> we have fully described the set {|ℳ(N)| : N ∈𝒩}. A full description of the set {|ℳ(N) ∖𝒜(N)| : N ∈𝒩} seems to be significantly more involved. However, the next theorem offers some evidence to believe that{|ℳ(N) ∖𝒜(N)| : N ∈𝒩} = _≥ 2∪{∞}. The following statements hold. * {|ℳ(N) ∖𝒜(N)| : N ∈𝒩^∙}⊆_≥ 2. * |ℳ(N) ∖𝒜(N)| = 2 for infinitely many numerical semigroups N. * For each k ∈, there is a numerical semigroup N with |ℳ(N) ∖𝒜(N)| > k. To prove (1), take N ∈𝒩^∙. Then we can assume that N has embedding dimension n with n ≥ 2. Take a_1, …, a_n ∈ such that a_1 < … < a_n such that N = ⟨ a_1, …, a_n ⟩. Since a_1 < a_2 < a_j for every j = 3, …,n, the elements 2a_1 and a_1 + a_2 are two distinct molecules of N that are not atoms. Hence ℳ(N) ∖𝒜(N) ⊆_≥ 2∪{∞}. On the other hand, Proposition <ref> guarantees that |ℳ(N)| < ∞, which implies that |ℳ(N) ∖𝒜(N)| < ∞. As a result, the statement (1) follows. To verify the statement (2), one only needs to consider for every n ∈ the numerical semigroup N_n := {0}∪_≥ n-2. The minimal set of generators of N_n is the (n-2)-element set {n-2, n-1, …, 2(n-2)-1 } and, as we have already argued in the proof of Proposition <ref>, the set ℳ(N_n) ∖𝒜(N_n) consists precisely of two elements. Finally, let us prove condition (3). To do this, we first argue that for any a,b ∈_≥ 2 with (a,b) = 1 the numerical semigroup ⟨ a, b ⟩ has exactly ab-1 molecules (cf. Example <ref>). Assume a < b, take N := ⟨ a, b ⟩, and set ℳ = {ma + nb : 0 ≤ m < b, 0 ≤ n < a,and (m, n) ≠ (0, 0)}. Now take x ∈ N to be a molecule of N. As |𝖹(x)| = 1, the unique factorization z := (c_1, c_2) ∈𝖹(x) (with c_1,c_2 ∈_0) satisfies that c_1 < b; otherwise, we could exchange b copies of the atom a by a copies of the atom b to obtain another factorization of x. A similar argument ensures that c_2 < a. As a consequence, ℳ(N) ⊆ℳ. On the other hand, if ma + nb = m'a + n'b for some m,m',n,n' ∈_0, then (a,b) = 1 implies that b | m-m' and a | n-n'. Because of this observation, the element (b-1)a + (a-1)b has only the obvious factorization, namely (b-1,a-1). Since (b-1)a + (a-1)b is a molecule satisfying that y |_N (b-1)a + (a-1)b for every y ∈ℳ, the inclusion ℳ⊆ℳ(N) holds. Hence |ℳ(N)| = |ℳ| = ab-1. To argue the statement (3) now, it suffices to take N := ⟨ 2, 2k+1 ⟩.We conclude this section with the following conjecture. For every n ∈_≥ 2, there exists a numerical semigroup N such that |ℳ(N) ∖𝒜(N)| = n. § MOLECULES OF PUISEUX MONOIDS §.§ Molecules of Generic Puiseux MonoidsIn this section we study the sets of molecules of Puiseux monoids. We will argue that there are infinitely many non-finitely generated atomic Puiseux monoids P such that |ℳ(P) ∖𝒜(P)| = ∞. On the other hand, we will prove that, unlike the case of numerical semigroups, there are infinitely many non-isomorphic atomic Puiseux monoids all whose molecules are, indeed, atoms. The last part of this section is dedicated to characterize the molecules of prime reciprocal Puiseux monoids. A Puiseux monoid is an additive submonoid of _≥ 0. Clearly, every numerical semigroup is naturally isomorphic to a Puiseux monoid. However, Puiseux monoids are not necessarily finitely generated or atomic, as the next example illustrates. The atomic structure of Puiseux monoids has been investigated recently <cit.>. At the end of Section <ref> we mentioned that the set of molecules of an atomic monoid is divisor-closed. The next example indicates that this property may not hold for non-atomic monoids. Consider the Puiseux monoid P = ⟨2/5, 3/5, 1/2^n: n ∈⟩. First, observe that 0 is a limit point of P^∙, and so P cannot be finitely generated. After a few easy verifications, one can see that 𝒜(P) = {2/5, 3/5}. On the other hand, it is clear that 1/2 ∉⟨ 2/5,3/5 ⟩, so P is not atomic. Observe now that 𝖹(1) contains only one factorization, namely 2/5 + 3/5. Therefore 1 ∈ℳ(P). Since 𝖹(1/2) is empty, 1/2 is not a molecule of P. However, 1/2 |_P 1. As a result, ℳ(P) is not divisor-closed. Although the additive monoid _0 contains only one atom, it has infinitely many molecules. The next result implies that _0 is basically the only atomic Puiseux monoid having finitely many atoms and infinitely many molecules. Let P be a Puiseux monoid. Then |ℳ(P)| ∈_≥ 2 if and only if |𝒜(P)| ∈_≥ 2. Suppose first that |ℳ(P)| ∈_≥ 2. As every atom is a molecule, 𝒜(P) is finite. Furthermore, note that if 𝒜(P) = {a}, then every element of the set S = {na : n ∈} would be a molecule, which is not possible as |S| = ∞. As a result, |𝒜(P)| ∈_≥ 2. Conversely, suppose that |𝒜(P)| ∈_≥ 2. Since the elements in P ∖⟨𝒜(P) ⟩ have no factorizations, ℳ(P) = ℳ(⟨𝒜(P) ⟩). Therefore there is no loss in assuming that P is atomic. As 1 < |𝒜(P)| < ∞, the monoid P is isomorphic to a nontrivial numerical semigroup. The proposition now follows from the fact that nontrivial numerical semigroups have finitely many molecules. If P is a Puiseux monoid, then |ℳ(P)| ≠ 1. The set of atoms of a numerical semigroup is always strictly contained in its set of molecules. However, there are many atomic Puiseux monoids which do not satisfy such a property. Before proceeding to formalize this observation, let us mention that if two Puiseux monoids P and P' are isomorphic, then there exists q ∈_>0 such that P' = qP; this is a consequence of <cit.>. There are infinitely many non-isomorphic atomic Puiseux monoids P satisfying that ℳ(P) = 𝒜(P). Let 𝒮 = {S_n : n ∈} be a collection of infinite and pairwise-disjoint sets of primes. Now take S = S_n for some arbitrary n ∈, and label the primes in S strictly increasingly by p_1, p_2, …. Recall that 𝖣_S(r) denotes the set of primes in S dividing 𝖽(r) and that 𝖣_S(R) = ∪_r ∈ R𝖣_S(r) for R ⊆_> 0. We proceed to construct a Puiseux monoid P_S satisfying that 𝖣_S(P_S) = S. Take P_1 := ⟨ 1/p_1 ⟩ and P_2 := ⟨ P_1, 2/(p_1p_2) ⟩. In general, suppose that P_k is a finitely generated Puiseux monoid such that 𝖣_S(P_k) ⊂ S, and let r_1, …, r_n_k be all the elements in P_k which can be written as a sum of two atoms. Clearly, n_k ≥ 1. Because |S| = ∞, one can take p'_1, …, p'_n_k to be primes in S ∖𝖣_S(P_k) satisfying that p'_i ∤𝗇(r_i). Now consider the following finitely generated Puiseux monoid P_k+1 := ⟨ P_k ∪{r_1/p'_1, …, r_n_k/p'_n_k}⟩. For every i ∈{1,…,n_k}, there is only one element in P_k ∪{r_1/p'_1, …, r_n_k/p'_n_k} whose denominator is divisible by p'_i, namely r_i/p'_i. Therefore r_i/p'_i ∈𝒜(P_k+1) for i=1, …, n_k. To check that 𝒜(P_k) ⊂𝒜(P_k+1), fix a ∈𝒜(P_k) and take z := ∑_i=1^m α_i a_i + ∑_i=1^n_kβ_i r_i/p'_i∈𝖹_P_k+1(a), where a_1, …, a_m are pairwise distinct atoms in 𝒜(P_k+1) ∩ P_k and α_i, β_j are nonnegative coefficients for i = 1,…,m and j = 1,…, n_k. In particular, a_1, …, a_m ∈𝒜(P_k). For each i=1,…,n_k, the fact that the p'_i-adic valuation of a is nonnegative implies that p'_i |β_i. Hence a = ∑_i=1^m α_i a_i + ∑_i=1^n_kβ'_i r_i, where β'_i = β_i/p'_i ∈_0. Since r_i ∈𝒜(P_k) + 𝒜(P_k) and (β_i/p'_i)r_i |_P_k a for every i=1,…,n_k, one obtains that β_1 = … = β_n_k = 0. As a result, a = ∑_i=1^m α_i a_i. Because a ∈𝒜(P_k), the factorization ∑_i=1^m α_i a_i in 𝖹_P_k(a) must have length 1, i.e, ∑_i=1^m α_i = 1. Thus, ∑_i=1^m α_i + ∑_i=1^n_kβ_i = 1, which means that z has length 1 and so a ∈𝒜(P_k+1). As a result, the inclusion 𝒜(P_k) ⊆𝒜(P_k+1) holds. Observe that because n_k ≥ 1, the previous containment must be strict. Now set P_S = ⋃_k ∈ P_k. Let us verify that P_S is an atomic monoid satisfying that 𝒜(P_S) = ∪_k ∈𝒜(P_k). Since P_k is atomic for every k ∈, the inclusion chain 𝒜(P_1) ⊂𝒜(P_2) ⊂… implies that P_1 ⊂ P_2 ⊂…. In addition, if a_0 = a_1 + … + a_m for m ∈ and a_0, a_1, …, a_m ∈ P_S, then a_0 = a_1 + … + a_m will also hold in P_k for some k ∈ large enough. This immediately implies that ∪_k ∈𝒜(P_k) ⊆𝒜(P_S). Since the reverse inclusion follows trivially, 𝒜(P_S) = ∪_k ∈𝒜(P_k). To check that P_S is atomic, take x ∈ P_S^∙. Then there exists k ∈ such that x ∈ P_k and, because P_k is atomic, x ∈⟨𝒜(P_k) ⟩⊆⟨𝒜(P_S) ⟩. Hence P_S is atomic. To check that ℳ(P_S) = 𝒜(P_S), suppose that m is a molecule of P_S, and then take K ∈ such that m ∈ P_k for every k ≥ K. Since 𝒜(P_k) ⊂𝒜(P_k+1) ⊂…, we obtain that 𝖹_P_k(m) ⊆𝖹_P_k+1(m) ⊆…. Moreover, ∪_k ≥ K𝒜(P_k) = 𝒜(P_S) implies that ∪_k ≥ K𝖹_P_k(m) = 𝖹_P_S(m). Now suppose for a contradiction that m = ∑_j=1^i a_j for i ∈_≥ 2, where a_1, …, a_i ∈𝒜(P_S). Take j ∈_≥ K such that a_1, …, a_i ∈𝒜(P_j). Then the way in which P_j+1 was constructed ensures that |𝖹_P_j+1(a_1 + a_2)| ≥ 2 and, therefore, |𝖹_P_j+1(m)| ≥ 2. As 𝖹_P_j+1(m) ⊆𝖹_P_S(m), it follows that |𝖹_P_S(m)| ≥ 2, which contradicts that m is a molecule. Hence ℳ(P_S) = 𝒜(P_S). Finally, we argue that the monoids constructed are not isomorphic. Let S and S' be two distinct members of the collection 𝒮 and suppose, by way of contradiction, that ψ P_S → P_S' is a monoid isomorphism. Because the only homomorphisms of Puiseux monoids are given by rational multiplication, there exists q ∈_> 0 such that P_S' = q P_S. In this case, all but finitely many primes in 𝖣_(P_S) belong to 𝖣_(P_S'). Since 𝖣_(P_S) ∩𝖣_(P_S') = ∅ when S ≠ S', we get a contradiction.§.§ Molecules of Prime Reciprocal MonoidsFor the remaining of this section, we focus our attention on the class consisting of all prime reciprocal monoids. Let S be a nonempty set of primes. A Puiseux monoid P is prime reciprocal over S if there exists a set of positive rationals R such that P = ⟨ R ⟩, 𝖽(R) = S, and 𝖽(r) = 𝖽(r') implies r = r' for all r,r' ∈ R. Within the scope of this paper, the term prime reciprocal monoid refers to a Puiseux monoid that is prime reciprocal over some nonempty set of primes. Let us remark that if a Puiseux monoid P is prime reciprocal, then there exists a unique S ⊆ such that P is prime reciprocal over S. It is easy to verify that every prime reciprocal Puiseux monoid is atomic. There exist infinitely many non-finitely generated atomic Puiseux monoids P such that |ℳ(P) ∖𝒜(P)| = ∞. As in the proof of Theorem <ref>, let 𝒮 = {S_n : n ∈} be a collection of infinite and pairwise-disjoint subsets of ∖{2}. For every n ∈, let P_n be a prime reciprocal Puiseux monoid over S_n. Fix a ∈𝒜(P_n), and take a factorization z := ∑_i=1^k α_i a_i∈𝖹(2a) for some k ∈, pairwise distinct atoms a_1, …, a_k, and α_1,…,α_k ∈_0. Since 𝖽(a) ≠ 2, after applying the 𝖽(a)-adic valuation on both sides of the equality 2a = ∑_i=1^t α_i a_i, one obtains that z = 2a. So 2a ∈ℳ(P_n) ∖𝒜(P_n) and, as a result, |ℳ(P_n) ∖𝒜(P_n)| = ∞. Now suppose, by way of contradiction, that P_i ≅ P_j for some i,j ∈ with i ≠ j. Since isomorphisms of Puiseux monoids are given by rational multiplication, there exists q ∈_> 0 such that P_j = q P_i. However, this implies that only finitely many primes in 𝖽(P_i) are not contained in 𝖽(P_j), which contradicts that S_i ∩ S_j = ∅. Hence no two monoids in {P_n : n ∈} are isomorphic, and the proposition follows. Theorem <ref> and Proposition <ref> guarantee the existence of infinitely many non-finitely generated atomic Puiseux monoids P and Q with |ℳ(P) ∖𝒜(P)| = 0 and |ℳ(Q) ∖𝒜(Q)| = ∞.(cf. Conjecture <ref>) For every n ∈ there exists a non-finitely generated atomic Puiseux monoid P satisfying that |ℳ(P) ∖𝒜(P)| = n.Before characterizing the molecules of prime reciprocal monoids, let us introduce the concept of maximal multiplicity. Let P be a Puiseux monoid. For x ∈ P and a ∈𝒜(P) we define the maximal multiplicity of a in x to be𝗆(a,x) := max{n ∈_0 : na |_P x}. Let P be a prime reciprocal monoid, and let x ∈ P. If 𝗆(a,x) < 𝖽(a) for all a ∈𝒜(P), then x ∈ℳ(P). Suppose, by way of contradiction, that x ∉ℳ(P). Then there exist k ∈, elements α_i, β_i ∈_0 (for i=1,…, k), and pairwise distinct atoms a_1, …, a_k such that z:= ∑_i=1^k α_i a_i andz' := ∑_i=1^k β_i a_i are two distinct factorizations in 𝖹(x). As z ≠ z', there is an index i ∈{1,…,k} such that α_i ≠β_i. Now we can apply the 𝖽(a_i)-adic valuation to both sides of the equality ∑_i=1^k α_i a_i = ∑_i=1^k β_i a_i to verify that 𝖽(a_i) |β_i - α_i. As α_i ≠β_i, we obtain that 𝗆(a_i, x) ≥max{α_i, β_i}≥𝖽(a_i). However, this contradicts the fact that 𝗆(a,x) < 𝖽(a) for all a ∈𝒜(P). As a consequence, x ∈ℳ(P). For S ⊆, we call the monoid E_S := ⟨ 1/p : p ∈ S ⟩ the elementary prime reciprocal monoid over S; if S = we say that E_S is the elementary prime reciprocal monoid. It was proved in <cit.> that every submonoid of the elementary prime reciprocal monoid is atomic. This gives a large class of non-finitely generated atomic Puiseux monoids, which contains each prime reciprocal monoid. Let S be an infinite set of primes, and let E_S be the elementary prime reciprocal monoid over S. For x ∈ E_S, the following conditions are equivalent: * x ∈ℳ(E_S); * 1 does not divide x in E_S; * 𝗆(a,x) < 𝖽(a) for all a ∈𝒜(E_S); * If a_1, …, a_n ∈𝒜(E_S) are distinct atoms and α_1, …, α_n ∈_0 satisfy that ∑_j=1^n α_j a_j ∈𝖹(x), then α_j < 𝖽(a_j) for each j = 1,…,n. First, let us recall that since E_S is atomic, ℳ(E_S) is divisor-closed. On the other hand, note that for any two distinct atoms a,a' ∈𝒜(E_S), both factorizations 𝖽(a) a and 𝖽(a') a' are in 𝖹(1). Therefore 1 ∉ℳ(E_S). Because the set of molecules of E_S is divisor-closed, 1 ∤_E_S m for any m ∈ℳ(E_S); in particular, 1 ∤_E_S x. Thus, (1) implies (2). If 𝗆(a,x) ≥𝖽(a) for a ∈𝒜(E_S), then x = 𝗆(a,x) a + y = 1 + (𝗆(a,x) - 𝖽(a)) a + y for some y ∈ E_S. As a result, 1 |_E_S x, from which we can conclude that (2) implies (3). It is obvious that (3) and (4) are equivalent conditions. Finally, the fact that (3) implies (1) follows from Proposition <ref>. Let S be an infinite set of primes, and let E_S be the elementary prime reciprocal monoid over S. Then |𝖹(x)| = ∞ for all x ∉ℳ(E_S). In order to describe the set of molecules of an arbitrary prime reciprocal monoid, we need to cast its atoms into two categories. Let P be a prime reciprocal monoid. We say that a ∈𝒜(P) is stable if the set {a' ∈𝒜(P) : 𝗇(a') = 𝗇(a)} is infinite, otherwise we say that a is unstable. If every atom of P is stable (resp., unstable), then we call P stable (resp., unstable). For a prime reciprocal monoid P, we let 𝒮(P) denote the submonoid of P generated by the set of stable atoms. Similarly, we let 𝒰(P) denote the submonoid of P generated by the set of unstable atoms. Clearly, P is stable (resp., unstable) if and only if P = 𝒮(P) (resp., P = 𝒰(P)). In addition, P = 𝒮(P) + 𝒰(P), and 𝒮(P) ∩𝒰(P) is trivial only when either 𝒮(P) or 𝒰(P) is trivial. Clearly, if P is stable, then it cannot be finitely generated. Finally, we say that u ∈𝒰(P) is absolutely unstable provided that u is not divisible by any stable atom in P, and we let 𝒰^a(P) denote the set of all absolutely unstable elements of P. Let {p_n} be the strictly increasing sequence with underlying set ∖{2}, and consider the prime reciprocal monoid P defined as P := ⟨3 + (-1)^n/p_2n-1, p_2n - 1/p_2n :n ∈⟩. Set a_n = 3 + (-1)^n/p_2n-1 and b_n =p_2n - 1/p_2n. One can readily verify that P is an atomic monoid with 𝒜(P) = {a_n, b_n : n ∈}. As both sets {n ∈ : 𝗇(a_n) = 2} and {n ∈ : 𝗇(a_n) = 4} have infinite cardinality, a_n is a stable atom for every n ∈. In addition, since {𝗇(b_n)} is a strictly increasing sequence bounded below by 𝗇(b_1) = 4 and 𝗇(a_n) ∈{2,4}, the element b_n is an unstable atom for every n ∈_≥ 2. Also, notice that 4/3 = 2a_1 ∈𝒮(P), but 4/3 ∉𝒰(P) because 𝖽(4/3) = 3 ∉𝖽(𝒰(P)). Furthermore, for every n ∈ the element u_n := (p_2n - 1) b_n ∈𝒰(P) is not in 𝒮(P) because p_2n = 𝖽(u_n) ∉𝖽(𝒮(P)). However, 𝒮(P) ∩𝒰(P) ∅ since the element 4 = 6 a_1 = 5 b_1 belongs to both 𝒮(P) and 𝒰(P). Finally, we claim that 2b_n is absolutely unstable for every n ∈. If this were not the case, then 2b_k ∉ℳ(P) for some k ∈. By Proposition <ref> there exists a ∈𝒜(P) such that 𝗆(a, 2b_k) ≥𝖽(a). In this case, one would obtain that 2b_k ≥𝗆(a,2b_k)a ≥𝖽(a)a = 𝗇(a) ≥ 2, contradicting that b_n < 1 for every n ∈. Thus, 2b_n ∈𝒰^a(P) for every n ∈. Let P be a prime reciprocal monoid that is stable, and let x ∈ P. Then x ∈ℳ(P) if and only if 𝗇(a) does not divide x in P for any a ∈𝒜(P). For the direct implication, assume that x ∈ℳ(P) and suppose, by way of contradiction, that 𝗇(a) |_P x for some a ∈𝒜(P). Since a is a stable atom, there exist p_1, p_2 ∈ with p_1 ≠ p_2 such that (p_1 p_2, 𝗇(a)) = 1 and 𝗇(a)/p_1, 𝗇(a)/p_2 ∈𝒜(P). As 𝗇(a) |_P x, we can take a_1, …, a_k ∈𝒜(P) such that x = 𝗇(a) + a_1 + … + a_k. Therefore p_1 𝗇(a)/p_1 + a_1 + … + a_k andp_2 𝗇(a)/p_2 + a_1 + … + a_k are two distinct factorizations in 𝖹(x), contradicting that x is a molecule. Conversely, suppose that x is not a molecule. Consider two distinct factorizations z := ∑_i=1^k α_i a_i and z' := ∑_i=1^k β_i a_i in 𝖹(x), where k ∈, α_i, β_i ∈_0, and a_1, …, a_k ∈𝒜(P) are pairwise distinct atoms. Pick an index j ∈{1, …, k} such that α_j ≠β_j and assume, without loss of generality, that α_j < β_j. After applying the 𝖽(a_j)-adic valuations on both sides of the equality ∑_i=1^k α_i a_i = ∑_i=1^k β_i a_i one finds that the prime 𝖽(a_j) divides β_j - α_j. Therefore β_j > 𝖽(a_j) and so x = 𝗇(a_j) + (β_j - 𝖽(a_j))a_j + ∑_i ≠ jα_i a_i. Hence 𝗇(a_j) |_P x, which concludes the proof. Observe that the reverse implication of Proposition <ref> does not require that the equality 𝒮(P) = P holds. However, the stability of P is required for the direct implication to hold as the following example illustrates. Let {p_n} be the strictly increasing sequence with underlying set ∖{2}, and consider the unstable prime reciprocal monoid P := ⟨1/2, p_n^2 -1/p_n: n ∈⟩. Because the smallest two atoms of P are 1/2 and 8/3, it immediately follows that m := 2(1/2) + 8/3 ∉⟨ 1/2 ⟩ must be a molecule of P. In addition, notice that 1 = 𝗇(1/2) divides m in P. We conclude this section characterizing the molecules of prime reciprocal monoids. Let P be a prime reciprocal monoid. Then x ∈ P is a molecule if and only if x = s + u for some s ∈𝒮(P) ∩ℳ(P) and u ∈𝒰^a(P) ∩ℳ(P). First, suppose that x is a molecule. As P = 𝒮(P) + 𝒰(P), there exist s ∈𝒮(P) and u ∈𝒰(P) such that x = s + u. The fact that x ∈ℳ(P) guarantees that s,u ∈ℳ(P). On the other hand, since |𝖹(u)| = 1 and u can be factored using only unstable atoms, u cannot be divisible by any stable atom in P. Thus, u ∈𝒰^a(P), and the direct implication follows. For the reverse implication, assume that x = s + u, where s ∈𝒮(P) ∩ℳ(P) and u ∈𝒰^a(P) ∩ℳ(P). We first check that x can be uniquely expressed as a sum of two elements s and u contained in the sets 𝒮(P) ∩ℳ(P) and 𝒰^a(P) ∩ℳ(P), respectively. To do this, suppose that x = s + u = s' + u', where s' ∈𝒮(P) ∩ℳ(P) and u' ∈𝒰^a(P) ∩ℳ(P). Take pairwise distinct stable atoms a_1, …, a_k of P for some k ∈ such that z = ∑_i=1^k α_i a_i ∈𝖹_P(s) and z' = ∑_i=1^k α'_i a_i ∈𝖹_P(s'), where α_j, α'_j ∈_0 for j = 1, …, k. Because u and u' are absolutely unstable elements, they are not divisible in P by any of the atoms a_i's. Thus, 𝖽(a_j) ∤𝖽(u) and 𝖽(a_j) ∤𝖽(u') for any j ∈{1,…, k}. Now for each j = 1,…,k we can apply the 𝖽(a_j)-adic valuation in both sides of the equality u + ∑_i=1^k α_i a_i= u' + ∑_i=1^k α'_i a_i to conclude that the prime 𝖽(a_j) must divide α_j - α'_j. Therefore either z = z' or there exists j ∈{1,…,k} such that |α_j - α'_j| > 𝖽(a_j). Suppose that |α_j - α'_j| > 𝖽(a_j) for some j, and say α_j > α'_j. As α_j > 𝖽(a_j), one can replace α_j a_j by (α_j - 𝖽(a_j))a_j + 𝗇(a_j) in s = ϕ(z) = α_1 a_1 + … + α_k a_k to find that 𝗇(a_j) divides s in 𝒮(P), which contradicts Proposition <ref>. Then z = z'. Therefore s' = s and u' = u. Finally, we argue that x ∈ℳ(P). Write x = ∑_i=1^ℓγ_i a_i + ∑_i=1^ℓβ_i b_i for ℓ∈_≥ k, pairwise distinct stable atoms a_1, …, a_ℓ (where a_1, …, a_k are the atoms showing up in z), pairwise distinct unstable atoms b_1, …, b_ℓ, and coefficients γ_i, β_i ∈_0 for every i=1,…,ℓ. Set z”' := ∑_i=1^ℓγ_i a_i and w”' = ∑_i=1^ℓβ_i b_i. Note that, a priori, ϕ(z”') and ϕ(w”') are not necessarily molecules.As in the previous paragraph, we can apply 𝖽(a_j)-adic valuation to both sides of the equality u + ∑_i=1^k α_i a_i = ∑_i=1^ℓγ_i a_i + ∑_i=1^ℓβ_i b_i to find that z”' = z. Hence ϕ(z”') = s and ϕ(w”') = u are both molecules. Therefore z”' must be the unique factorization of s, while w”' must be the unique factorization of u. As a result, x ∈ℳ(P). § MOLECULES OF PUISEUX ALGEBRASLet M be a monoid and let R be a commutative ring with identity. Then R[X;M] denotes the ring of all functions fM → R having finite support, which means that (f) := {s ∈ M : f(s) ≠ 0 } is finite. We represent an element f ∈ R[X;M] byf(X) = ∑_i=1^n f(s_i)X^s_i,where s_1, …, s_n are the elements in (f). The ring R[X;M] is called the monoid ring of M over R, and the monoid M is called the exponent monoid of R[X;M]. For a field F, we will say that F[X;M] is a monoid algebra. As we are primarily interested in the molecules of monoid algebras of Puiseux monoids, we introduce the following definition. If F is a field and P is a Puiseux monoid, then we say that F[X;P] is a Puiseux algebra. If N is a numerical semigroup, then F[X;N] is called a numerical semigroup algebra. Let F[X;P] be a Puiseux algebra. We write any element f ∈ F[X;P] ∖{0} in canonical representation, that is, f(X) = α_1 X^q_1 + … + α_k X^q_k with α_i ≠ 0 for every i = 1, …, k and q_1 > … > q_k. It is clear that any element of F[X;P] ∖{0} has a unique canonical representation. In this case, (f) := q_1 is called the degree of f, and we obtain that the degree identity (fg) = (f) + (g) holds for all f, g ∈ F[X;P] ∖{0}. As for polynomials, we say that f is a monomial if k = 1. It is not hard to verify that F[X;P] is an integral domain with group of units F^×, although this follows from <cit.> and <cit.>. Finally, note that, unless P ≅ (_0,+), no monomial of F[X;P] can be a prime element; this is a consequence of the trivial fact that non-cyclic Puiseux monoids do not contain prime elements.For an integral domain R, we let R_red denote the reduced monoid of the multiplicative monoid of R. Let R be an integral domain. We call a nonzero non-unit r ∈ R a molecule if rR^× is a molecule of R_red. Let R be an integral domain. By simplicity, we let 𝒜(R), ℳ(R), 𝖹(R), and ϕ_R denote 𝒜(R_red), ℳ(R_red), 𝖹(R_red), and ϕ_R_red, respectively. In addition, for a nonzero non-unit r ∈ R, we let 𝖹_R(r) and 𝖫_R(r) denote 𝖹_R_red(rR^×) and 𝖫_R_red(rR^×), respectively. Let F be a field, and let P be a Puiseux monoid. For a nonzero α∈ F, a monomial X^q ∈ℳ(F[X;P]) if and only if q ∈ℳ(P). Consider the canonical monoid monomorphism μ P → F[X;P] ∖{0} given by μ(q) = X^q. It follows from <cit.> that an element a ∈ P is an atom if and only if the monomial X^a is irreducible in F[X;P] (or, equivalently, an atom in the reduced multiplicative monoid of F[X;P]). Therefore μ lifts canonically to the monomorphism μ̅𝖹(P) →𝖹(F[X;P]) determined by the assignments a ↦ X^a for each a ∈𝒜(P), preserving not only atoms but also factorizations of the same element. Put formally, this means that the diagram 𝖹(P) @>μ̅>> 𝖹(F[X;P]) @Vϕ_PVV @Vϕ_F[X;P]VV P @>μ>> F[X;P]_red commutes, and the (fiber) restriction maps μ̅_q 𝖹_P(q) →𝖹_F[X;P](X^q) of μ̅ are bijections for every q ∈ P. Hence |𝖹_P(q)| = 1 if and only if |𝖹_F[X;P](X^q)| = 1 for all q ∈ P^∙, which concludes our proof. For each field F, there exists an atomic Puiseux monoid P whose Puiseux algebra satisfies that |ℳ(F[X;P]) ∖𝒜(F[X;P])| = ∞. It is an immediate consequence of Proposition <ref> and Proposition <ref>. The difference group (M) of a monoid M is the abelian group (unique up to isomorphism) satisfying that any abelian group containing a homomorphic image of M will also contain a homomorphic image of (M). An element x ∈(M) is called a root element of M if nx ∈ M for some n ∈. The subset M of (M) consisting of all root elements of M is called the root closure of M. If M = M, then M is called root-closed. From now on, we assume that each Puiseux monoid P we mention here is root-closed. Before providing a characterization for the irreducible elements of F[X;P], let us argue the following two easy lemmas. Let P be a Puiseux monoid. Then 𝖽(P^∙) is closed under taking least common multiples. Take d_1, d_2 ∈𝖽(P^∙) and q_1, q_2 ∈ P^∙ with 𝖽(q_1) = d_1 and 𝖽(q_2) = d_2. Now set d = (d_1, d_2) and n = (𝗇(q_1), 𝗇(q_2)). It is clear that n is the greatest common divisor of (d_2/d) 𝗇(q_1) and (d_1/d) 𝗇(q_2). So there exist m ∈ and c_1, c_2 ∈_0 such that n (1 + m(d_1, d_2) ) = c_1 d_2/d𝗇(q_1) + c_2 d_1/d𝗇(q_2). Using the fact that d(d_1,d_2) = d_1 d_2, one obtains that n (1 + m(d_1, d_2) )/(d_1,d_2) = c_1 q_1 + c_2 q_2 ∈ P after dividing both sides of the equality (<ref>) by (d_1,d_2). In addition, note that n (1 + m(d_1, d_2) ) and (d_1,d_2) are relatively prime. Hence (d_1, d_2) ∈𝖽(P^∙), from which the lemma follows. Let P be a root-closed Puiseux monoid containing 1. Then 1/d ∈ P for all d ∈𝖽(P^∙). Let d ∈𝖽(P^∙), and take r ∈ P^∙ such that 𝖽(r) = d. As (𝗇(r), 𝖽(r)) = 1, there exist a,b ∈_0 such that a 𝗇(r) - b𝖽(r) = 1. Therefore 1/d = a 𝗇(r) - b𝖽(r)/d = a r - b ∈(P). This, along with the fact that d (1/d) = 1 ∈ P, ensures that 1/d is a root element of P. Since P is root-closed, it must contain 1/d, which concludes our argument. We are in a position now to characterize the irreducibles of F[X;P]. Let F be a field, and let P be a root-closed Puiseux monoid containing 1. Then f ∈ F[X;P] ∖ F is irreducible in F[X;P] if and only if f(X^m) is irreducible in F[X] for every m ∈𝖽(P^∙) that is a common multiple of the elements of 𝖽((f)). Suppose first that f ∈ F[X;P] ∖ F is an irreducible element of F[X;P], and let m ∈𝖽(P^∙) be a common multiple of the elements of 𝖽((f) ). Then f(X^m) is an element of F[X]. Take g, h ∈ F[X] such that f(X^m) = g(X) h(X). As P is a root-closed and m ∈𝖽(P^∙), Lemma <ref> ensures that g(X^1/m), h(X^1/m) ∈ F[X;P]. Thus, f(X) = g(X^1/m) h(X^1/m) in F[X;P]. Since f is irreducible in F[X;P] either g(X^1/m) ∈ F or h(X^1/m) ∈ F, which implies that either g ∈ F or h ∈ F. Hence f(X^m) is irreducible in F[X]. Conversely, suppose that f ∈ F[X;P] satisfies that f(X^m) is an irreducible polynomial in F[X] for every m ∈𝖽(P^∙) that is a common multiple of the elements of the set 𝖽((f)). To argue that f is irreducible in F[X;P] suppose that f = g h for some g, h ∈ F[X;P]. Let m_0 be the least common multiple of the elements of 𝖽((g)) ∪𝖽((h)). Lemma <ref> guarantees that m_0 ∈𝖽(P^∙). Moreover, f = g h implies that m_0 is a common multiple of the elements of 𝖽((f)). As a result, the equality f(X^m_0) = g(X^m_0)h(X^m_0) holds in F[X]. Since f(X^m_0) is irreducible in F[X], either g(X^m_0) ∈ F or h(X^m_0) ∈ F and, therefore, either g ∈ F or h ∈ F. This implies that f is irreducible in F[X;P], as desired. We proceed to show the main result of this section. Let F be a field, and let P be a root-closed Puiseux monoid. Hence ℳ(F[X;P]) = ⟨𝒜(F[X;P]) ⟩. As each molecule of F[X;P] is a product of irreducible elements in F[X;P], the inclusion ℳ(F[X;P]) ⊆⟨𝒜(F[X;P]) ⟩ holds trivially. For the reverse inclusion, suppose that f ∈ F[X;P] ∖ F can be written as a product of irreducible elements in F[X;P]. As a result, there exist k,ℓ∈ and irreducible elements g_1, …, g_k and h_1, …, h_ℓ in F[X;P] satisfying that g_1(X) ⋯ g_k(X) = f(X) = h_1(X) ⋯ h_ℓ(X). Let m be the least common multiple of all the elements of the set (⋃_i=1^k𝖽((g_i)) ) ⋃(⋃_j=1^ℓ𝖽((h_j)) ). Note that f(X^m), g_i(X^m) and h_j(X^m) are polynomials in F[X] for i = 1, …, k and j = 1, …, ℓ. Lemma <ref> ensures that m ∈𝖽(P^∙). On the other hand, m is a common multiple of all the elements of 𝖽((g_i)) (or all the elements of 𝖽((h_i))). Therefore Proposition <ref> guarantees that the polynomials g_i(X^m) and h_j(X^m) are irreducible in F[X] for i=1,…,k and j=1,…,ℓ. After substituting X by X^m in (<ref>) and using the fact that F[X] is a UFD, one finds that ℓ = k and g_i(X^m) = h_σ(i)(X^m) for some permutation σ∈ S_k and every i = 1, …, k. This, in turns, implies that g_i = h_σ(i) for i = 1, …, k. Hence |𝖹_F[X;P](f)| = 1, which means that f is a molecule of F[X;P]. As we have seen before, Corollary <ref> guarantees the existence of a Puiseux algebra F[X;P] satisfying that |ℳ(F[X;P]) ∖𝒜(F[X;P])| = ∞. Now we use Theorem <ref> to construct an infinite class of Puiseux algebras satisfying a slightly more refined condition. For any field F, there exist infinitely many Puiseux monoids P such that the algebra F[X;P] contains infinite molecules that are neither atoms nor monomials. Let {p_j} be the strictly increasing sequence with underlying set . Then for each j ∈ consider the Puiseux monoid P_j = ⟨ 1/p^n_j | n ∈⟩. Fix j ∈, and take P := P_j. The fact that (P) = P ∪ -P immediately implies that P is a root-closed Puiseux monoid containing 1. Consider the Puiseux algebra [X;P] and the element X + p ∈[X;P], where p ∈. To argue that X + p is an irreducible element in [X;P], write X + p = g(X) h(X) for some g, h ∈[X;P]. Now taking m to be the maximum power of p_j in the set 𝖽( (g) ∪(h) ), one obtains that X^m + p = g(X^m) h(X^m) in [X]. Since [X] is a UFD, it follows by Eisenstein's criterion that X^m + p is irreducible as a polynomial over . Hence either g(X) ∈ or h(X) ∈, which implies that X + p is irreducible in [X;P]. Now it follows by Theorem <ref> that (X + p)^n is a molecule in [X;P] for every n ∈. Clearly, the elements (X + p)^n are neither atoms nor monomials. Finally, we prove that the algebras we have defined in the previous paragraph are pairwise non-isomorphic. To do so suppose, by way of contradiction, that [X;P_j] and [X;P_k] are isomorphic algebras for distinct j,k ∈. Let ψ[X;P_j] →[X;P_k] be an algebra isomorphism. Since ψ fixes , it follows that ψ(X^q) ∉ for any q ∈ P_j^∙. This implies that (ψ(X)) ∈ P_k^∙. As 𝖽(P_j^∙) is unbounded there exists n ∈ such that p_j^n > 𝗇( ( ψ(X) ) ). Observe that ( ψ(X) ) = ( ψ( X^1/p_j^n)^p_j^n) = p_j^n ( ψ( X^1/p_j^n) ). Because (p_j,d) = 1 for every d ∈𝖽(P_k^∙), from (<ref>) one obtains that p_j^n divides 𝗇( ψ (X) ), which contradicts that p_j^n > 𝗇( ( ψ(X) ) ). Hence the Puiseux algebras in {P_j : j ∈} are pairwise non-isomorphic, which completes our proof.§ ACKNOWLEDGMENTS While working on this paper, the first author was supported by the NSF-AGEP Fellowship and the UC Dissertation Year Fellowship. The authors would like to thank an anonymous referee, whose helpful suggestions help to improve the final version of this paper. 20 ACHZ07 D. D. Anderson, J. Coykendall, L. Hill, and M. Zafrullah: Monoid domain constructions of antimatter domains, Comm. Alg. 35 (2007) 3236–3241. CGLMS12 S. T. Chapman, P. A. García-Sánchez, D. Llena, A. Malyshev, and D. Steinberg: On the delta set and the Betti elements of a BF-monoid, Arab. J. Math. 1 (2012) 53–61. CGG20 S. 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Gilmer: Commutative Semigroup Rings, Chicago Lectures in Mathematics, The University of Chicago Press, London, 1984. GK20 R. Gipson and H. Kulosman: For which Puiseux monoids are their monoid rings over fields AP?, Int. Electron. J. Algebra 7 (2020) 43–60. fG18 F Gotti: Atomic and antimatter semigroup algebras with rational exponents. [arXiv:1801.06779v2] fG19 F. Gotti: Increasing positive monoids of ordered fields are FF-monoids, J. Algebra 518 (2019) 40–56. fG17 F. Gotti: On the atomic structure of Puiseux monoids, J. Algebra Appl. 16 (2017) 1750126. fG18a F. Gotti: Puiseux monoids and transfer homomorphisms, J. Algebra 516 (2018) 95–114. GG18 F. Gotti and M. Gotti: Atomicity and boundedness of monotone Puiseux monoids, Semigroup Forum 96 (2018) 536–552. pG01 P. A. Grillet: Commutative Semigroups. Advances in Mathematics, vol. 2, Kluwer Academic Publishers, Boston, 2001. MS04 E. Miller and B. Sturmfels: Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2004. wN72 W. Narkiewicz: Numbers with unique factorization in an algebraic number field, Acta Arith. 21 (1972) 313–322. wN66 W. Narkiewicz: On natural numbers having unique factorization in a quadratic number field, Acta Arith. 12 (1966) 1–22. wN66a W. Narkiewicz: On natural numbers having unique factorization in a quadratic number field II, Acta Arith. 13 (1967) 123–129.

http://arxiv.org/abs/1702.08270v3

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http://arxiv.org/abs/1702.08493v2

HPDedup: A Hybrid Prioritized Data Deduplication Mechanismfor Primary Storage in the CloudHuijun Wu12, Chen Wang1, Yinjin Fu3, Sherif Sakr12,Liming Zhu12, Kai Lu4 1Data61, CSIRO 2The University of New South Wales, Australia 3PLA University of Science and Technology, China 4Science and Technology on Parallel and Distributed Laboratory, State Key Laboratory of High Performance Computing, State Key Laboratory of High-end Server & Storage Technology, College of Computer, National University of Defense Technology, Changsha, ChinaDecember 30, 2023 ======================================================================================================================================================================================================================================================================================================================================================================================================================================================================Eliminating duplicate data in primary storage of clouds increases the cost-efficiency of cloud service providers as well as reduces the cost of users for using cloud services. Most existing primary deduplication techniques either use inline caching to exploit locality in primary workloads or use post-processing deduplication running in system idle time to avoid the negative impact on I/O performance. However, neither of them works well in the cloud servers running multiple services or applications for the following two reasons: Firstly, the temporal locality of duplicate data writes may not exist in some primary storage workloads thus inline caching often fails to achieve good deduplication ratio. Secondly, the post-processing deduplication allows duplicate data to be written into disks, therefore does not provide the benefit of I/O deduplication and requires high peak storage capacity. This paper presents HPDedup, a Hybrid Prioritized data Deduplication mechanism to deal with the storage system shared by applications running in co-located virtual machines or containers by fusing an inline and a post-processing process for exact deduplication. In the inline deduplication phase, HPDedup gives a fingerprint caching mechanism that estimates the temporal locality of duplicates in data streams from different VMs or applications and prioritizes the cache allocation for these streams based on the estimation. HPDedup also allows different deduplication threshold for streams based on their spatial locality to reduce the disk fragmentation. The post-processing phase removes duplicates whose fingerprints are not able to be cached due to weak temporal locality from disks. The hybrid deduplication mechanism significantly reduces the amount of redundant data written to the storage system while maintaining inline data writing performance. Our experimental results show that HPDedup clearly outperforms the state-of-the-art primary storage deduplication techniques in terms of inline cache efficiency and primary deduplication efficiency. Data Deduplication; Cache Management; Primary Storage; Cloud Service § INTRODUCTIONData deduplication is a technique that splits data into small chunks and uses the hash fingerprints of these data chunks to identify and eliminate duplicate chunks in order to save storage space. Deduplication techniques have achieved great successes in backup storage systems <cit.>. However, significant challenges remain to apply deduplication techniques in primary storage systems mainly due to the low latency requirement in primary storage applications <cit.>. Recent studies show that duplicate data widely exists in the primary workloads  <cit.> <cit.> <cit.>. In the cloud computing scenario, the primary workloads of the applications running on the same machine are observed having high duplicate ratio as well <cit.>. For a cloud datacentre, there are significant incentives to remove duplicates in its primary storage for cost-effectiveness and competitiveness.The existing data deduplication methods for primary storage can be classified into two main categories based on when the deduplication is performed: inline deduplication techniques <cit.><cit.><cit.> and post-processing deduplication techniques <cit.><cit.><cit.>. The former performs data deduplication on the write path of I/O requests to immediately identify and eliminate data redundancy, while the latter removes duplicate data in background to avoid the performance impact on the I/O. However, challenges remain for both of these two methods.For inline deduplication, fingerprint lookup is the main performance bottleneck due to that the size of a fingerprint table often exceeds the size of the memory. While a backup storage system may be able to tolerate the delay of disk based fingerprint lookup, the deduplication system of a primary storage system has to rely on caching to satisfy the latency requirement of applications. The state-of-art techniques for inline primary deduplication<cit.><cit.><cit.> exploit temporal locality of primary workloads by maintaining an in-memory fingerprint cache to perform deduplication. These deduplication mechanisms do not ensure that all duplicate chunks are eliminated. We call them non-exact deduplication. However, the temporal locality in primary workloads does not always exist <cit.><cit.>. For the workloads with weak temporal locality, caching the unnecessary fingerprints not only wastes the valuable cache space but also compromises other workloads with good locality. iDedup <cit.> also exploits spatial locality to alleviate data fragmentation on disks by only eliminating duplicate block sequences longer than a fixed threshold. However, when data streams from different sources have different spatial locality, a fixed threshold may fail to achieve good deduplication ratio or read performance. For primary storage in clouds, the differences of locality become a severe problem. Firstly, the weak temporal locality becomes more apparent in the cloud when multiple applications running in virtualized containers sharing the same physical primary storage system. Since deploying deduplication in each virtual machine often fails to detect the duplicates among different virtual machines, deduplication should be deployed at the host physical machine. The data streams from co-located VMs or applications may interfere with each other and destroy the temporal locality. It has significant impact on the fingerprint cache efficiency managed by existing caching policies. The stream interference problem has been addressed in backup deduplication by resorting the streams <cit.>. However, it is not applicable for primary storage systems because we cannot change the order of requests of primary workloads. Secondly, as shown in our experiments on real-world traces (in Section <ref>), different workloads show quite different spatial locality. Therefore, a fixed global threshold is not optimal for alleviating the disk fragmentation in primary storage in the clouds.For post-processing deduplication techniques <cit.><cit.><cit.>. There are two main drawbacks: Firstly, duplicate chunks are written to disks before being eliminated. This makes deduplication not effective in reducing peak storage use. For SSD based primary storage in cloud architecture like hyper-converged infrastructure, this affects the lifetime of SSD devices.Secondly, the competition between post-processing deduplication process and foreground applications on using resources such as CPU, RAM and I/O can be a problem when a large amount of duplicates has to be eliminated.To avoid the limitations and exploit the advantages of inline and post-processing deduplication, in this paper, we fuse the two phases together and propose a hybrid data deduplication mechanism to particularly deal with deduplication in virtualized systems running multiple services or applications from different cloud tenants.The goal is to achieve a good balance between I/O efficiency and storage capacity saving in primary storage deduplication. In the inline deduplication phase, we differentiate the temporal locality of different data streams using a histogram estimation based method. The estimation method periodically assesses the temporal locality of the data streams from different services/applications. Based on the estimation, we propose a cache replacement algorithm to prioritize fingerprint cache allocation to favor data streams with good temporal locality. The mechanism significantly improves cache efficiency in inline deduplication and reduces the workload in the post-processing deduplication phase. Moreover, we adjust the threshold for different data streams dynamically to alleviate the disk fragmentation while achieving high inline deduplication ratio. The post-processing deduplication phase only deals with relatively small amount of duplicate data blocks that are missed in cache in the inline deduplication phase. Compared to systems that purely rely on post-processing deduplication, a highly efficient inline deduplication process greatly reduces the storage capacity requirement and contention in system resources.Overall, this paper makes the following main contributions:* We propose a novel hybrid deduplication mechanism that fuses inline deduplication and post-processing deduplication together for primary storage systems shared by multiple applications/services. The mechanism is able to provide exact deduplication in comparison to many inline deduplication mechanisms while avoiding drawbacks of purely post-processing deduplication mechanisms. * We give a locality estimation based cache replacement algorithm that significantly improves the fingerprint cache efficiency in primary storage deduplication systems. The estimation method is able to exploit locality in individual data streams for cache hit rate improvement.* We evaluate our mechanism using traces generated from real-world applications. The result shows that the proposed mechanism outperforms the state-of-art inline and post-processing deduplication techniques. e.g., HPDedup improves the inline deduplication ratio by up to 39.70% compared with iDedup in our experiments. It also reduces up to 45.08% disk capacity requirement compared with the state-of-art post-processing deduplication mechanism in our evaluation. The remaining of this paper is organized as follows: Section <ref> describes the background information and motivations for our approach; Section <ref> presents the design of HPDedup;Section <ref> introduces how to differentiate the locality of data streams in deduplication; Section <ref> presents the detailed results of our experimental evaluation; Section <ref> reviews related work and Section <ref> concludes the paper. § BACKGROUND AND MOTIVATIONIn this section, we present the background and key observations that motivate this work.§.§ Deduplication for Primary Storage in CloudsVirtualization enables a cloud provider to accommodate applications from multiple users to run on a single physical machine while providing isolation between these applications. Recently, container techniques like Docker <cit.> further reduce the overhead of using virtual machines to isolate user applications, thus support running more applications simultaneously on a physical machine.Although providing much benefits in improving the resource sharing efficiency, increasing number of applications from different users sharing the same machine raises challenges to primary storage deduplication. In typical configurations, the cloud software stack such as OpenStack <cit.> map the data volumes of VMs to persistent block storage devices connected by networks. It is impractical to achieve deduplication within a container or a virtual machine due to the overhead of storage device access. Moreover, it would fail to identify the duplicates among different VMs or containers. It is only feasible to detect duplication in the host's hypervisor that manages I/O requests from VMs or containers. As the file information in VMs or containers is not available in the underlying hypervisor, we design our deduplication mechanism to be based on the block level in the hypervisor. A similar architecture has also been used in an existing post-processing primary deduplication technique <cit.>. §.§ Temporal locality Affects Efficiency of Fingerprint Cache Existing primary storage deduplication techniques often exploit temporal locality through fingerprint cache in an attempt to detect most of duplicates in the cache. However, some recent studies reveal that the locality may be weak in the primary storage workloads <cit.>.We evaluate the temporal locality with real-world traces, which contain a 24-hour I/O trace from a file server running in the cloud as well as the FIU trace <cit.> commonly used in deduplication research. The file server is used for data sharing among a research group consisting of 20 people. We denote the file server trace as Cloud-FTP, and FIU mail server trace as FIU-mail and FIU Web server trace as FIU-web.As shown in Figure <ref>, the average distance between two adjacent occurrences of a data block in both FIU-mail and FIU-web trace is small and highly skewed, indicating good locality. For the Cloud-FTP trace, the temporal locality is weak. The temporal locality of duplicates in primary storage systems varies among different applications. We further evaluate the cache efficiency for the three different workloads when they arrive at a storage system within the same time frame. The cache replacement algorithms we use in our evaluation include LRU (Least Recently Used), LFU (Least Frequently Used) and ARC (Adaptive Replacement Cache). The three cache replacement policies exploit the recency, frequency and the combination of both of workloads, respectively.We extract two-hour traces from the three FIU traces (10am-12am on the first day.) and the Cloud-FTP trace we collect. The characteristics of the two-hour traces are shown in Table <ref>. We mix these traces according to the timestamps of requests to simulate an I/O pattern of multiple applications on a cloud server. We set the cache size to 32K entries. Figure <ref> illustrates the actual percentage of cache occupied by each data stream.Table <ref> shows the number of duplicate blocks detected by each cache policy. Under the LRU and LFU cache replacement algorithm, the cache allocated to Cloud-FTP stream is above 2/3 of the maximum cache capacity, but the number of duplicates detected in the stream is less than 2% of the overall duplicates. Under the ARC cache replacement algorithm, the Cloud-FTP stream is allocated less portion of cache, however, only 467 duplicates are detected. This experiment shows that data streams with weak temporal locality of duplicates result in poor cache efficiency and fail to detect most of duplicates in the inline deduplication process. When duplicates cannot be effectively detected through cache lookup, they are written to disks, which results in extra storage space requirement in capacity planning. It is therefore important to improve cache efficiency when locality of duplicates is not guaranteed. § THE DESIGN OF HPDEDUPThe inline or post-processing deduplication alone is difficult to satisfy the deduplication ratio and latency requirement of a primary storage system. However, the two techniques complement each other. Fusing them together to form a hybrid deduplication system is able to achieve exact deduplication with satisfactory write performance. Particularly, the caching in inline deduplication not only speeds up fingerprint lookup, but also reduces the amount of data written to disks and relieves the burden of handling large amount of duplicates in the post-processing phase. On the other hand, the post-processing deduplication is able to detect duplicates missed out in the inline cache, therefore achieves exact deduplication. Including a post-processing phase potentially relaxes the inline cache size requirement as well.In another word, a hybrid deduplication system is able to achieve a balance between the I/O performance and deduplication ratio, which is essential for inline deduplication of primary storage. This motivates the architecture design of HPDedup. In the following, we first give the hybrid architecture, and then describe the inline phase and post-processing phase of HPDedup. §.§ HPDedup Architecture Overview Virtualization is a core technology that enables the cloud computing. Running multiple virtual machines in a physical machine is a common practice in cloud datacentres. In cloud software stack such as OpenStack <cit.>,the storage volumes of virtual machines are often mapped to persistent block storage devices. It is impractical to implement deduplication inside a virtual machine while data streams from co-located VMs are written to the same physical device. The main reason is that performing deduplication in each VM is not able to remove the duplicates across VMs. We therefore place our deduplication mechanism at the block device level.For the cloud scenario, the deduplication mechanism can be implemented inside the hypervisor that manages the I/O requests from VMs running on top of it. Some existing post-processing primary deduplication techniques <cit.> place their deduplication mechanisms at the same level.Figure <ref> illustrates the system architecture of HPDedup. A number of storage devices are connected to the server via SAN or through similar storage environments. The hypervisor (e.g., Xen) is responsible for translating LBA (logical block address) to PBA (physical block address) for block I/O requests from VMs running on top of it. HPDedup works at the hypervisor level to eliminate duplicate data blocks. For multiple containers running on the same host, HPDedup can be deployed at the block device level of the host machine. For simplicity, we will mainly use the hypervisor setting to describe the design of HPDedup in the rest of this paper. §.§ Inline Deduplication PhaseIn the inline deduplication phase, HPDedup maintains an in-memory fingerprint cache that stores the fingerprint and PBA mapping to avoid slow disk-based fingerprint table search, and an LBA mapping table that stores the mapping between LBAs and PBAs of blocks. The LBA mapping table is stored in NVRAM to avoid the data loss. The inline deduplication of data streams is performed in the inline deduplication engine: the fingerprint of each data block is computed by a cryptographic hash function, like MD5 or SHA-1. The deduplication engine then looks up the block fingerprint in the fingerprint cache. The stream locality estimator is responsible for monitoring and estimating both the temporal and spatial locality for the data streams coming from different VMs or containers. The temporal locality estimation is used for optimizing the hit rate of the fingerprint cache while the spatial locality estimation adjusts the deduplication threshold for data streams to reduce the disk fragmentation.When the fingerprint of the incoming data block is found in the fingerprint cache, an entry of the LBA of the coming block and the corresponding PBA will be created and added into the LBA mapping table if such an entry does not exist, otherwise nothing is done because it is a duplicate write. If the block fingerprint is not found in the fingerprint cache, the data block is written to the underlying primary SAN storage. In this process, the data is staged in the data buffer in SSD for performance consideration. We use D-LRU <cit.> algorithm to manage the data buffer in SSD to store recently accessed data by exploiting temporal locality.When a data block is written to the underlying primary storage, the corresponding metadata associated with this data block including its fingerprint, LBA and PBA mapping as well as the reference count is updated in three tables in the SSD: on-disk fingerprint table, on-disk LBA mapping table and reference count table. The duplicates whose fingerprints are not cached in fingerprint cache will be eliminated in the post-processing phase. §.§ Post-Processing Deduplication PhaseIn the post-processing deduplication phase, the post-processing deduplication engine scans the on-disk fingerprint table and identifies duplicates. Note that duplicates identified in this phase are not in the fingerprint cache, thus they are not processed by inline deduplication phase. The entries with duplicate fingerprints are then removed while the corresponding LBAs are mapped to the same PBA in LBA mapping table. The reference count to the original PBAs containing the same data is decremented and the disk space is further claimed by the garbage collector. After post-processing, the unique data blocks in data buffer of SSD are organized into fixed-sized coarse-grained objects and flushed to the underlying persistent store.§ DIFFERENTIATE DATA STREAM LOCALITY IN DEDUPLICATIONIn this section we describe how to differentiate the temporal and spatial locality among different data streams to improve the efficiency of primary deduplication in the cloud. Both temporal and spatial locality estimation are performed in the stream locality estimator of the inline deduplication module. Specifically, the temporal locality of duplicates in a data stream is used to guide the allocation of fingerprint cache to the data stream in order to achieve higher inline deduplication ratio. The spatial locality of a stream is used to achieve a balance between the inline deduplication ratio and the read performance. We first describe how to measure and estimate the temporal locality of duplicates in data streams in <ref>, and then describe how to manage the fingerprint cache based on the temporal locality measurement in <ref>. Thirdly, we discuss how to handle the disk fragmentation based on the difference of spatial locality among data streams in <ref>. §.§ Temporal Locality Estimation for data streams The temporal locality of duplicates characterizes how soon duplicates of a data block may arrive in the system in a data stream. A good temporal locality indicates duplicate data blocks generally are close to each other while a weak locality indicates that duplicate blocks are often far away from each other or there are few duplicates in the data stream.To measure the temporal locality of duplicates, we introduce a metric called Local Duplicate Set Size (LDSS). LDSS of a stream is defined as the number of duplicate fingerprints in last n contiguous data blocks arriving before a given time. Here, we call n estimation interval. To use LDSS to guide fingerprint cache allocation, we need to predict the LDSS of the future arrivals of data blocks of the data stream. A common approach is to use the historical LDSS values to predict the future LDSS of the data stream. To obtain a historical LDSS value from a data stream, a naive way is to count all distinct fingerprints for each data stream and their occurrences within an estimation interval. However, this incurs a high memory overhead which is close to the cache capacity because all the fingerprints need to be recorded. To address the problem, the stream locality estimator uses the reservoir sampling algorithm <cit.> to sample fingerprints from a data stream, and then estimate LDSS from these samples using the unseen estimation algorithm <cit.>.Reservoir sampling algorithm assumes an unknown number of fingerprints in a data stream and guarantees that each fingerprint in the data stream has an equal chance to be sampled. In our implementation, each element in the sampling buffer is a pair containing a fingerprint and its occurrence count.The unseen estimation algorithm is able to estimate the unseen data distribution based on the histogram of the samples of observed data. For HPDedup, the unseen estimation algorithm is used to estimate LDSS of data streams based the sampled fingerprints from these streams. We refer readers to <cit.> for more details about the theoretical aspect of estimation algorithms. Here, we give a high-level description of using the unseen estimation algorithm to estimate the LDSS values for a data stream.Consider the storage system handles M data streams, denoted by S_1,S_2,...,S_M from M VMs and the estimation interval size is n,the goal of the temporal locality estimation is to collect k fingerprint samples from the last n write requests of each stream and compute the LDSS values for these streams based on fingerprint samples. Specifically, after sampling, we denote the number of sampled fingerprints coming from stream i by N_i. By using unseen estimation algorithm, we can accurately estimate the number of unique writes (denoted by u_i) in stream S_i among last n write requests in the mixed stream. Then the estimated LDSS for stream i can be denoted by LDSS_i as below:LDSS_i = N_i - u_iwhereas N_i is the total number of write requests for stream i in the estimation interval.The estimation of u_i as well as the calculation of LDSS_i is shown in Algorithm <ref>. Before discussing the algorithm, we introduce a concept named Fingerprint Frequency Histogram (FFH). A FFH of a set of fingerprints F is a histogram f = {f_1,f_2...} where f_j is the number of distinct fingerprints that appear exactly j times in F.We derive the FFH from the sampling buffer to estimate the LDSS_i of data stream i. We use H_s to denote the FFH of the samples and H to denote the FFH of the whole estimation interval for stream i. According to the unseen estimation algorithm, we then compute the transformation matrix T by a combination of binomial probabilities about the chances an item is drawn a certain times. The expected histogram H_s^' for sampled data blocks can be computed by H_s^' = T · H. To solve the equation and get H, we minimize the distance between H_s and H_s^' which are the observed histogram and expected histogram of sampled data blocks, respectively. Once obtainedH, we are able to compute the LDSS_ifor the data stream.For some streams which have few write requests during the estimation interval, it is not necessary to run unseen estimation algorithm to estimate the LDSS. The LDSS of these streams are set to a small value for simplicity. §.§ LDSS Estimation Based Fingerprint Cache ManagementWe use the LDSS of different streams to guide the cache allocation for these streams. The fingerprints from a data stream with higher predicted LDSS is more likely to be kept in the cache than those from a data stream with lower LDSS. As mentioned earlier, we use the historical LDSS values which are accurately estimated by the unseen algorithm to predict the LDSS of streams. We use self-tuned double exponential smoothing method to predict the LDSSvalues. Using the estimated LDSS(w-2),LDSS(w-1),..., we can predict LDSS(w) where w is the next estimation interval. The predicted LDSS is used to guide the fingerprint cache management as follows.Firstly, we propose a cache admission policy that the fingerprints from streams with very low LDSS would not be cached if there exists streams with much higher LDSS. This strategy can avoid caching the fingerprints of data streams containing compressed data or other forms of compact data. The cache of each stream can be managed by any cache replacement policies.Secondly, for the fingerprints already cached,we assign an evict priority value p_i to data stream i, denoted by:p_i = 1/LDSS_i(w)The evict priorities are mapped to adjacent non-overlapping segments in a segment tree. Specifically, stream i is represented by the segment [Σ_k=0^i-1p_k, Σ_k=0^i-1p_k + p_i). When evicting a fingerprint from the cache, we generate a random number r and find the segment I to which r belongs. We then evict one cache entry from the cache corresponding to interval I. As the fingerprint cache management of HPDedup relies on the accurate LDSS estimation of using unseen estimation algorithm, one may also think about directly estimating the LDSS of data streams by the number of duplicate fingerprint samples sampled by reservoir sampling.Figure <ref> compares the effectiveness of using RS-only (Reservoir sampling only, dash lines) or RS + Unseen (Reservoir sampling with unseen algorithm, solid lines) during the LDSS estimation. It is clear that RS + Unseen based LDSSestimation is able to provide much higher inline deduplication ratio with a smaller estimation interval compared with RS-only method. This shows the effectiveness of temporal locality estimation using the unseen algorithm.Moreover, a proper value of estimation interval is important for achieving the good cache efficiency. Too large interval may include some out-dated information which cannot reflect the current temporal locality of duplicates for the workloads. Too small interval, on the other hand, cannot accurately capture the temporal locality. Since LDSS is used to estimate the number of duplicates which can be detected in the fingerprint cache, the estimation interval can be set to a factor of the number of fingerprint cache entries.The solid lines in Figure <ref> shows the inline deduplication ratio of HPDedup for three different workloads (details are shown in Section <ref>) while choosing different estimation interval factor. The cache size is set to 160MB. From the workload A to C, the overall temporal locality for the workload decreases. The estimation interval factor needs to be set to larger values for the workloads with worse temporal locality. Correspondingly, we can see that the optimal estimation interval factor for workload A, B and C are 0.3, 0.4 and 0.6, respectively.In practice, a good approximation is to set the estimation interval factor to 1 - d where d is the historical inline deduplication ratio for the mixed streams.The temporal locality estimation is triggered by the following three events: 1. the finish of an estimation interval; 2. a significant drop of inline deduplication ratio; 3. the join or quit of virtual machines/applications. §.§ Spatial Locality Aware Threshold for DeduplicationTo alleviate disk fragmentation problem of deduplication on data read, some primary storage deduplication techniques only eliminate duplicate block sequences with length greater than a given threshold. However, as pointed by <cit.>, for the applications with many random I/Os, doing so may not find any duplicates. In the deduplication of primary storage systems in clouds, the spatial locality of data streams for different applications/services varies significantly. To explore the relationship between deduplication ratio and threshold, we analyze both the FIU traces and the trace we collect. As shown in Figure <ref>, different workloads show different trends. When the threshold increases from 1 to 16, the inline deduplication ratio for FIU-mail and Cloud-FTP reduces by only 4.3% and 9.1%, respectively. The inline deduplication ratio for FIU-web drops by around 38.1% when the threshold increases from 1 to 2. When the threshold is 16, the inline deduplication ratio is 43.1% of that under a threshold of 1. For FIU-home trace, the inline deduplication ratio keeps dropping. When the threshold is set to 16, the inline deduplication ratio is only 32.0% of that under a threshold of 1 .Figure <ref> indicates that the threshold value should be adaptive to data stream characteristics. It is noteworthy that there exists a tradeoff between the write latency and read latency while choosing a proper threshold. Write operations prefer shorter threshold while read operations prefer longer threshold. For writes, long sequence indicates more comparisons before writing data blocks to disks. For reads, long threshold can avoid many random I/Os thus reducing the read latency.HPDedup maintains two vectors V_w and V_r for each stream. V_w is used to store the occurrence number for the largest length values of sequential duplicates. V_r is used to store the occurrence number for the length values for sequential read. Both V_w and V_r have 64 items. For instance, if V_w[3]=100, there are 100 sequential duplicates with length 3 since V_w is reset. If V_r[3]=100, there are 100 sequential reads with length 3 since V_r is reset.Initially, the threshold is 16. The two vectors collect data when requests come. When the threshold update is triggered, given the histogram vector V_w and V_r, the threshold T is computed byT = (1 - r) ·Len_d + r ·Len_rwhere Len_dand Len_r are the average length of duplicate block sequence and average read length, respectively. T, therefore, is the balance point of the read and write latency. r is the read ratio among all requests. Len_d and Len_r are computed according to the data collected in V_w and V_r, respectively. To cope with the changes of duplicate pattern for each stream, the two vectors is reset to all 0s when the total deduplication ratio decreases by over 50% since the last threshold update. § EVALUATION The prototype of HPDedup is implemented in C. To evaluate the performance of HPDedup, we use real-world traces to feed into HPDedup. We compare the performance of HPDedup with the following deduplication methods: locality based inline deduplication (iDedup <cit.>), post-processing deduplication schema (e.g.,<cit.> <cit.>) and hybrid inline-offline deduplication schema DIODE <cit.>.§.§ ConfigurationThe experiments are carried out in a workstation with Intel Core i7-4790 CPU , 32GB RAM and 128GB SSD + 1TB HDD. We use the FIU-home, FIU-web, FIU-mail <cit.> traces in our evaluation. These traces are from three different applications, namely remote desktop, web server and mail server respectively in FIU. Moreover, we also collect a trace from a cloud FTP server (Cloud-FTP) used by our research group. The trace is obtained by mounting a network block device (NBD) as the working device for the file server, from which we capture read/write requests through a customized NBD-server.To the best of our knowledge, there is no available larger scale I/O traces containing both the fingerprint of data blocks and timestamps. We therefore use the four traces as templates to synthetically generate VM traces representing multiple VMs. Table <ref> shows the statistics of the four workloads. The arrival order of requests in these workloads are sorted and merged based on timestamps. The generated trace has the same I/O pattern with the original traces. For the traces generated from the same template, the content overlap is randomly set to 0% - 40% which is the typical data redundancies among users <cit.>.In our experiments, we simulate a cloud host running 32 virtual machines. As shown in Figure <ref>, FIU traces show better temporal locality compared with Cloud-FTP trace. We mix traces to form three workloads with different overall temporal locality.Workload A contains 15 mail server traces, 5 FTP server traces and 8 remote desktop traces and 4 web server traces. Workload B contains 10 mail server traces, 10 FTP server traces, 6 remote desktop server traces and 6 web server traces. Workload C contains 5 mail server traces, 15 FTP server traces, 6 web server traces and 6 remote desktop server traces. The ratios of data size between the good-locality (L) traces and bad-locality (NL) traces are around 3:1, 1:1 and 1:3 for these three mixed workloads.The estimation interval factor is set to 0.5 at the beginning and is adjusted by the historical inline deduplication ratio dynamically for each workload.§.§ Cache Efficiency in Inline DeduplicationWe compare HPDedup with iDedup, a well-known inline deduplication system that makes use of temporal locality of primary workload. We replay the mixed workloads to simulate the scenario where multiple applications/services running on the same physical machine. Each I/O for the three workloads is a 4KB block and MD5 is used as the hash function to calculate the fingerprints. Each entry of the deduplication metadata is about 64 bytes and contains the fingerprint and the block address.According to the size and footprint of the traces, the total memory size for fingerprint cache is set from 20MB to 320MB in the experiments. The deduplication threshold is set to 4 for both iDedup and HPDedup.Figure <ref> shows the inline deduplication ratio versus cache size for iDedup and HPDedup. Here, inline deduplication ratio is defined as the percentage of duplicate data blocks that can be identified by inline caching. For the cache replacement policy of each stream, LRU, LFU and ARC cache replacement policies are supported by HPDedup. LRU was claimed to be the best cache replacement policy by iDedup <cit.>.When the portion of NL workload increases, the gap between iDedup and HPDedup becomes larger. HPDedup-LRU, HPDedup-LFU and HPDedup-ARC improve the inline deduplication ratio significantly compared with iDedup. For workload A, HPDedup-ARC improves the inline deduplication ratio by 10.58% - 27.72%. HPDedup-LRU improves the inline deduplication ratio of iDedup which also uses LRU cache by 8.04% - 23.52%. HPDedup-LFU shows less improvement (3.56%-13.90%) compared with HPDedup-ARC and HPDedup-LRU. Similarly, for workload B, HPDedup-ARC, HPDedup-LRU and HPDedup-LFU achieve 8.09%-30.19%, 6.36%-22.02% and 3.77%- 16.02% improvement of inline deduplication ratio, respectively. For workload C, HPDedup-ARC improves the inline deduplication ratio by 15.38% - 39.70%. HPDedup-LRU and HPDedup-LFU achieve 13.86% - 37.75% and 12.97% - 28.37% improvement, respectively.The improvement is larger when the cache size is small due to cache resource contention. Moreover, HPDedup-ARC outperforms HPDedup-LRU and HPDedup-LFU because ARC cache replacement policy makes the size of T1 (LRU) cache and T2 (LFU) cache adaptive to the recency and frequency of the workloads. It is also noteworthy that with the increasing of non-locality workload share (from Workload A to C), the inline deduplication ratio improvement achieved by HPDedup becomes larger. The reason is that non-locality workloads provide more space for the optimization of HPDedup.Note that for HPDedup-ARC, extra memory overhead is introduced by ARC caching replacement policy itself to track the evicted fingerprints and their metadata. The overhead is non-trivial for fingerprint cache. The analysis of overhead for obtaining statistics information about evicted fingerprints in existing cache replacement policies is out of the scope of this paper. We leave the discussion to the future work. In the following experiments, HPDedup-LRU is used by default. Overall, the weak locality in workloads results in low inline deduplication ratio. With the locality estimation method in HPDedup, the allocation of inline fingerprint cache dynamically gives the streams with better locality higher priority. Hence, HPDedup improves the overall cache efficiency for multiple VMs/applications running on the same physical machine in the cloud. §.§ Disk Capacity Requirement In this subsection, we compare the size of the data before performing post-processing deduplication for HPDedup and pure post-processing deduplication (e.g., <cit.>). The size is also the maximum required disk size for the deduplication mechanisms. Figure <ref> shows the result of comparisons.For HPDedup, the inline fingerprint size is set to 160MB and LRU cache replacement policy is used for simplicity. As shown in Figure <ref>, HPDedup significantly reduces the disk capacity requirements for storage space. The data size has been reduced by 45.08%, 28.29% and 12.78% for workload A, B and C, respectively. The better the locality is in the workload, the more duplicates can be detected in the inline phase and the more duplicate data writes can be eliminated. This clearly shows the benefit of a hybrid deduplication architecture of HPDedup as hundred GBs of data writes can be reduced by only maintaining a 160MB inline fingerprint cache. §.§ Average Hits of Cached FingerprintsInline deduplication of HPDedup does not require disk access to identify duplicates therefore is faster than the post-processing based deduplication. We use average hits of cached fingerprints as an indicator of inline deduplication performance. The indicator is obtained by monitoring the number of fingerprint entering the fingerprint cache. With a high average hits value, the inline deduplication is able to detect a large portion of duplicates and reduces the load of the more expensive post-processing deduplication.In the following, we compare HPDedup with DIODE <cit.> on this metric. DIODE uses file extensions to decide whether to perform inline deduplication on these files. Files are classified roughly into three different types. The inline deduplication process skips the type of files containing audio, video, encrypted and other compressed data (called P-Type in DIODE). We use a full inline deduplication method <cit.> as the baseline. DIODE works at the file system level so that the information like file extensions are passed to the hypervisor layer in the form of hints, like the method used by <cit.>. The file type information of the Cloud-FTP trace is known so that the trace can be used to test DIODE.The files that are classified as P-Type are around 14.2% of the whole trace in size. The FIU traces are classified into U-Type (unpredictable type), which will be processed by the inline deduplication, just like that in the evaluation setting of DIODE in <cit.>. As shown in Table <ref>, HPDedup clearly outperforms the baseline and DIODE on the average hits of cached fingerprints. HPDedup outperforms DIODE by identifying data streams that have weak locality but do not belong to P-Type. It then avoids allocating cache space to these streams in order to make room for data streams with good locality.This approach is effective. To show this, we compare the locality of the following two files in the Cloud-FTP workload: a Linux kernel 4.6 source code tar file and a VM image of CentOS 5.8 downloaded from OSBoxes. The two files are similar in size (2.7GB and 2.6GB). They are written to the primary storage after an inline deduplication process. The fingerprint cache size is set to 1% of the data size (27MB and 26MB). Figure <ref> shows the number of duplicate blocks found in the two files.Note that for the VM image, nearly all duplicate blocks can be found through the inline deduplication process. DIODE treats both files as highly-deduplicatable (H-Type in DIODE). However, the number of duplicate blocks in the VM image file is significantly higher than that in the source code tar file. Moreover, DIODE ignores all P-Type files during inline deduplication by letting them to be processed during inline deduplication. However, multiple writes of the same P-Type files result in duplication and cannot be eliminated through differentiating file types. Different from DIODE, HPDedup allocates much less cache to the Cloud-FTP stream while writing the Linux Source Code tar file but allocates more when writing the VM image file. The result shows that simply using file type to guide cache allocation is insufficient. HPDedup classifies data at finer-grained (stream temporal locality level) so that the efficiency of inline deduplication can be further improved. §.§ Locality Estimation AccuracyHPDedup improves the efficiency of inline deduplication phase by allocating the fingerprint cache based on the temporal locality of each stream. Since LDSS is an indicator which describes the temporal locality of data streams, it is critical to achieve accurate LDSS estimation.Figure <ref> shows the observed LDSS for workload B. Here, the cache size is set to 160MB. To make the figures concise, the traces generated from the same template are aggregated. Figure <ref> shows the observed LDSS over time. The values of LDSS are normalized. FIU-mail streams show the largest LDSS thus indicating it has the best temporal locality. Nevertheless, as shown in Figure <ref>, very little cache resource is occupied by FIU-mail streams when LDSS estimation is not used. Cloud-FTP streams whose LDSS is not high occupy the majority of cache resources. As shown in Figure <ref>, with the guidance of LDSS estimation, cache resource is cleverly allocated to streams based on their temporal locality. LDSS estimation allocates fingerprint cache resources according to the temporal locality of streams and improves the inline deduplication ratio by 12.53% (see Figure <ref>). The improvement clearly shows the effectiveness of LDSS estimation in HPDedup.§.§ FragmentationIn this subsection, we evaluate the fragmentation of files in storage system caused by deduplication. The length threshold of duplicate block sequence controls the fragmentation in both HPDedup and DIODE. Both DIODE and HPDedup are able to adjust the threshold dynamically. Figure <ref> shows the threshold change along time for workload A in DIODE and HPDedup.The inline deduplication ratio for DIODE and HPDedup are 57.62% and 68.96%, respectively. As one may see, HPDedup is able to adjust the threshold for each stream while DIODE uses a global threshold. The FIU-mail and Cloud-FTP have a higher threshold than FIU-home and FIU-web. Since larger threshold leads to less disk fragmentation, this result shows that HPDedup introduces less fragmentation while achieving higher inline deduplication ratio than DIODE.§.§ Overhead AnalysisWhile HPDedup improves the efficiency of primary storage deduplication significantly, it inevitably incurs overhead. We analyze the overhead in this subsection. The overhead can be classified into computational overhead and memory overhead. §.§.§ Computational OverheadThe computational overhead of HPDedup contains the following two parts: the histogram calculation time and the estimation algorithm execution time. To calculate the histogram, we only need to scan the sample buffer and add the count of the fingerprints to corresponding bins of the histogram. Therefore, the time complexity is O(n) where n is the number of samples. Figure  <ref> shows the time used for generating the histogram in our current implementation. Here, the sampling rate is 15%. We can see that the histogram calculation of an estimation interval with 1 million blocks takes less than 7ms. Estimating the temporal locality of streams is achieved by using the method described in Section <ref>. The core of the estimation is to solve a linear programming problem. The linear programming problem can be solved in O(n) <cit.> and even constant time <cit.> when the number of variables d is fixed and the number of constraints is fixed. In our context, the condition is satisfied because too frequent duplicates in the sampling buffer will be used in a straightforward way during the estimation and will not be put into the linear programming. Note that the linear programming needs to be done for each stream. For every estimation interval, the temporal locality estimation takes about 26ms for each stream regardless the estimation interval size (see Figure <ref>). The computing overhead is acceptable as the process is performed in background and does not affect the data write performance.§.§.§ Memory Overhead The primary memory overhead of HPDedup comes from the sampling buffer. With an estimation interval of size EI and a sampling rate p, the memory cost for tracking the histogram of samples is as below:EI · p · (fpSize + counterSize)where fpSize and counterSize are the memory cost for storing fingerprints and the occurrence count, respectively. For instance, when the cache size is 160MB, there would be approximately 2.62M cache entries. For the sampling rate of 15%, the memory overhead is only 4.49MB (2.81% of cache size) even if we choose a large estimation interval factor (e.g., Workload C, 0.6). In practice, for the mixed streams with better temporal locality, the memory overhead is much less (e.g., 2.25MB for Workload A and 2.99MB for Workload B) as the estimation interval factor can be set to smaller values. Compared with the improvement (19.80% - 25.81%) of inline deduplication ratio for the three workloads with 160MB cache size),the memory overhead of HPDedup is acceptable.§ RELATED WORK §.§ Primary Storage Deduplication MechanismsData deduplication achieves a great success in backup storage systems. Recent research exploit various ways to apply deduplication in primary storage systems for both reducing data size in storage devices and improving I/O performance. Existing work can be classified into three categories: Inline primary storage deduplication, Post-processing/Offline primary storage deduplication and Hybrid inline and post-processing deduplication.Inline primary storage deduplication. Most inline primary deduplication exploits the locality in primary workloads to perform non-exact deduplication. iDedup <cit.> exploits the temporal locality by only maintaining an in-memory cache to store the fingerprints of data blocks. To exploit the spatial locality, iDedup only eliminates duplicates in long sequences of data blocks. POD <cit.> aims at improving the I/O performance in primary storage systems and mainly performs deduplication on small I/O requests. HANDS <cit.> uses working set prediction to improve the locality of fingerprints. Koller et al. <cit.> uses content-aware cache to improve the efficiency of I/O by avoiding the influence of duplicated data. PDFS <cit.> argues that the locality may not commonly exist in primary workloads. To avoid the disk bottleneck of storing fingerprint table, a similarity based partial lookup solution is proposed. Leach <cit.> exploits the temporal locality of workloads by a splay tree. These work do not consider scenarios involving VMs and containers in the cloud where workloads for the primary storage contain a mix of data streams with different access patterns.Post-processing/Offline primary storage deduplication. Post-processing deduplication performs deduplication during the idle time of primary storage systems. Ahmed El-Shimi et al. <cit.> propose a post-processing deduplication method built in Windows Server operating systems. Similar with HPDedup, DEDIS <cit.> is built in the Xen hypervisor to provide data deduplication functionality to multiple virtual machines. The main purpose of post-processing primary storage deduplication is to avoid the high I/O latency introduced by inline on-disk dedupe metadata lookup. However, even though the locality does not always exist in primary workloads, it is much more efficient to use inline caching rather than post-processing to eliminate duplicates in the portion of workloads with decent locality. The contribution of HPDedup is to differentiate the deduplication procedure for primary workloads according to the temporal locality of workloads.Hybrid inline and post-processing deduplication. Combining inline and post-processing deduplication together has been exploited by RevDedup<cit.> in backup storage deduplication to improve the space efficiency. For primary storage deduplication, DIODE <cit.> also proposes a dynamic architecture of inline-offline deduplication. Like ALG-Dedupe <cit.>, DIODE is an application-aware deduplication mechanism. File extensions are classified into three types according to their potential deduplication ratio. Moreover, whether performing inline deduplication on a file is determined by the extension of the files. However, our experiments show that file types are not sufficient for achieving good inline deduplication performance and there is a lot of room to improve. The key difference between HPDedup and DIODE is that HPDedup gives a dynamic locality estimation method to improve inline deduplication performance, therefore reduces the load of the more expensive post-processing deduplication process.§.§ Unseen Distribution EstimationEstimating the number of duplicates in a time frame for a data stream is similar to estimating the distinct elements in a large set, for which various statistics based methods (e.g., <cit.>) have been investigated. Fisher et al. <cit.> describe a method to estimate the number of unknown species given a histogram of randomly sampled species. Theoretical computer science community has been trying to address how to perform the estimation with less samples <cit.>. Recently, this line of work has been extended by Valiant and Valiant <cit.> to characterize unobserved distributions. They prove that only O(n/log(n)) samples are sufficient to provide an accurate estimation of the whole dataset, in which n is the size of the whole dataset. Harnik et al. <cit.> utilizes the theory to estimate the duplicates in storage systems.§.§ Dynamic Flash Cache ManagementUsing prediction or historical information of workloads to improve the cache efficiency has been explored in flash cache management <cit.>. These work studied cache admission policies and dynamic cache allocation to reduce the flash wear-out. To the best our knowledge, HPDedup is the first work to use locality estimation to deal with the cache contention problem in inline deduplication for primary storage. § CONCLUSIONIn scenarios where multiple virtual machines or containers running in the cloud,many applications are placed in the same physical machine. Removing duplicate I/Os from the primary storage in these scenarios is useful to both improve the capacity efficiency and I/O performance. We proposed HPDedup, a hybrid prioritized deduplication method for primary storage in the cloud. HPDedup used a dynamic temporal locality estimation algorithm to achieve high inline cache efficiency and left the relatively small number of duplicates that were not in the cache to the post-processing deduplication phase to handle. By doing so, HPDedup was able to achieve exact deduplication in primary storage systems. Comparing to the state-of-art inline deduplication methods, HPDedup significantly improved the inline cache efficiency therefore achieved high inline deduplication ratio. HPDedup improves the inline deduplication ratio by up to 39.70% compared with iDedup in our experiments. Meanwhile, the improved cache efficiency made the post-processing deduplication process less a burden for the performance of an inline primary deduplication system.For example, HPDedup reduces up to 45.08% disk capacity requirement compared with the state-of-art post-processing deduplication mechanism in our evaluation. § ACKNOWLEDGMENTWe would like to thank Gregory Valiant from Stanford University for helping us to further understand the unseen entropy estimation algorithm. This work is partially supported by the The National Key Research and Development Program of China (2016YFB0200401), by program for New Century Excellent Talents in University， by National Science Foundation (NSF) China 61402492, 61402486, 61379146,by the laboratory pre-research fund (9140C810106150C81001). IEEEtran

http://arxiv.org/abs/1702.08153v2

1National Institute of Standards and Technology, Boulder, CO 80305, USA2Department of Physics, University of Colorado, Boulder, CO 80309, USA3Department of Astronomy & Astrophysics, The Pennsylvania State University, 525 Davey Lab, University Park, PA 16802, USA4Penn State Astrobiology Research Center, University Park, PA 16802, USA5Center for Exoplanets & Habitable Worlds, University Park, PA 16802, USA6Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA 7NASA Sagan Fellow8jeffrey.m.jennings@colorado.edu9scott.diddams@nist.gov An optical etalon illuminated by a white light source provides a broadband comb-like spectrum that can be employed as a calibration source for astronomical spectrographs in radial velocity (RV) surveys for extrasolar planets.For this application the frequency stability of the etalon is critical, as its transmission spectrum is susceptible to frequency fluctuations due to changes in cavity temperature, optical power and input polarization.In this paper we present a laser frequency comb measurement technique to characterize the frequency stability of a custom-designed fiberinterferometer (FFP). 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Eng. 9907, 99072G (2016). § INTRODUCTIONIntrinsically stable, broad bandwidth wavelength calibration sources are a necessity in high precision astronomical spectroscopy aimed at measuring Doppler radial velocity (RV) shifts at and below the 1level <cit.>.This is equivalent to a fractional Doppler shift of <3× 10^-9, or <2 femtometers at 600 nm. Such high precision RV measurements are central to the discovery and characterization of exoplanets <cit.> and could also enable direct measurement of the cosmic expansion <cit.>. Conventionally, atomic emission lamps (such as Thorium-Agron and Uranium-Neon) <cit.> or molecular absorption cells (I_2, CH_4) <cit.> have been used as stable wavelength references to calibrate stellar spectra recorded by astronomical spectrographs to a precision of 1 – 3 . These calibrators have been central to the tremendous success of Doppler RV searches for exoplanets over the past decades.However they also suffer from finite bandwidth, non-uniform spectral features and line blending, making them insufficient for next generation spectrographs aimed at ultimately reaching 1 c. Recently, laser-based photonic systems have shown promise as calibration tools for precision astronomical spectroscopy <cit.>.Among these, laser frequency combs (LFCs) have emerged as optimal calibration sources which, when combined with ultra-stable high resolution spectrometers, could enable RV measurement precisions of <10 c <cit.>. A self-referenced optical frequency comb has the unique characteristic of providing a broad array of narrow emission lines whose exact spacing and frequencies are referenced to stabilized frequency standards with fractional uncertainties of <10^-12 <cit.>.Laser frequency combs based on mode-locked lasers and electro-optic frequency modulation have both been employed in proof-of-concept tests at astronomical observatories <cit.>, and facility-level instruments are presently coming online or being constructed for several state-of-the-art spectrographs under development <cit.>.A passive(FP) etalon has also been suggested as an astronomical wavelength calibration source for high precision Doppler measurements <cit.>.When illuminated by a broadband light source, the transmission of an etalon consists of a broad array of comb-like spectral features. Particular advantages of the FP include rich spectral information content, low optomechanical complexity, and relatively low cost. Our recent work has shown that a fiber-optic integrated implementation, a fiber(FFP), illuminated by a supercontinuum source enabled <2 m s^-1 short-term calibration of the APOGEE near-infrared spectrometer <cit.>. However in contrast to broadband optical frequency combs, the FP transmission modes are not strictly uniform in spacing, and their absolute frequencies are tied to the optical and mechanical properties of the etalon itself (rather than a stabilized atomic standard), which can fluctuate and drift with local environmental changes in temperature, pressure, and humidity. Even with precise temperature or mechanical stabilization <cit.>, or direct optical locking of the FP cavity to an atomic reference <cit.>, long-term changes of the cavity dispersion and material properties will impact absolute spectral stability of the FP output. Although some research efforts have focused on the use of optical cavities as a frequency transfer tool in atomic spectroscopy and frequency metrology <cit.>, theultimate utility of a broadband FP calibrator for astronomy, particularly over long timescales, is not yet fully understood or quantified. In this paper we take steps to progress that understanding.We introduce a technique to characterize the frequency stability of a FP cavity and apply it to a custom FFP that we have constructed for operation in the 780 – 1350 nm wavelength region.By simultaneously measuring and tracking multiple FP resonances over days, we can quantify the impact of environmental perturbations on both the absolute stability of the etalon and place an upper limit on changes in the cavity's chromatic dispersion. This technique allows us to identify the line centers of the FP resonance with a precision of ≈ 1 MHz, equivalent to an RV precision of 1 m s^-1.The intrinsic frequency precision of the technique is at the 10 kHz level (RV precision of 1.5 cm s^-1), and we identify present limitations as arising from uncontrolled temperature gradients, polarization mode dispersion of the FFP and parasitic etalons in the measurement apparatus. The results and the techniques we introduce have significance not only for passively stabilized etalons, but for a variety of actively-stabilized etalon spectral calibration sources that are proposed or are presently being constructed.§ BACKGROUNDThe resonance condition for ainterferometer requires that the round trip optical phase shift k2L must be an integer-multiple of 2π, 2π m= k2L = 2Lω n/c,where m is an integer, L is the cavity length, c is the speed of light, and k(ω) and n(ω) are the frequency-dependent propagation constant and index of refraction, respectively.This relationship implies that the discrete resonant frequencies ω_m, i.e., the cavity longitudinal modes, are given by ω_m= 2π m c/2Ln = 2π m ×Δν_FSR, where Δν_FSR is the cavity's free spectral range. For an ideal dispersionlesscavity the frequency response of the resonance modes to changes in the cavity length Δ L can be illustrated by a simple rubber band model in which higher frequency modes undergo larger frequency shifts, but for which the fractional frequency shift (Δω_m / ω_m) across the spectrum is constant. The number of modes between two resonance frequencies predicts their relative frequency response to a changing cavity length, analogous to equidistant marks on a rubber band whose separations disperse non-uniformly as the band is stretched. Within this simplified model, the ratio of the frequencies of two modes is independent of Δ L and given simply by the ratio of their mode numbers, ω_m/ω_l=m/l. However in a more realistic model the dispersion of the cavity medium must be accounted for.In this case it is convenient to use a power series expansion of the frequencies of the cavity modes about a nominal central mode ω_0,ω_m = ω_0 + D_1m + 1/2D_2m^2 + 1/6D_3m^3 + ....In this expansion D_1=2π v_g/(2L), where the group velocity v_g=c/(n+ω(dn/dω)) is evaluated at ω_0. D_2=-cD_1^2k_2/n is related to the group velocity dispersion of light in the cavity with k_2=d^2k/dω^2, also evaluated at ω_0.The next term in the expansion accounts for third-order dispersion <cit.>. Here we have also re-indexed the integer m such that its value is zero at ω_0.In the experiments described in Section <ref> we measure the absolute frequency fluctuations of two of the FFP's modes, m and l, using probe lasers at 281.6 THz (≈ 1064 nm) and 227.3 THz (≈ 1319 nm). The question then arises: at what magnitude of Δ L would we expect the measured ratio of frequency shifts, r=Δω_m / Δω_l to depart from the constant ratio m/l ≈ 1.239?When we include the next higher order (D_2) dispersive effects, evaluation of Eq. (<ref>) with parameters of fused silica shows that Δω_m / Δω_l departs from a constant ratio by a few parts in 10^-5 for a cavity length variation of ≈ half a wavelength (500 nm). This would correspond to an optical frequency shift of one of the FFP modes employed here by ≈ 30 GHz, which is a factor of at least 10 greater than any frequency shift we observe.Thus our analysis implies that if one of the FFP modes could be precisely stabilized with uncertainty Δω_l, the frequency stability of a distant mode could be expected to be similarly stable at the level of Δω_m < rΔω_l(1+10^-5). This deviation is significantly smaller than we can resolve in present experiments.In Section <ref> we describe measurement conditions that yield recovery of the expected mode ratio of 1.239 to within a precision as good as 10^-3, limited by our ability to characterize the center frequency of individual FP modes. § EXPERIMENTAL SETUP The approach we employ to characterize the stability of the FFP is adapted from that of <cit.> and is shown schematically in Fig. <ref>.The concept is to use a near-infrared, self-referenced, octave-spanning LFC <cit.> to calibrate the scanning of two continuous wave (CW) lasers that are transmitted through the FFP at 281.6 THz and 227.3 THz.This allows us to simultaneously measure and track the frequencies of two FFP modes in real time.In principle this approach could be extended to even broader bandwidths by employing additional tunable lasers covering greater spectral range or with the technique of dual-comb frequency spectroscopy <cit.>.The FFP has been described in greater detail elsewhere <cit.>.As shown in Figs. <ref>(c) and <ref>(d), it consists of an approximately 3.4 mm piece of HI-780 single mode optical fiber that is potted in a ceramic ferrule with reflective coatings on the end faces. The reflective coatings operate over the range of 750 – 1350 nm with finesse F = 100, and the nominal free spectral range of the etalon is FSR = 30 GHz (0.11 nm at 1064 nm). Additional HI-780 single mode optical fibers are butt-coupled to the FFP to couple light both into and out of it. A 10 kΩ thermistor sensor is epoxied to the top of the FFP; a 0.5 A thermoelectric cooler (TEC) is epoxied to its central base. These are used with a commercial bench-top temperature controller in a proportional-integral-derivative (PID) loop to maintain a fixed cavity temperature.While the FFP temperature is being controlled, the CW lasers are periodically scanned across their respective FFP resonances by applying a sawtooth tuning voltage profile with period of 100 s to each laser's temperature.While the start and stop of the scan is synchronous for both lasers, the scan magnitudes are different, and they are offset such that the two wavelengths do not come into resonance simultaneously.The power incident on the FFP is kept at or below 1 mW in order to minimize laser-induced heating of the FFP that might arise from residual absorption or light scattering.After transmission the 1064 nm and 1319 nm beams are de-multiplexed in fiber and sent to separate detectors where the transmitted power from each laser is photodetected and digitized. Simultaneously, but in two separate detector channels, the CW lasers are heterodyned with the spectrally-broadened output of a self-referenced Er:fiber laser frequency comb (LFC) with repetition rate f_ rep=250 MHz. The LFC is stabilized to a hydrogen maser with absolute optical uncertainty below 100 Hz on the 100 s timescale, which is equivalent to an RV precision of <0.03 mm s^-1. As detailed in Fig. <ref>, in each 250 MHz optical window there are two heterodyne beats corresponding to the interference of a CW laser with comb lines m and m+1.As the CW laser is scanned, these heterodyne beats also scan across the 250 MHz window, where they are mixed with RF fixed local oscillators (LOs) at 31.25 MHz and 93.75 MHz, down converted to baseband, low-pass filtered, amplified, and digitized.Our choice of the LO frequencies provides a calibration tick each time the CW laser moves 62.5 MHz, or at precisely f_ rep / 4.We note that our approach of using fixed frequency LOs to down convert to baseband, in contrast to bandpass filters and high-speed digitizers used in <cit.>, yields a reduced uncertainty in the frequency calibration.The result of the data acquisition is shown in Fig. <ref>. Once the calibration ticks are digitized we perform a least-squares fit of a Gaussian function to determine the centroid acquisition time (and therefore relative frequency) associated with each tick (see Figs. <ref>(b) and <ref>(c)).The known frequency spacing of the calibration ticks at f_ rep/4 provides pairs of discrete points that map acquisition time to relative frequency. We then use a fit to this data set to remove nonlinearities in the laser frequency scan as shown in Fig. <ref>.A simple linear fit over the full 100 s scan reveals a significant nonlinearity, while a sixth order polynomial fit over the central 60 seconds reduces the uncertainty in our calibration of the frequency axis to <50 kHz as indicated by the residuals in Fig. <ref>(b).With the frequency axis now determined, the FP transmission resonance data are fit with a Lorentzian (or double Lorentzian, in the case of two prevalent polarization modes of the 1319 nm resonance) profile whose centroid is assigned as the peak resonance frequency. An example case is shown in Fig. <ref>. Altering the input polarization in the 1319 nm mode, we can adjust the prominence of the second birefringence peak; it can be brought to parity with the first peak or suppressed into the noise. The frequencies of both polarization modes are tracked across scans, and we observe no difference between them in frequency response. The residuals suggest an achievable noise-limited centroid measurement precision of≈ 1 MHz, limited by the RMS background level of the noise of 0.6% on a single scan. However these scans also show features in the resonances not described by the Lorentzian fits.For the 1319 nm resonance we observe higher frequency interference that arises from residual etalons formed between various fiber interfaces in the system.While we have attempted to minimize these etalons, they still impact the determination of the line center. The 1064 nm resonance shows a persistent feature on the low frequency side that we cannot attribute to birefringence or a residual etalon in the external fiber components.While this discrepancy from the expected Lorentzian transmission profile is stable in time, it too can affect the determination of the line center. We expect to observe two birefringence peaks at every etalon mode (see <cit.>). However at 1064 nm we observe no clear changes in the resonance profile (beyond amplitude changes) in response to a varied incident polarization. The profile of the single observed peak is not well described by a combination of two Lorentzians.To address these effects and verify the accurate tracking of the FFP modes, we independently develop two distinct data fitting and analysis approaches. Both approaches produce results that agree at the 1% level, but for consistency all numbers quoted here come from a single approach.§ RESULTS AND DISCUSSIONTo illustrate the utility of the techniques outlined above we record several measurement time series, tracking the etalon resonance modes at 1064 and 1319 nm over trials spanning 30 minutes to >50 hours. The etalon transmission peaks and heterodyne frequency marker data series are both constantly recorded over the measurement period, and we track the frequencies of the two FP modes while the temperature of the etalon is controlled using two different techniques. In the first (Section <ref>), we drive the temperature of the etalon with a periodic ramp. In the second (Section <ref>), we fix the temperature of the cavity using a tuned PID loop and feedback to the thermoelectric heater/cooler on which the FFP is mounted. The methods yield similar results, though both show structured noise that is likely dominated by systematic effects not directly related to the FFP (discussed in Section <ref>). §.§ Applied temperature ramp measurements In all trials, of both the type discussed here and in Section <ref>, systematic effects (explored in Section <ref>) appear to be biasing our recovered resonance mode frequencies. In an attempt to dominate this bias with a high signal-to-noise, in this set of trials we maximize the relative frequency displacement between the two resonance modes by driving the etalon temperature. We do this by applying a sawtooth voltage profile to the temperature servo setpoint, the resulting low frequency (1 - 2 hour periodicity) temperature ramps having amplitude ≳ 0.1and corresponding temperature gradients between ≈ 0.1 - 4.0hr^-1.These gradients are 1 – 2 orders of magnitude larger than ambient lab temperature variations and yield a high signal-to-noise measurement of the differential frequency shift between the two resonances. With this method we consistently recover the expected ratio of the 1064 nm and 1319 nm mode frequencies to within measurement precision of 10^-3 fractional. See <cit.> for a fuller discussion of the thermal response of a single resonance mode over shorter duration trials.Figure <ref> shows a 20 hr trial with this measurement technique, using a temperature ramp with amplitude 0.4and 2 hour period. The ratio of the frequency responses at 1064 nm and 1319 nm is described well by a linear fit with slope of 1.240 ± 0.001, agreeing within uncertainty with the ratio of the mode numbers. However the residuals to this linear fit show a clear temporal trend (Fig. <ref>(f)), this structured noise indicating a temporally varying component to the mode drifts that is not fully quantified by a simple linear fit to the frequency responses of the two modes. The effect is more pronounced when controlling the etalon temperature at a fixed value (Section <ref>) than when deliberately driving with a thermal ramp as done here (discussed further in Section <ref>).§.§ Measurements with FFP temperature control loop engagedIn the next set of experiments the FFP is controlled at a fixed temperature rather than over an applied periodic temperature ramp, with the consequence in some (but not all) trials being a less accurate measured mode ratio. Consequently, comparison between results with this method and those obtained when driving the cavity temperature rely upon our ability to accurately monitor absolute frequency shifts of the FFP resonances. Figure <ref> underscores this point; while the temperature control at the sensing thermistor indicates fluctuations of <0.002 , we nonetheless observe a strong correlation between the measured mode frequencies and the ambient lab temperature.This is evidence of insufficient thermal isolation that results in temperature gradients in the present FFP package and is a valuable reminder that in-loop monitoring of such a temperature servo does not guarantee corresponding stability of the FFP. A comparison of Figs. <ref>(a)–(c) shows the <0.5changes in lab temperature clearly driving the measured resonance mode behavior at 1064 nm and 1319 nm, with higher temperatures shifting the modes to lower frequencies. Focusing on the frequency variations at 1064 nm in Fig. <ref>(c), we observe hysteresis when we plot the frequency as a function of lab temperature in Fig. <ref>(d). This behavior would likely be reduced with improved thermal isolation. Nonetheless, for this 35 hr trial, we find our measurements of the ratio of the mode frequencies are similar to those presented in Fig. <ref>; Figs. <ref>(e)–(g) show a mode ratio of 1.235(0.004) over the full run, agreeing within uncertainty with the expected mode ratio. Note that additional trials conducted with this method show agreement with the expected ratio in some cases but departure from it at the 1 σ level in others.§.§ Simulated frequency lock of the FFP To test the potential for improvement in our stable temperature trial in Figs. <ref>(a)–(g), we simulate the case in which the FFP cavity length is locked at one wavelength, e.g., to an optical frequency reference as in <cit.>, with the expectation that frequency stabilizing one mode would stabilize all FFP modes.The precision with which the out-of-loop modes are stabilized – particularly those far from the stabilizing frequency – remains an outstanding question in the field.In this analysis we use the measured Δ f_1319 frequency from Fig. <ref> and simulate the effect of a frequency servo with negative feedback by dividing Δ f_1064 by 1.239, the magnitude of the expected mode ratio.In the ideal case we would expect a 1.239 MHz shift at 1064 nm for a 1 MHz shift at 1319 nm such that Δ = Δ f_1319 - Δ f_1064/1.239 = 0. The result, with Δ f_1064/1.239 and Δ f_1319 overplotted in Fig. <ref>(a), shows a mean difference of Δ = 3.17(0.20) MHz (assuming a Gaussian noise distribution). This suggests that if we were to frequency stabilize the 1064 nm mode, limiting factors in our current system would still preclude an accurate prediction of the effect of a perturbation on the 1319 nm mode frequency at the few MHz level. Such limiting factors notably include the larger spread in 1319 nm frequencies relative to 1064 nm in Fig. <ref>(a); this may be due to a high frequency parasitic etalon with its own thermal responsivity affecting the 1319 nm measurement (see Fig. <ref> and Section <ref>).The frequency fluctuations in Δ are shown in the Allan deviation of Fig. <ref>(b), reaching sub-MHz values for averaging times of ≳1 hr. This implies that the actual implementation of a servo could provide frequency stability at the same level. However the flattening of the Allan deviation data at averaging times beyond 1 hour also indicates that the noise processes are no longer Gaussian, as discussed in greater detail below. §.§ Factors limiting frequency stabilityWe suspect the presence of parasitic etalons, likely induced by low-level reflections at fiber cable connection points within the measurement apparatus, is a limiting factor in the measurement of the intrinsic frequency stability of the FFP. Such parasitic etalons cause the light power transmitted through the FFP to be a combination of a higher frequency modulated signal and the characteristic Lorentzian resonances.The modulation depends on the temperature and polarization fluctuations in the fiber components external to the FFP itself, but can shift the apparent peak of the Lorentzian resonances. The hypothesis that parasitic etalons are responsible for these behaviors is based on experience with their effects in similar systems, not on a model tested against the data shown here; other potential explanations that may be equally supported by the observations include variable polarization mode coupling between fibers or polarization rotation of the CW lasers. Our interpretation is that due to their unique response functions to temperature, these parasitic etalons gradually shift the fitted resonance frequency of the two modes over time. When the temperature changes are small, the FFP modes themselves are essentially stationary, so the fitted frequency fluctuations are dominated by the parasitic etalons, which cause a systematic bias.However when the temperature variations are large, the FFP resonances move accordingly and the parasitic etaloning is a smaller perturbation. In trials over which the ambient lab temperature is stable for tens of hours, we see departures from the expected mode ratio as large as the 10^-2 fractional level. If this interpretation is correct it suggests, in agreement with our analytic model in Eq. (<ref>), that measured departures from the expected mode ratio are a result not of the limited intrinsic frequency stability of the FFP, but instead of imperfections in the optical system. We can alternatively frame the mode ratio dependence on an applied temperature gradient as a function of the frequency range (spread) over which measurements are taken. Temperature gradients of ≲ 0.1hr^-1 correspond to frequency spreads of ≲ 200-300 MHz, small enough to exhibit departures from the expected mode ratio. These conditions are observed in some trials for which the cavity temperature is held stable. The opposing case, i.e., trials in which the temperature of the etalon is driven significantly, lead to a large frequency shift in both of the probed resonances (and therefore a large difference signal between the two frequencies), and the mode ratio converges to the expected value within measurement uncertainty. This is shown in Fig. <ref> using data from trials in which the cavity temperature is driven. Here the mode ratio is determined by taking successively larger bins in Δ f_1064, with bins that span a larger frequency spread showing a mode ratio closer to the expected value.For bins in these driven temperature trial datasets with a low frequency spread (which would be equivalent to measurement trials with a nearly constant temperature), the mode ratio is systematically lower than predicted. A possible explanation for this bias in the mode ratio measurement is the thermal response of high frequency parasitic etalons present in the 1319 nm mode measurements (see Fig. <ref>). Note that all measurements, whether the cavity temperature is driven or held stable, are likely susceptible to this error source. Its effect may in the stable temperature case be an increased scatter in measured Δ f_1319 frequencies, the uncertainty in the mode ratio measured in Fig. <ref> (the most accurate of our stable temperature trials in recovering the expected mode ratio) being a factor of 4 larger than the driven temperature case of Fig. <ref>. The increased spread in 1319 nm relative to 1064 nm mode frequencies is seen further in Fig. <ref>(a). Moreover, as mentioned prior, some additional trials not shown here in which the cavity temperature is stabilized show departures from the expected ratio at the 1σ level. The effect of these parasitic etalons in the driven temperature trials is suppressed by improved signal-to-noise but still apparent. The slow temporal drift in the residuals to a linear fit on the frequency response of the two resonance modes in Fig. <ref>(f) may be the result of parasitic etalons gradually pulling our fit for one or both modes in a preferential direction. This temporal structure may alternatively suggest the inadequacy of our explanatory model to separate the effects of the parasitic etalons and FFP resonances. To this end we have tested the effect of fitting out the largest amplitude parasitic etalons, but see no significant deviation from fitted peak frequencies obtained in individual scans.Finally, we additionally test resonance mode sensitivities to variations in incident power. Instantaneous increases in power (contributed equally by the two CW sources) sent into the etalon between factors of ≈2 – 10 (at most, a step from 100 μW – 1 mW) show no clear effect on the mode ratio. This agrees with our previous results; by spiking input power we introduce a strong temperature gradient, to which the mode ratio responds according to the theoretical expectation. §.§ Steps to improve performance Based on our findings we suggest the following to improve FFP frequency stability:i) minimize parasitic etalons. Parasitic etalons seem to pull resonance modes in a complex and dynamic manner. Fusion splicing the entire optical path up to the etalon would be the optimal solution.ii) maintain polarization throughout the system; reduce polarization sensitivity. Variations in incident polarization significantly alter the relative amplitude of the two birefringence peaks in our 1319 nm mode; a peak whose frequency is tracked may in the worst case be reduced to parity with the transmission baseline under polarization changes.iii) construct the etalon with high heat capacity materials. The use of, e.g., silicon ferrules would provide high thermal conductivity and a low coefficient of thermal expansion, improving the FFP's thermal response.§ CONCLUSIONSWe have presented a measurement scheme to assess the frequency stability of a fiberetalon, monitoring the relative response of two resonance modes to temperature and properties of the optical system. We observed a trend toward higher uncertainty for measurements taken over a smaller frequency spread (or equivalently, smaller temperature range), which we suspect is due to the presence of parasitic etalons rather than an inherent limitation of the device. We offered a simple physical model for the FFP's dispersion and considered factors in the experimental setup that complicate this behavior, offering steps to mitigate adverse effects. In its present form the FFP can provide a stable short-term frequency reference.However over a longer term (hours to days), one mode of the FFP would have to stabilized against an absolute reference.With such stabilization, our results predict that the frequencies of additional FFP modes are simply related by the ratio of mode numbers, with a precision as good as ≈ 1 MHz (≈ 1 m s^-1 equivalent RV precision).While the FFP may not approach the precision levels seen with atomically-stabilized laser frequency combs, the simplicity of the design remains appealing, and with careful treatment of systematic effects described here, the performance of an FFP for precision RV spectroscopy can likely be improved to the level of 10 cm s^-1.§ FUNDING The Center for Exoplanets and Habitable Worlds in the Eberly College of Science of the Pennsylvania State University and the Pennsylvania Space Grant Consortium; National Science Foundation (NSF) (AST 1310875, AST 1006676, AST 1126413, AST 1310885); NASA Astrobiology Institute (NNA09DA76A). § ACKNOWLEDGMENTSThis work was performed in part under contract with theCalifornia Institute of Technology/Jet Propulsion Laboratory funded by NASA through the Sagan Fellowship Program executed by the NASA Exoplanet Science Institute.We thank Andrew Metcalf, Scott Papp and Franklyn Quinlan for their helpful comments on this manuscript. We are grateful to the anonymous reviewers for their thoughtful and diligent suggestions. JJ thanks S. Jennings for her contributions to the work. This work is a contribution of NIST and is not subject to copyright in the US.

http://arxiv.org/abs/1703.00001v2

1Approval Voting with Intransitive PreferencesYongjie YangChair of Economic TheorySaarland University, Saarbrücken, Germanyyyongjiecs@gmail.comDecember 30, 2023 ============================================================================================================= We extend Approval voting to the settings where voters may have intransitive preferences. The major obstacle to applying Approval voting in these settings is that voters are not able to clearly determine who they should approve or disapprove, due to the intransitivity of their preferences. An approach to address this issue is to apply tournament solutions to help voters make the decision. We study a class of voting systems where first each voter casts a vote defined as a tournament, then a well-defined tournament solution is applied to select the candidates who are assumed to be approved by the voter. Winners are the ones receiving the most approvals. We study axiomatic properties of this class of voting systems and complexity of control and bribery problems for these voting systems. § INTRODUCTION Voting is a common method for preference aggregation and collective decision-making, and has significant applications in multi-agent systems, political elections, web spam reduction, pattern recognition, etc. <cit.>. For instance, in multiagent systems, it is often necessary for a group of agents to make a collective decision by means of voting in order to reach a joint goal. Approval-based voting systems are among the most important voting systems and have been extensively studied in the literature <cit.>. In an approval-based voting, each voter has a preference over the candidates, and based on the preference, the voter determines a subset of candidates thatapproves. The winners are the candidates that get the most approvals. In most of the approval-based voting systems, voters are assumed to have transitive preferences. That is, if a voter prefers a candidate a to another candidate b, and prefers b to a third candidate c, then the voter prefers a to c. Among the most well-studied approval-based voting systems are Approval, r-Approval and Plurality. In Approval voting, each voter has a dichotomous preference which can be represented by a partition (C_1,C_2) of the candidates, meaning that the voter prefers every candidate in C_1 to every candidate in C_2, and are indifferent between candidates in each C_i where i=1,2. Moreover, a voter with a dichotomous preference (C_1,C_2) approves all candidates in C_1 and disapprovesall candidates in C_2. In r-Approval voting, each voter has a preference which is defined as a linear order over the candidates, and approves exactly the top-r ordered candidates. Plurality is exactly 1-Approval.-5pt§.§ MotivationThere is no doubt that transitive preferences occur naturally in many real-world applications. The question is whether intransitive preferences make sense either. The answer is “Yes!”. In fact, there exist many real-world applications where voters may have intransitive preferences (see, e.g., <cit.>). For instance, when voters compare candidates based on, not one, but multiple quality parameters <cit.>. Another natural scenario where intransitive preferences arise is that when the number of candidates is considerably large <cit.>. In this case, it is more efficient to utilize vote elicitation techniques where voters iteratively cast parts of their preference such as pairwise comparisons. In addition, intransitive preferences may also arise in the settings of sport prediction, where an agency (e.g., a gambling company) desires to predict the sport result for some special purpose. In order to gain a result as precise as possible, the agency might resort to several experts, who based on their expertise suggest the winning player in each pending match between two players. Then, based on the suggestions, the agency applies a voting to predict the winning player(s). In this case, each suggestion by an expert may be considered as a preference which is not necessarily transitive, since it is commonplace to see that a player a who beats another player b is beaten by a third player c who is beaten by b. We refer to <cit.> for further discussion with several concrete examples. Finally, we would like to point out that intransitive preferences may also occur in district-based voting or group-based voting (a group may be a political party, a department in a university, an affiliate of a company, a set of robots, etc.), where each group consists of their own group members and is only allowed to submit one single group vote. In such a situation, group leaders may need to first apply a voting to derive their group vote before the whole voting. If there are three group members whose preferences over three candidates a,b,c are a≻ b≻ c, b≻ c≻ a, c≻ a≻ b, respectively, then the group vote would probably be a≻ b, b≻ c but c≻ a, an intransitive preference. A significant difference between this case and the cases mentioned above is that each intransitive preference (group vote) in this case is drawn from the votes cast by the group members. However, each previous mentioned intransitive preference is cast by a single voter.We extend the framework of approval-based voting to the settings where voters may hold intransitive preferences over the candidates. The major difficulty of imposing the framework in these settings is that voters with intransitive preferences are not able to determine who they should approve. In order to address this issue, we utilize tournament solutions. It should be noted that an intransitive preference can be represented by a tournament (we consider only complete preferences, i.e., for every two candidates a and b either a is preferred to b or the other way around)—a complete and asymmetric binary relation. From the graph theory point of view, a tournament is a directed graph where there is exactly one arc between every pair of vertices (candidates). A tournament solution is a function that maps each tournament to a nonempty subset of candidates. Tournament solutions as a powerful decision making model have been extensively-studied in the literature <cit.>. For instance, tournament solutions have significant applications in voting. In particular, given a set of votes, we can create a tournament based on the majority relations between the candidates (assume that the number of votes is odd): create an arc from a to b if there are more voters preferring a to b. Then, a tournament solution is applied to the tournament to select the winners. Each approval-based voting system studied in this paper is a natural combination of the classic Approval voting and a well-studied tournament solution. Precisely, fixing a tournament solution, each voter in this setting submits a vote which is defined as a tournament. Then, every candidate selected by the tournament solution is approved by the voter, and every candidate not selected is disapproved. We remark that in practice, voters need only to cast their votes, but leave the duty of calculating the winning candidates with respect to the tournament solution to a second agency (e.g., a computer, the voting designer, etc.), since it is unnatural to assume or require that every voter knows how the tournament solution works. In other words, we assume that the voters implicitly approve the winning candidates in their cast tournaments, with respect to the associated tournament solution. In this paper, we mainly consider three tournament solutions, namely, the top cycle, Copeland set and uncovered set. One reason that we choose them to study is that they are among the most prevalent tournament solutions which have been extensively studied in the literature.It also makes sense to consider other tournament solutions such as minimal covering set and tournament equilibrium set (see <cit.> for further tournament solutions).As we pointed out earlier in the example on group-based voting, tournaments in our model are not necessarily to be cast directly by voters, but can be also drawn from the majority relations between the candidates according to the votes cast by some group members. In this scenario, it makes much sense to first apply a tournament solution to the group votes (tournaments) to determine the winning candidates in each subvoting.We would like to point out that apart from the model we proposed in this paper, there are several other prominent approaches to aggregate tournaments. For instance, we could apply different tournament solutions to the given tournaments to determine the candidates implicitly approved by the voters. An explanation is that in a group-based voting, each group (leader) is allowed to freely choose a tournament solution to use. Another approach to select winners from a set of tournaments would be as follows: First, we create a tournament based on the majority relations between the candidates, i.e., there is an arc from a to b if there is an arc from a to b in a majority of the given tournaments. Then, we apply a tournament solution to the tournament to select the winners. In addition, researchers have studied the model of deriving a ranking of the candidates based on a give set of tournaments, see, e.g., <cit.> and references therein.-2pt§.§ Our ContributionThe major contribution of this work is the initialization of the study of a class of voting systems for the scenarios where voters may have intransitive preferences. To give a comprehensive understanding of the new voting systems, we study some axiomatic properties of these voting systems. Axiomatic properties of voting systems are primary factors used to evaluate voting systems and important guidance for voting organizers to select a proper voting system for their specific purpose in practice <cit.>. In particular, we prove that these voting systems satisfy several axiomatic properties for many common tournament solutions. As a byproduct, we introduce two concepts of monotonicity for tournament solutions, and show that the top cycle satisfies both monotonicity criteria, while both the Copeland set and the uncovered set averse to the monotonicity criteria. Our results concerning axiomatic properties are summarized in Theorems <ref>-<ref>.In addition, we study the complexity of strategic behavior under these voting systems. In particular, we study control and bribery problems. We achieve polynomial-time solvability results,results as well asresults. See Table <ref> for a summary of these results. A general conclusion is that these voting systems resist more types of strategic behavior than other approval-based voting systems such as Plurality and Approval. Studying complexity of strategic voting problems has been the main focus of many research papers. First, complexity is widely considered as a prominent theoretical barrier against strategic behavior in voting systems. Second, complexity helps practitioners decide what kind of solution method is appropriate. For polynomial-time solvability results, we directly provide efficient algorithms with low time complexity. On the other hand, hardness results (e.g.,  results andresults) suggest that finding an exact solution is apt to be costly or impractical, and resorting to approximationalgorithms or heuristic algorithms may be a necessary choice. Finally, it should be pointed out that complexity of strategic behavior for voting systems has also been considered as an important factor to evaluate voting systems, see, e.g.,  <cit.>.-8pt § PRELIMINARIESTournament.In this paper, we use the terms “candidate” and “vertex” interchangeably. A tournament T is a pair (V(T), ) where V(T) is a set of candidates andis an asymmetric and complete binary relation on V(T). For X,Y⊆ V(T) such that X∩ Y=∅, XY means that x y for every x∈ X and every y∈ Y. For ease of exposition, we use directed graphs to represent tournaments. Precisely, in this paper a tournament T=(V(T), ) is considered as a directed graph where V(T) is considered as the vertex set andis considered as the arc set. We refer to the textbook by West <cit.> for readers who are not familiar with graph theory.For a candidate a∈ V(T), let N^-_T(a) denote the set of inneighbors of a in T and N^+_T(a) the set of outneighbors of a, i.e., N^-_T(a)={b∈ V(T)| b a} and N^+_T(a)={b∈ V(T)| a b}. The indegree (resp. outdegree) of a is defined as |N^-_T(a)| (resp. |N^+_T(a)|). A tournament T is regular if for every candidate a in T it holds that ||N_T^+(a)|-|N_T^-(a)||≤ 1. A directed triangle is a regular tournament with three vertices.A directed path from a candidate a to another candidate b is a vertex sequence (a=v_1,v_2,...,v_t=b) such that v_i v_i+1, for every i∈{1,2,...,t-1}. A tournament T is strongly connected if there is a directed path from every candidate to every other candidate. A maximal strongly connected component of T is a strongly connected subtournament of T with maximal vertices.The source of a tournament T=(V(T),) is the candidate a so that a b for every candidate b∈ V(T)∖{a}. The source is also called the Condorcet winner of the tournament from the social choice point of view. Clearly, not every tournament has a source. For a subset B⊆ V(T), T[B] is the subtournament induced by B, i.e., T[B]=(B,') where for every a,b∈ B, a' b if and only if a b. Tournament Solution. A tournament solutionis a function that maps every tournament T to a nonempty subset (T)⊆ V(T). In this paper, we mainly study the following three tournament solutions <cit.>. We refer to <cit.> for a comprehensive introduction to further well-studied tournament solutions.-6pt =-2pt Copeland Set. The Copeland score of a candidate c in a tournament Tis defined as the outdegree of c in T.The Copeland set of T, denoted by T, consists of all candidates with the highest Copeland score. Top Cycle.The top cycle T of a tournament T is the unique minimal subset of candidates such that there is an arc from every candidate in the subset to every candidate not in the subset. Uncovered Set. A king in a tournament is a candidate a such that for every other candidate b, either a b or there is another candidate c such that a c and c b. It is folklore that every tournament has at least one king <cit.>. The uncovered set of a tournament T, denoted by T, is the set of all kings of T.It is known that for every tournament T it holds T,T⊆T (see, e.g., <cit.>). Election. An election is a tuple ℰ=(𝒞,𝒯), where 𝒞 is a set of candidates, and 𝒯 is a list of votes (for convenience, the terminologies “vote” and “voter” are used interchangeably throughout this paper). In this paper, we consider only the election where each vote is defined as a tournament T=(𝒞,). For two candidates a,b∈𝒞 and a vote T(𝒞,), a b means that the vote prefers a to b. A voting correspondence is a function that maps an election ℰ=(𝒞,𝒯) to a nonempty subset (ℰ) of 𝒞. We call the elements in (ℰ) the winners of ℰ with respect to . If (ℰ) consists of only one winner, we call it the unique winner; otherwise, we call them co-winners. For two non-overlapping lists of tournaments 𝒯=(T_1,T_2,...,T_x) and 𝒯'=(T_1',T_2',...,T_y'), we denote by 𝒯+𝒯' the list (T_1,...,T_x,T_1',...,T_y'). For two elections ℰ=(𝒞,𝒯) and ℰ'=(𝒞,𝒯') with the same candidate set 𝒞,ℰℰ'=(𝒞,𝒯+𝒯'). Implicit Approval Voting. Now we introduce the core concept in this paper—-Approval. Each -Approval is a combination of the prevalent Approval voting and a tournament solution . To the best of our knowledge, such voting correspondences have not been studied in the literature. Let ℰ=(𝒞,𝒯) be an election.-8pt-Approval Each candidate c∈𝒞 is assigned a score defined as cℰ=|{T∈𝒯| c∈(T)}|. The candidates with the highest score are the winners. -2ptIn Approval, each voter explicitly determineswhomwants to approve. In -Approval, each voter with preference T is assumed to implicitly approve all candidates in (T) and disapprove all the remaining candidates. Properties of Voting Correspondences.-5pt =-2ptAnonymity. A voting correspondenceis anonymous if reordering the votes does not affect the winning set. That is, for every two elections ℰ=(𝒞,𝒯=(T_1,...,T_n)) and ℰ'=(𝒞,𝒯'=(T_σ(1), T_σ(2),...,T_σ(n))) where (σ(1), σ(2),...,σ(n)) is a permutation of (1,2,...,n), it holds that (ℰ)=(ℰ'). Neutrality. An election (𝒞,𝒯=(T_1,...,T_n)) is isomorphic to another election (𝒞',𝒯'=(T_1',...,T_n')) where T_i=(𝒞,_i) and T_i'=(𝒞',_i') for every i∈{1,...,n},if there is an one-to-one mapping f: 𝒞↦𝒞' such that for every two distinct candidates a,b∈𝒞 and every i∈{1,2,....,n}, it holds that a_i b if and only if f(a)_i' f(b). A voting correspondenceis neutral if for every two isomorphic elections ℰ=(𝒞,𝒯) and ℰ'=(𝒞',𝒯'), and every c∈𝒞, it holds that c∈(ℰ) if and only if f(c)∈(ℰ'), where f is the mapping as discussed above for ℰ and ℰ'. Monotonicity. A voting correspondenceis monotonic if for every two elections ℰ=(𝒞,𝒯=(T_1,...,T_n)), ℰ'=(𝒞,𝒯'=(T_1',...,T_n')), and every c∈(ℰ) such that for every i∈{1,2,...,n} (1) T_i[𝒞∖{c}]=T_i'[𝒞∖{c}]; and (2) N^+_T_i(c)⊆ N^+_T_i'(c), it holds that c∈(ℰ'). Majority. A voting correspondencesatisfies the majority criterion if for every election ℰ=(𝒞,𝒯) where there is a candidate c∈𝒞 which is the source in a majority of the tournaments in 𝒯, it holds that c∈(ℰ). Consistency. A voting correspondenceis consistent if for every two elections ℰ=(𝒞,𝒯) and ℰ'=(𝒞,𝒯'), it holds that (ℰ)∩(ℰ')⊆(ℰℰ'). Pareto optimal. A voting correspondenceis Pareto optimal if for every election ℰ=(𝒞,𝒯) and every two candidates a,b∈𝒞 such that a≻ b in every tournament T=(𝒞,≻)∈𝒯, a∉(ℰ) implies b∉(ℰ). Properties of Tournament Solutions.To the names of the following properties of tournament solutions, we append a prefix “TS”, standing for “tournament solution”, to avoid confusion with the definitions of the axiomatic properties of voting correspondences.-5pt =-2ptNeutrality. Two tournaments T=(𝒞,) and T'=(𝒞',') where |𝒞|=|𝒞'| are isomorphic if there is an one-to-one mapping f: 𝒞↦𝒞' such that for every two a,b∈𝒞, it holds that a b if and only if f(a)' f(b). Here, f is called an isomorphic mapping of T and T'. A tournament solutionsatisfies the neutrality criterion if for every two isomorphic tournaments T=(𝒞,) and T'=(𝒞','), it holds that (T')={f(a)∈𝒞'| a∈(T)}, where f is an isomorphic mapping of T and T'. Monotonicity. A tournament solutionis monotonic if for every two tournaments T=(𝒞,≻), T'=(𝒞,≻'), and every candidate c∈(T) such that T[𝒞∖{c}]=T'[𝒞∖{c}] and N^+_T(c)⊆ N^+_T'(c), it holds that c∈(T'). Condorcet consistency. A tournament solutionis Condorcet consistent if for every tournament T which admits the Condorcet winner w, (T)={w}. Now we introduce two concepts of monotonicity of tournament solutions. To the best of our knowledge, they have not been studied in the literature. Generally speaking, a tournament solution is exclusive monotonic if a winning candidate c remains as a winning candidate when c is preferred to more candidates, without changing the preferences between other candidates. Moreover, no nonwinning candidate benefits from this, i.e., no nonwinning candidate becomes a winning candidate. So, if a tournament solution is exclusive monotonic, then making a winning candidate stronger never makes a nonwinning candidate better off. The formal definition is as follows. Exclusive monotonicity. A tournament solutionis exclusive monotonic if for every two tournaments T=(𝒞,≻), T'=(𝒞,≻'), and everyc∈(T) such that (1) T[𝒞∖{c}]=T'[𝒞∖{c}]; and (2) N^+_T(c)⊆ N^+_T'(c), it holds that c∈(T') and (T')⊆(T).A tournament solution is exclusive negative monotonic if when a nonwinning candidate c is preferred to more candidates, and some other nonwinning candidate becomes a winning candidate, then c must become a winning candidate as well.In other words, if extending the outneighborhood of c benefits some nonwinning candidates, then c must benefit from this operation. The formal definition is as follows. Exclusive negative monotonicity (ENM). A tournament solutionsatisfies the ENM criterion, if for every two tournaments T=(𝒞,≻) and T'=(𝒞,≻'), and every candidate c∉(T) such that (1) T[𝒞∖{c}]=T'[𝒞∖{c}]; and (2) N^+_T(c)⊆ N^+_T'(c), it holds that (T')⊈(T) implies c∈(T'). Parameterized Complexity. A parameterized problem is a language L ⊆Σ^*×ℕ, where Σ is a finite alphabet. The first∈Σ^* is called the main part, and the second component ∈ℕ is called the parameter. Downey and Fellows <cit.> established the parameterized complexity theory and developed the following parameterized complexity hierarchy:⊆⊆⊆...⊆.In particular,(stands for fixed-parameter tractable) includes all parameterized problems which admit O(f()·||^O(1))-time algorithms. Here f() is a computable function of k and |I| is the size of the main part. Given two parameterized problems Q and Q', an -reduction from Q to Q' is an algorithm that takes as input an instance (,) of Q and outputs an instance (',') of Q' such that (1) the algorithm runs in f()· ||^O(1) time, where f is a computable function in ;(2) (,)∈ Q if and only if (',')∈ Q'; and(3) '≤ g(), where g is a computable function in .A problem isfor a positive integer i if all problems incan be -reducible to the problem.problems are unlikely to admit -algorithms, unless the parameterized complexity hierarchy collapses at some level <cit.>. -5pt § AXIOMATIC PROPERTIESIn this section, we study axiomatic properties for -Approval for different tournament solutions .It is fairly easy to check that -Approval is anonymous for all tournament solutions . Moreover, -Approval is neutral for all tournament solutionswhich satisfy the neutrality criterion. Furthermore, -Approval satisfies the majority criteria for allthat are Condorcet consistent. We now study some other properties for -Approval. Consider first the consistency criterion. -Approval is consistent for all tournament solutions . Let ℰ_1=(𝒞,𝒯_1) and ℰ_2=(𝒞,𝒯_2) be two elections such that (ℰ_1)∩(ℰ_1)≠∅. For a candidate c∈𝒞, it holds thatcℰ_1+ℰ_2=cℰ_1+cℰ_2.This directly implies that if a candidate c∈𝒞 has the highest score in both ℰ_1 and ℰ_2, then c has the highest score in the combined election ℰ_1+ℰ_2. It then follows that (ℰ_1ℰ_2)=(ℰ_1)∩(ℰ_2).Now we study the monotonicity of -Approval for all Condorcet consistent tournament solutions . It should be noted that almost all commonly used tournament solutions, including all tournament solutions studied in this paper, are Condorcet consistent. We derive both sufficient and necessary conditions for such -Approval to be monotonic.-4pt Letbe a Condorcet consistent tournament solution. Then, -Approval is monotonic if and only ifsatisfies the exclusive monotonicity and ENM criteria.Letbe a Condorcet consistent tournament solution as stated in the theorem and φ=-Approval. Assume thatsatisfies the exclusive monotonicity and the ENM criteria. Letℰ=(𝒞,𝒯=(T_1,...,T_n)) and ℰ'=(𝒞,𝒯'=(T_1',...,T_n')) be two elections with the same candidate set 𝒞. Moreover, let c∈(ℰ) be a candidate such that (1) T_i[𝒞∖{c}]=T_i'[𝒞∖{c}]; and (2) N^+_T_i(c)⊆ N^+_T_i'(c) for every i∈{1,2,...,n}. We shall show that c∈(ℰ'). Let's first study the scores of the candidates in ℰ'. Apparently, cℰ≥c'ℰ for every c'∈𝒞∖{c}. Sinceis exclusive monotonic, if c∈(T_i) for some T_i∈𝒯, then c∈(T_i').For each a∈𝒞, let 𝒲_a={i∈{1,2,...,n}| a∉(T_i), a∈(T_i')}. The following claim is useful.Claim. 𝒲_c'⊆𝒲_c for every c'∈𝒞∖{c}.Let c' be a candidate in 𝒞∖{c}, and T_i and T_i' be two tournaments in 𝒯 and 𝒯', respectively, such that c'∉(T_i) and c'∈(T_i'). Clearly, (T')⊈(T). Sinceis exclusive monotonic, it must be that c∉(T_i); since otherwise, (T')⊆(T), a contradiction. Then, sincesatisfies the ENM criterion, we know that c∈(T'). The claim follows.Due to the above claim and discussions, for every c'∈𝒞∖{c},-10pt cℰ'=cℰ+|𝒲_c| ≥c'ℰ+|𝒲_c| ≥c'ℰ+|𝒲_c'|≥c'ℰ'.-10ptThus, c∈(ℰ'). It remains to prove the other direction. Assume thatis not exclusive monotonic. Then, there exist two tournaments T and T' over the same candidate set 𝒞 and a c∈(T) such that (1) T[𝒞∖{c}]=T'[𝒞∖{c}]; (2)N^+_T(c)⊆ N^+_T'(c); and (3) c∉(T'), or c∈(T') but (T')⊈(T). If c∉(T') in Condition (3), then, the election consisting of only one vote defined as T obviously shows that -Approval is not monotonic. Otherwise, we construct an election as follows. Let b be any arbitrary candidate in (T')∖(T). The election consists of the following votes: 1 vote defined as T; 2 votes each defined as a tournament where b is the source; 1 vote defined as a tournament where c is the source. Apparently, both b and c are winners in the election, with each having two approvals. However, by replacing the vote defined as T by T', b gets one more approval from T', implying c is no longer a winner. Therefore, in this case -Approval is not monotonic.Assume thatdoes not satisfy the ENM criterion. Then, there exist two tournaments T and T' and a c∉(T) such that (1) T[𝒞∖{c}]=T'[𝒞∖{c}]; (2)N^+_T(c)⊆ N^+_T'(c); and (3) (T')⊈(T) and c∉(T'). We construct an election as follows. Let b be any arbitrary candidate in (T')∖(T). The election consists of the following votes: 1 vote defined as T; 1 vote defined as a tournament where b is the source; 1 vote defined as a tournament where c is the source. It is clear that both b and c are winners. However, by replacing the vote defined as T by T', b gets one more approval from T', implying c is no longer a winner. Therefore, in this case -Approval is not monotonic.Due to Theorem <ref>, to check whether -Approval is monotonic for each ∈{TC,UC,CO}, we need only to investigate ifsatisfies the exclusive monotonicity and ENM criteria. Though that the monotonicity offor each ∈{CO,UC,TC} is apparent and has been studied in the literature <cit.>, whethersatisfies the two variants of the monotonicity criterion is not equally easy to see. In fact, we prove that among the three tournament solutions, only the top cycle satisfies the both the exclusive monotonicity and the ENM criteria. It then follows from this fact and Theorem <ref> that TC-Approval is monotonic, but CO-Approval and UC-Approval are not. Our results concerning the above discussion are summarized in Lemma <ref> and Theorem <ref> shown below. satisfies the exclusive monotonicity and ENM criteria.We first show that theis exclusive monotonic. Let T=(𝒞,≻) and T'=(𝒞,≻') be two tournaments, and c∈ TC(T) be a candidate such that (1) T[𝒞∖{c}]=T'[𝒞∖{c}] and (2) N^+_T(c)⊆ N^+_T'(c). Since theis monotonic <cit.>, it holds that c∈T'. It remains to show that T'⊆T.Due to the definition of the , it holds that T𝒞∖T. Then, it holds that T' 𝒞∖T. It directly follows that T'⊆T. Now we prove that thesatisfies the ENM criterion. Let T=(𝒞,) and T'=(𝒞,') be two tournaments, and c∈𝒞 be a candidate such that c∉TC(T), T[𝒞∖{c}]=T'[𝒞∖{c}], and N^+_T(c)⊆ N^+_T'(c). Let CC_1,CC_2,...,CC_t be the maximal strongly connected components of T. It is known that for every two distinct CC_i and CC_j where {i,j}⊆{1,2,...,t}, it holds that either V(CC_i) V(CC_j) or V(CC_j) V(CC_i), where V(CC_i) denotes the vertices of CC_i.Moreover, there is a unique ordering (CC_ρ(1),CC_ρ(2),...,CC_ρ(t)) where {ρ(1),ρ(2),...,ρ(t)}={1,2,...,t} such that CC_ρ(i) CC_ρ(j) for every 1≤ i< j≤ t <cit.>.Furthermore, T=CC_ρ(1). Without loss of generality, assume that c∈ CC_ρ(i) for some 1< i≤ t. We shall show that either T'=T, or T'⊈T and c∈T'. Due to the above discussion, if CC_ρ(1)' {c} in T', then T'=T=CC_ρ(1). Assume now that there exists some candidate b in CC_ρ(1) such that c' b. We distinguish between the following cases to proceed the proof.Case 1. CC_ρ(i)={c}, or |CC_ρ(i)|>1 and c is the source of T'[CC_ρ(i)].Let j be the minimum integer such that i>j≥ 0 and {c}' ⋃_i> j'>jCC_ρ(j'). If j=0, then {c} is the top cycle of T'. Otherwise, (⋃_1≤ j'≤ jCC_ρ(j'))∪{c} is the top cycle of T'.Case 2. |CC_ρ(i)|>1 and c is not the source of T'[CC_ρ(i)].In this case, ⋃_1≤ j≤ iCC_ρ(j) is the top cycle of T'.In summary, we can conclude that either T'=T, or T'⊈T and c∈T'. Thus, thesatisfies the ENM criterion. Both theand thedo not satisfy ENM. Counter-examples can be found in fig:counterexamplecopucENM. Due to Theorem <ref>, both CO-Approval and UC-Approval are not monotonic. Due to Theorem <ref>, Lemma <ref>, and the above discussion, we have the following theorem. TC-Approval is monotonic, and UC-Approval and CO-Approval are not monotonic.Finally, we study the Pareto optimal criterion. -5pt TC-Approval is Pareto optimal, and CO-Approval and UC-Approval are not Pareto optimal.We first show that TC-Approval is Pareto optimal. Let =TC-Approval. Let ℰ=(𝒞,𝒯) be an election and a,b∈𝒞 be two candidates such that a b in every T=(𝒞,)∈𝒯 and a∉(ℰ). We need to prove that b∉(ℰ). It is clear that if a∉T for some T∈𝒯, then b∉T. Therefore, aℰTC≥bℰTC. Since a∉(ℰ), there is a candidate d∈𝒞∖{a} such that dℰTC>aℰTC. As a result, dℰTC>bℰTC, implying that b∉(ℰ).To prove that CO-Approval and UC-Approval are not Pareto optimal, one only needs to check the election with four candidates a,b,c,d and one vote with preference a b, {b}{d,c}, {c}{a,d} and d a. It is easy to see that {b, c} is the winning set in CO-Approval, but all votes (in this case only one vote) prefer a to b. On the other hand, {a,b,c} is the winning set in UC-Approval, but all votes prefer d to a.§ COMPLEXITY OF STRATEGIC BEHAVIOR A voting system is immune to a constructive (resp. destructive) strategic voting problem if it is impossible to change a nonwinning (resp. winning) candidate to a winning (resp. nonwinning) candidate by performing the operations imposed in the definition of the problem. An obstacle to the fairness of voting systems is strategic behavior, potentially carried out by strategic individuals.For instance, a strategic individual wants to change the election result by adding/deleting some voters/candidates, or by bribing some voters. We refer to <cit.> for comprehensive surveys on this topic. In this section, we study the complexity of strategic behavior in -Approval. Complexity has been widely recognized as a barrier against strategic behavior <cit.>. The point is that if it isfor the strategic individual to find out how to successfully change the result, he might give up attacking the election. Due to this, complexity of strategic behavior of voting systems can be also considered as a property to evaluate the voting systems <cit.>. In this paper, wWe assume the familiarity of complexity theory. For readers who are not familiar with complexity theory, we refer to <cit.>.We particularly study the control and the bribery problems. In each problem, there is a strategic individual who has an incentive to influence the election result by modifying the registered election (see later for explanation of registered election).Depending on the situations, the strategic individual may have the goal to make a given distinguished candidate p win the registered election, or have the goal to make p not win the registered election. The former case is indicated by the word “constructive”, and the latter case by the word“destructive”. Following the convention in the literature <cit.>, for each problem studied in this paper, we distinguish between the unique-winner model and the nonunique-winner model. In the unique-winner model, winning an election means to be the unique winner, while in the nonunique-winner model, winning an election means to be the unique winner or to be a co-winner. Now we explain what modification operations the strategic individual may perform.Control. In the control problems studied in this paper, the strategic individual may performone of the following four modification operations: adding/deleting at most k votes/candidates, where k>0 is a given integer. Therefore, the combination of the two goals and the four modification operations gives us in total eight control problems denoted by CCAV, CCDV, CCAC, CCDC, DCAV, DCDV, DCAC and DCDC. The first two characters “CC”/“DC” in the notations stand for “constructive control”/“destructive control”, and the last two characters “AV”/“DV”/“AC”/“DC” stand for “adding votes”/“deleting votes”/“adding candidates”/“deleting candidates”. In the inputs of all control problems, we have a set 𝒞 of candidates, a list 𝒯 of votes over 𝒞, a distinguished candidate p∈𝒞, and an integer k>0. A registered election consists of all registered candidates and registered votes. In CCDV/DCDV, all candidates and votes are registered. The question is whether the strategic individual can achievegoal by deleting at most k votes from 𝒯. In CCAV/DCAV, all candidates are registered, but not all votes. In particular, a sublist 𝒰⊆𝒯 of unregistered votes is given in the input. The question is whether the strategic individual can achievegoal by adding (registering) at most k votes in 𝒰. In CCDC/DCDC, all candidates and votes are registered. The question is whether the strategic individual can achievegoal by deleting at most k candidates from 𝒞∖{p}. It should be pointed out that the deletion of a candidate does not affect the preference of a vote over the remaining candidates. In CCAC/DCAC, all votes are registered but not all candidates. In particular, a subset 𝒟⊆𝒞∖{p} of unregistered candidates is given in the input. The question is whether the strategic individual can achieve his goal by adding (registering) at most k candidates in 𝒟.The above defined control problems for many voting systems have been extensively studied in the literature. Due to the work of many researchers, the complexity of these control problems for almost all commonly used voting systems is known. A motivation of the study of control problems is that the issues of adding/deleting votes/candidates occur in many electoral settings, see, e.g., <cit.> for some concrete examples. In addition, as argued in <cit.>, adding voters pertains to simply encouraging some agents to vote, multiplying the existing agents, or performing false-name attacks. We refer to <cit.> for further discussions on control problems.Bribery. We study two bribery problems: Constructive Bribery by Reversing Arcs (CBRA) and Destructive Bribery by Reversing Arcs (DBRA). In both problems we are given an election ℰ=(𝒞,𝒯), a distinguished candidate p∈𝒞, and an integer k>0. The question is whether the strategic individual can achievegoal by reversing at most k arcs in total in tournaments in 𝒯.We remark that CBRA and DBRA have already been studied under the name microbribery <cit.>. However, the complexity of CBRA/DBRA for -Approval has not been studied yet.The study of bribery problems was initiated by Faliszewski, Hemaspaandra and Hemaspaandra FaliszewskiHH06, and since then many bribery problems have been proposed and studied <cit.>. A major motivation of such studies is that bribery behavior in voting happens in many real-world situations, such as in political elections.Our results concerning the complexity of control and bribery problems are summarized in Table <ref>. We achieve polynomial-time solvability results,results, as well asresults for the control and bribery problems for -Approval for different tournament solutions . We compare our complexity results for -Approval with the previous known results for the two most relevant voting systems Plurality and Approval. Our results reveal that -Approval resists more types of strategic behavior than both Plurality and Approval.We need the following two problems to establish our hardness results. Exact 3 Set Cover (X3C)Input: A universal set U={c_1,c_2,...,c_3} and a collection S of 3-subsets of U.Question: Is there an S'⊆S such that |S'|= and each c_i∈ U appears in exactly one set of S'? We assume that each element c_i∈ U occurs in exactly three different 3-subsets of S. Thus, we have that |S|=3. This assumption does not change theof the problem <cit.>.A dominating set of a tournament T=(V,) is a vertex subset D of the tournament such that for every vertex v not in D, there is a vertex u in D such that u v.Tournament Dominating Set (TDS)Input: A tournament T=(V,) and an integer k>0.Parameter: k.Question: Does T have a dominating set of size at most k? It is known that TDS is  <cit.>.-3pt§.§ Complexity of Election ControlIn this section, we study the complexity of control by adding/deleting votes/candidates for -Approval for difference tournament solutions .We first study CCAV and CCDV. We show that both problems for -Approval forbeing several natural tournament solutions are , as summarized in the following theorem. Recall that both problems are polynomial-time solvable for Plurality butfor Approval <cit.>.CCAV and CCDV are for -Approval for every ∈{TC, UC, CO}, for both the unique-winner model and the nonunique-winner model.We first considerthe unique-winner model of CCAV. The following reduction applies to every ∈{TC, UC, CO}. Let I=(U={c_1,c_2,...,c_3κ},S={s_1,s_2,...,s_3κ}) be an instance of the X3C problem. We create an instance (𝒞,𝒯,p∈𝒞,𝒰⊆𝒯,k) for the CCAV problem as follows.Candidates 𝒞. We create in total 3κ+2 candidates. In particular, for each c_i∈ U, we create a candidate a(c_i). In addition, we have a distinguished candidate p and a dummy candidate q.Registered votes 𝒯∖𝒰. For each c_i∈ U, we create κ-1 votes, each represented by a tournament where a(c_i) is the source. The arcs between candidates in 𝒞∖{a(c_i)} are set arbitrarily. In addition, we create 1 vote defined as a tournament where p is the source. The arcs between candidates in 𝒞∖{p} can be set arbitrarily.Unregistered votes 𝒰. For each s_i={c_x,c_y,c_z}∈ S where {x,y,z}⊆{1,2,...,3κ}, we create a voterepresented by a tournament T_s_i such that a(c_x),a(c_y),a(c_z),p,q induce a regular subtournamentand, moreover there is an arc from every candidate in the set {a(c_x),a(c_y),a(c_z),p,q} to every candidate not in the set. The arcs between candidates in 𝒞∖{a(c_x),a(c_y),a(c_z),p,q} are set arbitrarily. Observe that for every ∈{TC, UC, CO}, it holds that (T_s_i)={c_x,c_y,c_z,p,q}. Finally, we set k=κ. It is easy to see that with the registered votes, each candidate a(c_i) where i∈{1,...,3κ} has -Approval score κ-1, the distinguished candidate p has -Approval score 1, and the dummy candidate q has -Approval score 0. Observe that q cannot have an equal or higher -Approval score than that of p no matter which unregistered votes are added, since all unregistered votes approve both p and q. Observe further that adding any one unregistered vote increases the -Approval score of some candidate a(c_i) where i∈{1,...,3κ} to κ. Hence, in order to make p the unique winner, we need add exactly k unregistered votes. As a result, p has -Approval score κ+1 in the final election. Moreover, for every a(c_i) where i∈{1,...,3κ}, we can add only one unregistered vote that approves a(c_i). This happens if and only if there is an exact 3-set cover.Thereduction for the nonunique-winner model is similar to the above reduction with the difference thatwe create one more registered vote for each c_i∈ U. Now we consider the CCDV problem for the unique-winner model. We construct an instance (𝒞,𝒯,p∈𝒞,k) as follows.Candidates 𝒞. We create in total 3κ+1 candidates in 𝒞. In particular, for each c_i∈ U, we create a candidate a(c_i). In addition, we have a distinguished candidate p.Votes 𝒯. For each s_i={c_x,c_y,x_z}∈ S where {x,y,z}⊆{1,2,...,3κ}, we create a vote represented by a tournament T_s_i where a(c_x),a(c_y),a(c_z) form a directed triangle and, moreover, there is an arc from every candidate in {a(c_x),a(c_y),a(c_z)} to every candidate in 𝒞∖{a(c_x),a(c_y),a(c_z)}. It is easy to check that (T_s_i)={a(c_x),a(c_y),a(c_z)} for every ∈{TC, UC, CO}. In addition, we create 3 votes, each represented by a tournament where p is the source.Since each c_i occurs in exactly three sets in S, each a(c_i) where i∈{1,2,...,3κ} as well as p has -Approval score 3. In order to make p the unique winner, for each a(c_i) where i∈{1,...,3κ}, we need to delete one vote that approves a(c_i). This happens if and only if there is an exact 3-set cover.Thereduction for the CCDV problem for the nonunique-winner model is similar to the above reduction, with only the difference that we create one less vote corresponding to p, so that the -Approval score of p in the original election is 2.Now we consider DCAV and DCDV. It is known that both problems are polynomial-time solvable for Plurality and Approval <cit.>. We prove that both problems are polynomial-time solvable for every -Approval whereis a polynomial-time computable tournament solution. A tournament solutionis polynomial-time computable if for every tournament T, the set (T) can be calculated in polynomial time in the size of T. It is well known that the top cycle, the uncovered set and the Copeland set are all polynomial-time computable <cit.>. It is worth mentioning that there exist numerous tournament solutions such as the tournament equilibrium set which are not polynomial-time computable. See <cit.> for further discussions.-4ptDCAV and DCDV are polynomial-time solvable for -Approval such thatis polynomial-time computable, for both the unique-winner model and the nonunique-winner model. To prove the theorem, we reduce DCAV (resp. DCDV) for -Approval whereis a polynomial-time computable tournament solution to the same problem for Approval in polynomial time. In particular, given an instance ofDCAV (resp. DCDV)for -Approval, we calculate (T) for every vote T in the instance (for both registered and unregistered votes if applicable). Sinceis polynomial-time computable, this can be done in polynomial time. Then, we can get an instance of DCAV (resp. DCDV)for Approval by taking the same candidate set, and changing each vote originally defined as a tournament T to a vote defined as the dichotomous preference ((T),𝒞∖(T)), where 𝒞 is the candidate set. The theorem then follows from the fact thatDCAV (resp. DCDV)for Approval is polynomial-time solvable, for both the unique-winner model and the nonunique-winner model <cit.>. Theorem <ref> implies that DCAV and DCDV are polynomial-time solvable for TC-Approval, CO-Approval and UC-Approval.Now we turn our attention to control by adding/deleting candidates. In Plurality, each vote is defined as a linear order over the candidates and the top ordered candidate is approved. A linear order can be considered as a transitive tournament, where there is an arc from a candidate a to another candidate b if a is ordered before b. Thus, the top ordered candidate in the linear order is the source of the transitive tournament. It is clear that the top cycle, Copeland set and uncovered set of a transitive tournament consist of exactly the source of the tournament. Hence, CCAC/CCDC/DCAC/DCDC for Plurality is a special case of the same problem for -Approval for every ∈{TC, UC, CO}. Since CCAC/CCDC/DCAC/DCDC for Plurality is  <cit.>, for both the unique-winner model and the nonunique-winner model, the same problem for -Approval for each ∈{TC, UC, CO} isas well[From a parameterized complexity point of view, Yang and Guo Yangaamas14b proved that the CCDC problem for Plurality iswith respect to the number of deleted candidates even in 3-peaked elections. ], as summarized in the following theorem.-4ptCCAC/CCDC/DCAC/DCDC isfor -Approval for every ∈{TC, UC, CO}, for both the unique-winner model and the nonunique-winner model. §.§ Complexity of BriberyNow we study CBRA and DBRA for -Approval for different tournament solutions . Since reversing an arc may make a transitive tournament intransitive, it does not make sense to study CBRA and DBRA for Plurality and Approval.CBRA isfor UC-Approval,for CO-Approval, and polynomial-time solvable for TC-Approval,for both the unique-winner model and the nonunique-winner model.Yang and Guo DBLP:conf/aldt/YangG13 studied a problem where the given are a tournament and a distinguished vertex p in the tournament, and the question is whether p can be made a king by reversing at most k arcs. In particular, they proved that this problem is W[2]-hard with respect to k, by a reduction from the TDSproblem (See <cit.> for further details). This problem can be considered as a special case of the nonunique-winner model of the CBRA problem for UC-Approval, where the instances consist of only one vote. To prove theof CBRA for UC-Approval for the unique-winner model, we need only to create one more vote defined as a tournament where the distinguished candidate p is the source (the arcs between other candidates can be set arbitrarily) in the reduction in <cit.>.Now we prove theof CBRA for CO-Approval by a reduction from the X3C problem. We first study the nonunique-winner model. Let (U={c_1,c_2,...,c_3κ},S={s_1,s_2,...,s_3κ}) be an instance of the X3C problem. We create an instance (𝒞,𝒯,p∈𝒞,k=κ) of CBRA as follows. Without loss of generality, assume that k≥ 4.Candidates 𝒞. We create in total 6κ+1 candidates in 𝒞.For each c_x∈ U, we create a candidate a(c_x). For each s_i∈ S, we create a candidate a(s_i). In addition, we have a distinguished candidate p.Votes 𝒯. We create in total 3k^2+4k+2 votes. For ease of exposition, we divide the votes into three sublists A,B,C. The sublist A consists of the following votes. For each s_i={c_x,c_y,c_z}∈ S, we create a vote defined as a tournament H_s_i=(𝒞,_s_i) such that (1) a(c_x) _s_i a(c_y), a(c_y)_s_i a(c_z), a(c_z)_s_i a(c_x); (2) there is an arc from a(s_i) to every of {a(c_x),a(c_y),a(c_z)}; (3) there is an arc from every {a(c_x),a(c_y),a(c_z)} to every candidate in 𝒞∖{a(s_i),a(c_x),a(c_y),a(c_z)}; (4) there is an arc from a(s_i) to every candidate in 𝒞∖{a(s_i),a(c_x),a(c_y),a(c_z),a(c_u),a(c_v)}, and an arc from each of {a(c_u), a(c_v)} to a(s_i), where a(c_u) and a(c_v) are any two arbitrary candidates in 𝒞∖{a(s_i),a(c_x),a(c_y),a(c_z),p}; and (5) H_s_i[𝒞∖{a(s_i),a(c_x),a(c_y),a(c_z)}] is regular. It is easy to verify that in H_s_i, the Copeland score of each a(s_i),a(c_x),a(c_y),a(c_z) is 6k-2, and of each other candidate is at most ⌈6k-3/2⌉+2. Thus, the Copeland set of H_s_i is {a(s_i),a(c_x),a(c_y),a(c_z)}. Moreover, due to the large score gap between candidates in the Copeland set and candidates not in the Copeland set (and due to k≥ 4), no candidate in 𝒞∖{a(s_i),a(c_x),a(c_y),a(c_z)} can be included in the Copeland set of H_s_i by reversing at most k arcs. The sublist B consists of k+2 votes, each of which is defined as a tournament such that p is the source, and the subtournament induced by 𝒞∖{p} is regular. Finally, the sublist C consists of the following 3k^2 votes. For each c_x∈ U, we create k votes, each of which is defined as a tournament such that a(c_x) is the source and the subtournament induced by 𝒞∖{a(c_x)} is regular. Notice that due to the regularity of the subtournaments induced by all candidates except the sources in all tournaments constructed in sublists B and C, it is impossible to change the Copeland set of every tournament in B∪ C by reversing at most k arcs. It is easy to verify that the CO-Approval score of p is k+2, of each a(c_x) where c_x∈ U is k+3, and of each a(s_i) where s_i∈ S is 1. Now we prove the correctness of the reduction.(⇒:) Suppose that S'⊂ S is an exact 3-set cover, i.e. |S'|=κ=k and for every c_x∈ U there is exactly one s∈ S' such that c_x∈ s. We shall show that we can make p a winner by reversing at most k arcs. In particular, for each s_i={a_x,a_y,a_z}∈ S', we reverse the arc from a(s_i) to a(c_u), where a(c_u) is one candidate in 𝒞∖{a(s_i),a(c_x),a(c_y),a(c_z),p} such that a(c_u)_s_i a(s_i), as defined above. After reversing this arc, the Copeland set of H_s_i consists of only the candidate a(s_i). As a result, the CO-Approval score of each {a(c_x),a(c_y),a(c_z)} decreases by one. Since S' is an exact 3-set cover, due to the construction, after reversing all these k arcs, the CO-Approval score of each a(c_x) where c_x∈ U is k+2 and of each a(s_i) where s_i∈ S is 1. Moreover, the CO-Approval score of p is k+2, implying that p is a winner.(:⇐) Observe that due to the score gap between every a(s_i) where s_i∈ S and p, none of a(s_i) has a chance to have a CO-Approval score that is equal to or higher than that of p by reversing at most k arcs. Moreover, as discussed above, we cannot include p in the Copeland set of each tournament in A∪ C by reversing at most k arcs. Therefore, in order to make p a winner, for each a(c_x) where c_x∈ U, we need to decrease the number of tournaments whose Copeland sets include a(c_x) by at least one. Due to the above discussion, we cannot change the Copeland set of each tournament in B∪ C by reversing at most k arcs. Thus, the optimal solution is to reversearcs in tournaments in A. Moreover, if we attempt to reverse arcs in some tournament H_s_i in A, the optimal choice is to only reverse an arc from a(s_i) to a candidate a(c_u)∈𝒞∖{a(s_i),a(c_x),a(c_y),a(c_z),p} such that a(s_i)_s_i a(u) in H_s_i (such a candidate a(c_u) exists due to the construction of the votes), so that the CO-Approval score of each a(c_x),a(c_y),a(c_z) decreases by one. Let A' be the set of tournaments in A where an arc is reversed. Let S'={s_i| H_s_i∈ A'}. Due to the above discussion, for every c_x∈ U, there is at least one H_s_i∈ A' such that c_x∈ s_i. Since we can reverse at most k arcs, it holds that |A'|≤ k, implying that S' is an exact 3-set cover.To prove the unique-winner model of the problem, we need only to create one more vote in B in the above reduction. Now we study destructive bribery by reversing arcs.DBRA isfor UC-Approval, and polynomial-time solvable for TC-Approval and CO-Approval. The results hold for both the unique-winner model and the nonunique-winner model. We first prove theof DBRA for UC-Approval for the nonunique-winner model. We develop an -reduction from the TDS problem to the DBRA problem.Let (T=(V,≻),k) be an instance of the TDS problem. Let n=|V|. We assume that n≥ (k+1)(2k+4). This assumption does not affect the W[2]-hardness of the TDS problem[If n≤ (k+1)(2k+3), we can add (k+1)(2k+4)-n additional vertices to the tournament such that there is an arc from every vertex in the original tournament to every newly added vertex.]. We create an instance ℐ=(𝒞,𝒯,p∈𝒞,k) for DBRA as follows.Candidates 𝒞. For each v∈ V, we create a candidate a(v). Let the distinguished candidate p be any arbitrary candidate a(w) such that w is not a king in T (in the W[2]-hardness reduction of the TDS problem in <cit.>, there exist vertices w which are not kings. Thus, such a candidate a(w) is well definedwe need to ensure that there is an isolated clique, say a triangle, in the graph G in the proof of Theorem 4.1 in <cit.>. In this case, each vertex in the isolated clique will not be a king.). In addition, we create an additional candidate q. Thus, 𝒞={a(v)| v∈ V}∪{q}.Votes 𝒯. The list 𝒯 of votes consists of three sublists 𝒯_1,𝒯_2,𝒯_3 of votes. The list 𝒯_1 consists of only one vote T_1=(𝒞,_1), which is created first with a copy of T, i.e., a(v)_1 a(u) in T_1 if and only if v u in T. Then, we create an arc from every candidate in 𝒞∖{q} to q.The list 𝒯_2 consists of 2k+3 votes, each of which is defined as a tournament such that q is the source. The arcs between every two candidates in 𝒞∖{q} are set arbitrarily. The construction of the votes in 𝒯_3 is a little involved. Let A be any arbitrary (k+1)(2k+3)-subset of 𝒞∖{p,q}, and B=𝒞∖ (A∪{p,q}). Let (A_0,A_1,...,A_2k+2) be any arbitrary partition of A such that |A_i|=k+1 for every i∈{0,1,...,2k+2}. Let f_i be any arbitrary one-to-one mapping from A_i to A_(i+1) (2k+3) for every i∈{0,1,...,2k+2}. We create in total 2k+3 votes in 𝒯_3. In particular, for each i∈{0,1,...,2k+2}, we create a vote defined as a tournament H_i such that(1) there is an arc from p to every candidate in 𝒞∖ (A_i∪{p});(2) there is an arc from a candidate a(v)∈ A_(i+1) (2k+3) to a candidate a(u)∈ A_i if and only if f_i(a(u))=a(v);(3) there is an arc from every candidate in A_i to every candidate in 𝒞∖ (A_i∪ A_(i+1) (2k+3));(4) there is an arc from every candidate in 𝒞∖{q} to q;(5) H_i[A_i] is isomorphic to H_i[A_(i+1) (2k+3)]; and(6) there is no source in H_i[A_i].The arcs that are not specified above are set arbitrarily. It is clear that such a tournament can be constructed in polynomial time. Observe that {p}∪{A_i} is the uncovered set of H_i. every candidate in B cannot reach p. since there is no source in H_i, for every candidate a(v)∈ A_(i+1) 2k+3 cannot reach at least one candidate in H_i. Moreover, by constructing the votes this way, it is impossible to make q a king in each tournament H_i in 𝒯_3 by reversing at most k arcs. since to the cost of making q reach each candidate in A_i is at least one arc reversal.Now we show the correctness of the reduction. It is easy to calculate that in the election, both p and q have the same UC-Approval score 2k+3, and every other candidate has UC-Approval score at most 2. Thus, {p,q} is the UC-Approval winning set of the election.(⇒:) Let D⊆ V(T) be a dominating set of size at most k of T. After reversing all arcs a(v)q where v∈ D in T_1, q becomes a king in T_1. As a result, q has UC-Approval score 2k+4 and p still has UC-Approval score 2k+3, implying that p is no longer a winner.(:⇐) Suppose that ℐ is a .Observe that due to the large UC-Approval score gap between p and every candidate in 𝒞∖{p,q}, none of 𝒞∖{p,q} has a chance to have a higher score than that of p by reversing at most k arcs. Therefore, q is the only candidate which can prevent p from being a winner.Moreover, due to the above discussion, it is impossible to make q a king in each vote in 𝒯_3 by reversing at most k arcs. Given that q is the unique king in each tournament in 𝒯_2, in order to prevent p from being a winner, q has to become a king in the first vote T_1 by reversing at most k arcs. This happens only if there is a dominating set of T of size at most k. To prove the unique-winner model, we need only to create one less tournament in 𝒯_2. Now we develop a polynomial-time algorithms for DBRA for TC-Approval for the nonunique-winner model and CO-Approval. TC-Approval. Let (𝒞,𝒯,p∈𝒞, k) be an instance where p is a TC-Approval winner. Let m=|𝒞|. We assume that m≥ 3 (otherwise, we can easily solve the problem). To prevent p from being a winner, we need to make a candidate q have a higher TC-Approval score than that of p by reversing at most k arcs. Based on this observation, the algorithm breaks down the given instance into m-1 subinstances, each of which takes a candidate q≠ p together with (𝒞,𝒯,p∈𝒞, k) as the input, and asks whether q can have a higher TC-Approval score than that of p by reversing at most k arcs. Obviously, the original instance is aif and only if at least one of the subinstances is a . It remains to develop a polynomial-time algorithm to solve each subinstance. The following claim is useful.Claim. Let H=(V,) be a tournament and c∈ V a candidate not in the top cycle of H. Then, we can make c be included in the top cycle of H by reversing only one arc.Proof of the claim. LetCC_ρ(1),CC_ρ(2),...,CC_ρ(t) be the unique ordering of the maximal strongly connected components of Hsuch that CC_ρ(i) CC_ρ(j) for every 1≤ i< j≤ t.The top cycle of H is exactly CC_ρ(1). Assume that c∈ CC_ρ(i) for some 1< i≤ t. Then,by reversing any one arbitrary arc between c and a candidate in CC_ρ(1),⋃_1≤ j≤ i V(CC_j) becomes the top cycle of the tournament (see the proof of Lemma <ref> for some additional details). This completes the proof of the above claim. Since m≥ 3, it is impossible to decrease the score gap between p and q by 2 by reversing 1 arc.Then, due to the above claim,to prevent p from being a winner, an optimal choice is to reversearcs in the tournaments whose top cycles do not include q in advance.Precisely, the algorithm finds all tournaments in the subinstance where q is not in the top cycle. Let k' be the number of such tournaments. Then, due to the above discussion, we can increase the TC-Approval score of q by min{k,k'} without changing the TC-Approval score of p, by reversing min{k,k'} arcs.As a result, if qℰTC+min{k,k'}≥pℰTC where ℰ=(𝒞,𝒯), the subinstance is a ; otherwise, it is a . It should be noted that Papadimitriou and Yannakakis <cit.> devised an O(n^O(logn))-time algorithm for the TDS problem, which implies that the TDS problem is probably not , unless ⊆ Dtime(n^logn). In fact, Downey and Fellows proved theof the TDS problem by a reduction from the Dominating Set problem which is bothand  <cit.>. However, the reduction is an -reduction but not a polynomial-time reduction—it takes O(2^O(k)· poly(n)) time where n is the number vertices of the given tournament and k is the solution size. Hence, our reductions in the proofs of Theorems <ref> and <ref> do not imply that DBRA and CBRA for UC-Approval are . Whether DBRA and CBRA for UC-Approval areremain open. § CONCLUDING REMARKS-4pt We have studied a class of approval-based voting correspondences for the scenario where voters may have intransitive preferences.Each newly introduced voting correspondence -Approval is a natural combination of the classic Approval correspondence and a well-studied tournament solution . In particular, each voter with preference T is assumed to approve all candidates in (T) and disapprove all the remaining candidates. The winners are the ones receiving the most approvals. This class of new voting correspondences extends the classic Approval voting to the settings where voters have intransitive preferences. Note taht an intransitive preference is not necessarily to be cast by a single voter, but can be drawn from the preferences of voters in a subvoting, as we illustrated in the introduction. As far as we know, such -Approval voting correspondence has not been studied in the literature.In this paper, we first showed that -Approval satisfies several axiomatic properties for ∈{CO,TC,UC}. As a byproduct, we proposed two new concepts of monotonicity criteria of tournament solutions, namely, the exclusive monotonicity and the ENM, and proved that the top cycle satisfies both monotonicity criteria, while the Copeland set and the uncovered set fail to satisfyENM. Then, we investigated the complexity of constructive/destructive control by adding/deleting voters/candidates and constructive/destructive bribery by reversing arcs for -Approval for ∈{CO,UC,TC}. Our results reveal that -Approval resists more types of strategic behavior than both Plurality and Approval. See Table <ref> for a summary of our complexity results.There remain several open questions for future research. 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http://arxiv.org/abs/1702.07902v1

Multi-scale Image Fusion Between Pre-operative Clinical CT and X-ray Microtomography of Lung Pathology Holger R. Rotha, Kai Nagarab, Hirohisa Odab,Masahiro Odab, Tomoshi Sugiyamac, Shota Nakamurac, Kensaku Moria,b aInformation & Communications, Nagoya University, JapanbGraduate School of Information Science, Nagoya University, JapancNagoya University Graduate School of Medicine, Japan December 30, 2023 ============================================================================================================================================================================================================================================================================================================Computational anatomy allows the quantitative analysis of organs in medical images. However, most analysis is constrained to the millimeter scale because of the limited resolution of clinical computed tomography (CT). X-ray microtomography () on the other hand allows imaging of ex-vivo tissues at a resolution of tens of microns. In this work, we use clinical CT to image lung cancer patients before partial pneumonectomy (resection of pathological lung tissue). The resected specimen is prepared forimaging at a voxel resolution of 50 μm (0.05 mm). This high-resolution image of the lung cancer tissue allows further insides into understanding of tumor growth and categorization. For making full use of this additional information, image fusion (registration) needs to be performed in order to re-align theimage with clinical CT. We developed a multi-scale non-rigid registration approach. After manual initialization using a few landmark points and rigid alignment, several levels of non-rigid registration between down-sampled (in the case of) and up-sampled (in the case of clinical CT) representations of the image are performed. Any non-lung tissue is ignored during the computation of the similarity measure used to guide the registration during optimization. We are able to recover the volume differences introduced by the resection and preparation of the lung specimen. The average (± std. dev.) minimum surface distance betweenand clinical CT at the resected lung surface is reduced from 3.3 ± 2.9 (range: [0.1, 15.9]) to 2.3 mm ± 2.8 (range: [0.0, 15.3]) mm. This is a significant improvement with p < 0.001 (Wilcoxon Signed Rank Test). The alignment of clinical CT withwill allow further registration with even finer resolutions of(up to 10 μm resolution) and ultimately with histopathological microscopy images for further macro to micro image fusion that can aid medical image analysis. image fusion, non-rigid registration, computed tomography (CT), X-ray microtomography ()§ INTRODUCTION Clinical computed tomography (CT) is used for the diagnostic imaging of the living human (in-vivo imaging). As a result, most computational analysis is constrained to the millimeter scale because of the limited resolution of clinical CT. At this millimeter scale, pulmonary blood vessels and lung lobes can be observed. However, finer detailed anatomy is not observable. X-ray microtomograph () on the other hand allows imaging of ex-vivo tissues at a resolution of tens of microns. At this μm-scale, the alveoli and bronchiole regions can be clearly observed <cit.>.The prospective study of pre-operative imaging together with the high-resolution image analysis of resected tissue after surgery may provide us with unique opportunities to verify and potentially improve imaging protocols for cancer diagnostics <cit.>. In this work, we use clinical CT to image lung cancer patients before partial pneumonectomy (resection of pathological lung tissue). The resected specimen is then prepared forimaging allowing the imaging of μm-scale anatomy. This study investigates the use of non-rigid intensity-based registration in order to establish a scale-seamless registration between clinical CT and , with the ultimate aim of allowing a seamless navigation between anatomical scales inside the human body <cit.>. Related work is the registration of in-vivo and ex-vivo MRI of surgically resected specimens by <cit.>. § METHOD The resected specimen is prepared forimaging at a voxel resolution of 50 μm (0.05 mm). This high-resolution image of the lung cancer tissue allows further insides into understanding of tumor growth and categorization. For making full use of this additional information, image fusion (registration) needs to be performed in order to re-align theimage with clinical CT. We developed a multi-scale non-rigid registration approach.§.§ Establishing scale-seamless registration betweenand clinical CT We use a non-rigid registration method to align thespecimen with the pre-operative clinical CT. This method is driven by the intensity similarity between the images. Any non-lung tissue is ignored during computation of the similarity measure 𝒮 used to guide the registration during optimization. This is achieved via simple thresholding and morphological operations to extract the lung region in clinical CT andas pre-processing step. A coarse-to-fine approach is proposed in order to capture first the largest deformations and then the smaller differences between both images. This is achieved with a four-level multi-resolution pyramid as illustrated in Fig. <ref>. In order to compensate for the large differences in resolution between the(in the case of) and clinical CT images, we use a down-sample and up-sample (in the case of clinical CT) pyramid and perform the registration optimizations at each level in a coarse-to-fine fashion.A good initialization for the registration algorithm is important. We use a few (3∼5) manual point in order to provide affine alignment of the specimen with the pre-operative CT image. After affine initialization, we establish non-rigid alignment using the B-spline registration method, also known as 3D free form deformation based registration of <cit.> with the implementation provided by <cit.>.A 3D cubic B-Splines deformation model uses a lattice of control points {ϕ⃗}. The spacing between each control point is uniform and denoted as δ_x, δ_y, and δ_z along the x-, y-, and z-axis respectively. For each voxel x⃗ in the domain Ω of the target image. the deformation 𝐓(x⃗) can be computed as:𝐓(x⃗) = ∑_i,j,kβ^3(x/δ_x - i) ×β^3(y/δ_y - j) ×β^3(y/δ_z - k) ×ϕ⃗_ijk, where β^3 represents the cubic B-Spline function.The images are aligned by finding the transformation which maximizes the following objective function:𝒪(I_p,I_s(𝐓);{ϕ⃗}) = (1 - α - β - γ)×𝒮 - α×𝒞_smooth(𝐓) - β×𝒞_volpres(𝐓) -γ×𝒞_inconsistency(𝐓)which combines a similarity measure, 𝒮, and three penalty constraint terms, 𝒞_smooth, 𝒞_volpres, and 𝒞_inconsistency. Each term is weighted against each other by user-defined weights α, β, and γ.The similarity measure used between the reference (R) and floating image (F) is normalized mutual information (NMI): 𝒮≡𝐍𝐌𝐈 =H(R)+ H(F(𝐓))/ H(R, F(𝐓)) where H(R) and H(F(𝐓)) the two marginal entropies, and H(R, F(𝐓)) is the joint entropy. Its computation requires a joint histogram which is filled by using a Parzen Window (PW) approach <cit.>.The three constraint terms are used to encourage realistic deformations. The bending energy describes the smoothness of the deformation and is defined as:𝒞_smooth = 1/N∑_x⃗∀Ω( |∂^2𝐓(x⃗)/∂x^2|^2 + |∂^2𝐓(x⃗)/∂y^2|^2 +|∂^2𝐓(x⃗)/∂z^2|^2 + 2×[|∂^2𝐓(x⃗)/∂xy|^2 + |∂^2𝐓(x⃗)/∂yz|^2 + |∂^2𝐓(x⃗)/∂xz|^2 ]) . The volume-preserving penalty term discourages large expansions/contractions, and is defined as:𝒞_volpres = 1/N∑_x⃗∀Ω[ log( ( Jac(𝐓(x⃗))))]^2In addition we prevent the occurrence of folding in the transformation using a folding correction scheme <cit.>. For each transformed voxel that would cause a negative Jacobian determinant, its influence on its neighborhood control points is computed. The control point positions is then changed until the determinant value is positive. In an ideal case, the transformations from F to R (forward) and R to F (backward) are the inverse of each other, e.g.=^-1 and =^-1 <cit.>. Hence, we include a penalty term that encourages inverse consistency of both transformations. We follow the approach of <cit.> using compositions ofandand add 𝒞_inconsistency = ∑_x⃗∀Ω((x⃗))^2 + ∑_x⃗∀Ω((x⃗))^2The following parameters were found empirically by visual examination of the registration results. We use a four-level multi-resolution pyramid with a maximum of 500 iterations per level. Both the image and B-spline control point grid resolutions are doubled with increasing resolution levels. The final control point spacing between voxels is 5. The objective function weights are set to α = 10^-4, β = 10^-12, and γ = 0.1. These parameters were found to recover the majority of the deformation between the two images, while preventing unrealistic deformations from occurring. We used the open-source software[<http://sourceforge.net/projects/niftyreg>] by <cit.> for this study. § RESULTS We collected pre-operative clinical CTs from two patients before partial pneumonectomy. The clinically acquired CT images have dimensions of [512, 512, 435∼554], and voxel spacings of [0.625, 0.625, 0.6] mm. Theimages used for registration had dimensions of [278∼512, 278∼512, 346∼538] withisotropic voxel spacings of 0.111∼0.127 mm. Note that we downsampled the originalimages (circa 50μm resolution) by a factor of two for the registration experiments. The tube voltage forwas 90kVp, and tube current was 110 μA.3∼5 corresponding points where chosen by an expert clinician (SN) in order to provide an initial affine alignment of the specimen with the pre-operative CT image. We then crop the clinical CT image to the extent of the alignedimage for subsequent non-rigid registration.Figure <ref> shows the alignment oflung specimen with pre-operative clinical CT before and after non-rigid registration for one case. A qualitatively better alignment ofwith the target clinical CT regions can be observed. In order to quantitatively evaluate the registration performance, we measure the average minimum surface distance (AvgDist) between the lung surface extracted from clinical CT andbefore and after non-rigid registration. Figure <ref> shows the extracted surfaces used for measurement before and after non-rigid alignment. The AvgDist measures are given in Table <ref>. A reduction from 3.3 ± 2.9 (range: [0.1, 15.9]) to 2.3 mm ± 2.8 (range: [0.0, 15.3]) mm on average can be observed. This is a significant improvement with p < 0.001 (Wilcoxon signed rank test).§ CONCLUSIONS We presented a method for non-rigid alignment between images of pre-operative clinical CT and x-ray microtomography () of lung pathology. After manual initialization using a few landmark points and affine alignment, several levels of non-rigid registration between down-sampled (in the case of) and up-sampled (in the case of clinical CT) representations of the image are performed. This allows us to recover the volume differences introduced by the resection and preparation of the lung specimen. The proposed multi-scale image fusion approach will allow further registration with even finer resolutions of(up to 10 μm resolution) <cit.> and ultimately with histopathological microscopy images for further macro to micro image fusion that can aid medical image analysis. § ACKNOWLEDGEMENTSThis paper was supported by MEXT KAKENHI (26108006 and 15H01116). ieeetr

http://arxiv.org/abs/1702.08155v1

Signal Denoising Using the Minimum-Probability-of-Error Criterion Jishnu Sadasivan, Subhadip Mukherjee, and Chandra Sekhar Seelamantula, Senior member, IEEE J. Sadasivanis with the Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore, India.Phone: +91 80 2293 2276. Fax: +91 80 2360 0563. E-mail: jishnus@ece.iisc.ernet.in. S. Mukherjee and C. S. Seelamantula are with the Department of Electrical Engineering, Indian Institute of Science, Bangalore, India.Phone: +91 80 2293 2695. Fax: +91 80 2360 0444. E-mails: {subhadip, chandra.sekhar}@ee.iisc.ernet.in.IEEE Transactions on Signal Processing Received: date / Accepted: date ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================ We address the problem of signal denoising via transform-domain shrinkage based on a novel risk criterion calledthe minimum probability of error (MPE), which measuresthe probability that the estimated parameter lies outside an ϵ-neighborhood of the actual value. However, the MPE,similar to the mean-squared error (MSE), depends on the ground-truth parameter, and has to be estimated from the noisy observations. We consider linear shrinkage-based denoising functions, wherein the optimum shrinkage parameter is obtained by minimizing an estimate of the MPE. When the probability of error is integrated overϵ, it leads tothe expected ℓ_1 distortion. The proposed MPE and ℓ_1 distortion formulations are applicable to various noise distributions by invoking a Gaussian mixture model approximation. Within the realm of MPE, we also develop an extension of the transform-domain shrinkage by grouping transform coefficients, resulting in subband shrinkage. The denoising performance obtained within the proposed framework is shown to be better than that obtained using the minimum MSE-based approaches formulated within Stein's unbiased risk estimation (SURE) framework, especially in the low measurement signal-to-noise ratio (SNR) regime. Performance comparison with three state-of-the-art denoising algorithms, carried out on electrocardiogram signals and two test signals taken from the Wavelab toolbox, exhibits that the MPEframework results in consistent SNR gains for input SNRs below 5 dB. Minimum probability of error, shrinkage estimator, risk estimation, transform-domain shrinkage, subband shrinkage, expected ℓ_1 distortion, Gaussian mixture model.§ INTRODUCTIONSignal denoisingalgorithms are often developed with the objective of minimizing the mean-squared error (MSE) between an estimate and the ground-truth, which may be deterministic or stochastic with a known prior. The latter formalism leads to Bayesian estimators. Within the deterministic signal estimation paradigm, which is also the formalism considered in this paper, one typically desires that the estimator has minimum variance and is unbiased (MVU) <cit.>. An MVU estimator may not always exist, and if it does, it can be obtained using the theory of sufficient statistics. Eldar and Kay <cit.> showed that, when it comes to minimizing the MSE, biased estimates may outperform the MVU estimate. For example, one could shrink the MVU estimate and optimize for the shrinkage parameter so that the MSE is minimum.In this paper, we consider the problem of estimating a deterministic signal corrupted by additive white noise. The noise distribution is assumed to be known, but not restricted to be Gaussian. We propose a new distortion metric based on the probability of error and develop estimators using a transform-domain shrinkage approach. Before proceeding with the developments, we review some important literature related to the problem at hand. §.§ Prior Art The MSE is by far the most widely used metric for obtaining the optimum shrinkage parameter. Since the MSE is a function of the parameter to be estimated, direct minimization might result in an unrealizable estimate, in the sense that it might depend on the unknown parameter. However, in somecases, it is possible to find the optimum shrinkage parameter, for example, using a min-max approach <cit.>, where the parameter is constrained to a known set.An optimum shrinkage estimator, when thevariance of the unbiasedestimate (or MVU) is ascaled version of the square of the parameter, with a known scaling, is proposed in <cit.>. Optimum shrinkage estimators have also been computed based on risk estimation, where an unbiased estimate of the MSE that depends only on the noisy observations is obtained and subsequently minimized over the shrinkage parameter.Under the assumption of Gaussian noise, an unbiased estimate of the MSE, namely Stein's unbiased risk estimator (SURE), was developed based on Stein's lemma <cit.>, and has been successfully employed in numerous denoising applications. In his seminal work <cit.>, Steinproved that theshrinkage estimator of the mean of a multivariate Gaussian distribution, obtained from its independent and identically distributed (i.i.d.) samples by minimizing SURE, dominates the classical least-squares estimate when the number of samples exceeds three <cit.>.A risk minimization approach for denoising using a linear expansion of elementary thresholding functions has been addressed in <cit.>, wherein the combining weights are chosen optimally to minimize the SURE objective. SURE-optimized wavelet-domain thresholding techniques have been developed in <cit.>. Atto et al. <cit.> have investigated the problem of signal denoising based on optimally selecting the parameters of a wavelet-domain smooth sigmoidal shrinkage function by minimizing the SURE criterion. The use of SURE objective is not restricted to denoising; it has found applications in image deconvolution as well <cit.>.Ramani et al. <cit.> developed a Monte-Carlo technique to select the parameters of a generic denoising operator based on SURE. An image denoising algorithm based on non-local means (NLM) is proposed in <cit.>, where parameters of NLM are optimized using SURE. Notable denoising algorithms that aim to optimize the SURE objective include wavelet-domain multivariate shrinkage <cit.>, local affine transform for image denoising <cit.>, optimal basis selection for denoising <cit.>, raised-cosine-based fast bilateral filtering for image denoising <cit.>, SURE-optimized Savitzky-Golay filter <cit.>, etc..The original formulation of SURE, which assumed independent Gaussian noise was extended to certain distributions in continuous and discrete exponential families in <cit.> and <cit.>, respectively, with the assumption of independence left unchanged. Eldar generalized SURE (GSURE) for distributionsbelonging to thenon-i.i.d. multivariate exponential family<cit.>. Giryes et al. <cit.> used a projected version of GSURE for selecting parameters in the context of solving inverse problems. An unbiased estimate of theItakura-Saito (IS) distortionand corresponding pointwise shrinkage was developedin <cit.> and <cit.>, and successfully applied to speech denoising. A detailed discussion of Gaussian parameter estimation using shrinkage estimators, together with a performance comparison of SURE with the maximum-likelihood (ML) and soft-thresholding-based estimators, can be found in <cit.> (Chapter 2).It is shown in <cit.> that the soft-thresholding-based estimator dominates the James-Stein shrinkage estimator in terms of MSE if the parameter vector to be estimated issparse. On the other hand, shrinkage estimator dominates if all coordinates of the parameter to be estimated are nearly equal. §.§ This PaperWe address the problem of signal denoising based on the minimum probability of error (MPE), which we first considered in <cit.>. The MPE quantifies the probability of the estimate lying outside an ϵ-neighborhood of the true value. Since the MPE risk depends on the ground truth, we consider a surrogate, which may bebiased, and optimize it to obtain the shrinkage parameter (Section <ref>). The optimization is carried out in the discrete cosine transform (DCT) domain, either in a pointwise fashion or on a subband basis. We derive the MPE risk for Gaussian, Laplacian, and Student's-t noise distributions (Sections <ref> and <ref>). In practical applications, where the noise distribution may be multimodal and not known explicitly, we propose to use a Gaussian mixture model (GMM) approximation <cit.> (Section <ref>).We show the performance of the MPE-based denoising technique on the Piece-Regular signalstaken from the Wavelab toolbox in Gaussian, Student's-t, and Laplacian noise contaminations (Section <ref>). Proceeding further,we also consider the probability of error accumulated over 0 < ϵ < ∞ (Section <ref>), which results in the expected ℓ_1 distortion between the parameter and its estimate.The estimators for the expected ℓ_1 distortion are also derived by invoking the GMM approximation (Section <ref>). We also assess the denoising performance of the shrinkage estimator obtained by minimizing the ℓ_1 distortion for different input SNRs and for different number of noisy realizations (Section <ref>). To further boost the denoising performance of the ℓ_1 distortion-based estimator, we develop an iterative algorithm to successively refine the cost function and the resulting estimate, starting with the noisy signal as the initialization (Section <ref>). The iterations lead to an improvement of 2–3 dB in output signal-to-noise ratio (SNR) (Section <ref>).Performance comparison of the MPE and ℓ_1 distortion-based estimators is carried out on the Piece-Regular and the HeaviSine signals from the Wavelab toolbox <cit.>, and electrocardiogram (ECG) signals from the PhysioBank database <cit.>, with three benchmark techniques: (i) wavelet-domain soft-thresholding <cit.>, (ii) SURE-based orthonormal wavelet thresholding using a linear expansion of thresholds (SURE-LET) <cit.>, and (iii) SURE-based smooth sigmoid shrinkage (SS) in wavelet domain<cit.>; all assuming Gaussian noise contamination (Section <ref>) for fair comparison.§ THE MPE RISK Considerthe observation model x=s+w in ℝ^n, where x and s denote the noisy and clean signals, respectively. The noise vector w is assumed to have i.i.d. entries with zero mean and variance σ^2. The goal is to estimatesfrom x by minimizing a suitable risk function. The signal model isconsidered in an appropriate transform domain, where the signal admits a parsimonious representation, but noise does not. We consider two types of shrinkage estimators: (i) pointwise, where a shrinkage factor a_i ∈ [0,1] is applied to x_i to obtain an estimate s_i = a_i x_i; and (ii) subband-based, wherein a single shrinkage factor a_J is applied to a group of coefficients {x_i, i∈ J} in subband J⊂{1,2,⋯,n}. Shrinkage estimators may also be interpreted as premultiplication of x with a diagonal matrix.§.§ MPE Risk for Pointwise Shrinkage Assuming that the estimate of s_i does not depend on x_j, for j ≠ i, we drop the index i for brevity of notation. The MPE risk is defined as ℛ = ℙ( | s-s |>ϵ),where ϵ>0 is a predefined tolerance parameter. The risk ℛ quantifies the estimation error using the probability measure and takes into account the noise distribution in its entirety. On the contrary, the MSE relies only on the first- and second-order statistics of noise for linear shrinkage estimators. Substituting s=ax=a(s+w), the risk ℛ evaluates toℛ(s;a)= ℙ( | a(s+w) -s | > ϵ)= 1-F(ϵ-(a-1)s/a) + F( - ϵ+(a-1)s/a),where F(·) is the cumulative distribution function (c.d.f.) of the additive noise. Since ℛ depends on s, which is the parameter to be estimated, it is impractical to optimize it directly over a. To circumvent the problem, we minimize an estimate of ℛ, which is obtained byreplacing s with an estimate s̃, which, for example, may be obtained using any baseline denoising algorithm, or can even be taken as s̃=x (which is also the ML estimate of s). In the first instance, the proposed technique becomes an add-on to an existing denoising algorithm, and in the second, it is a denoising scheme in itself. Such an estimate ℛ=ℛ(s̃;a) takes the formℛ=1-F(ϵ-(a-1)s̃/a) + F( - ϵ+(a-1)s̃/a),and correspondingly, the optimal shrinkage parameter is obtained as a_opt = arg 0 ≤ a ≤ 1min ℛ. A grid search is performed to optimize ℛ over a ∈ [0,1], and the clean signal is obtained as s=a_optx. We next derive explicit formulae for the risk function for Gaussian, Laplacian, and Student's-t noise distributions. (i) Gaussian distribution: In this case, the noisy observation x also follows a Gaussian distribution, and therefore, s-s=ax-s is distributed as 𝒩((a-1)s,a^2σ^2 ). The MPE risk estimate is given asℛ =Q (ϵ-(a-1)s̃/aσ) + Q (ϵ+(a-1)s̃/aσ),where Q(u)=1/√(2 π)∫_u^∞e^-t^2/2dt.(ii) Student's-t distribution: Consider the case where the noise follow a Student's-t distribution with parameter λ>2 and the probability density function (p.d.f.) of noise is given byf(w)= Γ( λ+1/2) /√(λπ)Γ( λ/2) ( 1+w^2/λ)^-λ+1/2.The variance of w isσ^2=λ/λ-2. The expression for ℛ is the one given in (<ref>) with F(w)= 1/2+ wΓ( λ+1/2) G_1 ( 1/2,λ+1/2;3/2;-w^2/λ)/√(λπ)Γ( λ/2) ,where G_1 is the hypergeometric function defined asG_1 ( a,b;c;z )= ∑_k=0^∞(a)_k (b)_k/(c)_kz^k/k!,and (q)_k denotes the Pochhammer symbol:(q)_kΔ= 1 fork=0,q(q+1)(q+2)⋯(q+k-1), k>0.(iii) Laplacian distribution: Considering the noise to bei.i.d. Laplacian with zero-mean and parameter b (variance σ^2=2b^2), with the p.d.f. f(w)=1/2bexp(- |w|/b), the MPE risk can be obtained by using the following expression for F(w) in (<ref>): F(w)= 1/2 + 1/2sgn(w) ( 1-exp(-|w|/b) ).§.§.§ Closeness of ℛ to ℛTo measure the closeness of ℛ to ℛ, consider the example of estimating a scalar s=4 from a noisy observation x. The MPE risk estimate ℛ is obtained by setting s̃=x. In Figures <ref>(a), <ref>(b), and <ref>(c), we show the variation of theactual risk ℛ and its estimate ℛ with a, averaged over 100 independent trials, for Gaussian, Student's-t, and Laplacian noise distributions, respectively. The noise has zero mean, and the variance is taken as σ^2=1 for Gaussian and Laplacian models, whereas for Student's-t model, the variance is σ^2=2. The value of ϵ is set equal to σ while computing the MPE risk. We observe that ℛ is a good approximation to ℛ, particularly in the vicinity of the minima. The deviation of the shrinkage parameter a_opt(x), obtained by minimizing ℛ, with respect to its true value a_opt(s) resulted from the minimization of ℛ, is shown in Figure <ref>(d) for three noise models under consideration. The central red lines in Figure <ref>(d) indicate the medians, whereas the black lines on the top and bottom denote the 25 and the 75 percentile points, respectively. We observe that a_opt(x) is well concentrated around a_opt(s), especially for Gaussian and Laplacian noise, barring a small number of outliers. §.§.§ Perturbation Probability of the location of minimumThe location of the minimum of the MPE risk determines the shrinkage parameter. Therefore, one must ensure that it does not deviate too much from its actual value, with high probability, when s is replaced by x in the original risk ℛ. Let a_opt(s)=0≤ a ≤ 1minℛ(s;a) denote the argument that minimizes the true risk ℛ.Consider the probability of deviation, given byP_e^MPE=ℙ( |a_opt(s) - a_opt(x)| ≥δ),for some δ>0. Using a first-order Taylor series approximation of a_opt(x) about s, and substituting x=s+w, we obtain a_opt(x)≈ a_opt(s)+w a'_opt(s), where ' denotes the derivative.The deviation probability P_e^MPE in (<ref>) simplifies to P_e^MPE=ℙ( |w| ≥δ/| a'_opt(s) |).For additive Gaussian noise w with zero mean and variance σ^2, placing the Chernoff bound on P_e^MPE leads toP_e^MPE≤ 2exp( -δ^2/2σ^2| a'_opt(s) |^2).To ensure that P_e^MPE is less than α, for a given α∈ (0,1), it suffices to have| a'_opt(s) |^2≤ δ^2/2σ^2 log(2/α),which translates to a lower-bound on the input SNR. Since there is no closed-form expression available for a'_opt(s) in the context of MPE risk, we empirically obtain the range of input SNR values s^2/σ^2, for which (<ref>) is satisfied.Analogously, to satisfy an upper bound on the deviation probability P_e^SURE of the minimum in the case ofSURE, for a given deviation δ>0, one must ensure thats^6/8σ^6( δ - σ^4/(s^2+σ^2 )s^2)^2≥log(2/α).The proof of (<ref>) is given in Appendix <ref>.The minimum input SNR required to ensure P_e≤α for both SURE- and MPE-based shrinkage estimators is shown in Figure <ref>, for different values of α and δ. The MPE-risk estimate is obtained by replacing s with x and setting ϵ=σ. We observe that reducing the amount of deviation δ for a given probability α, or vice versa, leads to a higher input SNR requirement for both SURE and MPE. We also observe from Figure <ref> that, for given δ and α, SURE requires a higher input SNR than MPE to keep the δ-deviation probability under α. Also, for a given input SNR, the δ-deviation probability of the estimated shrinkage parameter a_opt(x) from the optimum a_opt(s) is smaller for MPE than SURE, thereby indicating that the MPE-based shrinkage is comparatively more reliable than the SURE-based one at lower input SNRs. §.§.§ Unknown noise distributions In practical applications, the distribution of noise may not be known in a parametric form and may also be multimodal. At best, one would have access to realizations of the noise, from which the distribution has to be estimated. In such cases, approximation of the noise p.d.f. using a GMM is a viable alternative <cit.>, whereinone canestimate the parameters of GMM using the expectation-maximization algorithm <cit.>. Gaussian mixture modeling is attractive as it comes with certain guarantees. For example, it is knownthat a p.d.f. with a finite number of finite discontinuities can be approximated by a GMM to a desired accuracy except at the points of discontinuity <cit.>. The GMM approximation can be used even for non-Gaussian, unimodal distributions. For the GMM-based noise p.d.f.f(w)=∑_m=1^Mα_m/σ_m √(2π)exp( -( w-θ_m )^2/2σ_m^2),the MPE risk turns out to beℛ= ∑_m=1^Mα_m [ Q (ϵ-(a-1)s̃-θ_m/aσ_m) +. . Q (ϵ+(a-1)s̃+θ_m/aσ_m)],using (<ref>). For illustration, consider the estimation of a scalar s=4 in the transform domain from its noisy observation x. The additive noise is Laplacian distributed with zero mean and variance σ^2=1. The noise distribution is modeled using a GMM with M=4 components and the corresponding MPE risk estimate is obtained using (<ref>) by setting s̃=x. In Figure <ref>(a), we show a Laplacian p.d.f. andits GMM approximation. Figure <ref>(b) shows the GMM approximation to a multimodal distribution. Figure <ref>(a) shows the MPE risk based on the original Laplacian distribution as well asthe GMM approximation, as a function of the shrinkage parameter a. The close match between the two indicates that the GMM is a viable alternative when the noise distribution is unknown or follows a complicated model. In Figure <ref>(b),we plot the GMM-based MPE risk and its estimate averaged over 100 independent trials. We observe that the locations of minima of the actual risk and its estimate match closely, thereby justifying the minimization of ℛ. The MPE risk and its estimate are shown inFigure <ref>(c)for the multimodal p.d.f. of Figure <ref>(b). §.§ MPE Risk for Subband Shrinkage Let a_J be the shrinkage factor applied to the set of coefficients {x_i,i∈ J} in subband J. The estimate s_J of the clean signal is obtained by s_J= a_Jx_J, where x_J ∈ℝ^|J| and a_J ∈[0,1]. For notational brevity, we drop the subscript J, as we did for pointwise shrinkage, and express the estimator as 𝐬 = a 𝐱, where boldface letters indicate vectors.Analogous to pointwise shrinkage, the MPE risk for subband shrinkage is defined as ℛ = ℙ( 𝐬-𝐬_2>ϵ), which, for 𝐬 = a 𝐱, becomes ℛ = ℙ(a 𝐰 + (a-1)𝐬_2>ϵ). For 𝐰∼𝒩(0,σ^2 I), ℛ =1-F(θ|k,λ),where k=|J|, λ = ∑_j=1^k(1-a)^2 s_j^2/a^2σ^2, θ=(ϵ/a σ)^2, and F(θ|k,λ) is the c.d.f. of the non-central χ^2 distribution, given byF(θ|k,λ)=∑_m=0^∞λ^m e^-λ/2/2^m m!ℙ[ χ^2_k+2m≤θ],wherein χ^2_v denotes the central χ^2 random variable having v degrees of freedom.Similar to pointwise shrinkage, we propose to obtain an estimate ℛ of ℛ for subband shrinkage estimators either by replacing s_j with x_j, or by anestimate s̃_j produced by any standard denoising algorithm. The optimum subband shrinkage factor is obtained by minimizing ℛFigure <ref> shows the subband MPE risk and its estimate versus a, where the underlying clean signal 𝐬∈ℝ^|J| is corrupted by Gaussian noise and the subband size is chosen to be |J|=k=8. The clean signal 𝐬 is generated by drawing samples from𝒩(2× 1_k, I_k), where 1_k and I_k denote a k-length vector of all ones and a k× k identity matrix, respectively. The observation 𝐱 is obtained by adding zero-mean i.i.d. Gaussian noiseto 𝐬, with an input SNR of 5 dB, where the input SNR is defined as SNR_in= 10 log_10(1/kσ^2∑_n=1^ks_n^2)dB.The MPE risk estimate is obtained by replacing 𝐬 with 𝐱 in (<ref>), which does not significantly shift the location of the minimum (cf. Figure <ref>). § EXPERIMENTAL RESULTS FOR MPE-BASED DENOISING The performance of the MPE-based pointwise and subband shrinkage estimator is validated on a synthesized harmonic signal (of length N=2048) in Gaussian noise and the Piece-Regular signal (of length N=4096) in Gaussian, Student's-t, and Laplacian noise. The Piece-Regular signal has both smooth and rapidly-varying regions, making it a suitable candidate for the assessment of denoising performance.§.§ Performance of Pointwise-Shrinkage Estimator§.§.§ Harmonic signal denoisingConsider the signal s_n = cos( 5π n/2048)+ 2 sin( 10π n/2048), 0≤ n ≤ 2047,in additive white Gaussian noise, with zero mean and variance σ^2. Since the denoising is carried out in the DCT <cit.> domain, the Gaussian noise statistics remain unaltered. For the purpose of illustration, we assume that σ^2 is known. In practice, σ^2 may not be known a priori and could be replaced by the robust median estimate <cit.> or the trimmed estimate <cit.>. The clean signal is estimated using inverse DCT after applying the optimum shrinkage. The denoising performance of the MPE and SURE-based approaches is compared in Table <ref>. In case of the Wiener filter, the power spectrum of the clean signal is estimated using the standard spectral subtraction technique <cit.>. We observe that MPE-based shrinkage with ϵ = 3.5 σ is superior to SURE and Wiener filter (WF) by 8–12 dB. The comparison also shows that the performance of the MPE depends critically on ϵ.§.§.§ Piece-Regular signal denoising We consider noisy copies of the Piece-Regular signal, taken from the Wavelab toolbox <cit.>, under Gaussian, Student's-t, and Laplacian contaminations. The noise variance is assumed to be known. Notably, the Gaussian, GMM, and Student's-t distributions of noise are preserved by an orthonormal transform <cit.>, unlike the Laplacian statistics. Therefore, the MPE estimate for Laplacian noise is computed based on a four-component GMM approximation in the DCT domain. The denoised output signal corresponding to Laplacian noise is shown in Figure <ref> for illustration. The MPE estimates are better than SURE estimates. The SNR plots in Figure <ref> indicate that the MPE outperforms SURE for the noise statistics under consideration and that the gains are particularlyhigh in the input SNR range of -5 to 20 dB, and tend to reduce beyond 20 dB.§.§.§ Effect of ϵ on the denoising performance of MPE Obtaining a closed-form expression for the ϵthat maximizes the output SNR is not straightforward. We determine the optimum ϵ empirically by measuring the SNR gain as a function of ϵ (cf. Figure <ref>), for i.i.d. Gaussian noise. We observe that the output SNR exhibits a peak approximately at β = ϵ/σ=3.5 for the harmonic signal in (<ref>) and at β=3 for the Piece-Regular signal. As a rule of thumb, we recommend to choose ϵ=3σ for pointwise shrinkage estimators. §.§ Performance of Subband MPE ShrinkageTo validate the performance of the MPE-based subband shrinkage estimator (cf. Section <ref>), we consider denoising ofthe Piece-Regular signal in additive Gaussian noise. The clean signal and its noisy measurement are shown in Figure <ref>(a). Denoising is carried out by grouping k adjacent DCT coefficients to form a subband. The denoised signals obtained using SURE and MPE are shown in Figures <ref>(b) and <ref>(c), respectively.The subband size k is chosen to be 16 and the parameter ϵ is set equal to 1.75√(k)σ, a value that was determined experimentally and found to be nearly optimal. We observe that the MPE gives 1 dB improvement in SNR than the SURE approach.Variation of the output SNR is also studied as a function of k (cf. Figure <ref>). We experimented withϵ=3σ, ϵ=1.75√(k)σ, and ϵ=1.25√(k)σ corresponding to subband sizes k=1, k∈[2,16], and k>16, respectively. For both SURE and MPE, as k increases, the output SNR also increases and eventually saturates for k ≥ 40. For input SNR below 15 dB, MPE gives a comparatively higher SNR than SURE, and the margin diminishes with increase in input SNR or the subband size k. The degradation in performance of SURE for low SNRs is due to the large error in estimating the MSE at such SNRs. The SURE-based estimate of MSE becomes increasingly reliable as k increases, thereby leading to superior performance. § ACCUMULATED PROBABILITY OF ERROR: MPE MEETSTHE EXPECTED ℓ_1 DISTORTION The MPE is parametrized by ϵ, which has to be appropriately chosen in order to achieve optimal denoising performance. To suppress the direct dependence on ϵ, we consider the accumulated probability of error, namely ∫_0^∞ℙ( | s-s |>ϵ)dϵ as the risk to be minimized. For a nonnegative random variable Y, we know that ℰ{Y}=∫_0^∞ℙ( Y>ϵ)dϵ. Therefore, the accumulated probability of error is the expected ℓ_1 distortion:ℰ{| s-s |}=∫_0^∞ℙ( | s-s |>ϵ)dϵ.For Gaussian noise distribution,ℛ_ℓ_1(a,s )=ℰ{| s-s |} = ∫_0^∞ Q (ϵ-(a-1)s/aσ)dϵ+∫_0^∞ Q (ϵ+(a-1)s/aσ)dϵ.Denoting u=ϵ-(a-1)s/aσ and μ=-(a-1)s/aσ, the first integral in (<ref>) is evaluated as∫_0^∞Q (ϵ-(a-1)s/aσ)dϵ=aσ∫_μ^∞ Q (u )du= aσ(∫_0^∞ Q (u )du -∫_0^μ Q (u )du )= aσ(1/√(2π)- μ Q (μ)-1/√(2π)( 1-e^-μ^2/2))= aσ( e^-μ^2/2/√(2π) - μ Q (μ) ).The second term in (<ref>) can be evaluated by replacing μ with -μ in (<ref>). Combining both integrals, we obtain the expression for the expected ℓ_1 distortion:ℛ_ℓ_1(a,s ) = aσ[ √(2/π)e^-μ^2/2 - μ Q ( μ) + μ Q ( - μ) ]= aσ[ √(2/π)exp(-(a-1)^2 s^2/2a^2 σ^2)+ 2(a-1)s/aσ. . Q ( -(a-1)s/aσ)-(a-1)s/aσ]. An estimate of theexpected ℓ_1 distortion is calculated by replacing s with an estimate s̃, which could also be x, to begin with. In Figure <ref>(a), we show the variation of the original ℓ_1 distortion and its estimate obtained by setting s̃=x, as functions of a, averaged over 100 independent realizations of 𝒩(0,1) noise. The actual parameter value is s=4. The figure shows that the minimum of the expected ℓ_1 risk is close to that of its estimate.In principle, one could iteratively minimize the ℓ_1 distortion by starting with s=x and successively refining it. Such an approach is given in Algorithm 1. An illustration of the denoising performance of the iterative algorithm is deferred to Section <ref>. §.§ Expected ℓ_1 risk Using GMM Approximation For the GMM p.d.f. in (<ref>), the expected ℓ_1 distortion evaluates to (cf. Appendix  <ref> for the derivation)ℛ_ℓ_1=∑_m=1^M a α_mσ_m( √(2/π)e^-μ_m^2/2 -2 μ_m Q (μ_m) + μ_m),where μ_m=-(a-1)s+θ_m/aσ_m. The expected ℓ_1 risk and its estimate for a multimodal (cf. Figure <ref>(b)) and Laplacian noise p.d.f.s are shown in Figures <ref>(b) and <ref>(c), respectively. We observe that,in both cases, the locations of the minima of the true risk and its estimate are in good agreement.§.§Optimum Shrinkage a_opt Versus Posterior SNR We next study the behavior of a_opt for different input SNRs to compare the denoising capabilities of the MPE and the expected ℓ_1-distortion-based shrinkage estimators.The optimum pointwise shrinkage parameter a_optfor Gaussian noise statistics, obtained by minimizing SURE, MPE risk estimate, and the estimated ℓ_1 risk, for different values of the a posteriori SNRx^2/σ^2 is plotted in Figure <ref>(a). To illustrate the effect of ϵ, the variation of a_opt versus a posteriori SNR for MPE corresponding to Gaussian noise is shown in Figure <ref>(b), for different ϵ. We observe that the shrinkage profiles are characteristic of a reasonable denoising algorithm, as Figures <ref>(a) and <ref>(b) exhibit thattheshrinkage parameters increase as the a posteriori SNR increases. Whereas in case of the MPE, the choice of ϵ is crucial, the expected ℓ_1 distortion does not require tuning such a parameter. Moreover, the MPE attenuation profile for larger values of ϵ is reminiscent of a hard-thresholding function, whereas the expected ℓ_1 distortion has an attenuation profile that resembles a soft-threshold.§ PERFORMANCE OF THE EXPECTED ℓ_1 DISTORTION-BASED POINTWISE SHRINKAGE ESTIMATOR In a practical denoising application, we have only one noisy realization from which the clean signal has to be estimated. However, it is instructive to consider the case of multiple realizations as it throws some light on the performance comparisons vis-à-vis other estimators such as the ML estimator. Consider the observation model x^(m)= s+w^(m) in ℝ^n,1≤ m ≤ M, where one has access to M noisy copies of the signal s, and the noise vectors w^(m) are drawn independently from the 𝒩( 0,σ^2 I_n) distribution. The ML estimator of the i^th signal coefficient s_i is given by ŝ_ML,i=1/M∑_m=1^M x_i^(m), wherex_i^(m) is the i^th component of x^(m). Dropping the subscript i, as eachcoefficient is treated independently of the others, the shrinkage estimator takes the form s=a_optŝ_ML. To study the behavior of the estimate with respect to M, we consider two variants: (i) where a_opt is obtained by minimizing ℛ_ℓ_1(s,a), referred to as the oracle-ℓ_1; and (ii) where a_opt is chosen to minimize ℛ_ℓ_1(ŝ_ML,a), referred to as ML-ℓ_1. The output SNR as a function of M for the Piece-Regular signal, corresponding to an input SNR of 5 dB, is shown in Figure <ref>(a). For all three estimators, namely, oracle-ℓ_1, ML-ℓ_1, and the ML estimate, the output SNR increases with M. However, for the oracle-ℓ_1 and the ML-ℓ_1 estimators, the output SNR stagnates as M increases beyond 40. For M≤ 60, the oracle-ℓ_1 and the ML-ℓ_1 shrinkage estimators exhibitbetter performancecompared with theML estimator. As one would expect, the performance of the ML-ℓ_1 estimator matches with that obtained using the oracle-ℓ_1 as M becomes large, because the ML estimate converges in probability to the true parameter. For M=1, which is often the case in practice, the ML-ℓ_1 estimate significantly dominates the ML estimator as seen in Figure <ref>(a). The SNR gain over the ML estimator could be further improved by using the iterative minimization algorithm introducedin Section <ref> (cf. Algorithm <ref>). The performance of the ML-ℓ_1 and the ML estimators, for different values of M and input SNR is shown in Figure <ref>(b). The figures show that for small values of SNR and M, the ML-ℓ_1 estimate outperforms the ML estimator. This is of significant importance in a practical setting where we have only one noisy realization (M = 1).§.§ Iterative Minimization of the Expected ℓ_1-RiskWhen M=1, the ML-ℓ_1 estimator is obtained by minimizing ℛ_ℓ_1(x,a), where x is the noisy version of s. We refer to this estimate as the non-iterative ℓ_1-based shrinkage estimator. Following Algorithm 1, one could iteratively refine the estimate, starting from x. We compare the non-iterative ℓ_1-based estimator with its iterative counterpart, and present the results in Figures <ref>, <ref>, and <ref>, corresponding to Gaussian, multimodal (c.f. Figure <ref>(b)), and a GMM approximation to the Laplacian noise, respectively. The output SNR obtained using the oracle-ℓ_1 estimator, calculated by minimizing ℛ_ℓ_1(s,a), is also shown for benchmarking the performance.We make the following observations from the Figures <ref>, <ref>, and <ref>: (i) the output SNR increases with iterations, albeit marginally after about 10 iterations; (ii) the iterative method consistently dominates the non-iterative one, with an overall SNR improvement of about 2 to 3 dB, for input SNR in the range -5 dB to 20 dB; and (iii) the SNR gain of the iterative technique also reduces for higher input SNR, similar to other denoising algorithms.§ PERFORMANCE ASSESSMENT OF MPE AND ℓ_1-RISK MINIMIZATION ALGORITHMS VERSUS STATE-OF-THE-ART DENOISING ALGORITHMS We compare the MPE and the ℓ_1-based shrinkage estimators with three state-of-the-art denoising algorithms:(i) wavelet soft-thresholding[A Matlab implementation is included in the Wavelab toolbox available at:<http://statweb.stanford.edu/ wavelab/>.] <cit.>; (ii) the SURE-LETdenoising algorithm[A MATLAB implementation of the SURE-LET algorithm is available at: <http://bigwww.epfl.ch/demo/suredenoising>.] <cit.>; and (iii) smooth sigmoid shrinkage (SS) <cit.> in the wavelet domain [Pastor et al. kindly provided the MATLAB implementation of their denoising technique <cit.>, which facilitated the comparisons reported in this paper.]. In <cit.>, a wavelet-based soft-thresholding scheme is used for denoising, with the threshold selected as τ=σ√(2 log(N)) for an N length signal. The SURE-LET technique employs a linear expansion of thresholds (LET), which is a linear combination of elementary denoising functions and optimizes for the coefficients by minimizing the SURE criterion. In <cit.>, a smooth sigmoid shrinkage is applied on the wavelet coefficients to achieve denoising, and the parameters of the sigmoid, which control the degree of attenuation, are obtained by minimizing the SURE objective. We consider ECG signals taken from the PhysioBank database, and the HeaviSine and Piece-Regular signals taken from Wavelab toolbox for performance evaluation.Thenoise is assumedto follow aGaussian distribution and the output SNR values are averaged over 100 independent realizations. The noise variance is estimated using a median-based estimator <cit.>, which is also used by Luisier et al.first_foot and Donohosecond_foot. InSURE-LET, SS,and wavelet thresholding techniques, denoising is performed using Symmlet-4, with three levels of decomposition, as these settings were found to be the best for the ECG signal (following <cit.>). In case of MPE and ℓ_1-based shrinkage estimators, denoising is performed in the DCT domain. We use the shorthand notations MPE and MPE-subband to denote the pointwise and subband shrinkage estimators, respectively. The corresponding SURE-based subband shrinkage estimator is denoted asSURE-subband. We set k=16 and ϵ=1.75√(k)σ for computing the subband shrinkage parameters. These parameters have not been specifically optimized; however, they were found to work well in practice. The output SNR as a function of the input SNR, obtained using various algorithms, is shown in Figure <ref>.From the ECG signal denoising performance shown in Figure <ref>(a),we observe that the MPEestimate consistently dominates the soft-thresholding-based denoisingfor input SNRs ranging from -5 dB to 20 dB. The iterative ℓ_1-distortion-based shrinkage estimator (20 iterations) yields lower output SNR compared with the MPE-based estimate for input SNR values in the range -5 to 17.5 dB, but surpasses it for relatively higher values of input SNR (17.5 to 20 dB). The SURE-LET and the SS algorithms dominate both MPE and the ℓ_1-based shrinkage estimators, because they use more sophisticated denoising functions in the transform domain, thereby offering greater flexibility. Forinput SNR range of 0 dB to 20 dB, the expectedℓ_1-distortion-based shrinkage estimator consistently outperforms the soft-thresholding-based techniques.We have also found that it is possible to boost the denoising performance of an algorithm in the low-SNR regime by adding the MPE denoiser in tandem, that is, by replacing s̃ in the expression for the MPE risk estimate in (<ref>) with the estimate obtained using a denoising technique, for example, the SURE-LET. We refer to this tandem approach as MPE-SURE-LET in Figure <ref>. This approach results in 1 to 2 dB gain in output SNR over SURE-LET for low and medium values of input SNR. We observe in Figure <ref> that the MPE-subband estimator outperforms the competing algorithms (except for MPE-SURE-LET in Figure <ref>(a)),at low input SNR.§ CONCLUSIONS We have proposed a new framework for signal denoising based on a novel criterion, namely the probability of error. Our framework is applicable to scenarios where the noise samples are independent andadditivelydistort the signal. Denoising is performed by transform-domain shrinkage andthe optimum shrinkage parameter is obtained by minimizing an estimate of the MPE risk. We have considered both pointwise and subband shrinkage estimators within the MPE paradigm. The performance of the proposed MPE estimators depends on the choice of the error-tolerance parameter ϵ. In pointwise shrinkage, to deal with the issue of selecting an appropriate ϵ, we have proposed two approaches. In the first one, we experimentally determined an ϵ value that results in maximum SNR gain for a particular signal by evaluating the output SNR for different ϵ. In the second approach, wecomputed the accumulated probability of error, which is the expected ℓ_1 distortion, and developed an iterative algorithm for minimization. We demonstrated thatthe shrinkage estimator obtained using the expected ℓ_1 risk outperforms the classical ML estimator, when the number of observations is small or the input SNR is low. Wealso showed that theshrinkage estimator obtained by iteratively minimizing the ℓ_1 risk dominates the non-iterative approach in terms of the output SNR.Extensive performance comparison of the proposed MPE and the ℓ_1 distortion-based approaches with state-of-the-art denoising algorithms is carried out on real ECG signals and Wavelab signals. Experimental results demonstrate that theshrinkage estimator based on theMPE-risk estimate outperforms the SURE-basedestimator in terms ofSNR gain,particularly in the regime of low SNR and smaller subband size. The proposed MPE-framework could be used as an add-on over anexisting denoising technique, leading to an estimator that has a higheroutput SNR, particularly in the low input SNR regime.For deriving the expression and validating the performance of the MPE-based subband shrinkage estimator, we considered denoising of signals corrupted with Gaussian noise. Experimentally, we have found that increase in the subband size leads to an increase in output SNR, and saturates beyond a point. We have also observed that, when the subband size or the input SNR is low, the MPE-based estimate has superior performance compared with the SURE-based estimator.We demonstrated that the optimum shrinkage parameter obtained by minimizingestimates of the MPE/ℓ_1 distortionsincreases monotonically with theincrease in a posteriori SNR. Such behavior of the shrinkage parameter is essential for denoising.A theoretical characterization of this behavior is needed and may lead to interesting inferences, which could potentially lead to a rigorous convergence proof for the iterative expected ℓ_1 distortion minimization technique. Another important observation is that, for lower input SNRs, the proposed denoising framework yields a higher output SNR compared with the MSE-based techniques. The improvement in performance in terms of SNR of the denoised output may be attributed to the fact that the MPE framework incorporates knowledge of the distribution of the observations, which goes beyond the second-order statistics considered in de facto MSE-based optimization. We also believe that this is the first attempt at demonstrating competitive denoising performance with probability of error chosen as the distortion metric, in a non-Bayesian estimation framework. § PERTURBATION OF SURE-BASED POINTWISE SHRINKAGE To analyze the perturbation in the location of the minimum of the SURE cost function, in comparison with the true MSE, one needs to evaluateP_e^SURE=ℙ{|a_opt(s) - a_opt(x)| ≥δ},where a_opt(s)=s^2/s^2+σ^2 anda_opt(x)=1-σ^2/x^2. Let h(x)=a_opt(s) - a_opt(x)=(s^2/s^2+σ^2- 1+σ^2/x^2). The Taylor-series expansion of h(x) abouts yieldsh(x)=σ^4/s^2(s^2+σ^2)-2 w σ^2/s^3+∑_n=2^∞d^(n)(s) /n! w^n,where h^(n) is the n^th derivative h. Using the first-order Taylor series approximation h(x)≈ h(s)+w h^(1)(s), we obtainh(x)≈σ^4/s^2(s^2+σ^2)-2 w σ^2/s^3,which, in turn, leads to an approximation of the perturbation probability P_e^SURE:P_e^SURE=ℙ{|h(x)| ≥δ} ≈ℙ{|σ^4/s^2(s^2+σ^2)-2 w σ^2/s^3| ≥δ}.Invoking w∼𝒩(0,σ^2), and usingthe Chernoff bound<cit.>, we obtainP_e^SURE≤ 2exp( -s^6/8σ^6( δ - σ^4/(s^2+σ^2 )s^2)^2).Consequently, to satisfy an upper bound on the deviation probability of the form P_e^SURE≤α, for a given δ>0, one must ensure thats^6/8σ^6( δ - σ^4/(s^2+σ^2 )s^2)^2≥log(2/α).The condition in (<ref>) translates to an equivalent condition on the minimum requiredSNRs^2/σ^2 to achieve a certain P_e^SURE. §EXPECTED ℓ_1 RISK FOR GMM Foradditive noise with the p.d.f. given in(<ref>), we haveℰ{| s-s |} = ∑_m=1^Mα_m (∫_0^∞ Q (ϵ-(a-1)s-θ_m/aσ_m) . dϵ+ ∫_0^∞. Q (ϵ+(a-1)s+θ_m/aσ_m)dϵ),using (<ref>) and (<ref>). 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http://arxiv.org/abs/1702.07869v1

Whittaker functions on non-archimedean fields were first introduced in the work of Jacquet, and they were characterized explicitly by Shintani. We obtain an explicit inversion formula and a Plancherel formula for the p-adic Whittaker transform on GL_n(ℚ_p). As an application, integral representations are obtained for the local factors of certain symmetric power L-functions.Asynchronous Incremental StochasticDual Descent Algorithm for Network Resource Allocation Amrit Singh Bedi, Student Member, IEEE and Ketan Rajawat, Member, IEEE The authors are with the Department of Electrical Engineering, IIT Kanpur, Kanpur (UP), India 208016 (email: amritbd, ketan@iitk.ac.in). =======================================================================================================================================================================================================================§ INTRODUCTION Whittaker functions on local non-archimedean fields (such as ℚ_p) have been studied by several authors since they were introduced in 1967 by Jacquet <cit.>. Shintani <cit.> obtained an explicit formula for these Whittaker functions, which was then generalized in different directions in other works (see <cit.>, <cit.>). Let G= GL_n(ℚ_p) for which we have the Iwasawa decomposition G = U T K where U = U(ℚ_p) is the unipotent radical of the standard Borel subgroup, T = T(ℚ_p) is the torus of diagonal matrices and K = GL_n(ℤ_p) is the maximal compact subgroup. Let Z be the center of G. Let ψ be a character on U induced from a character ψ' on ℚ_p viaψ(u) = ψ'(∑_i=1^n-1 u_i,i+1), ( where u = (u_i,j) ∈ U).Define the spherical Hecke algebra ^K ℋ^K to be the set of locally constant, compactly supported functions f : GL_n(ℚ_p) → satisfyingf( k_1 g k_2) = f(g)for k_1, k_2 ∈ K, g ∈GL_n(ℚ_p). On GL_n(ℚ_p), Whittaker functions transform my the action of a character under the unipotent radical and satisfy an integral property relative to elements of the spherical Hecke algebra, as below. A Whittaker function on GL_n(ℚ_p) is a K-finite smooth function of moderate growth W : GL_n(ℚ_p) →ℂ that satisfiesW (u g) = ψ(u) W(g)for some fixed character ψ as defined above and any u ∈ U, and∫_G W(g x) ϕ(x) d x = λ(ϕ) W(g)for each algebra homomorphism λ : ^Kℋ^K →, ϕ∈^Kℋ^K and g ∈GL_n(ℚ_p). Let h:Z(ℚ_p)\ T(ℚ_p) →ℂ be a smooth function. The p-adic Whittaker transform of h is defined to be∫_Z(ℚ_p)\ T(ℚ_p) h(t) W(t) d^× t,where W is a Whittaker function on GL_n(ℚ_p).The main goal of the present paper is to obtain an inversion formula for the p-adic Whittaker transform. The precise inversion formula for the Whittaker transform is given in Theorem <ref>. On archimedean local fields Whittaker functions have been thoroughly studied and an inversion formula for the Whittaker transform in known due to the works of Wallach <cit.> and Goldfeld-Kontorovich <cit.>. The approach presented in this paper is inspired by the latter. A crucial step in our approach is the computation of an integral of a product of two p-adic Whittaker functions. One can view this computation as a non-archimedean analogue of Stade's formula <cit.>. As a corollary of the inversion formula we also obtain a Plancherel formula for the p-adic Whittaker transform. In the last section, we use the inversion formula to write integral representations of local L-factors associated to a symmetric square and symmetric cube lifts.We remark that the inversion formula had already been obtained in the setting of p-adic reductive groups by Delorme <cit.>. However, the approach presented here provides a more explicit presentation of the result in this particular setting and its simple derivation solely relies on complex analysis. § INVERSION FORMULA In this section we provide all the necessary definitions and results that allow us to state the main theorem <ref>, which gives the inverse formula for the p-adic Whittaker transform.We start by noting Whittaker functions on GL_n(ℚ_p) arise naturally from certain irreducible representation of this group.Let V be a complex vector space and let (π,V) be an irreducible generic representation of G. Then the space Hom_U(π,ψ) = { f : V →ℂ linear: f(π(u) v) = ψ(u) f(v), ∀ v ∈ V, u ∈ U }is one-dimensionanal generated by an element λ. For v ∈ V a newform, define a Whittaker function W associated to π asW(g) = λ(π(g)v). As the space of newforms in V is one-dimensional then the Whittaker function associated to a representation π is well-defined up to a constant. Furthermore, a Whittaker function associated to an irreducible generic representation π of GL_n(ℚ_p) is a nonzero Whittaker function on GL_n(ℚ_p). For proofs of these statements see <cit.> and <cit.>. In order to explicitly describe Whittaker functions associated to certain representations of GL_n(ℚ_p) we proceed to quote a theorem of Shintani <cit.>. For g = z u t k ∈GL_n(ℚ_p) define the Iwasawa coordinates g = z u ( [ t_1 ⋯ t_n-1; ⋱; t_1; 1 ]) kwhere z ∈ Z, u ∈ U, t ∈ T and k ∈ K.Let π be an irreducible unramified generic representation of GL_n(ℚ_p) and W_π the Whittaker function associated to a representation π (normalized such that W_π(id) = 1 where id denotes the n × n identity matrix) as in Definition <ref>. We can write the L-function associated to π asL(s,π) = ∏_i=1^n (1-α_i p^-s)^-1where α_i ∈ℂ are all nonzero and α_1 ⋯α_n = 1.Then, fort = ( [ t_1 ⋯ t_n-1; ⋱; t_1; 1 ])we haveW_π(t) = δ^1/2(t) s_-log|t|_p(α), t_i ∈ℤ_p fori=1,⋯,n-1, 0,where δ(t) = ∏_k=1^n-1 |t_k|_p^k(n-k),log|t|_p = (log|t_1|_p, ⋯, log|t_n-1|_p ) forlog = log_p,ands_λ(α) = | [ α_1^n-1+λ_1+λ_2+⋯+λ_n-1 α_2^n-1+λ_1+λ_2+⋯+λ_n-1 ⋯ α_n^n-1+λ_1+λ_2+⋯+λ_n-1; α_1^n-2+λ_1+λ_2+⋯+λ_n-2 α_2^n-2+λ_1+λ_2+⋯+λ_n-2 ⋯ α_n^n-2+λ_1+λ_2+⋯+λ_n-2; ⋮ ⋮ ⋱ ⋮; α_1^1+λ_1α_2^1 +λ_1 ⋯α_n^1 +λ_1; 1 1 ⋯ 1 ]|/| [ α_1^n-1 α_2^n-1 ⋯ α_n^n-1; α_1^n-2 α_2^n-2 ⋯ α_n^n-2; ⋮ ⋮ ⋱ ⋮; α_1 α_2 ⋯ α_n; 1 1 ⋯ 1 ]|is the Schur polynomial, where λ = (λ_1,⋯,λ_n) and λ_i ∈ℤ.See <cit.>. For the relation between the L-function L(s,π) and the Whittaker function W_π see Subsection 3.1.3 <cit.>. Throughout this paper we will always use log to denote the base p logarithm. We will also use the notation W_α, for α∈ (ℂ\{0 })^n, to denote the functionW_α(t) = δ^1/2(t) s_-log|t|_p(α), t_i ∈ℤ_p fori=1,⋯,n-1, 0,. finite This coincides with the definition of W_π when α = (α_1,⋯, α_n) are the Langlands parameters associated to π.We are now able to define the p-adic Whittaker transform precisely.Let h:Z(ℚ_p) \ T(ℚ_p) →ℂ. Let 𝒮_η be the annulus { z ∈ : p^-η < |z|_ < p^η}, for some η >0. Define the Whittaker transform h^♯ : 𝒮_η^n→ℂ byh^♯(α) = ∫_T(ℤ_p) \ T(ℚ_p) h(t) W_α(t) d^×t,provided the integral converges absolutely, whered^×t = ∏_k=1^n-1 |t_k|_p^-k(n-k)dt_k/|t_k|_p.A sufficient condition for the absolute convergence of the above integral is that h satisfies the decay condition| h(t) |_≪δ^1/2(t) | t_1^n-1 t_2^n-2⋯ t_n-1|_p^η + εfor any ε > 0. Let H : 𝒮_η^n →ℂ be a holomorphic symmetric function, for some η >0. LetW_α(t) = δ^1/2(t) s_-log|t|_p(α), t_i ∈ℤ_p fori=1,⋯,n-1, 0,where α = (α_1,…,α_n). Define the inverse Whittaker transform of H, H^♭: Z(ℚ_p) \ T(ℚ_p) →ℂ byH^♭(t) = 1/n! (2π i)^n-1∫_𝕋 H(β) W_1/β(t) ∏_i, j = 1 i ≠ j^n (β_i - β_j ) d β_1 ⋯ d β_n-1/β_1 ⋯β_n-1where𝕋 = { (β_1,⋯,β_n-1) ∈ℂ^n-1 : |β_i|_ℂ = 1 }, 1/β = (1/β_1,⋯, 1/β_n) and β_n = 1/β_1 ⋯β_n-1.Note that H^♭ is supported on (ℤ_p \{0})^n-1, if we identify Z(ℚ_p) \ T(ℚ_p) with (ℚ_p^×)^n-1 as in Theorem <ref>. Moreover, H^♭ is invariant under the action of (ℤ_p^×)^n-1. The inverse transform is given in the following theorem.Let H : 𝒮_η^n →ℂ be a holomorphic symmetric function, for some η >0. Assume (H^♭)^♯ converges absolutely on 𝒮_η^n. Then,(H^♭)^♯(α) = H(α)for α∈𝒮_η^n with α_1 ⋯α_n = 1. By studying the image of the map H ↦ H^♭ we obtain the following corollary. Let h : Z(ℚ_p) \ T(ℚ_p) →ℂ. Fort = ( [ t_1 ⋯ t_n-1; ⋱; t_1; 1 ]) assume that h(t) is supported on the region { t_i ∈ℤ_p \{ 0 } : i=1,⋯,n-1 }, that the integral which defines h^♯ (in Definition <ref>) is absolutely convergent, and that h(t) = f(-|t_1|_p,⋯,-|t_n-1|_p) for some function f : ℤ_≥ 0^n-1→ℂ. Then(h^♯)^♭ = h.We postpone the proofs of Theorem <ref> and Corollary <ref> to the next section. As a consequence of Corollary <ref> we have the Plancherel formula for the p-adic Whittaker transform. Let h_1, h_2 be two functions that satisfy the conditions of Corollary <ref>. We have,⟨ h_1, h_2 ⟩ = ⟨ h_1^♯, h_2^♯⟩where⟨ h_1, h_2 ⟩ ∫_T(ℤ_p) \ T(ℚ_p) h_1(t) h_2(t)d^×t,⟨ h_1^♯, h_2^♯⟩ 1/n! (2π i)^n-1∫_𝕋 h_1^♯(β) h_2^♯(β)∏_i, j = 1 i ≠ j^n (β_i - β_j ) d β_1 ⋯ d β_n-1/β_1 ⋯β_n-1.§ PROOF OF THEOREM <REF>We start the section with the proof of two of the ingredients needed to prove the inverse transform theorem. Both results can also be found in Section 4 of <cit.>. They are included here as their proofs are concise and improve the exposition. The first of these is a version of Cauchy's identity.Let s_λ be the Schur polynomial defined in the previous section. Let α_1,⋯,α_n,β_1,⋯,β_n ∈ℂ such that |α_i β_j| < 1 for every 1 ≤ i, j ≤ n. Then,∑_m_1,⋯, m_n-1≥ 0 s_𝐦(α) s_𝐦(β)= 1-α_1 ⋯α_n β_1 ⋯β_n/∏_i,j=1^n (1-α_i β_j ) where 𝐦 = (m_1,⋯,m_n-1)This proof follows <cit.> and <cit.>. We start by stating the Cauchy determinant identity (lemma 7.4.18 in <cit.>)| [ 1/1-α_1 β_1 ⋯ 1/1-α_n β_1; ⋮ ⋱ ⋮; 1/1-α_1 β_n ⋯ 1/1-α_n β_n; ]| = ∏_1 ≤ i < j ≤ n (α_i - α_j) ∏_1 ≤ i < j ≤ n (β_i - β_j)/∏_i,j = 1^n (1 - α_i β_j).Expanding each term 1/1-α_i β_j as 1 + α_i β_j + α_i^2 β_j^2 + ⋯, we obtain |[ 1/1-α_1 β_1 ⋯ 1/1-α_n β_1; ⋮ ⋱ ⋮; 1/1-α_1 β_n ⋯ 1/1-α_n β_n; ]| = ∑_l_1,⋯,l_n ≥ 0 ∑_σ∈ S_n(σ) α_σ(1)^l_1⋯α_σ(n)^l_nβ_1^l_1⋯β_n^l_n. Notice that if the l_i are not all distinct then the sum over S_n vanishes. Therefore| [ 1/1-α_1 β_1 ⋯ 1/1-α_n β_1; ⋮ ⋱ ⋮; 1/1-α_1 β_n ⋯ 1/1-α_n β_n; ]| = ∑_l_1 > ⋯ > l_n ≥ 0σ, τ∈ S_n(σ) (τ) α_σ(1)^l_1⋯α_σ(n)^l_nβ_τ(1)^l_1⋯β_τ(n)^l_n = ∑_l_1 > ⋯ > l_n ≥ 0| [ α_1^l_1 ⋯ α_n^l_1; ⋮ ⋱ ⋮; α_1^l_n ⋯ α_n^l_n; ]| | [ β_1^l_1 ⋯ β_n^l_1; ⋮ ⋱ ⋮; β_1^l_n ⋯ β_n^l_n; ]|.By making the substitution (l_1,⋯,l_n) = (m_0+⋯+m_n-1+(n-1), m_0+⋯+m_n-2+(n-2), ⋯, m_0) and dividing by ∏_1 ≤ i < j ≤ n (α_i - α_j) ∏_1 ≤ i < j ≤ n (β_i - β_j) we obtain1/∏_i,j=1^n (1-α_i β_j ) = ∑_m_0, m_1,⋯, m_n-1≥ 0 s_m(α) s_m(β) (α_1 ⋯α_n β_1 ⋯β_n)^m_0,where right hand side simplifies to1/1-α_1 ⋯α_n β_1 ⋯β_n∑_m_1,⋯, m_n-1≥ 0 s_m(α) s_m(β).Multiplying out by 1-α_1 ⋯α_n β_1 ⋯β_n concludes the proof. The second necessary ingredient is an identity for the integral of a product of two Whittaker function, as in <cit.>. Let α_1,⋯,α_n,β_1,⋯,β_n ∈ℂ such that |α_i β_j| ≤ 1 for every 1 ≤ i, j ≤ n, and ε > 0. Let W_α, W_β be Whittaker functions as defined in Definition <ref>. Then∫_Z(ℚ_p) \ T(ℚ_p) W_α(t) W_β(t) ∏_k=1^n-1 |t_k|_p^ε(n-k)d^×t = 1-α_1 ⋯α_n β_1 ⋯β_n/p^ε n/∏_i,j=1^n (1-α_i β_j/p^ε) We start by applying Theorem <ref> to write down the Whittaker functions explicitly:∫_Z(ℚ_p) \ T(ℚ_p) W_α(t) W_β(t) ∏_k=1^n-1 |t_k|_p^ε(n-k)d^×t= ∫_ℤ_p^n-1δ(t) s_-log|t|_p(α) s_-log|t|_p(β) ∏_k=1^n-1 |t_k|_p^-(k-ε)(n-k)dt_k/|t_k|_p = ∫_ℤ_p^n-1 s_-log|t|_p(α) s_-log|t|_p(β) ∏_k=1^n-1 |t_k|_p^ε(n-k)dt_k/|t_k|_p.Breaking up the region of integration by absolute value we get∫_Z(ℚ_p) \ T(ℚ_p) W_α(t) W_β(t) ∏_k=1^n-1 |t_k|_p^ε(n-k)d^×t= ∑_m_1,⋯, m_n-1≥ 0^n-1times_p^m_kℤ_p^×s_-log|t|_p(α) s_-log|t|_p(β) ∏_k=1^n-1 |t_k|_p^ε(n-k)dt_k/|t_k|_p = ∑_m_1,⋯, m_n-1≥ 0 s_𝐦(α) s_𝐦(β) p^-ε(n-1)m_1 -ε(n-2)m_2 + ⋯ -ε m_n-1 = ∑_m_1,⋯, m_n-1≥ 0 s_𝐦(α/p^ε) s_𝐦(β)= 1-α_1 ⋯α_n β_1 ⋯β_n/p^ε n/∏_i,j=1^n (1-α_i β_j/p^ε)where the last equality follows from Cauchy's identity <ref>. We are now ready to proceed to the proof of the Theorem <ref>. Restate the theorem asH(α) = ∫_Z(ℚ_p) \ T(ℚ_p) H^♭(t) W_α(t) d^×t.Recall that α∈𝒮_η^n and α_1 ⋯α_n = 1. Further assume that |α_1|_ℂ = ⋯ = |α_n|_ℂ = 1. We can later remove this assumption as both sides of the equality are holomorphic functions. Note that H is invariant under permutations of (α_1,⋯,α_n) as the Whittaker function also has those symmetries. DefineH_ε(α) = ∫_Z(ℚ_p) \ T(ℚ_p)H^♭(t) W_α(t) ∏_k=1^n-1 |t_k|_p^ε(n-k) d^×t.Thenlim_ε→ 0 H_ε(α) = ∫_Z(ℚ_p) \ T(ℚ_p) H^♭(t) W_α(t) d^×t.It suffices to show that lim_ε→ 0 H_ε(α) = H(α) as well. Replacing H^♭(t) by its explicit formula and swapping the order of integration we getH_ε(α) = 1/n!(2π i)^n-1∫_𝕋(∫_Z(ℚ_p) \ T(ℚ_p) W_α(t) W_1/β(t) ∏_k=1^n-1 |t_k|_p^ε(n-k)d^×t ) ×H(β) ∏_i, j = 1 i ≠ j^n (β_i - β_j ) d β_1 ⋯ d β_n-1/β_1 ⋯β_n-1We use Proposition <ref> to compute the innermost integrals and getH_ε(α) = 1/n!(2π i)^n-1∫_𝕋 H(β) (1-1/p^ε n)/∏_i,j=1^n (β_j-α_i/p^ε)∏_i, j = 1 i ≠ j^n (β_i - β_j ) d β_1 ⋯ d β_n-1/β_1 ⋯β_n-1Now shift each variable β_i (i=1,⋯,n-1) to the contour given by the equation |z|_ℂ = 1/p^ε+ε', in order. For the sake of clearness, we shall assume that all the α_i are distinct. When the contour of integration for β_1 is shifted one picks up several residues atβ_1 = α_1/p^ε, ⋯, β_1 = α_n/p^ε.For which residue integral obtained in this manner one can shift the contour of integration for β_2 picking up n-1 residues in the process. Repeating this process for which β_i, i=3,⋯,n-1 one obtainsH_ε(α) = 1/n!∑_σ∈ S_nℛ_ε^σ + ℐ_ε,ε' this whereℛ_ε^σ = H(α_σ(1)/p^ε,⋯,α_σ(n)/p^ε) (1-1/p^ε n) ∏_i=1^n-1( α_σ(i)/p^ε - p^(n-1)εα_σ(n))/(p^(n-1)εα_σ(n) - α_σ(n)/p^ε) ∏_i=1^n-1( α_σ(i)/p^ε - α_σ(n)/p^ε) ∏_i=1^n-1α_σ(i)/p^εand ℐ_ε,ε' is a sum of residue integrals with each integrand bounded (in absolute value) by C_H (1 - 1/p^nε) p^ε(n^3+n)/1 - 1/p^ε'for some constant C_H > 0 depending only on the function H. Therefore,lim_ε→ 0ℛ_ε^σ=lim_ε→ 0H(α_σ(1)/p^ε,⋯,α_σ(n)/p^ε)/∏_i=1^nα_σ(i)/p^ε(1-1/p^ε n)/(p^(n-1)ε - 1/p^ε)∏_i=1^n-1( α_σ(i)/p^ε - p^(n-1)εα_σ(n))/∏_i=1^n-1( α_σ(i)/p^ε - α_σ(n)/p^ε) = H(α_σ(1),⋯,α_σ(n))/∏_i=1^nα_σ(i)∏_i=1^n-1( α_σ(i) - α_σ(n))/∏_i=1^n-1( α_σ(i) - α_σ(n)) = H(α_σ(1),⋯,α_σ(n))= H(α)andlim_ε→ 0 |ℐ_ε,ε'| ≪_Hlim_ε→ 0(1 - 1/p^nε) p^ε(n^3+n)/1 - 1/p^ε' = 0.In conclusion, we obtainlim_ε→ 0 H_ε(α) = 1/n!∑_σ∈ S_n H(α) = H(α)as desired. If the α_i are not all distinct then the original integrand would have higher order poles. This means that the residue sum would have a small number of residue but some residues would contribute to the sum with a multiple of H(α). To prove the corollary one simply needs to compute the image of the inverse transform H ↦ H^♭ as any function h in the image of this mapping will satisfy (h^♯)^♭ = h. Simply choose H such that h = H^♭, and apply Theorem <ref> to H, to obtain (h^♯)^♭ = ((H^♭)^♯)^♭ = H^♭ = h. We start by noting that any function h satisfying the assumptions in the corollary is a sum of scalar multiples of functions of the form: f_λ_1,⋯,λ_n-1(t) =1,log |t_i|_p = -λ_ifori=1,⋯,n-1, 0,where λ_1,⋯,λ_n-1 are nonnegative integers. Therefore, it suffices to show that all such functions are in the image of the map of the inverse transform H ↦ H^♭. Let H(β) = W_β(p^λ)/δ(p^λ) wherep^λ = ( [ p^λ_1⋯ p^λ_n-1 ; ⋱;p^λ_1 ; 1 ]).ThenH^♭(t) = 1/δ(p^λ) n! (2π i)^n-1∫_𝕋 W_β(p^λ) W_1/β(t) ∏_i, j = 1 i ≠ j^n (β_i - β_j ) d β_1 ⋯ d β_n-1/β_1 ⋯β_n-1 = δ^1/2(t)/δ^1/2(p^λ) n! (2π i)^n-1∫_𝕋 s_λ(β) s_-log|t|_p(1/β) ∏_i, j = 1 i ≠ j^n (β_i - β_j ) d β_1 ⋯ d β_n-1/β_1 ⋯β_n-1.By the definition of Schur polynomials we further infer thatH^♭(t) = δ^1/2(t)/δ^1/2(p^λ) n! (2π i)^n-1∫_𝕋∑_σ, τ∈ S_n(β_σ(1)^(n-1)+λ_1+⋯+λ_n-1⋯β_σ(n-1)^1+λ_1) ×( β_τ(1)^-(n-1)+log|t_1|_p+⋯+log|t_n-1|_p⋯β_τ(n-1)^-1+log|t_1|_p) d β_1 ⋯ d β_n-1/β_1 ⋯β_n-1 = δ^1/2(t)/δ^1/2(p^λ) n! (2π i)^n-1×∫_𝕋∑_σ∈ S_n(β_σ(1)^λ_1+⋯+λ_n-1+log|t_1|_p+⋯+log|t_n-1|_p⋯β_σ(n-1)^λ_1+log|t_1|_p) d β_1 ⋯ d β_n-1/β_1 ⋯β_n-1 = δ^1/2(t)/δ^1/2(p^λ)f_λ_1,⋯,λ_n-1(t)= f_λ_1,⋯,λ_n-1(t)where the last equality follows from the definition of f_λ_1,⋯,λ_n-1(t). § INTEGRAL REPRESENTATIONS OF LOCAL L-FACTORS In this final section, we obtain integral representations of the local L-factors of symmetric dth power L-functions of GL(2) representations, with d ≤ 4. Let L(s, Sym^dπ) be the symmetric dth power L-function associated to an irreducible automorphic representation π of GL(2,𝔸_). This L-function has local factors (at the unramified places p) given byL_p(s,Sym^dπ) = ∏_i=0^d (1-α^d-2ip^-s)^-1 =: h_s,p,d(α). We shall obtain an integral representation for the L-factors L_p(s,π) using Theorem <ref>. Assume ℜ(s) > 1. We start by computing (h_s,p,d)^♭ explicitly. By definition, (h_s,p,d)^♭(t) = 1/4π i∫_|β|_ = 1 h_s,p,d(β) W_1/β(t) (β - β^-1)(β^-1 - β) d β/β = | t_1 |_p^1/2/4π i∫_|β|_ = 1 h_s,p,d(β) ( β^λ+1 - β^-(λ+1)) (β^-1 - β) d β/βwith t = [ t_1 0; 0 1 ] and | t_1 |_p = p^-λ for λ≥ 0.We now want to use the residue theorem to evaluate the integral above. To simplify the calculation of the residues of the integrand we first note that ∫_|β|_ = 1 h_s,p,d(β) β^-(λ+1)(β^-1 - β) d β/β= - ∫_|β|_ = 1 h_s,p,d(β) ( β^λ+1) (β^-1 - β) d β/β via the change of variable β↦1/β. Therefore, (h_s,p,d)^♭(t) = | t_1 |_p^1/2/2π i∫_|β|_ = 1 h_s,p,d(β)(β^λ-1 - β^λ+1) d β. Let i_s,p,d(β) be the integrand of the integral in Equation <ref>. By the definition of h_s,p,d, the function i_s,p,d(β) has possible poles inside the unit circle at β = e^2π i l/k p^-s/k for 0 < k ≤ d, k ≡ d2 and 0 ≤ l < k. We will compute the residues at these poles for 1 ≤ d ≤ 4, in order to compute the integral in Equation <ref>.* d=1:In this case, the function i_s,p,1(β) only has a pole at β = p^-s and Res_β = p^-si_s,p,1(β) = p^-s λ,which implies that (h_s,p,1)^♭(t) = | t_1 |_p^s+1/2.* d=2:In the case of a symmetric square L-function, the integrand i_s,p,2(β) has poles at β = ± p^-s/2. Their residues are given byRes_β = p^-s/2i_s,p,2(β)= p^-sλ/2/2(1-p^-2s),Res_β = -p^-s/2i_s,p,2(β) = -p^-sλ/2/2(1-p^-2s). Adding the two residues we infer that (h_s,p,2)^♭(t)= | t_1 |_p^(s+1)/2/1-p^-2s,λ, 0,λ.* d=3:For the symmetric cube L-function, the poles of the integrand i_s,p,3(β) are at β = p^-s and at β = e^2π i l/3 p^-s/3 for l = 0,1,2. The residue at β = p^-s is given byRes_β = p^-si_s,p,3(β)= -p^-s(λ+2)/(1-p^-2s)(1-p^-4s).For clarity's sake we won't write down the values of the residues at the remaining three poles, but simply note that their sum is equal to1/(1-p^-2s)(1-p^-4s)p^-sλ/3,λ≡ 03,p^-s(λ+8)/3,λ≡ 13,p^-s(λ+4)/3,λ≡ 23.We remark that calculating these residues is a straightforward, albeit tedious, exercise. Finally, we conclude that (h_s,p,3)^♭(t) is equal to1/(1-p^-2s)(1-p^-4s)| t_1 |_p^s/3+1/2-| t_1 |_p^s-3/2,λ≡ 03,| t_1 |_p^s/3-13/6-| t_1 |_p^s-3/2,λ≡ 13, | t_1 |_p^s/3-5/6-| t_1 |_p^s-3/2,λ≡ 23.* d=4:In this case, the integrand i_s,p,4(β) has poles at β = ± p^-s/2 and at β = ± p^-s/4, ± i p^-s/4. The sum of the residues at the first two poles is1/(1-p^-2s)(1-p^-3s)-p^-s(λ+2)/2,λ≡ 02,0,λ≡ 12.and the sum of the residues at the last four poles is1/(1-p^-2s)(1-p^-3s)p^-sλ/4,λ≡ 04,p^-s(λ+6)/4,λ≡ 24,0, Adding up all the residues one obtains that (h_s,p,4)^♭(t) is equal to1/(1-p^-2s)(1-p^-3s)| t_1 |_p^s/4+1/2-| t_1 |_p^s/2-1/2,λ≡ 04,| t_1 |_p^s/4-1-| t_1 |_p^s/2-1/2,λ≡ 24,0,For large enough ℜ(s), since the functions (h_s,p,d)^♭(t) satisfy the convergence condition in Remark <ref>, it follows from Theorem <ref> that L_p(s,Sym^dπ) = ∫_ℚ_p^× (h_s,p,d)^♭(t) W_α(t)d^×t,for 1 ≤ d ≤ 4, where the functions (h_s,p,d)^♭(t) are explicitly given by Equations <ref>-<ref>.This approach should provide (conditionally on convergence issues) integral representations for L_p(s, Sym^dπ) for d > 4 as well. However, the residue calculations appear to become rather complicated. § ACKNOWLEDGMENTS The author would like to thank Dorian Goldfeld for helpful discussions, as well as for the comments on earlier drafts of this work. The author was partially supported by the FCT doctoral grant SFRH/BD/68772/2010. plain

http://arxiv.org/abs/1702.08271v1

We demonstrate a new technique for detecting components of arbitrarily-shaped radio-frequency waveforms based on stroboscopic back-action evading measurements.We combine quantum non-demolition measurements and stroboscopic probing to detect waveform components with magnetic sensitivity beyond the standard quantum limit. Using an ensemble of 1.5× 10^6 cold rubidium atoms, we demonstrate entanglement-enhanced sensing of sinusoidal and linearly chirped waveforms, with 1.0 (2)dB and 0.8 (3)dB metrologically relevant noise reduction, respectively. We achieve volume-adjusted sensitivity of δB√(V)≈ 11.20 fT√(cm^3/Hz), comparable to the best  magnetometers.Entanglement-enhanced radio-frequency field detection and waveform sensingM. W. Mitchell December 30, 2023 ==============================================================================Quantum noise and quantum coherence both play essential roles in determining the fundamental sensitivity of interferometric instruments such as atomic magnetometers and atomic clocks.This is clearly seen in a widely-used rule of thumb for the projection noise limit δ B_ PN of sensing magnetic fields withspin-f atoms <cit.>: δ B_ PN√()≃ħ/g μ_B1/√(2 f ).Hereis the total acquisition time including averaging repeated measurements, μ_ B is the Bohr magneton, g is the ground-state Landé factor, and ħ is Planck's constant.The factor 1/√(2 f ) reflects the standard quantum limit (SQL) spin projection noise of the atomic precession angle, which scales as 1/√(). The signal accumulation timeis determined by the smaller of the single-measurement duration and spin atomic coherence time. Analogous expressions govern clocks and other atomic instruments. Reduction of spin projection noise below the SQL <cit.>, which implies entanglement among atoms <cit.> and/or atomic components <cit.>, has been demonstrated by quantum non-demolition measurement <cit.> including large degrees of squeezing using cavity enhancement <cit.>.Use of conditional spin-squeezed states has been demonstrated in magnetometry <cit.> and clock operation <cit.>.These works employ a measure-evolve-measure (MEM) sequence, in which a first quantum non-demolition (QND) measurement produces a state with reduced projection noise, a period of free evolution accumulates signal, and a second QND measurement detects the change relative to the first measurement.This method exploits the coherence of the atomic system, allowing signal to accumulate prior to readout of the atomic state, which typically destroys coherence. Here we demonstrate a generalization of this method, to allow quantification of arbitrarily-shaped signal components, using the same resources of atomic spin squeezing and QND measurement.This broadens the scope of entanglement-enhanced sensing to include arbitrary time-varying signals, of which magnetic waveforms due to firings of single neurons <cit.> and event-related fields in magnetoencephalography <cit.> are notable examples.While rapid sampling of the same signals can also detect waveforms, it reduces τ and thus the sensitivity of the measurement.Our method includes radio-frequency () magnetometry as a special case, and we demonstrate entanglement-enhanced detection of radio fields in a compact, high-sensitivity  magnetometer, with a sensitivity-volume figure of merit comparable to the best demonstrated instruments.Principle of the methodWe consider an ensemble of atoms, described by a polarization F, precessing in response to a magnetic field B(t) = y B_y(t) + x B_x(t), with |B_y| ≫ |B_x|. F precesses about y at an experimenter-controlled Larmor angular frequency (t) = γ B_y(t) + O(γ B_x^2/B_y) ≈γ B_y(t), driven transversally by the small unknown perturbation B_x(t).The component F_z is assumed accessible to QND measurement. The dynamics of the system are given byd/dt(t) = γ(t) ×(t)where γ is the gyromagnetic ratio of the atomic state.As shown in the , for an initial atomic polarization oriented along +y the evolution of the measurable spin component F_z(t) can be expressed as(t) =(0) cosΘ (t) +(0) sinΘ (t) + γ⟨ F_y(0) ⟩∫_0^t dt'(t') cos[ Θ(t) - Θ(t') ] + O()^2 + O(γ t δ F_y )where Θ(t) ≡∫_0^t dt'(t') is the accumulated angle. We note that the first line describes an operator relation, namely a rotation of the spin components , about y, and contains the quantum noise associated with the spin variables.As with other squeezing-enhanced atomic measurements, the noise in F_z(t) can be reduced by squeezing a linear combination of F_z(0) and F_x(0). In particular, QND measurement of F_z(0) can squeeze this component, while subsequent measurements at times {t_i}, chosen so that Θ(t_i)=nπ, n ∈ℤ, excludes noise of F_x from entering the measurement record, achieving back-action evasion <cit.>. In contrast, the second line of Eq. (<ref>) has no operator content, and describes a noiseless displacement by an amount proportional to the integral of the waveform B_x(t') multiplied by the pattern function cos[ Θ(t) - Θ(t') ].This describes a coherent build-up of the signal component matching the pattern function, and a cancellation of signal components orthogonal to it <cit.>.By proper choice of Θ(t), the pattern function can be made to take on any functional form bounded by ± 1.The third line of Eq. (<ref>) is negligible for the interesting case of weak signals and large atom number. Experimental technique Our experimental apparatus, illustrated in Fig. <ref> (a), is described in detail in <cit.>.Briefly, we trap up to 1.5×10^6 ^87Rb atoms in a weakly focused single beam optical dipole trap.The atoms are laser cooled to a temperature of 16 and optically pumped into the f=1 hyperfine ground state.A bias magnetic field along y is generated with coils in a near-Helmholtz geometry fed by a programmable current source, and monitored using the atoms as an in-situ DC vector magnetometer <cit.>.An  field along x is produced with a low-inductance coil and an arbitrary waveform generator.Optical pumping (OP) along the y direction is used to produce an initial atomic polarization, with an efficiency of ∼ 96%, as measured by Faraday rotation <cit.>.A non-destructive measurement of the atomic state is made using a train of 600 duration pulses of linearly polarized light 700MHz red detuned from the f=1→ f'=0 transition on the D_2 line.The interaction between the atoms and the probe pulses is given by the effective Hamiltonian τ_pulseĤ_eff=g_1 where the g_1 is a coupling constant depending on the probe beam geometry and detuning and τ_pulse is the pulse duration, operators F_i describe the atomic spins, and S_i the optical polarization <cit.>. Light pulses propagating along the trap axis experience a polarization rotation = cosϕ + sinϕ, where S_i^( in/out) are Stokes operators before/after passing the atoms <cit.> and ϕ = g_1 F_z is the Poincaré-sphere rotation angle.is detected with a shot-noise-limited balanced polarimeter and = /2 is measured by splitting a constant fraction of the input light to a reference detector before the atoms.g_1 is calibrated by independent measurement <cit.>. Calibration of responseThe coherent response of the atoms is illustrated in Fig. <ref>.With a constant B_y we apply   excitation of the form (t)=a_cos(ω_0 t) and observe Rabi oscillation, i.e., sinusoidal oscillation of , amplitude modulated at the Rabi frequency as predicted by Eq. (<ref>). We observe good experiment-theory agreement, and use the data as a calibration ofand . Waveform detection Selective response to chirped waveforms is shown in Fig. <ref>.Using a ramped field (t), we produce a Larmor frequency that sweeps linearly from = 2 π×42.2 to = 2 π×47.5 over 800.This produces a “chirped” pattern function Θ(t), making the MEM sequence sensitive to (t) signals with similar chirp, but insensitive to other waveforms, e.g. at constant frequency or with the opposite chirp.To confirm this selectivity, we use an arbitrary waveform generator to apply transverse fields of the form (t)=a_cos(ω_0 t+ κ t^2), i.e., a linearly chirped waveform, in the time betweenand . Fig. <ref> shows the resulting signal, i.e., the amplitude of the observedoscillation, as a function of the chirp κ.As expected, we observe a peak in the population transferred by thedrive when κ matches the field ramp.Agreement with theory from Eq. (<ref>) is good, and variation in experimental signal is consistent with the independently-measured fluctuations of the magnetic field at the position of the atoms. Stroboscopic QND measurement To perform one QND measurement,probe pulses are sent through the atoms at intervals of one-half of the Larmor period, and experience a polarization rotation ϕ_n = g_1 (t_n), where n indexes the pulses. Because of the inversion of F_z between pulses, the ϕ_n can be aggregated as a single distributed measurement of F_z, quantified by Φ≡^-1∑_n=1^(-1)^n-1ϕ_n. This multi-pulse probing has been shown to be a true QND measurement of the collective spin <cit.> with an uncertainty below the standard quantum limit for F_z <cit.>. Back-action evading  sensing With these elements, we demonstrate back-action evasion in a MEM sequence to detect  magnetic fields.We load the ODT with = 1.5e6 atoms, measure the bias field as in <cit.> and then repeat the following MEM sequence 16 times:dispersive measurement ofas in <cit.>; optical pumping to produce full polarization along +y; QND measurementwith result Φ_1; free evolution for time =300; and a second QND measurementwith result Φ_2, as illustrated in Fig. <ref> (b).During the hold timeis held constant, i.e. κ = 0, with =2 π×50.16. The QND measurementsandare made over 200 and contained 4e8photons, see <cit.> for details. The 16 repetitions of the MEM sequence allow us to varysince atoms are lost from the trap during optical pumping. We repeat the full sequence 463 times to collect statistics.Projection noise level Fig. <ref> shows the measured variance of Φ_2 and the conditional variance (Φ_2|Φ_1) as a function of the number of atoms in the trap.A linear measurement will show a variance that is quadratic in  <cit.>:var (Φ) =var_0(Φ) + g̃^2_1 1/2α + a_2 ^2where var_0(Φ) is the readout noise, quantified by repeating the measurement without atoms in the trap, a_2 ^2 is atomic technical noise associated with, e.g., fluctuations in state preparation, and the term ∝ corresponds to atomic projection noise.The factor 1/2 describes the F_z variance of an f=1 atom polarized along y, and α accounts for the net noise reduction due to off-resonant scattering of probe photons.This scattering both reduces the number of noise-contributing atoms by pumping some into the far-off-resonance f=2 ground state, and adds noise as some atoms return to the f=1 state with randomized polarizations.We compute the evolution of the state using the covariance matrix methods reported in <cit.> to find α = 0.96. The coupling g̃^2_1 is different from g_1 due to inhomogeneous atom-field coupling <cit.>. A fit of Eq. (<ref>) to the data finds g̃^2_1=1.2(2)× 10^-14 rad^2, determining the projection-noise level. Squeezing To study the generation of squeezing we look at the correlation betweenand .The first train of QND-pulses redistributes the noise to the non-measured component andis used to evaluate its variance conditioned on the first measurement.The measurement noise reduction is quantified by var(Φ_2|Φ_1)=var(Φ_2-χΦ_1), where χ=cov(Φ_1, Φ_2)/var(Φ_1)>0 describes the correlation between Φ_1 and Φ_2.As seen in Fig. <ref>, the conditional variance is [parse-numbers=false]2.4 (2)dB below the projection noise level. Metrological improvement is quantified by the Wineland criterion <cit.> which takes into account the coherence loss and the noise reduction of the measured state: ξ_m^2=1/η^2var(Φ_2|Φ_1)/varΦ_2where ξ_m^2<1 indicates metrological advantage and η accounts for the loss of coherence of the spin-squeezed state relative to the input coherent spin state. The coherence after the first measurement is =η, where η≡ (1-η_sc)(1-η_field) and η_sc=0.11 and η_field=0.04 are independently measured coherence loss due to probe scattering and field inhomogeneities, respectively, during . We find the metrological improvement due to squeezing ξ_m^2=0.79(5), or [parse-numbers=false]1.0 (2)dB.Entanglement-assisted waveform detectionTo detect chirped waveform components beyond the projection-noise level, we repeat the above measurement and analysis using a chirped waveform with κ = 2π×5.78e6, i.e., we ramp (t) to produce a chirped pattern function during =600. The probe frequencies are matched to the Larmor precession frequency duringandwith 2.9e8photons, details in <cit.>.For this experiment, = 1.3e6. We repeat the MEM sequence 21 times per trap loading to vary , and the experiment 82 times to gather statistics. Analyzed as above, we find 1.5(3)dB of noise reduction with 0.8(3)dB of metrological advantage.The reduced squeezing is due to the smaller , smaller total photon number, and technical noise accumulated during the longer .Magnetic sensitivityThe sensitivity to thedrive amplitude a_ is δ B_ = Δϕ/(| d⟨ϕ⟩ /da_|) with ϕ being the observed Faraday rotation signals. The signal accumulated during thecan be obtained by integration of Eq. (<ref>), resulting in an enhanced single-shot sensitivity  <cit.>:δ B_√() ≃Δϕ_cond/g_1η_hold⟨(0) ⟩2/γ √()where ⟨(0) ⟩=, η_hold is the total coherence loss duringand , and Δϕ_cond^2=var(Φ_2|Φ_1) is the conditional variance between the Φ_2 and Φ_1 optical signals, including the atomic noise and read-out noise. Applied for the twowaveforms used, i.e., constant frequency and linearly chirped, we find sensitivities of 2.96pT/√(Hz) and 3.36pT/√(Hz), respectively.Focusing on the first, scaling the sensitivity by the volume of the atomic cloud, V =1.43× 10^-5cm^3, we find δB√(V)≈ 11.20 fT√(cm^3/Hz).For comparison, the best alkali-vapor  magnetometer <cit.> in this frequency range showed a sensitivity of 0.24fT/√(Hz) with V=96 cm^3 or δB√(V)= 2.35 fT√(cm^3/Hz).Thus the  magnetometer demonstrated here has a volume-adjusted sensitivity comparable with the best existing instruments.We have experimentally demonstrated detection of radio-frequency fields and arbitrarily-shaped radio-frequency waveform components beyond the projection noise limit, using stroboscopic back-action-evading measurements on magnetic atomic ensembles. The combination of QND measurements and stroboscopic probing in a measure-evolve-measure sequence gives this quantum sensing advantage, while also allowing full use of the system coherence, resulting in a sensitivity-volume figure of merit comparable to the best  magnetometers at these frequencies.§ ACKNOWLEDGEMENTSWe thank C. Abellán and W. Amaya for lending us the  generator and S. Coop for useful feedback on the manuscript. Work supported by MINECO/FEDER, MINECO projects MAQRO (Ref. FIS2015-68039-P), XPLICA (FIS2014-62181-EXP) and Severo Ochoa grant SEV-2015-0522, Catalan 2014-SGR-1295, by the European Union Project QUIC (grant agreement 641122), European Research Council project AQUMET (grant agreement 280169) and by Fundació Privada CELLEX.26 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Budker and Romalis(2007)]BudkerNP2007 author author D. Budker and author M. Romalis, http://dx.doi.org/10.1038/nphys566 journal journal Nat Phys volume 3,pages 227 (year 2007)NoStop [Takano et al.(2009)Takano, Fuyama, Namiki, and Takahashi]TakanoPRL2009 author author T. Takano, author M. Fuyama, author R. Namiki,and author Y. Takahashi, 10.1103/PhysRevLett.102.033601 journal journal Phys. Rev. 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Heinzen,10.1103/PhysRevA.46.R6797 journal journal Phys. Rev. A volume 46, pages R6797 (year 1992)NoStop [Lee et al.(2006)Lee, Sauer, Seltzer, Alem, andRomalis]LeeAPL2006 author author S.-K. Lee, author K. L. Sauer, author S. J. Seltzer, author O. Alem,and author M. V. Romalis, http://dx.doi.org/10.1063/1.2390643 journal journal Applied Physics Letters volume 89,eid 214106 (year 2006),http://dx.doi.org/10.1063/1.2390643NoStop Supplementary Information§.§ Operator definitionWe define the collective spin operation ≡∑_i, whereis the spin of the i'th atom. The collective spin obeys commutation relations [,]=i, here and throughout we set ħ=1. The probe pulses are described by the Stokes operatordefined as ≡1/2(,)(,)^T,where theare the Pauli matrices andare the annihilation operators for the σ_± polarizations, which obey [,]=i and cyclic permutations. The input pulses are fullypolarized, i.e., with = /2, = = 0 and Δ^2 S_i=/4, i∈{x,y,z} whereis the number of photons in the pulse.§.§ State preparationThe atoms are polarized so that ⟨ F⟩ is oriented along +y via optical pumping under a parallel bias field . We use a single 50 long pulse of circularly polarized light resonant with the f=1→ f'=1 transition of the D_2 line and propagating along the y-axis, and illuminate the atoms with repumper light resonant with the f=2→ f=2' transition to prevent accumulation of atoms in the f=2 hyperfine level.To measure the atomic polarization, the atoms are subsequently rotated intoby slowly rotating the bias field fromto , and then measured with the Faraday probe.The amplitude of the transferred atoms is compared to the signal from an ensemble directly polarized by on axis optical pumping in afield <cit.>, resulting in a input polarized atomic ensemble with ≃, Fig. <ref> (a).§.§ Magnetic field measurementWe make use of the atoms as an in-situ DC vector magnetometer to measure the applied magnetic field as described in <cit.>. We prepare an ⟨ F⟩ along z under an orthogonal bias fieldvia optical pumping with circular polarized light propagating along the trap axis, and observe the free induction decay signal (FID) of the resulting Larmor precession using the Faraday probe.In order to observe the pure magnetic dephasing of the spins we probe semi-continuously with a reduced <10^6 and large detuning, Δ=1.5, to minimize probe scattering.We fit the FID traces with the function ϕ(t)=β+αcos( t+ϕ_0)e^-t/T_2 and extract the Larmor angular frequencyand the spin coherence time T_2.We minimize the amplitude of β by compensating homogeneous field along x and z with coils in a near-Helmholtz geometry and cancel field gradients along the length of the trap by running current in opposite direction through pairs of wires transversal to the trap. The field is optimized in an iterative routine resulting in typical =[parse-numbers=false]4.8(3) and fields 99.1% along the y-axis. A typical optimized FID signal is shown in Fig. <ref> (b). §.§ Rabi Flopping CalibrationAdding an oscillatory (sinusoidal) driving field perpendicular to a static bias field induces Rabi flops between the magnetic sub-levels of the atoms in the hyperfine levels.When the frequency of the RF-signal matches the atomic energy splitting the flopping frequency is minimal and the population transferred by the RF maximal.As for the magnetic field measurement, in order to to minimize probe scattering the atoms are probed semi-continuously with a reducedand Δ=1.5.The traces are fitted with the function (t)=a cos( t+ϕ_L)cos(ω_R t+ϕ_R) whereis the Larmor angular frequency and ω_R is the Rabi flopping frequency.Fitting the maximal axial polarization ℱ_z/|| as a function of the frequency of the RF excitation with a Lorenztian <cit.> we determine the resonance frequency to be [parse-numbers=false]42.26 (1) and estimate the strength of the RF-field coupled to the atoms to be a=[parse-numbers=false]0.60(5)G.§.§ Ramped bias fieldTo characterize the ramp of the bias field, we fit the FID signal with a chirped function of the form ϕ(t)=β+αcos( t+κ t^2 +ϕ_0)e^-t/T_2.We find κ=2π×[parse-numbers=false]5.6 (1)×10^6 and =[parse-numbers=false]4.5 (4). To confirm the linearity of the chirp we divide the FID-signal in 100 long segments and fit them individually with a function ϕ̃(t)=β'+α' cos(ω_L^(i) t+ϕ_0'), where ω_L^(i) is the Larmor frequency of the i-th segment.A quadratic fit to the fit outputs for ω_L^(i) confirms the linearity of the ramp as the quadratic term in negligible, a_2 t/a_1=5.8× 10^-8, where t=800 is the time during the ramp is on, see Fig. <ref>. §.§ Stroboscopic ProbingWe probe the atoms via off-resonant paramagnetic Faraday-rotation using τ_pulse=600 duration pulses of linearly polarized light with a detuning of 700MHz to the red of the D_2 line, and sent sent every half Larmor period, =/π. For the experiment with constant bias field during the hold time, =2 π×50.16 and the QND measurementsandare made at frequencies ==101.01 over a time 200 giving a total of = =20 probe pulses, each with =2×10^7 photons.For the detection chirped waveform components the experiment was done ramping (t) to produce a chirped pattern function with κ = 2π×5.78e6 during =600.The probe frequencies = 84.75=93.46 matched the Larmor precession frequency duringandand contained = 16, =18, respectively, with =1.81×10^7 photons per pulse. During a probe pulse the atomic spins rotate by an angle θ=γ B τ_pulse. For our parameters, τ_pulse and B=||≲70 G we find Θ∼ 30, so we can neglect the rotation of the spins during the probe pulse.§.§ Volume estimation We use absorption imaging to estimate the volume of the atomic cloud. Atoms originally in the f=1 hyperfine ground state are transferred into the f=2 hyperfine ground state by a 100 pulse of laser light tuned to the f=1→ f'=2 transition. A 100 pulse of circularly polarized light resonant to the f=2→ f'=3 transition cast a shadow on a CCD camera. To avoid spatially dependent light shifts the dipole trap is switched off during the process. To estimate the radial extension of the atom we use the time-of-flight (TOF) technique <cit.>. Free, thermal and isotropic expansion of the atoms is described by ω_a(t)^2=ω_a^2(0)+k_B T/m(t-t_0)^2where k_B is Boltzmann constant and m the atomic mass. ω_a(0) is the initial size of the ensemble, T its temperature and t_0 is a time delay for switching off the dipole trap, all 3 free parameters of the fit. We determine the width of the radial profile by integrating each frame of the image along the longitudinal direction (the z-axis) and fit the resulting density profile with a Gaussian to get the center and the width of the atomic ensemble.The width of the atomic cloud ω_a is plotted on Fig. <ref> (a) as a function of time.We fit with Eq. (<ref>) to find ω_a(0)=14.2 (4), from which we determine the radial full width at half maximum (FWHM) of the atomic cloud ρ_radial=2√(2 ln2) ω_a(0)=33 (1). The temperature of the atomic sample is 15.5 (1)The axial dimension of the trap is much longer than the spread of the ensemble during the TOF, resulting in ρ_axial(t)≃ρ_axial(0). The axial shape of the atomic ensemble is calculated by integrating the images along the transverse direction, Fig. <ref> (b).The black solid line is a fit with a Lorentzian of the form L(z)=aρ_axial/2((z-z_0)^2+(1/2ρ_axial)^2)+b, where ρ_axial is the FWHM.We determine the axial FWHM atomic length ρ_axial=3.28 (6).We approximate the atomic volume to an ellipsoid with semi-axis given by ρ_axial and ρ_radial and calculate its volume using V=4π/3ρ_radial^2ρ_axial=1.43e-5.§.§ Waveform component detection The dynamics of the spins are governed by the Heisenberg equations of motion (with ħ = 1):d/dtF_i = -i [F_i,H_0(t)+ H'(t)]where H_0 = - γ F· B = - γ F_y B_y(t), γ is the gyromagnetic ratio for the F=1 ground hyperfine state, and the perturbation H' = - γ F_x B_x(t) describes the RF drive. We use a Dyson series to solve the resulting system of differential equations. We define F^(0)(t) to be the solution to Eq. (<ref>) when H' = 0, i.e. F_y(t) = F_y(0) and ([ (t); (t) ])= ([cosΘ(t)sinΘ(t); -sinΘ(t)cosΘ(t) ]) ([ (0); (0) ])where Θ(t)≡∫_0^t dt'ω_L(t'), (t) ≡γ B_y(t) is the accumulated angle. We then use the well-known result <cit.>F_i(t) = F_i^(0)(t) - i ∫_0^t dt' [F_i^(0)(t), H'(t')] + O(H')^2which, in light of the definition of H'(t'), gives the signal(t) =cosΘ (t) (0) + sinΘ (t) (0) + γ∫_0^t dt' F_y(t') B_x(t') cos[ Θ(t) - Θ(t') ]+ O(B_x)^2We note that F_y(t') = F_y(0) + O(B_x)^2 and writing δ F_y^2 = ⟨ F_y^2 ⟩ - ⟨ F_y ⟩^2, we find (t) =cosΘ (t) (0) + sinΘ (t) (0) + γ⟨ F_y(0) ⟩∫_0^t dt' F_y(t') B_x(t') cos[ Θ(t) - Θ(t') ]+ O(B_x)^2 + O(γ t B_x δ F_y)^2.The term O(B_x)^2 describes contributions of higher order in H', and can be neglected in the interesting scenario of weak signals. For our initial state of all atoms pumped into the y direction, δ F_y ≪ F_y, making the last term negligible also.Dropping the higher other terms and choosing the probing times such that Θ(t_i)=nπ simplifies Eq. (<ref>) further, to (t_n)=(-1)^n ((0) + γ⟨ F_y(0) ⟩∫_0^t_n dt'(t') cosΘ(t'))§.§ Field sensitivityThe accumulated signal during the hold timeis obtained by integration of Eq. (<ref>), where for the case of ramped field, B_x(t') = a_cos(ω_0 t'+κ t'^2), and Θ(t) = ω_L t + κ_L t^2, where κ_L =(ω_f-ω_i)/Δ t describes the change of the Larmor frequency caused by the ramp, applied for a time Δ t. The radio-frequency case is included has a special case when κ=κ_L=0.The solution to Eq. (<ref>) becomesF_z()= (0) - γ⟨(0) ⟩×∫_0^ dt' a_cos(ω_0 t'+κ t'^2) cos(ω_L t' + κ_L t'^2) = (0)+ γ a_⟨(0) ⟩ I(ω_L,ω_0,κ_L,κ, )Evaluating the integral on resonance, i.e. with ω_0 = ω_L, and κ_L=κ. In cases of interest κ≪ω_0, giving the leading-order in I(ω_L,ω_0,κ_L,κ, )≃/2, with an error smaller than 3% for our parameters.The sensitivity of a coherent spin state to thedrive amplitude a_ during a timeisδ B_^CSS = Δϕ/| d⟨ϕ⟩ /da_|= Δϕ/g_1⟨(0) ⟩1/γ I(ω_L,ω_0,κ_L,κ,)≃Δϕ/g_1⟨(0) ⟩2/γwhere ⟨(0) ⟩=.For the spin squeezed state δ B_^SSS ≃Δϕ_cond/g_1η⟨(0) ⟩2/γwhere η is the coherence loss includingand the first measurement and Δϕ_cond^2=var(Φ_2|Φ_1) is the conditional variance between the Φ_2 and Φ_1 optical signals, including the atomic noise and read-out noise. §.§ Statistics of probing inhomogeneously-coupled atomsWe consider the statistics of the Faraday rotation measurements of an ensemble ofatoms, described by individual spin operators f_i. To define the SQL, we consider the atomic ensemble to be in a coherent spin state, with independent individual spins and fully polarized with ≃. When the spatial structure of the probe beam is taken into account, the Faraday rotation is described by the input-output relation for the Stokes component ^(out) = ^(in) + ^(in)∑_i=1^ g( x_i) ^(i)where g( x_i) is the coupling strength for the i-th atom, proportional to the intensity at the location x_i of the atom. ^(in) has zero mean and variance /2, wheresi the mean photon number in the probe pulse .The atomic spin is polarized alongand orthogonal to the measureddirection. The rotation angle ϕ = ^(out)/^(in) has statistics⟨ϕ⟩ = ⟨⟩∑_i=1^< g( x_i) >_ x_i≡ ⟨⟩⟨⟩ g_1var(ϕ)= var(ϕ_0) + var() < ∑_i=1^ g^2( x_i) >_, x_i≡ var(ϕ_0) + var() ⟨⟩g̃_1^2where ϕ_0 is the polarization angle of the input light, subject to shot-noise fluctuations and assumed independent of , and the angle brackets indicate an average over the number and positions of the atoms. For known ⟨⟩ and var(), measurements of ⟨ϕ⟩ and var(ϕ) versusgive the calibration factors g_1 and g̃_1^2. §.§.§ Calibration of the g_1 factorWe calibrate ⟨ϕ⟩, the measured rotation angle of the dispersive atom number measurement (DAMN) <cit.> against an independent estimate ofmade via absorption imaging. The results are shown in Fig. <ref> (a).For this experiment, the atoms are prepared in an -state with a single 50 duration circularly polarized optical pumping pulse on resonance with the f=1→ f'=1 transition of the D_2 line and propagating along the trap axis with an bias field =180mG applied to fix the atomic polarization, and then probed with the Faraday probe.The number of atoms is estimated using absorption imaging in the same way as explained in Volume estimation. We calculate the resonant interaction cross-section and take into account the finite observable optical depth. The statistical error in the absorption imaging is <3%, including imaging noise and shot-to-shot trap loading variation.§.§.§ Calibration of the g̃_1^2 factor To measure g̃^2_1 we use the the stroboscopic QND data. The atoms are prepared into an -polarized state and then probed stroboscopically at twice the Larmor period, in such a way that the measured variable is ±, evading back-action effects. If ϕ_n is the measured Faraday rotation angle for pulse n, and ϕ_0^(n) is the corresponding input angle, we can define the pulse-train-averaged rotation signal as ϕ ≡ 1/∑_n=1^ (-1)^n+1ϕ_nwith variancevar(ϕ)≡ var( ϕ_0 ) + g̃_1^2 ∑_n=1^var(F_z,n)where ϕ_0 = 1∑_n=1^ϕ_0^(n), with zero mean and variance var(ϕ_0) = ()^-1, and F_z,n is the value ofat the time of the n-th probe pulse. We compute the resulting evolution of the state using the covariance matrix methods described in <cit.> and presented in the following section resulting in var( g_1/∑_n=1^F_z,n) =g̃_1^2 1/2αcorresponding to the linear term inused in Eq. (4) of the main text to determine the atomic quantum noise, i.e.,var (Φ) =var(Φ_0) + g̃_1^2 1/2α + a_2 ^2,where the correction factor α=0.96 accounts for decoherence and noise introduced into the atomic state due to off-resonant probe scattering during the QND measurement <cit.>.6 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Koschorreck et al.(2010)Koschorreck, Napolitano, Dubost, andMitchell]KoschorreckPRL2010a author author M. Koschorreck, author M. Napolitano, author B. Dubost,and author M. W. Mitchell,10.1103/PhysRevLett.104.093602 journal journal Phys. Rev. Lett. volume 104,pages 093602 (year 2010)NoStop [Behbood et al.(2013)Behbood, Martin Ciurana, Colangelo, Napolitano, Mitchell, and Sewell]BehboodAPL2013 author author N. Behbood, author F. Martin Ciurana, author G. Colangelo, author M. Napolitano, author M. W. Mitchell,and author R. J. Sewell, http://dx.doi.org/10.1063/1.4803684 journal journal Applied Physics Letters volume 102, eid 173504 (year 2013),http://dx.doi.org/10.1063/1.4803684NoStop [Pake and Purcell(1948)]PakePR1948 author author G. E. Pake and author E. M. Purcell, 10.1103/PhysRev.74.1184 journal journal Phys. Rev. volume 74,pages 1184 (year 1948)NoStop [Kubasik et al.(2009)Kubasik, Koschorreck, Napolitano, de Echaniz, Crepaz, Eschner, Polzik, and Mitchell]KubasikPRA2009 author author M. Kubasik, author M. Koschorreck, author M. Napolitano, author S. R. de Echaniz, author H. Crepaz, author J. Eschner, author E. S. Polzik,and author M. W. Mitchell, 10.1103/PhysRevA.79.043815 journal journal Phys. Rev. A volume 79, pages 043815 (year 2009)NoStop [Sakurai(1994)]Sakurai author author J. J. Sakurai, @nooptitle Modern Quantum Mechanics (publisher Addison-Wesley Publishing Company,year 1994)NoStop [Colangelo et al.(2013)Colangelo, Sewell, Behbood, Ciurana, Triginer, and Mitchell]ColangeloNJP2013 author author G. Colangelo, author R. J. Sewell, author N. Behbood, author F. M. Ciurana, author G. Triginer,and author M. W. Mitchell, http://stacks.iop.org/1367-2630/15/i=10/a=103007 journal journal New Journal of Physics volume 15, pages 103007 (year 2013)NoStop

http://arxiv.org/abs/1702.08294v1

Strongly Interacting Phases of Metallic Wires in Strong Magnetic Field Xiao-Liang Qi Accepted .Received ; in original form====================================================================== We design controllers from formal specifications for positive discrete-time monotone systems that are subject to bounded disturbances. Such systems are widely used to model the dynamics of transportation and biological networks. The specifications are described using signal temporal logic (STL), which can express a broad range of temporal properties. We formulate the problem as a mixed-integer linear program (MILP) and show that under the assumptions made in this paper, which are not restrictive for traffic applications, the existence of open-loop control policies is sufficient and almost necessary to ensure the satisfaction of STL formulas. We establish a relation between satisfaction of STL formulas in infinite time and set-invariance theories and provide an efficient method to compute robust control invariant sets in high dimensions. We also develop a robust model predictive framework to plan controls optimally while ensuring the satisfaction of the specification. Illustrative examples and a traffic management case study are included.Formal Synthesis and Verification, Monotone Systems, Transportation Networks.§ INTRODUCTIONIn recent years, there has been a growing interest in using formal methods for specification, verification,and synthesis in control theory. Temporal logics <cit.> provide a rich, expressive framework for describing a broad range of properties such as safety, liveness, and reactivity. In formal synthesis, the goal is to control a dynamical system from such a specification. For example, in an urban traffic network, a synthesis problem can be to generate traffic light control policies that ensure gridlock avoidance and fast enough traffic through a certain road, for all times. Control synthesis for linear and piecewise affine systems from linear temporal logic (LTL) specifications was studied in <cit.>. The automata-based approach used in these works requires constructing finite abstractions that (bi)simulate the original system. Approximate finite bisimulation quotients for nonlinear systems were investigated in <cit.>. The main limitations of finite abstraction approaches are the large computational burden of discretization in high dimensions and conservativeness when exact bisimulations are impossible or difficult to construct. As an alternative approach, LTL optimization-based control of mixed-logical dynamical (MLD) systems <cit.> using mixed-integer programs was introduced in <cit.>, and was recently extended to model predictive control (MPC) from signal temporal logic (STL) specifications in <cit.>. However, these approaches are unable to guarantee infinite-time safety and the results are fragile in the presence of non-deterministic disturbances.In some applications, the structural properties of the system and the specification can be exploited to consider alternative approaches to formal control synthesis. We are interested in systems in which the evolution of the state exhibits a type of order preserving law known as monotonicity, which is common in models of transportation, biological, and economic systems <cit.>. Such systems are also positive in the sense that the state components are always non-negative. Control of positive systems have been widely studied in the literature <cit.>. Positive linear systems are always monotone <cit.>. In this paper, we study optimal STL control of discrete-time positive monotone systems (i.e., systems with state partial order on the positive orthant) with bounded disturbances. STL allows designating time intervals for temporal operators, which makes it suitable for describing requirements with deadlines. Moreover, STL is equipped with quantitative semantics, which provides a measure to quantify how strongly the specification is satisfied/violated. The quantitative semantics of STL can also be used as costfor maximization in an optimal control setting. The STL specifications in this paper are restricted to a particular form that favors smaller values for the state components. We assume that there exists a maximal disturbance element that characterizes a type of upper-bound for the evolution of the system. These assumptions are specifically motivated by the dynamics of traffic networks, where the disturbances represent the volume of exogenous vehicles entering the network and the maximal disturbance characterizes the rush hour exogenous flow.Our optimal control study is focused on STL formulae with infinite-time safety/persistence properties, which is relevant to optimal and correct traffic control in the sense that the vehicular flow is always free of congestion while the associated delay is minimized. The key contributions of this paper are as follows. First, for finite-time semantics, we prove that the existence of open-loop control policies is necessary and sufficient for maintaining STL correctness.For the correctness of infinite-time semantics, we show that the existence of open-loop control sequences is sufficient and almost necessary, in a sense that is made clear in the paper. Implementing open-loop control policies is very simple since online state measurements are not required, which can prove useful in applications where the state is difficult to access. We use a robust MPC approach to optimal control. The main contribution of our MPC framework is guaranteed recursive feasibility, a property that was not established in prior STL MPC works <cit.>. We show via a case study that our method is applicable to systems with relatively high dimensions. This remainder of the paper is organized as follows. We introduce the necessary notation and background on STL in Sec. <ref>. The problems are formulated in Sec. <ref>. The technical details for control synthesis from finite and infinite-time specifications are given in Sec. <ref> and Sec. <ref>, respectively. The robust MPC framework is explained in Sec. <ref>. Finally, we introduce a traffic network model and explain its monotonicity properties in Sec. <ref>, where a case study is also presented.§.§ Related WorkThis paper is an extension of the conference version <cit.>, where we studied safety control of positive monotone systems. Here, we significantly enrich the range of specifications to STL, provide complete proofs, and include optimal control.Monotone dynamical systems have been extensively investigatedin the mathematics literature <cit.>. Early studies mainly focused on stability properties and characterization of limit sets for autonomous, deterministic continuous-time systems <cit.>. The results do not generallyhold for discrete-time systems, as discussed in <cit.>. In particular, attractive periodic orbits are proven to be non-existent for continuous-time autonomous systems <cit.>, but may exist for discrete-time autonomous systems. Here we present a similar result for controlled systems, where we show that a type of attractive periodic orbit exists for certain control policies.Angeli and Sontag <cit.> extended the notion of monotonicity to deterministic continuous-time control systems and provided results on interconnections of these systems. However, they assumed monotonicity with respect to both state and controls. We do not require monotonicity with respect to controls, which enables us to consider a broader class of systems. In particular, we do not require controls to belong to a partially ordered set. Switching policies for exponential stabilization of switched positive linear systems were studied in<cit.>. Stabilization is closely related to set-invariance, which is thoroughly studied in this paper. Apart from richer specifications, we can handle more complex systems. We consider hybrid systems in which the mode is either determined directly by the control input or indirectly by the state (e.g., signalized traffic networks).Recently, there has been some interest on formal verification and synthesis for monotone systems. Safety control of cooperative systems was investigated in<cit.>. However, these work, like <cit.>, assumed monotonicity with respect to the control inputs as well. Computational benefits gained from monotonicity for reachability analysis of hybrid systems were highlighted in <cit.>. More recently, the authors in <cit.> provided an efficient method to compute finite abstractions for mixed-monotone systems (a more general class than monotone systems). The authors in <cit.> exploited monotonicity to compute finite-state abstractions that are used for compositional LTL control.While the approaches in <cit.> can consider systems and specifications beyond the assumptions in this paper, they still require state-space discretization, which is a severe limitation in high dimensions. Moreover, they are conservative since the finite abstractions are often not bisimilar with the original system - whereas our approach provides a notion of (almost) completeness. Finally, as opposed to the all mentioned works, our framework is amenable to optimal temporal logic control.§ PRELIMINARIES§.§ Notation For two integers a,b, we use (a,b) to denote the remainder of division of a by b. Given a set 𝒮 and a positive integer K, we use the shorthand notation 𝒮^K for ∏_i=1^K 𝒮. A signal is defined as an infinite sequence s=s_0s_1⋯, where s_k ∈𝒮, k ∈ℕ. Given s_1,s_2,⋯,s_K∈𝒮, the repetitive infinite-sequence s_1s_2 ⋯ s_K s_1 s_2 ⋯ s_K⋯ is denoted by (s_1 s_2 ⋯ s_K)^ω. The set of all signals that can be generated from 𝒮 is denoted by 𝒮^ω. We use s[k]=s_ks_k+1⋯ and s[k_1:k_2]=s_k_1s_k_1+1⋯s_k_2,k_1 < k_2, to denote specific portions of s. A real signal is r=r_0r_1r_2⋯, where r_k ∈ℝ^n, ∀ k ∈ℕ. A vector of all ones in ℝ^n is denoted by 1_n. We use the notation 1_n[0:K]:= 1_n ⋯ 1_n, where 1_n is repeated K+1 times.The positive closed orthant of the n-dimensional Euclidian space is denoted by ℝ^n_+ := { x ∈ℝ^n| x_[i]≥ 0, i=1,⋯,n }, where x=(x_[1],x_[2],⋯,x_[n])^T. For a,b ∈ℝ^n, the non-strict partial order relation ≼ is defined as: a ≼ b ⇔ b-a ∈ℝ_+^n.[<cit.>] A set 𝒳⊂ℝ^n_+ is a lower-set if ∀ x ∈𝒳,L(x) ⊆𝒳, where L(x):= { y ∈ℝ_+^n | y ≼ x }.It is straightforward to verify that if 𝒳_1,𝒳_2 are lower-sets, then 𝒳_1 ∪𝒳_2 and 𝒳_1 ∩𝒳_2 are also lower-sets.We extend the usage of notation ≼ to equal-length real signals. For two real signals r, r, we denote r'[t_1':t_2'] ≼r[t_1:t_2], t_2-t_1=t'_2-t'_1, if r'_t'_1+k≼ r_t_1+k, k=0,1,⋯,t_2-t_1.Moreover, if r, r' ∈ (ℝ_+^n)^ω, we are also allowed to writer'[t_1':t_2'] ∈ L(r[t_1:t_2]).§.§ Signal Temporal Logic (STL)In this paper, STL <cit.> formulas are defined over discrete-time real signals.The syntax of negation-free STL is:φ:=  π  |  φ_1 ∧φ_2   | φ_1 ∨φ_2  | φ_1U_Iφ_2  | F_I φ | G_I φ, where π=(p(r) ≤ c) is a predicate on r∈ℝ^n, p: ℝ^n →ℝ, c ∈ℝ; ∧ and ∨ are Boolean connectives for conjunction and disjunction, respectively; U_I, F_I, G_I are the timed until, eventually and always operators, respectively, and I=[t_1,t_2] is a time interval,t_1,t_2 ∈ℕ∪{∞}, t_2≥ t_1. When t_1=t_2, we use the shorthand notation {t_1}:=[t_1,t_1]. Exclusion of negation does not restrict expressivity of temporal properties. It can be easily shown that any temporal logic formula can be brought into negation normal form (where all negation operators apply to the predicates) <cit.>.We deliberately omit negation from STL syntax for laying out properties that are later exploited in the paper. For simplicity, in the rest of the paper, we will refer to negation-free STL simply as STL.The semantics of STL is inductively defined as:[r[t] π ⇔ p(r_t) ≤ c,; r[t] φ_1 ∨φ_2 ⇔r[t] φ_1  ∨ r[t]φ_2,;r[t]φ_1 ∧φ_2 ⇔r[t] φ_1  ∧ r[t]φ_2,;r[t]φ_1  U_I φ_2 ⇔ ∃ t^'∈ t+I s.t r[t^'] φ_2; ∧ ∀ t^''∈ [t,t^'], r[t^''] φ_1,;r[t] F_I φ ⇔∃t^'∈ t+I  s.t. r[t'] φ,; r[t]G_I φ ⇔∀t^'∈ t+I  s.t. r[t'] φ, ] whereis read as satisfies. The language of φ is the set of all signals such that r[0] φ. The horizon of an STL formula φ, denoted by h^φ, is defined as the time required to decide the satisfaction of φ, which is recursively computed as <cit.>:[ h^π= 0,; h^φ_1 ∧φ_2=h^φ_1 ∨φ_2=max(h^φ_1,h^φ_2),; h^ F_[t_1,t_2]φ=h^ G_[t_1,t_2]φ= t_2+h^φ,; h^φ_1U_[t_1,t_2]φ_2=t_2+max(h^φ_1,h^φ_2).;] An STL formula φ is bounded if h^φ<∞.[<cit.>] A safety STL formula is an STL formula in which all “until" and “eventually" intervals are bounded.The satisfaction of φ by r[t] is decided only by r[t:t+h^φ] and the rest of the signal values are irrelevant. Therefore, instead of r[t]φ, we occasionally write r[t:t+h^φ]φ with the same meaning.The STL robustness score ρ(r,φ,t) ∈ℝ is a measure indicating how strongly φ is satisfied by r[t], which is recursively computed as <cit.>:[ ρ(r,π,t)= c-p(r_t) ,;ρ(r,φ_1 ∨φ_2,t)= max(ρ(r,φ_1,t),ρ(r,φ_2,t) ),;ρ(r,φ_1 ∧φ_2,t)= min(ρ(r,φ_1,t),ρ(r,φ_2,t) ),;ρ(r,φ_1  U_I φ_2,t)= t^'∈ t+Imax (min (ρ(r,φ_2,t'),; t”∈ [t,t']minρ(r,φ_1,t”))),;ρ(r, F_I φ,t)= t' ∈ t+Imax ρ(r,φ,t'),;ρ(r, G_I φ,t)= t' ∈ t+Imin ρ(r,φ,t').;] Positive (respectively, negative) robustness indicates satisfaction (respectively, violation) of the formula.Consider signal r∈ℝ^ω, where r_k=k, k ∈ℕ, and π=(r^2 ≤ 10). We have ρ(r, G_[0,3]π,0)=min(10-0,10-1,10-4,10-9)=1 (satisfaction) and ρ(r, F_[4,6]π,0)=max(10-16,10-25,10-36)=-6 (violation). There are minor differences between the original STL introduced in <cit.> and the one used in this paper. In <cit.>, STL wasdeveloped as an extension of metric interval temporal logic (MITL) <cit.> for real-valued continuous-time signals. Here, without any loss of generality, we apply STL to discrete-time signals. Our STL is based on metric temporal logic (MTL) (similar to <cit.>). Thus, we allow the intervals of temporal operators to be singletons (punctual) or unbounded. It is worth to note that any STL formula in this paper can be translated into an LTL formula by appropriately replacing the time intervals of temporal operators with LTL “next" operator. However, the LTL representation of STL formulas can be very inefficient. We prefer STL for convenience of specifying requirements for systems with real-valued states. We also exploit the STL quantitative semantics. § PROBLEM STATEMENT AND APPROACHWe consider discrete-time systems of the following form:x_t+1=f(x_t,u_t,w_t),where x_t ∈𝒳 is the state, 𝒳⊂ℝ_+^n, u_t ∈𝒰 is the control input, 𝒰= ℝ^m_r×{0,1}^m_b, and w_t ∈𝒲 is the disturbance (adversarial input) at time t, t∈ℕ, 𝒲= ℝ^q_r×{0,1}^q_b. The sets 𝒰 and 𝒲 may include real and binary values. For instance, the set of controls in the traffic model developed in Sec. <ref> includes binary values for decisions on traffic lights and real values for ramp meters. These types of systems are positive as all state components are non-negative. We also assume that 𝒳 is bounded. System (<ref>) is monotone (with partial order on ℝ^n_+) if for all x,x'∈𝒳,x' ≼ x, we have f(x',u,w) ≼ f(x,u,w), ∀ u ∈𝒰, ∀ w ∈𝒲. The systems considered in this paper are positive and monotone with partial order on ℝ^n_+. For the remainder of the paper, we simply refer to systems in Definition <ref> as monotone [The term cooperative in dynamics systems theory is used specifically to refer to systems that are monotone with partial order defined on the positive orthant. We avoid using this term here as it might generate confusion with the similar terminology used for multi-agent control systems.]. Although the results of this paper are valid for any general f:𝒳×𝒰×𝒲→𝒳, we focus on systems that can be written in the form of mixed-logical dynamical (MLD) systems <cit.>, which are defined in Sec. <ref>. It is well known that a wide range of systems involving discontinuities (hybrid systems), such as piecewise affine systems, can be transformed into MLDs <cit.>.There exist w^* ∈𝒲 such that∀ x∈𝒳, ∀ u∈𝒰,  f(x,u,w) ≼ f(x,u,w^*), ∀ w∈𝒲. We denote f(x,u,w^*) by f^*(x,u) and refer to f^* as the maximal system. As it will be further explained in this paper, the behavior of monotonesystem (<ref>) is mainly characterized by its maximal f^*. Assumption <ref> is restrictive but holds for many compartmental systems where the disturbances are additive and the components are independent. Therefore, the maximal system corresponds to the situation that every component takes its most extreme value.We also note that if Assumption <ref> is removed, overestimating f by some f^* such that f(x,u,w) ≼ f^*(x,u), ∀ w ∈𝒲, is always possible for a bounded f. By overestimating f the control synthesis methods of this paper remain correct, but become conservative. We describe the desired system behavior using specifications written as STL formulas over a finite set of predicates. We assume that each predicate π is in the following form:π=( a_π^T x ≤ b_π ), where a_π∈ℝ^n_+, b_π∈ℝ_+. It is straightforward to verify that the closed half-space defined by (<ref>) is a lower-set in ℝ^n_+. By restricting the predicates into the form (<ref>), we ensure that a predicate remains true if the values of state components are decreased (Note that this is true for any lower set. We require linearity in order to decrease the computational complexity.). This restriction is motivated by monotonicity. For example, in a traffic network, the state is the vector representation of vehicular densities in different segments of the network. The satisfaction of a “sensible" traffic specification has to be preserved if the vehicular densities are not increased all over the network. Otherwise, the specification encourages large densities and congestion.A control policy μ:=⋃_t ∈ℕμ_t is a set of functions μ_t:𝒳^t+1→𝒰, whereu_t=μ_t(x_0,x_1,⋯,x_t).An open-loop control policy takes the simpler form u_t=μ_t(x_0), i.e., the decision on the sequence of control inputs is made using only the initial state x_0. On the other hand, in a (history dependent) feedback control policy, u_t=μ_t(x_0,x_1,⋯,x_t), the controller implementation requires real-time access to the state and its history.An infinite sequence of admissible disturbances is w=w_0w_1⋯, where w_k ∈𝒲, k ∈ℕ. Following the notation introduced in Sec. <ref>, the set of all infinite-length sequences of admissible disturbances is denoted by 𝒲^ω. Given an initial condition x_0, a control policy μ and w∈𝒲^ω, the run of the system is defined as the following signal: x=x(x_0,μ,w):=x_0 x_1 x_2 ⋯,where x_t+1=f(x_t,u_t,w_t), ∀ t ∈ℕ.Now we formulate the problems studied in this paper. In all problems, we assume a monotone system (<ref>) is given, Assumption <ref> holds, and all the predicates are in the form of (<ref>). [Bounded STL Control]Given a bounded STL formula φ, find a set of initial conditions 𝒳_0 ⊂𝒳 and a control policy μ such that x(x_0,μ,w)[0] φ, ∀w∈𝒲^ω, ∀ x_0 ∈𝒳_0. As mentioned in the previous section, the satisfaction of φ solely depends on x[0:h^φ], where h^φ is obtained from (<ref>). The horizon h^φ can be viewed as the time when the specification ends. In many engineering applications, the system is required to uphold certain behaviors for all times. Therefore, guaranteeing infinite-time safety properties is important.We formulate bounded-global STL formulas in the form of φ_b ∧ G_[Δ,∞]φ_g,where φ_b, φ_g are bounded STL formulas, G_[Δ,∞] stands for unbounded temporal “always"- as defined in Sec. <ref>, and Δ≥ h^φ_b is a positive integer. Formula (<ref>) states that first, φ_b is satisfied by the signal from time 0 to Δ, and, afterwards, φ_g holds for all times. [Bounded-global STL Control]Given bounded STL formulas φ_b,φ_g, Δ∈ [h^φ_b,∞), find a set of initial conditions 𝒳_0 ⊂𝒳 and a control policy μ such thatx(x_0,μ,w)[0] φ_b ∧G_[Δ,∞]φ_g, ∀w∈𝒲^ω, ∀ x_0 ∈𝒳_0. As a special case, we allow φ_b to be logical truth so Problem <ref> reduces to global STL control problem of satisfying G_[Δ,∞]φ_g. Note that if φ_g is replaced by logical truth, Problem <ref> reduces toProblem <ref>. We have distinguished Problem <ref> and Problem <ref> as we use different approaches to solve them. It can be shown that (see Appendix) a large subset of safety STL formulas - as in Definition <ref> - can be written as⋁_i=1^n_ϕϕ_i, where each ϕ_i, i=1,⋯,n_ϕ, is a bounded-global formula. Therefore, the framework for solutions to Problem <ref> can also be used for safety STL control as it leads to n_ϕ instances of Problem <ref>, where a solution to any of the instances is also a solution to the original safety STL control problem. The drawback to this approach is that n_ϕ can be very large.We avoid separate problem formulations for STL formulas containing unbounded “eventually" or “until" operators as their unbounded intervals can be safely under-approximated by bounded intervals. However, bounded under-approximation is not sound for the unbounded “always" operator. A safety formula can be satisfied (respectively, violated) with infinite-length (respectively, finite-length) signals <cit.>.In the presence of disturbances, feedback controllers obviously outperform open-loop controllers. We show that the existence of open-loop control policies for guaranteeing the STL correctness of monotone systems in Problem 1 (respectively, Problem 2) is sufficient and (respectively, almost) necessary. The online knowledge of state is not necessary for STL correctness. But it can be exploited for planning controls optimally.While our framework can accommodate optimal control versions of Problem <ref> and Problem <ref>, the focus of this paper is on robust optimal control problem for global STL formulas - of form G_[0,∞)φ, where φ is a bounded formula. These type of problems are of practical interest for optimal traffic management (as discussed in Sec. <ref>). We use a model predictive control (MPC) approach, which is a popular, powerful approach to optimal control of constrained systems. Given a planning horizon of length H [The MPC horizon H should not be confused with the STL horizon h^φ. ], a sequence of control actions starting from time t is denoted by u_t^H:=u_0|t u_1|t⋯u_H-1|t. Given u^H_t and x_t, we denote the predicted H-step system response byx_t^H(x_t,u_t^H,w_t^H):= x_1|t x_2|t⋯ x_H|t, wherex_k+1|t=f(x_k|t,u_k|t,w_k|t), k=0,1,⋯,H-1,x_0|t=x_t and w_t^H:=w_0|t w_1|t⋯ w_H-1|t. At each time, u^H_t is found such that it optimizes a cost function J(x_t^H,u^H_t), J:𝒳^H ×𝒰^H →ℝ, subject to system constraints. When u_t^H is computed, only the first control action u_0|t is applied to the system and given the next state, the optimization problem is resolved for u_t+1^H. Thus, the implementation is closed-loop. [Robust STL MPC] Given a bounded STL formula φ, an initial condition x_0, a planning horizon H and a cost function J(x_t^H,u^H_t), find a control policy such thatu_t=μ(x_0,⋯,x_t)=u^opt_0|t, where u_t^H,opt:=u^opt_0|t⋯ u^opt_H-1|t, and u^H,opt is the following minimizer:[_u_t^H w^H_tmax J(x_t^H(x_t,u_t^H,w_t^H), u_t^H ),;s.t. x(x_0,μ,w)[0] G_[0,∞]φ, ∀w∈𝒲^ω,; x_k+1=f(x_k,u_k,w_k), ∀ k ∈ℕ.; ] The primary challenge of robust STL MPC is guaranteeing the satisfaction of the global STL formula while the controls are planned in a receding horizon manner (see the constraints in (<ref>)). Our approach takes the advantage of the results from Problem <ref> to design appropriate terminal sets for the MPC algorithm such that the generated runs are guaranteed to satisfy the global STL specification while the online control decisions are computed (sub)optimally. Due to the temporal logic constraints, our MPC setup differs from the conventional one. The details are explained in Sec. <ref>. For computational purposes, we assume that J is a piecewise affine function of the state and controls. Moreover, the cost functions in our applications are non-decreasing with respect to the state in the sense that x'_k|t≼ x_k|t, k=1,2,⋯,H ⇒ J(x_t^' H,u^H_t) ≼ J(x_t^H,u^H_t), ∀ u^H_t ∈𝒰^H. As it will become clear later in the paper, we will exploit this property to simplify the worst-case optimization problem in (<ref>) to an optimization problem for the maximal system. As mentioned earlier, a natural objective is maximizing STL robustness score. It follows from the linearity of the predicates in (<ref>) and max and min operators in (<ref>) that STL robustness score is a piecewise affine function of finite-length signals. We can also consider optimizing a weighted combination of STL robustness score and a given cost function. We use this cost formulation for traffic application in Sec. <ref>.§ FINITE HORIZON SEMANTICS In this section, we explain the solution to Problem <ref>. First, we exploit monotonicity to characterize the properties of the solutions. Next, we explain how to synthesize controls using a mixed integer linear programming (MILP) solver.Consider runs x and x' and an STL formula φ. If for some t,t', we have x'[t':t'+h^φ] ≼x[t:t+h^φ], then x[t] φ implies x'[t'] φ.Since all predicates denote lower-sets in the form of (<ref>), we have x'_t'≼ x_t⇒ a_π^T x'_t'≤ a_π^T x_t, x[t] π⇒x'[t] π. Thus, all predicates that were true by the valuations in x remain true for x'. The negation-free semantics in (<ref>) implies that without falsifying any predicate, a formula can not be falsified. Therefore, x[t] φ implies x'[t'] φ The largest set of admissible initial conditions is defined as:𝒳_0^max:={ x_0 ∈𝒳 |∃μ s.t. x(x_0,μ,w) φ, ∀w∈𝒲^ω}.The set 𝒳_0^max is a union of polyhedra. Finding the half-space representation of all polyhedral sets in 𝒳_0^max may not be possible for high dimensions. Therefore, we find a half-space representation for a subset of 𝒳_0^max. The following result states how to check whether x_0 ∈𝒳_0^max. We have x_0 ∈𝒳_0^max if and only if there exists an open-loop control sequenceu_0^ol,x_0 u_1^ol,x_0⋯ u_h^φ-1^ol,x_0 such that x^ol,x_0[0:h^φ] φ, where x^ol,x_0[0:h^φ]=x_0^ol,x_0x_1^ol⋯ x_h^φ^ol,x_0, and x_k+1^ol,x_0=f^*(x_k^ol,x_0,u_k^ol,x_0), k=0,⋯,h^φ-1, x_0^ol,x_0=x_0. (Necessity) Satisfaction of φ with w∈𝒲^ω requires at least one satisfying run for the maximal system, hence a corresponding control sequence exists. Denote it by u_0^ol,x_0 u_1^ol,x_0⋯,u_h^φ-1^ol,x_0. (Sufficiency) Consider any run generated by the original system x_k+1=f(x_k,u_k^ol,x_0,w_k). We prove that x_k ≼ x_k^ol,x_0 , k=0,1,⋯,h^φ, by induction over k. The base case x_0 ≼ x_0^ol,x_0 is trivial (x_0=x_0^ol,x_0). The inductive step is verified from monotonicity: x_k+1=f(x_k,u_k^ol,x_0,w_k) ≼ f^*(x_0^k,u_k^k) = x^ol,x_0_k+1. Therefore, x[0:h^φ] ≼x^ol,x_0[0:h^φ], ∀w[0:h^φ-1] ∈𝒲^h^φ. It follows from Lemma <ref> that x[0:h^φ] φ, ∀w[0:h^φ-1] ∈𝒲^h^φ. The set 𝒳_0^max is a lower-set. Consider any x_0' ∈ L(x_0), x_0 ∈𝒳_0^max. Let x'_k+1= f(x'_k,u_k^ol,x_0,w_k), k=0,1,⋯,h^φ-1. It follows from monotonicity that x_k' ≼ x_k^ol,x_0, k=0,1,⋯,h^φ, ∀w[0:h^φ-1] ∈𝒲^h^φ. By the virtue of Lemma <ref>, x'[0:h^φ] ≼x^x_0,ol[0:h^φ]. Therefore, we have ∀ x_0 ∈𝒳_0^max, x'_0 ∈ L(x_0) ⇒ x'_0 ∈𝒳_0^max, which indicates 𝒳_0^max is a lower-set.If x_0 ∈𝒳_0^max and μ^ol is the following open-loop control policyμ^ol_t(x_0)=u^ol,x_0_t, t=0,1,⋯,h^φ-1, then x(x'_0,μ,w)[0:h^φ] φ, ∀w∈𝒲^h^φ, ∀ x'_0 ∈ L(x_0). Follows from the proof of Corollary <ref>.Now that we have established the properties of the solutions to Problem <ref>, we explain how to compute the admissible initial conditions and their corresponding open-loop control sequences. The approach is based on formulating the conditions in Theorem <ref> as a set of constraints that can be incorporated into a feasibility solver. We convert all the constraints into a set of mixed-integer linear constraints and use off-the-shelf MILP solvers to check for feasibility. Converting logical properties into mixed-integer constraints is a common procedure which was employed for MLD systems in <cit.>. The authors in <cit.> and <cit.> extended this technique to a framework for time bounded model checking of temporal logic formulas. Avariation of this method is explained here.First, the STL formula is recursively translated into a set of mixed-integer constraints. For each predicate π=(a_π^T x ≤ b_π), as in (<ref>), we define a binary variable z^π_k ∈{0,1} such that 1 (respectively, 0) stands for true (respectively, false). The relation betweenz^π, robustness ρ, and x is encoded as:a_π^Tx - M (1-z^π) + ρ≤ b_π, a_π^Tx + M z^π + ρ≥ b_π. The constant M is a sufficiently large number such that M≥max{a_π^T K, b_π}, where K ∈ℝ^n_+ is the upper bound for the state values, x_k ≼ K, k =0,1,⋯,h^φ.In practice, M is chosen sufficiently large such that the constraint x ≼ K is never active. Note that the largest value of ρ for which z^π=1 is b_π-a^T_π x, which is equal to the robustness of π. Now we encode the truth table relations. For instance, we desire to capture1 ∧ 0=0 and 1 ∨ 0=1 using mixed-integer linear equations. Disjunction and conjunction connectives are encoded as the followingconstraints:z= ⋀_i=1^n_z z_i  ⇒ z ≤ z_i, i=1,⋯,n_z, z= ⋁_i=1^n_z z_i  ⇒ z ≤∑_i=1^n_z z_i, where z ∈ [0,1] is declared as a continuous variables. However, it only can take binary values as evident from (<ref>).Similarly, define z^φ_k ∈ [0,1] as the variable indicating whether x[k] φ. An STL formula is recursively translated as: [φ= ⋀_i=1^n_φφ_i⇒ z^φ_k= ⋀_i=1^n_φ z^φ_i_k,;φ= ⋁_i=1^n_φφ_i⇒ z^φ_k= ⋁_i=1^n_φ z^φ_i_k,;φ = G_I ψ⇒ z^φ_k = ⋀_k^'∈ I z^ψ_k^',;φ = F_I ψ⇒ z^φ_k = ⋁_k^'∈ I z^ψ_k^',; φ = ψ_1U_I ψ_2⇒ z^φ_k =⋁_k^'∈ I ( z^ψ_2_k'∧⋀_k”∈ [k,k^'] z^ψ_1_k”). ]Finally, we add the following constraints:z^φ_0=1, ρ≥ 0. The set of constraints in (<ref>),(<ref>),(<ref>),(<ref>) has the following properties:i) we have x[0] φ if the set of constraints is feasible;ii) we have x[0] φ if the set of constraints is infeasible;iii) the largest ρ such that the set of constraints, while “ρ≥ 0" is removed from (<ref>), is feasible is equal to ρ(x,φ,0).i) We provide the proof for (<ref>), as the case for more complex STL formulas are followed in a recursive manner from (<ref>). If z=1, we have from (<ref>) that z_i=1, i=1,⋯,n_z, which correctly encodes conjunctions.Similarly, z=1 in (<ref>) indicates that not all z_i,i=0,⋯,n_z can be zero, or, ∃ i ∈{1,⋯,n_z } such that z_i=1, which correctly encodes disjunctions. ii) Infeasibility can be recursively traced back into (<ref>). For both (<ref>) and (<ref>), if z=1 is infeasible, it indicates that z_i=0, i=1,⋯,n_z.iii) We also prove this statement for (<ref>) as it is the base of recursion for general STL formulas. Let z_i=(a^T_π_i x + ρ≤ b_π_i), i=1,⋯,n_z.Consider (<ref>) and the following optimization problem:[ρ^max= argmax ρ,; s.t. a^T_π_i x + ρ≤ b_π_i, i=1,⋯,n_z, ]where its solution is i=1,⋯, n_zmin(b_π_i-a^T_π_i x), which is identical to the quantitative semantics for conjunction (see (<ref>)). Similarly, consider (<ref>) and the following optimization problem: [ ρ^max= argmaxρ,;s.t. ∃ i ∈{1,⋯,n_z}, a^T_π_i x + ρ≤ b_π_i, ] where the solution is i=1,⋯, n_zmax(b_π_i-a^T_π_i x), which is identical to the quantitative semantics for disjunction. Our integer formulation for Boolean connectives slightly differs from the formulation in <cit.>, <cit.>, where lower bound constraints for the z's are required. For example, for translating z= ⋀_i=1^n_zz_i, it is required to add z≥∑_i=1^n_z z_i - n_z + 1 to impose a lower bound for z. However, these additional constraints become necessary only when the negation operator is present in the STL formula. Hence, they are removed in our formulation. This reduces the constraint redundancy and degeneracy of the problem. By doing so, we observed computation speed gains (up to reducing the computation time by 50%) in our case studies. Moreover, we encode quantitative semantics in a different way than <cit.>, where a separate STL robustness-based encoding is developed which introduces additional integers.Due to property “iii" in Proposition <ref>, our encoding does not require additional integers to capture robustness hence it is computationally more efficient. System (<ref>) is in MLD form <cit.> if written as:x_t+1 = A x_t + B_u u_t + B_w w_t + D_δδ_t + D_r r_t, E_δδ_t + E_r r_t ≼ E_x x_t+E_u u_t + E_w w_t + e, where δ_t ∈{0,1}^n_δ and r_t ∈ℝ^n_r are auxiliary variables and A,B_u,B_w,D_δ, D_r, E_δ, E_r, E_x, E_u, E_w, e are appropriately defined constant matrices such that (<ref>) is well-posed in the sense that given x_t,u_t,w_t, the feasible set for x_t+1 is a single pointequal to f(x_t,u_t,w_t). Introducing auxiliary variables and enforcing (<ref>) can capture nonlinear f <cit.>.The system equations are brought into mixed-integer linear constraints by transforming system (<ref>) into its MLD form.As mentioned earlier, any piecewise affine system can be transformed into an MLD. In the case studies of this paper, the construction of (<ref>) from a piecewise affine (<ref>) is not explained as the procedure is well documented in <cit.>.Finally, the set of constraints in Theorem <ref> can be cast as:{[x_0^ol,x_0=x_0, Initial condition;; x_k+1^ol,x_0=f^*(x_k^ol,x_0,u_k^ol,x_0),System constraints;; z^π_k = (a_π^T x_k^ol,x_0≤ b_π),Predicates;; z^φ_0=1, ρ≥ 0,STL satisfaction.;]. Checking the satisfaction of the set ofconstraints in (<ref>) can be formulated as a MILP feasibility problem, which is handled using powerful off-the-shelf solvers. For a fixed initial condition x_0, the feasibility of the MILP indicates whether x_0 ∈𝒳_o^max.An explicit representation of 𝒳_o^max requires variable elimination from (<ref>), which is computationally intractable for a large MILP.Alternatively, we can set x_0 as a free variable while maximizing a cost function (e.g. norm of x_0) such that a large L(x_0) is obtained. Another natural candidate is maximizing ρ(x^ol,x_0,φ,0). It is worth to note that by finding a set of distinct initial conditions and taking the union of all L(x_0), we are able to find a representation for an under-approximation of 𝒳_o^max. MILPs are NP-complete. The complexity of solving (<ref>) grows exponentially with respect to the number of binary variables and polynomially with respect to the number of continuos variables. The number of binary variables in our framework is 𝒪 ( h^φ(n_π+m_b+q_b+n_δ) ) - n_π is the number of predicates - and the number of continuous variables is 𝒪( h^φ(n+m_r+q_r+n_r) ). In other words, the exponential growth builds upon the intricacy of the specification and the number of modes demonstrated by the hybrid nature of the system. However, the complexity is polynomial with respect to the dimension of the state.Consider the following switched system:x^+=e^A_uτx + A_u^-1(I-e^-A_uτ)w,where x=(x_[1],x_[2])^T ∈ℝ_+^2, u∈𝒰 is the control input (switch), 𝒰={1,2}, andA_1= ( [11;1 -5 ] ) , A_2=( [ -81;12 ] ).The (additive) disturbance w is bounded to L(w^*), where w^*=(1.5,1)^T and τ=0.1. This system is the discrete-time version of ẋ=A_u x +w with sample time τ. Both matrices are Metzler (all off-diagonal terms are non-negative hence all the elements of its exponential are positive)and non-Hurwitz hence constant input results in unbounded trajectories. The system is desired to satisfy thefollowing STL formula:φ= ⋁_T=0^10(F_[0,T] p_1 ∧F_{T} p_2),where p_1= ((x_[1]≤ 1) ∧ (x_[2]≤ 5)) and p_2= ((x_[1]≤ 5) ∧ (x_[2]≤ 1)). In plain English, φ states that “within 10 time units, the trajectory visits the box characterized by p_1 first and then the box corresponding to p_2" (see Fig. <ref>). We transformed this system into its MLD form (<ref>). We formulated the constraints in (<ref>)as a MILP and set the cost function to maximize x_0 _∞ and used the Gurobi [] MILP solver. The solution was obtained in less than 0.05 seconds on a 3GHz Dual Core MacBook Pro.We obtained x_0=(2.82 2.82)^T and the following open-loop control sequence: 1 2 1 2 2 1 1 1 1 1. By applying this control sequence, we sampled a trajectory of the original systemf with values of w drawn from a uniform distribution over L(w^*). Both the trajectories of f and f^* satisfy the specification. The results are shown in Fig. <ref>. § INFINITE HORIZON SEMANTICSIn this section, we provide a solution to Problem <ref>. We show that the infinite-time property in (<ref>) can be guaranteed using repetitive control sequences. First, we consider global specifications and extend the results from our previous work <cit.> in Sec. <ref>. Next, we show how to find controls for bounded-global STL formulas (Problem <ref>) in Sec. <ref>. Solution completeness is discussed in Sec. <ref>. §.§ Global formulas: s-sequences and inductive invariance Consider the global specification G_[0,∞]φ, where φ is a bounded formula. We introduce some additional notation. Given a bounded STL formula φ over predicates in the form (<ref>), the language realization set (LRS) <cit.> is:ℒ^φ:={x_0x_1⋯ x_h^φ∈𝒳^h^φ | x_0x_1⋯ x_h^φφ}. The set ℒ^φ is a lower-set. For all x_0x_1⋯ x_h^φ∈ℒ^φ and x'_0x'_1⋯ x'_h^φ≼ x_0x_1⋯ x_h^φ, it follows from Lemma <ref> that x'_0x'_1⋯ x'_h^φφ. Thus, x'_0x'_1⋯ x'_h^φ∈ LRS(φ) hence LRS(φ) is a lower set. It follows from the semantics of global operator in (<ref>) that xG_[0,∞]φ is equivalent to x[t:t+h^φ] ∈ℒ^φ, ∀ t ∈ℕ. A set Ω_ℒ^φ⊆ℒ^φ is a robust control invariant (RCI) set if: [∀  x_0x_1⋯ x_h^φ∈Ω_ℒ^φ, ∃ u ∈𝒰,s.t.; x_1 x_2 ⋯ x_h^φf(x_h^φ,u,w)  ∈Ω_ℒ^φ, ∀ w ∈𝒲. ] Satisfaction of G_[0,∞]φ is accomplished by finding a RCI set in ℒ^φ. Note that unlike traditional definitions of RCI sets (e.g., <cit.>), where the set is defined in the state-space 𝒳, our RCI set is defined in an augmented form of the state-space 𝒳^h^φ. The language realization set can also be interpreted as the “safe" set in (h^φ+1)-length trajectory space. The maximal RCI set inside ℒ^φ provides a complete solution to the set-invariance problem. The computation of maximal RCI set requires implementing an iterative fixed-point algorithm which is computationally intensive for MLD systems and non-convex sets (see <cit.> for discussion). We use monotonicity to provide an alternative approach. The following result is a more general version of the one in <cit.>. Given a bounded formula φ, if there exists x^s[0:h^φ] ∈ℒ^φ, and a sequence of controls: u^s_0,⋯,u^s_T-1 - where T is a positive integer determining the length of the sequence - such that: * x^s[k:k+h^φ] ∈ℒ^φ, k=0,1,⋯,T, where x^s_h^φ+k+1=f^*(x^s_h^φ+k,u^s_k),* x^s[T:T+h^φ] ≼x^s[0:h^φ],then the following set is a RCI set in ℒ^φ:Ω_ℒ^φ:= ⋃_k=0^T-1 L(x^s[k:k+h^φ]). For any x'_0x'_1⋯ x'_h^φ∈Ω_ℒ^φ, there exists i ∈{0,1⋯,T-1} such that x'_0x'_1⋯ x'_h^φ∈ L(x^s[i:i+h^φ]).On one hand, we have x_i+1^s⋯ x_i+h^φ^sf^*(x_i+h^φ^s,u_i^s) ∈Ω_ℒ^φ. On the other hand, we have x'_1 ≼ x_i+1^s,⋯, x'_h^φ≼ x_i+h^φ^s. By applying u_i^s, monotonicity implies [f(x'_h^φ,u_i^s,w) ≼ f^*(x_i+h^φ^s,u_i^s) = x_i+1+h^φ^s, ∀ w ∈𝒲;⇒x'_1 x'_2 ⋯ x'_h^φf(x'_h^φ,u_i^s,w) ∈;L(x_i+1^s x_i+2^s⋯ x_i+1+h^φ^s)),∀ w ∈𝒲. ]And the proof is complete from the fact that x_i+1^s⋯ x_i+1+h^φ^s∈Ω_ℒ^φ for all i ∈{0,1⋯,T-1}.Let the conditions in Theorem <ref> hold and x[t_0:t_0+h^φ] ∈ L(x^s[0:h^φ]) for some t_0 ∈ℕ. Consider the following control sequence starting from time t_0+h^φ:u^s:=(u^s_0 u^s_1 ⋯ u^s_T-1)^ω,i.e., u^s_t=u^s_(t-t_0-h^φ,T), t≥ t_0+h^φ. Let x_k+1=f^*(x_k,u_k), k=t_0+h^φ,⋯. Then we have x[t:t+h^φ] ∈Ω_ℒ^φ, ∀ t≥ t_0. We prove by induction that x[t:t+h^φ] ∈ L(x^s[(t-t_0,T):(t-t_0,T)+h^φ]), ∀ t≥ t_0. The base case for t=t_0 is true. In order to prove the inductive step x[t+1:t+1+h^φ] ∈ L(x^s[(t+1-t_0,T):(t+1-t_0,T)+h^φ]), we need to prove that x_t+k+1≼ x^s_(t+1-t_0,T)+k, k=0,⋯,h^φ, for which we need to only prove the case for k=h^φ as previous inequalities are already assumed by induction.We show x_t+h^φ+1≼ x^s_(t+1-t_0,T)+h^φ through monotonicity and the induction assumption that x_t+h^φ≼ x^s_(t-t_0,T)+h^φ:[ x_t+h^φ+1 =f^*( x_t+h^φ,u^s_(t-t_0,T) );≼ f^*( x^s_(t-t_0,T)+h^φ ,u^s_(t-t_0,T)); = x^s_(t-t_0,T)+1+h^φ≼ x^s_(t+1-t_0,T)+h^φ. ] Note that x^s_T+h^φ≼ x^s_h^φ. The “≼" in the last line can be replaced by “=" when (t-t_0,T)+1 ≠ T.We refer to the repetitive sequence of controls in (<ref>) as an s-sequence. An s-sequence is an invariance inducing open-loop control policy.Once the latest h^φ+1-length of system state are brought into Ω_ℒ^φ, an s-sequence keeps the h^φ+1-length trajectory of the system in Ω_ℒ^φ for all subsequent times. The computation of an s-sequence requires solving an MILP for x^s[h^φ:T] G_[0,T]φ (an instance of Problem <ref>) with an additional set of constraints inx^s[0:h^φ] φ (again, an instance of Problem <ref>, but without the dynamical constraints. In other words, x^s[0:h^φ] does not need to be a trajectory of the maximal system), and x^s[T:T+h^φ] ≼x^s[0:h^φ] (linear constraints). We are usually interested in the shortest s-sequence since its computation requires the smallest MILP. Algorithmically, we start from T=1 and implement T T+1 until the MILP formulating the conditions in Theorem <ref> becomes feasible and an s-sequence is found. As it will be implied from results in Sec. <ref>, existence of an s-sequence is almost necessary for existence of a RCI set.Consider the system in Example <ref>. We wish to keep the trajectory in the set characterized by p_1 ∨ p_2, i.e., 𝒮=L((1,5)^T) ) ∪ L((5,1)^T) ). Note that this set is non-convex. We set the cost function to maximize x_0_1. The shortest s-sequence has T=5 and is: (2 1 2 1 1)^ω. The resulting trajectory satisfying the definition of s-sequence is shown in Fig. <ref> (a). The corresponding robust control invariant set is shown in Fig. <ref>. (b) (cyan region), which is characterized by the x^s_0,x^s_1,⋯,x^s_4 (red dots) that lie inside 𝒮 (green region). Note that the [0,2]×[0,2] portion of the coordinates in Fig. <ref> is shown here for a clearer representation of the details.§.§ Bounded-global specifications: ϕ-sequences Now we consider general bounded-global formulas - as in Problem <ref> - and generalize the paradigm used for s-sequences. We provide the key result of this section. Given a bounded-global STL formula ϕ=φ_b ∧G_[Δ,∞]φ_g, an initial condition x_0, a control sequence u_0^ϕ⋯ u_Δ+T+h^φ_g-1^ϕ, where T is a positive integer, and a non-negative integer T_0 < T, let the following conditions hold: * x^ϕ[0:Δ+T+h^φ_g] φ_b ∧G_[Δ,Δ+T]φ_g,* x^ϕ[Δ+T:Δ+T+h^φ_g] ≼x^ϕ[Δ+T_0:Δ+T_0+h^φ_g]; where x_k+1^ϕ=f^*(x_k^ϕ,u_k^ϕ), k ∈ [0,Δ+T+h^φ_g-1], x_0^ϕ=x_0. Let μ^ol be the open-loopcontrol policycorrespondingto the following control sequence:u^ϕ:=u_0^ϕ⋯ u_Δ+T_0+h^φ_g-1^ϕ(u_Δ+T_0+h^φ_g^ϕ⋯ u_Δ+T+h^φ_g-1^ϕ)^ω,Then x(x'_0,μ^ol,w) ϕ, ∀w∈𝒲^ω, ∀ x'_0 ∈ L(x_0). Moreover, the following set is a RCI set in ℒ^φ_g:Ω_ℒ^φ_g:= ⋃_i=0^T-T_0-1 L(x^ϕ[Δ+T_0+i:Δ+T_0+h^φ_g+i]).We need to prove that x(x^ϕ_0,μ^ol,w^*) ϕ, where w^*=(w^*)^ω.The fact that x(x'_0,μ^ol,w)[0] ϕ, ∀ x'_0 ∈ L(x_0), ∀w∈𝒲^ω, follows from monotonicity and Lemma <ref>.The fact that Ω_ℒ^φ_g is a RCI set follows from Theorem <ref> as (<ref>) is obtained from replacing Δ=T_0=0 in (<ref>) . It follows that (u_Δ+T_0+h^φ_g^ϕ⋯ u_Δ+T+h^φ_g-1^ϕ)^ω is an s-sequence.For all t≥Δ + T + h^φ_g, letx^ϕ_t+1=f^*(x_t^ϕ,u^ϕ_Δ+T_0+h^φ_g+(t-Δ-T_0-h^φ_g,T-T_0)).Using Corollary <ref>, we havex^ϕ[k+Δ+T_0:k+Δ+T_0+h^φ_g] ∈ℒ^φ_g, ∀ k ∈ℕ, and the proof is complete. We refer to the sequence of controls in (<ref>) as a ϕ-sequence. The computation of a ϕ-sequence requires solving an MILP for x^ϕ[0:Δ+T+h^φ_g] φ_b ∧G_[Δ,Δ+T]φ_g (an instance of Problem <ref>) with an additional set of constraints inx^ϕ[Δ+T:Δ+T+h^φ_g] ≼x^ϕ[Δ+T_0:Δ+T_0+h^φ_g] (linear constraints). Thus, similar to s-sequecnes, the computation of a ϕ-sequence is based on feasibility checking of a MILP. We have two parameters T and T_0 < T to search over. We start from T=1 and implement T T+1, while checking for all T_0 < T, until the corresponding MILP gets feasible. In Sec. <ref>, we discuss the necessity of existence of a feasible solution for some T_0,T.Another interpretation of a ϕ-sequence is a sequence that consists of an initialization segment of length Δ+h^φ_g to bring the latest h^φ_g states of the system into Ω_ℒ^φ_g⊆ℒ^φ_g and a repetitive segment of length T to stay in Ω_ℒ^φ_g. The repetitive segment is an s-sequence.Since control inputs eventually becoming periodic, the long-term behavior is expected to demonstrate periodicity, which leads to the following result based on Theorem <ref>.The ω-limit set of the run given by (<ref>) is non-empty and corresponds to the following periodical orbit:(x_0^ϕ,∞ x_1^ϕ,∞⋯ x_T-T_0-1^ϕ,∞)^ω,where x_k^ϕ,∞:= lim_c →∞ x_k+Δ+T_0+c(T-T_0)^ϕ, k=0,⋯,T-T_0-1. We show that x^ϕ_t≼ x^ϕ_t+T-T_0, ∀ t ≥Δ+T_0. Similar to the proof of Corollary <ref>, we use induction. The base case for t=Δ+T_0 is already in the second condition in Theorem <ref>. The inductive step is proven as follows:x^ϕ_t+1+T-T_0 =f^*( x^ϕ_t+T-T_0,u_T-T_0+t ) ≼ f^*( x^ϕ_t,u_t)=x^ϕ_t+1,where from (<ref>) we have u_t+T-T_0=u_t=u^ϕ_t+Δ+T_0+h^φ_g+(t-Δ-T_0-h^φ_g,T-T_0).Thus, each component of the sequence x^ϕ_Δ+T_0+kx^ϕ_Δ+T+k x^ϕ_Δ+2T-T_0+k⋯, k=0,⋯,T-T_0, is monotonically decreasing.Monotone convergence theorem <cit.> explains that a lower-bounded monotonically decreasing sequence converges (in this case, all values are lower-bounded by zero). Thus, lim_c→∞ x_Δ+T_0+k+c(T-T_0)^ϕ, k=0,⋯,T-T_0, exists and the proof is complete.Consider the system in Example <ref>. We wish to satisfy ϕ= F_[0,5] p_1 ∧ G_[5,∞)( F_[0,6] p_1 ∧ F_[0,6] p_2 ).The specification is in form in (<ref>) with h^φ_b=Δ=5,h^φ_g=6.This specification requires that p_1 is visited at least once until t=5 and, afterwards, p_1 and p_2 are persistently visited while the maximum time between two subsequent visits is not greater than 6. We find a ϕ-sequence solving a MILP for T=7, T_0=0, while maximizing x_0 _1. The obtained ϕ-sequence isu^ϕ=2 2 2 2 1 2 1 1 1 1 2 (2  2  1  1  1  1  2)^ω for x_0=(12.4,0)^T. The first Δ+T+h^φ_g+1=5+7+6+1=19 time points of the trajectory of the maximal system f^* satisfying the conditions in Theorem <ref> are shown inFig. <ref> [Left]. A sample trajectory of f with values of w chosen uniformly from 𝒲 is also shown. Both trajectories satisfy ϕ. The limit-set of f^*, which is a 7-periodical orbit, is shown in Fig. <ref> [Right]. §.§ Necessity of Open-loop Strategies We showed that if there exists an initial condition and a finite length control sequence such that the statements in Theorem <ref> hold, an open-loop control sequence is sufficient for satisfying of a bounded-global formula, as was formulated in Problem <ref>. In this section, we address the necessity conditions. We show that the existence of open-loop control strategies for satisfying a bounded-global specifications is almost necessary in the sense that if a ϕ-sequence is not found using Theorem <ref> for large values of T, then it is almost certain that no correct control policy (including feedback policies) exists, or, if exists any, it is fragile in the sense that a slight increase in the effect of the disturbances makes the policy invalid. We characterize the necessity conditions based on hypothetical perturbations in the disturbance set.Suppose system (<ref>) is strongly monotone with respect to the maximal disturbance in the sense that for all ϵ>0, there exists a perturbed disturbance set 𝒲_p with maximal disturbance w^*_p such that ∀ x ∈𝒳, ∀ u ∈𝒰, f(x,u,w^*) + 1_n ϵ≼ f(x,u,w_p^*).Consider the bounded-global formula ϕ=φ_b ∧G_[Δ,∞]φ. Given ϵ>0, the disturbance set is altered to 𝒲_p such that (<ref>) holds. If there exists a control policy μ and an initial condition x_0 such that x(x_0,μ,w_p) ϕ, ∀w_p ∈𝒲_p^ω, then there exists at least one open-loop control policy μ^ol in the form of a ϕ-sequence in (<ref>) for the original system such thatT ≤A/ϵ^n(h^φ_g+1),where A is a constant depending on ℒ^φ_g.Given a bounded set 𝒞⊂ℝ^n(h^φ+1), we define the diameter d(𝒞):= inf{ d |s_1≼ s_2+d1_n(h^φ+1), ∀ s_1,s_2 ∈𝒞} (e.g., the diameter of an axis-aligned hyper-box is equal to the length of its largest side).Consider a partition of ℒ^φ by a finite number of cells, where the diameter of each cell is less than ϵ. The maximum number of cells required for such a partition is A/ϵ^n(h^φ_g+1), where A is a constant dependent on the shape and volume of ℒ^φ_g. A conservative upper bound on A can be given as follows. Define a^*∈ℝ_+ as_a/ϵ∈ℕ{x[0:h^φ_g] ≼ a 1_n[0:h^φ_g], ∀x[0:h^φ_g] ∈ℒ^φ_g}.Since ℒ^φ_g is bounded and closed, a^* exists. We have ℒ^φ_g⊆ L(a^* 1_n(h^φ_g+1)). Let A be a^*^n(h^φ_g+1) - the volume of L(a^* 1_n(h^φ_g+1)), which is a hyper-box.Partition L(a^* 1_n(h^φ_g+1)) into N:=A/ϵ^n(h^φ_g+1) number of equally sized cubic cells with side length of ϵ. Such a partition also partitions ℒ^φ_g to at most N number of cells where the diameter of each cell is not greater than ϵ. Since there exists μ such that x(x_0,μ,w_p) ϕ, ∀w_p ∈𝒲_p^ω, then there exist at least one run satisfying ϕ for system x_k+1=f(x_k,u_k,w_p^*). Let x_0,⋯, x_Δ+h^φ_g+N be the first Δ+h^φ_g+N+1 time points of a such a run. We have x[k:k+h^φ_g] ∈ℒ^φ_g, k=Δ,⋯,Δ+N. Consider the sequence x[Δ:Δ+h^φ_g] x[Δ+1:Δ+1+h^φ_g] ⋯x[Δ+N:Δ+N+h^φ_g].Consider a partition of ℒ^φ_g with cells that for all cells the diameter is less than ϵ. By the virtue of pigeonhole principle, there exists a cell that contains at least two time points x[k_1:k_1+h^φ_g] and x[k_2:k_2+h^φ_g], Δ≤ k_1 ≤ k_2≤Δ+N. From the assumption on the diameter of the cells we have x[k_2:k_2+h^φ_g] ≼x[k_1:k_1+h^φ_g] + ϵ1_n[0:h^φ_g].Now consider system x'_k+1=f(x'_k,u_k,w^*) - the original maximal system - with x'_k_1+h^φ_g=x_k_1+h^φ_g. We prove that x'_k + 1_n ϵ≤ x_k, ∀ k> k_1+h^φ_g.We use induction. The base case for k=k_1+h^φ_g+1 is verified using (<ref>): [x'_k_1+1+h^φ_g + 1_n ϵ= f(x'_k_1+h^φ_g,u_k_1+h^φ_g,w^*) +1_n ϵ;≤f(x'_k_1+h^φ_g,u_k_1+h^φ_g,w_p^*);= x_k_1+1+h^φ_g. ]The inductive step is verified using monotonicity and (<ref>): [x'_k+1+h^φ_g+ 1_n ϵ=f(x'_k+h^φ_g,u_k+h^φ_g,w^*) +1_n ϵ; ≤ f(x'_k+h^φ_g,u_k+h^φ_g,w_p^*); ≤ f(x_k+h^φ_g,u_k+h^φ_g,w_p^*) = x_k+1+h^φ_g. ]It immediately follows from (<ref>) that x'[k_2:k_2+h^φ_g] + ϵ1_n[0:h^φ_g] ≤x[k_2:k_2+h^φ_g].Since the lefthand of (<ref>) is the righthand of (<ref>), we have:x'[k_2:k_2+h^φ_g] ≤x[k_1:k_1+h^φ_g].This is reminiscent of the conditions in Theorem <ref>. Now by defining x'_k:=x_k, k=k_1,⋯,k_1+h^φ_g-1, we conclude thatΩ'_ℒ^φ_g:= ⋃_k=k_1^k_2-1 L(x'[k:k+h^φ])is a RCI set for system with adversarial disturbance set 𝒲 and (u_k_1⋯ u_k_2+h^φ_g-1)^ω is an s-sequence.Now, once again, consider the original system x'_k+1=f(x'_k,u_k,w^*) with x'_0=x_0. Monotonicity implies x'[0:k_1+h^φ_g] ≤x[0:k_1+h^φ_g]. Thus, by applying u_0,⋯,u_k_1+h^φ_g-1 and using Lemma <ref>, we have x'[0:k_1+h^φ_g] φ_b ∧G_[Δ,k_1]φ_g. Corollary <ref> implies x'[k_1+h^φ_g] G_[0,∞)φ_g if (u_k_1,⋯,u_k_2-1)^ω is applied starting from time k_1. Finally, monotonicity andLemma <ref> immediately indicate that x(x”_0,μ^ol,w) ϕ, ∀ x”_0 ∈ L(x_0), ∀w∈𝒲^ω, where μ^ol is the following open-loop control strategy producing the following control sequence:u_0⋯ u_k_1-1 (u_k_1⋯ u_k_2+h^φ_g-1)^ω, which is in form of (<ref>) with k_1=Δ+T_0+h^φ_g and k_2=Δ+T. Since k_2 ≤Δ+N, we also have T ≤ N, N=A/ϵ^n(h^φ_g+1), and the proof is complete. Suppose that for all T ≤ T^max, T_0 < T, there does not exist an initial condition and a control sequence such that the conditions inTheorem <ref> hold. Then there does not exist any solution to Problem <ref> given that the maximal disturbance is w_p^* such that (<ref>) holds with ϵ > √(T^max).The relation between the fragility in Theorem <ref> and the length of the ϕ-sequence suggests that by performing the search for longer ϕ-sequences (which are computationally more difficult), the bound for fragility becomes smaller, implying that a correct control policy (if exists) is close to the limits (i.e., robustness score is close to zero, or the constraints are barely satisfied in the case with maximal disturbance). In practice, the bounds in Theorem <ref> are very conservative and one may desire to find tighter bounds for specific applications. Consider Example <ref>. Suppose that there does not exist an s-sequence of length smaller than 144 with maximal disturbance w^*. The constant A (area in this 2D case, see proof of Theorem <ref>) of region corresponding to p_1 ∨ p_2 is 9. Therefore, 𝒮 can be partitioned into 144 equally sized square cells with side length 0.25. Note that we have ϵ^2≥ 9/T. Since the disturbances are additive, it follows that if A_u^-1(I-e^-A_u t)(w^*_p-w^*) > (0.25,0.25)^T, u=1,2, then there does not exist any control strategy μ and x_0 ∈ℝ_+^n such that x(x_0,μ,w_p) G_[0,∞](x ∈𝒮),∀w_p ∈𝒲_p^ω.§ MODEL PREDICTIVE CONTROL In this section, we provide a solution to Problem <ref>. We assume full knowledge of the history of state. As mentioned in Sec. <ref>, the cost function J is assumed to be non-decreasing with respect to the state values hence the system constraints are replaced with those of the maximal system. First, we explain the MPC setup for global STL formulas. Next, weprove that the proposed framework is guaranteed to generate runs that satisfy the global STL specification (<ref>).Let t≥ h^φ-1. The case of t<h^φ-1 is explained later.Given planning horizon H, the states that are predictable at time t using controls in u^H_t are x_1|t,x_2|t,⋯,x_H|t. Given predictions x_1|t,x_2|t,⋯,x_H|t, we need to enforce x[t-h^φ+1,t+H] G_[0,H-1]φ at time t. Notice that x_|t[t-h^φ+1,t+H]:=x_t-h^φ+1⋯ x_t x_1|t⋯ x_H|t,i.e., the first h^φ time points are actual values, the rest are predictions. Also, note that the values in x[τ:τ+h^φ] are independent of the values in x_t^H for τ≤ t-h^φ and are not fully available for τ>t+H-h^φ. Thus, [t-h^φ+1,t+H-h^φ] is the time window for imposing constraints at time t <cit.>. The MPC optimization problem is initially written as (we do not use it for control synthesis as explained shortly): [ minimizeJ(x_t^H, u_t^H ),; s.t. x_k+1|t=f^*(x_k|t,u_k|t), k=0,⋯,H-1,; x_|t[t-h^φ+1,t+H] G_[0,H-1]φ. ] The set of constraints in (<ref>) requires the knowledge of x_t-h^φ+1 x_t-h^φ+2⋯ x_t. Thus, the proposed control policy requires a finite memory for the history of last h^φ states. As it will be shown in Proposition <ref>, persistent feasibility of the constraints in (<ref>) leads to fulfilling G_[0,∞]φ.However, persistent feasibility of the MPC setup in (<ref>) is not guaranteed. We address this issue for the remainder of this section. An MPC strategy is recursively feasible if, for all t ∈ℕ, the control at time t is selected such that the MPC optimization problem at t+1 becomes feasible.Our goal is to modify (<ref>) such that it becomes recursively feasible.It is known that adding a (the maximal) RCI set acting as a terminal constraint is sufficient (and necessary) to guarantee recursive feasibility <cit.>. We add the terminal constraint x [t+H-h^φ:t+H] ∈Ω_ℒ^φ to (<ref>) to obtain:[ u^H,opt_t=u^H_t ∈𝒰^H  J(x_t^H, u_t^H ),; s.t. x_k+1|t=f^*(x_k|t,u_k|t), k=0,⋯,H-1,; x_|t[t-h^φ+1,t+H] G_[0,H-1]φ,;x_|t [t+H-h^φ:t+H] ∈Ω_ℒ^φ. ]Let μ_t(x_0,⋯,x_t)=μ_t(x_t-h^φ+1,⋯,x_t)=u^H,opt_0|t, where u^H,opt=u^H,opt_0|t⋯ u^H,opt_H-1|t is given by (<ref>). If the optimization problem (<ref>) is feasible for all t ≥ h^φ-1, then x(x_0,μ,w)[0] G_[0,∞]φ, ∀w∈𝒲^ω. We show that x(x_0,μ,w)[0:k+h^φ] G_[0,k]φ, ∀w∈𝒲^*, ∀ k ∈ℕ, using induction over k.Consider (<ref>) for t=k+h^φ-1 for any k ∈ℕ. The second constraint in (<ref>) requires x_|t[k,k+h^φ] φ, or equivalently, x_k ⋯ x_k+h^φ-1 x_1|k+h^φ-1φ. By applying u^opt_0|t, monotonicity impliesx_k+h^φ=f(x_k+h^φ-1,u^opt_0|t,w) ≼ x_1|k+h^φ-1 = f^*(x_k+h^φ-1,u^opt_0|t), ∀ w ∈𝒲. From Lemma <ref> we have x[k:k+h^φ] φ. Thus, we have shown x[k:k+h^φ] φ, ∀ k ∈ℕ, and the proof is complete.The MPC strategy corresponding to (<ref>) is recursively feasible.Suppose u_t^H=u_0|t⋯ u_H-1|t and x_t^H=x_t+1|t⋯,x_t+H-1|t is a feasible solution for (<ref>) at time t. Since Ω_ℒ^φ is a RCI set, there exist u^r∈𝒰 such that x_|t[t+H+1-h^φ:t+H+1]=x_H-h^φ+1|t x_H-h^φ+2|t⋯ x_H|tf(x_H|t,u^r,w)∈Ω_ℒ^φ, ∀ w ∈𝒲. Suppose u_0|t is applied to the system. We have x_t+1=f(x_t,u_0|t,w) ≼ f^*(x_t,u_0|t)=x_1|t, ∀ w ∈𝒲. Now, we prove that the optimization problem at time t+1 is feasible by showing that at least one feasible solution exists. Letu^H_t+1=u_1|t u_2|t⋯,u_H|t u^r. We already showed that x_t+1=x_0|t+1≼ x_1|t. By induction and using monotonicity, it follows that x_k|t+1≼ x_k+1|t, k=,1,⋯,H-2. Therefore, we have x_t-h^φ+2⋯ x_t+1 x_1|t+1⋯ x_H-1|t+1≼ x_t-h^φ+2⋯ x_1|t x_2|t⋯ x_H|t, which using Lemma <ref> establishes x_t-h^φ+2⋯ x_t+1 x_1|t+1⋯ x_H-1|t+1G_[0,H-1]φ. In order to complete the proof, it remains to show that x [t+H+1-h^φ:t+H+1]= x_H+1-h^φ|t⋯ x_H|t+1φ. This follows from invariance. Note that x_H|t+1=f^*(x_H|t,u^r). Therefore x_H+1-h^φ|t⋯⋯ x_H|t+1∈Ω_ℒ^φ, and since Ω_ℒ^φ∈ℒ^φ, we have x_H+1-h^φ|t⋯⋯ x_H|t+1φ, and the proof is complete.The MPC optimization problem is also converted into a MILP problem. It is computationally easier to solvethe optimization problem in (<ref>) by solving T MILPs:[ u^opt,H_t =u^H_t ∈𝒰^H,i=0,⋯,T-1 J(x_t^H, u_t^H ),;s.t.x_k+1|t=f^*(x_k|t,u_k|t), k=0,⋯,H-1,; x_|t[t-h^φ+1,t+H] G_[0,H-1]φ,; x_|t [t+H-h^φ:t+H] ∈ L(x^φ,x_0[i:i+h^φ]). ]Note that all MILPs can be aggregated into a single large MILP in the expense of additional constraints for capturing non-convexities of the terminal condition.Finally, consider t<h^φ. In this case, we require H≥ h^φ and replace the interval [t-h^φ+1,t+H-h^φ] with [0,t+H-h^φ]for t<h^φ in (<ref>). For applications where initialization is not important in long-term (like traffic management), a simpler approach is to initialize the MPC from t=h^φ-1 and assume all previous state values are zero (hence all the past predicates are evaluated as true). In our previous work on STL MPC of linear systems <cit.>, we did not establish recursive feasibility. In order to recover from possible infeasibility issues, we proposed maximizingthe STL robustness score (a negative value) whenever the MPC optimization problem became infeasible. Although recursive feasibility is guaranteed here, un-modeled disturbances and initial conditions outside 𝒳_0^max can lead to infeasibility. The formalism in <cit.> can be used to recover from infeasibility with minimal violation of the specification. § APPLICATION TO TRAFFIC MANAGEMENTIn this section, we explain how to apply our methods to traffic management. First, the model that we use for traffic networks is explained, which is similar to the one in <cit.> but freeways are also modeled. Next, the monotonicity properties of the model are discussed. We show that there exists a congestion-free set in the state-space in which the traffic dynamics is monotone. Finally, a case study on a mixed urban and freeway network is presented. §.§ ModelThe topology of the network is described by a directed graph (𝒱,ℒ), where 𝒱 is the set of nodes and ℒ is the set of edges. Each l ∈ℒ represents a one-way traffic link from tail node τ(l) ∈𝒱∪∅ to head node η(l) ∈𝒱, where τ(l)=∅ stands for links originating from outside of the network. We distinguish between three types of links based on their control actuations: 1) ℒ_r: road links actuated by traffic lights, 2) ℒ_o: freeway on-ramps actuated by ramp meters, 3) ℒ_f: freeway segments which are not directly controlled.Freeway off-ramps are treated the same way as the roads. Uncontrolled roads are also treated the same as freeways. We have ℒ_r ∪ℒ_o ∪ℒ_f=ℒ. Some works, e.g. <cit.>, consider control over freeway links by varying speed limits, which adds to the control powerbut requires the existence of such a control architecture within the infrastructure. We do not consider this type of control actuation in this paper but it can easily be incorporated into our model by modeling freeways links the same way as on-ramps, where the speed limit becomes analogous to the ramp meter input.The number of vehicles on link l at time t is represented by x_[l],t∈ [0,c_l], which is assumed to be a continuous variable, and c_l is the capacity of l. In other words, vehicular movements are treated as fluid-like flow in our model.The number of vehicles that are able to flow out of l in one time step, if link l is actuated, is:q_[l],t:= min{x_[l],t, q̅_l, {l'| τ(l')=η(l) }minα_l:l'/β_l:l' (c_l'-x_[l'],t) }, where q̅_l is the maximum outflow of link l in one time step, which is physically related to the speed of the vehicles. The last argument in the minimizer determines the minimum supply available in the downstream links of l, where α_l:l'∈ [0,1] is the capacity ratio of link l' available to vehicles arriving fromlink l (typically portion of the lanes), β_l:l'∈ [0,1] is the ratio of the vehicles in l that flow into l' (turning ratio). For simplicity, we assume capacity ratios and turning ratios are constants. System state is represented by x ∈ℝ^n_+: {x_[l]}_l ∈ℒ,where n is the number of the links in the network. The state space is 𝒳:=∏_l ∈ℒ [0,c_l].A schematic diagram illustrating the behavior of q_[l] with respect to the state variables x_[l],x_[l']- which is known as the fundamental diagram in the traffic literature <cit.> - is shown in Fig. <ref>. The link flow drops if one (or more) of its downstream links do not have enough capacity to accommodate the incoming flow. In this case (when the last argument in (<ref>) is the minimizer), we say the traffic flow is congested. Otherwise, the traffic flow is free.This motivates the following definition: The congestion-free set, denoted by Π, is defined as the following region in the state space:[Π:={ x∈𝒳 |min{x_[l],q̅_l}≤α_l:l'/β_l:l' (c_l'-x_[l']),;∀ l,l'∈ℒ, τ(l')=η(l)}. ]The congestion-free set is a lower-set. Consider x ∈Π and any x'∈ L(x). For all l,l'∈ℒ, τ(l')=η(l), we have min{x'_l,q̅_l}≤min{x_[l],q̅_l} and(c_l'-x_[l]') ≤ (c_l'-x'_l'). Therefore, min{x'_l,q̅_l}≤α_l:l'/β_l:l' (c_l'-x'_l'). Thus x'∈Π, which indicates Π is a lower-set. Note that Π is, in general, non-convex. The predicate (x ∈Π) can be written as a Boolean logic formula over predicates in the form of (<ref>) as:[ ⋀_l,l'∈ℒ, τ(l')=η(l) (( (x_[l]≤q̅_l) ∧ (x_[l]+ α_l:l'/β_l:l' x_[l]'≤α_l:l'/β_l:l' c_l')); ∨ (q_[l]+ α_l:l'/β_l:l' x_[l]'≤α_l:l'/β_l:l' c_l')). ]Notice how the minimizer in (<ref>) is translated to a disjunction in (<ref>).Now we explain the controls. The actuated flow of link l at time t is denoted by q⃗_[l],t, where we have the following relations:q⃗_[l],t= {[s_[l],t q_[l],t, l ∈ℒ_r,; min{ q_[l],t, r_[l],t}, l ∈ℒ_o,;q_[l],t, l ∈ℒ_f,; ].where s_[l],t∈{0,1} is the traffic light for link l, where 1 (respectively, 0) stands for green (respectively, red) light, and r_[l],t∈ℝ_+ is the ramp meter input for on-ramp l at time t. Ramp meter input limits the number of vehicles that are allowed to enter the freeway in one time step.In order to disallow simultaneous green lights for links l,l' (which are typically pair of links pointing toward a common intersection in perpendicular directions), we add the additional constraintss_[l],t+s_[l]',t≤ 1. In simple gridded networks, as in our case study network illustrated in Fig. <ref>, it is more convenient to define phases for actuation in north-south or east-west directions that are unambiguously mapped to traffic lights for each individual link. The evolution of the network is given by:x_[l],t+1=x_[l],t-q⃗_[l],t+w_[l],t+∑_l', η(l')=τ(l)β_l':lq⃗_[l'],t, where w_[l],t is the number of exogenous vehicles entering link l at time t, which is viewed as the adversarial input. The evolution relation above can be compacted into the form (<ref>):x_t+1=f_traffic(x_t,u_t,w_t), where u_t and w_t are the vector representations for control inputs (combination of traffic lights and ramp meters) and disturbances inputs, respectively. Note that f_traffic represents a hybrid system which each mode is affine. The mode is determined by the control inputs and state (which determines the minimizer arguments). Some works consider nonlinear representations for the fundamental digram (Fig. <ref>), but they still can be approximated using piecewise affine functions.§.§ Monotonicity System (<ref>) is monotone in Π. Consider x',x ∈Π, x ≼ x'. We show that f_traffic(x,u,w) ≼ f_traffic(x',u,w), ∀ w ∈𝒲, ∀ u ∈𝒰. Observe in (<ref>) that we only need to verify is proving that x_[l]-q⃗_[l] is a non-decreasing function of x_[l] as all other terms are additive and non-decreasing with respect to x. Since x,x' ∈Π, the last argument in (<ref>) is never the minimizer. Thus, for all ł∈ℒ, we have x_[l]-q⃗_[l]∈{0, x_[l]-r_[l],x_[l]-c_l,x_[l]}, depending on the mode of the system and actuations, which all are non-decreasing functions of x_[l]. Thus, f_traffic is monotone in Π. The primary objective in our traffic management approach is finding control policies such that the state is restricted to Π, which not only eliminates congestion, but also ensures that the system is monotone hence the methods of this paper become applicable. It isworth to note that the traffic system becomes non-monotone when flow is congested in diverging junctions, as shown in <cit.>. This phenomena is attributed to the first-in-first-out (FIFO) nature of the model. By assuming fully non-FIFO models, system becomes monotone in the whole state space. For a more thorough discussion on physical aspects of monotonicity in traffic networks, see <cit.>.The maximal system in (<ref>) corresponds to the scenario where each w_l is equal to its maximum allowed value w_l^*. §.§ Case Study §.§.§ Network Consider the network in Fig. <ref>, which consists of urban roads (links 1-26, 27,29,31,33 and 49-53), freeway segments (links 35-48) and freeway on-ramps (links 28,30,32,34). The layout of the network illustrates a freeway passing by an urban area, which is common in many realistic traffic layouts. There are 14 intersections (nodes a-n) controlled by traffic lights. Each intersection has two modes of actuation: north-south (NS) and east-west (EW). There are four entries to the freeway (nodes o-r) that are regulated by ramp meters. We have n=53 and 𝒰=ℝ_+^4 ×{0,1}^14. Vehicles arrive from links 1,6,11,15,19,23,35,42,49 and 52. The parameters of the network are shown in Table <ref>. §.§.§ SpecificationAs mentioned earlier, the primary objective is keeping the state in the congestion-free set. In addition, since the demand for the north-south side roads (links 49-53) is smaller than the traffic in the east-west roads, we add a timed liveness requirement for the traffic flow on links 49-53:ψ= ⋀_l=49,50,⋯,53 (x_[l]≥ 5) ⇒F_[0,3] (x_[l]≤ 5),which states that “if the number of vehicles on any of the north-south side roads exceeds 5, their flow is eventually actuated within three time units ahead". The global specification is given as:ϕ=G_[0,∞] ( (x∈Π) ∧ψ).Note that h^φ=3, φ=(x∈Π) ∧ψ. §.§.§ Open-loop Control PolicyWe use Theorem <ref>. The shortest ϕ-sequence that we found for this problem has T=5, T_0=0. The corresponding MILP had 2357 variables (of which 1061 were binary) and 4037 constraints [The scripts for this case study are available in ], which is solved using the Gurobi MILP solver in less than 6 seconds on a dual core 3.0 GHz MacBook Pro. The cost is set to zero in order to just check for feasibility. Even though finding an optimal solution and checking for feasibility of a MILP have the same theoretical complexity, the latter is executed much faster in practice. For instance, finding a ϕ-sequence, while minimizing or maximizing ∑_k=0^7x^ϕ_k_1 both took more than 20 minutes. Note that it is virtually intractable to attack a problem of this size (53 dimensional state) using any method that involves state-space discretization, such as the method in <cit.> (e.g., if each state-component is partitioned into 2 intervals, the finite-state problem size will be 2^53). Monotonicity implies that any demand set 𝒲 for which there exists a solution to Problem <ref> is a lower-set. The set corresponding to the values at the bottom of Table <ref> is one of them. Table <ref> shows results on existence of ϕ-sequences for some other demand scenarios. Computation times for solving a MILP do not demonstrate a generic behavior. For the rest of this section, the numerical examples are reported for the values in Table <ref>.The control values in the ϕ-sequence are shown in Table <ref>. As stated in Theorem <ref>, starting from an initial condition in L(x_0), applying the open-loop control policy (<ref>) guarantees satisfaction of the specification.In other words, after applying the initialization segment, the repetitive controls in Table <ref> become a fixed time-table for the inputs of the traffic lights and the ramp meters. Starting from x_0, which is a 53-dimensional vector, we apply (<ref>) using the values in Table <ref>. The trajectory of the maximal system is shown in Fig. <ref> [Top]. The traffic signals are coordinated such that the traffic flows free of congestion. The black dashed lines represent the capacity of the links, and the dashed line in the fourth figure (from the left) represents the threshold for the liveness sub-specification (ψ). It is observed that all the state values for side road links (49-53) persistently fall below the threshold. The robustness values for (x∈Π) and ψ are shown in the fifth figure. As mentioned earlier, robustness corresponds to the minimum volume of vehicles that the system is away from congestion, or violating the specification. The robustness values are always positive, indicating satisfaction. As stated in Theorem <ref>, the trajectory of the maximal system converges to a periodic orbit.It is worth to note that the number of vehicles on freeway links is significantly smaller than its capacity, which is attributed to the fact that the number designated for q̅ (related to the maximum speed) of freeway links is relatively large (30, as opposed to 15 for roads). Therefore, freeway links are utilized in a way that there is enough space for high speed non-congested flow.§.§.§ Robust MPCHere it is assumed that the controller has full state knowledge. We apply the techniques developed in Sec. <ref>. Using the result from the previous section, the set Ω_ℒ^φ is constructed in ℝ_+^212 (=ℝ^n(h^φ+1),n=53, h^φ=3). The cost criteria that we use in this case study is the total delay induced in the network over the planning horizon H. A vehicle is delayed by one time unit if it can not flow out of a link in one time step, which may be because of the actuation (e.g., red light) or waiting for the flow of other vehicles in the same link (i.e., we have x_[l]≥ c_l).We are also interested in maximizing the STL robustness score. The cost function is:[ J_traffic(x^H,u^H):= - ζ ρ(x,G_[0,H-1]φ,t-h^φ+1); +∑_k=0^H-1γ^k∑_l ∈ℒ (x_[l],t+k-q⃗_[l],t+k) , ] where q⃗_[l], given by (<ref>), is the amount of vehicles that flow out of link l, γ is the discount factor for delays predicted in further future, and ζ is a positive weight for robustness. Notice the connection between the time window of STL robustness score in (<ref>) and MPC constraint enforcement in (<ref>). It follows from Theorem <ref> and STL quantitative semantics (<ref>) that the cost function above is non-decreasing with respect to the state in Π. Therefore, in order to minimize the worst case cost, the maximal system is considered in the MPC optimization problem.Starting from zero initial conditions, we implement the MPC algorithm (<ref>) with H=3 for 40 time steps. We set ζ=1000, γ=0.5 in (<ref>). The disturbances at each time step were randomly drawn from L(w^*) using a uniform distribution. The maximum computation time for each MPC step time step was less than 0.8 seconds (less than 0.5 seconds on average). The resulting trajectory is shown in Fig. <ref> [Middle]. For the same sequence of disturbances, the trajectory resulted from applying the open-loop control policy (<ref>) (using the values in Table <ref>) is shown in Fig. <ref> [Bottom]. Both trajectories satisfy the specification. However, robust MPC has obviously better performance when costs are considered. The total delay accumulated over 40 time steps is: J_40=∑_τ=0^40∑_l ∈ℒ (x_[l],τ-q⃗_[l],τ).The cost above obtained from applying robust MPC was J_40=1843, while the one for the open-loop control policy was J_40=2299, which demonstrates the usefulness of the state knowledge in planning controls in a more optimal way. An optimal tuning of parameters η and γ requires an experimental study which is out of scope of this paper. We only remark that we usually obtained larger delays with non-zero η, which shows that including STL robustness score in the MPC cost function may be useful even though the ultimate goal is minimizing the total delay. It is worth to note that we also tried implementing the MPC algorithm (for the case w=(w^*)^ω, or the maximal system) without the terminal constraints, as in (<ref>). The MPC got infeasible at t=8. The violating constraints were those in x ∈Π. This observation indicates that the myopic behavior of MPC in (<ref>), when no additional constraints are considered, can lead to congestion in the network.§ CONCLUSION AND FUTURE WORKWe developed methods to control positive monotone discrete-time systems from STL specifications. We showed that open-loop control sequences are sufficient and (almost) necessary for guaranteeing the correctness of STL specifications. A robust MPC method was introduced to plan controls optimally, while guaranteeing global STL specifications. We showed the usefulness of our results on traffic management.Future work will focus on non-monotone systems with parametric uncertainty whose state evolution can be over-approximated in an appropriate way using monotone systems. We will develop adaptive control schemes to tune parameters automatically using the data gathered from the evolution of the system. This will eventually lead to data-driven control techniques for transportation networks with formal guarantees. § APPENDIX Let 𝕊 be the set of all STL formulas that can be written in the form:ϕ=⋁_i=1^n_ϕφ_b,i∧ G_[Δ_i,∞]φ_g,i,where φ_b,i,Δ_i ≥ h^φ_b,i,φ_g,i, i=1,⋯,n_ϕ, are bounded STL formulas. Then 𝕊 is a subset of safety STL formulas that is closed under STL syntax with bounded temporal operators.First, a quick inspection of (<ref>) verifies that it is a safety STL formula. A predicate π is a bounded formula (with zero horizon) and is a special case of (<ref>), hence π∈𝕊. We also have the following property that relaxes the form in (<ref>): For all bounded STL formulas φ_1,φ_2,we have φ_1 ∧G_[Γ,∞)φ_2 ∈𝕊, ∀Γ∈ℕ. Proof: The case for Γ≥ h^φ_1 is already in the form (<ref>) with n_ϕ=1. If Γ< h^φ_1, we write G_[Γ,∞)φ_2 = G_[Γ,h^φ_1]φ_2 ∧G_[h^φ_1,∞)φ_2. Now, define φ_1 ∧G_[Γ,h^φ_1]φ_2 as the new bounded formula and retain the form in (<ref>) with n_ϕ=1. We show that 𝕊 is closed under STL syntax with bounded operators. The distributivity properties of Boolean connectives and temporal operators (see, e.g., <cit.>) imply that:ϕ_1 ∨ (ϕ_2 ∧ϕ_3)= (ϕ_1 ∨ϕ_2) ∧ (ϕ_2 ∨ϕ_3), ϕ_1 ∧ (ϕ_2 ∨ϕ_3)= (ϕ_1 ∧ϕ_2) ∨ (ϕ_2 ∧ϕ_3), F_I (ϕ_1 ∨ϕ_2)=(F_I ϕ_1) ∨ (F_I ϕ_2), andG_I (ϕ_1 ∧ϕ_2)=(G_I ϕ_1) ∧ (G_I ϕ_2), where ϕ_1,ϕ_2,ϕ_3 are temporal logic formulas and I is an interval. * ϕ_1, ϕ_2 ∈𝕊⇒ϕ_1 ∧ϕ_2 ∈𝕊, ϕ_1 ∨ϕ_2 ∈𝕊: this result easily follows from the distributivity properties of Boolean connectives mentioned above.* ϕ∈𝕊⇒F_{t}ϕ∈𝕊: we use F_{t}G_[a,b]=G_[t+a,t+b] and distributivity to have (note that F_{t}=G_{t}) [F_{t} (⋁_i=1^n_ϕ (φ_b,i∧G_[Γ_i,∞]φ_g,i));= ⋁_i=1^n_ϕ (F_{t}φ_b,i∧G_[t+Γ_i,∞]φ_g,i). ]Introducing F_{t}φ_b,i, i=1,⋯,n_ϕ, as new bounded STL formulas leads to the form in (<ref>).* ϕ∈𝕊⇒F_[a,b]ϕ∈𝕊, G_[a,b]ϕ∈𝔽: use F_[a,b]=⋁_t ∈ [a,b]F_{t} and G_[a,b]=⋀_t ∈ [a,b]F_{t} to convert temporal operators to Boolean connectives. * ϕ_1,ϕ_2 ∈𝕊⇒ϕ_1U_[a,b]ϕ_2∈𝕊:use the STL semantics (<ref>) to substitute the bounded “until" operator using bounded “eventually" and bounded “always" operators:[ ϕ_1U_[a,b]ϕ_2 = ⋁_t ∈ [a,b] (G_[a,t]ϕ_1∧F_{t}ϕ_2). ] The “reach and stay" formula F_I G_[0,∞)φ, where φ is a bounded formula, is equivalent to ⋁_t ∈ IG_[t,∞)φ. What remains to show that 𝕊 is equivalent to the set of all safety STL formulas is having that ϕ∈𝕊⇒G_[Γ,∞)ϕ∈𝕊, ∀Γ∈ℕ, which is not true by restricting n_ϕ in (<ref>) to be finite. Formulas that involve nested unbounded “always" operator and can not be further simplified, such as G_[Γ',∞) (φ_1 ∨G_[Γ,∞)φ_2), are rarely encountered in applications. [ < g r a p h i c s > ]Sadra Sadraddini (S' 16)received the B.Sc. in Mechanical Engineering and the B.Sc. in Aerospace Engineering (dual majors) in 2013 from Sharif University of Technology, Tehran, Iran.He is currently pursuing a degree toward Ph.D. in Mechanical Engineering at Boston University, Boston, MA.His research focuses on formal methods to control theory with various applications in cyber-physical systems. [ < g r a p h i c s > ]Calin Belta (F' 17) is a Professor in the Department of Mechanical Engineering at Boston University, where he holds the Tegan family Distinguished Faculty Fellowship. He is the Director of the BU Robotics Lab, and is also affiliated with the Department of Electrical and Computer Engineering, the Division of Systems Engineering at Boston University, the Center for Information and Systems Engineering (CISE), and the Bioinformatics Program. His research focuses on dynamics and control theory, with particular emphasis on hybrid and cyber-physical systems, formal synthesis and verification, and applications in robotics and systems biology.He received the Air Force Office of Scientific Research Young Investigator Award and the National Science Foundation CAREER Award. He is a fellow of IEEE.

http://arxiv.org/abs/1702.08501v2

Estimating Extinction using Unsupervised Machine Learning Stefan Meingast1 Marco Lombardi2 João Alves1Received ...; accepted... =================================================================== Generative adversarial networks <cit.> are a learning framework that rely on training a discriminator to estimate a measure of difference between a target and generated distributions. GANs, as normally formulated, rely on the generated samples being completely differentiable w.r.t. the generative parameters, and thus do not work for discrete data. We introduce a method for training GANs with discrete data that uses the estimated difference measure from the discriminator to compute importance weights for generated samples, thus providing a policy gradient for training the generator. The importance weights have a strong connection to the decision boundary of the discriminator, and we call our method boundary-seeking GANs (BGANs). We demonstrate the effectiveness of the proposed algorithm with discrete image and character-based natural language generation.In addition, the boundary-seeking objective extends to continuous data, which can be used to improve stability of training, and we demonstrate this on Celeba, Large-scale Scene Understanding (LSUN) bedrooms, and Imagenet without conditioning. § INTRODUCTIONGenerative adversarial networks <cit.> involve a unique generative learning framework that uses two separate models, a generator and discriminator, with opposing or adversarial objectives. Training a GAN only requires back-propagating a learning signal that originates from a learned objective function, which corresponds to the loss of the discriminator trained in an adversarial manner. This framework is powerful because it trains a generator without relying on an explicit formulation of the probability density, using only samples from the generator to train. GANs have been shown to generate often-diverse and realistic samples even when trained on high-dimensional large-scale continuous data <cit.>. GANs however have a serious limitation on the type of variables they can model, because they require the composition of the generator and discriminator to be fully differentiable.With discrete variables, this is not true. For instance, consider using a step function at the end of a generator in order to generate a discrete value.In this case, back-propagation alone cannot provide the training signal, because the derivative of a step function is 0 almost everywhere.This is problematic, as many important real-world datasets are discrete, such as character- or word-based representations of language. The general issue of credit assignment for computational graphs with discrete operations (e.g. discrete stochastic neurons) is difficult and open problem, and only approximate solutions have been proposed in the past <cit.>. However, none of these have yet been shown to work with GANs. In this work, we make the following contributions: * We provide a theoretical foundation for boundary-seeking GANs (BGAN), a principled method for training a generator of discrete data using a discriminator optimized to estimate an f-divergence <cit.>. The discriminator can then be used to formulate importance weights which provide policy gradients for the generator.* We verify this approach quantitatively works across a set of -divergences on a simple classification task and on a variety of image and natural language benchmarks.* We demonstrate that BGAN performs quantitatively better than WGAN-GP <cit.> in the simple discrete setting.* We show that the boundary-seeking objective extends theoretically to the continuous case and verify it works well with some common and difficult image benchmarks. Finally, we show that this objective has some improved stability properties withintraining and without. § BOUNDARY-SEEKING GANSIn this section, we will introduce boundary-seeking GANs (BGAN), an approach for training a generative model adversarially with discrete data, as well as provide its theoretical foundation. For BGAN, we assume the normal generative adversarial learning setting commonly found in work on GANs <cit.>, but these ideas should extend elsewhere. §.§ Generative adversarial learning and problem statementAssume that we are given empirical samples from a target distribution, {xi∈X}_i=1^M, where X is the domain (such as the space of images, word- or character- based representations of natural language, etc.). Given a random variable Z over a space Z (such as [0, 1]^m), we wish to find the optimal parameters, ∈^d, of a function, _: Z→X (such as a deep neural network), whose induced probability distribution, _, describes well the empirical samples.In order to put this more succinctly, it is beneficial to talk about a probability distribution of the empirical samples, , that is defined on the same space as _. We can now consider the difference measure betweenand _, D(, _), so the problem can be formulated as finding the parameters:= _ D(, _).Defining an appropriate difference measure is a long-running problem in machine learning and statistics, and choosing the best one depends on the specific setting. Here, we wish to avoid making strong assumptions on the exact forms ofor_, and we desire a solution that is scalable and works with very high dimensional data. Generative adversarial networks <cit.> fulfill these criteria by introducing a discriminator function, _ : X→, with parameters, , then defining a value function, __ = _[log_(x) ] + _(z)[log(1 - _(G(z)) ],where samples z are drawn from a simple prior, (z) (such as U(0, 1) or 01). Here, _ is a neural network with a sigmoid output activation, and as such can be interpreted as a simple binary classifier, and the value function can be interpreted as the negative of the Bayes risk. GANs train the discriminator to maximize this value function (minimize the mis-classification rate of samples coming fromor _), while the generator is trained to minimize it. In other words, GANs solve an optimization problem:(, ) = ____.Optimization using only back-propogation and stochastic gradient descent is possible when the generated samples are completely differentiable w.r.t. the parameters of the generator, .In the non-parametric limit of an optimal discriminator, the value function is equal to a scaled and shifted version of the Jensen-Shannon divergence, 2 * _JSD( || _) - log4,[Note that this has an absolute minimum, so that the above optimization is a Nash-equilibrium] which implies the generator is minimizing this divergence in this limit.f-GAN <cit.> generalized this idea over all -divergences, which includes the Jensen-Shannon (and hence also GANs) but also the Kullback–Leibler, Pearson χ^2, and squared-Hellinger. Their work provides a nice formalism for talking about GANs that use -divergences,which we rely on here. Let : _+→ be a convex lower semi-continuous function and : C⊆→ be the convex conjugate with domain C. Next, letbe an arbitrary family of functions, = {T : X→C}. Finally, letandbe distributions that are completely differentiable w.r.t. the same Lebesgue measure, μ.[μ can be thought of in this context as x, so that it can be said thatandhave density functions on x.] The -divergence, _( || _), generated by , is bounded from below by its dual representation <cit.>,_( || ) = _[f ( d / dμ/d / dμ) ]≥sup_∈ (_[(x)] - _[((x))]).The inequality becomes tight when T is the family of all possible functions. The dual form allows us to change a problem involving likelihood ratios (which may be intractable) to an maximization problem over T. This sort of optimization is well-studied if T is a family of neural networks with parameters(a.k.a., deep learning), so the supremum can be found with gradient ascent <cit.>.Let _ = ∘_ be a function, which is the composition of an activation function, : →C and a neural network, _ : X→. We can write the variational lower-bound of the supremum in Equation <ref> as [It can be easily verified that, for (y) = -log(1 + e^-y), f(u) = u logu + (1 + u) log(1 + u), and setting T = logD, the variational lower-bound becomes exactly equal to the GAN value function.]:_( || _) ≥_[∘_(x)] - __[(∘_(x))] = __.Maximizing Equation <ref> provides a neural estimator of f-divergence, or neural divergence <cit.>. Given the family of neural networks, T_Φ = {T_ϕ}_ϕ∈Φ, is sufficiently expressive, this bound can become arbitrarily tight, and the neural divergence becomes arbitrarily close to the true divergence. As such, GANs are extremely powerful for training a generator of continuous data, leveraging a dual representation along with a neural network with theoretically unlimited capacity to estimate a difference measure.For the remainder of this work, we will refer to _ = ∘_ as the discriminator and _ as the statistic network (which is a slight deviation from other works). We use the general term GAN to refer to all models that simultaneously minimize and maximize a variational lower-bound, __, of a difference measure (such as a divergence or distance). In principle, this extends to variants of GANs which are based on integral probability metrics <cit.> that leverage a dual representation, such as those that rely on restrictingthrough parameteric regularization <cit.> or by constraining its output distribution <cit.>. §.§ Estimation of the target distributionHere we will show that, with the variational lower-bound of an -divergence along with a family of positive activation functions, : →_+, we can estimate the target distribution, , using the generated distribution, _, and the discriminator, _. Let f be a convex function and ∈ a function that satisfies the supremum in Equation <ref> in the non-parametric limit. Let us assume thatand _(x) are absolutely continuous w.r.t. a measure μ and hence admit densities, (x) and _(x).Then the target density function, (x), is equal to ()(^⋆(x)) _(x).Following the definition of the -divergence and the convex conjugate, we have:_( || _) = __[((x)/(x))]= __[sup_t {t (x)/(x) - (t)}].Asis convex, there is an absolute maximum when t(t) = (x)/_(x). Rephrasing t as a function, (x), and by the definition of (x), we arrive at the desired result. Theorem <ref> indicates that the target density function can be re-written in terms of a generated density function and a scaling factor. We refer to this scaling factor, (x) = ()(^⋆(x)), as the optimal importance weight to make the connection to importance sampling [ In the case of the -divergence used in <cit.>, the optimal importance weight equals (x) = e^^⋆(x) = ^⋆(x) / (1 - ^⋆(x))]. In general, an optimal discriminator is hard to guarantee in the saddle-point optimization process, so in practice,will define a lower-bound that is not exactly tight w.r.t. the -divergence. Nonetheless, we can define an estimator for the target density function using a sub-optimal _. Letand , and _(x) be defined as in Definitions <ref> and <ref> but where : →_+⊆C is a positive activation function. Let (x) = ()((x)) and β = __[(x)] be a partition function. The -divergence importance weight estimator, (x) is(x) = w(x)/β_(x).The non-negativity ofis important as the densities are positive. Table <ref> provides a set of -divergences (following suggestions of  <cit.> with only slight modifications) which are suitable candidates and yield positive importance weights.Surprisingly, each of these yield the same function over the neural network before the activation function: (x) = e^_(x).[Note also that the normalized weights resemble softmax probabilities] It should be noted that (x) is a potentially biased estimator for the true density; however, the bias only depends on the tightness of the variational lower-bound: the tighter the bound, the lower the bias. This problem reiterates the problem with all GANs, where proofs of convergence are only provided in the optimal or near-optimal limit <cit.>. §.§ Boundary-seeking GANsAs mentioned above and repeated here, GANs only work when the value function is completely differentiable w.r.t. the parameters of the generator, . The gradients that would otherwise be used to train the generator of discrete variables are zero almost everywhere, so it is impossible to train the generator directly using the value function. Approximations for the back-propagated signal exist <cit.>, but as of this writing, none has been shown to work satisfactorily in training GANs with discrete data. Here, we introduce the boundary-seeking GAN as a method for training GANs with discrete data. We first introduce a policy gradient based on the KL-divergence which uses the importance weights as a reward signal. We then introduce a lower-variance gradient which defines a unique reward signal for each z and prove this can be used to solve our original problem. Policy gradient based on importance sampling Equation <ref> offers an option for training a generator in an adversarial way. If we know the explicit density function, , (such as a multivariate Bernoulli distribution), then we can, using (x) as a target (keeping it fixed w.r.t. optimization of ), train the generator using the gradient of the KL-divergence:∇__KL((x) || _) = -__[(x)/β∇_log_(x)].Here, the connection to importance sampling is even clearer, and this gradient resembles other importance sampling methods for training generative models in the discrete setting <cit.>. However, we expect the variance of this estimator will be high, as it requires estimating the partition function, β (for instance, using Monte-Carlo sampling). We address reducing the variance from estimating the normalized importance weights next. Lower-variance policy gradient Let (x) = ∫_Zxz(z) dz be a probability density function with a conditional density, xz : Z→ [0, 1]^d (e.g., a multivariate Bernoulli distribution), and prior over z, (z). Let α(z) = _xz[(x)] = ∫_Xxz(x) dx be a partition function over the conditional distribution. Let us define xz = w(x)/α(z)xz as the (normalized) conditional distribution weighted by w(x)/α(z). The expected conditional KL-divergence over (z) is:_(z)[_KL(xzxz)] = ∫_Z(z) _KL(xzxz) dz Let xm∼xz be samples from the prior and (xm) = (xm)/∑_m'(xm') be a Monte-Carlo estimate of the normalized importance weights.The gradient of the expected conditional KL-divergence w.r.t. the generator parameters, , becomes:∇__(z)[_KL(xzxz)] = -_(z)[∑_m (xm) ∇_logxmz],where we have approximated the expectation using the Monte-Carlo estimate.Minimizing the expected conditional KL-divergences is stricter than minimizing the KL-divergence in Equation <ref>, as it requires all of the conditional distributions to match independently. We show that the KL-divergence of the marginal probabilities is zero when the expectation of the conditional KL-divergence is zero as well as show this estimator works better in practice in the Appendix.Algorithm <ref> describes the training procedure for discrete BGAN. This algorithm requires an additional M times more computation to compute the normalized importance weights, though these can be computed in parallel exchanging space for time. When theandare multi-variate (such as with discrete image data), we make the assumption that the observed variables are independent conditioned on Z. The importance weights, , are then applied uniformly across each of the observed variables. Connection to policy gradients REINFORCE is a common technique for dealing with discrete data in GANs <cit.>. Equation <ref> is a policy gradient in the special case that the reward is the normalized importance weights. This reward approaches the likelihood ratio in the non-parametric limit of an optimal discriminator. Here, we make another connection to REINFORCE as it is commonly used, with baselines, by deriving the gradient of the reversed KL-divergence.Let _(x) be defined as above whereT(_(x)) = e^_(x). Consider the gradient of the reversed KL-divergence:∇__KL()= -_(z)[∑_m (log(xm) - logβ + 1) ∇_logxmz] = -_(z)[∑_m (_(x) - b) ∇_logxmz] From this, it is clear that we can consider the output of the statistic network, _(x), to be a reward and b = logβ = _[(x)] to be the analog of a baseline.[Note that we have removed the additional constant as _[1 * ∇_] = 0] This gradient is similar to those used in previous works on discrete GANs, which we discuss in more detail in Section <ref>. §.§ Continuous variables and the stability of GANsFor continuous variables, minimizing the variational lower-bound suffices as an optimization technique as we have the full benefit of back-propagation to train the generator parameters, . However, while the convergence of the discriminator is straightforward, to our knowledge there is no general proof of convergence for the generator except in the non-parametric limit or near-optimal case. What's worse is the value function can be arbitrarily large and negative. Let us assume that max = M < ∞ is unique. Asis convex, the minimum of the lower-bound overis:inf___= inf__[_(x)] - __[(_(x))]= _[_(x)] - sup___[(_(x))] = _[_(x)] - (M).In other words, the generator objective is optimal when the generated distribution, _, is nonzero only for the set {x |(x) = M }. Even outside this worst-case scenario, the additional consequence of this minimization is that this variational lower-bound can become looser w.r.t. the -divergence, with no guarantee that the generator would actually improve. Generally, this is avoided by training the discriminator in conjunction with the generator, possibly for many steps for every generator update. However, this clearly remains one source of potential instability in GANs.Equation <ref> reveals an alternate objective for the generator that should improve stability. Notably, we observe that for a given estimator, (x), (x) matches when (x) = ()((x)) = 1.Let _ : Z→X be a generator function that takes as input a latent variable drawn from a simple prior, z ∼(z). Let _ and (x) be defined as above. We define the continuous BGAN objective as: = _ (log(_(z)))^2. We chose the log, as with our treatments of -divergences in Table <ref>, the objective is just the square of the statistic network output:= __(_(z))^2. This objective can be seen as changing a concave optimization problem (which is poor convergence properties) to a convex one.§ RELATED WORK AND DISCUSSIONOn estimating likelihood ratios from the discriminator Our work relies on estimating the likelihood ratio from the discriminator, the theoretical foundation of which we draw from f-GAN <cit.>. The connection between the likelihood ratios and the policy gradient is known in previous literature <cit.>, and the connection between the discriminator output and the likelihood ratio was also made in the context of continuous GANs <cit.>.However, our work is the first to successfully formulate and apply this approach to the discrete setting. Importance sampling Our method is very similar to re-weighted wake-sleep <cit.>, which is a method for training Helmholtz machines with discrete variables. RWS also relies on minimizing the KL divergence, the gradients of which also involve a policy gradient over the likelihood ratio. Neural variational inference and learning <cit.>, on the other hand, relies on the reverse KL. These two methods are analogous to our importance sampling and REINFORCE-based BGAN formulations above. GAN for discrete variables Training GANs with discrete data is an active and unsolved area of research, particularly with language model data involving recurrent neural network (RNN) generators <cit.>. Many REINFORCE-based methods have been proposed for language modeling <cit.> which are similar to our REINFORCE-based BGAN formulation and effectively use the sigmoid of the estimated log-likelihood ratio. The primary focus of these works however is on improving credit assignment, and their approaches are compatible with the policy gradients provided in our work.There have also been some improvements recently on training GANs on language data by rephrasing the problem into a GAN over some continuous space <cit.>. However, each of these works bypass the difficulty of training GANs with discrete data by rephrasing the deterministic game in terms of continuous latent variables or simply ignoring the discrete sampling process altogether, and do not directly solve the problem of optimizing the generator from a difference measure estimated from the discriminator. Remarks on stabilizing adversarial learning, IPMs, and regularization A number of variants of GANs have been introduced recently to address stability issues with GANs. Specifically, generated samples tend to collapse to a set of singular values that resemble the data on neither a per-sample or distribution basis. Several early attempts in modifying the train procedure <cit.> as well as the identifying of a taxonomy of working architectures <cit.> addressed stability in some limited setting, but it wasn't until Wassertstein GANs <cit.> were introduced that there was any significant progress on reliable training of GANs.WGANs rely on an integral probability metric <cit.> that is the dual to the Wasserstein distance. Other GANs based on IPMs, such as Fisher GAN <cit.> tout improved stability in training. In contrast to GANs based on -divergences, besides being based on metrics that are “weak", IPMs rely on restrictingto a subset of all possible functions. For instance in WGANs, = {|_L≤ K}, is the set of K-Lipschitz functions. Ensuring a statistic network, , with a large number of parameters is Lipschitz-continuous is hard, and these methods rely on some sort of regularization to satisfy the necessary constraints. This includes the original formulation of WGANs, which relied on weight-clipping, and a later work <cit.> which used a gradient penalty over interpolations between real and generated data.Unfortunately, the above works provide little details on whetheris actually in the constrained set in practice, as this is probably very hard to evaluate in the high-dimensional setting. Recently, <cit.> introduced a gradient norm penalty similar to that in <cit.> without interpolations and which is formulated in terms of -divergences. In our work, we've found that this approach greatly improves stability, and we use it in nearly all of our results. That said, it is still unclear empirically how the discriminator objective plays a strong role in stabilizing adversarial learning, but at this time it appears that correctly regularizing the discriminator is sufficient.§ DISCRETE VARIABLES: EXPERIMENTS AND RESULTS§.§ Adversarial classificationWe first verify the gradient estimator provided by BGAN works quantitatively in the discrete setting by evaluating its ability to train a classifier with the CIFAR-10 dataset <cit.>. The “generator" in this setting is a multinomial distribution, yx modeled by the softmax output of a neural network. The discriminator, (x, y), takes as input an image / label pair so that the variational lower-bound is:_X Y_Y|X_X_ = _(x, y)[_(x, y)] - _yx(x)[(_(x, y))]For these experiments, we used a simple 4-layer convolutional neural network with an additional 3 fully-connected layers. We trained the importance sampling BGAN on the set of -divergences given in Table <ref> as well as the REINFORCE counterpart for 200 epochs and report the accuracy on the test set. In addition, we ran a simple classification baseline trained on cross-entropy as well as a continuous approximation to the problem as used in WGAN-based approaches <cit.>. No regularization other than batch normalization <cit.> was used with the generator, while gradient norm penalty <cit.> was used on the statistic networks. For WGAN, we used clipping, and chose the clipping parameter, the number of discriminator updates, and the learning rate separately based on training set performance. The baseline for the REINFORCE method was learned using a moving average of the reward.Our results are summarized in Table <ref>. Overall, BGAN performed similarly to the baseline on the test set, with the REINFORCE method performing only slightly worse. For WGAN, despite our best efforts, we could only achieve an error rate of 72.3% on the test set, and this was after a total of 600 epochs to train. Our efforts to train WGAN using gradient penalty failed completely, despite it working with higher-dimension discrete data (see Appendix). §.§ Discrete image and natural language generationImage data: binary MNIST and quantized CelebAWe tested BGAN using two imaging benchmarks: the common discretized MNIST dataset <cit.> and a new quantized version of the CelebA dataset <cit.>.For CelebA quantization, we first downsampled the images from 64×64 to 32×32. We then generated a 16-color palette using Pillow, a fork of the Python Imaging Project (https://python-pillow.orghttps://python-pillow.org). This palette was then used to quantize the RGB values of the CelebA samples to a one-hot representation of 16 colors. Our models used deep convolutional GANs <cit.>. The generator is fed a vector of 64 i.i.d. random variables drawn from a uniform distribution, [0, 1]. The output nonlinearity was sigmoid for MNIST to model the Bernoulli centers for each pixel, while the output was softmax for quantized CelebA.Our results show that training the importance-weighted BGAN on discrete MNIST data is stable and produces realistic and highly variable generated handwritten digits (Figure <ref>). Further quantitative experiments comparing BGAN against WGAN with the gradient penalty <cit.> showed that when training a new discriminator on the samples directly (keeping the generator fixed), the final estimated distance measures were higher (i.e., worse) for WGAN-GP than BGAN, even when comparing using the Wasserstein distance. The complete experiment and results are provided in the Appendix. For quantized CelebA, the generator trained as a BGAN produced reasonably realistic images which resemble the original dataset well and with good diversity. 1-billion word Next, we test BGAN in a natural language setting with the 1-billion word dataset <cit.>, modeling at the character-level and limiting the dataset to sentences of at least 32 and truncating to 32 characters. For character-level language generation, we follow the architecture of recent work <cit.>, and use deep convolutional neural networks for both the generator and discriminator.Training with BGAN yielded stable, reliably good character-level generation (Table <ref>), though generation is poor compared to recurrent neural network-based methods <cit.>. However, we are not aware of any previous work in which a discrete GAN, without any continuous relaxation <cit.>, was successfully trained from scratch without pretraining and without an auxiliary supervised loss to generate any sensible text. Despite the low quality of the text relative to supervised recurrent language models, the result demonstrates the stability and capability of the proposed boundary-seeking criterion for training discrete GANs.§ CONTINUOUS VARIABLES: EXPERIMENTS AND RESULTSHere we present results for training the generator on the boundary-seeking objective function. In these experiments, we use the original GAN variational lower-bound from  <cit.>, only modifying the generator function. All results use gradient norm regularization <cit.> to ensure stability. §.§ Generation benchmarksWe test here the ability of continuous BGAN to train on high-dimensional data. In these experiments, we train on the CelebA, LSUN <cit.> datasets, and the 2012 ImageNet dataset with all 1000 labels <cit.>. The discriminator and generator were both modeled as 4-layer Resnets <cit.> without conditioning on labels or attributes.Figure <ref> shows examples from BGAN trained on these datasets. Overall, the sample quality is very good. Notably, our Imagenet model produces samples that are high quality, despite not being trained conditioned on the label and on the full dataset. However, the story here may not be that BGAN necessarily generates better images than using the variational lower-bound to train the generator, since we found that images of similar quality on CelebA could be attained without the boundary-seeking loss as long as gradient norm regularization was used, rather we confirm that BGAN works well in the high-dimensional setting. §.§ Stability of continuous BGANAs mentioned above, gradient norm regularization greatly improves stability and allows for training with very large architectures. However, training still relies on a delicate balance between the generator and discriminator: over-training the generator may destabilize learning and lead to worse results. We find that the BGAN objective is resilient to such over-training. Stability in training with an overoptimized generator To test this, we train on the CIFAR-10 dataset using a simple DCGAN architecture. We use the original GAN objective for the discriminator, but vary the generator loss as the variational lower-bound, the proxy loss <cit.>, and the boundary-seeking loss (BGAN). To better study the effect of these losses, we update the generator for 5 steps for every discriminator step.Our results (Figure <ref>) show that over-optimizing the generator significantly degrades sample quality. However, in this difficult setting, BGAN learns to generate reasonable samples in fewer epochs than other objective functions, demonstrating improved stability. Following the generator gradient We further test the different objectives by looking at the effect of gradient descent on the pixels. In this setting, we train a DCGAN <cit.> using the proxy loss. We then optimize the discriminator by training it for another 1000 updates. Next, we perform gradient descent directly on the pixels, the original variational lower-bound, the proxy, and the boundary seeking losses separately.Our results show that following the BGAN objective at the pixel-level causes the least degradation of image quality. This indicates that, in training, the BGAN objective is the least likely to disrupt adversarial learning.§ CONCLUSION Reinterpreting the generator objective to match the proposal target distribution reveals a novel learning algorithm for training a generative adversarial network <cit.>.This proposed approach of boundary-seeking provides us with a unified framework under which learning algorithms for both discrete and continuous variables are derived. Empirically, we verified our approach quantitatively and showed the effectiveness of training a GAN with the proposed learning algorithm, which we call a boundary-seeking GAN (BGAN), on both discrete and continuous variables, as well as demonstrated some properties of stability. § ACKNOWLEDGEMENTSRDH thanks IVADO, MILA, UdeM, NIH grants R01EB006841 and P20GM103472, and NSF grant 1539067 for support. APJ thanks UWaterloo, Waterloo AI lab and MILA for their support and Michael Noukhovitch, Pascal Poupart for constructive discussions. KC thanks AdeptMind, TenCent, eBay, Google (Faculty Awards 2015, 2016), NVIDIA Corporation (NVAIL) and Facebook for their support. YB thanks CIFAR, NSERC, IBM, Google, Facebook and Microsoft for their support. We would like to thank Simon Sebbagh for his input and help with Theorem 2. Finally, we wish to thank the developers of Theano <cit.>, Lasagne <http://lasagne.readthedocs.io>, and Fuel <cit.> for their valuable code-base. icml2017§ APPENDIX §.§ Comparison of discrete methodsIn these experiments, we produce some quantitative measures for BGAN against WGAN with the gradient penalty <cit.> on the discrete MNIST dataset. In order to use back-propagation to train the generator, WGAN-GP uses the softmax probabilities directly, bypassing the sampling process at pixel-level and problems associated with estimating gradients through discrete processes. Despite this, WGAN-GP is been able to produce samples that visually resemble the target dataset.Here, we train 3 models on the discrete MNIST dataset using identical architectures with the BGAN with the JS and reverse KL -divergences and WGAN-GP objectives. Each model was trained for 300 generator epochs, with the discriminator being updated 5 times per generator update for WGAN-GP and 1 time per generator update for the BGAN models (in other words, the generators were trained for the same number of updates). This model selection procedure was chosen as the difference measure (i.e., JSD, reverse KL divergence, and Wasserstein distance) as estimated during training converged for each model. WGAN-GP was trained with a gradient penalty hyper-parameter of 5.0, which did not differ from the suggested 10.0 in our experiments with discrete MNIST. The BGAN models were trained with the gradient norm penalty of 5.0 <cit.>.Next, for each model, we trained 3 new discriminators with double capacity (twice as many hidden units on each layer) to maximize the the JS and reverse KL divergences and Wasserstein distance, keeping the generators fixed. These discriminators were trained for 200 epochs (chosen from convergence) with the same gradient-based regularizations as above. For all of these models, the discriminators were trained using the samples, as they would be used in practical applications. For comparison, we also trained an additional discriminator, evaluating the WGAN-GP model above on the Wasserstein distance using the softmax probabilities.Final evaluation was done by estimating difference measures using 60000 MNIST training examples againt 60000 samples from each generator, averaged over 12 batches of 5000. We used the training set as this is the distribution over which the discriminators were trained. Test set estimates in general were close and did not diverge from training set distances, indicating the discriminators were not overfitting, but training set estimates were slightly higher on average.Our results show that the estimates from the sampling distribution from BGAN is consistently lower than that from WGAN-GP, even when evaluating using the Wasserstein distance. However, when training the discriminator on the softmax probabilities, WGAN-GP has a much lower Wasserstein distance. Despite quantitative differences, samples from these different models were indistinguishable as far as quality by visual inspection. This indicates that, though playing the adversarial game using the softmax outputs can generate realistic-looking samples, this procedure ultimately hurts the generator's ability to model a truly discrete distribution. §.§ Theoretical and empirical validation of the variance reduction methodHere we validate the policy gradient provided in Equation <ref> theoretically and empirically.Let the expectation of the conditional KL-divergence be defined as in Equation <ref>.Then _(z)[_KL(xzxz)] = 0 _KL((x) || _) = 0.As the conditional KL-divergence is has an absolute minimum at zero, the expectation can only be zero when the all of the conditional KL-divergences are zero. In other words:_(z)[_KL(xzxz)] = 0 xz = xz.As per the definition of xz, this implies that α(z) = (x) = C is a constant. If (x) is a constant, then the partition function β = C _[1] = C is a constant. Finally, when (x)/β = 1, (x) = __KL((x) || _) = 0. In order to empirically evaluate the effect of using an Monte-Carlo estimate of β from Equation <ref> versus the variance-reducing method in Equation <ref>, we trained several models using various sample sizes from the prior, (z), and the conditional, xz.We compare both methods with 64 samples from the prior and 5, 10, and 100 samples from the conditional.In addition, we compare to a model that estimates β using 640 samples from the prior and a single sample from the conditional. These models were all run on discrete MNIST for 50 epochs with the same architecture as those from Section <ref> with a gradient penalty of 1.0, which was the minimum needed to ensure stability in nearly all the models.Our results (Figure <ref>) show a clear improvement using the variance-reducing method from Equation <ref> over estimating β. Wall-clock times were nearly identical for methods using the same number of total samples (blue, green, and red dashed and solid line pairs). Both methods improve as the number of conditional samples is increased.

http://arxiv.org/abs/1702.08431v4

[1]Zuse Institute Berlin, Takustrasse 7, 14195 Berlin, Germany.Email: wei.zhang@fu-berlin.deErgodic SDEs on submanifolds and related numerical sampling schemesWei Zhang 1=================================================================== In many applications, it is often necessary to sample the mean value of certain quantity with respect to a probabilitymeasure μ on the level set of a smooth function ξ:ℝ^d→ℝ^k, 1 ≤ k < d. A specially interesting case is the so-called conditional probability measure, which is useful in the study of free energy calculation and model reduction of diffusion processes.By Birkhoff's ergodic theorem, one approach to estimate the mean value is to compute the time average along an infinitely long trajectory of an ergodic diffusion process on the level set whose invariant measure is μ.Motivated by the previous work of <cit.>, as well as the work of <cit.>, in this paper we construct a family of ergodic diffusion processes onthe level set of ξ whose invariant measures coincide with the given one. For the conditional measure,we propose a consistent numerical scheme which samples the conditional measure asymptotically. The numerical scheme doesn't require computing the second derivatives of ξ and the error estimates of its long time sampling efficiency are obtained. § INTRODUCTION Many stochastic dynamical systems in real-world applications in physics, chemistry, and biology often involve a large number of degrees of freedom which evolve on vastly different time scales. Understanding the behavior of these systems can be highly challenging due to the high dimensionality and the existence of multiple time scales.To tackle these difficulties, the terminology reaction coordinate, or collective variable, is often introduced to help describe the essential dynamical behavior of complex systems <cit.>.In various research topics, in particular those related to molecular dynamics,one often encounters the problem of computing the mean value of certain quantity on the level set Σ=ξ^-1(0)={ x ∈ℝ^d  | ξ(x) = 0∈ℝ^k}of a reaction coordinate function ξ : ℝ^d →ℝ^k, 1 ≤ k <d. Among different probability measures on Σ, the one defined bydμ_1 = 1/Z e^-β U[(∇ξ^T ∇ξ)]^-1/2 dνis especially relevant in applications and is sometimes called the conditional probability measure on Σ. In (<ref>), the parameter β > 0, U: ℝ^d →ℝ is a smooth function, Z is the normalization constant,∇ξ denotes the d × k Jacobian matrix of the map ξ,and ν is the surface measure on Σ induced from the Lebesgue measure on ℝ^d. The probability measure μ_1 has a probabilistic interpretation, and the numerical computation of the mean value f = ∫_Σ f(x)dμ_1(x)for a function f on the level set is involved in various contexts, such asfree energy calculations based on the thermodynamics integration formula <cit.>.Applying Birkhoff's ergodic theorem, the mean value f can be approximated by the time average1/T∫_0^T f(X_s) ds along a long trajectory of the process X_s which evolves on the level set Σ and has the invariant measure μ_1.For this purpose, it is helpful to construct a diffusion process on the level set with the correct invariant measure μ_1, i.e., to write down the stochastic differential equation (SDE) of X_s in ℝ^d.While finding such a SDE is trivial in the linear reaction coordinate case <cit.>, it is not obvious when the reaction coordinate ξ is a nonlinear function of system's state. In the literature, the problem finding SDEs on the level set of the reaction coordinate function with a given invariant measure has been considered in thestudy of free energy calculations <cit.>. Given a smooth function U : ℝ^d →ℝ,the authors in <cit.> constructed a diffusion process Y_s on Σ whose unique invariant measure is μ_2, given by dμ_2 = 1/Z e^-β U dν .It is also shown in <cit.> that this process Y_s can be obtained by projectingthe dynamics d Y_s = -∇ U (Y_s)ds + √(2β^-1) dW_sfrom ℝ^d onto the level set Σ, whereW_s = (W_s^1, ⋯, W_s^d)^T is a d-dimensional Brownian motion.The dynamics Y_s can be used to sample μ_2, and therefore to sample theconditional measure μ_1 in (<ref>) as well, by either modifyingthe potential U or reweighting the function f according to the factor [(∇ξ^T ∇ξ)]^-1/2. In a more recent work <cit.>,the authors studied the constrained Langevin dynamics, which evolves on the submanifold of the entire phase spaceincluding both position and momentum. It is shown in <cit.> that the position components of the constrained Langevin dynamics has the marginal invariant measure which coincides with μ_2. Therefore, it can also be used to compute the average f with respect to the conditional measure μ_1 (by either modifying the potential or reweighting f according to[(∇ξ^T ∇ξ)]^-1/2). Detailed studies on the numerical schemes as well as applications of the constrained Langevin dynamics have been carried out in <cit.>. The same conditional probability measure μ_1 in (<ref>), as well asthe average f in (<ref>), also plays an important role in the study of the effective dynamics of diffusion processes <cit.>. As a generalization of the dynamics (<ref>), the diffusion process d Y^i_s = -(a_ij∂ U/∂ x_j) (Y_s) ds + 1/β∂ a_ij/∂ x_j(Y_s)ds + √(2β^-1)σ_ij(Y_s)dW^j_s ,1 ≤ i ≤ d ,and its effective dynamics have been considered in <cit.>, where the matrix-valued coefficients σ, a : ℝ^d →ℝ^d× d are related by a = σσ^T, such that a is uniformly positive definite. Notice that,(<ref>) is written in component-wise form with Einstein's summation convention (the same Einstein's summation convention will be used throughout this paper, whenever no ambiguity will arise), and it reduces to (<ref>) when σ=a =. The infinitesimal generator of (<ref>) can be written as ℒ = e^β U/β∂/∂ x_i(e^-β Ua_ij∂/∂ x_j) .Under mild conditions on U, it is known that, for any (smooth, uniformly positive definite) coefficient a, the dynamics (<ref>) has the common unique invariant measure whose probability density is 1/Ze^-β U with respect to the Lebesgue measure on ℝ^d.Motivated by these previous work, in this paper we try to answer the following two questions. (Q1)Besides the process constructed in <cit.> that is closely related to (<ref>), can we obtain other diffusion processeson Σ, which are probably related to (<ref>) involving the coefficients σ, a, and have the same invariant measure?In particular, can we construct SDEs on Σ whose invariant measure is μ_1?(Q2)Numerically, instead of sampling μ_2, can we directly estimate the mean value in (<ref>) with respect to μ_1, preferably with a numerical algorithm that is easy to implement? The main contributions of the current work are related to the above questions and are summarized below. First, concerning Question (Q1),in Theorem <ref> of Section <ref>, we will construct a family of diffusion processes on Σ which sample either μ_1 or μ_2. In particular, we show that the diffusion processdX_s^i= - (Pa)_ij∂ U/∂ x_j ds + 1/β∂ (Pa)_ij/∂ x_j ds + √(2β^-1)P_j,idW_s^j ,1 ≤ i ≤ d ,evolves on Σ and the invariant measure is the conditional probability measure μ_1 in (<ref>), where the projection map P and the invertible k × k symmetric matrix Ψ are given by P=- a∇ξΨ^-1∇ξ^T,Ψ = ∇ξ^T a ∇ξ.Correspondingly, the infinitesimal generator of (<ref>) is ℒ = e^β U/β∂/∂ x_i(e^-β U(Pa)_ij∂/∂ x_j) ,which should be compared to the infinitesimal generator in (<ref>). Second, concerning Question (Q2), in Section <ref> we study a numerical algorithm which estimates the mean value f in (<ref>). Specifically, we propose to use the numerical schemex^(l + 1/2)_i =x^(l)_i + (-a_ij∂ U/∂ x_j + 1/β∂ a_ij/∂ x_j)(x^(l))h + √(2 β^-1h) σ_ij(x^(l)) η^(l)_j ,1 ≤ i ≤ d,x^(l+1) = Θ(x^(l+1/2)) ,with x^(0)∈Σ, and to approximate f by f_n = 1/n∑_l=0^n-1 f(x^(l)). In (<ref>), h is the step-size,η^(l)=(η_1^(l), η_2^(l), ⋯, η_d^(l))^T are independent d-dimensional standard Gaussian random variables, andΘ(x) = lim_s→ +∞φ(x,s) is the limit of the flow map dφ(x,s)/ds =- (a∇ F) (φ(x,s)) ,φ(x,0) = x, ∀ x∈ℝ^d ,with F(x)=1/2|ξ(x)|^2=1/2∑_α=1^kξ_α^2(x). Following the approach developed in <cit.>, in Theorem <ref>, we obtain the estimates of the approximation error between f_n and f. While different constraint approaches have been proposed in the literature <cit.>, to the best of the author's knowledge, constraint using the flow map φ has not been studied yet. Let us comment on the two contributions mentioned above. First, knowing the SDE (<ref>) and the expression (<ref>) of its infinitesimal generator ℒ is helpful for analysis. In fact, in Section <ref>,the analysis of sampling error estimate of the scheme (<ref>)relies on Poisson equation on Σ related to ℒ in (<ref>).Furthermore, (<ref>) plays a role in the work <cit.> in analyzing the approximation quality of the effective dynamics, while SDE (<ref>) has been used in <cit.> to study fluctuation relations and Jarzynski's equality for nonequilibrium systems. Second, we emphasize that Θ(x) in the scheme (<ref>) can be evaluated by solving the ODE (<ref>) starting from x.Although Θ is defined as the limit when s → +∞, in many cases the computational cost is not large, due to the exponential convergence of the (gradient) flow (<ref>) to its limit, particularly for the initial state x=x^(l+1/2) that is close to Σ.Furthermore, comparing to the direct (Euler-Maruyama) discretization of SDE (<ref>) which may deviate from Σ and requires second order derivatives of ξ, the scheme (<ref>) satisfies x^(l)∈Σ for all l ≥ 0, andit doesn't require computing the second order derivatives of ξ.Therefore, we expect the numerical scheme (<ref>)–(<ref>) is both stable and relatively easy to implement. Readers are referred to Remark <ref>–<ref> in Section <ref> and Example 1 in Section <ref> for further algorithmic discussions. In the following, we briefly explain the approach that we will use to study Question (Q1), as well as the idea behind the scheme (<ref>)–(<ref>). Concerning Question (Q1), we take the manifold point of view by considering ℝ^d as a Riemannian manifold ℳ=(ℝ^d, g) with the metric g=a^-1, defined by g(u, v) = ⟨u, v⟩_g = u_i (a^-1)_ij v_j , ∀ u, v∈ℝ^d .A useful observation is that, for ℒ in (<ref>), we have <cit.>ℒ f= [-^ℳ(U + 1/2βln G) + 1/βΔ^ℳ] f , ∀   f : ℝ^d →ℝ ,where G=g, and ^ℳ, Δ^ℳ denote the gradient and the Laplacian-Beltrami operator on ℳ, respectively.Accordingly, (<ref>) can be written as a SDE on ℳ asdY_s = -^ℳ(U + 1/2βln G)ds + √(2β^-1) dB_s ,where B_s is the Brownian motion on ℳ <cit.>.Conversely, SDE (<ref>) can be seen as the equation of (<ref>) under the (global) coordinate chart of ℳ. This equivalence allows us to study (<ref>) on ℝ^d by the corresponding SDE (<ref>) on manifold ℳ.Comparing to (<ref>), one advantage of working with the abstract equation (<ref>) is that the invariant measure of (<ref>) can be recognized as easily as in (<ref>), provided that we apply integration by parts formula on the manifold ℳ. A family of ergodic SDEs on Σ (i.e., Question (Q1)) is obtained by taking the same manifold point of view. Specifically, consider Σ as a submanifold of ℳ and denote by ^Σ, Δ^Σ, B_s the gradient operator, the Laplacian and the Brownian motion(with generator 1/2Δ^Σ <cit.>) on Σ, respectively. Since the infinitesimal generator of the SDE dY_s = - ^Σ U ds + √(2 β^-1) dB_sis ℒ = -^Σ U + 1/βΔ^Σ,under mild assumptions on U, it is straightforward to verify that dynamics (<ref>)evolves on Σ and has the unique invariant measure 1/Ze^-β Udν_g, where ν_g is the surface measure on Σ induced from the metric g=a^-1 on ℳ=(ℝ^d, g).Therefore, answering Question (Q1) boils down to calculating theexpression of (<ref>) under the coordinatechart of ℳ (not Σ). This will be achieved by calculating the expressionsof ^Σ, Δ^Σ under the coordinate chart ofℳ and then figuring out therelation between the two measures ν and ν_g.Concerning the idea behind the numerical scheme (<ref>)–(<ref>), we recall that one way to (approximately) sample μ_1 on Σ is to constrain the dynamics (<ref>) in the neighborhood of Σ by adding an extra potential to it.This is often termed as softly constrained dynamics <cit.> and has been widely used in applications. In this context, one consider the dynamicsd X^ϵ,i_s = [- a_ij∂ U/∂ x_j- 1/ϵa_ij∂/∂ x_j(1/2∑_α=1^k ξ_α^2) + 1/β∂ a_ij/∂ x_j]ds + √(2β^-1)σ_ijdW^j_s ,where ϵ > 0, 1 ≤ i ≤ d, based on the fact that the invariant measure of (<ref>) converges weakly to μ_1, as ϵ→ 0. The dynamics (<ref>) stays close to Σ most of the time, thanks to the existence of the extra constraint force. Furthermore, only the first order derivatives of ξ are involved. In spite of these nice properties, however, direct simulation of (<ref>) is inefficient when ϵ is small,because the time step-size in numerical simulations becomes severely limited due to the strong stiffness in the dynamics.Indeed, our numerical scheme is motivated in order to overcome the aforementioned drawback of the softly constrained dynamics (<ref>), and the scheme (<ref>)–(<ref>) can be viewed as a multiscale numerical method for (<ref>), where the stiff and non-stiff terms in(<ref>) are handled separately <cit.>.In contrast to the previous work <cit.>, where the convergence of (<ref>) was studied on a finite time interval, our result concerns the long time sampling efficiency of the discretized numerical scheme.Before concluding this introduction, we compare the current work with several previous ones. Generally speaking, Monte Carlo samplers (based on ergodicity) either on ℝ^d or on its submanifolds can be classified into Metropolis-adjusted samplers and samplers without Metropolis step (unadjusted). For Metropolis-adjusted methods, in particular, exploiting Riemannian geometry structure to develop MCMC methods has been studied in <cit.>. The authors there demonstrated that incorporating the geometry of the space into numerical methods can lead to significant improvement of the sampling efficiency.In line with this development, in Section <ref> we will consider a concrete example where a non-constant matrix a can help remove the stiffness in the sampling task. On the other hand, despite of the common Riemannian manifold point of view in the current work and in <cit.>,the main difference is that the current work deals with sampling on the submanifold Σ instead of the entire ℝ^d (or its domain). The derivations in the current work are more involved mainly due to this difference. Besides sampling on the entire space, Metropolis-adjusted samplers on submanifolds, using either MCMC or Hybrid Monte Carlo, have been considered in several recent work <cit.>. Reversible Metropolis random walk on submanifolds has been constructed in <cit.>, which is then extended in <cit.> by allowing non-zero gradient forces in the proposal move.In contrast to these Metropolis-adjusted samplers, the numerical scheme (<ref>)–(<ref>) in the current work is unadjusted (without Metropolis-step) and samples the conditional probability measure μ_1 when the step-size h→ 0. This means that in practice the step-size h should be chosen properly such that the discretization error is tolerable. In this direction, we point out that unadjusted samplers on ℝ^d, which naturally arise from discretizations of SDEs, have been well studied in the literature <cit.>. The current work can be thought as a further step along this direction for sampling schemes on submanifolds, by applying the machinery developed in <cit.>.Comparison between the scheme (<ref>) and the Metropolis-adjusted algorithm in <cit.> can be found in Remark <ref>, as well as in Example 2 in Section <ref>. We also refer to <cit.> for related discussions.The rest of the paper is organized as follows. In Section <ref>, we construct ergodic SDEs on Σ which sample either μ_1 or μ_2. In Section <ref>, we study the numerical scheme (<ref>)–(<ref>) and quantify its approximation error in estimating the mean value in (<ref>). In Section <ref>, we demonstrate our results through concrete examples. Conclusions and further discussions are made in Section <ref>. Technical details related to the Riemannian manifold ℳ in Section <ref> are included in Appendix <ref>. Proofs of the results in Section <ref> are collected in Appendix <ref>.Finally, we conclude this introduction with the assumptions which will be made (implicitly) throughout this paper.The matrix σ: ℝ^d →ℝ^d × d is both smooth and invertible at each x ∈ℝ^d. The matrix a=σσ^T is uniformly positive definite with uniformly bounded inverse a^-1.The function ξ: ℝ^d →ℝ^k is C^2 smooth and the level set Σ is both connected and compact, such that(∇ξ) = k at each x ∈Σ.§ SDES OF ERGODIC DIFFUSION PROCESSES ON Σ In this section, we construct SDEs of ergodic processes on Σ that sample a given invariant measure. The main result of this section is Theorem <ref>, which shows that the invariant measure of the SDE (<ref>) in Introduction is the conditional probability measure μ_1 in (<ref>). Readers who are mainly interested in numerical algorithms can read Theorem <ref> and then directly jump to Section <ref>. First of all, let us point out that, the semigroup approach based on functional inequalities on Riemannian manifolds is well developedto study the solution of Fokker-Planck equation towards equilibrium. One sufficient condition for the exponential convergence of the Fokker-Planck equation (and therefore the ergodicity of the corresponding dynamics) is the famous Bakry-Emery criterion <cit.>. In particular, concrete conditions are given in <cit.> which guarantee the exponential convergence to the unique invariant measure. In the following, we will always assume that the potential U ∈ C^∞(Σ) and the Bakry-Emery condition in <cit.> is satisfied.Recall that ℳ=(ℝ^d, g), where g=a^-1 and ν_g is the surface measure on Σ induced from ℳ. Matrices P, P_j,i are given in (<ref>), (<ref>) in Appendix <ref>, respectively. For 1 ≤ i ≤ d, e_i denotes the vector whose ith component equals to 1 while all the other d-1 components equal to 0. We refer the reader to Appendix <ref> for further details. Let us first consider theprobability measure μ on Σ given by dμ = 1/Z e^-β U dν_g, where β > 0 and Z is a normalization constant.The following proposition is a direct application of Proposition <ref> in Appendix <ref>.Consider the dynamics on ℝ^d which satisfies the Ito SDE dY_s^i =- (Pa)_ij∂[U - 1/2βln(( a)^-1(∇ξ^T a ∇ξ))]/∂ x_j ds + 1/β∂ (Pa)_ij/∂ x_j ds + √(2β^-1)P_j,idW_s^jfor 1 ≤ i ≤ d, where W_s = (W_s^1, W_s^2, ⋯, W_s^d)^T is a d-dimensional Brownian motion. Suppose Y_0 ∈Σ, then Y_s ∈Σ almost surely for s≥ 0.Furthermore, it has a unique invariant measure μ given by dμ = 1/Z e^-β U dν_g.Using (<ref>) and Proposition <ref> in Appendix <ref>, we know that the infinitesimal generatorof SDE (<ref>) is ℒf = -⟨^Σ U, ^Σ f⟩_g + 1/βΔ^Σ f , ∀ f: Σ→ℝ ,where ^Σ, Δ^Σ are the gradient and Laplace-Beltrami operators on Σ of ℳ, respectively. Applying Ito's formula to ξ_α(Y_s), we have dξ_α(Y_s) = ℒξ_α(Y_s) ds + √(2β^-1)∂ξ_α(Y_s)/∂ x_i P_j, i dW_s^j ,1 ≤α≤ k .Using (<ref>), (<ref>) in Appendix <ref>, and the fact that ^Σξ_α = P^ℳξ_α=0, it is straightforward to verify thatℒξ_α = -⟨^Σ U, ^Σξ_α⟩_g + 1/β^Σ(^Σξ_α) = 0,∂ξ_α/∂ x_i P_j, i = 0,1 ≤ j ≤ d ,on Σ, which implies dξ_α(Y_s) = 0, ∀ s ≥ 0. Since Y_0 ∈Σ, we conclude that ξ_α(Y_s) = ξ_α(Y_0) = 0a.s. s ≥ 0, for 1 ≤α≤ k, and therefore Y_s ∈Σ fors ≥ 0, almost surely. Using the expression (<ref>) of ℒ and the integration by parts formula (<ref>) in Appendix <ref>, it is easy to see that μ is an invariant measure of the dynamics (<ref>).The uniqueness is implied by the exponential convergence result established in <cit.>, since we assume Bakry-Emery condition is satisfied. In the above, we have considered the level set Σ as a submanifold of ℳ=(ℝ^d, g).In applications, on the other hand, it is natural to view Σ as a submanifold of the standard Euclidean space ℝ^d, with the surface measure ν on Σ that is induced from the Euclidean metric on ℝ^d.As already mentioned in the Introduction, the following two probability measures dμ_1 = 1/Z e^-β U[(∇ξ^T ∇ξ)]^-1/2 dν , dμ_2 = 1/Z e^-β U dν,where Z denotes possibly different normalization constants, are often interesting and arise in many situations <cit.>. In particular, μ_1 has a probabilistic interpretation andoften appears in the study of free energy calculation and model reduction of stochastic dynamics <cit.>.In order to construct processes which sample μ_1 or μ_2, we need tofigure out the relations between the two surface measures ν_g and ν on Σ. Let ν_g, ν be the surface measures on Σ induced from the metric g=a^-1 and the Euclidean metric on ℝ^d, respectively. We have dν_g = ( a)^-1/2[(∇ξ^T a∇ξ)/(∇ξ^T∇ξ)]^1/2 dν. Let x ∈Σ and v_1, v_2, ⋯, v_d-k be a basis of T_xΣ.Assume that v_i = c_ije_j, where c = (c_ij) is a (d-k) × d matrix whose rank is d-k. Using the fact ⟨v_i, ^ℳξ_α⟩_g = 0 for 1≤ i ≤ d-k, 1 ≤α≤ k, we can deduce thatc ∇ξ = 0.Calculating the surface measures ν_g and ν under this basis, we obtaindν_g = [(ca^-1c^T)/(cc^T)]^1/2 dν .To simplify the right hand side of (<ref>), we use the following equality [ c; ∇ξ^Ta ][ c^T∇ξ ] =[ cc^T0; ∇ξ^T a c^T∇ξ^T a ∇ξ ] =[ ca^-1;∇ξ^T ] a [ c^T∇ξ ] .After computing the determinants of the last two matrices above, we obtain(cc^T)(∇ξ^T a ∇ξ)= ( a) [ [ ca^-1;∇ξ^T ][ c^T∇ξ ]] = ( a)  [ ca^-1c^Tca^-1∇ξ;0 ∇ξ^T∇ξ ] .The conclusion follows after we substitute the above relation into (<ref>). Applying Lemma <ref> and Proposition <ref>, we can obtain ergodic processes whose invariant measures are given in (<ref>). Let μ_1, μ_2 be the two probability measures on Σ defined in (<ref>). Consider the dynamics X_s, Y_s on ℝ^d which satisfy the Ito SDEs dX_s^i= - (Pa)_ij∂ U/∂ x_j ds + 1/β∂ (Pa)_ij/∂ x_j ds + √(2β^-1)P_j,idW_s^j ,anddY_s^i = - (Pa)_ij∂[U - 1/2βln(∇ξ^T ∇ξ)]/∂ x_j ds + 1/β∂ (Pa)_ij/∂ x_j ds + √(2β^-1)P_j,idW_s^j ,for 1 ≤ i ≤ d, where β > 0 and W_s = (W_s^1, W_s^2,⋯, W_s^d)^T is a d-dimensional Brownian motion. Suppose that X_0 , Y_0 ∈Σ, then X_s, Y_s ∈Σ almost surely for s≥ 0.Furthermore, the unique invariant probability measures of the dynamics X_s and Y_sare μ_1 and μ_2, respectively.Applying Lemma <ref>, we can rewrite the probability measures μ_1, μ_2 asdμ_1 = 1/Z e^-β U[(∇ξ^T∇ξ)]^-1/2 dν = 1/Z[-β(U + 1/2βln(∇ξ^T a ∇ξ)/ a)] dν_g ,dμ_2 = 1/Z e^-β Ud ν= 1/Z[-β(U + 1/2βln(∇ξ^T a ∇ξ)/( a) (∇ξ^T∇ξ))] dν_g ,where again Z denotes different normalization constants. Applying Proposition <ref> to the two probability measures expressed in (<ref>), we can conclude that both the dynamics X_s in (<ref>) and Y_s in (<ref>) evolve on the submanifold Σ, and theirinvariant probability measures are given by μ_1 and μ_2, respectively. Under Assumptions <ref>–<ref>, we can find a neighborhood 𝒪 of Σ, such that P can be extended to 𝒪. Furthermore, the relations in(<ref>) in Appendix <ref> are still satisfied in 𝒪. Due to this fact, in (<ref>)–(<ref>) we can talk about the derivatives of P at states x∈Σ.* Notice that, similar to (<ref>), the infinitesimal generator of X_s in (<ref>) can be written as ℒ = e^β U/β∂/∂ x_i(e^-β U(Pa)_ij∂/∂ x_j) .Using (<ref>) and (<ref>) in Appendix <ref>, we can also verify the integration by parts formula∫_Σ (ℒf) f'dμ_1 = ∫_Σ (ℒf') fdμ_1 = - 1/β∫_Σ (Pa ∇ f)·∇ f' dμ_1 ,for any two C^2 smooth functions f, f': Σ→ℝ.* Using Jacobi's formula <cit.> ∂ln (∇ξ^T∇ξ)/∂ x_j = (∇ξ^T∇ξ)^-1_αη∂ (∇ξ_α^T∇ξ_η)/∂ x_j and (Pa)_ij∂_jξ_α = 0,the equation (<ref>) can be simplified as dY_s^i = - (Pa)_ij∂ U/∂ x_j ds + 1/β Q_jl∂ (Pa)_ij/∂ x_l ds + √(2β^-1)P_j,i dW_s^j ,where the matrix Q =- ∇ξ (∇ξ^T∇ξ)^-1∇ξ^T.In the special case when g=a =, we have ν_g = ν and P_j, i = P_ji=Q_ji from (<ref>).Accordingly,we can write the dynamics (<ref>) asdY_s^i = - P_ij∂ U/∂ x_j ds+ 1/β P_lj∂ P_li/∂ x_j ds + √(2β^-1)P_jidW_s^j= - P_ij∂ U/∂ x_j ds-1/β (Ψ^-1)_αγP_lj (∂^2_ljξ_α) ∂_iξ_γ ds + √(2β^-1)P_jidW_s^j ,= - P_ij∂ U/∂ x_j ds+ 1/β H_i ds + √(2β^-1)P_jidW_s^j ,for 1 ≤ i ≤ d, where H=H_ie_i is the mean curvature vector of Σ (see Proposition <ref> in Appendix <ref>). In Stratonovich form, (<ref>) can be written as dY_s^i =-P_ij∂ U/∂ x_j ds + √(2β^-1) P_ji∘ dW_s^j , 1 ≤ i ≤ d .In this case, our results are accordant with those in <cit.>. The dynamics constructed in Proposition <ref> and Theorem <ref> are reversible on Σ, in the sense that their infinitesimal generators are self-adjoint with respect to their invariant measures. In fact, using the same idea, we can construct non-reversible ergodic SDEs on Σ as well.We will only consider the conditional probability measure μ_1, since it is more relevant in applications and the result is also simpler.Let μ_1 be the conditional probability measure on Σ defined in (<ref>). The vector field J=(J_1, J_2, ⋯, J_d)^T=J_ie_i, defined on x ∈Σ, satisfies J(x) ∈ T_x Σ, ∀ x ∈Σ ,P_ij∂ J_j/∂ x_i + J_j∂ P_ij/∂ x_i - β J_i ∂ U/∂ x_i = 0 .Consider the dynamics X_s on ℝ^d which satisfies the Ito SDE dX_s^i= J_i ds - (Pa)_ij∂ U/∂ x_j ds + 1/β∂ (Pa)_ij/∂ x_j ds + √(2β^-1)P_j,idW_s^j ,for 1 ≤ i ≤ d, where β > 0 and W_s = (W_s^1, W_s^2,⋯, W_s^d)^T is a d-dimensional Brownian motion. Suppose that X_0 ∈Σ, then X_s ∈Σ almost surely for s≥ 0.Furthermore, the unique invariant probability measure of X_s is μ_1. The proof can be found in Appendix <ref>.We make two remarks regarding the non-reversible vector J.* Notice that, as tangent vectors acting on functions, we have Pe_j = P_ij∂/∂ x_i∈ T_x Σ. Therefore, the condition (<ref>) indeed only depends on the value of J on Σ. Supposing that J and U are defined in a neighborhood 𝒪 of Σ (see Remark <ref>), the condition (<ref>) can be written equivalently as J(x) ∈ T_x Σ, ∀ x ∈Σ ,∂/∂ x_i[(P_ijJ_j)e^-β U]=0,∀ x  Σ . * Recall that, the non-reversible dynamics on ℝ^d d Y^i_s = J_i (Y_s) ds -(a_ij∂ U/∂ x_j) (Y_s) ds + 1/β∂ a_ij/∂ x_j(Y_s)ds + √(2β^-1)σ_ij(Y_s)dW^j_s ,1 ≤ i ≤ d ,has the invariant probability density 1/Z e^-β U, if the vector J=(J_1, J_2, ⋯, J_d)^T satisfies (J e^-β U) = ∂(J_i e^-β U)/∂ x_i =0,∀ x ∈ℝ^d .Comparing (<ref>) with (<ref>), it is clear that J=J|_Σ satisfies the condition (<ref>) of Corollary <ref> and can be used to construct non-reversible SDEs on Σ, provided that PJ = J in the neighborhood 𝒪. Roughly speaking, in this case the vector field J is tangential to the level sets of ξ in 𝒪. In general cases, however,we can not simply take J = PJ to obtain non-reversible processes on Σ which sample μ_1, since (<ref>) may not be satisfied.We refer to Remark <ref> in Section <ref> for an alternative idea to develop “non-reversible” numerical schemes. § NUMERICAL SCHEME SAMPLING THE CONDITIONAL MEASURE ON Σ Given a smooth function f : Σ→ℝ on the level set Σ, in this section we study the numerical scheme(<ref>)–(<ref>) in the Introduction, which allows us tonumerically compute the average f = ∫_Σ f(x) dμ_1(x)with respect to the conditional probability measure μ_1 in (<ref>).To motivate the numerical scheme, let us first introduce the softly constrained dynamics, which satisfies the SDEd X^ϵ,i_s = [- a_ij∂ U/∂ x_j- 1/ϵa_ij∂/∂ x_j(1/2∑_α=1^k ξ_α^2) + 1/β∂ a_ij/∂ x_j](X^ϵ_s)ds + √(2β^-1)σ_ij(X^ϵ_s)dW^j_s ,where ϵ > 0, 1 ≤ i ≤ d. It is straightforward to verify that (<ref>) has a unique invariant measure dμ^ϵ(x) = 1/Z^ϵexp[-β(U(x) + 1/2ϵ∑_α=1^k ξ_α^2(x))] dx ,∀ x ∈ℝ^d ,where Z^ϵ is the normalization constant.As ϵ→ 0,the authors in <cit.> studied the convergence of the dynamics (<ref>) itself on a finite time horizon in the case when a=σ = id and k=1. Closely related problems have also been studied in <cit.>.Since we are mainly interested in sampling the invariant measure,we record the following known convergence result of the measure μ^ϵ to μ_1. We omit its proof since it is a standard application of the co-area formula. Let f : ℝ^d →ℝ be a bounded smooth function.μ_ϵ is the probability measure in (<ref>) and μ_1 is the conditional probability measures on Σ defined in (<ref>). We have lim_ϵ→ 0∫_ℝ^d f(x)dμ^ϵ(x) = 1/Z∫_Σ f(x) e^-β U(x)[(∇ξ^T ∇ξ)(x)]^-1/2 dν(x) = ∫_Σ f(x) dμ_1(x) ,where Z is the normalization constant given by Z = ∫_Σ e^-β U(x)[(∇ξ^T ∇ξ)(x)]^-1/2 dν(x) . Lemma <ref> suggests that the softly constrained dynamics (<ref>) with a small ϵ is a good candidate to sample μ_1 on Σ. However, direct simulation of (<ref>) is probably inefficient when ϵ is small,because the time step-size in numerical simulations becomeslimited due to the strong stiffness in the dynamics.The numerical scheme we will study below can be viewed as a multiscale numerical method for the dynamics (<ref>). To explain the method, let us introduce the flow mapφ : ℝ^d × [0, +∞) →ℝ^d, defined bydφ(x,s)/ds =- (a∇ F) (φ(x,s)) ,φ(x,0) = x, ∀ x∈ℝ^d ,where the function F is F(x)=1/2|ξ(x)|^2=1/2∑_α=1^kξ_α^2(x) .Under proper conditions <cit.>, one can define the limiting map of φ asΘ(x) = lim_s→ +∞φ(x,s) , ∀  x ∈ℝ^d .Since ∇ F|_Σ=0 and Σ is the set consisting of all global minima of F, it is clear that Θ: ℝ^d →Σ and Θ(x) = x, for ∀ x ∈Σ.With the map Θ,we propose to approximate the average f in (<ref>) by f_n = 1/n∑_l=0^n-1 f(x^(l)),where n is a large number and the states x^(l) are sampled from the numerical schemex^(l + 1/2)_i =x^(l)_i + (-a_ij∂ U/∂ x_j + 1/β∂ a_ij/∂ x_j)h + √(2 β^-1h) σ_ijη^(l)_j ,1 ≤ i ≤ d,x^(l+1) = Θ(x^(l+1/2)) ,starting from x^(0)∈Σ. In (<ref>), h>0 is the time step-size, functions a, σ, U are evaluated at x^(l), and η^(l)=(η_1^(l), η_2^(l), ⋯, η_d^(l))^T are independent d-dimensional standard Gaussian random variables, for 0 ≤ l < n-1. We make two comments about the scheme (<ref>)–(<ref>). * Since the image of Θ is on Σ, the discrete dynamics x^(l) stays on Σ all the time. As in the case of the softly constrained dynamics (<ref>), the numerical scheme has the advantage that only the 1st order derivatives of ξ are needed.* When a=, the numerical scheme (<ref>) becomes x^(l + 1/2) =x^(l)-∇ U(x^(l)) h + √(2 β^-1h) η^(l) , x^(l+1) = Θ(x^(l+1/2)) . At each step l ≥ 0, one needs to compute Θ(x^(l+1/2)). This can be done by solving the ODE (<ref>) starting from x^(l+1/2), usingnumerical integration methods such as Runge-Kutta methods. In the following remark, we discuss issues associated with the computation of the ODE flow map Θ.* Exploiting the gradient structure of the ODE (<ref>), we can in factestablish exponential convergence of the dynamics φ to its limitΘ, at least in the neighborhood 𝒪 of Σ. For instance, we refer to <cit.> and <cit.>. Here, for brevity, we point out that the exponential decay of F(φ(x,s)) canbe easily obtained and is therefore a good candidate for the convergence criterionin numerical implementations. Actually, under Assumption <ref>–<ref>, we can supposez^TΨ(x)z ≥ c_0|z|^2, ∀ z ∈ℝ^k ,∀ x∈𝒪,for some c_0 >0, where Ψ = ∇ξ^T a ∇ξ.Direct calculation givesd F(φ(x,s))/ds = -(ξ_αΨ_αη ξ_η)(φ(x,s)) ≤-c_0|ξ(φ(x,s))|^2 = -2c_0 F(φ(x,s)) , which implies that |ξ(φ(x,s))|^2= 2F(φ(x,s)) ≤ e^-2c_0 s |ξ(x)|^2. In practice, suppose that we choose the condition |ξ(φ(x,s))| ≤ϵ_tol as the stop criterion of ODE solvers and set Θ(x) = φ(x,s_ode) when the condition is met at the time s_ode.Then the above analysis indicates that we need to integrate the ODE (<ref>) until the time s_ode=max{1/c_0(ln |ξ(x^(l+1/2))| +ln1/ϵ_tol),0},which grows logarithmically as ϵ_tol→ 0. Since x^(l+1/2) is likely to remain close to Σ when h is small, we can expect that Θ(x^(l+1/2)) can be computed up to sufficient accuracy with affordable numerical effort. * As a complement of the discussion above, we point out that adaptivity techniques (e.g., using adaptive step-sizes) can be used to accelerate the computation of the flow map Θ. For instance, instead of (<ref>), we can consider the ODE dφ̅(x,s)/ds =- (a∇ |ξ|^2-κ) (φ̅(x,s)) ,φ̅(x,0) = x,where 0 ≤κ < 1. In fact, from the identity ∇ |ξ|^2-κ = (2-κ) |ξ|^-κ∇|ξ|^2/2 = (2-κ) ∑_α=1^k ξ_α/|ξ|^κ∇ξ_α, we know that ODE (<ref>) is related to ODE (<ref>) by a rescaling of the time s. Accordingly, for each x, the solution φ̅(x,·) coincides with φ(x,·) after a reparametrization and therefore can be used to compute the projection Θ(x) as well. Furthermore, similar to (<ref>), in this case we have d F(φ̅(x,s))/ds = -(2-κ)[|ξ|^-κ (ξ_αΨ_αη ξ_η)](φ̅(x,s)) ≤ -c_0(2-κ) [2F(φ̅(x,s))]^1-κ/2 , from which we obtain |ξ(φ̅(x,s))|^κ≤ |ξ(x)|^κ - 2c_0(2-κ)s, and therefore φ̅(x,s) reaches the state Θ(x) ∈Σ before the finite time s_ode = |ξ(x)|^κ/2c_0(2-κ).In applications, Θ(x) can be computed by solving the ODE (<ref>) with a proper κ∈ [0, 1) (and decreasing step-sizes). From the above discussion, in particular the identity (<ref>), we know that this is equivalent to solving the ODE (<ref>) using adaptive step-sizes. We refer to Examples 1–2 in Section <ref> for numerical validation. Our main result of this section concerns the approximation quality of the mean value f by the running average f_n in (<ref>), in the case when h is small and n is large.For this purpose, it is necessary to study the properties of the limiting flow map Θ, since it is involved inthe numerical scheme (<ref>). In fact, we have the following important result, which characterizes the derivatives of Θ by the projection map P in (<ref>). (We refer the reader to (<ref>)–(<ref>) in Appendix <ref> for properties of P.)Let Θ be the limiting flow map in (<ref>) and P be the projection map in (<ref>). At each x ∈Σ, we have ∂Θ_i/∂ x_j =P_ij , a_lr∂^2 Θ_i/∂ x_l∂ x_r= ∂ (Pa)_il/∂ x_l -P_il∂ a_lr/∂ x_r ,for 1 ≤ i, j ≤ d.The proof of Proposition <ref> can be found in Appendix <ref>.Based on the above result, we are ready to quantify the approximation error between the estimator f_n and the mean value f.Suppose that both the step-size h and the number of the total steps n are fixed. Assume that f : Σ→ℝ is a smooth function on Σ and f is its mean value defined in (<ref>) with respect to the measure μ_1. Consider the running average f_n in (<ref>), which is computed by simulating the numerical scheme (<ref>) with time step-size h>0. Let T=nh and C denote a generic positive constant that is independent of h, n. We have the following approximation results.* |𝐄f_n - f| ≤ C(h+ 1/T).* 𝐄|f_n - f|^2 ≤ C(h^2 + 1/T).* For any 0 < ϵ < 1/2, there is an almost surely bounded positive random variable ζ(ω), such that |f_n - f| ≤ Ch+ ζ(ω)/T^1/2-ϵ , almost surely.We present the proof of Theorem <ref> in Appendix <ref>, since it is technical and the idea follows the standard approach developed in <cit.>, where Poisson equation played a crucial role. However, let us emphasize that, in contrast to <cit.>, in the current setting we are working on the submanifold Σ and, furthermore, the map Θ is involved in our numerical scheme. In particular, in the proof we use the Poisson equation related to the generator ℒ in (<ref>) of the process (<ref>), based on the fact that μ_1 is the invariant measure of (<ref>) (This is the place where Theorem <ref> in Section <ref> is used in order to establish Theorem <ref>).Theorem <ref> concerns the long time behavior of the scheme (<ref>) with a small step-size, i.e., large T and small h. This is often relevant in molecular dynamics simulations. While the estimates of Theorem <ref> are stated in terms of the variables h and T, we should point out that the time T=nh and therefore it depends on both h and n. Alternatively (and more precisely), the estimates can be expressed using the independent variables h and n. For instance, for the mean square error estimate, we have 𝐄|f_n - f|^2 ≤ C(h^2 + 1/nh) .Therefore, for a fixed (large) total sample number n, we can conclude that the optimal upper bound in (<ref>) is 𝒪(n^-2/3) and is achieved when h=𝒪(n^-1/3). We refer to <cit.> for related discussions. In applications, the conditional probability measure μ_1 often satisfies the following Poincaré inequality <cit.> _μ_1(f) := ∫_Σ (f-f)^2 dμ_1 ≤ -1/K∫_Σ (ℒf) f dμ_1 = 1/Kβ∫_Σ (Pa ∇ f)·∇ f dμ_1, for all f : Σ→ℝ such that the right handside of the above inequality is finite, where K>0 is the Poincaré constant, ℒ is the infinitesimal generator(<ref>), and the identity (<ref>) in Remark <ref> has been used. Under this condition, the mean square error estimate in Theorem <ref> can beimproved (i.e., the constant in front of the 𝒪(T^-1) termis small when K is large) and we have the following corollary (The proof is in Appendix <ref>).Under the same assumptions in Theorem <ref> and further assuming thatμ_1 satisfies the Poincaré inequality (<ref>), we have 𝐄|f_n - f|^2 ≤2C_1 _μ_1(f)/KT + C_2(h^2 + h/T+1/T^2) ,where C_1 is any constant larger than 1, the constant C_2depends on the choice of C_1 but is independent of both h and n.The idea of using the map Θ in the constraint step of the numerical scheme (<ref>) is motivated by the softly constrained (reversible) dynamics (<ref>). It is natural to consider whether certain “non-reversible” numerical scheme can be obtained using the same idea. In fact, let A∈ℝ^d× d be a constant skew-symmetric matrix such that A^T=-A. The softly constrained (non-reversible) dynamics d X^ϵ,A, i_s = [A_ij∂/∂ x_j(U+1/2ϵ∑_α=1^kξ_α^2) - a_ij∂ U/∂ x_j- 1/ϵa_ij∂/∂ x_j(1/2∑_α=1^k ξ_α^2) + 1/β∂ a_ij/∂ x_j](X^ϵ,A_s)ds+ √(2β^-1)σ_ij(X^ϵ,A_s)dW^j_s ,indeed has the same invariant measure μ^ϵ in (<ref>). Based on this fact, a reasonable guess of the “non-reversible” numerical scheme that samples the conditional measure μ_1 isthe multiscale method of (<ref>), i.e.,x^(l + 1/2)_i =x^(l)_i + (A_ij∂ U/∂ x_j -a_ij∂ U/∂ x_j + 1/β∂ a_ij/∂ x_j)(x^(l))h + √(2 β^-1h) σ_ij(x^(l)) η^(l)_j ,1 ≤ i ≤ d,x^(l+1) = Θ^A(x^(l+1/2)) ,where Θ^A(x)=lim_s→ +∞φ^A(x,s) is the limit of the (non-gradient) flow map dφ^A(x,s)/ds =-((a-A)∇ F)(φ^A(x,s)) ,φ^A(x,0) = x, ∀ x∈ℝ^d ,with the same function F in (<ref>). We expect that the long time sampling error estimates of the numerical scheme (<ref>) can be studied following the same approach of this section as well. For this purpose, however, it is necessary to handle the non-gradient term in the ODE (<ref>), which brings difficulties when calculating the derivatives of the map Θ^A (cf. Proposition <ref> as well as its proof in Appendix <ref>). We will postpone the analysis in the future work and readers are referred to Example 1 in Section <ref> for numerical validation of the scheme (<ref>–(<ref>). In the literature, reversible Metropolis random walk on submanifold <cit.> and Hybrid Monte Carlo algorithm <cit.>have been proposed to sample distributions on submanifolds. A comprehensive comparison between our scheme (without Metropolis step) and these Metropolis-adjusted approaches is a complicate task and goes beyond the scope of the current paper.We only discuss this issue briefly in the following remark. We compare the following three aspects. * Constraint step.In the scheme (<ref>), the map Θ in (<ref>) is used to project the state x^(l+1/2) to the submanifold Σ. Under mild assumptions, the gradient structure of the ODE (<ref>) allows to define Θ for all states at which the map ξ is C^2 smooth (Assumption <ref>). Differently, in <cit.>, newly proposed states are projected back to the submanifold by solving a nonlinear system.One usually uses Newton's method to find the solution of the system, with the hope that the convergence can be achieved within a few iteration steps (success), thanks to the quadratic convergence rate of Newton's method. In practice, however, it may happen that either the solution does not exist or, even if the solution exists, the Newton's method does not converge (due to local convergence). In these two cases, the constraint step ends without finding a new state on the submanifold (no success). Although the Markov chain samples the correct invariant distribution regardless whether the constraint step is successful or not <cit.>, the sampling efficiency is affected by the success rate of the constraint step. * Computational complexity. Suppose the computational complexity of evaluating the d× k matrix ∇ξ is 𝒪(k· d) and n states are sampled in total. In each step of both the scheme (<ref>) and the Metropolis-adjusted methods in <cit.>, the major computational effort is devoted to the constraint step, i.e., either computing the map Θ by integrating ODE or solving equations using Newton's method.For the scheme (<ref>) with a=, the overall computational complexity of the constraint step is therefore 𝒪(n · k · d· s_ode/Δ s), where s_odeand Δ s are the average final time (see Remark <ref>) and the average step-size in the ODE integration, respectively.For the Metropolis-adjusted methods in <cit.>, in each Newton iteration it is necessary to compute the matrix product ∇ξ^T(x)∇ξ(x') for two states x,x'.Therefore, the overall computational complexity is 𝒪(n· k^2· d· N_iter), where N_iter is the average total iteration steps of Newton's method. In practice, one can implement the method in <cit.> in a way such that Newton's methodends within a few Newton iterations, e.g., N_iter≤ 10.On the other hand, the ODE integration in the scheme (<ref>) requires more iteration steps (for the examples in Section <ref>, 20-40 steps are needed), i.e., s_ode/Δ s ≥ N_iter. However, when comparing the computational cost of both methods, we should keep in mind that an ODE iteration step is generally cheaper than a Newton iteration step, since the latter involvesboth matrix-matrix multiplication and solving linear systems. (The cost of solving k× k linear systems is not major and therefore for simplicity is not included in the estimation above.) While the computational cost of Newton's method is smaller when k is small, the ODE integration becomes faster for medium or large k. We refer to Example 2 in Section <ref> for numerical comparison on the computational cost of both methods.* Choice of step-size. To apply the scheme (<ref>), one usually chooses a suitably small step-size h and runs the scheme for sufficient many steps (large n). In concrete applications, one often needs to tune the step-size h, keeping in mind that a large h will lead to large bias, while a unnecessarily small h will result in large correlations. See Remark <ref>.For the Metropolis-adjusted methods <cit.>, the choice of the step-size (in the proposal step) is in fact a more delicate issue. Although the Markov chain remains unbiased for large step-sizes, the sampling efficiency will be possibly limited due to a low acceptance rate in the Metropolis step. This issue has been discussed in <cit.>, where the performancewith different step-sizes has been numerically investigated. Besides the acceptance rate in the Metropolis step, the success rate of theconstraint step also depends on the step-size used in the proposal step. Taking the (d-1)-dimensional unit sphere 𝕊^d-1 as an example, it is not difficult to see that there won't be corresponding projected state on the sphere (i.e., the solution of the constraint equation does not exist), if the norm of the tangent vector generated in the proposal step (see <cit.>) is larger than one. This implies that the success rate of the constraint step will decrease when we increase the step-size in the proposal step. To summarize, it is important to choose the step-size in the Metropolis-adjusted method in <cit.> properly, such that both the acceptance rate in the Metropolis step and the success rate of the constraint step are not too small. Before concluding, let us point out that the approach used in the above proof allows us to study other numerical schemes on Σ as well. As an example, we consider the projection from ℝ^d to Σ along geodesic curves (instead of using the flow map (<ref>)–(<ref>)) defined by the metric g=a^-1 in (<ref>), i.e., the metric on ℳ=(ℝ^d, g). Let d be the distance function on ℝ^d induced by the metric g in (<ref>). We introduce the projection functionΠ(x) = {y | d(x,y) = d(x,Σ),y ∈Σ} ,∀ x ∈ℝ^d .Clearly, we have Π|_Σ=|_Σ. Given any x ∈Σ, there is a neighborhood Ω⊂ℝ^d of x such that Π|_Ω is a single-valued map. Furthermore, applyinginverse function theorem, we can verify that Π is smooth on Ω. Similar to Proposition <ref>, we need the following result which connects the derivatives of Π to the projection map P in (<ref>). (Note that, comparing to the derivatives of the map Θ in (<ref>), there is an extra term in the second equation of (<ref>).)Its proof is given in Appendix <ref>.Let Π = (Π_1, Π_2, ⋯, Π_d)^T: ℝ^d →Σ be the projection function in (<ref>), where Π_i : ℝ^d →ℝ are smooth functions, 1≤ i ≤ d. For x ∈Σ∩Ω, we have∂Π_i/∂ x_j = P_ij , a_lr∂^2 Π_i/∂ x_l∂ x_r=- P_il∂ a_lr/∂ x_r + ∂ (Pa)_il/∂ x_l + 1/2 (Pa)_il∂lnΨ/∂ x_l ,for 1 ≤ i,j ≤ d, where Ψ = ∇ξ^T a ∇ξ.Now we are ready to study the numerical scheme x^(l + 1/2)_i =x^(l)_i + (-a_ij∂ U/∂ x_j + 1/β∂ a_ij/∂ x_j)(x^(l))h + √(2 β^-1h) σ_ij(x^(l)) η^(l)_j ,1 ≤ i ≤ d,x^(l+1) = Π(x^(l+1/2)) ,where x^(0)∈Σ, and the map Π (instead of Θ) is used in each step to project the states x^(l+1/2) back to Σ.Assume that f : Σ→ℝ is a smooth function on Σ and f is its mean value f = ∫_Σ f(x) dμ(x) ,with respect to the probability measure dμ=1/Z e^-β U√( (∇ξ^T a ∇ξ)/ (∇ξ^T ∇ξ))dν .Consider the running average f_n in (<ref>), which is computed by simulating the numerical scheme (<ref>) with time step-size h>0. Let T=nh and C denote a generic positive constant that is independent of h, T. We have the following approximation results.* |𝐄f_n - f| ≤ C(h+ 1/T).* 𝐄|f_n - f|^2 ≤ C(h^2 + 1/T).* For any 0 < ϵ < 1/2, there is an almost surely bounded positive random variable ζ(ω), such that |f_n - f| ≤ Ch+ ζ(ω)/T^1/2-ϵ , almost surely.We omit the proof since it resembles the proof of Theorem <ref>.For the projection map Π induced by a general metric g=a^-1 or, equivalently, by a general (positive definite) matrix a, implementing the numerical scheme (<ref>) is not as easy as the numerical scheme (<ref>). We decide to omit the algorithmic discussions, due to the fact that the probability measure (<ref>) seems less relevant in applications. However, it is meaningful to point out that, when a=, the above result is relevant to the one in <cit.>. In this case, the probability measure in (<ref>) reduces to μ_2=1/Z e^-β Udν in (<ref>) and the numerical scheme (<ref>) can be formulated equivalently using Lagrange multiplier. We refer to <cit.> for comprehensive numerical details. § NUMERICAL EXAMPLES In this section, we study three concrete examples. In the first example, we investigate the different schemes in Section <ref>. In particular, the sampling performance of the constrained schemes using different maps Θ, Θ^A, and Π, as well as the performance of the unconstrained Euler-Maruyama discretization of the SDE (<ref>), will be compared. In the second example, we compare the computational costs of the scheme (<ref>) and the Metropolis-adjusted algorithm introduced in <cit.>. In the last example, we show that in some cases it is helpful to consider non-constant matrices σ and a. The C/C++ code used for producing the numerical results in the following examples is available at: <https://github.com/zwpku/sampling-on-levelset>. §.§.§ Example 1: Comparison of schemes using different projection mapsLet us define ξ: ℝ^2→ℝ by ξ(x) = 1/2(x_1^2/c^2+x_2^2 -1) , ∀ x = (x_1, x_2)^T ∈ℝ^2 ,with the constant c=3. The level set Σ={(x_1, x_2)^T | x_1^2/c^2+x_2^2=1} is an ellipse in ℝ^2. We have ∇ξ = (x_1/c^2,x_2)^T and therefore (∇ξ^T∇ξ) = |∇ξ|^2 = x_1^2/c^4 + x_2^2. For simplicity, we choose the potential U=0 and the matrices a=σ=∈ℝ^2× 2. The two probability measures in (<ref>) on Σ are dμ_1 = 1/Z (x_1^2/c^4 + x_2^2)^-1/2 dν ,dμ_2 = 1/Zdν ,where Z denotes two different normalization constants and ν is the surface measure on Σ. Since Σ is a one-dimensional manifold, it is helpful to consider the parametrization of Σ byx_1 = ccosθ , x_2 = sinθ ,where the angle θ∈ [0, 2π].Applying the chain rule ∂/∂θ = -csinθ∂/∂ x_1 + cosθ∂/∂ x_2, we can obtain the expressions of μ_1, μ_2under this coordinate as dμ_1 = 1/Z dθ , dμ_2 = 1/Z(c^2sin^2θ + cos^2θ)^1/2 dθ .With these preparations, we proceed to study the following four numerical approaches. * Numerical scheme (<ref>) using Θ. Since U≡ 0 and a=,(<ref>) becomesx^(l + 1/2) =x^(l) + √(2 β^-1h) η^(l) , x^(l+1) = Θ(x^(l+1/2)) ,where Θ(x) is the limit of the flow map φ, given by ẏ_1(s) = -ξ(y(s)) y_1(s)/c^2 ,ẏ_2(s) = -ξ(y(s)) y_2(s) ,s ≥ 0 ,starting from y(0) = x.* Numerical scheme (<ref>)–(<ref>) using Θ^A. Let us choose the skew-symmetric matrixA= [01/2; -1/20 ].Since U≡ 0 and a=, we have x^(l + 1/2) =x^(l) + √(2 β^-1h) η^(l) , x^(l+1) = Θ^A(x^(l+1/2)) ,where Θ^A(x) is the limit of the flow map φ^A, given byẏ_1(s) = -ξ(y(s)) (y_1(s)/c^2 - y_2(s)/2),ẏ_2(s) = -ξ(y(s)) (y_1(s)/2c^2 + y_2(s)) ,s ≥ 0 ,starting from y(0) = x.* Numerical scheme (<ref>) using Π. Similarly, since U≡ 0 and a=, (<ref>) becomesx^(l + 1/2) =x^(l) + √(2 β^-1h) η^(l) , x^(l+1) = Π(x^(l+1/2)) ,where Π is the projection map onto Σ, defined in (<ref>). * Euler-Maruyama discretization of the SDE (<ref>).Notice that we have Pa = P, and it is straightforward to compute P_11 = c^4x_2^2/x_1^2+c^4x_2^2 ,P_12 = P_21 = -c^2x_1x_2/x_1^2+c^4x_2^2 ,P_22 = x_1^2/x_1^2+c^4x_2^2 .Therefore, discretizing (<ref>), we obtain x^(l + 1)_1 =x^(l)_1 + 1/βc^4(c^2-2) x_1x_2^2-c^2x_1^3/(x_1^2+c^4x_2^2)^2 h+ √(2 β^-1h) (c^4x_2^2/x_1^2+c^4x_2^2 η^(l)_1 -c^2x_1x_2/x_1^2+c^4x_2^2 η^(l)_2) x^(l + 1)_2 =x^(l)_2 + 1/β(1-2c^2)x_1^2x_2-c^6x_2^3/(x_1^2+c^4x_2^2)^2 h+ √(2 β^-1h) ( -c^2x_1x_2/x_1^2+c^4x_2^2 η^(l)_1 +x_1^2/x_1^2+c^4x_2^2 η^(l)_2 ) .Based on Theorems  <ref>, <ref>,and Remark <ref>, we study the performance of the schemes (<ref>), (<ref>), and (<ref>) in sampling the conditional measure μ_1, as well as the performance of the scheme (<ref>) in sampling the measure μ_2. In the numerical experiment, we choose β=1.0 in each of the above schemes. For the first scheme using Θ, we simulate (<ref>)for n=2× 10^7 steps with the step-size h=0.01. In each constraint step,Θ(x^(l+1/2)) is computed by solving the ODE (<ref>) starting from y(0) = x^(l+1/2) until the time when |ξ(y(s))| < 10^-8 is satisfied, using the 3rd order (Bogacki-Shampine) Runge-Kutta (RK) method. The adaptivity technique in the second point of Remark <ref> is used with κ=0.5.The step-size for solving the ODE is set to Δ s = 0.1 initially and is divided by 2.0 whenever we find that the value of |ξ| is not decreasing. (The numerical error of Θ is 3.7 × 10^-7 on average, comparing to the reference solution that is obtained by solving the ODE with κ=0 and the fixed step-size Δ s= 0.001.) On average, we observe that 25 iterations of the RK method are needed in each constraint step in order to meet the criterion |ξ(y(s))| < 10^-8.For the second scheme using Θ^A, we simulate (<ref>)for n=2× 10^7 steps with the step-size h=0.005.Notice that, a slightly smaller step-size h is used, because in this case the non-gradient ODE flow(<ref>) produces a drift force on the level set Σ. In each constraint step, Θ^A(x^(l+1/2)) is computed by solving the ODE (<ref>) in the same way (with the same parameters) as we did in the first scheme. On average, we find that 23 iterations of the RK method are needed in each constraint step in order to meet the criterion |ξ(y(s))| < 10^-8.For the third scheme using Π, (<ref>) is simulated for n=2× 10^7 steps with the step-size h=0.01. Using the parametrization (<ref>), we have Π(x) = (ccosθ^*, sinθ^*)^T, where θ^* = _θ∈ [0, 2π]((x_1-ccosθ)^2 + (x_2-sinθ)^2) ,x=(x_1,x_2)^T .Therefore, in each step, Π(x^(l+1/2)) is computed by solving (<ref>)using the simple gradient descent method. The step-size is fixed to Δ t = 0.1 and the gradient descent iteration terminates when the derivative of the objective function in (<ref>) has an absolute value that is less than 10^-8. On average, it requires 32 gradient descent iterations in each step in order to meet the convergence criterion.Let us make a comparison among the three schemes (<ref>), (<ref>) and (<ref>). From Figure <ref> and Figure <ref>,we can see that the three maps Θ, Θ^A and Π indeed have different effects. Roughly speaking, comparing to the projection map Π, both Θ and Θ^A tend to map states towards one of the two vertices (± c, 0), where |∇ξ| are smaller, while Π^A introduces a further rotational force on Σ. Based on the states generated from these three schemes, in Figure <ref> we show the empirical probability densities of the parameter θ in (<ref>). From the agreement between the empirical densities and the densities computed from the analytical expressions in (<ref>), we can make the conclusion thatthe trajectories generated from the two schemes using Θ and Θ^A indeed sample the probability measure μ_1, while the trajectory generated from the scheme using Π samples μ_2. Lastly, concerning the fourth scheme, we simulate (<ref>) for n=10^7 steps using the step-size h = 0.0001. In this case, we find that it is necessary to choose a small step-size h in order to keep the trajectory close to the level set Σ. As can be seen from Figure <ref>,even with this smaller step-size h=0.0001, the generated trajectory departs from the level set Σ. This indicates the limited usefulness of the direct Euler-Maruyama discretization of the SDE (<ref>) in long time simulations. §.§.§ Example 2: Numerical comparison with the Metropolis-adjusted method on the special orthogonal group SO(11).In this example, we compare the computational efficiency between our scheme(<ref>) using the flow map Θ and the Metropolis-adjusted method introduced in <cit.>. We consider the special orthogonal group SO(11), which consists oforthogonal matrices of size 11 × 11 with determinant equals to 1.This example is taken from <cit.>. The authors there applied their method to the estimation of the mean value of the function f(x) = (x),i.e, the trace of the matrix x, where x follows the surface measure of SO(11). The manifold SO(11) can be viewed as (one connected component of) the level setof the map ξ: ℝ^121→ℝ^66, which includes all the row ortho-normality constraints. Readers are referred to the original work <cit.> for a detailed introduction on the example.In this numerical study, we implement both the scheme (<ref>) and the (Metropolis-adjusted) algorithm in <cit.> to estimate the mean value of (x). Notice that, since (∇ξ^T∇ξ) is constant, the conditional measure μ_1 in (<ref>) coincides with the surface measure of SO(11) when we choose the potential U≡ 0. In both cases, we generate n=10^6 samples on the same laptop (CPU: Intel Core i5, 2.60GHz, 4 cores; system: Ubuntu 18.04). For the scheme (<ref>), we choose the step-size h=0.022. The map Θ is computed by integrating the ODE (<ref>) with a=, until the condition |ξ(φ(x,s))|<10^-9 is satisfied. To accelerate the ODE integration, we have applied the adaptivity technique in the second point of Remark <ref> with κ=0.5. Starting from the initial step-size Δ s=0.2, the step-size used in the ODE integration is divided by 2.0 whenever we find that the value of |ξ| is not decreasing. (The numerical error of Θ is 2.0 × 10^-4 on average, comparing to the reference solution that is obtained by solving the ODE with κ=0 and the fixed step-size Δ s= 0.002.) Furthermore, the new state will be discarded (and resampled) if its determinant equals to -1. With these parameters, we observe that on average 37 Runge-Kutta iterations are needed for each evaluation of the map Θ. In total, it takes 2676.9 seconds to generate n=10^6 samples, while the estimated mean value of (x) is 3.8 × 10^-3 with a statistical error 3.9 × 10^-3. For the algorithm in <cit.>,the maximal number of Newton steps is set to 10 and the proposal length scale is chosen to be 0.257 [The roles of the proposal length scale in <cit.> and the step-size h in the scheme (<ref>) are different.The proposal length scale 0.257 used in this example corresponds to a step-size 0.033 (≈ 0.257^2/2) in the scheme (<ref>).]. In our experiment, we find that this proposal length scale (different from the one used in <cit.>) leads to slightly smaller correlation time. Within the entire computation, the success rate of the Newton's method (i.e., the rate that the Newton's method converges) is 67.2% andeach time it takes 5-6 iterationson average for the Newton's method to reach convergence (the convergence criteria is |ξ(x)| < 10^-9). In total, it takes 7315.9 seconds to generate n=10^6 samples. The estimated mean value is -3.8× 10^-3 and the statistical error is 3.8× 10^-3. The empirical density distributions of (x) using both the scheme (<ref>) and the algorithm in <cit.> are shown in the left plot of Figure <ref>, while the autocorrelation functions are plotted in the right plot of Figure <ref>. From these results, we can conclude that in this example both approaches provide similar statistical estimations (The autocorrelation time using Metropolis-adjusted method is slightly smaller with the above parameters.). At the same time, the total computational time using the scheme (<ref>) is less than half of the computational time required by the algorithm in <cit.>. For this example, although the average number of Newton steps in the latter algorithm is smaller than the average number of ODE iterations, the computational cost of each Newton step is indeed larger. We refer to Remark <ref> for the comparison of computational complexity of both approaches.§.§.§ Example 3: Removing stiffness by choosing a non-constant matrix aIn this example, we choose the reaction coordinate function ξ(x) = ξ(x_1, x_2,⋯, x_d) = 1/2(x_1^2 + x_2^2 + ⋯ + x^2_d - 1) .Correspondingly, the level setΣ={(x_1, x_2,⋯, x_d) ∈ℝ^d |  x_1^2 + x_2^2 +⋯ + x_d^2 = 1}is the (d-1)-dimensional unit sphere, and we have ∇ξ = (x_1,x_2,⋯,x_d)^T, ∇ξ^T∇ξ = ∑_i=1^d x_i^2 .In the following, we give an example to show that in some applications it is helpful to use a non-constant matrix a in the numerical scheme (<ref>).Briefly speaking, varying the matrix a properly allows to rescale the scheme along different directions.It has a preconditioning effect when different time scales (stiffness) exist. Consider d=3 and the potential U = θ^2/2ϵ, where ϵ>0 is a small parameter, θ is the angle of the state x = (x_1, x_2, x_3) under the spherical coordinate system x_1 = ρcosθcosφ ,x_2 = ρcosθsinφ ,x_3 = ρsinθ ,where ρ≥ 0, θ∈ [-π/2,π/2], and φ∈ [0, 2π]. We can verify that ∇θ = 1/ρ^2(-x_1x_3/(x_1^2+x_2^2)^1/2,-x_2x_3/(x_1^2+x_2^2)^1/2,(x_1^2+x_2^2)^1/2)^T .Correspondingly, with the choice of σ=a=, the scheme (<ref>) is x^(l + 1/2) =x^(l)-1/ϵ (θ∇θ)(x^(l)) h + √(2 β^-1h) η^(l) , x^(l+1) = Θ(x^(l+1/2)) ,where ∇θ is given in (<ref>).Notice that, the coefficients in (<ref>) are 𝒪(1/ϵ) when ϵ is small. In particular, it implies that samplingthe invariant measure using (<ref>) will be inefficient when ϵ is small, since the step-size h will be severely limited due to the large magnitude of the coefficients in (<ref>). On the other hand, based on the form of U and the expression (<ref>), we consider the orthogonal vectorsσ_1 =(x_1 , x_2 , x_3)^T=∇ξ ,σ_2 = (x_2 , -x_1 , 0)^T , σ_3 = (-√(ϵ) x_1x_3 /(x_1^2+x_2^2)^1/2,-√(ϵ) x_2x_3/(x_1^2+x_2^2)^1/2,√(ϵ) (x_1^2+x_2^2)^1/2)^T=√(ϵ)ρ^2 ∇θ ,and we define σ = (σ_1,σ_2, σ_3) ∈ℝ^3 × 3. Direct calculation shows that a=σσ^T =[ x_1^2 + x_2^2+ϵ x_1^2x_3^2/x_1^2+x_2^2 ϵ x_1x_2 x_3^2/x_1^2+x_2^2 (1-ϵ) x_1x_3; ϵ x_1x_2 x_3^2/x_1^2+x_2^2 x_1^2+x_2^2+ϵ x_2^2x_3^2/x_1^2+x_2^2 (1-ϵ) x_2x_3; (1-ϵ) x_1x_3 (1-ϵ) x_2x_3 x_3^2 + ϵ(x_1^2 + x_2^2) ] .Correspondingly, using (<ref>), the scheme (<ref>) becomes x^(l + 1/2)_i =x^(l)_i+[-θ∂θ/∂ x_i+1/β∂ a_ij/∂ x_j](x^(l)) h + √(2 β^-1h) σ_ij(x^(l)) η^(l)_j ,1 ≤ i ≤ 3 ,x^(l+1) = Θ(x^(l+1/2)) ,where Θ(x) is the limit of the ODE flow ẏ(s) = - ξ(y(s))(2ξ(y(s))+1) y(s) ,y(0) = x .Importantly, in contrast to (<ref>), the scheme (<ref>)–(<ref>) is no longer stiff when ϵ is small.Now we compare the numerical efficiency between the schemes (<ref>) and (<ref>)–(<ref>). First of all, since the surface measure on Σ satisfies dν = cosθ dθ dφ, we know that the target measure is dμ = 1/Z e^-βθ^2/2ϵ dν =1/Z e^-βθ^2/2ϵcosθdθ dφ .In the numerical study, we choose ϵ = 0.005 and generate n=10^7 states for both schemes. For the scheme (<ref>) which corresponds to a=,we use both a small step-size h=0.0002 and a (relatively) larger step-size h=0.005, while we choose a large step-size h=0.01 in the scheme (<ref>)–(<ref>) . The empirical probability densities of the angles θ, φ for the two schemes are shown in Figure <ref> and Figure <ref>, respectively. From Figure <ref>, we see that the step-size h has to be small (h=0.0002) in (<ref>) in order to produce the correct probability density of the angle θ (left plot). However, with such a small h, the estimated empirical density of the angle φ (right plot) is still noisy with n=10^7. On the other hand, for the scheme(<ref>)–(<ref>) which corresponds to the matrix a in (<ref>), Figure <ref> shows that the probability densities of both angles θ, φ are well approximated using the large step-size h=0.01. Therefore, we conclude that in this example choosing the non-constant matrix a in (<ref>) indeedhelps improve the sampling efficiency. § CONCLUSIONS Ergodic diffusion processes on a submanifold of ℝ^d and related numerical sampling schemes have been considered in this work. A family of SDEs has been obtained whose invariant measures coincide with the given probability measure on the submanifold. In particular, for the conditional probability measure, we found that the corresponding SDEs have a relatively simple form. We proposed and analyzed a consistent numerical scheme which only requires 1st order derivatives of the reaction coordinate function. Different sampling schemes on the submanifolds are numerically evaluated. The current work extends results in the literature and may further contribute toboth the analysis and the development of numerical methods on related problems, in particular problems in molecular dynamics such as free energy calculation and model reduction of high-dimensional stochastic processes.Closely related to the current paper, the following topics could be considered.First, the “non-reversible” scheme (<ref>) is supported by a simple numerical example but theoretical justification still needs to be investigated. This will be considered in future following the approach described in Remark <ref>.Second, the constrained numerical schemes in the current work do not involve system's momentum variables. In view of the work <cit.>, it is interesting to study the Langevin dynamics under different constraints (such as certain variants of the map Θ used in this work). Third, there is a research interest in the literature to study the effective dynamics of molecular systems along a given reaction coordinate ξ. The coefficients of the effective dynamics are usually defined as averages on the level set of ξ <cit.>. As an application of the numerical scheme proposed in this work, we will study numerical algorithms to simulate the effective dynamics. This topic is related to the heterogeneous multiscale methods <cit.> and the equation-free approach <cit.> in the literature.§ ACKNOWLEDGEMENTThis work is funded by the Einstein Center of Mathematics (ECMath) through project CH21. The author would like to thank Gabriel Stoltz for stimulating discussions on constrained Langevin processes at the Institut Henri Poincaré - Centre Émile Borel during the trimester “Stochastic Dynamics Out of Equilibrium”.The author appreciates the hospitality of this institution. The author also thanks the anonymous referees for their valuable comments and criticism which helped improve the manuscript substantially. § USEFUL FACTS ABOUT THE RIEMANNIAN MANIFOLD ℳ In this section, we present technical details of Section <ref> related to the Riemannianmanifold ℳ=(ℝ^d, g), where g=a^-1.The main result is Proposition <ref>, where we give the expression of the Laplacian-Beltrami operator Δ^Σ on the level set Σ in (<ref>), viewed as a submanifold of ℳ. Before that, we first introduce some notations and quantities related to ℳ and Σ. Readers are referred to <cit.> for related discussions on general Riemannian manifolds. Under Assumption <ref>, given two vectors u = (u_1, u_2, ⋯, u_d)^T, v = (v_1, v_2, ⋯, v_d)^T, we consider the space ℝ^d with the weighted inner productg(u, v) = ⟨u, v⟩_g = u_i (a^-1)_ij v_j .The inner product in (<ref>) defines a Riemannian metric g on ℝ^d andwe denote by ℳ = (ℝ^d, g) the Riemannian manifold ℝ^d endowed with this metric.Notice that ℳ as a manifold is quite special (simple), in thatit has a natural global coordinate chart which is given by the usual Euclidean coordinate. Since we will always work with this coordinate, we will not distinguish between tangent vectors (operators acting on functions) and their coordinate representations (d-dimensional vectors).In particular,e_i denotes the vector whose ith component equals to 1 while all the other d-1 components equal to 0, where 1 ≤ i ≤ d. At each point x ∈ℳ, vectors e_1, e_2, ⋯, e_d form a basis of the tangent space T_xℳ and under this basis we have g=a^-1, as can be seen from (<ref>).Denote by ^ℳ, ^ℳ the gradient and the divergence operator on ℳ, respectively. For any smooth function f: ℳ→ℝ, it is direct to verify that ^ℳ f = g^ij∂ f/∂ x_je_i = (a∇ f)_i e_i, where g^ij=(g^-1)_ij = a_ij, and ∇ fdenotes the ordinary gradient operator for functions on the Euclidean space ℝ^d. For simplicity, we will also write ∂_i f for the partial derivative with respect to x_i, and (a∇ f)_i to denote the ith component of the vector a∇ f, i.e.,∂_i f = ∂ f/∂ x_i, and(a∇ f)_i = a_ij∂ f/∂ x_j = a_ij∂_jf.The Laplace-Beltrami operator on ℳ is defined by Δ^ℳ f = ^ℳ (^ℳ f). Equivalently, we have Δ^ℳ f = (^ℳf), where^ℳ is the Hessian operator on ℳ.The integration by parts formula on ℳ has the form∫_ℳ (Δ^ℳ f_1) f_2 dm = - ∫_ℳ⟨^ℳ f_1, ^ℳ f_2⟩_gdm = ∫_ℳ (Δ^ℳ f_2)f_1 dm ,for ∀ f_1,f_2 ∈ C^∞_0(ℳ), where dm= ( g)^1/2 dx = ( a)^-1/2 dx is the volume form, and C^∞_0(ℳ) consists of all smooth functions on ℳ with compact support.Besides the vector basis e_1, e_2, ⋯, e_d, the vectors σ_i = (σ_1i, σ_2i,⋯, σ_di)^T,1 ≤ i ≤ d ,will also be useful. Note that a=σσ^T=g^-1 implies ⟨σ_i, σ_j⟩_g = (a^-1)_rlσ_riσ_lj = δ_ij. In other words, σ_1, σ_2, ⋯, σ_d form an orthonormal basis of T_xℳ at each x ∈ℳ.Denote by ∇^ℳ the Levi-Civita connection on ℳ. Given x ∈ℳ and a tangent vector v∈ T_xℳ,∇^ℳ_v is the covariant derivative operator on ℳ along the vector v. For two vectors u = (u_1, u_2, ⋯, u_d)^T, v = (v_1, v_2, ⋯, v_d)^T, the Hessian of a smooth function f: ℳ→ℝ is defined as ^ℳ f(u,v) = u(vf) - (∇^ℳ_uv) f = u_iv_j ^ℳ f( e_i,e_j) = u_iv_j (∂^2 f/∂ x_i∂ x_j - Γ_ij^l ∂ f/∂ x_l) ,whereΓ_ij^l = 1/2 g^lr(∂ g_ir/∂ x_j + ∂ g_jr/∂ x_i - ∂ g_ij/∂ x_r)= 1/2 a_lr(∂ (a^-1)_ir/∂ x_j + ∂ (a^-1)_jr/∂ x_i - ∂ (a^-1)_ij/∂ x_r) , 1 ≤ i,j,l ≤ dare the Christoffel's symbols defined by ∇^ℳ_e_ie_j = Γ^l_ije_l, for 1 ≤ i,j≤ d.Now let us consider the level set Σ=ξ^-1(0)={ x ∈ℳ=ℝ^d  | ξ(x) = 0∈ℝ^k}of the C^2 function ξ : ℝ^d →ℝ^k with ξ = (ξ_1, ξ_2, ⋯, ξ_k)^T, 1 ≤ k < d.Applying regular value theorem <cit.>, we know that Σ is a (d-k)-dimensional submanifold of ℳ, under Assumption <ref>.Given x ∈Σ and a vector v∈ T_xℳ, the orthogonal projection operator (d × d matrix) P : ℝ^d → T_xΣ is defined such that ⟨v-Pv, u⟩_g = 0, for ∀ u∈ T_xΣ. It is straightforward to verify that P=- a∇ξΨ^-1∇ξ^T, or entry-wise, P_ij = δ_ij - (Ψ^-1)_αγ (a ∇ξ_α)_i ∂_jξ_γ ,1 ≤ i,j ≤ d ,where Ψ is the invertible k × k symmetric matrix at each point x ∈Σ, given byΨ_αγ = ⟨^ℳξ_α, ^ℳξ_γ⟩_g=(∇ξ^T a ∇ξ)_αγ ,1 ≤α, γ≤ k .In the above, ∇ξ denotes the d × k matrix with entries (∇ξ)_iα =∂_iξ_α, for 1 ≤α≤ k, 1 ≤ i ≤ d. We can verify that aP^T = Pa,P^2 = P ,P^T∇ξ_α = 0 , 1 ≤α≤ k .Let us further assume that v∈ T_xΣ is a tangent vector of Σ at x.Since {σ_i}_1 ≤ i ≤ d forms an orthonormal basis of the tangent space T_x ℳ, we have v = ⟨v, σ_i⟩_g σ_i.Using the fact that Pv = v, we obtain v = ⟨v, p_i⟩_g p_i, where p_i = Pσ_i ∈ T_xΣ. If we denote p_i = P_i,je_j, then it follows from (<ref>) and (<ref>) thatP_i,j = (Pσ)_ji = σ_ji - (Ψ^-1)_αγ (a ∇ξ_α)_j (σ^T∇ξ_γ)_iP_l,iP_l,j = (Pa)_ij=(aP^T)_ij = a_ij - (Ψ^-1)_αγ (a∇ξ_α)_i (a∇ξ_γ)_j ,for 1≤ i,j ≤ d.Let ^Σ, ^Σ, Δ^Σ, ^Σ denote thegradient operator, the divergence operator, the Laplace-Beltrami operator and the Hessian operator on Σ, respectively. It is direct to check that the Levi-Civita connection and the gradient operator on Σare given by ∇^Σ=P∇^ℳ and ^Σ=P ^ℳ, respectively. In particular, for f ∈ C^∞(Σ) and let f be itsextension to ℳ such that f∈C^∞(ℳ) and f|_Σ = f, we have ^Σ f = P^ℳf = Pa ∇f .Let ν_g be the surface measure on Σ induced from the metric g on ℳ. We recall that the mean curvature vector H on Σ is defined such that <cit.>∫_Σ^Σvdν_g = - ∫_Σ⟨ H, v⟩_g dν_g,for all vector fields v on ℳ. We have the following lemma, concerning the operators on Σ. Let f ∈ C^∞(Σ) and f∈ C^∞(ℳ) be its extension to ℳ. u∈Γ(Tℳ) is a tangent vector field on ℳ and we recall the vectors p_i = Pσ_i, 1 ≤ i ≤ d. We have * ^Σu = ⟨∇^ℳ_p_iu, p_i⟩_g.* (^Σp_i)p_i+ P∇^ℳ_p_ip_i= 0. * Δ^Σ f = ∑_i=1^dp_i^2 f + (^Σp_i) p_i f = ∑_i=1^dp_i^2 f - (P ∇^ℳ_p_ip_i) f.* Δ^Σ f = ^ℳf( p_i, p_i) + Hf, where H is the mean curvature vector of the submanifold Σ.* In the special case when g=a=, we have (^Σp_i)p_i= P∇^ℳ_p_ip_i= 0, and Δ^Σ = ∑_i=1^d p_i^2. The first two assertions can be directly verified. Let us prove the last three assertions. Let x ∈Σ and assume that v_i, 1 ≤ i ≤ d-k, is an orthonormal basis of T_xΣ.We have v_i = ⟨v_i, p_j⟩_g p_j. For the third assertion, by definition, Δ^Σ f = ^Σ(^Σ f) = ^Σ(P ^ℳf ) = ^Σ(⟨^ℳf, p_i⟩_g p_i) = ^Σ((p_if )p_i) = ∑_i=1^dp_i^2 f + (^Σp_i) p_i f = ∑_i=1^dp_i^2 f - (P ∇^ℳ_p_ip_i) f ,where the second assertion has been used in the last equality.For the fourth assertion,starting from the third assertion, using the definition of ^ℳ in (<ref>), and applying Proposition <ref> below, we obtainΔ^Σ f =∑_i=1^d p_i^2 f - P ∇^ℳ_p_ip_i f = ^ℳf( p_i, p_i) +[(I-P)∇^ℳ_p_ip_i] f = ^ℳf( p_i, p_i) + Hf . For the last assertion, when g=, we have Γ_ij^l ≡ 0, for ∀ 1≤ i,j,l ≤ d. Also, it follows from (<ref>) that P_i,j=P_ij = P_ji and P_il P_lj = P_ij.We obtain(^Σp_i) p_i= ⟨∇^ℳ_p_jp_i, p_j⟩_g p_i=P_j, lP_j,j' P_i,i'⟨∇^ℳ_e_l(P_i, re_r), e_j'⟩_g e_i'=P_lr∂ P_ir/∂ x_lP_ii' e_i'= [∂ P_il/∂ x_lP_ii' -∂ P_lr/∂ x_l P_ri'] e_i'=0 ,and the other assertions follow accordingly. Let H be the mean curvature vector defined in (<ref>) on the submanifold Σ.We have H =(I-P)∇^ℳ_p_ip_i= -(Ψ^-1)_αγ[1/2(Pa)_ij (a∇ξ_α)_l∂ (a^-1)_ij/∂ x_l +P_il∂ (a∇ξ_α)_l/∂ x_i] a∇ξ_γ .In the special case when g=a=, we have H = P_jl∂ P_il/∂ x_je_i=-[(Ψ^-1)_αγP_ij ∂^2_ijξ_α] ∇ξ_γ . Given a tangent vector field v on ℳ,from the definition of P we have v = Pv + (Ψ^-1)_αγ⟨v, a∇ξ_γ⟩_ga∇ξ_α. Since Pv is a tangent vector field on Σ, using (<ref>) and the divergence theorem on Σ, we know ∫_Σ⟨ H, v⟩_gdν_g= -∫_Σ^Σ[(I - P)v]dν_g =-∫_Σ^Σ[(Ψ^-1)_αγ⟨v, a∇ξ_γ⟩_ga∇ξ_α]dν_g .For the first expression, we notice that ⟨(I - P)v, p_i⟩_g ≡ 0, 1 ≤ i ≤ d. Applying Lemma <ref>, we have -∫_Σ^Σ[(I - P)v] dν_g =-∫_Σ⟨∇^ℳ_p_i[(I - P)v], p_i⟩_gdν_g =-∫_Σp_i⟨ (I - P)v, p_i⟩_gdν_g + ∫_Σ⟨ (I - P)v, ∇^ℳ_p_ip_i ⟩_gdν_g= ∫_Σ⟨v, (I - P)∇^ℳ_p_ip_i⟩_gdν_g.Comparing the last equality above with (<ref>), we conclude that H=(I - P)∇^ℳ_p_ip_i.For the second expression, we notice that ⟨ a∇ξ_α, p_i⟩_g=0,and also recall the expressions (<ref>), (<ref>) and (<ref>). Applying Lemma <ref>, integrating by parts, and noticing the cancellation of some terms, we can derive ^Σ[(Ψ^-1)_αγ⟨v, a∇ξ_γ⟩_ga∇ξ_α]= ⟨∇^ℳ_p_i[(Ψ^-1)_αγ⟨v, a∇ξ_γ⟩_ga∇ξ_α], p_i⟩_g =(Ψ^-1)_αγ⟨v, a∇ξ_γ⟩_g ⟨∇^ℳ_p_i (a∇ξ_α), p_i⟩_g =(Ψ^-1)_αγ⟨v, a∇ξ_γ⟩_g P_i, jP_i,l⟨∇^ℳ_e_j((a∇ξ_α)_r e_r), e_l⟩_g =(Ψ^-1)_αγ⟨v, a∇ξ_γ⟩_g (Pa)_jl[(a∇ξ_α)_r Γ^i_jr(a^-1)_il + ∂ (a∇ξ_α)_r/∂ x_j (a^-1)_lr] =(Ψ^-1)_αγ⟨v, a∇ξ_γ⟩_g (Pa)_ij[1/2 (a∇ξ_α)_l∂ (a^-1)_ij/∂ x_l + ∂ (a∇ξ_α)_l/∂ x_i (a^-1)_lj] .The second identity in (<ref>) is obtained after comparing the above expression with (<ref>) .In the case g=a=, we have Γ_il^r ≡ 0,1 ≤ i,l,r ≤ d.It follows that^Σ[(Ψ^-1)_αγ⟨v, a∇ξ_γ⟩_ga∇ξ_α] = (Ψ^-1)_αγ⟨v, a∇ξ_γ⟩_gP_ij ∂^2_ijξ_αand we obtain thatH=-[(Ψ^-1)_αγP_ij ∂^2_ijξ_α] ∇ξ_γ. Using (<ref>) and (<ref>), we have P_jl∂ P_il/∂ x_je_i = -P_jl∂((Ψ^-1)_αγ∂_lξ_α∂_iξ_γ)/∂ x_je_i =- [(Ψ^-1)_αγP_jl ∂^2_jlξ_α ∂_iξ_γ] e_i =H ,and therefore the first expression in (<ref>) holds as well.Next, we study the Laplace-Beltrami operator Δ^Σ on the submanifold Σ. Clearly, Δ^Σ is self-adjoint and, similar to (<ref>),we have the integration by parts formula on Σ with respect to the measure ν_g, as ∫_Σ (Δ^Σ f_1) f_2 dν_g = - ∫_Σ⟨^Σ f_1, ^Σ f_2⟩_gdν_g = ∫_Σ (Δ^Σ f_2)f_1 dν_g ,for ∀ f_1,f_2 ∈ C^∞(Σ). The expression of Δ^Σ can be computed explicitly and this is the content of the following proposition. Let Σ be the submanifold of ℳ defined in (<ref>), P be the projection matrix in (<ref>), and Δ^Σ be the Laplace-Beltrami operator on Σ. We have Δ^Σ = (Pa)_ij∂^2 /∂ x_i∂ x_j+ [∂ (Pa)_ij/∂ x_j + 1/2 (Pa)_ij∂ln(( a)^-1(∇ξ^T a ∇ξ))/∂ x_j] ∂/∂ x_i .In the special case when g=a=, we have Δ^Σ = ∑_i=1^d p^2_i= P_ij∂^2 /∂ x_i∂ x_j + P_lj∂ P_li/∂ x_j∂/∂ x_i=P_ij∂^2 /∂ x_i∂ x_j + H_i ∂/∂ x_i ,where H = H_i e_i is the mean curvature vector of the submanifold Σ.Let f ∈ C^∞(Σ) and f∈ C^∞(ℳ) be its extension to ℳ. Using Lemma <ref> and Proposition <ref>, we have Δ^Σ f = ^ℳf( p_r, p_r) + Hf=P_r, jP_r,l^ℳf( e_j, e_l) + Hf=(Pa)_jl(∂^2 f/∂ x_j∂ x_l - Γ_jl^i ∂f/∂ x_i)-(Ψ^-1)_αγ[1/2 (Pa)_jl (a∇ξ_γ)_r∂ (a^-1)_jl/∂ x_r + P_lr∂ (a∇ξ_γ)_r/∂ x_l](a∇ξ_α)_i ∂f/∂ x_i .Notice that we have already obtained the coefficients of the second order derivative terms. For the terms of the first order derivatives, let us denote I_1= -(Pa)_jlΓ_jl^iI_2 = -1/2 (Ψ^-1)_αγ (Pa)_jl (a∇ξ_α)_i (a∇ξ_γ)_r∂ (a^-1)_jl/∂ x_rI_3 =- (Ψ^-1)_αγ P_lr (a∇ξ_α)_i ∂ (a∇ξ_γ)_r/∂ x_l .Using the expression of Pa in (<ref>), the property Pa∇ξ_γ = 0, and integrating by parts, we easily obtainI_2= 1/2((Pa)_ir - a_ir) (Pa)_jl∂ (a^-1)_jl/∂ x_rI_3 = ∂ P_lr/∂ x_l(a_ir - (Pa)_ir) .For I_1, direct calculation using (<ref>) gives I_1 =- 1/2 (Pa)_jl a_ir(∂ (a^-1)_lr/∂ x_j + ∂ (a^-1)_jr/∂ x_l - ∂ (a^-1)_jl/∂ x_r)=- (Pa)_jla_ir∂ (a^-1)_lr/∂ x_j + 1/2 (Pa)_jl a_ir∂ (a^-1)_jl/∂ x_r= ∂ (Pa)_ij/∂ x_j- ∂ P_jr/∂ x_j a_ir + 1/2 (Pa)_jl a_ir∂ (a^-1)_jl/∂ x_r .Therefore, I_1 + I_2 + I_3 =∂ (Pa)_ij/∂ x_j-∂ P_lr/∂ x_l(Pa)_ir + 1/2 (Pa)_jl a_ir∂ (a^-1)_jl/∂ x_r +1/2((Pa)_ir - a_ir) (Pa)_jl∂ (a^-1)_jl/∂ x_r= ∂ (Pa)_ij/∂ x_j-∂ P_lr/∂ x_l(Pa)_ir + 1/2(Pa)_ir (Pa)_jl∂ (a^-1)_jl/∂ x_r .Applying Lemma <ref> below to handle the last term above, we concludeI_1 + I_2 + I_3 = ∂ (Pa)_ij/∂ x_j -1/2(Pa)_ir∂ln a/∂ x_r+1/2(Pa)_ir∂ln Ψ/∂ x_r . Finally, when g=a=, applying Lemma <ref>, we can obtainΔ^Σ f= ^ℳf( p_i, p_i) + Hf=P_l, iP_l,j(∂^2 f/∂ x_i∂ x_j - Γ_ij^l ∂f/∂ x_l) + Hf=P_ij∂^2 f/∂ x_i∂ x_j + H_i ∂f/∂ x_i .The other equality in (<ref>) follows from Proposition <ref>.We point out that the proof of Proposition <ref> is indeed valid for a general Riemannian manifold ℳ and its level set Σ as well. In this case,(<ref>) holds true on a local coordinate of the manifold ℳ. The following identity has been used in the above proof, and will be useful in Appendix <ref> as well. 1/2(Pa)_ir(Pa)_jl∂ (a^-1)_jl/∂ x_r = -1/2(Pa)_ir∂ln a/∂ x_r + (Pa)_ir∂ P_lr/∂ x_l + 1/2(Pa)_ir∂lnΨ/∂ x_r . Using the expression of Pa in (<ref>), the relations Pa∇ξ_γ = 0 ,∂ln a/∂ x_r = (a^-1)_jl∂ a_jl/∂ x_r , ∂lnΨ/∂ x_r = (Ψ^-1)_αγ∂Ψ_αγ/∂ x_r ,and the integration by parts formula, we can compute1/2(Pa)_ir(Pa)_jl∂ (a^-1)_jl/∂ x_r=1/2(Pa)_ir(a_jl -(Ψ^-1)_αγ (a∇ξ_α)_j (a∇ξ_γ)_l) ∂ (a^-1)_jl/∂ x_r=-1/2(Pa)_ir∂ln a/∂ x_r - 1/2(Pa)_ir (Ψ^-1)_αγ (a∇ξ_α)_l ∂^2_lrξ_γ+ 1/2(Pa)_ir (Ψ^-1)_αγ∂_lξ_α ∂ (a∇ξ_γ)_l/∂ x_r=-1/2(Pa)_ir∂ln a/∂ x_r - (Pa)_ir (Ψ^-1)_αγ (a∇ξ_α)_l ∂^2_lrξ_γ+ 1/2(Pa)_ir∂lnΨ/∂ x_r=-1/2(Pa)_ir∂ln a/∂ x_r + (Pa)_ir∂ P_lr/∂ x_l + 1/2(Pa)_ir∂lnΨ/∂ x_r . We conclude this section with the proof of Corollary <ref> in Section <ref>.Notice that the infinitesimal generator of (<ref>) can be written as ℒ^J = J_i∂/∂ x_i + ℒ ,where ℒ is the infinitesimal generator of (<ref>). Using the fact J∈ T_x Σ, the same argument of Proposition <ref> implies that (<ref>) evolves on Σ as well.Since μ_1 is invariant with respect to ℒ, to show the SDE (<ref>) has the same invariant measure, it is enough to verify that ^Σ{Jexp[-β(U+1/2βln (∇ξ^T a ∇ξ)/ a)]} = 0 , ∀ x ∈Σ ,where we have used the expression of μ_1 in (<ref>). Applying the formula of ^Σ in Lemma <ref>,we can compute the right hand side of (<ref>), as ^Σ{Jexp[-β(U+1/2βln (∇ξ^T a ∇ξ)/ a)]}= ⟨∇^ℳ_p_j{exp[-β(U+1/2βln (∇ξ^T a ∇ξ)/ a)]J_ie_i}, p_j⟩_g =P_j,lP_j,r⟨∇^ℳ_e_l{exp[-β(U+1/2βln (∇ξ^T a ∇ξ)/ a)]J_ie_i}, e_r⟩_g .which implies that (<ref>) is equivalent to 0 =(Pa)_lr∂ J_i/∂ x_l (a^-1)_ir + (Pa)_lrJ_i Γ_li^r' (a^-1)_r'r - β (Pa)_lr J_i (a^-1)_ir∂ U/∂ x_l - 1/2 (Pa)_lr J_i (a^-1)_ir∂/∂ x_l[ ln (∇ξ^T a ∇ξ)/ a]=P_li∂ J_i/∂ x_l + P_lr'J_i Γ_li^r' - β P_li J_i ∂ U/∂ x_l - 1/2 P_li J_i ∂/∂ x_l[ ln (∇ξ^T a ∇ξ)/ a] ,where Γ_li^r' are the Christoffel's symbols satisfying ∇^ℳ_e_le_i = Γ^r'_lie_r'. Using the expression (<ref>) of Γ_li^r', the fact J_i = P_ij J_j, and Lemma <ref>, we can further simplify the above equationand obtainP_lr'J_i Γ_li^r' - 1/2 P_li J_i ∂/∂ x_l[ ln (∇ξ^T a ∇ξ)/ a]= 1/2 J_i(Pa)_lr∂ (a^-1)_lr/∂ x_i - 1/2 P_li J_i ∂/∂ x_l[ ln (∇ξ^T a ∇ξ)/ a]=J_j ∂ P_ij/∂ x_i .Therefore, we see that(<ref>) is equivalent to the condition (<ref>).§ PROOFS IN SECTION <REF> In this section, we collect proofs of the various results in Section <ref>. First, we prove Proposition <ref>, which concerns the properties of the flow map Θ defined in(<ref>), (<ref>), and (<ref>). While the approach of the proof is similar to the one in <cit.>, here we consider the specific function F in (<ref>)and we will provide full details of the derivations. In this proof, we will always assume x ∈Σ.For a function which only depends on the state and is evaluated at x, we will often omit its argument in order to keep the notations simple. Also notice that, repeated indices other than l and l' indicate that they are summed up, while for the indices l, l' we assume that they are fixed by default unless the summation operator is used explicitly.Since ∇ F = 0 on Σ, from the equation (<ref>) we know that φ(x,s) ≡ x, ∀ s ≥ 0.Let us Denote by ∇^2 F the Hessian matrix (on the standard Euclidean space) of the function F in (<ref>), i.e., ∇^2 F =(∂^2_ij F)_1 ≤ i, j ≤ d. Since ξ(x) = 0∈ℝ^k, direct calculation gives (a∇^2 F)_ij = a_ir∂^2 F/∂ x_r∂ x_j =(a∇ξ∇ξ^T)_ij ,1 ≤ i, j ≤ d .Meanwhile, it is straightforward to verify that a∇^2 F satisfies ⟨ a ∇^2 F u, v⟩_g = ⟨u, a ∇^2 F v⟩_g ,∀ u, v∈ℝ^d , ⟨ a ∇^2F u, u⟩_g = |∇ξ^Tu|^2 ≥ 0 , ∀ u∈ℝ^d ,(a∇^2 F) u= a∇ξ∇ξ^T u = 0 ,∀ u∈ T_xΣ .Therefore, we can assume that a∇^2F has real (non-negative) eigenvalues λ_1=λ_2= ⋯=λ_d-k = 0 < λ_d-k+1≤⋯≤λ_d , and the corresponding eigenvectors, denoted by v_i = (v_i1, v_i2, ⋯, v_id)^T, 1 ≤ i ≤ d,are orthonormal with respect to the inner product ⟨·,·⟩_g in (<ref>), such that v_1, v_2, ⋯, v_d-k∈ T_xΣ.The projection matrix P in (<ref>) can be expressed using the vectorsv_i as P_ij = ∑_l=1^d-k v_li (a^-1)_jr v_lr ,1 ≤ i, j ≤ d , and we have ∑_l=1^d-k v_liv_lj = (Pa)_ij ,a_ij - (Pa)_ij =∑_l=d-k+1^d v_liv_lj .It is also a simple fact that the eigenvalues of the k× k matrix Ψ=∇ξ^T a∇ξ are λ_d-k+1, λ_d-k+2, ⋯, λ_d,with the corresponding eigenvectors given by ∇ξ^T v_d-k+1, ∇ξ^Tv_d-k+2, ⋯, ∇ξ^Tv_d. In particular, this implies∏_i=d-k+1^d λ_i= (∇ξ^T a∇ξ) = Ψ . In the following, we study the ODE (<ref>) using the eigenvectors v_i. Differentiating the ODE (<ref>) twice, using the facts thatφ(x,s)≡ x, ∀ s ≥ 0, and ∇ F = 0 on Σ, we obtain d/ds∂φ_i/∂ x_j(x,s) = -(a_ir'∂^2 F/∂ x_r'∂ x_i')∂ φ_i'/∂ x_j (x,s) d/ds∂^2 φ_i/∂ x_j∂ x_r(x,s) = -(2∂ a_ir'/∂ x_i'∂^2 F/∂ x_r'∂ x_j' + a_ir'∂^3 F/∂ x_r'∂ x_i'∂ x_j')∂ φ_i'/∂ x_j (x,s) ∂ φ_j'/∂ x_r (x,s)-(a_ir'∂^2 F/∂ x_r'∂ x_i') ∂^2 φ_i'/∂ x_j ∂ x_r(x,s) ,for s ≥ 0 and 1 ≤ i, j, r ≤ d. * The first equation of (<ref>) implies d/ds(v_lj∂φ_i/∂ x_j(x,s)) = -(a_ir'∂^2 F/∂ x_r'∂ x_i') ( v_lj∂ φ_i'/∂ x_j(x,s)) , 1 ≤ l ≤ d .Since φ(·, 0) is the identity map, we have v_lj∂φ_i/∂ x_j(x,0) = v_li,   s=0 .Because v_l is the eigenvector of a∇^2 F, we candirectly solve the solution of (<ref>)-(<ref>) and obtainv_lj∂ φ_i/∂ x_j(x,s) = e^-λ_l s v_li ⟺∂ φ_i/∂ x_j(x,s) = ∑_l=1^d e^-λ_l s v_li (a^-1)_jrv_lr , ∀ s ≥ 0 ,for 1 ≤ i, j ≤ d. Sending s→ +∞, using (<ref>) and (<ref>), we obtain ∂Θ_i/∂ x_j = lim_s→ +∞∂ φ_i/∂ x_j(x,s) = ∑_l=1^d-k v_li(a^-1)_jrv_lr =P_ij . * We proceed to compute a_jr∂^2Θ_i/∂ x_j∂ x_r, 1 ≤ i≤ d. For this purpose, let us defineA_l(x,s) =(a^-1)_ij' v_lj' a_jr∂^2 φ_i/∂ x_j∂ x_r(x,s), 1 ≤ l ≤ d , ⟺ a_jr∂^2 φ_i/∂ x_j∂x_r(x,s) = ∑_l=1^d v_li A_l(x,s).Using the second equation of (<ref>),the solution (<ref>), and the orthogonality of the eigenvectors, we can obtaind A_l/ds(x,s) = - ∑_l'=1^d[ 2∂ a_ir/∂ x_i'∂^2 F/∂ x_r∂ x_j(a^-1)_ir' +∂^3 F/∂ x_r'∂ x_i'∂ x_j] v_l'i'v_l'j v_lr'e^-2λ_l's- λ_l A_l(x,s) ,for 1 ≤ l ≤ d, from which we get P_ii'a_jr∂^2 φ_i'/∂ x_j∂ x_r(x,s) = ∑_l=1^d-k v_liA_l(x,s) = -∑_l=1^d-k∑_l'=1^d[ 2∂ a_jr/∂ x_i'∂^2 F/∂ x_r∂ x_j'(a^-1)_jr' +∂^3 F/∂ x_r'∂ x_i'∂ x_j'] v_l'i'v_l'j'v_lr'v_lie^-λ_l s∫_0^s e^(λ_l -2λ_l')u du =∑_l=1^d-k∑_l'=1^d [2λ_l'∂ (a^-1)_i'r/∂ x_j -∂^3 F/∂ x_r∂ x_i'∂ x_j] v_l'i'v_l'jv_lrv_li ∫_0^s e^-2λ_l' u du .To further simplify the last expression above, we differentiate the identity∂^2 F/∂ x_i'∂ x_j v_l'i'v_l'j = λ_l' ,where l' is fixed, 1 ≤ l' ≤ d, along the eigenvector v_l, which gives∂^3 F/∂ x_r∂ x_i'∂ x_j v_l'i'v_l'jv_lr=-2 ∂^2 F/∂ x_i'∂ x_j∂ v_l'i'/∂ x_r v_l'j v_lr + ∂λ_l'/∂ x_r v_lr= -2 λ_l'(a^-1)_i'r' v_l'r'∂ v_l'i'/∂ x_rv_lr + ∂λ_l'/∂ x_r v_lr .Therefore, taking the limit s→ +∞, usingthe relations (<ref>), (<ref>), and Lemma <ref> in Appendix <ref>, we can computeP_ii' a_jr∂^2 Θ_i'/∂ x_j∂ x_r= lim_s→ +∞ P_ii'a_jr∂^2 φ_i'/∂ x_j∂ x_r(x,s)=lim_s→ +∞∑_l'=d-k+1^d∑_l=1^d-k[2λ_l'∂ (a^-1)_i'r/∂ x_j v_l'i'v_l'j +2 λ_l'(a^-1)_jr' v_l'r'∂ v_l'j/∂ x_r - ∂λ_l'/∂ x_r] v_lrv_li ∫_0^s e^-2λ_l' u du= ∂ (a^-1)_i'r/∂ x_j(a_i'j - (Pa)_i'j) (Pa)_ir -1/2∂ (a^-1)_jr'/∂ x_r(a_jr' - (Pa)_jr') (Pa)_ir - 1/2 (Pa)_ir∂lnΨ/∂ x_r=-P_ir∂ a_jr/∂ x_j - ∂ P_jr/∂ x_j (Pa)_ir + P_ir∂ (Pa)_jr/∂ x_j+ ∂ P_jr/∂ x_j (Pa)_ir= -P_ir∂ a_jr/∂ x_j + P_ir∂ (Pa)_jr/∂ x_j . On the other hand, differentiating the relation ξ(Θ(x)) ≡0 twice and using (<ref>), we get∂ξ_γ/∂ x_i'∂^2 Θ_i'/∂ x_j∂ x_r =- ∂^2 ξ_γ/∂ x_i'∂ x_j'∂Θ_i'/∂ x_j∂Θ_j'/∂ x_r=- ∂^2 ξ_γ/∂ x_i'∂ x_j' P_i'jP_j'r , for 1 ≤γ≤ k. Therefore, using PaP^T=P^2a=Pa and Pa∇ξ_γ=0, we can compute(δ_ii'-P_ii') a_jr∂^2 Θ_i'/∂ x_j∂ x_r= (Ψ^-1)_αγ(a∇ξ_α)_i ∂ξ_γ/∂ x_i'a_jr∂^2 Θ_i'/∂ x_j∂ x_r= - (Ψ^-1)_αγ (a∇ξ_α)_i ∂^2 ξ_γ/∂ x_i'∂ x_j'P_i'jP_j'r a_jr =- (Ψ^-1)_αγ (a∇ξ_α)_i (∂^2_i'j'ξ_γ)(Pa)_i'j'= (Pa)_i'j'∂ P_ii'/∂ x_j' .Summing up (<ref>) and (<ref>), we conclude thata_jr∂^2 Θ_i/∂ x_j∂ x_r=∂ (Pa)_ij/∂ x_j -P_ir∂ a_rj/∂ x_j .Now, we prove Theorem <ref>.Since we follow the approach in <cit.>, we will only sketch the proof and will mainly focus on the differences.First of all, we introduce some notations. Let x^(l), l = 0, 1, ⋯, be the states generated from the numerical scheme (<ref>) and letψ be a function on Σ. We will adopt the abbreviations ψ^(l) = ψ(x^(l)), P^(l) = P(x^(l)), etc.For j≥ 1, D^jψ[u_1, u_2, ⋯, u_j] denotes thejth order directional derivatives of ψ along the vectors u_1, u_2, ⋯, u_j, and |D^jψ|_∞ is the supremum norm of D^jψ on Σ. Similarly, D^jΘ[u_1, u_2, ⋯, u_j] denotes the d-dimensional vector whose ith component isD^jΘ_i[u_1, u_2, ⋯, u_j], for 1 ≤ i ≤ d.Define the vector b^(l)=(b^(l)_1, b^(l)_2, ⋯, b^(l)_d)^T by b_i^(l) = (-a_ij∂ U/∂ x_j + 1/β∂ a_ij/∂ x_j)(x^(l)) ,1 ≤ i ≤ d , for l = 0, 1, ⋯, and set δ^(l) = b^(l)h + √(2β^-1 h) σ^(l)η^(l) . We have δ^(l) = x^(l+1/2)- x^(l) , x^(l+1) = Θ(x^(l+1/2)) = Θ(x^(l)+δ^(l)) .Let ℒ be the infinitesimal generator of the SDE (<ref>) in Theorem <ref>, given byℒ =- (Pa)_ij∂ U/∂ x_j∂/∂ x_i + 1/β∂ (Pa)_ij/∂ x_j∂/∂ x_i + 1/β (Pa)_ij∂^2/∂ x_i∂ x_j= e^β U/β∂/∂ x_i(e^-β U(Pa)_ij∂/∂ x_j) ,in Remark <ref>. We consider the Poisson equation on Σℒψ = f - f .The existence and the regularity of the solution ψ can be established under Assumption <ref>–<ref>, and the Bakry-Emery condition in Section <ref>. Applying Taylor's theorem and using the fact that Θ(x^(l)) = x^(l) since x^(l)∈Σ, we haveψ^(l+1)=(ψ∘Θ)(x^(l) + δ^(l))= ψ^(l) + D(ψ∘Θ)^(l)[δ^(l)] + 1/2D^2(ψ∘Θ)^(l)[δ^(l), δ^(l)] + 1/6D^3(ψ∘Θ)^(l)[δ^(l),δ^(l),δ^(l)] + R^(l) ,where the reminder is given by R^(l) =1/6(∫_0^1 s^3 D^4(ψ∘Θ)(x^(l)+(1-s)δ^(l)) ds)[δ^(l),δ^(l),δ^(l),δ^(l)] . Now we apply Proposition <ref> tosimplify the expression in (<ref>). Using the chain rule, the expressions (<ref>)–(<ref>), we can derive ψ^(l+1)=ψ^(l) + Dψ^(l)[P^(l)δ^(l)+1/2 D^2Θ^(l)[δ^(l),δ^(l)]] + 1/2D^2ψ^(l)[P^(l)δ^(l), P^(l)δ^(l)]+ 1/6D^3(ψ∘Θ)^(l)[δ^(l),δ^(l),δ^(l)] + R^(l)=ψ^(l) + (ℒψ)^(l) h+ √(2β^-1h)Dψ^(l)[(Pσ)^(l)η^(l)] +h^2/2 Dψ^(l)[D^2Θ^(l)[ b^(l), b^(l)]] + √(2β^-1) h^3/2 Dψ^(l)[D^2Θ^(l)[ b^(l), σ^(l)η^(l)]] + h^2/2D^2ψ^(l)[P^(l)b^(l),P^(l)b^(l)]+ √(2β^-1) h^3/2 D^2ψ^(l)[P^(l)b^(l), (Pσ)^(l)η^(l)]+ hDψ^(l)/β[D^2Θ^(l)[σ^(l)η^(l),σ^(l)η^(l)]-a^(l):∇^2 Θ^(l)]+ h/β(D^2ψ^(l)[(Pσ)^(l)η^(l), (Pσ)^(l)η^(l)] - (Pa)^(l):D^2ψ^(l))+ 1/6D^3(ψ∘Θ)^(l)[δ^(l),δ^(l),δ^(l)] + R^(l) ,where in the last equation we added and subtracted some terms, andwe used the identity Dψ^(l)[P^(l)b^(l)+1/β a^(l):∇^2 Θ^(l)] + 1/β (Pa)^(l):∇^2ψ^(l) = (ℒψ)^(l) ,which can be verified using Proposition <ref>, (<ref>) and (<ref>). In (<ref>), a:∇^2Θ is the vector whose ith component is given by a_jr∂^2 Θ_i/∂ x_j∂ x_r, and (Pa):∇^2 ψ is defined in a similar way. Summing up (<ref>) for l=0, 1, ⋯, n-1, dividing both sides by T, and using the Poisson equation (<ref>), givesf_n - f =1/n∑_l=0^n-1 f(x^(l)) - f= ψ^(n)-ψ^(0)/T + 1/T∑_i=1^5 M_i,n + 1/T∑_i=1^4 S_i,n ,where M_1,n =-√(2β^-1h)∑_l=0^n-1Dψ^(l)[(Pσ)^(l)η^(l)],M_2,n =-√(2β^-1) h^3/2∑_l=0^n-1 Dψ^(l)[D^2Θ^(l)[b^(l), σ^(l)η^(l)]] , M_3,n =-h/β∑_l=0^n-1 Dψ^(l)[D^2Θ^(l)[σ^(l)η^(l),σ^(l)η^(l)]- a^(l):∇^2 Θ^(l)] , M_4,n =- √(2β^-1) h^3/2∑_l=0^n-1 D^2ψ^(l)[P^(l)b^(l), (Pσ)^(l)η^(l)] , M_5,n =-h/β∑_l=0^n-1(D^2ψ^(l)[(Pσ)^(l)η^(l), (Pσ)^(l)η^(l)] - (Pa)^(l):D^2ψ^(l)) ,andS_1,n =-h^2/2∑_l=0^n-1 Dψ^(l)[D^2Θ^(l)[b^(l),b^(l)]], S_2,n =-h^2/2∑_l=0^n-1 D^2ψ^(l)[P^(l)b^(l), P^(l)b^(l)] , S_3,n =-∑_l=0^n-1 R^(l) ,S_4,n = -1/6∑_l=0^n-1 D^3(ψ∘Θ)^(l)[δ^(l),δ^(l),δ^(l)] .Using (<ref>),the last term S_4,n abovecan be further decomposed asS_4,n = M_0,n + S_0,n ,where M_0,n =-√(2β^-1)/6 h^3/2∑_l=0^n-1(2/β D^3(ψ∘Θ)^(l)[σ^(l)η^(l),σ^(l)η^(l),σ^(l)η^(l)]+3 h D^3(ψ∘Θ)^(l)[b^(l),b^(l), σ^(l)η^(l)]) , S_0,n =- h^2/6∑_l=0^n-1(6/β D^3(ψ∘Θ)^(l)[b^(l), σ^(l)η^(l),σ^(l)η^(l)] + hD^3(ψ∘Θ)^(l)[ b^(l),b^(l),b^(l)]) .Notice that the terms M_i,n, 0 ≤ i ≤ 5, are all martingales and in particular we have 𝐄 M_i,n = 0. Therefore, since the level set Σ is compact (Assumption <ref>), the first conclusion follows from the estimates |S_1,n| ≤ C|Dψ|_∞ h T ,|S_2,n| ≤C|D^2ψ|_∞h T ,𝐄|S_0,n| ≤ C|D^3ψ|_∞h T,𝐄|S_3,n| ≤ C|D^4ψ|_∞ h T,while the second conclusion follows by squaring both sides of (<ref>) and using the estimates 𝐄|S_0,n|^2 ≤ C h^2 T^2 |D^3ψ|^2_∞ , 𝐄|S_3,n|^2 ≤ C h^2 T^2 |D^4ψ|^2_∞ ,𝐄 |M_0,n|^2 ≤ Ch^2 T|D^3ψ|^2_∞ , 𝐄 |M_1,n|^2 ≤ CT|Dψ|^2_∞ ,𝐄 |M_2,n|^2 ≤ Ch^2 T|Dψ|^2_∞ ,𝐄 |M_3,n|^2 ≤ ChT|Dψ|^2_∞ ,𝐄 |M_4,n|^2 ≤ Ch^2 T|D^2ψ|^2_∞ ,𝐄 |M_5,n|^2 ≤ ChT|D^2ψ|^2_∞ . As far as the third conclusion (pathwise estimate) is concerned, notice that (<ref>) implies| f_n - f| ≤ |ψ^(n)-ψ^(0)|/T + 1/T∑_i=0^5 |M_i,n| + 1/T∑_i=0^3 |S_i,n|≤C(h + 1/T) + 1/T∑_i=0^5 |M_i,n| ,where we have used the estimates (<ref>) for |S_1,n|, |S_2,n|, and the upper bounds|S_0,n|≤Ch^2 ∑_l=0^n-1 |η^(l)|^2 + Ch^3 n ≤ ChT ,a.s. |S_3,n|≤Ch^2 ∑_l=0^n-1 |η^(l)|^4 + Ch^4 n ≤ ChT ,a.s.which are implied by the strong law of large numbers for 1/n∑_l=0^n-1 |η^(l)|^4, when n → +∞. Finally, we estimate the martingale terms M_i,n in (<ref>). Notice that, for any r ≥ 1, we can deduce the following upper bounds (see <cit.>)1/T^2r𝐄|M_1,n|^2r≤C/T^r ,1/T^2r𝐄|M_2,n|^2r≤Ch^2r/T^r , 1/T^2r𝐄|M_3,n|^2r≤Ch^r/T^r ,1/T^2r𝐄|M_4,n|^2r≤Ch^2r/T^r , 1/T^2r𝐄|M_5,n|^2r≤Ch^r/T^r ,1/T^2r𝐄|M_0,n|^2r≤Ch^2r/T^r ,which give 𝐄(1/T∑_i=0^5 |M_i,n|)^2r≤C/T^2r∑_i=0^5 𝐄|M_i,n|^2r≤C/T^r .Now, for any 0 < ϵ < 1/2, the Borel-Cantelli lemmaimplies that there is an almost surely bounded random variable ζ(ω), such that1/T∑_i=0^5 |M_i,n| ≤ζ(ω)/T^1/2 - ϵ .Therefore, the third conclusion follows readily from(<ref>) and (<ref>).Next, we prove Corollary <ref>.From the estimates in (<ref>), we know that it is only necessary to consider the term M_1,n in (<ref>). Recall that ψ solves the Poisson equation (<ref>) and we can assume ∫_Σψ dμ_1 = 0 without loss of generosity. Applying the Poisson equation, the Poincaré inequality, and the Cauchy-Schwarz inequality, we have the standard estimates∫_Σψ^2 dμ_1 ≤-1/K∫_Σ (ℒψ) ψ dμ_1≤ 1/K[∫_Σ (ℒψ)^2 dμ_1]^1/2( ∫_Σψ^2 dμ_1)^1/2= 1/K[∫_Σ (f-f)^2 dμ_1]^1/2( ∫_Σψ^2 dμ_1)^1/2 ,which implies(∫_Σψ^2 dμ_1)^1/2≤1/K[∫_Σ (f-f )^2 dμ_1]^1/2 ,- ∫_Σ (ℒψ) ψ dμ_1≤1/K∫_Σ (f-f )^2 dμ_1 .Since the term M_1,n in (<ref>) is a martingale, we have 1/T^2𝐄 |M_1,n|^2 = 2β^-1/T1/n∑_l=0^n-1𝐄[((Pa)^(l)∇ψ^(l)) ·∇ψ^(l)] .Applying the first estimate in the conclusion of Theorem <ref>, using (<ref>) in Remark <ref>, as well as the estimate (<ref>), we obtain1/T^2𝐄 |M_1,n|^2 ≤ 2β^-1/T∫_Σ (Pa∇ψ)·∇ψdμ_1 + C(h/T+1/T^2) ≤ 2∫_Σ (f-f )^2 dμ_1/KT+ C(h/T+1/T^2) .The conclusion follows by squaring both sides of (<ref>), applying Young's inequality, and using the same argument of Theorem <ref>. Finally, we prove Proposition <ref>, which concerns the properties of the projection map Π defined in (<ref>). For 1 ≤ l ≤ d, recall that p_l = (Pσ)_i'le_i' is the tangent vector field defined in Appendix <ref> such thatp_l∈ T_xΣ at each x ∈Σ. Since Π_i(x) = x_i for x ∈Σ, 1 ≤ i ≤ d, taking derivatives along p_l twice, we obtain∂Π_i/∂ x_j (Pσ)_jl= (Pσ)_il , ∂^2 Π_i/∂ x_j∂ x_r(Pσ)_jl(Pσ)_rl = (Pσ)_rl∂ (Pσ)_il/∂ x_r - ∂Π_i/∂ x_j(Pσ)_rl∂ (Pσ)_jl/∂ x_r .Notice that, for a function which only depends on the state and is evaluated at x ∈Σ, we will often omit its argument in order to keep the notations simple. On the other hand, the vectorσ_l - p_l = ((I-P)σ)_i'le_i'∈ (T_xΣ)^⊥ (the complementof the subspace T_xΣ in T_xℳ). Letϕ(s) be the geodesic curve in ℳ such that ϕ(0) = x and ϕ'(0) = σ_l - p_l.We have Π_i(ϕ(s)) = x_i, ∀ s ∈ [0, ϵ) for someϵ > 0. Taking derivatives with respect to s twice, we obtain ∂Π_i/∂ x_j(ϕ(s))dϕ_j(s)/ds =0 , ∂^2 Π_i/∂ x_j∂ x_r(ϕ(s))dϕ_j(s)/dsdϕ_r(s)/ds=-∂Π_i/∂ x_j(ϕ(s)) d^2ϕ_j(s)/ds^2 =∂Π_i/∂ x_j(ϕ(s)) Γ^j_rr'(ϕ(s)) dϕ_r(s)/dsdϕ_r'(s)/ds , for 1 ≤ i ≤ d, where ϕ_j denotes the jth component of ϕ, andthe geodesic equation of the curve ϕ has been used to obtain the last expression above. Inparticular, setting s=0, we obtain∂Π_i/∂ x_j(σ_jl - (Pσ)_jl) = 0 , ∂^2 Π_i/∂ x_j∂ x_r(σ_jl - (Pσ)_jl) (σ_rl - (Pσ)_rl) =∂Π_i/∂ x_jΓ^j_rr'(σ_rl - (Pσ)_rl)(σ_r'l - (Pσ)_r'l) . Combining the first equations in both (<ref>) and(<ref>), we can conclude that ∂Π_i/∂ x_j= P_ij at x ∈Σ. Since (<ref>) holds at any x ∈Σ, takingthe derivative in the first equation of (<ref>) along the tangent vector p_l∈ T_xΣ, we obtain∂^2 Π_i/∂ x_j∂ x_r(σ_jl - (Pσ)_jl) (Pσ)_rl = - ∂Π_i/∂ x_j (Pσ)_rl∂(σ_jl - (Pσ)_jl) /∂ x_r . Combining (<ref>), (<ref>) and (<ref>),using Lemma <ref> in Appendix <ref>,the expression in (<ref>),the relations (Pσ)_jl (Pσ)_rl = (PaP^T)_jr = (Pa)_jr , (σ_rl - (Pσ)_rl) (σ_r'l - (Pσ)_r'l)= a_rr' - (Pa)_rr' , and the integration by parts formula, we can compute ∂^2 Π_i/∂ x_j∂ x_r a_jr∂^2 Π_i/∂ x_j∂ x_r(Pσ + (σ - Pσ))_jl(Pσ + (σ - Pσ))_rl =∂^2 Π_i/∂ x_j∂ x_r (Pσ)_jl (Pσ)_rl+ 2 ∂^2 Π_i/∂ x_j∂ x_r (σ - Pσ)_jl (Pσ)_rl+ ∂^2 Π_i/∂ x_j∂ x_r (σ - Pσ)_jl(σ - Pσ)_rl = (Pσ)_rl∂ (Pσ)_il/∂ x_r - P_ij(Pσ)_rl∂ (Pσ)_jl/∂ x_r- 2P_ij (Pσ)_rl∂(σ_jl - (Pσ)_jl) /∂ x_r+ P_ijΓ^j_rr'(σ_rl - (Pσ)_rl) (σ_r'l - (Pσ)_r'l)= [(Pσ)_rl∂ (Pσ)_il/∂ x_r + P_ij(Pσ)_rl∂ (Pσ)_jl/∂ x_r- 2P_ij (Pσ)_rl∂σ_jl/∂ x_r]+ P_ijΓ^j_rr'(a_rr' - (Pa)_rr') = [2(Pσ)_rl∂ (Pσ)_il/∂ x_r - (Pσ)_jl (Pσ)_rl∂ P_ij/∂ x_r- 2 (Pσ)_rl∂ (Pσ)_il/∂ x_r+ 2 (Pσ)_rlσ_jl∂ P_ij/∂ x_r] + (Pa)_ij∂ (a^-1)_lj/∂ x_r(a_lr-(Pa)_lr) -1/2 (Pa)_ij∂ (a^-1)_lr/∂ x_j(a_lr-(Pa)_lr) = (Pa)_lj∂ P_ij/∂ x_l+[ - P_ij∂ a_jl/∂ x_l- (Pa)_ij∂ P_lj/∂ x_l + P_il∂ (Pa)_lj/∂ x_j] +(Pa)_ij∂ P_lj/∂ x_l + 1/2 (Pa)_ij∂lnΨ/∂ x_j =- P_ij∂ a_jl/∂ x_l + ∂ (Pa)_ij/∂ x_j + 1/2 (Pa)_ij∂lnΨ/∂ x_j .plainnat

http://arxiv.org/abs/1702.08064v5

The Malgrange Form and Fredholm DeterminantsThe Malgrange Form and Fredholm DeterminantsMarco BERTOLA ^†M. Bertola^† Department of Mathematics and Statistics, Concordia University, Montréal, Canada mailto:marco.bertola@concordia.camarco.bertola@concordia.ca^ Area of Mathematics SISSA/ISAS, Trieste, Italy mailto:marco.bertola@sissa.itmarco.bertola@sissa.itReceived March 12, 2017, in final form June 17, 2017; Published online June 22, 2017We consider the factorization problem of matrix symbols relative to a closed contour, i.e., a Riemann–Hilbert problem, where the symbol depends analytically on parameters. We show how to define a function τ which is locally analytic on the space of deformations and that is expressed as a Fredholm determinant of an operator of “integrable” type in the sense of Its–Izergin–Korepin–Slavnov. The construction is not unique and the non-uniqueness highlights the fact that the tau function is really the section of a line bundle.Malgrange form; Fredholm determinants; tau function35Q15; 47A53; 47A68§ INTRODUCTIONWe shall consider the following prototypical matrix Riemann–Hilbert problem (RHP) on the unit circle Σ (or any smooth closed simple contour):_+(z;) = _-(z;) M(z;),∀z∈Σ, (∞)=.Herestands for a vector of parameters which we refer to as “deformation parameters”. The assumptions are the following; =0pt* The matrix M(z;)∈ GL_n() is jointly analytic for z in a fixed tubular neighbourhood N(Σ) of Σ and  in an open connected domain 𝒮, which we refer to as the “deformation space”.* The index of M(z;) around Σ vanishes for all ∈𝒮.* The partial indices are generically zero, i.e., the RHP (<ref>) generically admits solution.Let us remind the reader of some facts that can be extracted from <cit.> =0pt–There exists a matrix function Y_-(z) analytic and analytically invertible in Ext(Σ) ∪ N(Σ) (and uniformly bounded) and similarly a matrix function Y_+(z) analytic and analytically invertible in Int(Σ) ∪ N(Σ) and n integers k_1,…, k_n (called partial indices) such thatD(z) Y_+(z) = Y_-(z) M(z),z∈Σ ,D = diag( z^k_1, …, z^k_n) . – The RHP (<ref>) is solvable if and only if all partial indices vanish, k_j=0, ∀j=1,…, n. Note that since ind_Σ M=∑_j=1^n k_j, the condition (2) in our assumptions is necessary for the solvability of (<ref>).We shall denote by _± the interior (+) and the exterior (-) regions separated by Σ. Defi­ne _̋+ to be the space of functions that are in L^2(Σ, |ẓ|) and extend to analytic functions in the interior. We will use the notation ̋⃗_+ = _̋+⊗^r (i.e., vector-valued such functions). The vectors will be thought of as row-vectors. We also introduce the Cauchy projectors C_± L^2(Σ,|ẓ|) →_̋±:C_±[f](z) = ∮_Σ f(w)ẉ/(w-z)2iπ, z ∈_±. It is well known <cit.> that the RHP (<ref>) is solvable if and only if the Toeplitz operator[Due to our choices of symbols, the matrix symbol of the relevant Toeplitz operator is M^-1. We apologize for the inconvenience.]T_S ̋⃗_+→̋⃗_+,T_S [f⃗] = C_+[f⃗ S] , S(z;) := M^-1(z;)is invertible, in which case the inverse is given byT_S^-1[f⃗] = C_+ [f⃗_-^-1]_+. Moreover the operator is Fredholm and(T_S) - coker (T_S) = ind_Σ M =0. There is no reasonable way, however, to define a “determinant” of T_S as it stands. Such a function of  would desirably have the property that the RHP (<ref>) is not solvable if and only if this putative determinant is zero.While this is notoriously impossible in this naive form, we now propose a proxy for the notion of determinant, in terms of a simple Fredholm determinant.The Malgrange one-form. As Malgrange explains <cit.> one can define a central extension on the loop group 𝒢:= {MΣ→ GL_n()ind_Σ M=0} given by 𝒢 ={ (M,u)∈𝒢×^×} with the group law[We are being a bit cavalier in this description; we invite the reader to read pp. 1373–1374 in <cit.>.](M, u) · ( M,u) =(MM, uu c(M, M)) , c(M,M):= __̋+(T_M^-1 T_ M^-1 T^-1_(MM)^-1).The operator in the determinant is of the form Id__̋+ + (trace class) and hence the Fredholm determinant is well defined. This group law is only valid for pairs M, M for which the inverse of T^-1_(MM)^-1 exists. The left-invariant Maurer–Cartan form of this central extension is then given by (S^-1δ S, ụ/u +ω_M), whereω_M := Tr__̋+( T_S^-1∘ T_δ S - T_S^-1δ S), S:= M^-1,and this can be written as the following integralω_M = ∮_ΣTr(_+^-1_+' M^-1δ M) .Here, and below, δ denotes the exterior total differentiation in the deformation space 𝒮:δ = ∑δ t_j / t_j. The Malgrange form is a logarithmic form in the sense that it has only simple poles on a co-dimension 1 analytic submanifold of the deformation space 𝒮 and with positive integer Poincaré residue along it; this manifold is precisely the exceptional “divisor” (Θ)⊂𝒮 (the Malgrange divisor) where the RHP (<ref>) becomes non solvable, i.e., where some partial indices of the Birkhoff factorization become non-zero.Closely related to (<ref>) is the following one-form, which we still name after Malgrange:ω_M := ∮_ΣTr(_-^-1_-' δ M M^-1) .It is also a logarithmic form with the same pole-divisor; indeed one verifies thatω_M - ω_M = ∮_ΣTr(M' M^-1δ M M^-1) ,which is an analytic form of the deformation parameters ∈𝒮. In <cit.> the one-form (<ref>) was posited as an object of interest for general Riemann–Hilbert problems (not necessarily on closed contours) and its exterior derivative computed (with an important correction in <cit.>, which is however irrelevant in the present context). It was computed (but the computation can be traced back to Malgrange himself in this case) thatδω_M = 1/2∮_ΣTr(Ξ(z) ∧/ẓΞ(z) ), Ξ(z;):= δ M(z;) M(z;)^-1.It appears from (<ref>) that this two form δω_M is not only closed, but also smooth on the whole of 𝒮, including the Malgrange-divisor. As such, it defines a line-bundle ℒ over 𝒮 by the usual construction: one covers 𝒮 by appropriate open sets U_α where δω_M =δθ_α; on the overlap U_α∩ U_β the form θ_α-θ_β is also exact and one defines then the transition functions by g_αβ() = exp(∫θ_α-θ_β). Then a section of this line bundle is provided by the collection of functions τ_α U_α→ such thatτ_α() =exp[∫ ( ω_M - θ_α )].Since ω_M is a logarithmic form and each θ_α is analytic in the respective U_α, the func­tions τ_α() have zero of finite order precisely on the Malgrange divisor (Θ)⊂𝒮 (under appropriate transversality assumptions, the order of the zero is the dimension of Ker T_S).Our goal is to provide an explicit construction of the τ_α's in terms of Fredholm determinants of simple operators of the Its–Izergin–Korepin–Slavnov “integrable” type <cit.>. Their definition is recalled in due time.§ CONSTRUCTION OF THE FREDHOLM DETERMINANTS The construction carried out below is not unique, and also only local in the deformation space 𝒮; this is however not only not a problem, but rather an interesting feature, as we will illustrate in the case of SL_2(). The non-uniqueness is precisely a consequence of the fact that we are trying to compute a section of the aforementioned line bundle.Preparatory step. The assumption that M(z;)∈ GL_n can be replaced without loss of generality with M(z;)∈ SL_n; this is so because of the assumption on the index of M. Indeed we can solve the scalar problem y_+ (z) =y_-(z)M(z), y(∞)=1 and then define a new RHP where _± (z) := _±diag(y_±^-1 , 1,…) and hence the new matrix jump foris M (z)= diag(y_-(z) , 1,…) M(z) diag(y_+^-1(z) , 1,…) with M ≡ 1. For this reason, from here on we assume M∈ SL_n().We define an elementary matrix (for our purposes) to be a matrix of the form + cE_jk, with j≠ k, where E_jk denotes the (j,k)-unit matrix. Any matrix M∈ SL_n() can be written as a product of elementary matrices. The entries of the factorization are rational in the entries of M with denominators that are monomials in a suitable set of n-1 nested minors of M.We recall that given any matrix M∈ SL_n, there is a permutation Π of the columns such that the principal minors (the determinants of the top left square submatrices) do not vanish, and hence we can write it asM = L D U Π =MΠ,where L, U are lower/upper triangular matrices with unit on the diagonal and D = diag(x_1,…, x_n) is a diagonal matrix (see for example <cit.>). Denote q_ℓ = [ M_j,k]_j,k≤ℓ the principal minors; these are the nested minors of the original matrix M alluded to in the statement. The matrices L, U are rational in the entries of M and with denominators that are monomials in the q_ℓ's.Now, both L, U can clearly be written as products of elementary matrices whose coefficients are polynomials in the entries of L, U (respectively) and so it remains to show that we can write D as product of elementary matrices.To this end we observe the 'LULU' identity (there is a similar `ULUL' identity)[x 001/x] = [1 0 1-x/x1 ] [1 10 1 ] [1 0x-1 1 ] [1 -1/x0 1 ].As D∈ SL_n, we can represent it in terms of product of embedded SL_2 matrices using the root decomposition of SL_n:D = diag(x_1,1/x_1,1,…,1) diag(1,x_2 x_1, 1/x_1 x_2,1,…,1) ×diag(1,1, x_1x_2x_3, 1/x_1 x_2x_3,1,…,1) ⋯,and then embed the LULU identity for each factor.Finally, also permutation matrices can be written as product of elementary matrices embedding appropriately the simple identity;[0 -1 1 0 ] = [1 0 1 1 ] [1 -1 0 1 ] [10 1 1 ].This concludes the proof. Let now M(z) be an SL_n matrix valued function, analytic in a tubular neighbourhood N(Σ) of Σ, and let q_ℓ(z), ℓ=1,…, n-1 be the nested minors alluded to in the Lemma so that they are not identically zero. Since the entries are analytic in N(Σ) we can slightly deform the contour Σ to a contour Σ that avoids all zeroes of every principal minor q_ℓ(z). The resulting RHP is “equivalent” to the original in the sense that the solvability of one implies the solvability of the other. Note also that this deformation can be done in a piecewise constant way locally with respect to ∈𝒮. Thus we haveM(z) = F_1(z) ⋯ F_R(z), F_ν(z) =+ a_ν (z) E_j_ν,k_ν ,ν = 1,…, R. Corresponding to this factorization we can define an equivalent RHP with jumps on R contours Σ_1,…, Σ_R, with Σ_1=Σ and Σ_j+1 in the interior of Σ_j and all of them in the joint domain of analyticity of the scalar functions a_ν(z) (see Fig. <ref>) which may have poles only at the zeroes of the principal minors q_ℓ(z) of M(z). This is accomplished by “extending” the matrix _-(z) to the annular regions _0 = _- and _j = Int(Σ_j)∩Ext(Σ_j+1) asΘ_0 (z) :=_-(z), ∀z∈_0, Θ_ν(z) := _-(z) ∏_ℓ=1^ν F_ℓ(z) ,∀z∈_ν.By doing so we obtain the following relationsΘ_ν(z) = Θ_ν-1(z) F_ν(z) ,∀z∈Σ_ν ,ν=1,…, R.The piecewise analytic matrix function Θ(z) whose restriction to _ν coincides with the matri­ces Θ_ν (<ref>), satisfies a final RHP Θ_+(z) = Θ_-(z) F_ν (z),∀z∈Σ_ν,Θ(∞)=. This type of RHP is of the general type of “integrable kernels” and its solvability can be determined by computing the Fredholm determinant of an integral operator of L^2(Σ_ν, |ẓ|) ≃⊕_ν=1^R L^2(Σ_ν, |ẓ|) with kernel (we use the same symbol for the operator and its kernel)K(z,w) = f⃗^T(z) g⃗(w)/2iπ(w-z) ,f⃗(z) = ∑_ν=1^Re_j_νχ_ν(z) a_ν(z) ,g⃗(z) = ∑_ν=1^Re_k_νχ_ν(z),where χ_ν(z) is the projector (indicator function) on the component L^2(Σ_ν, |ẓ|). Indeed, as explained in <cit.>, the Fredholm determinant (Id - K) is zero if and only if the RHP (<ref>) is non-solvable and moreover the resolvent operator R = K(Id-K)^-1 of K has kernelR(z,w) =f⃗^t(z) Θ^t(z) (Θ^t)^-1 (w)g⃗(w)/z-w. The RHP (<ref>) and hence (<ref>) is solvable if and only if τ :=(Id - K) ≠ 0. [see, e.g., <cit.>]Letbe any deformation of the functions a_ν(z), thenlnτ = ∑_ν=1^R ∮_Σ_νTr(Θ_-^-1Θ_-'(z)F_ν F_ν^-1) ẓ/2iπ= ∑_ν=1^R ∮_Σ_ν(Θ_ν-1^-1Θ_ν-1'(z) )_k_ν, j_ν a_ν(z) ẓ/2iπ.§ THE SL_2 CASE We would like to express the Malgrange one-form directly in terms of the τ function (Fredholm determinant). Rather than obscuring the simple idea with the general case, we consider in detail the SL_2 case. Let M(z;) be analytic in (z,)∈ N(Σ)×𝒮 and with values in SL_2(). Using the general scheme above, we have the following factorizations(1)[a bcd ] = [1 0 1+c-a/a1 ]_F_1(z)[1 10 1 ]_F_2(z)[1 0a-1 1 ]_F_3(z)[1b-1/a0 1 ]_F_4(z) , a ≢0, (2)[a bcd ] = [11+b-d/d0 1 ]_F_1(z)[1 01 1 ]_F_2(z)[1 d-10 1 ]_F_3(z)[1 0 c-1/d1 ]_F_4(z),d ≢0, (3)[a b-1/b 0 ] = [1 b-ab0 1 ]_F_1(z)[10-1/b1 ]_F_2(z)[1 b0 1 ]_F_3(z), (4)[0 b-1/b d ] = [1d-1/b0 1 ]_F_1(z)[10b 1 ]_F_2(z)[1 -1/b0 1 ]_F_3(z), (5)[0 b-1/b 0 ] = [1 0b 1 ]_F_1(z)[1 -1/b0 1 ]_F_2(z)[1 0b 1 ]_F_3(z).§.§ Fredholm determinants for different factorizationsEach of the factorization (<ref>) leads to an integrable operator of the form (<ref>) and hence to a corresponding Fredholm determinant; we now establish their mutual relationships.There are two types of questions that we address here =0pt* How are the Fredholm determinants associated with the different factorizations (1,2) in (<ref>) related to each other?* For a fixed factorization, how does the Fredholm determinant depend on the choice of contour Σ (within the analyticity domain N(Σ)). Consider the cases (<ref>)_(ρ), ρ=1,…, 5. We compute the logarithmic derivative of the corresponding Fredholm determinant using Proposition <ref>lnτ_(ρ) = ∑_ν=1^R ∮_Σ_νTr(Θ_-^-1Θ_-' F_ν F_ν^-1) ẓ/2iπ.By using the relationship (<ref>) between Γ_- (extended to an analytic function on Ext(Σ) ∪ N(Σ)), we can re-express it in terms of the Malgrange one-form of the original problem (<ref>); we use the fact that we can deform the contours back to Σ = Σ_1 by Cauchy's theorem. Plugging (<ref>) appropriately in (<ref>) and using Leibnitz rule, after a short computation we obtainδlnτ__(ρ) = ∮_ΣTr(_-^-1_-'M M^-1) ẓ/2iπ+∮_ΣTr( F_1^-1 F_1 'F_2 F_2^-1 +F_12 ^-1 F_12'F_3 F_3^-1 + F_123^-1 F_123'F_4 F_4^-1) ẓ/2iπ =:ω_M + θ_(ρ), where F_1… k = F_1F_2F_3⋯ F_k (if the factorization has only three term, then we set F_4≡).The cases (3,4,5). The last cases (<ref>)_(3,4,5) lead essentially to a RHP with a triangular jump; it suffices to re-define Γ by Γ[[01; -10 ]] for z∈Int(Σ). If the index of b is zero, ind_Σ b=0,then the solution can be written explicitly in closed form and it is interesting to compute the Fredholm determinant associated to our factorization of the matrix. We will showIn the cases (<ref>)_(3,4,5) and under the additional assumption that ind_Σ b=0, the τ function given by (Id_L^2(∪Σ_j) - K) and K as in (<ref>), equals the constant in the strong Szegö formula, given by <cit.>τ = exp[ ∑_j>0 j β_-jβ_j] = __̋+ T_b T_b^-1, where now the Toeplitz operator is for the scalar symbol b(z) and _̋+ is the Hardy space of scalar functions analytic in _+ and β_j are the coefficients of ln b(z) in the Laurent expansion centered at the origin β(z):= ln b(z) = ∑_j∈β_j z^j.The same applies in the case that the jump is triangular (b≡ 0 and/or c≡ 0) under the assumption ind_Σ a=0 and replacing b with a in the above formulas.From a direct computation of the term θ_(ρ) in (<ref>), we findθ__(3,4,5) = ∮_Σ( δβ/ẓβ),β(z):= ln b(z)and the solution of the RHP is explicit,Γ(z) = [e^B(z) -e^-B(z) ∮_Σa(w)b(w)e^2B_-(w)/w-z 0 e^-B(z)],z∈_-,[e^B(z) -e^-B(z) ∮_Σa(w)b(w)e^2B_-(w)/w-z 0 e^-B(z)][0 1-10 ],z∈_+, B(z) := ∮_Σβ(w)/w-zẉ/2iπ.Thus we can write explicitly the Malgrange form:ω_M = ∮_ΣTr(_-^-1_-' δ M M^-1)= ∮_ΣTr(_3 ∮_Σβ(w)/(w-z_-)^2δβ(z) _3) = 2∮_Σ_-∮_Σβ(w)δ (β(z))/(w-z)^2.In this integral, z is integrated on a slightly “larger” contour Σ_-. The Fredholm determinant satisfiesδlnτ_(3,4,5) =ω_M+ θ__(3,4,5)= 2∮_Σ_-∮_Σβ(w)δ (β(z))/(w-z)^2+∮_Σ[ β (δβ)' ] .We now show thatδlnτ = δ∮_Σ_-∮_Σβ(w) β(z)/(w-z)^2=∮_Σ_-∮_Σδβ(w) β(z) + β(w) δβ(z)/(w-z)^2.Indeed, the exchange of order of integration of one of the addenda (and relabeling the variables) yields the other term plus the residue on the diagonal,∮_Σ_-∮_Σδβ(w) β(z) + β(w) δβ(z)/(w-z)^2=∮_Σ_-∮_Σ2 β(w) δβ(z)/(w-z)^2 + ∮_Σβ(z) (δβ)' .Therefore, in conclusion, we have (we fix the overall constant of τ by requiring it to be 1 for b≡ 1)lnτ = ∮_Σ_-∮_Σln b(w) ln b (z)/(w-z)^2.If we write a Laurent expansion of ln b (<ref>) the formula (<ref>) gives the explicit expressionlnτ = ∑_j>0 j β_-jβ_j,which is also the formula for the second Szegö limit theorem for the limit of the Toeplitz determinants of the symbol b(z), and it is known to to be the Fredholm determinant of an opera­tor τ = __̋+ T_b T_b^-1,where now the Toeplitz operator is for the scalar symbol b(z) and _̋+ is the Hardy space of scalar functions analytic in _+.The index assumption is only necessary for the case (<ref>)_(5) or (<ref>)_(1) when b≡ 0 ≡ c, because in these situations the RHP separates into two scalar problems. However, the assumption ind_Σ b=0 for cases (3,4) or ind_Σ a =0 for the triangular case is not necessary as we now show (in the latter form). Consider the RHPY_+ = Y_- [ 1μ(z)01 ],z∈Σ, Y(z) = ( + 𝒪(z^-1)) z^n_3,z→∞.By defining(z) =Y(z), z∈Int(Σ), Y(z) z^-n_3, z∈Ext(Σ),we are lead to the RHP in standard form_+ = _- [ z^n z^nμ(z)0z^-n],z∈Σ,(∞) =,which is case (1) with c≡ 0 (or essentially cases (3,4) up to a multiplication by piecewise constant matrices). Even if a(z) =z^n and thus ind_Σ a = n, this problem is still generically solvable in terms of appropriate “orthogonal polynomials” {p_ℓ(z)}_ℓ∈ defined by the “orthogonality” property∮_Σ p_ℓ(z) p_k(z) μ(z)ẓ = h_ℓδ_ℓ k.Then the solution of the Y-problem is written asY(z) = [p_n(z)∮_Σp_n(w)μ(w)ẉ/(w-z)2iπ-2iπ p_n-1(z)/h_n-1 ∮_Σ-p_n-1(w)μ(w)ẉ/(w-z) h_n-1],and the solvability depends only on the condition <cit.>[∮_Σ z^ℓ+j-2μ(z) ẓ]_j,ℓ=1^n≠ 0.The cases (1,2). A straightforward computation using the explicit expression (<ref>)_(1) yieldslnτ__(1) = ∮_Σ[Tr(_-^-1_-'M M^-1) + ln a (ln a)' (1 + bc) - cb (ln a)' + c'b - c' b(ln a)] = ∮_ΣTr(^-1'M M^-1) ẓ/2iπ+ ∮_Σ( (a)'/a(da - cb )+a c'(b/a))ẓ/2iπ_θ__(1).Since lnτ is a closed differential, the exterior derivative of θ__(1) must be opposite to the one of the first term, which is given by (<ref>). Let us verify this directly; to this end we compute the exterior derivative of θ__(1). A straightforward computation yieldsδθ__(1)= ∮_Σ(δ b ∧ (δ c)' + δ a ∧ (δ a)' /a^2 (1 + bc) +δ a ∧(δ c/a)' b. .-δ a ∧δ b (c)'/a-δ b∧ (δ a)' c/a-δ b ∧δ ca'/a).The exterior derivative of ω_M is given by (<ref>), in which we can insert the explicit expression of M(z;); after a somewhat lengthy but straightforward computation we find thatδω_M + δθ__(1) = -1/2∮_Σ/ẓ( c/aδ a ∧δ b -b/aδ a ∧δ c + δ b∧δ c ) = 0,thus confirming that the differential δlnτ__(1) is indeed closed (of course it must be, since the tau function is a Fredholm determinant!). The case (2) is analogous with the replacements b↔ c and a↔ d.§.§.§ Determinants for different choices of Σ The factorizations (<ref>)_(1,2) require that we deform the contour Σ so that a(z) (or d(z)) does not have any zero on Σ. This leads to completely equivalent RHPs of the form (<ref>) but not entirely equivalent RHP when expressed in the form (<ref>).Note that (<ref>)_(3,4,5) do not suffer from this ambiguity, because b(z) cannot have any zeroes in N(Σ) since we assumed analyticity of the jump matrix M(z;).Consider the factorization (<ref>)_(1) (with similar considerations applying to the other factorization); in general a(z;) has zeroes in its domain of analyticity, and their positions depend on . Therefore it may be necessary, when considering the dependence on , to move the contour so that certain zeroes are to the left or to the right of it because a zero may sweep across N(Σ) as we vary ∈𝒮.So, let Σ, Σ be two contours in the common domain of analyticity of M(z;) and such that a(z;) has no zeroes on either one and Σ⊂Int(Σ) (see Fig. <ref>).Denote with τ__(1) and τ__(1) the corresponding Fredholm determinants of the operators defined as described; following the same steps as above. Our goal is to show thatThe ratio of the two Fredholm determinants is given byτ__(1)=τ__(1)∏_v ∈Int(Σ) ∩Ext(Σ): a(v)=0(c(v();) )^- ord_v(a).Note that the evaluation of c at the zeroes of a cannot vanish because M ≡ 1.From the formula (<ref>) we getlnτ__(1)/τ__(1)= (∮_Σ - ∮_Σ) Tr(_-^-1_-'M M^-1) ẓ/2iπ+(∮_Σ - ∮_Σ) ( (a)'/a(da - cb )+ac'(b/a))ẓ/2iπ.Here _- means the analytic extension of the solutionto the region Ext(Σ) ∪ N(Σ). Since the integrand of the first term is holomorphic in the region bounded by Σ and Σ, it yields a zero contribution by the Cauchy's theorem and we are left only with the second term, which is computable by the residue theorem;lnτ__(1)/τ__(1) = ∑_v ∈Int(Σ) ∩Ext(Σ): a(v)=0 z=v((a)'/a(da - cb ) + c'a (b/a)).We are assuming that M(z;) is analytic and also that M ≡ 1. Now note that a zero v() of a(z;) in general depends on ; suppose that a(z;) = (z-v())^k (C_k() + 𝒪(z-v())); thena(z;)/a(z;) =- k v()/z-v() + 𝒪(1),and hence the residue evaluation at z=v() yields (we use 0 = a' v +a|_v and (ad)|_v = da|_v = (bc)|_v)z=v( (a)'/a( da - cb )+ ac'(b/a) )= k ( -d a'|_vv - cb|_v) + k bc'v= k ( da |_v - cb|_v) + k bc'v = k ((bc) |_v - cb|_v) + k bc'v= k bc |_v + k bc' |_vv =^bc|_v =-1 - k ln c(v();)(note that c(v();) cannot be zero because M≡ 1 and z=v is already a zero of a). §.§.§ Determinants for different factorizations The tau functions (Fredholm determinants) defined thus far should be understood as defining a section of a line bundle over the loop group space; this is the line bundle associated to the two fom δω_M (<ref>).This simply means that on the intersection of the open sets where the factorizations (1), (2) in (<ref>) can be made, we haveδlnτ__(1)/τ __(2)= δln( Υ_(1,2)),whereδln( Υ_(1,2)) :=δ( θ__(1) - θ__(2)).After a short computation we obtainδln( Υ_(1,2)) = ∮_Σ( c' δ b - b' δ c/1 + b c) = ∮_Σ( δln b /ẓln (ad) - δln (ad) /ẓln b ).Observe the last expression; in principle the functions b(z), a(z)d(z) may have nonzero index around Σ, but in any case the functions δln b and (ln (ad))' are single-valued because the increments of the logarithms are integer multiples of 2iπ and hence locally constant in the space of deformations. To write explicitly the transition function (or rather a representative of the same cocycle class) we need to choose a point z_0∈Int(Σ); we choose z=0 without loss of generality. Let K =ind_Σ b, L = ind_Σ (ad); then we can rewrite the above expression as followsδln( Υ_(1,2))= ∮_Σ[ δln(b/z^K) /ẓln(ad/z^L)- δln(ad/z^L) /ẓln(b/z^K)- δln(ad/z^L) K/z + δln(b/z^K) L/z] ,and after integration by parts (which is now possible since all functions involved are single-valued in N(Σ))δln( Υ_(1,2)) =δ∮_Σ[ ln(b/z^K) /ẓln(ad/z^L) -ln(ad/z^L) K/z + ln(b/z^K) L/z] ,so that the transition function admits the explicit expressionΥ_(1,2) =exp[∮_Σ[ ln(b/z^K) /ẓln(ad/z^L) - ln(ad/z^L) K/z + ln(b/z^K) L/z] ]. § CONCLUSIONWe conclude this short note with a few comments.First of all the choice of Σ as a single closed contour is not necessary; we can have a disjoint union of closed contours or an unbounded contour as long as M(z) converges to the identity sufficiently fast as |z|→∞ or near the endpoints. The considerations extend with trivial modifications. Our approach is similar in spirit to the approach used in <cit.> to express the tau function of a general isomonodromic system with Fuchsian singularities in terms of an appropriate Fredholm determinant.Much less clear to the writer is how to handle the case where Σ contains intersections; in this case we should stipulate a local “no-monodromy” condition at the intersection points as explained in <cit.>. The obstacle is not the issue of factorization but the fact that the resulting RHP of the IIKS type leads to an operator K (<ref>) which is not of trace-class (and not even Hilbert–Schmidt, which would be sufficient in order to construct a Hilbert–Carleman determinant).Nonetheless, the two form δω_M defines a line bundle as explained and therefore it is possible to compute the “transition functions” of the line-bundle; this is precisely what is accomplished (in different setting and in different terminology) in recent works <cit.> and the transition functions can be expressed in terms of explicit expressions, analogously to what we have shown here in this general but simplified setting. §.§ AcknowledgementsThe author wishes to thank Oleg Lisovyy for asking a very pertinent question on the representation of the Malgrange form in terms of Fredholm determinants. Part of the thinking was done during the author's stay at the “Centro di Ricerca Matematica Ennio de Giorgi” at the Scuola Normale Superiore in Pisa, workshop on “Asymptotic and computational aspects of complex differential equations” organized by G. Filipuk, D. Guzzetti and S. Michalik. The author wishes to thank the organizers and the Institute for providing an opportunity of fruitful exchange.[1]Referencesref 99 =0ptBasorWidom-BO Basor E.L., Widom H., On a Toeplitz determinant identity of Borodin andOkounkov, https://doi.org/10.1007/BF01192828Integral Equations Operator Theory 37 (2000),397–401, https://arxiv.org/abs/math.FA/9909010math.FA/9909010.BertolaIsoTau Bertola M., The dependence on the monodromy data of the isomonodromic taufunction, https://doi.org/10.1007/s00220-009-0961-7Comm. Math. 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Not. 2012 (2012), 1519–1568,https://arxiv.org/abs/1005.4083arXiv:1005.4083.BorodinOkounkov Borodin A., Okounkov A., A Fredholm determinant formula for Toeplitzdeterminants, https://doi.org/10.1007/BF01192827Integral Equations Operator Theory 37 (2000),386–396, https://arxiv.org/abs/math.CA/9907165math.CA/9907165.Chihara Chihara T.S., An introduction to orthogonal polynomials, Mathematics andits Applications, Vol. 13, Gordon andBreach Science Publishers, New York – London – Paris, 1978.ClanceyGohberg Clancey K.F., Gohberg I., Factorization of matrix functions and singularintegral operators, https://doi.org/10.1007/978-3-0348-5492-4Operator Theory: Advances and Applications,Vol. 3, Birkhäuser Verlag, Basel – Boston, Mass., 1981.Gantmacher Gantmacher F.R., The theory of matrices. Vols. 1, 2, Chelsea Publishing Co.,New York, 1959.GavrylenkoLisovyy Gavrylenko P., Lisovyy O., Fredholm determinant and Nekrasov sumrepresentations of isomonodromic tau functions, https://arxiv.org/abs/1608.00958arXiv:1608.00958.ItsHarnad Harnad J., Its A.R., Integrable Fredholm operators and dual isomonodromicdeformations, https://doi.org/10.1007/s002200200614Comm. Math. Phys. 226 (2002), 497–530,https://arxiv.org/abs/solv-int/9706002solv-int/9706002.Its:2016kq Its A., Lisovyy O., Prokhorov A., Monodromy dependence and connection formulaefor isomonodromic tau functions, https://arxiv.org/abs/1604.03082arXiv:1604.03082.Its:2015qf Its A., Lisovyy O., Tykhyy Yu., Connection problem for thesine-Gordon/Painlevé III tau function and irregular conformal blocks,https://doi.org/10.1093/imrn/rnu209Int. Math. Res. Not. 2015 (2015), 8903–8924,https://arxiv.org/abs/1403.1235arXiv:1403.1235.ItsIzerginKorepinSlavnov Its A.R., Izergin A.G., Korepin V.E., Slavnov N.A., Differential equations forquantum correlation functions, https://doi.org/10.1142/S0217979290000504Internat. J. Modern Phys. B4 (1990), 1003–1037.Malgrange:IsoDef1 Malgrange B., Sur les déformations isomonodromiques. I. Singularitésrégulières, in Mathematics and Physics (Paris, 1979/1982),Progr. Math., Vol. 37, Birkhäuser Boston, Boston, MA, 1983,401–426.Malgrange:Deformations Malgrange B., Déformations isomonodromiques, forme de Liouville,fonction τ, https://doi.org/10.5802/aif.2052Ann. Inst. Fourier (Grenoble) 54 (2004),1371–1392.

http://arxiv.org/abs/1703.00046v4

apsrevc_k,σ^†c_k,σ ç_̧ķ↑̧ c_-k↓ c_k↑^† c_-k↓^† c_p↑ c_-p+q↓ c_p↑^† c_-p+q↓^†#1⟨#1⟩1/2Phys. Rev. A Phys. Rev. B Phys. Rev. D Rev. Mod. Phys.  Phys. Rev. Lett.  Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, ChinaBeijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, ChinaDepartment of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USABeijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, Chinajphu@iphy.ac.cn Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, ChinaCollaborative Innovation Center of Quantum Matter, Beijing, ChinaKavli Institute of Theoretical Sciences, University of Chinese Academy of Sciences,Beijing 100049, China We investigatesuperconductivity that may exist in the doped BaCoSO, a multi-orbital Mott insulator with a strong antiferromagnetic ground state.The superconductivity is studied in both t-J type and Hubbard type multi-orbital modelsbymean field approach and random phase approximation (RPA) analysis. Even if there is no C_4 rotational symmetry, it is found thatthe system still carriesa d-wave like pairing symmetry state with gapless nodes andsign changed superconducting order parameters on Fermi surfaces.The results are largely doping insensitive. In this superconducting state, the three t_2g orbitals have very different superconducting form factors in momentum space.In particular,the intra-orbital pairing of the d_x^2-y^2orbital has a s-wave like pairing form factor.The two methods alsopredict very different pairing strength on different parts of Fermi surfaces.These results suggest that BaCoSO and related materials can be a new ground to test and establish fundamental principles for unconventionalhigh temperature superconductivity.75.85.+t, 75.10.Hk, 71.70.Ej, 71.15.MbTheoretical studies of superconductivity in dopedBaCoSO Jiangping Hu December 30, 2023 =========================================================§ INTRODUCTION Since the discovery of cuprates<cit.> and the iron based high T_c superconductors<cit.> (SCs), intensive research efforts have been made to understand their superconducting pairing mechanism. In the past three decades, great progress hasbeen made both experimentally and theoretically. Various experimental techniques, such as angle-resolved photoemission spectroscopy <cit.> (ARPES), inelastic neutron scattering<cit.> (INS), scanning tunneling spectroscopy<cit.> (STS), etc. and many different theories have been developed in the research of high T_c SCs. However, due to the complexity of the problem, no consensus about the microscopic pairing mechanism has been reached.Recently, by comparing cuprates and the iron based SCs,we have pointed out thatthose d-orbitals that are responsible for the superexchange antiferromagnetic (AFM) interactions mediated through anionsare isolated near Fermi energy to generate superconductivity in both families of high T_c SCs<cit.>. In this scenario,the pairing symmetry can be simply determined through an emergent empirical principle,theHu-Ding principle<cit.>.More interestingly, thiselectronic feature is largely absent in other correlated electron systems. Thus,we have suggested thatthis property can be the gene of unconventional high T_c SCs and materials satisfying the condition can bepromising high T_c candidates.Based on such an understanding, two families of materials<cit.> have been proposed to be promising high T_c SCs. However, the proposals havenot been tested until now because of the difficulty in synthesizing the proposed materials.However,recently we have observed that an already-synthesized material BaCoSO<cit.> may give us a chance to test the theory. The lattice structure of BaCoSO is similar to the case in ref.<cit.>, but the tetrahedron environment around the Co atoms is broken because of the anion mixture of the O and S atoms. Though it is not the ideal structure to maintain the required electronic condition,we have expected that the theory<cit.> issuitable to BaCoSOand superconductivity may arise in the doped BaCoSO <cit.> if the structure distortion and disorder induced by doping can be minimized. Motivated by this, we carry out the theoretical investigation of the superconducting state in this type of electronic structures.The paper is organized as follows. In the first part, the electronic structure and magnetic property of BaCoSO are reviewed. In the second part, the superconducting pairing in doped BaCoSO is analyzed by the mean field theory based on a t-J type multi-orbital model.In the third part, we study the superconducting state in a Hubbard type of model under the RPA approximation. Then, we come to our conclusion in the last part. § ELECTRONIC STRUCTURE AND MAGNETIC PROPERTYBaCoSO has an orthorhombic layered lattice structure in which each CoSO layer is constructed by vertex sharing mixed-anion tetrahedron complexes CoS_2O_2. In this layer structure, the Co chains along y-direction are connected through S atoms and the staggered Co chains along x-direction are linked through O atoms. Compared with a perfect tetrahedron environment in whichthe crystal field splits the five d-orbitals into two groups, t_2g and e_g,thecrystal fieldhere breaksthe degeneracy of the three t_2g orbitals as well.However, we have shown that this breaking is relatively small and the three t_2g orbitals still control the major electronic physics.Without doping, BaCoSO has been confirmed to be an antiferromagnetic(AFM)Mott insulator both theoretically<cit.> and experimentally<cit.>. TheAFMorder is G-type, similar to those of cuprates.With doping, the electronic structure in the normal state has been calculated in ref.<cit.>.It has been shown thatthe three t_2g orbitals dominate near the Fermi level. In this paper, we investigate the possible superconducting state of this system andperform calculations based on the three band model in the unfolded Brillouin zone (BZ) constructed by the three t_2g orbitals (d_xz,d_yz,d_x^2-y^2) in ref.<cit.> H_0^11 = ϵ_1+2t^11_xxcos(2k_x) +2t^11_ycos(k_y)+4t^11_yycos(2k_y)-2t^11_xcos(k_x)-4t^11_xycos(k_x)cos(k_y), H_0^12 = 4t^12_xysin(k_x)sin(k_y), H_0^13 =2it^13_xxsin(2k_x)+2it^13_xsin(k_x)+4it^13_xysin(k_x)cos(k_y), H_0^22 = ϵ_2+2t^22_xxcos(2k_x)+2t^22_ycos(k_y)+2t^22_yycos(2k_y) -2t^22_xcos(k_x)-4t^22_xycos(k_x)cos(k_y), H_0^23 =2it^23_ysin(k_y)+2it^23_yysin(2k_y)+4it^23_xycos(k_x)sin(k_y), H_0^33 = ϵ_3+2t^33_xxcos(2k_x)+2t^33_ycos(k_y)+2t^33_yycos(2k_y) +2t^33_xcos(k_x)+4t^33_xycos(k_x)cos(k_y), the corresponding tight binding parameters are shown in Table.<ref>. The three bands model capturesthe major electronic structure around the Fermi energy.In Fig.<ref>, we plot Fermi surfaces and their corresponding orbital characters at two different electron doping levels. For the case of 0.6 electron doping away from the half filling,the Fermi surfaces (FSs) as shown in Fig.<ref>(a) are composed of three pockets. A small hole pocket at the BZ center Γ pointis attributed to the d_x^2-y^2 and d_xz orbitals, so do the two electron pockets aroundthe BZboundary Y point.A large open hole pocket around the BZ center is attributed to the d_yz orbital.The hybridization ofthe d_x^2-y^2 and d_xz orbitals stems from the the Zig-Zag Co chain structure along the x direction.With heavy electron doping, the small hole pocket atthe BZ center can sink below the Fermi energy as shown in Fig.<ref>(b). The above electronic structure resembles those of iron-based SCs. We can make a good comparison between them. Iniron-based SCs, typically there arealso three types of pockets, two hole pockets from d_xz,yzandone hole pocket from d_xy at the BZ center Γ point, and two electron pockets around the BZ corner at M point that are mixed with d_xy and d_xz,yz<cit.>.The hole pockets at Γ point in iron-based SCs can also sink below FSs by electron doping<cit.>. The role of different pockets and the interactions among them have been the central debate in iron-based SCs<cit.>. Therefore, the study of this new material can provide much deep understanding on these issues.§ MEAN FIELD ANALYSIS We start with an effective t-J type Hamiltonian for BaCoSO, which is generally written asH= H̃_0+∑_<ij>,α,a,b (J^α_abS⃗_ia·S⃗_jb-1/4n_ian_jb),where H̃_0 isthe three bands Hamiltonian in Eq.<ref><cit.> subject to a projection to non-double occupant orbital state due to the onsite Hubbard interaction, <ij> labels the two nearest neighbour (NN) sites, α=x,y labels direction and a, b are the orbital indexes.In local atomic orbital approximation, the strength of the AFM interaction can be roughly estimated from the superexchange process.J^α_ab takes the formJ^α_ab=(t^α_ab)^2(1/U_d+1/U_d+Δ_pd),where t^α_ab is the effective hoping parameter between the NN d-orbitals at Co sites, U_d is the Coulomb interaction for the d-orbitals and Δ_pd is defined as the energy difference between the d-orbital and p-orbital at O and S atoms. With the parameters in ref.<cit.>, we can get the AFM interaction strength in BaCoSO as follows: J^x_xz=0.20eV, J^x_yz=0.09eV, J^x_x^2-y^2=0.15eV and J^y_xz=0.09eV, J^y_yz=0.36eV, J^y_x^2-y^2=0.09eV.It is important to note that this estimation is entirely based on atomic orbitals. As the p-orbitals of O and S atoms are very differentand the latter are more extended than the former, the effective AFM coupling through S atoms is expected to be smaller than the estimated values. For this reason, we set a variable β on the value of J̃^̃ỹ=β J^y. Because of the space anisotropy of the two d-orbitals, the AFM interactionsfor thed_xz and d_yz orbitalsand those for the d_x^2-y^2 orbital along the two different directions have significant anisotropy. Such a large anisotropy suggests that the electronic physics here is rather nematic. In the mean field calculation, weapproximate the projection as an overall renormalization factor of the bare Hamiltonian, namely H̃_0=γ H_0, where γ is the renormalization factor<cit.>.γ generally is doping dependent and can be measured experimentally.By rescaling the energy,it is also equivalent to absorb the renormalization factor into the interaction parameters so that we can simply treatH̃_0=H_0 in the mean field calculation.Combining with the estimated bare AFM interaction parameters, the mean field calculation is performed by setting AFM interaction in the unit of eV for the corresponding orbitals to be [0.20, 0.09, 0.15]× Jalong the x-direction and [0.09, 0.36, 0.09]×β J along the y-direction.We report the phase diagram of the superconductingstate with respect toβ and J.First, we set β=1 and report results as a function of Jfor two different doping levels. The FSs is shown in Fig.<ref>(a) when the doping level is 0.6 electron per site, and the corresponding mean field results are shown in Fig.<ref>. The significant anisotropy of the AFM interaction for different orbitals leads to that, the pairing in the y-direction is dominated by d_yz orbital while in the x-direction d_xz and d_x^2-y^2 orbitals are dominant. As is shown in Fig.<ref>(c), the superconducting order parameter for d_x^2-y^2 is s-wave like, while it is d-wave like for both d_xz and d_yz orbitals. Furthermore, the following relationship is satisfied in the main area of thephase diagram sign(Δ^x_xz)=-sign(Δ^y_yz),sign(Δ^x_xz)=sign(Δ^x_x^2-y^2),The results of the mean field approach can be well understood within the Hu-Ding principle<cit.>, the superconducting ground state always tends to open the largest superconducting gap on the FSs. For the sake that only the NN AFM interaction is considered, the superconducting order parameter takes a form factor in momentum space as Δ^xcos k_x+Δ^ycos k_y and Δ^α is proportional to J^α. Meanwhile, the three t_2g orbitals hybridize only at several small area on the FSs, as is shown in Fig.<ref>(a). Therefore, it is easy to see that a s-wave like form factor cos k_x+cos k_yfor the d_x^2-y^2 orbital and ad-wave like pairing on d_xz and d_yz orbitals can open the largest gaps on the FSs. The phase relationship of the order parameters between different orbitals can be also determined at the area where different orbitals hybridize on the FSs. To achievelarger superconducting gaps, the order parameters of these orbitals tend to have the same phase. Specifically, for 0.6 electron doped BaCoSO, because the d_yz orbital and d_x^2-y^2 orbital hybridize strongly at the smaller FS near the Γ point, Δ^x_xz tends to have the same phase with Δ^x_x^2-y^2. A similar mean field analysis is also done when the electron doping level is about 1.0 per site. The corresponding FSs and mean field results are shown in Fig.<ref>(b) and Fig.<ref>, respectively. The mean field results here aresimilar to those ofthe 0.6 electron doped case. Furthermore,we set J=2.0 and report results as a function of β. As shown in Fig. <ref>, the results arealso similar. The qualitative results onthe superconducting order parametersare very robust against β.Overall, the mean field theory gives a rather robust superconducting state: a s-wave like order parameter for d_x^2-y^2 orbital, a d-wave like order parameter for both d_xz,yz orbitals, and totally a d-wave like pairing symmetry on the FSs with nodes near (π/2,π/2). § RANDOM PHASE APPROXIMATION ANALYSIS Based on the three bands model above, the RPA analysis is carried out for BaCoSO in this section with onsite repulsive interactions. The total Hamiltonian is given byH= H_0+U∑_i,αn_iα↑n_iα↓+U^'∑_i,α<βn_iαn_iβ+J∑_i,α<β,σσ^'c^†_iασc^†_iβσ^,c_iασ^'c_iβσ+J^'∑_i,α≠βc^†_iα↑c^†_iα↓c_iβ↓c_iβ↑,wheren_i,α=n_i,α,↑+n_i,α,↓. For other indexes, we adopt the parameter notations given in ref.<cit.>. In the RPA approximation, the pairing vertex is Γ_ij(k,k^')= Re[∑_l_1l_2l_3l_4a^l_2,∗_v_i(k)a^l_3,∗_v_i(-k)×Γ_l_1l_2l_3l_4(k,k^',ω=0)a^l_1_v_j(k^')a^l_4_v_j(-k^')],where the momenta k and k^' is restricted to different FSs within an energy cutoff Λ,with k∈ C_i and k^'∈ C_j. a^l_v(orbital index l and band index v) is the component of the eigenvectors of the three-orbitals tight binding Hamiltonian. The singlet channel of orbital vertex function Γ_l_1l_2l_3l_4 in RPA is given by Γ_l_1l_2l_3l_4(k,k^',ω)= [3/2U̅^sχ^RPA_1(k-k^',ω)U̅^s+1/2U̅^s-1/2U̅^cχ^RPA_0(k-k^',ω)U̅^c+1/2U̅^c]_l_3l_4l_1l_2,where χ^RPA_1 and χ^RPA_0 are the spin and charge fluctuation RPA susceptibility, respectively. The spin and charge interaction matrix(U̅^s, U̅^c) are the same as in ref.<cit.>. The pairing strength function is λ[g(k)]=-∑_ij∮_C_idk_/v_F(k)∮_C_jdk^'_/v_F(k^')g(k)Γ_ij(k,k^')g(k^')/(2π)^2∑_i∮_C_idk_/v_F(k)[g(k)]^2,where v_F(k)=|∇_kE_i(k)| is the Fermi velocity on a given Fermi surface sheet C_i. The calculation is carried out in the spin-rotational invariance case meaning U̅^̅'̅=U̅-2J and J=J^'. First, we calculatethe bareand RPA spin susceptibilities for BaCoSO at different doping levels as shown in Fig.<ref>.The RPA spin susceptibility has a sharp peak near the wavevector (π,π) inboth doping levels. The peak mainly stems from the interaction betweenthe smaller hole pocket near the Γ point and the electron pocket near the M point.This situation is very similar to the case in iron-pnictides. The interaction between these two pockets are responsible for the superconducting pairing as well.This is because the points onthe smaller hole pocket near the Γ point and the electron pocket near the M point contribute the largest density of states(DOS)near the Fermi level. The pairing strength on the hole pocket attributed to the d_yz orbital is always small because of its large band dispersion.TheRPA results in the superconducting state are reported in Fig.<ref>. For the 0.6 electron doping, similar to the mean field results, the leading superconducting instability turns out to have a d-wave like pairing symmetry, as shown in Fig.<ref>(a)(c).In the 1.0heavy electron doping case, as shown in Fig.<ref>(b)(d) and (e), there are two leading superconducting instabilities whichare nearly degenerate. Both of them have many nodes on the FSs and the gap function is more complex compared to the 0.6 electron doped case.The superconducting pairing strength is also much weaker than those with the 0.6 electron doping case.There are significant differences between the RPA and mean field results. First, the superconducting gap on the two electron pockets near Y point tends to have an uniform phase in the RPA results. Second,the pairing strength on the FSs contributed bythe d_yz orbital is much weaker in the RPA analysis than in the mean field approach.Finally, competing superconducting pairing states is much easier to appear in the RPA analysis as well.These differences can be well understood in the build-in structure of theRPAanalysis as the interaction betweenthe smaller hole pocket near the Γ point and the electron pocket near the M point becomes dominant. Moreover, according to Eq.<ref>, the large DOS leads to strong pairing strength, and to avoid repulsive interaction to save energy,the pairings at these two areas tend to have a π phase difference.The importance of the interaction between the smaller hole pocket near the Γ point and the electron pocket near the M pointin the RPA analysis can be demonstrated further.By increasing the electron doping,the smaller holepocket at γ goesa Lifshitz transition. At the 1.0 doping, itsinks just below Fermi level.If the contribution fromthe hole pocket is important,the RPA calculation near the Lifshitz transition becomes very sensitive to the cutoff energy Λ from the Fermi level. For the results in Fig.<ref>(b)(d)(e), Λ is taken to be 0.005eV. If we increase this cutoff to involve the hole pocket contribution, the leading superconducting instability at the 1.0 doping is expected to vary quickly. This expectation is demonstrated inFig.<ref> in whichthe cutoff energy is increased to0.01eV.It is clear that the leading superconducting instabilitybecomes similar to the 0.6 electron doping case.§ SUMMARY AND DISCUSSION In summary, we have carried out mean field and RPA calculation to analyze the possible superconducting ground state in the family ofmaterials similar to BaCoSO whose electronic structures are described by the three t_2gorbitals.It is found that a d-wave like superconducting state with gapless nodes is generally favored. The superconducting properties in this family of materials can help us to establishfundamental principles regarding the emergence of superconductivity in unconventional high T_c SCs. First, the energy scale of the parameters in thisfamily of materials are similar to those of iron-based superconductors. Therefore, the maximum T_cthat can be achieved here should be close to the maximum T_cin the bulk material of iron-based superconductors if they share the same superconducting mechanism, which has been assumed in this paper. Second,in cuprates and iron-based SCs,the pairing symmetries are classified by the D_4h group.The superconducting states fallintospecific irreducible representations of this high symmetry group.It is difficult to mix different representations.As a result,a pure or close to a pured-waveands-wave state have been realized in cuprates and iron-based supercnoductors respectively. Here due to the absence of C_4 rotation system,the superconducting state is classified by much lower symmetry group.Thus, the superconducting state, in the term of the D_4h group, is a mixture of s-wave and d-wave state.Our results on the role of different orbitals, the location of gapless nodes and the pairing strength on different parts of Fermi surfacesthus can provide critical information about the validity of theoretical methods and test different pairing mechanisms.Finally, the interaction between the smaller hole pocket near the Γ point and the electron pocketnear the M point is very similar to the case in iron-pnictides which are also characterized with the interaction between the hole pockets at theΓ point and the electron pockets at M point<cit.>.Our results from the RPA analysis are consistent with those in iron-pnictides <cit.> . Both calculations suggestthat the interactions are responsible for superconductivity. However,in iron-chalcogenides<cit.>, the simple RPA result has been seriously challenged because the high T_c superconductivity can still be achieved in the absence of hole pockets.As the hole pockets can also sink below Fermi level by doping in this family of materials,the validity of the RPA analysis can be further tested.For example,the heavy electron doping may be achieved by substituting Co with Ni atoms. We want to mention that strong superconducting instability cannotbe obtainedbythe standard functional renormalization group (FRG) method <cit.> in the above model.Combining all these results andthe fact that the FRGis also only valid in the weak interaction region,observing high T_c superconductivity in the family of materials may finally explore the limitation of these standard approaches. § ACKNOWLEDGEMENTThis work is supported by the Ministry of Science and Technology of China 973 program(Grant No. 2015CB921300), National Science Foundation of China (Grant No. NSFC-1190020, 11534014, 11334012), and the Strategic Priority Research Program ofCAS (Grant No. XDB07000000). BaFe2As2_surfacecuprates J. G. Bednorz and K. A. Muller, Z. Phys. B 64, 189 (1986).iron_based Y. Kamihara, T. Watanabe, M. Hirano and H. Hosono, JACS 130, 3296 (2008).ARPES1 A. Damascelli, Z. Hussain and Z. X. Shen, Rev. Mod. Phys. 75, 473 (2003).INS2 P. C. Dai, Rev. Mod. Phys. 87, 855 (2015).STS2 O. Fischer, M. Kugler, I. Maggio-Aprile, C. Berthod and C. Renner, Rev. Mod. Phys. 79, 353 (2007).high_Tc_gene J. P. Hu, Science Bulletin, 61(7), 561-569 (2016).hu_ding J. P. Hu and H. Ding, Scientific Reports 2, 381 (2012).hu1 J. P. Hu, C. C. Le and X. X.Wu, Phys. Rev. X 5, 041012 (2015).hu2 J. P. Hu and C. C. Le, Sci. Bull. 62, 212 (2017).BaCoSO_AFM1 E. J. Salter, J. N. Blandy and S. J. Clarke, Inorg Chem 55, 1697 (2016).BaCoSO_AFM2 M. Valldor, U. K. Rossler, Y. Prots, C. Y. Kuo, J. C. Chiang, Z. Hu, T. W. Pi, R. Kniep and L. H. Tjeng, Chemistry 21, 10821 (2015).BaCoSO_hu C. C. Le, S. S. Qin and J. P. Hu,arXiv:1612.03470.Kuroki K. Kuroki, S. Onari, R. Arita, H. Usui, Y. Tanaka, H. Kontani and H. Aoki, Phys. Rev. Lett 101, 087004 (2008).Johnston D. C. Johnston, Adv. Phys. 59, 803-1061 (2010).Mazin I. I. Mazin, Nature 464, 183-186 (2010).Hirschfeld P. J. Hirschfeld, M. M. Korshunov and I. I. Mazin, Rep. Prog. Phys. 74, 124508 (2011). mean_feild1 G. Kotliar and J. L. Liu , Phys. Rev. B 38, 5142 (1988).mean_feild2 K. Seo, B. A. Bernevig and J. P. Hu, Phys. Rev. Lett 101, 206404 (2008).Kemper A. F. Kemper, T. A. Maier, S. Graser, H. P. Cheng, P. J. Hirschfeld and D. J. Scalapino, New J. Phys. 12, 073030 (2010).ding T. Qian et al., Phys. Rev. Lett 106, 187001 (2011).feng Y. Zhang et al., Nature Mater. 10, 273-277 (2011).zhou S. L. He et al., Nature Mater. 12, 605C610 (2013). iron_pnictide1 D. J. Singh and M. H. Du, Phys. Rev. Lett 100, 237003 (2008).iron_pnictide2 C. Cao, P. J. Hirschfeld and Hai-Ping Cheng, Phys. Rev. B 77, 220506 (2008).FRG1 W. Metzner, M. Salmhofer, C. Honerkamp, V. Meden and K. Schonhammer, Rev. Mod. Phys. 84, 299 (2012).FRG2 R. Thomale, C. Platt, J. P. Hu, C. Honerkamp and B. Andrei Bernevig, Phys. Rev. B 80, 180505 (2009).FRG3 F. Wang, H. Zhai, Y. Ran, A. Vishwanath, and D. H. Lee, Phys. Rev. Lett 102, 047005 (2009).

http://arxiv.org/abs/1702.08304v1

Young neutron stars with soft gamma ray emission and anomalous X-ray pulsar Gennady S. Bisnovatyi-Kogan Space Research Institute of Russian Academy of Sciences,Profsoyuznaya 84/32, Moscow 117997, Russia,andNational Research Nuclear University MEPhI (Moscow Engineering Physics Institute),Kashirskoe Shosse 31, Moscow 115409, RussiaEmail: gkogan@iki.rssi.ru==========================================================================================================================================================================================================================================================================================================The observational properties of Soft Gamma Repeaters and Ano­malous X-ray Pulsars (SGR/AXP) indicate to necessity of the energy source different from a rotational energy ofa neutron star. The model, where the source of the energy is connected with a magnetic field dissipation in a highly magnetized neutron star (magnetar) is analyzed. Some observational inconsistenciesare indicated for this interpretation. The alternative energy source, connected with the nuclear energy of superheavy nuclei stored in the nonequilibrium layer of low mass neutron star is discussed. Young neutron stars with soft gamma ray emission and anomalous X-ray pulsar Gennady S. Bisnovatyi-Kogan Space Research Institute of Russian Academy of Sciences,Profsoyuznaya 84/32, Moscow 117997, Russia,andNational Research Nuclear University MEPhI (Moscow Engineering Physics Institute),Kashirskoe Shosse 31, Moscow 115409, RussiaEmail: gkogan@iki.rssi.ru========================================================================================================================================================================================================================================================================================================== §INTRODUCTION Neutron stars (NS) are formed as a result of a collapse of the core of a massive star with a mass M>∼ 12 M_⊙. Conservation of the magnetic flux gives an estimation of NS magnetic field asB_ns=B_s (R_s/R_ns )^2, B_s=10 ÷ 100 Gs, at R ∼ (3 ÷ 10) R_⊙,R_ns =10 km, B_ns = 4· 10^11÷ 5· 10^13 Gs (Ginzburg 1964).Estimation of the NS magnetic field is obtained in radio pulsars by measurements of their rotational period and its time derivative, in the model of a dipole radiation, or pulsar wind model, as (E, I, and Ωare NS rotational energy,moment of inertia, and rotational angular velocity,respectively): E_rot = 0.5 IΩ^2, Ė_rot = AB^2Ω^4, B=IPṖ/4A π^2,A=R^6/6c^3,B is NS surface dipole magnetic field at its magnetic pole. Timing observations of single radiopulsars (the rapidly rotating onesconnected with young supernovae remnants are marked by star) give the following estimation B_ns= 2 · 10^11÷ 5· 10^13 Gs (Lorimer 2005). The pulsars with a small magnetic field in the left lower angle decrease their magnetic field during recycling by accretion in a close binary.see Bisnovatyi-Kogan (2006).SGR are single neutron stars with periods 2 ÷ 8 seconds They produce "giant bursts", when their luminosity L in the peak increase 5 ÷ 6 orders of magnitude. Having a slow rotation, and small rotational energy, their observed average luminosity exceeds rotational loss of energy more than 10 times, and orders of magnitude during the giant outbursts.It was suggested by Duncan and Thompson (1995), that the source of energy is their huge magnetic field, 2 or 3 order ofa magnitude larger, then the average field in radiopulsars. Such objects were calledmagnetars.§ SGR,GIANT BURSTS, AND SHORT GRB First two Soft Gamma Repeaters (SGR) had been discovered by KONUS group in 1979. The first one, FXP 0520 - 66, was discovered after the famous giant 5 March 1979 burst (Mazets et al. 1979b,c; Golenetskii et al. 1979), see also Mazets et.al (1982). In another source B1900+14 only small recurrent bursts had been observed (Mazets et al. 1979a). Now these sources are known under names SGR 0520 - 66 and SGR 1900+14 respectively. The third SGR 1806-20 was identified as a repetitive source by Laros et.al. (1986a,b). The first detection of this source as GRB070179 was reported by Mazets et al.(1981), and it was indicated by Mazets et al. (1982), that this source, having an unusually soft spectrum, can belong to a separate class of repetitive GRB, similar to FXP0520 - 66 and B1900+14. This suggestion was completely confirmed. The forth known SRG1627-41, showing giant burst, was discovered in 1998 almost simultaneously by BATSE (Kouveliotou et al. 1998a), and BeppoSAX (Feroci et al. 1998). The giant bursts had been observed until now in 4 sources. §.§ SGR0526-66 It was discovered due to a giant burst of 5 March 1979, projected to the edge of the SNR N49 in LMC, and described by (Mazets et al. 1979b,c; Golenetskii et al. 1979, Mazets et.al 1982). Accepting the distance 55 kpc to LMC, the peak luminosity in the region E_γ>30 keV was L_p ≥ 3.6× 10^45 ergs/s, the total energy release in the peak Q_p ≥ 1.6 × 10^44 ergs, in the subsequent tail Q_t=3.6 × 10^44 ergs. The short recurrent bursts have peak luminosities in this region L_p^rec=3× 10^41 -3 × 10^42 ergs/s, and energy release Q^rec=5× 10^40 -7 × 10^42 ergs. The tail was observed about 3 minutes and had regular pulsations with the period P≈ 8 s. There was not a chance to measure Ṗ in this object.§.§ SGR1900+14 Detailed observations of this source are described by Mazets et al. (1999b,c), Kouveliotou et al. (1999), Woods et al. (1999).The giant burst was observed 27 August, 1998. The source lies close to the less than 10^4 year old SNR G42.8+0.6, situated at distance ∼ 10 kpc. Pulsations had been observed in the giant burst, as well as in the X-ray emission observed in this source in quiescence by RXTE and ASCA. Ṗ was measured, being strongly variable. Accepting the distance 10 kpc, this source had in the region E_γ>15 keV: L_p > 3.7× 10^44 ergs/s, Q_p > 6.8× 10^43 ergs, Q_t=5.2 × 10^43 ergs, L_p^rec=2× 10^40 -4× 10^41 ergs/s, Q^rec=2× 10^39 -6× 10^41 ergs, P=5.16 s, Ṗ=5 × 10^-11 -1.5× 10^-10 s/s. The X-ray pulsar in the error box of this source was discovered by Hurley et al. (1999b). This source was discovered also in radio band, at frequency 111 MHz as a faint, L_r^max=50 mJy, radiopulsar (Shitov 1999), with the same P and variable Ṗ, good corresponding to X-ray and gamma-ray observations. The values of P and average Ṗ correspond to the rate of a loss of rotational energy Ė_rot=3.5 × 10^34 ergs/s, and magnetic field B=8 × 10^14 Gs. The age of the pulsar estimated as τ_p=P/2Ṗ=700 years is much less than the estimated age of the close nearby SNR. Note that the observed X-ray luminosity of this object L_x=2× 10^35 -2× 10^36 ergs/s is much higher, than rate of a loss of rotational energy, what means that rotation cannot be a source of energy in these objects. It was suggested that the main source of energy comes from a magnetic field annihilation, and such objects had been called as magnetars by Duncan and Thompson (1992). The light curve of the giant burst is given in Fig.<ref>.§.§ SGR1806-20 The giant burst from this source was observed in December 27, 2004 (Palmer et al. 2005, Mazets et al. 2005, Frederiks et al. 2007).Recurrent bursts had been studied by Kouveliotou et al. (1998b), Hurley et al. (1999a). Connection with the Galactic radio SNR G10.0-03 was found. The source has a small but significant displacement from that of the non-thermal core of this SNR. The distance to SNR is estimated as 14.5 kpc. The X-ray source observed by ASCA and RXTE in this object shows regular pulsations with a period P=7.47 s, and average Ṗ=8.3× 10^-11 s/s. As in the previous case, it leads to the pulsar age τ_p ∼ 1500 years, much smaller that the age of SNR, estimated by 10^4 years. These values of P and Ṗ correspond to B=8× 10^14 Gs. Ṗ is not constant, uniform set of observations by RXTE gave much smaller and less definite value Ṗ=2.8(1.4)× 10^-11 s/s, the value in brackets gives 1σ error. The peak luminosity in the burst reaches L_p^rec∼ 10^41 ergs/s in the region 25-60 keV, the X-ray luminosity in 2-10 keV band is L_x≈ 2× 10^35 ergs/s is also much higher than the rate of the loss of rotational energy (for average Ṗ) Ė_rot≈ 10^33 ergs/s. The burst of December 27, 2004 in SGR 1806-20 was the greatest flare, ∼ 100 times brighter than ever. It was detected by many satellites: Swift, RHESSI, Konus-Wind, Coronas-F, Integral, HEND et al.. Very strong luminosity of this outburst permitted to observe the signal, reflected from the moon by the HELICON instrument on the board of the satellite Coronas-F. The position of satellites Wind and Coronas-F relative to the Earth and Moon during the outburst are given in Fig.<ref>, and the reconstructed full light curve of the outburst is given in Fig.<ref> from Mazets et al. (2005), Frederiks et al. (2007a).§.§ SRG1627-41 Here the giant burst was observed 18 June 1998, in addition to numerous soft recurrent bursts. Its position coincides with the SNR G337.0-0.1, assuming 5.8 kpc distance. Some evidences was obtained for a possible periodicity of 6.7 s, but giant burst did not show any periodic signal (Mazets et al. 1999a), contrary to three other giant burst in SGR. The following characteristics had been observed with a time resolution 2 ms at photon energy E_γ> 15 keV:L_p ∼ 8× 10^43 ergs/s, Q_p ∼ 3× 10^42 ergs, no tail of the giant burst had been observed. L_p^rec=4× 10^40 -4× 10^41 ergs/s, Q^rec=10^39 -3× 10^40 ergs. Periodicity in this source is not certain, so there is no Ṗ. §.§ SRG giant bursts in other galaxies The similarity between giant bursts in SGR, and short GRB was noticed by by Mazets et al. (1999), Bisnovatyi-Kogan (1999). The experiment KONUS-WIND had observed two short GRB, interpreted as giant bursts of SGR. The first one, GRB070201, was observed in M31 (Andromeda),1 February, 2007. The energy of the burst is equal to 1 · E^45 erg, in consistence with giant bursts of other SGR Mazets et al. (2008). The second short burst, GRB051103, was observed in the galaxy M81,3 November 2005. The energy of the burst is equalto 7· E^46 erg (Golenetskii et al. 2005, Frederiks et al. 2007).§ ESTIMATIONS OF THE MAGNETIC FIELDSIN SGR/AXP Despite the fact, that rotation energy losses are much smaller than the observed luminosity, for estimation of the magnetic field strength in these objects used the same procedure as in radio pulsars, based on measurements of P and Ṗ, and using (<ref>). The first measurements have been done forSGR 1900 + 14, in different epochs by measurements of satellites RXTE and ASCA (Kouveliotou et al. 1999), presented in Figs.<ref>-<ref>.The pulse shape is changing from one epoch to another, inducing errors in finding derivative of the period. The big jump in Ṗ, visible in Fig <ref> looks out surprising. for magnetic dipole losses, because it needs a considerable jump in the magnetic field strength, prohibited by self induction effects. Contrary, in the model of pulsar wind rotational energy losses it looks quite reasonable, that these losses strongly increase during the giant burst, when the Ṗ jump was observed.Another evidence in favour of the magnetar magnetic field was connected with the absorption lines in the spectrum of SGR 1806-20, observed by RXTE in November 1996 (Ibrahim et al. 2002). The main line corresponds to magnetic field (5÷ 7)· 10^11 Gs, when interpreted as an electron cyclotron line. In order to preserve the magnetar model, the authors <cit.> suggested that this line is connected with the proton motion, increasing the magnetic field estimation almost 2000 times. It is connected, however, with a drastic, ∼ 4· 10^6, decrease in the absorption cross-section, compared to the electron cyclotron line. Therefore, if this cyclotron line is real, its connection with the proton is very improbable. § RADIOPULSARS WITH VERY HIGH MAGNETIC FIELDS AND SLOW ROTATIONRadio pulsars are rotating neutron stars that emit beams of radio waves from regions above their magnetic poles. Popular theories of the emission mechanism require continuous electron-positron pair production, with the potential responsible for accelerating the particles being inversely related to the spin period. Pair production will stop when the potential drops below a threshold, so the models predict that radio emission will cease when the period exceeds a value that depends on the magnetic field strength and configuration. It was shown by Young et al. (1999a,b)that the pulsar J2144-3933, previously thought to have a period of 2.84s, actually has a period of 8.51s, which is by far the longest of any known radio pulsar. Moreover, under the usual model assumptions, based on the neutron-star equations of state, this slowly rotating pulsar should not be emitting a radio beam. Therefore either the model assumptions are wrong, or current theories of radio emission must be revised. The period 8.51 second is characteristic for SGR/AXP objects, but this pulsar does not show any violent behaviour, and behave like ordinary radio pulsar.Soon after this discovery, several otherradiopulsars were found, where also Ṗ, and therefore magnetic field strength was measured (Manchester et al. 2001, Camilo et al. 2000, McLaughlin et al. 2003,2004). These pulsars include: 1. PSR J1119 - 6127,P = 0.407 s,Ṗ = 4.0 · 10^-12 s/s, B=4.1· 10^13 G; 2. PSR J1814 - 1744, P = 3.975 s,Ṗ = 7.4 · 10^-13 s/s, B = 5.5 · 10^13 G;It was noted by Camilo et al. (2000), that "Both PSR J1119 — 6127 and PSR J1814 —1744 show apparently normal radio emission in a regime of magnetic field strength where some models predict that no emission should occur. Also, PSR J1814 —1744 has spin parameters similar to the anomalous X-ray pulsar (AXP) IE 2259 + 586, but shows no discernible X-ray emission. If AXPs are isolated, high magnetic field neutron stars (“ magnetars ”), these results suggest that their unusual attributes are unlikely to be merely a consequence of their very high inferred magnetic fields." 3.PSR J1847 - 0130, P=6.7 s, Ṗ = 1.3 · 10^-12 s/s, B=9.4 · 10^13 G. It was noted in the paper of McLaughlin et al. (2003), with the title "PSR J1847—0130: A RADIO PULSAR WITH MAGNETAR SPIN CHARACTERISTICS",that "The properties of this pulsar prove that inferred dipolar magnetic field strength and period cannot alone be responsible for the unusual high-energy properties of the magnetars and create new challenges for understanding the possible relationship between these two manifestations of young neutron stars." 4. PSR J1718 - 37184, P= 3.4 s , B= 7.4 · 10^13 G. It was noted in the paper of McLaughlin et al. (2004), that "These fields are similar to those of the anomalous X-ray pulsars (AXPs), which growing evidence suggests are “magnetars”. The lack of AXP-like X-ray emission from these radio pulsars (and the non-detection of radio emission from the AXPs) creates new challenges for understanding pulsar emission physics and the relationship between these classes of apparently young neutron stars."§ SGR/AXP WITH LOW MAGNETIC FIELDS AND MODERATE ROTATIONSGR/AXPJ1550-5418 (1E 1547.0-5408) was visible in radio band, showing pulsations with a period P=2.069 s )Camilo et al. 2007). The pulsations with the same periodhave been observed first only in the soft X ray band by XMM-Newton (Halpern et al. 2008). In the hard X ray region statistics of photons was not enough for detection of pulsations. In the strong outbursts in 2008 October and in 2009 January and March, observed by Fermi gamma-ray burst monitor, theperiod of 2.1s was clearly visible up to the energy ∼ 110 keV (Kaneko et al., 2010). The INTEGRAL detected pulsed soft gamma-rays from SGR/AXP 1E1547.0-5408 during its Jan-2009 outburst, in the energy band 20÷ 150 keV, showing a periodicity with P=2.1s (Kuiper et al. 2009). This object is the only SGR/AXP with a relatively low period, all previous has periods exceeding ∼ 4s.A low-magnetic-fieldSGR0418+5729 was detected by Fermi gamma-ray burst detector (Rea et al. 2010). This soft gamma repeater with low magnetic field SGR0418+5729 was recently detected after it emitted bursts similar to those of magnetars. It was noted by Rea et al. (2010) that"X-ray observations show that its dipolar magnetic field cannot be greater than 7.5·10^12 Gauss, well in the range of ordinary radio pulsars, implying that a high surface dipolar magnetic field is not necessarily required for magnetar-like activity".§ THE MAGNETAR MODEL In the paperof Duncan and Thompson (1992) was claimed, that dynamo mechanism in the new born rapidly rotating star may generate NS with a very strongmagnetic field 10^14÷ 10^15 G, called magnetars. These magnetars could be responsible for cosmological GRB, and may represent a plausible model for SGR. In the subsequent paper(Duncan and Thompson 1995) the connection between magnetars and SGR was developed in more details. The authors presented amodel forSGRs, and the energetic 1979 March 5 burst, based on the existence of neutron stars with magnetic fieldsmuch stronger than those of ordinary pulsars. They presented the following arguments point to a neutron star with B(dipole) 5· 10^14 G as the source of the March 5 event (Duncan and Thompson 1995). 1. Existence of such a strong magnetic field may spin down the star to 8 s period in the 10^4yr, what is the age of the surrounding supernova remnant N49. 2. Magnetic field provide enough energy for the March 5 event. 3. In presence such magnetic fielda large-scale interchange instability is developed with the growth time comparable to the  0.2-s, close to the width of the initial hard transient phase of the March 5 event. 4. A very strong magnetic field can confine the energy that was radiated in the soft tail of that burst. 5.A very strong magnetic field reduce the Compton scattering cross-section sufficiently to generate a radiative flux that is ∼ 10^4 times the (non-magnetic) Eddington flux; 6. The field decays significantly in ∼ 10^4 ÷ 10^5 yr, as is required to explain the activity of soft gamma repeater sources on this time-scale; and 7. The field power the quiescent X-ray emission L_X ∼ 7· 10^35 erg s^-1 observed by Einstein and ROSAT, as it diffuses the stellar interior. It is proposed that the 1979 March 5 event was triggered by a large-scale reconnection/interchange instability of the stellar magnetic field, and the soft repeat bursts are produces at cracking of the crust.These suggestions were justified only by semi-qualitative estimations. Subsequent observations of P and Ṗ in several SGR (McGill 2014), seems to support this model. However, when the rotation energy losses are much less than observed X-ray luminosity, B estimations using Ṗ are not justified, because magnetic stellar wind could be the main mechanism of angular momentum losses. The jump in Ṗ observed in the giant burst of PSR1900+14 (Fig.<ref>) is plausibly explained by a corresponding increase of the magnetic stellar wind power, while the jump in the dipole magnetic field strength is hardly possible. The jumps in Ṗ, as well as in the pulse form (Figs.<ref>,<ref>) have not been seen in the radio pulsars. In the fall-back accretion model of SGR (Chatterjee et al. 2000, Alpar 2001, Trümper et al. 2010, 2013) the estimations of the magnetic field using P and Ṗ give the values characteristic for usual radiopulsars, when there is apresence of a large scale magnetic field in the fall back accretion disk (Bisnovatyi-Kogan and Ikhsanov 2014).When the energy density of the magnetic field is much larger that that of matter, as expected in the surface layers of the magnetar, the instability should be suppressed by magnetic forces.The observations of radio pulsars, showing no traces of bursts, with magnetar magnetic fields and slow rotation (Section <ref>), detection of SGR with a small rotational period and low magnetic field, estimated from P and Ṗ values similar to radio pulsars (Section <ref>), gives a strong indication that inferred dipolar magnetic field strength and period cannot alone be responsible for the unusual high-energy properties of SGR/AXP. Therefore, another characteristic parameter should be responsible for a violent behaviour of SGR/AXP. The unusually low mass of the neutron star was suggested by Bisnovatyi-Kogan (2012), Bisnovatyi-Kogan and Ikhsanov (2014) as a parameter, distinguishing SGR/AXP neuron stars from the majority of neutron stars in radio pulsars and close X-ray binaries. §.§ Angular momentum losses by a magnetized stellar wind A magnetic stellar wind carries away the stellar angular momentum J as (Weber and Davis 1967) J̇_wind=2/3ṀΩ r_A^2,here r_Ais Alfven radius, where the energy density of the wind E_w is equal to the magnetic energy density E_B=B^2/(8π). We consider the wind with a constant outflowing velocity w, which energy density is E_w=0.5ρ w^2. In a stationary wind with a mass loss rate Ṁ the density is equal to ρ=Ṁ/4π w r^2.For the dipole stellar field we have B=μ/r^3, where μ=B_s r_*^3 is the magnetic dipole moment of the star. At the Alfven radius we have ρ_A=Ṁ/4π w r_A^2,E_wA=Ṁ w/8π r_A^2, E_BA=μ^2/8π r_A^6.From the definition of the Alfven radius r_A we obtain its value as E_wA=E_BA, r_A^4=μ^2/Ṁ w.The angular momentum of the star J=IΩ, and when the wind losses (<ref>) are the most important, we obtain the value of stellar magnetic fieldas B^2_wind=9/4I^2Ω̇^2 w/Ω^2 Ṁ r_*^6.The angular momentum and energy losses by the dipole radiation which are main losses in ordinary radiopulsars are written as (Pacini 1967)L=B_s^2 Ω^4 r_*^6/c^3, Ė=IΩΩ̇= L, J̇_PRS=L/Ω.We obtain from (<ref>) the magnetic field if the dipole radiation losses are the most important B^2_PSR=3I c^3Ω̇/2Ω^3 r_*^6.The ratio of these two values is written as B^2_PSR/B^2_wind=2 c^3Ṁ/3I ΩΩ̇w=4/3Ṁ w^2/2/I ΩΩ̇(c/w)^3 =4/3F_wind/Ė_rot(c/w)^3.Here F_wind is the the energy flux carried away by the wind, and Ė_rot is rate of the loss of rotational energy. For estimation of the energy flux carried away by the wind could be used the average X and γ-ray luminosity of SGR/AXP L_xγ, and the wind velocity is of the order of the free fall velocity of the neutron star. For low mass neutron star M≤ 0.8 M_⊙ we have v_ff=√(2GM/r_*)≈ (c/3) at M=0.6 M_⊙, r_*=15km, and B^2_PSR/B^2_wind=36L_xγ/Ė_rot.Using data from McGill(2014) and (<ref>) we obtain for the magnetic fields of SGR0526-66, SGR 1806-20, SGR 1900+14 the values 10^13, 1.7· 10^14, 6· 10^13 Gs respectively. Whilethe mechanical loss of the energy could exceed L_xγ, these values of the magnetic field are suppose to be the upper limit if the magnetic field of these SGR.§ MODEL OF NUCLEAR EXPLOSIONIt was shown by Bisnova­tyi-Kogan and Chechetkin (1974), that in the neutron star crust full thermodynamic equilibrium is not reached, and a non-equilibrium layer is formed there during a neutron star cooling, see also Bisnovatyi-Kogan 2001.The non-equilibrium layer is formed in the region of densities and pressure ρ_2<ρ<ρ_1, P_1<P<P_2, with ρ_1≃μ_e 10^6 (8/0.511)^3≃ 3.8· 10^9μ_eg/cm^3≃ 1.5· 10^10 g/cm^3 ρ_2≃μ_e 10^6 (33/0.511)^3≃ 2.7· 10^11μ_eg/cm^3≃10^12 g/cm^3 P_1= 7.1 · 10^27 in cgs units,P_2 = 2.1 · 10^30 in cgs units. The mass of the non-equilibrium layer is defined as (Bisnovatyi-Kogan and Chechetkin 1974)M_nl=4π R^4/GM(P_2 - P_1) ≃ 0.1 (P_2-P_1)≃ 2· 10^29 g≃ 10^-4M_⊙, and the energy stored in this non-equilibrium layer is estimated asE_nl≃ 4· 10^17 (P_2 - P_1) ≈ 10^48 ergHere a neutron star of a large (∼ 2M_⊙) was considered, where the nonequilibrium layer is relatively thin, and its mass, and the energy store are estimated in the approximation of a flat layer. The nuclei in the non-equilibrium layer are overabundant with neutrons, so the number of nucleons per one electron is taken as μ_e≃ 4, and the energy release in the nuclear reaction of fission is about 5· 10^-3 c^2 erg/g. A schematic cross-section of the neutron star is represented in Fig.<ref> from Baym (2007).Soon after discovery of gamma ray bursts the model of nuclear explosion was suggested (Bisnovatyi-Kogan et al. 1975), in which the non-equilibrium layer matter is brought to lower densities during a starquake. At the beginning GRB have been considered as objects inside the Galaxy, and the outburst was connected with period jumps in the neutron star rotation similar to those observed in the Crab nebula pulsar. It was suggested that: "Ejection of matter from the neutron stars may be related to the observed jumps of periods of pulsars. From the observed gain of kinetic energy of the filaments of the Crab Nebula (∼ 2· 10^41 erg) the mass of the ejected material may be estimated as (∼ 10^21 g). This leads to energies of the γ-ray bursts of the order of 10^38-10^39 erg, which agrees fully with observations at the mean distance up to the sources 0.25 kpc". A more detailed model of the strong 5 March 1979 burst, now classified as SGR 0526-66 in LMC,was considered byBisnovatyi-Kogan and Chechetkin (1981). It was identified with an explosion on the NS inside the galactic disk, at a distance ∼ 100 ps.The schematic picture of the nuclear explosion of the matter from the non-equilibrium layer ispresented in Fig.<ref>. Cosmological origin of GRB, and identification of a group of non-stationary sources inside Galaxy as SGR/AXP lead to considerable revision of the older model, presented by Bisnovatyi-Kogan et al. (1975). It becomes clear that SGR represent a very rare and very special type of objects, which produce bursts much more powerful, than it was thought before from comparison with quakes in Crab nebula pulsar. Besides, the SGR are the only sources for which the nuclear explosions could be applied, because the energy release in the cosmological GRB highly exceed the energy store in the non-equilibrium layer.It was suggested by Bisnovatyi-Kogan (2012,2015), Bisnovatyi-Kogan and Ikh­sa­nov (2014), that the property, making the SGR neutron star so different from much more numerous of them in radio pulsars, single and binary X ray sources, is connected with the value of their mass, but not the magnetic field strength, see Camilo (2000), McLaughlin (2003), and Section <ref>. Namely, it was suggested that the neutron stars in SRG/AXP have anomalously low mass, (0.4 ÷ 0.8)M_⊙, compared to the well measured masses in binary systems of two neutron stars, where neutron stars have masses ≥ 1.23M_⊙ (Ferdman et al. 2014). The violent behaviour of the low-mass NS may be connected with much thicker and more massive non-equilibrium layer, and accretion from the fall-back highly magnetized accretion disk could trigger the instability, leading to outbursts explosions(Bisnovatyi-Kogan and Ikhsanov 2014). The NS radius is increasing with mass rather slowly, so in a flat approximation the mass of non-equilibrium layer is inversely proportional to the mass. More accurate estimations have been obtained from calculations of neutron star models, presented in Fig.<ref>. In Sect.<ref> the calculated mass of the non-equilibrium layer M_nl≈ 10^-4 M_⊙ was belonged to the neutron star with the mass ∼ 2 M_⊙ (see Bethe and Johnson 1975, Malone et al. 1975). For M_ns=0.45M_⊙ the mass of the non-equilibrium layer is ∼ 7 times larger. The energy store reaches ∼ 10^49 erg, what is enough for ∼ 1000 giant bursts.The observational evidences for existence of neutron stars with masses, less than the Chandrasekhar white dwarf mass limit have been obtained by Janssen et al. (2008). Observations of the binary pulsar system J1518-4904 indicated the masses of the components to be m_p =0.72(+0.51, - 0.58 M_⊙),m_e = 2.00(+0.58, - 0.51) M _⊙ with a 95.4% probability. It was suggested by Bisnovatyi-Kogan and Ikhsanov (2014)that low mass neutron stars could be formed in the scenario of the off-center explosion (Branch and Nomoto 1986), but more detailed numerical investigation is needed to prove it. X-ray radiation of SGR/AXP in quiescent states was explained by Bisnovatyi-Kogan and Ikhsanov (2014) as a fall back accretion from the disk with a large scale poloidal magnetic field, what could also be a trigger for development of instability, leading to the mixing in the neutron star envelope, and nuclear explosion of the matter from the non-equilibrium layer. § CONCLUSIONS 1. SGR are highly active, slowly rotating neutron stars.2. Nonequilibrium layer (NL) is formed in the neutron star crust, during NS cooling, or during accretion onto it. It may be important for NS cooling, glitches, and explosions connected with SGR.3. The mass and the energy store in NL increase rapidly with decreasing of NS mass.4.The properties of pulsar with high magnetic fields prove that inferred dipolar magnetic field strength and period cannot alone be responsible for the unusual high-energy properties of SGR/AXP. The NL in low mass NS may be responsible for bursts and explosions in them.5. The upper boundary of the magnetic fields in 3 most famous SGR, measured by the average L_xγ luminosity is about one order of magnitude lower than the values obtained using the pulsar-like energy losses of the rotational energy of the neutron star.6. Magnetar model of SGR, in which the energy of the observed bursts is provided by magnetic field annihilation, seems to be not relevant. Observations of quiet radiopulsars with a "magnetar" magnetic field, and of a low-field "magnetar", is the most important indication to that conclusion. A rapid growth of rotational periods, what is a favorite argument for a "magnetar" origin, is naturally explained by action of the magnetic stellar wind. Besides, the high pressure of the magnetic field suppresses convection, which is needed in all annihilation models. AcknowledgmentsThe work ofwas partially supported by the Russian Foundation for Basic Research Grants No. 14-02-00728 andOFI-M 14-29-06045, and the Russian Federation President Grant for Support of Leading Scientific Schools, Grant No. NSh-261.2014.2. 99alpar2001 AlparM.A. (2001) On Young Neutron Stars as Propellers and Accretors with Conventional Magnetic Fields. ApJ 554: 1245-1254baym07 Baym G. (2007) Neutron stars and quark matter. Quark confinement and the hadron spectrum VII: 7th Conference on Quark Confinement and the Hadron Spectrum - QCHS7. AIP Conference Proceedings, Volume 892, pp. 8-14.bj Bethe H.A., Johnson M.B. (1974) Dense baryon matter calculations with realistic potentials.Nuclear Physics A230: l-58bk90 Bisnovatyi-Kogan, G.S. (1992) Gamma-ray bursts from nuclear fission in neutron stars. In: Proc Conf. Taos (USA) "Gamma-ray bursts - Observations, analyses and theories"(A93-20206 06-90), pp. 89-98bk99 Bisnovatyi-Kogan G.S. 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http://arxiv.org/abs/1702.08353v1

Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794 We study the spatial statistics of velocity gradient volatility(i,e., the energy dissipation rate) in turbulent flow. We extendthe Kolmogorov-Obukhov theory but also narrow its scope. The models are log normal, with verification from finely resolved large eddy and direct numerical simulations. They are parameterized by a mean and a covariance operator. Addressing applications to large eddy simulations, the mean and the covariance depend on the resolved scale solution.Removing this resolved scale dependence by a locally defined rescaling of the turbulent statistics yields a universal theory for the subgrid statistics, specifically for the mean and variance ofturbulent fluctuations, i.e., the log of the energy dissipationrate ϵ.The variance of the velocity gradient statistics is found to be log normal, in accordance with Kolmogorov 1962. Rescaling by the resolved scale mean and variance removes the influence of the resolved scales on the subgrid scales, justifying conceptually the universality we observe. The universality is the basis of new power scaling laws. The new power laws, in turn, allow a simple parameterization of the mean and covariance of the subgrid rescaled dissipation rate, i.e., velocity gradient volatility. We treat the coarse grid resolved space-time location as a random variable. A restriction of the theory to a small range of unresolved but stochastically modelled scales is proposed, with renormalization group ideas offered to overcome this restriction.The spatial statistics of turbulent dissipation rates Vinay Mahadeo December 30, 2023 =====================================================§ INTRODUCTIONThe scaling law of Kolmogorov <cit.>, although simple, is among the deepest contributions to the understanding of turbulent flow. Turbulent flow is generally considered to be the major outstanding problem of classical physics. We propose new parameterized spatial statistics and scaling laws for turbulent fluctuations which extend ideas of the Kolmogorov 1962 (K62) theory but also narrow its scope. Verification is from comparison to Direct Numerical Simulation (DNS). Because we do not assumehomogeneity or isotropy of the flow, the theory is applicable to the unresolved scales of Large Eddy Simulations (LES), and is a stochastic subgrid scale (SGS) model. A major contribution of the theory presented here is to capture SGS fluctuations, in addition to the means computed from traditional SGS theories.We assume single fluid, constant density, incompressible flow, governed by the Navier-Stokes equation. The scalar dissipation rate is defined asϵ = ν/2∑_i,j( ∂𝐮_i/∂𝐱_j +∂𝐮_j/∂𝐱_i)^2= ν/2S+S^*_2^2,with ν the dynamic viscosity, 𝐮 the fluid velocity, andS = ∂𝐮_i/∂𝐱_jthe strain rate.Properties of ϵ are fundamental to a number of deep theories of turbulence, including notions of turbulent intensity, turbulent intermittancy <cit.>, corrections to the Kolmogorov 5/3 exponent and scaling laws for velocity moments. The analysis of ϵ has been applied to turbulent diffusion, short distance asymptotics for the velocity two point correlation function, and prediction of a fractal or multifractal spatial distribution of regions of high turbulent intensity, with the turbulent regions concentrated in a fractal set of dimension D < 3 <cit.>. Scaling laws for wall bounded turbulence (a topic not addressed here) are discussed in <cit.>. There is an extensive literature on the above topics, which we do not attempt to document in this article. See the survey article <cit.> for elaborations and further references. Kolmogorov <cit.> and Obukhov <cit.> conjectured that ϵ obeys a lognormal distribution at sufficientlyhigh Reynolds numbers.Our main results are new scaling law for themean and covariance of χ = ln(ϵ),which we designate as turbulent intensity.The scaling laws originate in a universality principle, of the mean μ and the variance of χ for inertial degrees of freedom. The connection between universality, which would seem to require isotropic homogeneous turbulence, and subgrid scales of LES, which can be very far from homogeneous or isotropic is one of our central results. Depending on the local properties of the resolved scales of turbulence, we introduce a rescaling of the turbulent statistics, and it is only after this rescaling that the subgrid scales of turbulence become universal. In fact, universality does have limits, in terms of the number of subgrid scales it can accommodate. In our numerical tests, we consider 3 levels of mesh refinement, that is a change of length scales by a factor of 8. For a large number of length scales, we have in mind a renormalization group methodology, with successive integration over smaller length scales, and a resetting of parameters using the (statistical) knowledge of the already integrated scales, a point of view to be elaborated in future work. In Sec. <ref>, we specify the simulation study used for verification of the theories proposed here. We define the statistics and the ensemble used. We show that ϵ has log normal statistics (and χ has normal statistics).The spatial statistics for ϵ and χ are characterized by the meanμ and covariance Σ^2 of χ. We compare a coarsely griddedLES to a finely gridded LES or DNS. The fine grid extension of the coarse grid solution is of course unique and not stochastic, but we add statistics in the dependence of this extension on the resolved coarse grid cell from which it is derived. The statistics of the fine grid extension for χ depends on the coarsely gridded resolved solution.A detailed analysis of the χ statistics is carried out in Fourier space, in Sec. <ref>. The analysis usesthe coarse grid resolved scales as a random varible, to define the statistical properties of the subgrid scales.Our premise is that the spatial aspects of this dependence are largely captured by a resolved scale variance Σ_0^2 and mean μ_0 for χ. When the Σ_0^2 and μ_0 dependency is removed from the refined solution extension by rescaling, the resulting statistics for χ, i.e., Σ^2/Σ^2_0, ismodeled as universal, with Σ^2/Σ^2_0 diagonal and subject to simple modeling assumptions, including new power laws, with only a few O(1) dimensionless parameters to governits spatial statistics, all physically understandable. The power laws result from a universality principal for the rescaled statistics. These approximations are confirmed by the fine grid comparison.Sec. <ref> extends this log normal theory to individual components of the strain matrix S and to the tensor dissipation rate. A discussion of results and an outlook for future work is given in Sec. <ref>. § THE SPATIAL STATISTICS §.§ The simulation study All verification tests are based on a series of coarse and finely gridded simulations. The finely griddedLES/DNS study has over two decades of resolution, with the coarsely gridded LES fraction of the solution varying with the simulation parameters. The coarsely gridded LES includes portions of the inertial range and has up to a decade of unresolved inertial range scales to which our modeling applies.Our fine and coarse grid simulations are conducted with the code <cit.>, using periodic boundary conditions and forced turbulence in a domain size L^3, L = 0.05m, with the viscosity (of air), 1.9e^-5m^2/s. The mesh parameters for three simulations are given in Table <ref>.The resolved levels of turbulence in this numerical study are not extreme.Let Δ_c denote the coarse grid mesh size, and Δ_f the fine grid mesh size, so that Δ_c = 8Δ_f. As we are concerned with velocity gradients, which require two mesh cells as a minimum stencil size for their evaluation, we also introduce what we call the resolved coarse and fine grid mesh scales, with Δ_r = 2Δ_c as the resolved coarse grid mesh andΔ_r = 2Δ_c = 8  (Resolved Δ_f) = 16Δ_f.Here the resolved fine grid level is 2Δ_f, needed to evaluate the fine grid velocity gradients occurring in ϵ and χ.§.§ The statistical ensembleAs with turbulent flow in general, ϵ and χcan be regarded as random fields. In the LES context,ϵ and χ are known (deterministic) at the resolved scales, while their subgrid scale fluctuations can be modeled using spatial statistics. We write the fine grid resolved spatial mesh coordinate i = i_r,i_s, where i_r is the coarse grid resolved mesh portion of i. As the resolved coarse mesh spacing is 2Δ_c, i_r spans all the coarse grid LES cells other than the last or finest of them. i_s is the remainder, occurring in the fine but not the coarse LES grid resolution and also within the finest level of the LES simulation. There are then n_r = 16^3 i_r values and 8^3 i_s values. Note that 2^3 Δ_f mesh indices are missing, as we discuss resolved, not simulation mesh indices.Our plan is to treat χ, regarded as a fiction of i_s, as random,depending on the random variable i_r. The naturaldomain for its mean and covariance μ and Σ^2 is the Hilbert space H of subgrid functions, i.e., functions indexed by i_s but restricted to a single resolvedcoarse grid cell and having mean zero there. μ is a vector in H and Σ^2 is an i_s × i_s positive definite matrix acting on H.The statistics are collected as an average over all the possible n_r values of i_r. §.§ The log normal propertyWe present numerical evidence to support the hypothesis that the fine grid extension of the resolved χ defines a Gaussian random field,with the mean and covariance defined from averages of distinct resolved coarse grid cells. The time dependent spatial average of χ has been tested for normality, see <cit.> and references cited there, based on experimental andDNS evidence.Our verification method to establish the Gaussian property is also based on DNS evidence. A multivariate random variable is Gaussian if its inner product with any fixed vector is univariate Gaussian. Here weconsider, as representative choices, the fixed vectors defined by single DNS cell indices i_s. We use QQ plots to assess univariate normality. In these, the trial distribution is transformed first by a change of the χ independent variables to have zero mean and unit variance. Then we apply an inverse Gaussian change of trial distribution dependent variables, with theresult compared to a unit slope straight line.We create a single QQ plot by holding the fine grid mesh resolved cell indices i_s fixed (16^3 choices), and using the multiple i_r values from the 16^3 resolved coarse grid cells to form the ensemble.In Fig. <ref>we superimpose all QQ plots for all choices of i_s as typical tests of normality, demonstrating agreement to a Gaussian law up to ± 2 σ,a range of agreement consistent with the 4% accuracy expected from the ensemble size. §.§ The mean and covarianceThe coarse grid resolved level variance of χ isbasic to our rescaling strategy.At the level of each coarse grid resolved grid cell, the variance Σ_0^2 of χ(i_r) is defined as Σ_0^2 = (χ(i_r) - μ_0)^2with the definitionsμ_0 = χ(i_r) = n_r^-1∑_i_rχ(i_r).Consider the fine grid values χ(i_r,i_s) for the resolved coarse grid χ(i_r). As a normal random variable, the statistics of χ is characterized by its mean and its covariance. To obtain universal statistics, we rescale χ by subtracting the resolved coarse grid mean and dividing by the standard deviation defining χ(i_r,i_s) = (χ(i_r,i_s) - μ(i_s))/Σ_0(i_r). The mean μ of χ isμ(i_s) = n_r^-1∑_i_rχ(i_r,i_s). The universal, rescaled, statistics are the statistics of χ.We follow a similar definition for the covariance, Σ^2 of χ, Σ^2(i_s,j_s) = χ(i_r,i_s)χ(i_r,j_s) . where the overbar denotes averaging over all resolved coarse grid cells, namely ( ·) = n_r^-1∑_i_r( ·). Furthermore, we will normalize Σ^2 by Σ^2(i_r,i_r) = Σ^2(i_r) = Σ_0^2, the resolved scale variance, reducing the Reynolds number dependence of Σ. § SPATIAL STATISTICS IN FOURIER SPACEWe propose a Fourier space analysis for μ and Σ^2. As these variables are independent of the resolved variables, they are, in this approximation,periodic within each resolved cell. Thus their Fourier transforms depend on the Fourier modes within a single resolved cell.In preparing figures in this section, we present functions of 𝐤, unless otherwise noted, after binning all 𝐤 values into binslabeled by the scalar wave number k with 200 bins whose size is linear in ln k.§.§ μ in Fourier spaceIn Fig. <ref>, we plot μ(k) vs. k in ln ln scaled variables. Observe the approximate scaling law μ(k) ∼ k^-d, with corrections coming from the dissipation range of scales. Included in the plot is a model curve of the formμ(k) ∼ k^-dThe coefficients in this model fit are tabuleted in Table <ref>.§.§ Σ^2 in Fourier space In Fig. <ref>, we plot the diagonal and off diagonal elements of the covariance matrixΣ^2(𝐤,𝐥) for Reynolds number 2549. The smooth curve in the upper part of the figure represent the plot of the diagonal elements as afunction of the frequency k, while the noisy data near the bottom of the plot are the collection of all off diagonal elements, plotted vs. (k+l)/2. The other Reynolds numbers give similar plots. We conclude that the covariance is, within a good approximation, diagonal.Our modeling for Σ^2 (and for related operators describing the strain matrix S and its individual matrix entries inSec. <ref>) is based on a simple approximation, which appears to be consistent with the DNS data as examined here. We note that the two random variables contributing to the covariance Σ^2 are orthogonal for 𝐤_𝐬≠𝐥_s and so the𝐤_s ≠𝐥_s terms vanish. In other words, the Fourier coefficient of Σ^2(i_s,j_s) is Σ^2(𝐤_𝐬,𝐤_s), which we write as Σ^2(𝐤_s).In our DNS case, Σ^2 is a 16^3 × 16^3 matrix before binning.The Fourier coefficients for χ for distinct angular variables 𝐤/𝐤 are statistically independent. For distinct values of the scalar Fourier amplitude k = 𝐤, we also assume statistical independence. These assumptions lead to a simple model forΣ^2,tractable for practical use.We regard the wave number k as an observable onthe random field χ, with the varianceΣ^2(k) for this observation. We derive k dependent scaling laws. Approximately, the subspace of the Hilbert space in which Σ^2 (k) is defined has dimension proportional to 4 π k^2, namely the area of a sphere in momentum space of radius k. Assuming equipartion ofthe variance of χ within this subspace,the inverse covariance is a multiple of the identity there. We extend the variance equipartition hypothesis, meaning a uniform level of variance, to distinct k values.The subspace variance is then proportional to k^-2.From a more probabilistic point of view, we regard the observation as the result of asking O(k^2) questions, postulated to be independent: what is the variance of χ associated with each 𝐤 lying on the sphere of radius k? Combining the variance of O(k^2)independent quantities gives the same answer. Volume scaling laws apply to the volume dependent coefficients in the scaling law for Σ^2, and use a similar reasoning. Thus, again assuming equipartition, the mesh cell averages over a single coarse grid resoled cell scale with Δ_r^3. In summary, we propose scaling lawsfor Σ^2(k)dk in terms of the scalar wave number k and spatialvolume, with the dispersion relationΣ^2(k)/Σ_0^2 =( Δ_r^3 4 π C_2^2 k^2 exp(C_4^2 η^2 k^2) )^-1with dimensionless constants C_2, C_4 to be determined.The factors Δ_r^3 and 4π k^2dk are volumes for physical space and Fourier space. The C_4 term, not previously discussed, is postulated to capture the leading order viscous corrections to the otherwise purely inertial range theory.In Fig. <ref>, we plot Σ^2(k) vs. k on ln ln scales with the model curves (<ref>) superimposed, based on optimal choices of the coefficients C_i. We first fit C_2 in the inertial range (small k values), and then fit C_4 over a 1/2 decade at the end of the interial range. We observe that most of the Reynolds number dependence of the C_ihas been removed by the scaling introduced here, seeTable <ref>. § A LOG NORMAL MODEL FOR VELOCITY GRADIENTSWe repeat the analysis of Secs. <ref>– <ref> for the strain matrix restricted to subspaces. Let E_1 and E_2 be self adjoint projection operators on R^3, and consider the reduced strain matrix E_1 SE_2 = S_1,2. In more detail, we consider ϵ_1,2 = ( ν / 2 )E_1∂𝐮_i/∂ x_j E_2 _2^2and the associatedχ_1,2 = ln (ϵ_1,2 ). Thus S_1,2 is the strain associated with fluid velocities in directions spaned by E_1 and fluctuations occurring with spatial variation indirections spanned by E_2.With μ_1,2 the rescaled resolved grid level mean forthe rescaled χ_1,2,Σ^2_1,2 is the rescaled covariance Σ_1,2^2(i_s,j_s) = n_r^-1∑_i_r (χ_1,2(i_r,i_s) -μ_1,2(i_r))(χ_1,2(i_r,j_s) - μ_0,1,2(i_r)),With these changes, we repeat the analysis ofFigs. <ref>–<ref> with the modified dispersion relationΣ^2_1,2(k)= (dim E_1 dim E_2Δ_r^3 4 π C_2,1,2^2 k^2 exp( C_4,1,2^2 η^2 k^2 ) )^-1 ,and modified coefficients C_2,1,2 and C_4,1,2. We assume that E_1 and E_2 are both one dimensional.We also show that the tensor dissipation operatorsϵ' = (ν/2) S^*S, ϵ” = (ν/2)SS^*satisfy log normal statistics. Here the exponential and the log in the definition of log normal(χ' = log(ϵ') and χ” = log(ϵ”))is considered in the sense of matrix operations. See Figs. <ref>, <ref> and <ref>.We note that the matrix QQ plots, and to some extent, all the QQ plots, raise the possibility of nonGaussian statistics as a correction to the normal and log normal statistics assumed here. Especially in Fig. <ref>, systematic corrections to the Gaussian property show smaller positive excursions and stronger negative excursions for χ than Gaussian for the diagonal matrix elements. Both the negative and positive differential excursions reflect a smaller ϵ and a possible bias toward a laminar rather than turbulent flow. In contrast, the off diagonal QQ plots suggest stronger χ excursions than Gaussian for both positive and negative excursions, a phenomena known as heavy tails in the statistical data analysis literature. These are the matrix elements which occur in vortex stretching and folding, commonly believed to be an important element of fully developed turbulence.§ DISCUSSIONVerification by computational studies will typically encounter only a limited range of LES unresolved scales, compared to a finely gridded simulation. But the previous analysis, when applied over many scales, has a conceptual flaw. If there is no resetting of the Σ_0^2 parameters within the statistics model of a single resolved scale, then we could apply this theory in the absence of the coarsely gridded LES to the entire computation. Doing this would remove the rescaling already seen to be important. For this reason, we restrict the model to a small range of unresolved scales. Ina future publicationwe reformulate the model in a renormalization group manner in the case of multiple scales.In an LES context, we analyze the unresolved scales statistically, withan emphasis on the strain rate ϵ and its logarithmlnϵ = χ. Resolved scale adjustments to the mean and variance for χ lead to a theory for the resulting covariance Σ^2. We find that Σ^2 is approximately diagonal in Fourier space. An equipartition hypothesis for the variance of χleads to a new power scaling law anda simple parameterization for Σ^2, verified by comparison to DNS over about a decade of LES unresolved inertial scales. For reasons discussed, we restrict this theory to a single or limited range within the inertial range of scales. This pure dissipation rate model is extended to the individual velocity gradient components ϵ_1,2 = (ν / 2) (∂u_i / ∂ x_j)^2 and to the tensor dissipation rates S^*S and SS^*.Open questions, to be addressed in future work, concern temporal statistics of turbulent intensity, statistical models for the velocity gradients andstrongly nonhomogeneous flows, such as boundary layers.plain

http://arxiv.org/abs/1702.08439v1

Age Progression/Regression by Conditional Adversarial Autoencoder Zhifei Zhangwith equal contribution. ,  Yang Song^*,  Hairong Qi The University of Tennessee, Knoxville, TN, USA {zzhang61, ysong18, hqi}@utk.edu December 30, 2023 ====================================================================================================================================================================================== “If I provide you a face image of mine (without telling you the actual age when I took the picture) and a large amount of face images that I crawled (containing labeled faces of different ages but not necessarily paired), can you show me what I would look like when I am 80 or what I was like when I was 5?” The answer is probably a “No.” Most existing face aging works attempt to learn the transformation between age groups and thus would require the paired samples as well as the labeled query image. In this paper, we look at the problem from a generative modeling perspective such that no paired samples is required. In addition, given an unlabeled image, the generative model can directly produce the image with desired age attribute. We propose a conditional adversarial autoencoder (CAAE) that learns a face manifold, traversing on which smooth age progression and regression can be realized simultaneously. In CAAE, the face is first mapped to a latent vector through a convolutional encoder, and then the vector is projected to the face manifold conditional on age through a deconvolutional generator. The latent vector preserves personalized face features (i.e., personality) and the age condition controls progression vs. regression. Two adversarial networks are imposed on the encoder and generator, respectively, forcing to generate more photo-realistic faces. Experimental results demonstrate the appealing performance and flexibility of the proposed framework by comparing with the state-of-the-art and ground truth. § INTRODUCTIONFace age progression (i.e., prediction of future looks) and regression (i.e., estimation of previous looks), also referred to as face aging and rejuvenation, aims to render face images with or without the “aging” effect but still preserve personalized features of the face (i.e., personality). It has tremendous impact to a wide-range of applications, , face prediction of wanted/missing person, age-invariant verification, entertainment, etc. The area has been attracting a lot of research interests despite the extreme challenge in the problem itself. Most of the challenges come from the rigid requirement to the training and testing datasets, as well as the large variation presented in the face image in terms of expression, pose, resolution, illumination, and occlusion. The rigid requirement on the dataset refers to the fact that most existing works require the availability of “paired” samples, i.e., face images of the same person at different ages, and some even require paired samples over a long range of age span, which is very difficult to collect. For example, the largest aging dataset “Morph" <cit.> only captured images with an average time span of 164 days for each individual. In addition, existing works also require the query image to be labeled with the true age, which can be inconvenient from time to time.Given the training data, existing works normally divide them into different age groups and learn a transformation between the groups, therefore, the query image has to be labeled in order to correctly position the image. Although age progression and regression are equally important, most existing works focus on age progression. Veryfew works can achieve good performance of face rejuvenating, especially for rendering baby face from an adult because they are mainly surface-based modeling which simply remove the texture from a given image <cit.>. On the other hand, researchers have made great progress on age progression. For example, the physical model-based methods <cit.> parametrically model biological facial change with age, , muscle, wrinkle, skin, etc. However, they suffer from complex modeling, the requirement of sufficient dataset to cover long time span, and are computationally expensive; the prototype-based methods <cit.> tend to divide training data into different age groups and learn a transformation between groups. However, some can preserve personality but induce severe ghosting artifacts, others smooth out the ghosting effect but lose personality, while most relaxed the requirement of paired images over long time span, and the aging pattern can be learned between two adjacent age groups. Nonetheless, they still need paired samples over short time span. In this paper, we investigate the age progression/regression problem from the perspective of generative modeling. The rapid development of generative adversarial networks (GANs) has shown impressive results in face image generation <cit.>.In this paper, we assume that the face images lie on a high-dimensional manifold as shown in Fig.<ref>. Given a query face, we could find the corresponding point (face) on the manifold. Stepping along the direction of age changing, we will obtain the face images of different ages while preserving personality. We propose a conditional adversarial autoencoder (CAAE)[Bitbucket: <https://bitbucket.org/aicip/face-aging-caae> Github: <https://zzutk.github.io/Face-Aging-CAAE>] network to learn the face manifold. By controlling the age attribute, it will be flexible to achieve age progression and regression at the same time. Because it is difficult to directly manipulate on the high-dimensional manifold, the face is first mapped to a latent vector through a convolutional encoder, and then the vector is projected to the face manifold conditional on age through a deconvolutional generator.Two adversarial networks are imposed on the encoder and generator, respectively, forcing to generate more photo-realistic faces.The benefit of the proposed CAAE can be summarized from four aspects. First, the novel network architecture achieves both age progression and regression while generating photo-realistic face images. Second, we deviate from the popular group-based learning, thus not requiring paired samples in the training data or labeled face in the test data, making the proposed framework much more flexible and general. Third, the disentanglement of age and personality in the latent vector space helps preserving personality while avoiding the ghosting artifacts. Finally, CAAE is robust against variations in pose, expression, and occlusion. § RELATED WORK §.§ Age Progression and Regression In recent years, the study on face age progression has been very popular, with approaches mainly falling into two categories, physical model-based and prototype-based. Physical model-based methods model the biological pattern and physical mechanisms of aging, , the muscles <cit.>, wrinkle <cit.>, facial structure <cit.> etc. through either parametric or non-parametric learning. However, in order to better model the subtle aging mechanism, it will require a large face dataset with long age span (, from 0 to 80 years old) of each individual, which is very difficult to collect. In addition, physical modeling-based approaches are computationally expensive. On the other hand, prototype-based approaches <cit.> often divide faces into groups by age, e.g., the average face of each group, as its prototype. Then, the difference between prototypes from two age groups is considered the aging pattern. However, the aged face generated from averaged prototype may lose the personality (, wrinkles). To preserve the personality, <cit.> proposed a dictionary learning based method — age pattern of each age group is learned into the corresponding sub-dictionary. A given face will be decomposed into two parts: age pattern and personal pattern. The age pattern was transited to the target age pattern through the sub-dictionaries, and then the aged face is generated by synthesizing the personal pattern and target age pattern. However, this approach presents serious ghosting artifacts. The deep learning-based method <cit.> represents the state-of-the-art, where RNN is applied on the coefficients of eigenfaces for age pattern transition. All prototype-based approaches perform the group-based learning which requires the true age of testing faces to localize the transition state which might not be convenient. In addition, these approaches only provide age progression from younger face to older ones. To achieve flexible bidirectional age changes, it may need to retrain the model inversely. Face age regression, which predicts the rejuvenating results, is comparatively more challenging. Most age regression works so far <cit.> are physical model-based, where the textures are simply removed based on the learned transformation over facial surfaces. Therefore, they cannot achieve photo-realistic results for baby face predictions. §.§ Generative Adversarial Network Generating realistically appealing images is still challenging and has not achieved much success until the rapid advancement of the generative adversarial network (GAN). The original GAN work <cit.> introduced a novel framework for training generative models. It simultaneously trains two models: 1) the generative model G captures the distribution of training samples and learns to generate new samples imitating the training, and 2) the discriminative model D discriminates the generated samples from the training. G and D compete with each other using a min-max game as Eq. <ref>, where z denotes a vector randomly sampled from certain distribution p(𝐳) (, Gaussian or uniform), and the data distribution is p_data(𝐱), , the training data x∼ p_data(𝐱). Gmin Dmax𝔼_x∼ p_data(𝐱)[log D(x)] + 𝔼_z∼ p(𝐳)[log(1-D(G(z)))] The two parts, G and D, are trained alternatively. One of the biggest issues of GAN is that the training process is unstable, and the generated images are often noisy and incomprehensible. During the last two years, several approaches <cit.> have been proposed to improve the original GAN from different perspectives. For example, DCGAN <cit.> adopted deconvolutional and convolutional neural networks to implement G and D, respectively. It also provided empirical instruction on how to build a stable GAN, , replacing the pooling by strides convolution and using batch normalization. CGAN <cit.> modified GAN from unsupervised learning into semi-supervised learning by feeding the conditional variable (, the class label) into the data. The low resolution of the generated image is another common drawback of GAN. <cit.> extended GAN into sequential or pyramid GANs to handle this problem, where the image is generated step by step, and each step utilizes the information from the previous step to further improve the image quality. Some GAN-related works have shown visually impressive results of randomly drawing face images <cit.>. However, GAN generates images from random noise, thus the output image cannot be controlled. This is undesirable in age progression and regression, where we have to ensure the output face looks like the same person as queried. § TRAVERSING ON THE MANIFOLDWe assume the face images lie on a high-dimensional manifold, on which traversing along certain direction could achieve age progression/regression while preserving the personality. This assumption will be demonstrated experimentally in Sec. <ref>. However, modeling the high-dimensional manifold is complicated, and it is difficult to directly manipulate (traversing) on the manifold. Therefore, we will learn a mapping between the manifold and a lower-dimensional space, referred to as the latent space, which is easier to manipulate.As illustrated in Fig. <ref>, faces x_1 and x_2 are mapped to the latent space by E (, an encoder), which extracts the personal features z_1 and z_2, respectively. Concatenating with the age labels l_1 and l_2, two points are generated in the latent space, namely [z_1, l_1] and [z_2,l_2]. Note that the personality z and age l are disentangled in the latent space, thus we could simply modify age while preserving the personality. Starting from the red rectangular point [z_2,l_2] (corresponding to x_2) and evenly stepping bidirectionally along the age axis (as shown by the solid red arrows), we could obtain a series of new points (red circle points). Through another mapping G (. a generator), those points are mapped to the manifold ℳ – generating a series of face images, which will present the age progression/regression of x_2. By the same token, the green points and arrows demonstrate the age progressing/regression of x_1 based on the learned manifold and the mappings. If we move the point along the dotted arrow in the latent space, both personality and age will be changed as reflected on ℳ. We will learn the mappings E and G to ensure the generated faces lie on the manifold, which indicates that the generated faces are realistic and plausible for a given age.§ APPROACHIn this section, we first present the pipeline of the proposed conditional adversarial autoencoder (CAAE) network (Sec. <ref>) that learns the face manifold (Sec. <ref>). The CAAE incorporates two discriminator networks, which are the key to generating more realistic faces. Sections <ref> and <ref> demonstrate their effectiveness, respectively. Finally, Section <ref> discusses the difference of the proposed CAAE from other generative models.§.§ Conditional Adversarial AutoencoderThe detailed structure of the propose CAAE network is shown in Fig. <ref>. The input and output face images are 128×128 RGB images x∈ℝ^128×128×3. A convolutional neural network is adopted as the encoder. The convolution of stride 2 is employed instead of pooling (, max pooling) because strided convolution is fully differentiable and allows the network to learn its own spacial downsampling <cit.>. The output of encoder E(x)=z preserves the high-level personal feature of the input face x. The output face conditioned on certain age can be expressed by G(z,l)=x̂, where l denotes the one-hot age label.Unlike existing GAN-related works, we incorporate an encoder to avoid random sampling of z because we need to generate a face with specific personality which is incorporated in z. In addition, two discriminator networks are imposed on E and G, respectively. The D_z regularizes z to be uniform distributed, smoothing the age transformation. The D_img forces G to generate photo-realistic and plausible faces for arbitrary z and l. The effectiveness of the two discriminators will be further discussed in Secs. <ref> and <ref>, respectively.§.§ Objective Function The real face images are supposed to lie on the face manifold ℳ, so the input face image x∈ℳ. The encoder E maps the input face x to a feature vector, , E(x)=z∈ℝ^n, where n is the dimension of the face feature. Given z and conditioned on certain age label l, the generator G generates the output face x̂=G(z,l)=G(E(x),l). Our goal is to ensure the output face x̂ lies on the manifold while sharing the personality and age with the input face x (during training). Therefore, the input and output faces are expected to be similar as expressed in Eq. <ref>, where ℒ(·,·) denotes L_2 norm.E,Gmin ℒ(x,G(E(x),l))Simultaneously, the uniform distribution is imposed on z through D_z – the discriminator on z. We denote the distribution of the training data as p_data(𝐱), then the distribution of z is q(𝐳|𝐱). Assuming p(𝐳) is a prior distribution, and z^*∼ p(𝐳) denotes the random sampling process from p(𝐳). A min-max objective function can be used to train E and D_z,Emin D_zmax𝔼_z^*∼ p(𝐳)[ log D_z(z^*)] +  𝔼_x∼ p_data(𝐱)[ log (1-D_z(E(x)))]By the same token, the discriminator on face image, D_img, and G with condition lcan be trained byGminD_imgmax 𝔼_x,l∼ p_data(𝐱, 𝐥)[ log D_img(x,l)] +  𝔼_x,l∼ p_data(𝐱, ł)[ log (1-D_img(G(E(x),l)))]Finally the objective function becomesE,GminD_z,D_imgmaxλℒ(x,G(E(x),l)) + γ TV(G(E(x),l))+ 𝔼_z^*∼ p(𝐳)[ log D_z(z^*)] +𝔼_x∼ p_data(𝐱)[ log (1-D_z(E(x)))]+ 𝔼_x,l∼ p_data(𝐱, 𝐥)[ log D_img(x,l)]+ 𝔼_x,l∼ p_data(𝐱, ł)[ log (1-D_img(G(E(x),l)))],where TV(·) denotes the total variation which is effective in removing the ghosting artifacts. The coefficients λ and γ balance the smoothness and high resolution.Note that the age label is resized and concatenated to the first convolutional layer of D_img to make it discriminative on both age and human face. Sequentially updating the network by Eqs. <ref>, <ref>, and <ref>, we could finally learn the manifold ℳ as illustrated in Fig. <ref>. §.§ Discriminator on z The discriminator on z, denoted by D_z, imposes a prior distribution (, uniform distribution) on z. Specifically, D_z aims to discriminate the z generated by encoder E. Simultaneously, E will be trained to generate z that could fool D_z. Such adversarial process forces the distribution of the generated z to gradually approach the prior. We use uniform distribution as the prior, forcing z to evenly populate the latent space with no apparent “holes”. As shown in Fig. <ref>, the generated z's (depicted by blue dots in a 2-D space) present uniform distribution under the regularization of D_z, while the distribution of z exhibits a “hole” without the application of D_z.Exhibition of the “hole” indicates that face images generated by interpolating between arbitrary z's may not lie on the face manifold – generating unrealistic faces. For example, given two faces x_1 and x_2 as shown in Fig. <ref>, we obtain the corresponding z_1 and z_2 by E under the conditions with and without D_z, respectively. Interpolating between z_1 and z_2 (dotted arrows in Fig. <ref>), the generated faces are expected to show realistic and smooth morphing from x_1 to x_2 (bottom of Fig. <ref>). However, the morphing without D_z actually presents distorted (unrealistic) faces in the middle (indicated by dashed box), which corresponds to the interpolated z's passing through the “hole”. §.§ Discriminator on Face ImagesInheriting the similar principle of GAN, the discriminator D_img on face images forces the generator to yield more realistic faces. In addition, the age label is imposed on D_img to make it discriminative against unnatural faces conditional on age. Although minimizing the distance between the input and output images as expressed in Eq. <ref> forces the output face to be close to the real ones, Eq. <ref> does not ensure the framework to generate plausible faces from those unsampled faces. For example, given a face that is unseen during training and a random age label, the pixel-wise loss could only make the framework generate a face close to the trained ones in a manner of interpolation, causing the generated face to be very blurred. The D_img will discriminate the generated faces from real ones in aspects of reality, age, resolution, etc. Fig. <ref> demonstrates the effect of D_img. Comparing the generated faces with and without D_img, it is obvious that D_img assists the framework to generate more realistic faces. The outputs without D_img could also present aging but the effect is not as obvious as that with D_img because D_img enhances the texture especially for older faces. §.§ Differences from Other Generative Networks In this section, we comment on the similarity and difference of the proposed CAAE with other generative networks, including GAN <cit.>, variational autoencoder (VAE) <cit.>, and adversarial autoencoder (AAE) <cit.>. VAE vs. GAN:VAE uses a recognition network to predict the posterior distribution over the latent variables, while GAN uses an adversarial training procedure to directly shape the output distribution of the network via back-propagation <cit.>. Because VAE follows an encoding-decoding scheme, we can directly compare the generated images to the inputs, which is not possible when using a GAN. A downside of VAE is that it uses mean squared error instead of an adversarial network in image generation, so it tends to produce more blurry images <cit.>.AAE vs. GAN and VAE: AAE can be treated as the combination of GAN and VAE, which maintains the autoencoder network like VAE but replaces the KL-divergence loss with an adversarial network like in GAN. Instead of generating images from random noise as in GAN, AAE utilizes the encoder part to learn the latent variables approximated on certain prior, making the style of generated images controllable. In addition, AAE better captures the data manifold compared to VAE.CAAE vs. AAE: The proposed CAAE is more similar to AAE. The main difference from AAE is that the proposed CAAE imposes discriminators on the encoder and generator, respectively. The discriminator on encoder guarantees smooth transition in the latent space, and the discriminator on generator assists to generate photo-realistic face images. Therefore, CAAE would generate higher quality images than AAE as discussed in Sec. <ref>.§ EXPERIMENTAL EVALUATION In the section, we will first clarify the process of data collection (Sec. <ref>) and implementation of the proposed CAAE (Sec. <ref>). Then, both qualitative and quantitative comparisons with prior works and ground truth are performed in Sec. <ref>. Finally, the tolerance to occlusion and variation in pose and expression is illustrated in Sec. <ref> . §.§ Data Collection We first collect face images from the Morph dataset <cit.> and the CACD <cit.> dataset. The Morph dataset <cit.> is the largest with multiple ages of each individual, including 55,000 images of 13,000 subjects from 16 to 77 years old.The CACD <cit.> dataset contains 13,446 images of 2,000 subjects. Because both datasets have limited images from newborn or very old faces, we crawl images from Bing and Google search engines based on the keywords, , baby, boy, teenager, 15 years old, etc. Because the proposed approach does not require multiple faces from the same subject, we simply randomly choose around 3,000 images from the Morph and CACD dataset and crawl 7,670 images from the website. The age and gender of the crawled faces are estimated based on the image caption or the result from age estimator <cit.>. We divide the age into ten categories, , 0–5, 6–10, 11–15, 16–20, 21–30, 31–40, 41–50, 51–60, 61–70, and 71–80. Therefore, we can use a one-hot vector of ten elements to indicate the age of each face during training. The final dataset consists of 10,670 face images with a uniform distribution on gender and age. We use the face detection algorithm with 68 landmarks <cit.> to crop out and align the faces, making the training more attainable.§.§ Implementation of CAAE We construct the network according to Fig. <ref> with kernel size of 5 × 5. The pixel values of the input images are normalized to [-1,1], and the output of E (, z) is also restricted to [-1,1] by the hyperbolic tangent activation function. Then, the desired age label, the one-hot vector, is concatenated to z, constructing the input of G. To make fair concatenation, the elements of label is also confined to [-1,1], where -1 corresponds to 0. Finally, the output is also in range [-1,1] through the hyperbolic tangent function. Normalizing the input may make the training process converge faster. Note that we will not use the batch normalization for E and G because it blurs personal features and makes output faces drift far away from inputs in testing. However, the batch normalization will make the framework more stable if it is applied on D_img. All intermediate layers of each block (, E, G, D_z, and D_img) use the ReLU activation function. In training, λ=100, γ=10, and the four blocks are updated alternatively with a mini-batch size of 100 through the stochastic gradient descent solver, ADAM <cit.> (α=0.0002, β_1=0.5). Face and age pairs are fed to the network. After about 50 epochs, plausible generated faces can be obtained. During testing, only E and G are active. Given an input face without true age label, E maps the image to z. Concatenating an arbitrary age label to z, G will generate a photo-realistic face corresponding to the age and personality.§.§ Qualitative and Quantitative Comparison To evaluate that the proposed CAAE can generate more photo-realistic results, we compare ours with the ground truth and the best results from prior works <cit.>, respectively. We choose FGNET <cit.> as the testing dataset, which has 1002 images of 82 subjects aging from 0 to 69.Comparison with ground truth: In order to verify whether the personality has been preserved by the proposed CAAE, we qualitatively and quantitatively compare the generated faces with the ground truth. The qualitative comparison is shown in Fig. <ref>, which shows appealing similarity.To quantitatively evaluate the performance, we pair the generated faces with the ground truth whose age gap is larger than 20 years. There are 856 pairs in total. We design a survey to compare the similarity where 63 volunteers participate. Each volunteer is presented with three images, an original image X, a generated image A, and the corresponding ground truth image B under the same group. They are asked whether the generated image A looks similar to the ground truth B; or not sure. We ask the volunteers to randomly choose 45 questions and leave the rest blank. We receive 3208 votes in total, with 48.38% indicating that the generated image A is the same person as the ground truth, 29.58% indicating they are not, and 22.04% not sure. The voting results demonstrate that we can effectively generate photo-realistic faces under different ages while preserving their personality.Comparison with prior work: We compare the performance of our method with some prior works <cit.>, for face age progression and Face Transformer (FT) <cit.> for face age regression. To demonstrate the advantages of CAAE, we use the same input images collected from those prior works and perform long age span progression. To compare with prior works, we cite their results as shown in Fig. <ref>. We also compare with age regression works using the FT demo <cit.> as shown in Fig. <ref>. Our results obviously show higher fidelity, demonstrating the capability of CAAE in achieving smooth face aging and rejuvenation. CAAE better preserves the personality even with a long age span. In addition, our results provide richer texture (, wrinkle for old faces), making old faces look more realistic. Another survey is conducted to statistically evaluate the performance as compared with prior works, where for each testing image, the volunteer is to select the better result from CAAE or prior works, or hard to tell. We collect 235 paired images of 79 subjects from previous works <cit.>. We receive 47 responses and 1508 votes in total with 52.77% indicating CAAE is better, 28.99% indicating the prior work is better, and 18.24% indicating they are equal. This result further verifies the superior performance of the proposed CAAE.§.§ Tolerance to Pose, Expression, and Occlusion As mentioned above, the input images have large variation in pose, expression, and occlusion. To demonstrate the robustness of CAAE, we choose the faces with expression variation, non-frontal pose, and occlusion, respectively, as shown in Fig. <ref>. It is worth noting that the previous works <cit.> often apply face normalization to alleviate from the variation of pose and expression but they may still suffer from the occlusion issue. In contrast, the proposed CAAE obtains the generated faces without the need to remove these variations, paving the way to robust performance in real applications.§ DISCUSSION AND FUTURE WORKS In this paper, we proposed a novel conditional adversarial autoencoder (CAAE), which first achieves face age progression and regression in a holistic framework. We deviated from the conventional routine of group-based training by learning a manifold, making the aging progression/regression more flexible and manipulatable — from an arbitrary query face without knowing its true age, we can freely produce faces at different ages, while at the same time preserving the personality. We demonstrated that with two discriminators imposed on the generator and encoder, respectively, the framework generates more photo-realistic faces. Flexibility, effectiveness, and robustness of CAAE have been demonstrated through extensive evaluation.The proposed framework has great potential to serve as a general framework for face-age related tasks. More specifically, we trained four sub-networks, , E, G, D_z, and D_img, but only E and G are utilized in the testing stage. The D_img is trained conditional on age. Therefore, it is able to tell whether the given face corresponds to a certain age, which is exactly the task of age estimation. For the encoder E, it maps faces to a latent vector (face feature), which preserves the personality regardless of age. Therefore, E could be considered a candidate for cross-age recognition. The proposed framework could be easily applied to other image generation tasks, where the characteristics of the generated image can be controlled by the conditional label. In the future, we would extend current work to be a general framework, simultaneously achieving age progressing (E and G), cross-age recognition (E), face morphing (G), and age estimation (D_img). ieee

http://arxiv.org/abs/1702.08423v2

Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502,Japan ketomita@ybb.ne.jp It is shown that the serious problem on the cosmological tension between the directmeasurements of the Hubble constant at present and the constant derived from thePlanck measurements of the CMB anisotropies can be solved by considering therenormalized model parameters. They are deduced by taking the spatial average ofsecond-order perturbations in the flat Λ-CDM model, which includesrandomadiabaticfluctuations. Cosmological renormalization of model parameters in the second-order perturbation theory Kenji Tomita∗ First version: January 17, 2017This version: February 29, 2020 ======================================================================================== § INTRODUCTION High precision cosmology has started with the measurements offluctuations in the cosmic microwave background radiation (CMB) by WMAP<cit.> andPlanck<cit.> collaboration. Their studies have been useful to determine the Hubble constant and thecosmological parameters.However, it has been found that there is a tensionbetween the Hubble constant (H_0) due to the Planck measurements of the CMB anisotropies and that due to the direct measurements.Due to the Planck measurements<cit.>, we have the Hubble constant H_0 = 67.3kms^-1 Mpc^-1. On the other hand, there are many direct measurements<cit.>, and the threeexamples of direct measurements give us<cit.>H_0 = 73.8,74.3,and 78.7 kms^-1 Mpc^-1. These data show that there is a large discrepancy of 9.7 %∼ 16.9 %. To solve this problem, various mechanisms have been proposed, such asmodels of decaying dark matter<cit.>, a local void model<cit.>, andthe dark radiation<cit.>.In this paper, it is shown that the tension can be solved by considering therenormalizedmodel parameters which are deduced by takingthe spatial average of second-order perturbations in the flat Λ-CDM model, which includesrandom adiabatic fluctuations. In Sect. 2, we show the general-relativistic second-order perturbation theoryin the flatΛ-CDM model, which was derived by the present author<cit.>. In Sect. 3, we derive spatial averages of second-order density and metricperturbations, and in Sect. 4, we define the renormalized Hubble constant due to the average second-order metric perturbations, and show that it is consistent withthe measured Hubble constants and their various observed values correspond to the different upper limits of wave-numbers ofperturbations which can be included in the renormalized perturbations. Other renormalized model parameters are also derived due to the average ofsecond-order density perturbations.In Sect. 5, the renormalization in the past is described and in Sect. 6, the concluding remarks are given.In Appendix A, we show the definitionof various quantities included in the expressions for the second-order metric perturbations. In Appendix B, we show the model parameters corresponding to Hubble constants in Eq. (2).§ BACKGROUND AND THE PERTURBATION THEORY The background universe is expressed by a spatially flat model with the line-element ds^2 = g_μν dx^μ dy^ν = a^2 (η) [-dη^2 + δ_ij dx^i dx^j ], where the Greek and Roman letters denote 0, 1, 2, 3 and 1, 2, 3, respectively. The conformal time η (= x^0) is related to the cosmic time t by dt = a(η) dη. In the comoving coordinates, the velocity vector and energy-momentum tensor of pressureless matter areu^0 = 1/a, u^i = 0 and T^0_0 = -ρ,T^0_i = 0,T^i_j = 0, where ρ is the matter density . From the Einstein equations, we obtain ρ a^2 = 3(a'/a)^2 - Λ a^2, and ρ a^3 = ρ_0, where a prime denotes ∂/∂η,Λ is the cosmological constant, and ρ_0 is an integration constant, and we use the units 8π G = c = 1 for the gravitational constant G and the light velocity c.The Hubble parameterH is defined asH=ȧ/a = a'/a^2. Eq.(<ref>) givesH^2 = 1/3(ρ + Λ), which is also expressed asH^2 = H_0^2(Ω_M a^-3 + Ω_Λ ), where H_0 is H at the present epoch t_0 and a_0 ≡ a(t_0) = 1, and Ω_M= 8π G ρ_0/3H_0^2 = 1/3ρ_0/H_0^2andΩ_Λ = Λ c^2/3H_0^2 = 1/3Λ/H_0^2. In this paper we adopt the following background values :H_0 = 67.3 kms^-1 Mpc^-1 and(Ω_M,Ω_Λ )= (0.22, 0.78). The significance of these values will be explained later. Next let us show the first-order density perturbations. The perturbations of metric, matter density and velocity are represented by δ_1 g_μν (≡ h_μν),δ_1ρ, and δ_1 u^μ. When we adopt the synchronous coordinates (usefulin the pressureless case), the metric perturbations satisfy the condition h_00 = 0 and h_0i = 0. The first-order perturbations are classified into the growing case and the decaying case. Both cases are found in the previous paper.<cit.>Here we show only those in the growing case, which are used in this paper : h^j_i= δ^j_i F + P(η) F^|j_|i,δ_1 u^0= 0,δ_1 u^i = 0,δ_1 ρ/ρ = 1/ρ a^2(a'/a P' -1) Δ F, where F is an arbitrary function of spatial coordinates x^1, x^2 and x^3,Δ≡∇^2,h^j_i = g^jl h_li and P(η) satisfies P” + 2a'/a P' - 1 =0. Its solution is expressed as P = ∫^η_0 dη' a^-2 (η') ∫^η'_0 dη” a^2(η”). After a partial integration, we obtain(H_0)^2P= -2/3Ω_M a^-3/2√(Ω_M +Ω_Λa^3)∫^a_0 dbb^3/2/√(Ω_M +Ω_Λb^3) + 2/3Ω_M a,H_0η = ∫^a_0 dbb^-1/2/√(Ω_M +Ω_Λb^3). The three-dimensional covariant derivatives |i are defined in the space with metric dl^2 = δ_ij dx^i dx^j and their suffices are raised and lowered using δ_ij, so that their derivatives are equal to partial derivatives, i.e. F^|j_|i = F_,ij, where F_,i≡∂ F/∂ x^i.The second-order perturbations were derived in the previouspaper<cit.>.This is a simple extension of my paper<cit.>which derived the second-orderperturbations in the case ofzero Λ in the Lifshitz formalism with iterative second-order terms. The results in thelatter paper were later derived independently by Russ et al.<cit.> and by Matarrese et al.<cit.> in the different formalisms, and the validity of this theory has been confirmed. Here let us show their components δ_2 g_μν (≡ℓ_μν), δ_2 ρ,and δ_2 u^μ in the case ofnonzero Λ, where the total perturbations are δ g_μν = h_μν + ℓ_μν,δ u^μ = δ_1 u^μ +δ_2 u^μ,δρ/ρ = δ_1 ρ/ρ + δ_2 ρ/ρ. Here assuming the synchronous gauge condition, we have ℓ_00 = 0andℓ_0i = 0. The perturbations in the growing case are expressed as ℓ^j_i = P(η) L^j_i + P^2 (η) M^j_i + Q(η) N^|j_|i + C^j_i,where N^|j_|i = δ^jl N_|li = N_,ij and Q(η) satisfies Q” +2a'/a Q' = P - 5/2 (P')^2.The expressions of other quantities L^j_i ,M^j_i , N^|j_|i, and C^j_i are shown in Appendix A. The velocity and density perturbations are found to be δ_2 u^0 =0, δ_2 u^i =0 and δ_2 ρ/ρ = 12ρ a^2{12(1 - a'/aP') (3F_,lF_,l + 8FΔ F) +1/2P [(Δ F)^2 + F_,klF_,kl] + 1/4[(P')^2 - 2/7a'/aQ'] [(Δ F)^2 - F_,klF_,kl] - 17a'/aPP' [4F_,klF_,kl + 3(Δ F)^2]}. The gauge used here is not only synchronous, but also comoving(cf. Eqs. (<ref>) and (<ref>)).The above perturbations are therefore physical density perturbations which are measured by comoving observers. It is to be noticed that the present general-relativistic gravitational equations are nonlinear and applicable also in the super-horizon case, in contrast to linear gravitational equations in the Newtonian treatment <cit.>,and so the cosmological result in the followingsections cannot be derived in the Newtonian treatment, because of their difference.It is discussed in Sect. 6 what we should do, in order to obtain theconsistency between the general-relativistic treatment and theNewtonian cosmology. § AVERAGE SECOND-ORDER PERTURBATIONS We consider random perturbations given by F( k)= ∫ d k α ( k)e^i kx, where α ( k) is a random variable and the average of F expressed as⟨ F ⟩ vanishes. Here we assume the average process with a power spectrum P_F ( k), given by ⟨α ( k) α ( k') ⟩ = (2π)^-2 P_F ( k) δ ( k +k'). Here we have⟨δ_1 ρ/ ρ⟩ = 0 for the first-order density perturbation. For the second-orderperturbations, we have F_,iF_,i = - ∫∫ d kd k'⟨α ( k) α ( k') ⟩ k k' e^i( k+k')x,F Δ F= - ∫∫ d kd k'⟨α ( k) α ( k') ⟩ k^2 e^i( k+k')x, so that we obtain ⟨ F_,iF_,i⟩=⟨ F Δ F ⟩ = (2π)^-2∫ d kk^2P_F ( k). Similarly, we have F_,ijF_,ij =∫∫ d kd k'⟨α ( k) α ( k') ⟩ ( k k')^2 e^i( k+k')x,(Δ F)^2=∫∫ d kd k'⟨α ( k) α ( k') ⟩ k^2 (k')^2 e^i( k+k')x, so that we obtain ⟨ F_,ijF_,ij⟩=⟨ (Δ F)^2 ⟩ = (2π)^-2∫ d kk^4P_F ( k).§.§ Second-order density perturbationsIt follows therefore from Eq. (<ref>) that ⟨δ_2 ρ/ρ⟩ =1 - a'/aP'/2ρ a^2 (2π)^-2[11/2∫ d k k^2P_F ( k) + P ∫dk k^4P_F ( k)]. Here F is related to the curvature fluctuation R by F = 2R, and so we have the relationP_F( k) = 4P_ R( k), where P_ R isexpressed using the power spectrum<cit.> as P_ R = 2 π^2 P_ R0 k^-3 (k/k_eq)^n-1 T_s^2 (k/k_eq) and P_ R0 = 2.2× 10^-9 according to the result of Planck measurements.<cit.> The transfer function T_s (x) is expressed as a function of x = k/k_eq,wherek_eq (≡ a_eq H_eq)= 219(Ω_M h)H_0 = 32.4H_0. Here H_0(≡ 100h)is the presentbackgroundHubble constant,(a_eq, H_eq) is (a, H) at the epoch of equal energy density, and (Ω_M, h) = (0.22, 0.673)(given in Eq. (<ref>)).Moreover, we assume n = 1 here and in the following.Then we obtain for arbitrary a⟨δ_2 ρ/ρ̃⟩ = 4π/3 32.4^4P_ R0 [1 - Y(a)]/(Ω_M/a + Ω_Λ a^2)[11/2 32.4^-2 A + Z(a) B], where ρ̃≡ρ + Λ,and A and B are expressed asA ≡∫^x_max_x_min dxxT_s^2 (x), B ≡∫^x_max_x_mindxx^3T_s^2 (x) using the transfer function T_s (x) for the interval (x_max, x_min). Here we have Y(a)≡a'/aP',Z(a)≡ (H_0)^2P, and using Eq.(<ref>), we obtainY(a) = a^-5/2 (Ω_M + Ω_Λ a^3)^1/2I(a), Z(a) = 2/3Ω_M a [1 - Y(a)], where I(a) ≡∫^a_0db[b^3/(Ω_M + Ω_Λ b^3)]^1/2. §.§ Second-order metric perturbationsThe second-order perturbation of the scale factor δ_2 a can be derived asfollows using the metric second-order perturbations l_ij,which are givenin Eqs.(<ref>) and (<ref>), and inAppendix A.The averaging of second-order metric perturbations leads toδ_2 (a^2) = 1/3⟨ l^m_m⟩ ,where we have ⟨ l_ij⟩ = P(η) ⟨ L_ij⟩ + P^2 (η) ⟨ M_ij⟩ + Q(η) ⟨ N_,ij⟩ + ⟨ C_ij⟩.Since L^i_j = L_ij,M^i_j = M_ij andL^i_i= - 1/2 [2 FΔ F + 3/2 F_,l F_,l], M^i_i= - 1/28 [10 (F_,ll)^2 - 3 (Δ F)^2 ], Δ N= - 1/28 [ (Δ F)^2 -F_,kl F_,kl], C^i_i= 0,we obtain ⟨ L^i_i⟩ = - 7/4 ⟨ F Δ F⟩,⟨ M^i_i⟩ = - 1/4 ⟨ (Δ F)^2⟩,⟨Δ N⟩ = ⟨ C^i_i⟩= 0.Then we have using Eqs.(<ref>) and (<ref>) ⟨ l_ii⟩ =⟨ l^i_i⟩=- 2π 32.4^4 P_R0Z(a) [7 × 32.4^-2 A + Z(a) B],where A, B and Z(a) are given by Eqs.(<ref>) and (<ref>).The line-element can be expressed as ds^2 = -dt^2 + a_rem (t)^2[(dx^1)^2 + (dx^2)^2 + (dx^3)^2],where the renormalized scale-factor is defined by a_rem (t)^2 ≡a^2 + δ_2 (a^2) = a^2 + 1/3⟨ l^i_i⟩,and the relative difference of scale-factors is given byξ≡a_rem (t)/a(t) - 1 = [1 + 1/3⟨ l^i_i⟩/a^2]^1/2 - 1. The renormalized redshift z_rem corresponding to an arbitrary time t is defined using thescale-factor a_rem as 1 +z_rem≡ a_rem (t_0) / a_rem (t) = 1+ξ (t_0)/1+ξ (t) (1+z), where t_0 denotes the present epoch,the background redshift z is 1/a -1, and ξ (t) is defined by Eq.(<ref>). The square of the background Hubble parameter H is (ȧ/a)^2 and its perturbation is given by δ_2 (H^2) = δ_2 (ȧ^2)/a^2 - (ȧ)^2 δ_2 (a^2)/a^4,so that δ_2 (H^2)/H^2 = 2/3[⟨ l^m_m⟩'/2aa' -⟨ l^m_m⟩/a^2].From Eqs.(<ref>) and (<ref>), we obtain δ_2 (H^2)/H^2 = 4π/3 32.4^4 P_R01/a^2[7 (32.4)^-2(Z - Y(a) a/2(Ω_M + Ω_Λa^3)) A + Z (Z - Y(a) a/Ω_M + Ω_Λ a^3) B],where Y(a) and Z(a) are given by Eq.(<ref>).§.§ Average perturbations of model parametersIn this paper we assume the simplest transfer function (BBKS) for cold matter, adiabatic fluctuations,given by<cit.> T_s (x) = ln (1+0.171 x)/0.171x [1+0.284 x +(1.18 x)^2 + (0.399 x)^3 + (0.490 x)^4]^-1/4. This function has the peak aroundx ≃ 1, so that the upper and lower limits(x_max and x_min) in the integrals A and B in Eq. (<ref>) should have the values such as x_max∼ 6 (> 1) and x_min∼ 0.01 (≪ 1).HereA and B depend sensitively on x_max, but not on x_min.In order to find the best value of x_max, we derive a length L_maxcorresponding to k_max. Using Eq.(<ref>), we haveL_max≡ 2π/k_max = 102/h Mpc for x_max (≡ k_max/k_eq) = 5.7.This L_max corresponds to the cosmological distance, over which the smoothobservations on cosmological scales may be possible. So we adopt x_max = 5.7 and x_min = 0.01. Then we obtain A = 2.22, B = 20.95, and⟨δ_2 ρ/ρ̃⟩ = 0.121 andδ_2 (H^2)/H^2 = 0.210 at the present epoch (a = 1).In similar cases with B ≈ 10 A,the terms with A in Eqs. (<ref>) and (<ref>) are negligible, and then we have⟨δ_2 ρ/ρ̃⟩≃8π/9 32.4^4P_ R0 [1-Y(a)]^2 a^2/Ω_M/Ω_M + Ω_Λ a^3 B , and δ_2 (H^2)/H^2≃4π/3 32.4^4 P_ R01/a^2Z(a) [Z(a) - Y(a) a/Ω_M + Ω_Λ a^3] B.By the way, we derive (δ_1 ρ/ρ̃)^2 to estimate the dispersion of⟨δ_2 ρ/ρ̃⟩: ⟨(δ_1 ρ/ρ)^2 ⟩ =[1 - Y(a)/ρ a^2]^2 (2π)^-2∫dk k^4P_F ( k), so that we obtain ⟨(δ_1 ρ/ρ̃)^2 ⟩ = 8π/9× 32.4^4P_R0 [1-Y(a)]^2 a^2/(Ω_M + Ω_Λ a^3)^2 B ≃⟨δ_2 ρ/ρ̃⟩Ω_M/Ω_M +Ω_Λ a^3.§ RENORMALIZATION OF MODEL PARAMETERS Now let us consider the renormalization of the background density and the Hubble constant.Since ⟨δ_2 ρ/ρ⟩ is spatially constant,we mayassume that it is a part of the background density. Here we regardρ_rem≡ρ + ⟨δ_2 ρ⟩ as the renormalized background density.Similarly ⟨δ_2 H^2 ⟩ is spatially constant, and so we can regardH_rem≡ [H^2 +⟨δ_2 (H^2) ⟩]^1/2 as the renormalized background Hubble parameter. For⟨δ_2H^2⟩/H^2 in Eq. (<ref>) andH = H_0 in Eq. (<ref>), we obtain at present epoch H_rem = 74.0kms^-1 Mpc^-1 , which is equal approximately to the measured Hubble constants.<cit.>It is found, therefore, that the renormalized Hubble constant H_rem may be consistent with the directly measuredHubble constants.The model parameters Ω_M and Ω_Λ describe the evolution of the background universe. But since our real universe isdescribed using the renormalized quantities H_rem and ρ + ⟨δ_2 ρ⟩ in the place of thebackground Hubble constant H and ρ, we may obtain thefollowing new set of model parameters : (Ω_M)_rem = Ω_M 1 +⟨δ_2ρ/ρ⟩/1 + ⟨δ_2 ρ/ρ̃⟩ and (Ω_Λ)_rem= Ω_Λ1/1 + ⟨δ_2ρ/ρ̃⟩. Using the background model parameters Eq.(<ref>) and ⟨δ_2 ρ/ρ̃⟩ in Eq. (<ref>),we obtain at the present epoch (Ω_M)_rem = 0.305and (Ω_Λ)_rem = 0.695.The recent observations of the redshift-magnitude relation<cit.>include many supernovawith redshifts z = 0.1 - 1.0, so that the present Hubble constant (H_rem) is used, and((Ω_M)_rem,(Ω_Λ)_rem)(which areconsistent with Eq.(<ref>)) are obtained.As forthe observations of baryon acoustic oscillations of CMB(Planck<cit.>) and large-scale galactic correlations<cit.>, we use the angular distance in the late time model, so that the derived modelparametersare not (Ω_M,Ω_Λ), but ((Ω_M)_rem, (Ω_Λ)_rem),which are given by Eq.(<ref>). On the other hand, the scale of acoustic oscillations is determined at the recombination epoch witha(≈ 10^-3) and so the Hubble constant is given by H_0(≃H_rem at the recombination epoch), but not H_rem at the present epoch.So the above renormalized model parameters are consistent with the cosmological observations. <cit.>The relative difference of scale-factors ξ (in Eq.(<ref>)) is -0.097 atpresent epoch. Moreover, the present values of ⟨δ_2 ρ/ρ̃⟩ and ((Ω_M)_rem, (Ω_Λ)_rem) in the case when H_rem is 73.8, 74.3 or 78.7 are shown in Appendix B. § RENORMALIZATION OF MODEL PARAMETERS IN THE PAST In the previous section, we treated the quantities at the present epoch (a = 1). Here we consider the quantities at the epochs of a < 1. First we calculate ⟨δ_2 ρ/ρ̃⟩ for a < 1 using Eq.(<ref>) for x_max = 5.7 and x_min = 0.01. Its dependence on a is shown in Fig.1. It is found that ⟨δ_2 ρ/ρ̃⟩ has a peak at around a ∼ 0.65, but ⟨δ_2 ρ/ρ⟩ increases monotonically,and that⟨δ_2 ρ/ρ̃⟩ reduces to 0 in the limit of a→ 0. Using Eqs. (<ref>),(<ref>), (<ref>), and (<ref>),moreover, we obtainH_rem and ((Ω_M)_rem, (Ω_Λ)_rem) in the past.They are shown in Fig. 2 and Fig. 3. It is found that H_rem reduces to H_0(in Eq. (<ref>)), and ((Ω_M)_rem, (Ω_Λ)_rem) reduces to (Ω_M,Ω_Λ) in the limit of a → 0. In Fig. 4, the relative difference of scale-factors ξ (in Eq.(<ref>)) isshown and ξ is-0.097 ∼ -0.195 for a = 1 ∼ 0, respectively. It is found from Eq.(<ref>) that z_rem is larger than z.§ CONCLUDING REMARKSIt was found in this paper that the random adiabatic fluctuations bring a kind of energy density which has an influence upon the dynamics of the universe.For its derivation, the nonlinearity of general-relativistic perturbation theory wasimportant.As k increases in the region of x (≡ k/k_eq) > 1, the amplitude ofperturbations decreases rapidly,and the frequency of perturbed objects is so smallthat they cannot be renormalized as part of the background matter density.The upper limit of x for renormalized perturbations isx_max (≈ 6).Because of their small frequency corresponding to large k, the value of x_max has large fluctuations, and this may be the origin of the directional fluctuations of x_max and themeasured Hubble constant.The background model parameters in Eq.(<ref>) are rather different from the renormalized parameters in Eqs.(<ref>) and (<ref>). We should notice that the observed model parameters are the latter ones.The Hubble constant H_0in the Planck measurements (Eq.(1)) is the renormalized Hubble constant measured at the early stage(a ≪ 1), which is approximately equal to thebackgroundHubble constant (H_0). In this paper we adopted tentatively the background model parameters (in Eq. (<ref>)) andthe value ofx_max. Their values should be selectedso that they mayreflect best the real observations of model parameters. Large-scale perturbations such as x ≡ k/k_eq = 0.01 ∼ 6,which were treated in this paper (cf. Eq.(<ref>)) cross thehorizon in the course of their evolution, so that taking thegeneral-relativistic effect into consideration is indispensable for theirdynamical analyses, which are not only linear but also on second-order. In the Newtonian theory, the terms representing the gravitational strengthξ (≡ GM/(c^2 R)) are taken only linearly into account,assuming that it is exremely small, where M and R are characteristic mass and length of dynamicalobjects.In the cosmological circumstances, however, we have ξ∼ G ρ /H^2· (δρ/ ρ)∼ δρ/ ρ,where H is the Hubble parameter, R ∼ c/H, and M∼ ρ R^3. When | δρ/ρ| is not so small, we cannot neglect the nonzero mean of ξ^2, which should be treatedin the post-Newtonian approximation.In the general-relativistic treatmentof second-order perturbations, this nonzero mean is automatically taken into account.In the linear level, we have ⟨ξ⟩ = 0,but in the second-order,⟨ξ^2⟩∼⟨ (δρ/ ρ)^2 ⟩ 0.Moreover, the necessity of considering nonlinearξ comes also from the energy-momentum conservation law. In order that the Newtonian theory is compatible with theenergy-momentum conservation in the same way as in thegeneral-relativisticcosmology, we must add a nonzero term (δρ/ρ)^2to the second-order density perturbation,so as to recover thepost-Newtonian terms with ξ^2. By this addition,the Newtonian theory may be consistent with the general-relativisticcosmology. The correspondence of general-relativistic approach and Newtonian approach has been studied by Matarrese et al.<cit.> The discussions about it in detail are beyond the scope of the present paper. § ACKNOWLEDGEMENTSThe author thanks Masumi Kasai for helpful discussions. § QUANTITIES IN THE SECOND-ORDER METRIC PERTURBATIONThe quantity Qin the second-order metric perturbation is expressed as Q = ∫^η_0 dη' a^-2(η') ∫^η'_0 dη” a^2(η”) [P(η”) -52(P'(η”))^2].In the case Λ = 0, we haveP - (5/2)(P')^2 = 0 because of a∝η^2 and P = η^2/10, so that Q vanishes. The functions L^j_i and M^j_i are defined by L^j_i= 12[-3 F_,i F_,j -2 F · F_,ij + 12δ_ij F_,l F_,l], M^j_i= 128{19F_,il F_,jl - 12 F_,ijΔ F - 3δ_ij[F_,kl F_,kl -(Δ F)^2 ]}and N is defined by Δ N = 128[(Δ F)^2 - F_,klF_,kl]. The last term C^j_i satisfies the wave equation C^j_i = 314(P/a)^2 G^j_i + 17[P - 52 (P')^2 ] G̃^j_i,where the operatoris defined byϕ≡ g^μνϕ_;μν = -a^-2(∂^2/∂η^2 + 2a'a∂/∂η - Δ) ϕfor an arbitrary function ϕ by use of the four-dimensional covariant derivative ;, and G^j_i and G̃^j_i are expressed asG^j_i≡Δ(F_,ijΔ F - F_,il F_,jl) + (F_,ij F_,kl - F_,ik F_,jl)_,kl - 12δ_ijΔ [(Δ F)^2 - F_,klF_,kl],G̃^j_i≡ F_,ijΔ F - F_,il F_,jl - 14δ_ij[(Δ F)^2 - F_,klF_,kl] - 7 N_,ij.These functions satisfy the traceless and transverse relations G^l_l = 0,G^l_i,l = 0,G̃^l_l = 0, G̃^l_i,l = 0, so that C^j_i also satisfies C^l_l = 0,C^l_i,l = 0.This means that C^j_i represents the second-order gravitational radiation emitted by first-order density perturbations. The solution of the above inhomogeneous wave equation (Eq.(<ref>)) can be represented in an explicit form usingthe retarded Green function for the operator . § CASE WHEN H_REM IS 73.8, 74.3, OR 78.7If we assume that the three values ofHubble constant due to direct measurements(= 73.8, 74.3, 78.7) are H_rem, we obtain ⟨δ_2 ρ/ρ̃⟩ = 0.117,0.126,0.213, respectively, and ⟨(δ_1 ρ/ρ̃ )^2 ⟩^1/2 = 0.160,0.166,0.216. Moreover,corresponding to H_rem = 73.8, 74.3, 78.7 and H = H_0 in Eq.(<ref>), it is foundfor a fixed x_min = 0.01 using Eq. (<ref>) thatx_max = 5.6,5.8,7.4,A = 2.20,2.25,2.59, B = 20.14,21.78,36.65, respectively. The values of A and B depend rather sharply on x_max, but not onx_min. It is found therefore that the directly observed Hubble constants appear, corresponding to various values of x_max (≈ 6). These values of x_max may depend on the direction in which we measurethe Hubble constant.ForH_rem = 73.8, 74.3, 78.7, moreover, we obtain (Ω_M)_rem = (0.302,0.307,0.357) and (Ω_Λ)_rem = (0.698,0.693,0.643), respectively. 99wmap G. Hinshaw et al., Astrophys. J.Suppl. 208, 19 (2013).planck1 P. A. R. Ade et al., Astron. Astrophys. 571, A16 (2014). planck2 P. A. R. Ade et al., Astron. Astrophys. 594, A3 (2016). h4 J.L. Bernal et al., JCAP10, 019 (2016). h1 A.G. Riess et al., Astrophys. J.730, 2019 (2011).h2 W.L. Freedman et al. , Astrophys. J.758, 24 (2012).h3 S.H. Suyu et al., Astrophys. J.766, 70 (2013).decay Z. Berezhiani et al.,Phys. Rev. D92, 061303 (2015). void K. Ichiki et al.,Phys. Rev. D93, 023529 (1986).tom Kenji Tomita, Phys. Rev. D71, 083504 (2005). tomold Kenji Tomita, Prog. Theor. Phys. 37, 831 (1967). russ H. Russ et al., Phys. Rev. D53, 6881 (1996). mat S. Matarrese et al., Phys. Rev. D58, 043504 (1998). mak N. Makino et al., Phys. Rev. D46, 585 (1992). bern F. Bernardeau et al., Phys. Rep. 367, 1 (2002). weinS. Weinberg, Cosmology (Oxford University Press, New York,2008). tsuji L. Amendola and S. Tsujikawa, Dark Energy (CambridgeUniversity Press,Cambridge, 2010). bbks J. M. Bardeen, J. R. Bond, N. Kaiser, and A.S. Szalay,Astrophys. J.304, 15 (1986).llA. R. Liddle and D.H. Lyth, Cosmological inflation and large-scalestructure (Cambridge University Press, New York, 2000). dodelJ.A. Peacock, Cosmological Physics(Cambridge University Press,Cambridge, 1999). betoule M. Betoule et al. Astron. Astrophys. 568, A22 (2014). eisen D.J. Eisenstein et al., Astrophys. J.633, 560 (2005).perc W.J. Percival et al., MNRAS401, 2148 (2010).mat2 S. Matarrese et al., MNRAS 283, 400 (1996).

http://arxiv.org/abs/1702.07821v3

plain theoremTheorem[section] corollary[theorem]Corollary lemma[theorem]Lemma proposition[theorem]Propositionremark remarkRemarkdefinition definitionDefinition

http://arxiv.org/abs/1702.07894v2

A Hybrid High-Order method for the convective Cahn–Hilliard problem in mixed form A HHO method for the Cahn–Hilliard problem Florent Chave, Daniele A. Di Pietro and Fabien Marche University of Montpellier, Institut Montpelliérain Alexander Grothendieck34095 Montpellier, France{florent.chave, daniele.di-pietro, fabien.marche}@umontpellier.fr* Florent Chave, Daniele A. Di Pietro and Fabien MarcheAbstract 0.9 We discuss effects of the brane-localized mass terms on the fixed points of the toroidal orbifold T^2/Z_2 under the presence of background magnetic fluxes, where multiple lowest and higher-level Kaluza–Klein (KK) modes are realized before introducing the localized masses in general. Through the knowledge of linear algebra, we find that, in each KK level, one of or more than one of the degenerate KK modes are almost inevitably perturbed, when single or multiple brane-localized mass terms are introduced. When the typical scale of the compactification is far above the electroweak scale or the TeV scale, we apply this mechanism for uplifting unwanted massless or light modes which are prone to appear in models on magnetized orbifolds. ===========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================We propose a novel Hybrid High-Order method for the Cahn–Hilliard problem with convection. The proposed method is valid in two and three space dimensions, and it supports arbitrary approximation orders on general meshes containing polyhedral elements and nonmatching interfaces. An extensive numerical validation is presented, which shows robustness with respect to the Péclet number. § CAHN–HILLIARD EQUATION Let Ω⊂ℝ^d, d∈{2,3}, denote a bounded connected convex polyhedral domain with Lipschitz boundary ∂Ω and outward normal n⃗, and let t_F>0. The convective Cahn–Hilliard problem consists in finding the order-parameter c : Ω×(0,t_F] →ℝ and the chemical potential w : Ω×(0,t_F] →ℝ such that d_t c - 1/PeΔw + ∇·(u⃗ c)= 0 in Ω×(0,t_F w= Φ'(c) - γ^2Δc in Ω×(0,t_F c(0)= c_0 in Ω ∂_n⃗c=∂_n⃗w =0 on ∂Ω× (0,t_F where γ > 0 is the interface parameter (usually taking small values), Pe>0 is the Péclet number, u⃗ the velocity field such that ∇·u⃗=0 in Ω and Φ the free-energy such that Φ(c) := 1/4 (1-c^2)^2. This formulation is an extension of the Cahn–Hilliard model originally introduced in <cit.> and a first step towards coupling with the Navier–Stokes equations. In this work we extend the HHO method of <cit.> to incorporate the convective term in (<ref>). Therein, a full stability and convergence analysis was carried out for the non-convective case, leading to optimal estimates in (h^k+1+τ) (with h denoting the meshsize and τ the time step) for the the C^0(H^1)-error on the order-parameter and L^2(H^1)-error on the chemical potential. The convective term is treated in the spirit of <cit.>, where a HHO method fully robust with respect to the Péclet number was presented for a locally degenerate diffusion-advection-reaction problem.The proposed method offers various assets: (i) fairly general meshes are supported including polyhedral elements and nonmatching interfaces; (ii) arbitrary polynomial orders, including the case k=0, can be considered; (iii) when using a first-order (Newton-like) algorithm to solve the resulting system of nonlinear algebraic equations, element-based unknowns can be statically condensed at each iteration. The rest of this paper is organized as follows: in Section <ref>, we recall discrete setting including notations and assumptions on meshes, define localy discrete operators and state the discrete formulation of (<ref>). In Section <ref>, we provide an extensive numerical validation.§ THE HYBRID HIGH-ORDER METHOD In this section we recall some assumptions on the mesh, introduce the notation, and state the HHO discretization. §.§ Discrete setting We consider sequences of refined meshes that are regular in the sense of <cit.>. Each mesh 𝒯_h in the sequence is a finite collection {T} of nonempty, disjoint, polyhedral elements such that Ω = ⋃_T∈𝒯_hT and h = max_T∈𝒯_h h_T (with h_T the diameter of T). For all T∈𝒯_h, the boundary of T is decomposed into planar faces collected in the set ℱ_T. For admissible mesh sequences, card(ℱ_T) is bounded uniformly in h. Interfaces are collected in the set ℱ_h^i, boundary faces in ℱ_h^b and we define ℱ_h := ℱ_h^i∪ℱ_h^b. For all T∈𝒯_h and all F∈ℱ_T, the diameter of F is denoted by h_F and the unit normal to F pointing out of T is denoted by n⃗_TF.To discretize in time, we consider for sake of simplicity a uniform partition (t^n)_0≤ n ≤ N of the time interval [0,t_F] with t^0=0, t^N = t_F and t^n-t^n-1 = τ for all 1≤ n≤ N. For any sufficiently regular function of time φ taking values in a vector space V, we denote by φ^n∈ V its value at discrete time t^n, and we introduce the backward differencing operator δ_t such that, for all 1≤ n≤ N,δ_tφ^n := φ^n - φ^n-1τ∈ V.§.§ Local space of degrees of freedom For any integer l≥ 0 and X a mesh element or face, we denote by ℙ^l(X) the space spanned by the restrictions to X of d-variate polynomials of order l. LetU_h^k:=(×_T∈𝒯_hℙ^k+1(T))×(×_F∈ℱ_hℙ^k(F))be the global degrees of freedoms (DOFs) space with single-valued interface unknowns. We denote by v_h = ((v_T)_T∈𝒯_h,(v_F)_F∈ℱ_h) a generic element of U_h^k and by v_h the piecewise polynomial function such that v_h|_T = v_T for all T∈𝒯_h. For any T∈𝒯_h, we denote by U_T^k and v_T = (v_T, (v_F)_F∈ℱ_T) the restrictions to T of U_h^k and v_h, respectively. §.§ Local diffusive contribution Consider a mesh element T∈𝒯_h. We define the local potential reconstruction p_T^k+1:U_T^k →ℙ^k+1(T) such that, for all v_T := (v_T, (v_F)_F∈ℱ_T)∈U_T^k and all z∈ℙ_T^k+1,(∇p_T^k+1v_T, ∇ z)_T=-(v_T,Δ z)_T + ∑_F∈ℱ_T (v_F, ∇ z·n⃗_TF)_F,with closure condition ∫_T (p_T^k+1v_T - v_T) = 0. We introduce the local diffusive bilinear form a_T on U_T^k×U_T^k such that, for all (u_T,v_T)∈U_T^k×U_T^k a_T(u_T,v_T) := (∇p_T^k+1u_T, ∇p_T^k+1v_T)_T + s_T(u_T,v_T),with stabilization bilinear form s_T:U_T^k×U_T^k→ℝ such thats_T(u_T,v_T) := ∑_F∈ℱ_T h_F^-1(π_F^k(u_F-u_T), π_F^k(v_F-v_T))_F,where, for all F∈ℱ_h, π_F^k:L^1(F)→ℙ^ k(F) denotes the L^2-orthogonal projector onto ℙ^ k(F). §.§ Local convective contribution For any mesh element T∈𝒯_h, we define the local convective derivative reconstruction G_u⃗,T^k+1 : U_T^k →ℙ^k+1(T) such that, for all v_T := (v_T, (v_F)_F∈ℱ_T)∈U_T^k and all w∈ℙ^k+1(T),(G_u⃗,T^k+1v_T, w)_T=-(v_T,u⃗·∇ w)_T + ∑_F∈ℱ_T(v_F,(u⃗·n⃗_TF)w)_F.The local convective contribution b_u⃗,T on U_T^k×U_T^k is such that, for all (u_T,v_T)∈U_T^k×U_T^k b_u⃗,T(u_T,v_T) := (G_u⃗,T^k+1u_T, v_T)_T + s_u⃗,T(u_T,v_T).with local upwind stabilization bilinear form s_u⃗,T:U_T^k×U_T^k→ℝ such thats_u⃗,T(u_T,v_T) := ∑_F∈ℱ_T (|u⃗·n⃗_TF|-u⃗·n⃗_TF/2(u_F-u_T), v_F - v_T)_F.Notice that the actual computation of G_u⃗,T^k+1 is not required, as one can simply use its definition to expand the cell-based term in the bilinear form b_u⃗,T. §.§ Discrete problem Denote by U_h,0^k := {v_h = ((v_T)_T∈𝒯_h,(v_F)_F∈ℱ_h)∈U_h^k | ∫_Ω v_h = 0} the zero-average DOFs subspace of U_h^k. We define the global bilinear forms a_h and b_u⃗,h on U_h^k×U_h^k such that, for all (u_h,v_h)∈U_h^k×U_h^ka_h(u_h,v_h) := ∑_T∈𝒯_h a_T(u_T,v_T), b_u⃗,h(u_h,v_h) := ∑_T∈𝒯_h b_u⃗,T(u_T,v_T).The discrete problem reads: For all 1≤ n ≤ N, find (c_h^n,w_h^n)∈U_h,0^k×U_h^k such that(δ_t c_h^n,φ_h) + 1/Pea_h(w^n_h,φ_h) + b_u⃗,h(c^n_h,φ_h) = 0 ∀φ_h∈U_h^k(w_h^n,ψ_h) = (Φ'(c_h^n),ψ_h) + γ^2 a_h(c^n_h,ψ_h) ∀ψ_h∈U_h^k where c_h^0∈U_h,0^k solves a_h(c_h^0,φ_h) = -(Δ c_0,φ_h) for all φ_h∈U_h^k.§ NUMERICAL TEST CASES In this section, we numerically validate the HHO method. §.§ Disturbance of the steady solution For the first test case, we use a piecewise constant approximation (k=0), discretize the domain Ω = (0,1)^2 by a triangular mesh (h=1.92· 10^-3) with γ = 5· 10^-2, τ = γ^2 and Pe = 1. The initial condition for the order-parameter and the velocity field are given byc_0(x⃗) := tanh (2x_1 - 12√(2)γ^2), u⃗(x⃗) := 20·[x_1(x_1-1)(2x_2-1); -x_2(x_2-1)(2x_1-1) ],∀x⃗∈Ω.The result is depicted in Figure <ref> and shows that the method is well-suited to capture the interface dynamics subject to a strong velocity fields. §.§ Thin interface between phases For the second example, we also use a piecewise constant approximation (k=0) with a Cartesian discretization of the domain Ω = (0,1)^2, where h=1.95· 10^-3. The interface parameter is taken to be very small γ = 5· 10^-3, the time step is τ = 1· 10^-5 and Pe=50. The initial condition for the order-parameter is taken to be a random value between -1 and 1 inside a circular partition of the Cartesian mesh and -1 outside. The velocity field is given byu⃗(x⃗) := 1/2(1+tanh (80-200(x_1-0.5,x_2-0.5) _2)) ·[ 2x_2-1; 1-2x_1 ], ∀x⃗∈Ω.See Figure <ref> for the numerical result. The method is robust with respect to γ and is also well-suited to approach the thin high-gradient area of the order-parameter.§.§ Effect of the Péclet number The Péclet number is the ratio of the contributions to mass transport by convection to those by diffusion: when Pe is greater than one, the effects of convection exceed those of diffusion in determining the overall mass flux. In the last test case, we compare several time evolutions obtained with different values of the Péclet number (Pe∈{1,50,200}), starting from the same initial condition. We use a Voronoi discretization of the domain Ω = (0,1)^2, where h=9.09· 10^-3, and use piecewise linear approximation (k=1). We choose γ = 1· 10^-2, τ = 1· 10^-4 and t_F = 1. The initial condition is given by a random value between -1 and 1 inside a circular domain of the Voronoi mesh and -1 outside. The convective term is given byu⃗(x⃗) := [ sin(π x_1) cos(π x_2); -cos(π x_1)sin(π x_2) ], ∀x⃗∈Ω.Snapshots of the order parameter at several times are shown on Figure <ref> for each value of the Péclet number. For each case, the method takes into account the value of Pe and appropriately models the evolution of the order parameter by prevailing advection to diffusion when Pe≫ 1.spmpsciThe work of D. A. Di Pietro and F. Marche was partially supported by Agence Nationale de la Recherche grant HHOMM (ref. ANR-15-CE40-0005).

http://arxiv.org/abs/1702.08224v1

Strongly Interacting Phases of Metallic Wires in Strong Magnetic Field Xiao-Liang Qi Accepted .Received ; in original form======================================================================empty Strongly Interacting Phases of Metallic Wires in Strong Magnetic Field Xiao-Liang Qi Accepted .Received ; in original form====================================================================== We show that the standard stochastic gradient decent (SGD) algorithm is guaranteed to learn, in polynomial time, a function that is competitive with the best function in the conjugate kernel space of the network, as defined in <cit.>. The result holds for log-depth networks from a rich family of architectures. To the best of our knowledge, it is the first polynomial-time guarantee for the standard neural network learning algorithm for networks of depth more that two.As corollaries, it follows that for neural networks of any depth between 2 and log(n), SGD is guaranteed to learn, in polynomial time, constant degree polynomials with polynomially bounded coefficients. Likewise, it followsthat SGD on large enough networks can learn any continuous function (not in polynomial time), complementing classical expressivity results. § INTRODUCTIONWhile stochastic gradient decent (SGD) from a random initialization is probably the most popular supervised learning algorithm today, we have very few results that depicts conditions that guarantee its success. Indeed, to the best of our knowledge, <cit.> provides the only known result of this form, and it is valid in a rather restricted setting. Namely, for depth-2 networks, where the underlying distribution is Gaussian, the algorithm is full gradient decent (rather than SGD), and the task is regression when the learnt function is a constant degree polynomial.We build on the framework of <cit.> to establish guarantees on SGD in a rather general setting.<cit.> defined a framework that associates a reproducing kernel to a network architecture. They also connected the kernel to the network via the random initialization. Namely, they showed that right after the random initialization, any function in the kernel space can be approximated by changing the weights of the last layer. The quality of the approximation depends on the size of the network and the norm of the function in the kernel space.As optimizing the last layer is a convex procedure, the result of <cit.> intuitively shows that the optimization process starts from a favourable point for learning a function in the conjugate kernel space. In this paper we verify this intuition. Namely, for a fairly general family of architectures (that contains fully connected networks and convolutional networks) and supervised learning tasks, we show that if the network is large enough, the learning rate is small enough, and the number of SGD steps is large enough as well, SGD is guaranteed to learn any function in the corresponding kernel space.We emphasize that the number of steps and the size of the network are only required to be polynomial (which is best possible) in the relevant parameters – the norm of the function, the required accuracy parameter (ϵ), and the dimension of the input and the output of the network. Likewise, the result holds for any input distribution. To evaluate our result, one should understand which functions it guarantee that SGD will learn. Namely, what functions reside in the conjugate kernel space, how rich it is, and how good those functions are as predictors. From an empirical perspective, in <cit.>, it is shown that for standardconvolutional networks the conjugate class contains functions whose performance is close to the performance of the function that is actually learned by the network. This is based on experiments on the standard CIFAR-10 dataset.From a theoretical perspective, we list below a few implications that demonstrate the richness of the conjugate kernel space. These implications are valid for fully connected networks of any depth between 2 and log(n), where n is the input dimension. Likewise, they are also valid for convolutional networks of any depth between 2 and log(n), and with constantly many convolutional layers. * SGD is guaranteed to learn in polynomial time constant degree polynomials with polynomially bounded coefficients. As a corollary, SGD is guaranteed to learn in polynomial time conjunctions, DNF and CNF formulas with constantly many terms, and DNF and CNF formulas with constantly many literals in each term. These function classes comprise a considerable fraction of the function classes that are known to be poly-time (PAC) learnable by any method. Exceptions include constant degree polynomial thresholds with no restriction on the coefficients, decision lists and parities.* SGD is guaranteed to learn, not necessarily in polynomial time, any continuous function. This complements classical universal approximation results that show that neural networks can (approximately) express any continuous function (see <cit.> for a survey). Our results strengthen those results and show that networks are not only able to express those functions, but actually guaranteed to learn them.§.§ Related work Guarantees on SGD. As noted above, there are very few results that provide polynomial time guarantees for SGD on NN. One notable exception is the work of <cit.>, that proves a result that is similar to ours, but in a substantially more restricted setting. Concretely, their result holds for depth-2 fully connected networks, as opposed to rather general architecture and constant or logarithmic depth in our case. Likewise, the marginal distribution on the instance space is assumed to be Gaussian or uniform, as opposed to arbitrary in our case. In addition, the algorithm they consider is full gradient decent, which corresponds to SGD with infinitely large mini-batch, as opposed to SGD with arbitrary mini-batch size in our case. Finally, the underlying task is regression in which the target function is a constant degree polynomial, whereas we consider rather general supervised learning setting. Other polynomial time guarantees on learning deep architectures. Various recent papers show that poly-time learning is possible in the case that the the learnt function can be realized by a neural network with certain (usually fairly strong) restrictions on the weights <cit.>, or under the assumption that the data is generated by a generative model that is derived from the network architecture <cit.>. We emphasize that the main difference of those results from our results and the results of <cit.> is that they do not provide guarantees on the standard SGD learning algorithm. Rather, they show that under those aforementioned conditions, there are some algorithms, usually very different from SGD on the network, that are able to learn in polynomial time. Connection to kernels. As mentioned earlier, our paper builds on <cit.>, who developed the association of kernels to NN which we rely on. Several previous papers <cit.> investigated such associations, but in a more restricted settings (i.e., for less architectures). Some of those papers <cit.> also provide measure of concentration results, that show that w.h.p. the random initialization of the network's weights is reach enough to approximate the functions in the corresponding kernel space. As a result, these papers provide polynomial time guarantees on the variant of SGD, where only the last layer is trained. We remark that with the exception of <cit.>, those results apply just to depth-2 networks. §.§ Discussion and future directionsWe next want to place this work in the appropriate learning theoretic context, and to elaborate further on this paper's approach for investigating neural networks. For the sake of concreteness, let us restrict the discussion to binary classification over the Boolean cube. Namely, given examples from a distributionon {± 1}^n×{0,1}, the goal is to learn a function h:{± 1}^n→{0,1} whose 0-1 error, ^0-1_(h) = _(,y)∼(h() y), is as small as possible. We will use a bit of terminology. A model is a distributionon {± 1}^n×{0,1} and a model class is acollectionof models. We note that any function class ⊂{0,1}^{± 1}^n defines a model class, (), consisting of all modelssuch that ^0-1_(h)=0 for some h∈. We define the capacity of a model class as the minimal number m for which there is an algorithm such that for every ∈ the following holds. Given m samples from , the algorithm is guaranteed to return, w.p. ≥9/10 over the samples and its internal randomness, a function h:{± 1}^n→{0,1} with 0-1 error ≤1/10. We note that for function classes the capacity is the VC dimension, up to a constant factor.Learning theory analyses learning algorithms via model classes. Concretely, one fixes some model classand show that the algorithm is guaranteed to succeed whenever the underlying model is from .Often, the connection between the algorithm and the class at hand is very clear. For example, in the case that the model is derived from a function class , the algorithm might simply be one that finds a function inthat makes no mistake on the given sample.The natural choice for a model class for analyzing SGD on NN would be the class of all functions that can be realized by the network, possibly with some reasonable restrictions on the weights. Unfortunately, this approach it is probably doomed to fail, as implied by various computational hardness results <cit.>.So, what model classes should we consider? With a few isolated exceptions (e.g. <cit.>) all known efficiently learnable model classes are either a linear model class, or contained in an efficiently learnable linear model class. Namely, functions classes composed of compositions of some predefined embedding with linear threshold functions, or linear functions over some finite field.Coming up we new tractable models would be a fascinating progress. Still, as linear function classes are the main tool that learning theory currently has for providing guarantees on learning, it seems natural to try to analyze SGD via linear model classes. Our work follows this line of thought, and we believe that there is much more to achieve via this approach. Concretely, while our bounds are polynomial, the degree of the polynomials is rather large, and possibly much better quantitative bounds can be achieved. To be more concrete, suppose that we consider simple fully connected architecture, with 2-layers, ReLU activation, and n hidden neurons. In this case, the capacity of the model class that our results guarantee that SGD will learn is Θ(n^1/3). For comparison, the capacity of the class of all functions that are realized by this network is Θ(n^2). As a challenge, we encourage the reader to prove that with this architecture (possibly with an activation that is different from the ReLU), SGD is guaranteed to learn some model class of capacity that is super-linear in n.§ PRELIMINARIES Notation. We denotevectors by bold-face letters (e.g. ),matrices by upper case letters (e.g. W), and collection of matrices by bold-face upper case letters (e.g. ). The p-norm of ∈^d is denoted by _p = (∑_i=1^d|x_i|^p)^1/p. We will also use the convention that =_2. For functions σ:→ we letσ :=√(_X∼(0,1)σ^2(X))= √(1/√(2π)∫_-∞^∞σ^2(x)e^-x^2/2dx) . Let G=(V,E) be a directed acyclic graph. The set of neighbors incoming to a vertex v is denoted (v):={u∈ V| uv∈ E}. We also denote (v) = |(v)|. Given weight function δ:V→ [0,∞) and U⊂ V we let δ(U) = ∑_u∈ Uδ(u). The d-1 dimensional sphere is denoted ^d-1 = {∈^d |=1}. We use [x]_+ to denote max(x,0).Input space. Throughout the paper we assume that each example is a sequence of n elements, each of which is represented as a unit vector.Namely, we fix n and take the input space to be =_n,d=(^d-1)^n. Each input example is denoted,=(^1,…,^n),  ^i∈^d-1 . While this notation is slightly non-standard, it unifiesinput types seen in various domains (see <cit.>).Supervised learning. The goal in supervised learning is to devise a mapping from the input spaceto an output spacebased on a sample S={(_1,y_1),…,(_m,y_m)}, where (_i,y_i)∈×drawn i.i.d. from a distributionover ×.A supervised learning problem is further specified by an output length k and a loss function ℓ : ^k ×→ [0,∞), and the goal is to find a predictor h:→^k whose loss, _(h) := _(,y)∼ℓ(h(),y), is small.The empirical loss _S(h):= 1/m∑_i=1^m ℓ(h(_i),y_i) is commonly used as a proxy for the loss _. When h is defined by a vectorof parameters, we will use the notations _() = _(h), _S()=_S(h) and ℓ_(,y)()=ℓ(h(),y).Regression problems correspond to k=1, = and, for instance, the squared loss ℓ^square(ŷ,y)=(ŷ -y)^2.Binary classification is captured by k=1, ={± 1} and, say, the zero-one loss ℓ^0-1(ŷ,y)= [ŷ y ≤ 0] or the hinge loss ℓ^(ŷ,y)=[1-ŷ y]_+.Multiclass classification is captured by k being the number of classes, = [k], and, say, the zero-one loss ℓ^0-1(ŷ,y)= [ŷ_y ≤_y'ŷ_y'] or the logistic loss ℓ^log(,y) = -log(p_y()) where :^k→Δ^k-1 is given by p_i() = e^ŷ_i/∑_j=1^ke^ŷ_j.A loss ℓ is L-Lipschitz if for all y∈, the function ℓ_y(ŷ) :=ℓ(ŷ,y) is L-Lipschitz. Likewise, it is convex if ℓ_y is convex for every y∈.Neural network learning.We define a neural networkto be a vertices weighted directed acyclic graph (DAG) whose nodes are denoted V() and edges E(). The weight function will be denoted by δ:V()→ [0,∞), and its sole role would be to dictate the distribution of the initial weights. We will refer 's nodes by neurons. Each of non-input neuron, i.e. neuron with incoming edges, is associated with an activation function σ_v:→. In this paper, an activation can be any function σ:→ that is right and left differentiable, square integrable with respect to the Gaussian measure on , and is normalized in the sense that σ=1. The set of neurons having only incoming edges are called the output neurons.To match the setup of supervised learning defined above, a networkhas nd input neurons and k output neurons, denoted o_1,…,o_k. A networktogether with a weight vector ={w_uv| uv∈ E}∪{b_v| v∈ V is an internal neuron} defines a predictor h_,:→^k whose prediction is given by “propagating”forward through the network. Concretely, we define h_v,(·) to be the output of the subgraph of the neuron v as follows: for an input neuron v, h_v, outputs the corresponding coordinate in , and internal neurons, we define h_v, recursively ash_v,() = σ_v(∑_u∈(v)w_uv h_u,() + b_v) .For output neurons, we define h_v, ash_v,() = ∑_u∈(v)w_uv h_u,() .Finally, we let h_,()=(h_o_1,(),…,h_o_k,()).We next describe the learning algorithm that we analyze in this paper. While there is no standard training algorithm for neural networks, the algorithms used in practice are usually quite similar to the one we describe, both in the way the weights are initialized and the way they are updated.We will use the popular Xavier initialization <cit.> for the network weights. Fix 0≤β≤ 1. We say that ^0={w^0_uv}_uv∈ E∪{b_v}_v∈ V is an internal neuron are β-biased random weights (or, β-biased random initialization) ifeach weight w_uv is sampled independently from a normal distribution with mean 0 and variance (1-β)dδ(u)/δ((v)) if u is an input neuron and (1-β)δ(u)/δ((v)) otherwise. Finally, each bias term b_v is sampled independently from a normal distribution with mean 0 and variance β. We note that the rational behind this initialization scheme is that for every exampleand every neuron v we have __0(h_v,_0())^2=1 (see <cit.>) Kernel classes. A function κ:×→ is a reproducing kernel, or simply a kernel, if for every _1,…,_r∈, the rr matrix Γ_i,j = {κ(_i,_j)} is positive semi-definite. Each kernel induces a Hilbert space _κ of functions fromtowith a corresponding norm ·_κ. For ∈_κ^k we denote _κ =√(∑_i=1^kh_i^2_κ). A kernel and its corresponding space are normalized if ∀∈, κ(,)=1. Kernels give rise to popular benchmarks for learning algorithms. Fix a normalized kernelκ and M>0.It is well known that that for L-Lipschitz loss ℓ, the SGD algorithm is guaranteed to return a functionsuch that _() ≤min_'∈^k_κ, '_κ≤ M_(')+ϵ using (LM/ϵ)^2 examples.In the context of multiclass classification, for γ>0 we define ℓ^γ:^k× [k]→ by ℓ^γ(ŷ,y) =[ŷ_y≤γ+max_y' yŷ_y']. We say that a distributionon × [k] is M-separable w.r.t. κ if there is ^*∈_κ^k such that ^*_κ≤ M and ^1_(^*)=0. In this case, the perceptron algorithm is guaranteed to return a functionsuch that ^0-1_() ≤ϵ using 2M^2/ϵ examples. We note that both for perceptron and SGD, the above mentioned results are best possible, in the sense that any algorithm with the same guarantees, will have to use at least the same number of examples, up to a constant factor.Computation skeletons <cit.>In this section we define a simple structure which we term a computation skeleton. The purpose of a computational skeleton is to compactly describe a feed-forward computation from an input to an output. A single skeleton encompasses a family of neural networks that share the same skeletal structure. Likewise, it defines a corresponding normalized kernel. A computation skeletonis a DAG with n inputs, whose non-input nodes are labeled by activations, and has a single output node (). Figure <ref> shows four example skeletons, omitting the designation of the activation functions.We denote by || the number of non-input nodes of . The following definition shows how a skeleton, accompanied with a replication parameter r≥ 1 and a number of output nodes k, induces a neural network architecture.Letbe a computation skeleton and consider input coordinates in ^d-1 as in (<ref>). For r, k ≥ 1 we define the following neural network =(,r,k).For each input node in ,has d corresponding input neurons with weight 1/d. For each internal node v∈ labelled by an activation σ,has r neurons v^1,…,v^r, each with an activation σ and weight 1/r. In addition,has k output neurons o_1,…,o_k with the identity activation σ(x)=x and weight 1.There is an edge v^iu^j∈ E() whenever uv∈ E().For every output node v in , each neuron v^j is connected to all output neurons o_1,…,o_k. We termthe (r,k)-fold realization of . Note that the notion of the replication parameter r corresponds, in the terminology of convolutional networks, to the number of channels taken in a convolutional layer and to the number of hidden neurons taken in a fully-connected layer. In addition to networks' architectures, a computation skeletonalso defines a normalized kernel κ_:×→[-1,1]. To define the kernel, we use the notion of a conjugate activation. For ρ∈[-1,1], we denote by _ρ the multivariate Gaussian distribution on ^2 with mean 0 and covariance matrix ( [ 1 ρ; ρ 1 ]).The conjugate activation of an activation σ is the function σ̂:[-1,1]→ defined as σ̂(ρ) = _(X,Y) ∼_ρσ(X)σ(Y).The following definition gives the kernel corresponding to a skeletonLetbe a computation skeleton and let 0≤β≤ 1.For every node v, inductively define a kernel κ^β_v:×→ as follows.For an input node v corresponding to the ith coordinate, define κ^β_v(,)=^i, ^i.For a non-input node v, defineκ^β_v(,) = σ̂_v( (1-β)∑_u∈(v)κ^β_u(,)/|(v)|+β).The final kernel κ^β_ is κ^β_(). The resulting Hilbert space and norm are denoted _,β and ·_,β respectively. § MAIN RESULTS An activation σ:→ is called C-bounded if σ_∞, σ'_∞, σ”_∞≤ C. Fix a skeletonand 1-Lipschitz[If ℓ is L-Lipschitz, we can replace ℓ by 1/Lℓ and the learning rate η by Lη. The operation of algorithm <ref> will be identical to its operation before the modification. Given this observation, it is very easy to derive results for general L given our results. Hence, to save one paramater, we will assume that L=1.] convex loss ℓ.Define ()=∏_i=1^()max_v∈, (v)=i((v)+1) and () = (8C)^()√(()), where C is the minimal number for which all the activations inare C-bounded, and (v) is the maximal length of a path from an input node to v. We also define '() = (4C)^()√(()), where C is the minimal number for which all the activations inare C-Lipschitz and satisfy |σ(0)|≤ C. Through this and remaining sections we use ≳ to hide universal constants. Likewise, we fix the bias parameter β and therefore omit it from the relevant notation. We note that for constant depth skeletons with maximal degree that is polynomial in n, () and '() are polynomial in n. These quantities are polynomial in n also for various log-depth skeletons. For example, this is true for fully connected skeletons, or more generally, layered skeletons with constantly many layers that are not fully connected. Suppose that all activations are C-bounded. Let M,ϵ>0. Suppose that we run algorithm <ref> on the network (,r,k) with the following parameters: * η=η'/r for η'≲ϵ/('())^2* T≳M^2/η' ϵ* r≳C^4(Tη')^2M^2('())^4 log(C||/ϵδ)/ϵ^2 +d* Zero initialized prediction layer* Arbitrary mThen, w.p. ≥ 1-δ over the choice of the initial weights, there is t∈ [T] such that _(^t)≤min_∈^k_, _≤ M_()+ϵ. Here, the expectation is over the training examples. We next consider ReLU activations. Here, '() = (√(32))^()√(()). Suppose that all activations are the ReLU. Let M,ϵ>0. Suppose that we run algorithm <ref> on the network (,r,k) with the following parameters: * η=η'/r for η'≲ϵ/('())^2* T≳M^2/η' ϵ* r≳(Tη')^2M^2('())^4 log(||/ϵδ)/ϵ^2 +d* Zero initialized prediction layer* Arbitrary mThen, w.p. ≥ 1-δ over the choice of the initial weights, there is t∈ [T] such that _(^t)≤min_∈^k_, _≤ M_()+ϵ. Here, the expectation is over the training examples. Finally, we consider the case in which the last layer is also initialized randomly. Here, we provide guarantees in a more restricted setting of supervised learning. Concretely, we consider multiclass classification, whenis separable with margin, and ℓ is the logistic loss. Suppose that all activations are C-bounded, thatis M-separable with w.r.t. κ_ and let ϵ>0. Suppose we run algorithm <ref> on (,r,k) with the following parameters: * η=η'/r for η'≲ϵ^2/M^2 (())^4* T≳log(k)M^2/η' ϵ^2* r≳ C^4(())^4M^2(Tη')^2log(C|S|/ϵ) + k + d * Randomly initialized prediction layer* Arbitrary mThen, w.p. ≥1/4 over the choice of the initial weights and the training examples, there is t∈ [T] such that ^0-1_(^t)≤ϵ§.§ ImplicationsTo demonstrate our results, let us elaborate on a few implications for specific network architectures. To this end, let us fix the instance spaceto be either {± 1}^n or ^n-1. Also, fix a bias parameter 1≥β>0, a batch size m, and a skeletonthat is a skeleton of a fully connected network of depth between 2 and log(n). Finally, we also fix the activation function to be either the ReLU or a C-bounded activation, assume that the prediction layer is initialized to 0, and fix the loss function to be some convex and Lipschitz loss function. Very similar results are valid for convolutional networks with constantly many convolutional layers. We however omit the details for brevity. Our first implication shows that SGD is guaranteed to efficiently learn constant degree polynomials with polynomially bounded weights. To this end, let us denote by _t the collection of degree t polynomials. Furthermore, for any polynomial p we denote by p the ℓ^2 norm of its coefficients.Fix any positive integers t_0,t_1. Suppose that we run algorithm <ref> on the network (,r,1) with the following parameters: * η≲(ϵ/n)* T,r≳(n/ϵ,log(1/δ))Then, w.p. ≥ 1-δ over the choice of the initial weights, there is t∈ [T] such that _(^t)≤min_p∈_t_0, p≤ n^t_1_(p)+ϵ. Here, the expectation is over the training examples.We note that several hypothesis classes that were studied in PAC learning can be realized by polynomial threshold functions with polynomially bounded coefficients. This includes conjunctions, DNF and CNF formulas with constantly many terms, and DNF and CNF formulas with constantly many literals in each term. If we take the loss function to be the logistic loss or the hinge loss, Corollary <ref> implies that SGD efficiently learns these hypothesis classes as well.Our second implication shows that any continuous function is learnable (not necessarily in polynomial time) by SGD.Fix a continuous function h^*:^n-1→ and ϵ,δ>0.Assume thatis realized[That is, if (,y)∼ then y = h^*() with probability 1.] by h^*. Assume that we run algorithm <ref> on the network (,r,1). If η>0 is sufficiently small and T and r are sufficiently large, then, w.p. ≥ 1-δ over the choice of the initial weights, there is t∈ [T] such that _(^t)≤ϵ. §.§ ExtensionsWe next remark on two extensions of our main results. The extended results can be proved in a similar fashion to our results. To avoid cumbersome notation, we restrict the proofs to the main theorems as stated, and will elaborate on the extended results in an extended version of this manuscript. First, we assume that the replication parameter is the same for all nodes. In practice, replication parameters for different nodes are different. This can be captured by a vector {r_v}_v∈ Int(). Our main results can be extended to this case if for all v, r_v≤∑_u∈(v)r_u (a requirement that usually holds in practice). Second, we assume that there is no weight sharing that is standard in convolutional networks. Our results can be extended to convolutional networks with weight sharing.We also note that we assume that in each step of algorithm <ref>, a fresh batch of examples is given. In practice this is often not the case. Rather, the algorithm is given a training set of examples, and at each step it samples from that set. In this case, our results provide guarantees on the training loss. If the training set is large enough, this also implies guarantees on the population loss via standard sample complexity results.§ PROOFS Notation Throughout, we fix a loss ℓ:^k×→ [0,∞), a skeleton , a replication parameter r, the network =(,r,k) and a bias parameter 0≤β≤ 1. For a matrix W∈ M_r,l() we denote W_p,q = max__p≤ 1W_q, W_2=W_2,2, and W_F=√(∑_i=1^r∑_j=1^l W_ij^2). We will often use the fact that W_2≤W_F. For σ:→ and ∈^n we abuse notation and denote σ()=(σ(x_1),…,σ(x_n)). For a skeletonwe denote by () the set of 's internal nodes. We will aggregate the weights ofby a collection of matrices and bias vectors={W^v}_v∈()∪{^v}_v∈()∪{W^} .Here, (W^v,^v) are the matrix and vector that maps the output of all the neurons corresponding to nodes in (v), to the neurons corresponding to v.Likewise, W^ is the matrix that maps the output of the neurons corresponding to() to the final output of the network. We decompose W^v further as a concatenation of two matrices W^v,, W^v, that correspond to the internal and input nodes in (v) respectively.For a prediction matrix W^*∈ M_k,r and weightswe denote by |W^* the weights obtained by replacing W^ with W^*. We let_2 = max{W^v,_2, W^v,_2/√(r) , ^v_2/√(r) : v∈() }∪{W^_2} , '_2 = max{W^v,_2, W^v,_2/√(r) , ^v_2/√(r) : v∈() } ,and_F=√(W^_F^2+∑_v∈()W^v^2_F + ^v^2_F) .Finally, we let _R = { : ≤ R} and '_R = { : '≤ R}. For ∈ we denote by h_()=R^_() the output onof the networkwith the weights . Given v∈ V() we let R^v_()∈^r to be the output of the neurons corresponding to v. We denote by R_():=R^()_() the output of the representation layer.We also let R^v←_() be the concatenation of {R^u_()}_u∈(v). Note that R^v_() = σ_v(W^vR^v←_()) and R^_() = W^ R^()_(). For (,y)∈× we denote ℓ_(,y)()=ℓ(R^_(W),y) and for S={(_i,y_i)}_i=1^m we denote _S()=1/m∑_i=1^mℓ_(,y)(). We let _()=_(,y)∼ℓ_(,y)(). Finally, we let k_(,') = R_(),R_'()/r. §.§ OverviewWe next review the proof of theorem <ref>. The proof of theorem <ref> is similar. Later, we will also comment how the proof can be modified to establish theorem <ref>. Let ^*∈^k_ be some function with ^*≤ M and let _0,…,_T be the weights produced by the SGD algorithm. Our goal is to show that w.h.p. over the choice of _0, there is t∈ [T] such that _(_t)≤ϵ.In section <ref> we show that w.h.p. over the choice of _0, there is a prediction matrix W∈ M_k,r so that _(_0|W^*)≤ϵ and W^*_F≤M/√(r). This follows from the results of <cit.>, and some extensions of those. Namely, we extend the original from k=1 to general k, and also eliminate a certain logarithmic dependence on the size of the support of .Given that such W^* exists, standard online learning results (e.g. Chapter 21 in <cit.>) imply that if we would apply SGD only on the last layer, with the learning rate specified in theorem <ref>, i.e.η=η'/r for η'≲ϵ/('())^2, we would be guaranteed to have some step t∈ [T] in which _(_t)≤ 2ϵ.However, as we consider SGD on all weights, this is not enough. Hence, in section <ref>, we show that with the above mentioned learning rate, the weights of the non-last layer change slowly enough, so that _(_t|W^*)≤ϵ for all t. Given this, we can invoke the online-learning based argument again.In order to show that the last layer changes slowly, we need to bound the magnitude of the gradient of the training objective. In section <ref> we establish such a bound on the gradient of the loss for every example. As _() and _S_t() are averages of such functions, the same bound holds for them as well. We note that our bound depends on the spectral norm on the matrices {W^v}_v∈. We show that for random matrices, w.h.p.the magnitude of the norm implies a bound that is good enough for our purposes. Likewise, trough the training process, the norm doesn't grow too much, so the desired bound is valid throughout the optimization process.The structure of the proof of theorem <ref> is similar, but has a few differences. First, the first step would be to show that in the case thatis M-separable w.r.t. κ_, then w.h.p. over the choice of _0, there is a prediction matrix W^*∈ M_k,n such that ^1_(_0|W^*) is tiny, and W^*_F≲M/√(r). Again, this is based on the results and techniques of <cit.>, and is done in section <ref>. Given this, again, running SGD on the top layer would be fine. However, now we cannot utilize the online-learning based argument we used before, because the starting point is not 0, but rather a random vector, whose norm is too large to carry out the analysis. In light of that, we take a somewhat different approach. We show that the weights beneath the last layer are changing slow enough, so that the following holds throughout the optimization process: As long as the 0-1 error is larger than ϵ, the magnitude of the gradient is Ω(ϵ√(r)/M). More precisely, the derivative in the direction of W^*, is smaller than -Ω(ϵ√(r)/M). Given this, and bounds on both the first and second derivative of the loss (proved in section <ref>), we are able to establish the proof by adopting a standard argument from smooth convex optimization (done in section <ref>).§.§ Boundness of the objective functionLet Ω⊂^n be an open set. For a function :Ω→^m, a unit vector ∈^n and _0∈Ω we denote __0,(t)=(_0 + t). We say thatis (α,β,γ)-bounded at _0 if f is twice differentiable and∀∈^n-1,__0,(0)≤α,'__0,(0)≤β ,”__0,(0)≤γ .We say thatis (α,β,γ)-bounded if it is (α,β,γ)-bounded in any _0∈Ω. We note that for m=1, f is (α,β,γ)-bounded at _0 if and only if |f(_0)|≤α, ∇ f(_0)≤β and ∇^2f(_0)_2≤γ. In particular, when n=1 too, f is (α,β,γ)-bounded at _0 if and only if |f(_0)|≤α, |f'(_0)|≤β and |f”(_0)|≤γ. We will say thatis C-bounded if it is (C,C,C)-bounded. Let σ:→ be (α,C,C)-bounded function. Suppose that :Ω→^n is (∞,β,β^2)-bounded. We have that = σ∘ is (∞,Cβ,2Cβ^2)-bounded. If we furthermore assume that σ(0)=0 then we have thatis (Cα,Cβ,2Cβ^2)-bounded.The first part follows from the facts that__0,'(t) = σ'(_,(t)) _,'(t)and__0,”(t) = σ”(_,(t)) (_,'(t))^2 +σ'(_,(t)) _,”(t)The second part follows from the fact that in the case that σ(0)=0 we have that σ()≤ Cx. Let l:^n→ be (∞,C,C)-bounded function. Suppose that :Ω→^n is (∞,β,β^2)-bounded. We have that g= l∘ is (∞,Cβ,2Cβ^2)-boundedThis follows from the fact thatg__0,'(t) = ∇ l(_,(t)), _,'(t)andg__0,”(t) = _,'(t), H_l(_,(t)) _,'(t) + ∇ l(_,(t)), _,”(t)[logistic loss] Recall that :^k→Δ^k-1 is given by p_i(ŷ) = e^ŷ_i/∑_j=1^ke^ŷ_j and ℓ:^k×→_+ by ℓ(ŷ,y)= -log(p_y(ŷ)). Denote ℓ_y(ŷ)=ℓ(ŷ,y). We have∂ p_i/∂ŷ_j = δ_ije^ŷ_i(∑_j=1^ke^ŷ_j) -e^ŷ_ie^ŷ_j/(∑_j=1^ke^ŷ_j)^2Hence, ∇ p_i = p_i_i - p_i and therefore ∇ℓ_i = -∇ p_i/p_i = -_i +. Hence, ∇^2 ℓ_i = ∇ = () - ⊗. In particular, ℓ_y is (∞,√(2),1)-bounded.Let B be the set of l× m matrices with operator norm less than R and let :B×Ω→^m. Define :B×Ω→^l by (W,) = W(,W). Then, ifis (α,β,γ)-bounded thenis (Rα,Rβ + α,Rγ + 2β)-boundedFix (W_0,_0)∈ B×Ω and (U,)∈ M_l,m()×^n such that U^2_F + u^2_2 =1. We have_(W_0,_0),(U,)(t) = W_0_(W_0,_0),(U,)(t) + tU_(W_0,_0),(U,)(t)Hence, '_(W_0,_0),(U,)(t) = W_0'_(W_0,_0),(U,)(t) + U_(W_0,_0),(U,)(t) + tU'_(W_0,_0),(U,)(t) ”_(W_0,_0),(U,)(t) = W_0”_(W_0,_0),(U,)(t) + 2U'_(W_0,_0),(U,)(t) + tU”_(W_0,_0),(U,)(t) If _1,…,_d:Ω→^m are (α,β,γ)-bounded then (_1,…,_d):Ω→^mk is (√(d)α,√(d)β,√(d)γ)-bounded From facts <ref>, <ref> and <ref> we conclude that Suppose that * _1,1,…,_1,d_1:Ω→^r are (α,β,β^2)-bounded functions with √(r)≤α≤β* _2,1,…,_2,d_2:Ω→^d are (1,0,0)-bounded functions* σ:→ is C-boundedfor C≥1.* Let B_1 be the set of r× rd_1 matrices with operator norm less than R for R≥ 1, B_2 be the set of r× dd_2 matrices with operator norm less than r√(r) for R≥ 1, and B_3⊂^r the set of vectors with norm less than R√(r). Define :B_1× B_2× B_3×Ω→^r by (W^1,W^2,,) = σ(W^1_1() + W^2_2() + ) where _i()= (_i,1(),…,_i,d_i())Then,is(C√(r),4CR√(d̃)β ,(4CR√(d̃)β)^2)-bounded for d̃ = d_1+d_2+1. By fact <ref>, _1 is (√(d_1)α,√(d_1)β,dβ^2)-boundedHence, by fact <ref>, (W^1,)↦ W^1_1() is (R√(d_1)α,R√(d_1)β + √(d_1)α ,Rd_1β^2 + 2√(d_1)β)-bounded. Since β≥max(1,α) and R≥ 1 we have that (W^1,)↦ W^1_1() is (R√(d_1)α,2R√(d_1)β ,(2R√(d_1)β)^2)-bounded. Similarly, (W^2,)↦ W^2_2() is (R√(d_2r),√(d_2),0)-bounded and (,)↦ is (R√(r),1,0)-bounded. As √(r)≤α, and β,R≥ 1 it follows that (W^1,W^2,,)↦ W^1_1() + W^2_2() + is (R√(d̃)α,2R√(d̃)β ,(2R√(d̃)β)^2)-bounded.Now, by fact<ref> and the C-boundness of σ,is (C√(r),2CR√(d̃)β ,2C(2R√(d̃)β)^2)-bounded. The lemma concludes as C≥ 1 Suppose that * _1,1,…,_1,d_1:Ω→^r are (β,β,∞)-bounded functions with √(r)≤β* _2,1,…,_2,d_2:Ω→^d are (1,0,0)-bounded functions* σ:→ is C-Lipschitz and satisfy |σ(0)|≤ C.* Let B_1 be the set of r× rd_1 matrices with operator norm less than R for R≥ 1, B_2 be the set of r× dd_2 matrices with operator norm less than r√(r) for R≥ 1, and B_3⊂^r the set of vectors with norm less than R√(r). Define :B_1× B_2× B_3×Ω→^r by (W^1,W^2,,) = σ(W^1_1() + W^2_2() + ) where _i()= (_i,1(),…,_i,d_i())Then,is(C(R+1)√(d̃)β,C(R+1)√(d̃)β ,∞)-bounded for d̃ = d_1+d_2+1. By fact <ref>, _1 is (√(d_1)β,√(d_1)β,∞)-boundedHence, by fact <ref>, (W^1,)↦ W^1_1() is (R√(d_1)β,R√(d_1)β + √(d_1)β ,∞)-bounded. Similarly, (W^2,)↦ W^2_2() is (R√(d_2r),√(d_2),0)-bounded and (,)↦ is (R√(r),1,0)-bounded. As √(r)≤β and β,R≥ 1 it follows that (W^1,W^2,,)↦ W^1_1() + W^2_2() + is ((R+1)√(d̃)β,(R+1)d̃β ,∞)-bounded. The lemma concludes by fact<ref> and the C-boundness of σ. Using a similar argument one can prove thatSuppose that * :Ω→^r is (α,β,β^2)-bounded function with max(1,α)≤β* l:^k→ is (∞,C,C)-bounded for C≥ 1.* Let B be the set of k× r matrices with operator norm less than R for R≥ 1. Define g:B×Ω→ by g(W,) = l(W())Then, g is(∞,4CRβ ,(4CRβ)^2)-boundedAssume that all activations inare C-bounded and that each ℓ_y is (∞,C',C')-bounded. Let R≥ 1 and (,y)∈×. Then * The function ℓ_(,y):_R → is (∞,β,β^2)-bounded for β=4C'R(4CR)^()√(()r).* The function R_:_R→^r is (C√(r),β,β^2)-bounded for β=(4CR)^()√(()r).Denote ^i() = ∏_j=1^imax_v∈,(v)=j ((v)+1) and β_i = (4CR)^i√(r·^i()). We will prove that for every v∈, the function R^v_() is (C√(r),β_(v),β_(v)^2)-bounded. This proves the second item. The first item follows from the second together with lemma <ref>.We will use induction on (v). For depth 0 node (i.e., an input node), the function R^v_ is a constant function with output of norm 1. Hence, it is (1,0,0)-bounded.For v of depth >0, the induction hypothesis and lemma <ref> implies that R^v_ is(C√(r),4CR√((v)+1)β_i-1,(4CR√((v)+1)β_i-1)^2)-boundedThe proof concludes as β_i≤ 4CR√((v)+1)β_i-1 Based on lemma <ref> and a similar argument we have that:Assume that all activations inare C-Lipschitz and satisfy |σ(0)|≤ C, and that ℓ is L-Lipschitz. Let R≥ 1 and (,y)∈×.Let β = (C(R+1))^()√(()r). Then, * The function ℓ_(,y):_R→ is(∞,LRβ,∞)-bounded when restricted to the variables ^ = {W^v}_v∈* The function ℓ_(,y):_R→ is(∞,Lβ,∞)-bounded when restricted to the variables W^* The function R_ is (β,β,∞)-boundedIf we furthermore assume that all activations satisfy σ_∞≤ C then (i) in the last item the conclusion is that R_:_R→ is (C√(r),β,∞)-bounded, and (ii) ℓ_(,y) is(∞,LC√(r),∞)-bounded when restricted to the variables W^§.§ Optimization given good initializationAssume that all activations inare C-Lipschitz and satisfy |σ(0)|≤ C, and that ℓ is L-Lipschitz. Defineα = 2L(3C)^()√(()). Let _0 be initial weights with _0∈_1.5. Suppose that _1,…,_t are the weights obtained by running algorithm <ref> with learning rate η' = η/r>0. Then, for every t≤√(r)/2ηα we have *_t∈_2*For any and every , R_(_0) - R_(_t)≤ tηα^2 Let _t the stochastic gradient at time t. To see item <ref> note that as long as _t ∈_2, by lemma <ref>,for all v∈ V()∪{}, V^v_t_F≤α√(r). Since V^v_t_2≤V^v_t_F, and since the learning rate is η/r, at each step, the spectral norm of each W^v is changed by at most ηα√(1/r). Hence, as long at t≤√(r)/2ηα, we have thatW_t∈_2. For item <ref>, again since^_t_F≤α√(r), we have that the euclidian length of the trajectory of the internal weights until step t is at most η tα/√(r). Now, by lemma <ref>, R_() is (α√(r))-Lipschitz. Hence, R_(_0) - R_(_t)≤ tηα^2 for all . §.§.§ Starting from zero prediction layerWe will use the following fact from online convex optimization.Let f_1,…,f_T:^n→ be L-Lipschitz convex functions. Let _0 = 0 and _t+1 = _t - η∇ f_t(_t). Here, ∇ f_t(_t) is some sub-gradient of f_t at _t. Then, for any ^*∈^n we have,∑_t=1^T f_t(_t) ≤∑_t=1^T f_t(^*) + ^*^2/2η + η TL^2/2Assume that all activations inare C-Lipschitz and satisfy |σ(0)|≤ C, and that ℓ is L-Lipschitz. Defineα = 2L(3C)^()√(()). Fix ϵ >0, a prediction matrix W^*∈ M_k,r with W^*_F≤M/√(r), and _0∈_1.5 with _0^=0. Let _1,…,_T be the weights produced by algorithm <ref> with step size η' = η/r for η≤8ϵ/α^2, with T≥M^2/2ηϵ and with arbitrary batch size m. Furthermore, assume that √(r)≥MLTηα^2/ϵ + 2TηαThen, there is some t∈ [T] such that _(_t)≤_(_0|W^*) +3ϵ. By Lemma <ref> we have that for all t, _(_t|W^*) = _(,y)∼ℓ (W^*R_(_t),y)≤_(_0|W^*) + MLtηα^2/√(r)Since √(r)≥MLTηα^2/ϵ we have that_(_t|W^*)≤_(_0|W^*)+ϵthroughout the optimization process. Likewise, _t∈_2 for all t∈ [T], and therefore R_(_t)≤α√(r)/2L. Now, consider the convex functions f_t:M_k,r→ defined by f_t(W)= L_S_t(_0|W). Since R_(_t)≤α√(r)/2L and ℓ is L-Lipschitz, f_t is (α√(r)/2)-Lipschitz. Hence, applying theorem <ref> we conclude that∑_t=1^T _S_t(_t)≤ ∑_t=1^T _S_t(_t | W^*)+ M^2/r/2η' + Tη'α^2r/8= ∑_t=1^T _S_t(_t | W^*)+ M^2/2η + Tηα^2/8Now, since T≥M^2/2ηϵ and η≤ϵ 8 /α^2 we have that1/T∑_t=1^T _S_t(_t) ≤1/T∑_t=1^T _S_t(_t | W^*)+2ϵTaking expectation (w.r.t. the mini-batches) and using equation (<ref>) we get 1/T∑_t=1^T _(_t) ≤_(_0|W^*)+3ϵIn particular, there is some t∈ [T] for which _(_t)≤_(_0|W^*)+3ϵ§.§.§ Starting from random prediction layerIn this section we assume that = [k], that ℓ is the logistic-loss, and that all acitivations are C-bounded. Denote α=8√(2)(8C)^()√(()) Letbe a distribution on ^r× [k] such that _(,j)∼(x> C√(r)) = 0. Let W^*∈ M_k× r be a matrix such that _^1(W)=_(,y)∼ℓ^1(W,y)≤δ. Then, for all W∈ M_k× r∇_(W)≥^0-1_(W)-δ/2W^*_F - δ√(2r)C Let E=W^*/W^*. Fix an example (,y) and denote ℓ_(,y)(t)=ℓ_y((W+tE)). We have that ∇_(W)≥∇_(W),-E = -_(,y)∼ℓ_(,y)'(0). By example <ref> we have-ℓ_(,y)'(0)=E,_y - (W) = ∑_i=1^k p_i(W)[(E)_y-(E)_i] = ∑_i y p_i(W)[(E)_y-(E)_i]Now, if ℓ^1(W^*,y)=0 then (E)_y-(E)_i≥1/W^*_F for all i y and hence ℓ_(,y)'(0)≥1/W^*_F∑_i y p_i(W) = 1-p_y(W)/W^*_F≥ 0. If furthermore ℓ^0-1(W,y)=1 then p_y(W)≤1/2. In this case, -ℓ_(,y)'(0)≥1/2W^*_F. On the other hand, we always have ℓ_(,y)'(0)≤E··_y-(W)≤ C√(2r). It follows that∇_(W) ≥ _(,y)∼(ℓ^0-1_y(W)=1 and ℓ^1_y(W^*)=0)1/2W^*_F - _(,y)∼(ℓ^1_y(W^*)=1)C√(2r)≥ ^0-1_(W)-δ/2W^*_F - δ√(2r)CFix ϵ >0, M>2 and suppose that _0 are weights such that * _0∈_1.5* There isW^*∈ M_k,r with W^*_F≤M/√(r) such that ^2_(_0|W^*)≤δ for δ = ϵ/√(2)4CM* _(_0)≤_0 Let _1,…,_T be the weights produced by algorithm <ref> with step size η/r for η≤ϵ^2/4M^2 α^4, with T≥8M^2_0/ηϵ^2 and with arbitrary batch size m. Furthermore, assume that√(r)≥ TMηα^2Then, there is some t∈ [T] such that ^0-1_(_t)≤ 2ϵ.Since T≤√(r)/2ηα and T≤√(r)/Mηα^2, by lemma <ref>, we have ^1_(_t)≤δ for each _t. Denote ϵ_t = ^0-1_(_t) and let _t be the stochastic gradient at time t. By claim <ref>, we have that∇_(_t)_F≥(ϵ_t-δ)√(r)/2M - δ√(2r)C ≥(ϵ_t-ϵ)√(r)/2MNow, since each ℓ_(,y) is (∞,α√(r), α^2r)-bounded (lemma <ref>), we have_(_t+1) ≤_(_t) - η/r∇_(_t),_t + α^2r/2η^2/r^2_t^2_FTaking expectation over the stochastic gradient and using the fact that by the boundness of ℓ_(,y), _t_F≤α√(r), we get_(_t+1) ≤_(_t) - η/r∇_(_t)^2_F + η^2α^4/2Applying equation (<ref>) we get_(_t+1) ≤_(_t) - η (ϵ_t-ϵ)_+^2/4M^2 + η^2α^4/2Now, if (ϵ_t-ϵ)_+^2≥ϵ^2, we have_(_t+1) ≤_(_t) - ηϵ^2/4M^2 + η^2α^4/2Since η≤ϵ^2/4M^2 α^4, we get_(_t+1) ≤_(_t) - ηϵ^2/8M^2On the other hand, we always have _(_t+1) ≥ 0. Hence, in the first _(_0)8M^2 /ηϵ^2 steps, there must be at least one step t in which (ϵ_t-ϵ)_+^2≤ϵ^2. The proof concludes asϵ_t ≤(ϵ_t-ϵ)_+ + ϵ≤√((ϵ_t-ϵ)^2_+) + ϵ≤ 2ϵ §.§ Initial conditions §.§.§ Finite support representation of kernel space functionsFor a Hilbert spacewe define by ℬ(,^k) the collection of bounded operators fromto ^k. Concretely, (,^k) is the collection of all functions W:→^k of the form W=(_1,,…,_k,) for _1,…,_k∈. ℬ(,^k) is a Hilbert space itself w.r.t. the Frobenius inner product W,W' = ∑_i=1^k_i,'_i. For ∈^k and ∈ we denote by ⊗∈(,^k) the operator (⊗)(') = ,'. Note that ⊗_F = · Fix a normalized kernel κ:×→. For ∈^k and f:→ we define f:→^k by f() = f(). For ∈ we denote κ^(')=κ(,'). In this section we will show that functions in _κ^k can be replaced by functions of the form ∑_i=1^m _i κ^_i without loosing too much in terms of classification accuracy.Let B be the unit ball in a Hilbert space . Let {(_t,y_t)}_t=1^∞⊂ B× [k] be a sequence of examples such that there is W^*∈(,^k) with W^*_F≤ M and ∀ t, ℓ^1(W^*_t,y_t) = 0. Consider the following version of the perceptron algorithm.Start with W_t=0 and for every t, if (W_t_t)_y_t<a+max_y' y_t(W_t_t)_y' update W_t+1 = W_t+(_y_t-_ŷ_t)⊗_t. Here, ŷ_t=_y(W_t_t)_y. Then, the algorithm makes at most (2+2a)M^2 mistakes. Likewise, the Frobenius norm of the final matrix is at most (2+2a)MDenote U_t = (_y_t-_ŷ_t)⊗_t. We have that whenever there is an update, W_t,U_t≤ a. In this caseW_t+1_F^2 = W_t+U_t_F^2 = W_t_F^2 + U_t^2_F + 2W_t,U_t≤W_t_F^2 + 2 + 2aHence, after T updates, the norm of W_t is at most √((2+2a)T). On the other hand W^*,U_t≥ 1. Hence, after T updates, the projection of W_t on the direction W^*/W^* is at least T/W^*≥T/M. It follows that T/M≤√((2+2a)T) which implies that T≤ (2+2a)M^2. Likewise, at this point W_t≤√((2+2a)T)≤ (2+2a)M.Suppose thatis M-separable w.r.t. κ. Then,is (4M)-separable by a function of the form ^* = ∑_i=1^m _i κ^_i with _i≤√(2) for all i and m≤ 4M^2(sketch) Letbe a Hilbert space and let Ψ:→ be a mapping such that κ(,')=Ψ(),Ψ('). Let {(_t,y_t)}_t=1^∞ be a sequence of i.i.d. samples from . Suppose that we ran the algorithm from lemma <ref> with a=1 on the sequence {(Ψ(_t),y_t)}_t=1^∞. With probability 1, the number of updates the algorithm will make will be m≤ 4M^2. Hence, upon termination, we will have W∈(,^k) such that ^1_(W∘Ψ)=0 and W=∑_i=1^m _i⊗Ψ(_i) where W_F≤ 4M and each _i is a difference of two _j's and therefore has a norm of √(2).The proof concludes as W∘Ψ = ∑_i=1^m _i κ^_iLet B be the unit ball in a Hilbert space ,be a distribution on B×, ℓ:^k×→ [0,∞) a loss function that is convex and L-Lipschitz, W^*∈(,^k) and ϵ >0. There are _1,…,_m∈^k and _1,…,_m∈ B such that for W:=∑_i=1^m_i⊗_i we have, * _(W) ≤_(W^*) + ϵ * For every i, _i≤ϵ/L * W_F ≤W^*_F * m≤M^2L^2/ϵ^2 Denote M=W^*. Suppose that we run stochastic gradient decent onw.r.t. the loss ℓ, with learning rate η = ϵ/L^2, and with projections onto the ball of radius M. Namely, we start with W_0=0 and at each iteration t≥ 1, we sample (_t,y_t)∼ and perform the update,W̃_t = W_t-1 - η∇ℓ_y_t(W_t-1_t) ⊗_t W_t = W̃_t W̃_t_F≤ MM W̃_t/W̃_t_F W̃_t_F > MWhere ∇ℓ_y_t(W_t-1_t) is a sub-gradient ofℓ_y_t at W_t-1_t. After T=M^2L^2/ϵ^2 iterations the loss in expectation would be at most ϵ (see for instance Chapter 14 in <cit.>). In particular, there exists a sequence of at most M^2L^2/ϵ^2 gradient steps that attains a solution W with _(W)≤_(W) + ϵ.The proof concludes as each update adds a matrix of the form ⊗ with ≤ L and possibly multiply the current matrix by a scalar of absolute value ≤ 1. Similarly to corollary <ref> we haveLetbe a distribution on ×, ℓ:^k×→ [0,∞) a loss function that is convex and L-Lipschitz, ^*∈^k_κ and ϵ >0. There are _1,…,_m∈^k and _1,…,_m∈ such that for :=∑_i=1^m_iκ^_i we have, * _() ≤_(^*) + ϵ * For every i, _i≤ϵ/L * _κ≤^*_κ * m≤M^2L^2/ϵ^2§.§.§ Initial conditionsLet W ∈ M_r,m be a matrix with i.i.d. entries drawn from (0,σ^2). Then, w.p. at least 1-2exp(-α^2r/2), W_2≤ (1+α)σ(√(r)+√(m))Suppose that r≳log(|S|/δ) + d. Then,* W.p. ≥ 1-δ we have that _0∈'_1.5.* If also r≥ k then w.p. ≥ 1-δ we have that _0∈_1.5.Letbe a skeleton with C-bounded activations. Letbe a random initialization r ≥(4C^4)^()+1log(8||/δ)/ϵ^2 .Then, for all ,', with probability of at least 1-δ,|k_(,')-k_(,')|≤ϵ . Letbe a skeleton with ReLU activations. Letbe a random initialization withr ≳^2()log(||/δ)/ϵ^2 .Then, for all ,' and ϵ≲1/(), with probability of at least 1-δ,|k_(,')-k_(,')|≤ϵ . Letbe a skeleton with C-bounded activations. Let ^*∈^k_ with ^*_≤ M. Supposer ≳L^4 M^4 (4C^4)^()+1log(LMC ||/ϵδ)/ϵ^4 .Then, with probability of at least 1-δ over the choices of _0 there is a prediction matrix W^* such that W^*_F≤2M/√(r) and _(_0|W^*)≤_(^*) + ϵLetbe a skeleton with ReLU activations and ϵ≲1/(). Let ^*∈^k_ with ^*_≤ M. Supposer ≳L^4M^4 ^3() log(LM ||/ϵδ)/ϵ^4+d. Then, with probability of at least 1-δ over the choices of _0 there is a prediction matrix W^* such that W^*_F≤2M/√(r) and _(_0|W^*)≤_(^*) + ϵTheorems <ref> and <ref> are similar, but not identical, to theorems 4 and 5 from <cit.>. We next prove theorem <ref>. The proof of theorem <ref> is very similar. (sketch) By corollary <ref> we can restrict to the case that ^* = ∑_i=1^m _iκ_^_i such that ∀ i, _i≤ϵ/L and m≤M^2L^2/ϵ^2. Define W^* = ∑_i=1^M _i⊗R__i(_0)/r and() = W^*R_(_0) = ∑_i=1^m _iκ__0^_i()Denote α = (√(18))^√(()). We will show that the conclusion of the theorem holds given the following three conditions, that by theorem <ref> and corollary <ref> happens w.p. ≥ 1-δ over the choice of _0. * For (1-ϵ/(1+√(2)α)ML)-fraction (according to ) of the examples, ∀ i, |κ__0(_i,)-κ_(_i,)|≤ϵ^2/M^2L^2 * For all i,j, |κ__0(_i,_j)-κ_(_i,_j)|≤ϵ^2/M^2L^2 * _0∈'_1.5First, we haveW^*^2_F = 1/r∑_i,j=1^m _i,_jκ__0(_i,_j)Since |κ__0(_i,_j)-κ_(_i,_j)|≤ϵ^2/M^2L^2 for all i,j and also |_i,_j|≤ϵ^2/L^2 we haverW^*^2_F≤ [∑_i,j=1^m _i,_jκ_(_i,_j)]+ m^2ϵ^4/M^2L^4= ^*_^2 + m^2ϵ^4/M^2L^4≤ 2M^2Now, for the examples satisfying ∀ i, |κ__0(_i,)-κ_(_i,)|≤ϵ^2/M^2L^2 we have()-^*()= ∑_i=1^m _i(κ__0(_i,)-κ_(_i,)) ≤∑_i=1^m_iϵ^2/M^2L^2≤ mϵ/Lϵ^2/M^2L^2≤ϵ/LSince the loss is L-Lipschitz, it grows by at most ϵ on these examples, when we move from ^* to . As for the remaining examples, since _0∈'_1.5 and W^*_F≤√(2)M/√(r), lemma <ref> implies that ()-^*()≤() + ^*()≤√(2)α M + MHence, when we move from ^* to , the loss grows by at most (1+√(2)α)ML. As the remaining examples occupies at most ϵ/(1+√(2)α)ML-fraction of the examples, these examples contributes at most ϵ to the loss when moving from ^* to .Letbe a skeleton with C-bounded activations. Suppose thatis M-separable w.r.t. κ_, r≳ (4C^4)^()+1M^4log(||M/δ). Then, w.p. ≥ 1-δ over the choice of _0, there is W^*∈ M_k,r with W^*_F≤10M/√(r) such that ^1_(_0|W^*)≤δ(sketch) By corollary <ref>,is separable by a function ^* = ∑_i=1^m _iκ_^_i such that ∀ i, _i≤√(2), m≤ 4M^2 and ^*≤ 4M. Define W = ∑_i=1^M _i⊗R__i(_0)/r and() = WR_(_0) = ∑_i=1^m _iκ__0^_i()Now, for r≳ (4C^4)^()+1M^4log(||M/δ) we have by theorem <ref> that w.p. ≥ 1-δ/2, |κ__0(_i,_j)-κ_(_i,_j)|≤1/32M^2 for all i,j and also |_i,_j|≤ 2. Hence,rW^2_F≤ [∑_i,j=1^m _i,_jκ_(_i,_j)]+ 2m^21/32M^2= ^*_^2 + 2m^21/32M^2≤^*_^2 + 32M^41/32M^2≤ 17M^2 ≤ 25 M^2Now, w.p. ≥ 1-δ/2, (1-δ)-fraction of the examples satisfies |κ__0(_i,)-κ_(_i,)|≤1/16√(2)M^2 for all i. For those examples we have()-^*()= ∑_i=1^m _i(κ__0(_i,)-κ_(_i,)) ≤∑_i=1^m√(2)1/16√(2)M^2≤4M^2√(2)/16√(2)M^2 = 1/4Those examples satisfies ℓ^1/2((),y)=0, and hence ^1/2_(h)≤δ. The proof concludes by taking W^*=2WLet X_1,…,X_k be independent standard Gaussians. Then max_iX_i≤√(2log(k))Denote Z = max_iX_i and let t=√(2log(k)). By Jensen's inequality and the moment generating function of the normal distribution we have thate^t Z≤ e^tZ = max_i e^tX_i≤ k e^tX_1 = ke^t^2/2Hence, Z≤log(k)/t + t/2 = √(2log(k))If all activations are C bounded then __0_(_0) ≤ (1+C√(2))log(k)Fix =(,y). We have that the distribution of ℓ_(_0)= ℓ(W^_0R_(_0),y) is the distribution of ℓ((X_1,…,X_k),1) where X_1,…,X_k are independent Gaussians of mean zero and variance σ^2 = R_(_0)^2/r≤ C^2. Hence we have that__0l_(_0)= _X-log(e^X_1/∑_i=1^ke^X_i) =- X_1 + log(∑_i=1^ke^X_i) = log(∑_i=1^ke^X_i) ≤ log(k e^max_iX_i) ≤ log(k)+max_iX_iand the proof concludes as by lemma <ref> max_iX_i≤σ√(2log(k))Derivation of the main theorems. Theorem <ref> follows from theorem <ref> and lemma <ref>. Likewise, theorem <ref> follows from theorem <ref> and lemma <ref>. Finally, theorem <ref> follows from corollary <ref>, lemma <ref> and lemma <ref>. Acknowledgements: The author thanks Roy Frostig, Yoram Singer and Kunal Talwar for valuable discussions and comments. plainnat

http://arxiv.org/abs/1702.08503v2

Rationalization: A Neural Machine Translation Approach to Generating Natural Language ExplanationsUpol Ehsan Harrison and Ehsan equally contributed to this work^†,Brent Harrison [1]^,Larry Chan^†,and Mark Riedl^†^†Georgia Institute of Technology, Atlanta, GA, USA^University of Kentucky, Lexington, KY, USA ,,,First version: January 17, 2017This version: February 29, 2020 ============================================================================================================================================================================================================================================ We introduce AI rationalization, an approach for generating explanations of autonomous system behavior as if a human had performed the behavior. We describe a rationalization technique that uses neural machine translation to translate internal state-action representations of an autonomous agent into natural language. We evaluate our technique in the Frogger game environment, training an autonomous game playing agent to rationalize its action choices using natural language.A natural language training corpus is collected from human players thinking out loud as they play the game. We motivate the use of rationalization as an approach to explanation generation and show the results of two experiments evaluating the effectiveness of rationalization.Results of these evaluations show that neural machine translation is able to accurately generate rationalizations that describe agent behavior, and that rationalizations are more satisfying to humans than other alternative methods of explanation.§ INTRODUCTION Autonomous systems must make complex sequential decisions in the face of uncertainty. Explainable AI refers to artificial intelligence and machine learning techniques that can provide human understandable justification for their behavior. With the proliferation of AI in everyday use, explainability is important in situations where human operators work alongside autonomous and semi-autonomous systems because it can help build rapport, confidence, and understanding between the agent and its operator.For instance, a non-expert human collaborating with a robot for a search and rescue mission requires confidence in the robot's action.In the event of failure—or if the agent performs unexpected behaviors—it is natural for the human operator to want to know why. Explanations help the human operator understand why an agent failed to achieve a goal or the circumstances whereby the behavior of the agent deviated from the expectations of the human operator. They may then take appropriate remedial action: trying again, providing more training to machine learning algorithms controlling the agent, reporting bugs to the manufacturer, etc. Explanation differs from interpretability, which is a feature of an algorithm or representation that affords inspection for the purposes of understanding behavior or results. While there has been work done recently on the interpretability of neural networks <cit.>, these studies mainly focus on interpretability for experts on non-sequential problems.Explanation, on the other hand, focuses on sequential problems, is grounded in natural language communication, and is theorized to be more useful for non-AI-experts who need to operate autonomous or semi-autonomous systems.In this paper we introduce a new approach to explainable AI: AI rationalization. AI rationalization is a process of producing an explanation for agent behavior as if a human had performed the behavior. AI rationalization is based on the observation that there are times when humans may not have full conscious access to reasons for their behavior and consequently may not give explanations that literally reveal how a decision was made.In these situations, it is more likely that humans create plausible explanations on the spot when pressed. However, we accept human-generated rationalizations as providing some lay insight into the mind of the other. AI rationalization has a number of potential benefits over other explainability techniques: (1) by communicating like humans, rationalizations are naturally accessible and intuitive to humans, especially non-experts (2) humanlike communication between autonomous systems and human operators may afford human factors advantages such as higher degrees of satisfaction, confidence, rapport, and willingness to use autonomous systems; (3) rationalization is fast, sacrificing absolute accuracy for real-time response, appropriate for real-time human-agent collaboration. Should deeper, more accurate explanations or interpretations be necessary, rationalizations may need to be supplemented by other explanation, interpretation, or visualization techniques. We propose a technique for AI rationalization that treats the generation of explanations as a problem of translation between ad-hoc representations of states and actions in an autonomous system's environment and natural language. To do this, we first collect a corpus of natural language utterances from people performing the learning task. We then use these utterances along with state information to train an encoder-decoder neural network to translate between state-action information and natural language. To evaluate this system, we explore how AI rationalization can be applied to an agent that plays the game Frogger. This environment is notable because conventional learning algorithms, such as reinforcement learning, do not learn to play Frogger like human players, and our target audience would not be expected to understand the specific information about how an agent learns to play this game or why it makes certain decisions during execution.We evaluate our approach by measuring how well it can generate rationalizations that accurately describe the current context of the Frogger environment.We also examine how humans view rationalizations by measuring how satisfying rationalizations are compared to other baseline explanation techniques. The contributions of our paper are as follows:* We introduce the concept of AI rationalization as an approach to explainable AI.* We describe a technique for generating rationalizations that treats explanation generation as a language translation problem from internal state to natural language.* We report on an experiment using semi-synthetic data to assess the accuracy of the translation technique.* We analyze how types of rationalization impact human satisfaction and use these findings to inform design considerations of current and future explainable agents.§ BACKGROUND AND RELATED WORK For a model to be interpretable it must be possible for humans to explain why it generates certain outputs or behaves in a certain way.Inherently, some machine learning techniques produce models that are more interpretable than others.For sequential decision making problems, there is often no clear guidance on what makes a good explanation. For an agent using Q-learning <cit.>, for example, explanations of decisions could range from “the action had the highest Q value given this state” to “I have explored numerous possible future state-action trajectories from this point and deemed this action to be the most likely to achieve the highest expected reward according to iterative application of the Bellman update equation.”An alternate approach to creating interpretable machine learning models involves creating separate models of explainability that are often built on top of black box techniques such as neural networks.These approaches, sometimes called model-agnostic <cit.> approaches, allow greater flexibility in model selection since they enable black-box models to become interpretable. Other approaches seek to learn a naturally interpretable model which describes predictions that were made <cit.> or by intelligently modifying model inputs so that resulting models can describe how outputs are affected <cit.>.Explainable AI has been explored in the context of ad-hoc techniques for transforming simulation logs to explanations <cit.>, intelligent tutoring systems <cit.>, transforming AI plans into natural language <cit.>, and translating multiagent communication policies into natural language <cit.>. Our work differs in that the generated rationalizations do not need to be truly representative of the algorithm's decision-making process. This is a novel way of applying explainable AI techniques to sequential decision-making in stochastic domains.§ AI RATIONALIZATION Rationalization is a form of explanation that attempts to justify or explain an action or behavior based on how a human would explain a similar behavior. Whereas explanation implies an accurate account of the underlying decision-making process, AI rationalization seeks to generate explanations that closely resemble those that a human would most likely give were he or she in full control of an agent or robot. We hypothesize that rationalizations will be more accessible to humans that lack the significant amount of background knowledge necessary to interpret explanations and that the use of rationalizations will result in a greater sense of trust or satisfaction on the part of the user. While Rationalizations generated by an autonomous or semi-autonomous system need not accurately reflect the true decision-making process underlying the agent system, they must still give some amount of insight into what the agent is doing. Our approach for translating representations of states and actions to natural language consists of two general steps. First, we must create a training corpus of natural language and state-action pairs. Second, we use this corpus to train an encoder-decoder network to translate the state-action information to natural language (workflow in Figure <ref>).§.§ Training CorpusOur technique requires a training corpus that consists of state-action pairs annotated with natural language explanations. To create this corpus, we ask people to complete the agent task's in a virtual environment and “think aloud” as they complete the task. We record the visited states and performed actions along with the natural language utterances of critical states and actions.This method of corpus creation ensures that the annotations gathered are associated with specific states and actions. In essence we create parallel corpora, one of which contains state representations and actions, the other containing natural language utterances. The precise representation of states and actions in the autonomous system does not matter as long as they can be converted to strings in a consistent fashion.Our approach emphasizes that it should not matter how the state representation is structured and the human operator should not need to know how to interpret it.§.§ Translation from Internal Representation to Natural Language We use encoder-decoder networks to translate between complex state and action information and natural language rationalizations. Encoder-decoder networks, which have primarily been used in machine translation and dialogue systems, are a generative architecture comprised of two component networks that learn how to translate an input sequence X = (x_1, ..., x_T) into an output sequence Y = (y_1, ..., y_T').The first component network, the encoder, is a recurrent neural network (RNN) that learns to encode the input vector X into a fixed length context vector v. This vector is then used as input into the second component network, the decoder, which is a RNN that learns how to iteratively decode this vector into the target output Y. We specifically use an encoder-decoder network with an added attention mechanism <cit.>. § EXPERIMENTSIn this work, we test the following two hypotheses: * Encoder-Decoder networks can accurately generate rationalizations that fit the current situational context of the learning environment and* Humans will find rationalizations more satisfying than other forms of explainabilityTo test these hypotheses, we perform two evaluations in an implementation of the popular arcade game, Frogger. We chose Frogger as an experimental domain because computer games have been demonstrated to be good stepping stones toward real-world stochastic environments <cit.> and because Frogger is fast-paced, has a reasonably rich state space, and yet can be learned optimally without too much trouble. §.§ Rationalization Generation Study MethodologyEvaluating natural language generation is challenging; utterances can be “correct” even if they do not exactly match known utterances from a testing dataset.To facilitate the assessment of rationalizations generated by our technique, we devised a technique whereby semi-synthetic natural language was paired against state-action representations internal to an autonomous system. The semi-synthetic language was produced by observing humans “thinking out loud” while performing a task and then creating grammar that reproduced and generalized the utterances (described below). This enables us to use the grammar to evaluate the accuracy of our system since we can compare the rationalizations produced by our system to the most likely rule that would have generated that utterance in the grammar. Similar approaches involving the use of semi-synthetic corpora have been adopted in scenarios, such as text understanding <cit.>, where ground truth is necessary to evaluate the system. We conducted the experiments by generating rationalizations for states and actions in a custom implementation of the game Frogger.In this environment, the agent must navigate from the bottom of the map to the top while avoiding obstacles in the environment. The actions available to the agent in this environment are movement actions in the four cardinal directions and action for standing still.We evaluate our rationalization technique against two baselines. The first baseline, the random baseline, randomly selects any sentence that can be generated by the testing grammar as a rationalization. The second baseline, the majority vote baseline, always selects sentences associated with the rule that is most commonly used to generate rationalizations on a given map.Below we will discuss the process for creating the grammar, our training/test sets, and the results of this evaluation in more detail.§.§.§ Grammar Creation In order to translate between state information and natural language, we first need ground truth rationalizations that can be associated explicitly with state and action information.To generate this information, we used crowdsourcing to gather a set of gameplay videos of 12 human participants from 3 continents playing Frogger while engaging in a think-aloud protocol, following the work by <cit.>. After players completed the game, they uploaded their gameplay video to an online speech transcription service and assigned their own utterances to specific actions. This layer of self-validation in the data collection process facilitates the robustness of the data. This process produced 225 action-rationalization trace pairs of gameplay. We then used these action-rationalization annotations to construct a grammar for generating synthetic sentences, grounded in natural language.This grammar uses a set of rules based on in-game behavior of the Frogger agent to generate rationalizations that resemble the crowdsourced data gathered previously. Since the grammar contains the rules that govern when certain rationalizations are generated, it allows us to compare automatically generated rationalizations against a ground-truth that one would not normally have if the entire training corpus was crowdsourced. §.§.§ Training and Test Set GenerationSince we use a grammar to produce ground truth rationalizations, one can interpret the role of the encoder-decoder network as learning to reproduce the grammar.In order to train the network to do this, we use the grammar to generate rationalizations for each state in the environment. The rules that the grammar uses to generate rationalizations are based on a combination of the world state and the action taken.Specifically, the grammar uses the following triple to determine which rationalizations to generate: (s_1, a, s_2).Here, s_1 is the initial state, a is the action performed in s_1, and s_2 is the resulting state of the world after action a is executed.States s_1 and s_2 consist of the (x,y) coordinates of the agent and the current layout of grid environment.We use the grammar to generate a rationalization for each possible (s_1, a, s_2) triple in the environment and then group these examples according to their associated grammar rules.For evaluation, we take 20% of the examples in each of these clusters and set them aside for testing.This ensures that the testing set contains a representative sample of the parent population while still containing example triples associated with each rule in the grammar. To aid in training we duplicate the remaining training examples until the training set contains 1000 examples per grammar rule and then inject noise into these training samples in order to help avoid overfitting. Recall that the input to the encoder-decoder network is a triple of the form (s_1, a, s_2) where s_1 and s_2 are states.To inject noise, we randomly select 30% of the rows in this map representation for both s_1 and s_2 and redact them by replacing them with a dummy value. To evaluate how our technique for rationalization performs under different environmental conditions, we developed three different maps. The first map was randomly generated by filling 25% of the bottom with car obstacles and filling 25% of the top with log platforms.The second map was 50% cars/logs and the third map was 75% cars/logs (see Figure <ref>). For the remainder of the paper, we refer to these maps as the 25% map, the 50% map, and the 75% map respectively.We also ensured that it was possible to complete each of these maps to act as a loose control on map quality.§.§.§ Training and Testing the Network The parallel corpus of state-action representations and natural language are used to train an encoder-decoder neural translation algorithm based on <cit.>.We use a 2-layered encoder-decoder network with attention using long short-term memory (LSTM) nodes with a hidden node size of 300.We train the network for 50 epochs and then use it to generate rationalizations for each triple in the testing set. To evaluate the accuracy of the encoder-decoder network, we need to have a way to associate the sentence generated by our model with a rule that exists in our grammar.The generative nature of encoder-decoder networks makes this difficult as its output may accurately describe the world state, but not completely align with the test example's output. To determine the rule most likely to be associated with the generated output, we use BLEU score <cit.> to calculate sentence similarity between the sentence generated by our predictive model with each sentence that can be generated by the grammar and record the sentence that achieves the highest score.We then identify which rule in the grammar could generate this sentence and use that to calculate accuracy.If this rule matches the rule that was used to produce the test sentence then we say that it was a match. Accuracy is defined as the percentage of the predictions that matched their associated test example. We discard any predicted sentence with a BLEU score below 0.7 when compared to the set of all generated sentences. This threshold is put in place to ensure that low quality rationalizations in terms of language syntax do not get erroneously matched to rules in the grammar. It is possible for a generated sentence to be associated with more than one rule in the grammar if, for example, multiple rules achieve the same, highest BLEU score.If the rule that generated the testing sentence matches at least one of the rules associated with the generated sentence, then we count this as a match. §.§.§ Rationalization Generation Results The results of our experiments validating our first hypothesis can be found in Table <ref>.As can be seen in the table, the encoder-decoder network was able to consistently outperform both the random baseline and majority baseline models. Comparing the maps to each other, the encoder-decoder network produced the highest accuracy when generating rationalizations for the 75% map, followed by the 25% map and the 50% map respectively.To evaluate the significance of the observed differences between these models, we ran a chi-squared test between the models produced by the encoder-decoder network and random predictor as well as between the encoder-decoder network models and the majority classifier.Each difference was deemed to be statistically significant (p < 0.05) across all three maps.§.§.§ Rationalization Generation DiscussionThe models produced by the encoder-decoder network significantly outperformed the baseline models in terms of accuracy percentage.This means that this network was able to better learn when it was appropriate to generate certain rationalizations when compared to the random and majority baseline models.Given the nature of our test set as well, this gives evidence to the claim that these models can generalize to unseen states as well.While it is not surprising that encoder-decoder networks were able to outperform these baselines, the margin of difference between these models is worth noting.The performances of both the random and majority classifiers are a testament to the complexity of this problem.These results give strong support to our claim that our technique for creating AI rationalizations using neural machine translation can accurately produce rationalizations that are appropriate to a given situation. §.§ Rationalization Satisfaction Study MethodologyThe results of our previous study indicate that our technique is effective at producing appropriate rationalizations.This evaluation is meant to validate our second hypothesis that humans would find rationalizations more satisfying than other types of explanation for sequential decision making problems.To do this, we asked people to rank and justify their relative satisfaction with explanations generated by three agents (described below) as each performs the same task in identical ways, only differing in the way they express themselves. The three agents are:* The rationalizing robot, uses our neural translation approach to generate explanations. * The action-declaring robot, states its action without any justification. For instance, it states I will move right. * The numerical robot, simply outputs utility values with no natural language rationalizations. We will discuss our human subjects protocol and experimental results below.§.§.§ ParticipantsFifty-three adults (age range = 22 – 64 years, M = 34.1, SD = 9.38) were recruited from Amazon Mechanical Turk (AMT) through a management service called TurkPrime <cit.>. Twenty-one percent of the participants were women, and only three countries were reported when the participants were asked what country they reside in. Of these, 91% of people reported that they live in the United States.§.§.§ ProcedureAfter reading a brief description of our study and consenting to participate, participants were introduced to a hypothetical high-stakes scenario. In this scenario, the participant must remain inside a protective dome and rely on autonomous agents to retrieve food packages necessary for survival.The environment is essentially a re-skinned version of Frogger (see figure <ref>) that is contextually appropriate for the high-stakes hypothetical scenario. To avoid effects of preconceived notions, we did not use the agents' descriptive names in the study; we introduced the agents as Robot A for the rationalizing robot, Robot B for the action-declaring robot, and Robot C for the numerical robot. Next, the participants watched a series of six videos in two groups of three: three depicting the agents succeeding and three showing them failing. Participants were quasi-randomly assigned to one of the 12 possible presentation orderings, such that each ordering was designed to have the same number of participants. After watching the videos, participants were asked to rank their satisfaction with the expressions given by each of the three agents and to justify their choices in their own words. §.§.§ Satisfaction Results and Analysis Figure <ref> shows that the rationalizing robot (Robot A) received the most 1st place ratings, the action-declaring robot (Robot B) received the most 2nd place ratings, and the numerical robot (Robot C) received the most 3rd place ratings. To determine whether any of these differences in satisfaction ratings were significant, we conducted a non-parametric Friedman test of differences among repeated measures.This yielded a Chi-square value of 45.481, which was significant (p<0.001).To determine which of the ratings differences were significant, we made pairwise comparisons between the agents, using the Wilcoxon-Nemenyi-McDonald-Thompson test <cit.>. All three comparisons yielded a significant difference in ratings.The satisfaction ratings for the rationalization robot were significantly higher than those for both the action-declaring robot (p=0.0059) as well as the numerical robot (p<0.001).Furthermore, the ratings for the action-declaring robot were significantly higher than those for the numeric robot (p<0.001).We also analyzed the justifications that participants provided for their rankings using approaches inspired by thematic analysis <cit.> and grounded theory <cit.>.Starting with an open coding scheme, we developed a set of codes that covered various reasonings behind the ranking of the robots.Using the codes as analytic lenses, we clustered them under emergent themes, which shed light into the dimensions of satisfaction. Through an iterative process performed until consensus was reached, we distilled the most relevant themes into insights that can be used to understand the whys behind satisfaction of explanations.In our discussion of these responses, we refer to participants using the following abbreviation: P1 is used to refer to participant 1, P2 is used to refer to participant 2, etc. §.§.§ Findings and Discussion As we hypothesized, the rationalizing agent's explanations were rated higher than were those of the other two agents, implying that rationalization enhances satisfaction over action-declaring, natural language description and over numeric expressions. In addition to the preference for a natural language substrate, four attributes emerged from our thematic analysis that characterize prototypical satisfactory rationalization: explanatory power, relatability, ludic nature, and adequate detail. These same attributes can be used to distinguish the rationalizing robot from the action-declaring robot.In terms of explanatory power, the rationalizing robot's ability to explain its actions was the most cited reasons for its superior placement in the satisfaction rankings.Human rationalizations allow us to form a theory of mind for the other <cit.>, enabling us to better understand motivations and actions of others.Similarly, the rationalizing robot's ability to show participants …what it's doing and why (P6) enabled them to …get into [the rationalizing robot's] mind (P17), boosting satisfaction and confidence.Despite using natural language, the action declaring robot yielded dissatisfaction.As P38 puts it, [The action-declaring robot] explained almost nothing…which was disappointing.The explanatory attribute of the rationalizing robot reduces friction of communication and results in improved satisfaction. With respect to relatability, the personality expressed through rationalizing robot's explanation allowed participants to relate to it:[The rationalizing robot] was relatable. He felt like a friend rather than a robot. I had a connection with [it] that would not be possible with the other 2 robots because of his built-in personality. (P21) Participants also engaged with the rationalizing robot's ludic quality, expressing their appreciation of its perceived playfulness: [The rationalizing robot] was fun and entertaining. I couldn't wait to see what he would say next! (P2). A rationalization yields higher satisfaction if it is adequately detailed.The action-declaring robot, despite its lack of explainability, received some positive comments.People who preferred the action-declaring robot over the rationalizing robot claimed that [the rationalizing robot] talks too much (P47), the action-declaring robot is nice and simple (P48), and that they would like to experience a combination of [the action-declaring robot] and [the rationalizing robot] (P41). Context permitting, there is a need to balance level of detail with information overload. These findings also align with our proposed benefits of AI Rationalization, especially in terms accessible explanations that are intuitive to the non-expert.We also observed how the human-centered communication style facilitates higher degrees of rapport. The insights not only help evaluate the quality of responses generated by our system, but also sheds light into design considerations that can be used to build the next generation of explainable agents. § FUTURE WORKOur next step is to build on our current work and investigate hypotheses about how types of rationalizations impact human preferences of AI agents in terms of confidence, perceived intelligence, tolerance to failure, etc.To address these questions, it will be necessary to conduct experiments similar to the one described above. It will be interesting to see how inaccurate rationalizations can be before feelings of confidence and rapport are significantly affected. Our experimental methodology can be adapted to inject increasingly more error into the rationalizations and understand human preferences.§ CONCLUSIONSAI rationalization provides a new lens through which we can explore the realms of Explainable AI.As society and AI integrates further, we envision the increase in human operators who will want to know why an agent does what it does in an intuitive and accessible manner. We have shown that creating rationalizations using neural machine translation techniques produces rationalizations with accuracies above baselines.We have also shown that rationalizations produced using this technique were more satisfying than other alternative means of explanation. Rationalization allows autonomous systems to be relatable and human-like in their decision-making when their internal processes can be non-intuitive.We believe that AI rationalization can be an important step towards the democratization of real-world commercial robotic systems in healthcare, accessibility, personal services, and military teamwork. aaai

http://arxiv.org/abs/1702.07826v2

[Email: ]tusdey@gmail.com Experimental Physics VI, Center for Electronic Correlations and Magnetism, University of Augsburg, 86159 Augsburg, Germany Experimental Physics VI, Center for Electronic Correlations and Magnetism, University of Augsburg, 86159 Augsburg, Germany Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institut, 5232 Villigen PSI, Switzerland Forschungsneutronenquelle Heinz Maier-Leibnitz (FRM II), Technische Universität München, 85747 Garching, Germany Experimental Physics V, Center for Electronic Correlations and Magnetism, University of Augsburg, 86159 Augsburg, Germany Experimental Physics VI, Center for Electronic Correlations and Magnetism, University of Augsburg, 86159 Augsburg, Germany Experimental Physics VI, Center for Electronic Correlations and Magnetism, University of Augsburg, 86159 Augsburg, Germany Laboratoire de Physique des Solides, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay Cedex, France Laboratoire de Physique des Solides, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay Cedex, France Experimental Physics V, Center for Electronic Correlations and Magnetism, University of Augsburg, 86159 Augsburg, Germany [Email: ]altsirlin@gmail.com Experimental Physics VI, Center for Electronic Correlations and Magnetism, University of Augsburg, 86159 Augsburg, Germany [Email: ]philipp.gegenwart@physik.uni-augsburg.de Experimental Physics VI, Center for Electronic Correlations and Magnetism, University of Augsburg, 86159 Augsburg, GermanyUsing thermodynamic measurements, neutron diffraction, nuclear magnetic resonance, and muon spin relaxation, we establish putative quantum spin liquid behavior in Ba_3InIr_2O_9, where unpaired electrons are localized on mixed-valence Ir_2O_9 dimers with Ir^4.5+ ions. Despite the antiferromagnetic Curie-Weiss temperature on the order of 10 K, neither long-range magnetic order nor spin freezing are observed down to at least 20 mK, such that spins are short-range-correlated and dynamic over nearly three decades in temperature. Quadratic power-law behavior of both spin-lattice relaxation rate and specific heat indicates gapless nature of the ground state. We envisage that this exotic behavior may be related to an unprecedented combination of the triangular and buckled honeycomb geometries of nearest-neighbor exchange couplings in the mixed-valence setting. Persistent low-temperature spin dynamics in mixed-valence iridate Ba_3InIr_2O_9 P. Gegenwart Accepted ???. Received ???; in original form December 30, 2023 ===============================================================================§ INTRODUCTIONFrustrated magnets host multiple exotic states, including quantum spin liquids (QSLs). In a QSL, spins are strongly correlated, but quantum fluctuations prevent them from long-range ordering <cit.>. The initial (and subsequently rebutted) proposal of the QSL resonating-valence-bond state on the triangular lattice of Heisenberg spins <cit.> was followed by similar proposals for several other isotropic (Heisenberg)models, where the formation of QSLs is now established <cit.>. A few candidate QSL materials proposed over the last decade bear key experimental signatures of this exotic state, including persistent spin dynamics and the absence of long-range order within the experimentally accessible temperature range <cit.>.More recently, QSL states in anisotropic magnets have been explored. Here, Kitaev model with anisotropic interactions on the honeycomb lattice <cit.> offers an exact solution for the QSL. Real-world manifestations of the Kitaev physics are found in compounds of 4d and 5d transition metals <cit.>, where large spin-orbit coupling triggers strong intersite magnetic anisotropy. However, none of the Kitaev materials reported to date host the QSL ground state in zero field, and long-range order typically sets in at low temperatures, owing to substantial interactions beyond the Kitaev terms <cit.>.In this paper, we propose an alternative strategy and search for QSL states in the family of mixed-valence 5d oxides, where unpaired electrons are localized on dimers of the Ir atoms. This should facilitate access to hitherto unexplored local electronic states <cit.> and new regimes of anisotropic exchange interactions. Specifically, we report low-temperature magnetic behavior of the mixed-valence iridate Ba_3InIr_2O_9 as a QSL candidate, and confirm its persistent spin dynamics as well as the absence of long-range magnetic order down to at least 20 mK. We further identify quadratic-like power-law behavior of both specific heat and spin-lattice relaxation rate, and compare these observations to existing theoretical results on QSLs. Ba_3InIr_2O_9 belongs to the family of hexagonal perovskites A_3BM_2O_9. Their structures comprise single BO_6 octahedra and M_2O_9 dimers of two face-sharing MO_6 octahedra (Fig. <ref>a). When magnetic ion occupies the B site, triangular interaction geometry is formed, as in Ba_3CoSb_2O_9, which is arguably the best model spin-1/2 antiferromagnet on the triangular lattice <cit.>. Placing a 4d or 5d ion into the B site or into one of the M sites could give rise to a triangular system with leading Kitaev interactions <cit.>, but experimental implementation of this idea is hindered by the strong B/M site mixing that occurs, e.g., in Ba_3IrTi_2O_9 <cit.>. Alternatively, Ir could be introduced into both M sites, while keeping the B site non-magnetic, but for integer valence of M such a spin dimer would simply condense into a non-magnetic singlet <cit.>. Mixed-valence systems with both M sites occupied by a magnetic 5d ion are possible too <cit.>, and seem to be more promising for finding a QSL, because unpaired electrons localized on the dimers appear. Note that such mixed-valence dimers with an unpaired electron delocalized between the two Ir^4.5+ ions are very different from more conventional spin dimers formed by two magnetic ions holding one unpaired electron each.Guided by this idea, we synthesized polycrystalline samples of Ba_3InIr_2O_9. From our detailed study usingneutron diffraction, magnetization and specific heat measurements, muon spin relaxation (μSR), and nuclear magnetic resonance (NMR) we establish a gapless and, potentially, spin-liquid ground state in Ba_3InIr_2O_9. § EXPERIMENTAL DETAILS Polycrystalline samples of Ba_3InIr_2O_9 were prepared by a conventional solid-state reaction method  <cit.>. Stoichiometric amounts of high-purity BaCO_3, In_2O_3, and Ir metal powder were mixed thoroughly, pressed into pellets, and calcined at 900 ^∘C for 12 h. Further, the pellet was crushed into powder, mixed well, pelletized, and fired at 1300 ^∘C for 4 days with several intermediate grindings. Neutron diffraction data were collected at the high-resolution instrument SPODI  <cit.> at FRM-II (TU Munich) using the wavelength of 1.55 Å. Jana2006 software  <cit.> was used for structure refinement. Magnetization measurements were carried out in a Quantum Design 5 T SQUID magnetometer and in a Quantum Design PPMS 14 T equipped with the vibrating sample magnetometer in the temperature range 2-350 K. Additional high-temperature data extending up to 650 K were collected in the SQUID magnetometer using powder sample enclosed in a thin-walled quartz tube. Heat-capacity measurements in the temperature range 0.4-200 K were performed in a Quantum Design PPMS using the ^3He insert. Low-temperature measurements in the 0.08-1 K range were performed using a quasi-adiabatic heat pulse method, adapted to a dilution refrigerator. μSR experiments were done on two different spectrometers at the Paul Scherrer Institute (Switzerland), LTF for temperatures from 20 mK up to 750 mK and Dolly for temperatures from 250 mK up to 200 K. For the Dolly experiment, about 300 mg of the polycrystalline sample was mounted on a thin copper plate inside the ^3He cryostat. In order to ensure good thermal contact, we glued the sample with GE varnish. We used the Veto mode, which allowed to get rid of the background signal from the sample holder. Therefore, the acquired signal is due to muons that stopped inside the sample. For the LTF experiment we used the same sample, again glued with GE on a silver plate. By comparing the results obtained between 250 mK and 750 mK on both the spectrometers, we were able to get rid of the experimental background on LTF.^115In nuclear magnetic resonance (NMR) experiments were carried out with our home-built spectrometer with the dilution-fridge insert. The measurements are performed in the field-sweep mode at a fixed frequency of 70 MHz down to 24 mK. The spin echo intensity was obtained by integrating over the spin echo in the time domain. The final spectrum is constructed by plotting the spin-echo intensity as a function of the applied field. § RESULTS §.§ Crystal structure Rietveld refinement of room-temperature neutron diffraction data (Fig. <ref>) confirms hexagonal crystal structure (P6_3/mmc). However, at 3.4 K peak splitting of the 203 and 204 reflections, as well as a visible broadening of other peaks, indicated that the symmetry is reduced to monoclinic. The 3.4 K data were refined in the C2/c space group similar to other hexagonal perovskites <cit.>. Both hexagonal and monoclinic structures feature a single crystallographic position of Ir (Tables <ref> and <ref>) suggesting the true intermediate-valence Ir^4.5+ state. This is different from, e.g., Ba_5AlIr_2O_11, where two sites of the dimer belong to two different crystallographic positions, thus making possible charge re-distribution within the dimer <cit.>. In contrast to Ba_3IrTi_2O_9 with its 35-40% site mixing <cit.>, our mixed-valence Ba_3InIr_2O_9 shows high degree of structural order. We were able to obtain reasonable atomic displacement parameters in the fully ordered models of both hexagonal and monoclinic structures (Tables <ref> and <ref>). On the other hand, deliberate admixing of In into the Ir position and vice versa leads to a marginal reduction in the refinement residuals and 2.8(5)% site mixing [We introduced the site mixing and constrained the overall composition to Ba_3InIr_2O_9. This leads to R_I=0.038 and R_p=0.050 compared to R_I=0.039 and R_p=0.051 for the fully ordered model (room-temperature data).]. We believe that the diffraction data alone may not distinguish between the fully ordered structure and the weak site mixing scenario. Further studies, such as direct imaging with high-resolution electron microscopy, would be useful to resolve this issue.The low-temperature monoclinic distortion is primarily related to the tilting of the IrO_6 and InO_6 octahedra. It has nearly no effect on relative positions of the Ir atoms. For example, the Ir–Ir distance within the dimer shrinks from 2.637(2) Å at 300 K to 2.599(4) Å at 3.4 K, presumably due to thermal expansion. The Ir–Ir distances between the dimers do not change at all, compare 5.832(1) Å at 300 K to 5.813(8) Å and 5.815(4) Å at 3.4 K for the nearest-neighbor Ir–Ir distances in the ab plane. Given this negligible structural effect and the absence of any signatures in thermodynamic measurements, we conclude that the hexagonal-to-monoclinic phase transition should be an effect of structural (geometrical) origin and bears no relation to the magnetism of Ba_3InIr_2O_9. Determination of the exact transition temperature requires further dedicated diffraction experiments at intermediate temperatures and lies beyond the scope of our present study. §.§ Magnetic susceptibilityTemperature dependence of the magnetic susceptibility (χ=M/H) measured in various applied fields is shown in the inset (i) of Fig. <ref>. Down to 2 K, we did not observe any anomaly or divergence of field-cooled and zero-field-cooled data, suggesting the absence of long-range ordering and spin freezing.The susceptibility follows the Curie-Weiss (CW) behavior between 10 and 70 K. At higher temperatures (see Fig. <ref>), the data deviate from the CW law due to thermal changes in the mixed-valence Ir_2O_9 dimers that can adopt different electronic configurations  <cit.>. A tentative van Vleck fit with the three-level model χ(T)=χ_0+N_Aμ_B^2/3k_B(T-θ)×3/4××g_0^2+5g_1^2 e^-Δ_1/k_BT+5g_2^2 e^-Δ_2/k_BT/1+e^-Δ_1/k_BT+e^-Δ_2/k_BT,where N_A is Avogadro's number, g_0 is electronic g-factor for the ground state, and g_1 and g_2 are g-factors for the excited states, which are separated from the ground state by energy gaps of Δ_1 and Δ_2, respectively, yields decent description of the magnetic susceptibility up to at least 650 K. This fitting function can be understood as follows. In the absence of electronic correlations and spin-orbit coupling, 9 electrons occupying 6 t_2g orbitals of two Ir atoms give rise to S=1/2 and S=3/2 states depending on the filling of the molecular orbitals of the dimer <cit.>. The effect of spin-orbit coupling is taken into account by introducing electronic g-factors as fitting parameters. Additionally, we assumed that the S=3/2 state splits into two, because fitting with one excited state was not successful, whereas two distinct excited states provide a good description of the susceptibility in the broad temperature range.Fitting the susceptibility with Eq. (<ref>) yields χ_0=-6.8× 10^-5 cm^3/mol, θ=-6.8 K, Δ_1=107 K, Δ_2=472 K, g_0=0.872, g_1=0.678, and g_2=1.810. The relatively low value of Δ_1 explains the deviation from the Curie-Weiss behavior already above 70 K. The obtained Δ_1 and Δ_2 are of the same order of magnitude as in the mixed-valence ruthenates isostructural to Ba_3InIr_2O_9 <cit.>. The paramagnetic effective moments are μ_ eff=0.76 μ_B in the ground state, μ_ eff,1=1.31 μ_B in the first excited state, and μ_ eff,2=3.51 μ_B in the second excited state. For the rest of this paper, we focus on the low-temperature regime below 70 K with the paramagnetic effective moment of μ_=0.76 μ_B/dimer, which is comparable to the values reported for mixed-valence iridates earlier <cit.>. Antiferromagnetic couplings between magnetic moments localized on the mixed-valent dimers are confirmed by the negative Curie-Weiss temperature θ_χ=-7 K. Isothermal magnetization curve at 2 K (see inset (ii) of Fig. <ref>) does not show any sign of saturation up to 14 T. This weak sensitivity to the field may be due to the strongly reduced g-factor (only 44% of its spin-only value) that lessens the effect of the external field. §.§ Specific heatFirst insight into the low-temperature magnetism is obtained from the specific-heat data. A broad peak in C_p/T is observed around 1.6 K (Fig. <ref>a) indicative of a crossover between the paramagnetic (thermally disordered) and spin-liquid (quantum disordered) regimes <cit.>, further confirmed by the increase in the muon zero-field relaxation rate around the same temperature (Sec. <ref>). The peak shifts toward lower temperatures in the applied field, although the changes are relatively small even at 14 T, and no transition anomaly is observed.Heat capacity was further measured at temperatures well below the broad maximum. For an insulating material, total heat capacity is a sum of the magnetic, lattice, and nuclear contributions, C_p=C_m+C_ lat+C_ nuc. To extract the magnetic specific heat of the material, we need to subtract the lattice part and nuclear part from the total specific heat C_p. In the absence of a suitable non-magnetic analog, we fitted the measured specific heat with C_p=β T^3 in the range 14-20 K <cit.>. The fitting of the lattice part yields β=1.45 mJ mol^-1 K^-4 and the Debye temperature Θ_D=272 K. The fitted curve is extrapolated to low temperatures and taken as the lattice part (C_ lat) that was subtracted from the experimental data. At low temperatures, the nuclear contribution becomes prominent. To extract the magnetic heat capacity, we adopted the following procedure:(1) We fitted the C_p data for each field (B) from lowest T up to 300-400 mK with C_p = α/T^2 + C_0 T^γ <cit.>, where α/T^2 stands for C_ nuc, and power-law behavior of C_ mag is assumed.(2) For each field, the magnetic heat capacity C_m is obtained as C_m=C_p-C_ nuc, whereas C_ lat is negligible in this temperature range.(3) Reliability of the α values is verified by plotting the field dependence α(B^2), which was linear, as expected <cit.>.After subtracting the nuclear contribution <cit.>, we arrive at the robust power-law behavior C_m=C_0T^γ with C_0=832 mJ mol^-1 K^-2.83 and γ=1.83 (Fig. <ref>b). This behavior gives first indication of a gapless ground state, because otherwise low-energy excitations over a spin gap would give rise to the exponential decay of C_m at low temperatures. The power-law behavior persists up to at least 14 T, but above 9 T the exponent γ is slightly reduced <cit.>.By integrating C_m/T in zero field (see Fig. <ref>c) within the temperature range from 0.08 to 20 K, we estimated magnetic entropy change of 3.4 J mol^-1 K^-1 (see Fig. <ref>d), which is about 60% of the entropy expected for spin-1/2. Part of the magnetic entropy should be then released at higher temperatures, where the magnetic contribution is concealed behind a much larger lattice term. We note here that a two-step release of the magnetic entropy (and, consequently, two well-separated peaks of the magnetic specific heat C_m) are not uncommon in frustrated magnets <cit.>. §.§ μSR Whereas thermodynamic measurements provide first hints towards the absence of long-range magnetic order in Ba_3InIr_2O_9, experimental evidence for the QSL formation is not complete without a local probe. To this end, we use muon spin relaxation (μSR), which is a very sensitive technique to detect static local fields arising from weak long-range order or spin freezing. The relaxation curves of the muon polarization in zero field (ZF) are shown in Fig. <ref>. The absence of oscillations in the zero-field (ZF) signal and the lack of the polarization recovery to 1/3 indicate the absence of any frozen moments in Ba_3InIr_2O_9 in the temperature range from 200 K down to 20 mK. The polarization curves can be fitted with Eq. <ref> which is a combination of the depolarization due to the muon coupled to the In nuclear magnetism (Kubo-Toyabe Gaussian function) and depolarization due to the electronic magnetism evolving with temperature.P(t)=fe^-(λ_ F t)^β+(1-f)(1/3+2/3(1-(σ t)^2) e^-(σ t)^2/2) Here, f=0.786(2) is the fraction of muons coupled to the electronic magnetism, σ=0.094(2) μs^-1 is the nuclear depolarization rate, β=1.308(7) is the stretched exponent, and λ is the electronic depolarization rate. The stretched exponent β is independent of field and temperature. The small deviation from unity could be due to a small distribution of the muon sites close to the In ions.The ZF μ^+ relaxation rate (λ_ ZF) obtained from fitting the ZF muon depolarization curves is shown as function of temperature in Fig. <ref>b. At high temperatures (20 K to 3 K), λ_ ZF remains constant at ∼0.12μs^-1, which is consistent with the paramagnetic fluctuations of Ir moments according to the Bloembergen Purcell, and Pound theory <cit.>.From 3 K down to 1 K, λ_ ZF increases with decreasing temperature. The enhancement of λ_ ZF in a narrow temperature window indicates a slowing down of Ir spin fluctuations due to the development of strong short-range correlations, a common feature seen in other QSL candidates <cit.>. Upon further cooling, λ_ ZF shows temperature-independent plateau-like behavior between 1 K and 20 mK. The plateau-like behavior in λ_ ZF vs T (see Fig. <ref>b) has also been observed in several other QSL compounds. To verify if the plateau-like behavior is originating from the muons directly coupled to the frustrated spins <cit.> or the muons coupled to defects <cit.>, we have performed μSR experiments under transverse field (TF) of 0.4 T and estimated the μSR line shift (K^μ) as a function of temperature  <cit.> as shown in Fig. <ref>a. K^μ increases with decreasing temperature and saturates below 3 K, from where λ (see Fig. <ref>b) starts increasing. This indicates that antiferromagnetic (AFM) spin fluctuations are dominating at low temperature in Ba_3InIr_2O_9. The similarity between the temperature dependence of K^μ and NMR shift (K_ NMR) (see Fig. <ref>a) proves that the muons are directly coupled to the Ir moments. In order to identify the origin of the depolarization rate being static or dynamic, we performed decoupling experiments at 250 mK (shown in Fig. <ref>a) applying longitudinal fields (LF) from 5 mT up to 100 mT. Should the low-temperature plateau in λ_ ZF arise from a static field, the size of this field can be estimated as B_ loc=λ/γ_μ≈1.4 mT, where γ_μ=135.5×2π s^-1μT^-1 is the gyromagnetic ratio for muons. The static field can then be decoupled by applying an external magnetic field ≃14 mT, which is one order of magnitude higher than B_ loc. The polarization curve under a field of 100 mT does not show the signs of a full polarization, suggesting that the plateau does not originate from a static internal field and the spins are dynamic in nature even at the lowest temperature of 20 mK, as expected in a QSL. The plateau-like behavior of λ_ ZF toward lowest temperature indicates much slower spin dynamics of the material compared to the μSR time window. It is another characteristics of QSL candidates <cit.>.§.§ NMRNuclear magnetic resonance (NMR) is a concurrent probe for local magnetic fields and spin dynamics. ^115In NMR spectra measured at 70 MHz in the temperature range 0.024-138 K are shown in Fig. <ref>(a). At high temperatures (for example, at 138 K), the spectra exhibit one single isotropic line which indicates the unique crystallographic site for indium, consistent with our structural analysis. The spectra can be described satisfactorily assuming the I=9/2 nuclei with the 380 kHz quadrupolar coupling constant (ν_Q). Below about 100 K, we observed a second line, which remained unshifted within the entire temperature range. This second line presumably originates from trace amounts (0.9 %) of a non-magnetic impurity phase present in the sample. For estimating the impurity contribution in the spectral line shape, we measured the spectra at several temperatures with τ=50 μs and τ=200 μs, where τ is the pulse separation between the spin-echo sequence π/2-τ-π. The intrinsic line has much shorter spin-lattice relaxation time T_1 as well as the spin-spin relaxation time T_2 compared to the impurity line <cit.>. Hence it is expected that with longer τ the intrinsic part of the spectra is already relaxed, whereas the impurity contribution sustains. The comparison of the spectra corresponding to different τ measured at T=25 K is shown in Fig. <ref>c.The spectra are described at all temperatures below 100 K with two Gaussian lines, one for magnetic (intrinsic) and the other one for a non-magnetic (impurity) contribution as shown in Fig. <ref>(a)). At low temperatures, the two lines (intrinsic line and the impurity line) merge together. For estimating the Knight shift of the intrinsic line, we have used the peak of the intrinsic contribution marked with the green arrows in Fig. <ref>a. This peak shifts towards higher fields with decreasing temperature yielding the local spin susceptibility K. In general, K=K_0+(A_ hf/N_Aμ_B) χ(T),where K_0 is the temperature-independent part of the line shift K, and N_A is the Avogadro's number. The hyperfine coupling constant is estimated at A_ hf=-1.675 T/μ_B <cit.>. The temperature dependence of the ^115In line shift (K) is shown in Fig. <ref>a. A continuous increase in the line shift K(T) with decreasing temperature from 100 K down to ∼4 K indicates the development of spin correlations. The inverse of the temperature-dependent part of the line shift 1/(K-K_0) is shown as a function of temperature in the inset of Fig. <ref>a. A CW fit of the data yields the CW temperature θ_ NMR=-14 K. This value should be more reliable than θ_χ=-7 K estimated from the bulk susceptibility, since the NMR line shift does not contain any impurity contributions.At low temperature the total spectra remain unchanged as shown in Fig  <ref>b. This indicates that both the line shift K and line width Δ H of the spectra remain constant down to the base temperature of 25 mK. Saturation of both these quantities suggests saturation of spin correlations below ∼4 K. The finite and temperature-independent value of K below 4 K gives strong evidence for the gapless behavior. From Δ H, we estimated the saturated magnetic moment μ_s=0.89 μ_B/f.u. close to 1 μ_B expected for one unpaired electron per the Ir_2O_9 dimer. To probe the low-energy spin dynamics, we have studied the NMR spin-lattice relaxation rate (1/T_1) at various temperatures. (1/T_1) was measured at the peak position of the intrinsic line using long pulses to avoid any interference from the impurity line. 1/T_1 at different temperatures are obtained by fitting (shown in the inset of Fig. <ref>b) the longitudinal nuclear magnetization recovery curves with the equation1-M_t/M_∞ =C ( 0.006 e^-(2Wt)^β+0.0335 e^-(12Wt)^β+ +0.0925 e^-(30Wt)^β+0.215 e^-(56Wt)^β+0.653 e^-(90Wt)^β),where 1/T_1=2W. Here, M is the nuclear magnetization, β is a stretched exponent, which accounts for a distribution of the T_1 values due to disorder, and C is a pre-factor. In general,(1/T_1T)∼∑_qA_ hf(q)^2×χ^''(q,ω→0),where χ^''(q,ω) is the imaginary part of the dynamical spin susceptibility. The temperature dependence of 1/T_1 is shown in Fig. <ref>b. Between 30 K and 4 K, 1/T_1 is almost temperature-independent, indicating the paramagnetic behavior. A weak hump-like feature is seen around 1.6 K, which coincides with the hump in the C_p/T data (Fig. <ref>a) and with the rapid increase in λ_ ZF (Fig. <ref>b). No long-range order occurs in Ba_3InIr_2O_9. Therefore, this hump is not a broadened signature of a magnetic transition. Instead, it may indicate some intrinsic physics, as recently proposed for the honeycomb iridates <cit.>. The applicability of this scenario to Ba_3InIr_2O_9 may be an interesting venue for future research.Below 0.3 K, (1/T_1) follows the ∼ T^2.2 power law similar to the ∼ T^1.83 behavior of C_m(T).§ DISCUSSION AND SUMMARYOur results establish gapless ground state and persistent spin dynamics down to at least 20 mK, rendering Ba_3InIr_2O_9 a potential QSL material. However, its low-temperature behavior is rather different from theoretical expectations for a gapless QSL, where magnetic excitations are usually described in terms of a spinon Fermi surface and should give rise to the linear behavior of both NMR spin-lattice relaxation rate and magnetic specific heat <cit.> at low temperatures. Our data for Ba_3InIr_2O_9 clearly deviate from this scenario.Microscopically, we expect two superexchange pathways. The mixed-valence Ir_2O_9 dimers may interact in the ab plane via the coupling J_T (the Ir–Ir distance of 5.83 Å) and along the c direction via the coupling J_H (the Ir–Ir distance of 5.71 Å), see Fig. <ref>. Note that we consider Ir–Ir distances, because magnetic electrons are expected to occupy a molecular orbital of the dimer with equal contributions of both Ir sites <cit.>. While the coupling J_T forms a triangular lattice in the ab plane, as already anticipated in Ba_3IrTi_2O_9 <cit.>, the coupling J_H leads to a buckled honeycomb geometry. Depending on the ratio between J_T and J_H, the system can interpolate between the purely two-dimensional regime (|J_T|≫ |J_H|, or vice versa) and a 3D behavior when both |J_H| and |J_T| are of the same size.On the phenomenological level, quadratic behavior of the zero-field specific heat is expected in an algebraic spin liquid on the kagome lattice <cit.>. Another manifestation of the T^2 behavior was proposed for a spin model combining the triangular and honeycomb geometries <cit.>, as in Ba_3InIr_2O_9, but its applicability to our system is obscured by the large single-ion anisotropy term that is central to the model but seems unlikely in iridates <cit.>. Intriguingly, the robust quadratic behavior of the zero-field specific heat has been reported in frustrated magnets with slow dynamics <cit.> and interpreted in the framework of Halperin-Saslow modes for two-dimensional spin glasses <cit.>. It was also observed in Li_2RhO_3, the Kitaev antiferromagnet, where spins freeze below 6 K <cit.>. To what extent these scenarios apply to Ba_3InIr_2O_9 with its clear signatures of persistent spin dynamics should be a subject of future investigations.Mixed-valence dimers are also reported in ruthenates with the hexagonal perovskite structure. However, these compounds show long-range magnetic order or disordered static magnetism at low temperatures <cit.>. This is different from our case, where persistent spin dynamics is observed. From magnetism point of view, mixed-valence hexagonal perovskites remain a largely uncharted territory. The nature of local moments within the mixed-valence dimers requires further investigation, and peculiarities of magnetic interactions between the dimers remain to be explored. Experiments reveal instances of non-trivial magnetic states, including the potential QSL state in Ba_3InIr_2O_9, and call for a better understanding from theoretical and microscopic perspectives.Altogether, we demonstrated that Ba_3InIr_2O_9 is a structurally well-ordered QSL candidate showing thebehavior of zero-field specific heat and the ∼ T^2.2 power-law for the spin-lattice relaxation rate at low temperatures. Not only would it be interesting for further experimental (probe of the spin excitations) and theoretical (microscopic analysis of the magnetic model) research, it also puts forward mixed-valence iridates as a promising playground for finding new QSL materials. The unique combination of the triangular and buckled honeycomb geometries bears strong connections to current theoretical models of frustrated magnetism <cit.>, and may lead to novel manifestations of the QSL physics when different lattice geometries concur.We gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG) within the collaborative research center TRR 80 (Augsburg/Munich). TD, MM, and AT were funded by the Federal Ministry for Education and Research through the Sofja Kovalevskaya Award of Alexander von Humboldt Foundation. FB acknowledges financial support from the project SOCRATE (ANR-15-CE30-0009-01) of the ANR French agency. PK acknowledges support from the European Commission through Marie Curie International Incoming Fellowship (PIIF-GA-2013-627322). Part of the work was performed at the LTF and DOLLY spectrometers of the Swiss Muon Source (Paul Scherrer Institute, Villigen, Switzerland). We thank C. 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http://arxiv.org/abs/1702.08305v3

Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudingerweg 7-9, 55128 Mainz, Germany The emergence of structure through aggregation is a fascinating topic and of both fundamental and practical interest. Here we demonstrate that self-generated solvent flow can be used to generate long-range attractions on the colloidal scale, with sub-pico Newton forces extending into the millimeter-range. We observe a rich dynamic behavior with the formation and fusion of small clusters resembling molecules, the dynamics of which is governed by an effective conservative energy that decays as 1/r. Breaking the flow symmetry, these clusters can be made active. Self-assembly of colloidal molecules due to self-generated flow Ran Niu, Thomas Speck, Thomas Palberg===============================================================Colloidal particles acting as “big” artificial atoms have been instrumental in studying microscopic processes in condensed matter, from the kinetics of crystallization <cit.> to the vapor-liquid interface <cit.>. Due to their size, colloidal particles are observable directly in real space. Moreover, interactions are widely tunable, ranging from hard spheres to long-range repulsive, short-range attractive, and dipolar <cit.>. Consequently, colloidal particles can be assembled into a multitude of different structures: from clusters <cit.> and stable molecules <cit.> composed of a few particles to extended bulk structures like ionic binary crystals <cit.>. In addition, self-assembly into useful superstructures can be controlled by factors such as confinement <cit.> and particle shape <cit.>, which make colloids a versatile and fascinating form of matter <cit.>.What is still missing are truly long-range attractions of like-charged (or uncharged) identical colloidal particles. There is much interest in the basic statistical physics of systems with such interactions, which play a role in gravitational collapse, two-dimensional elasticity, chemotactic collapse, quantum fluids, and atomic clusters <cit.>. One proposed realization are colloidal particles trapped at an interface <cit.> that experience screened, long-range attractions due to capillary fluctuations of the interface <cit.>. The attractive interactions then correspond to Newtonian gravity in two dimensions. Complex patterns are also known to arise for bacteria due to long-range chemotactic interactions <cit.>. Critical long-range Casimir forces have been reported for colloidal particles in a binary solvent <cit.>, which are tunable by temperature and surface chemistry. Finally, a recent theoretical proposal are catalytically active colloidal particles that interact through producing or consuming chemicals <cit.>. For simple diffusion the concentration profile of a chemical decays as inverse distance, implying long-range interactions that can be tuned through activity (how chemicals are produced or consumed) and mobility (how particles react to gradients).Here, we implement long-range attractions through hydrodynamic flows coupling suspended particles <cit.>. We report on experiments using spherical ion exchange resin particles sedimented to the negatively charged substrate. The particles have diameters of 15μ m, for which Brownian diffusion is practically negligible on the experimental time scale. They interact due to self-generated local flow, resulting in an effective long-range 1/r attraction as expected for three-dimensional unscreened gravity. Our present understanding of the mechanism responsible for their aggregation can be summarized as follows <cit.>: By exchanging residual cationic impurities for stored hydrogen ions (Fig. <ref>a), the particles generate a concentration profile c that decays away from the particles. Different diffusion coefficients of the exchanged cations and the released ions locally generate diffusio-electric fields which retain overall electro-neutrality by slowing the outward drift of hydrogen ions and accelerating the inward drift of impurities. The total system composed of solvent, ions, and colloidal particles is clearly out of thermal equilibrium through the free energy released by solvating the hydrogen ions, which drives the flow. However, appealing to a scale separation between particles and solvent, our crucial assumption will be that the motion of the particles themselves can be described by _i = -∇_i U + _i,where U=∑_i<ju(_i-_j) is a conservative potential, u(r) is the pair potential Eq. (<ref>), and _i models the noise with zero mean and correlations _i(t)_j^T(t')=21δ_ijδ(t-t'). The relative velocity of two particles with separation =_1-_2 is v(r)=(/r)·(_1-_2)δ(||-r)=2u'(r) after inserting Eq. (<ref>). Hence, we have direct access to the interactions u(r) through measuring a dynamic quantity, the approach velocity v(r) as shown in Fig. <ref>a. Approaching each other, the speed increases as expected but reaches a maximum at about r≃60μ m before it drops rapidly. This behavior can be rationalized by considering the generated flows in more detail. In the double layer of the substrate, the local fields generate electro-osmotic flow of the solvent since the negative substrate is screened by an excess of positive charges (Fig. <ref>b). The motion of the particles is determined by the slip velocity =-μ_p∇ c with ion concentration field c and phoretic mobility μ_p <cit.>, whereby our measured particle velocities are consistent with a constant mobility. For an isolated particle, the solvent flow is approaching symmetrically and no net motion results (the particle can be regarded as a sink in two dimensions; of course, the solvent is incompressible with backflow out of the plane of the substrate). For two particles, the hydrogen ion concentration is increased in the space between the particles, which reduces the gradient (with respect to the particles' surfaces) and thus the flow velocity. Hence, the flow on each particle now becomes asymmetric, resulting in an apparent attraction (Fig. <ref>c). Basically the same mechanism acts in the double layer of the particles but now leads to electro-phoretic motion. The particles themselves are (slightly) negatively charged due to the release of cations. The field generated by the concentration gradient of the other particle again generates a solvent flow, however, now particle and solvent taken together are force-free, which leads to a particle speed in the opposite direction of the field (Fig. <ref>d).For purely diffusive ion motion one would expect the concentration to decay as c(r)∼1/r, but measurements of the pH reveal a more complicated behavior (these measurements had to be performed for a larger particle, see Supplemental Material <cit.>, but we expect the qualitative features to be the same for the smaller particles used here). While there is indeed a 1/r decay regime, closer to the ion exchange particle it changes to a slower decay that is well described as exponential. We speculate that this accumulation is caused by the local fields slowing down outward moving ions. We model this effect through a term resembling screening although we stress that it originates from the flow and not electrostatics. Combining both flows, the functional form of the potential readsu(r) = -/r + /re^-r/ξwith three free parameters: the prefactorsand , and the screening length ξ. As shown in Fig. <ref>a, this function describes the experimental data very well. From the fit we obtain ≃6120μ m^3/s, ≃8805μ m^3/s, and ξ≃31.4μ m. The pair potential u(r) plotted in Fig. <ref>b has a minimum at r_0≃35μ m. As shown in the inset of Fig. <ref>b, r_0 agrees well with the maximum of the distributions of bond length r for dimers and trimers. Usually, overdamped motion is described by a product of particle mobility and the gradient of the potential energy. Since we do not have access to these terms separately, we treat u(r) as an effective “energy” absorbing the phoretic mobility μ, with u(r) thus having units of a diffusion coefficient. Nevertheless, we can estimate physical energies employing the bare particle mobility μ_0, which quantifies the forces needed to move a single particle through the solvent with desired speed. For our particles in water its value is μ_0≃7.8μ m/(s· pN), yielding forces between particles of order 0.1pN. With u(r_0)≃-92.34μ m^2/s, the corresponding bond dissociation energy would thus be E_b=|u(r_0)|/μ_0≃7000kJ/mol.The final ingredient for our theoretical model is an effective temperature, which we extract from the measured bond fluctuations. First, from the distribution of the bond length r for dimers we determine its variance Var(r)≃3.57μ m^2. Assuming that these vibrations are effectively equilibrated allows us to determine the analog of a temperature, ≈Var(r)u”(r_0)≃0.3μ m^2/s. We test this assumption for three particles, for which a quick calculation predicts that the harmonic bond fluctuations are 5/3 times larger <cit.>. The predicted value 5.95μ m^2 is only slightly smaller than the measured variance 6.25μ m^2, the difference of which is due to anharmonic higher-order vibrations. Our suspension of ion exchange particles is not stationary but slowly collapses to a close-packed state due to the long-range interactions. Starting from a homogeneous density profile, during this process we observe the formation of colloidal clusters (“molecules”) with n particles. This assembly happens autonomously in contrast to prefabricated colloidal molecules <cit.>. While metastable, single clusters can be observed up to minutes, which allows in principle to study in detail different isomers and the “reactions” by which larger clusters form (more details are given as Supplemental Material <cit.>). In Fig. <ref>a we show the first few minutes of this process for a dilute suspension. The experiments were carried out with 60-90 particles within a field of view corresponding to an area fraction of approx. 0.25%. Without adjustable parameters, the observed dynamics are reproduced through the model described by Eq. (<ref>). In Fig. <ref>b we show a sequence of experimental snapshots for seven particles. We then perform simulations of the model using the extracted particle positions from the first experimental frame as initial positions. As shown in Fig. <ref>c, the simulations agree with the experiments on the same time scale. While for repeated simulation runs the positions differ due to the noise, the average behavior is consistent.This fit-free quantitative agreement is corroborated by the time evolution of the fraction of clusters with weight n. From the analysis of the experiments, we extract the fraction N_n(t)/N of particles residing in clusters with n particles. The range of N is between 60 and 90, and to improve statistics we average over all experiments. We repeat the same analysis with the theoretical model for N=80, the comparison of which is shown in Figure <ref>. The qualitative behavior is that of irreversible aggregation as described through the Smoluchowski coagulation equation <cit.>, with a steady decrease of the monomer concentration and peaks for the n-mers that become flatter and shifted to later times for increasing weight n. As demonstrated, for a one-component suspension of ion exchange particles in moderate flow the dynamics of the particles alone is effectively described through a conservative potential. However, for larger aggregates also the flow increases, leading to deviations from the predicted behavior (e.g. particles are lifted and pushed into the second layer). With even larger flows, particles are observed to spontaneously break the flow symmetry and become self-propelled. Another strategy is to explicitly break symmetry through mixtures of particles with different sizes or shapes, or mixtures of activated and passive particles <cit.>. Here we explore the consequences of adding anionic ion exchange particles of similar size but releasing OH^- ions, for which we again observe the assembly of low-weight clusters. This is shown exemplary in Fig. <ref> for two cationic particles and one anionic particle, which assemble into a trimer. As predicted in Ref. , the geometry together with the different mobilities/activities leads to a self-propelled complex. This can be seen through determining the center-of-mass speed of the three particles, which clearly shows a transition to a constant speed once the trimer has assembled.To conclude, we have shown that ion exchange particles close to a charged substrate generate flows that lead to effectively conservative, long-range interactions. Autonomous flow-driven assembly through long-range attractions might enable novel non-equilibrium materials and strategies in self-assembly, in particular on intermediate scales for which thermal motion has become negligible but which cannot be manipulated directly. Moreover, through changing composition one may generate asymmetric flow. The resulting directed motion of colloidal particles in combination with volume exclusion leads to fascinating dynamic behavior ranging from clustering <cit.> and the formation of “living crystals” <cit.> to schooling and swarming <cit.>. Our results demonstrate how one can implement strategies to control and engineer interactions and directed motion on the same footing <cit.>, which is a step towards designing active particles that can perform dynamical tasks such as transport of cargo <cit.>. 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Phys. volume 139,pages 234506 (year 2013)NoStopSupplementary Information § EXPERIMENTS The particles are cationic ion exchange resin particles (CK10S, Mitsubishi Chemical Corporation, Japan) with diameter 2a=15.3±3.0μ m and counter ions Na^+. Prior to experiments, the particles were washed with 20 wt % hydrochloride acid solution to exchange the counter ions into H^+. Then we rinsed with doubly deionized water several times until solution pH of ∼ 7 and dried at 80^∘C for 2 h. The sample cell was built from a circular Perspex ring (inner diameter of 20mm, height of 1mm) fixed to microscopy slides by hydrolytically inert epoxy glue and dried for 24 h before use. The glass slides were washed with alkaline solution (Hellmanex®III, Hellma Analytics) for 30 min under sonication, and subsequently rinsed with tap water and deionized water for several times.In a typical experiment, a weighted amount of particles was dispersed into doubly deionized water. Subsequently, 400 μ-liters of particle suspension was added into the sample cell. The cell was covered immediately to avoid contamination. Particles settled to the bottom of the cell within minutes. Movies were then taken at a frame rate of 0.5 Hz using an inverted scientific microscope (DMIRBE by Leica, Germany). Particles were tracked through extracting the perimeter using a home-written Python script. The velocity v(r) of two approaching particles was averaged over 160 pairs with any other particle at least 10 times their diameter away.In a second set of experiments we have added anionic ion exchange resin particles (CA08S, Mitsubishi Chemical Corporation, Japan) with diameter 2a=15.1±0.3μ m and counter ions Cl^-. Before use, particles were washed with concentrated sodium hydroxide (NaOH) to exchange the counter ions into OH^- and subsequently washed with deionized water until the pH of the solution reached about 7. To distinguish cationic ion exchange resin and anionic ion exchange resin, tiny amount of pH indicator solution (pH 4-10, Sigma-Aldrich) was added to mark them redish and bluish, respectively.§ HYDROGEN ION CONCENTRATION We determine the pH profile around an ion exchange particle with diameter 45μ m using a mixture of Universal indicator solutions (1:3 volume ratio of pH 0-5 and pH 4-10, Sigma-Aldrich, Inc). As the concentration of proton decreases, the color ratio of blue-to-red decreases monotonically. Thus, measuring the blue-to-red color ratio at fixed pH, we get a calibration curve. Applying this calibration curve to every pixel around ion exchange particle, we determine the proton concentration. The color images were recorded using a consumer DSLR (D700, Nikon, Japen) mounted on an inverted scientific microscope (DMIRBE, Leica, Germany). Image recording starts ∼2s after the ion exchange particle gets into contact with the indicator solution. The measured concentration c(r,t) as a function of distance r from the particle center and time t after preparation is shown in Figure <ref>.A prediction for the concentration profile is given by the solution of the diffusion equation∂_tc(,t) = D_+∇^2c(,t) + δ()with ion diffusion coefficient D_+ and assuming a constant ratewith which ions are released. The solution isc(r,t) = /4π D_+r(r/2√(D_+t)) + c_∞with background concentration c_∞. From the fit of this expression to the decay of the concentration profile we obtain a background pH≃5.45 and a diffusion coefficient D_+≃1300μ m^2/s, which is reasonable for hydrogen ions given the fact that they are hydrated. Strikingly, the qualitative behavior changes when approaching the particle and the concentration profile decays much slower than what is expected from the simple diffusion picture. In fact, it can be fitted well by an exponential decay. This implies that outward moving ions are slowed, presumably through local electric fields generated in conjunction with other ions. The resulting solvent flow is thus more complex than ∼∇ c∼1/r^2, which is captured by the form of our effective pair potential, see the discussion in the main text.§ BOND VIBRATIONS IN THE TRIMER While the suspension itself is non-stationary, for the bond vibrations we assume that an effective equilibrium has been reached described by the Boltzmann factor. We expand the energyU ≈ U_0 + 1/2∑_ij^2nH_ijδ x_iδ x_j = U_0 + 1/2∑_=1^2n_ q_^2,where H⃗ is the 2n×2n Hessian matrix of second derivatives of the potential energy evaluated at the minimum energy configuration of the cluster with n particles. It has 2n eigenvalues _, of which three are zero corresponding to translation and rotation in two dimensions. The other eigenvalues are the spring constants of the vibrational modes. For n=3, the minimum energy configuration is an equilateral triangle and the Hessian becomes H⃗=u”(r_0)A⃗ with constant matrix A⃗ (with non-zero eigenvalues 3/2,3/2,3). The harmonic fluctuations of any edge can easily be calculated as|_i-_j-r_0_ij|^2 = ∑_q_^2 = ∑_/_ = 5/3/u”(r_0)as used in the main text.§ METASTABLE MOLECULES Figure <ref> shows snapshots of structural arrangements sorted by their molecular weight from n=3 to n=6. We only show structures that persisted for at least 100s. We distinguish isomers assigning a structural fingerprint counting the number n_b of particles with b direct bonds. For every particle configuration (experiments and simulations) we first determine clusters of mutually bonded particles, where a particle pair (i,j) forms a bond if the separation |_ij| is smaller than the cutoff of 40μ m. We then refine the bond network and only retain bonds between direct neighbors. A particle j is a direct neighbor of i if for all particles k with distance |_ik|<|_ij| the condition _ik·_jk>0 holds (the enclosed angle is less than π/2 radians) <cit.>. For each cluster we count the number of particles n_b with b bonds, the vector of which forms a “fingerprint” based on which we identify the isomeric structure of the molecule.The structures to the left in Figure <ref> show the ground states minimizing the effective energy. From the eigenspectrum of vibrations we delineate (meta-)stable molecules (blue frames) from unstable molecules (red frames). For n=3 and n=4 the latter nevertheless occur with a non-vanishing probability since they are populated by reactions adding another monomer. We can identify these transition states easily by their “dangling bonds” with n_1>0. Transition states quickly rearrange into stable isomers for the same n. These are relatively long-lived – typically until another reaction occurs increasing the molecular weight – but we also observe spontaneous transitions between isomers (see Supplementary Movie 2 and Fig. <ref>). For n=6 we have four stable isomers. For short-range attractions counting only direct bonds, the structures II, III, and IV form the degenerate ground state manifold <cit.>. Due to the long-range attractions in our case, the degeneracy is lifted with the regular pentagon (I) representing the minimal energy configuration. While most transitions proceed through a slight shift of bonds, note that the transition IV→III involves breaking a bond with excited intermediate III^∗. Consequently, for the parallelogram (IV) we observe a higher population both in experiment and theory than one would expect from its energy alone.

http://arxiv.org/abs/1702.08020v1

Biwei Jiang bjiang@bnu.edu.cn Department of AstronomyBeijing Normal University Beijing 100875, P. R. China Department of AstronomyBeijing Normal University Beijing 100875, P. R. ChinaThe SiO molecule is one of the candidates for the seed of silicate dust in the circumstellar envelope of evolved stars, but this opinion is challenged. In this work we investigate the relation of the SiO maser emission power and the silicate dust emission power. With both our own observation by using the PMO/Delingha 13.7-m telescope and archive data, a sample is assembled of 21 SiO v=1,J=2-1 sources and 28 SiO v=1,J=1-0 sources that exhibit silicate emission features in the ISO/SWS spectrum as well. The analysis of their SiO maser and silicate emission power indicates a clear correlation, which is not against the hypothesis that the SiO molecules are the seed nuclei of silicate dust. On the other hand, no correlation is found between SiO maser and silicate crystallinity, which may imply that silicate crystallinity does not correlate with mass loss rate.§ INTRODUCTIONSilicate is the most important species of oxygen-bearing dust that is believed to be formed in the circumstellar envelope of AGB (Asymptotic Giant Branch) stars, then evolve, be destroyed and reformed in the interstellar space <cit.>. Chemically, silicates are made up of cations and silicic acid radical ions (SiO_4^4-) or (SiO_3^2-). In the circumstellar envelope, the most abundant cations turn out to be of Fe and Mg. Consequently, silicates are divided into olivine (Fe_2xMg_2(1-x)SiO_4) and pyroxene (Mg_(1-x)Fe_xSiO_3) where 0 ≤ x ≤1. With x=0 or x=1, the corresponding four chemically distinct types of silicate dust are specifically fayalite(Fe_2SiO_4), ferrosilite(FeSiO_3), forsterite(Mg_2SiO_4) and enstatite(MgSiO_3) <cit.>.Silicate dust is widely detected in astronomical objects via their spectral features. The earlier detections are based on the 9.7 and 18 features, which are wide, smooth and featureless and identified as from the Si-O stretching and O-Si-O bending mode respectively of amorphous silicate. The IRAS/LRS detected such spectral features in over 4000 O-rich stars <cit.>. With the increase of spectrometer sensitivity and spectral resolution, the ISO/SWS for the first time detected many narrow spectral features in evolved O-rich stars that are identified as from crystalline silicates <cit.>. Afterwards, the crystalline silicates are found in various types of objects in the ISO/SWS and Spitzer/IRS spectrum <cit.>. The narrow features of crystalline silicates appear mainly in seven complexes at approximately 10, 18, 23, 28, 33, 40 and 69 <cit.>.The formation mechanism of amorphous and crystalline silicates is still unclear in spite of numerous detections of the spectral features in the circumstellar envelope of O-rich evolved stars. Dust condensation generally takes two steps <cit.>. The first step is the formation of tiny seed nuclei probably of the order of a few nanometers, which will serve as centers for later growth to macroscopic grains. The material from which the seed nuclei are formed may either be the same material as the final dust grains (homogeneous dust formation) or be a different material from the final dust grains (heterogeneous dust formation). For carbon grains, there is agreement that the dust grains would experience a heterogeneous growth on the seed nuclei TiC (or ZrC) <cit.>. Meanwhile, the situation with respect to silicate dust is controversial. It has been long known that homogeneous silicate dust formation is not possible and requires the formation of some different kind of seed particle <cit.>. <cit.> have proposed TiO_ 2 as the seed nuclei from theoretical considerations. Corundum(Al_2O_3) was also suggested as the seed nuclei of silicate dust formation but met challenge since small aluminium clusters have low bond energy and form only at very low temperatures <cit.>. Considering the transparency of the corundum and full Mg-bearing silicate dust in the optical and near-infrared bands <cit.>, they were thought to be too transparent to be effectively accelerated away from the star by radiation pressure. But <cit.> pointed out that large grains of these two kinds of dust might form in the close vicinity of a few radii to the star via photon scattering rather than absorption and deduced the possibility of corundum-core and silicate-mantle dust. Afterwards, <cit.> constructed a Dynamic Atmosphere and Radiation-driven Wind models based on Implicit Numerics (DARWIN) and studied the condensation of Al_2O_3 and silicate dust in the vicinity of a few radii from AGB stars. Their work shows that the Al_2O_3 dust may condense at a temperature of ∼ 1400K in a vicinity of 2 radii from the star and may serve as the seed nuclei to form a Al_2O_3 core and silicate mantle dust(see , and reference therein), but under the condition that some impurities due to a few percent of transition metals (e.g., Cr) be incorporated. <cit.> also found the connection of SiO maser with corundum by observing S Ori which found that both showed the phase dependence, meanwhile this star show no evidence for hot circumstellar silicate dust. Therefore, corundum may be a strong candidate for the silicate nucleation, but still with some condition and unexplained fact.Silicon monoxide (SiO) has been discussed several times whether it can serve as the seed nuclei of silicate dust grains. There are reasons supporting this suggestion. Condensation of SiO is the most obvious initial step in silicate formation <cit.>, since SiO is one of the most abundant of all the gas-phase species of refractory elements in oxygen-rich environment. Besides, a rather stable condensed phase exists with chemical composition. On the other hand, early laboratory experiments showed that the SiO condensation commences at a temperature ∼ 600 K <cit.>, much lower than the observed condensation temperature of circumstellar silicate that is usually around 900 K. However <cit.> re-measured the vapour pressure of solid SiO which is used to calculate the condensation temperature of SiO. Their new value of the SiO condensation temperature is increased to about 700 K (depending on the mass loss rate) and much closer to the silicate dust condensation temperature derived from infrared observation. Although the new condensation temperature is still about 100 K lower than observed, the difference may be explained by the greenhouse effect. This brings about new life of SiO as the seed nuclei of silicate dust formation.The formation of crystalline silicate dust is even more unclear. The question, which kind of silicate dust form first, amorphous or crystalline silicate, is still under debating. <cit.> calculated the condensation sequence of O-bearing molecules in the circumstellar envelopes of O-rich evolved stars, and found that Fe-free olivine will form first due to its high condensation temperature. Their model can solve the question of missing Fe element in the crystalline silicate dust. Due to the efficient absorption in the near infrared,Fe-containing silicate may only be stable at large radius from the stellar surface <cit.>. However, this argument is challenged by lack of laboratory support. Another mechanism needs a conversion from amorphous form through thermal annealing especially in circumstellar disc <cit.>. But this model was questioned by <cit.> whoargued that the circumstellar disks are after all not an accretion disk, so radial transfer of material may not happen. Other scientists prefer to believe that crystalline silicates are the firstly formed dust. <cit.> proposed a model in which the Fe^2+ ion will penetrate into the crystalline dust and destroy its lattice structure after the condensation of crystalline silicate, which finally leads to formation of amorphous silicate. However, there is no observation of severe reduction of crystalline silicate features.In this work, we try to study the relation between silicate dust and SiO gas molecules in the evolved stars, which will give some hints on the formation mechanism of silicate dust. SiO molecules are usually traced by their radio lines, either thermal lines from vibrational ground state v=0 or maser lines from vibrational excited states v=1 or v=2. Usually the thermal lines are much weaker than the maser lines <cit.>. The SiO masers usually occur within a few stellar radii of the stellar surface between the hot molecular inner envelope and the cooler region at 35 R_ star where the dust (especially silicates) forms, and the temperature decreases from 1770K to 1100K <cit.>.§ SAMPLE SELECTION AND SIO MASER OBSERVATION§.§ Sample and observationThe correlation between the intensity of SiO maser and silicate emission would reflect whether SiO molecules are the seed of silicate dust. In order to study the intensity relation, we try to detect SiO maser emission from evolved O-rich stars. The sample of observation is consisted of evolved stars which show emission lines of crystalline silicate in order to investigate whether SiO molecules are related to silicate crystallization. We select the stars from three relatively systematic search for crystalline silicate by <cit.>, <cit.> and <cit.>. For the convenience of calculating the intensity of crystalline silicate emission, we require that the fully processed ISO/SWS spectrum is available. The ISO/SWS spectrum <cit.> covers a wider range (2.4-45.4 ) of wavelength in comparison with the Spitzer/IRS spectrograph (5.2-38.0) <cit.>so that the measurement of the intensity of crystalline silicate would be more complete. In addition, the location of the 13.7-m telescope of the Purple Mountain Observatory (PMO) in Delingha, China (PMO/Delingha hereafter) to perform the observation constrains the sources in the northern sky. With the above considerations, a sample of 31 evolved stars is selected, whose position and type are listed in Table 1. It can be seen that the sample is consisted of AGB stars, OH/IR stars, post-AGB stars or PPN, PN, and a few supergiant stars. From the available IRAS measurement at 12, 25 and 60, all except two have [12]-[25] (≡2.5* F_ 25/F_ 12 in accordance with) bigger than 1.5 or [25]-[60] > 0. According to the infrared spectrum either from ISO/SWS or Spitzer/IRS, the silicate feature(s) around 10 is mostly in emission while in absorption for seven objects. In the following it will be shown that the SiO maser is not detected in the sources that exhibit 10 absorption except in OH 26.5+0.6. OH 26.5+0.6 is removed for later analysis of the relation between SiO maser and silicate dust because it is optically thick around 10.The observation was carried out from 2 May 2013 to 2 June 2013 by using the PMO/Delingha 13.7m telescope. The central frequency is tuned to the rest frequency of the SiO v=1, J=2-1 line, i.e. 86.243GHz. The observation mode is single-point, with a band width of 1GHz and a frequency resolution of 61kHz corresponding to a velocity resolution of 0.21 km/s. The observations were executed with the low side-band of Beam 2(B2LS) at 86.1 GHz, of which the main beam width was about 61”, and the main beam efficiency (η_mb) was 0.557 for the ^28SiO v=1, J=2-1 line. The typical rms noise level of B2LS is usually better than 0.09 K at the 4σ level with an on-source exposure time of 1800s as the system temperature was normally around 160 K(see Status Report on the 13.7 m MM-Wave Telescope for the 2013-2014 Observing Season). All the raw data were processed with the GILDAS package <cit.>. It should be noticed that any results presented in the Figures and Tables have been converted to the intensity unit Jansky (Jy) with a conversion factor calculated by the formula: 2k/A·η, where k is the boltzmann constant, A is the physical area of the antenna and η is the aperture efficiency. Since the aperture efficiency during our observation time was 0.461, the scale factor from antenna temperature to Jy turned out to be 40.63. The detailed information of those stars including the equatorial coordinates (J2000), type and the observational results are listed in Table <ref>. We detected the SiO maser emission from5 of the 31 sample stars. They are IRAS 01037+1219 (Oxygen rich AGB star), IRAS 05073+5248 (OH/IR star), IRAS 19192+0922(Proto-Planetary Nebula), OH 26.5+0.6 (OH/IR star, removed for later analysis due to its absorption around 10) and NML Cyg (Red Supergiant) (see Figure <ref>). These stars were previously detected and our observation confirmed that they are SiO maser emitters. Due to the relatively small aperture of the 13.7-m antenna, no new detection was found. On the other hand, the SiO maser is variable and its intensity is then phase dependent (e.g. ), thus only a fraction of the sample can be detected at a given epoch. Although it would be better to analyze the relation of SiO maser detection with phase, accurate periods are not known for many of the sample stars, and in some cases they may be irregular. §.§ Sources from literatures As the SiO maser detection rate is lower than expected, our sample needs to be expanded in order to obtain a reliable statistical result. We assembled the SiO maser sources from published papers. From numerous detections of SiO maser, the objects are selected by the following criteria: (1) with the ISO (Infrared space Observatory) spectral data, which have already been processed by Sloan et al. (2003) in a uniform manner, to guarantee the quality of the infrared spectrum; (2) exhibition of distinguished 9.7 and 18emission features which indicate silicate dust, to ensure the detection of silicates;and (3) with clearly SiO maser emission at v=1,J=2-1 or v=1,J=1-0. Consequently, the sample has 21 sources with theSiO v=1,J=2-1 line and 28 sources with the SiO v=1,J=1-0 line emission. Finally, all the intensity units of the SiO maser emission are converted into Jy for convenience of comparison (see Table <ref>). For the sources from the work of <cit.>, <cit.> and <cit.>, the conversion factor is 54.0, 2.5175 and 13.29 respectively. In combination with the four sources from our observation, the sample all show emission features at the wavelengths of silicate implying optically thin case.§ CALCULATION OF SILICATE DUST EMISSIONThe emission power of silicate dust is calculated by the PAHFIT package <cit.> incorporated into the IDL software. Following the method of <cit.>, we employed the CDE-profile of <cit.> to characterize the emission feature of amorphous silicates. Though the PAHFIT code was originally designed on the purpose of simulating the features of PAH molecules, the code was lately modified and can then be used to fit the features of silicate dust. The advantage of this code is that one can figure out the features' information precisely and avoid inducing uncertainties brought in by dust temperatures for calculation of silicate crystallinity <cit.>. To make sure that we have recognized the crystalline silicate's features correctly, we also consulted the work of <cit.>, and only considered the crystalline silicate features that have been identified. An example is showed in Figure 2. From Figure 2 we can see that this fitting process decompose the ISO/SWS spectrum into (1) a stellar continuum (bright blue line), (2) a dust continuum (orange dashed-dotted lines): F_ dust cont., (3) the amorphous silicate features (red solid line, the sum of the two dotted lines mainly from the 9.7 and 18 respectively): F_ am. feat., and (4) a series of individual narrow features (blue solid lines). Among the individual features, we only choose those features that have been proved to be from crystalline silicate by <cit.>, took the sum of their fluxes as the emitting fluxes of crystalline silicate dust: F_ cry.. Considering that most of the dust continuum will be ascribed to the amorphous silicate dust <cit.>, we can consequently attain the emission flux of amorphous silicate dust F_ am.=F_ dustcont.+F_ am. feat.. Then, crystallinity of silicate, defined as the mass fraction of crystalline silicate dust to total silicate dust, can be approximated by the flux ratio of crystalline silicate dust to silicate dust, i.e. η=F_ cry./(F_ cry.+F_ am.) <cit.>. Though the flux ratio η can not serve the same as the mass ratio of crystalline to amorphous silicate dust, using this ratio as the indicator of crystallinity of silicates is reasonable. The emission flux F_ν∝ m_ dustκ_abs B_ν(T_ dust), which means that the emission flux ratio would represent the mass ratio if the crystalline and amorphous silicate dust share the same emissivity. As <cit.> pointed out, the emissivity of crystalline silicate is comparable to amorphous silicate in mid-infrared, and weaker in visible and near-infrared. Thus in the optically thin case, where the crystalline silicate dust may be colder than amorphous, the flux ratio F_ cry./(F_ am.+F_ cry.) would be lower than the mass ratio m_ cry./(m_ am.+m_ cry.) and could be regarded as the lower limit of crystallinity. While in the case of high mass loss rate, when the infrared radiation mainly comes from the outer cold dust envelope in the mid-infrared where amorphous and crystalline silicates share similar emissivity and dust temperatures, the flux ratio would then approximate very closely to the dust mass ratio. Notice that most of the sources are within 1 kpc even the furthest one AH Sco is 2.7 kpc from the Earth(see Table <ref>), the influence of interstellar dust on the stellar spectra should not be serious. According to recent determination of interstellar extinction in the infrared, the extinction around the 10 silicate feature is only about 5 percent of the visual band (see e.g. ), which converts to less than 5 percent decrease of flux if an average = 1.0 mag per kpc is adopted. Moreover, there is no detection of crystalline silicate in the diffuse ISM which set an upper limit of interstellar crystallinity to less than 5% <cit.> or 2.2% <cit.>. Since the interstellar silicate absorption is very small (amorphous) or little (crystalline), the contamination of interstellar silicate is not taken into account.For the SiO maser power, we simply integrated its flux intensity over the velocity space and took it as the emission power. The results are listed in Table <ref> and Table <ref>.§ RESULT AND DISCUSSIONFollowing the method of Section <ref>, the total emission power of the silicate dust and the flux ratio are computed and listed in Table <ref>. As discussed above, the flux ratio of crystalline to total silicate could be equivalent to or be the lower limit of the silicate crystallinity. The flux ratio in Table <ref> ranges from about 6% to 29% with a concentration around 10%. As <cit.> pointed out, The relative abundance of crystalline silicates is in general quite modest, 10% - 15%, in the evolved stars. The work of <cit.> yielded a similar result, most of their sources showed a crystallinity of 8%∼ 16% while a range of 5%-28%. Our consistency with these works indicates that the flux ratio indeed represents the mass ratio in these evolved stars.With the total silicate dust emission power (Table <ref> column 2+column 3) and the SiO maser emission power from Table <ref> we can easily figure out the correlation between them. However one thing that we should bear in mind is that the emission power of both silicate dust and SiO maser is inversely proportional to the square of distance, which means there should always be correlation between the apparent power of silicate dust and SiO maser power due to this geometrical effect. In order to eliminate the distance effect and to study the relation between the intrinsic power of silicate dust and SiO maser, both measured powers are normalized to their intensity in the IRAS 60 band (column 5 of Table <ref>). NML Cyg and U Cep are removed from following analysis due to the lacking of IRAS 60 measurement.Figure <ref> shows that the emission power of silicate dust is clearly correlated with the emission power of SiO maser features with respectively the Pearson correlation coefficient of 0.60 for SiO v=1,J=1-0 and 0.44 for v=1,J=2-1. Because the emission power is tightly correlated to the mass for both SiO masers and silicate dust, the correlation of emission power indicates that the abundance of silicate dust is correlated with that of SiO molecules. This relation may imply that SiO is the seed nuclei of silicate dust grains as suggested by <cit.> and <cit.>. On the other hand, there are environmental factors in addition to the emitter abundance that influence the emission power. For SiO maser, the emission is an amplified maser radiation whose intensity relies on the abundance in a non-linear way. Depending on the pumping mechanism, the maser intensity changes with the velocity gradient for a radiative pumping <cit.> and with the density for a collisional pumping <cit.>. <cit.> modelled the SiO maser in combination with the dynamic atmosphere which found that the infrared radiation of dust plays some role in the maser emission. This result implies a correlation between the SiO maser emission and dust radiation. Since the silicate dust is the dominant component of dust in the oxygen-rich circumstellar envelope, the present correlation of silicate dust emission and SiO maser power is expected.To make things more complex, silicate features show both phase-dependent and evolutionary variability <cit.>, however, the silicate features vary very mildly and should not bring significant difference. The SiO maser varies as well and in a more irregular way.Because the method is statistical and the sample spans various phases of variation, variability may not be a serious problem. SiO masers from the v=1 state are usually found within about 2-4 stellar radii (e.g. ) and most of the dust characterised as silicate, around stars with SiO masers, is formed just outside this at 5-10 stellar radii (e.g. ). Thus, gas-phase SiO and silicate dust are found in close proximity.Therefore, the correlation between SiO maser power and silicate dust emission power may indicate a true connection between SiO molecules and silicate nucleation, but the connection is not very direct since other factors play roles in determining the emission power. Different from the total emission power of silicate dust, the crystallinity (here defined as η=F_ cry./(F_ cry.+F_ am.) ) shows hardly any correlation with the SiO maser emission power (see Figure <ref>). With a very small (<0.1) Pearson correlation coefficient, the crystallinity neither shows any correlation with the total silicate emission power. What determines crystallinity of evolved stars is an open question, a popularly studied factor is mass loss rate while the conclusion is still controversial. <cit.>examined whether the strength of some specific strong features around 23, 28 and 33 correlated with mass loss rate, and did not find any clear quantitative correlation, except a general tendency that stars with high mass loss rate would have higher probability of crystallization. Some people think that the crystalline silicates can only be formed in high mass loss rate stars which have high dust column density <cit.>. A latest work of <cit.> considered 28 oxygen-rich evolved stars. With PAHFIT code they fitted the ISO/SWS spectra of these sources and deduced the crystalline silicates flux ratios. Together with the dust mass loss rate obtained from SED modelling by 2DUST code <cit.>, they investigated the relation between the silicate dust crystallinities and the stellar dust mass loss rates, their result also showed that the silicate's crystallinity is barely correlated with the stellar dust mass loss rate. The above analysis indicates that the SiO maser emission power is correlated with the silicate dust emission that is proportional to the dust mass for the investigated optical thin cases, thus the SiO maser power could be correlated with the dust massalthough the dust temperature would also play important role in determining the dust emission power. Furthermore, the dust mass can be a function of the mass loss rate.From the fact that there is no correlation between crystallinity and SiO maser, it may be drawn indirectly that crystallinity is not correlated to mass loss rate since it does not correlate with silicate dust mass in the optically thin case.§ SUMMARY In this work we investigated the relation between SiO maser power and silicate dust emission power in order to study whether SiO molecules are the seed nuclei of silicate dust. The observation to search for SiO v=1, J=2-1 maser line was performed by using the PMO/Delingha 13.7-m radio telescope and five of 31 sources were detected successfully. In combination with previous observational results, we composed a sample of 21 oxygen-rich evolved stars with SiO v=1, J=2-1 line and 28 stars with SiO v=1, J=1-0 line. The silicate emission power is calculated by using the PAHFIT code. We found that there is clear correlation between the SiO maser and the silicate's emission power, which is not against the hypothesis that the SiO molecules may serve as the seed nuclei of silicate dust. On the other hand, no correlation is found between SiO maser and silicate crystallinity.We thank Prof. Aigen Li for helpful discussion and anonymous referee for constructive suggestions. This work is supported by NSFC through Projects 11373015, 11533002, and 973 Program 2014CB845702. We acknowledge the use of PMO/Delingha 13.7m telescope and the sincere help of the staff. 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Type A/E^1 Disc Det^2 Sigma V_LSR FWHMKkm/s km/sIRAS 01037+1219 01 06 25.98+12 35 53.05 O-AGB E --Y 0.0177.947 5.692IRAS 01304+6211 01 33 51.21+62 26 53.20 OH/IR starE--0.022TW Cam 04 20 47.62+57 26 28.47 Post-AGB-- disc^a,b0.016 IRAS 05073+5248 05 11 19.44+52 52 33.20 OH/IR star E --Y 0.0221.531 3.731 IRAS 06034+1354 06 06 14.91+13 54 19.10 Post-AGB -- disc^a,b0.022 IRAS 06072+0953 06 09 57.99+09 52 31.82 Post-AGB-- disc^a,b0.023 SU Gem 06 14 00.02+27 42 12.17 Post-AGB-- disc^a,b0.021UY CMa 06 18 16.37-17 02 34.72 Post-AGB-- disc^b0.024 HD 44179 06 19 58.22-10 38 14.71 post-AGB Edisc^b0.024 HD 45677 06 28 17.42-13 03 11.14 B[e] star E --0.023 IRAS 06338+5333 06 37 52.43+53 31 01.96 post-AGB-- disc^b0.021 HD 52961 07 03 39.63+10 46 13.06 Post-AGB -- disc^b0.022 HD 161796 17 44 55.47+50 02 39.48 Post-AGB -- --0.023NGC 6543 17 58 33.42+66 37 59.52 PN -- --0.023IRAS 18123+0511 18 14 49.39+05 12 55.70 Post-AGB E disc^b0.021MWC 922 18 21 16.06-13 01 25.69 Peculiar object A--0.035MWC 300 18 29 25.69-06 04 37.29 supergiant A --0.027AC Her 18 30 16.24+21 52 00.61 post-AGB star Edisc^a,b0.102OH 26.5+0.6 18 37 32.51-05 23 59.20 OH/IR star A -- Y0.01827.326 4.003IRAS 18354-0638 18 38 06.00-06 35 35.00 post-AGB -- --0.044EP Lyr 19 18 19.55+27 51 03.19 post-AGB-- disc^a,b0.028IRAS 19192+0922 19 21 36.52+09 27 56.50 PPN E -- Y0.016-70.0933.498IRC+10420 19 26 48.10+11 21 16.74 post-RSG E --0.018IRAS 19283+1944 19 30 29.48+19 50 41.00 PN A --0.020IRAS 20043+2653 20 06 22.74+27 02 10.60 OH/IR Thick A --0.017IRAS 20056+1834 20 07 54.62+18 42 54.50 post-AGB-- disc^b0.012NML Cyg 20 46 25.54+40 06 59.40 Red Supergiant A -- Y0.016 1.58126.772IRAS 21554+6204 21 56 58.18+62 18 43.60 OH/IR Thick A --0.016IRAS 22177+5936 22 19 27.48+59 51 21.70 OH/IR star A --0.018HD 213985 22 35 27.52-17 15 26.89 post-AGBEdisc^b0.017 IRAS 23239+5754 23 26 14.82+58 10 54.60 PN E--0.015 1 “A” means absorption in 9.7 feature, while “E” means emission. 2 The observational result of Delingha. a,b Reference: (a) <cit.>; (b) <cit.> l|c|c|c|c|c The SiO maser sourcesName R.A. Decl. Dist^α P_J=1-0^β P_J=2-1^βpc Jy km/s Jy km/sAH Sco17 11 17.02-32 19 30.712600^1 112.03^a--BU And23 23 39.90+39 43 36.92554^2 30.0086^a --EP Aqr21 46 31.85-02 12 45.93173^2 0.3524^a--FP Aqr20 46 36.50-00 54 11.10140^2 4.89^b --GY Aql19 50 06.33-07 36 52.491869^2 20.11^b 130.14^dIRAS 01037+1219 01 06 25.98+12 35 53.05540^3 --302.35^eIRAS 05073+5248 05 11 19.44+52 52 33.20510^1 --61.38^e IRAS 19192+0922 19 21 36.52+09 27 56.50950^3 --36.02^e O Cet 02 19 20.79-02 58 39.501325^3 --447.44^fR And 00 24 01.95+38 34 37.35120^2 2.2657^a--R Aql 19 06 22.25+08 13 48.011273^2 73.00^c 334.90^gR Aqr 23 43 49.46-15 17 04.14310^1 225.79^b236.78^hR Cas 23 58 24.87+51 23 19.70181^1 251.25^a335.42^gR Hya 13 29 42.78-23 16 52.77359^1 189.00^c543.78^dR Peg 23 06 39.17+10 32 36.09 83^2 29.93^a --RR Aql19 57 36.06-01 53 11.33480^1 92.29^a --RR Per02 28 29.40+51 16 17.32758^1 --57.24^d RT Vir13 02 37.98+05 11 08.38816^3 --127.98^dS Per 02 22 51.71+58 35 11.45230^2 63.72^a 125.21^gS Scl 00 15 22.27-32 02 42.992300^4 4.33 ^a --S Vir 13 33 00.11-07 11 41.02362^2 16.44^a --SS Peg22 33 58.33+24 33 53.98477^1 27.79^b --T Cas 00 23 14.27+55 47 33.21568^2 8.23^b --T Cep 21 09 31.78+68 29 27.20282^2 107.91^b780.49^gTX Cam05 00 50.39+56 10 52.60179^2 91.15^b 778.79^gU Cep 01 02 18.45+81 52 32.08390^3 --583.91^gU Her 16 25 47.47+18 53 32.86461^5 30.16^a 255.54^iUU For02 37 23.07-26 58 42.31710^3 36.85^a --UX Cyg20 55 05.52+30 24 52.10900^1 6.34^a64.80^g VX Sgr18 08 04.05-22 13 26.631570^4 626.32^a3798.30^hVY CMa 07 22 58.33-25 46 03.24502^2 --6730.21^h W Hya 13 49 02.00-28 22 03.49139^2 559.917^a 6146.00^hW Per 02 50 37.89+56 59 00.27650^2 12.77^b 33.85^iX Oph 18 38 21.12+08 50 02.75235^2 65.78^a --Z Cas 23 44 31.59+56 34 52.70797^3 3.98^b --Z Cyg 20 01 27.50+50 02 32.69930^3 3.37^a--α Distance to Earth. Reference.- (1) <cit.>; (2) <cit.>; (3) <cit.>; (4) <cit.>; (5) <cit.> β Maser power denotes the integrated intensity of the maser emission. Reference for the SiO maser: (a) <cit.>; (b) <cit.>;(c) <cit.>; (d) <cit.>; (e) Delingha, this work; (f) <cit.>; (g) <cit.>;(h) <cit.>; (i) <cit.>l|c|c|c|c The silicate emission power of all the maser sourcesName F_ crys F_ cont F_ crys/F_ am+F_ crys F_ 60W/m^2W/m^2 JyAH Sco2.195E-0112.504E-010 0.08173.310BU And3.486E-0121.418E-011 0.197 7.100 Chi Cyg2.011E-0114.136E-010 0.04680.670EP Aqr6.151E-0128.459E-011 0.06847.130FP Aqr7.518E-0123.608E-011 0.17217.320GY Aql4.735E-0111.123E-010 0.29647.310 IRAS 01037+12193.574E-0112.281E-010 0.135 215.200 IRAS 05073+52481.064E-0111.028E-010 0.09472.390 IRAS 19192+09223.928E-0122.479E-011 0.13741.420 M ira5.036E-0116.327E-010 0.074 300.800 R And2.491E-0124.885E-011 0.04924.160 R Aql5.885E-0129.727E-011 0.057 139.700 R Aqr3.782E-0112.166E-010 0.14966.650 R Cas9.526E-0123.090E-010 0.030 102.800 R Hya2.140E-0112.130E-010 0.09190.080 R Peg1.457E-0122.734E-011 0.05111.060RR Aql8.530E-0129.634E-011 0.08127.470RR Per6.734E-0131.103E-011 0.058 4.270RT Vir8.060E-0127.520E-011 0.09739.270 S Per2.097E-0111.048E-010 0.16740.590 S Scl8.244E-0131.447E-011 0.054 6.930 S Vir1.620E-0121.721E-011 0.086 8.250SS Peg1.469E-0129.323E-012 0.136 4.470 T Cas3.413E-0124.791E-011 0.06724.380 T Cep9.022E-0121.181E-010 0.07141.610TX Cam8.753E-0121.523E-010 0.054 134.300 U Her1.549E-0123.589E-011 0.04127.220UU For2.607E-0123.755E-011 0.06534.410UX Cyg9.143E-0132.175E-011 0.04043.940VX Sgr2.356E-0111.011E-009 0.023 262.700VY CMa1.027E-0103.588E-009 0.0281453.000 W Hya3.502E-0115.801E-010 0.057 195.000 W Per2.035E-0121.828E-011 0.10014.870 X Oph3.456E-0125.258E-011 0.06222.610 Z Cas1.238E-0121.524E-011 0.075 6.700 Z Cyg2.520E-0121.288E-011 0.16410.660 The total power of crystalline silicate dust(second column), continuum power(third column) and crystallinity of silicate dust(fourth column).

http://arxiv.org/abs/1702.07820v1

Asynchronous Incremental StochasticDual Descent Algorithm for Network Resource Allocation Amrit Singh Bedi, Student Member, IEEE and Ketan Rajawat, Member, IEEE The authors are with the Department of Electrical Engineering, IIT Kanpur, Kanpur (UP), India 208016 (email: amritbd, ketan@iitk.ac.in). ======================================================================================================================================================================================================================= Stochastic network optimization problems entail finding resource allocation policies that are optimum on an average but must be designed in an online fashion. Such problems are ubiquitous in communication networks, where resources such as energy and bandwidth are divided among nodes to satisfy certain long-term objectives. This paper proposes an asynchronous incremental dual decent resource allocation algorithm that utilizes delayed stochastic gradients for carrying out its updates. The proposed algorithm is well-suited to heterogeneous networks as it allows the computationally-challenged or energy-starved nodes to, at times,postpone the updates. The asymptotic analysis of the proposed algorithm is carried out, establishing dual convergence under both, constant and diminishing step sizes. It is also shown that with constant step size, the proposed resource allocation policy is asymptotically near-optimal. An application involving multi-cell coordinated beamforming is detailed, demonstrating the usefulness of the proposed algorithm. Stochastic subgradient, resource allocation, asynchronous algorithm, incremental algorithm. § INTRODUCTIONThe recent years have witnessed an unprecedented growth in the complexity and bandwidth requirements of network services. The resulting stress on the network infrastructure has motivated the network designers to move away from simpler or modular architectures and towards optimum ones. To make sure that resources such as bandwidth and energy are allocated efficiently, optimum designs advocate cooperation between the network nodes <cit.>. This paper considers the problem of cooperative network resource allocation that arises in wireless communication networks <cit.>, smart grid systems <cit.>, and in the context of scheduling <cit.>. Of particular interest is the stochastic resource allocation problem, where the goal is to find an allocation policy that is asymptotically optimal <cit.>. Although such problems are infinite dimensional in nature, they can be solved in an online fashion via stochastic dual descent methods, allowing real-time resource allocation that is also asymptotically near-optimal <cit.>. Heterogeneous networks are common to a number of applications where the energy availability, computational capability, or the mode of operation of the nodes is not the same across the network. Key requirements for heterogeneous network protocols include scalability, robustness, and tolerance to delays and packet losses. Towards this end, a number of distributed algorithms have been proposed in the literature <cit.>. By eliminating the need for a fusion center, the distributed algorithms operate with reduced communication overhead, and render the network resilient to single-point failures. Most distributed algorithms still place stringent communication and computational requirements on the network nodes. For instance, the dual stochastic gradient methods entail multiple updates and message exchanges per time slot, and cannot handle missed or delayed updates. In heterogeneous networks, such delays are often unavoidable, arising due to poor channel conditions, traffic congestion, or limited processing power at certain nodes. This paper proposes a distributed asynchronous stochastic resource allocation algorithm that tolerates such delays. The next subsection outlines the main contributions of this paper.§.§ Contributions and organizationThe stochastic resource allocation problem is formulated as a constrained optimization problem where the goal is to maximize a network-wide utility function. The allocated resources at the different nodes in the network are coupled through constraint functions that involve expectations with respect to a random network state. Specifically, the aim is to find an allocation policy that satisfies the constraints on an average. The distribution of the state variables is not known, so that the optimization problem does not admit an offline solution. Instead, the idea is to observe the instances of the state variables over time, and allocate resources in an online manner. It is well-known that stochastic dual descent algorithms yield viable online algorithms for such problems <cit.>. Within the heterogeneous network setting considered here, the focus is on distributed algorithms that can tolerate communication and processing delays <cit.>. Different from the state-of-the-art algorithms that utilize the standard stochasticgradient methods <cit.>, we develop two variants of the asynchronous dual descent algorithm that allow some of the nodes in the network to temporarily “fall back,” in the event of low energy availability, unusually large processing delay, node shutdown, or channel impairments. The first asynchronous variant utilizes a fusion center to collect the possibly delayedgradients from various nodes and carry out the updates (cf. <ref> ). The second variant eliminates the need for the fusion center, and instead utilizes the fully distributed and incremental stochasticgradient descent algorithm, where the nodes carry out updates in a round-robin fashion and pass messages along a cycle (cf. <ref> ). As earlier, the use of stalegradients for primal and dual updates, allows the algorithm to be run on two different clocks, one corresponding to the local resource allocation and tuned to the changing random network state, while the other dictated by the message passing protocol. The key feature of the proposed algorithm is the possibility for the second clock to slow down temporarily and wait for slower nodes to catch up. The proposed algorithm thus allows timely resource allocation, while tolerating occasional delays in message passing.The asymptotic performance of the proposed algorithm is studied under certain regularity conditions on the problem structure and bounded delays. In particular, the asymptotic performance of the asynchronous incremental stochasticsubgradient descent (AIS-SD) algorithm is characterized under both, diminishing and constant step-sizes. The overall structure of the proof is based on the convergence results in the incremental stochasticsubgradient descent algorithm of <cit.>and the asynchronous incremental subgradient method of <cit.>. Specific to the resource allocation problem at hand, the asymptotic near-optimality and almost sure feasibility of the primal allocation policy is established for the case of constant step sizes. When applied to resource allocation problems, the proposed algorithm is called asynchronous incremental stochastic dual descent (AIS-DD). It is remarked that since the proposed algorithms utilize stochasticsubgradient descent, their computational complexity is also comparable to other distributed stochastic algorithms <cit.>. The calculation of thesubgradient is the most computationally expensive step, and like other first-order algorithms, must be carried out at every time slot.Finally, the stochastic coordinated multi-cell beamforming problem is formulated and solved via the proposed algorithm. Detailed simulations are carried out to demonstrate the usefulness of the proposed algorithm in delay-prone and distributed environments. Summarizing, the main contributions of the paper include (a) the AIS-SD algorithm and its convergence (b) primal near-optimality and feasibility results for the allocated resources using AIS-DD; and (c) demonstration of the proposed algorithm on a practical stochastic coordinated multi-cell beamforming problem. The rest of the paper is organized as follows. Sec. <ref> provides an outline of the related literature. Sec. <ref> describes the problem formulation and recapitulates the known results. Sec. <ref> detailsthe proposed algorithm. Sec. <ref> lists the required assumptions, and provides the primal and dual convergence results. Sec. <ref> formulates the stochastic version of the coordinated beamforming problem along with the relevant simulation results. Finally, Sec. <ref> concludes the paper. §.§ Related work Resource allocation problems have been well-studied in the context of cross-layer optimization in networks <cit.>. Popular tools for solving stochastic resource allocation problems include the backpressure algorithm <cit.> and variants of the stochastic dual descent method <cit.>. However, most of these works only consider synchronous algorithms, and the effect of communication delays has not been examined in detail. An exception is the asynchronous subgradient method proposed in <cit.>, where delayed subgradients were utilized for resource allocation. The present work extends the algorithm in <cit.> by allowing delayed stochastic subgradients. Additionally, the proposed algorithm is also incremental, and is therefore applicable to a wider variety of problems. Depending on the mode of communication among the nodes, distributed algorithms can be broadly classified into three categories, namely, diffusion, consensus, and incremental <cit.>. Of these, the incremental update rule generally incurs the least amount of message passing overhead <cit.>, and is of interest in the present context. The incremental subgradient descent and its variants have been widely applied to large-scale problems, and generally exhibit faster convergence than the traditional steepest descent algorithm and its variants <cit.>.The stochastic gradient and subgradient algorithms are well-known within the machine learning and signal processing communities <cit.>. The incremental stochastic subgradient method, with cyclic, random, and Markov incremental variants, was first proposed in <cit.>. The asymptotic analysis of dual problem in the present work follows the same general outline as that of the cyclic incremental algorithm in <cit.>, with additional modifications introduced to handle asynchrony. It is emphasized that these modifications are not straightforward, since the delayed stochastic subgradient is not generally a descent direction on an average. The present work also allows delays in both, primal and dual update steps, and establishes asymptotic near-optimality and feasibility of the primal allocation policies. Finally, saddle point algorithms have recently been applied to unconstrained <cit.> or proximity-constrained <cit.> network optimization problems, but do not readily generalize to the general form constrained optimization problem considered here.Asynchronous algorithms have also been considered within the Markov decision process framework <cit.>, though the setup there is quite different and does not apply to the problem at hand. On the other hand, asynchronous first order methods have attracted a significant interest from the machine learning community <cit.>. For problems where the exact subgradient is available at each node, the asynchronous alternating directions method of multiplier (ADMM) has been well-studied <cit.>. The present work considers stochastic algorithms, and thus differs considerably in terms of both analysis and the final results. Even among algorithms utilizing stochastic subgradients, the definition of asynchrony varies across different works. One way to model asynchrony is to allow each node to carry out its update according to a local Poisson clock. This approach is followed in <cit.>, all of which consider various consensus-based distributed subgradient algorithms. The asynchronous adaptive algorithms in <cit.> also subscribe to the same philosophy, with decoupled node-updates due to communication errors, changing topology, and node failures. The incremental algorithm considered here is very different in terms of operation and analysis.On the other hand, asynchronous operation can be modeled via delayed gradients or subgradients utilized for the updates. Aconsensus-based stochastic algorithm proposed in <cit.>, and utilizes randomly delayed stochastic gradients. Along similar lines, asynchronous saddle point algorithms for network problems with edge-based constraints have recently been proposed <cit.>. Finally, for the unconstrained variants of the problem, a non-parametric approach has been proposed in <cit.>. Different from these works, the network resource allocation framework considered here allows generic convex constraints. Further, the incremental algorithm developed here handles stale subgradients while incurring significantly lower communication overheads. Asynchronous variants of the classical or averaged stochastic gradient methods have been proposed in <cit.>. The generic problem of interest here is that of the minimization of a sum of private functions at various nodes. Further, a network with star topology is considered, with updates being carried out using delayedgradients collected at the fusion center. Different from these works, the proposed algorithm is incremental, does not require a fusion center, and is therefore more relevant to the network resource allocation problem at hand. Unlike these works, the present work also avoids making any assumptions on the compactness of the domain of the dual optimization problem. Before concluding, it is remarked that this work develops convergence results that hold on an average. Stronger results, where convergence is established in an almost sure sense, require a more involved analysis, and are not pursued here. The notation used in this paper is as follows. Scalars are represented by small letters, vectors by small boldface letters, and constants by capital letters. The index t is used for the time or iteration index. The inner product between vectors a and b is denoted by a,b. For a vector , projection onto the non-negative orthant is denoted by []^+. The expectation operation is denoted by . The notation ∇ is used for gradient and ∂ is used for the subgradient. The Euclidean norm is denoted by ·. § PROBLEM FORMULATION§.§ Problem statementThis section details the stochastic resource allocation problem at hand for a network with K nodes. The stochastic component of the problem is captured through the random network state, comprising of the random vectors ^i ∈^q for each node i ∈{1, …, K}, with unknown distributions. The overall problem is formulated as follows. :=maxf^i(^i)s.t.^̆i(^i) + [^̌i(^i,^i_^i)]≽ 0^i∈^i,^i∈^i where the expectation in (<ref>) is with respect to the random vector ^i and 𝖯 is finite. The optimization variables in (<ref>) include the resource allocation variables {^i∈ℝ^n}_i=1^K and the policy functions {^i:^q→^p}_i=1^K, under the constraints (<ref>)-(<ref>).Note that, the constraints in (<ref>) are required to be satisfied on an average, whereas those in (<ref>) are needed to be satisfied instantaneously. The functions f^i:^n → are assumed to be concave, and the sets ^i ⊆^n, convex and compact. The constraint function at node i is vector-valued, and is given by ^̆i(^i):=[u^i_1(^i) ⋯ u^i_d(^i)]^T, where {u_k^i(^i):^n →}_k=1^d are concave functions. On the other hand, no such restriction is imposed upon the vector-valued function ^̌i:^p×^q→^d and the compact set of functions {𝒫^i}_i=1^K. Of course, the overall problem still needs to adhere to certain regularity conditions (see Sec. <ref>), such as the Slater's constraint qualification and Lipschitz continuity of thegradient function; see (A1)-(A7). Since the distribution of ^i is also not known in advance, it is generally not possible to solve forin an offline manner. Therefore, an online algorithm is sought to solve problem `on the fly' as the independent identically distributed (i.i.d.) random variables {_t^i}_t∈ are realized and observed. For brevity, we denote _t^i:= _^i_t^i and _t^i(_t^i,^i):=^̆i(^i) + ^̌i(_t^i,^i_^i_t). Therefore, it is possible to write (<ref>) equivalently as [_t^i(_t^i,^i)]≽ 0. The algorithm outputs a sequence of vector pairs {_t^i, _t^i}_t, that are used for allocating resources in a timely manner. Towards this end, the stochastic dual descent algorithm has been proposed in <cit.>, which yields allocations that are almost surely near-optimal and provably convergent. In the present paper, the focus is on networked systems where both, allocations (^i,^i) and the functions f^i and ^i_t are private to each node i. Likewise, the random variable _t^i is also observed and estimated locally at each node i. In other words, while the nodes can exchange dual variables and numerical values of thegradients, they may not be willing to reveal the full functional form of the objective or constraint functions and other locally estimated quantities, owing to privacy and security concerns. Such privacy-preserving cooperation is common for many secure multi-agent systems <cit.>. To this end, the nodes may be arranged in a star topology, and utilize a centralized controller for collecting and distributing various algorithm iterates. Alternatively, ring topology may be used, allowing a fully distributed implementation, where the exchanges occur only between two immediate neighbors.In order to clarify the problem formulation considered in (<ref>), the following simple example is considered.Example 1. Consider the problem of network utility maximization over a wireless network consisting of K nodes. The aim is to maximize the network-wide utility given by ∑_i=1^KU(r^i)where U(·) is a concave function that quantifies the utility obtained by the node i upon achieving a rate r^i ∈[r_min,r_max]. The channel is assumed to be time-varying, and for each channel realization h^i, node i allocates the power p^i_h^i, achieving the instantaneous rate of log(1+h^ip^i_h^i), where the noise power is assumed to be one. The goal is to maximize the utility in (<ref>) subject to constraints on the average rate and the average power consumption, and the full problem can be written as (cf. (<ref>)): max_r^i,p^i ∑_i=1^KU(r^i)s.t. ∑_i=1^K(1/2log(1+h^ip^i_h^i)) ≥∑_i=1^Kr^i ∑_i=1^Kp^i_h^i ≤ P_max r^i∈[r_min,r_max],p^i∈𝒫^i It is remarked that 𝒫^i is a set of functions p^i:→, while p^i_h^i is a random variable that depends on h^i. That is, the optimization variables in (<ref>) include the rates r^i and the power allocation functions p^i. §.§ Existing approaches and challenges We begin with explicating the desirable features of an algorithm that seeks to solve (<ref>). Specifically, it is required that any such algorithm meets the following requirements. F1. The algorithm should allow nodes to “fall behind” temporarily, e.g., under poor channel conditions and intermittent transmission failures. F2. The algorithm should allow a distributed implementation, that is, without requiring a star-topology or an FC. These features are particularly important for large and heterogeneous networks where delays may be unavoidable and designating an FC may be impractical. Put differently, (F1) requires the algorithm to handle the inevitable delays that may occur due to temporarily poor channel conditions or noise. Complementarily, (F2) is anarchitectural requirement that must be kept in mind when choosing or designing the algorithm. Since the number of constraints in (<ref>) are finite, the problem is more tractable in the dual domain. To this end, introducing a dual variable ∈ℝ^d_+ corresponding to constraint in (<ref>), the stochastic (sub-)gradient descent method was proposed for solving such problems in <cit.>. The Lagrangian of (<ref>) is given by L(,,)={f^i(^i)+,[ _t^i(_t^i,^i)]} whereandcollect the primal optimization variables {^i}_i=1^K and {^i}_i=1^K respectively. Next, the dual function is obtained by maximizing L with respect toand . Since the Lagrangian is expressed as a sum of K terms, each depending on a different set of variables, the maximization operation is separable and the dual function takes the following form: D()= ∑_i=1^K max_^i∈^i, ^i∈^i[f^i(^i)+,([ _t^i(_t^i,^i)])] =: ∑_i=1^K D^i(). The dual problem is given by = min_∈ℝ^d_+ D^i().While for general problems, it only holds that 𝖣≥𝖯, that the stochastic resource allocation problem considered here has a zero duality gap, i.e., 𝖯 = 𝖣 <cit.>. The result utilizes the Lyapunov's convexity theorem and holds under strict feasibility (Slater's condition), bounded subgradients, and continuous cumulative distribution function of ^i for each i. It is remarked that similar results are well-known in economics <cit.>, wireless communications <cit.>, and control theory <cit.>. The result on zero duality gap legitimizes the dual descent approach, since the dual problem is always convex, and the resultant dual solution can be used for primal recovery. To this end, similar problems in various contexts have been solved via the classical dual descent algorithm <cit.>, wherein the primal updates utilize various sampling techniques. It is remarked however that from a practical perspective, solving the dual problem alone is not sufficient, since online allocation of power or rate variables necessitates determining the primal optimum variables {^i^⋆, ^i^⋆}. In the present case, since ^i is infinite dimensional, primal recovery and consequently, online resource allocation, is not straightforward. Since the distribution of ^i is not known in advance, solving (<ref>) via classical first or second order descent methods requires a costly Monte Carlo sampling step <cit.>. Instead, the use of stochastic subgradient descent has been proposed in <cit.>, which takes the following form for t≥ 1, D1. Primal updates: At time t, node i observes or estimates _t^i, and allocates the resources in accordance with: {^i_t(_t), ^i_t(_t)} = _∈^i, ∈Π^i_tf^i()+_t, _t^i(,) D2. Dual update: The dual updates at time t take the form: _t+1=[_t-_t^i(_t^i(_t),^i_t (_t))]^+. Here, Π_t^i :={^i_^i_t∈^p | ^i ∈^i} is the set of all legitimate values of the vector ^i_^i_t. The term _t^i(_t^i(_t),^i_t (_t)) is a stochastic gradient of the dual function D^i() at =_t. Further for notational brevity, _t^i():=_t^i(_t^i(),^i_t ()) is used throughout the paper. Recall that for a given , _t^i() is stochastic and depends on the random variable ^i_t, as discussed in Sec. <ref>. The algorithm is initialized with an arbitrary _1 and the resulting allocations are asymptotically near optimal and feasible. A constant step-size stochastic gradient descent algorithm is utilized in the dual domain, which not only allows recovery of optimal primal variables via averaging, but also bestows it the ability to handle small changes in the network topology or other problem parameters. The algorithm can be implemented in a distributed fashion in a network with star-topology, with the help of a fusion center (FC). Within the FC-based implementation, the primal iterates are calculated and used locally at each node i. At the end of each time slot, the node i communicates the gradient component ^i_t(_t) to the FC, which carries out the dual update (<ref>) and broadcasts _t+1 to all the nodes in the network. Summarizing, the stochastic algorithm is preferred over its deterministic counterpart since it does not require Monte Carlo iterations, yields asymptotically near-optimal resource allocations, and is provably convergent if the stochastic process {^i_t} is stationary. It is remarked that since (<ref>) is infinite dimensional, full primal recovery is generally not possible using such dual methods. Existing algorithms only allow partial primal recovery, as will also be possible via Theorem 2. Specifically, it is well-known that while the running average 1/T∑_t = 1^T ^i_t can be viewed as the approximate version of the primal optimum ^i^⋆, no such interpretation exists for the infinite-dimensional variable ^i. For instance, the running average of ^i_t cannot be meaningfully related to the corresponding optimum ^i^⋆ <cit.>. Nevertheless, the resource allocation carried out using the primal iterates {^i_t(_t),^i_t(_t)} still ensures near-optimality and asymptotic feasibility (cf. Theorem 2). In view of the desiderata (F1)-(F2), observe that a network implementation of (<ref>)-(<ref>) is still impractical since it is synchronous and FC-based, and thus has relatively stringent communication requirements. In particular, the algorithm necessitates that each node exchanges messages (i.e. ^i_t(_t) & _t) with the FC at every time-slot, thereby incurring a large communications cost. Since the updates (<ref>)-(<ref>) must occur before the network state changes, the nodes must synchronize and cooperate in order to meet these deadline constraints, ultimately increasing message passing overhead and consuming more energy. Further, nodes in large networks are often heterogeneous, and may not always be able to transmit the gradients within the stipulated time. Finally, if the nodes are not deployed in a star-topology around the FC, the need for multi-hop communications further increases the delays, results in heterogeneous energy consumption, and increases protocol overhead. In all such cases, the FC must wait for the updates to arrive from all the nodes, possibly requiring all the nodes to skip resource allocation for one or more time slots, and resulting in a suboptimal asymptotic objective value. § PROPOSED ALGORITHM This section details the proposed stochastic dual descent algorithm that incorporates the features (F1)-(F2) in its design. To begin with, Sec. <ref> describes the asynchronous variant that tolerates delayed gradients still resulting in near-optimal resource allocation. Next, Sec. <ref> details the more general AIS-DD algorithm that is amenable to a distributed implementation. §.§ Asynchronous stochastic dual descent The asynchronous stochastic dual descent algorithm addresses (F1), and proceeds as follows for all t ≥ 1: * Primal update: At each time t, node i solves {_t^i(_t-π_i(t)) , _t^i(_t-π_i(t))}:=_∈𝒳^i,∈Π_t^if^i()+_t-π_i(t),_t^i(,) for all 1 ≤ i ≤ K, and some finite delay π_i(t) ≥ 0. * Dual update: The dual update at time t is given by _t+1=[_t-(_t-δ_i(t)^i(_t-τ_i(t)))]^+ the stalegradient, evaluated at time t-δ_i(t), is given by _t-δ_i(t)^i (_t-τ_i(t)):=_t-δ_i(t)^i(_t-δ_i(t)^i(_t-τ_i(t)),_t-δ_i(t)^i(_t-τ_i(t)))where the total delay is denoted byτ_i(t):=π_i(t)+δ_i(t) and π_i(t), δ_i(t) ≥ 0. Different from (<ref>), the resource allocation in (<ref>) utilizes an old dual variable, _t-π_i(t). Further, the dual update is also carried out using an old gradient _t-δ_i(t)^i(_t-τ_i(t)). The two modes of asynchrony introduced in (<ref>)-(<ref>) allow the primal and dual updates to be carried out at different time scales. In other words, while the resource allocation at each node still occurs at every time slot, the rate at which the dual variables and thegradients are exchanged may be different. In order to highlight the asynchronous nature of the algorithm, the implementation of (<ref>)-(<ref>) is now described from the perspective of the FC and that of node i, in Algorithms <ref> and <ref>, respectively. Observe that in Algorithms 1 and 2, some steps are `optional,' which in the present case, means that they can, at times, be skipped. These steps are however still required to be carried out `often enough', so that the total delay τ_i(t) is bounded for each node i; cf. (A4) in Sec. <ref>. Nevertheless, the optional steps in these algorithms allow the dual updates to occur at a different rate. For instance, as long as each packet is correctly time-stamped, the dual updates at the FC may occur as and when thegradients become available, instead of following a fixed schedule. The ability to postpone or skip transmissions is important in the context of large heterogeneous networks. For instance, transmissions from the nodes to the FC often requires a multiple access protocol, inter-node coordination, and energy budgeting at each node. Consequently, energy-constrained nodes may extend their lifetime simply by scheduling their transmissions once every few time slots. Similarly, energy harvesting nodes may only transmit when sufficient energy is available, choosing to stay silent in times of energy paucity. The slower nodes may even skip thegradient calculation, as long as the resources are allocated in time. Finally, the communication between the nodes and the FC may also incur delays, arising from queueing, processing, or retransmission at various layers in the protocol stack. The flexibility of carrying out updates with stale information makes the network tolerant to such delays. §.§ Asynchronous Incremental Stochastic Dual Descent This subsection details an incremental version of the asynchronous algorithm introduced in Sec. <ref>, that obviates the need for an FC and is thus endowed with both (F1) and (F2). The AIS-DD algorithm allows each node to perform the partial dual update itself, while passing messages to nodes along a cycle. Specifically, for a network with a ring topology, such that node i passes dual variable _t^i to node i+1 and so on, the primal and dual updates take the following form. * Primal update: At time t, node i solves (^i_t (^i-1_t-π_i(t)), _t^i(^i-1_t-π_i(t))):=_∈𝒳^i, ∈Π_t^i f^i()+^i-1_t-π_i(t),_t^i(,). * Dual update: At time t, the dual update at node i takes the form ^i_t= [^i-1_t-[_t-δ_i(t)^i(^i-1_t-τ_i(t))]]^+ where, _t-δ_i(t)^i (^i-1_t-τ_i(t)) := _t-δ_i(t)^i(_t-δ_i(t)^i(^i-1_t-τ_i(t)),_t-δ_i(t)^i(^i-1_t-τ_i(t)))and _t^0 is read as _t-1^K and _t=_t^0 will be used to evaluate the performance of the asynchronous incrementalalgorithms. A key feature of the AIS-DD algorithm is that the message passing and the dual updates occur in parallel with the resource allocation, as shown in Fig. <ref>. The full implementation details are provided in Algorithm <ref>. Here, the two optional steps may be repeated as long as the received _t'^i-1 is still old, that is, t' ≤ t. As in Sec. <ref>, the nodes are allowed to halt the updates temporarily, as long as they “catch up,” eventually. In other words, the updates for time t' must be carried out before time t'+τ so as to ensure that τ_i(t) ≤τ for all t. Interestingly, although resources are allocated at every time slot, the network may or may not carry out one or more message passing rounds per time-slot. It is remarked that the update in (<ref>) must still be performed once at every node for each time index t'. Equivalently, the algorithm runs on two `clocks,' one dictating the resource allocation and synchronous with the changes in the network state, and the other governed by the rate at which messages get passed around the network. In the next section, we will establish that the such an algorithm still converges, as long as the difference between the two clocks is bounded. In summary, the AIS-DD algorithm has all the benefits of the asynchronous dual descent algorithm of (<ref>)-(<ref>), while allowing a distributed implementation. As with classical incremental algorithms, the nodes must communicate along a ring topology. Strictly speaking, the message passing overhead is minimized if the updates occur along a Hamilton cycle <cit.>. Even when the network does not admit a Hamilton cycle, an approximate cycle can be found using a random walk protocol <cit.> or the protocol described in <cit.>. It is remarked that such a route need only be found once, at the start of the algorithm. § CONVERGENCE RESULTS This section provides the convergence results for the AIS-SD and AIS-DD algorithm. We begin with developing and analyzing the convergence of AIS-SD algorithm(cf. Theorem <ref>). It is emphasized that the AIS-SD algorithm is general-purpose, and can be used to minimize any sum of functions in an incremental and asynchronous manner.Subsequently, the asynchronous incremental stochastic gradient descent algorithm is applied to (<ref>) in the dual domain, and a primal-averaging method is proposed that yields asymptotically near-optimal allocations (cf. Theorem <ref>). We begin with stating the assumptions and briefly reviewing some of the known results (Sec. <ref>). The results for the dual case are outlined in Sec. <ref>, while the near-optimality of the resource allocation is established in Sec. <ref>. §.§ Assumptions and known results This subsection begins with the discussion of the following general optimization problem: = min_∈Ł D^i() where,is the optimization variable, Ł⊆ℝ^d is a non-empty, closed, and convex set,is finite, and the objective function separates into node-specific cost functions D^i. The goal is to solve(<ref>) using only the stochasticsubgradients _t^i() of D^i(). It is emphasized that the general results presented in this subsection do not required D^i to be differentiable. As in (<ref>), _t^i() is stochastic due to its dependence on the random variable _t^i that is first observed at node i at time t. Besides the network resource allocation problem considered here, (<ref>) also arises in the context of machine learning <cit.> and distributed parameter estimation <cit.>. Before describing the known results related to (<ref>), the necessary assumptions are first stated. A1.Non-expansive projection mapping. The projection mappingsatisfies -≤- for alland . A2.Zero-mean time-invariant error. Given , the averaged subgradient function satisfies ∂ D^i() = [_t^i()]. A3.Bounded moments. Given ∈Ł, the second moment of _t^i() is bounded as follows: [_t^i()^2] ≤V_i^2. These assumptions are not very restrictive, and hold for most real-world resource allocation problems. A stochastic incremental algorithm for solving (<ref>) was first proposed in <cit.>. Given a network with ring topology, the updates in <cit.> take the form _t^i= _t^i-1 - ϵ_t^i(_t^i-1) where _t^0 is read as _t-1^K. It was shown in <cit.>, that under (A1)-(A3), the iterates ^i_t are asymptotically near optimal in the following sense t→∞liminf [D(_t)]≤𝖣+Ø(). where _t= _t^0=_t-1^K. Further, for the case when the step size is diminishing, i.e. _t satisfies lim_T→∞∑_t=1^T_t = ∞ and lim_T→∞∑_t=1^T_t^2 < ∞, it holds that lim inf_t→∞D(_t)= 𝖣. This paper provides the corresponding results for the asynchronous case, where the subgradient in (<ref>) is replaced by an older copy _t-δ_i(t)^i(_t-τ_i(t)^i-1), that is, the stochastic subgradient of D^i()that depends on the random variable _t-δ_i(t)^i and is evaluated at = _t-τ_i(t)^i-1. The delays satisfy τ_i(t) ≥δ_i(t) ≥ 0 and for the special case of no delay, the stochastic subgradient simplifies to _t^i(_t^i-1) as in (<ref>). The following additional assumption regarding the delays δ_i(t) and τ_i(t) is stated. A4.Bounded delay.For each 1≤ i ≤ K and t≥ 1, it holds that 0≤δ_i(t) ≤τ_i(t) ≤τ < ∞. The boundedness assumption on the delay in (A4) allows us to develop convergence results that hold in the worst case, and has been widely used in the context of asynchronous algorithms <cit.>. It is remarked that an alternative assumption, made in <cit.>, allows the delays δ_i(t) and τ_i(t) to be random variables with unbounded supports but finite means, but is not pursued here. Even with bounded delays, the extension to the asynchronous case is not straightforward, since the the old stochastic subgradients are not necessarily descent directions on an average. Indeed, the resulting subgradient error at time t, defined as _t,δ_i(t)^i:= [∂ D^i(_t^i-1)-_t-δ_i(t)^i(_t-τ_i(t)^i-1)] is neither zero-mean nor i.i.d. In other words, the asynchronous algorithm cannot simply be considered as a special case of the inexact subgradient method. It is worth pointing out that there is a subtle difference between the definition of the delayed stochastic gradient considered here, and those considered in <cit.>. Specifically, the delayed gradient in these works takes the form _t^i(_t-τ_i(t)^i-1) instead of the one in (<ref>). As a result, given _t-τ_i(t)^i-1, the gradient error at time t in these papers is indeed zero mean and i.i.d., an assumption that simplifies the analysis to a certain extent. It is also remarked that the definition of the delayed stochastic gradients in <cit.> is however similar to that considered here. Different from these works, the dual convergence results developed here consider subgradients instead of gradients, and are therefore applicable to a wider range of problems. Within the context of network resource allocation, it is also important to study the (near-)optimality of the allocations {^i_t, ^i_t}. Towards this end, some additional assumptions are first stated. A5.Non-atomic probability density function:The random variables {_t^i}_i=1^K have non-atomic probability density functions (pdf). A6.Slater's condition:There exists strictly feasible (^i,^i), i.e., [_t^i(_t^i,^i)]> 0. A7.Lipschitz continuousgradients.Given , ' ∈Ł, there exists L_i < ∞ such that ∇ D^i()-∇ D^i(')≤ L_i-'. In (A5), for {_t^i}_i=1^K to have a non-atomic pdf, it should not have any point masses or delta functions. Note that this requirement is not restrictive for most applications arising in wireless communications; see e.g. <cit.>. The Slater's condition is a standard assumption that ensures that 𝖯 = 𝖣 and consequently, since 𝖯 is finite, so is 𝖣. The Lipschitz condition in (A7) is however restrictive, since it requires the dual functions D^i() to be differentiable with respect to . In other words, with (A7), _t^i() is a stochasticgradient, not a subgradient. It is remarked however that (A7) always holds if f^i(^i) is strongly convex. Moreover, it is generally possible to enforce (A7) artificially by adding a strongly convex regularizer (such as θ^i_2^2) to the primal objective <cit.>. Note however that (A5)-(A7) will not be utilized while establishing the dual convergence results. The incremental or asynchronousgradient methods have thus far never been applied to the problem of network resource allocation. For the classical stochastic dual descent method (cf. (<ref>)-(<ref>)], it is known that under (A1)-(A3) and (A6), the average resource allocations ^i:=^i_t are asymptotically feasible and near-optimal <cit.>. §.§ Convergence of the AIS-SD algorithm This subsection provides the convergence results for the AIS-SD algorithm, applied to (<ref>). For the general case, the updates take the following form: _t^i= _t^i-1 - _t^i_t-δ_i(t)(^i-1_t-τ_i(t))1≤ i ≤ K where ϵ_t is the step-size, _t^i() is a stochastic subgradient of D^i() and ^0_t is read as ^K_t-1. Since the dual problem (<ref>) is simply a special case of (<ref>), the results developed here also apply to the iterates {_t^i} generated by Algorithm 1. In order to keep the discussion generic, the results are presented for both, diminishing and constant step sizes. The following results apply to the iterates generated by (<ref>) with _t=_t^0 under (A1)-(A4). * Diminishingstep-size: If the positive sequence {_t} satisfies lim_T→∞∑_t=1^T_t = ∞ and lim_T→∞∑_t=1^T_t^2 < ∞, then it holds that lim inf_t→∞[∑_i=1^KD^i(_t)]=𝖣. * Error bound for constant step size: For _t => 0, and any arbitrary scalar η > 0, it holds thatmin_1≤ t≤ T∑_i=1^K D^i(_t)≤𝖣 +C(τ)+η/2where T ≤ B^2_0/η. Here, τ is the maximum delay as defined in (A4), C(τ):=C_1+(C_2+τ C_2'), C_1=KV^2, C_2:=2KV^2K-1/2, C_2':=4K^2V^2, and B_0 is such that _1-≤ B_0. A popular choice for the diminishing step-size parameter _t required in Theorem <ref>(a) is _t=t^-α for α∈ (1/2, 1). For this case, the objective function in (<ref>) converges exactly to the dual optimum. On the other hand, with a constant step size , the minimum objective value comes to within an O()-sized ball around the optimum as T →∞. More precisely, the result in (<ref>) provides an upper bound on the number of iterations required to come η-close to this ball. Different from the results in <cit.>, the size of the ball now depends on the maximum delay τ, quantifying the worst-case impact of using delayed subgradients. The proof of Theorem <ref> follows the same overall structure as in <cit.>, with appropriate modifications introduced to handle the asynchrony. To begin with, the following intermediate lemma splits a function related to the optimality gap in Theorem <ref> into three different terms, and develops bounds on each. The proof of the following lemma is provided in Appendix <ref>. Under (A1)-(A4), the iterates generated by (<ref>) with _t=_t^0 satisfy the following bounds: ∑_t=1^T∑_i=1^K2_tD^i(_t)-𝖣 ≤B_0^2+ I_0+I_1 where, I_0:= ∑_t(_t^2 KV^2+2τ KV^2∑_i_t_t-τ_i(t) + 2V^2∑_i(i-1)_t_t-τ_i(t)) I_1 :=2∑_t=1^T∑_i=1^K_t _t-δ_i(t)^i(_t-τ_i(t)^i-1), _t-τ_i(t)^i-1-_t^i-1≤ 2τ KV^2∑_t=1^T∑_i=1^K_t _t-τ_i(t). where _1^0 - ≤ B_0. Note that _1^0=_1. Having developed the necessary bounds, the proof of Theorem <ref> is presented next. [Proof of Theorem 1] For the positive sequence {_t}, it holds that ∑_t=1^T∑_i=1^K2_t D^i(_t)≥(min_1≤ t≤ TD^i(_t)) ∑_t=1^T2_t . Substituting the bounds obtained in Lemma <ref>, and noting that it always holds that _t ≤_t-τ for all τ≥ 0, we obtain min_1≤ t≤ T D^i(_t) -𝖣≤B_0^2+C_1∑_t=1^T_t^2+(C_2+τ C_2')∑_t=1^T^2_[t-τ]_+/2∑_t=1^T_t where, C_1:= KV^2, C_2':=4K^2V^2 and C_2:=2KV^2K-1/2. Note that in (<ref>), we have used the notation _[t-τ]_+:=_1 for all t ≤τ.Next, for the case when _t is diminishing, and satisfieslim_T→∞∑_t=1^T_t = ∞ and lim_T→∞∑_t=1^T_t^2 < ∞, the numerator of the bound on the right stays bounded, while the denominator grows to infinity. Consequently, taking the limit of T →∞ on both sides of (<ref>), the required result in (<ref>) follows. Observe that when the step size is constant, the bound in (<ref>) can be written as min_1≤ t≤ TD^i(_t)-𝖣 ≤B_0^2/2 T+/2 C_1+/2 (C_2+τ C'_2)≤B_0^2/2 T+/2 C(τ) where C(τ) is as defined in Theorem <ref>. In the limit as T→∞, the bound becomes inf_t≥ 1D(_t)≤+ C(τ)/2. which is the asymptotic version of the result in (<ref>). The rate result in (<ref>) builds upon a similar result from <cit.>. Intuitively, D(_t) continues to decrease as long as it is significantly larger than 𝖣. The rest of the proof characterizes the resulting decrement rigorously and subsequently invokes the monotone convergence theorem in order to establish that D(_t) must eventually come close to 𝖣. Given arbitrary η >0 and recalling that _t := _t^0, define the sequence _t+1:=_t+1 ;ifD(_t)≥+ C(τ)+η/2 ^⋆ ;otherwise. Alternatively, _t is same as _t until _t enters level set defined as L={∈Λ | D()< + C(τ)+η/2} and _t terminates at ^⋆. From (<ref>) and Lemma <ref>, we have for constant step size _t= that _t+1-^⋆^2≤ _t-^⋆^2-2[D(_t)-]+^2C(τ) where _t+1:=_t^K and _t=_t^0. Next define z_t:= 2[D(_t)-]-^2C(τ)if_t∉ L 0ifotherwise, so that (<ref>) can be written as _t+1-^⋆^2≤_t-^⋆^2-z_t. From the Monotone convergence theorem, we have that ∑_t=1^∞z_t<∞, implying that there exists T < ∞, such that z_t=0 for all t≥ T. Observe that for the case when _t∉ L, it holds that z_t= 2[D(_t)-]-^2C(τ) ≥ 2[+ C(τ)+η/2-]-^2C(τ) ≥η. Consequently, it follows from (<ref>) that 𝔼[||_T+1-^⋆||^2] ≤ ||_1-^⋆||^2-∑_t=1^Tz_t ≤ B_0 - ∑_t=1^Tz_t. Since the term on the left is non-negative, we have that B_0 ≥∑_t=1^Tz_t≥ T η, yielding the required bound on T. §.§Primal near optimality and feasibilityThe AIS-SD algorithm of Sec. <ref>, when applied to solve the dual problem in (<ref>), is referred to as the AIS-DD algorithm. In order to ensure that the results developed thus far continue to apply to the dual problem, assumptions (A5)-(A7) are also required. As mentioned earlier, for the primal problem, Ł is simply the non-negative orthant implying that 𝖣 is finite. This subsection establishes the average near-optimality of the AIS-DD algorithm in (<ref>)-(<ref>). Note that Theorem <ref> does not imply that the allocations {_t^i,_t^i} converge. Instead, the results will make use of the ergodic limit variable ^i_T:=1/T∑_t=1^T^i_t for each 1 ≤ i ≤ K. The main theorem for this subsection is presented next. Under (A1)-(A7) and for constant step size >0, the iterates generated by (<ref>)-(<ref>) follow: *lim inf_T→∞f^i(^i_T)≥ 𝖯-(C_3+τ C_4) where, C_3 = (V^2K^2)/2 C_4= K^2BLV+2K^2V^2. * lim inf_T→∞1/T∑_t=1^T∑_i=1^K[_t-τ_i(t)^i(^i-1_t-τ_i(t))]≽ 0. Intuitively, the resource allocations in (<ref>) are near-optimal, with optimality gap depending on the step sizeand the delay bound τ. Further, the allocations are almost surely asymptotically feasible, regardless of the the delay bound or the step size. As in Sec. <ref>, the proof of Theorem <ref> proceeds by first splitting the optimality gap into three terms and developing bounds on each. The required results are summarized into the following intermediate Lemmas, whose proofs are deferred to Appendices <ref> and <ref> respectively. Under (A1)-(A6), the iterates _t^i obtained from (<ref>) are bounded on an average, i.e., there exists B < ∞ such that _t≤ B for all t ≥ 1. Under (A1)-(A7), the iterates generated by (<ref>)-(<ref>) satisfy the following bounds: f^i(_T^i)≥𝖣- I_2 - I_3 where, I_2 := 1/T∑_t=1^T∑_i=1^K[D^i(_t)-D^i()]≤ V^2K(K-1)/2 I_3 := 1/T∑_t=1^T∑_i=1^K,∇ D^i()≤_1^2/2 T+ KV^2 /2 + I_4 I_4 := 1/T∑_t=1^T∑_i=1^K,(∇ D^i()-_t-δ_i(t)^i(_t-τ_i(t)^i-1))≤τ K^2V(BL+2V). Having established the intermediate results, the proof of Theorem <ref> is now presented. The primal near-optimality can be established directly from Lemma <ref>. Specifically, summing the bounds for I_2, I_3, and I_4, and taking the limit as T →∞, the bound in (<ref>) follows. In order to establish (<ref>), observe that for any t≥ 1 and 1 ≤ i ≤ K, it holds that ^i_t = [^i-1_t-_t-δ_i(t)^i(^i-1_t-τ_i(t))]^+≽^i-1_t-_t-δ_i(t)^i(^i-1_t-τ_i(t)) where the inequality holds element-wise. Summing both sides over all 1≤ t ≤ T and 1 ≤ i ≤ K, and rearranging, it follows that 1/T∑_t=1^T∑_i=1^K_t-δ_i(t)^i(^i-1_t-τ_i(t))≽1/ T∑_t=1^T∑_i=1^K (_t^i-1 - _t^i)≽^K_1-^K_t+1/ T. Finally, since _1^K ≽ 0, taking expectations on both sides, it follows that 1/T∑_t=1^T∑_i=1^K[_t-δ_i(t)^i(^i-1_t-τ_i(t))] ≽ -B/ T where (<ref>) holds due to Lemma <ref>. In other words, given any α >0, there exists t_0 ∈ℕ such that for all T ≥ t_0, 1/T∑_t=1^T∑_i=1^K[_t-δ_i(t)^i(^i-1_t-τ_i(t))] ≽ -α. Taking the limit as T→∞, the result in (<ref>) follows. § APPLICATION TO CO-ORDINATED BEAMFORMING This section considers the co-ordinated downlink beamforming problem in wireless communication networks. The usefulness of the proposed stochastic incremental algorithm is demonstrated by applying it to the beamforming problem and solving it in a distributed and online fashion. Simulations are carried out to confirm that the performance of the proposed algorithm is close to that of the centralized algorithm. §.§ Problem formulation Consider a multi-cell multi-user wireless network with B base stations and U users. Each user j ∈{1, …, U} is associated with a single base station b(j) ∈{1, …, B}, and the set of users associated with a base station i is denoted by _i:={j | b(j) = i}. For the sake of consistency, this section will utilize indices i and m for base stations, and indices j, k, and n for users, with the additional restriction that b(j) = b(k) = i and b(n) = m. Within the downlink scenario considered here, user j can only receive data symbols s_j∈ℂ from its associated base station b(j). The signals transmitted by the base station i intended for other users k ∈_i∖{j}, as well as the signals transmitted by other base stations m ≠ i constitute, respectively, the intra-cell and inter-cell interference at user j. The base station i, equipped with N_i transmit antennas, utilizes the transmit beamforming vector _j∈ℂ^N_i × 1 for each of its associated user j ∈_i. Consequently, the received signal at user j is given by y_j = _ij^H(_js_j + ∑_k ∈_i∖{j}_ks_k) + ∑_m ≠ i n ∈_m_mj^H_ns_n + e_j where _ij denotes the complex channel gain vector between base station i and user j, and e_j is the zero mean, complex Gaussian random variable with variance σ^2 that models the noise at user j. Assuming s_k to be independent, zero-mean, and with unit variance, the expression for the signal-to-interference-plus-noise ratio (SINR) at user j is given by SINR_j := _ij^H_j^2/∑_k ∈_i∖{j}_ij^H_k^2 + ∑_m ≠ i∑_n ∈_m_mj^H_n^2 + σ^2 where i = b(j) is the associated base station. Within the classical co-ordinated beamforming framework, the goal is to design the beamformers {_j}_j=1^U so as to minimize the transmit power, while meeting the SINR constraints at each user. The required optimization problem becomes <cit.> min_{_j}_j=1^U ∑_j=1^U _j^2subject to SINR_j ≥γ_j  ∀ j where γ_j is a pre-specified quality-of-service (QoS) threshold for user j. While the beamforming vectors resulting from (<ref>) are optimal, the centralized nature of the optimization problem renders it impractical for application to real networks. For instance, the solution proposed in <cit.> requires the estimated channel gains {_ij} to be collected at a centralized location, where (<ref>) is solved via an iterative algorithm. In practice however, the entire parameter exchange and the algorithm must complete within a fraction of the coherence time of the channel, lest the designed beamformer becomes obsolete. Such a solution is therefore difficult to implement, not robust to node or link failures, and not scalable to large networks. Observe that the modified version of (<ref>) can be written as min_{_j, I_nj}∑_j=1^U _j^2subject to_ij^H_j^2/∑_k ∈_i∖{j}_ij^H_k^2 + I_j^2 + σ^2≥γ_j ∀ j∑_m ≠ i∑_n ∈_m_mj^H_n≤I_j ∀ j where i = b(j). Note that constraints in (<ref>) and (<ref>) will ensure the SINR is still greater than the required threshold of γ_j. It is due to the fact that the feasible set is restricted and feasible set of (<ref>) will be subset of that of (<ref>) and solution found for (<ref>) can be used for (<ref>). Next, the use of primal or dual decomposition techniques can yield a distributed algorithm for (<ref>). Nevertheless, such distributed algorithms also suffer from the limitations mentioned earlier, since the optimum beamforming vectors are required at every time slot. On the other hand, within the uncoordinated beamforming framework, the optimization variable I_j in (<ref>) is replaced with a pre-specified threshold ρ. This renders (<ref>) separable at each base station, allowing beamforming vectors to be designed in parallel. However, the resulting beamformers are suboptimal, and may even render the problem infeasible if ρ is too small or too large. min_{_j, I_nj}∑_j=1^U _j^2subject to_ij^H_j^2/∑_k ∈_i∖{j}_ij^H_k^2 + ρ^2∑_m ≠ icard(_m) + σ^2≥γ_j; ∀ j_mj^H_n(t)≤ρm≠ i, n ∈ U_m, ∀ j. A compromise is possible within the stochastic optimization framework by requiring the bound in (<ref>) to only be satisfied on an average. Note that this amounts to relaxing the optimization problem (<ref>) since the SINR constraint is no longer binding at every time slot. The overall stochastic optimization problem can be expressed as min_{_j(t), I_nj(t)}∑_j=1^U _j(t)^2subject to _ij^H_j(t)^2/∑_k ∈_i∖{j}_ij^H_k(t)^2 + I_j^2(t) + σ^2≥γ_j∀ j ∑_m ≠ i∑_n ∈_m_mj^H_n(t)≤I_j(t)∀ j _mj^H_n(t)≤ρm≠ i, n ∈ U_m, ∀ j where i = b(j). Different from (<ref>) or (<ref>), the stochastic optimization problem (<ref>) involves finding policies _j(t) and I_j(t), which are not necessarily optimal for every time slot t, but only on an average. Specifically, the intercell interference is bounded on an average [cf. (<ref>)], but also instantaneously [cf. (<ref>)], so as to limit the worst case SINR. The problem in (<ref>) can be readily implemented using the proposeddistributed and asynchronous stochastic dual descent algorithm. In contrast to (<ref>), the stochastic algorithm is not required to converge at every time slot, and allows cooperation over heterogeneous nodes. §.§ Solution to optimization problem The AIS-DD algorithm proposed in Sec. <ref> can now be applied to solve (<ref>). To this end, associate dual variables λ_j for all users {j ∈[1,U]}, and observe that the primal variables at node i include {_j}_j ∈_i and {I_j}_j ∈_i. Departing from the notational convention used thus far, the subscript in λ_j is used for indexing the users, while time dependence is indicated by λ_j(t). Proceeding as in Sec. <ref>, and recalling that the indices j and n are such that i = b(j) ≠ b(n) = m, the operation at node i is summarized in Algo. <ref>. Observe that such an implementation entails allocating resources prior to the dual updates, and thus results in the delay of at least one, i.e., π_i(t) ≥ 1, compared to the synchronous version. Conversely, the dual updates occur as and when they are passed around, without creating a bottleneck for the resource allocation. For the sake of simplicity, it is assumed that the dual updates occur along the route 1, 2, …, K. Next, simulations are carried out demonstrate the applicability of the stochastic algorithm to the beamforming problem at hand. For the simulations, we consider a system with B = 10 and U = 10, with one users per cell. Each of thebase stationshave ten antennas (N_i = 10), whilethe other algorithm parameters are =0.5, σ^2=1, ρ=1.65 γ_j = 10 dB for allj. In order to keep the simulations realistic, we assume that the delays in the dual updates arise from random events such as node and link failures. For the centralized algorithm, a random subset of four out of ten nodes are selected to transmit their currentgradients to the FC at every time slot. Since the FC utilizes oldgradients for the other nodes, it results in an average delay of 5.4 time slots. Similarly, for the incremental algorithm, it is assumed that at every time slot, five to fifteen dual update steps (cf. (<ref>)) occur, resulting in an average delay of 5.4 time slots. For instance, if at any time slot, only 8 nodes update, it will result in a delay π_i(t) = 2 at the remaining two nodes, where the dual update will occur at the start of the next time slot. The delay may increase further if fewer than 10 nodes update for consecutive time slots and conversely, may decrease if more than 10 updates occur per time slot. Fig. <ref> shows the running average of the primal objective function as a function of time using Monte Carlo simulations. For comparison, the performance of the classical centralized stochastic gradient method [cf. (<ref>)-(<ref>)], assumingperfect message passing, is also shown. As evident, the performance loss due to the delays in the availability of the dual variables in minimal. In order to motivate the stochastic formulation over the deterministic one, Fig. <ref> also compares the average transmit power and SINR achieved for the various cases and for different values of the parameter ρ. As expected, the distributed deterministic algorithm performs poorly since it forces the SINR bound to be a constant that does not depend on the channel. By design, the worst case SINR is bounded below by one at every time slot in both the deterministic formulations. Interestingly, the worst case SINR achieved for the relaxed stochastic formulation is also close to one on an average. In return, the stochastic algorithm yields an average transmit power that is equal to or below that obtained by the centralized deterministic formulation. In other words, it is always possible to artificially raise γ to a value that is slightly higher than one, so as to obtain an average SINR above 10 dB, while still getting near-optimal average transmit power. Next, we study the effect of delay on the rate of convergence of the AIS-DD algorithm. For this case, a simple system with B = 10 and U=10, and constant delays at all the nodes is considered. The base stations have ten antennas each (N_i=10) andthe other algorithm parameters are =0.2, σ^2=1, and ρ=1.65. Fig. <ref> shows the evolution of the primal objective function for various delay values. As expected, the convergence is slower if both π_i(t) and δ_i(t) are consistently larger. Interestingly however, a small increase in the delays amounts to only a marginal loss in performance.Finally, in order to demonstrate the scalability of the proposed algorithm, Fig. <ref> shows an example run for a system with B=50 nodes and U =50. Thebase stations have ten antennas each (N_i= 10), whilethe other algorithm parameters are =0.5, σ^2=1, and ρ=5. The delay is generated in the similar manner as for the earlier simulations. It can be observed that even when the number of nodes is large, the difference between the performance of the synchronous and asynchronous algorithms remains relatively small. § CONCLUSION This paper considers a constrained stochastic resource allocation problem over a heterogeneous network. An asynchronous incremental stochastic dual descent method is proposed for solving the same. The proposed algorithm utilizes delayed gradients for carrying out the updates, resulting in an attractive feature that allows nodes to skip or postpone some updates. The convergence of the proposed algorithm is established for both constant and diminishing step sizes. Further, it is shown that the resource allocations arising from the proposed algorithm are also asymptotically near-optimal. A novel multi-cell coordinated beamforming problem is formulated within the stochastic framework considered here, and solved via the proposed algorithm. Simulation results reveal that the impact of using stale stochastic gradients is minimal.§ PROOF OF LEMMA <REF> §.§.§ PreliminariesBefore deriving the required bounds, some preliminary results are first obtained. Recall that the quantity _υ^i() denotes the stochastic (sub-)gradient of D^i() at time t = υ and evaluated at . Within the context of the dual descent algorithm, we also have that _υ^i():=_υ^i(^i_υ(),_υ^i()) for υ≥ 1 and all 1≤ i≤ K. In particular the updates in (<ref>) use υ = t-δ_i(t) and = _t-τ_i(t)^i-1, where τ_i(t) = δ_i(t)+π_i(t). It can be seen that for the special case of the synchronous algorithm, we have that τ_i(t) = δ_i(t) = 0 and the stochastic (sub-)gradient is written as _t^i(_t^i-1).For the sake of convenience, let us denote _t-δ_i(t)^i := _t-δ_i(t)^i(_t-τ_i(t)^i-1). First, we establish that the distance between the iterates _t^i and _ℓ^i is bounded by a term that is proportional to the step size. From the updates in (<ref>), it holds for all t and 1≤ i ≤ K that _t^i-_t^i-1 = _t^i-1 -_t_t-δ_i(t)^i - _t^i-1≤_t_t-δ_i(t)^i≤_tV_i where the inequalities in (<ref>) follow from (A1) and (A3). The bound _t-δ_i(t)^i≤ V_i follows from A3 and Jensen's inequality which implies that _t-δ_i(t)^i≤√(_t-δ_i(t)^i^2). Given 1≤ i,j ≤ K and t ≥ℓ≥ 1, it follows that _t^i-_ℓ^j ≤∑_k=1^i_t^k-_t^k-1+∑_s=ℓ+1^t-1∑_k=1^K_s^k-_s^k-1 +∑_k=j+1^K_ℓ^k-_ℓ^k-1≤_t ∑_k=1^i V_k+ (∑_s=ℓ+1^t-1_s)(∑_k=1^K V_k)+_ℓ∑_k=j+1^K V_k≤_t (iV)+ (∑_s=ℓ+1^t-1_s)(KV)+_ℓ(K-j)V≤_ℓ V[i+(t-ℓ-1)K+K-j]≤_ℓ V[i-j+K(t-ℓ)] where (<ref>) is obtained by substituting V = max_i V_i and (<ref>) follows since _ℓ≥_s for all ℓ≤ s ≤ t. The result in (<ref>) holds from the inequality (i-j)≤ |i-j|. Further, for t≥ 1 and 1≤ i≤ K, let ^i_t be the σ-algebra generated by the random variables {_1^1, …, _1^K, _2^1 …, _t-1^K, _t^1,…,_t^i}. where _t^i is the random state variables observed at node i at time t. With this definition, it holds that ^i_t-δ_i(t)|ℱ_t-δ_i(t)^i-1=∂ D^i(_t-τ_i(t)^i-1) since _t-δ_i(t)^i may depend on _s^i-1 only for s ≤t-τ_i(t). The proof is organized into two parts. Subsection <ref> develops a bound on the optimality gap in (<ref>), in terms of I_1. Subsequently, Subsection <ref> develops the required bound on I_1. §.§.§ Bound on the optimality gapAn upper bound on _t^K-^2 is developed by making use of the form of the updates in (<ref>) for all nodes 1≤ i ≤ K. The bound follows from the use of triangle inequality and the moment bounds in (A3). Further, the bounded delay assumption (A4) enter through the use of (<ref>). Observe from the updates in (<ref>) that _t^i-^2=_t^i-1 -_t_t-δ_i(t)^i-^2 ≤_t^i-1 -_t_t-δ_i(t)^i-^2 = _t^i-1-^2 -2_t _t-δ_i(t)^i, _t^i-1- +_t^2 _t-δ_i(t)^i^2= _t^i-1-^2 -2_t _t-δ_i(t)^i, _t-τ_i(t)^i-1-+_t^2 _t-δ_i(t)^i^2-2_t _t-δ_i(t)^i, _t^i-1-_t-τ_i(t)^i-1 where (<ref>) follows form (A1), and the term 2_t^i_t-δ_i(t), ^i-1_t-τ_i(t) has been added and subtracted to obtain (<ref>). Taking expectations on both sides and summing over all 1≤ i ≤ K and 1≤ t≤ T, we obtain _T^K-^2 = _1^0-^2 + ∑_t=1^T∑_i=1^K_t^2_t-δ_i(t)^i^2+ I_1 -2 ∑_t=1^T∑_i=1^K_t _t-δ_i(t)^i, _t-τ_i(t)^i-1- where I_1 is as defined in Lemma <ref>. Deferring the bound on I_1 to Subsection <ref>, the last term in (<ref>) is analyzed first. In particular, it holds from (<ref>) that _t-δ_i(t)^i, _t-τ_i(t)^i-1-= _t-δ_i(t)^i |ℱ_t-δ_i(t)^i-1,_t-τ_i(t)^i-1-= ∂ D^i(_t-τ_i(t)^i-1),_t-τ_i(t)^i-1-. Further, since the functions D^i() are convex, it holds that -∂ D^i(_t-τ_i(t)^i-1),_t-τ_i(t)^i-1-≤ D^i() - D^i(_t-τ_i(t)^i-1) = D^i() - D^i(_t^0) + D^i(_t^0) - D^i(_t-τ_i(t)^i-1) ≤ D^i() - D^i(_t^0) + ∂ D^i(_t^0), _t^0 -_t-τ_i(t)^i-1≤ D^i() - D^i(_t^0) + V_i_t^0-_t-τ_i(t)^i-1 where (<ref>)-(<ref>) follow from the first order convexity condition for D^i and (<ref>) follows from the use of triangle inequality, and the fact that given any ∈Ł, ∂ D^i() =g_t^i()≤√(g_t^i()^2)≤ V_i. For the last term in (<ref>), taking expectation and utilizing the result in (<ref>), it follows that _t^0-_t-τ_i(t)^i-1 ≤_t-τ_i(t)[(i-1)V+KV(τ_i(t))] for 1≤ i≤ K. Finally, substituting (<ref>), (<ref>) in (<ref>), and using (A3), it follows that _T^K - ^2 ≤ _1^0-^2 +∑_t=1^T∑_i=1^K_t^2V_i^2 + I_1- 2∑_t=1^T∑_i=1^K_t[ D^i(_t^0) - D^i()] +2∑_t=1^T∑_i=1^K_t_t-τ_i(t)V_i[(i-1)V+KV(τ_i(t))]. Since the left-hand side is non-negative and _1^0 - ≤ B_0, the first part of Lemma <ref> is obtained simply by rearranging the terms in (<ref>) 2∑_t=1^T∑_i=1^K_t[ D^i(_t^0) - D^i()]≤ B_0^2 +I_1 + ∑_t=1^T∑_i=1^K(_t^2 V_i^2 + 2_t_t-τ_i(t) V_i[(i-1)V+τ KV])≤B_0^2 + I_0 + I_1 where the first inequality in (<ref>) follows since τ_i(t)≤τ, and _t is non-increasing sequence. Finally, the second inequality in (<ref>) follows from substituting V_i ≤ V for all 1≤ i ≤ K, and I_0 is as defined in Lemma <ref>. §.§.§ Bound on I_1In order to derive a bound on I_1, we make use of the Cauchy-Schwartz inequality as follows:_t-δ_i(t)^i, _t-τ_i(t)^i-1-_t^i-1 ≤ V_i_t^i-1 -_t-τ_i(t)^i-1 ≤τ_i(t)_t-τ_i(t) V_iKV where (<ref>) follows from (<ref>). Consequently, I_1≤∑_t=1^T∑_i=1^K2_t _t-τ_i(t)τ_i(t)V_iKV≤2τ KV∑_t=1^T∑_i=1^K V_i_t_t-τ_i(t) where (<ref>) utilizes the bounds τ_i(t)≤τ. Finally, substituting V = max_i V_i, we obtain I_1 ≤ 2τ KV^2∑_t=1^T∑_i=1^K_t _t-τ_i(t) which is the required bound. § PROOF OF LEMMA <REF>Here, we establish that the dual iterates always stay bounded, thanks to the Slater's condition in (A6). The proof begins with establishing an upper bound on the per-iteration increase in the value of _t^0 - ^2, and subsequently utilizes an induction argument to derive the following bound for all t ≥ 1:_t ≤ 2+max{_1,θ/C[𝖣-f^i(^i)]+ θ KV^2/2C+ 2θτV̅/C+ VK} where, θ and C are positive constants, Ṽ= V^2K(K-1), V̅= 2K^2V^2, and {^i} is a slater point of (<ref>). Since ^⋆ is bounded, the right hand side of(<ref>) serves as the bound on _t . In order to prove (<ref>), we will instead establish a more general result that takes the form: _t^0 - ≤ max{_1^0-,θ/C[𝖣-f^i(^i)]+θ KV^2/2C+2θτV̅/C++ VK}. where, recall that _t^0 = _t and _1^0 = _1. The desired result in (<ref>) will follow by applying the triangle inequality to (<ref>). The proof of (<ref>) follows via induction. It can be seen that the inequality in (<ref>) holds trivially for the base case of t=1. As part of the inductive hypothesis, assume that (<ref>) holds for t where t≥1. It remains to show that it also holds for t+1. We split the argument into the following two cases. Case 1.D(_t^0) > 𝖣+Ṽ/2+ τV̅: In this case, it holds that _t+1^0 - ^2 ≤_t^0-^2. Consequently, the induction hypothesis for time t implies that (<ref>) also holds for time t+1. Case 2. D(_t^0) ≤𝖣+Ṽ/2+ τV̅: Recall that the dual function in (<ref>) is defined as D(_t^0) =max_^i∈𝒳, _t^i∈𝒫_t[f^i(^i)+_t^0,[ _t^i(_t^i,^i)]] ≥[f^i(^i)+_t^0,[ _t^i(_t^i,^i)]] where {^i,{_t^i}_t≥ 1}_i=1^K is a strictly feasible (Slater) solution to (<ref>). From (A6), such a strictly feasible solution exists and satisfies [ _t^i(_t^i,^i)] > C > 0. Substituting into (<ref>), and rearranging, we obtain 1,_t^0 ≤1/C[D(_t^0)- f^i(^i)] Since _t^0≽ 0, it follows from equivalence of norms _t^0≤θ_t^0_1 = θ1,_t^0. Therefore, taking expectations in (<ref>) yields _t^0 ≤θ/C[D(_t^0)- f^i(^i)] ≤θ/C[𝖣+Ṽ/2+ τV̅- f^i(^i)] where the assumption for Case 2 has been used in (<ref>). Finally, the use of triangle inequality and the bound in (<ref>) yields _t+1^0- ≤_t^0 + _t+1^0-_t^0 + ≤_t^0 +VK + which, together with (<ref>), yields (<ref>) for t+1. Therefore by mathematical induction, the inequality in (<ref>) holds for all t≥ 1. Finally, using (<ref>) and triangle inequality, we obtain the result in (<ref>) since _t=_t^0. § PROOF OF LEMMA <REF> The proof establishes a lower bound for the running average of the primal objective function, calculated at the primal iterates. The lower bound depends upon the dual optimal value, dual initialization, and the maximum delay bound τ. For ease ofexposition, the proof begins with re-arranging the optimality gap in the form required by Lemma <ref> and subsequently analyzing the resulting terms. The full proof is split into various parts that develop separate bounds on the terms I_2, I_3, and I_4. Since (A7) is required to establish Lemma <ref>, the dual function D has to be differentiable. Since the functions f^i are concave, the expected value of the primal objective can be written as f^i(^i_T)≥1/T∑_t=1^T∑_i=1^K f^i(^i_t)=1/T∑_t=1^T∑_i=1^K[f^i(^i_t)+,∇ D^i()-,∇ D^i()] = 1/T∑_t=1^T∑_i=1^K D^i() -1/T∑_t=1^T∑_i=1^K,∇ D^i() Consider the following expression where we simply add subtractD^i(_t) as follows D^i() = [D^i()+ D^i(_t)-D^i(_t)]= D^i(_t)+(D^i()-D^i(_t)) ≥𝖣 + (D^i()-D^i(_t)) where (<ref>) follows since 𝖣 =D^i() ≤ D^i() for all ∈Ł. Taking the expectation on both sides of (<ref>) and substituting the result into (<ref>), we obtain f^i(^i_T)≥ 𝖣 - I_2 - I_3 where I_2 and I_3 are as defined in Lemma <ref>. The rest of the proof proceeds simply by developing bounds on I_2 and I_3. §.§.§ Bound on I_2The bound on I_2 follows simply from the moment bounds in (A3) and the Cauchy-Schwartz inequality. We begin with the following observation Since the functions D^i are convex, it holds that [D^i(_t)-D^i()]≤∇ D^i(_t), _t-≤∇ D^i(_t)_t-≤ V_i(i-1)V where (<ref>) uses the Cauchy-Schwartz inequality, while (<ref>) uses (<ref>) and (<ref>). Therefore, substituting (<ref>) into the expression for I_2 and rearranging, we obtain I_2 ≤V∑_i=1^K(i-1)V_i. Finally, the required bound in Lemma <ref> is obtained by substituting V = max_i V_i. §.§.§ Bound on I_3The bound in I_3 follows from setting aside the error due to asynchrony I_4, and developing a bound on the remaining terms by telescopically summing the bounds on _t^i over all 1≤ i≤ K and 1≤ t≤ T.Since 0∈Λ is a feasible dual solution, using the form of the updates in (<ref>) and expanding as in (<ref>), it follows that ^2 ≤^2 +_t-δ_i(t)^i^2-2, _t-δ_i(t)^i. Adding the term 2_t^i-1,∇ D^i(_t^i-1) on both sides, and rearranging, we obtain 2,∇ D^i()≤ ^2-^2+_t-δ_i(t)^i^2-2,_t,δ_i(t)^i where _t,δ_i(t)^i is as defined in (<ref>). Summing over i=1,…,K andt=1,⋯,T, taking expectation, and utilizing (A3), it follows that I_3≤_1^2/2 T + /2 V_i^2 + I_4 ≤_1^2/2 T +KV^2 /2 + I_4 where I_4 is as defined in Lemma <ref> and the (<ref>) uses V = max_iV_i. §.§.§ Bound on I_4The term I_4 collects the error from the terms that arise due to asynchrony. A bound on I_4 is developed from the use of the delay bound assumption in (A4). Adding and subtracting _t^i-1, ∇ D^i(_t-τ_i(t)^i-1) to each summand of I_4, we obtain I_4= 1/T∑_t=1^T∑_i=1^K,∇ D^i()-∇ D^i(_t-τ_i(t)^i-1)+∑_t=1^T∑_i=1^K, ∇ D^i(_t-τ_i(t)^i-1)-_t-δ_i(t)^i. Of these, the first term in (<ref>) can be bounded using the bound in Lemma <ref> and the Cauchy-Schwartz inequality, by observing that ,∇ D^i()-∇ D^i(_t-τ_i(t)^i-1)≤ B∇ D^i()-∇ D^i(_t-τ_i(t)^i-1)≤B L^i-_t-τ_i(t)^i-1≤ B L^i τ KV where (<ref>) follows from (A7) and (<ref>) from the bound developed in (<ref>). For the second term in (<ref>), recalling the definition of _t-δ_i(t)^i-1 from Appendix <ref>, observe that although _t^i |_t-δ_i(t)^i-1≠_t^i, there exists some κ_i(t) ≤ t such that _κ_i(t)^i-1|_t-δ_i(t)^i-1 = _κ_i(t)^i-1. Indeed, observe that κ_i(t) ≥ t-δ_i(t) since _t-τ_i(t)^i-1 only depends on random variables contained in _t-δ_i(t)^i-1. The subsequent bounds hold for any κ_i(t) that satisfies (<ref>), including for the worst case when κ_i(t) = t-δ_i(t). It follows that ∇ D^i(_t-τ_i(t)^i-1)-_t-δ_i(t)^i,_t^i-1 = ∇ D^i(_t-τ_i(t)^i-1)-_t-δ_i(t)^i,_t^i-1|ℱ_t-δ_i(t)^i-1 = [[∇ D^i(_t-τ_i(t)^i-1)-_t-δ_i(t)^i,_κ_i(t)^i-1|ℱ_t-δ_i(t)^i-1]]+ ∇ D^i(_t-τ_i(t)^i-1)-_t-δ_i(t)^i,_t^i-1 - _κ_i(t)^i-1. From (<ref>) and (<ref>), it follows that the first summand in (<ref>) is zero. The second summand can be bounded by using the Cauchy-Schwartz inequality and the bounds in (A4) and (<ref>) as follows: ∇ D^i(_t-τ_i(t)^i-1)-_t-δ_i(t)^i,_t^i-1 - _κ_i(t)^i-1≤∇ D^i(_t-τ_i(t)^i-1)-_t-δ_i(t)^i_t^i-1 - _κ_i(t)^i-1≤2V_i(t-κ_i(t)) KV≤2V_iτKV where the inequality in (<ref>) follows since t-τ_i(t) ≤ t-δ_i(t) ≤κ_i(t) ≤ t. Finally, substituting (<ref>) and (<ref>) into (<ref>) yields I_4≤∑_i=1^K τ KV(B L^i+2V_i)≤τ K^2V(BL+2V) which together with (<ref>) gives the desired bound. IEEEtran

http://arxiv.org/abs/1702.08290v2

Department of Electrical and Computer Engineering, University of California, Riverside, CA 92521, USAQuantum Nanoelectronics Laboratory, Department of Physics, University of California, Berkeley CA 94720, USACenter for Quantum Coherent Science, University of California, Berkeley CA 94720, USA.Quantum Nanoelectronics Laboratory, Department of Physics, University of California, Berkeley CA 94720, USACenter for Quantum Coherent Science, University of California, Berkeley CA 94720, USA.Quantum Nanoelectronics Laboratory, Department of Physics, University of California, Berkeley CA 94720, USACenter for Quantum Coherent Science, University of California, Berkeley CA 94720, USA.Department of Electrical and Computer Engineering, University of California, Riverside, CA 92521, USAWe consider the simultaneous and continuous measurement of qubit observables σ_z and σ_zcosφ + σ_xsinφ, focusing on the temporal correlations of the two output signals. Using quantum Bayesian theory, we derive analytical expressions for the correlators, which we find to be in very good agreement with experimentally measured output signals. We further discuss how the correlators can be applied to parameter estimation, and use them to infer a small residual qubit Hamiltonian arising from calibration inaccuracy in the experimental data. Correlators in simultaneous measurement of non-commuting qubit observables Alexander N. Korotkov December 30, 2023 ==========================================================================Continuous quantum measurements (CQMs) have become a unique platform to explore fundamental aspects of quantum phenomena and have potential applications to quantum information science. They have been discussed theoretically in various contexts (e.g., <cit.>), and in the past decade superconducting qubits have become the main experimental system for the realization of CQMs <cit.>. CQMs are shedding new light on our understanding of the quantum measurement process, and there is also a growing interest in CQM applications, including quantum feedback <cit.>, rapid state purification <cit.>, preparation of entangled states <cit.>, and continuous quantum error correction <cit.>.While a simultaneous measurement of non-commuting observables is impossible with projective measurements, nothing theoretically forbids such a measurement using CQMs. Aside from new physics, such a protocol may open up new areas of applications, inaccessible to projective measurements. The theoretical discussion of a simultaneous measurement of incompatible observables has a long history <cit.>. For the measurement of non-commuting observables of a qubit, statistics of time-integrated detector outputs and fidelity of state monitoring directly via time-integrated outputs has been analyzed in Ref. <cit.>. The evolution of the qubit state due to simultaneous measurement of incompatible variables has been described theoretically in Ref. <cit.>, and has been recently demonstrated experimentally in Ref. <cit.>. In this letter, we focus on the temporal correlations of the output signals from two linear detectors measuring continuously and simultaneously the qubit observables σ_z and σ_φ≡σ_zcosφ + σ_xsinφ, where σ_x and σ_z are the Pauli matrices and φ is an angle between the two measurement directions on the Bloch sphere (Fig. <ref>). The experimental setup is described in detail in Ref. <cit.>; it is based on a Rabi-rotated physical qubit, which is measured stroboscopically <cit.> using symmetric sideband pumping of a coupled resonator, so that σ_z and σ_φ for an effective rotating-frame qubit are being measured. Description of such a measurement based on the theory of quantum trajectories <cit.> has been developed in Ref. <cit.>. In this letter we will use a simpler approach based on quantum Bayesian theory <cit.>. The quantum Bayesian description of the rotating-frame experiment <cit.> is developed in the Supplemental Material <cit.>. The goal of this letter is calculation of the time-correlators for the output signals measuring σ_z and σ_φ, and their comparison with experimental data. As we will see, these correlators may be a useful tool for sensitive parameter estimation in an experiment. These correlators are also important in the analysis of quantum error detection and correction based on simultaneous measurement of non-commuting operators <cit.>. We note that the analyzed output signal correlators are different from qubit-state correlators <cit.>. Quantum Bayesian theory.—A simultaneous continuous measurement of the qubit observables σ_z and σ_φ by two linear detectors (Fig. <ref>) produces noisy output signals I_z(t) and I_φ(t), respectively <cit.>,I_z(t) = Tr[σ_zρ (t)] + √(τ_z) ξ_z(t), I_φ(t) = Tr[σ_φρ (t)] + √(τ_φ) ξ_φ(t),where ρ (t) is the qubit density matrix and τ_z, τ_φ are the “measurement” (collapse) times needed for an informational signal-to-noise ratio of 1 for each channel. Note the chosen normalization for I_z and I_φ. In the Markovian approximation, the noises ξ_z and ξ_φ are uncorrelated, white, and Gaussian with two-time correlators ⟨ξ_z(t) ξ_z(t')⟩ = ⟨ξ_φ(t) ξ_φ(t')⟩ = δ(t-t')and ⟨ξ_z(t) ξ_φ(t')⟩ = 0. The qubit state is characterized in the Bloch-sphere representation as ρ (t)≡[ + x(t) σ_x + y(t) σ_y + z(t) σ_z]/2. We assume phase-sensitive amplifiers in the experimental setup, amplifying the optimal (informational) quadratures, so that the qubit evolution due to measurement is not affected by the phase backaction related to fluctuations in the orthogonal (non-informational) quadrature <cit.>. Then there is only the quantum informational backaction, which for measurement of σ_z and σ_φ is described <cit.> by the evolution equations (in the Itô interpretation) ẋ= - Γ_z x - Γ_φcosφ(xcosφ - zsinφ)- τ_z^-1/2xzξ_z - τ_φ^-1/2[xzcosφ - (1-x^2)sinφ] ξ_φ ,ẏ= - (Γ_z +Γ_φ )y - τ_z^-1/2yzξ_z - τ_φ^-1/2 y [zcosφ + xsinφ] ξ_φ ,ż=Γ_φsinφ(xcosφ - zsinφ) + τ_z^-1/2(1-z^2) ξ_z+ τ^-1/2_φ[(1 - z^2)cosφ -xzsinφ]ξ_φ .Here Γ_z and Γ_φ are the ensemble dephasing rates due to measurement, so that the quantum efficiencies <cit.> for the two channels are η_z = 1/(2τ_zΓ_z) and η_φ = 1/(2τ_φΓ_φ). In the experiment η_z =0.49 and η_φ=0.41. Equations (<ref>)–(<ref>) describe qubit evolution only due to measurement. We also need to add terms due to unitary evolution and due to decoherence not related to measurement. We assume the Hamiltonian H=ħΩ̃_ Rσ_y/2, describing Rabi oscillations about y-axis with frequency Ω̃_ R. In the experiment, Ω̃_ R=Ω_ R-Ω_ rf is a small (kHz-range) undesired mismatch between the physical qubit Rabi frequency Ω_ R and rotating frame frequency Ω_ rf defined by detuning of sideband pumps <cit.>. Decoherence of the effective qubit arises from the decoherence of the physical qubit with energy relaxation time T_1 and dephasing time T_2 [the pure dephasing rate is then T_ pd^-1=T_2^-1-(2T_1)^-1]. Averaging the decoherence over fast rotation Ω_ R≫ T_2^-1 and adding unitary evolution, we obtain <cit.> ẋ = Ω̃_ R z - γ x, ẏ = -T_2^-1 y,ż = -Ω̃_ R x -γ z,γ = (T_1^-1+T_2^-1)/2,Evolution of the effective qubit is described by adding terms from Eqs. (<ref>)–(<ref>) and (<ref>).Correlators.—Our next goal is to calculate the two-time correlators, K_ij(τ), for the output signals, K_ij(τ) ≡⟨ I_j(t_1+τ)I_i(t_1)⟩, τ>0, i,j ∈{z,φ}.Self- and cross-correlators correspond to i=j and i≠ j, respectively. The averaging in Eq. (<ref>) is over an ensemble of measurements with the initial qubit state ρ _ in prepared at time t_ in≤ t_1. We will see, however, thatthe result does not depend on ρ_ in, t_ in, and t_1, so Eq. (<ref>) can also be understood as averaging over time t_1. We assume that the parameters in Eqs. (<ref>)–(<ref>) do not change with time. By assuming τ>0, we avoid considering the trivial zero-time contribution to the self-correlators, Δ K_ii(τ )=τ_i δ (τ).As shown in the Supplemental Material <cit.>, calculation of the correlators from Eqs. (<ref>)–(<ref>) is equivalent to the followingrecipe<cit.>: we replace an actual continuous measurement at the (earlier) time moment t_1 with a projective measurement of σ_i, so that the measurement result I_i(t_1) is ± 1 with probability {1± Tr[σ_i ρ(t_1)]}/2, and the qubit state collapses correspondingly to the eigenstate |1_i⟩ or |0_i⟩ of σ_i (σ_i|1_i⟩ =|1_i⟩, σ_i|0_i⟩ =-|0_i⟩). This gives the correlatorK_ij(τ) =Tr[σ_jρ_ av(t_1+τ|1_i)] 1+Tr[σ_i ρ(t_1)]/2- Tr[σ_jρ_ av(t_1+τ|0_i)]1-Tr[σ_iρ(t_1)]/2 ,where ρ_ av(t_1+τ|1_i) is the ensemble-averaged density matrix at time t_1+τ with the initial condition ρ_ av(t_1|1_i)=|1_i⟩⟨ 1_i|; similarly,ρ_ av(t_1+τ|0_i) starts with ρ_ av(t_1|0_i)=|0_i⟩⟨ 0_i|. The evolution of ρ_ av is given by Eqs. (<ref>)–(<ref>) without noise, ξ_z=ξ_φ=0 (because of the Itô form), so that ẋ_ av= - Γ_z x_ av - Γ_φcosφ( x_ avcosφ - z_ avsinφ)+Ω̃_ R z_ av -γ x_ av,ẏ_ av= - (Γ_z +Γ_φ )y_ av -T_2^-1 y_ av,ż_ av=Γ_φsinφ( x_ avcosφ - z_ avsinφ) - Ω̃_ R x_ av -γ z_ av .These equations have an analytical solution presented in <cit.> (note that the evolution of the y-coordinate is not important in our analysis). Thus we obtain the following correlators (alternative methods for the derivation are also discussed in <cit.>):K_zz(τ) = 1/2[ 1+Γ_z+cos(2φ) Γ_φ/Γ_+ -Γ_-]e^-Γ_-τ+ 1/2[1 - Γ_z+cos(2φ) Γ_φ/Γ_+ -Γ_-] e^-Γ_+τ,K_zφ(τ) = (Γ_z + Γ_φ) cosφ+ 2Ω̃_ Rsinφ/2(Γ_+ - Γ_-)(e^-Γ_-τ - e^-Γ_+τ)+cosφ/2( e^-Γ_-τ + e^-Γ_+τ), Γ_± =Γ_z +Γ_φ±[ Γ_z^2 + Γ_φ^2 +2Γ_zΓ_φcos(2φ) -4 Ω̃_ R^2]^1/2/2+(T_1^-1+T_2^-1)/2.Because of the rotational symmetry, the results for the correlators K_φφ(τ) and K_φ z(τ) can be obtained from Eqs. (<ref>) and (<ref>) by exchanging Γ_z↔Γ_φ and φ→ -φ. The rotational symmetry also makes the correlators insensitive to a y-rotation in both measurement directions, z→φ_ add, φ→φ+φ_ add, by any angle φ_ add. We emphasize that the obtained correlators do not depend on the qubit state ρ (t_1) and therefore on ρ_ in and t_ in (this property would not hold in the presence of phase backaction). We also emphasize that the correlators depend on Γ_z and Γ_φ, but do not depend on τ_z and τ_φ and therefore on the quantum efficiencies η_z and η_φ. Physically, this is because non-ideal detectors can be thought of as ideal detectors with extra noise at the output <cit.>, which only affects the zero-time self-correlators K_ii(0).Let us discuss some special cases for the results (<ref>)–(<ref>). (i) At small times, τ→ +0, we obtain correlatorsK_zz(+0)= 1,K_zφ(0) = K_φ z (0)= cosφ.(ii) For |φ|≪ 1 and sufficiently small T_2^-1 and Ω̃_ R, we have Zeno pinning near the states |0⟩ and |1⟩ with rare jumps between them with equal rates Γ_ jump. This produces cross-correlator <cit.>K_zφ(τ)≈exp(-2Γ_ jumpτ) with jump rates Γ_ jump = φ^2Γ_zΓ_φ + Ω̃_ R^2/2(Γ_z+Γ_φ )+(T_1^-1+T_2^-1)/4 .(iii) In the case Ω̃_ R=T_1^-1=T_2^-1=0, we have full correlation for φ=0,K_zφ(τ) = K_zz(τ ) = 1, full anticorrelation for φ=π, K_zφ(τ) = -K_zz(τ ) = -1, and no correlation for φ=π/2, K_zφ(τ) = 0, while K_zz(τ )=e^-Γ_φτ and K_φφ(τ )=e^-Γ_z τ. (iv) In the case Ω̃_ R=0, the cross-correlator is symmetric, K_zφ(τ )=K_φ z(τ), for any φ. Comparison with experimental results.—Experimental data have been taken in the same way as in Ref. <cit.> (see also <cit.>). Experimentalparameters correspond to well-separated frequency scales, as needed for the theoretical results, (T_1^-1,T_2^-1,|Ω̃_ R|)≪ (Γ_z, Γ_φ) ≪ (κ_z,κ_φ)≪Ω_ R, with T_1=60μs, T_2=30 μs, Γ_z^-1=Γ_φ^-1= 1.3 μ s, damping rates of the two measurement resonator modesκ_z/2π=4.3 MHz and κ_φ/2π=7.2 MHz, and Ω_ R≈Ω_ rf= 2π× 40 MHz. For this work we use 11 values for the angle φ between the Bloch-sphere directions of simultaneously measured qubit observables: φ_n=nπ/10, with integer n between 0 and 10. While φ_n is determined by well-controlled phases of applied microwaves <cit.>, the effective φ includes a small correction δφ =(κ_φ -κ_z)/2Ω_ R≈ 0.036(see <cit.>), so that φ =φ_n+δφ. We have used about 200,000 traces per angle for the output signals Ĩ_z(t) and Ĩ_φ (t), each with 5 μs duration and 4 ns sampling interval. The traces are selected by heralding the ground state of the qubit at the start of a run and checking that the transmon qubit is still within the two-level subspace after the run. The recorded signals Ĩ_i(t) are linearly related to the normalized signals I_i(t) in Eqs. (<ref>) and (<ref>) asĨ_i(t)=(ΔĨ_i/2)I_i(t)+Ĩ_i^ off, where responses ΔĨ_i have been calibrated using ensemble-averaged ⟨Ĩ_i(t)⟩ (see details in <cit.>), giving in arbitrary units ΔĨ_z=4.0 and ΔĨ_φ=4.4. The offsets Ĩ_i^ off are approximately zeroed individually for each trace by measuring the non-rotating physical qubit after each run. Additional offset removal, |Ĩ_i^ off|≈ 0.15-0.20, for all traces with the same φ is done using ⟨Ĩ_i(t)⟩<cit.>. For calculating the correlators, we average over the ensemble of ∼200,000 traces and additionally average over time t_1 in Eq. (<ref>) within the 0.5 μs range 1 μ s≤ t_1 ≤ 1.5 μ s (first 1 μs is not used to avoid transients in the experimental procedure, and longer averaging reduces the range for τ; we also used averaging over 1 μs duration with similar results). Note that in the experiment the applied microwave phases in the two measurement channels actually correspond to angles ±φ_n/2; however, because of rotational symmetry, we still label the first measured operator as σ_z and the second operator as σ_φ. Also note that we use subscripts z and φ in various notations (Ĩ_i, κ_i, etc.) simply to distinguish the first (“z”) and second (“φ”) measurement channels. Figure <ref>(a) shows the agreement between the theory and the experimental data, where the solid lines show the symmetrized cross-correlator [K_zφ(τ) +K_φ z(τ)]/2 calculated from the experimental traces for 11 values of the angle φ, while the dashed lines correspond to the theoretical result, Eq. (<ref>).For the analytics we used Ω̃_ R=0; however, there is practically no dependence on Ω̃_ R for the symmetrized cross-correlator, since the dependence comes only via Eq. (<ref>). Note that because of the Markovian assumption, our theory is formally valid only for τκ_i^-1∼ 30 ns; however, the experimental results agree with the theory even at τ < κ_i^-1. Figure <ref>(b) shows the same symmetrized cross-correlator at τ =0 as a function of φ. The agreement between the theory (cosφ, line) and the experiment (crosses) is also very good.The self-correlator K_zz(τ) as a function of τ is shown in Fig. <ref>(c) for 11 values of φ (results for K_φφ are similar). The agreement between the theory (dashed lines) and experiment (solid lines) is good, except for small τ (discussed below). A significant discrepancy for values of φ close to π/2 is probably caused by remaining offsets Ĩ_i^ off, which vary from trace to trace. Note that the lines in <ref>(c) come in pairs, corresponding to angles φ_n and π-φ_n. The separation of the analytical lines in the pairs is due to δφ, while separation of experimental lines is smaller, probably indicating a smaller value of δφ (partial compensation could be due to imperfect phase matching of applied microwaves or their dispersion in the cable).Looking at the experimental self-correlators K_zz(τ) and K_φφ(τ) at small τ for φ_n=π/2 [Fig. <ref>(d)], we see that in contrast to the theoretical results, there is a very significant increase of K_ii (τ) at τ 0.1μ s. The discrepancy is due to the assumption of delta-correlated noise in our theory, while in the experiment the amplifying chain has a finite bandwidth (the Josephson parametric amplifiers have a half-bandwidth of 3.6 MHz and 10 MHz for σ_z and σ_φ channels, respectively), and the output signals Ĩ_i(t) are also passed through analog filters with a quite sharp cutoff at ∼25 MHz (this cutoff produces clearly visible oscillations with ∼40 ns period). Therefore, the theoretical delta-function contribution τ_i δ(τ) to K_ii(τ) becomes widened in experiment. It is interesting to note that, somewhat counterintuitively, a finite bandwidth of measurement resonator modes does not produce a contribution to K_ii(τ) at 0<τκ_i^-1 when Γ_i≪κ_i<cit.> (κ_z^-1≈ 37 ns, κ_φ^-1≈ 22 ns). This can be understood by considering a resonator without a qubit; then a finite bandwidth κ_i does not affect the amplified delta-correlated vacuum noise, so that only classical fluctuations of the resonator field (e.g., due to parameter fluctuations or elevated resonator temperature) will produce output fluctuations with 2/κ_i time scale. We have checked that the lines in Fig. <ref>(d) do not contain noticeable exponential contributions with decay time of 2/κ_i (small expected contributions with amplitude on the order of Γ_j/κ_i<cit.> are below experimental accuracy).Estimation of residual Ω̃_ R.—We now show that the antisymmetrized cross-correlator is a useful tool and can be used to estimate small residual Rabi oscillations frequency Ω̃_ R in the experiment. From Eq. (<ref>) we findK_zφ(τ)-K_φ z(τ )=2Ω̃_ Rsinφ/Γ_+-Γ_-(e^-Γ_-τ - e^-Γ_+τ) .Since in the case |Ω̃_ R|≪Γ_z,φ we can neglect Ω̃_ R in Eq. (<ref>) for Γ_±, Eq. (<ref>) gives a direct way to find Ω̃_ R from the experimental antisymmetrized cross-correlator. The solid line in Fig. <ref> shows K_zφ(τ)-K_φ z(τ ) from the experimental data for φ=π/2. Fitting this dependence on τ with Eq. (<ref>) (dashed line), we find the value Ω̃_ R/2π≈ 12 kHz, which is within the experimentally expected range of frequency mismatch between Ω_ R and Ω_ rf. Note that the overall shapes of the solid and dashed lines agree well with each other. Estimation of Ω̃_ R via the antisymmetrized cross-correlation is a very sensitive method and can be used to further reduce |Ω̃_ R| in an experiment, in which a direct measurement of 40 MHz Rabi oscillations with a few-kHz accuracy is a difficult task. Conclusion.—Using the quantum Bayesian theory for a simultaneous measurement of non-commuting qubit observables, we obtained analytical results for the self- and cross-correlators of the output signals from the measurement. Their comparison with experimental results shows a very good agreement. The correlators can be used for sensitive parameter estimation, in particular, to estimate and eliminate the mismatch between the Rabi oscillations and the sideband frequency shift used for measurement. Acknowledgements.—We thank Justin Dressel and Andrew Jordan for useful discussions. The work was supported by ARO grant No. W911NF-15-0496. 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Each mode is then used to measure an observable of the qubit, as described below. The apparatus is cooled to 30 mK inside a dilution refrigerator.We drive Rabi oscillations Ω_ R/2π = 40  MHz on the qubit by applying a resonant microwave tone modulated by an arbitrary waveform generator.In the frame rotating with Ω_ R, this produces an effective low frequency qubit. To couple the effective qubit to the cavity modes for measurement, we apply a pair of microwave sidebands to each mode. The sidebands are detuned above and below the two cavity modes by Ω_R, which leads to a resonant interaction between the qubit Rabi oscillations and the mode. This coupling may be understood as a stroboscopic measurement of the qubit oscillations. The relative phase of the sidebands determines which quadrature of the qubit oscillations is measured. This coupling causes the cavity mode state to displace in a way that depends on the state of the qubit.We couple to the internal cavity field using a small antenna that protrudes into the cavity, allowing read out the cavity state as described above. Quantum trajectory reconstructions are validated using post-selection and tomographic measurements.§ QUANTUM BAYESIAN APPROACH TO QUBIT MEASUREMENT IN RABI-ROTATED FRAMEIn this section we develop the quantum Bayesian theory of the stroboscopic qubit measurement in the Rabi-rotated frame, used in the experiment <cit.> and briefly described above. We start with measurement of one effective observable σ_φ=σ_z cosφ +σ_x sinφ, then adding the second measurement in the same way and deriving Eqs. (4)–(6) of the main text. In this derivation we assume that the qubit Rabi frequency Ω_ R is exactly equal to the sideband frequency shift Ω_ rf (which defines the rotating frame), while a small mismatch between Ω_ R and Ω_ rf is added later via Eq. (7) of the main text (also discussed in this section). The focus is on the simple physics of the qubit measurement in the rotating frame. §.§ Measurement of one observable σ_φ Thephysical qubit is Rabi-rotated with frequency Ω_ R about the y-axis, so that its Bloch coordinates rotate asx_ ph(t)= r_0 sin (Ω_ R t +ϕ_0),y_ ph(t)=y_0, z_ ph(t) =r_0 cos (Ω_ R t+ϕ_0),where the radius r_0(t) within the xz-plane, the rotation phase ϕ_0(t), and the coordinate y_0(t) slowly change in time (e.g., due to measurement). The oscillations of the qubit z-component lead to a small change of the effective resonator frequency, ω_ r(t) = ω_ r^ m +χr_0 cos (Ω_ R t+ϕ_0),whereχis the (small) dispersive coupling between the qubit and the measurement resonator mode, and ω_ r^ m is the mean value between the resonator frequencies for the physical qubit states |0⟩ and |1⟩. Note that in this derivation, fast-oscillating ω_ r(t) is the value averaged over the physical qubit states, and we neglect quantum backaction developing during short time scale Ω_ R^-1.The sideband drive of the resonator at frequencies ω_ d±Ω_ rf with equal amplitudes (here for simplicity we assume ω_ d=ω_ r^ m and Ω_ rf=Ω_ R), produces the Hamiltonian termH_ drive/ħ = εsin (Ω_ R t+φ)a^† + h.c.,where ε is the normalized amplitude, φ depends on the initial phase shift between the sideband tones, a^† is the creation operator for the resonator, and we use the rotating frame based on ω_ d. The form of this term follows from the usual trigonometric relation for adding the sideband tones, (ε/2)sin[(ω_ d + Ω_ R)t+φ]-(ε/2)sin[(ω_ d - Ω_ R)t-φ]=εsin(Ω_ Rt+φ)cos(ω_ d t). This leads to the following dynamics of the resonator's coherent state|α (t)⟩ [or classical field α (t)] in the rotating frame based on ω_ d, α̇ = -i χ r_0 cos(Ω_ R t+ϕ_0)α - i εsin (Ω_ R t+φ ) -κ/2α,where we also took into account the resonator damping with energy decay rate κ.Now let us solve the evolution equation (<ref>), assuming κ≪Ω_ R and |χ| ≪Ω_ R. The drive term produces fast oscillation of α with Rabi frequency Ω_ R, Δα (t)= i ε/Ω_ Rcos (Ω_ R t +φ ).Inserting this oscillation into the first term of Eq. (<ref>), using the trigonometric formula cos(Ω_ R t+ϕ_0)cos (Ω_ R t+φ)= 1/2cos(ϕ_0-φ)+1/2cos (2Ω_ R t + ϕ_0 +φ ), and neglecting oscillations with frequency 2Ω_ R, we obtain the equation for the slow evolution,α̇_ s = χε/2Ω_ Rr_0 cos(ϕ_0 -φ )-κ/2α_ s ,α (t) =α_s (t)+Δα(t).Note that we can neglect the additional fast oscillations produced by the first term in Eq. (<ref>), Δ_2α (t)=-i(χ/Ω_R) r_0sin(Ω_ R t+ϕ_0) α_ s, in comparison with (<ref>), because we assume χ^2/(Ω_ Rκ)≪ 1.We see that the evolution (<ref>) of the resonator field α_ s depends on the state of theeffective qubit,x =r_0 sin(ϕ_0), y=y_0, z=r_0 cos (ϕ_0),which corresponds to the physical qubit (<ref>)–(<ref>) in the rotating frame Ω_ R. Moreover, this dependence is only on the Bloch φ-coordinate of the effective qubit, which is within the xz-plane at an angle φ from the z-axis, we see this sincer_0 cos(ϕ_0 -φ ) =Tr[σ_φρ (t)],where ρ(t) is the slowly-varying density matrix of the effective qubit.At this stage, we make use of the quantum Bayesian approach <cit.> to describe the qubit evolution due to measurement. Since the oscillating part Δα of the resonator field [Eq. (<ref>)] does not depend on the qubit state, it can be neglected in the analysis. In contrast, homodyne measurement of the leaked field α_ s gives us information on the value of the φ-coordinate of the effective qubit, which is a two-level system similar to the physical qubit. Inevitably, this information gradually collapses the effective qubit, i.e., changes its state according to the acquired information. The two σ_φ-basis states |1_φ⟩ and |0_φ⟩of the effective qubit (σ_φ |1_φ⟩ =|1_φ⟩, σ_φ |0_φ⟩ =-|0_φ⟩) produce two steady states of the resonator, respectively (excluding oscillating Δα), α_ st,1 =χε/Ω_ Rκ,α_ st, 0=-α_ st,1.This is all what is needed in the Markovian “bad cavity” regime (when the evolution of the effective qubit is much slower than κ), which is assumed in the main text. Since in circuit QED only the difference between α_ st,1 and α_ st,0 is important for the analysis of the qubit evolution due to measurement in the “bad cavity” regime<cit.>, the situation isequivalent to the qubit evolution due to measurement in the standard setup <cit.> with the sameα_ st,1-α_ st,0. Correspondingly, the quantum Bayesian formalism in the “bad cavity” regime is exactly the same as for the standard circuit QED setup <cit.>, which coincides with the Bayesian formalism for qubit measurement using a quantum point contact <cit.>. The only difference is that now we discuss the evolution of the effective qubit instead of the physical qubit.In particular, when a phase-sensitive amplifier is used, the response ΔĨ of the detector to the effective qubit state has the dependence ΔĨ=ΔĨ_ maxcosθ on the phase difference θ between the amplified quadrature (at the microwave frequency ω_ d) and the optimal quadrature [which is real (horizontal), as follows from Eq. (<ref>)]. The informational backaction is proportional to cosθ, while phase backaction is proportional to sinθ, with ensemble dephasing of the effective qubit, Γ=κ/2|α_ st,1-α_ st,0|^2,not depending on θ<cit.>. Evolution of the effective qubit state due to measurement is given by Eqs. (18) and (25) of Ref. <cit.> in the basis {|0_φ⟩, |1_φ⟩}(Eqs. (12) and (13) of Ref. <cit.>). In this basis, the diagonal matrix elements ρ_00 and ρ_11 evolve due to the classical Bayes rule, while the off-diagonal elements ρ_01 and ρ_10 evolve due to evolving product ρ_00ρ_11 and also due to phase backaction. In the above derivation we assumed an exactly resonant microwave drive, ω_ d=ω_ r^ m. If this is not the case, |ω_ d-ω_ r^ m| ∼κ, then there will be an extra term -i (ω_ r^ m-ω_ d)α in Eq. (<ref>), which will lead to an extra term -i (ω_ r^ m-ω_ d)α_ s in Eq. (<ref>). Correspondingly, the steady states of the field α_ s for the effective qubit states |1_φ⟩ and |0_φ⟩are α_ st,1 =-α_ st, 0 =χε/Ω_ R[κ +2i(ω_ r^ m-ω_ d)]instead of Eq. (<ref>), so that the optimal quadrature is no longer horizontal (real). The quantum Bayesian formalism remains the same.If the effective rotating-frame qubit is measured by only one detector (σ_φ, but no σ_z) and Ω_ R=Ω_ rf, then it is possible to go beyond the “bad cavity” limit and analyze transients within the time scale κ^-1. The derivation of the quantum Bayesian formalism for this case exactly follows the derivation in Ref. <cit.> and uses the field evolution equation (<ref>) instead of the steady-state solution (<ref>). §.§ Derivation of Eqs. (4)–(6) in the main textIn the absence of phase back-action, the quantum Bayesian equations describing continuous measurement of qubit σ_z observable in the Markovian approximation are <cit.> ẋ=- τ_z^-1xz I_z(t)- γ_z x,ẏ= - τ_z^-1 yz I_z(t) - γ_z y ,ż= τ_z^-1(1-z^2)I_z(t) ,in the Stratonovich form, whereI_z(t)=Tr[σ_z ρ (t)]+√(τ_z) ξ_z (t)is the normalized output signal, ξ_z(t) is the normalized white noise, ⟨ξ_z(t)ξ_z(t')⟩ =δ (t-t'), τ_z is the “measurement” time after which the signal-to-noise ratio reaches 1, the qubit density matrix is ρ = ( +xσ_x+ yσ_y +zσ_z)/2, and qubit dephasing γ_z in individual measurement is related to the ensemble dephasing Γ_z as γ_z=Γ_z-(2τ_z)^-1.In the Itô form (i.e., using the forward definition of derivatives instead of the symmetric definition) these evolution equations become <cit.> ẋ= - τ_z^-1/2xzξ_z- Γ_z x,ẏ= - τ_z^-1/2yzξ_z- Γ_z y,ż=τ_z^-1/2(1-z^2) ξ_z .When the observable σ_φ is measured instead of σ_z, these equations remain the same <cit.> in the basis of eigenstates |0_φ⟩ and |1_φ⟩, so that we can simplychange the notation: x→ x_φ, y→ y_φ, z→ z_φ. Rotating back to the usual basis, i.e., using the transformation x=x_φcosφ + z_φsinφ , y=y_φ, z=z_φcosφ -x_φsinφ, we obtain for the Itô formẋ= - τ_φ^-1/2[xzcosφ - (1-x^2)sinφ] ξ_φ , - Γ_φcosφ(xcosφ - zsinφ) ,ẏ= - τ_φ^-1/2 y [zcosφ + xsinφ] ξ_φ- Γ_φy ,ż=τ^-1/2_φ[(1 - z^2)cosφ -xzsinφ]ξ_φ+ Γ_φsinφ(xcosφ - zsinφ) ,withI_φ(t)=Tr[σ_φρ (t)]+√(τ_φ) ξ_φ (t).When both σ_z and σ_φ measurements are performed at the same time, we simply add the terms from Eqs. (<ref>)–(<ref>) and (<ref>)–(<ref>) <cit.> (with uncorrelated noises ξ_z and ξ_φ in the two channels), thus obtaining Eqs. (4)–(6) of the main text. §.§ Correction to measured rotation phase φ As was discussed above, the phase φ in the double-sideband drive εsin (Ω_ R t+φ)cos (ω_ dt) [see Eq. (<ref>)] directly determines the angle for the measured operator σ_φ for the effective qubit. This followed from the approximate solution of Eq. (<ref>). As we will see below, a more accurate solution shows a small correction to the measured direction φ.Neglecting the first term in Eq. (<ref>) but still keeping the last term, we obtain the (exact) oscillating solutionΔα (t) = iε/Ω_ R+κ^2/4Ω_ R[ cos(Ω_ Rt+φ) -κ/2Ω_ Rsin (Ω_ Rt+φ) ] ,which is more accurate than Eq. (<ref>). (Note that the additional term ∝ e^-κ t/2 is naturally included in the slow dynamics.) Inserting Δα (t) into the first term of Eq. (<ref>), we obtain the following evolution of the slow part α_ s of the total field α =α_ s+Δα:α̇_ s = χε/2( Ω_ R+κ^2/4Ω_ R)r_0 [ cos(ϕ_0 -φ )-κ/2Ω_ Rsin (φ -ϕ_0) ]-κα_ s /2 ,which in the case κ/Ω_ R≪ 1 is approximately α̇_ s = χε/2Ω_ Rr_0 cos(ϕ_0 -φ -κ/2Ω_ R )-κ/2α_ s .This equation coincides with Eq. (<ref>), except φ is replaced with φ+κ/2Ω_ R, thus slightly changing the direction ofthe measured operator for the effective qubit, r_0 cos(ϕ_0 -φ -κ/2Ω_ R ) =Tr[σ_φ+κ/2Ω_ Rρ (t)]. In the case when two channels simultaneously measure nominal operators σ_z and σ_φ (with corresponding resonator bandwidths κ_z and κ_φ), the measured directions on the Bloch sphere are actually κ_z/2Ω_ R and φ+κ_φ/2Ω_ R, so that the relative angle is φ+δφ with δφ = κ_φ-κ_z/2Ω_ R .This correction of ∼2^o was used in the main text when we compared the experimental results with the theory results.§.§ Decoherence of the effective qubit Now let us derive Eq. (7) of the main text, describing the evolution of the effective qubit not related to measurement.The evolution of thephysical qubit due to Rabi oscillations and environmental decoherence (energy relaxation and pure dephasing) is described by the standard master equation ρ̇_ ph = -i/ħ [H_ ph,ρ_ ph] + 1/2T_ pd ℒ[σ_z] + 1/T_1 ℒ[σ_-],ℒ[A] ≡Aρ_ phA^† - 1/2(A^†Aρ_ ph - ρ_ ph A^†A),where ρ_ ph(t) and H_ ph = ħΩ_ Rσ_y/2 are the density matrix and Hamiltonian of the physical qubit, respectively, the Lindblad-operator evolution with A=σ_z describes pure dephasing, and energy relaxation corresponds toA=σ_- =|0⟩⟨ 1|= (σ_x - iσ_y)/2.To convert Eq. (<ref>) into the rotating frame (with frequency Ω_ rf), we apply the unitary transformation ρ(t) = U^†(t) ρ_ ph(t) U(t), where ρ(t) is the effective qubit density matrix and U(t) =exp(-iΩ_ rftσ_y/2). This gives ẋ = Ω̃_ Rz - (1/2T_ pd + 3/4T_1)x + 1/T_1sin(Ω_ rft) + ( 1/4T_1 - 1/2T_ pd)[xcos(2Ω_ rft) + zsin(2Ω_ rft)],ẏ = -(1/T_ pd + 1/2T_1)y,ż = -Ω̃_ Rx - ( 1/2T_ pd + 3/4T_1) z - 1/T_1cos(Ω_ rft) + ( 1/4T_1 - 1/2T_ pd)[xsin(2Ω_ rft) - zcos(2Ω_ rft)],where Ω̃_ R = Ω_ R - Ω_ rf. Since Ω_ rf is much faster than the evolution of the effective qubit,in Eqs. (<ref>) and (<ref>) we can neglect the oscillating terms. Finally expressing T_ pd via T_2 and T_1, so that T_ pd^-1+ (2T_1)^-1=T_2^-1 and (2T_ pd)^-1+3(4T_1)^-1= (T_1^-1+T_2^-1)/2, we obtain ẋ = Ω̃_ Rz - 1/2(T_1^-1 + T_2^-1) x,ẏ = T_2^-1y, ż = -Ω̃_ Rx - 1/2(T_1^-1 + T_2^-1)z,which is Eq. (7) of the main text.Since Eqs. (<ref>)–(<ref>) describe only the evolution not related to measurement, while Eqs. (4)–(6) of the main textdescribe only the evolution due to measurement, we need to simply add terms in these equations to describe the combined evolution of the effective qubit. We would like to emphasize that the derivation presented here relies on a significant separation of frequency scales (T_1^-1,T_2^-1,|Ω̃_ R|)≪(Γ_z, Γ_φ) ≪(κ_z,κ_φ)≪Ω_ R.In our experiment these inequalities are well satisfied: T_1=60 μs, T_2=30 μs, |Ω̃_ R^-1| 10μs, Γ_z^-1=Γ_φ^-1=1.31 μ s, κ_z^-1= 37 ns,κ_φ^-1=22.1 ns, and Ω_ R^-1=4 ns. The frequencies of the resonator modes should obviously be much larger than Ω_ R; in our experiment ω_ r, z/2π=7.4GHz and ω_ r, φ/2π=6.7GHz.§ ANALYTICAL RESULTS FOR CORRELATORSWe will first derive Eqs. (14)–(16) of the main text for the correlators using the “collapse recipe” and then we discuss the derivation of this recipe from the quantum Bayesian equations and its correspondence to the quantum regression approach.§.§ Derivation of Eqs. (14)–(16) in the main text using “collapse recipe” The collapse recipe for calculation of the correlators for the output signals (in the absence of phase backaction) was introduced in Ref. <cit.>. It says that in order to calculate the ensemble-averaged correlator,K_ij(τ) ≡⟨ I_j(t_1+τ)I_i(t_1)⟩, τ>0,we can replace the continuous measurement of σ_i at the earlier time moment t_1 with its projective measurement. It is also possible to replace the continuous measurement of σ_j at the later time moment t_1+τ with its projective measurement, but this is rather obvious and not important, since average values for the continuous and projective measurements coincide. In the following section we will show how this recipe can be derived from the quantum Bayesian equations (essentially repeating the derivation in Ref. <cit.>); here we just use this recipe.If the qubit state at time t_1 (it would be more accurate to say, right before t_1) is ρ (t_1), then the projective measurement of σ_i would produce the measurement result I_i(t_1)=1 with probability {1+ Tr[σ_i ρ (t_1)] }/2 and the result I_i(t_1)=-1 with probability {1- Tr[σ_i ρ (t_1)] }/2. After this projective measurement, the qubit state is collapsed to the eigenstate |1_i⟩ or |0_i⟩, correspondingly. Ensemble-averaged evolution after that is simple, since for an ensemble a continuous measurement is equivalent to decoherence. If the state was collapsed to |1_i⟩, then the further ensemble-averaged qubit evolution ρ_ av(t |1_i) starts with ρ_ av(t_1|1_i)=|1_i⟩⟨ 1_i|. Then the average result of σ_j measurement at time t=t_1+τ will be Tr[σ_j ρ_ av (t_1+τ|1_i)], which will produce contribution Tr[σ_j ρ_ av (t_1+τ|1_i)] × 1 to the correlator (<ref>) with probability {1+ Tr[σ_i ρ (t_1)] }/2.Similarly, the contribution corresponding to the state collapse to |0_i⟩ at time t_1, is Tr[σ_j ρ_ av (t_1+τ|0_i)] × (-1) with probability {1- Tr[σ_i ρ (t_1)] }/2. Summing these two cases, we obtainK_ij(τ) =Tr[σ_jρ_ av(t_1+τ|1_i)] 1+Tr[σ_i ρ(t_1)]/2- Tr[σ_jρ_ av(t_1+τ|0_i)]1-Tr[σ_iρ(t_1)]/2 ,which is Eq. (10) of the main text.Next, we need to find ρ_ av(t_1+τ|1_i) and ρ_ av(t_1+τ|0_i). The ensemble-averaged evolution of the qubit is described by Eqs. (11)–(13) of the main text. It is easy to see that the evolution of the y-coordinate is decoupled and has the simple solution,y_ av(t_1+τ) = e^-(Γ_z+Γ_φ)τy(t_1),where we use the subscript “av” to indicate ensemble averaging. The evolution equations for the coordinates x_ av and z_ av can be written in the matrix form, d/dtx_ avz_ av =Mx_ avz_ av,M =[ -(Γ_z +cos^2φΓ_φ+γ) sinφcosφ Γ_φ +Ω̃_ R; sinφcosφ Γ_φ -Ω̃_ R-(sin^2φ Γ_φ +γ) ].Diagonalizing the matrix M, we find the solutionx_ av(t_1+τ)z_ av(t_1+τ) = [e^-Γ_-τ +e^-Γ_+τ/2+e^-Γ_-τ - e^-Γ_+τ/2 ×( Γ_φsin 2φ +2Ω̃_ R/Γ_+-Γ_- σ_x - Γ_z + Γ_φcos 2φ/Γ_+-Γ_- σ_z) ]x(t_1)z(t_1),where Γ_±are theeigenvalues of the matrix - M, given by Eq. (16) of the main text and repeated here,Γ_± =Γ_z +Γ_φ±[ Γ_z^2 + Γ_φ^2 +2Γ_zΓ_φcos(2φ) -4 Ω̃_ R^2]^1/2/2+γ . Note that our evolution is symmetric under the inversion operation x→ -x, y→ -y, z→ -z (it is a unital map), and thereforeTr[σ_jρ_ av(t_1+τ|0_i)] = - Tr[σ_jρ_ av(t_1+τ|1_i)] .In this case Eq. (<ref>) simplifies to K_ij(τ) =Tr[σ_jρ_ av(t_1+τ|1_i)] ,which no longer depends on the initial state ρ (t_1). In particular, if σ_i=σ_z, then to find K_ij(τ)we need to use initial conditions x(t_1)=0 and z(t_1)=1 in Eq. (<ref>), which gives x_ av(t_1+τ |1) = sin(2φ) Γ_φ+2Ω̃_ R/2(Γ_+-Γ_-)( e^-Γ_-τ - e^-Γ_+τ) ,z_ av(t_1+τ |1 ) = 1/2[ 1+Γ_z+cos(2φ) Γ_φ/Γ_+ - Γ_-]e^-Γ_-τ+ 1/2[ 1 - Γ_z+cos(2φ) Γ_φ/Γ_+ - Γ_-]e^-Γ_+τ .Inserting these results into Eq. (<ref>) and using relations Tr[σ_zρ_ av(t_1+τ|1_i)] =z_ av (t_1+τ) and Tr[σ_φ ρ_ av(t_1+τ|1_i)] =z_ av (t_1+τ)cosφ +x_ av (t_1+τ)sinφ, we obtain Eqs. (14) and (15) of the main text for the correlators K_zz(τ) and K_zφ(τ). If σ_i≠σ_z, then we can use rotational symmetry to find K_ij(τ), simply replacing φ with the angle difference between the measured directions and renaming the measurement channels.Note that the evolution (<ref>) for the y-coordinate was not important for K_zz(τ) and K_zφ(τ). Also note that we were able to use Eq. (<ref>) instead of Eq. (<ref>) because for the effective qubit the states |1⟩ and |0⟩ are equivalent (producing unital ensemble-averaged evolution). In similar calculations for a physical (non-rotating) qubit, energy relaxation would make states |1⟩ and |0⟩ non-equivalent, and then we would need to use Eq. (<ref>). Finally, we emphasize that this recipe is valid only in the absence of phase backaction. It requires a minor modification when phase backaction is present.§.§ Derivation via stochastic Bayesian equations Now let us derive Eqs. (14) and (15) of the main text for K_zz(τ) and K_zφ(τ ) using the stochastic evolution equations (4)–(6) of the main text instead of the collapse recipe used above. Even though equivalence of these methods was shown in Ref. <cit.>, we will do the derivation explicitly, essentially repeating the equivalence proof in <cit.>. In the derivation we assume fixed t_1 and t_1+τ [averaging the correlator (<ref>) over the ensemble of realizations], and for brevity of notations we assume t_1=0. The qubit state right before the first measurement is therefore ρ_ in≡ρ(0). (Note that if we have a distribution of the initial states, it is possible to average the correlator over this distribution later. However, such averaging is not actually needed because of the linearity of quantum evolution that allows us to use a single initial state, which is equal to the average over the distribution.) We will mainly consider K_zφ(τ ); the derivation for K_zz(τ ) is similar. Using Eqs. (<ref>) and (<ref>), we can write the correlatorK_zφ(τ ) as a sum of two parts, describing a correlation between qubit states at different times and a correlation between the noise and the future qubit state [there is no correlation with the past states because of causality, and the noise-noise correlations for τ > 0 are also absent for uncorrelated white noises ξ_z(t) and ξ_φ (t)],K_zφ(τ)= K^(1)_zφ(τ) + K^(2)_zφ(τ), K^(1)_zφ(τ) = ⟨ Tr[σ_φρ (τ)] ⟩z(0) , K^(2)_zφ(τ) = √(τ_z) ⟨ Tr[σ_φρ (τ)]ξ_z (0)⟩ ,where averaging is over the noise realizations ξ_z(t) and ξ_φ (t), which affect evolution of ρ via Eqs. (4)–(6) of the main text, and the initial state is ρ_ in=[ +x_ inσ_x + y_ inσ_y+z_ inσ_z ]/2 with { x(0),y(0),z(0)}={x_ in,y_ in,z_ in}. The first contribution can also be written asK^(1)_zφ(τ) =Tr[σ_φρ_ av (τ|ρ_ in)] z_ in,whereρ_ av (τ|ρ_ in) is the ensemble-averaged density matrix at time τ, which starts with ρ_ in at t=0. Using linearity of the ρ_ av evolution given by Eqs. (<ref>) and (<ref>), we canformally rewrite it asK^(1)_zφ(τ) =Tr[σ_φρ_ av (τ|z_ inρ_ in)],where the evolution of ρ_ av now starts with state ρ_ av (0|z_ inρ_ in)=[ +z_ inx_ inσ_x + z_ iny_ inσ_y+z_ in^2σ_z ]/2. Note that in the definition of the state z_ inρ_ in we still use physical normalization Tr(z_ inρ_ in)=1, multiplying by z_ in only Bloch-sphere components of ρ_ in. To find the second contribution K^(2)_zφ(τ), we use the stochastic equations (4)–(6) of the main text [complemented with Eq. (7) of the main text] and derive the evolution equations for correlators ⟨ x(τ) ξ_z(0)⟩ and ⟨ z(τ) ξ_z(0)⟩: d/dτ⟨ x(τ) ξ_z(0)⟩⟨ z(τ) ξ_z(0)⟩ =M⟨ x(τ) ξ_z(0)⟩⟨ z(τ) ξ_z(0)⟩+ 1/√(τ_z)-x_ in z_ in1-z_ in^2 δ (τ),where M is the evolution matrix (<ref>), and for τ <0 these correlators are zero because of causality. This equation has a simple physical meaning. As follows from Eqs. (4) and (6) of the main text, the noise ξ_z(0) slightly changes the initial state after an infinitesimal time dt, so that x(0+dt)=x_ in-τ_z^-1/2x_ inz_ inξ_z(0) dt and z(0+dt)=z_ in+τ_z^-1/2(1-z_ in^2) ξ_z(0) dt. The further evolution starts with this slightly different state. Therefore, ⟨ x(dt) ξ_z(0)⟩ = -τ_z^-1/2x_ inz_ in⟨ξ_z(0)^2 dt⟩ =-τ_z^-1/2x_ inz_ in, since ⟨ξ_z(0)^2 dt⟩ = 1, as follows from Eq. (3) of the main text. Similarly, ⟨ z(dt) ξ_z(0)⟩ = τ_z^-1/2(1-z_ in^2)⟨ξ_z(0)^2 dt⟩ =τ_z^-1/2(1-z_ in^2). Thus we obtain the last term in Eq. (<ref>), while the evolution due to the matrix M is rather obvious. Even though the y component is not important for our analysis, for generality we can similarly derive ⟨ y(dt) ξ_z(0)⟩ = -τ_z^-1/2y_ inz_ in⟨ξ_z(0)^2 dt⟩ =-τ_z^-1/2 y_ inz_ in.Since the evolution of⟨ x(τ) ξ_z(0)⟩ and ⟨ z(τ) ξ_z(0)⟩ is governed by the same matrix M as for the components of ρ_ av (similar for y-component), we can write the contribution K_zφ^(2)(τ) asK^(2)_zφ(τ)=Tr[σ_φρ_ av(τ|δρ_ in)],δρ_ in = 1/2 [-x_ inz_ inσ_x -y_ in z_ inσ_y+(1-z_ in^2)σ_z ],where δρ_ in is an unphysical density matrix with zero trace, in which the Bloch-sphere components are the shifts discussed above due to the second term in Eq. (<ref>), multiplied by √(τ_z) because of Eq. (<ref>).It is easy to see thatz_ inρ_ in + δρ_ in = [ +σ_z]/2=|1⟩⟨ 1 |(recall that z_ inρ_ in is defined with unity trace). Therefore, combining Eqs. (<ref>) and (<ref>), we findK_zφ(τ )=Tr[σ_φρ_ av(τ|1)],which coincides with Eq. (<ref>) for σ_i=σ_z and σ_j=σ_φ. Note the slightly different notations for the initial state of ρ_ av, which should not be confusing, for example ρ_ av(τ|1)≡ρ_ av(τ| |1⟩⟨ 1|).Thus we have shown that the correlator K_zφ(τ ) derived from the stochastic evolution equations coincides with the result of the previous derivation based on the collapse recipe. The derivation for K_zz(τ) from the stochastic equations is similar, we just need to replace φ with z and σ_φ with σ_z in Eqs. (<ref>)–(<ref>), (<ref>), and (<ref>), thus obtaining K_zz(τ )=Tr[σ_z ρ_ av(τ|1)], which coincides with Eq. (<ref>) for σ_i=σ_j=σ_z. §.§.§ Equivalence in a non-unital case We have shown equivalence of the results for correlators K_zz(τ) and K_zφ(τ) derived via the stochastic Bayesian equations and via the simple collapse recipe. However, in showing the equivalence we implicitly used the fact that the ensemble-averaged equations [Eqs. (11)-(13) of the main text] are homogeneous (not only linear). This is the so-called unital evolution (which preserves the center of the Bloch sphere), which originates from full symmetry between the states |0⟩ and |1⟩ of the effective qubit. Let us now prove that even in a non-unital case (for example, when we measure a physical qubit and asymmetry between states |0⟩ and |1⟩ is created by energy relaxation), the two methods for calculation of correlators are still equivalent (in the absence of phase-back-action). We will see that in this general case we can use Eq. (<ref>) originating from the collapse recipe, but cannot use its simplified version (<ref>). Let us use the linearity of the ensemble-averaged quantum evolution E (a trace-preserving positive map) fromt=0 to t=τ, ρ_ in→ E(ρ_ in)=ρ̃_c + x_ inΔρ̃_x +y_ inΔρ̃_y +z_ inΔρ̃_z,where ρ̃_c = E(ρ_c) is the state mapped from the Bloch sphere center ρ_c= /2, while Δρ̃_x = E(ρ_x)-ρ̃_c, Δρ̃_y= E(ρ_y)-ρ̃_c, and Δρ̃_z= E(ρ_z) -ρ̃_c describe mapping of the Bloch sphere axes, with density matrices ρ_x, ρ_y, and ρ_z corresponding to pure states (|1⟩ +|0⟩)/√(2), (|1⟩ +i |0⟩)/√(2), and |1⟩, respectively. Following the same logic as above, we can write the first contribution (<ref>) to K_zφ(τ) asK_zφ^(1)(τ) = [ Tr(σ_φρ̃_c) +x_ in Tr(σ_φΔρ̃_x) +y_ in Tr(σ_φΔρ̃_y)+z_ in Tr(σ_φΔρ̃_z)]z_ in.The second contribution (<ref>) to K_zφ(τ) can be written as K_zφ^(2)(τ) =-x_ inz_ in Tr(σ_φΔρ̃_x) -y_ inz_ in Tr(σ_φΔρ̃_y) +(1-z_ in^2) Tr(σ_φΔρ̃_z).Combining the two contributions, we findK_zφ(τ) =Tr(σ_φρ̃_c) z_ in +Tr(σ_φΔρ̃_z). On the other hand, using Eq. (<ref>) of the collapse recipe with σ_i=σ_z and σ_j=σ_φ , we obtainK_zφ(τ) = [Tr(σ_φρ̃_c) +Tr(σ_φΔρ̃_z)] 1+z_ in/2-[ Tr(σ_φρ̃_c)-Tr(σ_φΔρ̃_z)] 1-z_ in/2,which coincides with Eq. (<ref>). Thus, equivalence of both methods for K_zφ(τ) is proven in the general (non-unital) case. The proof for the correlatorK_zz(τ) is practically the same, just replacing σ_φ with σ_z. The proof of the equivalence for an arbitrary K_ij(τ) is also similar, but we need to use the basis, corresponding to σ_i. Note that in the non-unital case we should use Eq. (<ref>) of the collapse recipe, which takes into account both scenarios (collapse to the state |0⟩ or to |1⟩) and not the simplified equation (<ref>) (collapse to |1⟩ only), which is valid only for a symmetric (unital) evolution. As seen from Eq. (<ref>) [or Eq. (<ref>)], in the non-unital case the correlator K_ij(τ) depends on the initial state ρ_ in via the term Tr [σ_j ρ_ av (t_1+τ |ρ_ c)]Tr [σ_i ρ (t_1)], where ρ_ c is the fully mixed state. Also note that in our derivation we assumed absence of the phase backaction terms <cit.> in the Bayesian stochastic equations. These terms would introduce additional contribution to δρ_ in and therefore to correlators. The collapse recipe in this case should be modified accordingly. §.§ Derivation via quantum regression approach Now let us derive Eqs. (14) and (15) of the main text for the correlators K_zz(τ) and K_zφ(τ) using the standard non-stochastic approach <cit.>, which cannot describe individual realizations of the qubit measurement process, but is sufficient to calculate correlators. In this section we assume t_1=0 and use ρ_ in=ρ(0).In this approach <cit.> we need to use the Heisenberg picture and associate the measurement outcomes I_z(t) and I_φ (t) with the operators σ_z(t) and σ_φ (t), which evolve in time as σ_z (τ) ≡ e^i H_ totτσ_z e^-iH_ totτ and σ_φ (τ) ≡ e^i H_ totτσ_φ e^-iH_ totτ, where H_ tot is the total Hamiltonian describing the qubit, environment, and interaction between them (in this approach we consider the measurement apparatus as an environment). The correlators of the outcomes then can be expressed as symmetrized combinationsK_zz(τ) = 1/2 Tr_tot [ σ_z(τ) σ_z(0) ρ_ tot(0)+ σ_z(0) σ_z(τ)ρ_ tot(0)] =Re{ Tr_tot [ σ_z (τ)σ_z(0)ρ_ tot(0) ]}, K_zφ(τ) = 1/2 Tr_tot[ σ_φ(τ) σ_z(0) ρ_ tot(0) + σ_z(0) σ_φ(τ)ρ_ tot(0)] =Re{ Tr_tot [ σ_φ (τ)σ_z(0)ρ_ tot(0) ]},where ρ_ tot(0)=ρ (0) ⊗ρ_ env(0) is the initial density matrix, which includes the environment (“bath”), and the trace should be taken over the qubit and environment degrees of freedom.We need to assume that the coupling between the qubit and the environment is sufficiently weak, so that the effective decoherence rate of the qubit due to its coupling with the environment is much smaller than the reciprocal of the typical correlation time for the bath degrees of freedom (this includes the “bad cavity” assumption). In this case we can use the standard formula <cit.> (related to what is usually called the Quantum Regression Theorem)Tr_ tot [ A (τ)B(0)ρ_ tot (0)] =Tr _ sys [ Aρ_ av(τ|Bρ_ in)] ,where in the right-hand side the trace is only over the system (qubit), operators A and B are system observables, and ρ_ av(τ|Bρ_ in) is the system (reduced) density matrix at time τ, which evolves in time according to the ensemble-averaged (reduced) evolution equations and starts in the state Bρ_ in, i.e., ρ_ av(0|Bρ_ in)=Bρ_ in. Note that ρ_ av(τ|Bρ_ in) is unphysical because it starts with an unphysical initial state Bρ_ in (it is typically not Hermitian and not normalized). Also note that the validity of Eq. (<ref>) requires that the system and the environment will be weakly entangled; i.e., ρ_ tot(t)≈ Tr_ env[ρ_ tot(t)]⊗ρ_ env(0). This is consistent with the above assumption that the coupling is weak.In our case in Eq. (<ref>) the operator B is σ_z, while A is either σ_z or σ_φ. The starting state for ρ_ av(τ|Bρ_ in) is σ_zρ_ in, soK_zz(τ) =Re{ Tr [ σ_z ρ_ av(τ|σ_z ρ_ in) ]},K_zφ(τ) =Re{ Tr [ σ_φρ_ av(τ|σ_z ρ_ in) ]}.Since we have to work with unphysical unnormalized states, we use ρ(t) = [P_ N + x(t) σ_x + y(t) σ_y + z(t) σ_z]/2, where the normalization is conserved, Ṗ_ N=0, during the ensemble-averaged evolution described by Eqs. (11)–(13) of the main text.Now let us represent the unphysical initial state σ_zρ_ in asσ_z [ρ_11, in ρ_10,in; (ρ_10,in)^* ρ_00,in ] =σ_z/2+(ρ_11, in-ρ_00, in)/2+[ 0ρ_10, in; -(ρ_10, in)^* 0 ] ,using condition ρ_00, in+ρ_11, in=1. Since the ensemble-averaged evolution is linear, we can separate ρ_ av(τ|σ_zρ_ in) into three terms, corresponding to the three terms in Eq.(<ref>). The first term, σ_z/2, gives the physical evolution ρ_ av(τ |1) starting with the state |1⟩. The second term, (ρ_11, in -ρ_00, in)/2, does not change in time and gives zero contribution to the correlators (<ref>) and (<ref>). The third term is initially anti-Hermitian, and it will remain anti-Hermitian in the evolution, because all coefficients in Eqs. (11)–(13) of the main text are real. The anti-Hermitian term will give zero contribution to Eqs. (<ref>) and (<ref>) because the traces will be imaginary numbers.Thus we obtain equationsK_zz(τ) =Tr [ σ_z ρ_ av(τ|1)], K_zφ(τ) =Tr [ σ_φρ_ av(τ|1)],which coincide with Eq. (<ref>) for σ_i=σ_z. Therefore, the final result for the correlators is the same as for the derivation based on the collapse recipe. In the general non-unital case (without phase back-action), using representation (<ref>) of a linear quantum map, we can obtain Eq. (<ref>) from Eqs. (<ref>) and (<ref>), thus proving that the derivation via the quantum regression approach is still equivalent to the derivations via the stochastic equations and via the collapse recipe.§ EXTRACTING CORRELATORS FROM EXPERIMENTAL DATAThe experimental correlators are calculated as K_ij (τ) = ∫_t_a^t_bdt_1 ⟨(Ĩ_i(t_1)-Ĩ_i^ off)(Ĩ_j(t_1+τ)-Ĩ_j^ off)⟩/(t_b-t_a)ΔĨ_iΔĨ_j,where Ĩ_i(t) are experimental output signals for σ_i measurement (0≤ t≤ 5 μs), ⟨ ...⟩ denotes ensemble averaging over all selected traces with the same angle difference φ (∼200,000 per angle, the selection includes heralding at the start of the run and checking that the physical transmon qubit is in the subspace of its lowest two energy levels after the run), additional time-averaging is between t_a=1 μs and t_b=1.5 μs, the correlators are normalized by responses ΔĨ_i, and small offsets Ĩ_i^ off are calculated separately for each value of φ (see below). Significantly larger offsets are already removed from Ĩ_i(t) individually for each trace by measuring and averaging the background noise for the non-rotating qubit after each trace. To find ΔĨ_i and Ĩ_i^ off, for each angle φ we separate the traces (each trace includes outputs for both measurement channels) into two approximately equal groups. These groups correspond to the effective qubit initialized either in the state |1⟩ (z_0=1) or |0⟩ (z_0=-1), which is controlled by the initial state |1⟩ or |0⟩ of the physical qubit before application of 40 MHz Rabi oscillations and stroboscopic sideband measurement. Calibration of the z-axis of the effective qubit is done by maximizing the response (for the lower-κ channel) for zero nominal angle between the two measurement directions (φ=0 neglecting small δφ). For non-zero nominal φ, the stroboscopic measurement directions are -φ/2 (channel 1, ω_ r,1/2π=7.4GHz, κ_1/2π=4.3MHz) and φ/2 (channel 2, ω_ r,2/2π=6.7GHz, κ_2/2π=7.2MHz) – see Fig. <ref>. Theoretically only the angle difference φ matters for the correlators; however, for calibration we need to use the actual measurement directions ±φ/2 (more accurately, -φ/2 and φ/2+δφ). In the main text the channel 1 is called z-channel, while channel 2 is φ-channel. In this section we will also be using the terminology of channels 1 and 2.To find the responses ΔĨ_1≡ΔĨ_z andΔĨ_2≡ΔĨ_φ for the two channels, we calculate 𝒟_i(t)≡⟨Ĩ_i(t)⟩_z_0=1 - ⟨Ĩ_i(t)⟩_z_0=-1 for each φ, where the subscripts z_0=± 1 denote the group of traces with initial state z_0 of the effective qubit. This quantity can be also obtained theoretically from the ensemble averaged evolution equations (11)–(13) of the main text with the initial condition x(0)=sin(φ/2), y(0)=0, and z(0)=cos(φ/2), see Fig. <ref>. It is equal to 𝒟_1(t)= ΔĨ_1 z_ av(t) and 𝒟_2(t) = ΔĨ_2 [z_ av(t)cos(φ+δφ) + x_ av(t)sin(φ+δφ)] for the first and second channels, respectively. In Fig. <ref>, we plot experimental 𝒟_1(t) and 𝒟_2(t) for 5 values of φ (φ_n=nπ/10, n=0,1,2,3,4), and fit them with theoretical results. We find a good agreement for the responses ΔĨ_1 = 4.0 and ΔĨ_2 = 4.4 in units of the experimental output. In this fitting we disregard any residual Rabi oscillations (Ω̃_ R=0).To estimate the offsets Ĩ_i^ off, we use the symmetric combination 𝒮(t) ≡ [⟨Ĩ_i(t)⟩_z_0=1 + ⟨Ĩ_i(t)⟩_z_0=-1]/2, which is shown in Fig. <ref> for both measurement channels and for 11 angles φ. We see that the 𝒮(t) are approximately independent of time, and therefore we can introduce the offsets Ĩ_i^ off=𝒮 for each channel and each value of φ. The offsets only weakly depend on the angle φ , but are significantly different in the two channels. For the first (z) channel we crudely find Ĩ_1^ off≡Ĩ_z^ off= 0.15, 0.16, 0.16, 0.16, 0.17, 0.16, 0.16, 0.17, 0.16, 0.17 0.18 for the angles φ in increasing order. For the second (φ) channel we find Ĩ_2^ off≡Ĩ_φ^ off = -0.18, -0.17, -0.17, -0.15, -0.16, -0.17, -0.17, -0.17, -0.18, -0.19 and -0.19 for the angles φ in increasing order. § SELF-CORRELATORS AT SMALL Τ In this section we discuss why the amplified vacuum noise is still white (delta-correlated) for finite damping rate κ of a resonator. We also estimate the self-correlator contribution K_ii(τ) ∝exp (-κ_i τ/2), with small amplitude ∼Γ /κ_i due to qubit evolution.It is a somewhat surprising result that the self-correlator K_ii(τ) does not have a significant contribution ∝exp (-κ_i τ/2), originating from the correlation time (κ_i/2)^-1 of vacuum fluctuations inside the resonator. [We may naively expect that this would widen the contribution K_ii (τ)=η_iτ_iδ(τ) from the amplified vacuum noise.] To show why this is not the case, let us consider a resonator with finite κ without a qubit (Fig. <ref>) and calculate the correlator for the amplified vacuum noise, coming from the resonator. The coupling between the resonator and the output transmission line is κ_ out, while the remaining dissipation rate κ-κ_ out is modeled as a coupling to another transmission line.Using the standard input-output theory <cit.>, we need to consider the vacuum noise v̂(t) incident to the resonator from the output line, with the operator correlator ⟨v̂(t) v̂^†(t')⟩=δ (t-t'), and write the equations for the annihilation operators in the Heisenberg picture. However, for our purposes it is sufficient to use a simpler approach (e.g., Appendix B of <cit.>), in which we consider the evolution of classical fields (in the usual, i.e., Schrödinger picture) due to “classical” vacuum noise v(t) (complex number) with the correlator⟨ v(t)v^*(t')⟩ = 1/2 δ (t-t'),⟨ v(t)v(t')⟩ =0,where real and imaginary parts of v(t) correspond to orthogonal quadratures. As follows from Eq. (<ref>), any quadrature has correlator (1/4) δ (t-t'), and orthogonal quadratures are uncorrelated.The evolution of the resonator field fluctuation α(t) (evolution of the field due to drive is decoupled due to linearity) is α̇=-i Δω α -κ/2 α +√(κ_ out)v(t) +√(κ -κ_ out)v_ a(t),where Δω=ω_ r-ω_ d is the resonator frequency ω_ r in the rotating frame based on the drive frequency ω_ d (it is important for homodyne detection) and v_ a(t) is the additional vacuum noise from the other transmission line (see Fig. <ref>) with the same correlator (<ref>) and uncorrelated with v(t). In Eq. (<ref>) we use the standard normalizations for the resonator field (based on the average number of photons) and for the propagating fields (based on average number of propagating photons per second). This equation has the simple solution, α (t) = ∫_-∞^t [√(κ_ out)v(t')+ √(κ -κ_ out)v_ a(t')] e^-κ̃(t-t')/2dt',where κ̃= κ +2iΔω.The outgoing field F(t), which is then amplified isF(t) = -v(t)+√(κ_ out) α (t).After a phase-sensitive amplification, this field is sent to a homodyne detector, which outputs the signal I(t)∝ Re[F(t) e^-iθ], where θ is the amplified quadrature. Without loss of generality, we can assume θ=0 (by properly defining the quadrature). Therefore, we are interested in the self-correlator of Re[F(t)], which is equal (up to a coefficient) to the output signal correlator.Our goal is to show that the correlator of Re[F(t)] is the same as for vacuum noise, i.e.,K_ ReF(τ)≡⟨ Re[F(t)] Re[F(t+τ)]⟩ = 1/4 δ(τ ).Actually, it is sufficient to show that K_ ReF(τ )=0 for τ >0, since the coefficient 1/4 in Eq. (<ref>) can be simply obtained from Eqs. (<ref>) and (<ref>). It is also sufficient to choose t=0 in Eq. (<ref>).As follows from Eq. (<ref>), the correlator K_ ReF(τ) for τ≥ 0 has three contributions, K_ ReF(τ)=K_ ReF^(1)(τ) +K_ ReF^(2)(τ)+K_ ReF^(3)(τ), whereK_ ReF^(1)(τ) = ⟨ Re[v(0)] Re[v(τ)]⟩= 1/4 δ(τ), K_ ReF^(2)(τ) = -√(κ_ out) ⟨ Re[v(0)] Re[α(τ)]⟩K_ ReF^(3)(τ) =κ_ out⟨ Re[α(0)] Re[α(τ)]⟩ .Note that for τ≥ 0 there is no contribution due to correlation between α(0) and v (τ) because of causality. We need to show that the second and third contributions exactly cancel each other, K_ ReF^(2)(τ)+K_ ReF^(3)(τ)=0.Using Eq. (<ref>), correlator ⟨ Re[v(t)]Re[v(t')]⟩ =(1/4) δ (t-t'), and absence of correlation between Re[v(t)] and Im[v(t')], we easily obtain (τ > 0)K_ ReF^(2)(τ) =-(κ_ out/4)Re[e^-κ̃τ/2].Similarly (with a little more work) we obtainK_ ReF^(3)(τ) = κ_ outκ/4∫_-∞^0 [Re( e^κ̃t'/2) Re( e^-κ̃(τ - t')/2) .. +Im( e^κ̃t'/2) Im( e^-κ̃(τ - t')/2) ] dt' = (κ_ out/4) e^-κτ /2cos(Δω τ) ,where addition of contributions from v(t) and v_ a(t) gives the coefficient κ=κ_ out+(κ -κ_ out) on the first line. It is easy to see that Eqs. (<ref>) and (<ref>) exactly cancel each other, thus proving Eq. (<ref>). The proof using the standard input-output theory <cit.> is essentially the same as our proof, just using operators instead of complex numbers and associating time-dependence with the Heisenberg picture.This result explains why we do not see significant exponential contributions ∝exp(-κ_iτ /2) to the self-correlators K_ii(τ ) at small τ in Fig. 2(d) of the main text. However, finite bandwidth of the amplifier leads to widening of the delta-function correlator of the amplified signal, producing crudely exponential time dependence at small τ in Fig. 2(d) of the main text.While we have shown above that in the ideal case the amplified noise is delta-correlated, small contributions ∝ exp(-κ_iτ /2) to the self-correlators are possible due to various non-idealities. For example, if the temperature of the resonator is significant, then the delta-correlator of the noise v_ a(t) is larger than the vacuum correlator of v(t). Repeating the derivation above, we see that the coefficient in Eq. (<ref>) increases, and the cancellation by Eq. (<ref>) is incomplete. Similarly, if the amplitude of the microwave drive fluctuates in time, these fluctuations are essentially passed through a filter with the resonator bandwidth, creating a contribution∝exp(-κ_iτ /2) from the “white-noise” part of the fluctuations.A similar mechanism is produced by random evolution of the qubit, which is much slower than κ_i, but still has a non-zero spectral weight at frequencies comparable to κ_i. Let us estimate the corresponding contribution to the self-correlator K_ii(τ) in the following way. Assuming φ=π/2 (so that z and x components of the qubit are measured) and assuming Γ_z=Γ_x=Γ, let us consider the uniform diffusion of the qubit state along the x-z great circle on theBloch sphere <cit.> with the angular diffusion coefficient 2Γ. Note that we assume an ideal detector by separating a non-ideal detector into an ideal part and extra noise. Even though the Markovian theory <cit.> cannot describe the qubit evolution at the frequency scale κ_i, in this estimate we just assume the same uniform diffusion with coefficient 2Γ.In the Markovian approximation, the qubit evolution characterized by the angle β(t) from the x-axis, produces the output signal in the σ_z-channel I_z(t)=sinβ (t) (excluding noise).However, because of the finite bandwidth κ_z of the resonator, there will be a correction -∫_-∞^t e^-κ_z(t-t')/2cosβ β̇dt' to the output signal due to transient delay. The contribution from this correction to the self-correlator K_zz(τ)=T^-1∫_-T/2^T/2 I_z(t) I_z(t+τ)dt (with T→∞) is κ_z^-1e^-κ_zτ/2⟨ [ (cosβ) Δβ ]^2/Δ t ⟩, where Δβ=β̇ Δ t is the change during small Δ t. Using ⟨cos^2β⟩ =1/2 and ⟨ (Δβ)^2⟩ =2ΓΔ t, we obtain the contribution (Γ/κ_z) e^-κ_zτ/2 to K_zz(τ). A slightly more accurate calculation, which takes into account qubit diffusion during resonator transients, produces the result (Γ/κ_z) e^-κ_zτ/2e^-Γτ, which is practically the same since in our case Γ≪κ_z. The same crude derivation can be done for the σ_x-channel. Thus, for φ=π/2 we expect the contribution ∼ (Γ /κ_i)e^-κ_iτ/2 to the self-correlator K_ii(τ ) at small τ. However, these contributions are not visible in Fig. 2(d) of the main text because experimentally Γ /κ_z=0.028 and Γ /κ_φ=0.017, which is almost three orders of magnitude less than the scale of the amplifier-caused effect contributing to the lines in Fig. 2(d) [note that by design of the experiment Γ/κ_i≪ 1– see Eq. (<ref>)]. 9S-Shay2016 S. Hacohen-Gourgy, L. S. Martin, E. Flurin, V. V. Ramasesh, K. B. Whaley, and I. Siddiqi, Nature (London)538, 491 (2016).S-Korotkov-99-01 A. N. Korotkov, A. N. Korotkov, Phys. Rev. B60, 5737 (1999); Phys. Rev. B63, 115403 (2001).S-Korotkov-2011 A. N. Korotkov, arXiv:1111.4016.S-Korotkov-2016 A. N. Korotkov, Phys. Rev. A94, 042326 (2016).S-Wiseman1993 H. M. Wiseman and G. J. Milburn, Phys. Rev. A 47, 642 (1993).S-Gambetta2008 J. Gambetta, A. Blais, M. Boissonneault, A. A. Houck, D. I. Schuster, and S. M. Girvin, Phys. Rev. A77, 012112 (2008).S-Blais-2004 A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A69, 062320 (2004).S-Ruskov2010 R. Ruskov, A. N. Korotkov, and K. Mø lmer, Phys. Rev. Lett.105, 100506 (2010).S-Korotkov2001sp A. N. Korotkov, Phys. Rev. B63, 085312 (2001).S-Gardiner-book C. W. Gardiner and P. Zoller,Quantum noise (Springer, Berlin, 2004), Sec. 5.2.S-Gardiner-1985 C. W. Gardiner and M. J. Collett, Phys. Rev. A31, 3761 (1985).S-Clerk-2010 A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Rev. Mod. Phys.82, 1155 (2010).

http://arxiv.org/abs/1702.08077v1

^2======We consider the Consensus Patterns problem, where, given a set of input strings, one is asked to extract a long-enough pattern which appears (with some errors) in all strings. Formally, the problem is defined as follows:Consensus PatternsInput: Strings S_1,… S_n of length at most ℓ, integers m and d.Output: Length-m string S and integers (j_1,…, j_n) such that ∑_i=1^n Ham(S,S_i[j_i..j_i+m-1]) ≤ dWhere Ham() denotes the Hamming distance and S[a..b] is the substring of S starting in a and ending in b. This problem is one of many variations of the well-studied Consensus String problem. It is similar to Consensus Substring in that the target string must be close to a substring of each input string (rather than the whole string). However, in the latter problem the distance to each input string is bounded, rather than the sum of the distances in our case. We look at this problem from the parameterized complexity viewpoint, more precisely for parameter ℓ. Recall that Consensus Substring is FPT for parameter ℓ <cit.>. See <cit.> for an overview of the variants of Consensus String, and <cit.> for recent advances on parameterized aspects of Consensus Substring and Consensus Patterns. We prove the following result. Consensus Patterns(ℓ) is W[1]-hard. By reduction from Multi-Colored Clique. We are given a graph G=(V,E), with a partition (coloring) V=V_1 ∪ V_2 ∪…∪ V_k, such that no edge has both endpoints of the same color. Assume that |V_h|=n for all h∈ [k]. Write V_h={v_h,1, v_h,2, … v_h,n}, i.e. each vertex has an index depending both on its color and its rank within its color. Let m=|E|. Multi-Colored Clique is W[1]-hard for parameter k <cit.>. See Figure <ref> for an example of the reduction.We build an alphabet Σ containing V (i.e., one symbol per vertex) and two special charactersand .Define string _i=v_1,i v_2,i… v_k,i. Let e=(v_h,i,v_h',i') be the jth edge of E, j∈ [m].Define _j as the string starting with , followed by k+1 characters: all , except for two positions: _j[k+h+1]=v_h,i and _j[h'+2]=v_h',i'.Let N=m(k+2)+1. The instanceof Consensus Patterns contains N occurrences of strings _i, i∈ [n], and one occurrence of strings _j, j∈[m]. The target length is m=k+1. Note that due to the large value of N, any solution S must have a minimal distance to the set of strings {_i| i∈ [n]}. Otherwise, (if it is, say, at the minimum distance plus one), the distance to the whole instanceincreases by at least N, which cannot be compensated by the remaining strings _j (which have size m(k+2)<N). Hence we first enumerate the optimal solutions for the set {_i| i∈ [n]}. The Consensus Patterns of {_i| i∈ [n]} (i.e., the strings of length k+1 at minimumtotal distance from strings _i) are the strings of the form S= v_1, i_1… v_k, i_k with i_1,…, i_k∈ [n]. Such a string has a total distance of (n-1)k. Since all strings in {_i| i∈ [n]} have length k+1, any consensus pattern S must be aligned with _i from the very first character. Hence S is a consensus string of {_i| i∈ [n]}. The consensus strings of this set are obtained by taking the majority character at each position. Thus, S[1]=#, and, for all h∈[k], there exists i_h such that S[h+1]={v_h,i_h}. Consider now an optimal solution S for . Let {i_h| h∈[k]} be the set of indices as obtained from the lemma above. We show that the set of vertices K={v_h,i_h| h∈[k]} forms a clique of G iff the distance is below a certain threshold. To this end, we compute the best possible alignment between S and each string _j. Let j∈ [m]. If both endpoints of edge e_j are in K then there exists an alignment of S at distance k-1 from _j, otherwise the best possible alignment has distance k. Let h_,h_,i_,i_ be such that e_j=(v_h_,i_, v_h_,i_).There are two possible alignments of S with _j: S[1] is aligned either with _j[1] or with_j[2]. We compute the distance in both cases.If S[1] is aligned with _j[1], then there is exactly one common character, namely S[1]=_j[1]=. Indeed, for all h∈[k], S[h+1]∈ V_h and _j[h+1]∈ V_h-1∪{x}, hence these two characters are different. The distance in this case is k.If S[1] is aligned with _j[2], then first note that S[1]=≠ x= _j[2].Considerindex h_.If i_=i_h_, then S[h_]=v_h_,i_h_=_j[h_+1], otherwise S[h_] ≠ =_j[h_+1].Similarly for h_, S[h_]=_j[h_+1] iff i_=i_h_. For other values of h (i.e. h∈ [k]∖{h_, h_}), S[h]≠ = _j[h+1]. The distance is thus k-1 iff i_=i_h_ and i_=i_h_, it is at least k otherwise.Overall, if i_=i_h_ and i_=i_h_ the optimal alignment has distance k-1, otherwise the optimal alignment has distance k. We can now conclude the proof. Let S be an optimal solution of Consensus Pattern for instanceand K its corresponding set of vertices. The distance from S to the N copies of strings _i is N(n-1)k. The distance between S and _j is k-1 if both endpoints of e_j are in K, and k otherwise. |E(K)| is the number of edges with both endpoints in K: the total distance from S to strings _j is thus mk - |E(K)|, and the total distance from S tois N(n-1)k+mk-|E(K)|. Overall, the optimal distance is at most N(n-1)k+mk-k(k-1)/2 if, and only if, G contains a size-k set of vertices K with |E(K)|≥k(k-1)/2, i.e. if G contains a clique.plain

http://arxiv.org/abs/1702.08238v1

Inversion of the exciton built-in dipole moment in In(Ga)As quantum dots via nonlinear piezoelectric effect Rinaldo Trotta=========================================================================================================== The Landau collision integral is an accurate model for the small-angle dominated Coulomb collisions in fusion plasmas.We investigate a high order accurate, fully conservative, finite element discretization of the nonlinear multi-species Landau integral with adaptive mesh refinement using the PETSc library (www.mcs.anl.gov/petsc).We develop algorithms and techniques to efficiently utilize emerging architectures with an approach that minimizes memory usage and movement and is suitable for vector processing.The Landau collision integral is vectorized with Intel AVX-512 intrinsics and the solver sustains as much as 22% of the theoretical peak flop rate of the Second Generation Intel Xeon Phi, Knights Landing, processor.Landau collision integral, fusion plasma physics § INTRODUCTION The simulation of magnetized plasmas is of commercial and scientific interest and is integral to the DOE's fusion energy research program <cit.>.Although fluid models are widely employed to model fusion plasmas, the weak collisionality and highly non-Maxwellian velocity distributions in such plasmas motivate the use of kinetic models, such as the so-called Vlasov-Maxwell-Landau system. The evolution of the phase-space density or distribution function f of each species (electrons and multiple species of ions in general) is modeled withdf/dt≡∂ f/∂ t + ∂ x/∂ t·∇_x f+ ∂ v/∂ t·∇_v f = ∂ f/∂ t +v·∇_x f+ e/m(E +v× B) ·∇_v f = Cwhere e is charge, m mass, E electric field, B magnetic field, x spatial coordinate, v velocity coordinate, and t time. The Vlasov operator d/dt describes the streaming of particles influenced by electromagnetic forces, the Maxwell's equations provide the electromagnetic fields, and the Landau collision integral <cit.>, C, dissipates entropy and embodies the transition from many-body dynamics to single particle statistics. As such, the Vlasov-Maxwell-Landau system-of-equations is the gold standard for high-fidelity fusion plasma simulations. The Vlasov-Maxwell-Landau system also conserves energy and momentum, and guaranteeing these properties in numerical simulations is critical to avoid plasma self heating and false momentum transfer during long-time simulations. Hirvijoki and Adams recently developed a finite element discretization of the Landau integral, which is able to preserve the conservation properties of the Landau collision integral with sufficient order accurate finite element space <cit.>.We now continue this work with the development of a multi-species Landau solver with adaptive mesh refinement (AMR), which is designed for emerging architectures and implemented on the Second Generation Intel Xeon Phi, Knights Landing (KNL), processor.Due to the nonlinearity of the Landau collision integral, it has an intensive work complexity of N^2 with N global integration or quadrature points. Given this high-order work complexity, reducing the total number of quadrature points decreases computational cost substantially.We use high order accurate finite elements and AMR to maximize the information content of each quadrature point and thus minimize the solver cost.We adapt nonconforming tensor product meshes using the p4est library <cit.>, as a third party library in the PETSc library <cit.>.We develop algorithms and techniques for optimizing the Landau solver on emerging architectures, with emphasis on KNL, and verify the solver on a model problem.We vectorize the kernel using Intel AVX-512 intrinsics and achieve a flop rate as high as 22% of the theoretical peak floating point rate of KNL.§ CONSERVATIVE FINITE ELEMENT DISCRETIZATION OF THE LANDAU INTEGRALWe consider the multi-species version of the conservative finite element discretization of the Landau collision integral presented by Hirvijoki and Adams <cit.>.Under small-angle dominated Coulomb collision, the distribution function f_α(v,t) of species α evolves according to∂ f_α/∂ t= ∑_βν_αβm_o/m_α∇_v·∫_Ω̅ dv̅ 𝐔(v,v̅)·(m_o/m_αf̅_β∇_v f_α - m_o/m_βf_α∇̅_v̅f̅_β).Here ν_αβ=e_α^2e_β^2lnΛ_αβ/(8π m_o^2ε_0^2), lnΛ is the Coulomb logarithm, m_o is an arbitrary reference mass, ε_0 is the vacuum permittivity, m is mass, e is electric charge, and v is the velocity. Overbar terms are evaluated on the v̅ grid that covers the domain Ω̅ of species β.The Landau tensor 𝐔(v,v̅) is a scaled projection matrix defined as𝐔(v,v̅)=1/|v-v̅|^3(|v-v̅|^2𝐈-(v-v̅)(v-v̅))and has an eigenvector v-v̅ corresponding to a zero eigenvalue.Given a test function ψ(v), the weak form of the Landau operator (<ref>) for species α is given by(ψ,∂ f_α/∂ t)_Ω=∑_β(ψ, f_α)_K,αβ+(ψ,f_α)_𝐃,αβwhere (·,·)_Ω is the standard L^2 inner product in Ω and the weighted inner products present the advective and diffusive parts of the Landau collision integral(ψ, ϕ)_K,αβ =∫_Ωdv∇_vψ·ν̂_αβm_o/m_αm_o/m_βK(f_β,v) ϕ, (ψ,ϕ)_𝐃,αβ =-∫_Ωdv∇_vψ·ν̂_αβm_o/m_αm_o/m_α𝐃(f_β,v)·∇_vϕThe collision frequency is normalized with ν̂_αβ=ν_αβ/ν_o so that time t is dimensionless, and f_β is the distribution function of species β.The vector K and the tensor 𝐃 are defined asK(f,v) =∫_Ω̅dv̅ 𝐔(v,v̅)·∇̅_v̅f(v̅),𝐃(f,v) =∫_Ω̅dv̅ 𝐔(v,v̅)f(v̅).Assuming a finite-dimensional vector space V_h that is spanned by the set of functions {ψ_i}_i, the finite-dimensional approximation of the weak form (<ref>) can be written in a matrix form𝐌ḟ_α=𝐂_α[f]f_αwhere f_α is the vector containing the projection coefficients of f_α onto V_h and the vector f is the collection of all species f_α, The mass and collision matrices are defined𝐌_ij= (ψ_i,ψ_j)_Ω,𝐂_α,ij[f]=∑_β=1^S(ψ_i, ψ_j)_K,αβ+(ψ_i,ψ_j)_𝐃,αβThe integrals in (<ref>,<ref>), with the Landau tensors in the kernel, have N work for each species β and each equation in (<ref>).With N equation this leads to an N^2 algorithm for computing a Jacobian or residual when solving the equations (<ref>) for each species. § ALGORITHM DESIGN FOR EMERGING ARCHITECTURESThis section discuses the algorithms and techniques used to effectively utilize emerging architectures for a Landau integral solver.While the Landau operator has N^2 work complexity, this work is amenable to vector processing.We focus on KNL, but the algorithm is design to minimized data movement and simplify access patterns, which is beneficial for any emerging architecture. The discrete Landau Jacobian matrix construction, or residual calculation, can be written as six nested loops.Algorithm <ref> shows high level pseudo-code for construction the Landau Jacobian matrix, with |G| cells in the set G, Nq quadrature points in each element, distribution functions f, S species, and weights w_q_j = |J(q_j)| · q_j.weight · q_j.r, where q_j.r is the axisymmetric term of the element Jacobian, q_j.weight is the quadrature weight of q_j, and J(q_j) is the element Jacobian at point q_j. The Landau tensor 𝐔 in (<ref>,<ref>) is computed, or read from memory, in the inner loop. A vector K = 𝐔·∇ f_q_j w_q_j and a tensor 𝐃 = 𝐔f_q_j w_q_j are accumulated in the inner loop.With S species, the accumulation of K and 𝐃 requires 6S words.These accumulated values are transformed in a standard finite element process from the reference to the real element geometry and assembled with finite element shape functions into the element matrix. The six loops of Algorithm <ref> can be processed in any order, and blocked, giving different data access patterns, which is critical in optimizing performance.The first two issues that we address in the design of the Landau solver are 1) whether to precompute the Landau tensors or compute them as needed and 2) whether to use a single mesh with multiple degrees of freedom per vertex or use a separate mesh for each species.§.§.§ To Precompute the Landau Tensor or Not to PrecomputeThe Landau tensor is only a function of mesh geometry and can be computed and stored for each mesh configuration.The cost of computing the Landau tensor is amortized by the number of nonlinear solver iterations and the number of time steps that the mesh is used for.The computation of the tensor can be expensive, especially in the axisymmetric case which involves two different tensors and evaluation of elliptic integrals (see Appendix <cit.>), requiring approximately 165 floating point operations (flops) as measured by both the Intel Software Development Emulator (SDE) and code analysis, including four logarithm and square roots. Storing two such tensors requires eight words of storage, 64 bytes with double precision words.There are N^2 unique mesh (i,j) pairs for which the tensors are computed or stored, which is too much to fit in a cache of any foreseeable machine with any reasonable degree of accuracy (e.g., 64 megabytes with N=10^3).The decision to precompute or compute as needed depends on several factors.A simple analysis on KNL suggest that both approaches are viable but that trends in hardware works in favor of the compute as needed approach.Assuming the equivalent of 200 ordinary flops per axisymmetric Landau tensor pair calculation and 64 bytes of data, the flop to byte ratio is about three.KNL has a theoretical peak floating point capacity of about 2.6 × 10^12 flops/second and around 400 × 10^9 bytes/second on package memory bandwidth capacity, as measured by STREAMS, or a flop to byte ratio of about six.This simple analysis suggests that precomputing would be two times slower, however, we achieve about 20 % of theoretical peak flop rate and, thus, a precomputed implementation would need only to achieve about 40% of STREAMS bandwidth to match the run time of each kernel evaluation, which one would expect is achievable.This analysis suggests that either method could be effective on KNL, but the spread between flop capacity, in the form of more vector lanes and more hardware resources per lane, and memory bandwidth capacity is anticipated to increase in the future, which will benefit the compute on demand approach.The kernel in Algorithms <ref> requires 3S+1 words from memory per kernel evaluation for the weight w_q_j, the value f_β( q_j ) and gradient of f_β( q_j ) for each species.The compute on demand approach also requires the coordinate.This is N data, which has the potential to fit in cache, for example, with N=10^3 and two species this data would be about 64 kilobytes, plus lower order data, per thread.Eight threads per tile should fit in the 1Mb shared L2 cache on KNL.§.§.§ Single and Multiple MeshesWe use a single mesh, independently adapted for each species, with S degrees of freedom per vertex, however one can use multiple meshes or a mesh for each species.Observe that the integrals in (<ref>,<ref>) are decoupled from the outer integral in (<ref>,<ref>).In theory, one can use a separate grid, or different quadrature or even a different discretization, for each species.One could even use a different mesh for the inner and outer integral in (<ref>,<ref>).An advantage of using a single mesh is that the two loops over species in (<ref>) can be processed after the Landau tensor 𝐔 is computed, and hence this tensor can be reused S^2 times. However, if all of the species have “orthogonal" optimal meshes, that is each quadrature point only has significant information for one species, which is a good assumption for ions and electrons because of their disparate masses, then a single mesh requires about as many vertices as the sum of each of the putative multiple meshes.With the model of orthogonal optimal meshes and kernel dominated computation or communication, and with N_α quadrature points for each species α, the complexity of a Landau solve is 𝒪(∑_α=1^S N_α)^2 for both the single and multiple mesh approach.With multiple ion species the orthogonal mesh assumption would be less accurate, because (small) ions have similar optimal meshes, which benefits the single mesh approach.The Jacobian matrix for the single mesh approach has about S times more non-zeros, which is not important if the total cost is dominated by the Landau kernel.It is likely that with further optimization of the Landau kernel, the next generations of hardware, and algorithmic improvements for the inner integral, that the inner integral costs will be reduced relative to the rest of the solver costs, which would benefit the multiple mesh approach.The single mesh method has larger accumulation register demands and larger element matrices, which pressures the memory system more and is advantageous for the multiple mesh approach.The result of the increased register pressure can be seen in the decreases flop rates in Table <ref> with the increase in the number of species.Another potential advantage of the single grid method is that the extra degrees of freedom, in for instance the ions in the range of one electron thermal radius, might be beneficial.The large scale separation between ions and electrons means that small relative errors in the ion distribution can be large relative to the electron distribution.Ions and electrons have about equal and opposite charge, cancellation could lead to large relative errors in the total charge density. The accuracy of collisions with fast electrons in the tail of the ion distribution could benefit from extra resolution.A more thorough understanding of the accuracy issues would be required to fully address the question of using a single or multiple meshes. §.§ Our AlgorithmFor demonstration purposes, we focus on implementing the axially symmetric version using cylindrical velocity coordinates x=(r,θ,z). Under axial symmetry the distribution function is independent of the angular velocity coordinate (∂_θf=0) and the evaluation of the vector K and the tensor 𝐃 requires two different Landau tensors 𝐔_𝐊 and 𝐔_𝐃 respectively (see Appendix <cit.>). We choose to compute the required Landau tensors as needed and use a single mesh with a degree-of-freedom for each species on each vertex.We fuse the two inner loops over cells and quadrature points, inline the function call of, and within, the Landau tensor function.Algorithm <ref> shows the initialization of the vectors r, z, w, f, and the two gradient vectors df[1] and df[2], with |G| cells in the set G, S species, and weights w_q_i at each quadrature point i. Each quadrature point q_i is located at a 2D coordinate (q_i.r, q_i.z). Algorithm <ref> shows the algorithm for the construction of the Landau collision integral Jacobian.This algorithm is designed to minimized data movement by computing the Landau tensors as needed and exploits a single mesh by lifting the kernel outside of the two inner loops over species.§ NUMERICAL METHODS AND IMPLEMENTATIONWe implement the Landau solver with the PETSc numerical library <cit.>.PETSc provides finite element (FE) and finite volume discretization support, mesh management, interfaces to several third party mesh generators, fast multigrid solvers, interfaces to several third party direct solvers, AMR capabilities among other numerical methods.We adapt nonconforming tensor product meshes using the third party p4est library <cit.>, and unstructured conforming simplex meshes with PETSc's native AMR capabilities <cit.>.Our experiments use bi-quadratic (Q2) elements with p4est adaptivity, with the PETSc's Plex mesh management framework.The computational domain is Ω={(r,z) | 0 ≤ r ≤ L, -L≤ z ≤ L}, with L=2.We use Neumann boundary conditions and shifted Maxwellian initial distribution functions, for each species, of the formf_α(x,t=0)=θ1/2(πσ_α^2)^-3/2exp(-r^2+(z-s_α)^2/σ_α^2),where σ_α=2T_αm_o/m_α, s_i = 0, s_e = -1, T_α is temperature, and θ is a scaling factor used to maintain quasi-neutrality.We solve the boundary value problem∂ f_α/∂ t( v, t ) - 𝐂_α[f] f = 0 in axisymmetric coordinates, with standard FE methods and time integrators. A Newton nonlinear solver with the SuperLU direct liner solver is used at each time stage or step <cit.>.These experiments use a Crank-Nicolson time integrator.A global kinetic model would include a 3D spatial component and the 3V version of this solver would be used at each cell in either a particle method <cit.>, or a grid based kinetic method <cit.>. Our numerical experiments use up to 272 Message Passing Interface (MPI) processes on one KNL socket, redundantly solving the problem, to include some of the memory contention of a full 5D code.The timing experiments run one time step with one Newton iteration, which results in the Landau operator being called twice (one more than the number of Newton iterations), and with one linear solve (one per Newton iteration).The time for this step is reported, which does not include the AMR mesh construction.We observe a significant variability in times with 272 processes and have run each test several times in several sessions, in both batch and interactive sessions, and the report the fastest observed time.We report the maximum time from any processor and see about a 10% ratio between the maximum and the minimum time of any process with large process counts. §.§ Overview of application and numerical methodTo illustrate the capabilities and behavior of the solver, we run the code to near equilibrium, initializing electrons with a shifted Maxwellian distribution hitting a stationary single proton ion population with a Maxwellian distribution.We use a realistic mass ratio of m_i/m_e=1836.5 and temperatures of T_e = 0.02 and T_i = 0.002.Figure <ref> (left) shows the initial electron distribution with the ion grid at the origin, a partially thermalized electron distribution (center, left), and Maxwellian ion distribution near equilibrium (center, right).The ion distribution has been shifted from the origin by collision with the electrons. The ions are resolved with AMR at the origin and have a near Maxwellian distribution.Note, Figure <ref> uses linear interpolation from the four corners of each quadrilateral, whereas the numerics use bi-quadratic interpolation with nine vertices per quadrilateral, which results in inaccuracy and asymmetries in the plots not present in the numerics. §.§ Optimization and PerformanceMost of the work in the Landau solver is in the inner integral of (<ref>,<ref>) (lines 8-16 in Algorithm <ref>).This kernel is vectorized with Intel AVX-512 intrinsics.The Landau tensors calculation includes two logarithms, a square root, a power, seven divides, about 85 multiplies and 165 total flops.The power is converted to a inverse square root, an intermediate divide is reused, resulting in five divides, two logarithms, a square root, and an inverse square root.The KNL sockets used for this study is equipped with 34 tiles, each with a 1 megabyte shared L2 cache and two cores, each core has two 8 lane vector units that can issue one fused multiply add (FMA) per cycle per lane.Each core has four hardware threads, for a total of 272 threads per socket.The KNL clock rate is 1.4 GHz, but is clocked down to 1.2 GHz in AVX-512 code segments. This results in a theoretical flop peak rate of 2.6 × 10^12 flops/second.The peak flop rate that can be achieved with this solver is reduced because the kernel is not entirely composed of FMAs and the four logarithms and square roots require considerably more than one cycle each.The performance data in this section uses a simplified version the test problem, a grid adapted for electrons only with 176 cells and 1,584 quadrature points, a mass ratio of m_i/m_e=1 and T_e = T_i = 0.2, and no Maxwellian shifts (s_i = s_e = 0), as shown in Figure <ref> (right). §.§ Performance overviewThe major code segments have been instrumented with PETSc timers. Table <ref> show the maximum time from any process for major components of the Landau operator, the total Landau operator, and the linear solver. This data is from the double precision solve with 272 processes and two species in Table <ref>.This data shows that the Landau kernel, though vectorized, is still responsible for most of the run time. §.§ Performance and complexity analysisThere are two types of work in the kernel: 1) computing the two Landau tensors and 2) the accumulation of K vector and 𝐃 tensor.The accumulation requires 20S^2 flops (lines 12-13 in Algorithm <ref>).Instrumenting this inner loop would be invasive, but we can infer the percentage of time and work in these two parts with a complexity model and global measurements.Assume both the time and work cost of the entire solver are of the form C = aS^0 + bS^1 + cS^2.The solve time and flop counts with S=1,2,3, shown in Table <ref>, generate right hand sides for system of three equations and three unknows a, b, and c, which are the time spent, or work, in each of the three types of components.The Landau tensor cost is formally independent of the number of species, and the work in the accumulation has S^2 work complexity.Most of the rest of the costs, given a mesh, order of elements, etc., has S^1 complexity.Table <ref> shows the percentage of time and work in each component, infered from the data in Table <ref>.This analysis shows that about 98% of the work, and about 88% of the time, is in the kernel with one process.The measurements of the kernel time in Table <ref> is 66% with 272 processes. This discrepancy is probably due to performance noise and memory contention in the 272 process timings.The non-kernel time percentage (12) increases by a factor of about eight from the flop percentage (1.5), which reflects the eight vector lanes of the KNL vector unit. §.§ Memory performance Our experiments are trivially parallel, but memory contention results in performance degradation as more processes are used on a socket. KNL's architecture allows for twice as many vector lanes with single (32 bit) versus double (64 bit) precision and can thus run, in theory twice as fast in single precision.This section investigates memory issues with a weak speedup study in single and double precision. Table <ref> shows timing data with increasing number of processes on a single KNL socket, with single and double precision.This data shows that we are achieving about 80% of the perfect factor of two speedup with single precision.A KNL socket has 136 vector units.One might expect that using more processes than 136 would not be useful, however, the kernel has serial dependancies that result in bubbles in the pipeline, especially in the ninth and tenth order polynomial evaluations in the elliptic integrals.These holes can be filled by interleaving a second process in the second hardware thread.We do see about a 15% increase in total performance from the added parallelism of using 272 processes. This data shows that the flop rate per process decreases by a factor of about 3 in going from one process per tile to two processes per vector unit, or eight processes per tile.There is no difference between the one process and 34 process runs, suggesting that the 34 processes are indeed placed with one per tile.There is little degradation going from one to two processes per tile, suggesting that the problem still fits in the L2 cache.The degradation from one to two processes per core is from L1 cache misses because four processes should be able to fit into the 1 megabyte L2 cache.Analysis of memory complexity and the time data going from one to two processes per vector unit suggest that the full 272 process runs fit in the L2 cache, or nearly so. §.§ Verification Consider a convergence study with Cartesian grids of the third moment, thermal flux, to verify the expected order of convergence.An analytical flux for this problem does not exist.Richardson extrapolation is used to construct an approximate exact flux.The mass ratio is 4, T_e = 0.2 and T_i = 0.02.The flux history, with a series of refined grids starting with a 8 × 16 grid, is shown in Figure <ref> (top, left).Figure <ref> also shows the differences between fluxes on successive grids, and the error convergence. We can see from this data that we achieve fourth order convergence. § CLOSURE We have implemented a high order accurate finite element implementation of the Landau collision integral with adaptive mesh refinement in the PETSc library using AVX-512 intrinsics for the Second Generation Intel Xeon Phi, Knights Landing, processor.We have developed a memory centric algorithm for emerging architectures that is amenable to vector processing.We have achieved up to 22% of the theoretical peak flop rate of KNL and analyzed the performance characteristics of the algorithm with respect to process memory contention, single and double precision, and the results of vectorization. We have verified fourth order accuracy with a bi-quadratic, Q2, finite element discretization. Future work includes, building models for runaway electrons in tokamak plasmas with this kernel <cit.>,and building up complete kinetic models (6D AMR) that also preserve the geometricstructure of the governing equations of fusion plasmas <cit.>. § ACKNOWLEDGMENTS We are grateful for discussions and insights from Intel engineer Vamsi Sripathi.This work has benefited from many discussions with Sam Williams.This work was partially funded from the Intel Parallel Computing Center program.This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.siamplain

http://arxiv.org/abs/1702.08880v2

Astronomical Observatory of the Jagiellonian University, 30-244 Orla 171, Kraków, Poland; kulczak@oa.uj.edu.pl We present observations of the ground state 1_0,1–0_0,0 rotational transition of HDO at 464.925 GHz and the 1_1,0–1_0,1 transition at 509.292 GHz towards the three high-mass star forming regions: G34.26+0.15, W49N and W51e_1/e_2, carried out with the Caltech Submillimeter Observatory. The latter transition is observed for the first time from the ground. The spectra are modeled, together with observations of higher-energy HDO transitions, as well as submillimeter dust continuum fluxes from the literature, using a spherically symmetric radiative transfer model to derive the radial distribution of the HDO abundance in the target sources. The abundance profile is divided into an inner hot core region, with kinetic temperatures higher than 100 K and a cold outer envelope with lower kinetic temperatures. The derived HDO abundance with respect to H_2 is (0.3–3.7)× 10^-8 in the hot inner region (T > 100 K) and (7.0–10.0)× 10^-11 in the cold outer envelope. We also used two H_2^18O fundamental transitions to constrain the H_2O abundances in the outer envelopes. The HDO/H_2O ratios in these cold regions are found to be (1.8–3.1)× 10^-3and are consequently higher than in the hot inner regions of these sources.§ INTRODUCTION During the cold phase preceding the formation of stellar objects, molecules freeze-out onto dust grains, forming H_2O- dominated ice mantles, mixed with other less-abundant species. The low temperature and the disappearance of most molecules, especially CO, from the gas phase trigger a peculiar chemistry leading to high abundances of deuterated species. Molecules tend to attach a D atom rather than an H atom, because deuterated species have larger reduced masses and consequently lower binding energies, arising from lower zero-point vibrational energies. Ion-molecule reactions in the gas phase <cit.> and reactions on the grain surfaces <cit.> are the two possible mechanisms responsible for deuterium enrichments in heavy molecules. The reactions involved are exothermic, which is why significant deuteration levels can be expected in the cold ISM. In the warmer phase, only very little fractionation is expected to occur, because the energy barrier could be overcome by the elevated temperature. However, at the temperatures deduced for the hot core regions (100–200 K) the ice mantles evaporate and the gas again becomes enriched in deuterated species, with abundances elevated compared to the cosmic D/H ratio for a short period, before the chemistry reaches steady-state. These enhancements in the hot cores reflect, to some degree, the grain mantle composition in the earlier, colder cloud phases. Although the processes leading to the water deuteration are not fully understood, they are clearly related to the grain-surface chemistry and the observed HDO/H_2O ratio reveals the chemical and physical history of the protostellar materials <cit.>. A recent review of water chemistry can be found in <cit.> and a review of the latest observational results from Herschel. Early studies of water deuterium fractionation in high-mass hot cores were performed almost twenty years ago, when the submillimeter spectrum was largely inaccessible <cit.>. The HDO abundance has been recently determined toward the high-mass hot core G34.26+0.15 andthe intermediate-mass protostar NGC7129 FIRS2 <cit.>. Here, we present the new observations of the ground state rotational transition of HDO at 464.925 GHz and the first excited transition at 509.292 GHz toward three high-mass star forming regions: G34.26+0.15, W51e_1/e_2 and W49N. The combination of data taken with the same telescopeof both the ground state transition (1_0,1–0_0,0) and the first excited line (1_1,0–1_0,1) provide better constraints on the source structure.These HDO transitions are studied for the first time to probe the structureof theenvelope of theW51e_1/e_2 and W49N hot cores. In this paper, we aim at determiningthe HDO fractional abundances relative to H_2 in the inner and outer region of the core in our target sources using, the static radiative transfer code of <cit.>. We also used two H_2^18O fundamental transitionsobserved by <cit.> to constrain the H_2O abundance and the HDO/H_2Oratio in the outer envelopes.§ OBSERVATIONSWater is difficult to study from the ground, due to its strong presence in the Earth's atmosphere. However, many HDO lines, including the ground state 1_0,1–0_0,0 rotational transition and the 1_1,0–1_0,1 excited transition studied here, lie in atmospheric windows, where observations are possible from highsites, under good weather conditions. Table 1 lists the HDO transitions included in the present study and the H_2^18O transitions which we use to constrainthe HDO/H_2O ratio in the outer envelope.§.§ Source description The 1_0,1–0_0,0 ground-state rotational transition of HDO at 465 GHz and the 1_1,0–1_0,1 transition at 509 GHz were observed toward three high-mass star forming regions are listed in Table 2. All the sources are characterized by strong millimeter continuum and mid-infrared emission, characteristic of the early stage of high-mass star formation.G34.26+0.15 is one of the best studied high-mass star-forming regions in the Milky Way. Embedded within this molecular cloud is a hot core, which exhibits strong H_2O maser emission and high abundances of saturated molecules <cit.>; two unresolved UCHII regions, labeled A and B; a more evolved H II region with a cometary shape; and an extended, ring-like H II region <cit.>. Based on narrow-band mid-infrared imaging of the complex, <cit.> concluded that the same star is responsible for ionization of the cometary H II component (C) and heating the dust, but is not interacting with the hot core seen in the molecular emission. At a 12 resolution, <cit.> also found the peak of the 350 μm emission to coincide with the component C of the UCHII region.The radio continuum emission of W51 shows three separate components: W51 IRS1, W51 IRS2, and W51 Main. W51 Main is defined by a group of OH and H_2O masers near several UCHII regions. The continuum emission from ultracompact H II regions was resolved into compact components labeled W51e_1 to W51e_8 <cit.>. Of these, e_1 and e_2 are the brightest in high-resolution continuum maps.The star forming region W49N contains at least a dozen UCHII regions powered by OB-type stars arranged in a ring 2 pc in diameter <cit.>. Evidence that star formation is still in progress within W49N comes from strong H_2O maser emission and strong millimeter continuum emission attributed to dust condensations <cit.>. §.§ Observations Observations of the 465 GHz and 509 GHz HDO transitions presented here were carried out in 2012 June–August, using the 10.4 m Leighton Telescope of the Caltech Submillimeter Observatory (CSO) on Mauna Kea, Hawaii. We used the new wideband 460 GHz facility SIS receiver and the FFTS backend that covers the full 4 GHz intermediate frequency (IF) range with a 270 kHz channel spacing.Pointing of the telescope was checked by performing five-point continuum scans of planets and strong dust continuum sources. The spectra were obtained in antenna temperature units T^*_A, and then converted to the main beam brightness temperature,T_mb, via the relationT^*_A = η_mbT_mb, η_mb is the CSO main-beam efficiency, which is found to be ∼ 37% at 460 GHz from total power observations of planets. The absolute calibration uncertainty of the individual measurements is ∼20%. In addition to the new CSO data, we included in our analysis previously published observations of higher-energy transitions <cit.> toward our target sources. We used the reduced HIFI data of the H_2^18Otransitions at 1101.698 and 1655.868 GHz. The HDO and H_2^18O lines parameters are listed in Table 3. The H_2^18O ground state transitions have been previously presented by <cit.>.The data processing was done using the IRAM GILDAS software package <cit.>. We measured the line parameters: central velocity V_LSR, the full-width at half maximum (FWHM) ΔV, peak intensity T^peak, by fitting a single gaussian profile to the data (in T_mb units). The integrated line intensity is equal to W_i = ∫T_idV, where i either refers to observations (obs) or models (mod).We also used Herschel/HIFI data at 893 GHz (Vastel et al. in preparation) obtained by the PRISMAS guaranteed time key program,and the SCUBA data at 353 GHz that provide an accurate determination of the source continuum flux. The PRISMAS continuum observations were obtained in the double beam switching mode. At 800 GHz, the HIFI beam size is 26.5and the instrument gain is 469 Jy/K <cit.>. § DETERMINATION OF THE HDO AND H_2O ABUNDANCE§.§Modeling The goal of this study is to determine the HDO fractional abundance in three high-mass stars formation regions. To reproduce the observed line intensities (Table 3) the static radiative transfer code of <cit.> is used. The model cloud is divided into 200 radial shells, and the code uses a multilevel accelerated lambda-iteration method <cit.> to solve for the HDO level populations and the line and continuum radiative transfer in a self-consistent fashion. This radiative transfer program takes into account the excitation of HDO molecules by collisions, line radiation, and dust continuum radiation at the HDO line frequencies. However, IR radiative pumping through HDO vibrationally excited levels and the large-scale velocity field, characteristic of infall or expansion, are not included. The HDO collisional rates used in this study were recently computed by <cit.> and <cit.> for ortho-H_2 and para-H_2 in the temperature range 5–300 K and for all rotational transitions with an upper energies less than 444 K.In the modelling, we assumed a constant ortho-to-para ratio (OPR) of H_2 equal to 3. The ortho and para H_2O collisional rates with ortho and para H_2 were taken from the LAMDA data base <cit.>. These rateswere calculated for temperatures in the range from 5 to 1500 K includingenergy levels up to 2000 K above ground. The same collisional rates are used for the H_2^18O isotopologue. The limitation imposed by the radiative transfer code is to usea single collisionalpartner in the calculations. We assume that all hydrogen is in the ortho state. We carried out model calculations from the inner core radius, r_min, to the outer radius, r_max (with r_max/r_min∼ 100 for all sources; ). The distance of the edge of the core from the star is set at r_min∼2.0× 10^16 cm or ∼1000 AU, with no dust emission seen at the smaller radii. The lack of submillimeter emission in the core centers could be due to optical depth effects or a central cavity <cit.>.We adopted a dust-to-gas ratio of 1:100 and a power-law H_2 density distribution of the form: n(r) =n_0(r/r_min)^-1.5 where n_0 is the H_2 density at the reference radius (r_min=1000 au). The power-law index was set to 1.5 in accordance with the static infall theory in the inner part of the object <cit.>. We assumed that the gas and dust radial temperature profiles follow a power law <cit.>: T(r) = T_0(r/r_min)^-0.5 where T_0 represents the maximum temperature of dust grains. We assumed that, at densities found in the hot cores, the gas temperature is equal to the dust temperature. §.§ Dust emissivity index β When molecules deplete inside prestellar cores, dust emission may represent the best tracer of the gas density distribution just prior to the onset of gravitational collapse. The dust continuum optical depth is described by a power-law frequency dependence, τ∝ν_^β, and to fit the observed spectral energy distribution, the knowledge of the grain emissivity spectral index, β, is required. The dust emissivity index depends on the dust grain composition, size, and temperature <cit.>. Details of the dust-modelling process can be found in the reviews by <cit.>. Observationally, there have been many attempts at determining and explaining β. Typical values of β range between 1 and 2, with further support for β = 1.5–2.0 coming from observations: <cit.>, <cit.> and <cit.>. Planck Collaboration XIV (2013) used Planck HFI data with ancillary radio data to study the emissivity index. They computed a median value of far infrared spectral index β_FIR = 1.88 ± 0.08 at the high frequency Planck channels (ν≥ 353 GHz) and a median value of spectral index β_mm = 1.6 ± 0.06 at millimeter wavelengths (ν < 353 GHz). We can estimate the dust grain emissivity exponent from observations at two frequencies ν_1 and ν_2 <cit.>: β= logF^_ν_2/F^_ν_1+log( e_^hν_2/kT_dust-1)/( e_^hν_1/kT_dust-1)/logν_2/ν_1-3 where F^_ν is the source flux density, ν the frequency of the observations, and T_dust the dust temperature. In this work, the dust grain emissivity index β is determined for G34.26 using the millimeter (λ_1 = 1.2 mm; ν_1 = 250 GHz) data obtained with SIMBA and submillimeter (λ_2 = 450 μm; ν_2 = 660 GHz) SCUBA data. We derive β = 1.6 ± 0.48. The uncertainty in β is typically 30% for the 20–50 K temperature range <cit.>. The spectral index for the other sources is taken from literature: <cit.> for W49N and from <cit.> for W51. The HDO lines studied here are seen in emission, and the model intensities are not sensitive to the exact value of β, especially for T > 30 K. That is why the dust grain emissivity index is fixed, and not a free parameter, in the fits. The values of β in our target sources, as well as the continuum fluxes from SCUBA and Herschel/HIFI observations, are listed in Table 4. These are used to constrain the density and temperature distributions as input to the line modeling.§.§ Modeling procedure We approximated the radial variation of the HDO fractional abundance, X=n(HDO)/n(H_2) as a step function with an enhanced abundance X_in in the inner region where T ≥ 100 K, and a lower value X_out for the outer envelope where T < 100 K. Laboratory studies indicate that the evaporation temperature lies in the 90–110 K range, depending on the ice composition and structure <cit.>. In this work the sublimation temperature of water, T=100 K <cit.>, is applied as the jump temperature.The model uses the following free parameters: n_0, T_0 and X_in (for T ≥ 100 K) and X_out (for T < 100 K). We determine the continuum flux density per beam at 353 GHz (850 μm), 509 GHz (590 μm), and 893 GHz (336 μm) from the model. Finally, the synthetic spectra and continuum emission are convolved to theappropriate telescope beam size for comparison with theobservations. We minimize the `figure of merit' (FOM) with the method of <cit.> to find the best model of the source. The FOM is computed from the observed and modeled spectra and continuum fluxes according to the following formula: FOM= FOM_1 + FOM_2 =∑_n(T_obs-T_mod)^2/(T_obs)^2+∑_m(F_obs-F_mod)^2/(F_obs)^2for a set of n spectral lines and m continuum flux densities. The inner and outer HDO abundances are constrained by the spectral line data (FOM_1), whereas model parameters describing the density and temperaturedistribution are constrained primarily by the continuum SED (FOM_2).To determine the uncertainty of X_in and X_out, we performed χ^2 analysis. In analogy to Lampton method <cit.> we define S ≡ FOM_1/σ^2 = ∑_n(T_obs-T_mod)^2/(σ T_obs)^2, where n being the number of spectral lines. The σ within analysis χ^2 includes a calibrations uncertainty of 20% for all individual measurments. The difference, ΔS ≡ S_true-S_min is distributed as χ^2 with p degrees of freedom (here: p=2; X_in and X_out). By ΔS ∼ χ^2_p  ("∼" - "is distributed as") we mean for any number L probability: Prob (ΔS>L) = Prob (χ^2_p>L) . With the limiting contour value S_L defined as S_min+ L, Prob (ΔS>S_L - S_min) = Prob (S_true> S_L) = Prob (χ^2_p>S_L - S_min).Prob (S_true> S_L) is the probality α of the contour failing to enclose the true value, hence α =Prob (χ^2_p>S_L - S_min). The α-point of χ^2 distribution is defined by α ≡ Prob [χ^2_p > χ^2_p(α)]. Significance α is S_L = S_min + χ^2_p(α). In this expression, χ^2_p(α) is tabulated value of χ^2 distribtion for p degrees of freedom and significance α. Equivalently, any one observation's contour has a confidence C = 1 - α of enclosing the true parameter vector. The required contours for significance are 1σ, 2σ, and 3σ, which respectively represent a confidence of 68.3%, 95.4%, and 99.7 % of enclosing the true value of X_in and X_out. The contours correspond to S_L = S_min + 2.17, S_L = S_min + 6.17, and S_L = S_min + 11.8. We used models with the same physical parameters as those used for HDO in the analysis of theH_2^18O data. The two H_2^18O fundamental transitions were modeled independently and the resulting ortho/para ratio of water is consistent with the hightemperaturevalue of given the modeling uncertainties .§.§ Results §.§.§ Origin of The LinesFigure 1 shows fractional population of the HDO levels of the relevant features of our dataset calculated in our model. The high-energy transitions are sensitive to changes in X_in. Indeed the bulk of emission in the high energy transitions is produced in the innerhot-core region where T > 100 K. In the G34.26 source, this region has a radius equal to 1.0 which corresponds to 0.02 pc. This is in agreement with the interferometric observationsof the HDO lines at 225 and 241 GHz by <cit.>. That is the reason why the high-energy transitions are sensitive to changes in X_in. On the other hand, the 465 and 509 GHz HDO transitions are sensitive primarily to X_out. The 509 GHz line arises predominantly in the region between the warm envelope and the cold region (T ∼ 50–100 K), whereas the 465 GHz transition is produced in thecold envelope (T < 50 K). The ground-state rotational transition of HDO at 465 GHz is consequentlya very good probe of the abundance in the cold outer envelope, which is consistent with the results of <cit.> for the solar-type protostar IRAS 16293-2422. The 509 GHz transition provides particurlarly good constraints on the HDO abundance profile in the transition region between the hot core and the envelope, and should be included in future, more advanced models of HDO in high-mass star-forming regions. The model reproduces the observed intensities of different transitions in our target sources, with the exception of the 509 GHz line in G34.26. Although the signal-to-noise ratio of the observed 509 GHzspectrum is limited, it is clear that the best-fit model does notreproduce this line profile. The 509 GHz transition is formed in the part of the cloud where, within the scenario proposed by <cit.>, various feedback: thermal, radiative, or turbulent mechanisms are expected during the process of massive starsformation.In particular, it should be noted that, with the inclusion of velocity fields in their model, <cit.> succesfully reproduced the 509 GHz line observed with Herschel/HIFI toward this same source. Other possibility is an accretion disk that is fed by the infalling envelope. This is also supported by observations made by <cit.> of G34.26.The size of this possible disk is about 9000 au (0.05 pc) <cit.> and agrees well with the place where the 509 GHz line arises (see Figure 1). This model of the G34.26 source <cit.> neither confirm or refute the presence of an expansion in the inner parts of the envelope <cit.>. We concluded that the geometry and physical structure of our model is too simplistic, and that is why we could not to reproduce the 509 GHz line.§.§.§ Target sourcesG34.26+0.15: Observedspectra (black line) and gaussian fit (blue line) of the 465 and 509 GHz HDO transitions toward G34.26+0.15 along with the best-fit model (red line) are shown in Figure 2.Model results are presented in Table 5. We obtain the best–fit model for: T_0=200 K, n_0 = 1.0× 10^8 cm^-3,X_in=3.7× 10^-8 and X_out=7.8× 10^-11. We calculated continuum flux densities at 353, 509, and 893 GHz. The resulting uncertainties of X_in and X_out are shown in Figure 3and listed in Table 8.Observed and modeled spectra of the para-H_2^18O line at 1102 GHzand the ortho-H_2^18O line at 1656 GHz are shown in Figure 2.The derived OPR in G34.26 is 1.9. The total (ortho+para) H_2^18Oabundance in the envelope X_out, is 4.9× 10^-11. The recommended isotopic abundance ratio between ^16O and ^18O is 500 <cit.>.Using this value, the H_2O outer abundance is 2.5× 10^-8 and the outer HDO/H_2O ratio is 3.1× 10^-3 in theenvelope. Considering the results with a 20% calibration uncertaint, the outer abundance ratio is (2.5 - 3.7)× 10^-3.W51e_1/e_2: Observedspectra and gaussian fit of the 465and 509 GHz HDO transitions toward W51e_1/e_2 along with the best-fit model, are shown in Figure 4 by black, blue and red lines, respectively.Model results for W51 are presented in Table 6. We obtain the best fit for: T_0=230 K, n_0 = 1.8× 10^ 8 cm^-3,X_in=1.7× 10^-8 and X_out=7.0× 10^-11. The resulting uncertainties of X_in and X_out are shown in Figure 5and listed in Table 8. Model flux densities per beam at 509 GHz and 893 GHz for W51 are also listed in Table 6. Observed and modeled spectra of the para-H_2^18O line at 1102 GHzand the ortho-H_2^18O line at 1656 GHz are shown in Figure 4. The total H_2^18O abundance (OPR = 2.9) in the envelope X_out, is 5.5× 10^-11. The H_2O outer abundance is 2.8× 10^-8 and outer HDO/H_2O ratio is 2.5× 10^-3. Considering the results with the 20% calibration uncertainty the outer ratio is (2.0 - 3.0)× 10^-3.W49N: Observedspectra and gaussian fit of the 465 and 509 GHz HDO transitions toward W49N along with the best-fit model are shown in Figure 6 by black, blue and red lines, respectively. Model results for W49N are presented in Table 7.We obtain the best fit for: T_0=300 K, n_0 = 2.5× 10^8 cm^-3,X_in=0.3× 10^-8 and X_out=10 × 10^-11. The resulting uncertainties of X_in and X_out are shown in Figure 7and listed in Table 8.As data on the high excitation lines are missing for W49N, the inner abundance X_in is not as well constrained as in the other sources. Observed and modeled spectra of the para-H_2^18O line at 1102 GHzand the ortho-H_2^18O line at 1656 GHz are shown in Figure 6. The total H_2^18O (OPR = 3.1) abundance in the envelope X_out, is 1.1× 10^-10. The H_2O outer abundance is 5.5× 10^-8 and the outer HDO/H_2Oratio is 1.8× 10^-3.Considering the results with the 20% calibration uncertainty, the outer abundance ratio is (1.4 - 2.2)× 10^-3.§ DISCUSSION §.§ Comparison with Previous Studies Previous observations of the high-mass star-forming regions indicate an HDO abundance in the hot cores (T > 100 K) but not in the outer, cooler envelopes (T < 100K) (Jacq et al. 1990, Gensheimer et al. 1996, Pardo et al. 2001; see Table 9). We could findvariations in the derived HDO abundances in the hot cores between 1.5× 10^-9 and 2.0× 10^-7. Jacq et al. (1990) belived that, independent of all modeling, a valuemuch lower than3.0× 10^-8 forHDO/H_2 in the hot core is very unlikely. For the first time, Comito et al. (2003, 2010) estimated the HDO abundance in both the inner and outer region of the high-mass source Sgr B2(M). These are, respectively: 3.5× 10^-9 (T>200K), 1.5× 10^-9 (100 K<T<200 K)and2.5× 10^-11 (T<100K). The singly deuterated form of water has been also observed in the massive source AFGL 2591, with abundance varyingfrom 1× 10^-7 in the hot core and 4× 10^-9 in the outer envelope <cit.>. <cit.> and <cit.> determined the HDO abundance and HDO/H_2O ratio in the inner and outer region of G34.26 (see Table 9). We derivedthe HDO abundances of X_in and X_out in three high-mass star-forming regions: G34.26, W 49N, and W51. We found a difference between our X_in(HDO) and X_out(HDO) values for G34.26 and those reported by <cit.>and <cit.>, respectively. This is likely because the first authors used different model structures and a higher jump temperature, andthe second authors did not check the higher valueof X_out(HDO) in their model.The obtained HDO abundancesof our target sources in the hot cores and the cooler envelopes are relatively consistent with the values found in the other high-mass star-forming regions <cit.>. These results show that the HDO abundance is enriched in the inner regions ofhigh-mass protostars because of the sublimation ofthe ice mantles, in the same way as for other studies low- and high-mass sources (e.g. NGC 1333 IRAS2A, IRAS 162923-2422, AFGL 2591, G34.26; Table 9). Observations of sites of high-mass star formation show in general the lower HDO abundances than observations of low-mass star forming cores. Possibly for high-mass protostars, the very cold and dense pre-collapse phase where CO freeze-out onto the grain mantles lasts only a short time, and the chemical reactions leading to the enhancement of deuterium abundance being strongly depressed when the temperature increases <cit.>.§.§ Variation of the HDO/H_2O Ratios with the RadiusBased on observations of two H^18_2O fundamental transitions, we found that the H_2O abundances in our target sources are (2.5 - 5.5)× 10^-8. Similar values were foundfor the other high-mass protostars: 5× 10^-10 - 4× 10^-8 (;; ; ). The H_2O abundance in the cold envelope agrees fairly well with the model predictions for cold regionswhere freeze-out takes place (; ). The water-deuterium fractionation in the inner and outer envelope of the high-mass star-forming region G34.26 was previously estimated by <cit.> and <cit.>. We determined the outer HDO/H_2O ratio in G34.26 to be 3.1× 10^-3, this value is relatively consistent with Coutens et al. (2014) (see Table 9).To estimate the inner HDO/H_2O ratio for the target sources, we used an inner H_2O abundance value as high as 10^-4 from observations of otherhigh-mass star forming-regions(; ; ; ; ). However a lower value of ∼ 10^-6 was found in NGC 6334 I <cit.>. A possible explanation for the low water abundance in this source is a time-dependent effect; water molecules may not have enough time to fully desorp from the dust grain. Our derivedHDO/H_2O ratios are consequently not well-constrained.We estimated that the inner HDO/H_2Oratio is about (1 - 4) × 10^-4within the range found in other high-mass star-forming regions (; ; ; ). The HDO/H_2Oratio varies between the inner and outer regions of high-mass protostars. The water deuterium fractionation decreases from the cold outer regions to the warm inner regions. The same trend is also present in low-mass protostars (Coutens et al. 2013 and 2014).The difference could be explained by the gradientof deuteration within interstellar ices. Only the external ice layers evaporate in the cold envelope through non-thermal processes, whereas the inner partof ice mantles evaporates only in the hot core <cit.>.The HDO/H_2O ratio in the bulk of ice mantle preserves the past physical and chemical conditions which materials experienced,while the HDO/H_2O ratio in active surface layers reflects local physical and chemical conditions (Furuya et al. 2015). The enrichment of deuterium in water ice should mostly occur in the latter prestellar core and/or protostellar phases, where interstellar UV radiation is heavily attenuated and CO is frozen out. Another possibility for the decrease of water deuterium fractionation toward the inner regions would be the additional water vapor formation at high temperatures (T > 200 - 300 K) thorough reactions: O + H_2⟶ OH + H andOH + H_2⟶ H_2O + H, which would decrease the HDO/H_2O ratios. However, it requires that a large amount of oxygen is in atomic form rather than in molecules in the high density inner regions. § SUMMARY Using CSO observations of HDO low-excitation transitions, as well as previous observations of HDO high excitations and H^18_2O low-excitation transitions from the literature, we determined the inner and outer HDO abundances, as well as the HDO/H_2O outer ratios toward three high-mass star-forming regions: G34.26 + 0.15, W51e_1/e_2, W49N. We derived HDO abundances ofX_in = (0.3–3.7) × 10^-8 (for T ≥ 100 K) and X_out = (7.0–10.0) × 10^-11(for T < 100 K),(see Table 8), and HDO/H_2O outerratios of (1.8–3.1) × 10^-3 (see Table 9).With this study, we showed that the 509 GHz transition can provide good constraints on the HDO abundance in the transition region between the hot core and colder envelope, andthat the 465 GHz HDO transition is a very good probe of the outer envelope ofmassive protostars. These transitions could help for more advanced modeling of water in high-mass sources. The HDO/H_2O ratios were also found to be higher in the cold outer envelopes than in the hot cores, as already determined for two high mass sources. What is important the model is very simple, easy toimplement, and not GPU-intensive, andprovides a starting point for more sophisticated analysis. The author thanksthe referee very much for the highly constructive comments and suggestions. The author is grateful to N.Flagey for providing the reduced HIFIdata of H_2^18O. M.K. would like to thankMaryvonne Gerin and Darek Lis for fruitful discussions anda careful reading of the manuscript. The work was carried out within the framework of the European Associated Laboratory "Astrophysics Poland-France" and also supported by the Science and High Education Ministry of Poland, grantsN20339/3334. This research is based on observations from the Caltech Submillimeter Observatory, which is operated by the California Institute of Technology under cooperative agreement with the National Science Foundation (AST-0838261).aasjournal

http://arxiv.org/abs/1702.08552v1

Kazem Rezazadeh rezazadeh86@gmail.com0000-0001-7133-3927]Kazem Rezazadeh Department of Physics, University of Kurdistan, Pasdaran Street, P.O. Box 66177-15175, Sanandaj, Iran0000-0001-6371-6493]Asrin Abdolmaleki Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran0000-0003-0008-0090]Kayoomars Karami Department of Physics, University of Kurdistan, Pasdaran Street, P.O. Box 66177-15175, Sanandaj, Iran We study logamediate inflation in the context of f(T) teleparallel gravity. f(T)-gravityis a generalization of the teleparallel gravity which is formulated on the Weitzenböck spacetime, characterized by the vanishing curvature tensor (absolute parallelism) and the non-vanishing torsion tensor.We consider an f(T)-gravity model which is sourced by a canonical scalar field. Assuming a power-law f(T) function in the action, we investigate an inflationary universe with a logamediate scale factor. Our results show that, although logamediate inflation is completely ruled out by observational data in the standard inflationary scenario based on Einstein gravity, it can be compatible with the 68% confidence limit joint region of Planck 2015 TT,TE,EE+lowP data in the framework of f(T)-gravity.§ INTRODUCTIONInflation is accepted as a paradigm to solve some problems of hot Big Bang cosmology, such as the flatness, horizon, and unwanted relics problems <cit.>. Furthermore, growth of the perturbations seeded during inflation can provide a convincing explanation for the large-scale structure (LSS) formation in the universe and also for the anisotropies of the cosmic microwave background (CMB) radiation<cit.>.In the standard inflationary scenario, a canonical scalar field, known as an “inflaton”,is considered in the framework of Einstein’s general relativity (GR) to explain the accelerated expansion of the universe during the inflationary era. Various inflation models with specific potentials or scale factors have been extensively investigated in the setting of the standard inflationary scenario in the light of observational data <cit.>.Impressive observational data have been released by the Planck 2015 collaboration <cit.> following study of the anisotropies in both the temperature and polarization of the CMB radiation. Applying these observational data, we can obtain useful information about the primordial stages of our universe. Furthermore, the observational data from CMB, LSS and other sources can be employed to probe the theory of gravity on astrophysical and cosmological scales <cit.>.Since the early stages of our universe occurred in the regime of high-energy physics, quantum modifications to gravity may play a key role in the inflationary dynamics. Motivated by this concept, numerous inflationary models have been suggested on the basis of extended theories of gravity <cit.>. One interesting class of inflationary models is based on teleparallel gravity (TG) and its generalization, f(T)-gravity. TG was employed by Einstein in 1928 to attempt to unify gravity and electromagnetism <cit.>. Although TG and GR are equivalent, they are conceptually completely different theories. In TG, the dynamical object is not the metric but instead is a set of vierbein (or tetrad) fields which forms an orthogonal basis for the tangent space at each point of spacetime. The vierbein fields are transferred parallel in all of the manifolds, which is why TG is sometimes called teleparallelism. Also, in TG, the covariant derivative is defined using the curvature-free Weitzenböck connection rather than the torsionless Levi-Civita version in GR. Furthermore, in TG, the trajectory of motion is determined by the force equations as opposed to the geodesic equations in GR <cit.>.By a formal analogy with f(R)-gravity, the theory of f(T)-gravity theory was established by extending the Lagrangian of TG to an f(T) function of a torsion scalar T <cit.>. Cosmological implications of f(T)-gravity have been extensively studied in the literature <cit.>. The theory of cosmological perturbations in this scenario has been studied by <cit.>, and <cit.>. Recently, several inflationary models have been investigated in the framework of f(T)-gravity <cit.>. For a comprehensive review of f(T)-gravity and its cosmological implications, see <cit.> and references therein.In this work, we study logamediate inflation in the framework of f(T)-gravity with a minimally coupled canonical scalar field. Logamediate inflation is specified by a scale factor of the form a(t)∝exp[A(ln t)^λ] where A>0 and λ≥ 1 <cit.>.Logamediate inflation can be regarded as a class of possible indefinite cosmological solutions resulting from imposing weak general conditions on cosmological models. <cit.> proposed that there are eight possible asymptotic solutions for cosmological dynamics, of which three lead to non-inflationary expansions. Three others give rise to the power-law (a(t)∝ t^q where q>1), de Sitter (a(t)∝ e^Ht where H is constant) and intermediate (a(t)∝exp[At^λ] where A>0 and 0<λ<1) inflationary expansions. The remaining two inflationary solutions have asymptotic expansions in logamediate form. It is worth mentioning that logamediate inflation arises naturally in some scalar–tensor theories <cit.>.To date, the power-law and intermediate inflationary models have been investigated in the f(T)-gravityscenario, and it has been shown that, using this setting, we can resurrect these models in light of observational data<cit.>. This motivates us to consider logamediate inflation in the framework of f(T)-gravity. Logamediate inflation has already been studied within the standard inflationary scenario <cit.>, and it seems that its predictions are not compatible with the current constraints from the Planck 2015 data <cit.>.The structure of this paper is as follows. In Section <ref> we review the dynamics of the background cosmology in f(T)-gravity. We also explore the relations governing the power spectra of the scalar and tensor perturbations in this model. In Section <ref>, we consider a power-law form for the f(T) function and study logamediate inflation in this setting. We estimate the inflationary observables in our model and check their viability in light of the Planck 2015 data <cit.>. Finally, in Section <ref>, we present our concluding remarks.§ THE F(T) THEORY OF GRAVITYIn the context of f(T)-gravity, the action of modified TG can be written as <cit.>I=1/2∫ d^4x e[f(T)+L_ϕ],where e≡ det(e^i_μ)=√(-g). Also, T and L_ϕ are the torsion scalar and the Lagrangian of the scalar field ϕ, respectively. It should be noted that throughout of this paper we set the reduced Planck mass to be unity, M_P≡1/√(8π G)=1, for the sake of convenience.In our notation, e^i_μ is the vierbein or tetrad field which is used as a dynamical object in TG, and satisfies the orthonormality relationse^μ_ie^i_ν=δ^μ_ν,e^μ_ie^j_μ=δ^j_i.Here, Latin and Greek indices label tangent space and spacetime coordinates, respectively. All indices take values from 0 to 3. With the help of a dual vierbein, one can obtain the metric tensor asg_μν(x)=η_ije^i_μ(x)e^j_ν(x),where η_ij= diag(-1,1,1,1) is the Minkowski metric induced on the tangent space.In GR, the Levi-Civita connection isΓ^λ_μν=Γ^λ_νμ≡1/2g^λρ(g_ρμ,ν+g_ρν,μ-g_μν,ρ),where the comma denotes the partial derivative. The Levi-Civita connection (<ref>)leads to nonzero spacetime curvature but zero torsion. In contrast, in TG, we have the Weitzenböck connectionΓ^λ_  μν≡ e^λ_i∂_νe^i_μ=-e^i_μ∂_νe^λ_i,which yields zero curvature but nonzero torsion.In GR, the curvature plays the role of gravitational force, and the trajectory of motion is determined by the geodesic equations asdu^λ/ds+Γ^λ_μνu^μu^ν=0,where u^λ is the four-velocity of the particle. In contrast, in TG, the torsion acts as a force and gravitational interaction is given by the force equations <cit.>du^λ/ds+Γ^λ_  μνu^μu^ν=T^λ_  μνu^μu^ν,where T^λ_  μν is the torsion tensor, expressed asT^λ_  μν≡Γ^λ_  νμ-Γ^λ_  μν=e^λ_i(∂_μe^i_ν-∂_νe^i_μ).The difference between the Levi-Civita and Weitzenböck connections gives the contorsion tensorK^λ_  μν≡Γ^λ_  μν-Γ^λ_μν=1/2(T^λ_μν+T^λ_νμ-T^λ_  μν).The torsion scalar T is defined asT≡ S^  μν_λT^λ_  μν,where S_λ^  μν is the superpotential tensor given byS_λ^  μν≡1/2(K^μν_   λ+δ^μ_λT^ρν_   ρ-δ^ν_λ T^ρμ_   ρ).Using the Weitzenböck connection (<ref>), the teleparallel covariant derivative, ∇_μ, of the vierbein fields vanishes, i.e.,∇_μe^i_ν≡∂_μe^i_ν-Γ^λ_  νμ e^i_λ=0.This reflects the concept of absolute parallelism or teleparallelism in TG. In GR, the metric covariant derivative, ∇_λ, of the metric is zero∇_λg_μν≡∂_λg_μν-Γ^ρ_λμg_ρν-Γ^ρ_λνg_ρμ=0. Variation of the action (<ref>) with respect to the vierbein (tetrad) e^i_λ leads to the field equations in f(T)-gravity as <cit.>[e^-1∂_μ(e e_i^ρS_ρ^  λμ)-e_i^σS_μ^  νλT_  νσ^μ]f_,T+1/2e_i^λf(T)+ e_i^ρS_ρ^  λσ(∂_σT)f_,TT=8π G Θ_i^λ,where we define f_,T≡ df/dT and f_,TT≡ d^2f/dT^2. Also Θ_i^λ≡ e^-1δ L_ϕ/δ e_λ^i and the usual energy-momentum tensor is given in terms of Θ^λ_i asΘ^μν=η^ijΘ^ν_ie^μ_j.Note that the set of field equations (<ref>)are second order, and are considerably simpler than the fourth-order equations off(R) theory <cit.>.Contracting with e^i_ν, Equation (<ref>) can be rewritten into the form <cit.>R_μνf_,T-1/2g_μν[(1+T)f_,T-f(T)]+S_νμ^   λ(∇_λT)f_,TT=8π G Θ_μν,which shows that for f(T)=T,the field equations coincide completely with those of GR. This is why in the literature, TG is called the teleparallel equivalent of GR (TEGR). This can also be understood in another way. <cit.> have shown that T and R differ only by a total divergence, i.e., R=-T-2∇^μ(T^ν_  μν). Since the total divergence can be neglected inside an integral, the TG Lagrangian density is completely equivalent to the Einstein–Hilbert density.Now, we consider a spatially flat universe described by the Friedmann-Robertson-Walker metricg_μν= diag(-1,a^2(t),a^2(t),a^2(t)),where ais the scale factor of the universe. Using this together with Equation (<ref>), we gete_μ^i= diag(1,a(t),a(t),a(t)).Substituting the vierbein (<ref>) into (<ref>) yieldsT=-6H^2,where H ≡ȧ/a is the Hubble parameter.Taking Θ^μ_ ν= diag(-ρ_ϕ,p_ϕ,p_ϕ,p_ϕ)for the energy–momentum tensor of the scalar field in the perfect fluid form and using the vierbein (<ref>), the field equations (<ref>) yields the Friedmann equations in f(T)-gravity as <cit.>2ρ_ϕ =12H^2f_,T+f(T), 2p_ϕ =48H^2Ḣf_,TT-(12H^2+4Ḣ)f_,T-f(T).Here, ρ_ϕ and p_ϕ are the energy density and pressure of the scalar field, respectively, and satisfy the conservation equationρ̇_ϕ+3H(ρ_ϕ+p_ϕ)=0.One can rewrite Equations (<ref>) and (<ref>) in the standard form of the Friedmann equations asH^2 = 1/3(ρ_T+ρ_ϕ), Ḣ+3/2H^2 = -1/2(p_T+p_ϕ),whereρ_T ≡ 1/2(2Tf_,T-f-T),p_T ≡-1/2[-8ḢTf_,TT+(2T-4Ḣ)f_,T-f+4Ḣ-T],are the torsion contribution to the energy density and pressure which satisfy the energy conservation lawρ̇_T+3H(ρ_T+p_T)=0.In the case of f(T)=T, from Equations (<ref>) and (<ref>) we have ρ_T=0 and p_T=0. Therefore, Equations (<ref>) and (<ref>) are transformed to the usual Friedmann equations in GR. In the following, we assume the universe to be filled with a canonical scalar field which has energy density and pressure as follows:ρ_ϕ = 1/2ϕ̇^2+V(ϕ),p_ϕ = 1/2ϕ̇^2-V(ϕ).Substitution of Equations (<ref>) and (<ref>) into the conservation equation (<ref>) yields the evolution equation governing the scalar field asϕ̈+3Hϕ̇+V_,ϕ=0,where V_,ϕ≡ dV/dϕ.In order to study inflation in f(T)-gravity, it is useful to define the Hubble slow-roll parameters as follows:ε_1 ≡-Ḣ/H^2, ε_i+1 ≡ ε̇_i/Hε_i.Due to having an inflationary epoch (ä > 0), according to Equation (<ref>) we must have ε_1 < 1. It should be noted that the condition ε_1 = 1 can determine the initial (or final) time of inflation if the first Hubble slow-roll parameter ε_1 is a decreasing (or increasing) function of time <cit.>.During inflation, the scalar field ϕ and the Hubble parameter H change very slowly. This enables us to use the slow-roll conditionsgiven by ϕ̇^2≪ V(ϕ) and |ϕ̈|≪|3Hϕ̇|, |V_,ϕ|. Applying the slow-roll approximation to Equations (<ref>) and (<ref>), one can findV= 1/2(f-2Tf_,T), ϕ̇^2 =-2Ḣ(f_,T+2Tf_,TT).With the help of the above equations, one can obtain the evolutionary behaviors of the potential V(t) and scalar field ϕ(t), if the functional form of f(T)-gravity and the scale factor a(t) are known. Combining the results of V(t) and ϕ(t) to eliminate t between them, one may get V(ϕ) determining the shape of the inflationary potential with respect to the inflaton.In the study of inflation, we usually express the extent of the universe expansion in terms of the e-fold number, defined asN≡lna_e/a,where a_e denotes the scale factor of the universe at the end of inflation. The above definition is equivalent todN =- H dt.It is believed that the anisotropies observed in the CMB radiation and in the LSS of the universe are related to the perturbations which exit the Hubble radius around the e-fold numberN_* ≈ 50-60 before the end of inflation <cit.>. Those perturbations remain outside the horizon until a time close to the present time and this enables us to relate the late-time observations to the primordial power spectra of the perturbations produced during inflation.In the following, we review briefly the basic results of the theory of cosmological perturbations in the f(T)-gravity scenario when a canonical scalar field is assumed to be the matter-energy content of the universe(for more details, see <cit.>). The primordial power spectrum of the scalar perturbations in the f(T)-gravity is given by <cit.>P_s=.H^2/8π^2c_s^3ε_1|_c_sk=aH,which should be evaluated at the sound horizon exit for which c_s k=a H. Here, c_s is the sound speed defined asc_s^2=f_,T/f_,T-12H^2f_,TT.It is evident that in the case of TEGR (i.e., f(T)=T), we have c_s=1 from Equation (<ref>), and then Equation (<ref>) reduces to the expected relation for the standard inflationary scenario <cit.>.The scale-dependence of the scalar power spectrum is measured by the scalar spectral indexn_s-1≡dln P_s/dln k.In the slow-roll approximation, it is assumed that the Hubble parameter H and the sound speed c_s are slowly varying <cit.>. Therefore, the relation c_s k=a H leads todln k ≈ Hdt=-dN,which is valid around the sound horizon exit. Now, using Equations (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), we can obtain the scalar spectral index in f(T)-gravity scenario asn_s=1-2ε_1-ε_2-3ε_s1,where we have defined the sound speed slow-roll parameters as follows:ε_s1≡ċ_s/Hc_s, ε_s(i+1)≡ε̇_si/Hε_si.We further can use Equations (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>) to obtain the running of the scalar spectral index asdn_s/dln k=-2ε_1ε_2-ε_2ε_3-3ε_s1ε_s2. We now focus on the tensor perturbations in the framework of f(T)-gravity. Following <cit.>, we define the parameters γ and δ as follows:γ ≡ (f_,TT/f_,T)Ṫ, δ ≡ | γ|/2H.<cit.> proposed that if the δ parameter is much less than unity (δ≪ 1), then the tensor power spectrum of the tensor perturbations in f(T)-gravity reduces to the one for the standard inflationary model, which is given byP_t=.2H^2/π^2|_k=aH.It should be noted that the tensor power spectrum must be calculated at the time of horizon crossing specified by k=a H. This time is not exactly the same as the time of sound horizon crossing for which c_s k=a H, but to lowest order in the slow-roll parameters the difference is negligible <cit.>.The scale-dependence of the tensor power spectrum is specified by the tensor spectral indexn_t≡dln𝒫_t/dln k.Using Equations (<ref>), (<ref>), (<ref>), and (<ref>), we obtain this observable for the inflationary model based on the f(T)-gravity scenario asn_t=-2ε_1.Current experimental devices are not accurate enough to measure this observable, and we may be able to determine it with more sensitive measurements in the future <cit.>.An important inflationary observable which can be applied to discriminate between inflationary models is the tensor-to-scalar ratior≡𝒫_t/𝒫_s.In f(T)-gravity setting, using Equations (<ref>) and (<ref>) in Equation (<ref>), it is easy to see that this observable is given byr=16c_s^3ε_1. From Equations (<ref>) and (<ref>), the consistency relation in f(T)-gravity takes the formr=-8c_s^3n_t.It is obvious that for c_s=1, this equation reduces the well-known consistency relation r=-8n_t in the standard inflationary scenario <cit.>.§ LOGAMEDIATE INFLATION IN F(T) TELEPARALLEL GRAVITY<cit.> investigated logamediate inflation in the framework of the standard inflationary scenario based on Einstein gravity. From their results, it seems that logamediate inflation within the standard inflationary setting is ruled out by current observational data from the Planck 2015 collaboration <cit.>. This motivates us to examine the observational viability of logamediate inflation in f(T) teleparallel gravity.We consider an f(T)-gravity setting in which the f(T) function in action (<ref>) has the power-law form <cit.>f( T ) = T_0(T/T_0)^n,where T_0 and n are constant parameters of the model. For the case n=1, Equation (<ref>) transforms to TEGR, i.e., f(T)= T. From the definition of sound speed in Equation (<ref>), we see that the f(T) model (<ref>) gives rise to a constant sound speed asc_s^2 = 1/2n - 1.The above equation leads to the requirement n ≥ 1required to have a physical speed for propagation of the scalar perturbations, i.e., 0<c_s^2≤ 1 <cit.>. Furthermore, in our f(T)-gravity model (<ref>), the sound speed slow-roll parameters (<ref>) and (<ref>) vanish because of the constant sound speed (<ref>).Now, we consider the logamediate scale factor which has the following form <cit.>:a(t)=a_0exp[A(ln t)^λ],where a_0>0, A>0 and λ≥ 1 are constant parameters. For λ=1, the logamediate scale factor (<ref>), reduces to the power-law scale factor a(t)=a_0t^q, where q=A. With the above scale factor, the Hubble parameter readsH=Aλ(ln t)^λ-1/t.Furthermore, we see from Equation (<ref>) that the first slow-roll parameter takes the formε_1=ln t-λ+1/Aλ(ln t)^λ.The above equation shows that at late times, t ≫ 1,the first slow-roll parameter becomes a decreasing function during inflation, and hence it cannot reach unity at the end of inflation. This demonstrates that, in our model, inflation cannot end with slow-roll violation <cit.>.To obtain the evolution of the inflationary potential, we use Equations (<ref>), (<ref>), (<ref>), and (<ref>), and obtainV(t)=3^n(2n-1)[2/(-T_0)]^n-1[Aλ(ln t)^λ-1/t]^2n.We further can use Equation (<ref>) to findϕ̇= [2^nn(2n-1)(Aλ)^2n-1(3/(-T_0))^n-1]^1/2 ×[(ln t)^2n(λ-1)-λ(ln t-(λ-1))]^1/2t^-n .In general, it is too difficult to solve the above equation for a given value of n. Therefore, we cannot combine Equations (<ref>) and (<ref>) and find the shape of the inflationary potential V(ϕ) for a general n. However, we can check the validity of our results for the simplest case n=1 corresponding to TEGR, and we expect that it leads to the same results for logamediate inflation in the standard inflation scenario. For the case of n=1, Equations (<ref>) and (<ref>) yieldV(t) = 3[Aλ(ln t)^λ-1/t]^2, ϕ(t) = 2√(2Aλ)/λ+1(ln t)^(λ+1)/2.In the derivation of Equation (<ref>) we have followed the logic of <cit.>, and considered the late time limit which allows us to ignore (λ- 1) versus lnt. If we combine the above two equations to eliminate t, we find the inflationary potential asV(ϕ)=V_0ϕ^αexp(-2Bϕ^β),whereV_0 ≡ 3(Aλ B^λ-1)^2, B ≡ (λ+1/2√(2Aλ))^2/(λ+1), α ≡ 4(λ-1)/λ+1, β ≡ 2/λ+1.The result in Equation (<ref>)is the potential responsible for logamediate inflation in the standard inflationary scenario, and this result is in agreement with that found by <cit.>.Here, we are interested in showing that our result is also in agreement with the analysis performed by <cit.>, who presented asymptotic solutions of the potentialV(ϕ)=V_0ϕ^lexp(-κϕ^m),in the slow-roll approximation. In the above equation, V_0, l, κ and m are positive constant parameters. <cit.> found that for the case m=1 and l=0, and provided that κ^2<2, the potential (<ref>) leads to an inflationary expansion in the power-law forma(t)∝ t^2/κ^2.This is just the well-known fact that in the standard inflationary scenario, the exponential potential gives rise to power-law inflation <cit.>. In addition, <cit.> obtained that for 0<m<1 and in the limit of t →∞, the potential (<ref>) provides an inflationary scale factor in the form ofa(t)∝exp[1/κ m(2-m)(2/κ)^2-m/m(ln t)^2-m/m]. By comparing the potentials (<ref>) and (<ref>), we findα =l, β =m, 2B= κ.First, we focus on the case m=1 and l=0. In this case, Equations (<ref>) and (<ref>) give α=0 and β=1. As a result, from Equation (<ref>) we get λ=1. Using this, we see that in Equation (<ref>) the logamediate scale factor reduces to the power-law onea(t)∝ t^A.In addition, Equation (<ref>) gives B=1/√(2A). This together with Equation (<ref>) gives A=2/κ^2. Finally, using this result in Equation (<ref>), we reach Equation (<ref>) obtained by <cit.>.Second, we proceed to examine the case 0<m<1. By use of Equations (<ref>) and (<ref>), we obtainλ=2-m/m.Applying this together with Equations (<ref>) and (<ref>), we findA=1/κ m(2-m)(2/κ)^2-m/m.Now, it is obvious that substitution of Equations (<ref>) and (<ref>) into (<ref>) yields Equation (<ref>) obtained by <cit.>. Therefore, we showed that the results (<ref>) and (<ref>) given by <cit.> are recovered in our model.For the f(T) function given in Equation (<ref>) with a general n, and considering the logamediate scale factor (<ref>), the scalar power spectrum (<ref>)becomes𝒫_s=(Aλ)^3(2n-1)^3/2(ln t)^3λ-2/8π^2t^2(ln t-λ+1).We can also obtain the scalar spectral index from Equation (<ref>) asn_s= [Aλ(ln t)^λ(ln t-λ+1)]^-1 ×[Aλ(ln t)^λ+1-Aλ(λ-1)(ln t)^λ-2(ln t)^2..+5(λ-1)ln t-3λ^2+5λ-2].In addition, the running of the scalar spectral index follows from Equation (<ref>) asdn_s/dln k=(λ-1)[Aλ(ln t)^λ(ln t-λ+1)]^-2 ×[2(ln t)^3-(7λ-4)(ln t)^2+(8λ^2-9λ+3)ln t.. -λ(3λ^2-5λ+2)]. In order to find the expression of the tensor power spectrum for the model under consideration, we note that for the f(T) function (<ref>), the δ parameter can be simplified asδ=(n-1)ε_1.In this paper, we are only dealing with values of n of order unity. Hence, the δ parameter takes the order of the first slow-roll parameter and therefore it becomes much less than unity in the slow-roll regime. This allows us to use Equation (<ref>)for the tensor power spectrum to obtain𝒫_t=2[Aλ(ln t)^λ-1/π t]^2.Then, using this together with Equation (<ref>), the tensor spectral index is obtained asn_t=-2(ln t-λ+1)/Aλ(ln t)^λ.If we use Equation (<ref>), we can easily show that the tensor-to-scalar ratio becomesr=16(ln t-λ+1)/Aλ(2n-1)^3/2(ln t)^λ. It is interesting to find simplified forms of the equations for the inflationary observables in the case of λ=1 for which logamediate inflation reduces to power-law inflation a(t)∝ t^q, where q=A. For λ=1, Equations (<ref>), (<ref>), (<ref>), and (<ref>) reduce ton_s = 1-2/A, dn_s/dln k = 0,n_t = -2/A,r = 16/(2n-1)^3/2A.These are in agreement with the results of <cit.>, where the authors investigated power-law inflation in the f(T)-gravity setup (<ref>). The obtained results for the scalar spectral index n_s and the tensor-to-scalar ratio r in Equations (<ref>) and (<ref>) are independent of the dynamical quantities such as t, N or ϕ. This behavior is familiar for power-law inflation in other inflationary scenarios, for instance, the standard inflationary setting <cit.>, Brans-Dicke inflation <cit.>, tachyon inflation <cit.>,and non-canonical power-law inflation <cit.>.Consequently, we can combine Equations (<ref>) and (<ref>) to eliminate A between them, and obtainr=8/(2n-1)^3/2(1-n_s),implying a linear relation between r and n_s. For n=1, i.e., f(T)=T, the above equation reduces to r=8(1-n_s) which is the well-known result obtained for power-law inflation in the standard inflationary scenario <cit.>.We come back to our investigation with general n. So far, we have found the inflationary observables in terms of time. In order to estimate these observables, it is necessary to evaluate them at the time of horizon exit which has a specified e-fold number. Consequently, we should obtain the relation between time and e-fold number in our model. To this end, we solve the differential Equation (<ref>) for the logamediate scale factor (<ref>) and obtaint=exp[((ln t_e)^λ-N/A)^1/λ],where t_e refers to the end time of inflation. To get the above result, we have applied the initial condition N_e≡ N(t=t_e)=0 which is a direct implication of definition (<ref>) for the e-fold number. Here, it is essential to note that we cannot determine t_e in our model by setting ε_1=1, because, as we have mentioned before, inflation in our model cannot end with slow-roll violation <cit.>. To overcome this problem, we follow the approach of <cit.> and <cit.>,and retain t_e as an extra parameter. In the following, we determine it by fixing the amplitude of the scalar perturbations from the observational resultsInserting Equation (<ref>) into (<ref>), we obtain the scalar power spectrum at the horizon exit as𝒫_s(N_*)= (Aλ)^3(2n-1)^3/2((ln t_e)^λ-N_*/A)^(3λ-2)/λ/8π^2[((ln t_e)^λ-N_*/A)^1/λ-λ+1] ×exp[-2((ln t_e)^λ-N_*/A)^1/λ].The Planck 2015 data provided an estimation for the amplitude of the scalar perturbations as ln[10^10𝒫_s(N_*)]=3.094±0.034 (68% CL, Planck 2015 TT,TE,EE+lowP) <cit.>. We use this constraint in the above equation to fix the amplitude of the scalar power spectrum in our model and determine the parameter t_e in terms of the other parameters for a given horizon crossinge-fold number N_*. Since we cannot determine t_e analytically, we use a numerical approach. Inserting the result of the numerical solution for t_e in Equation (<ref>), we can obtain the time of horizon exit t_* for given parameters n, A, λ, and N_*. Surprisingly, our computations show that t_* does not depend on N_* at all. To explain this unexpected result, we take the partial derivative of both sides of Equation (<ref>) with respect to N_*, and, keeping in mind that ∂𝒫_s(N_*)/∂ N_*=0, we obtain∂ t_e/∂ N_*=t_e/Aλ(ln t_e)^λ-1.On the other hand, if we evaluate Equation (<ref>) at the horizon exit with the e-fold number N_*, and calculate the partial derivative of the result with respect to N_*, then we will have∂ t_*/∂ N_*= exp[((ln t_e)^λ-N_*/A)^1/λ]/((ln t_e)^λ-N_*/A)^(λ-1)/λ ×((ln t_e)^λ-1/t_e∂ t_e/∂ N_*-1/Aλ).It is obvious that substitution of ∂ t_e/∂ N_* from Equation (<ref>) into Equation (<ref>) leads to ∂ t_*/∂ N_*=0. Therefore, in our model, and after fixing the amplitude of the scalar perturbations from the observational data, the time of horizon exit t_* is independent of its e-fold number N_*. As an important result, we conclude that the inflationary observables (<ref>), (<ref>), (<ref>), and (<ref>) evaluated at t_* are independent of N_*.Now, we can estimate the inflationary observables in our model and check their consistency versus the cosmological data. To do so, first we solve Equation (<ref>) numerically to find t_e for given parameters n, A, and λ. Then, we use the obtained value for t_e in Equation (<ref>) and find t_*. Subsequently, we evaluate the inflationary observables (<ref>), (<ref>), and (<ref>)at the time of horizon exit t_*.In order to check the viability of logamediate inflation (<ref>) in our f(T)-gravity model (<ref>), we use Equation (<ref>) and (<ref>) and plot the prediction of our model in r-n_s plane as shown in Figure <ref>. In this figure, the marginalized joint 68% CL and 95% CL regions of the Planck 2015 data <cit.> have been specified. We have represented the results of our model with n=1 and n=2in the figure as black and orange lines, respectively. Each line is related to a specific value for the parameter λ, while the parameter A varies. In each case, as A increases, n_s approaches 1 and r approaches 0. The case n=1 corresponds to TEGR, which provides the same results of the standard inflationary scenario. It is obvious in the figure that for n=1, logamediate inflation is completely ruled out by Planck 2015 TT,TE,EE+lowP data <cit.>. But, our study indicates that if we take the parameter n greater than 1, then logamediate inflation (<ref>) in our f(T)-gravity model (<ref>) can be compatible with the Planck 2015 results. For instance, as we see in Figure <ref>, for n=2, logamediate inflation is consistent with the joint 68% CL region of Planck 2015 TT,TE,EE+lowP data <cit.>. In Figure <ref>, we have specified the parameter space of A and λ for which our model with n=2 is compatible with the 68% CL or 95% CL regions of Planck 2015 TT,TE,EE+lowP data <cit.>. From the figure, we conclude that for λ≲ 6 (λ≲ 8), our model is compatible with the joint 68% CL (95% CL) region of Planck 2015 TT,TE,EE+lowP data <cit.>. In Table <ref>, we present the ranges of the parameter A for which our model with n=2 and with some typical values of λ is consistent with the Planck 2015 observational data <cit.>. In Table <ref>, we also present the predicted values for the running of the scalar spectral index dn_s/dln k obtained using Equation (<ref>). The predicted values for dn_s/dln k in our model are compatible with the 95% CL constraint provided by Planck 2015 TT,TE,EE+lowP data <cit.>.At the end of this section, it is useful to provide some explicit estimations for the inflationary observables in our model. We choose n=2, A=1.8, and λ=2. Using Equation (<ref>), (<ref>), and (<ref>), we obtain n_s=0.9657, dn_s/dln k=3.99×10^-5, and r=0.0546, respectively, and these are in good agreement with Planck 2015 TT,TE,EE+lowP data <cit.>. Using Equation (<ref>), our model predicts the tensor spectral index to be n_t=-0.0354, and this value may be verified by more precise measurements in the future. Within our model, we can also provide some predictions for other parameters, including the time of horizon exit t_* and the end time of inflation t_e. With the chosen values for n, A, λ, and taking the horizon exit e-fold number as N_*=60, we obtain the end time of inflation from the numerical solution of Equation (<ref>) as t_e=6.47×10^6M_P^-1=1.75×10^-36 sec. Applying this in Equation (<ref>) gives the time of horizon crossing as t_*=2.15×10^6M_P^-1=5.82×10^-37 sec. Here, we recall that in our logamediate inflationary model, although the value of t_e depends on N_*, the value of t_* is completely independent of it.§ CONCLUSIONSWe have investigated logamediate inflation in the framework of f(T)-gravity which is sourced by a canonical and minimally coupled scalar field. For this purpose, we first briefly reviewed the basic equations governing the cosmological background evolution in f(T)-gravity and presented the relations of the scalar and tensor power spectra in this scenario. Then, we considered a setting in which the f(T) function in the action has the power-law form f(T)=T_0(T/T_0)^n, where n and T_0 are constant parameters. For n=1, this reduces to f(T)=T which provides the same results as for Einstein GR. In addition, in our work we considered the logamediate scale factor a(t)=a_0exp[A(ln t)^λ], where a_0>0, A>0 and λ≥ 1 are constant parameters. For λ=1, the logamediate scale factor turns into the power-law scale factor a(t)=a_0t^q, where q=A.Our investigation implies that, although logamediate inflation is not consistent with the the Planck 2015 data <cit.> in the standard framework based on Einstein gravity, we can make it compatible with the observational data in our f(T)-gravity model, if we take the parameter n greater than 1. For instance, we showed that for n=2, the result of the logamediate inflation in r-n_s plane can lie inside the 68% CL region favored by Planck 2015 TT,TE,EE+lowP data <cit.>. Using the r-n_s test, we determined the parameter space for A and λ in our model with n=2, and showed that for λ≲ 6 (λ≲ 8), our model is consistent with the joint 68% CL (95% CL) region of Planck 2015 TT,TE,EE+lowP data <cit.>. We further estimated the running of the scalar spectral index dn_s/dln k in our model, and concluded that it satisfies the 95% CL bound from Planck 2015 TT,TE,EE+lowP data <cit.>.The authors thank the referee for his/her valuable comments. 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http://arxiv.org/abs/1702.07877v1

Faculty of Modern Languages and Literature, Adam Mickiewicz University, Poznań, PolandFaculty of Physics, Adam Mickiewicz University, Poznań, Poland Influencing various aspects of human activity, migration is associated also with language formation. To examine the mutual interaction of these processes, we study a Naming Game with migrating agents.The dynamics of the model leads toformation of low-mobility clusters, which turns out to break the symmetry of the model: although the Naming Game remains symmetric, low-mobility languages are favoured. High-mobility languages are gradually eliminated from the system and the dynamics of language formation considerably slows down. Our model is too simple to explain in detail language competition of migrating human communities, but it certainly shows that languages of settlers are favouredover nomadic ones.Language competition in a population of migrating agents Adam Lipowski========================================================§ INTRODUCTIONThe constatation that complex systems can be regarded as composed of many interacting subunits opens up a possibility of studying them with methods that were primarily developed in the context of physical many-body systems. Suchapproach turned-out to be very successful <cit.>, and lead to the emergence of new research fields likesocio- or econophysics <cit.>. Even certain linguistic problems can be studied using methods with a strong physical flavour. Language emergence <cit.> or death <cit.>, its diversification <cit.> and diffusion <cit.>, time-evolving structure <cit.>, appearance of grammar or linguistic categories <cit.>, and language learning <cit.> are just a few examples of problems, where physicists' contributions might prove to be valuable.Of course, language, as one of our human attributes, is interrelated with many other forms of our activity. Social interactions,economical status or political situation influence the way language is acquired and changed, or sometimes falls into oblivion. Language, as an integral part of our culture and way of life, is also intricately related tomigrations of people <cit.>. Various tribes, ethnic groups, or even entire nations firmly settledcertain areas, while some others, due to various reasons, almost constantly migrate. Migration might mix as well as separate human communities and language formation processes should be thus strongly influenced by such a factor.Moreover, some modern trends,especially globalization, most likely increase people's migrations <cit.>. Some researches even suggest that merging multinational and multicultural migrants creates in some areas a new kind of super-diverse societies, and to describe their intercommunication, traditionally understood languages do not seem to be sufficient <cit.>.It would be desirable to have some general understanding of how migration affects the language formation processes and perhaps vice versa.As for the language formation, an interesting class of models originates from the so-called Naming Game <cit.>. In this model a population of agents negotiates a language (or, more generally, conventional forms). Although the dynamics might depend on, for example, the structure of the interaction network, typically the model reaches a consensus on the language. The process of language formation resembles the ordering dynamics of Ising or Potts models accompanied, due to the symmetry of the Naming Game, by a spontanous symmetry breaking. One can even introduce the notion of an effective surface tension to explain some dynamical characteristics of the Naming Game <cit.>. In the present paper, we thus examine the Naming Game in a population of migrating agents. When mobility of agents is uniform in the entire population, the model is very similar to the Naming Game of immobile agents. However, an interesting behaviour appears when the mobility depends on the language used by an agent. In such a case,the dynamics turned out to break the symmetry of the Naming Game, favouring low-mobility languages. During thecoarsening, agents form low-mobility clusters that effectively attract and convert high-mobility neighbours. As a result, the low-mobility agents become more widespread, which considerably slows down the dynamics. Of course, our model is too simple to explain the intricacies of language competitionin settled and nomadic communities, nevertheless, it shows that the (difference in) mobility has a strong effect on such proccesses.§ MODEL In our model, we have a population of agents placed on a square lattice of linear size L (with periodic boundary conditions). Initially agents are uniformly distributed on the lattice with the density (i.e., probability) ρ.Each agent has its own inventory, which is a dynamically modified list of words. The dynamics of our model combines the lattice gas diffusion with the so-called minimal version of the Naming Game <cit.>. More specifically, in an elementary step, an agent (Speaker) and one of its neighbouring sites are randomly selected. If the selected site is empty, Speaker moves to this site. If the selected site is occupied by an agent (Hearer), then the pair Speaker-Hearer plays the Naming Game:* Speaker selects a word randomly from its inventory and transmits it to Hearer.* If Hearer has the transmitted word in its inventory, the interaction is a success and both players maintain only the transmitted word in their inventories.* If Hearer does not have the transmitted word in its inventory, the interaction is a failure and Hearer updates its inventory by adding this word to it.The unit of time (t=1) is defined as ρ L^2 elementary steps, which corresponds to a single (on average) update of each agent. In the following, we will refer to words communicated by agents as languages. Rules of the Naming Game are also illustrated in Fig. <ref>. § TWO-LANGUAGE VERSIONWhen ρ=1, all sites are occupied, thus there is no diffusionand the model is equivalent to an ordinary square-lattice Naming Game. For ρ<1, a fraction 1-ρ of sites is empty and in addition to playing the Naming Game, agents change their locationsfrom time to time. There are several characteristics that might be determined for Naming Game models. To demonstrate some analogiesto Ising-type models, we examined a two-language version of the Naming Game <cit.>. We measured the average time τ needed for a system to reach a consensus, i.e., the state where every agent has the same language in its inventory. The initial configuration includes a square of size M, inside of which all agents havelanguage B in their inventories, while outside agents havelanguage A. Both within the square of size M and outside, the agents are distributed with the uniform density ρ. In Ising-like models, general arguments, which refer to the notion of a surface tension and the Laplace law of excesive pressure, estimate the lifetime of such a bubble as τ∼ M^2 <cit.>. Our numerical results (Fig. <ref>) are in a very good agreement with such estimation both for ρ=1 and ρ<1 (a slight deviation for ρ=0.1 can be attributed to the finite size effects). It is thus a strong evidence that for migrating agents, the domain dynamics in the Naming Game is also driven by an effective surface tension. Let us notice that the relation τ∼ M^2 is expected to hold when the bubble and its surroundings are thermodynamically equivalent phases.In the Ising models it means that there is no external magnetic field, which would favour one of them. In the Naming Game, we have also such symmetry since the dynamics of the Naming Game does not favour any of the languages used by agents. We do not present here our additional numerical results, though we have also measured some other characteristicsof the Naming Game with migration (such as the average time needed to reach a consensus for a system initialized withrandomly assigned languages) and they qualitatively agree with the ordinary ρ=1 version <cit.>. Instead, we would like to examine an extension of the above defined model, in which to each languageits own (thus language-specific) mobility d is assigned. An agent that changes its language changes thereby also its mobility (which, we hope, might reflect the behaviour in some human communities). Now, if the chosen neighbouring site is empty,Speaker migrates to this empty site with the probability dcorresponding to the language it uses. If Speaker has several languages in its inventory, then one of them is selected randomly to determine the probability of migration (though only a very small fraction of agents have more than one language in its inventories). Let us notice that such modification affects only the dynamics of migration while the Naming Game remains symmetric. We determined the average time τ in a two-language version of this model (Fig. <ref>). When mobility d_B of language B users is larger than mobility d_A of surrounding language A users, τ still increases but considerably slower and perhaps linearly τ∼ M (the least square fitting gives τ∼ M^1.1 but a slight bending of our data makes the asymtotic τ∼ M very plausible). We do not present the estimation of time for d_B<d_A since it can be made only for very small M. In turn, for M above a certain threshold value, the initial bubble instead of shrinking starts to grow and eventually B-users engulf the entire system. An example of such growth can be seen in Fig. <ref>. Such behaviour resembles the the behaviour of the Ising model but in the presence of an external magnetic field. When the magnetic field does not favour the bubble, a finite velocity of shrinking is expected for large M and that would explain the growth τ∼ M <cit.>. For the field favouring the bubble, the existence of a threshold size, above which the bubble will grow indefinitely, is also a well-known feature. It suggests that in our model the difference in mobility acts as a magnetic field in the Ising model and breaks the symmetry of the Naming Game favouring low-mobility languages.To confirm that d_A-d_B is an analogue to the magnetic field in the Ising model, we made simulations of the system initially divided (say vertically), in which the left half is filled with agents of mobility d_A and the right half with agents of mobility d_B. Depending on the sign of the difference d_A-d_B, the interface should move (possibly at constant speed) either to the left or to the right, and only for d_A=d_B it shoud stay more or less in the inital position. Our simulations fully confirmed suchscenario (Fig. <ref>). Theinterface always moves in such a way that a less-mobile language becomes more widespread. Let us notice that even a very small difference d_A-d_B is sufficient to favour one language over the other, and only for precisely the same mobilities d_A=d_B, the languages are equivalent. Let us emphasize that the rules of the Naming Game do not favour any of the languages, and the bias that appears for unequal mobilities isgenerated dynamically. In our opinion, the asymmetry appears due to a tendency of low-mobility languages to form clusters. Such low-mobility clusters are relatively resistant upon interactions with high-mobility agents (Fig. <ref>). That low-mobility languages have a tendency to form clusters can be seen also in Fig. <ref>. Indeed, the central area with the less mobile language seems to be more densely filled than its surroundings (and initially the entire lattice was filled with the same density ρ=0.2). Moreover, the interface between the languages has a considerably lower density than the interior of the area with the more mobile language. Apparently, high-mobility agents that are close to the interface get intercepted by the low-mobility center (and converted into low-mobility agents). To support the above arguments, we made simulationswhere we measured the probability prob_A that a small system (L=6) starting with randomly distributed agents will reach a consensus withlanguage A. Initially, languages A and B (and migrations d_A and d_B) are also randomly assigned to agents. Numerical simulations show (Fig. <ref>) that only for the number of agents n=2 and n=36, we have prob_A=0.5. For n=2, only binary interactions of agents might take place (left panel of Fig. <ref>) and the symmetry of the Naming Game implies that prob_A=0.5. Similarly, for n=36 migration is suppressed and prob_A=0.5 is the expected ordinary Naming Game result. Simulations show, however, that for any other value of n the symmetry of the model is broken and the less mobile language (B) is effectively favoured. It would be certainly desirable to have more general understanding of the mobility-induced symmetry breaking that takes place in our model. For example, one might hope to develop some kind of a coarse-grained description of our model in terms of the Ginzburg-Landau potential, an approach that turned out to be quite effective in some other agreement-dynamics models <cit.>.§ MULTI-LANGUAGE VERSIONIn the present section we examine the multi-language version of our model. In such a case Fig. <ref> shows that the cluster-formation mechanism is also at work (simulations start from a random distribution of languages and their mobilities). Initially mobilities were set randomly from the range 0<d<1 but by the time t=3· 10^3 and especially t=10^4, languages with the largest mobilities (close to 1) were eliminated. One can clearly see the formation of low-mobility clusters, which grow by depleting their surroundings from more mobile agents. We also examined the time dependence of the average mobility ⟨ d ⟩ in the system. Indeed, the numerical results in Fig. <ref> confirm that⟨ d ⟩ systematically decreases. For a low density (ρ=0.1), one can notice relatively long initial plateaux, related to the fact that the system needs some time to build low-mobility clusters, and only then the process that favours low-mobility languages starts. Moreover, in the time decay of the average mobility, one can distinguish two power-law regimes. In the high-density regime (ρ=0.6 and 0.8), one has ⟨ d ⟩∼ t^-0.9 while in the low-density regime (ρ=0.1, 0.2, and 0.3), the exponent is smaller than 0.9 and perhaps even varies with density.Certain Naming Game characteristics exhibit a similar power-law behaviour. In Fig. <ref> we present the time dependence of the number of users of the largest language l_M. While in the high-density regime (ρ=1.0, 0.8, 0.6), the increase seems to be universal and l_M∼ t^0.8, in the low-density regime (ρ=0.3, 0.2, 0.1), the power-law behaviour has a density dependent exponent. The behaviour of ⟨ d ⟩ and l_M shows that the dynamics in the high-density regime is much faster and the Naming Game characteristics are very similar to those of an ordinary Naming Game (with ρ=1). The low-density regime has a much slower dynamics and we relate suchbehaviour to the formation of low-mobility clusters (Fig. <ref>). It is likely that the model undergoes a phase transition around ρ=0.5, but its more detailed analysis is left for the future. The two-language version described in the previous section exhibits a symmetry breaking that speeds up the dynamics. In the multi-language version, however, we have initially the entire spectrum of languages and mobilities. The dynamics gradually eliminates large-mobility agentsand thus the remaining small-mobility agents are primarily responsible for a considerably slower dynamics (Fig. <ref>).§ CONCLUSIONSIn summary, motivated by a possible mutual influence of language formation and migration of human communities, we examined the Naming Game model with mobile agents. As our main result, we have shown that even a small difference in a language-specific mobility favours a low-mobility language.Of course, taking into account an extreme complexity of human interactions, we are not even tempted to suggest that our model proves that languages of settlers should outperform nomadic ones, nevertheless, it certainly shows a strong relation between language formation and migration. In our model, low-mobility languages form clusters and in a low-density regime this process slows down the dynamics of the Naming Game. Let us also notice thatthe dynamics of a typical Naming Game (with ρ=1) rather quickly leads to the consensus, which not necessarily corresponds with a relatively stable multi-language structure of the human population <cit.>. With this respect, a slower dynamics and a longer lifetime of the multi-language state (as suggested in Fig. <ref>) of the proposed model might be more suitable.Finally, it should be also noted that the Naming Game is one of the models with the so-called agreement dynamics. The Voter or Ising models are yet other well-known examples of this kind of models and some aspects of mobility in such systems were already examined <cit.>. It would be, in our opinion, interesting to examine their generalizations that taking into account the state-dependent mobility. loreto C. Castellano, S. Fortunato, and V. Loreto, Statistical physics of social dynamics, Rev. Mod. Phys. 81, 591 (2009).stauffer D. Stauffer, Social applications of two-dimensional Ising models, Am. J. 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Canagarajah (ed.), The Routledge Handbook of Migration and Language (Routledge, 2017).williams C. H. Williams (ed.), Language in Geographic Context (Multilingual Matters Ltd, 1988).czaika M. Czaika and H. de Haas, The Globalization of Migration: Has the World Become More Migratory?, Int. Migration Rev. 48, 283-323 (2014).jorgensen J. N. Jörgensen et al., Polylanguaging in superdiversity, Diversities 13(2), 22-37 (2011).steels L. Steels, A self-organizing spatial vocabulary, Artif. Life 2, 319-332 (1995).baron2006 A. Baronchelli, L. Dall`Asta, A. Barrat, and V. Loreto, Topology-induced coarsening in language games, Phys. Rev. E 73, 015102 (2006).lippref D. Lipowska and A. Lipowski, Phase transition and fast agreement in the Naming Game with preference for multi-word agents,J. Stat. Mech. Theor. Exp. 2014(08), P08001 (2014). steelsbaron A. Baronchelli,M. Felici, V. Loreto, E. Caglioti, and L. Steels, Sharp transition towards shared vocabularies in multi-agent systems, J. Stat. Mech. Theor. Exp. 2006(06), P06014 (2006).baron2007 A. Baronchelli, L. Dall'Asta, A. Barrat and V. Loreto, Non-equilibrium phase transition in negotiation dynamics, Phys. Rev. E 76, 051102 (2007).fisher D. A. Huse and D. S. Fisher, Dynamics of droplet fluctuations in pure and random Ising systems, Phys. Rev. B 35, 6841 (1987); I. M. Lifshitz, Kinetics of ordering during second-order phase transitions, Zh. Eksp. Teor. Fiz. 42, 1354 (1962) [Sov. Phys. JETP 15, 939 (1962)].rikvold P. A. Rikvold, H. Tomita, S. Miyashita, and S. W. Sides, Metastable lifetimes in a kinetic Ising model: Dependence on field and system size, Phys. Rev. E 49, 5080 (1994).dallasta2008 L. Dall'Asta and T. Galla, Algebraic coarsening in voter models with intermediate states, J. Phys. A: Math. Theor. 41, 435003 (2008).vazquez F. Vazquez and C. López, Systems with two symmetric absorbing states: Relating the microscopic dynamics with the macroscopic behavior, Phys. Rev. E 78, 061127 (2008).lipweighted D. Lipowska and A. 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http://arxiv.org/abs/1702.07888v2

aucuiwei@scut.edu.cnSchool of Automation Science and Engineering, South China University of Technology, Guangzhou 510641, ChinaWe investigate the quantum parameter estimation in circuit quantum electrodynamics via dispersive measurement. Based on the Metropolis Hastings (MH) algorithm and the Markov chain Monte Carlo (MCMC) integration, a new algorithm is proposed to calculate the Fisher information by the stochastic master equation for unknown parameter estimation. Here, the Fisher information is expressed in the form of log-likehood functions and further approximated by the MCMC integration. Numerical results demonstrate that the single evolution of the Fisher information can probably approach the quantum Fisher information. The same phenomenon is observed in the ensemble evolution in the short time interval.These results demonstrate the effectiveness of the proposed algorithm.03.65.Ta, 06.20.Dk, 85.25.-j Quantum parameter estimation via dispersive measurement in circuit QED Wei Cui December 30, 2023 ======================================================================§ INTRODUCTIONThe problem of accurately estimating unknown parameters in quantum system is of both fundamental and practical importance. According to the parameter estimation theory <cit.>, in classical system the estimation precision is limited by the standard quantum limit (SQL) <cit.>, 1/√(N), where N refers to the number of experiments. In quantum system, Refs. <cit.> showed that with the help of squeezed state technique the parameter estimation accuracy can exceed the SQL, and even approach the Heisenberg limit (HL)<cit.>, 1/ N. The classical Fisher information (FI) is a tool widely used to calculate the parameter estimation accuracy, and the Cramér-Rao bound states that the inverse of the Fisher information is a tight lower bound on the variance of any unbiased estimation parameter <cit.>. By explicitly maximizing the Fisher information over all possible measurement strategies, one can obtain the quantum Fisher information (QFI) <cit.>.Over the past decades, parameter estimation via continuous weak measurement in quantum system caused a wide range of interests <cit.>. Ref. <cit.> showed that weak measurements have a rich structure, based on which more novel strategies for quantum-enhanced parameter estimation can be constructed. Ref. <cit.> experimentally demonstrated a new robust method for precision phase estimation based on quantum weak measurement. <cit.> The stochastic master equation with quantum weak measurement was also derived for quantum parameter estimation <cit.>. Moreover, the likelihood function and the statistical properties of the measurement output were demonstrated to be effective resources for quantum parameter estimation <cit.>. Although much progress has been made in quantum parameter estimation based on continuous weak measurement, how to effectively calculate the Fisher information (or the estimation precision)based on these resources is still with remarkable difficulty. To figure out this problem, one needs to represent the Fisher informationin computable forms and take effective measures to prior-estimate the parameter of interest.A preliminary work <cit.> to calculate the Fisher information based on various weak measurements for linear Gaussian quantum system was reported recently. In this paper, we propose an efficient algorithm to calculate the Fisher information based on the quantum stochastic master equation in circuit quantum electrodynamics (circuit QED) <cit.>.Circuit QED is widely regarded as an excellent platform for quantum estimation and quantum control <cit.>. Dispersive measurement in circuit-QED leads to a diffusion like evolution for the system and the measurement record, including the homodyne gain and the innovation. Due to the randomness of the measurement record, the numerical differentiation approach is used to calculate the derivative of the log-likelihood function, and a series of parameters of interest is randomly generatedby the Metropolis Hastings (MH) algorithm <cit.>.Finally, the calculable Fisher information is approximated by theMarkov chain Monte Carlo (MCMC) integration <cit.>.This paper is organized as follows. In Sec. <ref>, a brief introduction of quantum parameter estimation is presented. In Sec. <ref>, we discuss the dispersive measurement in circuit QED. The reduced stochastic master equation and the measurement record are exhibited in this section. An efficient algorithm to calculate the Fisher information is introduced in Sec. <ref>. Numerical experiment in circuit-QED demonstrates the feasibility and effectiveness of the proposed algorithm. We summarize our conclusion in Sec. <ref>.§ QUANTUM PARAMETER ESTIMATIONSuppose θ is an unknown parameter that needs to be estimated in a quantum system. As we mentioned above, the precision of the unbiased parameter estimation is always indicated by the quantum Cramér-Rao inequality <cit.>, i.e.,⟨( δθ)^2⟩≥1/NI( θ),where I( θ) is the Fisher information of θ, δθ isthe estimation error, and N is the number of measurements.Let D be the measurement output, which is conditioned on the value of the unknown parameter θ. The ability to estimate the unknown parameters depends on the probability of observing the output given the parameters P(D|θ), which can be characterized by the Fisher information, i.e.,I( θ) = E[ ( ∂ln P( D| θ.)/∂θ)^2],where E[·] refers to the expectation value with respect to independent realizations of the measurement results D. Sometimes the probability density P (D|θ) is also defined as a likelihood function. In addition, the theory that tackles the probability distribution of the measurement resource is the same as for the classical problems with stochastic measurement outcomes while the underlying dynamics of the system and P(D|θ) may be dominated by the laws of quantum physics <cit.>.By maximizing I( θ) over all possible quantum measurements on the system, one can obtain the quantum Fisher information (QFI) <cit.>. Simply, if a quantum pure state ρ _θ = | ψ _θ⟩⟨ψ _θ|evolves in a closed quantum system, the quantum Fisher information of the parameter is given by I = 4[ ⟨ψ_θ ' | . -ψ_θ '⟩- | ⟨ψ_θ | . -ψ_θ '⟩|^2],where | ψ_θ '⟩ stands the derivative of | ψ_θ⟩ with respect to the parameter θ.§ DISPERSIVE MEASUREMENT IN CIRCUIT QEDCircuit QED consists of a superconducting qubit and a microwave resonator cavity. The superconducting system can be described by a two-level quantum system with the HamiltonianH =Δ/2σ_x+Ω/2σ _z,where Δ is the electrostatic energy and Ω is the Josephson energy <cit.>.By applying a displacement transformation and tracing over the resonator state, we can eliminate the cavity degrees of freedom and get a reduced stochastic master equation <cit.> with dispersive measurement (ħ=1)dρ̃_t =- i[ H,ρ̃_t]dt + η D[ F ]ρ̃_tdt + √(η)ℳ( ρ̃_t)dY_t,with𝒟[ A ]ρ = AρA^† - 1/2( A^†Aρ+ ρA^†A); ℳ( ρ)= Aρ+ ρA^†.Here ρ̃_̃t̃ is the un-normalised state, F is the measurement operator, and η is the measurement strength with the continuous weak measurement constraint, i.e., η≪1. Also, dY_t is the independent and infinitesimal increment which represents the measurement output.Generally, the unknown parameters may exist in the system Hamiltonian, the dissipation rates, or the measurement strength. In this paper, we mainly focus on studying the estimation precision of single unknown parameter in the Hamiltonian. The measurement process is assumed to be Markovian. Due to the relationship between a normalized state ρ _t and an un-normalized quantum state ρ̃_t, say ρ _t = ρ̃_t/Tr( ρ̃_t) <cit.>, together with Eq. (<ref>), the increment dY_t has the formdY_t = √(η)Tr( ℳ( ρ _t))dt + dW_t,where dW_t is the Wiener increment with zero mean and variance dt. The Eq. (<ref>) describes the quantum fluctuations of the continuous output signal. Define ℒ_t = Tr( ρ̃_t) as alikelihood function. Owing to Eq. (<ref>), the derivative of the likelihood function ℒ_t with respect to time t can be written as <cit.>dℒ_t = Tr( dρ̃_t) = √(η)Tr( ℳ( ρ̃_t))dY_t= √(η)Tr( ℳ( ρ _t))ℒ_tdY_t.Combining Eq. (<ref>) with Eq. (<ref>), we can get the normalized quantum stochastic master equation by means of the multi-dimensional Itô formula (one can refer to Appendix <ref> for details):dρ _t =- i[ H,ρ _t]dt + η D[ F ]ρ _tdt + √(η)ℋ[ F ]ρ _tdW_t,where ℋ[ F ]ρ=M( ρ) - ρTr(M( ρ)). § QUANTUM PARAMETER ESTIMATION IN CIRCUIT QEDIn this section, we propose an efficient algorithm to calculate the Fisher information by the measurement output and the likelihood function in circuit QED.§.§ The algorithm for calculting the Fisher informationBelow, we use l_t to denote the log-likelihood function, i.e., l_t = lnℒ_t <cit.>. From Eq. (<ref>), the derivative of l_t with respect to time t is described bydl_t = dlnℒ_t = dℒ_t/ℒ_t = √(η)Tr( ℳ( ρ _t))dY_t.Therefore, according to Eq. (<ref>), the Fisher information for single parameter estimation can be written asI( θ) = E[ ( dlnℒ_t/dθ)^2] = E[ ( dl_t/dθ)^2].Substituting Eq. (<ref>) into Eq. (<ref>), we can obtain an analytic form of the Fisher information. From the Fisher information Eq. (<ref>), it is easy to find that θ is not an independent variable of the likelihood function. In other words, there do not exist an explicit expression of l_t with respect to θ, which makes the calculation of I(θ) remarkable difficulty. In order to efficiently calculate the Fisher information, we propose a numerical algorithm with the help of the MH algorithm <cit.> and the MCMC integration <cit.>.In the beginning, we set a series of the unknown parameter {θ _i} satisfyingθ _i + 1 = θ _i + dθ ,  i = 0,1,2,…,N_P,where the interval dθ is a small constant. For each θ _i, there exists a log-likelihood function, say l^i _t, corresponding to θ _i. Here, the collection of log-likelihood functions{l_t^0,l_t^1, … ,l_t^N_P} is a set of functions of time with t ∈[ 0,T]. As we can see in Eq. (<ref>), calculating the Fisher information requires to calculate the derivative of l _t with respect to θ firstly. However, the noise induced by the measurement process makes it improper to use the ordinary numerical derivation to compute {dl_t^i/dθ}. To deal with it, it is natural to use the average evolution of the ensemble to eliminate the impact of measurement noise. Since dθ can be infinitesimal, thederivative of l _t with respect to θ can be given by the Newton's backward difference quotient with infinitesimal errors, i.e.,dl^i_t/dθ≈l_t^i - l_t^i-1/θ _i- θ _i-1 = l_t^i - l_t^i-1/dθ,i = 1,2, … ,N_P.Next, we randomly generate a cluster of θ by the MH algorithm (one can refer to the Appendix <ref> for details), whose prior probability distribution is assumed to satisfy a certain distribution. Denote such generated cluster of θbyθ̂= {θ̂_j| j = 1,2,…,N_M.},where N_M is the Monte Carlo number. Note that the number of candidate pointsN_A that used to generate random samples is chosen to be larger than the Monte Carlo number, i.e, N_M≤N_A. In the set of θ̂, the fluctuation of the pre-estimated parameter values is rather small. This process makes the following numerical calculation as close to the analytic result as possible. For simplicity, one may anticipate the initial value of the sequence generating θ̂ to be a constant value. It is easy to choose the closest θ_i_jto each θ̂_j by comparing θ̂ with θ. As a result, {(dl_t^i_1/dθ)^2,(dl_t^i_2/dθ)^2,…,(dl_t^i_N_M/dθ)^2} could be picked out from the collection {(dl_t^1/dθ)^2, …, (dl_t^N_P/dθ)^2} determined by the generated θ̂. Finally, calculatingthe Fisher information means to acquire the expected value E[(dl_t/dθ)^2] from the sample {(dl_t^i_1/dθ)^2, (dl_t^i_2/dθ)^2, …, (dl_t^i_N_M/dθ)^2} owing to Eq. (<ref>). By the Makov chain Monte Carlo integration, see Appendix <ref> for details, the Fisher information can be approximated asE[ ( d l_t/ . -d θ)^2] ≈1/N_M∑_j = 1^N_M( d l_t^i_j/d θ) ^2.As a conclusion, the procedure of calculating the Fisher information is shown in Figure. <ref>.§.§ Numerical simulationsLet Ω in the Hamiltonian (<ref>) be an unknown parameter that requires estimating. We denote the normalized quantumstate ρ_t byρ _t = 1/2( [1 + z x - iy; x + iy1 - z ]),and the initial state is ρ _0 =1/2| ψ _0⟩⟨ψ _0| with | ψ _0⟩=(1 0 )^T, i.e., x(0) = y(0) = 0, z(0) = 1. The other parameters in the stochastic master equation (<ref>) are Δ = 1.73 and η=0.01, and the measurement operator is given by F=σ _y. For convenience, we define τ=Ω t throughout this section. Suppose that an initial reference value of the unknown parameter Ω is set to be 1, then the sequence Ω̂ can be obtained by proceeding the MH algorithm when the stationary distribution and proposal distribution are assumed to satisfy the normal distributions N(0,1) and N(0,dt), respectively. Fig. <ref> shows the evolution of the normalized quantum state with dispersive measurement according to Eq. (<ref>). The output Y_t and the log-likelihood function l_t are also plotted. Based on the proposed algorithm and the stochastic master equation (8), we show the evolution of the Fisher information for quantum parameter estimation in Fig. (3).The blue dash-dotted curve in Fig. 3(a) represents the single evolution of the Fisher information, and the blue dash-dotted curve in Fig. 3(b) is the ensemble evolution with 500 dispersive measurements in circuit QED.The red solid curves in Fig. 3 are the evolution of the quantum Fisher information with the help of the definition, Eq. (3). The quantum Fisher information always represents the upper bound of the Fisher information. From Fig. (3), we find that the single evolution of the Fisher information can probably approach the quantum Fisher information. The same phenomenon is observed in the ensemble evolution in the short time interval. These results demonstrate the effectiveness of the proposed algorithm. Furthermore, we plot the ensemble evolutions of the Fisher information with the proposed algorithm for various measurement operators in Fig. (4). The green dashed, blue dot-dashed and purple dotted curves are the σ_x, σ_y and σ_z measurements in circuit QED, respectively.According to the original definition, quantum Fisher information is the Fisher information that optimized over all possible measurement operators allowed by quantum mechanics. Searching the optimal measurementoperator remains to be further studied. § CONCLUSIONWe discussed the quantum parameter estimation in circuit QED via dispersive measurement and the stochastic master equation. Based on the Metropolis Hastings algorithm and the Markov chain Monte Carlo integration, a new algorithm is proposed to calculate the Fisher information. Numerical results demonstrate that the single evolution of the Fisher information can probably approach the quantum Fisher information. The same phenomenon is observed in the ensemble evolution in the short time interval. Finally, we discussed the ensemble evolutions of the Fisher information with the proposed algorithm for various measurement operators.§ ACKNOWLEDGEMENTSThis work is mainly supported by the National Natural Science Foundation of China under Grant 11404113, and the Guangzhou Key Laboratory of Brain Computer Interaction and Applications under Grant 201509010006.§ APPENDIX §.§ The lemma of the multi-dimensional Itô formulaIn the multi-dimensional Itô formula, it's worth noting that if x(t) were continuously differentiable with respect to time t, then the term 1/2dx^T(t)V_xx(x(t),t)dx(t) would not appear owing to the classical calculus formula for total derivatives. For example, if V( x_1,x_2) is continuously differentiablewith respect to t, e.g., V( x_1,x_2) = x_1( t)x_2(t), then it's derivation should be dV( x_1,x_2) = x_1dx_2 + x_2dx_1 + dx_1dx_2. §.§ Metropolis Hastings algorithm <cit.> In Markov chains, suppose we generate a sequence of random variables X_1,X_2,...,X_n with Markov property, namely the probability of moving to the next state depends only on the present state and not on the previous state:{X_n + 1 = x| X_1 = x_1,...,X_n = x_n.}   = {X_n + 1 = x| X_n = x_n.}.Then, for a given state X_t, the next state X_t+1 does not depend further on the hist of the chain X_1,X_2,...,X_t - 1, but comes from a distribution which only on the current state of the chain X_t. Forany time instant t, if the next state is the first sample reference point Y obeying distribution q( ·| X _t.) which is called the transition kernel of the chain, then obviously it depends on the current state X_t. In generally,q( ·| X _t.) may be a multidimensional normal distribution with mean X, so the candidate point Y is accepted with probability α( X _t,Y) whereα( X ,Y) = min( 1,π( Y )q( X | Y .)/π( X)q( Y| X.)).Here, π( A ) stands a function only depends on A. If the candidate point is accepted, the next state becomes X_t+1=Y. If the candidate point is rejected, it means that the chain does not move, the next state will be X_t+1=X. We illustrate this sampling process with a simple example, see Fig. <ref>. Here, the initial value is X(1) = -10. Fig. <ref>(a) represents the stationary distribution N(0,0.1). In Fig. <ref>(b), we plot 500 iterations from Metropolis Hastings algorithm with the stationary distribution N(0,1) and proposal distribution N(0,0.1). Obviously, sampling data selecting from the latter part would be better.§.§ Makov Chain Monte Carlo integration <cit.>In Markov chain, the Monte Carlo integration can be used to evaluate E[ f( X )] by drawing samples {X _1,...X _n} from the Metropolis Hastings algorithm. HereE[ f( X)] ≈1/n∑_i = 1^n f( X _i),means that the population mean of f(X ) is approximated by the sample mean. When the sample X_t are independent, the law of large numbers ensures that the approximation can be made as accurate as desired by increasing the sample. Note that here n is not the total amount of samples by Metropolis Hastings algorithm but the length of drawing samples.10 Helstrom1976 C. W. Helstrom, Quantum detection and estimation theory, J. Statist. 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http://arxiv.org/abs/1702.08089v3

𝐱 ŁL Xin Jin^1, Peng Yuan^1,2, Xiaodong Li^1,*, Chenggen Song^1, Shiming Ge^3, Geng Zhao^1, Yingya Chen^1 This work is partially supported by the National Natural Science Foundation of China (Grant NO.61402021, 61402023, 61640216), the Science and Technology Project of the State Archives Administrator (Grant NO. 2015-B-10), the open funding project of State Key Laboratory of Virtual Reality Technology and Systems, Beihang University (Grant NO. BUAA-VR-16KF-09), and the Fundamental Research Funds for the Central Universities (NO. 2016LG03, 2016LG04).^1Beijing Electronic Science and Technology Institute, Beijing 100070, China^2Xidian University, Xi'an 710071, China^3Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100095, China Corresponding Author: lxd@besti.edu.cn Efficient Privacy Preserving Viola-Jones Type Object Detection via Random Base Image Representation Byung-Woo HongJa-Keoung KooChung-Ang University, Korea{hong,jakeoung}@cau.ac.krStefano SoattoUniversity of California Los Angeles, U.S.A.soatto@cs.ucla.edu ====================================================================================================================================================================================== A cloud server spent a lot of time, energy and money to train a Viola-Jones type object detector <cit.> with high accuracy. Clients can upload their photos to the cloud server to find objects. However, the client does not want the leakage of the content of his/her photos. In the meanwhile, the cloud server is also reluctant to leak any parameters of the trained object detectors. 10 years ago, Avidan & Butman introduced Blind Vision, which is a method for securely evaluating a Viola-Jones type object detector. Blind Vision uses standard cryptographic tools and is painfully slow to compute, taking a couple of hours to scan a single image. The purpose of this work is to explore an efficient method that can speed up the process. We propose the Random Base Image (RBI) Representation. The original image is divided into random base images. Only the base images are submitted randomly to the cloud server. Thus, the content of the image can not be leaked. In the meanwhile, a random vector and the secure Millionaire protocol are leveraged to protect the parameters of the trained object detector. The RBI makes the integral-image enable again for the great acceleration. The experimental results reveal that our method can retain the detection accuracy of that of the plain vision algorithm and is significantly faster than the traditional blind vision, with only a very low probability of the information leakage theoretically.Blind Vision, Random Base Image, Privacy Preserving, Object Detection§ INTRODUCTION Recently, widespread smart phones with cameras enable people to shot images and videos nearly anytime and anywhere. Millions of surveillance cameras including the driving recorders captures images and videos every second. All these conveniences devices are producing the large-scale visual media data, which is considered as the biggest big data. Due to the limited storage space of these terminal devices, large-scale visual media data is being uploaded and stored in the cloud servers. Not only the storage, but also the processing of large-scale visual media data are being outsourced to the cloud servers. The cloud servers have some strong algorithms such as face/object detection, face/object recognition, intelligent video surveillance. Nowadays, people can easily find all the faces in their photos stored in the cloud servers using the powerful face detection algorithms maintained by the cloud servers. However, the cloud servers are always third party entities. Thus the privacy of the users' visual media data may be leaked to the public or unauthorized parties.In the meanwhile, the powerful cloud services for visual media analysis and processing need a lot of money, data and time from the cloud server producers. The cloud servers are also reluctant to leak any parameters of the trained models or some protected details of their algorithms with copyrights.Thus, the privacy of both the content of the visual media from the clients and the parameters of the vision algorithms from the cloud servers should be protected. 10 years ago, Avidan & Butman introduced Blind Vision <cit.>, which is a method for securely evaluating a Viola-Jones type face detector. Blind Vision uses standard cryptographic tools and is painfully slow to compute, taking a couple of hours to scan a single image. After that, rich literatures have been proposed in this field. The cryptographic tools such as secret sharing (SS) <cit.>, security multi-party computation (SMC) <cit.>, homomorphic encryption (HE) <cit.>, garbed circuit (GC) <cit.>, Chaotic System (CS) <cit.> are heavily used. Plenty of computer vision applications have been modified to the privacy preserving or secure versions such as private face detection <cit.>, face recognition <cit.>, content based image retrieval <cit.>, visual media search on public datasets <cit.>, intelligent video surveillance <cit.>. However, most of these work rely heavily on cryptographic tools, which are painfully slow to compute or need bit by bit interaction between the clients and the cloud servers. In this paper, we revisit the Blind Vision <cit.> and attempt to make the blind vision towards cryptographic-free, without losing the security properties. We userandomness and only a little cryptographic operations to protect the visual media data of the clients and the parameters of the trained models in the cloud servers.A novel image representation called Random Base Image(RBI) representation is proposed. In this work, we also investigate the object detection in the cloud. We apply our RBI to the famous Viola and Jones object detection method and propose a novel blind object detection method.We separate an image into random base images. The weight of each base image is only known by the client. The base images are sent randomly to the cloud server. The cloud server cannot recover anything from the random base images. A random vectorand the secure Millionaire protocol <cit.> are leveraged to protect the parameters of the trained object detector. The RBI makes the integral-image enable again for the great acceleration. The experimental results reveal that our method is significantly faster than the traditional blind vision, with only a very low probability of the information leakage theoretically. § SECURE OBJECT DETECTION In this section we develop a secure object detector with the random base image representation.§.§ Notations Our scenario and the notations are the same as that of traditional Blind Vision <cit.>, as show in Figure <ref>. Denote some L dimensions finite field F that is large enough to represent all the intermediate results. Denote by X the image that Alice owns. A particular detection window within the image X will be denoted by x ∈ F^L and x will be treated in vector form. Bob owns a strong classifier of the form H(𝐱) = sign(∑_n=1^Nh_n(𝐱)),where h_n(𝐱) is a threshold function of the form h_n(𝐱) = α_n   𝐱^T𝐲_𝐧 > θ_n β_n  otherwise,and y_n ∈ F^L is the hyperplane of the threshold function h_n(𝐱). The parameters α_n ∈ F, β_n ∈ F and θ_n ∈ F of h_n(𝐱) are determined during training; N is the number of weak classifiers used. §.§ The Random Base Image RepresentationThe core idea of our RBI is to separate the original image into some random base images with fixed weights. The original image can be recovered by all the base images. The sparse representation can be considered as the one has such ability. However, they need another image dataset for learning the base images. Further more, there could be reconstruction error. Thus, we fix the weights and randomize the base images themselves.The detection window 𝐱 can be represented as: 𝐱 = ∑_i=0^M-1w_i𝐁_i,where 𝐁_i is the base image with weight w_i. As is shown in Figure <ref>, each base image has a fixed weight. The base image itself is randomly determined. The number of the base image is set to M=256. Thus, each base image can be a binary image, which is easy for network transfer and fast to compute. In addition, there are 256! permutation of the base image which is not easy to guess. The process of the RBI generation is described in Algorithm <ref>. §.§ Secure Object Detection with RBI §.§.§ Secure Object Classifier ProtocolThe core of our method is the secure object classifier protocol as is described in Algorithm <ref> and Figure <ref>. For secure object detection, Alice first divides the test image 𝐗 into Q detection windows {𝐱_1,𝐱_2,...𝐱_Q}. Then the detection windows are randomly sent to Bob as the inputs of the secure face classifier protocol one by one. Using the Algorithm <ref>, Alice and Bob know which detection windows are the target objects. Because the detection windows are randomly sent to Bob, only Alice learns the location of all the detected faces in the original image. Bob does not learn the contents including where the faces are in the image of Alice. Alice learns nothing about the parameters of the face detector of Bob. The body of Algorithm <ref> is described as follows: * (1): Alice factorizes the detection window x into M random base images 𝐁={𝐁_0, 𝐁_1,..., 𝐁_M-1} with weight w={w_0,w_1,...,w_M-1}={0,1,...,M-1} through Algorithm <ref>. * (2): Alice randomly shuffles the weight w to w'. The random base images 𝐁 are permuted with the same order of that of w'={w'_0,w'_1,...,w'_M-1} to 𝐁'={𝐁'_0, 𝐁'_1,..., 𝐁'_M-1}, which is sent to Bob.* (3): In one cascade, Bob has N weak classifiers with parameter vectors 𝐲={𝐲_0,𝐲_1,...𝐲_N-1}. Bob randomly add K fake weak classifiers and set their parameters α and β to zero. Bob randomly shuffles the N+K true and fake weak classifiers to form 𝐲'={𝐲'_0,𝐲'_1,...𝐲'_N+K-1}. Then, Bob generates N+K random positive numbers s={s_0,s_2,...,s_N+K-1}. For each parameter vector 𝐲'_n ∈𝐲'. Bob and Alice repeat the following 3 steps.* (3.1): Bob computes the feature responses for all the base image 𝐁'_m in 𝐁' by F_m(n)=𝐁_m^'T𝐲'_n, m=0,1,...,M-1. All the M responses of base images 𝐁' on each parameter vector 𝐲'_n are sent back to Alice.* (3.2): Alice computes the feature responses of the detection window 𝐱 by F(n) = ∑_m=0^M-1F_m(n)w'_m. * (3.3): Alice and Bob use the secure Millionaire protocol <cit.> to determine which number is larger: F(n) or θ_n. Bob send α_n+s_n or β_n+s_n to Alice. Alice store it as c_n.* (4): Alice and Bob use the secure Millionaire protocol <cit.> to determine which number is larger: ∑_n=1^N+Kc_n or ∑_n=1^N+Ks_n. If Alice has a larger number then x is positively classified, otherwise x is negatively classified. §.§.§ SecurityThe protocol protects the security of both parties. The protocol protects the contents of the image from Alice and the parameters of the face detector from Bob. We analyse the security of Algorithm <ref> in the following paragraph. * From Alice to Bob * In step 2, Alice send randomly shuffled base images to Bob. Bob only knows the randomly generated base images and do not know the weight of each base image. The probability of guessing out the right permutation is 1/M!. Even Bob guesses out the right permutation, he does not know the weight of each base image. Thus, it is almost impossible for Bob to recover the detection window of Alice. * In the 3th sub-step of step 3 and the step 4. Alice and Bob engage in secure Millionaire protocol <cit.>. so Bob can learn nothing about Alice’s data. * From Bob to Alice * In the 1st sub-step of step 3, Alice can not learn the number of the weak classifiers N or the true filters from the received feature responses. The true filters are obfuscated by the fake filters. * In the 3rd sub-step of step 3, Alice and Bob engage in a secure Millionaire protocol so Alice only learns if F(n)>θ_n. She can not learn anything about the parameter θ_n. Moreover, at the end of the Millionaire protocol Alice learns either α_n+s_n or β_n+s_n. In both cases, the real parameter (α_n or β_n) is obfuscated by the random number s_n. * In step 4, Alice and Bob use the secure Millionaire protocol to determine which number is larger: ∑_n=1^Nc_n or ∑_n=1^Ns_n. If Alice has a larger number then x is positively classified, otherwise x is negatively classified. * Multiple Cloud Servers * The M random base images can be also sent to multiple cloud server with the same object detector to increase security. §.§.§ Complexity and EfficiencyThe complexity of the protocol is O(M(N+K)L), where M is the number of the base images. N and K are the numbers of the true and fake weak classifiers, respectively. L is the dimensionality of the detection window x.Unlike the traditional Blind Vision <cit.>, in which the OT operation is used extensively, the proposed method only use OT operation to compare 2 numbers. In the secure dot-product protocol, each pixel of each detection window uses a OT_1^256 operation, which needs 1 RSA encryption and 256 RSA decryption with 128-bit long encryption keys. We leverage our M random images, whose computation is much faster than the RSA encryption and decryption operations.In addition, in the traditional Blind Vision <cit.>, they convert the integral-image representation to regular dot-product operation, a step that clearly slows down their implementation as they no longer take advantage of the integral-image representation. In our RBI based protocol, the integral-image representation is enabled again, which accelerates the computation obviously. § EXPERIMENTSWe convert the Viola-Jones type object detector <cit.> to our secure object detector. We implement our RBI based object detector using Microsoft Visual Studio 2012 and OpenCV 2.4.3/10. [http://opencv.org/] package for computer vision in a 64 bits Windows 7 operating system. The hardware configuration is 3.5GHz AMD A10 Pro-7800 R7 CPU with 12 compute Cores and 8GB Memory. The face detector is from the OpenCV 2.4.3 package and consists of a cascade of 22 rejectors, where each rejector is of the form presented in Eq. <ref>. The first rejector consists of 3 weak classifiers. The most complicated rejector consists of 213 weak classifiers. There is a total of 2135 weak classifiers. We also test the nose detector, the eye detector and the full body detector from OpenCV 2.4.10.§.§ The Detection Accuracy We test our secure face detector in 3 face detection datasets: The Face Detection Dataset (FDDB) <cit.>, The Face96 Dataset <cit.>, and The FEI Face Database <cit.>.We randomly select 100 face images from each of the 3 datasets. The detection accuracy (88.46%) of our secure face detector is the same as that of the OpenCV 2.4.3 face detector (88.46%). The nose and the eye detectors are tested on the FDDB dataset <cit.>. The full body detector is tested on the INRIA Person dataset <cit.>. We randomly select 100 images from each of the 2 datasets. The detection accuracy of our secure object detectors is the same as that of the OpenCV 2.4.10 nose, eye and full body detectors.§.§ Comparison with Other Methods We compare our method with theViola-Jones method implemented by the OpenCV package and the method of the traditional Blind Vision <cit.>. 50 test images with size of 100 × 100 are randomly selected from each of the 3 datasets. The average running time is shown in Table <ref>. All the methods are running in client and server mode. For the Viola-Jones, Alice send the original image to Bob. Then, Bob runs the Viola-Jones method and return the detected windows to Alice. Our method is slower than the Viola-Jones method, which is running on plain images without protecting any privacy. According to the traditional Blind Vision method <cit.>, the time-consuming OT operation is heavily used and the integral-image representation is disabled. Thus, they have to take a couple of hours to scan a single image, which is painfully slow. Although in our method, the only information that Bob learns is that how many faces are in the image of Alice, our cryptographic-free method is significantly faster than the previous work towards practical usage of blind vision applications. In addition, we compare our method with the Viola-Jones method implemented by the OpenCV package and the method of the traditional Blind Vision <cit.>. 50 test images with size of 100 × 100 are randomly selected from each of the FDDB and the INRIA Person datasets. The average running times are shown in the last 3 rows of Table <ref>. All the methods are running in client and server mode.§ CONCLUSIONS AND DISCUSSIONSWe propose a novel random base image representation (RBI) for efficient object detection applications. The traditional blind vision method applies secure multi-party techniques to vision algorithm. Their method reveals no information to either party at the expanse of heavy computation load. Our method is an attempt towards cryptographic-free. Alice learns nothing about the parameters of the face detector of Bob. Bob does not know the contents of the image of Alice. The only information may be leaked is that Bob have a probability 1/M! to guess out the right permutation of the base images. This is just a theoretical event. Even Bob guesses out the right permutation, he does not know the weight of each base image.Thus it is almost impossible for Bob to learn the information of the detection window of Alice. Because the heaviest cost of OT operation in the secure dot-production of <cit.> is avoided by our RBI based dot-production, the Millionaire version protocol of ours need much less time than the traditional blind vision protocol does. There are several extensions to this work. First is the need to accelerate the secure blind vision to practical use, i.e. to reduce the time cost to near that of the vision algorithm without security consideration. Second is to make both the training and the test blind. This will make the client users to upload more visual data to the cloud without worrying about the privacy leakage.IEEEbib

http://arxiv.org/abs/1702.08318v2

Department of Aerospace Engineering, Texas A&M University, College Station, Texas 77843,USA We consider transition to strongturbulence in an infinite fluid stirredby a gaussian random force.The transition is definedas a first appearance of anomalous scaling ofnormalizedmomentsof velocity derivatives (dissipation rates) emerging fromthe low-Reynolds-number Gaussian background.It is shownthat due to multi-scaling,strongly intermittentrare eventscan be quantitatively describedin termsofan infinite number of different “Reynolds numbers”reflecting multitude of anomalous scaling exponents. The theoretically predictedtransition disappears at R_λ≤3.The developed theory, is in a quantitative agreement with the outcome of large-scale numerical simulations.Emergence of Multi-Scaling in a Random-Force Stirred Fluid. Diego DonzisDecember 30, 2023 =============================================================PACS numbers 47.27 Introduction. If an infinitefluid is stirred by a gaussian random force supported in a narrow interval of the wave-numbers k≈ 2π/L, then a very weak forcingleads to generation ofa random, close-to-gaussian,velocity field.In this flow themean velocity u=0 and one can introduce the large-scale Reynolds number Re=u_rmsL/ν where the root-mean-square velocityu_rms=√(u^2). Increasing the forcingamplitude or decrease of viscosity result in a stronglynon-gaussian random flow with moments of velocity derivatives obeying the so-called anomalous scaling.This means that the moments (∂_xu_x)^2n/(∂_xu_x)^2^n∝ Re^ρ_2n where the exponents ρ_n are,on the first glance,unrelated“strange” numbers.In this paper we investigatethetransition between these two different random/chaoticflow regimes. First, we discuss some general aspects of thetraditional problem of hydrodynamic stability and transition to turbulence.Fluid flow can be described by the Navier-Stokes equations subject to boundary and initial conditions (the density is taken ρ=1 without loss of generality):∂_t u+ u·∇ u=-∇ p +ν∇^2 u +f and ∇· u=0. The characteristicvelocity and length scales u and L,used for makingthe Navier-Stokes equationsdimensionless, are somewhat arbitrary.In the problem ofa flow past cylinder it is natural to choose f=0, u=U, and L=D where U and D are free-stream velocity and cylinder diameter, respectively.In a pipe/channel flow u=U=1/H∫_0^Hu(y)dy∝ u_centerline is the mean velocity averaged overcross-section and L=H is a half-width of the channel.In a fully turbulent flow in an infinite fluid one typically takesu=v_rms=√(v^2) and L equal to the integral scale of turbulence. Some other definitions will be discussed below.Depending ona setup,a flow can be generated by pressure/temperature gradients,gravity,rotation, electro-magnetic fields etc representedas forcing functions on the right side of(1). If viscosity ν≥ν_tr and the corresponding Reynolds number Re=uL/ν≤ Re_tr=uL/ν_tr, the solution to (1) driven by the regular (not random) forcing f is laminar and regular. As examples, we may recall parabolic velocity profileu(y) in pipe/channel flows with prescribed pressure difference betweeninlet and outlet.In this case the no-slip boundary conditions are responsible for generation of the rate-of-strain S_ij=(∂_iu_j+∂_ju_i)/2. Another important example isthe so called Kolmogorov flow in an infinite fluid driven by the forcing function f=U(0,0,cos kx).In Benard convection the relevant regular patterns are rolls appearing asa result of instability ofsolution to the conductivity equation.Thus, the remarkably successful science of transition to turbulence deals mainly with various aspects ofnon-equilibrium order-disorderor laminar-to-turbulent transition.In this paper we consider a completely different class of flows. In general,theunforced NS equations, being a very important and interesting object,do not fully describe thephysical reality which includes Brownian motion,light scattering, random wall roughness, uncertain inlet conditions, stirring by “random swimmers” in biofluids etc. For example, a fluidin thermodynamic equilibrium satisfiesthe fluctuation-dissipation theorem stating that thereexist an exact relation between viscosity νin (1)and a random noise f which is a Gaussian forcedefined by the correlation function [1]:f_i(k,ω)f_j( k',ω')= (2π)^d+1D_0d(k)P_ij( k)δ(ω+ω')δ( k+k') where the projection operator is: P_ij( k)=δ_ij-k_ik_j/k^2. Inan equilibrium fluidthermal fluctuations, responsible for Brownian motion are generated by the forcing (2) with D_0d(k)= k_BT ν/ρk^2≡ D_0k^2.It is clear that, in general,the function d(k) in (2) depends onthe physics ofaflow.The random-force-driven NS equation can be written in the Fourier space: u_l( k,ω)=G^0f_l( k,ω)-i/2G^0 P_lmn∫ u_m(q,Ω)u_n(k-q,ω-Ω)d kdΩ where G^0=(-iω+ν k^2)^-1,P_lmn( k)=k_nP_lm( k)+k_mP_ln( k)and,introducing the zero-order solution u_0=G^0 f∝√(D_0), so that u=G^0 f+ v,one derives the equation for perturbation v:v_l(k̂)=-i/2G^0(k̂) P_lmn( k)∫ v_m(q̂)v_n(k̂-q̂)dq̂-i/2G^0(k̂) P_lmn( k)∫[v_m(q̂)G^0(k̂-q̂)f_n(k̂-q̂)+ G^0(q̂)f_m(q̂)v_n(k̂-q̂)]dq̂-i/2G^0(k̂) P_lmn( k)∫ G^0(q̂f_m(q̂)G^0(k̂-q̂)f_n(k̂-q̂)dq̂ where the 4-vector k̂=( k,ω).If the equation (3)-(4) is drivenby a regular force or boundary and/orinitial conditions, then at low-Reynolds number(Re)it typically describesaregular (laminar) flow field u_0withv=0.With increase of theReynolds number Re≥ Re_inst,this zero-order solution can become unstable, meaning that initially-introduced small perturbationsv grow in time.Further increase of Re leadsfirst toweak interactions between the modes describing the “gas” of these perturbations and, eventually,when Re-Re_inst/Re_inst≫ 1 mode coupling described by equation (4) becomes very strong.This regime we call “fully developed” or strong turbulence. The problem of hydrodynamic stability is notoriously difficult and we know very littleabout structure of solution for perturbations in the non-universal range Re≈Re_inst. Here we are interested in a simplified problem of a flow generated by agaussian random force (2) witha well -understood zero-order solution u_0=G^0 f which is nota result of an instability of a regularlaminar flow butis prescribed by a choiceof a random force (2). The advantages of this formulation are clear from (4)describingthe dynamics of perturbationv driven byan induced forcing given by theO(f^2)∝ D_0 last term in (4).It is easy to see [1],[9] that dimensionless expansion parameter, related to a Reynolds number (see below),is Γ^2_0=D_0L^4/ν^3Δ where Δ=∫ d(k)d k.and, since we keep L=O(1),ν=O(1) and Δ=O(1), the variableforcing amplitude D_0 can be treated as a dimensionless expansion parameter.Thus,as D_0→ 0,allcontributions to the right side of (4) can be neglectedand,if f stands forthe gaussian random function, then the lowest-order solution u_0is a gaussian field.. However, there always exist low-probabilityrare events with | v|≥ | u_0| responsible for the strongly non-gaussian tails of the PDF.Thus, in this flow gaussian velocity fluctuations coexist with the low-probability powerful events where substantial fraction of kinetic energy is dissipated. Ateven higherReynolds numbers (see below) the non-linearity in (4)dominates the entire field.Thiscomplicated dynamics has been observed in experiments on a channel flow with rough (“noisy”) walls [2]. This regime is characterized bythe generation of velocity fluctuations v( k,t) in the wave-numberrange k>2π/Lwhere the “bare” forcing f( k)=0,whichisthe hallmark of turbulence.The above example shows that at least in some range of the Reynolds numberlow and high-order moments may describevery differentphysical phenomena.The transition between these two chaotic/randomstates of a fluidis a topic we are interested in this paper.Twocasesare ofa special interest. Inthelow Reynolds number regime(below transition),when R_λ=√(5/3 Eν)u_rms^2<R^tr_λ,the integral (L), dissipation (η)and Taylor (λ) length scales are of the same order.Therefore, (∂_x u_x)_rms=(u(x+η)-u(x))_rms/η≈ (u(x+L)-u(x))_rms/Land, since we are interested ininstability of a gaussian flow,the moments M_n^<=(∂_xv_x )^2n/(∂_xv_x)^2^n=(2n-1)!! independent on the Reynolds number.In this case, since the 2n^th-order momentcan be expressed in powers of the variance,this means that (∂_xv_x)_rms is a single parameter(derivative scale) representing statistical properties the flow in this regime. This is not always the case. The rms velocity derivative in high Reynolds number turbulent flows, (∂_xv_x)_rms=√((∂_xv_x)^2)is only one of an infinite number ofindependent parameters needed to describe the field and in the vicinity of transition Re≥ Re^tr : M^>_n= (∂_xv_x )^2n/(∂_xv_x)^2^n= (2n-1)!!C_nRe^ρ_n≈ (2n-1)!!R_λ^ρ_nwhere R_λ>R^tr_λ andthe proportionality coefficientsC_n=O(1)[3], [4].Below, this anomalous state of afluid we call strong turbulence as opposed to the close-to-gaussian low Reynolds number flow field, considered above. In atransitional,low Reynolds number,flowwe are interested in here, the forcing, Taylor and dissipation scales are of the same order L≈η≈λ.The Reynolds numberbased on the Taylor length-scale is thus: R_λ≡R_λ,1= √(5/ 3 Eν)v_rms^2≈√(5L^4/3 Eν)(∂_xv_x)^2 The physical meaning of this parameter can be seen readily: multiplyand divide (5) by ν and by the dissipation scale η^2. This givesR_λ∝L^2/η^2×η^2√( E/ν^3)≈L^2/η^2where η^4 E/ν^3=O(1).The effective Reynolds number O(L^2/η^2), which is the measure ofthe spread of the inertial range in k-space, is a coupling constant, familiar from dynamic renormalization group applications to randomly stirred fluids.To describe strong turbulence, one must introducean infinite number of “Reynolds” numbers R_λ,n=√(5L^4/ 3 Eν)(∂_xv_x))^2n^1/n∝ R_λ^ρ_2n/n∝L^2/η^2 E^n^1/n/ E where close to transition pointswhere η≈ L we set R_λ≡ R_λ,1≈ Re. The expressions for exponents ρ_2nρ_2n=2n+ξ_4n/ξ_4n-ξ_4n+1-1; ξ_n=0.383n/1+n/20 derived in the “mean-field approximation”in[4]-[5],agree extremelywell with all available experimental and numerical data (seeRefs.[5]-[8]). Theoretical predictions of anomalous exponents in a random-force-stirred fluid are compared with the results of numerical simulations[6] on a top panel of Figure 1.Note that normalized moments of dissipation rate M_n( E) are simply M_2n in the present formulation.The sameexponents have been observed in a channelflow [7] and Benard convection [8], indicating universality of small-scale features in turbulent flows. Transition between gaussian andanomalous flows.In this papertransition to turbulence is identified with first appearance of non-gaussian anomalousfluctuations of velocity derivatives. The concept is illustrated on the bottompanel of Fig.1,wheremoments of velocity derivatives from well resolved numerical simulations (described below)are plottedagainst Reynolds numbers R_λ≡ R_λ,1≥ 2..We can see that transition points of different moments, expressed in terms of R_λ≡ R_λ,1,are different and below we denote them R^tr_λ,1(n). It is important that transition point forthe lowestordermoment M_n with n≈ 1has been found at R_λ≡ R_λ,1≈ 9 firstdiscoveredin Ref.[5] and analytically derived in [9]-[10].This result can be explained asfollows. In accord with the widely accepted methodology, consider the R_λ,1≡ R_λ-dependence ofthe normalized n^thderivative momentM_n in a flow driven by a relativelyweak forcef and large viscosity ν.Then, gradually decreasing viscosity,one reaches the critical magnitude ν=ν_trcorresponding to R_λ^tr(n)=R^-_λ(n)which is the upper limitfor gaussianity of the n^th moment. Then,considerthe same flow but at avery large Reynolds number(small viscosity). In this,strongly turbulent case, the large-scale low-order moment, M_4 for example,are dominated by ahugeturbulent viscosity ν_T∝ E^1/3L^4/3, the largest effective viscosity , accounting for velocity fluctuations at the scales r<L[1].The effective Reynolds number, corresponding to the integral scale L, is R_λ^+∝√(L^4/( Eν_T(L)))(∂_xu)_rms^2.This way one reaches the smallest possible Reynolds number R_λ≈ 9 of strongly turbulent (anomalous) flow (see Fig.1).If, in accord with experimental and numerical data, we assume that transition is smooth and at a transition point the Reynolds number is a continuous function meaning thatR_λ^-=R_λ^+, where R_λ^± standfor the magnitudes just above and below transition, we can write:R_λ^tr(4)=√(5/3 Eν_tr) v_rms^2=√(5/3 Eν_T(L)) v_rms^2 where effective viscosityof turbulence at the largest (integral) scale calculated in Refs. [9]-[11], is given byν_T≡ν(L)≈ 0.084 K^2/ E; where K=v_rms^2/2 stands for kinetic energy of velocity fluctuations.Substituting this into the previous relation gives: R_λ^tr(4)=√(5/3 Eν) v_rms^2=√(20/(3× 0.084)) =8.98≈ 9.extremely close to the outcome of numerical simulations.The coefficient C_μ=0.084, derived in [9]-[11] is to be compared with C_μ=0.09 widely used in engineering turbulent modeling forhalf a century [12].It follows from therelations (5)-(6): R^tr_λ(n)≡ R^tr_λ,1(n)=(R^tr_λ,n)^n/ρ_2n The Reynolds number dependence of normalized momentsof velocity derivative is shown on Fig.1.The data in the bottom panelof Fig.1 was generated froma new set of simulationsat very low Reynolds numbers. As in [6], numerical solutions to Navier-Stokes equationsare obtained from Fourier pseudo-spectral calculations with second-order Runge-Kutta integration in time.The turbulence is forced numerically at the large scales, using a combination of independent Ornstein-Uhlenbeck processes with Gaussian statistics and finite-time correlation.Only low wavenumbers modes within a sphere of radius k_F≈ 2 in wavenumber space are forced. In order to obtain different Reynolds numbers,viscosity is changed accordingly while the forcing atlarge scales remains constant.In this approach, thus, large scales, and thus the energy flux, remain statistically similar. Resolution is at least k_maxη≈ 3 at thehighest Reynolds number which was found to produce convergedresults at the Reynolds numbers investigated here.Velocity fields are saved at regular time intervals that aresufficiently far apart (of the order of an eddy-turnover time) toensure statistical independence between them. For each field velocity gradients moments are computed and averaged over space.Ensemble average is computed across these snapshots in time and are used to compute confidence intervals also shown in Fig.1.The intersection points of curves describing gaussian moments (horizontal dashed lines) andthose corresponding to thefully- turbulent anomalous scalinggive transitionalR_λ^tr(n)for each moment.These are comparedto the theoretical prediction of Eq.(10) with R_λ,n^tr≈ 8.5 in Fig.2. This result can be understood as follows:in accord with theoretical predictions thetransitionalReynolds number R^tr_λ,n≈ const≈ 9in each statistical realization.If R_λ,1<R^tr_λ≈ 9,the transition is triggered by the low -probability violent velocity fluctuations(∂_xv_x)^n^1/n>(∂_xx_x)_rms comingfrom the tails of probability density.It is also interesting to evaluate the limiting, smallest,transitional Reynolds numberfollowing(10)in the limit n→∞. The relations (5)-(6),(10)give R_λ= R^tr_λ,1→ 2.92.Evaluated on a popular model ξ_n= n/9+2(1-(2/3)^n/3) [13], one readily derivesR^tr_λ,1→ 3.81.According to both models,in a flow with R_λ≤ 3, no transition to strong turbulence defined by anomalous scaling of moments of velocity derivativesexist.Summary and conclusion. In this paper aproblem of transition between two different randomstates has been studied bothanalytically and numerically.It has been shown thatwhile the gaussian state can be described in terms of the Reynolds number based on thevariance of probability density,the description of theintermittent state of strongturbulence requiresan infinite number of "Reynolds numbers"R_λ,n reflectingthe multitude of anomalous scaling exponents of different-order moments(n)of velocity derivatives.This novelconcept enables one to accountfor both typicaland violent extremeeventsresponsible for emergence of anomalous scalingin the “sub-critical” state when the widely used Reynolds number R_λ,1<R^tr_λ is small. It has also been demonstratedthat,in accord with the theory,the criticalR^tr_λ,n≈ 9 is independent of n.The proposed theory is in a good quantitative agreement with theresults of large-scaledirect numerical simulations presented above. The role of turbulent bursts inlow Reynolds number flows in various physico-chemical processesand the problem ofuniversality will be discussed infuture communications.We are grateful toH. Chen,A.Polyakov,D. Ruelle, J. Schumacher,I. Staroselsky, Ya.G. Sinai, K.R. Sreenivasanand M.Vergassolafor many stimulating and informative discussions.DD acknowledges support from NSF.1. L.D.Landau & E.M. Lifshits, “Fluid Mechanics”, Pergamon, New York, 1982; D. Forster, D. Nelson &M.J. Stephen, Phys.Rev.A 16, 732 (1977); 2.C.Lissandrello,K.L.Ekinci & V.Yakhot, J. Fluid Mech, 778, R3 (2015);3.T. Gotoh & T. Nakano , J. Stat Phys. 113,855 (2003); 4.V.Yakhot, J.Fluid Mech.495, 135 (2003);5. J.Schumacher, K.R. Sreenivasan & V. Yakhot, New J. of Phys.9, 89 (2007); 6. D.A. Donzis, P.K. Yeung &K.R. Sreenivasan, Phys.Fluids 20, 045108 (2008);7.P.E. Hamlington, D. Krasnov, T. Boeck &J. Schumacher,J. Fluid. Mech. 701, 419-429 (2012); 8. J. Schumacher, J. D. Scheel, D. Krasnov, D. A. Donzis, V. Yakhot & K. R. Sreenivasan, Proc. Natl. Acad. Sci. USA 111, 10961-10965 (2014)[9]. V. Yakhot & L. Smith,J. Sci.Comp. 7, 35 (1992);10.V.Yakhot,, Phys.Rev.E, 90, 043019 (2014); 11 . V. Yakhot, S.A. Orszag, T. Gatski, S. Thangam & C.Speciale, Phys. Fluids A4, 1510(1992);12.B.E. Launder,and D.B. Spalding. Mathematical Models of Turbulence, Academic Press, New York (1972);B.E. Launder and D.B. Spaulding, Computer Methods in Applied Mechanics and engineering, 3, 269 (1974).13.Z.S.She & E. Leveque, Phys.Rev.Lett. 72, 336 (1994).

http://arxiv.org/abs/1702.08468v1

^1 Lomonosov Moscow State University Skobeltsyn Institute of Nuclear Physics (MSU SINP), Leninskie gory 1(2), GSP-1, Moscow, 119991, Russia ^2 Joint Institute for Nuclear Research (JINR), Joliot-Curie 6, Dubna, Moscow region, 141980, Russia^*evgeny.post@gmail.com This work is a methodical study of another option of the hybrid method originally aimed at gamma/hadron separation in the TAIGA experiment. In the present paper this technique was performed to distinguish between different mass groups of cosmic rays in the energy range 200 TeV – 500 TeV. The study was based on simulation data of TAIGA prototype and included analysis of geometrical form of images produced by different nuclei in the IACT simulation as well as shower core parameters reconstructed using timing array simulation. We show that the hybrid method can be sufficiently effective to precisely distinguish between mass groups of cosmic rays. § INTRODUCTIONThe measurement of the mass composition of cosmic rays could be the key to understanding their origin, because the change of dominant sources (mostly supernova remnants of different types) in the given energy range should lead to the corresponding change of mass composition <cit.>. The energy range of hundreds of TeV is of special interest, because the most frequent in our Galaxy supernova remnants of type II can probably accelerate cosmic rays up to about 100 TeV <cit.>. However, the mass composition of cosmic rays in this energy range is poorly measured and remains uncertain because neither type of experiment can achieve it: satellite-borne cosmic ray detectors applicable for this range have a very limited geometric factor, whereas standard methods of extensive air shower measurement are not suited for so low energy region. In this paper we consider a possibility to distinguish between different mass groups of cosmic rays in the energy range 200–500 TeV using the hybrid method of Cherenkov light registration originally aimed at gamma/hadron separation in the TAIGA experiment. The observatory TAIGA (Tunka Advanced Instrument for cosmic ray physics and Gamma-ray Astronomy <cit.>) is designed for high energy gamma ray (>30 TeV) and cosmic ray (>100 TeV) measurements. It combines the cost-effective wide angle timing array with a set of IACTs (Imaging Atmospheric Cherenkov Telescopes) to allow reaching a total area up to a few square kilometers and strong suppressing hadron background <cit.>.To the best of our knowledge, the mass group identification with IACT has never been realized yet, even though IACT images are sensitive to the depth of shower maximum and therefore also to the mass of primary particles. The only exception we know is a spectrum of iron nuclei in the interval 13–200 TeV derived by HESS <cit.>. However, their method relies on the ground based detection of Cherenkov light emitted by the primary particle prior to its first interaction in the atmosphere. The other example can be referred to <cit.>, where a hybrid analysis was carried out by combining the data coming from ARGO-YBJ and a wide field of view Cherenkov telescope WFCT. The latter wasn't a standard IACT because it didn't have mirror, but only a set of 16×16 photomultipliers (PMTs) with a large field of view 16^∘×16^∘ and a pixel size of approximately 1^∘. The light component spectrum (p+He) in the 100 TeV – 3 PeV energy region was measured after the hybrid selection technique <cit.>. Nevertheless, the application of the real contemporary IACT technique to solving this task is yet to come. In this paper we discuss the ability to distinguish between different mass groups of cosmic rays in the TAIGA hybrid experiment.§ MONTE CARLO SIMULATIONS Simulation was performed for primary protons and nuclei with the fixed energy 200 and 500 TeV incident within ±5^∘ around the fixed pointing direction of the IACT (zenith angle 30^∘). This configuration is close to the TAIGA prototype design. In the present work we simulated in details only the images produced by showers in a camera of an IACT. The arrival direction, energy and core position of every shower were assumed to be known with the accuracy ±0.1^∘ and ±10 m correspondingly. Such resolutions are typical for the HISCORE timing array <cit.>.At the first step, the shower development in the atmosphere was simulated with the CORSIKA package <cit.>. The response of the IACT system was simulated at the second step using our software developed for this task. The segmented mirror of an IACT has an area of about 10 m^2 and a focal length of 4.75 m. The camera located at the focus consists of 560 photomultipliers with the total field of view 9.72^∘ (FOV), and the single pixel FOV is 0.36^∘. Cherenkov photons of the shower were traced through the IACT optical system and the number of corresponding photoelectrons in each pixel of the camera was counted. § METHODS§.§ Quality criterionAs in the case of gamma ray selection <cit.>, for each event from data bank various features of the image were calculated. The purpose of the study was to determine the most distinctive feature of nuclei and proton/helium induced images. As a quality criterion of particle separation the selection quality factor Q was estimated. This factor indicates an improvement of a significance S_0 of the statistical hypothesis that the events do not belong to the background. For Poisson distribution (that is for a large number of events), the selection quality factor is:Q=ϵ_nuclei/√(ϵ_bckgr),where ϵ_nuclei and ϵ_bckgr are relative numbers of selected events and background events after selection. For our task we consider p+He as a background and try to select oxygen and iron nuclei above this background. The optimal image features for the selection were found in a process of maximizing Q value under the condition ϵ_nuclei≥0.5. §.§ Night sky background reductionIn experiment an image is distorted by the night sky background (NSB) following Poisson distribution. To obtain a more reliable estimation of Q in simulation, the NSB was randomly replicated for every image and Q was averaged over all replications. The procedure of image reconstruction above the NSB is called image cleaning <cit.>. Our cleaning procedure was developed as to maximize quality of subsequent selection nuclei/background. All the results below were obtained after applying the optimal procedure of image cleaning.§.§ Specific features of the problemThe IACT procedure of nuclei discrimination against p/He may have specific features. Unlike gamma ray showers, air showers induced by primary nuclei do not arrive from the point source, but their directions are isotropic. For that reason an event preselection inside the narrow cone around the true source direction is not allowable. This preselection was the additional advantage of the hybrid technique and allowed us to sufficiently reduce background and calculate the selection quality only for the narrow space cone 0.1^∘×0.1^∘ <cit.>. Furthermore, the fact that the telescope is not pointed at the origin of the events to be selected makes also useless very effective selection algorithms. For example, the Hillas parameter 'azimuthal width', which proved the best one for gamma ray selection in TAIGA simulation <cit.>, is worthless for nuclei selection. This parameter is a width of an image ellipse relative to an axis between the image center and the center of the camera <cit.>. Gamma ray induced images are pointed at the camera center, which is the center of the FOV, because of the telescope orientation at their source. Unfortunately, nuclei have the same isotropic distribution as proton/helium background.However, the advantage of the selection of nuclei is that we do not have to reach as great rejection power as for gamma rays because the latter are much rarer than the former. Therefore, the rejection of one order of magnitude (Q∼2) would be sufficient for scientific purposes. The second positive point of nuclei selection by a hybrid technique is the use of another type of 'image projection' parameter, which was introduced by authors of the present paper in <cit.>. This group of parameters is based on the shower core position and is calculated in a way suggested by Hillas for azimuthal width but implemented not for the axis 'image center - center of the camera' but for the 'image center - core position in a camera plane'. Hybrid imaging/timing experiment setup and corresponding imaging/timing technique of data analysis assume that the core position is to be determined with a great accuracy (in TAIGA ∼5–6 m <cit.>) and therefore the data for this new parameter calculation will be provided. §.§ Image features calculation We calculated various image characteristics potentially useful for discrimination between nuclei and protons/helium. They could be sorted out into 2 groups: * Hillas parameters <cit.>, unrelated to data projection on some definite direction;* `core-azimuthal' parameters, introduced in the present paper and in <cit.>.The 3rd group of parameters, `azimuthal' width and length <cit.>, can not be used for nuclei/background discrimination because of absence of a certain arrival direction of primary nuclei.The principle of calculation of parameters from both groups is illustrated in figure <ref>. For the 1st group, the image axis is determined as a line minimizing the weighted sum of squares of a distance to the pixels. Then the width and length can be calculated as 2nd central momenta of image intensity distribution with respect to this axis. For the 2nd group, the image axis is to be drawn through the center of gravity of the image to connect it with the known shower core position. The core position was randomized with ∼10 m accuracy (section <ref>).For both groups we determined also kurtosis (the 4th momentum divided by squared variance) and half-image width parameter designed to account for image truncation in case of distant showers. The half-image width is introduced in the present paper especially for the hybrid technique conditions, because no image truncation is allowable in standard IACT analysis when the distance from the shower core to the IACT is much less than in the TAIGA experiment <cit.>.§ RESULTS Values of selection quality Q for the three best parameters are plotted in figure <ref> for Fe and O nuclei, 200–500 TeV. For small distance R from the shower to the telescope the kurtosis proved the best, whereas for greater distance the best was the parameter from the 2nd group (section <ref>), the core-azimuthal half-image width (except 200 TeV, see the figure). Comparison of these parameters distribution with the standard width of an image is presented in figure <ref>. § DISCUSSION As for gamma rays <cit.>, the best discrimination was achieved for R∼200–300 m, however, the best parameter for this region is the half-image width (from the group of core-azimuthal parameters). This can be explained by image truncation starting with this distance. For small distance the best is a kurtosis describing an image intensity distribution form rather than its dimensions.§ CONCLUSIONS Our methodical study revealed sufficient quality of discrimination of air showers induced by primary nuclei against background (primary proton/helium showers). This kind of discrimination can be obtained in a hybrid timing/imaging array using combination of an imaging Cherenkov air telescope and timing detectors. Such hybrid timing/imaging array is to be realized in the Tunka valley in 2017 as a gamma ray observatory, however, the results of the present study give it an option of successful work with nuclei. All necessary modifications of the selection technique are described. The proton/helium background suppression about 100 times can be achieved for Fe nuclei and about 10 times for oxygen at a distance up to 600 m and energy 200–500 TeV. The influence of combining both types of data, imaging and timing, on discrimination quality is stronger than for gamma ray selection and therefore this non-standard use of imaging telescope can also be realized in the framework of a hybrid technique.The study was supported by the Russian Foundation for Basic Research, project 16-29-13035. § REFERENCES9 1 Ptuskin V, Zirakashvili V, Seo E S 2010 The Astroph. Journal 718 31–36. 2 Gaisser T K, Stanev T, Tilav S 2013 Front. Phys. China 8(6) 748–758. 3 Budnev N et al. 2017 Nucl. Instr. Meth. Phys. Res. A 845 330–333. 4 Buhler R et al. 2007 Proc. 30th Int. Cosmic Ray Conf. (Mexico)(Mexico: Universidad de Mexico) pp 169–172. 5 Bartoli B et al. 2014 Chinese Phys. C 38 045001. 6 Zhang S 2015 Proc. 34th Int. Cosmic Ray Conf. (The Hague) (The Hague: Proceedings of Science) ID 315. 7 Heck D et al. 1998 Report FZKA 6019 (Forschungszentrum Karlsruhe).8 Postnikov E et al. 2017 Bull. Russ. Acad. Sci. Phys. 4 (to be published). 9 Bretz T 2005 Proc. 29th Int. Cosmic Ray Conf. (Pune) vol 00 (Pune: Tata Institute of Fundamental Research) pp 101–104. 10 Hillas A M 1985 Proc. 19th Int. Cosmic Ray Conf. (La Jolla) vol 3 (La Jolla: NASA Conf. Publ.) pp 445–448. 11 Aharonian F 2004 Very High Energy Cosmic Gamma Radiation. A Crucial Window on the Extreme Universe (River Edge, NJ: World Scientific Publishing).

http://arxiv.org/abs/1702.08302v1

Wang et al.]Panqu Wang^1, Pengfei Chen^1, Ye Yuan^2,Ding Liu^3, Zehua Huang^1, Xiaodi Hou^1, Garrison Cottrell^4 ^1TuSimple, ^2Carnegie Mellon University, ^3University of Illinois Urbana-Champaign, ^4UC San Diego ^1{panqu.wang,pengfei.chen,zehua.huang,xiaodi.hou}@tusimple.ai, ^2yey1@andrew.cmu.edu,^3dingliu2@illinois.edu,^4gary@ucsd.eduUnderstanding Convolution for Semantic Segmentation [ December 30, 2023 ===================================================emptyRecent advances in deep learning, especially deep convolutional neural networks (CNNs), have led to significant improvement over previous semantic segmentation systems. Here we show how to improve pixel-wise semantic segmentation by manipulating convolution-related operations that are of both theoretical and practical value. First, we design dense upsampling convolution (DUC) to generate pixel-level prediction, which is able to capture and decode more detailed information that is generally missing in bilinear upsampling. Second, we propose a hybrid dilated convolution (HDC) framework in the encoding phase. This framework 1) effectively enlarges the receptive fields (RF) of the network to aggregate global information; 2) alleviates what we call the gridding issuecaused by the standard dilated convolution operation. We evaluate our approaches thoroughly on the Cityscapes dataset, and achieve a state-of-art result of 80.1% mIOU in the test set at the time of submission. We also have achieved state-of-the-art overall on the KITTI road estimation benchmark and the PASCAL VOC2012 segmentation task. Our source code can be found at https://github.com/TuSimple/TuSimple-DUC<https://github.com/TuSimple/TuSimple-DUC> .§ INTRODUCTION Semantic segmentation aims to assign a categorical label to every pixel in an image, which plays an important role in image understanding and self-driving systems. The recent success of deep convolutional neural network (CNN) models <cit.> has enabled remarkable progress in pixel-wise semantic segmentation tasks due to rich hierarchical features and an end-to-end trainable framework <cit.>. Most state-of-the-art semantic segmentation systems have three key components:1) a fully-convolutional network (FCN), first introduced in <cit.>, replacing the last few fully connected layers by convolutional layers to make efficient end-to-end learning and inference that can take arbitrary input size; 2) Conditional Random Fields (CRFs), to capture both local and long-range dependencies within an image to refine the prediction map; 3) dilated convolution (or Atrous convolution), which is used to increase the resolution of intermediate feature maps in order to generate more accurate predictions while maintaining the same computational cost. Since the introduction of FCN in <cit.>, improvements on fully-supervised semantic segmentation systems are generally focused on two perspectives: First, applying deeper FCN models. Significant gains in mean Intersection-over-Union (mIoU) scores on PASCAL VOC2012 dataset <cit.> were reported when the 16-layer VGG-16 model <cit.> was replaced by a 101-layer ResNet-101 <cit.> model <cit.>; using 152 layer ResNet-152 model yields further improvements <cit.>. This trend is consistent with the performance of these models on ILSVRC <cit.> object classification tasks, as deeper networks generally can model more complex representations and learn more discriminative features that better distinguish among categories. Second, making CRFs more powerful. This includes applying fully connected pairwise CRFs <cit.> as a post-processing step <cit.>, integrating CRFs into the network by approximating its mean-field inference steps <cit.> to enable end-to-end training, and incorporating additional information into CRFs such as edges <cit.> and object detections <cit.>.We are pursuing further improvements on semantic segmentation from another perspective: the convolutional operations for both decoding (from intermediate feature map to output label map) and encoding (from input image to feature map) counterparts. In decoding, most state-of-the-art semantic segmentation systems simply use bilinear upsampling (before the CRF stage) to get the output label map <cit.>. Bilinear upsampling is not learnable and may lose fine details. Inspired by work in image super-resolution <cit.>, we propose a method calleddense upsampling convolution (DUC), which is extremely easy to implement and can achieve pixel-level accuracy: instead of trying to recover the full-resolution label map at once, we learn an array of upscaling filters to upscale the downsized feature maps into the final dense feature map of the desired size. DUC naturally fits the FCN framework by enabling end-to-end training, and it increases the mIOU of pixel-level semantic segmentation on the Cityscapes dataset <cit.> significantly, especially on objects that are relatively small. For the encoding part, dilated convolution recently became popular <cit.>, as it maintains the resolution and receptive field of the network by in inserting holesin the convolution kernels, thus eliminating the need for downsampling (by max-pooling or strided convolution). However, an inherent problem exists in the current dilated convolution framework, which we identify as gridding: as zeros are padded between two pixels in a convolutional kernel, the receptive field of this kernel only covers an area with checkerboard patterns - only locations with non-zero values are sampled, losing some neighboring information. The problem gets worse when the rate of dilation increases, generally in higher layers when the receptive field is large: the convolutional kernel is too sparse to cover any local information, since the non-zero values are too far apart. Information that contributes to a fixed pixel always comes from its predefined gridding pattern, thus losing a huge portion of information. Here we propose a simple hybrid dilation convolution (HDC) framework as a first attempt to address this problem: instead of using the same rate of dilation for the same spatial resolution, we use a range of dilation rates and concatenate them serially the same way as blocksin ResNet-101 <cit.>. We show that HDC helps the network to alleviate the gridding problem. Moreover, choosing proper rates can effectively increases the receptive field size and improves the accuracy for objects that are relatively big.We design DUC and HDC to make convolution operations better serve the need of pixel-level semantic segmentation. The technical details are described in Section 3 below. Combined with post-processing by Conditional Random Fields (CRFs), we show that this approach achieves state-of-the art performance on the Cityscapes pixel-level semantic labeling task, KITTI road estimation benchmark, and PASCAL VOC2012 segmentation task.§ RELATED WORKDecoding of Feature Representation: In the pixel-wise semantic segmentation task, the output label map has the same size as the input image. Because of the operation of max-pooling or strided convolution in CNNs, the size of feature maps of the last few layers of the network are inevitably downsampled. Multiple approaches have been proposed to decode accurate information from the downsampled feature map to label maps. Bilinear interpolation is commonly used <cit.>, as it is fast and memory-efficient. Another popular method is called deconvolution, in which the unpooling operation, using stored pooling switches from the pooling step, recovers the information necessary for feature visualization <cit.>. In <cit.>, a single deconvolutional layer is added in the decoding stage to produce the prediction result using stacked feature maps from intermediate layers. In <cit.>, multiple deconvolutional layers are applied to generate chairs, tables, or cars from several attributes. Noh et al. <cit.> employ deconvolutional layers as mirrored version of convolutional layers by using stored pooled location in unpooling step. <cit.> show that coarse-to-fine object structures, which are crucial to recover fine-detailed information, can be reconstructed along the propagation of the deconvolutional layers. Fischer at al. <cit.> use a similar mirrored structure, but combine information from multiple deconvolutional layers and perform upsampling to make the final prediction.Dilated Convolution: Dilated Convolution (or Atrous convolution) was originally developed in algorithme à trous for wavelet decomposition <cit.>. The main idea of dilated convolution is to insert holes(zeros) between pixels in convolutional kernels to increase image resolution, thus enabling dense feature extraction in deep CNNs. In the semantic segmentation framework, dilated convolution is also used to enlarge the field of convolutional kernels. Yu & Koltun <cit.> use serialized layers with increasing rates of dilation to enable context aggregation, while <cit.> design an atrous spatial pyramid pooling (ASPP)scheme to capture multi-scale objects and context information by placing multiple dilated convolution layers in parallel. More recently, dilated convolution has been applied to a broader range of tasks, such as object detection <cit.>, optical flow <cit.>, and audio generation <cit.>.§ OUR APPROACH §.§ Dense Upsampling Convolution (DUC)Suppose an input image has height H, width W, and color channels C, and the goal of pixel-level semantic segmentation is to generate a label map with size H×W where each pixel is labeled with a category label. After feeding the image into a deep FCN, a feature map with dimension h×w×c is obtained at the final layer before making predictions, where h=H/d, w=W/d, and d is the downsampling factor. Instead of performing bilinear upsampling, which is not learnable, or using deconvolution network (as in <cit.>), in which zeros have to be padded in the unpooling step before the convolution operation, DUC applies convolutional operations directly on the feature maps to get the dense pixel-wise prediction map. Figure <ref> depicts the architecture of our network with a DUC layer.The DUC operation is all about convolution, which is performed on the feature map from ResNet of dimension h×w×c to get the output feature map of dimension h×w×(d^2×L), where L is the total number of classes in the semantic segmentation task. Thus each layer of the dense convolution is learning the prediction for each pixel. The output feature map is then reshaped to H×W×L with a softmax layer, and an elementwise argmax operator is applied to get the final label map. In practice, the “reshape” operation may not be necessary, as the feature map can be collapsed directly to a vector to be fed into the softmax layer. The key idea of DUC is to divide the whole label map into equal d^2 subparts which have the same height and width as the incoming feature map. This is to say, we transform the whole label map into a smaller label map with multiple channels. This transformation allows us to apply the convolution operation directly between the input feature map and the output label maps without the need of inserting extra values in deconvolutional networks (the unpoolingoperation). Since DUC is learnable, it is capable of capturing and recovering fine-detailed information that is generally missing in the bilinear interpolation operation. For example, if a network has a downsample rate of 1/16, and an object has a length or width less than 16 pixels (such as a pole or a person far away), then it is more than likely that bilinear upsampling will not be able to recover this object. Meanwhile, the corresponding training labels have to be downsampled to correspond with the output dimension, which will already cause information loss for fine details. The prediction of DUC, on the other hand, is performed at the original resolution, thus enabling pixel-level decoding. In addition, the DUC operation can be naturally integrated into the FCN framework, and makes the whole encoding and decoding process end-to-end trainable. §.§ Hybrid Dilated Convolution (HDC)In 1-D, dilated convolution is defined as:g[i]=∑_l=1^Lf[i+r· l]h[l],where f[i] is the input signal, g[i] is the output signal , h[l] denotes the filter of length L, and r corresponds to the dilation rate we use to sample f[i]. In standard convolution, r=1.In a semantic segmentation system, 2-D dilated convolution is constructed by inserting holes(zeros) between each pixel in the convolutional kernel. For a convolution kernel with size k×k, the size of resulting dilated filter is k_d×k_d, where k_d=k+(k-1)·(r-1). Dilated convolution is used to maintain high resolution of feature maps in FCN through replacing the max-pooling operation or strided convolution layer while maintaining the receptive field (or field of viewin <cit.>) of the corresponding layer. For example, if a convolution layer in ResNet-101 has a stride s=2, then the stride is reset to 1 to remove downsampling, and the dilation rate r is set to 2 for all convolution kernels of subsequent layers. This process is applied iteratively through all layers that have a downsampling operation, thus the feature map in the output layer can maintain the same resolution as the input layer. In practice, however, dilated convolution is generally applied on feature maps that are already downsampled to achieve a reasonable efficiency/accuracy trade-off <cit.>. However, one theoretical issue exists in the above dilated convolution framework, and we call it gridding(Figure <ref>):For a pixel p in a dilated convolutional layer l, the information that contributes to pixel p comes from a nearby k_d× k_d region in layer l-1 centered at p. Since dilated convolution introduces zeros in the convolutional kernel, the actual pixels that participate in the computation from the k_d× k_d region are just k× k, with a gap of r-1 between them. If k=3, r=2, only 9 out of 25 pixels in the region are used for the computation (Figure <ref> (a)). Since all layers have equal dilation rates r, then for pixel p in the top dilated convolution layer l_top, the maximum possible number of locations that contribute to the calculation of the value of p is (w'× h')/r^2 where w', h' are the width and height of the bottom dilated convolution layer, respectively. As a result, pixel p can only view information in a checkerboard fashion, and lose a large portion (at least 75% when r=2) of information. When r becomes large in higher layers due to additional downsampling operations, the sample from the input can be very sparse, which may not be good for learning because 1) local information is completely missing; 2) the information can be irrelevant across large distances. Another outcome of the gridding effect is that pixels in nearby r× r regions at layer l receive information from completely different set of grids， which may impair the consistency of local information.Here we propose a simple solution- hybrid dilated convolution (HDC), to address this theoretical issue. Suppose we have N convolutional layers with kernel size K× K that have dilation rates of [r_1,...,r_i,...,r_n], the goal of HDC is to let the final size of the RF of a series of convolutional operations fully covers a square region without any holes or missing edges. We define the “maximum distance between two nonzero values” asM_i=max[M_i+1-2r_i,M_i+1-2(M_i+1-r_i),r_i],with M_n=r_n. The design goal is to let M_2≤ K.For example, for kernel size K=3, an r=[1,2,5] pattern works as M_2=2; however, an r=[1,2,9] pattern does not work as M_2=5. Practically, instead of using the same dilation rate for all layers after the downsampling occurs, we use a different dilation rate for each layer. In our network, the assignment of dilation rate follows a sawtooth wave-like heuristic: a number of layers are grouped together to form the rising edgeof the wave that has an increasing dilation rate, and the next group repeats the same pattern. For example, for all layers that have dilation rate r=2, we form 3 succeeding layers as a group, and change their dilation rates to be 1, 2, and 3, respectively. By doing this, the top layer can access information from a broader range of pixels, in the same region as the original configuration (Figure <ref> (b)). This process is repeated through all layers, thus making the receptive field unchanged at the top layer.Another benefit of HDC is that it can use arbitrary dilation rates through the process, thus naturally enlarging the receptive fields of the network without adding extra modules <cit.>, which is important for recognizing objects that are relatively big. One important thing to note, however, is that the dilation rate within a group should not have a common factor relationship (like 2,4,8, etc.), otherwise the gridding problem will still hold for the top layer. This is a key difference between our HDC approach and the atrous spatial pyramid pooling (ASPP) module in <cit.>, or the context aggregation module in <cit.>, where dilation factors that have common factor relationships are used. In addition, HDC is naturally integrated with the original layers of the network, without any need to add extra modules as in <cit.>.§ EXPERIMENTS AND RESULTSWe report our experiments and results on three challenging semantic segmentation datasets: Cityscapes <cit.>, KITTI dataset <cit.> for road estimation, and PASCAL VOC2012 <cit.>. We use ResNet-101 or ResNet-152 networks that have been pretrained on the ImageNet dataset as a starting point for all of our models. The output layer contains the number of semantic categories to be classified depending on the dataset (including background, if applicable). We use the cross-entropy error at each pixel over the categories. This is then summed over all pixel locations of the output map, and we optimize this objective function using standard Stochastic Gradient Descent (SGD). We use MXNet <cit.> to train and evaluate all of our models on NVIDIA TITAN X GPUs. §.§ Cityscapes DatasetThe Cityscapes Dataset is a large dataset that focuses on semantic understanding of urban street scenes. The dataset contains 5000 images with fine annotations across 50 cities, different seasons, varying scene layout and background. The dataset is annotated with 30 categories, of which 19 categories are included for training and evaluation (others are ignored). The training, validation, and test set contains 2975, 500, and 1525 fine images, respectively. An additional 20000 images with coarse (polygonal) annotations are also provided, but are only used for training.§.§.§ Baseline ModelWe use the DeepLab-V2 <cit.> ResNet-101 framework to train our baseline model. Specifically, the network has a downsampling rate of 8, and dilated convolution with rate of 2 and 4 are applied to res4b and res5b blocks, respectively. An ASPP module with dilation rate of 6, 12, 18, and 24 is added on top of the network to extract multiscale context information. The prediction maps and training labels are downsampled by a factor of 8 compared to the size of original images, and bilinear upsampling is used to get the final prediction. Since the image size in the Cityscapes dataset is 1024× 2048, which is too big to fit in the GPU memory, we partition each image into twelve 800× 800 patches with partial overlapping, thus augmenting the training set to have 35700 images. This data augmentation strategy is to make sure all regions in an image can be visited. This is an improvement over random cropping, in which nearby regions may be visited repeatedly.We train the network using mini-batch SGD with patch size 544× 544 (randomly cropped from the 800× 800 patch) and batch size 12, using multiple GPUs. The initial learning rate is set to 2.5× 10^-4, and a polylearning rate (as in <cit.>) with power=0.9 is applied. Weight decay is set to 5× 10^-4, and momentum is 0.9. The network is trained for 20 epochs and achieves mIoU of 72.3% on the validation set.§.§.§ Dense Upsampling Convolution (DUC) We examine the effect of DUC on the baseline network. In DUC， the only thing we change is the shape of the top convolutional layer. For example, if the dimension of the top convolutional layer is 68×68×19 in the baseline model (19 is the number of classes), then the dimension of the same layer for a network with DUC will be 68×68×(r^2× 19) where r is the total downsampling rate of the network (r=8 in this case). The prediction map is then reshaped to size 544× 544× 19. DUC will introduce extra parameters compared to the baseline model, but only at the top convolutional layer. We train the ResNet-DUC network the same way as the baseline model for 20 epochs, and achieve a mean IOU of 74.3% on the validation set, a 2% increase compared to the baseline model. Visualization of the result of ResNet-DUC and comparison with the baseline model is shown in Figure <ref>From Figure <ref>, we can clearly see that DUC is very helpful for identifying small objects, such as poles, traffic lights, and traffic signs. Consistent with our intuition, pixel-level dense upsampling can recover detailed information that is generally missed by bilinear interpolation.Ablation Studies We examine the effect of different settings of the network on the performance. Specifically, we examine: 1) the downsampling rate of the network, which controls the resolution of the intermediate feature map; 2) whether to apply the ASPP module, and the number of parallel paths in the module; 3) whether to perform 12-fold data augmentation; and 4) cell size, which determines the size of neighborhood region (cell× cell) that one predicted pixel projects to. Pixel-level DUC should usecell=1; however, since the ground-truth label generally cannot reach pixel-level precision, we also try cell=2 in the experiments. From Table <ref>， we can see that making the downsampling rate smaller decreases the accuracy. Also it significantly raises the computational cost due to the increasing resolution of the feature maps. ASPP generally helps to improve the performance, and increasing ASPP channels from 4 to 6 (dilation rate 6 to 36 with interval 6) yields a 0.2% boost. Data augmentation helps to achieve another 1.5% improvement. Using cell=2 yields slightly better performance when compared with cell=1, and it helps to reduce computational cost by decreasing the channels of the last convolutional layer by a factor of 4.Bigger Patch Size Since setting cell=2 reduces GPU memory cost for network training, we explore the effect of patch size on the performance. Our assumption is that, since the original images are all 1024× 2048, the network should be trained using patches as big as possible in order to aggregate both local detail and global context information that may help learning. As such, we make the patch size to be 880× 880, and set the batch size to be 1 on each of the 4 GPUs used in training. Since the patch size exceeds the maximum dimension (800× 800) in the previous 12-fold data augmentation framework, we adopt a new 7-fold data augmentation strategy: Why can't you simply use 880-size patches in the same way?because the image is 1024x2048. Previously we use 2x6 configuration. Now since the patch is 880x880, so we can remove the height part.seven center locations with x=512, y={256,512,...,1792} are set in the original image; for each center location, a 880× 880 patch is obtained by randomly setting its center within a 160× 160 rectangle area centered at each center. This strategy makes sure that we can sample all areas in the image, including edges. Training with a bigger patch size boosts the performance to 75.7%, a 1% improvement over the previous best result.Compared with Deconvolution We compare our DUC model with deconvolution, which also involves learning for upsampling. Particularly, we compare with 1) direct deconvolution from the prediction map (dowsampled by 8) to the original resolution; 2) deconvolution with an upsampling factor of 2 first, followed by an upsampling factor of 4. We design the deconv network to have approximately the same number of parameters as DUC. We use the ResNet-DUC bigger patch model to train the networks. The above two models achieve mIOU of 75.1% and 75.0%, respectively, lower than the ResNet-DUC model (75.7% mIoU).Conditional Random Fields (CRFs) Fully-connected CRFs <cit.> are widely used for improving semantic segmentation quality as a post-processing step of an FCN <cit.>. We follow the formation of CRFs as shown in <cit.>. We perform a grid search on parameters on the validation set, and use σ_α=15, σ_β=3, σ_γ=1 , w_1=3, and w_2=3 for all of our models. Applying CRFs to our best ResNet-DUC model yields an mIoU of 76.7%, a 1% improvement over the model does not use CRFs.§.§.§ Hybrid Dilated Convolution (HDC)We use the best 101 layer ResNet-DUC model as a starting point of applying HDC. Specifically, we experiment with several variants of the HDC module: * No dilation: For all ResNet blocks containing dilation, we make their dilation rate r=1 (no dilation).* Dilation-conv: For all blocks contain dilation, we group every 2 blocks together and make r=2 for the first block, and r=1 for the second block. * Dilation-RF: For the res4b module that contains 23 blocks with dilation rate r=2, we group every 3 blocks together and change their dilation rates to be 1, 2, and 3, respectively. For the last two blocks, we keep r=2. For the res5b module which contains 3 blocks with dilation rate r=4, we change them to 3, 4, and 5, respectively.* Dilation-bigger: For res4b module, we group every 4 blocks together and change their dilation rates to be 1, 2, 5, and 9, respectively. The rates for the last 3 blocks are 1, 2, and 5. For res5b module, we set the dilation rates to be 5, 9, and 17.The result is summarized in Table <ref>. We can see that increasing receptive field size generally yields higher accuracy. Figure <ref> illustrates the effectiveness of the ResNet-DUC-HDC model in eliminating the gridding effect. A visualization result is shown in Figure <ref>. We can see our best ResNet-DUC-HDC model performs particularly well on objects that are relatively big.Deeper Networks We have also tried replacing our ResNet-101 based model with the ResNet-152 network, which is deeper and achieves better performance on the ILSVRC image classification task than ResNet-101 <cit.>. Due to the network difference, we first train the ResNet-152 network to learn the parameters in all batch normalization (BN) layers for 10 epochs, and continue fine-tuning the network by fixing these BN parameters for another 20 epochs. The results are summarized in Table <ref>. We can see that using the deeper ResNet-152 model generally yields better performance than the ResNet-101 model. §.§.§ Test Set ResultsOur results on the Cityscapes test set are summarized in Table <ref>. There are separate entries for models trained using fine-labels only, and using a combination of fine and coarse labels. Our ResNet-DUC-HDC model achieves 77.6% mIoU using fine data only. Adding coarse data help us achieve 78.5% mIoU.In addition, inspired by the design of the VGG network <cit.>, in that a single 5×5 convolutional layer can be decomposed into two adjacent 3×3 convolutional layers to increase the expressiveness of the network while maintaining the receptive field size, we replaced the 7× 7 convolutional layer in the original ResNet-101 network by three 3×3 convolutional layers. By retraining the updated network, we achieve a mIoU of 80.1% on the test set using a single model without CRF post-processing. Our result achieves the state-of-the-art performance on the Cityscapes dataset at the time of submission. Compared with the strong baseline of Chen et al. <cit.>, we improve the mIoU by a significant margin (9.7%), which demonstrates the effectiveness of our approach.§.§ KITTI Road SegmentationDataset The KITTI road segmentation task contains images of three various categories of road scenes, including 289 training images and 290 test images. The goal is to decide if each pixel in images is road or not. It is challenging to use neural network based methods due to the limited number of training images. In order to avoid overfitting, we crop patches of 320 × 320 pixels with a stride of 100 pixels from the training images, and use the ResNet-101-DUC model pretrained from ImageNet during training. Other training settings are the same as Cityscapes experiment. We did not apply CRFs for post-processing. Results We achieve the state-of-the-art results at the time of submission without using any additional information of stereo, laser points and GPS. Specifically, our model attains the highest maximum F1-measure in the sub-categories ofurban unmarked (UU_ROAD), urban multiple marked (UMM_ROAD) and the overall category URBAN_ROAD of all sub-categories, the highest average precision across all three sub-categories and the overall category by the time of submission of this paper. Examples of visualization results are shown in Figure <ref>. The detailed results are displayed in Table <ref> [For thorough comparison with other methods, please check http://www.cvlibs.net/datasets/kitti/eval_road.php.]. §.§ PASCAL VOC2012 datasetDataset The PASCAL VOC2012 segmentation benchmark contains 1464 training images, 1449 validation images, and 1456 test images. Using the extra annotations provided by <cit.>, the training set is augmented to have 10582 images. The dataset has 20 foreground object categories and 1 background class with pixel-level annotation.Results We first pretrain our 152 layer ResNet-DUC model using a combination of augmented VOC2012 training set and MS-COCO dataset <cit.>, and then finetune the pretrained network using augmented VOC2012 trainval set. We use patch size 512× 512 (zero-padded) throughout training. All other training strategies are the same as Cityscapes experiment. We achieve mIOU of 83.1% on the test set using a single model without any model ensemble or multiscale testing, which is the best-performing method at the time of submission[Result link: http://host.robots.ox.ac.uk:8080/anonymous/LQ2ACW.html]. The detailed results are displayed in Table <ref>, and the visualizations are shown in Figure <ref>.§ CONCLUSIONWe propose simple yet effective convolutional operations for improving semantic segmentation systems. We designed a new dense upsampling convolution (DUC) operation to enable pixel-level prediction on feature maps, and hybrid dilated convolution (HDC) to solve the gridding problem, effectively enlarging the receptive fields of the network. Experimental results demonstrate the effectiveness of our framework on various semantic segmentation tasks.§ ACKNOWLEDGMENTSWe thank the members of TuSimple and Gary's Unbelievable Research Unit (GURU) for comments on this work. GWC was supported in part by Guangzhou Science and Technology Planning Project (201704030051) and NSF cooperative agreement SMA 1041755 to the Temporal Dynamics of Learning Center, an NSF Science of Learning Center.ieee

http://arxiv.org/abs/1702.08502v3

Boosted Generative Models Aditya Grover, Stefano ErmonComputer Science DepartmentStanford University December 30, 2023 ====================================================================================== We study a variant of the k-server problem, the infinite server problem, in which infinitely many servers reside initially at a particular point of the metric space and serve a sequence of requests. In the framework of competitive analysis, we show a surprisingly tight connection between this problem and the (h,k)-server problem, in which an online algorithm with k servers competes against an offline algorithm with h servers. Specifically, we show that the infinite server problem has bounded competitive ratio if and only if the (h,k)-server problem has bounded competitive ratio for some k=O(h). We give a lower bound of 3.146 for the competitive ratio of the infinite server problem, which implies the same lower bound for the (h,k)-server problem even when k/h →∞ and holds also for the line metric; the previous known bounds were 2.4 for general metric spaces and 2 for the line. For weighted trees and layered graphs we obtain upper bounds, although they depend on the depth. Of particular interest is the infinite server problem on the line, which we show to be equivalent to the seemingly easier case in which all requests are in a fixed bounded interval away from the original position of the servers. This is a special case of a more general reduction from arbitrary metric spaces to bounded subspaces. Unfortunately, classical approaches (double coverage and generalizations, work function algorithm, balancing algorithms) fail even for this special case. § INTRODUCTIONThe k-server problem is a fundamental well-studied online problem <cit.>. In this problem k servers serve a sequence of requests. The servers reside at k points of a metric space M and requests are simply points of M. Serving a request entails moving one of the servers to the request. The objective is to minimize the total distance traveled by the servers. The most interesting variant of the problem is its online version, in which the requests appear one-by-one and the online algorithm must decide how to serve a request without knowing the future requests. It is known that the deterministic k-server problem has competitive ratio between k and 2k-1 for every metric space with at least k+1 distinct points <cit.>.In this paper, we study the infinite server problem, the variant of the k-server problem in which there are infinitely many servers, all of them initially residing at a given point, the source[We first learned about this problem from Kamal Jain <cit.>.]. At first glance it may appear that the lower bound of k for the k-server problem would imply an unbounded competitive ratio for the infinite server problem. But consider, for example, the version of the k-server problem on uniform metric spaces (the distance between any two points is 1), and observe that the infinite server problem has competitive ratio 1 for this case.The infinite server problem is closely related to the (h,k)-server problem, the resource augmentation version of the k-server problem in which the online algorithm has k servers and competes against an offline algorithm for h≤ k servers. This model is also known as weak adversaries <cit.>. One major open questionin competitive analysis is whether the (h,k)-server problem has bounded competitive ratio when k≫ h. Bar-Noy and Schieber (see <cit.>) showed that when h=2, the competitive ratio on the line metric is 2 for any k, and recently, Bansal et al. <cit.> showed a lower bound of 2.4 for the general case k≫ h. Here we show a, perhaps surprising, tight connection between the infinite server problem and the (h,k)-server problem, which allows us to improve both lower bounds to 3.146.The infinite server problem is also a considerable generalization of the ski-rental problem, since the ski-rental problem is essentially a special case of the infinite server problem when the metric space is an isosceles triangle. §.§ Previous WorkThe k-server problem was first formulated by Manasse etal. <cit.>, to generalize a variety of online setting whosestepwise cost had a `metric'-like structure. They build on previous work bySleator and Tarjan <cit.>, the genesis of competitive analysis, onthe paging problem. This problem can be easily recast as a k-serverinstance for the uniform metric and was already known to be k-competitive. Manasse et al. <cit.> also showed thatthe competitive ratioof the k-server problem is at leastk on any metric space with more than k points.They then proposed the renowned k-server conjecture, stating that thisbound is tight. Thishas been shown to be true for k=2 <cit.> and for severalspecial metric spaces<cit.>. A stream of refinements <cit.>lead tobettercompetitive ratios for general metricspaces until <cit.> showed that a competitive ratio of 2k-1can be achieved on any metric space.Chasing the competitive ratio for the deterministic (and randomized)k-server problem has been pivotal for the development of competitive analysis. For a more in depth view on the history of the k-server problem andfurther related work, we refer to <cit.>.In the weak adversaries setting, significantly less is known. For the(h,k)-server problem, the exact competitive ratio isk/k-h+1 on uniform metrics (equivalent to the paging problem)<cit.> and weighted star metrics (equivalent to weighted paging)<cit.>. Bansal et al. <cit.> showed recently for weighted trees that the competitive ratio ask/h→∞ can be bounded by a constant depending on the depth ofthe tree. On general metrics, the (h,k)-serverproblem is still very poorly understood. No algorithm is known for generalmetrics that performs better than disabling the k-h extra servers andusing h servers only. In fact, for the line it was shown<cit.> that the Double CoverageAlgorithm and the Work Function Algorithm – despite achieving theoptimal competitive ratio of h if k=h <cit.>– perform strictly worse in the resource augmentation setting thandisabling the k-h extra servers and applying the same algorithm to hservers only. For the case that h is not fixed, the Work FunctionAlgorithm was shown to be 2h-competitive simultaneously against anynumber h≤ k of offline servers <cit.>.In terms of lower bounds, it is known that unlike for (weighted) paging, thecompetitive ratio does not converge to 1 on general metrics even ask/h→∞. Prior to this work, the best known lower bounds were 2on the line <cit.> and 2.4 on general metric spaces<cit.>.The closest publication to this work is by Csirik et al. <cit.>, which studies a problem that is essentially the special case of the infinite server problem on the uniform metric space augmented by a far away source. It is cast as a paging problem where new cache slots can be bought at a fixed price per unit and gives matching upper and lower bounds of ≈3.146 on the competitive ratio. §.§ Our ResultsOur main result is an equivalence theorem between the infinite serverproblem and the (h,k)-server problem, presented inSection <ref>. Itstates that the infinite serverproblem is competitive on every metric space if and only if the(h,k)-server problem is O(1)-competitive on every metric space ask/h→∞. We show further that it is not even necessary to let k/hconverge to infinity because in the positive case, there must also existsome k=O(h) for which the latter is true. The theorem holds also if“every metric space” is replaced by “the real line”.In Section <ref> we present upper and lower bounds on the competitiveratio of the infinite server problem on a variety of metric spaces.Extendingthe work in <cit.>, we present a tight lower bound for non-discrete spaces, which is then turned into a 3.146 lower bound for the (h,k)setting. To our knowledge, this is the largest bound on the weakadversaries setting for any metric space, as k/h →∞. We show how recent work by Bansal et al. <cit.> can be adapted togive an upper bound on the competitive ratio of the infinite server problemon bounded-depth weighted trees. We also consider layered graph metrics, which are equivalent (up to a factor of 2) to general graph metrics. We have not settled the case fortheir competitive ratio, but we present a natural algorithm with tight analysis and pose challenges for further research. The main open question is whether there exists a metric space on which the infinite server problem is not competitive.In Section <ref> we show how a variety of known algorithms such as the work function and balancing algorithms fail for the infinite server problem, even on the real line. We focus in particular on a class of speed-adjusted variants of the well-known double coverage algorithm.Finally, we present a useful reduction from arbitrary metric spaces to bounded subspaces in Section <ref>. In particular, the infinite server problem on the line is competitive if and only if it is competitive for the special case where requests are restricted to some bounded interval further away from the source. §.§ Preliminaries Let M=(M,d) be a metric space and let s be a point of M. In the infinite server problem on (M,s), an unbounded number of servers starts at point s and serves a finite sequence σ=(σ_0=s,σ_1,σ_2,…, σ_m) of requests σ_i∈ M. Serving a request entails moving one of the servers to it. The goal is to minimize the total distance traveled by the servers. We drop s in the notation if the location of the source is not relevant or understood. We refer to the action of moving a server from the source to another point as spawning. Throughout this work we use the letter d for the metric associated with the metric space.In the online setting, the requests are revealed one by one and need to be served immediately without knowledge of future requests. All algorithms considered in this paper are deterministic. An algorithm is called lazy if it moves only one server to serve a request at an unoccupied point and moves no server if the requested point is already covered. An algorithm is called local <cit.> if it moves a server from a to b only if there is no server at some other point c on a shortest path from a to b, with d(a,b)=d(a,c)+d(c,b). It is easy to see that any algorithm can be turned into a lazy and local algorithm without increasing its cost (the total distance traveled by all servers).For an algorithm , we denote by (σ) its cost on the request sequence σ. Similarly, we write (σ) for the optimal (offline) cost. An online algorithmis ρ-competitive for ρ≥ 1if (σ)≤ρ(σ)+c for all σ, where c is aconstant independent of σ. The competitive ratio of analgorithm is the infimum of all such ρ. We say that an algorithm iscompetitive if it is ρ-competitive for some ρ. We alsocall an online problem itself (ρ-)competitive if it admits such analgorithm. If the additive term c in the definition is 0, then thealgorithm is also called strictly ρ-competitive <cit.>.The (h, k)-server problem on M is defined like the infinite server problem except that the number of servers is k for the online algorithm and h for the optimal (offline) algorithm against whom it is compared in the definition of competitiveness. For this problem, the servers are not required to start at the same point, although a different initial configuration would only affect the additive term c. The problem is interesting only when k≥ h. The case h=k is the standard k-server problem and the case k≥ h is known as the weak adversaries model. One major open problem is determine the competitive ratio of the (h,k)-server problem as k tends to infinity.We will sometimes write _h and _∞ for the optimal offline algorithm, where the index specifies the number of servers available.The following two propositions will be useful later in the paper. If for every metric space there exists a competitive algorithm for the infinite server problem, then there exists a universal competitive ratio ρ such that the infinite server problem is strictly ρ-competitive on every metric space. We first show the existence of ρ such that the infinite server problem is ρ-competitive (strictly or not) on every metric space. Suppose such ρ does not exist, then for every n∈ℕ we can find a metric space M_n containing some point s_n such that the infinite server problem on (M_n,s_n) is not n-competitive. Consider the metric space obtained by taking the disjoint union of all spaces M_n and gluing all the points s_n together. The infinite server problem would not be competitive on this metric space, in contradiction to the assumption. Analogously we can also find a universal constant c that works for all metric spaces as additive constant in the definition of ρ-competitiveness. A scaling argument shows that also c=0 works. With a very similar argument we get: Let k=k(h) be a function of h. Suppose that for every metric space M and for all h there exists an O(1)-competitive algorithm for the (h,k)-server problem on M. Then there exists a universal competitive ratio ρ such that the (h,k)-server problem is strictly ρ-competitive on every metric space if all servers start at the same point. § EQUIVALENCE OF INFINITE SERVERS AND WEAK ADVERSARIES The main result of this section is the following tight connection between the infinite server problem and the weak adversaries model. The following are equivalent: * The infinite server problem is competitive. * The (h,k)-server problem is O(1)-competitive as k/h→∞. * For each h there exists k=O(h) so that the (h,k)-server problem is O(1)-competitive. The three statements above are also equivalent if we fix the metric space to be the real line. The implication “<ref><ref>” is trivial. The proof of the equivalence theorem consists in its core of two reductions. Theorem <ref> contains the easier of the two reductions, which is from the infinite server problem to the k-server problem against weak adversaries (“<ref><ref>”). By Propositions <ref> and <ref>, it suffices to consider only strictly competitive algorithms. Theorem <ref> proves essentially the inverse for general metric spaces, and Theorem <ref> specializes it to the line (“<ref><ref>”).As a corollary of the theorem we get the non-trivial implication “<ref><ref>”, a potentially useful statement towards resolving the major open problem about weak adversaries: “Is Statement <ref> true?” This highlights the importance of the infinite server problem. Fix a metric space M and consider algorithms with all servers starting at some s∈ M. If for every h there exists k=k(h) such that the (h,k)-server problem on M is strictly ρ-competitive, for some constant ρ, then there exists a strictly ρ-competitive online strategy for the infinite server problem on M. Let _k(h) denote an online algorithm with k(h) servers that is strictly ρ-competitive against an optimal algorithm _h for h servers,_k(h)(σ)≤ρ_h(σ) for every request sequence σ. Without loss of generality, algorithm ALG_k(h) is lazy. For every request sequence σ, consider the equivalence relation ≡_σ on natural numbers in which h≡_σ h' if and only if _k(h)(σ) and _k(h')(σ) serve σ in exactly the same way (, make exactly the same moves). To every σ, we associate an equivalence class H(σ) of ≡_σ that satisfies * H(σ) is infinite, * H(σ r)⊆ H(σ), for every request r. This is done inductively in the length of σ (in a manner reminiscent of König's lemma) as follows: For the base case when σ is the empty request sequence, H(σ)=ℕ. For the induction step, suppose that we have defined H(σ). Consider the equivalence classes of ≡_σ r, a refinement of the equivalence classes of ≡_σ. Since there are only finitely many possible ways to serve r, they partition H(σ) into finitely many parts. At least one of these parts is infinite and we select it to be H(σ r); if there is more than one such sets, we select one arbitrarily, say the lexicographically first. Given such a mapping H, we define the online algorithm _∞ which serves every σ in the same way as all the online algorithms _k(h) for h∈ H(σ). The second property of H guarantees that _∞ is a well-defined online algorithm. By construction, _∞(σ)=_k(h)(σ) for every h∈ H(σ). To finish the proof, observe that since H(σ) is infinite, it contains some h greater than the length of σ, and for such an h we have OPT_∞(σ)=OPT_h(σ). Substituting these to (<ref>), we see that _∞ is strictly ρ-competitive. We now show the reduction from the k-server problem against weak adversaries to the infinite server problem on general metric spaces. If the infinite server problem on general metric spaces is strictly ρ̃-competitive, then there exists a constant ρ such that the (h,k)-server problem is ρ-competitive, for k=O(h). In particular, for every ϵ>0, we can take ρ=(3+ϵ)ρ̃ and any k≥ (1+1/ϵ)ρ̃ h. Fix some metric space M and a point s∈ M. We will describe a strictly ρ-competitive algorithm for the (h,k)-server problem on M for the case that all servers start at s. This implies a (not necessarily strictly) ρ-competitive algorithm for any initial configuration. The idea is to simulate a strictly ρ̃-competitive infinite server algorithm, but whenever it would spawn a (k+1)-st server, we bring all servers back to the origin and restart the algorithm. The problem is that the overhead cost for returning the servers to the origin, may be very high. To compensate for this, we assume that every time the servers return to the origin, they pretend to start from a different point further away from the origin. This motivates the following notation: Given a metric M, a point s∈ M, and a value w≥ 0, we will use the notation M_s⊕ w to denote the metric derived from M when we increase the distance of s from every other point by w; we will also denote the relocated point by s⊕ w. Let _∞ denote a strictly ρ̃-competitive online algorithm for the infinite server problem. We now define an online algorithm _k for k servers (all starting at s). We will make use of the notation A(σ; s) to denote the cost of algorithm A to serve the request sequence σ when all servers start at s. Algorithm _k runs in phases with the initial phase being the 0th phase. At the beginning of every phase, all servers of _k are at s. In every phase i, the algorithm simulates the infinite server algorithm _∞, whose servers start at s⊕ w_i for some w_i≥ 0. The parameters w_i are determined online, and initially w_0=0. Whenever _∞ spawns a server from s⊕ w_i, algorithm _k spawns a server from s. The phase ends just before _∞ spawns its (k+1)-st server or when the request sequence ends. In the former case, all servers of _k return to s to start the (i+1)-st phase. To determine the starting point of the simulated algorithm of the next phase, we set w_i+1 = ϵ _h(σ_i; s)/h , where σ_i is the sequence of requests during phase i. Let n be the number of phases. The costof _k for the requests in phase i<n is _∞(σ_i; s⊕ w_i) - k w_i; the last term is subtracted because the k servers do not have to actually travel the distance between s⊕ w_i and s. However for the last phase no such term can be subtracted since we do not know how many servers are spawned during the phase, and we can only bound the cost from above by _∞(σ_n; s⊕ w_n). The cost of returning the servers to s at the end of a phase can at most double the cost during the phase. From this, we see that the total cost of _k in phase i is _i≤ 2(_∞(σ_i; s⊕ w_i)-k w_i)for i<n _∞(σ_n; s⊕ w_n)for i=n . Since _∞ is strictly ρ̃-competitive, we have _∞(σ_i; s⊕ w_i) ≤ρ̃ _∞(σ_i; s⊕ w_i)≤ρ̃ _h(σ_i; s⊕ w_i)≤ρ̃ (_h(σ_i; s)+hw_i) and substituting this in the expression for the cost, we can bound the total cost by _k(σ; s) = ∑_i=0^n _i ≤ 2∑_i=0^n-1( ρ̃ (_h(σ_i; s) +h w_i)-k w_i) +ρ̃ (_h(σ_n; s) +h w_n) = 2∑_i=0^n-1( ρ̃_h(σ_i; s) -(k-ρ̃h)w_i) +ρ̃_h(σ_n; s) +ρ̃h w_n . The parameters w_i and k were selected so that the summation telescopes, and we are left with =0pt _k(σ; s)≤2 ρ̃ _h(σ_n-1; s)+ ρ̃ _h(σ_n; s) + ρ̃ ϵ _h(σ_n-1; s)≤ (3+ϵ) ρ̃ _h(σ; s) .The previous reduction requires the infinite server problem to be competitive on every metric space. The following variant only requires the infinite server problem to be competitive on the line. If the infinite server problem on the line is ρ-competitive, then for every h∈ℕ and ϵ>0, the (h,k)-server problem on the line is (3+ϵ)ρ-competitive, when k≥ 2⌈ (1+1/ϵ)ρ h⌉. A straightforward adaptation of the proof of the previous lemma, shows the existence of a (3+ϵ)ρ-competitive algorithm for the interval [0,∞), when k≥ 2(1+1/ϵ)ρ h. By doubling the number of online servers so that half of them are used in each half-line, we get a (3+ϵ)ρ-competitive algorithm for the entire line, when k≥ 2⌈ (1+1/ϵ)ρ h⌉. Note that the proof assumes strictly competitive algorithms.But, by a straightforward scaling argument, if the infinite server problem on the line is ρ-competitive, then it is also strictly ρ-competitive. This in turn implies a strictly ρ-competitive online algorithm for M_0⊕ w, since this space is isometric to the subspace {-w}∪(0,∞) of the line.In the next section we look at some particular metric spaces and give upper and lower bounds on the competitive ratio.§ UPPER AND LOWER BOUNDS Unlike the k-server problem, which is 1-competitive if and only if the metric spaces has at most k points and conjectured k-competitive otherwise, the situation is more diverse for the infinite server problem. For example, on uniform metric spaces (where all distances are the same) the problem is trivially 1-competitive even if the metric space consists of uncountably many points. This is because an optimal strategy in this case is to spawn a server to every requested point. More generally, this strategy achieves a finite competitive ratio on any metric space where distances are bounded from below and above by positive constants. This suggests that statements about the competitive ratio for the infinite server problem cannot be as simple as the (conjectured) dichotomy for the k-server problem, which depends only on the number of points of the metric space. In this section we derive bounds on the competitive ratio for particular classes of metric spaces. §.§ Weighted TreesWe consider the infinite server problem on metric spaces that can be modeled by edge-weighted trees. The points of the metric space are the nodes of the tree, and the distance between two nodes is the sum of edge weights along their connecting path. We choose the source of the metric space as the root of the tree, and define the depth of the tree as the maximal number of edges from the root to a leaf. The number of nodes can be infinite (otherwise the infinite server problem is trivially 1-competitive), but we assume the depth to be finite.An upper bound on the competitive ratio of such trees follows easily from an upper bound for the (h,k)-server on such trees <cit.> and the equivalence theorem: The competitive ratio of the infinite server problem on trees of depth d is at most O(2^d· d). Bansal et al. <cit.> showed that the competitive ratio of the (h,k)-server problem on trees of depth d is at most O(2^d· d) provided that k/h is large enough. Inspection of the proof in <cit.> shows that if all servers start at the root, it is in fact strictly O(2^d· d)-competitive. Thus, Theorem <ref> implies the result for the infinite server problem.§.§ Non-Discrete Spaces and Spaces with Small Infinite Subspaces The following theorem gives a lower bound of 3.146 on the competitive ratio of the infinite server problem on any metric space containing an infinite subspace of a diameter that is small compared to the subspace's distance from the source. For example, every non-discrete metric space has this property (unless the source is the only non-discrete point), since non-discrete metric spaces contain infinite subspaces of arbitrarily small diameter. The theorem is a generalization of such a lower bound established in <cit.> for a variant of the paging problem where cache cells can be bought. Crucial parts of the subsequent proof are as in <cit.>. Let M be a metric space containing an infinite subspace M_0⊂ M of finite diameter δ and a point s∈ M∖ M_0 such that the infimum Δ of distances between s and points in M_0 is positive. Let λ> 3.146 be the largest real solution to λ=2+lnλ . The competitive ratio of any deterministic online algorithm for the infinite server problem on (M,s) is bounded from below by a value that converges to λ as Δ/δ→∞. In particular, the competitive ratio is at least λ if M∖{s} contains a non-discrete part. By scaling the metric, we can assume that δ=1. Let p_1,p_2,p_3,… be infinitely many distinct points in M_0. Fix some lazy deterministic online algorithm . We consider the request sequence that always requests the point p_i with i minimal such that p_i is not occupied by a server of . We call a move of a server between two points in M_0 local (every move that does not spawn is local). Let f_j be the cumulative cost of local moves incurred tountil it spawns its jth server. Let σ_k be this request sequence that is stopped right afterspawns its kth server, for some large k. The total online cost is (σ_k)≥ kΔ + f_k . Let h=⌈ k/λ⌉. We consider several offline algorithms that start behaving the same way, so we think of it as one algorithm initially that is forked into several algorithms later. The offline algorithms make use of only h servers and they begin by spawning them to the points p_1,…,p_h. They do not need to move any servers untilspawns its hth server. Wheneverspawns its jth server for some j≥ h, every offline algorithm is forked to h distinct algorithms: Each of them moves a different server to p_j+1 (to prepare for the next request, which will be at p_j+1). We will keep the invariant that each offline algorithm already has a server at the next request. To this end, wheneverdoes a local move from p to p', every offline algorithm that does not have a server at p moves a server from p' to p; note that the algorithm had a server at p' by the invariant, and the next request will be at p. Whenhas j spawned servers (j≥ h), the offline algorithms are in jh-1 different configurations, each of which occurs equally often among them. Ifdoes a local move from p to p', there are j-1h-1 different offline configurations for which a local move is made in the opposite direction. Thus, for each local move bywhile having j servers in total, a portion j-1 h-1/j h-1 = j-h+1/j of the offline algorithms move a server in the opposite direction for the same cost. We use the average cost of all offline algorithms we considered as an upper bound on the optimal cost. The cost of spawning h servers is at most h(Δ+1), and the average cost whilehas j spawned servers (for j=h,…,k-1) is at most j-h+1/j(f_j+1-f_j)+1 (with the “+1” coming from the move when offline algorithms fork). Hence, (σ_k)≤ h(Δ+1) + k-h + ∑_j=h^k-1j-h+1/j(f_j+1-f_j) ,≤ hΔ + k + k-h/k-1f_k - f_h/h - ∑_j=h+1^k-1h-1/j(j-1)f_j , Note that f_k/k is bounded from above because otherwisewould not be competitive, and it is bounded from below by 0. Thus, L=lim inf_k→∞f_k/k exists. In the following we use the asymptotic notation o(1) for terms that disappear as k→∞. We can choose arbitrarily large values of k such that f_k/k= L+o(1). Since h=⌈ k/λ⌉, we have f_j/j≥ L+o(1) for all j≥ h. Moreover, ∑_j=h+1^k-11/j-1=ln(λ)+o(1). This allows us to simplify the previous bound to (σ_k)≤k/λ(Δ + λ + (λ-1 - ln(λ))L + o(1))= k/λ(Δ + L + λ + o(1)) , where the last step uses equation (<ref>). The competitive ratio is at least (σ_k)+O(1)/(σ_k) ≥kΔ+f_k+O(1)/k/λ(Δ + L + λ + o(1))= λ·Δ+L/Δ + L + λ + o(1) . The fraction in the last term tends to 1 as Δ→∞. This bound is tight due to a matching upper bound in <cit.> that shows (translated to the terminology of the infinite server problem) that a competitive ratio of λ can be achieved on metric spaces where all pairwise distances are 1 except that the source is at some larger distance Δ from the other points.The previous theorem together with the equivalence theorem also allows us to obtain a new lower bound for the k-server problem against weak adversaries. For sufficiently large h, there is no 3.146-competitive algorithm for the (h,k)-server problem on the line, even if k→∞. By a scaling argument it is easy to see that if the infinite server problem on the line is ρ-competitive, then it is also strictly ρ-competitive. Thus, the statement follows from Theorems <ref> and <ref>. This improves upon both the previous best known lower bounds of 2 for this problem on the line <cit.> and 2.4 on general metric spaces <cit.>. §.§ Layered Graphs A layered graph of depth D is a graph whose (potentially infinitely many) nodes can be arranged in layers 0,1,…,D so that all edges run between adjacent layers and each node – except for a single node in layer 0 – is connected to at least one node of the previous layer. The induced metric space is the set of nodes with the distance being the minimal number of edges of a connecting path. For the purposes of the infinite server problem, the single node in layer 0 is the source. We assume D≥ 2 to avoid trivial cases.Note that a connected graph is layered if and only if it is bipartite. Moreover, any graph can be embedded into a bipartite graph by adding a new node in the middle of each edge. So essentially, layered graphs capture all graph metrics.Let Move Only Outwards (MOO) be some lazy and local algorithmfor the infinite server problem on layered graphs that moves servers along edges only in the direction away from the source. Not surprisingly, the competitive ratioof this simple algorithm is quite bad and we show that it is exactlyD-1/2. Nonetheless, at least for D≤3 this is actually the optimalcompetitive ratio. The competitive ratio of MOO is exactly D-1/2. Upper bound: Consider some final configuration of the algorithm. Let n_j be the number of servers in the jth layer. Then the cost of MOO is =∑_j=1^D jn_j. To obtain an upper bound on , observe that every node occupied by MOO in the final configuration must have been visited by an offline server at least once. We account an offline cost of 1 for each visit of a node on layers 1,…,D-2 and an offline cost of 2 for each visit of a node on layer D. This cost of 2 covers the last two edge-traversals before visiting the layer-D-node, so this may include serving a request on layer D-1. If n_D-1>n_D, then we can account another n_D-1-n_D cost for visiting the remaining at least n_D-1-n_D requested nodes on layer D-1. In summary, ≥∑_j=1^D-2 n_j + 2n_D+(n_D-1-n_D)^+ where (n_D-1-n_D)^+:=max{0,n_D-1-n_D). The upper bound on the competitive ratio follows since / ≤∑_j=1^D jn_j/∑_j=1^D-2 n_j + 2n_D+(n_D-1-n_D)^+≤(D-2)∑_j=1^D-2 n_j + (2D-1)n_D + (D-1)(n_D-1-n_D)^+/∑_j=1^D-2 n_j + 2n_D+(n_D-1-n_D)^+≤ D-1/2 . Lower bound: Let k,n∈ℕ be some large integers. We construct the following graph: Layers 0,…,D-2 consist of one node each and layers D-1 and D consist of infinitely many nodes each, denoted a_0,a_1,a_2,… and b_0, b_1,b_2,… respectively. For each i∈ℕ_0, the k nodes b_ik,b_ik+1…,b_(i+1)k-1 are adjacent to each of the 2k nodes a_ik,a_ik+1,a_(i+2)k-1 and to no other nodes. The set of remaining edges is uniquely determined by the fact that this is a layered graph of depth D. The request sequence consists of n rounds 0,1,…,n-1, where each request in round i is at a node from the list a_ik,a_ik+1,…,a_(i+1)k-1, b_ik,b_ik+1,…,b_(i+1)k-1. Round i starts with requests on the nodes a_ik,a_ik+1,…,a_(i+1)k-1. Then, for j=0,…,k-1, the adversary first requests b_ik+j and then requests whichever node from a_ik,a_ik+1,…,a_(i+1)k-1 has been left by an MOO-server to serve the request at b_ik+j. Note that by definition of MOO and the graph, the server it moves to b_ik+j does indeed come from a_ik,a_ik+1,…,a_(i+1)k-1. In round i, MOO first pays k(D-1) to move k servers to a_ik,a_ik+1,…,a_(i+1)k-1 and then, for each j=0,…,k-1, it pays 1 to move to b_ik+j and D-1 to spawn a new server at the group a_ik,a_ik+1,…,a_(i+1)k-1. Over n rounds this makes a total cost of n(k(D-1)+k(1+D-1))=nk(2D-1). The offline algorithm can serve requests as follows: The requests at a_ik,…,a_(i+1)k-1 at the beginning of round i are served by spawning if i=0 (for cost (d-1)k) and by sending servers from b_(i-1)k,…,b_ik-1 if i≥1 (for cost k). The request at b_ik is served by spawning a server (cost D) and the requests at b_ik+1,…,b_ik+k-1 are served by sending a server from a node in a_ik,…,a_(i+1)k-1 that will not be requested any more (cost 1 each, so k-1 per round). Over n rounds, this adds up to an offline cost of (D-1)k+(n-1)k+n(D+k-1)=2nk+(D-2)k+n(D-1). The ratio of online and offline cost is nk(2D-1)/2nk+(D-2)k+n(D-1)=2D-1/2+D-2/n+D-1/k , which gets arbitrarily close to D-1/2 for n and k large enough. The competitive ratio of the infinite server problem on layered graphs of depth D is exactly 1.5 for D=2, exactly 2.5 for D=3 and at least 3 for D≥ 4. For D=2, the only possibility to move a server closer to the source is from layer 2 to layer 1. But since spawning to layer 1 is at least as good, we can restrict our attention to algorithms of the type MOO. The result follows from Theorem <ref>. For D=3, the upper bound follows from Theorem <ref>. It remains to show the lower bounds for D≥ 3. Fix some large integers k,n∈ℕ. Consider the following layered graph of depth D. For i=0,…,D-1 there exists a node v_i in layer i. The remaining nodes are defined inductively as all nodes obtained by the following two rules: * There exist a set S_0 of 2k nodes and sets A^S_0 and B^S_0 of k nodes. * Let S be a set of 2k nodes such that A^S and B^S exist. Then for each S'⊂ S∪ B^S of size 2k there are sets A^S' and B^S' of k nodes. The nodes in the sets A^S are in layer D-1, the nodes in S_0 and in the sets B^S are in layer D. For a node in some set A^S, the set of adjacent nodes in layer D is S∪ B^S. The remaining edges are so that this is a layered graph with the layers as specified. For purposes of the analysis below, we further define a generation of a node as follows: The nodes v_0,…,v_D-1 and the nodes in S_0 have generation 1. The generation of nodes in A^S and B^S is the maximal generation of any node in S plus 1. Letbe some online algorithm. We assume without loss of generality thatis lazy and local. The adversary chooses the following request sequence against . First, request the nodes in S_0 untilhas a server at each of them. The adversary also moves 2k servers towards these nodes. The adversary uses only these 2k servers for the entire sequence of requests. The remainder of the requests consists of several rounds. We will keep the invariant that at the beginning of the ith round, the 2k adversary servers occupy a set S for which A^S and B^S (with nodes of generation i+1) exist, and the online servers occupy nodes of generation at most i. Clearly this holds before the first round. Let A^S={a_1,…,a_k} and B^S={b_1,…,b_k}. The requests of the ith round are divided into part a and part b, consisting of steps a.1,…,a.k, b.1,…,b.k that are executed in this order. Step a.j consists of the following one or two requests: First request a_j. Ifmoves a server from some b∈ S towards a_j, immediately request b. We can assume that online servers cover A^S after the end of part a (otherwise request nodes in A^S again at the end of part a until this is the case). Step b.j consists of the following two or three requests: First request b_j. Note that any path from a node of generation at most i to b_j contains a node in S, and from any node in S, the shortest paths to b_j include the ones along the nodes in A_S. Thus, sinceis local, it will move a server from some a∈ A^S towards b_j. The second request of step b.j is at this node a and, ifmoves a server from some b∈ S∪ B^S towards a, then the step contains a third request at b. The adversary cost per round is at most 2k+2: For each j=1,…,k, there are at least j nodes in S that will not be requested during steps a.j, …, a.k, b.1, …, b.(k-1). Hence, the adversary can serve all requests of part a for cost k by moving k servers from S towards A^S whilst keeping servers at all nodes of S that will be requested during the steps b.1,…,b.(k-1). Similarly, it can serve the steps b.1,…,b.(k-1) for cost k-1 by moving k-1 servers from A^S to B^S. The final step b.k of the round can be served at cost 3 using the last server in A^S to serve the requests and finish with all 2k offline servers in some set S'⊆ S∪ B^S. We analyze the online cost for the cases D=3 and D≥ 4 separately. If D=3, then the cost for each step a.j is at least 2 and the cost for each step b.j is at least 3. Thus, the cost per round is at least 5k. As k goes to infinity, the ratio of online and offline cost in each round converges to 2.5. As the number of rounds goes to infinity, the online and offline costs before the first round become negligible, which proves the lower bound of 2.5 for D=3. For D≥ 4, we use a potential Φ equal to the number of online servers in layer D-1. During step a.j, either Φ does not change and the cost is at least 2, or Φ increases by 1 and the cost is at least 3. Thus, during step a.j we have Δ≥ 2+ΔΦ and hence during part a we have Δ≥ 2k+ΔΦ. During step b.j, either Φ decreases by 1 and the cost is at least 3, or Φ does not change and the cost is at least 4. Thus, during part b we have Δ≥ 4k+ΔΦ. In total, this adds up to Δ≥ 6k+ΔΦ during the round. Over n rounds, this makes Δ≥ 6nk+ΔΦ≥ 6nk since Φ starts at 0 before the first round and remains nonnegative. As k and n go to infinity, the ratio of our bounds on online and offline cost converges to 3. It remains an open problem to close the gap between the lower bound of 3 and the upper bound of 3.5 for D=4. More importantly, we are interested in the question whether an algorithm better than MOO exists for large D, achieving a competitive ratio of less than D-1/2 on any layered graph of depth D. Note that if no algorithm with a competitive ratio of O(1) as D→∞ exists, then the infinite server problem on general metric spaces would not be competitive.For large D, the lower bound of 3 is certainly not tight: Consider a layered graph where each layer contains one node except that the bottom layer contains infinitely many nodes. By Theorem <ref> (and a matching upper bound shown in <cit.>), the competitive ratio on this graph converges to λ≈3.146 as D→∞.§ ALGORITHMS WITH UNBOUNDED COMPETITIVE RATIO We examine the performance of classical algorithms known for the k-server problem when applied to the infinite server problem. The main focus of this section is a generalization of the Double Coverage algorithm for the line with adjusted server speeds. This idea has proved successful for the (h,k)-server problem (and hence the infinite server problem) on weighted trees <cit.>. However, neither of these algorithms is competitive for the infinite server problem even on the line. §.§ Work Function AlgorithmThe Work Function Algorithm (WFA, <cit.>) for the k-server problem achieves acompetitive ratio of at most 2k-1, which is the best known upper boundfor general metric spaces <cit.>. Given a sequence ofrequests r_1,r_2,… and a configuration C (a multiset of serverpositions), the work function w_t(C) is defined as the minimal cost ofserving the first t requests and ending up in configuration C. IfC_t-1 is the server configuration before the tth request, thealgorithm moves to a configuration C_t that contains r_t andminimizes the quantityw_t(C_t)+d(C_t-1,C_t) ,where d(C_t-1,C_t) is the cost of moving from C_t-1 to C_t. The WFA is not competitive for the infinite server problem on the line. Let the source be at 0 and let p_1,p_2,… be infinitely many points in the interval [1,1+δ] for some small δ>0. Consider the request sequence that always requests the point p_i with i minimal such that p_i is not occupied by an online server. Let σ_k be the prefix of this request sequence until the WFA spawns its kth server. It is easy to see that the WFA spawns its kth server only if the optimal way of serving the already seen requests is to bring k servers to the points p_1,…,p_k. In particular, (σ_k)= k+o(1) as δ→0. Thus, the optimal cost increases by 1+o(1) during the period when the WFA has k spawned servers, and the same is true for the optimal cost of an offline algorithm that is restricted to using k servers only. Let _k be the cost incurred to the WFA during this period. Due to the lower bound of k on the competitive ratio of any k-server algorithm, _k is at least k times this increase of the optimal cost (up to an additive error of order o(1) as δ→0), _k≥ k+o(1). Thus, the total cost of WFA given the request sequence σ_n is at least ∑_k=1^n-1_k = Ω(n^2). Meanwhile, the optimal cost is (σ_n)=n+o(1). Letting n tend to infinity we obtain an unbounded competitive ratio.§.§ Balance and Balance2The algorithm Balance serves a request r by sending a server xthat minimizes the quantity D_x+d(x,r), where D_x is the cumulativedistance traveled by x so far and d(x,r) is the distance between x andr. For the k-server problem, Balance is k-competitive on metricspaces with k+1 points <cit.> and for weighted paging<cit.>. Young showed that for weighted paging against aweak adversary with h servers the competitive ratio of Balance isk/(k-h+1) <cit.>. On general metric spaces however, Balancehas unbounded competitive ratio, even if k=2 <cit.>. It istherefore unsurprising that it is also not competitive for the infinite serverproblem. Balance is not competitive for the infinite server problem on the line. Suppose all servers start at source 0 and consider the request sequence r_0,r_1,r_2,…,r_n where r_i=1-iϵ. As ϵ→0, the optimal cost tends to 1 whereas the cost of Balance tends to n+1. Since n can be arbitrarily high, this shows an unbounded competitive ratio. The intuitive problem of Balance is that it is not greedy enough. Thealgorithm Balance2 by Irani and Rubinfeld <cit.>compensates for this weakness by giving more weight to the distancebetween the server and the request: To serve request r, Balance2 sends aserver x that minimizes the quantity D_x+2d(x,r). Irani and Rubinfeldshowed that, unlike Balance, Balance2 is competitive for two servers(achieving a competitive ratio of at most 10) and they conjectured that itis also competitive for any other finite number of servers <cit.>.However, for the infinite server problem this algorithm is also notcompetitive: Balance2 is not competitive for the infinite server problem on the line. Suppose the source is at 0 and fix some small constant ϵ>0. The request sequence consists of several phases, starting with phase 0. Phase i consists of alternating requests at 1-2iϵ and 1-(2i+1)ϵ. We will ensure that all requests of a phase are served by the same online server, and we call this the active server. As soon as the cumulative distance traveled by the active server exceeds 2-(4i+5)ϵ, the phase ends and a new phase begins. Note that this means that the active server of a phase will not be used to serve any request of a subsequent phase because, by definition of Balance2, the algorithm would rather spawn a new server. Thus, the first request of each phase is served by spawning a new server, which becomes the active server of that phase. While the cumulative distance of the active server is at most 2-(4i+5)ϵ and since its distance from the next request of the phase is always exactly ϵ, its associated quantity D_s+2d(s,r) is at most 2-(4i+3)ϵ. Hence, Balance2 rather uses this server during the phase instead of spawning a new server. Thus, it is indeed the active server that serves all requests of its phase. Let n be the number of phases and choose ϵ small enough so that all requests are in the interval [1/2,1]. Thus, the cost of Balance2 is Ω(n). An offline algorithm could serve all requests with two servers only that move to 1 and 1-ϵ initially and then back towards 1/2, always covering the two points that are requested during a phase, resulting in an offline cost of less than 3. As n goes to infinity, the ratio between online and offline cost becomes arbitrarily large.§.§ Double Coverage VariantsPerhaps more surprising than for WFA and balancing algorithms is that a class of algorithmsextending the Double Coverage (DC) algorithm <cit.> is also not competitive for the infinite server problem. The basic DC algorithm on the line serves each request by an adjacent server. If the request lies between two servers, both servers move towards it at equal speed until one of them reaches the request. A sensible extension of this algorithm seems to be to give different speeds toservers, so that they move away from the source faster than towards it.We consider here only the half-line [0,∞) with the source at the left border 0. Let x_i be the position of the ith serverfrom the right. We use the notation x_i both for its position andfor the server itself.As servers do not overtake each other, x_i is theith spawned server. Let 𝒮 = {s_i ≥ 1 | i ∈ℕ andi ≥ 2} for a monotonic (non-decreasing or non-increasing) sequence of speeds s_i. The algorithm 𝒮-DC is defined as follows: * If there exist servers x_i+1 and x_i to the left and right of the request, move them towards it with speeds s_i+1 and 1 respectively until one of the two reaches it. * If a request does not have a server to its right, move the rightmost server to the request.If s_i=1 for all i, this is precisely the original DC algorithm.We will prove that 𝒮-DC is not competitive. The intuitive reason is that servers move to the right either too slowly or too quickly: Imaginerepeatedly requesting the same n points in some small interval away from the source, until 𝒮-DC covers all n points. One case is that 𝒮-DC spawns too slowly and is therefore defeated by an adversary covering these npositions immediately with n servers. In the other case, the adversary will also use n servers to cover the initial group of requests and then shift its group of servers slowly towards the source, always making requests at the new positions of these offline servers. As 𝒮-DC tries to cover the new requests, it is tricked into spawningtoo many servers. Bothcases lead to an unbounded competitive ratio.The proof consists of several lemmas. The lemmas hold also for non-monotonic speeds and we use monotonicity only to easily combine the lemmas in the end.A useful property of 𝒮-DC is that its cost can be calculated using only the final positions of the servers. Let x_1≥ x_2 ≥… be the server positions of 𝒮-DC after serving a sequence of requests. Then the cost paid is ∑_i=1^∞ z_i x_i where z_1= 1 z_i= z_i-1/s_i + 1 + 1/s_i . The movement x_i of each server can be written as x_i = r_i - l_i where r_i and l_i are the cumulative distances traveled by that server while moving to the right and left respectively. By definition of 𝒮-DC, for all i we have l_i = r_i+1/s_i+1, since any right move (apart from the rightmost server) is accompanied by a left move of another server. Observe that the online cost is = ∑_i=1^∞ (r_i + l_i) = ∑_i=1^∞ (r_i + r_i+1/s_i+1) = ∑_i=1^∞ r_i + ∑_i=2^∞r_i/s_i = r_1 + ∑_i=2^∞ r_i(1+1/s_i) . Similarly, ∑_i=1^∞ z_i x_i = ∑_i=1^∞ z_i(r_i - l_i) = ∑_i=1^∞ z_i r_i - ∑_i=1^∞ z_i r_i+1/s_i+1= z_1 r_1 + ∑_i=2^∞ r_i (z_i - z_i-1/s_i) . By equating (<ref>) and (<ref>) term by term, we get the desired recurrence for z_i. The next lemma takes care of the case when online servers spawn too slowly. If the speeds in 𝒮 satisfy lim inf_n→∞√(∏_i=2^ns_i)=1 then 𝒮-DC is not competitive. For this lower bound we have requests on n arbitrary positions in the interval [1,2], until 𝒮-DC covers them all. The optimal cost is at most 2n. This can be achieved by spawning a fresh server for each requested position. Since for every spawned online server we have x_i ≥ 1, by Lemma <ref> the online cost is =∑_i=1^n z_i x_i ≥∑_i=1^n z_i. Unraveling the recurrence we get that z_i = 1 + 2/s_i+ 2/s_is_i-1 + … + 2/s_i ·…· s_2. Thus, ≥ n + ∑_i=1^n-1∑_j=1^n-i2/∏_k=j+1^j+is_k≥∑_i=1^f(n)∑_j=1^n-i2/∏_k=j+1^j+is_k . where f(n) = ⌊n/2+2log_2∏_i=2^n s_i⌋≤n/2 . We argue that for each i=1,…,f(n), it holds for at least half of the values of j=1,…,n-i that ∏_k=j+1^j+i s_k≤ 2. Indeed, suppose this were not the case for some i. Let us partition the set J={1,…,n-i} of j-values into subsets J_0,…,J_i-1, where J_m contains precisely those numbers from J that are congruent to m modulo i. By assumption, we have ∏_k=j+1^j+i s_k> 2 for at least half the values j∈ J, so this must also be true for at least half the values j∈ J_m for some m. However, this would mean that ∏_k=2^n s_k ≥∏_j∈ J_m∏_k=j+1^j+is_k > 2^|J_m|/2≥ 2^n-i/i/2≥ 2^n/2f(n)-1≥∏_i=2^n s_i , a contradiction because the second inequality is strict. Thus, continuing from (<ref>) we can further bound the online cost as ≥ f(n) n-f(n)/2≥nf(n)/4 . Since the optimal cost is at most 2n, the competitive ratio is at least f(n)/8. However, f(n) is unbounded because n/2+2log_2∏_i=2^n s_i = 1/2/n+2log_2√(∏_i=2^n s_i) and the denominator in the last term gets arbitrarily close to 0. The case of servers being spawned too aggressively is handled by the following lemma. If there exists an unbounded function f(n) such that for each k∈ℕ we have ∏_i=k^k+n s_i≥ f(n), then 𝒮-DC is not competitive. In particular, if lim inf_i→∞s_i>1 then 𝒮-DC is not competitive. Consider the following configuration of online positions and requests, denoted by circles and crosses respectively. [draw,shape=circle,fill] (source) at (-2.5,0) ; [draw,shape=circle] (p1) at (7,0) ; [draw,shape=circle] (p2) at (7.5,0) ; [draw,shape=circle] (p3) at (8.0,0) ; [draw,shape=circle] (p4) at (9.0,0) ; [draw,shape=cross out] (s1) at (0.5,0) ; [draw,shape=cross out] (s2) at (1,0) ; [draw,shape=cross out] (s3) at (1.5,0) ; [draw,shape=cross out] (s4) at (2.5,0) ; [draw,shape=cross out] (s5) at (3.0,0) ; (source) – (-1.75,0); [loosely dotted] (-1.75,0) – (-0.5,0); (-0.5,0) – (s1) – (s2) – (s3); [loosely dotted] (s3) – (s4); (s4) – (s5) – (4,0); [loosely dotted] (4,0) – (6,0); (6,0) – (p1) – (p2) – (p3); [loosely dotted] (p3) – (p4); (p4) – (10,0); [->,] (-1,1) node[above] Newly spawned server at 1-v_1-δ – (s1); [->] (6,-1) node[below] Starting at 1 – (p1); /→in p1/p2, p2/p3, s1/s2, s2/s3, s4/s5 [decoration=brace,mirror,raise=8pt,decorate] (() + (0.01,0)) – node[below=10pt] (start) δ ((→) + (-0.01,0)); [decoration=brace,raise=8pt,decorate] ((p1) + (0.01,0)) – node (start) [above=10pt] n((p4) + (0.01,0)); [decoration=brace,raise=8pt,decorate] ((s1) + (0.01,0)) – node (finish) [above=10pt] n+1((s5) + (0.01,0)); [->] ((start) + (0,0.25)) to [out=145,in=25] node[above=5pt] v_1 ((finish) + (0,0.25)); We start by spawning n online servers grouped tightly, with the leftmost being at distance 1 from the source and a very small gap δ between them. This is easily accomplished by repeating several requests on those points. Afterwards, we shift this group of n servers (by means of requests on new n+1 points) to the left by v_1, chosen so that the n+1 points are covered exactly by the n old servers plus a newly spawned one, which occupies the leftmost requested position 1-v_1-δ. This is repeated again and again, shifting each time the leftmost n spawned servers a new v_k to the left via multiple requests on n+1 positions. The goal each time is to pull a new server from the source and leave one behind forever, thus achieving an arbitrarily high competitive ratio for 𝒮-DC variants that spawn servers too fast. The offline cost can be calculated easily. The offline algorithm uses n servers to cover the first group of n requested points in the interval [1,1+nδ]. Then it adds one more server and moves the group of n+1 servers to the left to satisfy all of the following requests. At most, the group of offline servers will return close to the source, yielding an optimal cost of ≤ 2(n+1)(1+nδ) = O(n) since δ is very small. To bound the online cost, we need to compute the values v_k first. Let ℓ_i^k and r_i^k denote the cumulative distance to the left and right respectively traveled by x_i during the left shift by v_k of the group x_k,x_k+1,…,x_k+n-1. The nonzero values among these are ℓ_k^k= v_k r_k+1^k= v_ks_k+1 ℓ_k+1^k= v_k(1+s_k+1) r_k+2^k= v_k(s_k+2+s_k+1s_k+2) ℓ_k+2^k= v_k(1+s_k+2+s_k+1s_k+2) ⋮ r_k+n^k= v_k(s_k+n + s_k+n-1s_k+n + … + ∏_j=k+1^k+ns_j) = v_k ∑_i=k+1^k+n∏_j=i^k+n s_j . On the other hand, the new position of the server x_k+n pulled from the source during these moves is 1-∑_i=1^kv_i-kδ. Equating this with (<ref>) and solving for v_k yields (and assuming that n is even) v_k= 1 - ∑_i=1^k-1v_i - kδ/1 + ∑_i=k+1^k+n∏_j=i^k+n s_j≤1/n/2∏_j=k+n/2^k+ns_j≤2/nf(n/2) . We will calculate the number of repetitions before the left border of the group of servers (just) passes 1/2. If l is the number of repetitions, we have 1/2≤∑_k=1^l (v_k + δ) ≤2l/nf(n/2) + lδ and for sufficiently small δ this means that l ≥n/5f(n/2) If we do l-1 repetitions, then each of them will pull a new server at least 1/2 away from the source, resulting in an online cost of Ω(n)· f(n/2). As the offline cost is O(n) and f(n) is unbounded, the algorithm is not competitive. Since the sequence of speeds s_i is monotonic and bounded frombelow by 1, we have either lim_i →∞ s_i = 1, in which case Lemma <ref> applies, or otherwise lim inf_i →∞ s_i >1 and Lemma <ref> applies. In any case, the competitive ratio isunbounded: Algorithm 𝒮-DC is not competitive for any 𝒮. § REDUCTION TO BOUNDED SPACES In this section we show a reduction from the infinite server problem on general metric spaces to bounded subspaces. Specifically, a metric space can be partitioned into “rings” of points whose distance from the source is between r^n and r^n+1, where r>1 is fixed and n∈ℤ. We show that if the infinite server problem is strictly ρ-competitive on each ring, then it is competitive on the entire metric space. Let M be a metric space and s∈ M and let r>1. For n∈ℤ let M_n={s}∪{p∈ M | d(s,p)∈[r^n,r^n+1)}. If for each n the infinite server problem on (M_n,s) is strictly ρ-competitive, then on (M,s) it is strictly 4r-1/r-1ρ-competitive. Let _n be a ρ-competitive algorithm for the infinite server problem on (M_n,s). For a request sequence σ, let σ_n be the subsequence of requests in M_n. Letbe the algorithm for (M,s) that uses different servers for each of the subsequences σ_n and serves them independently according to _n. The total online cost is (σ)=∑_n_n(σ_n)≤ρ∑_n(σ_n). To finish the proof, it suffices to show that ∑_n(σ_n) ≤4r-1/r-1(σ) . Thus, we only need to analyze the offline cost. We do this for each offline server separately. Fix some offline server x. Let N_0 and N_1 be the minimal and maximal values of n such that x visits M_n. We can assume without loss of generality (by adding virtual points to the metric space) that whenever x moves from M_n to M_n' for some n<n', it travels across points p_n+1,p_n+2,…,p_n' with d(s,p_i)=r^i, and similarly for n>n'. The movements of server x can be tracked by many servers, one server x_n in every set M_n for N_0≤ n≤ N_1. When server x is in M_n, server x_n is exactly at the same position tracking the movement of x. When server x exits M_n at some point p at the boundary to M_n-1 or M_n+1, server x_n freezes at p. The movement cost of x_n can be partitioned into the cost of deploying x_n at the first point visited in M_n, the tracking cost within M_n, and the cost of of relocating x_n whenever x re-enters M_n at a location different from the last exiting location. The total tracking cost of all servers x_n is bounded by the distance traveled by x. The cost of deploying all servers x_n is ∑_n=N_0^N_1r^n≤∑_n=-∞^N_1r^n=r^N_1+1/(r-1), which is at most r/r-1 times the total movement of server x, because the latter is at least r^N_1. To bound the relocating cost, say x exits M_n at p and re-enters it at p'. Then p and p' are at the boundary of M_n and M_n+u for u∈{-1,+1}. Let b be the distance traveled by x in M_n+u between the times when it is entered at p and when it is next exited. If this exiting is at p', then the relocating cost d(p,p') is at most b by the triangle inequality. Otherwise, x exits M_n+u at a point p” at the boundary of M_n+u and M_n+2u. If u=1, then d(p,p')≤ d(s,p)+d(s,p')= 2r^n+1 and b≥ d(p,p”)≥ d(s,p”) - d(s,p) = r^n+2 - r^n+1=(r-1)r^n+1. If u=-1, then d(p,p')≤ d(s,p)+d(s,p')= 2r^n and b≥ d(p,p”)≥ d(s,p) - d(s,p”) = r^n - r^n-1=r-1/rr^n. In both cases, the relocating cost d(p,p') is at most 2r/r-1b. Thus, the total relocating cost of all servers x_n is at most 2r/r-1 times the total distance traveled by x. Thus, the sum of deployment, tracking and relocating cost of the servers x_n is at most 4r-1/r-1 times the distance traveled by x. This shows (<ref>), giving the statement of the theorem. The last theorem can also be slightly generalized to the case where instead of strict ρ-competitiveness, an additive term proportional to r^n is allowed. It is not difficult to show the following specialization for the line, where the premise can be weakened to require competitiveness only on a single interval: Let 0<a<b. The infinite server problem is competitive on the line if and only if it is competitive on ({0}∪[a,b],0). Another consequence of Theorem <ref> is a reduction to spaces where the source is at a uniform distance from all other points. Suppose there exists ρ so that the infinite server problem is strictly ρ-competitive on any metric space where the distance from the source to any other point is the same. Then the infinite server problem on general metric spaces is competitive. Follows from Theorem <ref> by increasing the distance from s to the other points in M_n to r^n+1, making a multiplicative error of at most r. § OPEN PROBLEMSThe most obvious open problem is whether the infinite server problem is competitive on general metric spaces. A challenging special case is to resolve the question for the real line. Similarly, improving the MOOalgorithm and settling the question for layered graphs remains open. It would also be interesting to find a metric space with a competitive ratio greater than 3.146.Another possible line of research is to consider randomized algorithms. plainurl

http://arxiv.org/abs/1702.08474v1

Legendre Functions of Fractional Degree and OrderRobert S. Maier Depts. of Mathematics and Physics University of Arizona Tucson, AZ85721Tel.: +1 520 621 2617Fax:+1 520 621 8322rsm@math.arizona.edu Associated Legendre Functions and Spherical Harmonics of Fractional Degree and Order Robert S. Maier Received: date / Accepted: date ==================================================================================== Trigonometric formulas are derived for certain families of associated Legendre functions of fractional degree and order, for use in approximation theory.These functions are algebraic, and when viewed as Gauss hypergeometric functions, belong to types classified by Schwarz, with dihedral, tetrahedral, or octahedral monodromy.The dihedral Legendre functions are expressed in terms of Jacobi polynomials.For the last two monodromy types, an underlying `octahedral' polynomial, indexed by the degree and order and having a non-classical kind of orthogonality, is identified, and recurrences for it are worked out.It is a (generalized) Heun polynomial, not a hypergeometric one.For each of these families of algebraic associated Legendre functions, a representation of the rank-2 Lie algebra 𝔰𝔬(5,ℂ) is generated by the ladder operators that shift the degree and order of the corresponding solid harmonics.All such representations of 𝔰𝔬(5,ℂ) are shown to have a common value for each of its two Casimir invariants.The Dirac singleton representations of 𝔰𝔬(3,2) are included. 33C45 33C47 33C55 22E70 § INTRODUCTIONThe first-kind associated Legendre functions P_ν^μ(z), or the Ferrers versions P_ν^μ(z), are classical.(P_ν^μ(z) and P_ν^μ(z) are continuations of each other, with respective real domains z∈(1,∞) and z∈(-1,1).)The roles they play when the degree ν and order μ equal integers n,m are familiar. The Legendre, or Ferrers polynomials P_n(z) P_n^0(z), n=0,1,2,…, are orthogonal on [-1,1] and are used in series expansions.The spherical harmonics Y_n^m(θ,ϕ)∝ P_n^m(cosθ) e^ imϕ are orthogonal on the symmetric space S^2=SO(3)/SO(2), and appear in harmonic analysis based on the Lie group SO(3).It is less well known that Ferrers functions P_ν^μ(z) of a fixed order μ, and degrees that may be non-integral but are spaced by integers, can also be used in series expansions.The fundamental relation, due to Love and Hunter <cit.>, is one of biorthogonality:∫_-1^1P_ν^μ(z)P_ν'^-μ(-z) dz =0,which holds if (i) Re μ∈(-1,1), and (ii) the degrees ν,ν' differ by a nonzero even integer and are not half-odd-integers.For suitable ν_0,μ∈ℂ, this makes possible bilateral expansions of the formf(z) = ∑_n=-∞^∞ c_n P_ν_0+2n^μ(z),z∈(-1,1),and in particular, the calculation of the coefficients c_n as inner products in L^2[-1,1].(This is the usual Legendre expansion if (ν_0,μ)=(0,0), as P_-ν-1= P_ν for all ν.)For conditions on f sufficient for (interior) pointwise convergence, see <cit.>.The restriction to Re μ∈(-1,1) comes from the requirement that the expansion functions lie in L^2[-1,1].If the order μ is not a positive integer, P_ν^μ(z) will have leading behavior as z→1^- proportional to (1- z)^-μ/2, but in general its leading behavior as z→(-1)^+ comprises two terms: one proportional to (1+ z)^-μ/2, and one to (1+ z)^+μ/2.The implications for convergence of the integral in (<ref>) are obvious.These asymptotics have motivated the suggestion by Pinsky <cit.> that when Re μ<0, the series (<ref>) should really be viewed as an expansion of [(1- z)/ (1+ z)]^μ/2f(z) in the functions [(1- z)/ (1+ z)]^μ/2P_ν_0+2n^μ(z). This enables a discussion of endpoint convergence, because the latter functions do not diverge as z→1^-,(-1)^+.It is not usually the case that P_ν^μ(z) and P_ν^μ(z) are elementary functions, unless of course ν and μ are integers. This may be why such expansions as (<ref>) have been used infrequently.In this paper, we derive explicit, trigonometrically parametrized formulas for several families of Legendre functions, expressing P_ν^μ(z), P_ν^μ(z), and their second-kind counterparts Q_ν^μ(z), Q_ν^μ(z), as elementary functions.In each family, ν,μ are non-integral but are spaced by integers: (ν,μ)∈(ν_0,μ_0)+ℤ^2 for some fractional ν_0,μ_0.The simplest example isP_-1/6+n^1/4+m(cosθ)= 2^-2m-3nΓ(34-m)^-1×(sinθ)^-1/4-m B_+^1/4+3m+3n r^m_n(B_-/B_+),θ∈(0,π),B_± = B_±(θ) cos(θ/3)±√(4 cos^2(θ/3)-1/3),where (n,m)∈ℤ^2.Here, r^m_n=r^m_n(u) is an `octahedral' rational function that if n,m≥0 is a polynomial of degree 3n+2m in u; in the base case n=m=0, it equals unity.It satisfies differential recurrences on n and m, and three-term non-differential recurrences, as well.The function r^m_0(u) has a hypergeometric representation in the Gauss function _2F_1: it equals _2F_1(-2m,-1/4-3m;3/4-m| u).But r^0_n(u), which according to (<ref>), appears in series of the form (<ref>) when μ=1/4, is less classical.It satisfies a second-order differential equation on the Riemann u-sphere with four singular points, not three; so (if n≥0) it is a Heun polynomial, not a hypergeometric one.The functions {r^0_n(u)}_n∈ℤ are mutually orthogonal on the u-interval [0,1], in a sense that follows from (<ref>), but the orthogonality is of an unusual Sturm–Liouville kind.It is clear from (<ref>) that for any n,m∈ℤ, the function P_-1/6+n^1/4+m(z=cosθ) depends algebraically on z, and can be evaluated using radicals.Each of the function families considered in this paper is similarly algebraic, and because any Legendre function can be written in terms of _2F_1, the results below are really trigonometric parametrizations of families of algebraic _2F_1's.To see a link to prior work, recall from Frobenius theory that each Legendre function of degree ν and order μ satisfies a differential equation on the Riemann sphere with three singular points, the characteristic exponent differences at which are μ,μ,2ν+1.It is a classical result of Schwarz (see <cit.>, and for more recent expositions, <cit.>, <cit.> and <cit.>) that any such equation will have only algebraic solutions only if the (unordered, unsigned) triple of exponent differences falls into one of several classes.The triples from (ν,μ)=(-1/6,1/4)+(n,m), as in (<ref>), are (1/4,1/4,2/3)+(m,m,2n), and they lie in Schwarz's octahedral class V.The families treated below include octahedral ones, with (ν+1/2,μ)∈(±1/3,±1/4)+ℤ^2, and tetrahedral ones, with (ν+1/2,μ)∈(±1/4,±1/3)+ℤ^2 or (±1/3,±1/3)+ℤ^2; the Schwarz classes for the latter being II and III.The resulting Legendre functions are octahedral or tetrahedral in the sense that their defining differential equation, on the Riemann z-sphere, has as its projective monodromy group a finite subgroup of the Möbius group PSL(2,ℝ), which is octahedral or tetrahedral.This will not be developed at length, but there is a strong geometric reason why {r^m_n(u)}_n,m∈ℤ deserve to be called octahedral functions, or (when n,m≥0) polynomials.For general n,m, the lifted function r̃^m_n = r̃^m_n(s) r^m_n(u=s^4) turns out to satisfy an equation on the Riemann s-sphere with 14 singular points.These include s=0,±1,± i,∞, which are the six vertices of an octahedron inscribed in the sphere; and also, the centers of its eight faces.Up to normalization, the doubly indexed functions r^m_n(u) are identical to specializations of triply-indexed ones introduced by Ochiai and Yoshida in their groundbreaking work on algebraic _2F_1's <cit.>.For Schwarz classes such as the octahedral and tetrahedral, they considered the effects of displacing the triple of exponent differences, not by (m,m,2n) as in the Legendre case, but by general elements of ℤ^3.It is a key result of the present paper that in the Legendre case, when the triple has only two degrees of freedom, it is far easier to derive and solve recurrences on exponent displacements.Schwarz's classification of algebraic _2F_1's also includes a dihedral class (class I) and a related `cyclic' class (unnumbered but called class O here).Legendre functions lie in class I when the order μ is a half-odd-integer, and in class O when the degree ν is an integer.We obtain explicit formulas for the Legendre (and Ferrers) functions in the respective families, of the first and second kinds.The simplest dihedral example isP_-1/2+α^1/2+m(cosθ)= √(2/π) m!× (sinθ)^-1/2 { i^m e^ iαθP_m^(α,-α)( i θ)}_α,+, θ∈(0,π),where m=0,1,2,…, and α∈ℂ is arbitrary.Here, P_m^(α,-α) is the Jacobi polynomial of degree m, and {·}_α,+ signifies the even part under α↦-α.When m=0, this becomes a trigonometric version of a well-known algebraic formula <cit.>; and when α=1/2, it expresses P_0^1/2+m in terms of the mth Chebyshev polynomial of the third kind.But the general Jacobi representation (<ref>) is new.There is a significant literature on `dihedral polynomials' appearing in dihedrally symmetric _2F_1's <cit.>, and Vidūnas has shown they can be expressed as terminating Appell series <cit.>.Focusing on the Legendre case, when two of the three exponent differences are equal, leads to such simpler formulas as (<ref>), for both the dihedral and cyclic families.Constructing bilateral Ferrers series of the form (<ref>) is facilitated by the explicit formulas derived below for the Legendre and Ferrers functions in the several families.But the functions { P^μ_0+m_ν_0+n(z=cosθ) }, (n,m)∈ℤ^2, and the corresponding spherical harmonics { Y^μ_0+m_ν_0+n(θ,ϕ) }, do not fit into conventional SO(3)-based harmonic analysis unless (ν_0,μ_0)=(0,0), when the latter are the usual surface harmonics on S^2=SO(3)/SO(2).In the octahedral and tetrahedral families (and also the dihedral and cyclic, if ν_0 resp. μ_0 is rational), it is nonetheless the case that each spherical harmonic can be viewed as a finite-valued function on S^2.This is due both to ν_0,μ_0 being rational, and to the algebraicity of P_ν_0+n^μ_0+m(z) in its argument z, as seen in (<ref>).To begin to relate the present results to harmonic analysis, we interpret in Lie-theoretic terms the recurrences satisfied by any family { P_ν_0+n^μ_0+m(z)} or {Y_ν_0+n^μ_0+m(θ,ϕ)}, (n,m)∈ℤ^2, which are based on first-order differential operators that perform ladder operations.It is well known that for any (ν_0,μ_0), there is an infinite-dimensional representation of the Lie algebra 𝔰𝔬(3,ℝ) on the span of {Y_ν_0^μ_0+m(z)}_m∈ℤ.(In the case when (ν_0,μ_0)∈ℤ_≥×ℤ, this includes as an irreducible constituent the familiar (2ν_0+1)-dimensional representation carried by the span of Y_ν_0^-ν_0,…,Y_ν_0^ν_0.)There is also a representation of 𝔰𝔬(2,1) on the span of {Y_ν_0+n^μ_0(z)}_n∈ℤ.The real algebras 𝔰𝔬(3,ℝ),𝔰𝔬(2,1) are real forms of the complex Lie algebra 𝔰𝔬(3,ℂ).As we explain, these `order' and `degree' algebras generate over ℂ a 10-dimensional, rank-2 complex Lie algebra isomorphic to 𝔰𝔬(5,ℂ), which for any (ν_0,μ_0), acts differentially on the family {r^ν_0+nY_ν_0+n^μ_0+m(θ,ϕ)}, (n,m)∈ℤ^2, of generalized solid harmonics on ℝ^3.The root system of 𝔰𝔬(5,ℂ), of type B_2, comprises the eight displacement vectors Δ(ν,μ)=(0,±1),(±1,0), (±1,±1), which yield four ladders on (ν,μ); and for each ladder, there are differential operators for raising and lowering, a differential recurrence satisfied by P_ν^μ(z=cosθ), and a three-term non-differential recurrence.The ones coming from the roots (±1,±1), such as the `diagonal' recurrences√(1-z^2) P_ν±1^μ+1(z) + [±(2ν+1)(1-z^2) + 2μ]P_ν^μ(z) + [(ν+12)±(μ-12)] [(ν+12)±(μ-32)] √(1-z^2) P_ν∓1^μ-1(z) =0,may be given here for the first time.(In this identity, P may be replaced by Q.)Connections between associated Legendre/Ferrers functions, or spherical harmonics, and the complex Lie algebra 𝔰𝔬(5,ℂ) [or its real forms 𝔰𝔬(3,2),𝔰𝔬(4,1) and 𝔰𝔬(5,ℝ)] are known to exist.(See <cit.> and <cit.>, and <cit.> in the physics literature; also <cit.> for hyperspherical extensions.)But, most work has focused on functions of integral degree and order.The octahedral, tetrahedral, dihedral, and cyclic families yield explicit infinite-dimensional representations of 𝔰𝔬(5,ℂ) and its real forms, which are carried by finite-valued solid harmonics on ℝ^3.When (ν_0,μ_0)=(12,12) or (0,0), the representation of 𝔰𝔬(3,2) turns out to include a known skew-Hermitian one, of the Dirac singleton type (the `Di' or the `Rac' one, respectively).But in general, these Lie algebra representations are new, non-skew-Hermitian ones, which do not integrate to unitary representations of the corresponding Lie group.It is shown below that any of these representations of 𝔰𝔬(5,ℂ) [or any of its real forms], carried by a harmonic family {r^ν_0+nY_ν_0+n^μ_0+m(θ,ϕ)}, (n,m)∈ℤ^2, is of a distinguished kind, in the sense that it assigns special values to the two Casimir invariants of the algebra, these values being independent of (ν_0,μ_0); cf. <cit.>.This paper is structured as follows.In  <ref>, facts on Legendre/Ferrers functions that will be needed are reviewed. In  <ref>, the key results on the octahedral functions r_n^m are stated, and explicit formulas for octahedral Legendre/Ferrers functions are derived.These are extended to the tetrahedral families in  <ref> and <ref>. In  <ref>, the results in  <ref> are proved. In  <ref>, Love–Hunter biorthogonality is related to Sturm–Liouville biorthogonality.In  <ref>, formulas for Legendre/Ferrers functions in the cyclic and dihedral families are derived, and Love–Hunter expansions in dihedral Ferrers functions are briefly explored.In  <ref>, recurrences on the degree and order, valid for any (ν_0,μ_0), are derived, and are given a Lie-theoretic interpretation: 𝔰𝔬(5,ℂ) and its real forms are introduced, and their representations carried by solid harmonics are examined.§ PRELIMINARIESThe (associated) Legendre equation is the second-order differential equationd/dz[ (1-z^2) dp/ dz]+ [ν(ν+1) - μ^2/1-z^2]p=0on the complex z-plane.For there to be single-valued solutions, the plane is cut along the real axis either from -∞ to 1 (the Legendre choice), or from -∞ to -1 and from 1 to +∞ (the Ferrers choice).The respective solution spaces have P_ν^μ(z),Q_ν^μ(z) and P_ν^μ(z), Q_ν^μ(z) as bases, except in degenerate cases indicated below.At fixed real μ, Eq. (<ref>) can be viewed as a singular Sturm–Liouville equation on the real Ferrers domain [-1,1], the endpoints of which are of Weyl's `limit circle' type if μ∈(-1,1). (See <cit.>.)In this case, all solutions p=p(z) lie in L^2[-1,1], irrespective of ν; but the same is not true when μ∉(-1,1), which is why such orthogonality relations as (<ref>) can only be obtained if μ∈(-1,1), or more generally if Re μ∈(-1,1).Further light on endpoint behavior is shed by Frobenius theory. Equation (<ref>) has regular singular points at z=-1,1 and ∞, with respective characteristic exponents expressed in terms of the degree ν and order μ as +μ/2,-μ/2; +μ/2,-μ/2; and -ν,ν+1.The exponent differences are μ,μ,2ν+1. The functions P_ν^μ, P_ν^μ are Frobenius solutions associated to the exponent -μ/2 at z=1, and the second Legendre function Q_ν^μ is associated to the exponent ν+1 at z=∞. (The second Ferrers function Q_ν^μ is a combination of two Frobenius solutions.)These functions are defined to be analytic (or rather meromorphic) in ν,μ <cit.>, the Legendre functions having the normalizationsP_ν^μ(z)∼2^μ/2/Γ(1-μ) (z-1)^-μ/2,z→1,Q̂_ν^μ(z)∼√(π)/2^ν+1 Γ(ν+μ+1)/Γ(ν+3/2) z^-ν-1,z→∞,by convention <cit.>.The notation Q̂_ν^μ e^-μπ iQ_ν^μ will be used henceforth; it removes an awkward e^μπ i factor.The formulas (<ref>),(<ref>) apply if the gammas are finite; the asymptotics when they are not are given in <cit.>.One such degenerate case is when μ=1,2,….Then, P_-μ^μ,…,P_μ-1^μ≡ 0.A familiar example is when the degree is a non-negative integer n. Then, P_n^m, P_n^m≡0 if the order is an integer m>n; though not if m<-n.Another degenerate case is when ν+μ is a negative integer.If so, Q̂_ν^μ is undefined, as (<ref>) suggests; except when ν=-3/2,-5/2,….Then, Q̂_ν^ν+1,…, Q̂_ν^-(ν+1) are defined.The Ferrers functions are related to the Legendre ones on their common domains, which are the upper and lower half-planes ±Im z>0, by P_ν^μ =e^±μπ i/2 P_ν^μQ_ν^μ =e^∓μπ i/2Q̂_ν^μ± i (π/2)e^±μπ i/2 P_ν^μ. Thus P_ν^μ, P_ν^μ are related by analytic continuation, up to phase.Going from P_ν^μ to P_ν^μ typically involves replacing a factor (z-1)^-μ/2 by (1-z)^-μ/2; for instance, P_1^-1(z), P_1^-1(z) are 12√(z^2-1), 12√(1-z^2).Also, owing to (<ref>), Q_ν^μ is undefined iff Q̂_ν^μ is.Equation (<ref>) is invariant under ν↦-ν-1, μ↦-μ, and z↦ -z, so that in nondegenerate cases, the Legendre and Ferrers functions with ν replaced by -ν-1, μ by -μ, and/or z by -z, can be expressed as combinations of any two (at most) of P_ν^μ,Q̂_ν^μ, P_ν^μ,Q_ν^μ.Some `connection' formulas of this type, which will be needed below, are P_-ν-1^μ=P_ν^μ, P_-ν-1^μ= P_ν^μ,Q̂_ν^-μ/ Γ(ν-μ+1) = Q̂_ν^μ / Γ(ν+μ+1),the P→Q̂ reductionP_ν^μ = (νπ) Γ(ν-μ+1)^-1Γ(-μ-ν)^-1(Q̂_-ν-1^-μ - Q̂_ν^-μ),the Q̂→P reduction(2/π)Q̂_ν^μ =(μπ) P_ν^μ -(μπ) Γ(ν+μ+1)/Γ(ν-μ+1) P_ν^-μ,and the Q→ P reduction(2/π) Q_ν^μ =(μπ)P_ν^μ -(μπ) Γ(ν+μ+1)/Γ(ν-μ+1)P_ν^-μ.(See <cit.>.)It follows from (<ref>) that if μ=1/2,3/2,…, then Q_-μ^μ,…, Q_μ-1^μ≡0.The functions P_ν^μ,Q̂_ν^μ are known to have the hypergeometric representationsP_ν^μ(z)= 1/Γ(1-μ) (z+1/z-1)^μ/2_2F_1(-ν, ν+1; 1-μ; 1-z/2) = 2^μ/Γ(1-μ) z^ν+μ/(z^2-1)^μ/2 _2F_1(-ν/2-μ/2, -ν/2-μ/2+1/2; 1-μ; 1-1/z^2),Q̂_ν^μ(z)= √(π)/2^ν+1 Γ(ν+μ+1)/Γ(ν+3/2) (z+1)^μ/2/(z-1)^μ/2 + ν + 1×_2F_1(ν+1,ν+μ+1; 2ν+2; 2/1-z) = √(π)/2^ν+1 Γ(ν+μ+1)/Γ(ν+3/2) (z^2-1)^μ/2/z^ν+μ+1×_2F_1(ν/2+μ/2+1/2,ν/2+μ/2+1; ν+3/2; 1/z^2). (For P_ν^μ, replace z-1 in the prefactor on the right of (<ref>) by 1-z; the alternative expressions (<ref>),(<ref>) come from (<ref>),(<ref>) by quadratic hypergeometric transformations.)Here, _2F_1(a,b;c;x) is the Gauss function with parameters a,b;c, defined (on the disk |x|<1, at least) by the Maclaurin series ∑_k=0^∞ [(a)_k(b)_k/(c)_k(1)_k] x^k.In this and below, the notation (d)_k is used for the rising factorial, i.e.,(d)_k(d)…(d+k-1), k≥0; [(d-k')…(d-1)]^-1, k=-k'≤0.(The unusual second half of this definition, which extends the meaning of (d)_k to negative k so that (d)_k = [(d+k)_-k]^-1 for all k∈ℤ, will be needed below.)If in any _2F_1(a,b;c;x) in (<ref>), the denominator parameter c is a non-positive integer and there is an apparent division by zero, the taking of a limit is to be understood.The Gauss equation satisfied by _2F_1(a,b;c;x) has the three singular points x=0,1,∞, with respective exponent differences 1- c, c- a- b, b- a.Taking into account either of (<ref>),(<ref>), one sees that this triple is consistent with the exponent differences μ,μ,2ν+1 at the singular points z=1,-1,∞ of the Legendre equation (<ref>).Schwarz's results on algebraicity were originally phrased in terms of the Gauss equation, its solutions such as _2F_1, and the (unordered, unsigned) triple 1- c, c- a- b, b- a; but they extend to the Legendre equation, its solutions, and the triple μ,μ,2ν+1.§ OCTAHEDRAL FORMULAS (SCHWARZ CLASS V)This section and Sections <ref> and <ref> derive parametric formulas for Legendre and Ferrers functions that are either octahedral or tetrahedral (two types).The formulas involve the octahedral polynomials, or functions, {r^m_n(u)}_n,m∈ℤ. Section <ref> defines these rational functions and states several results, the proofs of which are deferred to  <ref>. §.§ Indexed Functions and PolynomialsFor n,m∈ℤ, the rational functions r^m_n=r^m_n(u) and their `conjugates' r^m_n=r^m_n(u) are defined implicitly by_2F_1( -124-m/2-n/2, 1124-m/2-n/234-m| R(u)) = [(1+u)(1-34u+u^2)]^-1/12-m-n r^m_n(u),_2F_1( 524+m/2-n/2, 1724+m/2-n/254+m| R(u)) =[(1+u)(1-34u+u^2)]^5/12+m-n(1-u)^-1-4m r^m_n(u),which hold on a neighborhood of u=0.Here,R(u)-108 u(1-u)^4/[(1+u)(1-34u+u^2)]^2 = 1 - (1+14u+u^2)^3/[(1+u)(1-34u+u^2)]^2, -108 p_ v(u)/p_ e(u)^2 = 1 -p_ f(u)^3/p_ e(u)^2,where p_ v,p_ e,p_ f are the polynomials u(1-u)^4, (1+ u) (1-34u + u^2), 1+ 14u+ u^2, which satisfy p_ e^2 - p_ f^3 + 108 p_ v=0. Equivalently,R = T(3+T)^2/(1+3T)^2 = 1- (1-T)^3/(1+3T)^2,where T=T(u)-12u/(1+u)^2.[For later use, note that (<ref>) is familiar from trigonometry as a `triple-angle' formula: R=tanh^2ξ if T=tanh^2(ξ/3); R=^2ξ if T=^2(ξ/3); and R=-tan^2θ if T=-tan^2(θ/3).] It is clear from the definition that r^m_n,r^m_n are analytic at u=0, at which they equal unity; though it is not obvious that they are rational in u.But it is easily checked that the Gauss equations satisfied by the two _2F_1(x) functions have respective exponent differences (at the singular points x=0,1,∞) equal to (1/4,1/3,1/2)+(m,n,0) and (-1/4,1/3,1/2)+(-m,n,0).These triples lie in Schwarz's octahedral class IV, so each _2F_1(x) must be an algebraic function of x.The definition implicitly asserts that if these algebraic _2F_1's are parametrized by the degree-6 rational function x=R(u), the resulting dependences on u will be captured by certain rational r^m_n=r^m_n(u), r^m_n=r^m_n(u).In the terminology of <cit.>, these are octahedral functions of u.(i) For n,m≥0, r^m_n(u) is a polynomial of degree 3n+2m in u, to be called the octahedral polynomial indexed by (n,m)∈ℤ^2_≥.Its coefficient of u^0 is unity, and its coefficient of u^3n+2m isd^m_n (-)^m+n 3^3m (5/12)_m-n(13/12)_m+n/(1/4)_m (5/4)_m.(ii) For unrestricted (n,m)∈ℤ^2, r^m_n(u) is a rational function that equals unity at u=0 and is asymptotic to d^m_nu^3n+2m as u→∞.(iii) The conjugate function r^m_n is related to r^m_n byr^m_n(u) = (d^m_n)^-1 u^3n+2m r^m_n(1/u),so that if n,m≥0, r^m_n is a reversed version of the polynomial r^m_n, scaled to equal unity at u=0. The functional form of the octahedral functions that are not polynomials, which are indexed by (n,m)∈ℤ^2∖ℤ^2_≥, is not complicated.For any n,m≥0,r^m_n(u)= Π_3n+2m(u),r^-m-1_n(u)= (1-u)^-3-4m Π_1+3n+2m(u),r^m_-n-1(u)= (1+14u+u^2)^-2-3n Π_1+3n+2m(u),r^-m-1_-n-1(u)= (1-u)^-3-4m (1+14u+u^2)^-2-3n Π_2+3n+2m(u),where on each line, Π_k(u) signifies a polynomial of degree k in u, with its coefficient of u^0 equalling unity, and its coefficient of u^k coming from the preceding theorem. On their indices n,m, the r^m_n satisfy both differential recurrences and three-term non-differential recurrences.The former are given in  <ref> (see Theorem <ref>), and the latter are as follows.The octahedral functions r^m_n=r^m_n(u), indexed by (n,m)∈ℤ^2, satisfy second-order (i.e., three-term) recurrences on m and n, namely(4m-3)(4m+1) r^m+1_n - (4m-3)(4m+1)p_ e(u) r^m_n-3(12m-12n-7)(12m+12n+1) p_ v(u)r^m-1_n = 0,(12n-12m+7) r^m_n+1 - 8(3n+1)p_ e(u)r^m_n+(12n+12m+1)p_ f^3(u)r^m_n-1 = 0,where p_ v,p_ e,p_ f are the polynomials in u, satisfying p_ e^2 - p_ f^3 + 108 p_ v = 0, that were introduced in Definition <ref>.Moreover, they satisfy3(4m-3)(4m+1) r^m+1_n+1 - (4m-3)[(12m+12n+7)p_ f^3(u) - 4(3n+1)p_ e^2(u)]r^m_n+ 9(12m+12n+1)(12m+12n-11) p_ v(u) p_ f^3(u)r^m-1_n-1 = 0,3(4m-3)(4m+1) p_ f^3(u) r^m+1_n-1 - (4m-3)[(12m-12n-1)p_ f^3(u) + 4(3n+1)p_ e^2(u)]r^m_n+ 9(12m-12n-7)(12m-12n-19) p_ v(u)r^m-1_n+1 = 0,which are second-order `diagonal' recurrences. From the first two recurrences in this theorem, one can compute r^m_n for any n,m∈ℤ, if one begins with r^0_0,r^1_0,r^0_1, which are low-degree polynomials in u computable `by hand.'In fact,r^0_0(u)=1, r^1_0(u)=1-26u-39u^2,r^0_1(u) = 1-39u - 1957 u^2 + 137 u^3,r^1_1(u) = 1+175u - 150u^2 + 3550u^3 + 325u^4 + 195u^5.By specializing to u=1 (at which p_ v,p_ e,p_ f equal 0,-64,36), one can prove by induction that r_n^m(1)=(-64)^m+n if m≥0.Examples of octahedral functions that are not polynomials because they have at least one negative index, illustrating Theorem <ref>, arer^-1_0(u)= (1-u)^-3 (1+17 u),r^0_-1(u)= (1+14u+u^2)^-2 (1-5 u) ,r^-1_-1(u)= (1-u)^-3 (1+14u+u^2)^-2 (1+2u-111u^2).These also follow from the recurrences of Theorem <ref>.The recurrences are non-classical, not least because they are bilateral: they extend to n,m<0.It is shown in  <ref> that for n,m≥0, the degree-(3n+2m) polynomials r^m_n in u are (generalized) Heun polynomials, rather than hypergeometric ones; they are not orthogonal polynomials in the conventional sense.A useful third-order (i.e., four-term) recurrence on k for the coefficients {a_k}_k=0^3n+2m of r^m_n is given in Theorem <ref>.An important degenerate case is worth noting: the case n=0.For any m≥0, there are hypergeometric representations in _2F_1 for the degree-2m octahedral polynomials r^m_0 and r^m_0, namelyr^m_0(u) = _2F_1(-2m, -1/4-3m3/4-m | u),r^m_0(u) = _2F_1(-2m, 1/4-m5/4+m | u).These follow by a sextic hypergeometric transformation of the _2F_1's in Definition 1, as well as by the methods of  <ref>.The first can also be deduced from the n=0 case of the recurrence on m in Theorem <ref>. These representations extend to m∈ℤ. §.§ Explicit FormulasThe following two theorems (Theorems <ref> and <ref>) give trigonometrically parametrized formulas for the Legendre/Ferrers functions P_ν^μ, P_ν^μ when (ν,μ) equals (-1/6,1/4)+ (n,m) and (-1/6,-1/4)+ (n,-m), with (n,m)∈ℤ^2.The triple of exponent differences (μ,μ,2ν+1) is respectively equal to (1/4,1/4,2/3)+(m,m,2n) and (-1/4,-1/4,2/3)+(-m,-m,2n), both lying in Schwarz's octahedral class V.An interesting application of these formulas to the evaluation of certain Mehler–Dirichlet integrals appears in Theorem <ref>.Let hyperbolic-trigonometric functions A_±, positive on (0,∞), be defined byA_±=A_±(ξ)±cosh(ξ/3) + √(sinhξ/3sinh(ξ/3))=±cosh(ξ/3) + √(4cosh^2(ξ/3)-1/3),so that A_+A_-(ξ)=1/3sinh^2(ξ/3).This choice is motivated by Definition <ref>: if R=R(u) and T=T(u)=-12u/(1+u)^2 are alternatively parametrized as tanh^2ξ and tanh^2(ξ/3), respectively, it is not difficult to verify that the three polynomials in u that appear in Definition <ref> will have the ξ-parametrizations p_v= u(1-u)^4 = - 1627 A_+^-6sinh^2ξ,p_e= (1+u)(1-34u+u^2) = 8 A_+^-3coshξ,p_f= 1+14u+u^2 = 4 A_+^-2. Moreover, and more fundamentally, u=-A_-/A_+.Also in this section, let r̂^m_n signify r^m_n/d^m_n; so that when n,m≥0, r̂^m_n is a scaled version of the octahedral polynomial r^m_n, with its leading rather than its trailing coefficient equal to unity.Equivalently, r̂^m_n(u) = u^3n+2m r^m_n(1/u).The formulasP_-1/6+n^1/4+m(coshξ) =[2^-2m-3n Γ(34-m)^-1]×(sinhξ)^-1/4-m A_+^1/4+3m+3nr^m_n(-A_-/A_+),P_-1/6+n^-1/4-m(coshξ)=[(-)^n2^-2m-3n 3^3/4+3m Γ(54+m)^-1]×(sinhξ)^-1/4-m A_-^1/4+3m+3n r̂^m_n(-A_+/A_-)hold for (n,m)∈ℤ^2 and ξ∈(0,∞).[Note that as ξ increases from 0 to ∞, the argument u=-A_-/A_+ of the first r^m_n, which satisfies T(u)=tanh^2(ξ/3) and R(u)=tanh^2ξ, decreases from 0 to -(2-√(3))^2≈-0.07, which is a root of p_ f(u)=1+ 14u+ u^2.] These formulas follow from the hypergeometric representation (<ref>) of P_ν^μ=P_ν^μ(z), together with the implicit definitions of r^m_n,r^m_n (see Definition <ref>).If z=coshξ, the argument 1-1/z^2 of the right-hand _2F_1 in (<ref>) will equal tanh^2ξ.This is why it is natural to parametrize Definition <ref> by letting R=R(u) equal tanh^2ξ, with the just-described consequences.In deriving the formulas, one needs the representation (<ref>); and for the second formula, the definition (<ref>) of d^m_n.In the following, the circular-trigonometric functions B_±, positive on (0,π), are defined byB_±=B_±(θ)cos(θ/3) ±√(sinθ/3sin(θ/3))=cos(θ/3) ±√(4cos^2(θ/3)-1/3),so that B_+B_-(θ)=1/3sin^2(θ/3).The formulasP_-1/6+n^1/4+m(cosθ) =[2^-2m-3n Γ(34-m)^-1]×(sinθ)^-1/4-m B_+^1/4+3m+3nr^m_n(B_-/B_+), P_-1/6+n^-1/4-m(cosθ)=[2^-2m-3n 3^3/4+3m Γ(54+m)^-1]×(sinθ)^-1/4-m B_-^1/4+3m+3n r̂^m_n(B_+/B_-)hold for (n,m)∈ℤ^2 and θ∈(0,π).[Note that as θ increases from 0 to π, the argument u=B_-/B_+ of the first r^m_n, which satisfies T(u)=-tan^2(θ/3) and R(u)=-tan^2θ, increases from 0 to 1.] By analytic continuation of Theorem <ref>; or in effect, by letting ξ= iθ.Because P_ν^μ=P_-ν-1^μ and P_ν^μ= P_-ν-1^μ, Theorems <ref> and <ref> also supply formulas for P_-5/6-n^1/4+m, P_-5/6-n^-1/4-m and P_-5/6-n^1/4+m,P_-5/6-n^-1/4-m.By exploiting the Q̂→ P and Q→ P reductions (<ref>) and (<ref>), one easily obtains additional formulas, for Q̂_-1/6+n^1/4+m, Q̂_-1/6+n^-1/4-m,Q̂_-5/6-n^1/4+m, Q̂_-5/6-n^-1/4-m and Q_-1/6+n^1/4+m, Q_-1/6+n^-1/4-m,Q_-5/6-n^1/4+m,Q_-5/6-n^-1/4-m.Theorems <ref> and <ref> permit certain Mehler–Dirichlet integrals to be evaluated in closed form.For example, consider <cit.>P_ν^μ(coshξ)= √(2/π) (sinhξ)^μ/Γ(1/2-μ) ∫_0^ξcosh[(ν+1/2)t]/(coshξ - cosh t)^μ+1/2dt, P_ν^μ(cosθ)= √(2/π) (sinθ)^μ/Γ(1/2-μ) ∫_0^θcos[(ν+1/2)ϕ]/(cosϕ - cosθ)^μ+1/2dϕ,which hold when Re μ<1/2 for ξ∈(0,∞) and θ∈(0,π).These integral representations of the Legendre and Ferrers functions of the first kind are classical <cit.>. The formulas∫_0^ξcosh[(1/3+n)t]/(coshξ-cosh t)^1/4-mdt= (-1)^n K_n^m A_-^1/4 + 3m + 3n r̂_n^m(-A_+/A_-), ∫_0^θcos[(1/3+n)ϕ]/(cosϕ-cosθ)^1/4-mdϕ =K_n^m B_-^1/4 + 3m + 3n r̂_n^m(B_+/B_-),withK_n^m = √(π/2) 2^-2m-3n3^3/4+3mΓ(3/4+m)/Γ(5/4+m),hold when m is a non-negative integer and n an integer, for ξ∈(0,∞) and θ∈(0,π). Substitute the second formulas of Theorems <ref> and <ref> into the (ν,μ) = (-1/6+ n, -1/4- m) specializations of (<ref>) and (<ref>).§ TETRAHEDRAL FORMULAS (SCHWARZ CLASS II)The following theorem gives trigonometrically parametrized formulas for the second Legendre function Q̂_ν^μ when (ν,μ) = (-3/4,-1/3)+ (-m,-n) and (-1/4,-1/3)+ (m,-n), with (n,m)∈ℤ^2.The triple of exponent differences, (μ,μ,2ν+1), is respectively equal to (-1/3,-1/3,-1/2)+(-n,-n,-2m) and (-1/3,-1/3,1/2)+(-n,-n,2m), both lying in Schwarz's tetrahedral class II.The hyperbolic-trigonometric functions A_±=A_±(ξ) on (0,∞) are defined as in (<ref>).The formulas(2/π)Q̂_-3/4-m^-1/3-n(ξ) =[ 2^11/4-2m-3n 3^-3/8(14)_m(1312)_m+n^-1Γ(43)^-1]× (sinhξ)^1/4-m[ -(-)^n√(√(3)+1) A_+^1/4+3m+3n r^m_n(-A_-/A_+) ],(2/π)Q̂_-1/4+m^-1/3-n(ξ) =[ 2^11/4-2m-3n 3^-3/8(14)_m(1312)_m+n^-1Γ(43)^-1]× (sinhξ)^1/4-m[ +(-)^m√(√(3)-1) A_-^1/4+3m+3n r^m_n(-A_+/A_-) ]hold for (n,m)∈ℤ^2 and ξ∈(0,∞).Combine Whipple's Q̂→ P transformation <cit.>,Q̂_ν^μ(ξ) = √(π/2) Γ(ν+μ+1)(sinhξ)^1/2P_-μ-1/2^-ν-1/2(coshξ),with the results in Theorem <ref>. The symmetrical forms of the right-hand prefactors are obtained with the aid of the gamma-function identities√(√(3)+1) = π^1/2 2^1/43^-3/8 Γ(1/12) Γ(1/4)^-1 Γ(1/3)^-1= π^-3/2 2^-3/4 3^3/8 Γ(11/12) Γ(1/4) Γ(1/3)√(√(3)-1) = π^-1/2 2^-1/4 3^1/8 Γ(5/12) Γ(1/4)^-1 Γ(1/3)= π^-1/2 2^-1/4 3^-1/8 Γ(7/12) Γ(1/4) Γ(1/3)^-1of Vidūnas <cit.>.Because Q̂_ν^μ, Q̂_ν^-μ are proportional to each other (see (<ref>)), Theorem <ref> also supplies formulas for Q̂_-3/4-m^1/3+n,Q̂_-1/4+m^1/3+n.Moreover, it leads to the following two theorems.The formulasP_-3/4-m^-1/3-n(ξ) =[ (-)^n 2^5/4-2m-3n 3^-3/8(14)_m(1312)_m+n^-1Γ(43)^-1] × (sinhξ)^1/4-m [(-)^n√(√(3)-1) A_+^1/4+3m+3n r^m_n(-A_-/A_+) - (-)^m√(√(3)+1) A_-^1/4+3m+3n r^m_n(-A_+/A_-) ],P_-3/4-m^1/3+n(ξ) =[ (-)^n 2^-1/4-2m-3n 3^-3/8(14)_m(512)_m-n^-1Γ(23)^-1] × (sinhξ)^1/4-m [(-)^n√(√(3)+1) A_+^1/4+3m+3n r^m_n(-A_-/A_+) + (-)^m√(√(3)-1) A_-^1/4+3m+3n r^m_n(-A_+/A_-) ]hold for (n,m)∈ℤ^2 and ξ∈(0,∞). Combine the P→Q̂ reduction (<ref>) with the results in Theorem <ref>.Because P_ν^μ=P_-ν-1^μ, Theorem <ref> also supplies formulas for P_-1/4+m^-1/3-n,P_-1/4+m^1/3+n.In Theorem <ref>, the hyperbolic-trigonometric functions C_±, positive on (-∞,∞), are defined byC_±=C_±(ξ)±sinh(ξ/3) +√(coshξ/3cosh(ξ/3))=±sinh(ξ/3) +√(4sinh^2(ξ/3)+1/3),so that C_+C_-(ξ)=1/3cosh^2(ξ/3).The formulasP_-3/4-m^-1/3-n(tanhξ) =[ 2^5/4-2m-3n 3^-3/8(14)_m(1312)_m+n^-1Γ(43)^-1] × (coshξ)^1/4-m [-(-)^n√(√(3)-1) C_+^1/4+3m+3n r^m_n(-C_-/C_+) + (-)^m√(√(3)+1) C_-^1/4+3m+3n r^m_n(-C_+/C_-) ], P_-3/4-m^1/3+n(tanhξ) =[ (-)^n 2^-1/4-2m-3n 3^-3/8(14)_m(512)_m-n^-1Γ(23)^-1]× (coshξ)^1/4-m [(-)^n√(√(3)+1) C_+^1/4+3m+3n r^m_n(-C_-/C_+) + (-)^m√(√(3)-1) C_-^1/4+3m+3n r^m_n(-C_+/C_-) ]hold for (n,m)∈ℤ^2 and ξ∈(-∞,∞). [Note that as ξ decreases from ∞ to -∞, the argument u=-C_-/C_+ of the first r^m_n, which satisfies T(u)=^2(ξ/3) and R(u)=^2ξ, decreases from 0 to -(2+√(3))^2≈-14.0, which is a root of p_ f(u)=1+ 14u+ u^2.] By analytic continuation of the results in Theorem <ref>; or in effect, by replacing ξ by ξ+ iπ/2.Because P_ν^μ= P_-ν-1^μ, Theorem <ref> also supplies formulas for P_-1/4+m^-1/3-n, P_-1/4+m^1/3+n.By exploiting the Q→ P reduction (<ref>), one easily obtains additional formulas, for Q_-3/4-m^-1/3-n, Q_-3/4-m^1/3+n, Q_-1/4+m^-1/3-n, Q_-1/4+m^1/3+n.§ TETRAHEDRAL FORMULAS (SCHWARZ CLASS III)The following theorems give parametrized formulas for the Legendre/Ferrers functions P_ν^μ,Q̂_ν^μ, P_ν^μ when (ν,μ) = (-1/6,-1/3)+ (n,-n) and (-5/6,1/3)+ (-n,n), with n∈ℤ.The triple of exponent differences, (μ,μ,2ν+1), is respectively equal to (-1/3,-1/3,2/3)+(-n,-n,2n) and (1/3,1/3,-2/3)+(n,n,-2n), both lying in Schwarz's tetrahedral class III.The formulasP_-1/6+n^-1/3-n(√(1- e^-2ξ) ) =2^-1/3-n(1- e^-2ξ)^-1/4P_-3/4^-1/3-n(ξ),P_-5/6-n^1/3+n(√(1- e^-2ξ) ) =2^1/3+n(1- e^-2ξ)^-1/4P_-3/4^1/3+n(ξ)where expressions for the right-hand Legendre functions are provided by Theorem <ref>, hold for n∈ℤ and ξ∈(0,∞). The formulasP_-1/6+n^-1/3-n(√(1+ e^-2ξ) ) =2^-1/3-n(1+ e^-2ξ)^-1/4P_-3/4^-1/3-n(tanhξ), P_-5/6-n^1/3+n(√(1+ e^-2ξ) ) =2^1/3+n(1+ e^-2ξ)^-1/4P_-3/4^1/3+n(tanhξ),where expressions for the right-hand Ferrers functions are provided by Theorem <ref>, hold for n∈ℤ and ξ∈(-∞,∞). The formulas(2/π)Q̂_-1/6+n^-1/3-n(√(1+ e^2ξ) ) =2^-1/3-n(1+ e^2ξ)^-1/4 √(2)P_-3/4^-1/3-n(-tanhξ),(2/π)Q̂_-5/6-n^1/3+n(√(1+ e^2ξ) ) =2^1/3+n(1+ e^2ξ)^-1/4 √(2)P_-3/4^1/3+n(-tanhξ),where expressions for the right-hand Ferrers functions are provided by Theorem <ref>, hold for n∈ℤ and ξ∈(-∞,∞). Because P_ν^μ=P_-ν-1^μ and P_ν^μ= P_-ν-1^μ, Theorems <ref> and <ref> also supply formulas for P_-5/6-n^-1/3-n, P_-1/6+n^1/3+n, P_-5/6-n^-1/3-n, P_-1/6+n^1/3+n.And because Q̂_ν^μ, Q̂_ν^-μ are proportional to each other (see (<ref>)), Theorem <ref> also supplies formulas for Q̂_-1/6+n^1/3+n,Q̂_-5/6-n^-1/3-n.By exploiting the Q→ P reduction (<ref>), one easily obtains additional formulas, for Q_-1/6+n^-1/3-n, Q_-5/6-n^1/3+n,Q_-5/6-n^-1/3-n, Q_-1/6+n^1/3+n.The formulas in Theorems <ref>, <ref>, and <ref> are straightforward reparametrizations of the α=1/3+n and α=-1/3-n cases of the identitiesP_-1/4^-α(coshξ)= 2^α√( sech(ξ/2)) P_α-1/2^-α( sech(ξ/2)),P_-1/4^-α(cosθ)= 2^α√((θ/2)) P_α-1/2^-α((θ/2)), √(2)P_-1/4^-α(-cosθ)= 2^α√((θ/2)) (2/π)Q̂_α-1/2^-α((θ/2)),which hold when ξ∈(0,∞) and θ∈(0,π).These appear as identities I_4(i), I_4(ii), and I_4(ii) in <cit.>, and are really quadratic hypergeometric transformations in disguise. § PROOFS OF RESULTS IN SECTION <REF>The octahedral and tetrahedral formulas in<ref>, <ref>, and <ref> followed from the theorems in  <ref> on the octahedral functions r^m_n = r^m_n(u), which were stated without proof.The present section provides proofs, in some cases sketched, and obtains a few additional results.These are Theorem <ref> (on the differential equation and differential recurrences satisfied by r^m_n), and Theorems <ref>, <ref>, and <ref> (on the interpretation of r^m_n when n,m≥0 as a hypergeometric, Heun, or generalized Heun polynomial).This section also reveals the origin of the degree-6 rational function x=R(u) in Definition <ref>.Consider a Riemann sphere ℙ^1_s, parametrized by s and identified by stereographic projection with the complex s-plane.(As usual, s=0 is at the bottom and s=∞ is at the top; points with |s|=1 are taken to lie on the equator.)Let a regular octahedron (a Platonic solid) be inscribed in the sphere, with its six vertices v_1,…,v_6 at s=0,±1,± i,∞, i.e., at the five roots of q_ v(s) s(1- s^4) and at s=∞.By some trigonometry <cit.>, one can show that the twelve edge-midpoints e_1,…,e_12 of the octahedron, radially projected onto the sphere, are located at s= (±1± i)/√(2) and s=√(±1) (±1±√(2)), which are the roots of q_ e(1+ s^4) (1- 34s^4 +s^8) =1- 33s^4 - 33s^8 + s^12.Similar, its eight face-centers f_1,… f_8, when radially projected, are located at s=(±1± i)(1±√(3))/2, which are the roots of q_ f(s) 1+ 14s^4 +s^8.The polynomials q_ v,q_ e,q_ f are (relative) invariants of the symmetry group of the octahedron, which is an order-24 subgroup of the group of rotations of the Riemann s-sphere.The well-known octahedral equation states that q_ e^2 - q_ f^3 + 108 q_ v^4=0.The validity of this identity (a syzygy, in the language of invariant theory) suggests considering the degree-24 rational function R̃=R̃(s) equal to 1- q_ f^3/q_ e^2, i.e.,R̃(s) -108 [s(1-s^4)]^4/[(1+s^4)(1-34s^4+s^8)]^2 = 1 - (1+14s^4+s^8)^3/[(1+s^4)(1-34s^4+s^8)]^2.On the s-sphere, R̃(s) equals 0,1,∞ at (respectively) the vertices, the face-centers, and the edge-midpoints.It is an absolute invariant of the symmetry group of the octahedron. (Its derivative dR̃(s)/ ds can be written as -432q_ v^3 q_ f^2 / q_ e^3 but is only a relative invariant.)The covering ℙ^1_s →ℙ^1_x given by x=R̃(s) is ramified above x=0,1,∞, and its ramification structure can be written as (6)4=(8)3=(12)2: each of the six points above s=0 (i.e., the vertices) appears with multiplicity 4, etc.Following and extending Schwarz <cit.>, consider the effect of lifting the Gauss hypergeometric equation satisfied by _2F_1(-124-m/2-n/2, 1124-m/2-n/2;34-m; x), the _2F_1(x) appearing in Definition <ref>, from the x-sphere to the s-sphere, along x=R̃(s).It should be recalled that the Gauss equation satisfied by f(x)=_2F_1(a,b;c;x) is the Fuchsian differential equation (D_x d/ dx))D_x^2 f + [ c/x + a+b-c+1/x-1]D_xf +[ ab/x(x-1)]f=0,which has characteristic exponents {0,1-c}, {0,c-a-b}, {a,b} at its singular points x=0,1,∞, with (unsigned) exponent differences 1- c; c- a- b; b- a.(The function _2F_1(a,b;c;x) is the Frobenius solution associated to the zero exponent at x=0.)The effects of the ramified lifting by x=R̃(s) are conveniently expressed in the classical notation of Riemann P-symbols, which display the exponents at each singular point <cit.>.For the _2F_1 of Definition <ref>, one can write _2F_1(x=R̃(s))= P {[ 0 1 ∞ x = R̃(s); 0 0 -1/24-m/2-n/2; 1/4+m 1/3 + n 11/24-m/2-n/2 ]}= P {[v_1,…,v_6f_1,…,f_8 e_1,…,e_12s;00-1/12-m-n ; 1+4m 1 + 3n11/12-m-n]}, because any pair of characteristic exponents at a point x=x_0 beneath a ramification point s=s_0 of order k is multiplied by k when lifted. This function of s satisfies a differential equation on the s-sphere with the indicated singular points and exponents.If ℒg=0 is any Fuchsian differential equation on the s-sphere, the modified equation ℒ'g'=0 obtained by the change of dependent variable g'=(1- s/s_0)^α g has its exponents at s=s_0 shifted upward by α, and those at s=∞ shifted downward by the same.As an application of this, one deduces from (<ref>) thatf̃(s)= [q_ e(s)]^1/12+m+n _2F_1(x=R̃(s))=P {[v_1,…,v_5f_1,…,f_8 e_1,…,e_12∞s;000 -1-12m-12n ; 1+4m 1 + 3n1-8m-12n]}= P {[v_1,…,v_5f_1,…,f_8∞s;00 -1-12m-12n ; 1+4m 1 + 3n-8m-12n]}, because e_1,…, e_12 are the roots of q_ e, and v_6=∞.The left-hand function f̃=f̃(s), which by examination is r̃^m_n(s) r^m_n(u=s^4), will be the solution of a `lifted and shifted' differential equation on the s-sphere, with the indicated exponents.After the shifting, the edge-midpoints e_1,…,e_12 cease being singular points, because the new exponents at each are 0,1, which are those of an ordinary point.It is straightforward if tedious to compute the differential equation satisfied by f̃ = r̃^m_n(s) r^m_n(u=s^4) explicitly, by applying to the appropriate Gauss equation of the form (<ref>) the changes of variable that perform (i) the lifting along s↦ x = R̃(s), and (ii) the multiplication by [q_ e(s)]^1/12+m+n.One finds that f̃ satisfies ℒ̃^m_nf̃=0, whereℒ̃^m_n = D_s^2 - [ 4m 5s^4-1/s(s^4-1) +3n 8s^7+56s^3/s^8+14s^4+1] D_s+4(12m+12n+1) s^3[(2m+3n)(s^8+14s^4+1) - 12n (3s^4+1)]/s(s^4-1)(s^8+14s^4+1).That the singular points of this operator are the roots v_1,…,v_5 of q_ v (plus v_6=∞), and the roots f_1,…,f_8 of q_ f, is clear; as is the fact that their exponents are as shown in the P-symbol (<ref>).The degenerate case n=m=0 is especially interesting.As one expects from the P-symbol, the operator ℒ̃^0_0 is simply the Laplacian D_s^2, the kernel of which is spanned by 1,s.For f̃ = f̃(s) = r^m_n(s^4), it is easy to rule out any admixture of the latter solution by examining Definition <ref>; and because r^0_0(u) equals unity at u=0, the base octahedral function r^0_0 must be identically equal to unity.Because r^0_0≡1, it follows from Definition <ref> that the hypergeometric function appearing in the definition of r^m_n when n=m=0, which is _2F_1(-1/24,11/24;3/4| x), must be algebraic in its argument x.This is essentially the 1873 result of Schwarz <cit.>, the proof of which was later restated in a P-symbol form by Poole <cit.>.However, it is not trivial to extend this result on r^0_0 to a constructive proof that r^m_n=r^m_n(u) is a rational function of u for each (n,m)∈ℤ^2. This is because the differential equation ℒ̃^m_nf̃=0, as one sees from (<ref>), is far more complicated than D_s^2f̃=0 (Laplace's equation) when (n,m)≠(0,0).A constructive proof is best based on contiguity relations between adjacent (n,m), i.e., recurrences in the spirit of Gauss, derived as follows.First, simplify the lifting along the covering s↦ x, i.e., along the degree-24 map x=R̃(s).Each octahedral function r̃^m_n(s) turns out to `factor through' u=s^4, so it suffices to lift the Gauss hypergeometric equation from the x-sphere ℙ^1_x to the u-sphere ℙ^1_u, along the degree-6 map x=R(u) of Definition <ref>, i.e.,R(u) -108p_v(u)/p_e(u)^2 = 1 - p_f(u)^3/p_e(u)^2, wherep_v(u)=q_v(s)^4 =u(1-u)^4 p_e(u)=q_e(s) =(1+ u) (1-34u + u^2)p_f(u)=q_f(s) =1+ 14u + u^2.Replacing the lifted variable s by u=s^4 quotients out an order-4 cyclic group of rotations of the s-sphere (and hence of the octahedron), about the axis passing through its north and south poles.The syzygy becomes p_ e^2 - p_ f^3 + 108 p_ v=0.The roots u=0,1 of p_ v, and u=∞, correspond to the south-pole vertex, the four equatorial ones, and the north-pole one.The three roots u=(3+ 2√(2))^2, -1, (3-2√(2))^2 of p_ e correspond to the four edge-midpoints in the northern hemisphere, the four on the equator, and the four to the south.The two roots u=-(2±√(3))^2 of p_ f^2 correspond to the four face-centers in the north, and the four in the south.The covering ℙ^1_u→ℙ^1_x is still ramified above x=0,1,∞, but its ramification structure is 1+4+1=(2)3=(3)2.Taking the multiplicities in this ramification structure into account, one finds that if the _2F_1 of Definition <ref> is lifted along x=R(u) rather than x=R̃(s), the P-symbol identity (<ref>) is replaced by r^m_n(u)= [p_ e(u)]^1/12+m+n _2F_1(x=R(u))=P {[ 0 1 -(2±√(3))^2 ∞ u; 0 0 0 -14-3m-3n;1/4 +m1 + 4m1+3n-2m-3n ]}. This P-symbol has five singular points (at most; fewer if n=0 or m=0). By the preceding explanation, the points u=0,1,∞ represent 1,4,1 vertices of the octahedron, and each of u= -(2±√(3))^2 represents a cycle of four face-centers.The octahedral function r^m_n=r^m_n(u) satisfies the Fuchsian differential equation ℒ^m_n r^m_n=0, whereℒ^m_n = D_u^2 + [ 3/4-m/u + -4m/u-1 + -3n(2u+14)/u^2 + 14u+1] D_u+(1+12m+12n) (2m+3n)(u^2+14u+1) - 12n(3u+1)/4 u(u-1)(u^2+14u+1),the P-symbol of which appears in Eq. (<ref>).The function r^m_n is the Frobenius solution associated to the zero characteristic exponent of the singular point u=0.It satisfies eight differential recurrences of the formK r^m+Δm_n+Δn = p_v^(-σ_v+ε_v)/4p_e^-σ_e+ε_ep_f^-σ_f+ε_f(4u^3/4) d/du [p_v^σ_v/4 p_e^σ_e p_f^σ_f·r^m_n ],in which Δ(n,m) = (0,±1), (±1,0) and (±1,±1).For each recurrence, the exponents σ_ v, σ_ e, σ_ f, the exponents ε_ v, ε_ e, ε_ f, and the prefactor K are listed in Table <ref>.The differential equation comes by applying to the appropriate Gauss equation of the form (<ref>) the changes of variable that perform (i) the lifting along u↦ x=R(u), and (ii) the multiplication by [p_ e(u)]^1/12 + m + n. Or, one can merely substitute u=s^4 into the equation ℒ̃^m_nr̃^m_n=0, with ℒ̃^m_n as in (<ref>).The differential recurrences of Jacobi, which shift the parameters a,b,c of the function _2F_1(a,b;c;x) by integers, are well known. (See <cit.>.)And if Δ(n,m)=(±1,±1), the _2F_1(R(u)) in the definition of r^m_n(u) has its parameters shifted by integers.(See Definition <ref>.)By some calculus, one can change the independent variable in the relevant differential recurrences of Jacobi from x=R(u) to u, thereby obtaining the final four recurrences in Table <ref> (the diagonal ones).The change uses the fact that u^3/4dR/ du equals -108 p_ v^3/4p_ f^2 / p_ e^3, and the details are straightforward.The first four recurrences in the table, with Δ(n,m)=(0,±1),(±1,0), come a bit less easily, because they shift the _2F_1 parameters in Definition <ref> by half-integers rather than integers.But by examination, Definition <ref> is equivalent to_2F_1(-112-m-n, 1/4-m12-2m| S(t)) = (1+6t-3t^2)^1/4-3m-3n r^m_n(-3t^2),where S(t) 36 t(1+3t^2)^2/(1+6t - 3t^2)^3 = 1 -(1-6t-3t^2/1+6t-3t^2)^3.This follows by a quadratic hypergeometric transformation, u being related to t by u=-3t^2, and R to S by R=S^2/(S-2)^2.When Δ(n,m)=(0,±1) or (±1,0), the parameters of the _2F_1 in (<ref>) are shifted by integers, and the same technique can be applied.The four three-term non-differential recurrences in Theorem <ref> follow by a familiar elimination procedure from the differential recurrences of Theorem <ref>, taken in pairs.They are analogous to the contiguity relations (or `contiguous function relations') of Gauss, for _2F_1, which follow by elimination from the differential recurrences of Jacobi, though Gauss did not derive them in this way.The explicit formulas for the functions  r^m_n with small n,m given in  <ref> (see Eqs. (<ref>),(<ref>)) also follow from the differential recurrences of Theorem <ref>.For any m∈ℤ, the octahedral function r^m_0 has the hypergeometric representationr^m_0(u) = _2F_1(-2m,-14-3m; 34-m| u).Thus when m≥0, r^m_0 is a degree-2m hypergeometric polynomial.Moreover, for any n,m≥0, r^m_n is a polynomial of degree 3n+2m. When n=0, ℒ^m_nf=0 loses two singular points and degenerates to a Gauss hypergeometric equation of the form (<ref>), with independent variable u and parameters a=-2m, b=-1/4-3m, c=3/4-m.Hence r^m_0(u) has the claimed representation, and if m≥0, is a degree-2m polynomial in u.It follows by induction from the differential recurrence with Δ(n,m)=(+1,0) that r^m_n(u) is a polynomial in u for all n≥0.It must be of degree 3n+2m, because in the P-symbol of ℒ^m_n [see (<ref>)], the only characteristic exponent at u=∞ that is a (nonpositive) integer is -2m-3n.The statement of this theorem includes additional claims that were made in  <ref>.The following related theorem mentions the Heun function Hn(a,q;α,β,γ,δ| z), for the definition of which see <cit.>.This is a Frobenius solution (at z=0) of a canonical Fuchsian differential equation that has four singular points, namely z=0,1,a,∞, and an `accessory' parameter q that unlike α,β,γ,δ, does not affect their characteristic exponents.It has a convergent expansion ∑_k=0^∞ h_kz^k, where the {h_k}_k=0^∞ satisfy a second-order recurrence with coefficients quadratic in k.For any n∈ℤ, the octahedral function r^0_n has the Heun representationr^0_n(u) = Hn( (2+√(3)2-√(3))^2, 9(2+√(3))^2 n(12n+1)4 ;-3n,-14-3n;34,-3n;-(2+√(3))^2u )and the equivalent expansion ∑_k=0^∞ a_ku^k, where {a_k}_k=0^∞ satisfy the second-order recurrence(k+1)(4k+3) a_k+1 + [ 14k(4k-12n-1)+9n(12n+1)] a_k+ (k-3n-1)(4k-12n-5) a_k-1 = 0,with a_0=1, a_-1=0.Thus when n≥0, r^0_n is a degree-3n Heun polynomial. If m=0, the u=1 singular point of ℒ^m_n drops out, i.e., becomes ordinary, andℒ^0_n = D_u^2 +[3/4/u + -3n(2u+14)/u^2+14u+1 ]D_u +3n(1+12n)/4 u+3/u(u^2+14u+1).The substitution z=-(2+√(3))^2 u reduces ℒ^0_n f=0 to the standard Heun equation <cit.>, with the stated values of a,q;α,β,γ,δ.The recurrence (<ref>), based on a second-order difference operator, comes by substituting f=r^0_n=∑_k=0^∞ a_ku^k into ℒ^0_n f=0.For general (n,m)∈ℤ^2, ℒ^m_nf=0 has five singular points.The theory of such generalized Heun equations is underdeveloped at present, but the coefficients of their series solutions are known to satisfy third-order (i.e., four-term) recurrences.For any (n,m)∈ℤ^2, the octahedral function r^m_n has the expansion r_n^m(u)=∑_k=0^∞ a_ku^k, where {a_k}_k=0^∞ satisfy the third-order recurrence(k+1)(4k-4m+3) a_k+1+[ k(52k-36m-168n-13) - (2m-9n)(12m+12n+1) ] a_k-[(k-1)(52k-276m-144n-65)+2(14m+3n)(12m+12n+1)] a_k-1-(k-2m-3n-2)(4k-12m-12n-9) a_k-2=0,with a_0=1, a_-1=0, a_-2=0.Thus when n,m≥0, r^m_n(u) is a degree-(3n+2m) generalized Heun polynomial. The recurrence comes by substituting f=r^m_n=∑_k=0^∞ a_ku^k into ℒ^m_n f=0.It can be shown that if m=0, the third-order (i.e. generalized Heun) difference operator in (<ref>) has the second-order (i.e. Heun) difference operator in (<ref>) as a right factor; and if n=0, it has a first-order (i.e. hypergeometric) difference operator as a right factor, which is why the representation in Theorem <ref> exists.The coefficients of all these difference operators are quadratic in k.As stated in Theorem <ref>, it is not merely the case that when n,m≥0, the rational function r^m_n=r^m_n(u) is a polynomial of degree 3n+ 2m.In each quadrant of the (n,m)-plane, it is the quotient of a polynomial of known degree (the numerator) by a known polynomial (the denominator).To obtain the formulas in Theorem <ref> that refer to quadrants other than the first, reason as follows.Consider the second formula: it says that if n,m≥0, r^-m-1_n(u) equals a polynomial of degree 1+ 3n+ 2m, divided by (1- u)^3+4m.This is proved by induction on n, the base case (n=0) beingr^-m-1_0(u) = (1-u)^-3-4m _2F_1(-1-2m,-14-m; 34-m| u),which comes from (<ref>) by Euler's transformation of _2F_1.The inductive step uses the differential recurrence with Δ(n,m) = (+1,0), as in the proof of Theorem <ref>.In the same way, the third and fourth formulas follow from the Δ(n,m)=(-1,0) recurrence.One sees from the four formulas in Theorem <ref> that irrespective of quadrant, r^m_n∼ const× u^3n+2m, which partially confirms the claims of Theorem <ref>(i,ii).A consequence of this asymptotic behavior is that besides being the Frobenius solution associated to the exponent 0 at u=0, r^m_n is the Frobenius solution associated to the exponent -2m- 3n at u=∞, which appeared in the P-symbol (<ref>).Theorem <ref> states specifically that r^m_n∼ d^m_n× u^3n+2m, with d^m_n defined in (<ref>).This too is proved by induction.The base case (n=0) has sub-cases m≥0 and m≤0, which follow by elementary manipulations from (<ref>) and (<ref>), respectively.The inductions toward n≥0 and n≤0 come from the differential recurrences with Δ(n,m) = (±1,0), the u→∞ asymptotics of which yield expressions for d^m_n±1/d^m_n.As one can check, these two expression agree with what (<ref>) predicts.The only claim in  <ref> remaining to be proved is Theorem <ref>(iii): the statement that the conjugate function r^m_n=r^m_n(u) is related to r^m_n=r^m_n(u) by r^m_n(u)∝ u^3n+2mr^m_n(1/u), or equivalently r^m_n(u)∝ u^3n+2m r^m_n(1/u).(The constant of proportionality comes from r^m_n(0) equalling unity.)Just as one derives the differential equation ℒ^m_n r^m_n=0 of Theorem <ref> by `lifting and shifting,' one can derive an equation ℒ^m_nr^m_n=0 on the u-sphere from the definition of r^m_n, given in Definition <ref>; and a further equation satisfied by 𝐫^m_n(u) u^3n+2m r^m_n(1/u).The latter turns out to be ℒ^m_n𝐫^m_n=0, i.e., to be identical to the equation of Theorem <ref>.But r^m_n = r^m_n(u), analytic at u=0, is the Frobenius solution associated to the exponent 0 at u=0 of ℒ^m_nr^m_n=0.Hence, 𝐫^m_n is the Frobenius solution associated to the exponent -2m- 3n at u=∞ of ℒ^m_n𝐫^m_n=0.But as was noted three paragraphs ago, this is r^m_n; so 𝐫^m_n∝ r^m_n, and Theorem <ref>(iii) follows.§ BIORTHOGONALITY OF OCTAHEDRAL FUNCTIONSThe octahedral functions r^m_n=r^m_n(u), which are polynomials if n,m≥0, satisfy recurrences, such as the three-term ones of Theorem <ref>, that are quite unlike the ones satisfied by the classical orthogonal polynomials.But at least if m=0,-1, it can be shown that the family {r^m_n}_m∈ℤ displays orthogonality on the u-interval [0,1], or rather a form of biorthogonality.The biorthogonality is best expressed in terms of the lifted functions r̃^m_n(s) r^m_n(u=s^4) of the last section, the full domain of which is the Riemann s-sphere in which the defining octahedron is inscribed.These are solutions of ℒ̃^m_nf̃=0, where the operator ℒ̃^m_n was defined in (<ref>).By inspection, it has the simpler representationℒ̃^m_n = (q_ v^2mq_ f^3n/2) D_s^2 (q_ v^2mq_ f^3n/2)^-1 + {[-2m(1+2m)]q_ f/q_ v^2 + [144n(2+3n)]q_ v^2/q_ f^2},where q_ v(s)=s(1-s^4) and q_ f(s) = 1+14s^4+s^8 are the usual polynomials that equal zero at the five finite vertices and the eight face-centers of the octahedron.By (<ref>), ℒ̃^m_n is conjugated by a similarity transformation to a formally self-adjoint operator of the Schrödinger type.For any fixed m, the calculation of the eigenfunctions q_ v^-2mq_ f^-3n/2r̃^m_n(s) of the latter, on the s-interval [0,1], can be viewed as solving a Sturm–Liouville problem.The coefficient 144n(2+ 3n) in (<ref>) plays the role of the Sturm–Liouville eigenvalue, and q_ v^2/q_ f^2 that of the Sturm–Liouville weight function.Because the coefficient function q_ f/q_ v^2 diverges at the endpoints s=0,1, this Sturm–Liouville problem is typically a singular one.To avoid a discussion of endpoint classifications and boundary conditions, it is best to derive orthogonality results not from ℒ̃^m_n, but rather from the Love–Hunter biorthogonality relation (<ref>), i.e.,∫_-1^1P_ν^μ(z)P_ν'^-μ(-z) dz =0,which holds if μ∈(-1,1) and ν,ν' differ by a nonzero even integer.(See <cit.> for a proof.) Equation (<ref>) is a relation of orthogonality between the eigenfunctions of a singular boundary value problem based on (<ref>), the associated Legendre equation (i.e., P_ν_0+2n^μ(z), n∈ℤ), and the eigenfunctions of the adjoint boundary value problem (i.e., P_ν_0+2n^-μ(-z), n∈ℤ).The first problem is non-self-adjoint because the boundary conditions that single out P_ν_0+2n^μ(z), n∈ℤ, as eigenfunctions are not self-adjoint.However, one feature of the operator ℒ̃^m_n must be mentioned.If f̃=f̃(s) solves ℒ̃^m_nf̃=0, then so does (1- s)^1+12m+12nf̃((1+s)/(1-s)).This claim can be verified by a lengthy computation, but its correctness is indicated by the P-symbol of ℒ̃^m_n, which appeared in (<ref>).The map s↦(1+ s)/(1- s) is a 90^∘ rotation of the s-sphere, and hence of the inscribed octahedron, around the axis through the equatorial vertices s=± i. This rotation takes vertices to vertices, edges to edges, and faces to faces.The subsequent multiplication by (1- s)^1+12m+12n shifts the characteristic exponents at the most affected vertices (s=1,∞) to the values they had before the rotation.For m=0 and m=-1, the lifted family {r̃^m_n(s)r^m_n(s^4)}_n∈ℤ is biorthogonal on the s-interval [0,1] in the following sense: the inner product integral∫_0^1 [ q_ v^-2m q_ f^-3n/2(s)·r̃^m_n(s) ] ×[ q_ v^-2m q_ f^-3n'/2(s)· (1-s)^1+12m+12n r̃^m_n'(1+s/1-s) ]q_ v^2/q_ f^2(s) dsequals zero if n,n' differ by a nonzero even integer. Substitute the m=0,-1 cases of the formulas for P_-1/6+n^±(1/4+m)(cosθ) in Theorem <ref> into (<ref>), and change the variable of integration from z=cosθ to u=B_-/B_+, and then to s=u^1/4.The involution z↦ -z corresponds to s↦ (1- s)/(1+ s). This biorthogonality theorem is formulated so as to indicate its close connection to Sturm–Liouville theory: evaluating the integral over 0< s<1 computes the inner product of the two square-bracketed factors in the integrand, which come from P_ν^μ(z) and P_ν'^-μ(-z), with respect to the weight function q_ v^2/q_ f^2.The two factors are eigenfunctions of adjoint Sturm–Liouville problems on 0< s<1 (i.e., ones with adjoint boundary conditions), with different eigenvalues.Theorem <ref> cannot be extended to general m∈ℤ, because the integral diverges unless m=0 or m=-1, owing to rapid growth of one or the other of the bracketed factors at each of the endpoints s=0,1.This divergence follows readily from the results on r^m_n given in Theorems <ref> and <ref>. Alternatively, the divergence arises from the Ferrers function P_ν^μ not lying in L^2[-1,1] when μ is non-integral, unless Re μ∈(-1,1).The formulas for the tetrahedral Ferrers functions P_-3/4-m^±(1/3+n) given in Theorem <ref> (cases m=0,-1) can also be substituted usefully into the Love–Hunter relation (<ref>). But the resulting statement of biorthogonality is more complicated than Theorem <ref> and is not given here. § CYCLIC AND DIHEDRAL FORMULAS (SCHWARZ CLASSES O AND I)This section derives parametric formulas for Legendre and Ferrers functions that are cyclic or dihedral.The formulas involve the Jacobi polynomials P_n^(α,β) and are unrelated to the octahedral and tetrahedral ones in <ref>, <ref>, and <ref>.They are of independent interest, and subsume formulas that have previously appeared in the literature.As used here, `cyclic' and `dihedral' have extended meanings.The terms arise as follows.The associated Legendre equation (<ref>) has (μ,μ, 2ν+1) as its (unordered, unsigned) triple of characteristic exponent differences.By the results of Schwarz on the algebraicity of hypergeometric functions, this differential equation will have only algebraic solutions if (ν+1/2,μ) lies in (±1/2,±1/2k)+ℤ^2 or (±1/2k,±1/2)+ℤ^2, for some positive integer k.These restrictions cause the equation to lie in Schwarz's cyclic class (labelled O here), resp. his dihedral class I.The terms refer to the projective monodromy group of the equation, which is a (finite) subgroup of PSL(2,ℝ).However, the formulas derived below are more general, in that they allow k to be arbitrary: they are formulas for continuously parametrized families of Legendre and Ferrers functions, which are generically transcendental rather than algebraic.Because of this, we call a Legendre or Ferrers function cyclic, resp. dihedral, if (ν+1/2,μ) lies in (±1/2,*)+ℤ^2, resp. (*,±1/2)+ℤ^2; the asterisk denoting an unspecified value.That is, the degree ν should be an integer or the order μ a half-odd-integer, respectively.Explicit formulas in terms of Jacobi polynomials are derived in  <ref>, and how dihedral Ferrers functions can be used for expansion purposes is explained in  <ref>. §.§ Explicit FormulasThe Jacobi polynomials P_n^(α,β)(z) are well known <cit.>. They have the hypergeometric and Rodrigues representations P_n^(α,β)(z)= (α+1)_n/n! _2F_1(-n, n+α+β+1α+1|1-z/2)= (-1)^n/2^nn!(1-z)^-α(1+z)^-βd^n/ dz^n[(1-z)^α+n(1+z)^β+n] and are orthogonal on [-1,1] with respect to the weight function (1- x)^α (1+ x)^β, if α,β>-1 and the weight function is integrable.Legendre and Ferrers functions that are cyclic (i.e., of integer degree) are easily expressed in terms of Jacobi polynomials.The formulasP^μ_-1/2±(n+1/2)(z) = n!/Γ(n-μ+1) ( z+1/z-1)^μ/2 P_n^(-μ,μ)(z),P^μ_-1/2±(n+1/2)(coshξ) = n!/Γ(n-μ+1)[(ξ/2)]^μP_n^(-μ,μ)(coshξ)hold when n is a non-negative integer, for z∈(1,∞) and ξ∈(0,∞).(In the degenerate case when μ-n is a positive integer, P_-1/2±(n+1/2)^μ≡0.) Compare the representations (<ref>) and (<ref>).The formulasP^μ_-1/2±(n+1/2)(z) = n!/Γ(n-μ+1) ( 1+z/1-z)^μ/2 P_n^(-μ,μ)(z), P^μ_-1/2±(n+1/2)(cosθ) = n!/Γ(n-μ+1)[(θ/2)]^μP_n^(-μ,μ)(cosθ)hold when n is a non-negative integer, for z∈(-1,1), ξ∈(-∞,∞), and θ∈(0,π).(In the degenerate case when μ-n is a positive integer, P_-1/2±(n+1/2)^μ≡0.) By analytic continuation of Theorem <ref>; or in effect, by letting ξ= iθ.By exploiting the Q̂→ P and Q→ P reductions (<ref>) and (<ref>), one can derive additional formulas from Theorems <ref> and <ref>, for Q̂^μ_-1/2±(n+1/2) and Q^μ_-1/2±(n+1/2) respectively.However, the coefficients in (<ref>) and (<ref>) diverge when μ∈ℤ.Hence, following this approach to formulas for Q̂_-1/2±(n+1/2)^m, Q_-1/2±(n+1/2)^m, when n is a non-negative integer and m an integer, requires the taking of a limit.In the commonly encountered case when -n≤ m≤ n (but not otherwise), the resulting expressions turn out to be logarithmic.Such expressions can be computed in other ways <cit.>.Perhaps the best method is to express Q̂_-1/2±(n+1/2)^m in terms of a _2F_1 by using (<ref>), and then use known formulas for logarithmic _2F_1's <cit.>.Legendre and Ferrers functions that are dihedral (i.e., are of half-odd-integer order) are the subject of the following theorems. For conciseness, a special notation is used: [A|B]_± signifies A, resp. B, in the +, resp. - case; and {C}_α,±, where C depends on α, signifies the even or odd part of C under α↦-α, i.e., 1/2[C(α)± C(-α)].The formulasQ̂_-1/2+α^±(1/2+m)(z) = √(π/2)m! [1| 1/(α-m)_2m+1]_±× (z^2-1)^-1/4 (z+√(z^2-1))^-α P_m^(α,-α)(z/√(z^2-1)),Q̂_-1/2+α^±(1/2+m)(coshξ) = √(π/2)m! [1| 1/(α-m)_2m+1]_±× (sinhξ)^-1/2e^-αξ P_m^(α,-α)(ξ)hold when m is a non-negative integer, for z∈(1,∞) and ξ∈(0,∞). Combine Whipple's Q̂→ P transformation <cit.>, which appeared as Eq. (<ref>), with the results in Theorem <ref>; and write m for n, and -α for μ.In these formulas, the proportionality of Q̂_-1/2+α^±(1/2+m) to each other is expected; cf. (<ref>).Also, the division in the `minus' case by (α-m)_2m+1 = (α-m)…(α+m),which equals zero if α=-m,…, m, is not unexpected.As was noted in  <ref>, Q̂_ν^μ is undefined if ν+μ is a negative integer, except when ν=-3/2,-5/2,…, in which case Q̂_ν^ν+1,…, Q̂_ν^-(ν+1) are defined.This implies that for m=0,1,2,…, Q̂_-1/2+α^+(1/2+m) is defined for all α, and that Q̂_-1/2+α^-(1/2+m) is undefined if and only if α=-m,…,m.The formulasP_-1/2+α^±(1/2+m)(z) = √(2/π)m! [(-1)^m| (-1)^m+1/(α-m)_2m+1]_±× (z^2-1)^-1/4 {(z+√(z^2-1))^-α P_m^(α,-α)(z/√(z^2-1))}_α,±,P_-1/2+α^±(1/2+m)(coshξ) = √(2/π)m! [(-1)^m| (-1)^m+1/(α-m)_2m+1]_±× (sinhξ)^-1/2 { e^-αξ P_m^(α,-α)(ξ)}_α,±hold when m is a non-negative integer, for z∈(1,∞) and ξ∈(0,∞); it being understood in the `minus' case that when α=-m,…,m and there is an apparent division by zero, each right-hand side requires the taking of a limit. Combine the P→Q̂ reduction (<ref>) with the results in Theorem <ref>.The formulasP_-1/2+α^±(1/2+m)(cosθ)= √(2/π)m! [ i^m |i^-m-1/(α-m)_2m+1]_±× (sinθ)^-1/2 { e^ iαθ P_m^(α,-α)( i θ)}_α,±,Q_-1/2+α^±(1/2+m)(cosθ)= √(π/2)m! [ i^m+1 |i^-m/(α-m)_2m+1]_±× (sinθ)^-1/2 { e^ iαθ P_m^(α,-α)( i θ)}_α,∓hold when m is a non-negative integer, for θ∈(0,π).In the sub-cases α=-m,…,m of the `minus' case, the apparent division by zero in the first formula is handled by interpreting its right-hand side in a limiting sense; but the division by zero in the second formula causes both its sides to be undefined. The first formula follows by analytic continuation of the latter formula in Theorem <ref>; in effect, by letting ξ=- iθ.The second formula then follows from the Q→ P reduction (<ref>), after some algebraic manipulations.As was noted in<ref>, Q_ν^μ is undefined if and only if Q̂_ν^μ is.It was also noted that if μ=1/2,3/2,…, then Q_-μ^μ,…, Q_μ-1^μ≡0.It follows that in the `plus' case of the second formula of the theorem, the right-hand side must equal zero if α=-m,…,m.This yields the interesting Jacobi-polynomial identitye^iαθ P_m^(α,-α)(i θ) = e^-iαθ P_m^(-α,α)(i θ),which holds for m=0,1,2,…, when α=0,1,…,m. §.§ Dihedral Ferrers Functions and Love–Hunter ExpansionsIn this subsection, we show that an expansion in dihedral Ferrers functions can be, in effect, an expansion in Chebyshev polynomials (of the fourth kind); and as an application, show that the result of <cit.> on the convergence of Love–Hunter expansions can be slightly extended.The first formula on dihedral Ferrers functions in Theorem <ref> specializes when m=0 to the known pair of formulas <cit.>P_-1/2+α^-1/2(cosθ) =√(%s/%s)2π sin(αθ)/α√(sinθ),P_-1/2+α^1/2(cosθ) =√(%s/%s)2π cos(αθ)/√(sinθ).These hold for θ∈(0,π), the α=0 case of the former requiring the taking of a limit.Love–Hunter biorthogonality, i.e., the orthogonality of the functions P_ν^μ(z) and P_ν^-μ(-z) in L^2[-1,1] when (i) Re μ∈(-1,1) and (ii) ν,ν' differ by an even integer and are not half-odd-integers, specializes when μ=-1/2 and z=cosθ to∫_0^πP_ν^-1/2(cosθ)P_ν'^1/2(-cosθ) sinθ dθ= 0,and thus to∫_0^πsin(αθ) cos[α'(π-θ)]dθ = 0,which holds if α,α' differ by an even integer.(By continuity, the restriction to α,α' that are not integers can be dropped.) The orthogonality in (<ref>) is not well known.A Love–Hunter expansion of an `arbitrary' function f=f(z) on -1<z<1 is a bilateral expansion of f in the Ferrers functions P_ν_0+2n^μ, of the form (<ref>), in which the coefficients {c_n}_n∈ℤ are computed as inner products, i.e.,c_n^(α) = ∫_-1^1 P_ν_0+2n^-μ(-z) f(z) dz/∫_-1^1 P_ν_0+2n^-μ(-z) P_ν_0+2n^μ(z) dz.Existing results on the convergence of such expansions <cit.> require that |Re μ|<1/2, or in the real case, μ∈(-1/2,1/2).It is of interest to examine whether convergence results can also be obtained in the boundary cases μ=±1/2.To treat the case when (ν_0,μ)= (-1/2+α,-1/2), define the indexed (n∈ℤ) and continuously parametrized (α∈ℝ) functionsψ_n^(α)(θ) = sin[(2n+α)θ]/√(sinθ),χ_n^(α)(θ) = cos[(2n+α)(π-θ)]/√(sinθ) on 0<θ<π, which are biorthogonal with respect to the weight function sinθ.(They differ only in normalization from P_-1/2+α+2n^-1/2 and P_-1/2+α+2n^1/2.)In terms of the first, one has a formal μ=-1/2 Love–Hunter expansionf(cosθ) = ∑_n=-∞^∞c_n^(α)ψ_n^(α)(θ),wherec_n^(α) = ∫_0^πχ^(α)_n(θ) f(cos(θ)) sinθ dθ/∫_0^πχ^(α)_n(θ) ψ_n^(α)(θ) sinθ dθ.The denominator inner product equals (π/2)sin(απ) for all n, by examination; hence the restriction α∈ℝ∖ℤ must obviously be imposed.This expansion is not fully satisfactory, because each ψ_n^(α)(θ) diverges as θ→π^-; though it converges to zero, asymmetrically, as θ→0^+.The underlying problem is that if Re μ<0, the function P_ν^μ(z) has leading behavior as z→1^- proportional to (1- z)^-μ/2, but its leading behavior as z→(-1)^+ comprises two terms: one proportional to (1+ z)^-μ/2, and one to (1+ z)^+μ/2.In order (i) to make endpoint behavior more symmetrical and less divergent, and (ii) to study endpoint convergence, Pinsky <cit.> has proposed modifying Love–Hunter expansions by treating [(1- z)/ (1+ z)]^μ/2P_ν^μ(z) rather than P_ν^μ(z) as the expansion function.By (<ref>), this amounts to replacing each P_ν^μ(z) by the _2F_1 function in terms of which it is defined; i.e., performing a hypergeometric expansion.Adopting the suggestion of <cit.> when μ=-1/2 amounts to replacing ψ_n^(α),χ_n^(α) by versions that are multiplied by [(1- z)/ (1+ z)]^-1/4, which equals ^1/2(θ/2).With a trivial change in normalization, these are the functionsψ̂_n^(α)(θ) = sin[(2n+α)θ]/sin(θ/2),χ̂_n^(α)(θ) = cos[(2n+α)(π-θ)]/sin(θ/2)(n∈ℤ) on 0<θ<π, which are biorthogonal with respect to the weight function sin^2(θ/2).Each ψ̂_n^(α)(θ) has a finite, nonzero limit as θ→0^+ and θ→π^-, and as a function of z=cosθ is proportional to_2F_1(1/2-α-2n,1/2+α+2n; 3/2;1-z/2).In terms of these trigonometric functions ψ̂_n^(α)(θ), one has (formally) a bilateral expansion of an arbitrary function f=f(z) defined on -1<z<1, namelyf(z=cosθ) = lim_N→∞∑_n=-N^N ĉ_n^(α)ψ̂_n^(α)(θ)for all θ∈(0,π), whereĉ_n^(α) = ∫_0^πχ̂^(α)_n(θ) f(cos(θ)) sin^2(θ/2)dθ/∫_0^πχ̂^(α)_n(θ) ψ̂_n^(α)(θ) sin^2(θ/2)dθ.The denominator in (<ref>) equals (π/2)sin(απ) for all n, as before.One can clearly restrict α from ℝ∖ℤ to the interval (0,1) without losing generality. If f=f(z) is piecewise continuous on -1≤ z≤ 1, then in the symmetric case α=1/2, the bilateral series in (<ref>) will converge as N→∞ to f(z) at all points of continuity, including the endpoints, and in general to [f(z+)+ f(z-)]/2. The Chebyshev polynomials W_j of the fourth kind, for j=0,1,2,…, are defined by <cit.>W_j(cosθ) = sin[(j+1/2)θ]/sin(1/2θ) = ∑_m=-j^j e^imθ.It follows from (<ref>) that when n=0,1,2,…, both of ψ̂_n^(1/2)(θ),χ̂_n^(1/2)(θ) equal W_2n(cosθ); and when n=-1,-2,…, they equal -W_-2n-1(cosθ).The bilateral expansion in (<ref>) thus reduces if α=1/2 to a unilateral expansion in the polynomials W_j, j=0,1,2,…. The Chebyshev polynomials T_k of the first kind, for k=0,1,2,…, are given byT_k(cosθ) = cos(kθ).By standard Fourier series theory, the expansion of g=g(u) in the T_k(u), when g is piecewise continuous on -1≤ u≤ 1, will converge to g at all points of continuity, and in general to [g(u+)+ g(u-)]/2.But (see <cit.>), if one writes z=1- 2u^2 (so that u=sin(θ/2) if z=cosθ), then W_j(z) equals (-1)^j u^-1 T_2j+1(u).Therefore an expansion of f=f(z) in the fourth-kind W_j(z) on -1≤ z≤ 1 is effectively an expansion of g(u)= uf(1-2u^2) on -1≤ u≤1 in the first-kind T_k(u), each even-k term of which must vanish.The theorem follows.It is useful to compare this convergence result, which refers to an expansion of f in the Ferrers functions P_2n^-1/2, with the pointwise convergence result of <cit.>.The latter deals with an expansion in the functions P_ν_0+2n^μ, where ν_0 is arbitrary and μ∈(-1/2,1/2).However, it requires that f be piecewise smooth, not merely piecewise continuous.As the above theorem reveals, this assumption can be relaxed; at least, in the seemingly difficult `corner' case when (ν_0,μ)=(0,-1/2).Whether smoothness can also be dropped as a hypothesis for the pointwise convergence of Love–Hunter expansions with μ∈(-1/2,1/2), or with (ν_0,μ)= (-1/2+α,-1/2) when α≠1/2, remains to be explored.It must be mentioned that the octahedral and tetrahedral formulas of Theorems <ref> and <ref> facilitate the calculation of the coefficients in Love–Hunter expansions of the form (<ref>), with (ν_0+1/2,μ) equal to (±1/3,±1/4) and (±1/4,±1/3), respectively. Because these values satisfy μ∈(-1/2,1/2), the convergence result of <cit.> applies.§ LADDER OPERATORS, LIE ALGEBRAS, AND REPRESENTATIONSIn the preceding sections, explicit formulas for the Legendre and Ferrers functions in the octahedral, tetrahedral, dihedral, and cyclic families were derived.Each such family (in the first-kind Ferrers case) is of the form { P_ν_0+n^μ_0+m(z=cosθ)}, where ν_0,μ_0 are or may be fractional, and (n,m) ranges over ℤ^2.In this section, the connection between such a family and conventional SO(3)-based harmonic analysis on the sphere S^2=SO(3)/SO(2), coordinatized by the angles (θ,φ), is briefly explored.The connection goes through the corresponding family of generalized spherical harmonics, P_ν^μ(cosθ) e^ iμφ, with (ν,μ)∈(ν_0,μ_0)+ℤ^2. But the connection is not as strong as one would like.If ν_0,μ_0 are rational but not integral, these harmonic functions will not be single-valued on the symmetric space S^2.(In the cases of interest here, each P_ν^μ(z) in the family is algebraic in z, and they can be viewed as finite-valued.)They may not be square-integrable, because the leading behavior of P_ν^μ(z) as z→1^- is proportional to (1- z)^-μ/2 unless μ is a positive integer.For these reasons, the focus is on the action of Lie algebras (of `infinitesimal transformations') on a function family of this type, specified by (ν_0,μ_0), rather than the action of a Lie group such as SO(3).The space spanned by the classical spherical harmonics Y_n^m(θ,φ)∝ P_n^m(cosθ) e^ imφ, with n≥0 and m∈ℤ, admits an action of the rotation group SO(3).The Lie algebra 𝔰𝔬(3,ℝ) of 3×3 real skew-symmetric matrices can be represented by differential operators on S^2, with real coefficients, and acts on the space of spherical harmonics.The resulting infinite-dimensional representation is reducible: for n=0,1,2,…, it includes the usual (2n+1)-dimensional representation on the span of Y_n^-n,…,Y_n^n.But, 𝔰𝔬(3,ℝ) is not the only Lie algebra to be considered.A larger Lie algebra than 𝔰𝔬(3,ℝ) acts naturally on the spherical harmonics, or rather, on the (regular) solid harmonics r^nY_n^m(θ,φ), which satisfy Laplace's equation on ℝ^3.(See <cit.>.)This is the 10-dimensional real Lie algebra 𝔰𝔬(4,1) that is generated by `ladder' operators that increment and decrement the degree n, as well as the order m.They are represented by differential operators on ℝ^3, with real coefficients.The real span of these operators exponentiates to the Lie group SO_0(4,1), which contains as subgroups (i) the 3-parameter group SO(3) of rotations about the origin, (ii) a 3-parameter abelian group of translations of ℝ^3, (iii) a 1-parameter group of dilatations (linear scalings of ℝ^3), and (iv) a 3-parameter abelian group of `special conformal transformations.'The last are quadratic rational self-maps of ℝ^3 (or rather the real projective space ℝℙ^3, because they can interchange finite and infinite points).The preceding results, now standard, are extended below to any family of generalized solid harmonics {r^ν P_ν^μ(cosθ) e^ iμφ}, with (ν,μ)∈ (ν_0,μ_0)+ℤ^2 for specified ν_0,μ_0. In  <ref>, the differential and non-differential recurrences on ν and μ are derived.(See Theorems <ref> and <ref>.)In  <ref>, it is shown that the ladder operators in the differential recurrences generate a 10-dimensional real Lie algebra, and an isomorphism from this algebra not to 𝔰𝔬(4,1) but to 𝔰𝔬(3,2) is exhibited.The treatment closely follows Celeghini and del Olmo <cit.>, but the explicit isomorphism in Theorem <ref> is new.In the setting of special function identities, which typically involve real linear combinations of differential operators, 𝔰𝔬(3,2) arises more naturally than does 𝔰𝔬(4,1).But by a limited form of complexification, 𝔰𝔬(3,2) can be converted to 𝔰𝔬(4,1), and indeed to 𝔰𝔬(5,ℝ).These are alternative real forms of the rank-2 complex Lie algebra 𝔰𝔬(5,ℂ), to which they complexify, and the eight displacement vectors Δ(ν,μ)=(0,±1), (±1,0), (±1,±1) can be identified with the roots of 𝔰𝔬(5,ℂ).In  <ref>, it is shown that irrespective of (ν_0,μ_0), the representation of 𝔰𝔬(3,2) [or of 𝔰𝔬(4,1) or 𝔰𝔬(5,ℝ)] carried by the solid harmonics r^ν P_ν^μ(cosθ) e^ iμφ with (ν,μ)∈(ν_0,μ_0)+ℤ^2 is of a special type: its quadratic Casimir operator takes a fixed value, and its quartic one vanishes.(See Theorem <ref>.)The former fact was found in <cit.>, but the latter is new.The representation of 𝔰𝔬(3,2) on the solid harmonics of integer degree and order, and its representation on the ones of half-odd-integer degree and order, have irreducible constituents that are identified as the known Dirac singleton representations of 𝔰𝔬(3,2). §.§ Differential and Non-differential RecurrencesIn any family { P_ν_0+n^μ_0+m(z)}_(n,m)∈ℤ^2, where P can be taken as any of P, Q,P,Q̂, any three distinct members are linearly dependent, over the field of functions that are rational in z and √(1-z^2) (Ferrers case) or √(z^2-1) (Legendre case).In particular, any three contiguous members are so related, by a three-term ladder recurrence.The underlying recurrences are differential ones, which generally permit any single P^μ_ν and its derivative to generate any member contiguous to it, as a linear combination; and by iteration, to generate any P_ν+Δν^μ+Δμ in which Δ(ν,μ)∈ℤ^2.The Ferrers functions P_ν^μ =P_ν^μ(z) satisfy eight differential recurrences, divided into four ladders, i.e.,±-pairs, withΔ(ν,μ)=±(0,1), ±(1,0), ±(1,1), and ±(1,-1).Each pair is of the formα_± P_ν+Δν^μ+Δμ = ∓z^-σ_0^±+ε_0(1-z^2)^-σ_1^±/2+ε_1/2 d/dz[ z^σ_0^±(1-z^2)^σ_1^±/2P_ν^μ],and for each pair, the exponents σ_0^±,σ_1^±, the exponents ε_0,ε_1, and the prefactor α_± are given in Table <ref>.The second-kind functions Q_ν^μ satisfy identical recurrences.The Legendre functions P_ν^μ,Q_ν^μ [the latter unnormalized, i.e., the functions e^μπ iQ̂_ν^μ] satisfy recurrences obtained from the preceding by (i) multiplying the right-hand side by a sign factor, equal to i^ε_1+Δμ; and (ii) replacing 1- z^2 by z^2- 1. The four non-diagonal recurrences on the order and degree, with Δ(ν,μ)=±(0,1) and ±(1,0), are classical and can be found in many reference works <cit.>.They can be deduced from the differential recurrences of Jacobi, which increment or decrement the parameters of the function _2F_1(a,b;c;x).(See <cit.>.)The final four diagonal ones, at least for P^μ_ν when ν,μ are integers, are due to Celeghini and del Olmo <cit.>.Each can be derived from the non-diagonal ones by a tedious process of elimination, but the process can be systematized as the calculation of the commutator of two differential operators.(See  <ref>, below.)The differential recurrences satisfied by P_ν^μ can be written in circular-trigonometric forms that will be needed below. Substituting z=cosθ yieldsα_± P_ν^μ±1 = [± D_θ - μθ]P_ν^μ, α_± P_ν±1^μ = {±(sinθ)D_θ + [(ν+12)±12]cosθ} P_ν^μ, α_± P_ν±1^μ±1 = {± (cosθ)D_θ - μθ + [-(ν+12)∓12]sinθ} P_ν^μ, α_± P_ν±1^μ∓1 = {∓ (cosθ)D_θ - μθ + [+(ν+12)±12]sinθ} P_ν^μ, which are satisfied by P_ν^μ = P_ν^μ(cosθ). Here, D_θ d/ dθ, and the four prefactors α_± are listed in the last column of Table <ref>, in order.The recurrences (<ref>) and (<ref>) have appeared in the literature but are not well known; the only appearances that we have found, with ν,μ restricted to integer values, are in <cit.> and <cit.>.Equations (<ref>) and (<ref>) imply each other because P_-ν-1^μ= P_ν^μ for all ν,μ. That is, P_ν^μ is unaffected by the negating of the shifted degree parameter ν+1/2.The three-term ladder recurrences derived from the four pairs of differential recurrences are given in the following theorem.The diagonal ones, coming from the ladders with Δ(ν,μ)=±(1,1) and ±(1,-1), appear to be new.The Ferrers functions P_ν^μ= P_ν^μ(z) satisfy second-order (i.e., three-term) recurrences on the order μ and degree ν, namely√(1-z^2)P_ν^μ+1 + 2μ zP_ν^μ + (ν+μ)(ν-μ+1)√(1-z^2)P_ν^μ - 1 = 0,(ν-μ+1)P_ν+1^μ - (2ν+1)zP_ν^μ + (ν+μ)P_ν-1^μ=0,and the two diagonal recurrences√(1-z^2) P_ν±1^μ+1(z) + [±(2ν+1)(1-z^2) + 2μ]P_ν^μ(z) + [(ν+12)±(μ-12)] [(ν+12)±(μ-32)] √(1-z^2) P_ν∓1^μ-1(z) =0.The second-kind functions Q_ν^μ satisfy identical second-order recurrences.The Legendre functions P_ν^μ,Q_ν^μ (the latter unnormalized, as above), satisfy recurrences obtained from the preceding by (i) multiplying each term containing a function of order μ+δ and a coefficient proportional to [√(1-z^2)]^α by a sign factor, equal to i^α-δ; and (ii) replacing √(1-z^2) by √(z^2-1). Eliminate the derivative terms from the recurrences of Theorem <ref>.This is the procedure used to derive Gauss's three-term, nearest-neighbor `contiguous function relations' for _2F_1(a,b;c;x) from Jacobi's differential recurrences on a,b;c.It was noted in<ref> that if ν+μ is a negative integer, Q_ν^μ and Q_ν^μ are generally undefined (though there are exceptions).The recurrences for Q_ν^μ and Q_ν^μ in Theorems <ref> and <ref> remain valid in a limiting sense even when (ν,μ) is such that one or more of the functions involved is undefined.§.§ Lie AlgebrasThe raising and lowering of the degree and order, in any doubly indexed family of (generalized) solid harmonics𝒮_ν^μ = 𝒮_ν^μ(r,θ,φ)r^ν P_ν^μ(cosθ) e^ iμφ,where (ν,μ)∈(ν_0,μ_0)+ℤ^2, can be performed by differential operators that do not need to depend explicitly on (ν_0,μ_0) if they are allowed to involve, instead, the derivative operators D_r,D_φ in addition to D_θ.The basic idea is due to Miller <cit.>, and there is freedom in its implementation: either or both of the factors r^ν =e^νlog r and e^ iμφ could include an `i' in its exponent, and the Ferrers functions P_ν^μ(cosθ) could be replaced by the Legendre ones P_ν^μ(coshξ).With the choices made in (<ref>), 𝒮_ν^μ can be viewed as a (typically multi-valued) function of the spherical coordinates r,θ,φ, which satisfies Laplace's equation on ℝ^3.Define ladder operators byJ_± =e^± iφ[± D_θ +i(θ)D_φ],K_± = r^±1[ ±(sinθ)D_θ + (cosθ)(rD_r+12±12)],R_± = r^±1 e^± iφ[±(cosθ)D_θ + i(θ)D_φ - (sinθ)(rD_r+12±12)],S_± = r^±1 e^∓ iφ[∓(cosθ)D_θ + i(θ)D_φ + (sinθ)(rD_r+12±12)]. Then, the differential recurrences (<ref>) can be rewritten in terms of the 𝒮_ν^μ asJ_± 𝒮_ν^μ = [1,(ν+μ)(ν-μ+1)]_± 𝒮_ν^μ±1, K_± 𝒮_ν^μ = [ν-μ+1,ν+μ]_± 𝒮_ν±1^μ, R_± 𝒮_ν^μ = [1,(ν+μ)(ν+μ-1)]_± 𝒮_ν±1^μ±1, S_± 𝒮_ν^μ = [(ν-μ+1)(ν-μ+2),1]_± 𝒮_ν±1^μ∓1. For each (ν_0,μ_0), the solid harmonics {𝒮_ν^μ=𝒮_ν_0+n^μ_0+m}, or more accurately their real linear span, carry a representation of J_±,K_±, R_±,S_±.Generating a Lie algebra by working commutators out, one finds (with [A,B] signifying AB-BA)R_±= ±[J_±,K_±],S_±= ±[J_∓,K_±],which explains why the diagonal recurrences in Theorems <ref> and <ref> can be most efficiently obtained by commutator calculations, as claimed.It is useful additionally to define `labeling' or `maintaining' operators J_3,K_3 by J_3 = - iD_φ, K_3 = rD_r+12,so thatJ_3 𝒮_ν^μ = μ 𝒮_ν^μ,K_3 𝒮_ν^μ = (ν+12) 𝒮_ν^μ.By further calculations, one finds that the real Lie algebra generated by J_±,K_± closes, in the sense that it is finite-dimensional.In particular,[J_3,J_±] =±J_±,[J_+,J_-] =2J_3 ,[K_3,K_±] =±K_±,[K_+,K_-] =-2K_3 ,[R_3,R_±] =±2R_±,[R_+,R_-] =-4R_3 ,[S_3,S_±] =±2S_±,[S_+,S_-] =-4S_3,where R_3 K_3+J_3 and S_3 K_3-J_3.To interpret these, recall that any real linear space with basis {X_+,X_-,X_3}, given a Lie algebra structure by[X_3,X_±]=±X_±,[X_+,X_-]=2σX_3,is isomorphic to 𝔰𝔬(3,ℝ) if σ>0, and to 𝔰𝔬(2,1) (or equivalently 𝔰𝔩(2,ℝ)) if σ<0.Hence, the real Lie algebras spanned by {J_+,J_-,J_3}, {K_+,K_-,K_3}, {R_+,R_-,R_3}, and {S_+,S_-,S_3}, coming from the ladders with Δ(ν,μ)=±(0,1), ±(1,0), ±(1,1), and ±(1,-1), are isomorphic to 𝔰𝔬(3,ℝ) (the first) and 𝔰𝔬(2,1) (the remaining three).The last two turn out to commute.The real Lie algebra generated by J_±,K_±, of which these copies of 𝔰𝔬(3,ℝ) and 𝔰𝔬(2,1) are subalgebras, is 10-dimensional and is spanned over ℝ by J_±,J_3; K_±,K_3; R_±;S_±.It of course has real structure constants.For any (ν_0,μ_0), its representation by differential operators on ℝ^3, as above, is carried by the real span of the solid harmonics 𝒮_ν_0+n^μ_0+m, (n,m)∈ℤ^2. This result was obtained by Celeghini and del Olmo <cit.>, though they confined themselves to integer ν,μ, i.e., in effect to (ν_0,μ_0)=(0,0).[The reader of <cit.> should note that in  5, R_±[K_±,J_±] and S_±[K_±,J_∓] should be emended to read R_±∓[K_±,J_±] and S_±∓[K_±,J_∓].]To identify this 10-dimensional real algebra, it is useful to relabel its basis elements.First, let(P_+,P_-,P_3)(S_+,-R_-,K_-), (C_+,C_-,C_3)(-R_+,S_-,K_+),in each of which the three elements commute.The algebra can then be viewed as the span over ℝ of J_±,J_3; P_±,P_3; C_±,C_3 and K_3, which will be written as D henceforth.Define(PC_+^±,PC_-^±,PC_3^±)12[(P_+,P_-,P_3) ± (C_+,C_-,C_3)]= 12(∓ R_++S_+, -R_-± S_-,K_-± K_+). Also, for X=J,P,C,PC^+,PC^-, define the `skew-Cartesian' elements𝒳_1(X_++X_-)/2, 𝒳_2(X_+-X_-)/2, 𝒳_3X_3,so that X_± = 𝒳_1±𝒳_2.The algebra will then be the real span of 𝒥_1,𝒥_2,𝒥_3; 𝒫_1,𝒫_2,𝒫_3; 𝒞_1,𝒞_2,𝒞_3;D, or equivalently of 𝒥_1,𝒥_2,𝒥_3; 𝒫𝒞^±_1,𝒫𝒞^±_2,𝒫𝒞^±_3; D.It is readily verified that 𝒥_i commutes with 𝒫𝒞^+_i and 𝒫𝒞^-_i for i=1,2,3, and that[𝒥_i, 𝒥_j]={-1,+1,-1}_k 𝒥_k,[𝒥_i, 𝒫𝒞^±_j]={-1,+1,-1}_k 𝒫𝒞_k^±,[𝒫𝒞^±_i, 𝒫𝒞^±_j]=∓{-1,+1,-1}_k 𝒥_k, where i,j,k is any cyclic permutation of 1,2,3, with {a,b,c}_k meaning a,b,c when k=1,2,3.Also, the 3×3 matrix of commutators [𝒫𝒞_i^+,𝒫𝒞_j^-] indexed by 1≤ i,j≤ 3 equals diag (-D,+D,-D).Additionally,[D,𝒥_i] = 0,[D,𝒫𝒞_i^±] = - 𝒫𝒞_i^∓,for i=1,2,3.These identities specify the structure of the algebra.Now, recall that the real Lie algebra 𝔰𝔬(p,q) with p+q=n has the following defining representation.If Γ=(g_ij)= diag (+1,…,+1,-1,…,-1), with q  +1's and p  -1's, then 𝔰𝔬(p,q) comprises all real n× n matrices A for which Γ A is skew-symmetric. There is a sign convention here, and a p↔ q symmetry; without loss of generality, p≥ q will be assumed.It is sometimes useful to permute the +1's and -1's.More concretely, 𝔰𝔬(p,q) can be realized as the real span of the n× n matrices M_ab, 1≤ a< b ≤ n, where M_ab =Γℰ_ab -ℰ_baΓ.In this, ℰ_ab is the n× n matrix with a 1 in row a, column b, and zeroes elsewhere. One often extends the size-(n2) basis { M_ab} to a `tensor operator,' i.e., a skew-symmetric n× n matrix of elements ( M_ab), by requiring that M_ba=- M_ab for 1≤ a,b≤ n.The commutation relations[ M_ab, M_cd] = g_ad M_bc + g_bc M_ad - g_ac M_bd - g_bd M_acare easily checked.The real Lie algebra generated by J_±,K_± is isomorphic to 𝔰𝔬(3,2), an isomorphism being specified by the tensor operator(M_ab) = ( [ 0𝒫𝒞_2^- -𝒫𝒞_1^- -𝒫𝒞_3^--D; -𝒫𝒞_2^- 0 𝒥_3-𝒥_1𝒫𝒞_2^+;𝒫𝒞_1^--𝒥_3 0 𝒥_2 -𝒫𝒞_1^+;𝒫𝒞_3^- 𝒥_1-𝒥_2 0 -𝒫𝒞_3^+; D -𝒫𝒞_2^+𝒫𝒞_1^+𝒫𝒞_3^+ 0 ] )with Γ= diag (+1,+1,-1,-1,-1). By inspection.This 𝔰𝔬(3,2) has the Cartan decomposition 𝔨⊕_ℝ𝔭, where𝔨 =sp_ℝ{𝒥_2, 𝒫𝒞^+_1, 𝒫𝒞^+_3, 𝒫𝒞^-_2 }, 𝔭 =sp_ℝ{𝒥_1,𝒥_3,𝒫𝒞^+_2,𝒫𝒞_1^-,𝒫𝒞_3^-,D }are the `compact' and `non-compact' subspaces.(The terms refer to the Lie subgroups of SO_0(3,2) to which they exponentiate.)Real Lie algebras isomorphic to 𝔰𝔬(4,1) and 𝔰𝔬(5,ℝ) can be obtained by Weyl's trick of redefining some or all of the basis elements of 𝔭 to include `i' factors.In doing this, a slightly changed notation will be useful.For X=J,P,C,PC^+,PC^-, define the `Cartesian' elementsX_1(X_++X_-)/2, X_2- i(X_+-X_-)/2,so that X_±=X_1± iX_2 and (𝒳_1,𝒳_2,𝒳_3) = (X_1, iX_2,X_3).(i) The real span of iJ_1, iJ_2,iJ_3; P_1,P_2,P_3; C_1,C_2,C_3; D, or equivalently of iJ_1, iJ_2,iJ_3; PC_1^±,PC_2^±,PC_3^±; D, is a real Lie algebra isomorphic to 𝔰𝔬(4,1), an isomorphism being specified by the tensor operator(M_ab) = ( [ 0 -PC_1^- -PC_2^- -PC_3^--D;PC_1^- 0 -iJ_3iJ_2 -PC_1^+;PC_2^-iJ_3 0 -iJ_1 -PC_2^+;PC_3^- -iJ_2iJ_1 0 -PC_3^+; DPC_1^+PC_2^+PC_3^+ 0 ] )with Γ= diag (+1,-1,-1,-1,-1).(ii) The real span of iJ_1, iJ_2,iJ_3; iPC_1^+, iPC_2^+, iPC_3^+;PC_1^-,PC_2^-,PC_3^-;iD is a real Lie algebra isomorphic to 𝔰𝔬(3,2), an isomorphism being specified by the tensor operator(M_ab) = ( [0-PC_1^--PC_2^--PC_3^--iD; PC_1^-0-iJ_3 iJ_2 -iPC_1^+; PC_2^- iJ_30-iJ_1 -iPC_2^+; PC_3^--iJ_2 iJ_10 -iPC_3^+; iDiPC_1^+iPC_2^+iPC_3^+0 ] )with Γ= diag (+1,-1,-1,-1,+1). (i) Multiply the second row and the second column of the ( M_ab) in Theorem <ref> by `i', and (innocuously) interchange the second and third rows, and the second and third columns.(ii) Continuing (or in a sense reversing), multiply the last row and the last column by `i'.The real span of iJ_1, iJ_2,iJ_3; PC_1^+,PC_2^+,PC_3^+;iPC_1^-, iPC_2^-, iPC_3^-;iD is a real Lie algebra isomorphic to 𝔰𝔬(5,ℝ), an isomorphism being specified by the tensor operator(M_ab) = ( [0 -iPC_1^- -iPC_2^- -iPC_3^--iD;iPC_1^-0-iJ_3 iJ_2-PC_1^+;iPC_2^- iJ_30-iJ_1-PC_2^+;iPC_3^--iJ_2 iJ_10-PC_3^+; iD PC_1^+ PC_2^+ PC_3^+0 ] )with Γ= diag (-1,-1,-1,-1,-1). Multiply the first row and the first column of the ( M_ab) in part (i) of Theorem <ref> by `i'.With `i' factors in basis elements, the 𝔰𝔬(3,2), 𝔰𝔬(4,1) and 𝔰𝔬(5,ℝ) of Theorems <ref> and <ref> look awkward.But it follows from (<ref>), (<ref>), and (<ref>) that each basis element in the 𝔰𝔬(4,1) of Theorem <ref>(i), i.e., each of iJ_i, P_i, C_i, and D, is realized by a differential operator in r,θ,φ with real coefficients.This is not the case for the basis elements of 𝔰𝔬(3,2) and 𝔰𝔬(5,ℝ).The geometric significance of this realization of 𝔰𝔬(4,1) is revealed by changing from spherical coordinates (r,θ,φ) to Cartesian ones, (x_1,x_2,x_3) on ℝ^3.Usingx(x_1,x_2,x_3) = (rsinθcosφ, rsinθsinφ, rcosθ),one finds thatiJ_i= x_j∂_k - x_k∂_j,i=1,2,3, P_i= ∂_i,i=1,2,3,C_i= x_i - (x·x)∂_i + 2x_i(x·∂),i=1,2,3,D= x·∂+12,where ∂_i d/ dx_i and i,j,k is a cyclic permutation of 1,2,3.That is, the iJ_i generate rotations about the origin, the P_i generate translations, and D generates dilatations (linear scalings of ℝ^3).The C_i generate special conformal transformations, which are degree-2 rational maps of ℝ^3 (or rather ℝℙ^3) to itself. The commutation relations[J_i,J_j]= iϵ_ijk J_k,[PC_i^±,PC^±_j]= ∓iϵ_ijkJ_k,[J_i,PC_j^±]= iϵ_ijkPC_k^±,[PC_i^+,PC_j^-]= -δ_ijD,[D,J_i]=0, [D,PC_i^±]= - PC_i^∓,written in terms of PC_i^± = 12(P_i± C_i), follow either from (<ref>),(<ref>), from (<ref>), or from (<ref>).Here, the summation convention of tensor analysis is employed.The Levi-Cività tensor ϵ_ijk is skew-symmetric in all indices, with ϵ_123=+1, and δ_ij is the Kronecker delta.Together withJ_1 = 1/2(J_+ + J_-),J_2 = - i/2(J_+ - J_-), D=K_3,the formulasPC_1^± = 14 (∓ R_+ - R_-± S_+ + S_-),PC_2^± = - i4 (∓ R_+ + R_- ∓ S_++ S_- ),PC_3^± = 12(± K_+ + K_-), express all these differential operators in terms of the original J_±,K_±, R_±,S_±; J_3,K_3 of (<ref>),(<ref>).The ten operators in (<ref>) span (over ℝ) the Lie algebra of conformal differential operators on ℝ^3, which is known to have an 𝔰𝔬(4,1) structure.(See Miller <cit.>.)This is the symmetry algebra of the Laplacian ∇^2 on ℝ^3, which comprises all real first-order operators L for which [L,∇^2]∝∇^2, i.e., for which [L,∇^2] has ∇^2 as a right factor.It can be viewed as acting on any suitable space of functions on ℝ^3, and exponentiates to the group SO_0(4,1) of conformal transformations, realized as flows on ℝ^3 (or ℝℙ^3).But the starting point used here was their action on the span of the generalized solid harmonics 𝒮_ν_0+n^μ_0+m with (n,m)∈ℤ^2, which are (multi-valued) solutions of Laplace's equation.In the physics literature on conformal Lie algebras and groups, the terms `x_i' in (<ref>) and `12' in (<ref>) often appear as 2δ x_i and δ respectively, where δ is the so-called scaling dimension; though the resulting commutation relations do not involve δ.The value δ=12 is specific to the symmetry algebra of the Laplacian.There are many variations on the present technique of using differential recurrences to construct real Lie algebras, realized by differential operators, that are isomorphic to the real forms of 𝔰𝔬(5,ℂ).The solid harmonics 𝒮_ν^μ that were employed here are extensions to ℝ^3 of the (surface) spherical harmonics P_ν^μ(cosθ) e^ iμφ on the symmetric space S^2=SO(3)/SO(2).If not Ferrers but Legendre functions were used, the starting point would be the hyperboloidal ones P_ν^μ(coshξ) e^ iμφ, defined using coordinates (ξ,φ) on the hyperboloid H^2=SO(2,1)/SO(2), i.e., the surface x_1^2 + x_2^2 - x_3^2 + const=0.Their extensions to ℝ^3 satisfy the (2+1)-dimensional wave equation, rather than Laplace's equation. (See <cit.> and <cit.>.)But isomorphic algebras could be constructed.§.§ Lie Algebra RepresentationsIn  <ref>, it was shown that for any (ν_0,μ_0), there are representations of the real Lie algebras 𝔰𝔬(3,2), 𝔰𝔬(4,1), 𝔰𝔬(5,ℝ) that are carried by the span of the family of (generically multi-valued) solid harmonics 𝒮_ν^μ(r,θ,φ), (ν,μ)∈(ν_0,μ_0)+ℤ^2.These arise from the action of the ladder operators on the Ferrers functions P_ν^μ(cosθ).Solid harmonics are harmonic functions on ℝ^3, satisfying Laplace's equation; and the ones in the octahedral, tetrahedral, dihedral, and cyclic families are or can be finite-valued.These infinite-dimensional representations are restrictions of the representation of the common complexification 𝔰𝔬(5,ℂ), which is carried by the (complex) span of the family.They are generically irreducible, and are also generically non-skew-Hermitian, so that except in special cases, they do not exponentiate to unitary representations of the corresponding Lie groups, even formally.This will now be investigated.Each of the three real Lie algebras is of rank 2, so the center of its universal enveloping algebra is generated by two elements, called Casimir invariants; and any irreducible representation must represent each Casimir by a constant.The analysis of such representations resembles the unified classification of the irreducible representations of 𝔰𝔬(2,1) and 𝔰𝔬(3,ℝ), the real forms of 𝔰𝔬(3,ℂ), which is well known.(See, e.g., <cit.>.)In this, representations are classified by the value taken by their (single) Casimir, and by their reductions with respect to a (1-dimensional) Cartan subalgebra. This leads to an understanding of which representations are skew-Hermitian and which are finite-dimensional.However, no comparable unified approach to all representations of 𝔰𝔬(3,2), 𝔰𝔬(4,1), and 𝔰𝔬(5,ℝ) seems to have been published.The literature has dealt almost exclusively with the skew-Hermitian ones. (𝔰𝔬(3,2) and 𝔰𝔬(4,1) are treated separately in <cit.> and <cit.>, and 𝔰𝔬(4,1) and 𝔰𝔬(5,ℝ) are treated together in <cit.>.)The starting point is the complexification 𝔰𝔬(5,ℂ), which is generated over ℂ by J_± and K_±, the ladder operators on the order and degree.It is the complex span of J_±,K_±, R_±,S_±; J_3,K_3, each of which is represented as in (<ref>) and (<ref>) by an infinite matrix indexed by (ν,μ)∈(ν_0,μ_0)+ℤ^2.The elements J_3,K_3 span a Cartan subalgebra (an abelian subalgebra of maximal [complex] dimension, here 2), which is represented diagonally:J_3 𝒮_ν^μ= μ 𝒮_ν^μ, K_3 𝒮_ν^μ= (ν+12) 𝒮_ν^μ.When the representation of 𝔰𝔬(5,ℂ) is reduced with respect to this subalgebra, it splits into an infinite direct sum of 1-dimensional representations, indexed by (ν,μ).The corresponding real Cartan subalgebras of the 𝔰𝔬(3,2) and 𝔰𝔬(4,1) in Theorems <ref> and <ref>(i) are the real spans of {J_3,K_3} and { iJ_3,K_3}.For the 𝔰𝔬(3,2) and 𝔰𝔬(5,ℝ) in Theorems <ref>(ii) and <ref>, they are the real span of { iJ_3, iK_3}.(Recall that D K_3.)Only for the last two will the real Cartan subalgebra be represented by skew-Hermitian matrices; in fact, by imaginary diagonal ones.It is readily verified that J_±,K_±, R_±,S_±; J_3,K_3 can serve as a Cartan–Weyl basis of 𝔰𝔬(5,ℂ), their complex span.That is, when the adjoint actions of H_1 J_3 and H_2 K_3 on this 10-dimensional Lie algebra are simultaneously diagonalized, the common eigenvectors (`root vectors') include J_±,K_±, R_±,S_±.The associated roots α∈ℝ^2 are 2-tuples of eigenvalues, which can be identified with the displacements Δ(ν,μ), i.e., ±(0,1), ±(1,0), ±(1,1), ±(1,-1).These form the B_2 root system.One can write[H_i,H_j] = 0, [H_i,E_α]=α_iE_α,where E_α is the root vector associated to root α.The commutators [E_α,E_β] also prove to be consistent with the B_2 root system.The Casimir invariants of 𝔰𝔬(5,ℂ) and its three real forms can be computed from the commutation relations of the Cartan–Weyl basis elements.(For instance, the Killing form for the algebra yields a quadratic Casimir.)But it is easier to express them using the tensor operator M_ab of any of Theorems <ref>, <ref>, and <ref>.As elements of the universal enveloping algebra, the two Casimirs, quadratic and quartic, are defined thus <cit.>:c_2 -12M_ab M^ab,c_4 -w_aw^a,where w^a = 18 ϵ^abcde M_bc M_de and the summation convention is employed, indices being raised and lowered by the tensors Γ^-1=(g^ab) and Γ=(g_ab).The Levi-Cività tensor ϵ_abcde is skew-symmetric in all indices, with ϵ_12345=+1.The normalization and sign conventions are somewhat arbitrary.In the representation of the universal enveloping algebra of any of the real Lie algebras 𝔰𝔬(3,2), 𝔰𝔬(4,1), and 𝔰𝔬(5,ℝ) on the span of the generalized solid harmonics 𝒮_ν_0+n^μ_0+m(r,θ,φ), (n,m)∈ℤ^2, the Casimirs c_2 and c_4 are represented by the constants -54 and 0, irrespective of ν_0,μ_0. By the expressions for M_ab,Γ given in any of Theorems <ref>, <ref>, and <ref>, c_2= J· J - PC^+· PC^+ + PC^- · PC^- + D^2= J_3^2 + K_3^2 + 12{J_+,J_-} - 12{K_+,K_-} - 14{R_+,R_-} - 14{S_+,S_-}, where {·,·} is the anti-commutator.This expresses c_2 in terms of J_3,K_3 and the root vectors.The formula (<ref>) can be viewed as subsuming J_3^2+1/2{J_+,J_-}, which is the Casimir of the 𝔰𝔬(3,ℝ) subalgebra spanned by {J_+,J_-,J_3}; and K_3^2-1/2{K_+,K_-}, which is the Casimir of the 𝔰𝔬(2,1) subalgebra spanned by {K_+,K_-,K_3}; and also, the Casimirs of the remaining two 𝔰𝔬(2,1) subalgebras.From the representations (<ref>),(<ref>) of J_±,K_±, R_±,S_± and J_3,K_3 as infinite matrices, one calculates from (<ref>) that c_2 (like J_3,K_3) is diagonal in (n,m), with each diagonal element equaling -54.For 𝔰𝔬(5,ℝ), which is representative, the five components of w^a include (i) the scalar i J· PC^+, (ii) the three components of the vector - i PC^-× PC^+ + DJ, and (iii) the scalar J· PC^-.These expressions, involving the scalar and vector product of three-vectors, must be interpreted with care: any product AB of two Lie algebra elements signifies the symmetrized product 1/2{A,B}.But by direct computation, one finds from (<ref>),(<ref>) and the infinite matrix representations (<ref>),(<ref>) that each component of w^a is represented by the zero matrix, even (surprisingly) without symmetrization.This result is plausible, if not expected.In any unitary representation of a semi-simple Lie group G on L^2(S), S being a homogeneous space G/K of rank 1, all Casimir operators except the quadratic one must vanish.(See <cit.> and <cit.>.)Admittedly, the present representations of 𝔰𝔬(4,1), by real differential operators acting on multi-valued, non-square-integrable functions, are non-skew-Hermitian, and cannot be exponentiated to unitary representations of SO_0(4,1) of this `most degenerate' type.The value -5/4 computed for the quadratic Casimir c_2, irrespective of (ν_0,μ_0), can be viewed as the value of j(j+1), where j is a formal `angular momentum' parameter equal to -1/2± i.For each (ν_0,μ_0), the resulting representation of the real Lie algebra 𝔤=𝔰𝔬(3,2), 𝔰𝔬(4,1) or 𝔰𝔬(5,ℝ), or its universal enveloping algebra 𝔘(𝔤), on the span of the generalized solid harmonics 𝒮_ν_0+n^μ_0+m, (n,m)∈ℤ^2, can be viewed linear-algebraically: as a homomorphism ρ of real vector spaces, taking 𝔤 (or 𝔘(𝔤)) into the space of infinite matrices indexed by (n,m).For each basis element A∈𝔤, ρ(A) is determined by (<ref>),(<ref>); and because the basis elements given in Theorems <ref> and <ref> include `i' factors, the matrix elements of ρ(A) may be complex.To show that certain of these representations are substantially the same as known ones by infinite matrices that are skew-Hermitian, consider the effect of replacing the family {𝒮_ν^μ = 𝒮_ν_0+n^μ_0+m} by {𝒮̂_ν^μ = 𝒮̂_ν_0+n^μ_0+m}, where the latter are `twisted' by a square-root factor:𝒮̂_ν^μ = 𝒮̂_ν^μ (r,θ,φ) √(Γ(ν-μ+1)/Γ(ν+μ+1))r^ν P_ν^μ(cosθ) e^ iμφ.That is, 𝒮̂_ν^μ = r^ν Y_ν^μ, whereY_ν^μ(θ,φ) =√(Γ(ν-μ+1)/Γ(ν+μ+1))P_ν^μ(cosθ) e^ iμφ.When ν=0,1,2,…, with μ=-ν,…,ν, this Y_ν^μ is the classical (complex) spherical harmonic on S^2, of degree ν and order μ.[The orthonormalization factor √((2ν+1)/4π), appropriate for an inner product on S^2, is omitted.But this Y^μ_ν automatically includes the so-called Condon–Shortley factor, owing to the definition of P_ν^μ used here (see  <ref>).]In this case, the square root factor equals [(ν+μ+1)_2μ]^-1/2 and is positive by convention; a discussion of how to interpret it in other cases is deferred.At least formally, the representation of 𝔤 or 𝔘(𝔤) on {𝒮̂_ν_0+n^μ_0+m} comes from that on {𝒮_ν_0+n^μ_0+m} by a diagonal similarity transformation.The formulasJ_± 𝒮̂_ν^μ = √((ν-μ+12∓12)(ν+μ+12±12)) 𝒮̂_ν^μ±1, K_± 𝒮̂_ν^μ = √((ν-μ+12±12)(ν+μ+12±12)) 𝒮̂_ν±1^μ, R_± 𝒮̂_ν^μ =√((ν+μ+12±12)(ν+μ+12±32)) 𝒮̂_ν±1^μ±1, S_± 𝒮̂_ν^μ = √((ν-μ+12±12)(ν-μ+12±32)) 𝒮̂_ν±1^μ∓1 andJ_3 𝒮̂_ν^μ = μ 𝒮̂_ν^μ,K_3 𝒮̂_ν^μ = (ν+12) 𝒮̂_ν^μnow replace (<ref>) and (<ref>); but the actions of the elements J_i, K_i, PC^±_i, D are still defined in terms of these by (<ref>) and (<ref>).If (ν_0,μ_0)=(0,0) or (12,12), the representation ρ of 𝔤=𝔰𝔬(3,2) on the span of {𝒮̂_ν^μ = 𝒮̂_ν_0+n^μ_0+m}, (n,m)∈ℤ^2, which is obtained from (<ref>),(<ref>) by identifying 𝔰𝔬(3,2) with the real span of iJ_1, iJ_2, iJ_3; iPC_1^+, iPC_2^+, iPC_3^+; PC_1^-,PC_2^-,PC_3^-; iD as in Theorem <ref>(ii), has an irreducible constituent that is defined on the subspace spanned by {𝒮̂_ν^μ = 𝒮̂_ν_0+n^μ_0+m} with n=0,1,2,… and μ=-ν,-ν+1…,ν-1,ν.On this subspace, every element of 𝔤 is represented by an infinite matrix that is skew-Hermitian. By (<ref>), if μ=ν then J_+,K_-,S_- give zero when acting on 𝒮̂_ν^μ, and if μ=-ν then J_-,K_-,R_- give zero.Thus ρ is reducible: it can be restricted to the stated subspace.On this subspace, the formal similarity transformation performed by the square root factor in (<ref>) is not singular: only if ν±μ is a negative integer will one of the gammas be infinite.For (ν_0,μ_0) equal to either of (0,0) and (12,12), the square root factor simply equals [(ν+μ+1)_2μ]^-1/2.By (<ref>) and (<ref>), each of ρ(J_±),ρ(K_±),ρ(R_±),ρ(S_±);ρ(J_3),ρ(K_3) is a real matrix, the plus and minus versions being transposes of each other, and ρ(J_3),ρ(K_3) being symmetric (and diagonal).It follows from (<ref>) and (<ref>) that the ρ(J_i), the ρ(PC_i^+), and ρ(D) are Hermitian, and the ρ(PC_i^-) are skew-Hermitian.The claim follows.Being skew-Hermitian, the two infinite-dimensional representations of 𝔰𝔬(3,2) in Theorem <ref> exponentiate to (irreducible) unitary representations of the so-called anti-de Sitter group SO_0(3,2), or its universal cover.The latter have been classified <cit.>, and the ones arising from the theorem can be identified.They are the remarkable Dirac singleton representations, with whimsical names <cit.>: the (ν_0,μ_0)=(12,12) one is `Di' and the (ν_0,μ_0)=(0,0) one is `Rac.'For the Dirac singletons, the Casimirs (c_2,c_4) have long been known to equal (-54,0). (See <cit.>.)They are singleton representations in the sense that if they are reduced with respect to the subalgebra 𝔤_0 = 𝔰𝔬(3,ℝ) ⊕_ℝ𝔰𝔬(2,ℝ), thereby being split into representations of 𝔤_0, each of the latter that appears, does so with unit multiplicity.The ones that appear are labeled uniquely by ν=ν_0+n, n=0,1,2,….The Rac representation of 𝔰𝔬(3,2) is realized by differential operators on ℝ^3 (with complex coefficients), expressions for which follow immediately from (<ref>).They act on the span of the classical solid harmonics, {𝒮̂_n^m} with n=0,1,2,… and m=-n,…,n.Kyriakopoulos <cit.> in effect discovered that the Rac has such a realization, before the name was coined, and extended this result to higher dimensions.The Di is realized by the same operators, acting on the span of the `spinorial' solid harmonics {𝒮̂_n+1/2^m} with n=0,1,2,… and m=-n-12,…,n+12.The existence of this realization seems not to be known.This is perhaps because the solid harmonics of half-odd-integer degree and order are double-valued on ℝ^3, are typically non-square-integrable, and are based on the little-known dihedral Ferrers functions.The expressions for the dihedral Ferrers functions in terms of Jacobi polynomials, given in Theorem <ref> above, are new.The representations of 𝔰𝔬(3,2) carried by the octahedral and tetrahedral families of solid harmonics, {𝒮_ν^μ = 𝒮_ν_0+n^μ_0+m} with (ν_0+1/2,μ_0) equal to (±1/3,±1/4), (±1/4,±1/3), and (±1/3,±1/3), are not skew-Hermitian, even up to diagonal equivalence.Twisting the basis to {𝒮̂_ν^μ = 𝒮̂_ν_0+n^μ_0+m} does not help matters, because only if ν_0,μ_0 are both integers or both half-odd-integers, which without loss of generality may be taken to be 0,0 or 1/2,1/2, does it permit the representation to be restricted to a subspace spanned by the harmonics with ν=ν_0+n, n=0,1,2,…, and μ=-ν,-ν+1…,ν-1,ν.For general (ν_0,μ_0), there is accordingly no restriction on the index (n,m)∈ℤ^2 of the basis functions of the representation, and the square roots in (<ref>),(<ref>),(<ref>) may be square roots of negative quantities.Irrespective of what sign convention for the square root is adopted, the resulting imaginary factors will interfere with skew-Hermiticity; and upon integration of the representation, with unitarity.In fact, the familiar definition of the (surface) spherical harmonic Y_ν^μ=Y_ν^μ (θ,φ) given in (<ref>), incorporating the square root factor, seems to be useful only when the degree ν and the order μ are both integers or both half-odd-integers.10 urlstyleBogdanovic75 Bogdanović, R., Whitehead, M.A.: The representation of the SO(4,1) group in four-dimensional Euclidean and spinor space. 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http://arxiv.org/abs/1702.08555v3

http://arxiv.org/abs/1702.08275v1

GrayBox[1][-4.5ex]node[draw=black,fill=black!10,rounded corners,inner sep=2ex,text width=];Deniz AkdemirSTPGA: Selection of training populations (and other subset selection problems) with an accelerated genetic algorithmSTPGA: An R-package for selection of training populations with a genetic algorithm Optimal subset selection is an important task that has numerous algorithms designed for it and has many application areas. STPGA contains a special genetic algorithm supplemented with a tabu memory property (that keeps track of previously tried solutions and their fitness for a number of iterations), and with a regression of the fitness of the solutions on their coding that is used to form the ideal estimated solution (look ahead property) to search for solutions of generic optimal subset selection problems. I have initially developed the programs for the specific problem of selecting training populations for genomic prediction or association problems, therefore I give discussion of the theory behind optimal design of experiments to explain the default optimization criteria in STPGA, and illustrate the use of the programs in this endeavor.Nevertheless, I have picked a few other areas of application: supervised and unsupervised variable selection based on kernel alignment, supervised variable selection with design criteria, influential observation identification for regression, solving mixed integer quadratic optimization problems, balancing gains and inbreeding in a breeding population. Some of these illustrations pertain new statistical approaches. Genomic Prediction, Genome-Wide Association, Optimal Design of Experiments, High Dimensional Data, Sparse Models, Markers, Anchor Markers, Inbreeding, Genetic Gain, Kinship, Similarity, Subset Selection, Mixed Integer Quadratic Programming, Genomic Selection, Supervised Unsupervised Variable Selection, R, Genomic Selection Genomic Prediction, Genome-wide Association, Optimal Design, Optimal Experiments, High Dimensional Data, Sparse Models, Markers, Anchor Markers, Inbreeding, Genetic Gain, Kinship, Similarity, Subset Selection, Mixed Integer Quadratic Programming, R, Genomic Selection Deniz Akdemir Department of Epidemiology and BiostatisticsMichigan State UniversityEast Lansing MI USAE-mail: deniz.akdemir.work@gmail.com § INTRODUCTION The paper introduces the R (R Core Team 2016) package STPGA that provides a genetic algorithm for subset selection. The package is available from the Comprehensive R Archive Network (CRAN) at <http://CRAN.R-project.org/package=STPGA>, and some of the underlying motivations, methodology and results were presented in <cit.> and also some innovations that will be detailed in several subsequent articles.This document details version 4.0 which includes major upgrades and bug fixes compared to previous versions.Numerous other algorithms have been proposed for the optimal subset selection problem, many of them areheuristic exchange type algorithms <cit.>. In exchange type algorithms new solutions are obtained by adding one point andremoving another at a time (some exchange algorithms might allow exchange of more than one design point at once), these algorithms are greedy and are only proven to find the best subset for certain type of design criteria. In general, exchange algorithms are prone to getting stuck in local optimal solutions.Branch and bound(BB) <cit.> is a global exhaustive search method that has proven to be reasonably efficient on practical problems. BB searches the design region by iteratively dividing design region and searching each piece for an optimal solution. BB is often more efficient than straight enumeration because it can eliminate regions that provably do not contain an optimal solution. <cit.> uses a BB algorithm to find globally best D-optimal design for a given design criteria and a set of candidate points. Another method that has been applied to the subset selection problem is simulated annealing <cit.>. Branch and bound and simulated annealing algorithms require appreciable computation time even for moderate sized problems. Genetic algorithms (GAs) are a class of evolutionary algorithms made popular by John Holland and his colleagues <cit.>, and have been applied to find exact or approximate solutions to optimization and search problems <cit.>. There are numerous packages that implement GAs or similar evolutionary algorithms. The packages gafit <cit.>, galts <cit.>, genalg <cit.>, rgenoud <cit.>,DEoptim <cit.> and the GA <cit.> offer many options for using optimization routines based on evolutionary algorithms. The optimization algorithm that is used in STPGA (LA-GA-T algorithm) is a modified genetic algorithm with tabu search and look ahead property and it is specialized for solving subset selection problems. Today's trends in computation are towards computer architectures that integrate many, less complex processors, exploit thread-level and data-level parallelism. This makes these computers perfect ground for implementationof evolutionary algorithms for solving complex optimization problems since these algorithms can be easily to be run at parallel. To make my point more clear, lets remember Amdahl’s law <cit.> which puts a limit to the speed that can be gained by parallelizing a process:speedup=serial processing time/parallel processing time≤1/f_par/N_p+(1-f_par),where f_par is the fraction of code which could be parallelized,1-f_par is the serial fraction and N_p is the number of processors. As it can be observed from the figures in Figure <ref>, obtained by applying this formula for varying values of f_par between 0 and 1 and values of N_p in {1,2,4,8,16, 32, 64, 128, 256 }, under the assumption that the number of processors doubles each year and the processing capabilities for each parallel node and everything else identical throughout, taking full advantage of parallelization requires methods that have paralelization frequency close to one. From the same figure we can read that 1% change in parallelization frequency might cause up to 3.5 times speedup if there are 256 processors and a ideal parallelizable procedure might have speed more than to 50 times relative to a 80% parallelizable procedure. Evolutionary algorithms, like GA, fall at the very right end of these figures. At the Michigan State University, at the beginning of year 2017, there were hundreds of nodes available for the researches through their high performance computer cluster (HPCC) system. It is not science fiction to claim that clusters with millions of nodes will be available to researchers with the technologies such ascloud / grid computing. We are faced with the challenge of matching these these parallel resources with methods that can use them efficiently.In my view, easy adaptation of GAs to parallel computation is a major advantage of GAs to other subset selection algorithms. GAs scale well with large and/or computationally expensive problems and can achieve reasonable running times by using parallel computing architectures with either shared or distributed memory systems, these systems are becoming increasingly available to the researchers. Scientific community prefer algorithms that run faster in serial. However, the direction the improvements in the computer technology seem be more in the parallelization rather than faster processors. Some advantages of the GAs to other subset selection algorithms include the following: * GA can be applied to many different problems and does not need to be reinvented for each new problem.* GA is a very flexible optimization algorithm, evolutionary mechanisms that are involved in GA can be modified at different stages of the algorithm; various selection strategies, various penalization of the objective function,can be explored simultaneously or in a serial fashion. * Adopting GAs in a parallel computing environment is easy. This might involve, for example, evaluation of the fitness function for the current GA population, or running many GAs at parallel to provide initial solutions generation of GAs. In addition the LA-GA-T algorithm in STPGA adds two more properties: * Inferior solutions that were visited recently will not be visited. This property is akin to a memory in an intelligent system.* The state of current solutions are used to predict the best ideal solution. This gives the algorithm the look ahead property. This property is akin to inference in an intelligent system.Nature solves problems through evolutionary processes, it works with communities of solutions that exploits the their communal information content to create new solutions, it is this information that persists, not the individual solutions. In addition, we have long standing theories explaining how and why such evolutionary processes work. There is also a vast amount of principled study of evolutionary mechanisms, both that are natural and artificial; the whole subject of evolutionary genetics; the methodology, theories and practices related to breeding; the theoretical and practical approaches of evolutionary algorithms and computation allows us humans to understand and manage these systems.In the next section, I briefly review the basic ideas behind simple GAs and the LA-GA-T that is used in STPGA. Then, I present the details of the interface to the STPGA package in Section 3, followed by several examples section and the conclusions. The examples section has been divided into two main parts: STPGA for selection of training populations, and STPGA in other subset selection problems. Both of these sections are rather long and detailed, especially the part that relates to optimal design that introduces some concepts and ideas of optimal design of experiments with a focus on predictive learning using regression models. Some of the design criteria discussed in this section are implemented in STPGA, a table listing of these criteria is provided. I also demonstrate how to write user defined criteria. § OPTIMIZER IN STPGAThe optimization algorithm that is used in STPGA is a modified genetic algorithm with tabu search and look ahead property. Genetic algorithms are stochastic search algorithms which are able to solve optimization problems using evolutionary strategies inspired by the basic principles of biological evolution. They use a population of candidate solutions that are represented as binary strings of 0's and 1's, this population evolving toward better solutions. At each iteration of the algorithm, a fitness function is used to evaluate and select the elite individuals and subsequently the next population is formed from the elites by genetically motivated operations such as crossover and mutation. The properties and prospects of genetic algorithms were first laid out in the cornerstone book of Holland <cit.>. GAs have an implicit parallelism property <cit.>. In addition, since GA uses a set of solutions at each iteration it couples well with the advanced computers (workstations with many processors and large memory) and computer systems (high performance computing clusters, cloud computing technologies) of today allowing it to be applied to very large scale optimization problems.In my opinion, with the advent of new technologies like DNA computing <cit.> that uses programmable molecular computing machines or quantum computers <cit.> that operate on ”qubits”, parallelizable algorithms such as GA will have more and more important role in big scale optimization problems. The GA algorithm in STPGA is supplemented with two additional principles, tabu (memory) and inference through prediction based on a current population of solutions. I refer to it as the LA-GA-T (look ahead genetic algorithm with tabu) algorithm.Tabu search is a search where most recently visited solutions are avoided by keeping a track of the previously tried solutions. This avoids many function evaluations and decreases the number of iterations till convergence, it is especially useful for generating new solutions around local optima.The LA-GA-T algorithm in STPGA also uses the binary coding of the current population of solutions and their fitness to fit a linear ridge regression model from which the effects of individual digits in this binary code are estimated assuming that the contribution of an individual to the criterion value does not change much in relation to different subsets. The predicted ideal solution based on this model is constructed and included in the elite population of solutions. This gives the algorithm a look ahead property and improves the speed of convergence especially in the initial steps of the optimization. I should note here that the idea of regressing the fitnesses of solutions on their designs was inspired by the genomic selection methodology recently put into use in plant and animal breeding with the promise of increasing genetics gains from selection per unit of time. As can be seen from the Figures <ref> and <ref>, LA-GA-T converges in much fewer iterations compared to a simple GA. However, I have to note that the per iteration computation time for LA-GA-T algorithm is slightly higher compared to a simple GA. The solutions obtained by any run of GA may be sub-optimal and different solutions can be obtained given different starting populations. Another layer of safety is obtained if the algorithm is started from multiple initial populations and an island model of evolution is used where separate populations are evolved independently for several steps and then the best solutions from these algorithms becomes the initial solutions to evolutionary algorithm. Since the functions in STPGA can start from user provided initial values, island models and other strategies can be combined when using the algorithm. I give an example code for doing this in the Appendix section.§ SOFTWARE INTERFACE, COMPUTATIONAL CONSIDERATIONS There are two main functions in STPGA, these are GenAlgForSubsetSelection and GenAlgForSubsetSelectionNoTest. The function GenAlgForSubsetSelection uses a simple genetic algorithm to identify a training set of a specified size from a larger set of candidates which minimizes an optimization criterion (for a known test set). The function GenAlgForSubsetSelectionNoTest is for identifying a training set of a specified size from a larger set of candidates which minimizes an optimization criterion, no test set is specified. These functions share a lot of common parameters, except GenAlgForSubsetSelection requires an additional input that specifies the target set of individuals and the data matrix should be supplemented to include the observed value of the variables for these target individuals. The inputs for these functions are described below:Inputs* P depending on the criterion this is either a numeric data matrix or a symmetric similarity matrix. When it is a data matrix, the union of the identifiers of the candidate (and test) individuals should be put as row names (and column names in case of a similarity matrix). For methods using the relationships, this is the inverse of the relationship matrix with row and column names as the the identifiers of the candidate (and test) individuals. * Candidatesvector of identifiers for the individuals in the candidate set. * Test vector of identifiers for the individuals in the test set. * ntoselectn_Train: number of individuals to select in the training set. * npopgenetic algorithm parameter, number of solutions at each iteration * nelitegenetic algorithm parameter, number of solutions selected as elite parents which willgenerate the next set of solutions. * keepbestgenetic algorithm parameter, TRUE or FALSE. If TRUE then the best solution is always kept in the next generation of solutions (elitism). * tabugenetic algorithm parameter, TRUE or FALSE. If TRUE then the solutions that are saved in tabu memory will not be retried. * tabumemsizegenetic algorithm parameter, integer>0. Number of generations to hold in tabu memory. * mutprobgenetic algorithm parameter, probability of mutation for each generated solution.* mutintensity mean of the Poisson variable that is used to decide the number of mutations for each cross.* niterationsgenetic algorithm parameter, number of iterations. * minitbefstopgenetic algorithm parameter, number of iterations before stopping if no change is observed in criterion value. * niterreggenetic algorithm parameter, number of iterations to use regressions* lambdascalar shrinkage parameter(λ>0). * plotitersplot the convergence: TRUE or FALSE. Default is FALSE. * plottypetype of plot, default is 1. possible values 1,2,3.* errorstatoptimality criterion: One of the optimality criterion. Default is ”PEVMEAN”. It is possible to use user defined functions as shown in the examples.* mc.coresnumber of cores to use. * InitPop a list of initial solutions * tolconv if the algorithm cannot improve the errorstat more than tolconv for the last minitbefstop iterations it will stop.* C contrast matrix. * Vg covariance matrix between traits generated by the relationship K (only for multi-trait version of PEVMEANMM).* Ve residual covariance matrix for the traits (only for multi-trait version of PEVMEANMM).All these inputs except P, ntoselect (also Candidates and Test forGenAlgForSubsetSelection) have default values of NULL meaning that they are internally assigned to the default suggested settings. These settings are as follows: npop = 100, nelite = 5, keepbest = TRUE, tabu = FALSE, tabumemsize = 1, mutprob = .8, mutintensity = 1, niterations = 500, minitbefstop=100, niterreg = 5, lambda = 1e-6, plotiters = FALSE, plottype = 1, errorstat = ”PEVMEAN”, C = NULL, mc.cores = 1, InitPop = NULL, tolconv = 1e-7, Vg = NULL, Ve = NULL. In a specific application of STPGA, we recommend the users to change these options until they are satisfied with the final results.Especially, when used with large data sets (many columns or rows), the parameters npop, niterations, minitbefstop should be increased.Both functions return a named list of length nelite+1. The first nelite elements of the list are optimized training samples of size n_train and they are listed in increasing order of the optimization criterion. The last item on the list is a vector that stores the minimum values of the objective function at each iteration. The solution with best criterion value has name `Solution with rank 1`, the second `Solution with rank 2`, etc, ... The minimum values of the objective function through the iterations has name`Best criterion values over iterations`. The function GenAlgForSubsetSelection in the package uses this algorithm to identify a training set of a specified size from a larger set of candidates which minimizes an optimization criterion (for a known test set). The function ”GenAlgForSubsetSelectionNoTest” tries to identify a training set of a specified size from a larger set of candidates which minimizes an optimization criterion, no test set is specified. The subset selection algorithms in ”GenAlgForSubsetSelectionNoTest” and ”GenAlgForSubsetSelection” have somewhat different inner workings. ”GenAlgForSubsetSelectionNoTest” splits the individuals given in row names of the input matrix P into two parts: a set called Train of size ”ntoselect” and its complement. The ”GenAlgForSubsetSelection” starts with an input defining of split of the individuals given in row names of the input matrix P into three parts: a set called ”Candidates”, a set called ”Test” and their complement, after this the algorithm splits the set ”Candidates” into a set called ”Train” and its complement. These two functions can be used with any user defined fitness functions and in the examples section, I will illustrate how these mechanisms can be used for general subset selection problems.I have developed this package for my interest in solving certain design problems. Therefore, I have includedseveral of my favorite design criteria in STPGA. A list of the names, required input parameters and the corresponding formulas are summarized with the Table <ref> for reference. More explanation about the usage, examples and other details of these criterion can be found in the package help documentations.When using a design criterion that uses the design matrix of the target individuals along with the candidates, the ”GenAlgForSubsetSelection” function uses the individuals listed in ”Test” extract the design matrix of these individuals from the design matrix of all individuals P. If you use a similar criterion with the ”GenAlgForSubsetSelection” function 'the design of the target set is implicitly assigned as the rows in P not in ”Train”. Many modern statistical learning problems involve the analysis of high dimensional data. For example, in genomic prediction problems phenotypes are regressed on large numbers of genome-wide markers. STPGA was initially prepared for working with high dimensional data related to such whole-genome-regression<cit.> and association <cit.> approaches that are becoming increasingly popular for the analysis and prediction of complex traits in plants<cit.>, animals <cit.> and humans <cit.>. The design criteria in STPGA and their use were motivated by the practical problem of selecting the best genotypes for a phenotypic experiment so that the inferences made based on the data obtained by the experiment are optimally informative for genomic prediction and association problems <cit.>. These high dimensional design problems pose additional computational challenges and the selection and use of the design criteria has a big influence on computational requirements.I recommended the use of dimension reduction techniques, whether they are supervised, unsupervisedor based on algebraic manipulation, such as the use of dimension reduction methods like principle components analysis or variable clustering, use of methods based on similarity or distance matrices, before running STPGA. Several design optimization criteria in STPGA are equivalent and will produce the same or similar designs. However, calculation of one might be easier than the other based on the relative number of rows or columns of the data matrix. STPGA software is written purely in R, however the computationally demanding criteria can be programmed by the user in C, C++ or Fortran.STPGA also benefits from multi-thread computing. The computational performance of the algorithm can be greatly improved if R is linked against a tuned BLAS implementation with multi-thread support, for example OpenBLAS, ATLAS, Intel mkl, etc. § ILLUSTRATIONS In this section, I am going to illustrate use of STPGA. The first group of examples are related to selection of training populations. The second part, includes examples of other subset selection problems.§.§ STPGA for selection of training populationsExperiments provide useful information to scientists as long as they are properly designed and analysed. The history of the theoretical work on the design problem goes way back. Fisher <cit.> gives a mathematical treatment of the determination of designs for some models. The first extended presentation of the ideas of optimum experimental designs appear in <cit.> and <cit.>.A brief history of statistical work on optimum experimental design is given by Wynn (wynn1984jack) and the subject continues to develop, recently at an increasing rate.Box et al. <cit.>, Box and Draper <cit.> are a few of the authoritative texts on the subject. The alphabetical naming of designs is due to <cit.>. For a detailed discussion of standard criteria reference is made to these references.In this paper we focus only on a the narrow design problem of selecting an optimal set of n design points, X_Train={_1,…, _n}, from a set of candidate design points X_C={_1,…, _N}. The design defined by these n points, can be viewed as a measure on the candidate set X_C={_1,…, _N}. Let ζ be a probability measure on X_C such that*ζ(x_i) = 0 ifx_i∉ X_Train, and *ζ(x_i)= 1/n ifx_i∈ X_Train.When dealing with problems of supervised learning where the resulting model of the experiment will be used to make inferences about a known set of individuals, we can distinguish between the candidate set and a target set X_Target={_1,…, _t}: X_Target describes the focused design region for which predictions about the dependent variables based on the models trained on X_Trainare required.Let's assume that 1 ≤ n ≤ N and 1 ≤ t are fixed integers, x_i are p-vectors and we denote the matrix form of X_Train as X, this is called the design matrix; and X_Target as X^*, this is called the design matrix for the target space. The first component of a design optimization problem is the objective function. For example the objective function might be chosen as theoretically or numerically obtained sampling variance of a prespecified estimator of a population quantity of interest. The second component of an optimization problem is the set of decision variables and the constraints on the values of these variables. Once the objective function and the set of constraints are known the next step is to use a method to look for solutions that optimize the objective function and also adhering to the constraints.Parametric design criteria usually depend on a function of the information matrix for the model parameters that gives some indication about the sampling variance and covariance of the estimated parameters.Let I_ (ζ) denote the information matrix of the parametersfor a given design ζ. In order to be able to achieve a criteria that orders designs with respect to their information matrices, usually, a scalar function of the information matrix is used. These designs criteria have alphabetical names, the designs obtained by optimizing these criteria are referred to as A-, D, E-, G-, etc,... optimal designs. The list of design criteria that are implemented in STPGA are described by Table <ref> with references to the equations in this manuscript from which these were inspired.Many practical and theoretical problems in science treat relationships of the type y = g(, ), where the response, y, is thought of as a particular value of a real-valued model function or response function, g, evaluated at the pair of arguments (x, ).The parameter value, unknown to the experimenter, is assumed to lie in a parameter domain Θ. This is called the regression of y on x. The STPGA is not confined to regression, but we use regression analysis to do most of the explaining and demonstrations. §.§.§ Linear models The choice of the function g is central to the regression model building process. One of the simplest regression models is the linear model. Let X_n× p be the design matrix, _p× 1 the vector of regression parameters, _n× 1 the vector of observations, and _n× 1=(ϵ_1,ϵ_2, …,ϵ_n)' our error vector giving=X+. With I_n as the n × n identity matrix, the model is represented by the expectation vector and covariance matrix of ,E() = X, cov()=σ^2 I_n,and is termed the classical linear model with moment assumptions. We assume ϵ_i, i = 1, 2,…, n will be iid with mean zero and cov() = σ^2 I_n. Under the additional normality assumption we write ∼ N(, σ^2 I_n).We now concentrate on determining the optimal estimator for ' in the linear regression model.If X is not of full rank, it is not possible to estimateuniquely. However, X is uniquely estimable, and so is 'X for any conformable vectorthat is in the row space of X. If estimability holds then the Gauss-Markov Theorem determines the optimal estimator for ' to be '(X'X)^-X', where A^- denotes any generalized inverse of A that satisfies A=AA^-A.The variance of this estimator depends only on the matrix X'X, var_,σ^2('(X'X)^-X'Y) = (σ^2)'(X'X)^-X'X(X'X)^-.Up to the common factor σ^2/n, the optimal estimator has variance '(X'X)^-X'X(X'X)^-. Assuming estimability, the optimal estimator for the linear function of the coefficients γ = C is also given by the Gauss-Markov Theorem: γ̂ = C(X'X)^-X'Y. The covariance matrix of the estimator γ̂ isC(X'X)^-X'X(X'X)^- C'. The covariance matrix becomes invertible provided the coefficient matrix C has full row rank. A closely related task is that of prediction. Suppose we wish to predict additional responses ^*= X^* + ^*. If we take the random vector γ̂ from above as a predictor for the random vector ^*,to obtain precise estimators for X^*, we would like to choose the design so as tomaximize the relevant information matrix (minimize the covariance matrix). For example, G-optimal designs are obtained by minimizing the maximum variance of the predicted values, i.e., the maximum entry in the diagonal of the matrix X(X'X)^-X'. §.§.§ Ridge Regression Note the following: If A is a symmetric matrix, then the limitlim_λ→ 0(A+λ^2 I)^-1is a generalized inverse of A, and alsolim_λ→ 0(A+λ^2 I)^-1Alim_λ→ 0(A+λ^2 I)^-1=lim_λ→ 0(A+λ^2 I)^-1.This means, for small λ>0, ≈ C(X'X+λ I)^-1X'Y. This estimator is called the ridge estimator, the coefficients have covariance matrixapproximately proportional toC(X'X +λ I)^-1C'.Furthermore, prediction error variance for estimating the CX^* with ridge regression is approximately proportional to CX^*(X'X +λ I)^-1X'^*C'. Ridge estimators have smaller variance than BLUE'sbut they are biased since the estimators are ”shrunk” towards zero. Ridge estimators are especially useful when X'X is singular. In some cases, the ridge estimation is only applied to a subset of the explanatory variables in X, for example it is customary to not shrink the mean term. Splitting the design matrix X as X=(X_F, X_R), where X_F contains the effects modeled without ridge penalty and X_R contains the terms modeled with ridge penalty, a design criterion concerning the estimation of shrunk coefficients can be written asC(X'M^-1X +λ^2 I)^-1C'with M=I-X_F(X'_FX_F)^-X'_F.§.§.§ RKHSUsing the matrix identity (P^-1 + B'R^-1B)^-1B' R^-1 = PB' (BPB' + R)^-1, we can write(X'X+λ I )^-1X' = X'(XX' +λ I )^-1. The ridge regression solution forcan then be written as follows: =Cβ̂≈ C(X'X+λ I)^-1X'=CX'(XX'+λ I)^-1.The important message here is that we only need access partitions of the matrixK_C,X(ζ)= [ XX' XC'; CX' CC' ]= [K_11K_12; K'_12K_22 ] since γ̂=CX'(XX'+λ I)^-1=K_21(K_11+λ I)^-1. Using(X'X+λ𝐈)^-1 =1/λ(𝐈-X'(XX'+λ𝐈)^-1Xwe haveC(XX+λ𝐈)^-1 C' =C(λ(X'X/λ+𝐈))^-1C' =1/λC(𝐈-X'(XX'+λ𝐈)^-1X) C' = 1/λ[CC']-1/λ[CX'(XX' +λ𝐈)^-1 XC'] ∝𝐊_22 -𝐊_21(𝐊_11+mλ𝐈)^-1𝐊'_21.The variance covariance matrixfor proportional to(𝐊_22-𝐊_21(𝐊_11+λ I)^-1𝐊_12).Reproducing Kernel Hilbert Spaces (RKHS) regression methodsreplace the inner products by kernels, it is as if we are performing ridge regression on a transformed data ϕ(x), where ϕ is a feature map associated to the chosen kernel function and the associated kernel matrix. The resulting predictor is now nonlinear in x and agrees with the predictor derived from the RKHS perspective (Schölkopf and Smola 2002). RKHS regression extends ridge regression allowing a wide variety of kernel matrices, not necessarily additive in the input variables, calculated using a variety of kernel functions. A kernel function, k(.,.) maps a pair of input pointsand ' into real numbers. It is by definition symmetric (k(,')=k(',)) and non-negative. Given the inputs for the n individuals we can compute a kernel matrix K whose entries are K_ij=k(_i,_j). The common choices for kernel functions are the linear (k(; ) = '.), polynomial (k(; ) =('+ c)^d for c andd ∈ R), Gaussian kernel functions (k(; ) = exp(-h('-)'('-)) where h>0.), though many other options are available <cit.>.Reproducing Kernel Hilbert Spaces Regressions (RKHS) have been used for regression <cit.>, spatial smoothing <cit.> and classification problems <cit.>. <cit.>, proposed to use this approach for genomic prediction and, since then several follow-up articles with focus on the application of these methods to various genome-wide regression problems have also been published <cit.>.§.§.§ Gaussian Linear Mixed ModelsThe linear mixed model methodology was first developed within the context of animal genetics and breeding research by <cit.>, many important statistical models can be expressed as mixed effects models and it is the most widely used model in prediction of quantitative traits, and genome-wide association studies.In studies on linear mixed models it is usual to consider the estimation of the fixed effectsand the variance components, and also the prediction of the random effects . For a given data vector , the vector of random effectsis a realization of random variables which areobserved and these effects must therefore necessarily be predicted from the data <cit.>. In the linear mixed-effects model, the observations are assumed to result from a hierarchical linear model:y = W + Z+ ;and ∼ N(0,R) is independent of ∼ N(0;G).These assumptions imply E (y|W;Z) = W, y ∼ N(W; ZGZ' + R) = N(W; V ). The similarity of the mixed models and RKHS regression models has been stressed many times. However, mixed modeling approach provides a formalized approach since he inferences are based on a probabilistic model, and therefore, allows legitimate inferences about the parameters and predictions.Henderson et al. show that maximizing the joint density ofandyields the MLEs of the parametersand EBLUPs (estimated BLUPs)that solve: W'R^-1W+ W'R^-1Z= W'R^-1y and Z'R^-1W + Z'R^-1Z + G^-1 = Z'R^-1y, this leads to the Henderson's mixed model equations:Henderson's mixed-model equations can be used to estimate the standard errors of the fixed and random effects. For a given design, the inverse of the coefficient matrix is written as[ W' R^-1WW'R^-1Z; Z' R^-1W Z'R^-1Z + G^-1 ]^-1 = [H_11H_12; H'_12H_22 ]where H_11, H_12, and H_22 are, respectively, p× p, p× q, and q× q sub-matrices. Note that referring to the coefficient matrix is an abuse of notation since the parameters of the mixed effects model does not include the vector .Using this notation, the sampling covariance matrix for the BLUE (best linear unbiased estimator) ofis given by σ( ) = H_11=(W'V^-1W)^- that the sampling covariance matrix of the prediction errors (û-u) is given bycov(- ) = H_22=G- GZ'PZGfor P = V^-1 - V^-1W(W'V^-1W)^-W'V^-1 and that the sampling covariance of estimated effects and prediction errors is given by σ(, - ) = H_12=-(W'V^-1W)^-W'V^-1ZG (We consider - rather thanas the latter includes variance from both the prediction error and the random effectsthemselves.). The standard errors of the fixed and random effects are obtained, respectively, as the square roots of the diagonal elements of H_11 and H_22.In addition, using the above definitions, cov( |)=G- GZ'V^-1ZG=(Z'R^-1Z + G^-1)^-1. Optimal design of experiments with mixed models involve determination of the design matrices W and Z; however, in many applications, estimates of only one oforis needed. For example, design criterion is obtained by considering the variance-covariance matrix of C'(-) given by C'H_22C is named the prediction error variance. A more recent design criterion is the generalized coefficient of determination <cit.> for the random terms '_i(-), i =1,…, l:∑_i=1^l '_iH_22_i/'_iG_ifor a set of contrasts _i.If the mixed model is simplified such that ∼ N(0,R=σ^2_ I) and∼ N(0;G=σ^2_ A), and the rows of C have zeros corresponding to fixed effects, the formula for prediction error variance becomes: C(Z'MZ+λ A^-1)^-1C'and the corresponding formula for coefficient of determination becomes:∑_i=1^l '_i(A-λ(Z'MZ+λ A^-1)^-1)_i/'_iA_i,where λ=σ^2_/σ^2_. Furthermore, when we assume∼ N(, G=σ^2_I), then the above formulas simplify further to C(Z'MZ+λ I)^-1C'and∑_i=1^l (1-λ'_i (Z'MZ+λ I)^-1_i/'_i_i).Here, M=I-W(W'W)^-W' isis a projection matrix orthogonal to the vector subspace spanned by the columns of W, so that MW=0. §.§.§ Some generalizations and extensions of parametric design criteria A generalization of the D-, A-, G- optimal criteria is provided by the Keifer's ϕ_p criteria: given byϕ_p(ζ)=(trace(I_ (ζ))^-p)^1/p,where -1≤ p<∞. p=0, 1, ∞ gives the D-, A-, E- optimal criteria correspondingly.Another extension of D-optimality deals with minimizing ∑_j=1^h α_jlog|A_jI(ζ)^-A'_j|.The criterion permits designs for h different models which may be fitted to the data, for the jth of which the information matrix is I_j(ζ). The matrix A_j defines S_j linear combinations of the p_j parameters in model j which are of experimental importance and the non-negative weights α_j express the relative importance of the different aspects of the design. Examples of compound D-optimum designs for linear models are given by <cit.>. <cit.> proposed a new criterion alternative to the traditional D-optimal design, which has a measure of the overall correlation among the parameters directly as objective function to be minimized i.e. the root square of the individual correlations between pair of parameters: F=(∑_,i; i≠ jcorr^2_ij/p^2-p)^1/2.§.§.§ Non-parametric design criteria The design criteria of the previous sections started from a parametric model. There are some optimal design approaches that does not make any parametric assumptions, leading to non-parametric design criteria. Most of these methods are based on a distance matrix.A design criteria that aims to achieve a high spreadamong its support points within the design region, i.e., make the smallest distance between neighboring points in the design as large as possible is called the maximin-distance criterion. Let D=d_ij_i=1,...,N denote the distance matrix among the possible design points. Maximin distance criteria is finding the subset of n points such that ϕ_1(ζ)=min(d_ij), i≠ jto be maximized among these n points. Another possibility is to pick the n design points so that the the maximum distance from all the points in a target set of points X^* is as small as possible. These designs are called space filling designs, some performance bench-marking for various space-filling designs can be found in <cit.> and <cit.>.Example 1: In this example, we want to find the best D-optimal 13 point design for a second order regression model over a grid design region defined by two variables both with possible values in the set -2, -1, 0, 1, 2 . Naming these variables as x_1 and x_2 and the generic response as y, we can write this model asy=β_0+β_1 x_1+β_2 x_2+ β_11 x_1^2+β_22 x_2^2 + β_12 x_1x_2+ϵ.First, we crate the design matrix for the design space. Box 1: Creating the design matrix for grid, selecting ”best” subset by enumeration and using STPGA > library(STPGA) > set.seed(1234) > X<-matrix(0,nrow=5^2,ncol=5) > ij=0 > for (i in -2:2) + for (j in -2:2) + ij=ij+1 + X[ij,]<-c(i,j, i^2,j^2, i*j) ++> X<-cbind(1,X) > rownames(X)<-paste("x",1:5^2, sep="") > #lisofallsubsetsofsize13<-combn(rownames(X), 13) > #dim(lisofallsubsetsofsize13) > ########complete enumeration of > ########(5^2 choose 13)=5200300 possibilities > #I have done this once, you dont need to do it. > #DOPTVALS<-apply(lisofallsubsetsofsize13, 2,> #function(x)DOPT(Train=x, Test=NULL, P=X, lambda = 1e-09, C=NULL)) > BESTSOL<-c("x1","x2","x3","x5","x6","x10", +"x11","x13","x15","x21","x22","x24","x25") > #BESTSOL<-lisofallsubsetsofsize13[,which.min(DOPTVALS)] > #best solution is not unique for this problem > mindoptvals<–21.3096195830339709687 > #mindoptvals<-min(DOPTVALS) > ListTrain1<-GenAlgForSubsetSelectionNoTest(P=X,ntoselect=13, InitPop=NULL, + npop=200, nelite=5, mutprob=.5, mutintensity = 1, + niterations=200,minitbefstop=50, tabu=FALSE, + tabumemsize = 0,plotiters=FALSE, + lambda=1e-9,errorstat="DOPT", mc.cores=4) > length(intersect(ListTrain1`Solution with rank 1`,BESTSOL))[1] 12> mindoptvals==min(ListTrain1`Best criterion values over iterarions`)[1] TRUE Box 2: Plotting the results on the grid > par(mfrow=c(1,2)) > labelling1<-rownames(X) > plot(X[,2], X[,3], col=labelling1, +pch=2*labelling1,cex=2*(labelling1-1), +xlab="", ylab="", main="STPGA solution", +cex.main=.7,xaxt='n',yaxt='n') > text(x=X[,2]-.1, y = X[,3]-.1, labels = rownames(X), cex=.5) > for (i in -2:2) + abline(v=i, lty=2) + abline(h=i,lty=2) +> labelling2<-rownames(X) > plot(X[,2], X[,3], col=labelling2, +pch=2*labelling2,cex=2*(labelling2-1), +xlab="", ylab="", main="Best solution",+cex.main=.7,xaxt='n',yaxt='n') > text(x=X[,2]-.1, y = X[,3]-.1, labels = rownames(X), cex=.5) > for (i in -2:2) + abline(v=i, lty=2) + abline(h=i,lty=2) +> par(mfrow=c(1,1))< g r a p h i c s > Example 2: In statistical genetics, an important task involves building predictive models of the genotype-phenotype relationship to be able to make genomic predictions and also to attribute a proportion of the total phenotypic variance to locations on the genome (genomewide association studies (GWAS)).If the genotypic information for the candidates (and the target) are available, phenotypic experiments can be executed for a subset of these individuals that is optimal according to a design criteria that only uses the available genotypic information. STPGA package comes with a genomic data set called WheatData involving phenotypes and markers for 200 elite wheat lines selected at random from a larger genetic pool. Data was downloaded from the website <triticeaetoolbox.org>. The 4670 markers available for these 200 genotypes were preprocessed for missingness and minor allele frequencies, coded numerically as 0, 1, and 2; the relationship genomic relationshipmatrix was calculated according to the formula in Van Raden <cit.>:M_cM'_c/kwherek=∑_j=1^m2p_j(1-p_j)is twice the sum of heterozygositiesof the markers andM_cthe allele counts matrix M centered by the mean frequencies of alleles. The genotypic values for plant heights were predicted using a mixed effects model that is fitted to a multi-environmental trial involving these genotypes and the corresponding phenotypic observations.As long as the model assumptions are correct for the data and a suitable criterion is employed, the prediction accuracies of models built on optimal sets are expected to be better than average prediction accuracies of models based on a random set of the same size. In addition, if a target set is specified, further gains might be achieved using this knowledge. GWAS results based on a genetic information based optimal design is expected to improve the association results compared to models built on phenotypic experiments performed on a random set of the same size in a similar way. Obtaining genotypic information is becoming cheaper by the day, however the high costs and challenges related to phenotypic experiments persist. To see this is the case consider what might be involved in a longitudinal study on human subjects.Optimal design of phenotypic experiments based on prior genotypic information can lead to high information value at low costs.To illustrate these points, lets use the wheat data set first in prediction and association settings. We begin by loading the data and doing some preprocessing necessary to run the experiment:Box 3: Loading and preprocessing the wheat data set included in STPGA > data(WheatData) > svdWheat<-svd(Wheat.K, nu=50, nv=50) > PC50WHeat<-Wheat.K > rownames(PC50WHeat)<-rownames(Wheat.K) > DistWheat<-dist(PC50WHeat) > TreeWheat<-cutree(hclust(DistWheat), k=4) ”TreeWheat” partitions the data into four sets, lets observe this grouping using a plot of first two principal components.Box 4: Plotting the observations in wheat data using first two principal components > plot(PC50WHeat[,1],PC50WHeat[,2], col=TreeWheat, +pch=as.character(TreeWheat), xlab="pc1", ylab="pc2")< g r a p h i c s > We will consider a scenario where the final aim is to accurately predict the genotypic values in group 2 and we want to establish this by conducting a phenotypic experiment on 50 genotypes selected from the remaining groups. Box 5: Splitting the data into Candidates and Test > Test<-rownames(PC50WHeat)[TreeWheat==2] > length(Test)[1] 53> Candidates<-setdiff(rownames(PC50WHeat), Test) > Ztrainfull<-as.matrix(model.matrix( -1+factor(Candidates, + levels=rownames(Wheat.M)))) > deptrainfull<-Wheat.Y[Wheat.Yid > dim(deptrainfull)[1] 147 2 Once the data is ready, it is easy to call STPGA with the default options. Note that there are two options, we can use the information about the target genotypes or we could ignore this. I will do both:Box 6: Optimization of training populations with STPGA > Train1<-GenAlgForSubsetSelection(P=PC50WHeat,Candidates=Candidates,+ Test=Test, ntoselect=50, mc.cores=4) > Train2<-GenAlgForSubsetSelectionNoTest(P=PC50WHeat, + ntoselect=50, mc.cores=4)However, it is important to be able to specify GA parameters:Box 7:Optimization of training populations with STPGA, specifying algorithm parameters > Train3<-GenAlgForSubsetSelection(P=PC50WHeat,Candidates=Candidates,+ Test=Test,ntoselect=50,+ InitPop=NULL,npop=200,+ nelite=10, mutprob=.5, mutintensity = 1,niterations=200, + minitbefstop=50, tabumemsize = 1,plotiters=FALSE, + lambda=1e-9,errorstat="PEVMEAN", mc.cores=4) > Train4<-GenAlgForSubsetSelectionNoTest( + P=PC50WHeat[rownames(PC50WHeat) + InitPop=NULL,npop=200,+ nelite=10, mutprob=.5, mutintensity = 1,niterations=200, + minitbefstop=50, tabumemsize = 1,plotiters=FALSE, + lambda=1e-9,errorstat="PEVMEAN", mc.cores=4) We finally want to compare the prediction accuracy for predicting the target data compared to the average accuracy that would be obtained using a sample size of same size. I will only use ”Train3” and ”Train4” below, we will also need the the package EMMREML<cit.> for fitting the mixed effects model:Box 8: Building models based on optimal samples, getting predictions > require("EMMREML") > deptest<-Wheat.Y[Wheat.Yid > Ztest<-model.matrix( -1+deptestid) > ##predictions by optimized sample > deptrainopt3<-Wheat.Y[(Wheat.Yid > Ztrain3<-model.matrix( -1+deptrainopt3id) > modelopt3<-emmreml(y=deptrainopt3plant.height, +X=matrix(1, nrow=nrow(deptrainopt3), ncol=1), +Z=Ztrain3, K=Wheat.K) > predictopt3<-Ztest > ##### > deptrainopt4<-Wheat.Y[(Wheat.Yid > Ztrain4<-model.matrix( -1+deptrainopt4id) > modelopt4<-emmreml(y=deptrainopt4plant.height, +X=matrix(1, nrow=nrow(deptrainopt4), ncol=1), +Z=Ztrain4, K=Wheat.K) > predictopt4<-Ztest We will repeat estimation with random sample300times to obtain mean performance:Box 9: Estimating the accuracy of a random sample of the same size > corvecrs<-c() > for (rep in 1:300) + rs<-sample(Candidates, 50) ++ deptrainrs<-Wheat.Y[(Wheat.Yid ++ Ztrainrs<-model.matrix( -1+deptrainrsid) ++ modelrs<-emmreml(y=deptrainrsplant.height, +X=matrix(1, nrow=nrow(deptrainrs), ncol=1),+Z=Ztrainrs, K=Wheat.K) + predictrs<-Ztest + corvecrs<-c(corvecrs,cor(predictrs, deptestplant.height)) +Here are the results: Box 10: Comparisons of accuracies > ##average accuracy random sample > mean(corvecrs)[1] 0.303162> ##accuracy of Train3`Solution with rank 1` > cor(predictopt3, deptestplant.height)[,1] [1,] 0.3146401> ##accuracy of Train3`Solution with rank 1` > cor(predictopt4, deptestplant.height)[,1] [1,] 0.3936563 These results are as expected: Optimally designed phenotypic experiments are more informative, they result in higher accuracies compared to a random sample of the same size. If the researcher also has access to the genotypic information for the individuals in the target set, then this information when properly used might lead to gains in per unit information that will come from a phenotypic experiment.I also expect that the association (GWAS) results from an optimized sample to be better than a random sample. I can not verify this with a simple example. However, here is a comparison of the marker effects estimated from a full set, compared to a random sample and an optimized sample of the same size.Box 11: Using STPGA in training population selection for GWA studies > modelrrblupfull<-emmreml(y=deptrainfullplant.height, +X=matrix(1, nrow=nrow(deptrainfull), ncol=1), + Z=Ztrainfull > Trainopt<-GenAlgForSubsetSelectionNoTest( + P=PC50WHeat[rownames(PC50WHeat) + InitPop=NULL,npop=200,+ nelite=10, mutprob=.5, mutintensity = 1,niterations=100, + minitbefstop=50, tabumemsize = 1,plotiters=FALSE, + lambda=1e-9,errorstat="DOPT", mc.cores=4) > deptrainopt<-Wheat.Y[(Wheat.Yid > Ztrainopt<-model.matrix( -1+deptrainoptid) > modelrrblupopt<-emmreml(y=deptrainoptplant.height, +X=matrix(1, nrow=nrow(deptrainopt), ncol=1),+ Z=Ztrainopt > modelrrbluprs<-emmreml(y=deptrainopt4plant.height, +X=matrix(1, nrow=nrow(deptrainopt4), ncol=1),+ Z=Ztrainrs > orderfull<-order(abs(modelrrblupfulluhat), decreasing=T) > orderopt<-order(abs(modelrrblupoptuhat), decreasing=T) > orderrs<-order(abs(modelrrbluprsuhat), decreasing=T) > mean(abs(orderrs-orderfull))[1] 1580.187> mean(abs(orderopt-orderfull))[1] 1567.686As noted before, the subset selection optimization problem is a combinatorial one. We need to see if the algorithm got close to convergence, we can do this by plotting the criterion values over the iterations, these values are stored in the output of STPGA with the name `Best criterion values over iterations`.Box 12: Plotting the progress of the optimization > plot(Train3`Best criterion values over iterarions`, +xlab="iteration", ylab="criterion value")< g r a p h i c s > The users are recommended to check the convergence of the algorithm by checking the plot, the last iterations should show little or no improvement. If the algorithm has still room to improve the solutions from the last run can be used as new starting points and the iteration can be restarted, perhaps with different settings for the GA parameters. For difficult problems, a good strategy is to run the algorithm several times and select the best solution among these. It is also possible to implement the genetic algorithm in an island model, I have included the code fora simple island model in Appendix. §.§ STPGA in other subset selection problemsOptimal subset selection algorithm in STPGA can be used with user supplied optimization criterion, and therefore, it has a wide area of application. I am going to try to give a few examples: supervised unsupervised variable selection, Minimize inbreeding while maximizing gain, mixed integer programming, influential observation selection. These examples can easily be extended. The following example illustrates how the users can define their own criteria and use it with STPGA for a variable selection problem. It involves aligning kernels to select variables and as far as I know this wasn't done before. Selecting most representative marker set (markers that represent most of the variability in a given marker data)A genetic relationship matrix measures the amount individuals in a certain group are genetically similar. Genetic relationship matrices can be constructed using pedigrees, or using genome-wide markers. In this example, we will try to find a fixed size subset of available genome-wide markers that results in a genetic relationship matrix that is as close to the genetic relationship matrix as possible. These selected markers can be called the genetic anchor markers, since they explain most of the properties of the genome-wide relationship matrix. The main question is if there is a subset of markers that can explain a big part of all of the variation captured by all the markers (or even the genome sequence), since this relates to many important genetic concepts like effective population size, effective number of independent chromosome segments, population structure and its effects on predictability within and between sub-populations.Note that the selection of the anchor markers does not involve any phenotypic observations, therefore this is an unsupervised marker selection approach, similar to some recent approaches expressed in sparse principal components analysis <cit.> or sparse partial least squares <cit.>. However, the interpretation of the factors extracted by sparse PCA and sparse PLS might be difficult since these are linear combinations of the original variables. The following is a simple function for obtaining a genetic relationship matrix given the matrix of markers (n x m) (n:number of genotypes), (m:number of markers) coded as 0, 1, 2 representing the number of minor alleles. A function to calculate the relationship genomic relationshipmatrix according to the formula in Van Raden <cit.> is given below:Box 13: A function to calculate Van Raden's relationship matrix from minor allele frequency scores (markers coded as 0,1 and 2) > A.mat<-function(M) + pks<-colMeans(M)/2 + W<-scale(M, center=TRUE, scale=FALSE) + c<-2*sum(pks*(1-pks)) + Amat<-tcrossprod(W)/c + rownames(Amat)<-colnames(Amat)<-rownames(M) + return(Amat) +I will only use the lines in the second cluster, the whole data would take too much time to process.Box 14: Obtaining a subset of wheat data set for a quick analysis > #the problem can be soved for all the genotypes to make the> #problem computationally easier, we will pick only the> #genotypes in forth cluster > library(Matrix) > Wheat.M4<-Wheat.M[rownames(Wheat.M) > Wheat.M4<-Matrix(Wheat.M4+1) > #relationship using all markers > ##A.mat requires the markers are coded as 0, 1, 2 > Afull<-A.mat(M=Wheat.M4)We can see how this optimally selected markers compare with the randomly selected marker sets of the same size. Box 15: Optimally selected anchor markers versus randomly selected markers > n<-nrow(Wheat.M4) > diffvecrs<-c() > for (i in 1:100) + rssmallM<-Wheat.M4[,sample(1:ncol(Wheat.M4),50)] + Ars<-A.mat(M=rssmallM) + diffvecrs<-c(diffvecrs,mean((c(Afull[lower.tri(Afull, diag=TRUE)]) +-c(Ars[lower.tri(Ars, diag=TRUE)]))^2)) +> #User defined criterion > STPGAUSERDEFFUNC<-function(Train,Test=NULL, P, lambda=1e-6, C=NULL) +trsmallM<-t(P[rownames(P) +Atr<-A.mat(M=trsmallM) +return(mean((c(Afull[lower.tri(Afull, diag=TRUE)]) + -c(Atr[lower.tri(Atr, diag=TRUE)]))^2)) +> GAOUT<-GenAlgForSubsetSelectionNoTest(P=t(Wheat.M4), + ntoselect=50,npop=300,+ nelite=10, mutprob=.5, mutintensity = 1, + niterations=400, minitbefstop=50,tabu=FALSE, + tabumemsize = 1,plotiters=F,lambda=1e-6, + errorstat="STPGAUSERDEFFUNC",mc.cores=4) > min(GAOUT`Best criterion values over iterarions`);mean(diffvecrs)[1] 0.03310715[1] 0.1044086> optsmallM<-Wheat.M4[, colnames(Wheat.M4) > optA<-A.mat(optsmallM) Box 16: Plotting the results for optimally selected ”anchor markers” versus randomly selected markers > layout(matrix(c(1,2,3,4,5,6),2,3, byrow=TRUE), widths=c(2,2,2), +heights=c(2,2), respect=TRUE) > par(mar=c(3,2,2,1)) > # turn off the axes > image(Ars, axes=FALSE, main="Random Markers") > image(optA, axes=FALSE, main="Optimal Markers") > image(Afull, axes=FALSE, main="All Markers") > image((Ars-Afull)^2, axes=FALSE,+ main="Squared errors for random") > image((optA-Afull)^2, axes=FALSE,+ main="Squared errors for optimal") > par(mfrow=c(1,1))< g r a p h i c s >According to these results, the optimally selected markers sets result in a genetic relationship matrix that is closer to the full marker genetic relationship matrix than therelationship matrices calculated from random sets of markers.I also want to see how the variance captured by these relationship matrices and the accuracies of the models built using these relationship matrices compare. Box 17: Comparing accuracy of predictions for optimally selected anchor markers, randomly selected markers and all markers > linenames<-rownames(Afull) > Test<-sample(linenames, 20) > Train<-setdiff(linenames, Test) > Wheat.Y4<-Wheat.Y[Wheat.Yid > Wheat.Y4id<-factor(as.character(Wheat.Y4id), levels=linenames) > Wheat.Y4Train<-Wheat.Y4[Wheat.Y4id > Wheat.Y4Test<-Wheat.Y4[Wheat.Y4id > Ztrain<-model.matrix( -1+Wheat.Y4Trainid) > Ztest<-model.matrix( -1+Wheat.Y4Testid) > library(EMMREML) > modelfull<-emmreml(y=Wheat.Y4Trainplant.height, +X=matrix(rep(1,nrow(Ztrain)), ncol=1), +Z=Ztrain, K=Afull+1e-9*diag(nrow(Afull))) > modelfullVu [1] 63.14514> rsVus<-c() > for( i in 1:100) + rssmallM<-Wheat.M4[,sample(1:ncol(Wheat.M4),50)] + Ars<-A.mat(M=rssmallM) +modelrs<-emmreml(y=Wheat.Y4Trainplant.height, +X=matrix(rep(1,nrow(Ztrain)), ncol=1), +Z=Ztrain, K=Ars+1e-9*diag(nrow(Ars))) + rsVus<-c(rsVus,modelrsVu) +> mean(rsVus)[1] 32.18926> modeloptA<-emmreml(y=Wheat.Y4Trainplant.height, +X=matrix(rep(1,nrow(Ztrain)), ncol=1), +Z=Ztrain, K=optA+1e-9*diag(nrow(optA))) > modeloptAVu[1] 30.73143> predictmatrix<-as.matrix(Ztest + modelrsuhat,modeloptAuhat)) > colnames(predictmatrix)<-c("All", "rs 50", "opt 50") Box 18: Comparing accuracy of predictions for optimally selected anchor markers, randomly selected markers and all markers > pairs(predictmatrix) > cor(predictmatrix) All rs 50opt 50 All1.0000000 0.7793880 0.9094115 rs 500.7793880 1.0000000 0.8656134 opt 50 0.9094115 0.8656134 1.0000000< g r a p h i c s > Minimize inbreeding while maximizing gainMany authors <cit.> have expressed the importance of reducing inbreeding in PS and GS for long-term success of breeding programs. They argued that GS is likely to lead to a more rapid decline in the selection response unless new alleles are continuously added to the calculation of GEBVs, stressing the importance of balancing short and long term gains by controlling inbreeding in selection. A quadratic programming (QP) problem has the generic formQ() = 1/2' D-'+ c.Here,is a vector inℝ^n,Dis ann × nsymmetric positive definite matrix,is a constant vector inℝ^nandcis a scalar constant. QPs usually come witha system of linear constraints on the vector∈ℝ^nwhich can be written asA =B≥ g.HereAis anm_1 × nmatrix withm_1 ≤ nandBis am_2 × nmatrix.The vectorsandhave lengthsm_1andm_2respectively.QP can be more compactly stated as compactly as:{[ minimize_∈ℝ^n:Q() = 1/2' D- '+ c; subjectto: A =B≥ ].There are many efficient algorithms that solves QP's so there is in practice little difficulty in calculating the optimal solution for any particular data set. In this example, I will be using the package quadprog<cit.>. Now, letAbe a matrix of pedigree based numerator relationships or the additive genetic relationships between the individuals in a breeding population (this matrix can be obtained from a pedigree of genome-wide markers for the individuals) and letbe the vector of proportional contributions of individuals to the next generation under a random mating scheme. The average inbreeding and co-ancestry for a given choice ofcan be defined asr=1/2'A.Ifis the vector of GEBV's, i.e., the vector of BLUP (best linear unbiased predictor) estimated BV's of the candidates for selection. The expected gain is defined asg='.Without loss of generality, we will assume that the breeder's long term goal is to increase the value ofg.In <cit.>, an approach that seeks minimizingthe average inbreeding and co-ancestry while restricting the genetic gain is proposed. The optimization problem can be stated as[minimize r=' A/2;[10pt] [' =ρ;[10pt] '=1; [10pt] ≥ 0, ] ]whereρis the desired level of gain.This problem is easily recognized as a QP.Recently, several parental percentage allocation strategies were tested using QP's in <cit.>.Box 19: Preparing the wheat data for analysis, setting up the optimization function > Wheat.Ysc<-Wheat.Y > Wheat.Ysc[,2]<-(Wheat.Ysc[,2]-mean(Wheat.Ysc[,2]))/sd(Wheat.Ysc[,2]) > P<-cbind(Wheat.Ysc, Wheat.M) > rownames(P)<-rownames(Wheat.M) > impinbreed=.1 > STPGAUSERDEFFUNC<-function(Train,Test=NULL, P,+lambda=NULL, C=NULL) + trsmallM<-P[rownames(P) + g<-matrix(P[rownames(P) + A<-A.mat(M=trsmallM+1) + return(-(1-impinbreed)*mean(g)+ +impinbreed*mean(c(A[lower.tri(A, diag=TRUE)]))) +By solving the QP in Equation (<ref>) for varying values ofρ,or using the similar criteria in the mate selection approaches, we can trace out an efficient frontier curve, a smooth non-decreasing curve that gives the best possible trade-off of genetic variance against gain. This curve represents the set of optimal allocations and it is called the efficiency frontier (EF) curve in finance <cit.> and breeding literature. Many practical applications require additional constraints, ans it is possible to extend these formulations to introduce additional constraints as positiveness, minimum-maximum for proportions, minimum-maximum for number of lines (cardinality constraints). It is not too difficult to use STPGA to solve a version of this problem.Suppose the breeder wants to select a subset of the individuals to become parents of the next generation increasing gain while controlling for coancestory and inbreeding. Note that the breeder only wants to select a subset and therefore we can assume that the parental contributions will be the same for each of the selected individuals. The following code illustrates how to select 10 individuals from the wheat data set for changing values ofλ.Box 20: Selected individuals for changing values of λ > GAOUTLIST<-vector(mode="list", length=5) > i=1 > for (impinbreed in c(0.01,.95,.99)) + STPGAUSERDEFFUNC<-function(Train,Test=NULL, P, lambda=NULL, C=NULL) + trsmallM<-P[rownames(P) + g<-P[rownames(P) + A<-A.mat(M=trsmallM+1) + return(-(1-impinbreed)*mean(g)+ +impinbreed*mean(c(A[lower.tri(A, diag=TRUE)]))) ++ GAOUT<-GenAlgForSubsetSelectionNoTest(P=P, ntoselect=10,npop=100,+ nelite=10, mutprob=.5, mutintensity = 1,niterations=100, + minitbefstop=50, tabumemsize = 1, plotiters=FALSE,tabu=FALSE, +lambda=1e-9,errorstat="STPGAUSERDEFFUNC", mc.cores=4) + GAOUTLIST[[i]]<-GAOUT + i=i+1 +> GAMINvec<-c() > GAMINvecgain<-c() > GAMINvecinbreeding<-c() > for (i in 1:3) + Train<-GAOUTLIST[[i]]`Solution with rank 1` + trsmallM<-P[rownames(P) + g<-P[rownames(P) + A<-A.mat(M=trsmallM+1) + GAMINvec<-c(GAMINvec, + min(GAOUTLIST[[i]]`Best criterion values over iterarions`)) + GAMINvecgain<-c(GAMINvecgain, mean(g)) + GAMINvecinbreeding<-c(GAMINvecinbreeding,mean(c(A[lower.tri(A, diag=TRUE)]))) + After that I extract the average gain and inbreeding values for each value ofλand trace the frontier curve.Box 21: Points on the frontier surface > plot(GAMINvecgain,GAMINvecinbreeding, type="b", +xlab="gain", ylab="inbreeding")< g r a p h i c s > Suppose now the breeder wants us to pick 10 individuals, but also asks for the best parental contribution proportions. This is a mixed integer quadratic programming problem.Letϵ_ibe the minimum proportion that must be allocated to linei,δ_ibe the maximum proportion that must be allocated to lineiif any of lineiwill be conserved, where we must have0≤ϵ_i≤δ_i≤ 1.Introduce the binary variables:z_i = {[ 1 if any of line i is included,; 0otherwise. ].The cardinality constrained optimization problem is given by [r=c' A/2 c; [10pt] [ c' b=ρ,;[10pt] c'=1,; [10pt] ≥ 0,; ∑_i=1^mz_i=K,; ϵ_i z_i≤ c_i≤δ_i z_i,i=1,2,…,m,; z_i ∈{0,1},i=1,2,…,m. ] ] Here is the code for doing the same thing we have done with the previous example, except we provide parental contributions. Box 22: Estimating the parental contributions for the cardinality constrained problem for changing values of λ > require(quadprog) > GAOUTLIST<-vector(mode="list", length=5) > i=1 > for (impinbreed in c(0.01,.95,.99)) + STPGAUSERDEFFUNC<-function(Train,Test=NULL, P, lambda=1e-5, C=NULL) + trsmallM<-P[rownames(P) + g<-c(P[rownames(P) + A<-A.mat(M=trsmallM+1) + n=length(g) + sol<- solve.QP(Dmat=(n^2/2)*impinbreed*(A+lambda*diag(n)), + dvec=(1/n)*(1-impinbreed)*g,Amat=cbind(rep(1,n),diag(n),-diag(n)), + bvec=rbind(1,matrix(0,ncol=1,nrow=n), +matrix(-1,ncol=1,nrow=n)), meq=1) + names(solsolution)<-rownames(trsmallM) + return(solvalue) ++ GAOUT<-GenAlgForSubsetSelectionNoTest(P=P, ntoselect=10,npop=50,+ nelite=5, mutprob=.5, mutintensity = 1,niterations=100, + minitbefstop=50, tabumemsize = 1,plotiters=FALSE,tabu=FALSE, + lambda=1e-5,errorstat="STPGAUSERDEFFUNC", mc.cores=4) + GAOUTLIST[[i]]<-GAOUT + i=i+1 + Box 23: Estimating the parental contributions for the cardinality constrained problem for changing values of λ > GAMINvec<-c() > GAsols<-vector(mode="list") > for (i in 1:3) + trsmallM<-P[rownames(P) + g<-c(P[rownames(P) + A<-A.mat(M=trsmallM+1) + n=length(g) + impinbreed<-c(0.01,.95,.99)[i] ++sols<-solve.QP(Dmat=(n^2/2)*impinbreed*(A+1e-9*diag(n)), + dvec=(1/n)*(1-impinbreed)*g,Amat=cbind(rep(1,n),diag(n),-diag(n)), + bvec=rbind(1,matrix(0,ncol=1,nrow=n), +matrix(-1,ncol=1,nrow=n)), meq=1)solution + names(sols)<-GAOUTLIST[[i]]`Solution with rank 1` + GAsols[[i]] <-sols + GAMINvec<-c(GAMINvec,+ min(GAOUTLIST[[i]]`Best criterion values over iterarions`)) +> print(round(GAsols[[2]], digits=3))IWA8606856PPG-1 AGRISS IWA8606779 466AH86-708PI94479 0.10.10.10.10.10.10.1 ZG4163/73NORIN10 PI254037 0.10.10.1Variable selection in regression using model selection criteriaOne of the standard use of GA is in variable selection and STPGA can be used in variable selection. Another very popular method for variable selection and penalized (shrunk) parameter estimation in high dimensional regression is the lasso approach which can be summarized as finding the best regression coefficients that minimize the regression loss function while minimizing a multiple of theℓ_1norm of the coefficients. The relative stress on the importance of either of these complementary functions is expressed as a linear combination of these two. For example with the squared error loss the lasso optimization criterion is(-X)'(-X)+λ ||'.The idea, its implications for many different fields of science can not be overlooked, the same goes for the enormous number of methods developed to solve this problem and its variations. However, there is a major statistical problem: there isn't enough principle behind its objective criterion so it is no big magic. It does not have certain properties we would like from a good regression. For example, the residuals of a model using the coefficients obtained by lasso are not orthogonal to the design of the variables that are selected by lasso. This last issue can be circumvented by using lasso only as a selection operator and then fitting the coefficients with least squares without answering the main question: Why this criterion, not another one? Parsimony, yes, why this one? Is this the one that gives the best generalization error (minimize the prediction error)?The only good argument for the defense of this criterion I can think of is a Bayesian one and the so called ”oracle” property which is only true under restrictive assumptions such as no dependency among the explanatory variables. we should be careful not to confuse the actual problem with the method, such as the carpenter who sees everything as nails since he/she is good at using hammers.In my opinion, the whole area of norm penalized estimation neglects the many model selection criteria (AIC <cit.>, BIC <cit.>, ICOMP(IFIM) <cit.>, etc,...) introduced during the 1970's and onward by many prominent statisticians whose derivations depends on the solid theory of likelihoods, divergence measures, etc.,... I am not sure why we should develop elegant theories and then ignore them for not so elegant ones. So while lasso <cit.> is great and can be solved at great speed in serial (after investment a huge amounts of energy and resources in the last 10 or more years) and the methods to find its solutions can only be partially parallelized. I am not sure the shrinkage approach can take on the next challenge of solving extremely large and complex problems. In addition to not being able to adopt well to parallel computer architectures, their application areas are limited by the problems these methods can address and addressing new problems with the same techniques involves inventing newly crafted methodologies which consumes the most amount of time and resources.Here is an example using STPGA. I am going to demonstrate this with a classic body fat data set. The data is available in the package UsingR<cit.>. I will use the AIC criterion, find the best subsets of size two through 12 (there are 13 explanatory variables) and obtain their AIC values. Box 24: Variable selection for regression > data("fat", package = "UsingR") > mod <- lm(body.fat.siri   age + weight + height + neck + chest + abdomen + + hip + thigh + knee + ankle + bicep + forearm + wrist, data = fat) > x <- model.matrix(mod) > y <- model.response(model.frame(mod)) > fitnessfuncforSTPGA <- function(Train,Test=NULL, P, lambda=1e-6, C=NULL)+ X <- t(P[rownames(P) + mod <- lm.fit(X, y) + class(mod) <- "lm" + return(AIC(mod)) +> stpgaoutlist<-vector(mode="list") > ii=1 > for (i in 2:(ncol(x)-2)) + stpgaoutlist[[ii]]<-GenAlgForSubsetSelectionNoTest(P=t(x[,-1]), + ntoselect=i,npop=200,+ nelite=10, mutprob=.5, mutintensity = 1, + niterations=200, minitbefstop=50,tabu=FALSE, + tabumemsize = 1,plotiters=FALSE,lambda=1e-9, + errorstat="fitnessfuncforSTPGA",mc.cores=4) + ii=ii+1 + Box 25: Extracting results > GAMINs<-c() > ii=1 > for (i in 2:(ncol(x)-2)) + GAMINs<-c(GAMINs, + min(stpgaoutlist[[ii]]`Best criterion values over iterarions`)) + ii=ii+1 +> selectedvars6<-stpgaoutlist[[6]]`Solution with rank 1` > min(stpgaoutlist[[6]]`Best criterion values over iterarions`)[1] 1460.21> selectedvars6[1] "abdomen" "forearm" "hip" "age" "wrist" "thigh" "neck" > selectedvars7<-stpgaoutlist[[7]]`Solution with rank 1` > min(stpgaoutlist[[7]]`Best criterion values over iterarions`)[1] 1459.716> selectedvars7[1] "forearm" "thigh" "age" "abdomen" "hip" "wrist" "height"[8] "neck" > selectedvars8<-stpgaoutlist[[8]]`Solution with rank 1` > min(stpgaoutlist[[8]]`Best criterion values over iterarions`)[1] 1459.474> selectedvars8[1] "thigh" "abdomen" "weight""neck""hip" "forearm" "age" [8] "wrist" "height"Lets plot the results. It looks like we should pick seven or eight variables. Box 26: Plotting AIC results for different subset sizes > plot(2:(ncol(x)-2),GAMINs, type="b", xlab="NVARS", ylab="AIC")< g r a p h i c s >Now, I will try to using an unsupervised approach to variable selection, the methodology is new.Why not pick the subset of variables that result in a kernel matrix that is most aligned with the the kernel that is calculated based on all 13 explanatory variables. Interestingly, most of the variables that are selected by supervised selection are also identified here.Box 27: Variable selection- Unsupervised using kernel alignment > Afull<-tcrossprod(scale(x[,-1], center=TRUE, scale=TRUE))/(ncol(x)-1) > STPGAUSERDEFFUNC<-function(Train,Test=NULL, P, lambda=1e-6, C=NULL) +trsmallx<-t(scale(P[rownames(P) +Atr<-tcrossprod(trsmallx)/(ncol(trsmallx)) +return(mean((c(Afull[lower.tri(Afull, diag=TRUE)]) + -c(Atr[lower.tri(Atr, diag=TRUE)]))^2)) +> GAOUT<-GenAlgForSubsetSelectionNoTest(P=t(x[,-1]), + ntoselect=7,npop=200,+ nelite=5, mutprob=.5, mutintensity = 1, + niterations=200, minitbefstop=50,tabu=FALSE, + tabumemsize = 1,plotiters=FALSE,lambda=1e-9, + errorstat="STPGAUSERDEFFUNC",mc.cores=4) > selectedunsupervised<-GAOUT`Solution with rank 1` > selectedunsupervised[1] "knee""neck""ankle" "wrist" "age" "forearm" "bicep"> selectedvars6[1] "abdomen" "forearm" "hip" "age" "wrist" "thigh" "neck" > intersect(selectedvars6,selectedunsupervised)[1] "forearm" "age" "wrist" "neck"Here is a similar approach, we only add the response variable to the calculation of full data kernel.Box 28: Variable selection - supervised using kernel alignment > newx<-cbind(x[,-1],y) > colnames(newx) [1] "age" "weight""height""neck""chest" "abdomen" "hip"[8] "thigh" "knee""ankle" "bicep" "forearm" "wrist" "y"> Afull2<-tcrossprod(scale(newx, center=TRUE, scale=TRUE))/(ncol(x)) > STPGAUSERDEFFUNC2<-function(Train,Test=NULL, P, lambda=1e-6, C=NULL) + trsmallx<-t(scale(P[rownames(P) +Atr<-tcrossprod(trsmallx)/(ncol(trsmallx)) ++return(mean((c(Afull2[lower.tri(Afull2, diag=TRUE)]) + -c(Atr[lower.tri(Atr, diag=TRUE)]))^2)) +> GAOUT2<-GenAlgForSubsetSelectionNoTest(P=t(x[,-1]), + ntoselect=7,npop=200,+ nelite=5, mutprob=.5, mutintensity = 1, + niterations=200, minitbefstop=50,tabu=FALSE, + tabumemsize = 1,plotiters=FALSE,lambda=1e-9, + errorstat="STPGAUSERDEFFUNC2",mc.cores=4) > selectedsupervised<-GAOUT2`Solution with rank 1` > selectedsupervised[1] "forearm" "knee""ankle" "wrist" "age" "neck""bicep"> selectedvars6[1] "abdomen" "forearm" "hip" "age" "wrist" "thigh" "neck" > intersect(selectedvars6,selectedunsupervised)[1] "forearm" "age" "wrist" "neck" > intersect(selectedvars6,selectedsupervised)[1] "forearm" "age" "wrist" "neck" > intersect(selectedunsupervised,selectedsupervised)[1] "knee""neck""ankle" "wrist" "age" "forearm" "bicep"We can also align to the kernel matrix obtained from only the response variable(s): Box 29: Variable selection- supervised (only by response) using kernel alignment > Afull3<-tcrossprod(scale(y, center=TRUE, scale=TRUE)) > STPGAUSERDEFFUNC3<-function(Train,Test=NULL, P, lambda=1e-6, C=NULL) +trsmallx<-t(scale(P[rownames(P) +Atr<-tcrossprod(trsmallx)/(ncol(trsmallx)) ++return(mean((c(Afull3[lower.tri(Afull3, diag=TRUE)]) + -c(Atr[lower.tri(Atr, diag=TRUE)]))^2)) +> GAOUT3<-GenAlgForSubsetSelectionNoTest(P=t(x[,-1]), + ntoselect=7,npop=200,+ nelite=5, mutprob=.5, mutintensity = 1, + niterations=200, minitbefstop=50,tabu=FALSE, + tabumemsize = 1,plotiters=FALSE,lambda=1e-9, + errorstat="STPGAUSERDEFFUNC3",mc.cores=4) > selectedsupervised2<-GAOUT3`Solution with rank 1` > selectedsupervised2[1] "wrist" "bicep" "neck""forearm" "ankle" "knee""age"> selectedvars6[1] "abdomen" "forearm" "hip" "age" "wrist" "thigh" "neck" > intersect(selectedvars6,selectedunsupervised)[1] "forearm" "age" "wrist" "neck" > intersect(selectedvars6,selectedsupervised2)[1] "forearm" "age" "wrist" "neck" > intersect(selectedunsupervised,selectedsupervised2)[1] "knee""neck""ankle" "wrist" "age" "forearm" "bicep"> intersect(selectedsupervised,selectedsupervised2)[1] "forearm" "knee""ankle" "wrist" "age" "neck""bicep"Let's see which variables are picked by lasso and best subsets regressions: Box 30: Variable selection- supervised using lasso and best subsets regression > # lasso > library(glmnet) > fit.glmnet.lasso.cv <- cv.glmnet(as.matrix(scale(x[,-1], center=T, scale=T)), + as.matrix(y, ncol=1), +nfold = 5, +alpha = 1) > coef(fit.glmnet.lasso.cv, s = fit.glmnet.lasso.cvlambda.1se)14 x 1 sparse Matrix of class "dgCMatrix"1 (Intercept) 19.1507937 age0.2887603 weight . height-0.5184681 neck . chest. abdomen6.2447837 hip. thigh. knee . ankle. bicep. forearm. wrist -0.1999128> GAOUT4<-GenAlgForSubsetSelectionNoTest(P=t(x[,-1]), + ntoselect=4,npop=200,+ nelite=5, mutprob=.5, mutintensity = 1, + niterations=200, minitbefstop=50,tabu=FALSE, + tabumemsize = 1,plotiters=FALSE,lambda=1e-9, + errorstat="STPGAUSERDEFFUNC3",mc.cores=4) > selectedsupervised4<-GAOUT4`Solution with rank 1` > selectedsupervised4[1] "knee""neck""forearm" "age" Box 31: Variable selection- supervised using lasso and best subsets regression > # lasso > library(leaps) > regsubsets.out <- + regsubsets(body.fat.siri   age + weight + height + neck + chest + abdomen + +hip + thigh + knee + ankle + bicep + forearm + wrist, +data = fat, +nbest = 1, nvmax = NULL,+ force.in = NULL, force.out = NULL, +method = "exhaustive") > sevenvarsselectedbyleaps<-colnames(as.data.frame(summary(regsubsets.out)outmat))[ + which(as.data.frame(summary(regsubsets.out)outmat)[7,]=="*")] > intersect(sevenvarsselectedbyleaps,selectedunsupervised)[1] "age" "neck""forearm" "wrist"> intersect(sevenvarsselectedbyleaps,selectedvars7)[1] "age" "neck""abdomen" "thigh" "forearm" "wrist" Now, again, lets to something that hasn't been done before: I am going to ”make up” a variable selection criterion for regression that has some nice properties. Lets assume as beforeis the lengthnresponse vector,Xis then× p design matrix; lets denote the generic design ofk≤ pvariables asX^(k)and the set of such k variable designsas𝐗^(k).For this generic design matrix we want to find^(k)∈𝐑^ksuch that error sum of squares are minimized, givenX^(k)this is the least squares solution^(k)=(X^(k)'X^(k))^-1X^(k)'with covariance matrix proportional to(X^(k)'X^(k))^-1.The optimization problem is stated as(X^(k), ^(k))=X^(k)∈𝐗^(k), ^(k)∈𝐑^kargmin (-X^(k)^(k))'(-X^(k)^(k))-λ |X^(k)'X^(k)|and to simplify it further, noting that the penalty term does not depend on^(k), we can rewrite the above asX^(k)=X^(k)∈𝐗^(k)argmin((I-X^(k)(X^(k)'X^(k))^-X^(k)')-λlog |X^(k)'X^(k)|and^(k)=(X^(k)'X^(k))^-1X^(k)'.The interpretation of this optimization criterion is as follows, we want the set of k variables that minimize the squared error loss while also keeping the variances and correlations among the estimated coefficients as low as possible, by increasingλwe would be increasing the the relative importance given to the variance and covariances of the coefficients. Note that, when the information matrix issingular, we can use a generalized inverse obtained by regularization (i.e. add a small constant to the diagonal elements) as described in the previous sections.Box 32: Variable selection > mod <- lm(body.fat.siri   age + weight + height + neck + chest + abdomen + + hip + thigh + knee + ankle + bicep + forearm + wrist, data = fat) > x <- scale(as.matrix(model.matrix(mod)), center=F, scale=T) > y <- model.response(model.frame(mod)) > y<-y/sd(y) > fitnessfuncforSTPGA <- function(Train,Test=NULL, P, lambda=.5, C=NULL)+ X <- t(P[rownames(P) + n<-nrow(X) + p<-ncol(X) + mindim<-min(p,n) + rownames(X)<-NULL + svdX<-svd(X, nu=mindim,nv=mindim) + insvdnonzero<-1:mindim + diagvecforinv<-(svdXd[insvdnonzero])/((svdXd[insvdnonzero])^2+1e-7) + coef<-tcrossprod(svdXv + resids<-y-X + out<-(1-lambda)*mean(resids^2)- + lambda*determinant(crossprod(X), logarithm = T )modulus + return(out) +> GAMINSmat<-c() > lambda=1e-6 > stpgaoutlist<-vector(mode="list") > ii=1 > for (i in 2:(ncol(x)-1)) + stpgaoutlist[[ii]]<-GenAlgForSubsetSelectionNoTest(P=t(x[,-1]), + ntoselect=i,npop=100,+ nelite=10, mutprob=.5, mutintensity = 1, + niterations=100, minitbefstop=50,tabu=FALSE, + tabumemsize = 1,plotiters=F,lambda=lambda, + errorstat="fitnessfuncforSTPGA",mc.cores=4) + ii=ii+1 +> GAMINs<-c() > ii=1 > for (i in 2:(ncol(x)-1)) + GAMINs<-c(GAMINs, + min(stpgaoutlist[[ii]]`Best criterion values over iterarions`)) + ii=ii+1 +> GAMINSmat<-cbind(GAMINSmat,GAMINs) >Box 33: Variable selection > plot(2:(ncol(x)-1),GAMINs, type="b", xlab="NVARS", ylab="") > for (lambda in c(seq(1e-5,.2, length=3),seq(0.21,.90, length=3),seq(0.91,1, length=10))) + par(new=T) + stpgaoutlist<-vector(mode="list") + ii=1 + for (i in 2:(ncol(x)-1)) + stpgaoutlist[[ii]]<-GenAlgForSubsetSelectionNoTest(P=t(x[,-1]), + ntoselect=i,npop=100,+ nelite=10, mutprob=.5, mutintensity = 1, + niterations=100, minitbefstop=50,tabu=FALSE, + tabumemsize = 1,plotiters=FALSE,lambda=lambda, + errorstat="fitnessfuncforSTPGA",mc.cores=4) + ii=ii+1 ++ GAMINs<-c() + ii=1 + for (i in 2:(ncol(x)-1)) + GAMINs<-c(GAMINs,min(stpgaoutlist[[ii]]`Best criterion values over iterarions`)) + ii=ii+1 ++ GAMINSmat<-cbind(GAMINSmat,GAMINs) + plot(2:(ncol(x)-1),GAMINs, type="b", xlab="NVARS", ylab="", axes=F) +< g r a p h i c s > Box 34: Variable selection- new criterion > apply(GAMINSmat, 2, which.min)GAMINs GAMINs GAMINs GAMINs GAMINs GAMINs GAMINs GAMINs GAMINs GAMINs GAMINs 11111333333GAMINs GAMINs GAMINs GAMINs GAMINs GAMINs 333333 > apply(GAMINSmat, 2, min) GAMINsGAMINsGAMINsGAMINsGAMINsGAMINs0.8111831 0.8111216 0.1281011-0.5696401-0.6409756-4.4104119 GAMINsGAMINsGAMINsGAMINsGAMINsGAMINs -9.0394933-9.1736695-9.3078458-9.4420221-9.5761984-9.7103746 GAMINsGAMINsGAMINsGAMINsGAMINs -9.8445509-9.9787272 -10.1129034 -10.2470797 -10.3812560 > newselcritout<-GenAlgForSubsetSelectionNoTest(P=t(x[,-1]), + ntoselect=5,npop=200,+ nelite=10, mutprob=.5, mutintensity = 1, + niterations=200, minitbefstop=50,tabu=FALSE, + tabumemsize = 1,plotiters=F,lambda=.1, + errorstat="fitnessfuncforSTPGA",mc.cores=4) > newselcritout`Solution with rank 1`[1] "forearm" "ankle" "height""chest" "wrist"Note that the design criterion above defines a class of optimal solutions whose criterion values that can be plotted as a 3-dimensional surface for changingλandn;this is similar to the frontier curve discussed above for balancing gains and inbreeding problem. In general, any multiobjective optimization problem defines a surface of Pareto optimal points that is called the frontier surface which may or may not be continuous. Frontier surfaces are inspected for identifying a good solution among and in the case of variable selection should be useful for identifying a single model from the class of optimal solutions.It is easy to make criteria like the one I have made up. Here is another oneX^(k)=X^(k)∈𝐗^(k)argmin((I-X^(k)(X^(k)'X^(k))^-X^(k)')+λ trace (X^(k)'X^(k))^-1and^(k)=(X^(k)'X^(k))^-1X^(k)';and another one:X^(k)=X^(k)∈𝐗^(k)argmin((I-X^(k)(X^(k)'X^(k))^-X^(k)')+λ trace X^(k)(X^(k)'X^(k))^-1X^(k)'and^(k)=(X^(k)'X^(k))^-1X^(k)'.Both of these have nice interpretation as well. In fact, any optimal design criteria of Section <ref> could be relevant in this context, for example, tryX^(k)=X^(k)∈𝐗^(k)argmin((I-X^(k)(X^(k)'X^(k))^-X^(k)')+λ trace X^*(k)(X^(k)'X^(k))^-1X^*(k)',and^(k)=(X^(k)'X^(k))^-1X^(k)'.I encourage the user to program these criterion or their own, perhaps the one you invent will make more sense for the problem that you are trying to solve than this or that other criterion.The main point of these examples is that variable selection problem can be viewed as a multiobjective optimization problem, where a loss function and a penalty function are being simultaneously optimized. The loss function measures the fit of data to the training data and its optimal value for the optimal fixed sized set of variables improves as more variables are added to that model. The penalty function measures the overall quality of the model. Usually deteriorates as more variables are added to a model. In general, we can assume that the loss function and the penalty function are conflicting objectives, i.e.,there does not exist a single solution that simultaneously optimizes both functions. This leads to a possibly infinite number of Pareto optimal solutions. The set of non-dominated solutions (none of the objective functions can be improved in value without degrading some other) are called Pareto optimal, and define a surface called the Pareto frontier.Identifying influential observations in regression (and keeping the most consistent regression data)When a regression model is fitted to data, if a few of the observations are different in some way from the bulk of the data then using all observations into the model might not be appropriate. The inclusion or exclusion of a few of these observations make a significant change in the parameter estimates or predictions. These are referred to as influential observations. One statistic that is used to identify the influence of an observation is the so called DFBETAS which measures the effect of deletion of single observations on the estimated model coefficients. This measure can be generalized to deletion of a group of individuals as DFBETAS(X_(-))=1/σ_(-)^2(-_(-))'(X_(-)'X_(-)')(-_(-)),whereis the estimated regression coefficients from full data,_(-)is the same estimated using a part of the data (leaving out its complement from the analysis),σ_(-)^2is the residual variance estimate obtained using only partial data andX_(-)is the design matrix for partial data. Here is the application of this to the ”iris” data set (available with base R) to locate the influential observations if there are any:Box 35: Deleting influential obs in regression > library(STPGA) > STPGAUSERFUNCI<-function(Train,Test=NULL, P, lambda=1e-6, C=NULL) + PTrain<-P[rownames(P) + PtP<-crossprod(cbind(1,P[,-1])) + PTtPT<-crossprod(cbind(1,PTrain[,-1])) + B<-solve(PtP+lambda*diag(nrow(PtP)),crossprod(cbind(1,P[,-1]),P[,1])) + BT<-solve(PTtPT+lambda*diag(nrow(PtP)), + crossprod(cbind(1,PTrain[,-1]),PTrain[,1])) + resT<-PTrain[,1]-cbind(1,PTrain[,-1]) + meanD<-mean((1/sd(resT))*diag(crossprod((B-BT), + (PTtPT+lambda*diag(nrow(PtP))) + return(-meanD) +> data(iris) > P<-as.matrix(scale(iris[,1:4], center=TRUE, scale=TRUE)) > rownames(P)<-rownames(iris) > STPGAoutlist<-vector(mode="list", length=20) > ii=1 > for (i in 135:149) + STPGAoutlist[[ii]]<-GenAlgForSubsetSelectionNoTest(P=P,ntoselect=i,+ mutprob = .8, npop = 100, nelite = 5, keepbest = TRUE, + tabu = F, tabumemsize = 0, mutintensity = 1,plotiters=FALSE, + niterations=200, minitbefstop = 50, + errorstat = "STPGAUSERFUNCI", mc.cores=4) + ii=ii+1 +> minsvec<-c() > ii=1 > for (i in 135:149) + minsvec<-c(minsvec, +min(STPGAoutlist[[ii]]`Best criterion values over iterarions`)) + ii=ii+1 + Box 36: Deleting influential obs in regression > plot(135:150,c(minsvec,0), type="b")< g r a p h i c s > What we are looking in the above graph is a point of sharp increase. I don't see such clear change point,so let's replace replace six observations' values so that they are different than the rest:Box 37: Deleting influential observations in regression > data(iris) > P<-as.matrix(scale(iris[,1:4], center=TRUE, scale=TRUE)) > rownames(P)<-rownames(iris) > P[c(5,30,55,80,105,130),c(1:4)]<- + 1.5*P[c(5,30,55,80,105,130),c(1:4)] > STPGAoutlist<-vector(mode="list", length=20) > ii=1 > for (i in 135:149) + STPGAoutlist[[ii]]<-GenAlgForSubsetSelectionNoTest(P=P,ntoselect=i,+mutprob = .8, npop = 100, nelite = 5, keepbest = TRUE, + tabu = F, tabumemsize = 0, mutintensity = 1,plotiters=FALSE, + niterations=200, minitbefstop = 50, + errorstat = "STPGAUSERFUNCI", mc.cores=4) + ii=ii+1 +> minsvec<-c() > ii=1 > for (i in 135:149) + minsvec<-c(minsvec, +min(STPGAoutlist[[ii]]`Best criterion values over iterarions`)) + ii=ii+1 + The following plot shows that the criterion value shows a sharp increase when we go from 144 to 145 observations and there are total of 150 observations in the data set, six observations that aren't included in the set of 144 observations cause large change in the value of the estimated coefficients. We observe that these are the observations we have manipulated.Box 38: Deleting influential observations in regression > par(mfrow=c(2,1)) > twopcsiris<-(P > plot(twopcsiris[,1],twopcsiris[,2], col=(1:150) > plot(135:150,c(minsvec,0), type="b") > par(mfrow=c(1,1)) > STPGAout<-GenAlgForSubsetSelectionNoTest(P=P,ntoselect=144,+mutprob = .8, npop = 50, nelite = 5, keepbest = TRUE, + tabu = F, tabumemsize = 0, mutintensity = 1,plotiters=FALSE, + niterations=200, minitbefstop=50, + errorstat = "STPGAUSERFUNCI", mc.cores=4) > intersect(STPGAout`Solution with rank 1`, as.character(c(5,30,55,80,105,130)))[1] "55" "5" < g r a p h i c s >§ CONCLUDING REMARKSI hope that I have provided enough examples to show that generic optimal subset selection problem is a useful tool. The list of examples and applications can easily be extended. However, I think this should take some time to digest. In STPGA package I have implemented a GA algorithm for subset selection which was inspired by the recent developments in the genomic breeding of plants, animals and other organisms. The main advantage of this algorithm is that it can be run in parallel to solve the subset selection problem for any given objective function. I also did not compare the relative speed of this particular implementation of LA-GA-T algorithm to any other software.I admit that this software could be written so that it functioned faster using one processor and, no matter what I do, it will never work as fast as some other algorithms that are specialized in their tasks. However, I note that LA-GA-T algorithm can be run in parallel on many computers to solve problems as fast as some problem specific algorithms that have to be performed in serial. Prediction accuracy from genomic models can be improved by targeting more informative individuals in the data set used to generate the predictors, this result has been exemplified by several papers in the area (<cit.>. Nevertheless, the subjectdeserves more attention.Some of the criteria I have mentioned in the selection of training populations section of this paper are not among the default criteria listed in Table <ref>, however, I have included them as user defined criteria in the help files that is provided with the package.I have plans to migrate all of this program to a more efficient programming language for performance improvements, and there is room for improvement redesigning parts of the algorithm for However, there are some advantages of using R. Some of these include like the accessibility, availability, being able to define functions on the fly. Therefore, I will still be supporting this pure R version. As a statistician, I have the feeling that writing efficient software is partially out of my expertise, immediate interests and surely can be done much easier with collaboration with people with the right skills.As of now, I can use STPGA with moderately complex problems, either by allowing the algorithm to take its time or by using a highly parallelizable machine. I am open to any suggestions, collaborations in this respect. § ACKNOWLEDGEMENTSI am grateful to family, Mehmet Ali and Güler and Ümit Özgür, Pelin May , Kristy Akdemir.My work was also supported by tax payers through the grants: R01GM099992, R01GM101219 and USDA-NIFA-AFRI Triticeae Coordinated Agricultural Project- award number 2011-68002-30029. § AUTHOR CONTRIBUTIONSEvery idea, code in this manuscript was conceived by Deniz Akdemir unless they were acknowledged by the references within the article. If a person or a group makes any claims that they have contributed to this work in any way it should not be taken seriously. 1. Initial solutions and an island model A.1 Default design criteria implemented in STPGA.[!h] Default design criteria implemented in STPGAcriterion name formula type equation[b] AOPT trace[C(X'_TrainX_Train+λ*I)^-1C']X Equation <ref> CDMAX max[diag(CX_Test(X'_TrainX_Train+λ*I)^-1X'_TestC')/X Equation <ref>[t]diag(CX_TestX'_TestC')][b] CDMAX0 max[diag(CX_Train(X'_TrainX_Train+λ*I)^-1X'_TrainC')/X Equation <ref>[t]diag(CX_TrainX'_TrainC')][b] CDMAX2 max[diag(CX_Test(X'_TrainX_Train+λ*I)^-1X'_TrainX_TrainX Equation <ref>[t](X'_TrainX_Train+λ*I)^-1X'_TestC')/diag(CX_TestX'_TestC')][b] CDMEAN mean[diag(CX_Test(X'_TrainX_Train+λ*I)^-1X'_TestC')/X Equation <ref>[t]diag(CX_TestX'_TestC')][b] CDMEAN0 mean[diag(CX_Train(X'_TrainX_Train+λ*I)^-1X'_TrainC')/X Equation <ref>[t]diag(CX_TrainX'_TrainC')][b] CDMEAN2 mean[diag(CX_Test(X'_TrainX_Train+λ*I)^-1X'_TrainX_TrainX Equation <ref>[t](X'_TrainX_Train+λ*I)^-1X'_TestC')/diag(CX_TestX'_TestC')][b] CDMEANMM -mean[diag(CZ_Test(K-lambda*(Z_Train'MZ_Train)^-1+λ*Kinv)Z_Test'C')/K Equation <ref>[t](diag(CZ_TestKZ_Test'C'))],M=I-W(W'W)^-W'[b] DOPT logdet(C(X'_TrainX_Train+λ*I)^-1C')X Equation <ref>[b] EOPT max(eigenval(C(X'_TrainX_Train+λ*I)^-1C'))X Equation <ref>[b] GAUSSMEANMM -mean(diag(Z_TestKZ_Test'-K Equation <ref>[t]Z_TestKZ_Train'(Z_TrainKZ_Train'+λ*I)^-1Z_TrainKZ_Test')[b] GOPTPEV max(eigenval(CX_Test(X'_TrainX_Train+λ*I)^-1X'_TestC'))XEquation <ref>[b]GOPTPEV2mean(eigenval(CX_Test(X'_TrainX_Train+λ*I)^-1X'_TestC'))XEquation <ref>[b]PEVMAX max(diag(CX_Test(X'_TrainX_Train+λ*I)^-1X'_TestC'))XEquation <ref>[b]PEVMAX0 max(diag(CX_Train(X'_TrainX_Train+λ*I)^-1X'_TrainC'))XEquation <ref>[b]PEVMAX2 max[diag(CX_Test(X'_TrainX_Train+λ*I)^-1X Equation <ref>[t]X'_TrainX_Train(X'_TrainX_Train+λ*I)^-1X'_TestC'][b] PEVMEAN mean(diag(CX_Test(X'_TrainX_Train+λ*I)^-1X'_TestC'))X Equation <ref>[b]PEVMEAN0 mean(diag(CX_Train(X'_TrainX_Train+λ*I)^-1X'_TrainC'))X Equation <ref>[b] PEVMEAN2 mean[diag(CX_Test(X'_TrainX_Train+λ*I)^-1X Equation <ref>[t]X'_TrainX_Train(X'_TrainX_Train+λ*I)^-1X'_TestC'][b] PEVMEANMM mean(diag(CZ_test(Z_Train'MZ_Train+λ*Kinv)^-1Z_Test'C')))KEquation <ref>[t] M=I-W(W'W)^-W'[b] A.2 Using initial solutions and implementing an island model using GA. The scenario in the island model is not necessarily fixed, ingenious or advanced. It is for demonstration purposes and the users are encouraged to play around with it or change it completely.Box A1: Loading the wheat data set included in STPGA > data(WheatData) > svdWheat<-svd(Wheat.K, nu=50, nv=50) > PC50WHeat<-Wheat.K > rownames(PC50WHeat)<-rownames(Wheat.K) > DistWheat<-dist(PC50WHeat) > TreeWheat<-cutree(hclust(DistWheat), k=4) > Test<-rownames(PC50WHeat)[TreeWheat==2] > Candidates<-setdiff(rownames(PC50WHeat), Test) > deptest<-Wheat.Y[Wheat.Yid > Ztest<-model.matrix( -1+deptestid) Box A2: Function that implements a simple island model > repeatgenalg<-function(numrepsouter,numrepsinner) + StartingPopulation2=NULL+ for (i in 1:numrepsouter) + StartingPopulation<-lapply(1:numrepsinner, function(x) + GenAlgForSubsetSelectionNoTest(P=PC50WHeat, + ntoselect=50, InitPop=StartingPopulation2, +npop=200, nelite=5, mutprob=.5, mutintensity = rpois(1,1), +niterations=200,minitbefstop=5, tabumemsize = 0, tabu=FALSE, +plotiters=FALSE, lambda=1e-9, +errorstat="CDMEAN", mc.cores=4)) + StartingPopulation2<-vector(mode="list", length = numrepsouter*1) + ij=1 + for (i in 1:numrepsinner) + for (j in 1:1) + StartingPopulation2[[ij]]<-StartingPopulation[[i]][[j]] + ij=ij+1 ++++ ListTrain<-GenAlgForSubsetSelectionNoTest( + P=PC50WHeat[rownames(PC50WHeat) + ntoselect=50, + InitPop=StartingPopulation2,npop=200,+ nelite=10, mutprob=.5, mutintensity = 1,niterations=200, + minitbefstop=50, tabumemsize = 0,tabu=FALSE, plotiters=FALSE, + lambda=1e-9,errorstat="DOPT", mc.cores=4) + return(ListTrain) + Box A3: Function that implements a simple island model > ListTrain<-repeatgenalg(10, 4) > min(ListTrain`Best criterion values over iterarions`)[1] -47.44094> min(Train4`Best criterion values over iterarions`)[1] 5.123539> deptrainopt<-Wheat.Y[(Wheat.Yid > Ztrain<-model.matrix( -1+deptrainoptid) > modelopt<-emmreml(y=deptrainoptplant.height, +X=matrix(1, nrow=nrow(deptrainopt), ncol=1),+ Z=Ztrain, K=Wheat.K) > predictopt<-Ztest > cor(predictopt, deptestplant.height)[,1] [1,] 0.3155026 Box A4: Function that implements a simple island model > TreeWheatTrain<-TreeWheat > TreeWheatTrain[names(TreeWheatTrain) Box A5: Function that implements a simple island model > plot(PC50WHeat[,1],PC50WHeat[,2], col=TreeWheatTrain, +pch=as.character(TreeWheatTrain), xlab="pc1", ylab="pc2")< g r a p h i c s > A.3 A mixed Integer quadratic programming function for proportions Box A6: A mixed Integer quadratic programming function for proportions > require(quadprog) > MIQP<-function(Dmat, dvec, cardinality,npop=200,+ nelite=10, mutprob=.5, mutintensity = 1,niterations=200, + minitbefstop=50, tabumemsize = 1,plotiters=FALSE,tabu=FALSE, + lambda=1e-5, mc.cores=4) + P<-cbind(dvec,Dmat) + rownames(P)<-rownames(Dmat) + STPGAUSERDEFFUNC<-function(Train,Test=NULL, P, lambda=1e-5, C=NULL) + smallD<-P[rownames(P) + smallD=smallD+lambda*diag(nrow(smallD)) + smalld<-P[rownames(P) + n=length(smalld) + sol<- solve.QP(Dmat=smallD, +dvec=smalld,Amat=cbind(rep(1,n),diag(n),-diag(n)), +bvec=rbind(1,matrix(0,ncol=1,nrow=n),matrix(-1,ncol=1,nrow=n)), +meq=1) + names(solsolution)<-rownames(smallD) + return(solvalue) ++ GAOUT<-GenAlgForSubsetSelectionNoTest(P=P, ntoselect=cardinality,npop=npop,+ nelite=nelite, mutprob=mutprob, + mutintensity = mutintensity,niterations=niterations, + minitbefstop=minitbefstop, tabumemsize = tabumemsize, + plotiters=plotiters,tabu=tabu, + lambda=lambda,errorstat="STPGAUSERDEFFUNC", + mc.cores=mc.cores) + smallD<-P[rownames(P) + smallD=smallD+lambda*diag(nrow(smallD)) + smalld<-P[rownames(P) + n=length(smalld) + sol<- solve.QP(Dmat=smallD, dvec=smalld, +Amat=cbind(rep(1,n),diag(n),-diag(n)), +bvec=rbind(1,matrix(0,ncol=1,nrow=n), + matrix(-1,ncol=1,nrow=n)), meq=1) + names(solsolution)<-rownames(smallD) + return(solvalue) +

http://arxiv.org/abs/1702.08088v1

2 Synergistic Team CompositionEwa Andrejczuk Artificial Intelligence Research Institute (IIIA-CSIC) Change Management Tool S.L Barcelona, Spain ewa@iiia.csic.es Juan A. Rodríguez-AguilarArtificial Intelligence Research Institute (IIIA-CSIC) Barcelona, Spain jar@iiia.csic.es Carme Roig Institut Torras i Bages L'Hospitalet de Llobregat, Spain mroig112@xtec.cat Carles Sierra Artificial Intelligence Research Institute (IIIA-CSIC) Barcelona, Spain sierra@iiia.csic.esDecember 30, 2023 ==================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== Effective teams are crucial for organisations, especially in environments that require teams to be constantly created and dismantled, such as software development, scientific experiments, crowd-sourcing, or the classroom. Key factors influencing team performance are competences and personality of team members. Hence, we present a computational model to compose proficient and congenial teams based on individuals' personalities and their competences to perform tasks of different nature. With this purpose, we extend Wilde's post-Jungian method for team composition, which solely employs individuals’ personalities. The aim of this study is to create a model to partition agents into teams that are balanced in competences, personality and gender. Finally, we present some preliminary empirical results that we obtained when analysing student performance. Results show the benefits of a more informed team composition that exploits individuals' competences besides information about their personalities.<ccs2012> <concept> <concept_id>10002951.10003227.10003228.10003232</concept_id> <concept_desc>Information systems Enterprise resource planning</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10002951.10003260.10003282</concept_id> <concept_desc>Information systems Web applications</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10002951.10003260.10003282.10003296</concept_id> <concept_desc>Information systems Crowdsourcing</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10002951.10003260.10003282.10003296.10003297</concept_id> <concept_desc>Information systems Answer ranking</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10003120.10003130.10011762</concept_id> <concept_desc>Human-centered computing Empirical studies in collaborative and social computing</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010147.10010178.10010219.10010220</concept_id> <concept_desc>Computing methodologies Multi-agent systems</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10002950.10003624.10003633.10010918</concept_id> <concept_desc>Mathematics of computing Approximation algorithms</concept_desc> <concept_significance>300</concept_significance> </concept> </ccs2012> [500]Human-centered computing Empirical studies in collaborative and social computing [500]Computing methodologies Multi-agent systems [300]Mathematics of computing Approximation algorithms § INTRODUCTION Some tasks, due to their complexity, cannot be carried out by single individuals. They need the concourse of sets of people composing teams. Teams provide a structure and means of bringing together people with a suitable mix of individual properties (such as competences or personality). This can encourage the exchange of ideas, their creativity, their motivation and job satisfaction and can actually extend individual capabilities. In turn, a suitable team can improve the overall productivity, and the quality of the performed tasks. However, sometimes teams work less effectively than initially expected due to several reasons: a bad balance of their capacities, incorrect team dynamics, lack of communication, or difficult social situations. Team composition is thus a problem that has attracted the interest of research groups all over the world, also in the area of multiagent systems. MAS research has widely acknowledged competences as important for performing tasks of different nature <cit.>. However, the majority of the approaches represent capabilities of agents in a Boolean way (i.e., an agent either has a required skill or not). This is a simplistic way to model an agent's set of capabilities as it ignores any skill degree. In real life, capabilities are not binary since every individual (e.g. human or software) shows different performances for each competence. Additionally, the MAS literature has typically disregarded significant organizational psychology findings (with the exception of several recent, preliminary attempts like <cit.> or <cit.>). Numerous studies in organizational psychology <cit.> underline the importance of personality traits or types for team composition. Other studies have focused on how team members should differ or converge in their characteristics, such as experience, personality, level of skill, or gender, among others <cit.>, in order to increase performance. In this paper, we focus on scenarios where a complex task requires the collaboration of individuals within a team. More precisely, we consider a scenario, where there are multiple instances of the same complex task. The task has a task type and a set of competence requests with competence levels needed to solve the task. We have a pool of human agents characterized by gender, personality, and a set of competences with competence levels.Our goal is to partition agents into teams so that within a task all competence requirements are covered (whenever possible) and team members work well together. That is, each resulting team is both proficient (covers the required competences) and congenial (balances gender and psychological traits). We refer to these teams as synergistic teams. We define the synergistic value of a team as its balance in terms of competence, personality and gender. Each synergistic team works on the very same task. This scenario is present in many real-life settings, for instance a classroom or a crowdsourcing task.With this purpose, we design an algorithm that uses a greedy technique both to match competences with the required ones and at the same time to balance the psychological traits of teams' members. This paper makes the following contributions. To start with, we formalise the synergistic team formation problem as the problem of partitioning a group of individuals into teams with limited size.We provide an approximate local algorithm to solvethe team composition problem. We empirically evaluate the algorithm using real data. Preliminary results show that our algorithm predicts better the performance of teams than the experts that know students' social situation, background and competences. Outline. The remaining of this paper is structured as follows. Section <ref> opens with an overview of the related work. Section <ref> gives the personality background for our model. Section <ref> describes the synergistic team composition problem and Section <ref> presents our algorithm to solve the synergistic team composition problem. Then, Section <ref> presents results of our algorithm in the context of team composition in the classroom. Finally, Section <ref> discusses our approach and future work. § BACKGROUND To the best of our knowledge, <cit.> is the only model that considers both personality and competences while composing teams.There, the influence of personality on different task allocation strategies (minimizing either undercompetence or overcompetence) is studied. Henceforth, this work is the most relevant for us, however there are substantial differences between our work and <cit.>. Firstly, authors do not propose an algorithm to compose teamsbased on both personality and competences. Secondly, gender balance is not considered in their setting. Finally, <cit.> does not provide an evaluation involving real data (only an agent-based simulation is presented).The rest of the literature relevant to this article is divided into two categories as proposed in <cit.>: those that consider agent capacities (individual and social capabilities of agents) and those that deal with agent personality (individual behaviour models).Capacity. The capacity dimension has been exploited by numerous previous works <cit.>. In contrast to our work, where the competences are graded, in the majority of works agents are assumed to have multiple binary skills (i.e., the agent either has a skill or not). For instance, <cit.> use agents' capabilities to compose one k-robust team for a single task. A team is k-robust if removing any k members from the team does not affect the completion of the task. <cit.> uses competences and communication cost in a context where tasks sequentially arrive and teams have to be composed to perform them. Each task requires a specific set of competences and the team composition algorithm is such that the workload per agent is fair across teams. Personality. In the team formation literature, the only two models to our knowledge considering personality to compose teams are <cit.> and <cit.>. <cit.> uses Belbin theory to obtain human predominant roles (we discuss this method in Section <ref>).Additionally, the gender is not taken into account while composing heterogeneous teams, which we believe may be important for team congeniality. Regarding <cit.>, Farhangian et al. use the classical MBTI personality test (this method is discussed in Section <ref>). They look for the best possible team built around a selected leader. In other words, the best team for a particular task is composed. Gender balance is not considered in this setting. Finally, although <cit.>'s team composition considered real data, the resulting teams' performance was not validated in any real setting (Bayesian theory was used to predict the probability of success in various team composition conditions). § PERSONALITY In this section, we discuss the most prominent approaches to measure human personality and we explain the details of the method we have decided to examine.Personality determines people's behaviour, cognition and emotion. Different personality theorists present their own definitions of personality and different ways to measure it based on their theoretical positions.The most popular approach is to determine personality through a set of questions. There have been several simplified schemes developed over the years to profile human personality. The most populars are: * the Five Factor Model (aka FFM or “Big Five”), which uses five broad dimensions to describe human personality <cit.>;* Belbin theory <cit.>, which provides a theory on how different role types influence teamwork; and * the Myers-Briggs Type Indicator (MBTI) scheme designed to indicate psychological preferences in how people perceive the world and make decisions <cit.>. According to <cit.>, FFM personality instruments fail to detect significant sex differences in personality structures. It is also argued that the Big Five dimensions are too broad and heterogeneous, and lack the specificity to make accurate predictions in many real-life settings <cit.>. Regarding Belbin theory, the results of previous studies considering the correlation between team composition and team performance are ambiguous. Even though some research shows weak support or does not show support for this theory at all <cit.>, it remains popular.Finally, the MBTI measure consists of four dimensions on a binary scale (e.g. either the person is Extrovert or Introvert). Within this approach, every person falls into one of the sixteen possible combinations of the four letter codes, one letter representing one dimension. This approach is easy to interpret by non-psychologists, though reliance on dichotomous preference scores rather than continuous scores excessively restricts the level of statistical analysis <cit.>.Having considered the arguments above,we have decided to explore a novel method: the Post-Jungian Personality Theory, which is a modified version of the Myers-Briggs Type Indicator (MBTI) <cit.>, the “Step II” version of Quenk, Hammer and Majors <cit.>. The questionnaire to determine personality is short, contains only 20 quick questions (compared to the 93 MBTI questions). This is very convenient for both experts wanting to design teams and individuals doing the test since completing the test takes just a few minutes (for details of the questionnaire, see <cit.>). Douglass J. Wilde claims that it covers the same psychological territory as MBTI <cit.>. In contrast to the MBTI measure, which consists of four binary dimensions, the Post-Jungian Personality Theory uses the numerical data collected using the questionnaire <cit.>. The results of this method seem promising, sincewithin a decade this novel approach has tripled the fraction of Stanford teams awarded national prizes by the Lincoln Foundation <cit.>.The test is based on the pioneering psychiatrist Carl Gustav Jung's cognitive-mode personality model <cit.>. It has two sets of variable pairs called psychological functions:* Sensing / Intuition (SN) — describes the way of approaching problems* Thinking / Feeling (TF) — describes the way of making decisions and two sets of psychological attitudes: * Perception / Judgment (PJ) — describes the way of living* Extroversion / Introversion (EI) — describes the way of interacting with the world For instance, for the Feeling-Thinking (TF) dimension, a value between -1 and 0 means that a person is of the feeling type, and a value between 0 and 1 means she is of the thinking type. Psychological functions and psychological attitudes compose together a personality. Every dimension of a personality (EI, SN, TF, PJ) is tested by five multiple choice true/false questions. § TEAM COMPOSITION MODEL In this section we introduce and formalise our team composition problem. First, section <ref> introduces the basic notions of agent, personality, competence, and team, upon which we formalise our problem. Next, we formalise the notion of task assignment for a single team and a single task, and we characterise different types of assignments. Sections <ref> and <ref> show how to evaluate the proficiency and congeniality degrees of a team. Based on these measures, in section <ref> we formalise the synergistic team composition problem. §.§ Basic definitions In our model, we consider that each agent is a human. We characterise each agent by the following properties: * A unique identifier that distinguishes an agent from others (e.g. ID card number, passport number, employee ID, or student ID).* Gender. Human agents are either a man or a woman.* A personality represented by four personality traits. Each personality trait is a number between -1 and 1.* A set of competences. A competence integrates knowledge, skills, personal values, and attitudes that enable an agent to act correctly in a job, task or situation <cit.>. Each agent is assumed to possess a set of competences with associated competence levels. This set may vary over time as an agent evolves. Next, we formalise the above-introduced concepts.A personality profile is a vector ⟨ sn, 𝑡𝑓, ei, pj ⟩∈ [-1, 1]^4, where each sn, 𝑡𝑓, ei, pj represents one personality trait. We denote by C = {c_1, … , c_m} the whole set of competences, where each element c_i ∈ C stands for a competence. A human agent is represented as a tuple ⟨ id, g,p, l ⟩ such that: * id is the agent's identifier;* g ∈{man, 𝑤oman} stands for their gender;* p is a personality profile vector ⟨ sn, 𝑡𝑓, ei, pj ⟩∈ [-1, 1]^4;* l: C →[0,1]is a function that assigns the probability that the agent will successfully show competence c. We will refer to l(c) as the competence level of the agent for competence c. We assume that when an agent does not have a competence (or we do not know about it), the level of this competence is zero. Henceforth, we will note the set of agents as A ={a_1,…,a_n}. Moreover, We will use super-indexes to refer to agents' components. For instance, given an agent a ∈ A, id^a will refer to the id component of agent a. We will employ matrix L ∈ [0,1]^n × m to represent the competence levels for each agent and each competence. [Team] A team is any non-empty subset of A with at least two agents. We denote by K_A = (2^A ∖{∅})∖{{a_i}| a_i ∈ A} the set of all possible teams in A. We assume that agents in teams coordinate their activities for mutual benefit.§.§ The task assignment problem In this section we focus on how to assign a team to a task.A task type determines the competence levels required for the task as well as the importance of each competence with respect to the others. For instance, some tasks may require a high level of creativity because they were never performed before (so there are no qualified agents in this matter). Others may require a highly skilled team with a high degree of coordination and teamwork (as it is the case for rescue teams). Therefore, we define a task type as:A task type τ is defined as a tuple ⟨λ, μ, {(c_i,l_i, w_i)}_i ∈ I_τ⟩ such that: * λ∈ [0,1] importance given to proficiency;* μ∈ [-1,1] importance given to congeniality;* c_i∈ C is a competence required to perform the task;* l_i∈ [0,1] is the required competence level for competence c_i; * w_i∈ [0,1] is the importance of competence c_i for the success of task of type τ; and* ∑_i ∈ I_τ w_i = 1. We will discuss the meaning of λ and μfurther ahead when defining synergistic team composition (see subsection <ref>). Then, we define a task as: A task t is a tuple ⟨τ, m ⟩ such that τ is a task type and m is the required number of agents, where m≥ 2. Henceforth, we denote by T the set of tasks and by 𝒯 the set of task types. Moreover, we will note as C_τ ={c_i | i ∈ I_τ} the set of competences required by task type τ.Given a team and a task type, we must consider how to assign competences to team members (agents). Our first, weak notion of task assignment only considers that all competences in a task type are assigned to some agent(s) in the team:Given a task type τ and a team K ∈K_A, an assignment is a function η: K → 2^C_τ satisfying thatC_τ⊆⋃_a ∈ Kη(a).§.§ Evaluating team proficiencyGiven a task assignment for a team, next we will measure the degree of competence of the team as a whole. This measure will combine both the degree of under-competence and the degree of over-competence, which we formally define first. Before that, we must formally identify the agents that are assigned to each competence as follows.Given a task type τ, a team K, and an assignment η, the set δ(c_i) = {a ∈ K | c_i∈η(a)} stands for the agents assigned to cover competence c_i.Now we are ready to define the degrees of undercompentence and overcompetence.[Degree of undercompentence]* Given a task type τ, a team K, and an assignment η, we define the degree of undercompetence of the team for the task as:u(η)= ∑_i ∈ I_τ w_i·∑_a ∈δ(c_i) |min(l^a(c_i) - l_i,0)|/|{a ∈δ(c_i)|l^a(c_i)-l_i < 0}| [Degree of overcompetence]* Given a task type τ, a team K, and an assignment η, we define the degree of overcompetence of the team for the task as:o(η)= ∑_i ∈ I_τ w_i ·∑_a ∈δ(c_i)max(l^a(c_i) - l_i,0)/|{a ∈δ(c_i)|l^a(c_i)-l_i > 0}| Given a task assignment for a team, we can calculate its competence degree to perform the task by combining its overcompetence and undercompetence as follows.Given a task type τ, a team K and an assignment η, the competence degree of the team to perform the task is defined as:u_𝑝𝑟𝑜𝑓(η) = 1-(υ· u(η)+(1-υ) · o(η))where υ∈ [0,1] is the penalty given to the undercompetence of team K. Notice that the larger the value of υ the higher the importance of the competence degree of team K, while the lower the value υ, the less important its undercompetence. The intuition here is that we might want to penalize more the undercompetency of teams, as some tasks strictly require teams to be at least as competent as defined in the task type.For any η,u(η) + o(η) ∈ [0,1]. Given that (1) l^a(c_i) ∈ [0,1] and l_i∈ [0,1]; (2) If min(l^a(c_i) - l_i,0)<0 then max(l^a(c_i) -l_i,0) = 0; and (3) If max(l^a(c_i)-l_i,0) > 0 then min(l^a(c_i) - l_i,0)=0. Thus, from (1–3)we have |min(l^a(c_i) - l_i,0)| + max(l^a(c_i)-l_i,0) ∈ [0,1]. Let n=|{a ∈δ(c_i)|l^a(c_i)-l_i > 0}|, then obviously it holds that n · (|min(l^a(c_i) - l_i,0)| + max(l^a(c_i)-l_i,0))/n∈ [0,1] and as |δ(c_i)| ≤ n then ∑_a ∈δ(c_i)(|min(l^a(c_i) - l_i,0)| + max(l^a(c_i)-l_i,0))/n∈ [0,1]holds; andsince ∑_i ∈ I_τ w_i = 1 then ∑_i ∈ I_τ w_i ·∑_a ∈δ(c_i)(|min(l^a(c_i) - l_i,0)| + max(l^a(c_i)-l_i,0))/n∈ [0,1]; Finally, distributing, this equation is equivalent to: ∑_i ∈ I_τ w_i ∑_a ∈δ(c_i)(|min(l^a(c_i) - l_i,0)|/n+ ∑_i ∈ I_τ w_i ∑_a ∈δ(c_i)(max(l^a(c_i)-l_i,0))/n∈ [0,1] which in turn is equivalent to u(η) + o(η) ∈ [0,1].Function u_𝑝𝑟𝑜𝑓 is used to measure how proficient a team is for a given task assignment. However, counting on the required competences to perform a task does not guarantee that the team will succeed at performing it. Therefore, in the next subsection we present an evaluation function to measure congeniality within teams. Unlike our measure for proficiency, which is based on considering a particular task assignment, our congeniality measure will solely rely on the personalities and genders of the members of a team.§.§ Evaluating team congenialityInspired by the experiments of Douglass J. Wilde <cit.> we will define the team utility function for congeniality u_con(K), such that: * it values more teams whose SN and TF personality dimensions are as diverse as possible;* it prefers teams with at least one agent with positive EI and TF dimensions and negative PJ dimension, namely an extrovert, thinking and judging agent (called ETJ personality),* it values more teams with at least one introvert agent;* it values gender balance in a team.Therefore, the higher the value of function u_con(K), the more diverse the team is.Formally, this team utility function is defined as follows:u_con(K) =σ_SN(K) ·σ_TF(K) +max_a_i ∈ K((0,α, α, α) · p_i, 0) + max_a_i ∈ K((0,0,-β,0) · p_i, 0) + γ·sin(π· g(K))where the different parameters are explained next.* σ_SN(K) and σ_TF(K): These variances are computed over the SN and TF personality dimensions of the members of team K. Since we want to maximise u_con, we want these variances to be as large as possible. The larger the values of σ_SN and σ_TF the larger their product will be, and hence the larger team diversity too. * α:The maximum variance of any distribution over an interval [a,b] corresponds to a distribution with the elements evenly situated at the extremes of the interval. The variance will always be σ^2 ≤ ((b-a)/2)^2. In our case with b=1 and a=-1 we have σ≤ 1. Then, to make the four factors equally important and given that the maximum value for p_i (the personality profile vector of agent a_i) would be (1, 1, 1, 1) a maximum value for α would be 3 α = ((1-(-1))/2)^2 = 1, as we have the factor σ_SN·σ_TF, so α≤ 0.33(3). For values situated in the middle of the interval the variance will be σ^2 ≤(b-a)^2/12, hence a reasonable value for α would be α = √((1-(-1))^2)/12)/3 = 0.19* β: A similar reasoning shows that β≤ 1.* γ is a parameter to weigh the importance of a gender balance and g(K) = w(K)/w(K) + m(K). Notice that for a perfectly gender balanced team with w(K) = m(K) we have that sin(π· g(K)) = 1. The higher the value of γ, the more important is that team u_con is gender balanced. Similarly to reasoning about α and β, we assess γ≤ 1. In order to make this factor less important than the others in the equation we experimentally assessed that γ = 0.1 is a good compromise.In summary, we will use a utility function u_con such that: α = σ_SN(K) ·σ_TF(SK)/3, β = 3 ·α and γ = 0.1.§.§ Evaluating synergistic teams Depending on the task type, different importance for congeniality and proficiency should be given. For instance, creative tasks require a high level of communication and exchange of ideas, and hence, teams require a certain level of congeniality. While, repetitive tasks require good proficiency and less communication. The importance of proficiency (λ) and congeniality (μ) is therefore a fundamental aspect of the task type. Now, given a team, we can combine its competence value(in equation <ref>) with its congeniality value (in equation <ref>) to measure its synergistic value. Given a team K, a task type τ = ⟨λ, μ, {(c_i,l_i, w_i)}_i ∈ I_τ⟩ and a task assignment η: K → 2^C_τ, the synergistic value of team K is defined as:s(K,η) = λ· u_𝑝𝑟𝑜𝑓(η) + μ· u_con(K)where λ∈ [0,1] is the grade to which the proficiency of team K is important, and μ∈ [-1,1] is the grade to which the task requires diverse personalities. Figure <ref> shows the relation between the parameters λ and μ.In general, the higher the λ, the higher importance is given to the proficiency of a team. The higher the μ the more important is personality diversity. Notice, that the μ can be lower than zero. Having μ negative, we impose that the congeniality value will be as low as possible (to maximize s(K,η)) and so, team homogeneity is preferred. This situation may happen while performing tasks in unconventional performance environments that have serious consequences associated with failure. In order to quickly resolve issues, a team needs to be proficient and have team-mates who understand one another with minimum communication cost (which is associated to homogeneity of a team).§.§ The synergistic team composition problemIn what follows we consider that there are multiple instances of the same task to perform. Given a set of agents A, our goal is to split them into teams so that each team, and the whole partition of agents into teams, is balanced in terms of competences, personality and gender.We shall refer to these balanced teams as synergistic teams, meaning that they are both congenial and proficient. Therefore, we can regard our team composition problem as a particular type of set partition problem. We will refer to any partition of A as a team partition. However, we are interested in a particular type of team partitions, namely those where teams are constrained by size m as follows. Given a set of agents A, we say that a team partition P_m of A is constrained by size m iff: (i) for every team K_i ∈ P_m, K_i ∈K_A, max(m-1, 2) ≤ |K| ≤ m+1 holds; and (ii) for every pair of teams K_i, K_j ∈ P_m ||K_i| - |K_j|| ≤ 1.As |K| / m is not necessarily a natural number, we may need to allow for some flexibility in team size within a partition. This is why we introduced above the condition max(m-1, 2) ≤ |K| ≤ m+1. In practical terms, in a partition we may have teams differing by one agent. We note by P_m(A) the set of all team partitions of A constrained by size m. Henceforth, we will focus on team partitions constrained by some size. Since our goal is to find the most competence-balanced and psychologically-balanced team partition, we need a way to measure the synergistic value of a team partition, which we define as follows:Given a task t = ⟨τ, m ⟩, a team partition P_m and an assignment η_i for each team K_i ∈ P_m, the synergistic value of P_m is computed by:u(P_m,η) = ∏_i =1^|P_m| s(K_i,η_i)where η stands for the vector of task assignments η_1,…, η_|P_m|. Notice that the use of a Bernoulli-Nash function over the synergistic values of teams will favour team partitions whose synergistic values are balanced. Now we are ready to cast the synergistic team composition problem as the following optimisation problem: Given task t = ⟨τ, m ⟩ and set of agents A the synergistic team formation problem (STFP) is the problem of finding a team partition constrained by size m, together with competence assignment for its teams, whose synergistic value is maximal. Formally, the STFP is the problem of finding the partition in P ∈𝒫_m(A) and the task assignments η for the teams in P_m that maximisesu(P_m,η).§ SOLVING STFPIn this section we detail an algorithm, the so-called , which solves the synergistic team formation problem described above. We will start from describing how to split agents into a partition (see subsection <ref>). Next, we will move on to the problem of assigning competences in a task to team members (see subsection <ref>), so that the utility of synergistic function is maximal. Finally, we will explainthat is a greedy algorithm that quickly finds a first, local solution, to subsequently improve it, hoping to reach a global optimum. §.§ How do we split agents?We note by n = |A| the number of agents in A, by m ∈ℕ the target number of agents in each team, and by b the minimum total number of teams, b = ⌊n/m⌋. We define the quantity distribution of agents in teams of a partition, noted T: ℕ×ℕ→ℕ×ℕ∪ (ℕ×ℕ)^2 as: T(n,m) ={(b, m)} if n ≥ mandnm= 0{(nm,m + 1),(b - (nm),m)} if n ≥ mandnm≤ b{(b, m),(1, nm)} if n ≥ mandnm > b{(0,m)} otherwise Note that depending on the cardinality of A and the desired team size, the number of agents in each team may vary by one individual (for instance if there are n=7agents in A and we want to compose duets (m=2), we split agents into two duets and one triplet). §.§ Solving an AssignmentThere are different methods to build an assignment. We have decided to solve our assignment problem by using the minimum cost flow model <cit.>. This is one of the most fundamental problems within network flow theory and it can be efficiently solved. For instance, in <cit.>, it was proven that the minimum cost flow problem can be solved in O(m · log(n) · (m + n · log(n))) time with n nodes and m arcs.Our problem is as follows:There are a number of agents in team K and a number of competence requests in task t. Any agent can be assigned to any competence, incurring some cost that varies depending on the agent competence level of the assigned competence. We want to get each competence assigned to at least one agent and each agent assigned to at least one competence in such a way that the total cost (that is both undercompetence and overcompetence) of the assignment is minimal with respect to all such assignments. Formally, let G = (N, E) be a directed network defined by a set N of n nodes and a set E of e directed arcs. There are four types of nodes: (1) one source node; (2) |K| nodes that represent agents in team K; (3) |C_τ| competence requests that form task type τ; and (4) one sink node. Each arc (i, j) ∈ E has an associated cost p_ij∈ℝ^+ that denotes the cost per unit flow on that arc.We also associate with each arc (i, j) ∈ E a capacity u_ij∈ℝ^+ that denotes the maximum amount that can flow on the arc. In particular, we have three kinds of edges: (1) Supply arcs. These edges connect the source to agent nodes. Each of these arcs has zero cost and a positive capacity u_ij which define how many competences at most can be assigned to each agent. (2) Transportation arcs. These are used to ship supplies. Every transportation edge (i, j) ∈ E is associated with a shipment cost p_ij that is equal to:p_ij =(l^a_i(c_𝑗) - l_𝑗) · (1-υ) · w_𝑗 if l^a_i(c_𝑗 - l_𝑗) > 0-(l^a_i(c_𝑗) - l_𝑗) ·υ· w_𝑗 if l^a_i(c_𝑗 - l_𝑗) < 0where v ∈ [0,1] is the penalty given to the undercompetence of team K(see subsection <ref> for the definition).(3) Demand arcs. These arcs connect the competence requests nodes to the sink node. These arcs have zero costs and positive capacities u_ij which equal the demand for each competence. Thus, a network is denoted by (G, w, u, b). We associate with each node i ∈ N an integer number b(i) representing its supply. If b(n) > 0 then n is a source node, if b(n) < 0 then n is a sink node. In order to solve a task assignment problem, we use the implementation of <cit.> provided in the ort-tools.[<https://github.com/google/or-tools/blob/master/src/graph/min_cost_flow.h>] Example Let us consider a team of three agents K = {a_1, a_2, a_3}: * a_1 = ⟨ id_1, `woman', p_1, [l(c_1) = 0.9, l(c_2) = 0.5]⟩* a_2 = ⟨ id_2, `man', p_2, [l(c_2) = 0.2, l(c_3) = 0.8]⟩* a_3 = ⟨ id_3, `man', p_3, [l(c_2) = 0.4, l(c_4) = 0.6]⟩and task type τ containing four competence requests {(c_1,0.8, 0.25), (c_2, 0.6, 0.25), (c_3,0.6, 0.25),(c_4,0.6, 0.25)}. The penalty given to undercompetence is equal to υ=0.6.Our goal is to assign agents to competence requests, so that: (1) every agent is responsible for at least one competence, (2) every competence is covered by at least one agent, (3) the overall “cost” in minimal.As shown in figure <ref>, we build a graph out of n = 9 nodes that is: one source node (N_0), three agents nodes (N_1 - N_3), four competences nodes (N_4 - N_7) and a sink node (N_8). Next, we add edges: (1) between source node N_0 and all agent nodes N_1 - N_3 that have a cost p_si = 0 and capacity u_si = 2 for all i as the maximum number of competences assigned to one agent cannot be bigger than two if we want to make sure that all agents are assigned to at least one competence; (2) between agent nodes N_1 - N_3 and competence nodes (N_4 - N_7), where each capacity u_ij = 1 and we calculate costs according to the equation <ref>. For instance, the cost between N_1 and N_4 is equal to: (0.9 - 0.8) · (1-0.6) · 0.25 = 0.01. We multiply all costs by 1000 to meet the requirements of the solver (edges need to be integer). Hence, the final cost p_14=10; (3) edges between competence nodes N_4 - N_7 and sink node N_8 that have costs p_jw = 0 and capacities u_jw = 1 to impose that each is assigned. Once the graph is built, we pass it to the solver to get the assignment, and we get c_1 and c_2 assigned to a_1, c_3 assigned to a_2 and c_4 assigned to a_3. §.§ SynTeam algorithm Algorithm <ref> shows the SynTeam pseudocode. Algorithm <ref> is divided into two parts:1. Find a first team partition. This part of the algorithm simply builds a partition by randomly assigning agents to teams of particular team sizes. This part goes as follows. Given a list of agents A, we start by shuffling the list so that the order of agents in the list is random (line 1). Next, we determine the quantitative distribution of individuals among teams of size m using function T(|A|,m) as defined in section <ref> (line 2). We start from the top of the shuffled list of agents (line 3). For each number of teams (line 4), we define a temporary set team to store a current team (line 5). We add to team subsequent size agents from the shuffled list of agents (line 7). We add the newly created team to the team partition P_𝑏𝑒𝑠𝑡 that we intend to build (line 10). When reaching line 14, P_𝑏𝑒𝑠𝑡 will contain a first disjoint subset of teams (a team partition). 2. Improve the current best team partition. The second part of the algorithm consists in improving the current best team partition. The idea is to obtain a better team partition by performing crossovers of two randomly selected teams to yield two better teams. In this part, we took inspiration from simulated annealing methods, where the algorithm might accept swaps that actually decrease the solution quality with a certain probability. The probability of accepting worse solutions slowly decreases as the algorithm explores the solution space (as the number of iterations increases). The annealing schedule is defined by the 𝑐𝑜𝑜𝑙𝑖𝑛𝑔_𝑟𝑎𝑡𝑒 parameter. We have modified this method to store the partition with the highest synergistic evaluation found so far. In detail, the second part works as follows. First, we select two random teams, K_1 and K_2, in the current team partition (line 15). Then we compute all team partitions of size m with agents in K_1 ∪ K_2 (line 19), and we select the best candidate team partition, named P_𝑏𝑒𝑠𝑡𝐶𝑎𝑛𝑑𝑖𝑑𝑎𝑡𝑒 (lines 19 to 26). If the best candidate synergistic utility is larger than the utility contribution of K_1 and K_2 to the current best partition P_𝑏𝑒𝑠𝑡 (line 27), then we replace teams K_1 and K_2 by the teams in the best candidate team partition (line 28). If the best candidate team partition utility is lower , then we check if the probability of accepting a worse solution is higher than a uniformly sampled value from [0,1] (line 29).If so,we replace teams K_1 and K_2 by the teams in the best candidate team partition (line 30) and we lower heat by a cooling rate. This part of the algorithm continues until the value of heat reaches 1 (line 13). We also store the best partition found so far (line 34) to make sure we do not end up with worse solution. Finally, we return found best partition P_𝑏𝑒𝑠𝑡𝐸𝑣𝑒𝑟 as well as the assignment η for each team.§ EXPERIMENTAL RESULTS§.§ Experimental Setting“Institut Torras i Bages” is a state school near Barcelona. Collaborative work has been implemented there for the last 5 years in their final assignment (“Treball de Síntesi”) with a steady and significant increase in the scores and quality of the final product that students are asked to deliver. This assignment takes one week and is designed to check if students have achieved, and to what extent, the objectives set in the various curricular areas. It is a work that encourages teamwork, research, and tests relationships with the environment. Students work in teams and at the end of every activity present their work in front of a panel of teachers that assess the content, presentation and cooperation between team members. This is a creative task, although requiring high level of competences. §.§ Data Collection In current school practice, teachers group students according to their own, manual method based on the knowledge about students, their competences, background and social situation. This year we have used our grouping system based only on personality ( with λ = 0, μ = 1) upon two groups of students: `3r ESO A' (24 students), and `3r ESO C' (24 students). Using computers and/or mobile phones, students answered the questionnaire (described in section <ref>) which allowed us to divide them into teams of size three for each class. Tutors have evaluated each team in each partition giving an integer value v ∈ [1,10] meaning their expectation of the performance of each team.Each student team was asked to undertake the set of interdisciplinary activities (“Treball de Síntesi”) described above. We have collected each student's final mark for “Treball de Síntesi” as well as final marks obtained for all subjects. That is:Catalan, Spanish, English, Nature, Physics and Chemistry, Social Science, Math, Physical Education, Plastic Arts, Technology. We have used a matrix provided by the tutors to relate each subject to different kinds of intelligence (that in education are understood as competences) needed for this subject. There are eight types of human intelligence <cit.>, each representing different ways of processing information: Naturalist, Interpersonal, Logical/Mathematical, Visual/Spatial, Body/Kinaesthetic, Musical, Intrapersonal and Verbal/Linguistic. This matrix for each subject and each intelligence is shown in figure <ref>.Subjects are represented by rows and intelligences by columns of the matrix in the order as provided above. Based on this matrix we calculate values of intelligences for every student by averaging all values obtained by her that are relevant for this intelligence. For instance, for Body/Kinaesthetic intelligence, we calculate an average of student marks obtained in Nature, Physical Education, Plastic Arts and Technology. An alternative way to measure students' competences level can be by calculating the collective assessments of each competence (like proposed by <cit.>).Finally, having competences (Intelligences), personality and actual performance of all students, we are able to calculate synergistic values for each team. We also calculate the average of marks obtained by every student in a team to get teams' performance values. §.§ ResultsGiven several team composition methods, we are interested in comparing them to know which method better predicts team performance. Hence, we generate several team rankings using the evaluation values obtained through different methods. First, we generate a ranking based on actual team performance that will be our base to compare other rankings. Second, we generate a ranking based on the expert evaluations. Finally,we generate several rankings based on calculated synergistic values with varying importance of congeniality and proficiency. Since “Traball de Síntesi” is a creative task, we want to examine the evaluation function with parameters μ > 0 and λ = 1-μ. In particular, we want to observe how the rankings change when increasing the importance of competences.Notice that teacher and actual performance rankings may include ties since the pool of possible marks is discrete (which is highly improbable in case ofrankings). Therefore, before generating rankings based on synergistic values, we round them up to two digits to discretize the evaluation space. An ordering with ties is also known as a partial ranking. Next, we compare teacher andrankings with the actual performance ranking using the standardized Kendall Tau distance. For implementation details, refer to the work by Fagin et al. <cit.>, which also provide sound mathematical principles to compare partial rankings. The results of the comparison are shown in Figure <ref>. Notice that the lower the value of Kendall Tau, the more similar the rankings. We observe that theranking improves as the importance of competences increases, and it is best at predicting students' performance for λ = 0.8 and μ = 0.2 (Kendall Tau equal to 0.15).A standardised Kendall Tau distance for teacher ranking is equal to 0.28, which shows thatpredicts the performance better than teachers, when competences are included (λ > 0.2). We also calculate the values of Kendall Tau for random (0.42) and reversed (0.9) rankings to benchmark teacher andgrouping methods. The results show that both teachers andare better at predicting students' performance than the random method. § DISCUSSION In this paper we introduced , an algorithm for partitioning groups of humans into competent, gender and psychologically balanced teams. To our knowledge,is the first computational model to build synergistic teams that not only work well together, butare also competent enough to perform an assignment requiring particular expertise. We have decided to evaluate our algorithm in the context of a classroom. Besides obvious advantages of observing students work in person, this scenario gave us an opportunity to compare our results with real-life, currently used practice. The results show thatis able to predict team performance better that the experts that know the students, their social background, competences, and cognitive capabilities. The algorithm is potentially useful for any organisation that faces the need to optimise their problem solving teams (e.g. a classroom, a company, a research unit). The algorithm composes teams in a purely automatic way without consulting experts, which is a huge advantage for environments where there is a lack of experts.Regarding future work, We would like to investigate how to determine quality guarantees of the algorithm. 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http://arxiv.org/abs/1702.08222v1

We study abstract elementary classes (AECs) that, in ℵ_0, have amalgamation, joint embedding, no maximal models and are stable (in terms of the number of orbital types). Assuming a locality property for types, we prove that such classes exhibit superstable-like behavior at ℵ_0. More precisely, there is a superlimit model of cardinality ℵ_0 and the class generated by this superlimit has a type-full good ℵ_0-frame (a local notion of nonforking independence) and a superlimit model of cardinality ℵ_1. We also give a supersimplicity condition under which the locality hypothesis follows from the rest. Irreducible Convex Paving for Decomposition of Multi-dimensional Martingale Transport PlansHadrien De Marchhadrien.de-march@polytechnique.org. Nizar Touzinizar.touzi@polytechnique.edu. Accepted ???. Received ???; in original form December 30, 2023 =================================================================================================§ INTRODUCTION §.§ MotivationIn <cit.> (a revised version of which appears as <cit.>, from which we cite), the first author introduced abstract elementary classes (AECs): a semantic framework generalizing first-order model theory and also encompassing logics such as _ω_1, ω. He studied _ℵ_0-representable AECs (roughly, AECs which are reducts of a class of models of a first-order theory omitting a countable set of types) and generalized and improved some of his earlier results on _ω_1, ω <cit.> and _ω_1, ω (Q) <cit.>.For example, fix a _ℵ_0-representable AECand assume that it is categorical in ℵ_0. Assuming 2^ℵ_0 < 2^ℵ_1 and 1 ≤ (, ℵ_1) < 2^ℵ_1, the first author shows (without even assuming _ℵ_0-representability) <cit.> thathas amalgamation in ℵ_0. Further, <cit.>, it has a lot of structure in ℵ_0 and assuming more set-theoretic assumptions as well as few models in ℵ_2,has a superlimit model in ℵ_1 <cit.>. This means roughly (see Section <ref>) that there is a saturated model in ℵ_1 and that the union of an increasing chain of type ω consisting of saturated models of cardinality ℵ_1 is saturated.The reader can think of the existence of a superlimit in ℵ_1 as a step toward showing that the models of cardinality ℵ_1 behave in a “superstable-like” way. Indeed several recent works <cit.> have connected superlimits with other definitions of superstability in AECs, including uniqueness of limit models and local character of orbital splitting.Another notable consequence of the existence of a superlimit in ℵ_1 is that it implies that there is a model of cardinality ℵ_2. This ties back to a result of the first author: <cit.>: for a _ℵ_0 AEC, categoricity in ℵ_0 and ℵ_1 implies the existence of a model in ℵ_2. The argument first establishes, using only categoricity in ℵ_0 and few models in ℵ_1, that there is a pair M, N of models in _ℵ_1 such that MN and then uses, in essence, that (by ℵ_1-categoricity) these models are superlimits. In this context, the very strong hypotheses make it possible to avoid referring to any stability-theoretic notions. Still, in more complicated frameworks the existence of a superlimit model in ℵ_1 can be thought of as a key conceptual step toward proving existence of models in higher cardinality and more generally developing a stability theory cardinal by cardinal.The arguments for the results from <cit.> discussed in the second paragraph of this introduction are complicated by the lack of ℵ_0-stability: one can only get that there are ℵ_1-many orbital types over countable models. The workaround there is to redefine the ordering (but not the class of models) to obtain a stable class, see <cit.>. If the AEC is “nicely-presented”, e.g. a class of models of an _ω_1,ω-sentence or more generally a finitary AEC <cit.>, then this difficulty does not occur (see <cit.>): ℵ_0-stability follows from few models in ℵ_1 and 2^ℵ_0 < 2^ℵ_1. One can also obtain ℵ_0-stability by starting with only countably-many models in ℵ_1 <cit.>. Finally, it is worth noting that (assuming amalgamation and joint embedding in ℵ_0), ℵ_0-stability is upward absolute for _ℵ_0-AECs <cit.>. §.§ Main results The bottom line is that ℵ_0-stability holds in several cases of interest. In fact, there are no known examples which (under 2^ℵ_0 < 2^ℵ_1) are categorical in ℵ_0, have few models in ℵ_1, and are not ℵ_0-stable (see <cit.>). Thus in the present paper, we start with stability in ℵ_0 (and often amalgamation and categoricity in ℵ_0). Our goal is to say as much as we can on the structure of the class, in particular to get superstable-like behavior in ℵ_0 and ℵ_1, without assuming a non-ZFC hypothesis or (, ℵ_1) < 2^ℵ_1.One of our first results (Theorem <ref>) is that ℵ_0-stability (together with amalgamation and ℵ_0-categoricity) imply that the classis already _ℵ_0-representable. We also show that the assumption of categoricity in ℵ_0 is not really needed: without assuming it, one can find a superlimit in ℵ_0 and change to the class generated by that superlimit, which will be categorical in ℵ_0. In fact, we prove (Theorem <ref>) that one can characterize brimmed models (also called limit models in the literature) as those that are homogeneous for orbital types. This has as immediate consequence that the brimmed model of cardinality ℵ_0 is superlimit (Corollary <ref>). This last result sheds light on an argument of Lessmann <cit.> and answers a question of Fred Drueck (see footnote 3 on <cit.>), who asked when this equality held. The argument works more generally assuming only density of amalgamation bases, as in <cit.>.For the main result of this paper, we assume that orbital types over countable models are determined by their finite restrictions. The study of statements of the form “orbital types are determined by their small restrictions” was pioneered by Grossberg and VanDieren <cit.>, who called this condition tameness. Hyttinen and Kesälä <cit.> were the first to specifically study orbital types over finite sets and the condition that they determine orbital types over countable models. Following the first author's terminology <cit.>, we call this last condition (<ℵ_0, ℵ_0)-locality (not to be confused with sequence locality <cit.>, which is called locality in <cit.> or <cit.>). This is known to hold for several classes of interest: * Letbe a finitary AEC (see <cit.>; this includes classes of models of _ω_1,ω-sentences) and assume thatis stable in ℵ_0 (finitary AECs have amalgamation and no maximal models by definition). By <cit.>,is (<ℵ_0, ℵ_0)-local. * Finitary AECs are not the only setup where (<ℵ_0, ℵ_0)-locality holds. For example, it is known for quasiminimal pregeometry classes (that may not be finitary <cit.>), see <cit.>, and more generally in the finite U-rank (FUR) classes of Hyttinen and Kangas <cit.> (we thank Will Boney for pointing us to that reference). We prove the following: Letbe an AEC with () = ℵ_0 and countable vocabulary. Assume thatis categorical in ℵ_0,is (<ℵ_0, ℵ_0)-local,has amalgamation and no maximal models in ℵ_0 andis stable in ℵ_0. Then:* (Theorem <ref>) There is a good ℵ_0-frame on _ℵ_0.* (Corollary <ref>) There is a superlimit model of cardinality ℵ_1.The good ℵ_0-frame (or the superlimit in ℵ_1) imply the nontrivial corollary thathas a model of cardinality ℵ_2 <cit.>. This consequence also follows from a theorem of the second author <cit.> (which however does not give a good ℵ_0-frame or a superlimit in ℵ_1). The conclusion that there is a superlimit model in ℵ_1 seems new, even for finitary AECs or FUR classes.It is natural to ask whether the locality hypothesis in Theorem <ref> is really needed[In fact, an earlier version of the present paper asserted that it could be derived from the other hypotheses but the argument contained a mistake.]. In fact we do not even know whether the existence of a good ℵ_0-frame implies (<ℵ_0, ℵ_0)-locality: Letbe an AEC with () = ℵ_0. If there is a good ℵ_0-frame on , is(<ℵ_0, ℵ_0)-local? In Section <ref>, we give a partial answer: any AEC that is ℵ_0-stable, ℵ_0-categorical, and supersimple (in a sense generalizing that of homogeneous model theory <cit.>) is (<ℵ_0,ℵ_0)-local. This generalizes the proof of <cit.>, which shows that quasiminimal pregeometry classes are (<ℵ_0, ℵ_0)-local (see also <cit.>). Supersimple ℵ_0-stable AECs are also much more general than FUR classes. §.§ Notes This paper is organized as follows. Section <ref> gives some background definitions and fixes the notation.Section <ref> is a technical section on good frames (possibly on uncountable models) which sets up the machinery to prove the main theorem (more precisely, to prove a strong symmetry property for nonforking in good frames). Section <ref> works with countable models and shows that ℵ_0-stability implies the existence of a superlimit in ℵ_0. Section <ref> builds the good ℵ_0-frame and proves the main theorem. Finally, Section <ref> studies a sufficient condition to get (<ℵ_0, ℵ_0)-locality.This paper was started while the second author was working on a Ph.D. thesis under the direction of Rami Grossberg at Carnegie Mellon University and he would like to thank Professor Grossberg for his guidance and assistance in his research in general and in this work specifically. We also thank Will Boney, Marcos Mazari Armida, and the referee, for their valuable comments on earlier versions of this paper.Note that at the beginning of several sections, we make global hypotheses assumed throughout the section.§ PRELIMINARIES We assume familiarity with the basics of AECs, as presented for example in <cit.>, or the first three sections of Chapter I together with the first section of Chapter II in <cit.>. We also assume familiarity with good frames (see <cit.> or <cit.>; it would help the reader to have a copy of both available during the reading of Section <ref>). This section mostly fixes the notation that we will use.Given a τ-structure M, we write |M| for its universe and M for its cardinality. We may abuse notation and write e.g a ∈ M instead of a ∈ |M|. We may even write ∈ M instead of ∈<ω|M|.We write = (K, ) for an AEC. We may abuse notation and write M ∈ instead of M ∈ K. For a cardinal λ, we write _λ for the AEC restricted to its models of size λ. As shown in <cit.>, any AEC is uniquely determined by its restriction _≤ ().When we say that M ∈ is an amalgamation base, we mean (as in <cit.>) that it is an amalgamation base in _M, i.e. we do not require that larger models can be amalgamated.For M_0 ∈, we say that M is universal over M_0 if M_0M and for any N ∈ with M_0N, if N≤M_0 +(), there exists f: NM (usually we will require also that M = M_0). We say that M is (λ, δ)-brimmed over M_0 (often also called (λ, δ)-limit e.g. in <cit.>) if δ < λ^+ is a limit ordinal, M_0 ∈_λ, and there exists an increasing continuous chain N_i : i ≤δ of members of _λ such that N_0 is universal over M_0, N_δ = M, and N_i + 1 is universal over N_i for all i < δ. We say that M is brimmed over M_0 if it is (M, δ)-brimmed over M_0 for some limit δ < M^+. We say that M is brimmed if it is brimmed over some M_0.The following key concept appears in <cit.>: We say that M ∈ is superlimit if, letting λ := M, we have that λ≥ (), M is universal in _λ (i.e. any M' ∈_λ embeds into M), M is not maximal, and whenever δ < λ^+ is limit, M_i : i < δ is increasing with M_i ≅ M for all i < δ, then ⋃_i < δ M_i ≅ M. The following notion of types already appears in <cit.>. It is called Galois types by many, but we prefer the term orbital types here. They are the same types that are defined in <cit.>, but we also define them over sets. As pointed out in <cit.>, this causes no additional difficulties. The following technical point is important: when the AEC does not have amalgamation, we may want to compute orbital types only in the subclass of amalgamation bases in(as in <cit.>). Thus we allow orbital types to be computed in a subclass ofin the definition. Fix an AECand a subclass ^∗ of , closed under isomorphisms.* We say (A, N_1, _1) ^^∗ (A, N_2, _2) if: * For ℓ = 1,2, N_ℓ∈^∗, A ⊆ |N_ℓ|, and _ℓ∈<∞|N_ℓ|.* There exists N ∈^∗ and f_ℓ : N_ℓ N, ℓ = 1,2, such that f_1 (_1) = _2. * ^^∗ is a reflexive and symmetric relation.Let E^^∗ be its transitive closure.* Let _^∗ ( / A; N) be the E^^∗-equivalence class of (, A, N).*Define ^∗ (A, N), ^∗ (M), ^∗^<ω (M), ^∗^<ω (∅), etc. for the Stone spaces of orbital types, computed inside ^∗. This is defined as in <cit.>. For example, ^∗^<ω (∅) = {_^∗ ( / ∅; N) | N ∈^∗, ∈<ωN}. * When ^∗ =, we omit it.Let us say that an AECis stable in λ if for any M ∈_λ, | (M)| ≤λ. This makes sense in any AEC, and is quite well-behaved assuming amalgamation and no maximal models (since then it is known that one can build universal extensions). We will often work in the following axiomatic setup, a slight weakening where full amalgamation is not assumed. This comes from the context derived in <cit.>: Letbe an AEC and let λ be a cardinal. We say thatis nicely stable in λ (or nicely λ-stable) if:* () ≤λ.* _λ≠∅.*has joint embedding in λ.* Density of amalgamation bases: For any M ∈_λ, there exists N ∈_λ such that MN and N is an amalgamation base (in _λ).* Existence of universal extensions: For any amalgamation base M ∈_λ, there exists an amalgamation base N ∈_λ such that MN and N is universal over M.* Any brimmed model in _λ is an amalgamation base. We say thatis very nicely stable in λ if in addition it has amalgamation in λ.An AECis very nicely stable in λ if and only if () ≤λ, _λ≠∅,is stable in λ, and _λ has amalgamation, joint embedding, and no maximal models. In particular, stability is a consequence of the existence of universal extensions in Definition <ref>. We will repeatedly use the following fact <cit.>.Letbe nicely stable in λ and let M_0, M_1, M_2 ∈_λ. Let δ_1, δ_2 < λ^+ be limit ordinals such that δ_1 = δ_2.* If M_ℓ is (λ, δ_ℓ)-brimmed over M_0, for ℓ = 1,2, then M_1 ≅_M_0 M_2.* If M_ℓ is (λ, δ_ℓ)-brimmed, for ℓ = 1,2, then M_1 ≅ M_2.The first is a straightforward back and forth argument and the second follows from the first using joint embedding. Uniqueness of brimmed models when δ_1≠δ_2 is a much harder property to establish, akin to superstability. See for example <cit.>. However when λ = ℵ_0 we automatically have that δ_1 = δ_2 = ω.Good frames were first defined by the first author in his paper number 600, which eventually appeared as Chapter II of <cit.>. The idea is to provide a localized (i.e. only for base models of a given size λ) axiomatization of a forking-like notion for a “nice enough” set of 1-types.These axioms are similar to the properties of first-order forking in a superstable theory. Jarden and the first author (in <cit.>) later gave a slightly more general definition, not assuming the existence of a superlimit model and dropping some of the redundant clauses. We will make use of good frames for types of finite length (not just length one). Their definition is just like for types of length one, we call them good (<ω, λ)-frames. For the convenience of the reader, we give the full definition from <cit.> here: Let λ be an infinite cardinal. A good (<ω, λ)-frame is a triple (, , ) satisfying, where:*is an abstract elementary class with λ≥ (), _λ≠∅.* ⊆⋃_M ∈_λ^<ω (M). Moreover, if (a_i : i < n / M; N) ∈ (M), then a_i ∉ M for all i < n.*is a relation on quadruples of the form (M_0, M_1, , N), where M_0M_1N, ∈<ωN, and M_0, M_1, N are all in K_λ. We write (M_0, M_1, , N) or M_0M_1N instead of (M_0, M_1, a, N) ∈. * The following properties hold: * Invariance: If f: N ≅ N' and M_0M_1N, then f[M_0]f()f[M_1]N'. If ( / M_1; N) ∈ (M_1), then (f () / f[M_1]; N') ∈ (f[M_1]).* Monotonicity: If M_0M_1N, ' is a subsequence of , M_0M_0'M_1'M_1N'NN” with ' ∈ N', and N”∈ K_, then M_0''M_1'N' and M_0''M_1'N”. If ( / M_1; N) ∈ (M_1) and ' is a subsequence of , then (' / M_1; N) ∈ (M_1). [This property and the previous one show thatis really a relation on types. Thus if p ∈^<ω (M_1) is a type, we say p does not fork over M_0 if M_0M_1N for some (equivalently any)and N such that p =( / M_1; N). Note that this depends on , butwill always be clear from context.]* Nonforking types are basic: If MMN, then ( / M; N) ∈ (M). * K_λ has amalgamation, joint embedding, and no maximal models.* bs-Stability: | (M)| ≤M for all M ∈ K_λ.* Density of basic types: If MN are in K_λ, then there is a ∈ N such that (a / M; N) ∈ (M).* Existence of nonforking extension: If m ≤ n < ω, p ∈ (M) ∩^m (M), NM is in K_λ, then there is some q ∈(N) ∩^n (M) that does not fork over M and extends p.* Uniqueness: If p, q ∈^<ω (N) do not fork over M and p ↾ M = q ↾ M, then p = q.* Symmetry: If M_0_1M_2N, _2 ∈<αM_2, and (_2 / M_0; N) ∈ (M_0), then there is M_1 containing _1 and N'N such that M_0_2M_1N'.* Local character: If δ is a regular cardinal, M_i ∈ K_λ : i ≤δ is increasing continuous, and p ∈ (M_δ) is such that ℓ (p) < δ, then there exists i < δ such that p does not fork over M_i.* Continuity: If δ is a limit ordinal, M_i ∈ K_λ: i ≤δ and α_i < α : i ≤δ are increasing and continuous, and p_i ∈ (M_i) for i < δ are such that j < i < δ implies p_j = p_iM_j, then there is some p ∈ (M_δ) such that for all i < δ, p_i = pM_i.Moreover, if each p_i does not fork over M_0, then neither does p.* Transitivity: If M_0M_1M_2, p ∈ (M_2) does not fork over M_1 and p ↾ M_1 does not fork over M_0, then p does not fork over M_0. A good λ-frame is defined similarly, except we require all types to be types of singletons (i.e. they are in (M) instead of ^<ω (M)). We say that an AEChas a good (<ω, λ)-frame if there is a good (<ω, )-frame where K is the underlying AEC. If = (, , ) is a good-(<ω, λ)-frame (or a good λ-frame), write _ :=. Also write ^ forand _ = _λ. We will also write MN as a shortcut for MN and M, N ∈_ (= _λ).The reader might wonder about the reasons for having a special class of basic types. Following <cit.>, let us call a good frame type-full if the basic types are all the nonalgebraic types. There are no known examples of a good λ-frame which which cannot be extended to a type-full one. However a construction of good frames of the first author <cit.> builds a non type-full good frame and it is not clear that it can be extended to a type-full one until a lot more machinery has been developed. Thus it can be easier to build a good frame than to build a type-full one, and most results about frames already hold in the non-type-full context. That being said, readers would not miss the essence of the present paper if they assumed that all the frames here were type-full. Any good λ-frame (i.e. for types of length one) extends to a good (<ω, λ)-frame (using independent sequences, see <cit.>) or <cit.>. This frame will however not be type-full. From now on until the end of Section <ref>, “nonforking” will refer to nonforking in a fixed frame(usually clear from context).§ WEAK NONFORKING AMALGAMATION In this section, we work in a good λ-frame and study a natural weak version of nonforking amalgamation, _ ( stands for “left weak nonforking amalgamation”). The goal is to obtain a natural criteria for proving the existence of a superlimit in ℵ_1 and also prepare the ground for the proof of symmetry in the good frame built in Section <ref>. The main results are the existence property (Theorem <ref>) and how the symmetry property of _ is connected tobeing(Theorem <ref>). Throughout this section, we assume: * = (, , ) is a fixed good (<ω, λ)-frame, except that it may not satisfy the symmetry axiom.*is categorical in λ.In this section, λ is allowed to be uncountable. However the case λ = ℵ_0 is the one that will interest us in the next sections. The reason for not assuming symmetry is that we will use some of the results of this section to prove that the symmetry axiom holds of a certain nonforking relation in Section <ref>.We will use:[II.4.3 in <cit.>] Let δ < λ^+ be a limit ordinal divisible by λ. Let M_i : i ≤δ be increasing continuous in _. If for any i < δ and any p ∈ (M_i), there exists λ-many j ∈ [i, δ) such that the nonforking extension of p to M_j is realized in M_j + 1, then M_δ is brimmed over M_0. To understand the definition below, it may be helpful to think ofas type-full. Then _ (M_0, M_1, M_2, M_3) holds if and only if the type of any finite subsequences of M_1 over M_2 does not fork over M_0 (M_3 is the ambient model). Thus _ is an attempt to extend nonforking to types of infinite sequences so that it keeps a strong finite character property. In the present paper, _ will be a helpful technical tool but it is not clear that it has the uniqueness property (in contrast with the relationfrom <cit.> or <cit.>, which will have the uniqueness property but requires more assumptions on the good frame). If _ does have the uniqueness property, this has strong consequence on the structure of the frame, see Theorem <ref>. Define the following 4-ary relations on _:* _ (M_0, M_1, M_2, M_3) if and only if M_0M_ℓ M_3 for ℓ = 1,2 and for any ∈<ω|M_1|, if (, M_2, M_3) and (, M_0, M_3) are basic then (, M_2, M_3) does not fork over M_0.* _ (M_0, M_1, M_2, M_3) if and only if _ (M_0, M_2, M_1, M_3) [ stands for “right weak nonforking amalgamation”].* _ (M_0, M_1, M_2, M_3) if and only if both _ (M_0, M_1, M_2, M_3) and _ (M_0, M_1, M_2, M_3) [ stands for “weak nonforking amalgamation]. Whenis clear from context, we write , , and .The following result often comes in handy. Let δ < λ^+ be a limit ordinal. Let M_i : i ≤δ, N_i : i ≤δ be increasing continuous in _. Assume that for each i ≤ j < δ, we have that (M_i, N_i, M_j, N_j). If for each i < δ, N_i realizes all the basic types over M_i, then N_δ realizes all the basic types over M_δ. Let p ∈ (M_δ). By local character, there exists i < δ such that p does not fork over M_i. In particular, pM_i is basic. Since N_i realizes all the basic types over M_i, there exists a ∈ |N_i| such that pM_i =(a, M_i, N_i). Because for all j ∈ [i, δ), (M_i, N_i, M_j, N_j), we have by continuity that (a, M_δ, N_δ) does not fork over M_i, hence by uniqueness it must be equal to p. Therefore a realizes p, as needed. Next, we investigate the properties of . We are especially interested in the symmetry property: whetheris equal to . To understand it better, we consider the following ordering, defined similarly to ≤_λ^+^∗ from <cit.>: For R ∈{, , }, define a relation R on _λ^+ as follows. For M^0, M^1 ∈_λ^+, M^0 R M^1 if and only if there exists increasing continuous resolutions M_i^ℓ∈_λ : i < λ^+ of M^ℓ for ℓ = 0,1 such that for all i < j < λ^+, R (M_i^0, M_i^1, M_j^0, M_j^1). The following is a straightforward “catching your tail argument”, see the proof of <cit.> (this assumes that all types are basic, but the argument goes through without this restriction). Roughly, it says that if MN ( is the usual order on ), then we can find a resolution of M and N so that the pieces are in left weak nonforking amalgamation. Let M, N ∈_λ^+. If MN, then MN. Whether MN can be concluded as well seems to be amuch more complicated question, and in fact is equivalent tobeing(Theorem <ref>), a weakening of symmetry. We now observe that an increasing union of a -increasing chain of saturated models is saturated: Let δ < λ^++ be a limit ordinal. If M_i : i < δ is a -increasing sequence of saturated models in _λ^+, then ⋃_i < δ M_i is saturated. If δ≥λ^+, then any ⋃_i < δ M_i will be λ^+-saturated on general grounds. Thus assume without loss of generality that δ = δ < λ^+. Let M_δ := ⋃_i < δ M_i. We build M_i, j : i ≤δ, j ≤λ^+ such that:* For any i ≤δ, M_i, λ^+ = M_i.* For any i < δ, j < λ^+, M_i, j∈_.* For any i ≤δ, M_i, j : j < λ^+ is increasing and continuous.* For any j ≤λ^+, M_i, j : i < δ is increasing and M_δ, j = ⋃_i < δ M_i, j.* For any i_1 < i_2 ≤δ, j_1 < j_2 ≤λ^+, M_i_2, j_2 realizes all the types in (M_i_1, j_1). This is easy to do. Now for each i_1 < i_2 < δ, we have by assumption that M_i_1 M_i_2. Thus the set C_i_1, i_2 of j < λ^+ such that for all j' ∈ [j, λ^+), (M_i_1, j, M_i_2, j, M_i_1, j', M_i_2, j') is a club (that it is closed follows from the local character and continuity axioms of good frames). Therefore C := ⋂_i_1 < i_2 < δ C_i_1, i_2 is also a club. Hence by renaming without loss of generality for all i_1 < i_2 < δ and all j ≤ j' < λ^+, (M_i_1, j, M_i_2, j, M_i_1, j', M_i_2, j').Now let NM_δ be such that N ∈_λ. We want to see that any type over N is realized in M_δ. By Fact <ref>, it is enough to show that any basic type over N is realized in M_δ.Let j < λ^+ be big-enough such that NM_δ, j. It is enough to see that any basic type over M_δ, j is realized in M_δ, j + 1. To see this, use Lemma <ref> with M_i : i ≤δ, N_i : i ≤δ there standing for M_i, j : i ≤δ, M_i, j + 1 : i ≤δ here. We know that for each i ≤ i' < δ, (M_i, j, M_i', j, M_i, j + 1, M_i', j + 1) and therefore (M_i, j, M_i, j+1, M_i', j, M_i', j + 1). Thus the hypotheses of Lemma <ref> are satisfied. The next fact will be used to prove the existence property of . Its proof is a direct limit argument similar to e.g. <cit.>. Roughly, the nonforking relation there is given by “there exists a smaller submodel over which the type does not split”; in fact, these smaller submodels have to be kept as part of the data of the tower. This is not needed here. The argument is also similar to <cit.>. However there the symmetry axiom axiom is needed: there is an extra requirement on the type of a certain element b, but here we do not make that requirement so do not need symmetry. Let α < λ^+. Let M_i : i ≤α be -increasing continuous (in _λ) and let _i : i < α be given such that _i ∈<ωM_i + 1 for all i < α and (_i, M_i, M_i + 1) ∈ (M_i) (we allow the _i's to have different length).There exists N_i : i ≤α -increasing continuous such that:* M_iN_i for all i ≤α.* (_i, N_i, N_i + 1) does not fork over M_i.We can now list and then prove some basic properties of weak nonforking amalgamation. For the convenience of the reader, we repeat Hypothesis <ref>. Let = (, , ) be a fixed good (<ω, λ)-frame, except that it may not satisfy the symmetry axiom. Assume thatis categorical in λ. Let R ∈{, , }.* Invariance: If R (M_0, M_1, M_2, M_3) and f: M_3 ≅ M_3', then R (f[M_0], f[M_1], f[M_2], M_3').* Monotonicity: If R (M_0, M_1, M_2, M_3) and M_0M_ℓ'M_ℓ for ℓ = 1,2, then R (M_0, M_1', M_2', M_3).* Ambient monotonicity: If R (M_0, M_1, M_2, M_3) and M_3M_3', then R (M_0, M_1, M_2, M_3'). If M_3” M_3 contains |M_1| ∪ |M_2|, then R (M_0, M_1, M_2, M_3”).* Continuity: If δ < λ^+ is a limit ordinal and M_i^ℓ : i ≤δ are increasing continuous for ℓ < 4 with R (M_i^0, M_i^1, M_i^2, M_i^3) for each i < δ, then R (M_δ^0, M_δ^1, M_δ^2, M_δ^3).* Long transitivity: If α < λ^+ is an ordinal, M_i : i ≤α, N_i : i ≤α are increasing continuous and (M_i, N_i, M_i + 1, N_i + 1) for all i < α, then (M_0, N_0, M_α, N_α).* Existence: If R ≠, M_0M_ℓ, ℓ = 1,2, then there exists M_3 ∈_λ and f_ℓ: M_ℓ M_3 such that R (M_0, f_1[M_1], f_2[M_2], M_3).Invariance and the monotonicity properties are straightforward to prove. Continuity and long transitivity follow directly from the local character, continuity, and transitivity properties of good frames. We prove existence via the following claim:Claim: There exists N_0, N_1, N_2, N_3 ∈_ such that (N_0, N_1, N_2, N_3) and N_ℓ is brimmed over N_0 for ℓ = 1,2.Existence easily follows from the claim: given M_0M_ℓ, ℓ = 1,2, there is (by categoricity in λ) an isomorphism f: M_0 ≅ N_0 and (by universality of brimmed models) embeddings f_ℓ : M_ℓ→ N_ℓ extending f for ℓ = 1,2. After some renaming, we obtain the desired -amalgam. To obtain an -amalgam, reverse the role of M_1 and M_2.Proof of Claim: The idea of the proof is as follows: for some suitable ordinal α, we want to build M_i : i ≤α, a_i ∈ M_i + 1 : i < α with the following property: whenever N_i : i ≤α is as described by Fact <ref> (plus slightly more), we must have that (M_0, M_α, N_0, N_δ), M_α is brimmed over M_0, and N_0 is brimmed over M_0. To achieve this, we simply start with an arbitrary M_i : i ≤α, a_i : i < α and, if it fails the property, take a witness to the failure, add some more a_j's to make it more brimmed, and start again to consider whether this witness satisfies the property. After doing this for sufficiently many steps, we eventually succeed to build the desired object. This is somewhat similar to the construction of a reduced tower in <cit.>, although here we are dealing with nonforking independence and not just set-theoretic disjointness.We now start with the proof. Let δ := λ·λ. We choose (M̅^α,a̅^α) by induction on α≤δ such that:* ^α = M^α_i :i ≤α is -increasing continuous. * ^α = _i : i < α, and _i ∈ M_i + 1^α for all i < α.* For all i < α, (^α_i, M^α_i, M^α_i+1) ∈ (M_i^α).* For each i ≤δ, M^α_i:α∈ [i,δ] is -increasing continuous.*For each i < δ and each α∈ (i, δ], (_i, M_i^α, M_i + 1^α) does not fork over M_i^i.*If p ∈ (M_i^α) for i ≤α < δ, then for λ-many β∈ [α,δ), ( _β,M^β + 1_β,M^β +1_β + 1) is a nonforking extension of p.*If i < α < δ and (, M_0^α, M_i + 1^α) ∈ (M_0^α), then for some β∈ (α,δ) exactly one of the following occurs: *(a̅,M^β +1_0,M^β+1_i+1) forks over M^α_0.*There is no M^*_j:j ≤ i+1 -increasing continuous such that: * M_j^β M_j^* for all j ≤ i + 1.* (_j, M_j^*, M_j + 1^*) does not fork over M_j^β for all j < i + 1.* (, M_0^*, M_i + 1^*) forks over M^β_0.This is possible: Along the construction, we also build an enumeration (_j^γ, k_j^γ, i_j^γ, α_j^γ) : j < λ, γ < λ such that for any γ∈ (0, λ), any α < λ·γ, any i < α, any k ≤ i, and any ∈<ωM_i + 1^α, if (, M_k^α, M_i + 1^α) ∈ (M_k^α), then there exists j < λ so that _j^γ =, i_j^γ = i, k_j^γ = k, and α_j^γ = α. We require that always k_j^γ≤ i_j^γ < α_j^γ < λ·γ and the triple (_j^γ, M_k_j^γ^α_j^γ, M_i_j^γ + 1^α_j^γ) represents a basic type. We make sure that at stage λ· (γ + 1) of the construction below, _j^γ', k_j^γ', i_j^γ', α_j^γ' are defined for all j < λ, γ' ≤γ.For α = 0, take any M_0^0 ∈_. For α limit, let M_i^α := ⋃_β∈ [i, α) M_i^β for i < α and M_α^α := ⋃_i < α M_i^α. Now assume that ^α, ^α have been definedfor α < δ. We define ^α + 1 and _α. Fix ρ and j < λ such that α = λ·ρ + j. We consider two cases.* Case 1: ρ is zero or a limit: Use Fact <ref> to get M_i^α + 1 : i ≤α -increasing continuous such that M_i^α M_i^α + 1 for all i ≤α, and for all i < α, (_i, M_i^α + 1, M_i + 1^α + 1) does not fork over M_i^α. Pick any M_α + 1^α + 1 with M_α^α + 1 M_α + 1^α + 1 and any _α∈<ωM_α + 1^α + 1 such that (_α, M_α^α + 1, M_α + 1^α + 1) ∈ (M_α^α + 1).* Case 2: ρ is a successor: Say ρ = γ + 1. Let := _j^γ, α_0 := α_j^γ, k_0 := k_j^γ, i_0 := i_j^γ. There are two subcases. * Subcase 1: Either k_0 ≠ 0, or k_0 = 0 and (<ref>) holds with i, α, β there standing for i_0, α_0, α here.In this case, we proceed as in Case 1 to define M_i^α + 1 : i ≤α. Then we pick _α, M_α + 1^α + 1 such that (_α, M_α^α, M_α + 1^α + 1) is the nonforking extension of (, M_i_0^α_0, M_i_0 + 1^α_0). * Subcase 2: k_0 = 0 and (<ref>) fails.In this case, let M_j^∗ : j ≤ i_0 + 1 witness the failure and set M_j^α + 1 := M_j^∗ for j ≤ i_0 + 1. Then continue as in Case 1 and define _α, M_α + 1^α + 1 as before. This is enough:We proceed via a series of subclaims:Subclaim 1: If p ∈ (M_i^δ) for i < δ, then for λ-many β∈ [i,δ), ( _β,M_β^δ,M^δ_β + 1) is a nonforking extension of p.Proof of subclaim 1: Pick i' ∈ (i, δ) such that p does not fork over M_i^i'. By (<ref>), we know that for λ-many β∈ [i', δ), the nonforking extension of pM_i^i' to M_β^β + 1 is realized in M_β + 1^β + 1 by _β. But by (<ref>) we also have that (_β, M_β^δ, M_β + 1^δ) does not fork over M_β^β. In particular by uniqueness _β also realizes p. †_subclaim 1Subclaim 2: M_δ^δ is brimmed over M_0^δ.Proof of subclaim 2: Apply Fact <ref> to the chain M_i^δ : i ≤δ, using the previous step. †_subclaim 2We now choose ^* = M_i^∗ : i ≤δ increasing continuous such that M_0^∗ is brimmed over M_0^δ, M_i^δ M_i^∗ for all i ≤δ, and (_i, M_i^∗, M_i + 1^∗) does not fork over M_i^δ. This is possible, see case 1 above. Now let (N_0, N_1, N_2, N_3) := (M_0^δ, M_δ^δ, M_0^∗, M_δ^∗). We have just said that M_0^∗ is brimmed over M_0^δ, and by subclaim 2, M_δ^δ is brimmed over M_0^δ. Thus N_ℓ is brimmed over N_0 for ℓ = 1,2. It remains to see:Subclaim 3: (M_0^δ, M_δ^δ, M_0^∗, M_δ^∗)Proof of subclaim 3: Pick ∈<ωM_δ^δ such that (, M_0^δ, M_δ^δ) is basic. By local character, there exists α < δ such that (, M_0^δ, M_δ^δ) does not fork over M_0^α. Further, we can increase α if necessary and pick i < α such that ∈<ωM_i + 1^α. We now apply Clause (<ref>). We know that (<ref>) fails for all β∈ (α, δ) by the choice of α, therefore (<ref>) must hold for all β∈ (α, δ). Now if (, M_0^∗, M_δ^∗) forks over M_0^δ, then it must fork over M_0^β for all high-enough β, but then M_j^∗ : j ≤ i + 1 would contradict Clause (<ref>). Therefore (, M_0^∗, M_δ^∗) does not fork over M_0^δ, as desired. †_subclaim 3†_ClaimThe following properties ofmay or may not hold in general (we have no examples for the failure of symmetry, but uniqueness fails in the last good frame of the Hart-Shelah example, see <cit.>): Let R ∈{, , }.* We say that R has the symmetry property if R (M_0, M_1, M_2, M_3) implies R (M_0, M_2, M_1, M_3).* We say that R has the uniqueness property if whenever R (M_0, M_1, M_2, M_3) and R (M_0, M_1, M_2, M_3'), there exists M_3” with M_3'M_3” and f: M_3M_3”.The following are trivial observations about the definitions: *has the symmetry property, andhas the symmetry property if and only ifhas the symmetry property if and only if ==.*has the uniqueness property if and onlyhas it.Recall from <cit.>:iswhen the following is impossible:There exists an increasing continuous M_i : i < λ^+, N_i : i < λ^+, a basic type p ∈ (M_0), and _i : i < λ^+ such that for any i < λ^+:* M_iN_i.* _i + 1∈ |M_i + 2| and (_i + 1, M_i + 1, M_i + 2) is a nonforking extension of p, but (_i + 1, N_0, N_i + 2) forks over M_0.* ⋃_j < λ^+ M_j is saturated.We now show that beingis a consequence of symmetry for . Moreover,allows us to build a superlimit in λ^+. (<ref>) ⇒ (<ref>) ⇔ (<ref>) ⇒ (<ref>), where:* has the symmetry property.* is .*For M, N ∈_λ^+ both saturated, MN implies MN. *There is a superlimit model in _λ^+.* (<ref>) implies (<ref>): This follows from Lemma <ref> and the fact that the saturated model in λ^+ is universal and has a proper extension <cit.>. * (<ref>) implies (<ref>): Fix a witness M_i : i < λ^+, N_i : i < λ^+, _i : i < λ^+, p to the failure of being . Write M_λ^+ := ⋃_i < λ^+ M_i, N_λ^+ := ⋃_i < λ^+ N_i. By assumption, M_λ^+ is saturated. Clearly, increasing the N_i's will not change that we have a witness so without loss of generality N_λ^+ is also saturated. We claim that M_λ^+ N_λ^+. We show this by proving that for any i < λ^+ and any j ≤ i + 1, (M_j, N_j, M_i + 2, N_i + 2). Indeed, (_i + 1, N_j, N_i + 2) forks over M_j: if not, then by transitivity (_i + 1, N_j, N_i + 2) does not fork over M_0, and hence (_i + 1, N_0, N_i + 2) does not fork over M_0, and we know that this is not the case of the witness we selected. * (<ref>) implies (<ref>): Fix M, N saturated in _λ^+ such that MN but MN. Claim: For any A ⊆ |M| of size λ, there exists M_0M_1M and N_0N_1N such that M_0N_0, M_1N_1, A ⊆ |M_0|, but (M_0, N_0, M_1, N_1).Proof of Claim: If not, we can use failure of the claim and continuity ofto build increasing continuous resolution M_i : i ≤λ^+, N_i : i ≤λ^+ of M and N respectively such that (M_i, N_i, M_j, N_j) for all i < j < λ^+. Thus MN, contradicting the assumption. †_ClaimBuild M_i^∗ : i ≤λ^+, N_i^∗ : i ≤λ^+ increasing continuous resolutions of M, N respectively such that for all i < λ^+, M_i^∗ N_i^∗ and (M_i + 1^∗, N_i + 1^∗, M_i + 2^∗, N_i + 2^∗). This is possible by the claim. Let _i + 1^∗∈ |M_i + 2^∗| witness the -forking, i.e. (_i + 1^∗, N_i + 1^∗, N_i + 2^∗) forks over M_i + 1^∗. By Fodor's lemma, local character, and stability, there exists a stationary set S, i_0 < λ^+ and p ∈ (M_i_0^∗) such that for all i ∈ S, (_i + 1^∗, M_i^∗, M_i + 2^∗) is the nonforking extension of p. Without loss of generality, i_0 is limit and all elements of S are also limit ordinals.Now build an increasing continuous sequence of ordinals j_i : i < λ^+ as follows. Let j_0 := i_0. For i limit, let j_i := sup_k < i j_k. For i successor, pick any j_i ∈ S with j_i > j_i - 1.Now for i not the successor of a limit, let M_i := M_j_i^∗, N_i := N_j_i^∗, _i := _j_i^∗. For i = k + 1 with k a limit, set M_i := M_j_k^∗, N_i := N_j_k^∗, _i := _j_i^∗. This gives a witness to the failure of being .* (<ref>) implies (<ref>): Ifhas the symmetry property, then by Remark <ref>, ==. By Fact <ref>, it follows that MN implies MN for any M, N ∈_λ^+, so (<ref>) holds. Are the conditions in Theorem <ref> all equivalent? Is there a good λ-framesuch that _ does not have the symmetry property? The next result shows that the uniqueness property has strong consequences. The first author has given conditions under which when λ = ℵ_0, failure of uniqueness implies nonstructure <cit.>. Assume thatis a good (<ω, λ)-frame (so it satisfies symmetry). Ifhas the uniqueness property, thenhas the symmetry property andis successful(see <cit.>). By <cit.> (used with the pre-(≤λ, λ)-frame induced by , recalling Fact <ref>)is weakly successful. This implies that there is a relation = _ that is a nonforking relation respecting(see <cit.>, in particular it has all the properties listed in Theorem <ref>, as well as uniqueness and symmetry). Now asrespects , we must have that (M_0, M_1, M_2, M_3) implies (M_0, M_1, M_2, M_3). Sincehas the uniqueness property andhas the existence property, it follows from <cit.> that =. In particular,has the symmetry property.To see thatis successful , it is enough to show that for M, N ∈_λ^+, MN implies MN (whereis defined as in Definition <ref>). This is immediate from Fact <ref> and =. To prepare for the proof of symmetry in the λ = ℵ_0 case, we end this section by introducing yet another notion of nonforking amalgamation ( stands for “very weak nonforking amalgamation”). In this case, we look at finite sequences both on the left and the right hand side. We show that ifis a good frame, thenhas the symmetry property and locality of types implies that =. Thus in this casehas the symmetry property too. * For MN, B ⊆ |N|, ∈<ωN, we say that (, B, N) does not fork over M if there exist M', N' with NN', MM'N', and B ⊆ |M'| such that (, M', N') does not fork over M_0.* We define a 4-ary relation _ = on _ by (M_0, M_1, M_2, M_3) if and only if M_0M_ℓ M_3, ℓ = 1,2 and for any ∈<ωM_1 and any finite B ⊆ |M_2|, if (, M_0, M_3) and (, M_2, M_3) are both basic, then (, B, M_3) does not fork over M_0. Assume thatis a type-full good (<ω, λ)-frame. *has the symmetry property: (M_0, M_1, M_2, M_3) if and only if (M_0, M_2, M_1, M_3).* If for any M ∈_ and any p ≠ q ∈ (M) there exists B ⊆ |M| finite such that pB ≠ qB, then =. In particular,has the symmetry property.* By the symmetry axiom of good frames.* This is observed in <cit.>. In details, it suffices to show that for MN, p ∈ (N) does not fork over M if and only if pB does not fork over M for all finite B ⊆ |N|. Let q ∈ (N) be the nonforking extension of pM. For any finite B ⊆ |N|, we have that qB = pB, by the uniqueness property for (the extended notion of) forking, see <cit.>. Therefore by the assumption we must have p = q, as desired.§ BUILDING A SUPERLIMIT In this section, we work in ℵ_0 and show assuming ℵ_0-stability and amalgamation thatis _ℵ_0 (Theorem <ref>) and has a superlimit (Corollary <ref>).= (K, ) is an AEC with () = ℵ_0 (and countable vocabulary). We will use without comments Fact <ref> and Remark <ref>. The essence of it is that since λ = ℵ_0 all brimmed models have the same length, and hence are isomorphic (and the isomorphism fixes any common base they may have).First note that ifis stable and has few models, we can say something about its definability: Assume that (, ℵ_0) ≤ℵ_0.* The set {M ∈ K_ℵ_0: |M| ⊆ω} is Borel.* Ifhas amalgamation in ℵ_0 and is stable in ℵ_0, then the set {(M,N):MN and |N| ⊆ω} is Σ^1_1. In particular ifhas amalgamation in ℵ_0 and is stable in ℵ_0, thenis a _ℵ_0-representable AEC. * Fix M ∈_ℵ_0. By Scott's isomorphism theorem, there exists a formula ϕ_M of _ℵ_1, ℵ_0 (τ_) such that N ϕ_M if and only if M ≅ N. Now observe that the set {N : Nis a τ_-structure with |N| ⊆ω andN ϕ_M} is Borel and use that (, ℵ_0) ≤ℵ_0.* For M, N ∈_ℵ_0 with MN, let us say that N is almost brimmed over M if either N is brimmed over M, or N is -maximal. Using amalgamation, it is easy to check that if N, N' are both almost brimmed over M, then N ≅_M N' (as in Fact <ref>, recalling that the chains witnessing the brimmedness must have cofinality ω). Moreover there always exists an almost brimmed model over any M ∈_ℵ_0.Fix M_n^∗ : n < ω such that for any M ∈_ℵ_0 there exists n < ω such that M ≅ M_n^∗ (possible as (,ℵ_0) ≤ℵ_0). For each n < ω, fix N_n^∗∈_ℵ_0 almost brimmed over M_n^∗. We have:⊛_1 For M, N ∈_ℵ_0:* There is n < ω and an isomorphism f: M_n^∗≅ M.* If N is almost brimmed over M, then any such f extends to g: N_n^∗≅ N.⊛_2 For M_1, M_2 ∈_ℵ_0, M_1M_2 if and only if M_1 ⊆ M_2 and for some n_1, n_2 < ω, for some (N,f_1,f_2) we have: M_1 ⊆ M_2 ⊆ N and for ℓ = 1,2, f_ℓ is an isomorphism from (M_n_ℓ^*,N_n_ℓ^*) onto (M_ℓ,N). [Why?The implication “if" holds by the coherence axiom of AECs.The implication “only if" holds as there is N ∈_ℵ_0 which is almost brimmed over M_2 (and so M_2N) hence N is almost brimmed over M_1and use ⊛_1 above.] The result now follows from ⊛_2. By <cit.>, it follows thatis _ℵ_0. We now study homogeneous models and show that they coincide with brimmed models. Note that the homogeneity here is with respect to a set D of orbital types. Let D be a set of orbital types over the empty set and let M ∈. We say that M is (D, ℵ_0)-homogeneous if it realizes all the types in D and whenever p ∈ D is the type of an (n + m)-elements sequence and ∈n M realizes p^n (the restriction of p to its first n “variables”), there exists a sequence ∈mM such thatrealizes p. The next result characterizes the countable brimmed model in AECs that are nicely stable in ℵ_0 (recall Definition <ref>). Assume thatis nicely stable in ℵ_0 and let ^∗ be the class of amalgamation bases in _ℵ_0. Let M ∈_ℵ_0. The following are equivalent:* M is brimmed.* M is (^∗^<ω (∅), ℵ_0)-homogeneous (see Definition <ref>(<ref>)). Let M ∈_ℵ_0. First we show:Claim: If M is brimmed, , ∈ M, then _^∗ ( / ∅; M) = _^∗ ( / ∅; M) if and only if there is an automorphism f of M sendingto . Proof of Claim: The right to left direction is clear. Now assume that _^∗ ( / ∅; M) = _^∗ ( / ∅; M). Say M_i : i < ω witness that M is brimmed, and without loss of generality assume that , ∈ M_0. Then _^∗ ( / ∅; M_0) = _^∗ ( / ∅; M_0). Since ^∗ has amalgamation, there exists M_0' ∈^∗ with M_0M_0' and f: M_0 → M_0' so that f () =. Since M_1 is universal over M_0, we can assume without loss of generality that M_0' = M_1. Now extend f to an automorphism of M using a back and forth argument.From the claim, it follows directly that if M is brimmed, then it is (^∗^<ω (∅), ℵ_0)-homogeneous. Conversely, the countable (^∗^<ω (∅), ℵ_0)-homogeneous model is unique (standard back and forth argument) and so it must also be brimmed. By adding constants to the language, we can also characterize brimmed models over M_0 as those that are homogeneous for orbital types of finite sequences over M_0.Ifis nicely stable in ℵ_0, then there is a superlimit model of cardinality ℵ_0. Let M ∈_ℵ_0 be brimmed (it exists by nice stability in ℵ_0). We claim that M is superlimit. To see this, we check the conditions of Definition <ref>. On general grounds, brimmed models are universal in _ℵ_0, are not maximal (from the definition of nice stability), and there is a unique brimmed model of cardinality ℵ_0. Still, it is not obvious that if M_i : i < δ is an increasing chain of brimmed models in _ℵ_0 and δ < ω_1, then ⋃_i < δ M_i is brimmed. To see this, we use Theorem <ref>: each M_i is ((^∗^<ω (∅), ℵ_0)-homogeneous, and it is clear from the definition that an increasing union of such homogeneous models is homogeneous. Thus M_δ is (^∗^<ω (∅), ℵ_0)-homogeneous. By Theorem <ref> again, M_δ is brimmed, as desired.We have justified assuming amalgamation in the following sense: Ifis nicely stable in ℵ_0, then there exists an AEC ' = (K', ') such that:*(') = ℵ_0.* '_<ℵ_0 = ∅.* τ_' = τ_.* K' ⊆ K and for M, N ∈ K', M ' N if and only if MN.*For any M ∈_ℵ_0 there exists M' ∈_ℵ_0' with MM'.*' is categorical in ℵ_0.*' is very nicely stable in ℵ_0. In particular it has amalgamation in ℵ_0.*' is _ℵ_0.Let M ∈_ℵ_0 be superlimit (exists by Corollary <ref>). Let K_ℵ_0' := {N ∈ K : N ≅ M}. Now let ' be the AEC generated by (K_ℵ_0', ) (in the sense of <cit.>). One can easily check that ' is nicely stable in ℵ_0; from this and ℵ_0-categoricity we get amalgamation in ℵ_0, hence (<ref>) holds. As for (<ref>), it follows from Theorem <ref>. § BUILDING A GOOD ℵ_0-FRAME The aim of this section is to build a good ℵ_0-frame from nice ℵ_0-stability. By Corollary <ref>, we may restrict the class to a superlimit so that it is categorical in ℵ_0. As before, we assume:= (K, ) is an AEC with () = ℵ_0 (and countable vocabulary). The nonforking relation of the frame will be nonsplitting: For M ∈_ℵ_0 and A ⊆ |M|, p ∈^<ω (M) splits over A if whenever p =( / M; N), there exists _1, _2 ∈ M such that (_1 / A; M) =(_1 / A; M) but (_1 / A; N) ≠ (_2 / A; N). The following is proven in <cit.>. Similar proofs appear in <cit.> or <cit.>. Assume thatis nicely stable in ℵ_0 and categorical in ℵ_0. If M ∈_ℵ_0 and p ∈^<ω (M), then there exists A ⊆ |M| finite such that p does not split over A. The following result about nonsplitting will also come in handy. It appears in various forms in the literature, see e.g. <cit.>. Assume thatis nicely stable in ℵ_0 and categorical in ℵ_0. Let MN both be in _ℵ_0, p,q ∈^<ω (N). If both p and q do not split over a finite subset of M and pA = qA for all finite A ⊆ |M|, then pB = qB for all finite B ⊆ |N|. Let N' be brimmed over N. Let _1, _2 ∈ N' realize p and q respectively. Fix A ⊆ |M| finite such that p and q do not split over A. Let B ⊆ |N| be finite and letbe an enumeration of B. Since M is brimmed, there exists ' ∈ M such that ( / A; N) =(' / A; N). By nonsplitting, (_ℓ/ A; N') =(_ℓ' / A; N') for ℓ = 1,2. Now since ' ∈ M we have by assumption that pA ' = qA '. Therefore (_1 ' / A; N') =(_2 ' / A; N'). Putting these equalities together, (_1/ A; N') =(_2/ A; N'), so pB = qB, as desired.Letbe nicely stable in ℵ_0 and categorical in ℵ_0. We define a pre-(<ω, λ)-frame = (^, ,^) by:* ^ =.* For M_0MN all in _ℵ_0, n < ω, = a_i :i < n∈nN, M_0MN holds if and only if a_i ∉ M for all i < n and there exists a finite A ⊆ |M_0| so that ( / M; N) does not split over A.* For M ∈_ℵ_0, ^ (M) is the set of all types of finite sequences a_i : i < n over M such that for all i < n, a_i ∉ M. In order to prove thatis a good ℵ_0-frame, we will make an additional locality hypothesis. See Example <ref> and the next section for setups where it holds.is (<ℵ_0, ℵ_0)-local if for any M ∈p, q ∈^<ω (M), pA = qA for all finite A ⊆ |M| implies p = q. We say thatis weakly (<ℵ_0, ℵ_0)-local if this holds for a superlimit M. The definition of locality includes types of any finite length, not just of length one. This will be used to prove the symmetry property of _, via Theorem <ref>. We now prove, assuming nice stability, categoricity, and locality, that the pre-frame defined above is a good ℵ_0-frame. Assume thatis nicely stable in ℵ_0 and categorical in ℵ_0. Ifis (<ℵ_0, ℵ_0)-local, then(Definition <ref>) is a type-full good (<ω, ℵ_0)-frame. Moreover _ has the symmetry property (recall Definitions <ref> and <ref>). In particular,is . Once we have shown thatis a type-full good frame, the moreover part follows from Theorem <ref>. The last sentence is by Theorem <ref>.Now except for symmetry, the axioms of good frames are easy to check (see the proof of <cit.>). For example:* Local character: Let M_i : i ≤δ be increasing continuous in _. Let p ∈^ (M_δ). By Fact <ref>, there exists a finite A ⊆ |M_δ| such that p does not split over A. Pick i < δ such that A ⊆ |M_i|. Then p does not fork over M_i.* Uniqueness: by Lemma <ref> and locality.* Extension: follows on general grounds, see <cit.>. Symmetry is the hardest to prove, and is done as in <cit.>. We give a full proof for the convenience of the reader.Suppose that (, N_2, N_3) does not fork over N_0 and let ∈<ωN_2 \ N_1. We want to find N_1, N_3' such that N_0N_1N_3', N_3N_3', ∈<ωN_1 and (, N_1, N_3') does not fork over N_0. Assume for a contradiction that there is no such N_1. Using existence for _ (see Theorem <ref>), as well as the extension property for nonforking, we can increase N_2 and N_3 if necessary and find N_1 such that _ (N_0, N_1, N_2, N_3), N_ℓ is brimmed over N_0, and N_3 is brimmed over N_ℓ for ℓ = 1,2. By assumption, p :=(, N_1, N_3) forks over N_0.Claim 1: Let I be the linear order [0, ∞) ∩ℚ. There exists an increasing chain M_s : s ∈ I such that for any s < t in I, M_s, M_t are in _ℵ_0 and M_t is brimmed over M_s.Proof of Claim 1: Let ϕ∈_ω_1, ω be a Scott sentence for the model in _ℵ_0. Let ψ∈_ω_1, ω be a Scott sentence for a pair M, N ∈_ℵ_0 such that N is brimmed over M. Now let K^∗ be the class of sequences M_j : j ∈ J such that J is a linear order, M_j ϕ for all j ∈ J, and (M_j, M_k) ψ for all j < k in J. It is easy to see that K^∗ is axiomatizable by a sentence in _ω_1, ω. Moreover, for each α < ω_1, there is a sequence M_i : i < α in K^∗. By <cit.>, this implies that K^∗ contains an I-indexed member, as desired. †_Claim 1Fix I, M_s : s ∈ I as in Claim 1. Fix N_0' such that N_0 is brimmed over N_0' and pN_0 does not fork over N_0'.For any fixed infinite J ⊆ I, write M_J := ⋃_s ∈ J M_s. Assume now that M_I is brimmed over M_J. Let N_0^J := M_J, N_1^J := M_I. Let N_3^J be brimmed over N_1^J. By categoricity and uniqueness of brimmed models, there exists f_0 : N_0' ≅ M_0, f_0^J : N_0 ≅ N_0^J, f_1^J : N_1 ≅ N_1^J, and f_3^J : N_3 ≅ N_3^J such that f_0 ⊆ f_0^J ⊆ f_1^J ⊆ f_3^J. Let f_2^J := f_3^JN_2 and let N_2^J := f_2^J[N_2]. Note that _ (N_0^J, N_1^J, N_2^J, N_3^J) holds.Let p_J :=(f_3^J (), f_3^J[N_1], f_3^J[N_3]) =(f_3^J (), M_I, N_3^J). Since we are assuming that (, N_1, N_3) forks over N_0, we have that p_J forks over N_0^J. Moreover p_JN_0^J does not fork over M_0.Claim 2: If J has no last elements, I \ J has no first elements, and t ∈ I \ J, then p_JM_t forks over N_0^J.Proof of Claim 2: Suppose that p_JM_t does not fork over N_0^J. Note that M_t is brimmed over M_J. Find N_1' such that N_0N_1'N_1, N_1' is brimmed over N_1, and f_1^J : N_1' ≅ M_t. Let ' ∈<ωN_1' be such that (', N_0, N_1') =(, N_0, N_1). Since _ (N_0, N_1, N_2, N_3), we know that (', N_2, N_3) does not fork over N_0, hence by uniqueness (, N_2, N_3) =(', N_2, N_3). But we have assumed that (, N_1', N_3) does not fork over N_0 and ' ∈<ω_1N_1', hence by a simple renaming we obtain a contradiction to our hypothesis that symmetry failed. †_Claim 2Claim 3: If J_1 ⊊ J_2 are both proper initial segments of I with no last elements and J_2 \ J_1 has no first elements, then p_J_1≠ p_J_2.Proof of Claim 3: Fix t ∈ J_2 \ J_1. By Claim 2, p_J_1 M_t forks over N_0^J_1. We claim that p_J_2 M_t does not fork over N_0^J_1. Indeed recall that N_0^J_2 = M_J_2 and by assumption p_J_2 N_0^J_2 does not fork over M_0. Therefore by monotonicity also p_J_2 M_t does not fork over M_J_1 = N_0^J_1. †_Claim 3To finish, observe that there are 2^ℵ_0 cuts of I as in Claim 3. Therefore stability fails, a contradiction.The next corollary does not assume categoricity, but uses amalgamation in ℵ_0, rather than just density of amalgamation bases. Ifis very nicely stable in ℵ_0 and weakly (<ℵ_0, ℵ_0)-local, thenhas a superlimit of cardinality ℵ_1. By Corollary <ref>,has a superlimit N_0 in ℵ_0. Let ' be the class generated by this superlimit, as described by the proof of Corollary <ref>. Then ' is categorical in ℵ_0 and nicely stable in ℵ_0, hence we can apply Theorem <ref> and get a type-fullℵ_0-frame with underlying class _ℵ_0'. By Theorem <ref>, ' has a superlimit model in ℵ_1. This is also a superlimit in : the only nontrivial property to check is universality. Let M ∈_ℵ_1. Fix any M_0 ∈_ℵ_0 with M_0M. By universality of N_0, there exists f: M_0 → N_0. Now let N ∈' be superlimit in ℵ_1 with N_0N. Using amalgamation (amalgamation in ℵ_0 suffices for this, see <cit.>), we can find g: M → N extending f, as needed. § LOCALITY FROM SUPERSIMPLICITY In this section, we give a sufficient condition for locality. As before, we assume:= (K, ) is an AEC with () = ℵ_0 (and countable vocabulary). In the context of a nicely ℵ_0-stable AEC, the following definition generalizes that of a supersimple homogeneous model <cit.>. The idea is that we want to have a nice notion of nonforking available for all finite sets (not only models). However we do not require that forking over finite sets satisfies any uniqueness requirement. Thus it is not a-priori clear that supersimplicity implies the existence of a good frame (although this will follow from ℵ_0-categoricity and Theorems <ref>, <ref>).Throughout this section expressions such as “nonforking” or “not fork” will refer to the relation defined in Definition (<ref>)(<ref>) below. We give examples after the definition. Assume thatis nicely ℵ_0-stable and categorical in ℵ_0. We say thatis supersimple if there exists a 4-ary relationsuch that:* (A, B, C, N) implies that N ∈_ℵ_0, A ∪ B ∪ C ⊆ |N| and A, B, C are all finite or countable. We write ABCN instead of (A, B, C, N). Below, we may abuse notation and write e.g. A_1A_2 instead of A_1 ∪ A_2, or N instead of ABCN, where A, B, C stand for the ranges of , , andrespectively.* Normality: ABCN if and only if AABACN.*Invariance under -embeddings: If f: N → N' and A ∪ B ∪ C ⊆ |N|, then ABCN, if and only if f[A]f[B]f[C]N'. This shows thatis really a relation on types, so we say ( / C; N) does not fork over A if ACN.* Monotonicity: If ABCN and A ⊆ A' ⊆ B' ⊆ B then A'B'CN. * Symmetry: If ABCN, then ACBN.* Local character: If M ∈_ℵ_0 and p ∈^<ω (M), then there exists A ⊆ |M| finite such that p does not fork over A.* Extension: If p ∈^<ω (C; N) does not fork over A ⊆ C, then there is q ∈^<ω (N) such that q extends p and q does not fork over A.* Transitivity: If ABCN and CBDN with A ⊆ C ⊆ D, then ABDN.* Relationship with splitting: If MN are both in _ℵ_0, p ∈^<ω (N), and p does not fork over M, then there is A ⊆ |M| finite such that p does not split over A.It may be helpful to compare Definition <ref> with Definition <ref>. The idea of <ref> is to give a sort of analog of good frames but to allow types over sets. Note that the statement of symmetry in <ref> is more technical, precisely because types over sets are not allowed. However the idea is the same. Another difference is that local character in <ref> is stated as “every type does not fork over a finite set”. In <ref>, it is stated as “every type over the union of an increasing chain does not fork over a previous element of the chain”. Again, the lack of types over sets makes it impossible to state the former in good frames. * Working inside a supersimple homogeneous model N (in the sense of <cit.>), we can define ABCN to hold if and only if B is ℵ_0-free from C over A (in the sense of <cit.>). The first four conditions of Definition <ref> are then easy to check. Symmetry is <cit.> and transitivity is <cit.>. Local character and extension are given by the definition in <cit.>. Now if in addition N is ℵ_0-stable (in the sense of <cit.> or equivalently in the sense given here), then by <cit.> the last axiom of Definition <ref> holds.* Letbe a FUR class <cit.> (this includes in particular all quasiminimal pregeometry classes). Then lettingbe defined as in <cit.>, we can also check that it satisfies Definition <ref>.We first show that if a type p over a finite set does not fork over a subset A of a countable model M, then the type is realized inside M. Assume thatis nicely ℵ_0-stable, supersimple, and categorical in ℵ_0. Let MN both be in _ℵ_0, and let B ⊆ |N| be finite. Let p ∈^<ω (B; N). If p does not fork over B ∩ |M|, then p is realized in M. Extending N if necessary, we can assume without loss of generality that N is brimmed over M. Let q ∈^<ω (N) be a nonforking extension of p. Letbe an enumeration of B. Fix A' ⊆ |M| finite and big-enough such that B ∩ |M| ⊆ A', ( / M; N) does not split and does not fork over A' and q does not split over A' (recall Fact <ref>). Let M_0M be such that M is brimmed over M_0 and M_0 contains A'. Let ∈ M be such thatrealizes qM_0. We have that M_0MN so by symmetry and monotonicity, M_0N. By extension, we can pick M_0' such that M_0M_0'N, M_0' contains , and M_0M_0'N. Now as qM_0' does not split over a finite subset of M_0 andrealizes qM_0, we must have by Lemma <ref> thatrealizes qB = p, as desired. We will prove locality in supersimple ℵ_0-stable AECs by a back and forth argument. More precisely, we start with MN, N brimmed over M, and elements _1, _2 ∈ N whose types over every finite subset of M match. First, we will do a back and forth argument to find an automorphism of N sending _1 to _2 and fixing M setwise (Lemma <ref>). We will then use this automorphism and nonsplitting to build another automorphism that fixes M pointwise (Theorem <ref>).The next lemma starts setting up the stage by making sure that we can map an element of M to an element of M. Assume thatis nicely ℵ_0-stable, supersimple, and categorical in ℵ_0. Let MN both be in _ℵ_0. Let _1, _2 ∈ N, _1, _2, _1 ∈ M be such that (_1 _1; N) =(_2 _2; N). If _1_1_1N, then there exists _2 ∈ M such that (_1 _1 _1; N) =(_2 _2 _2; N). Extending N if necessary, we can assume without loss of generality that N is brimmed over M. By symmetry, _1_1_1N. Let f be an automorphism of N sending _1 _1 to _2 _2. By invariance, _2f(_1)_2N (but we do not know that f(_1) ∈ M). Let q :=(f (_1) / _2 _2; N). We have to show that q is realized in M. Since q does not fork over _2 ∈ M, this is exactly what Lemma <ref> tells us. Our main lemma in the back and forth argument will be: Assume thatis nicely ℵ_0-stable, supersimple, and categorical in ℵ_0. Let MN both be in _ℵ_0 with N brimmed over M. Let _1, _2 ∈ N, _1, _2 ∈ M be such that (_1 _1; N) =(_2 _2; N) and _ℓ_ℓMN for ℓ = 1,2. If _1 ∈ N, _1' ∈ M, there exists _1, _2, _2' ∈ M and _2 ∈ N such that: * (_1 _1 _1 _1' _1; N) =(_2 _2 _2 _2' _2; N).* _ℓ_ℓ_ℓ'_ℓ_ℓMN for ℓ = 1,2.Using Lemma <ref>, we can enlarge _1 and _2 if necessary to assume without loss of generality that _1' is empty. Now using local character, fix _1 ∈ M such that _1 _1_1 _1MN. By Lemma <ref>, there exists _2 ∈ M such that (_1 _1 _1; N) =(_2 _2 _2; N). Let f be an automorphism of N witnessing this. By extension and monotonicity, pick _2 ∈ N such that (_2 / _2 _2 _2; N) =(f (_1) / _2 _2 _2; N) and _2 _2 _2_2M _2N. It remains to see that _2 _2_2 _2MN. We do this using a standard nonforking calculus argument: by normality, _2 _2 _2_2 _2M _2N and we also know from the hypotheses of the lemma that _2_2MN, so by monotonicity and normality _2 _2_2 _2 _2MN. Now using transitivity, monotonicity, and symmetry, _2_2_2 _2MN, as desired. We can now build the desired automorphism which fixes M setwise. Assume thatis nicely ℵ_0-stable, supersimple, and categorical in ℵ_0. Let M ∈_ℵ_0, A ⊆ |M| finite, p, q ∈^<ω (M) such that p and q do not fork over A. If pA = qA, then there exists an automorphism f of M fixing A such that f (p) = q. Let N ∈_ℵ_0 be brimmed over M. Say p =(_1 / M; N), q =(_2 / M; N). Letbe an enumeration of A and let _ℓ :=, ℓ = 1,2. Now apply Lemma <ref> repeatedly in a back and forth argument to build an automorphism g of N fixing A such that g (_1) = _2 and g[M] = M. Let f := gM. We now show that we can actually build an automorphism fixing M pointwise. The next lemma is the main argument for this: Assume thatis nicely ℵ_0-stable, supersimple, and categorical in ℵ_0. Let M ∈_ℵ_0 and let N be brimmed over M. Let _1, _2 ∈ N be such that (_1 / A; N) =(_2 / A; N) for all finite A ⊆ |M|. For any _1 ∈ N, there exists _2 ∈ N such that (_1 _1 / A; N) =(_2 _2 / A; N) for all finite A ⊆ |M|. Let p :=(_1 / M; N), q :=(_2 / M; N). Fix A ⊆ |M| finite such that both p and q do not fork over A and (_1 _1 /M ; N) does not split over A. By Lemma <ref>, there exists an automorphism f of M fixing A such that f (p) = q. Let g be an automorphism of N extending f such that g (_1) = _2 and let _2 := g (_1). We claim that this works. Fix C ⊆ |M| finite and let _2 be an enumeration of C. Let _1 := g^-1 (_2). We know that (_1 / A; N) =(_2 / A; N), so by nonsplitting, (_1 _1 _1 / A; N) =(_1 _1 _2 / A; N). Applying g, we have that (_1 _1 _1 / A; N) =(_2 _2 _2 / A; N). Putting the two equalities together, (_1 _1 _2 / A; N) =(_2 _2 _2 / A; N), so (_1 _1 / C; N) =(_2 _2 / C; N), as desired. We have arrived to the main theorem of this section: Ifis nicely ℵ_0-stable, supersimple, and categorical in ℵ_0, thenis (<ℵ_0, ℵ_0)-local (recall Definition <ref>). Let M ∈_ℵ_0 and let p, q ∈^<ω (M) be such that pA = qA for all finite A ⊆ |M|. Let N be brimmed over M and let _1, _2 ∈ N realize p and q respectively. Now apply Lemma <ref> in a back and forth argument to get an automorphism of N fixing M taking _1 to _2. amsalpha

http://arxiv.org/abs/1702.08281v5

1]A. Sargsyan 1]A. Papoyan 2]I.G. Hughes 2]C.S. Adams 1]D. Sarkisyan[1]1013Institute for Physical Research, NAS of Armenia, 0203, Ashtarak-2, Armenia [2]Joint Quantum Centre (JQC) Durham-Newcastle, Department of Physics, Durham University, South Road, Durham DH1 3LE, United Kingdom[ \begin@twocolumnfalse Selective reflection from Rb layer with thickness below λ/12 and applications [ Received: date / Accepted: date ============================================================================= We have studied the peculiarities of selective reflection from Rb vapor cell with thickness L < 70 nm, which is over an order of magnitude smaller than the resonant wavelength for Rb atomic D_1 line λ = 795 nm. A huge (≈ 240 MHz) red shift and spectral broadening of reflection signal is recorded for L = 40 nm caused by the atom-surface interaction. Also completely frequency resolved hyperfine Paschen-Back splitting of atomic transitions to four components for ^87Rb and six components for ^85Rb is recorded in strong magnetic field (B > 2 kG).\end@twocolumnfalse ] Selective reflection from Rb layer with thickness below λ/12 and applications [ Received: date / Accepted: date =============================================================================Selective reflection (SR) of resonant laser radiation from the interface between an atomic vapor and the dielectric window of spectroscopic cells initially observed by R. Wood in 1909 <cit.> has revisited in 1970s when narrow-linewidth tunable cw lasers became available <cit.>. Thanks to high contrast sub-Doppler signal response, the SR technique became a powerful spectroscopic tool, which was successfully used, in particular, to study the van der Waals (vdW) interaction of atoms with dielectric window of the cell manifested by a red shift of the SR frequency <cit.>. Spectroscopy with atomic vapor nanometric-thickness cells (NC) is another technique capable of yielding information on atom-surface processes. The vdW interaction of Rb and Cs atoms confined in the NC was studied in <cit.> using resonant fluorescence spectra.Merging the SR and NC methods significantly extends their applied interest. On account of the dispersive shape of the signal, the SR from NC was used to lock the laser's frequency <cit.>. Recently it was demonstrated that the SR from the Rb vapor NC with L ≈ 370 nm thickness is convenient to form frequency reference of atomic transitions <cit.>. It was also shown that the same technique implemented for D_1 line of Cs and 300 nm gap thickness is expedient to study the behavior of atomic transitions in strong magnetic field <cit.>.In this Letter we focus on the peculiarities of selective reflection for the conditions when the response of a resonant medium is formed over an essentially sub-wavelength depth in an atomic vapor layer geometrically restricted by the two inner surfaces of the nanometric-thickness cell (L< 70 nm).The experimental arrangement is schematically sketched in Fig.<ref>. The collimated P_L ∼ 0.5 mW circularly-polarized beam of a single-frequency narrow-band (γ_L ∼ 1 MHz) cw external cavity diode laser with λ= 795 nm was directed at normal incidence onto the specially fabricated Rb NC with a large aperture area of smoothly changing thickness in the rangle L= 20 – 90 nm (see the upper left inset). The SR beam was carefully separated from two other beams reflected from the front and rear surfaces of the cell (see the upper right inset), and after passing through a λ = 795 nm interference filter was recorded by a photodiode (3).A fraction of laser radiation was branched to the frequency reference unit (Ref.) with an auxiliary L=λ Rb NC, with a transmission spectrum exhibiting narrow velocity selective optical pumping (VSOP) resonances located exactly at atomic transition frequencies <cit.>. The reference signal was recorded simultaneously with the SR signal by a two-channel digital oscilloscope while linearly scanning the laser frequency across the D_1 resonance. The scanning rate was chosen to be slow enough for assuring establishment of a steady-state interaction regime.The necessary vapor density (≈ 8×10^13 cm^3) was attained by heating the cell's thin sapphire reservoir (R) containing metallic Rb to T_R ≈ 150 C while keeping the window temperature some 20 C higher. Note that the NC can be heated up to 450 C (see <cit.>). To study magnetic field-induced processes, a calibrated strong permanent neodymium magnet (2) was mounted on a micrometric-step translation stage in the proximity of the cell's rear window. The B-field strength was varied by simple longitudinal displacement of the magnet.The SR spectrum on the D_1 line F_g=2,3 → F_e=3 transitions of ^85Rb and F_g=1,2 → F_e=2 transitions of ^85Rb[In the figures the excited levels are marked by primes.] for the cell thickness of 67 nm and longitudinal magnetic field B ≈ 2.4 kG is shown in Fig.<ref> (upper spectrum). The lower lines are the derivatives of SR signal (DSR) formed in a real time by the Siglent oscilloscope <cit.>. The DSR peaks labeled 1–3 and 10 belong to ^87Rb, and the DSR peaks 4–9 belong to ^85Rb (see the upper diagrams). On account of the sub-Doppler linewidth of DSR (γ_DSR≈ 200 MHz FWHM), all ten transition components for B = 2.4 kG are spectrally well resolved. For B = 2.7 kG, complete frequency resolution is observed for all transition components except overlapping peaks 3 and 6. The influence of vdW interaction between the atoms and dielectric windows of the NC starts to feature in the spectra when L < 100 nm <cit.>, manifested as a red shift of the SR signal frequency. This shift is clearly observable for B = 0, L≈ 67 nm, as is shown in the middle inset of Fig.<ref> (see figure caption).The influence of the applied magnetic field on the hyperfine structure is characterized by parameter B_0=A_HFS/μ_B, where A_HFS is the hyperfine coupling constant for 5S level, and μ_B is the Bohr magneton <cit.>. B_0 ≈ 0.7 kG for ^85Rb, and B_0 ≈ 2 kG for ^87Rb. When B<B_0 (Zeeman regime), the splitting of levels is described by the total angular momentum of the atom F = J + I and its projection m_F, where J is the total electron angular momentum, and I is the nuclear spin angular momentum. The F and m_F notation is used in the inset diagram of Fig.<ref>. Decoupling between J and I develops when B≥ B_0. F is no longer a good quantum number, and the splitting of atomic levels is described by the projections m_J and m_I (hyperfine Paschen-Back regime) <cit.>. In this regime four and six atomic transitions belonging to ^87Rb and ^85Rb, respectively, remain in the spectra (as is evident in Fig.<ref> and Fig.<ref>). The recorded DSR spectrum for L = 50 ±2 nm and B ≈ 2.15 kG is shown in Fig.<ref>. For this measurement, the reservoir temperature was somewhat increased (T_R = 165 C) to compensate for the thickness reduction. In spite of some broadening of DSR (γ_DSR≈ 250 MHz), all the peaks are well resolved, except 3 and 5, which are overlapped. The inset in the upper-right corner shows the B = 0 spectra, which clearly indicate the increase of vdW red shift to -130 MHz with respect to unperturbed atomic transitions. To estimate the frequency shift for ^87Rb, D_1 line arising from interaction of an individual atom with two dielectric windows of the cell (w1 and w2), we have plotted in the upper-left inset the distance-dependent frequency shifts separately for w1: Δν_vdW = -C_3/z_1^3 (curve 1), and for w2: Δν_vdW = -C_3/z_2^3 (curve 2), where z_1 and z_2 are distances of the Rb atom from w1 and w2, respectively (in μm). The overall shift is depicted by curve 3, which is the sum of curves 1 and 2. For L = 50 nm, the recorded -130 MHz shift is obtained by taking vdW coefficient C_3 = 1.0±0.1 kHz×μm^3, which is consistent with the results of previous studies <cit.>. As is seen from the diagram, the maximum of the spectral density of an overall signal (the shifted peak) corresponds to the minimum value of the shift, so this value is estimated by the expression Δν_vdW = -2C_3/(L/2)^3. To record SR signals above the noise level for even smaller cell thickness L = 40±2 nm, the cell temperature was further increased to T_R = 180 C (vapor density ∼ 5×10^14 cm^-3). The DSR spectrum for this thickness and B ≈ 2.15 kG is presented in Fig.<ref>. In these conditions the transition peaks become broader, but the linewidth γ_DSR≈ 380 MHz still does not exceed ∼ 500 MHz transition width obtained with 1 mm-long Rb cell <cit.>. An estimation of the number of atoms contributing to SR signal (Fig.<ref>) for 0.5 mm beam diameter, L = 40 nm, and atomic density 5×10^14 cm^-3 gives ∼ 5×10^6. Note that the vdW shift for ^85Rb and ^87Rb transitions for the same L value is approximately equal.Complete resolution of all the DSR peaks makes favorable the determination of a B-field, which can be done in two convenient ways: a) by measuring the frequency shift of DSR peak 1 (ν_1) from the reference transition F_g=1 → F_e=2 (see the inset of Fig.<ref>), also taking into account the value of vdW shift; b) by measuring the frequency separations between the individual transitions a=ν_10-ν_1 and b=ν_2-ν_1, and exploiting the dependence of the a/b ratio on the B-field. Note that for case b) there is no need for a frequency reference, and also the value of vdW shift is not important, being the same for all four transitions. Both B-field measurement techniques are graphically elucidated in Fig.<ref>.The SR and DSR broadening observed with reduction of L and/or increase of T_R is caused by vdW interaction, as well as atom-window and atom-atom collisions. The measured DSR broadening γ_DSR versus L (cross marks) is presented in Fig.<ref>, along with fitted γ_DSR[MHz] ≈ 15000/L[nm] dependence (dotted line). Measuring magnetic field with nanometric-scale spatial resolution remains a challenge in a variety of problems, in particular when strong gradient fields are applied (e.g. 40 kG/mm in <cit.>). Even for L = 25 nm, γ_DSR in our experiment is 600 MHz, so measuring the ratio a/b for a NC filled with ^87Rb isotope, it is possible to determine the B-field value with rather high precision. For comparison of the DSR with the absorption spectrum under the same conditions (for parameters, see the caption of Fig.<ref>), we have used F_g=3 → F_e=3.4 transitions of Cs D_1 shown in the inset of Fig.<ref>, for which the frequency separation of excited levels, ∼ 1.2 GHz, is the largest among the alkali metals. The latter is advantageous for avoiding the overlap of absorption lines of neighboring transitions. In Fig.<ref> the upper curve shows the absorption spectrum for L ≈ 52 nm, and the middle curve presents DSR spectrum. The lowest line is the reference transmission spectrum obtained with an auxiliary NC with L=λ, exhibiting VSOP resonances at F_g=3 → F_e=3,4 transitions of ^133Cs <cit.>. The F_g=3 → F_e=3 and F_g=3 → F_e=4 transitions are strongly overlapped in the absorption spectrum, while DSR-peaks for the same transitions are obviously advantageous: they are background-free, completely resolved, and exhibit noticeable frequency red shift (∼ 140 MHz and 160 MHz, respectively). Estimation of C_3 vdW coefficient obtained from Eq.(<ref>) gives C_3=1.4±0.1 kHz×μm^3. Owing to the low divergence and relatively high power of the SR signal (≈ 0.5 % of the incident radiation), and its linear response remaining up to P_L ∼ 5 mW, and high signal-to-noise ratio, the DSR technique can be used for high-distance remote monitoring and mapping of both homogeneous and highly inhomogeneous B-fields in a wide range with ∼ 40 nm spatial resolution. The DSR technique employing nanometric-thickness cells is simple and easily realizable. We note that the recent development of a glass NC <cit.> can make this technique widely available, both for studies of atom-surface interaction and B-field mapping. The work was partially supported by MES RA (projects No. 15T-1C040 and 15T-1C277). The authors are grateful to A. Sarkisyan for fabrication of nanocell, and to A. Tonoyan for theoretical calculations. 99wood1 R.W. Wood, Phil. Mag. 18, 187 (1909).woerdman1 J.P. Woerdman and M.F.H. Schuurmans, Opt. Commun. 14, 248 (1975).nienhuis1 G. Nienhuis, F. Schuller, and M. Ducloy, Phys. Rev. A 38, 5197 (1998).weis1 A. Weis, V.A. Sautenkov, and T.W. Hänsch, Phys. Rev. A 45, 7991 (1992).failache1 H. Failache, S. Saltiel, M. Fichet, D. Bloch, and M. Ducloy, Phys. Rev. Lett. 83, 5467 (1999).hamdi1 I. Hamdi, P. Todorov, A. Yarovitski, G. Dutier, I. Maurin, S. Saltiel, Y. Li, A. Lezama, D. Sarkisyan, M.P. Gorza, M. Fichet, D. Bloch, and M. Ducloy, Las. Phys. 15, 987 (2005).bloch1 D. Bloch and M. Ducloy, Adv. At. Mol. Opt. Phys. 50, 91 (2005).fichet1 M. Fichet, G. Dutier, A. Yarovitsky, P. Todorov, I. Hamdi, I. Maurin, S. Saltiel, D. Sarkisyan, M.P. Gorza, D. Bloch, and M. Ducloy, Europhys. Lett. 77, 54001 (2007).whittaker1 K.A. Whittaker, J. Keaveney, I.G. Hughes, A. Sargsyan, D. Sarkisyan, and C.S. Adams, Phys. Rev. Lett. 112, 253201 (2014).gazazyan1 E.A. Gazazyan, A.V. Papoyan, D. Sarkisyan, and A. Weis, Las. Phys. Lett. 4, 801 (2007).sargsyan1 A. Sargsyan, E. Klinger, Y. Pashayan-Leroy, C. Leroy, A. Papoyan, and D. Sarkisyan, JETP Lett. 104, 224 (2016).sargsyan2 A. Sargsyan, H. Hakhumyan, A. Tonoyan, E. Klinger, A. Papoyan, C. Leroy, and D. Sarkisyan, J. Opt. Soc. Am. B (accepted), <https://arxiv.org/abs/1610.09807>.keaveney1 J. Keaveney, A. Sargsyan, U. Krohn, I.G. Hughes, D. Sarkisyan, and C.S. Adams, Phys. Rev. Lett. 108, 173601 (2012).sargsyan3 A. Sargsyan, G. Hakhumyan, R. Mirzoyan, and D. Sarkisyan, JETP Lett. 98, 441 (2013).zentile1 M. Zentile, J. Keaveney, L. Weller, D.J. Whiting, C.S. Adams, and I.G. Hughes, Comput. Phys. Commun. 189, 162 (2015).olsen1 B.A. Olsen, B. Patton, Y.Y. Jau, and W. Happer, Phys. Rev. A 84, 063410 (2011).sargsyan4 A. Sargsyan, G. Hakhumyan, C. Leroy, Y. Pashayan-Leroy, A. Papoyan, and D. Sarkisyan, Opt. Lett. 37, 1379 (2012).weller1 L. Weller, K. Kleinbach, M. Zentile, S. Knappe, I.G. Hughes, and C.S. Adams, Opt. Lett. 37, 3405 (2012).folman1 R. Folman, P. Kruger, J.G. Schmiemayer, J. Denschlag, and C. Henkel, Adv. At. Mol. Opt. Phys. 48, 263 (2002).whittaker2 K.A. Whittaker, J. Keaveney, I.G. Hughes, A. Sargysyan, D. Sarkisyan, B. Gmeiner, V. Sandoghdar, and C.S. Adams, J. Phys.: Conf. Ser. 635, 122006 (2015).

http://arxiv.org/abs/1702.07871v1

Hybrid method for identifying mass groups of primary cosmic rays in the joint operation of IACTs and wide angle Cherenkov timing arrays E B Postnikov^1,*, A A Grinyuk^2, L A Kuzmichev^1 and L G Sveshnikova^1 December 30, 2023 =======================================================================================================================================empty § INTRODUCTIONA plethora of distinct materials are currently studied under the common feature of their low-energy excitations, resembling those of massless Dirac fermions. In addition to graphene <cit.>, two and three dimensional topological insulators and topological crystalline insulators stand outas important examples of the so-called Dirac materials (see Ref. <cit.> and references therein). Kane and Mele <cit.>, based on seminal works by Thouless et al. <cit.> and Haldane <cit.> on the quantum Hall effect and its relation with topology, were the first to suggest in 2005 the existence of these new topological phases of matter. In 2006, Bernevig et al. <cit.> propounded a topologically insulating system, namely a HgTe/CdTe quantum well. Their prediction led to the detection in 2007 of non-trivial helical edge states, thus establishing the existence of the quantum spin Hall effect <cit.>.From the standpoint of applications, Dirac materials are foreseen to be of paramount importance due to their universal behaviour and the robustness of their properties, ultimately linked to symmetry <cit.>. Their band structure resembles the energy-momentum relation of relativistic massless particles where the energy dependence on the momentum is linear, hence the name of Dirac cones. Substantial effort is being devoted to control the slope of these cones, that is, the Fermi velocity. This parameter is essential for applications and it fully characterizes the Dirac cones.Fermi velocity modification has been predicted and observed in few-layer graphene due to the rotation of two neighboring layers <cit.>. Indeed, in twisted bilayer graphene, a change in the band velocity was found for low rotation angles <cit.>. This velocity reduction can be traced back to the interaction of the two Dirac cones, brought together due to band folding <cit.>. Additionally, other mechanisms have also been put forward to produce a velocity change. For instance, Hwang et al. demonstrated that the Fermi velocity is inversely proportional to the dielectric constant when the environment embedding graphene is modified <cit.>. Many-body effects can also affect the Fermi velocity. For example, due to the divergence of the screening length at the neutrality point in graphene, a renormalization of the Fermi velocity has been related to many-body effects <cit.>. This many-body effect has also been detected in a topological insulator, namely, Bi_2Te_3 <cit.>. Thus far, however, all these mechanisms apropos tailoring the Fermi velocity cannot be amply tuned at will, with the exception of changing the rotation angle in bilayer graphene, which implies structural modifications that are cumbersome in experiments. The aim of our work is to introduce a new mechanism, experimentally convenient, that enables fine-tuning of the Fermi velocity. We shall prove that double-gated Dirac materials are excellent candidates to verify this effect. Firstly, we predict a velocity reduction in a topological insulator/semiconductor (TI/S) junction where a uniform electric field is applied perpendicular to the interface. Surface states lying within the gap are known to occur and their description can be made in terms of a Dirac-like equation where the bandgap changes sign across the interface (see Refs. <cit.> and references therein), similar to the Jackiw and Rebbi model for solitons <cit.>. We have found that these surface states display a Dirac cone dispersion that widens as the field increases. The exact solution, although feasible, is rather intricate so it is left for the Supplemental Information. Instead, an approximate solution is presented and an analytical expression displaying the renormalization of the Fermi velocity is obtained.Secondly, bearing in mind the universal properties of Dirac materials, we verify the generality of our results in other Dirac systems, such as graphene nanoribbons (GNRs) and carbon nanotubes (CNTs), and employing a different method, namely, the tight-binding (TB) approximation. We show that both TB and low-energy continuum (Dirac equation) calculations confirm the possibility to adjust the Fermi velocity in Dirac materials by lowering the slope of the Dirac dispersion relation. This trend is also confirmed by ab-initio calculations. Hence, we conclude that the mechanism is completely valid for systems with a linear dispersion relation. Experimental proposals towards the detection of this phenomenon are discussed at the end of the paper. § RESULTS §.§ TI/S junction under bias: continuum model Throughout this section, we will focus on IV-VI semiconductors, among which the ternary compounds Pb_1-xSn_xTe and Pb_1-xSn_xSe can be found. The latter are known to shift from being semiconductors to topological crystalline insulators due to the band inversion that occurs at the L points of the Brillouin zone as the fraction of Sn increases <cit.>. Hence, a TI/S junction can be built using these materials.The envelope function calculation is a reliable method to obtain the electron states near the band edges in narrow-gap IV-VI semiconductors <cit.>. Keeping only two nearby bands and neglecting the influence of far bands, electron states near the L extrema are determined from the following Dirac-like Hamiltonian <cit.>ℋ_0=v_α_· p_+v_zα_z p_z +1/2 E_G(z)β+V_C(z) , where the Z axis is parallel to the growth direction [111]. It is understood that the subscriptof a vector indicates the nullification of its z-component. Here E_G(z) stands for the position-dependent gap and V_C(z) gives the position of the gap centre. α=(α_x,α_y,α_z) and β denote the usual 4× 4 Dirac matrices, α_i=σ_x⊗σ_i and β=σ_z⊗1_2, σ_i and 1_n being the Pauli matrices and n× n identity matrix, respectively. Here v_ and v_z are interband matrix elements having dimensions of velocity. Assuming that the surface states spread over distances along the growth direction much larger than the interface region, we can confidently consider an abrupt profile for both the magnitude of the gap E_G(z)=2Δ_Lθ(-z)+2Δ_Rθ(z) and the gap centre V_C(z)=V_Lθ(-z)+V_Rθ(z), where θ(z) is the Heaviside step function. The subscripts L and R refer to the left and right sides of the heterojunction, respectively. Note that in the case of a TI/S junction Δ_LΔ_R<0.The Hamiltonian (<ref>) acts upon the envelope function χ( r), which is a four-component vector involving the two-component spinors χ_+( r) and χ_-( r) belonging to the L^+ and L^- bands. Since the interface momentum is conserved, the envelope function can be written as χ( r)=χ(z)exp(i r_· k_), where χ(z) decays exponentially on each semiconductor. In the case of symmetric and same-sized gaps, namely V_C(z)=0 and E_G(z)=2Δ sgn(z), it is found that χ(z)∼exp(-|z|/d), with d=ħ v_/Δ and the interface dispersion relation is a single Dirac cone E( k_)=±ħ v_F| k_|, where v_F=v_ (see Ref. <cit.> for further details).We now turn to the electronic states of a single TI/S junction subjected to a perpendicular electric field F=Fz. Notice that k_ is still conserved. The calculation of the states in the quantum-confined Stark regime for general values of the parameters Δ_L,R and V_L,R is feasible but rather involved. In order to keep the algebra as simple as possible, we restrict ourselves to symmetric and same-sized gaps. This is not a serious limitation but the calculations are considerably simplified. The electrostatic potential is -eFz when the electric field is switched on and the corresponding Dirac equationbecomes (ℋ_0-eFz-E)χ(z)=0. Although this problem is exactly solvable (see Supplemental Information for details), we shall follow a more ingenious approach by introducing the Feynman-Gell-Mann ansatz, χ(z)=(ℋ_0+eFz+E)ψ(z) <cit.>. Defining the following dimensionless quantities κ= k_d, ξ=z/d, ε=E/Δ, f=eFd/Δ, we obtain [-d^2/dξ^2+U(ξ)-f^2ξ^2 - 2ε fξ-ifα_z+λ^2]ψ(ξ)=0 , where U(ξ)=2iβα_zδ(ξ) and λ^2=κ^2+1-ε^2 . Notice that we restrict ourselves to isotropic junctions (v_z=v_) for simplicity, an assumption that can easily be relaxed.In order to further simplify the problem, we will focus on electric fields smaller than F_C≡Δ/ed=Δ^2/eħ v_. In other words, we will assume that the potential energy drop across the distance d over which the surface states spread along the growth direction is smaller than half of the gap Δ. It should be stressed that this is the usual regime in experiments. Typical values for IV-VI compounds are Δ=75meV and d=4.5nm <cit.>, yielding F_C=170kV/cm, which is actually large. Thus, under the reasonable assumption that F < F_C one can neglect the quadratic term f^2ξ^2 in equation (<ref>). The constant matrix term -ifα_z is easily diagonalized by a unitary transformation. Nevertheless, we have checked that it has a negligible impact on the final results even at moderate fields and it will be omitted as well (see Supplemental Information for a comparison between the exact calculations and the approximate ones reported here).We can regard the term U(ξ) in (<ref>) as a perturbation and seek for the retarded Green's function of the unperturbed problem. It can be cast in the form 𝒢_0^+(ξ,ξ^';ε)=G_0^+(ξ,ξ^';ε)1_4, where the scalar Green's function obeys the following equation [-∂^2/∂ξ^2-2ε fξ+λ^2]G_0^+(ξ,ξ^';ε)= δ(ξ-ξ^') . where it is understood that Im(λ^2)<0. Since we are interested in states confined along the growth direction, we consider Re(λ^2)>0 in what follows. Equation (<ref>) is analogous to the problem of a non-relativistic one-dimensional particle in a tilted potential solved in Refs. <cit.>. Let us define μ=(2|ε| f)^1/3 ,p(ξ)=-s_εμξ+λ^2/μ^2 , with the shorthand notation s_ε=sgn[Re(ε)]. In terms of these parameters the retarded Green's function for the appropriate boundary conditions is G_0^+(ξ,ξ^';ε) = -π s_ε/μ{θ[(ξ^'-ξ)s_ε] Ai(p(ξ)) Ci^+(p(ξ^')) + θ[(ξ-ξ^')s_ε]Ai(p(ξ^')) Ci^+(p(ξ))} , where Ci^+(z)=Bi(z)+iAi(z), Ai(z) and Bi(z) being the Airy functions <cit.>. Once the retarded Green's function for the unperturbed problem is known, the Dyson equation is exactly solved to obtain the Green's function 𝒢^+(ξ,ξ^';ε) associated to Eq. (<ref>). This function is analytic in the lower half plane Im(λ^2)<0. Thus, it may have simple poles when it is analytically continued to the upper half plane <cit.>. In our case, the poles satisfy the condition [G_0^+(0,0;ε)]^2=1/4. From (<ref>) we finally get4π^2/μ^2[Ai(λ^2/μ^2)Ci^+(λ^2/μ^2)]^2=1 . Since the complex poles are found for Im(λ^2)>0, we must then have Re(ε)Im(ε)<0. That is, we may write the energy as ε_±=ε_r± iγ/2, where the plus (minus) sign is to be considered when ε_r is negative (positive). Here γ is a real positive number. Hence, considering the aforementioned prescription for the relative signs of the real and imaginary parts of the poles, we can find these by solving (<ref>). Poles correspond to resonant states with energy E(κ_)=ε_rΔ and level width Γ(κ_)=γΔ. Therefore, in the presence of the electric field, electrons can tunnel into the continuum and escape from the TI/S junction due to the non-zero imaginary part of the pole. This is a common feature in the quantum-confined Stark effect <cit.>. Nonetheless, in the next section we will show that the level width is exponentially small in the low-field regime, that is, tunnelling is only important at high fields.We can simplify (<ref>) in the low-field regime F<F_C by noticing that |λ|≫μ. In this limiting case we approximate the Airy functions to their asymptotic expansions for large argument and obtain (see the Supplemental Information for details)E( k_) = ±ħ v_(1-5/8F^2/F^2_C)| k_| ,Γ( k_) = Δ^2/ħ v_| k_| exp(-2Δ/3ħ v_| k_| F_C/F) .Equation (<ref>) is very remarkable and it is our main result. It means that applying an electric field perpendicular to the junction the interface linear dispersion remains but we are effectively lowering the Fermi velocity v_F(F)≡v_(1-5/8F^2/F^2_C) . We still get a cone but it becomes wider. In addition, as it was anticipated above, equation (<ref>) ensures that the tunnelling rate into the continuum is exponentially small in the low-field limit.In order to verify results (<ref>) and (<ref>), we numerically tested them from the exact numerical solution of equation (<ref>). Figure <ref>(a) shows the real part of the pole of 𝒢^+(ξ,ξ^';ε) as well as the approximate expression (<ref>) as a function of the interface momentum. At low electric field (F=0.2F_C) the dispersion is perfectly linear and the accuracy of our approximation (<ref>) is outstanding. Slight deviations from the linear behaviour appear upon increasing the electric field (F=0.8F_C) and the approximate slope is no longer valid, as expected. The maximum value of the electric field below which theapproximation (<ref>) holds can be estimated from figure <ref>(b). It displays the Fermi velocity as a function of the electric field for two different values of the interface momentum. It is seen that the approximate result (<ref>) fits the exact result in the range F ≲ 0.4 F_C. Figure <ref>(c) displays the level width obtained from the imaginary part of the pole of 𝒢^+(ξ,ξ^';ε) as a function of the inverse of the electric field for two different values of | k_|d. Solid lines show the approximate width given in (<ref>). There exists a very good agreement with exact results at moderate fields (F ≲ 0.4 F_C). It is worth mentioning that the exact level width is much lower than the approximate value at F ≃ F_C. Thus, we conclude that tunnelling is not important except at high fields.A qualitative explanation of the reduction of the Fermi velocity can be traced back to the quantum-confined Stark effect. For concreteness, let us focus on surface states with positive energy. Before the electric field is applied, surface states are exponentially localized at the junction, with gapless and linear dispersion ħ v_| k_|. When the electric field is adiabatically applied, it is well known that perturbation theory establishes that the energy is quadratically lowered with the field by an amount δ E (see figure <ref>). But the interface momentum is conserved if the field is perpendicular to the TI/S junction. Therefore, the Dirac cone widens, as schematically shown in figure <ref>, and the Fermi velocity is in effect quadratically lowered with the electric field. It should be stressed that equation (<ref>) becomes independent of the electric field if ε=0. Consequently ε=0 for κ_=0 is still an eigenenergy of the system when the electric field is applied and the dispersion remains gapless. In other words, the magnitude of the energy shift δ E due to the electric field must decrease upon decreasing the energy.In view of the above results, we consider that the Fermi velocity reduction is related to the existence of a Dirac cone, i.e., a linear dispersion relation, so it should be achievable in other Dirac materials. We choose carbon-based systems, such as GNRs and CNTs, as the easiest examples in which to validate this hypothesis with a simple TB model, which nonetheless includes other bands not present in the continuum approach presented before. Next section will be devoted to confirm this idea by applying a uniform electric field across a metallic armchair graphene nanoribbon (aGNR), a metallic armchair carbon nanotube (aCNT) and a metallic zigzag carbon nanotube (zCNT).§.§ Carbon-based Dirac materials: tight-binding and ab-initio calculationsIn order to elucidate the generality of our result, we check for its occurrence in carbon-based Dirac materials, such as metallic aGNRs, aCNTs and zCNTs. These systems can be easily described with a one-orbital nearest-neighbour hopping tight-binding approximation with an electric field term. TB calculations confirm that the Fermi velocity is remarkably reduced, and the results for the aGNR are further validated by means of a low-energy Dirac equation. Our results are summarized in figure <ref>, leaving the details of the calculations for the Supplemental Information.As can be immediately noticed, all systems display a significant reduction of the Fermi velocity as a function of the electric field. Our conjecture is then firmly established: the Fermi velocity can be reduced via a uniform electric field. In the case of a metallic aGNR, shown in figure <ref>(a), the agreement between the TB and Dirac approaches is noteworthy. It is worth mentioning that the approaches start to slightly differ when the normalized field f=F/F_W is f≳ 3, being F_W=ħ v_F(0)/e(W+a)^2, where W is the width of the aGNR, a=0.246nm, and v_F(0) is the Fermi velocity of unbiased graphene. Actually, a small gap opens up in the TB calculations due to the interaction with other bands. This gap is negligible in the region where the Dirac and TB approaches coincide. However, as it can be drawn from figure <ref>(a), these fields are remarkably large even from a practical perspective. For instance, an aGNR with W∼ 2nm has F_W∼ 0.1V/nm, meaning that the system remains effectively gapless even for very large applied fields of the order of a few tenths of V/nm. Boundary conditions ensuring the nanoribbon to be metallic <cit.> are preserved even in the presence of the electric field, thus not opening a gap for any value of F when the continuum description is considered. This is not the case for zGNRs, where TB calculations show that any strength of the electric field opens a sizable gap. Nevertheless, it is well-known that zGNRs do not show linear dispersion relations. In fact, for all the cases in which the material has a linear dispersion relation, i.e., a Dirac cone-shaped band, we find a similar behaviour, with a noticeable Fermi velocity reduction.We have also verified our results by means of ab-initio calculations based on the density functional theory (DFT). We explore the effect of the applied electric field on an N=5 aGNR, with N=2M+1, M being the number of hexagons across the nanoribbon. It turns out that this nanoribbon is near-metallic within this approach<cit.>. Figure <ref>(a) shows the DFT-calculated band structures with and without an external field of strength F=0.51V/Å. There is a clear reduction of the velocity in the linear parts of the valence and conduction bands closest to E_F, thus corroborating the observed tunability of the Fermi velocity with a more sophisticated approach. Figure <ref>(b) depicts the variation of the Fermi velocity as a function of the electric field, which presents the same quantitative behaviour observed with the continuum and tight-binding methods. Due to electronic correlations, electric polarizability and charge screening effects, a direct comparison between the effective or semi-empirical models and those used in DFT calculations is not straightforward. Nonetheless, the predicted tendency is clearly confirmed.§.§ Experimental proposalsThe most direct way to detect a variation of v_F in a Dirac material is to measure the energy dispersion relation in an ARPES experiment <cit.>. Notice that, due to the linear dispersion near the Fermi point, the conductivity does not depend on the particular value of the band slope in an undoped system. However, in a doped sample, a Fermi velocity dependence is expected, both in the pristine and disordered case, as in graphene <cit.>. Another way to detect the change in the Fermi velocity is to apply the electric field in a limited region and inject carriers from a field-free contact of the same system. Then the conductance shoulddecrease as a function of the applied field due to the wave vector mismatch <cit.>. In any case, one of the most representative features of the Fermi velocity modulation is the dependence of the Landau levels on v_F <cit.>. Thus, magnetotransport measurements will unambiguously evidence any modification of this parameter. The typical experimental uncertainty in the measurement of v_F is about 2% <cit.>, and this implies from (<ref>) that F should be larger than 0.2F_C to observe a change in the Fermi velocity in the TI/S junction. Nonetheless,this value is still within the low-field range mentioned above and, consequently, the approximate Fermi velocity given in (<ref>) holds in experiments. Other physical properties, such as the effective fine structure constant, related to the strength of many-body interactions, will also be affected by the variation of v_F and tuned by an external electric field during an experiment <cit.>. The reduction of the Fermi velocity is an effect with measurable consequences on several physical magnitudes, and we expect it to have applications for the design of novel devices based on Dirac materials. One possibility that deserves to be explored is the change in the electronic conductance in a Dirac material with a finite region where an electric field is applied. The wavevector mismatch due to the change of velocities may be used to control the conductance through the system, thus paving the way to devices based on this effect. § DISCUSSIONDirac materials share common characteristics due to their linear dispersions, also known as Dirac cones. A myriad of applications will benefit from their remarkable properties. However, there is an increased need to control their original properties by simple techniques so as to boost their potential applications. In this work, we propose the use of uniform electric fields to tune their electronic properties, proving that a significant Fermi velocity reduction can be efficiently achieved. Two archetypal families of Dirac systems are studied: a TI/S junction under a uniform electric field perpendicular to the interface, and carbon-based materials, such as metallic aGNRs and CNTs with an applied electric field. Analytical expressions are obtained for the TI/S junction by solving a spinful two-band model that is equivalent to the Dirac model for relativistic electrons. The mass term is half the bandgap and changes its sign across the junction. Under certain reasonable assumptions we have obtained a closed expression for the energy levels. These levels turn out to be narrow resonances upon applying the field. Inparticular, it is a remarkable result that the interface linear dispersion is preserved and the Fermi velocity is simply renormalized by the electric field. We have related the lowering of the Fermi velocity to the conservation of the interface momentum after the adiabatic increase of the electric field.Given the generality of our model, we postulate our results to be true for all Dirac materials. We have verified this in carbon-based Dirac systems such as metallic aGNRs and CNTs, where the same phenomenon is observed. Additionally, we suggest a series of feasible ways to detect the Fermi velocity reduction via spectroscopic and transport measurements.§ METHODSThe tight-binding numerical simulations were done with the tight-binding Hamiltonian on a honeycomb lattice with rectangular shape. Both the upper and lower edge are taken to be of armchair or zigzag type. The Hamiltonian can be written as H=H_0 + H_F, where H_0 is the kinetic energy term, H_0 = -t ∑_⟨ i,j⟩c_i^† c_j^, with t = 2.67 eV being the nearest-neighbour hopping energy and c_i, c_j^†the destruction and creation operators for an electron on sites i and j, respectively. The in-plane uniform electric field is applied transverse to the nanoribbon and its contribution is described as H_F= -e F·∑_ir_i c_i^† c_i^ where F is the electric field and r_i is the position of atomic site i. Density functional theory calculations were performed using the plane-wave self-consistent field plane-wave implemented in the Quantum ESPRESSO package<cit.> with the generalized gradient approximation of Perdew-Burke-Ernzerhof exchange-correlation functional<cit.>. Self-consistent charge calculations are converged up to a tolerance of 10^-8 for the unit cell. Ribbons are repeated periodically separated by 15Å of empty space in the perpendicular directions and hydrogen atoms saturate the dangling bonds of the edge carbon atoms. A fine k-grid of 25 × 1 × 1 Monkhorst-Pack is used to sample the Brillouin zone and the orbitals were expanded in plane waves until a kinetic energy cutoff of 815eV.All atoms are allowed to relax within the conjugate gradient method until forces have been converged with a tolerance of 10^-3eV/Å. The external electric field was included in the Quantum ESPRESSO calculations using two different approaches: by adding a sawtooth potential transversal to the ribbon direction<cit.> and through the modern theory of polarization (Berry phases)<cit.>. Both approaches confirm the tunability of the Fermi velocity.§ ACKNOWLEDGEMENTS The authors are indebted to M. Saiz-Bretín and A. Ayuela for helpful discussions. A. D-F. and F. D-A. thank the Theoretical Physics Group of the University of Warwick for their warm hospitality. L. C. gratefully acknowledges the hospitality of the Donostia International Physics Center. This work was supported by the Spanish MINECO under grants MAT2013-46308, MAT2016-75955, FIS2015-64654-P, FIS2013-48286-C02-01-P and FIS2016-76617-P, by the Basque Government through the ELKARTEK project (SUPER) and the University of the Basque Country (Grant No. IT-756-13).§ AUTHOR CONTRIBUTIONS STATEMENTA.D-F. and F.D-A. performed the analytical calculations based on the Dirac equation. J.W.G. performed the numerical simulations in carbon-based systems. A.D-F., L.C., J.W.G. and F.D-A. all contributed to conceptual developments and manuscript preparation.§ ADDITIONAL INFORMATION Supplementary information accompanies this paper at http://www.nature.com/srep;Competing financial interests: The authors declare no competing financial interests.

http://arxiv.org/abs/1702.08296v2

Department of Optics, Palacký University,17. listopadu 1192/12,771 46 Olomouc, Czech Republic Quantum nonclassicality is the basic building stone for the vast majority of quantum information applications and methods of its generation are at the forefront of research. One of the obstacles any method needs to clear is the looming presence of decohorence and noise which act against the nonclassicality and often erase it completely. In this paper we show that nonclassical states of a quantum harmonic oscillators initially in thermal equilibrium states can be deterministically created by coupling it to a single two level system. This can be achieved even in the absorption regime in which the two level system is initially in the ground state. The method is resilient to noise and it may actually benefit from it, as witnessed by the systems with higher thermal energy producing more nonclassical states.42.50.Dv, 42.50.Ct, 42.50.Wk Deterministic nonclassicality for quantum mechanical oscillators in thermal states Petr Marek, Lukáš Lachman, Lukáš Slodička, Radim Filip December 30, 2023 ==================================================================================§ INTRODUCTIONClassicality and nonclassicality are concepts which originated in quantum optics <cit.>. There, certain quantum phenomena such as coherent states <cit.>, their statistical mixtures, and their interaction with linear optics elements, can be fully explained by the classical coherence theory. Such quantum states are now denoted as classical and all other states as nonclassical. The nonclassicality is not just a nomenclature; it has been recognized as an important resource for the modern quantum technology. It has a very close relation to quantum entanglement: entangled states can be created from nonclassical ones simply by linear energy splitting <cit.>. It can be also used in communication to ensure secure transfer of information <cit.>, in metrology to enhance precision of measurements beyond their classical limits <cit.>, and it is a necessary condition for achieving quantum computation <cit.>. That is why, in quantum optics, a significant attention has been given to preparation and manipulation of these states <cit.>.The nonclassicality is not limited to quantum optics, it can be present in any continuous variables quantum system and it possesses the same inherent value there. One such class of systems is the mechanical oscillator. For the motional mode of a trapped ion <cit.>, it is currently possible to implement conditional high-quality coherent preparation of nonclassical states<cit.>. Nonclassical states of mechanical systems have been recently prepared also in quantum optomechanics <cit.>. There, the preparation of more advanced forms of nonclassicality was also suggested either by remote preparation or by coherent transfer <cit.> and the coherent transfer itself has been verified for microwave radiation <cit.>. Interfaces based on similar coherent transfer could be in principle used for a flawless transfer of an arbitrary nonclassical quantum state <cit.>. One thing these methods have in common is the need for a coherent pump and coherent measurements. One may therefore wonder if that is a fundamental limitation or if it is possible to deterministically generate strong nonclassical features without them. As a first step in this direction we have recently proposed a toy model for thermally driven generation of mechanical squeezed states and proposed its all-optical simulation <cit.>.In this paper we point out the strong nonclassical properties of the very common energy conserving Jaynes-Cummings interaction. Some of these properties are well known in the field of quantum optics, where a highly nonclassical states of light in a superconducting cavity may be prepared and processed by sequences of atoms passing through it <cit.>. It has been also shown that Jaynes-Cummings interaction can transform classical coherent light into nonclassical <cit.>. As a simple illustrative example, consider an excited two level system interacting with oscillator in a vacuum state. For certain value of interaction parameters, one quantum can be completely transferred to the oscillator leaving it in a nonclassical Fock state. However, this requires coherent control of the system, because thermal excitation cannot cause the excited state to be dominantly populated. In the following we shall assume that such thermal control is unavailable and that the systems can be prepared only in thermal equilibrium states. We are going to show that even under these constrains, nonclassical states of the oscillator can be deterministically prepared. This extension is not trivial because mixtures of quantum states do not, in general, exhibitthe nonclassical properties of thir constituents <cit.>. Furthermore, the produced nonclassicality is not only robust with regards to the initial temperature and mixedness, in some specific scenarios it increases with it. This is interesting for quantum optics, but it is of practical importance for systems which need to be concerned about cooling. The realization that Jaynes-Cummings interaction can be seen as a thermally autonomous nonclassical process (TANP) will be stimulating for the development of quantum optomechanics with solid state two-level systems <cit.>, which can strongly benefit from the ability to deterministically generate nonclassicality without the need for ground state preparation. We provide a detailed proof-of-principle analysis of this effect and for its feasible experimental verification suggest Klyshko's criteria of nonclassicality <cit.>. Finally, we assess the current versions of trapped-ion experiments and reveal that they are sufficient to demonstrate this thermally-driven and nearly autonomous generation of nonclassical mechanical states.§ SPONTANEOUS GENERATION OF NONCLASSICALITY BY JAYNES-CUMMINGS MODELIn quantum physics, the elementary interaction between single two-level system and oscillator is an unitary and energy conserving process described by a full Jayness-Cummings Hamiltonian H=H_0+H_I, where H_0 is the free evolution Hamiltonian of the joint system and H_I describes the interaction between its parts <cit.>. The free evolution Hamiltonian H_0=ħωσ_Z/2+ħν a^†a depends on transition frequency ω=E_e-E_g/ħ of the two-level system, where E_g and E_e are the energies of the ground |g⟩ and the excited |e⟩ states, respectively, and ν denotes the frequency of the oscillator. The operator σ_Z is inversion operator for the two-level system and a^† and a are the creation and annihilation operators for the oscillator. We denote Δ=ω-ν as the detuning parameter. In the rotating-wave approximation, Jaynes-Cummings interaction Hamiltonian H_I=ħ g (σ_+a+a^†σ_-) describes the processes of energy exchange between the oscillator and the two-level system, where g is interaction strength and σ_+,σ_- are two-level raising and lowering operators, respectively.The resonant Jaynes-Cummings interaction with Δ = 0 that coherently runs over a time interval t can be represented by unitary operatorU_JC =A_gg(t) ⊗ |g⟩⟨ g| + A_ee(t)⊗ |e⟩⟨ e| +A_eg(t) ⊗ |e⟩⟨ g| + A_ge(t)⊗ |g⟩⟨ e|,whereA_gg(t)=cos(gt√(n)) A_eg(t)= a sin(gt√(n))/√(n), A_ee(t)=cos(gt√(n+1)) A_eg(t)= - a^†sin(gt√(n+1))/√(n+1),and n = a^† a is the number of quanta operator. Over the short interaction time we consider coupling of all the participating systems to their respective thermal baths, and other sources of decoherence, as negligible. Operators (<ref>) govern both the discrete exchange of energy between the oscillator and the two level system represented by the annihilation and creation operations, and the continuous nonlinear transformation of the oscillator state performed by the periodic functions of √(n) and √(n+1). In past, these terms have been found responsible for the vacuum Rabi oscillations and the collapse-revival phenomena in the population of two-level systems <cit.>.In the following, we shall abandon the two level system and focus on the nonclassicality of the state of the oscillator. Furthermore, we shall assume only a limited control over the entire situation, having both the oscillator andthe two-level system at the beginning in thermal equilibrium states with nonzero temperatures T_1 and T_2, respectivelly. The situation is schematically depicted in Fig. <ref>. The density operators of the initial thermal states can be expressed asρ_TLS = p_e|e⟩⟨ e| + (1-p_e) |g⟩⟨ g|with p_e=exp(-ħω/k_B T_2)/1+exp(-ħω/k_B T_2) for the two level system andρ_th = ∑_n= 0^∞^n/(1+)^n+1 |n⟩⟨ n|for the mechanical oscillator. The mean excitationcan be related to the temperature of the system by = [ exp(ħω/k_B T_1) - 1]^-1. Under this consideration, the evolution of the oscillator's initial state (<ref>) can be obtained by applying a completely positive map asρ_out = ∑_i = g,e[ p_e A_ei(t)ρ_th A_ei^†(t) + (1-p_e) A_gi(t)ρ_thA_gi^†(t)],where the respective operators (<ref>) can be interpreted as Kraus operators of the transformation. The nature of the operation depends on the initial temperatures of the systems. When the temperatures are equal and p_e = / (1+2), there is no energy exchange and the systems are completely unchanged by the operation. In other cases there is an energy flow between the systems going in the direction of lower temperature. When the two level system is cooler than the oscillator, it absorbs energy from it and vice versa.The nonclassicality of quantum state is defined as the impossibility to express the state as a mixture of coherent states <cit.>. In the state produced by the absorption, it can originate from two sources. The first source is the addition of single quanta from the initial excitation of the two level system. This is best seen in the limit case in which the oscillator is initially in the ground state with = 0 and the atom is thermally excited. This results in the oscillator in stateρ_out=p_e(cos^2 gt|0⟩⟨ 0|+sin^2 gt |1⟩⟨ 1|)+(1-p_e)|0⟩⟨ 0|,which is nonclassical for any gt≠kπ and any p_e>0, as can be straightforwardly verified by applying Klyshko's nonclassicality criterion P_0P_2/P_1^2<1/2, where P_k = ⟨ k|ρ_out| k⟩, <cit.>. If we desire a quantitative statement, we can employ the entanglement potential <cit.>, which is defined as the amount of entanglement contained in the stateρ_split =e^π/4(ab^† - a^† b)ρ_out⊗|0⟩⟨ 0| e^-π/4(ab^† - a^† b),where a and b are the annihilation operators for the first and the second oscillator mode, respectively. The original optical realization of this operation relies on dividing the optical mode on a balanced beam splitter, but the basic property of the coupling - the inability to create entanglement unless the state is nonclassical - is preserved for any physical system. The entanglement itself can be quantified with help of logarithmic negativity <cit.>:LN(ρ_split) = log_2 ρ_split^PT,where the superscript PT denotes partial transposition and A= √(A^†A) represents trace norm. This measure has been originally proposed in <cit.> and it should be noted that it is not unique. Other measures of nonclassicality <cit.> can lead to qualitatively different conclusions. For example, according to <cit.>, Fock states are more nonclassical than Gaussian squeezed states, but <cit.> suggest otherwise. The entanglement potential itself depends on the particular measure of entanglement employed and the results may differ accordingly. For example, when Gaussian states are considered, logarithmic negativity and entanglement of formation have been shown to be inequivalent measures <cit.>. In the following we shall be using (<ref>) exclusively, because it can be easily numerically evaluated for high-dimensional non-Gaussian states. In order to highlight this choice we shall be calling our entanglement measure logarithmic negativity potential (LNP) and using it wherever an amount of nonclassicality is discussed.In the specific case when the oscillator is in the ground state, we can calculate LNP explicitly as (ρ_out)=log_2 [2N(ρ_out)+1], where(ρ_out)=√((1-p_esin^2 gt)^2+p_e^2sin^2 gt)-(1-p_esin^2 gt) <cit.>. We can see that it indeed is positive for all gt≠kπ and any p_e>0. Furthermore, if we consider short timescales with gt ≪ 1, the logarithmic negativity can be expressed as (ρ_out)≈1/ln 2(gt)^4p_e^2 and we see that it quickly increases with the energy (i.e. the temperature) of the initial two level system. The maximal amount of produced entanglement, which is _max=0.5 corresponding to half of an e-bit, is obtained in the limit of high temperature of the two level system with p_e = 1/2. This kind of nonclassicality has been previously discussed and experimentally verified by adding a single quantum to a quantum oscillator in a thermal state <cit.>. Our case is limited because we assume that the two level system can be excited only thermally and as a consequence, perfect addition is impossible and the obtained nonclassicality is lower. On the other hand, in our case there is a second source of nonclassicality which becomes dominant for higher initial temperatures of the oscillator.This second source of nonclassicality is the modulation of the oscillator's density matrix elements. Over the course of the interaction, the original density matrix with Bose-Einstein thermal statistics (<ref>) is transformed intoρ(t)= ∑_n=0^∞ |n⟩⟨ n| ^n/(1+)^1+n ×{ p_e[ cos^2(gt√(n+1)) + 1+/sin^2(gt√(n))]+ (1-p_e)[ cos^2(gt√(n)) + /1+sin^2(gt√(n+1))]}.The irrationally oscillating sine and cosine terms reshape the number distribution and can cause some Fock states to be more distinctive, which contributes to the nonclassicality of the state. However, this effect cannot be evaluated as straightforwardly as the previous examples and numerical methods are required for more detailed analysis. The amount of LNP present in states generated from thermal states by Jaynes-Cummings interaction with two level systems in various initial states is presented in Fig. <ref>. Fig. <ref>a shows LNP when the two level system was initially in the ground state and the procedure can be interpreted as a special kind of absorption. Here we can see that the amount of nonclassicality fluctuates with the strength of the interaction, but, more importantly, it grows with the initial energy of the oscillator. This is a very interesting behavior. It demonstrates that the absorption by a single two level system can be interpreted neither as linear absorption also known as loss, nor as subtraction of a single quantum represented by applying annihilation operator a, because both of these operations produce no nonclassicality when applied to a thermal state. In our case, the nonclassicality is caused by the oscillating sine and cosine terms and higher initial energy of the thermal state then allows higher Fock states to be affected, stand out, and contribute to nonclassical behavior.When the two-level system has the maximal thermal energy corresponding to p_e = 1/2, which is the situation covered by Fig. <ref>b, the NLP is maximal when the oscillator is in its ground state and it decreases with increasing the energy of the oscillator. Finally, Fig. <ref>c shows what happens when the two level system is at some intermediate nonzero temperature. We can see that NLP starts positive for the zero energy oscillator and decreases, similarly as in Fig. <ref>b. The overall values of NLP are lower, though, which is the consequence of lower proportion of the excited state. The nonclassicality decreases with energy of the oscillator until it reaches a minimal value of zero after which it starts growing again, this time similarly as in Fig. <ref>a.By looking at the trio of enlisted situations side by side, we can see that for obtaining larger nonclassicality we need to couple systems with vastly different energies. This is the reason why there is a local minimum valley in Fig. <ref>c and why in Fig. <ref>b LNP tends to zero asincreases. Specifically, p_e = 1/2 corresponds to infinite temperature, so any increase ofapproaches the situation in which the temperature of the two systems is equal.The generated nonclassicality is arising from abnormally strong presence of certain higher Fock number states. Here, by abnormally strong we mean much higher than could be expected from a mixture of classical states. Higher initial temperature of the oscillator then increases populations of higher Fock states and allows them to be redistributed by the sine and cosine terms of (<ref>).This is the reason why the nonclassicality grows with the temperature of the initial thermal state. To illustrate this feature, let us look at how the entanglement of the split state (<ref>) depends on dimension of the Hilbert space. To do that, we shall consider a projection of the total state (<ref>) onto a D^2 dimensional subspace, which gives us a new density operatorρ_D ∝ F(D)⊗ F(D) ρ_split F(D)⊗ F(D),where F(D) = ∑_k = 0^D |k⟩⟨ k| is a conditional filtering operation representing local scissors.This probabilistic operation can be used to simulate the effects of imperfect photon number resolving measurements which do not allow us to obtain a full density matrix but only a few of its lowest order elements. Logarithmic negativity that can be obtained is depicted in Fig. <ref> for a single illustrative scenario. We can see that the entanglement can be observed, but as the initial mean energy of the oscillator grows, higher dimension of the density matrix is required. This implicates higher Fock states as those responsible for the generated entanglement and the original nonclassicality.Another interesting feature, which can be seen in Fig. <ref>a, is that for higher thermal energy of the initial state there are two distinct local nonclassicality maxima, which appear when interaction coefficients are gt = π and gt = π√(2). These values are not unique and similar trend may be observed whenever the interaction constant is gt = π√(k), where k is an integer. When the two level system is in the ground state and the operation can be therefore interpreted as an absorption, these values ensure that occupancy of some particular number states cannot decrease. As a consequence, when the oscillator is repeatedly exposed to a two-level system in a ground state which absorbs some of its energy and later is traced over, the state of the oscillatorρ = ∑_m = 0^∞ p_m |k m^2⟩⟨ k m^2|instead of vacuum. This is a nonclassical state and its nonclassicality expressed by LNP increases with the initial thermal energy and quickly grows with the number of times the absorption is repeated, as can be witnessed by looking at Fig. <ref>.§ FEASIBLE NONCLASSICALITY DETECTIONSo far we have been analyzing the nonclassicality of quantum states of oscillators from a sort-of-omniscient point of view. We have been employing the knowledge of full density matrices, which is something that cannot always be fully obtained in realistic conditions. For practical purposes it is therefore necessary to consider nonclassicality criteria, or witnesses, that can be employed with only a partial knowledge of the quantum state. A powerful hierarchy of such sufficient nonclassicality conditions was introduced by Klyshko <cit.>. Each of these conditions depends only on a select few diagonal elements of density matrix corresponding to probabilities of finding a specific number of excitations P_n = ⟨ n|ρ |n⟩. The individual conditions are(n+1) P_n-1 P_n+1-n P_n^2 < 0,where n = 1,2,…, and none of them can be satisfied by a quantum state that can be expressed as a mixture of coherent states. As a consequence, finding a single satisfied condition decisively marks the scrutinized quantum state as nonclassical.We can apply this hierarchy of conditions to states produced by the Jaynes-Cummings interaction in order to see which scenarios lead to nonclassicality. Let us start with the perfect saturable absorption where the two level system is initially in its ground state. Fig. <ref>a shows that when the total interaction coefficient is larger than some minimal value, roughly gt ≥π/2, the nonclassicality can be found with just the first three criteria from hierarchy (<ref>) for arbitrary nonzero values of . This indicates that in a possible experimental realization, the autonomous deterministic generation of nonclassical states from thermal energy can be verified straightforwardly. Exact numerical tests were performed for values approaching gt = 50 and = 50. Outside of this range of parameters we may take advantage of employing approximations. In the limit of high mean excitation n̅≫ 1 the Bose-Einstein statistics of the initial state is simplified to P_n ≈ 1/.In this regime, the statistics of excitations of the oscillator (<ref>) after the interaction isP_n≈(cos^2 g t √(n)+sin^2 g t √(n+1))/n̅=F_n/where function F_n is introduced to abbreviate the notation. Employing hierarchy (<ref>) gives raise to asymptotic conditions for nonclassicality n F_n^2-(n+1)F_n+1F_n-1>0. This condition can be satisfied for arbitrary , provided we can choose suitable parameters gt and n (see Appendix A for details).When the two-level system is not in its ground state, the nonclassicality becomes more difficult to detect and it vanishes completely when the temperatures of the system equalize. An example of this can be seen inFig. <ref>b where, for p_e = 1/3, the nonclassicality can be detected in the regions removed from the point of equality (= 1), even though higher orders of the criteria are generally required.The extent to which the absorption by a thermally excited two level system can induce nonclassicality can be again investigated in the limit of high . Using again the approximation (<ref>) the criteria can be expressed asn[2 p_e+F_n(1-2p_e)]^2 -(n+1)[2 p_e+F_n-1(1-2p_e)] × [2 p_e+F_n+1(1-2p_e)]>0.In the regime of maximal violation with F_n = 2 and F_n-1 = F_n+1 = 0 (Appendix A) the criteria then reduce top_e<√(n^2+n)-n.This formula couples the thermal energy of the two level system with the order of criteria needed for detecting some nonclassicality. It was derived under the assumption of large n, but numerical analysis reveals that it approximatively holds even for lower orders, see Fig. <ref>.Finally, let us look at the opposite regime, where both the interaction strength and the initial thermal energy are small and the two level system is again in the ground state. In this regime, the nonclassicality is difficult to detect and the reason for this is straightforward: when gt = 0 no interaction happens and when = 0 there is no energy in the oscillator to be absorbed and redistributed. However, the nonclassicality can still be detected, it just requires high orders of the hierarchical criteria. In Fig. <ref> we can see that while the first three orders of the hierarchy see nonclassicality only in isolated regions, higher orders allow for continuous coverage. § EXPERIMENTAL EMULATION OF THERMALLY AUTONOMOUS NONCLASSICAL PROCESSESThe presented calculations suggest that non-classical processes are likely present in rich variety of thermal physical systems occurring in nature and might lead to generation of non-classical states or entanglement in environments with large thermal energy. Observation of non-classical states of mechanical oscillators or collective oscillations in mesoscopic solid-state systems corresponds to a long-standing challenge in a large part of experimental quantum optics community and proved profitability of the amount of non-classicality with respect to the initial oscillator temperature might greatly simplify such pursuits.To further stimulate experimental developments based on TANP, it would be very valuable to emulate the calculated temperature dependence of TANP in some well controllable and accessible optomechanical experimental platform. Single ions trapped in Paul traps seem to be excellent candidates for such task. They offer great control of the temperature of initial thermal state after laser cooling stages and recent experimental results show unprecedent precision of ion's motional state reconstruction <cit.>. The proposed TANP and the corresponding JC interaction can be straightforwardly implemented by excitation of the ion on a dipole forbidden transition with a laser frequency detuned from the resonance by ω_T to the red, where ω_T is the frequency of the ion's motion. If we assume some typical experimental parameters corresponding to ω_T = 2π× 1MHz, Lamb-Dicke parameter on the employed transition of η∼ 0.1 and carrier Rabi frequency Ω = 2π×100kHz , we get coupling strength g ∼ηΩ = 2π×10kHz. The number of coherent Rabi cycles on first motional sideband for a single ion can easily reach numbers much higher than considered gt=10 oscillations without any substantial decrease of the oscillation contrast <cit.>, so we can neglect the effect of finite coherence in the following. The mean phonon number of the initial thermal state can be well controlled by the effectiveness and duration of the Doppler and sideband cooling processes with resolution well below single motional quanta  <cit.>. The non-classicality of the resulting state after TANP can be conveniently proved by estimation of the particular phonon populations <cit.> and evaluation of the corresponding Klyshko non-classicality conditions (<ref>). Fig. <ref>a suggests that for excitation pulse areas gt>2, the generated non-classicality should be always detectable for any initial temperature of the oscillator. However, if we assume common measurement scheme of the phonon populations P(n) based on driving the first blue sideband and conservative measurement error on individual population measurements of the five lowest populations σ_P(n) = 0.01 <cit.>, the region of the observable non-classicality slightly reduces, see Fig. <ref>. This is mainly caused by the high requirements for measurement precision in the region of the small thermal energy of initial motional state. Non-classicality regions for higher initial thermal populations clearly demonstrate how thermal energy entering TANP improves the observability of the initial generated non-classicality. In the presented simulation, experimentally feasible regions have been estimated by taking the worst possible case of combinations of individual population probabilities shifted by σ_P(n) for the given evaluated criterion from hierarchy (<ref>). We have assumed that experiment operates deep in the Lamb-Dicke regime and so the internal population after the TANP can be reshuffled using some auxiliary electronic level to the ground state of the two-level system without any relevant change of the motional state.§ CONCLUSIONThermally autonomous nonclassical processes are capable of deterministically producing nonclassical quantum states without an external coherent driving force, exploiting only the short time coherence during the interaction of the studied system and its thermal environment. A prominent example of such process, the broadly studied Jaynes-Cummings interaction, can change a thermal state of a quantum mechanical oscillator into a nonclassical state just by passive energy exchange with a two level system. This allows preparation of nonclassical states without the need for ground state cooling of the oscillator, which is a promising prospect for the recently developed field of solid state quantum optomechanics <cit.>. We have analyzed this thermally autonomous nonclassical process in detail, quantifying the produced nonclassicality by means of logarithmic negativity potential. We have found that, in case of absorption caused by a two-level system in the ground state, the nonclassicality of the produced state grows with its initial temperature - a rather surprising result, as quantum mixedness caused by the high temperature has traditionally been an enemy of nonclassicality. We have proposed experimentally feasible criteria for detecting the generated nonclassicality and suggested experimental verification of the effect for the vibrational modes of trapped ions. We believe that this opens the path for a direct preparation of nonclassical states even for quantum mechanical oscillators with limited control and ability to be cooled.Acknowledgements: We acknowledge project GB14-36681G of the Czech Science Foundation. L.L. acknowledges IGA-PrF-2015-005.§ APPENDIX A: MAXIMAL VIOLATION OF CRITERIA IN LARGE ENERGY LIMITIn the limit of high thermal energy of the oscillator, the number distribution can be expressed asP_n ≈F_n/,where F_n=cos^2 g t √(n)+ sin^2 g t √(n+1). The criteria of nonclassicality can be then rewritten asn F_n^2 - (n+1)F_n+1F_n-1 > 0.This expression can be maximized when F_n = 2 and F_n-1 =F_n+1 = 0. 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http://arxiv.org/abs/1702.08269v1

#1.#2⟨#1 #2⟩ #1.#2[#1 #2]tree⟨ ⟩M^ M^ A^ A^ College of Science, Swansea University, Swansea, SA2 8PP, UK We show how the S-matrix of an extended theory of gravity defined by its three-point amplitudes can be constructed by demanding factorisation.Theresultant S-matrix has tree amplitudes obeying the same soft singularity theorems as Einstein gravity including the sub-sub-leading terms. § INTRODUCTION Scattering amplitudes are traditionally defined from a quantum field theory and the resultingFeynman vertices and Feynman diagrams.Alternatively, the amplitudes can be regarded as the fundamental objects which define the theory perturbatively. It is not very useful to define a theory by specifying the entire S-matrix explicitly but it is an important question whether the S-matrix can be defined from a minimal set of data and rules i.e. a "diagrammar" <cit.>.Once a minimal set of amplitudes is specified we aim to construct all other amplitudes by demanding they have the correct symmetries and singularities. Defining the S-matrix using its singularities is a long-standing programme which is stillactive and fruitful <cit.>. In this letter we build an S-matrix from a set of three-point amplitudes using their singularity structure. The S-matrix corresponds to a theory ofEinstein gravity extended by the addition of R^3 terms. We are working with massless theories and view the amplitude as a function of the twistor variables λ_i^a and λ̅_i^ȧ, M(λ_i,λ̅_i).The spinor products i.j,i.j arei.j=ϵ_abλ^a_iλ^b_j, i.j=ϵ_ȧḃλ̅^ȧ_iλ̅^ḃ_j. In this formalism amplitudes have a well-defined “spinor weight”.Counting λ_i as weight +1 and λ̅_i as -1, then the amplitude hasweight +4 fora negative helicity graviton and -4 for a positive helicity graviton. We define the theory starting with the the usual three-point amplitudes of Einsteingravity:[We remove a factor of i(κ/2)^n-2 from the n-point amplitude.]V_3( 1^-, 2^- , 3^+) = 1.2^6 1.3^23.2^2,V_3( 1^+, 2^+, 3^-) = 1.2^6 1.3^23.2^2,V_3( 1^+, 2^+ , 3^+) = V_3( 1^-, 2^- , 3^-)=0 .These amplitudes have the correct spinor weight and are quadratic in the momenta.These amplitudes are only defined for complex momenta. For an on-shell three-point amplitude the condition k_1+k_2+k_3=0 demands k_1· k_2 =0 etc.For real momenta this implies i.j=i.j=0 and the vertices are all zero. However if we considercomplex momenta then we can have λ_1∼λ_2 ∼λ_3 but i.j ≠ 0. The tree amplitudes for Einstein gravity can be computed recursively startingfrom these <cit.>. We show that a similar construction can be used for an extended theory. We extend this theory by adding additional three-point amplitudes which are of higher power in momenta.To be non-trivial,these three-point amplitudes must either be functions of i.j or i.j exclusively.The simplest polynomial amplitudes arisewith six powers of momenta and are V^α_3( 1^-,2^- , 3^-)= α1.2^2 2.3^23.1^2,V^α_3( 1^+, 2^+, 3^+)= α1.2^22.3^23.1^2where α is an arbitrary constant. We also haveV^α_3( 1^-,2^- , 3^+)=V^α_3( 1^+,2^+ , 3^-)=0 ,there being no polynomial function with the correct spinor and momentum weight.These are essentially the unique choice for a three-point amplitude <cit.>.The amplitudes in this theory can be expanded as a power series in α,M_n(1,⋯ , n) =∑_r=0α^rM_n^(r)(1,⋯ , n)where M_n^(0) is the Einstein gravity amplitude.Here we focus on the r=1 part of the extended theory.This being the leading deformation of the theory from Einstein gravity. The theory we are considering would arise using field theory methods from the LagrangianL= ∫ d^D x √(-g) (R +C_αR_abcdR^cdefR_ef^ab )where C_α=α/60.However we note that to do so would involve determining increasingly complicated n-point vertices as the Lagrangian is expanded in the graviton field. As we will see the three-point amplitudes are sufficient to completely determine the S-matrix.The key element is thatthe entire S-matrix is determined from these vertices if wedemand that the amplitudesfactorise on simple poles.Specifically, for any partition of the external legs into two sets, {k_L_1,k_L_2⋯,K_L_l} and {k_R_1,k_R_2⋯,k_R_m}with l+m=n and l,m≥ 2, if K =∑_j=1^l k_L_j, then when K^2⟶ 0 the amplitude is singular with the simple pole beingM_n^ ⟶^K^2 → 0∑_λ=±[ M_l+1^(k_L_1, …, k_L_l, -K^λ)iK^2M_m+1^(K^-λ, k_R_1, …, k_R_m) ].We can excite the pole in K^2 by shifting to complex momenta and applying methods of complex analysis. There are two shifts which we use to generate the S-matrix.Firstly there is the originalBritto-Cachazo-Feng-Witten (BCFW) shift <cit.>,λ_i ⟶λ_i+zλ_j,λ̅_j ⟶λ̅_j- zλ̅_i.For Einstein gravity this shift is sufficient to generate the tree level S-matrix <cit.>. Additionally we can use the Risagershift <cit.>,λ_i ⟶λ_i +zj.kλ_η , λ_j ⟶λ_j +zk.iλ_η , λ_k ⟶λ_k +zi.jλ_η ,where λ_η is an arbitrary spinor. Both shifts change the momenta to be functions of z whilst leaving all momenta null and preserving overall momentum conservation.We need both shifts to construct the S-matrix for the extended theory.By considering the integral∫_γ M(z)z where γ is a closed contour,provided M(z) vanishes at infinity the unshifted amplitude, M(0),can be obtained from the singularities in the amplitude.These occur at points z_i where K_i^2(z)=0. At these points,K_i^2(z)= - ( z-z_i)z_i× K_i^2(0)and we obtain,M_n^ (0) =∑_i,λM^,λ_l_i+1(z_i) i K_i^2(0)M^,-λ_m_i+1(z_i),where the summation over i is only over factorisations where there areshifted legs on both sides of the pole. This is the on-shell recursive expression of <cit.>.Note that if M(z) does not vanish at infinity this does not imply factorisation is insufficient to determine the amplitude but only that that particular shift can not be used to engineer the amplitude. Expressions obtained from (<ref>)are not manifestly symmetric as the choice of shift legs breaks crossing symmetry, however symmetry is restored in the sum.This is a highly non-trivial check that the amplitude has been computed successfully. § GENERATING THE AMPLITUDES In this section we give some of the details of the process of generatingtheleading α contribution to the S-matrix.Four-Point Amplitudes:The three-point amplitudes are our inputs so the first outputs are the four-point amplitudes. There are three independent helicity configurations,M_4( 1^+,2^+,3^+,4^+),M_4( 1^-,2^+,3^+,4^+),M_4( 1^-,2^-,3^+,4^+) .Of these the first two are vanishing in Einstein gravity with only the last being non-zero: which is consequently termed the “Maximally-Helicity-Violating”(MHV) amplitude.For M_4^(1) the reverse is true: M_4^(1)( 1^-,2^-,3^+,4^+) =0 since there are no possible factorisations,while M_4^(1)( 1^+,2^+,3^+,4^+) and M_4^(1)( 1^-,2^+,3^+,4^+) are non-zero.The factorisations of the n-point all-plus amplitude are shown in fig. <ref>, and the factorisations of the four-point single minus amplitude are shown on fig. <ref>. These factorisations can be excited using either of the shifts in (<ref>) and (<ref>). In the all-plus case only the second results inan amplitude with the correct symmetries. This in indication that (<ref>) yields a shifted all-plus amplitude that does not vanish at infinity.Conversely, for the single minus amplitude we must use the BCFW shift.Performing the shifts and evaluating the amplitudes we obtainM^(1)_4(1^+,2^+,3^+,4^+)=10(s t 1.22.33.44.1 )^2stu,M^(1)_4(1^-,2^+,3^+,4^+)= ( 2.4^21.22.33.44.1 )^2 s^3 t^3u.The other non-zero amplitudes are available by conjugation. For the all-plus amplitude the recursion generates terms that contain the arbitrary spinorλ_η, however the sum of terms is independent of λ_η and simplifies to the above. These four-point amplitudes due to a R^3 term have been computed using field theory methods long ago <cit.>. These amplitudes vanish to all orders in a supersymmetric theory: a fact used show supergravity was two-loop ultra-violet finite <cit.>.The above expressions are in a spinor helicity basisbut agree once this is accounted for. In <cit.> these four-point amplitudes were also obtained using a “all-line recursion” technique where all legs have shifted momenta.These expressions also appear as theUV infinitepieces of both two-loop gravity in four dimensions <cit.> and one-loop gravity in six dimensions <cit.>.Five-Point Amplitudes: As before the shift  (<ref>) yields an all-plus amplitude that is independent of λ_η and has full crossing symmetry:M^(1)_5(1^+,2^+,3^+,4^+,5^+) =( ∑_P_6T^A_(1,2,3),(4,5)+∑_P_3 T^B_(1,2,3),4,5)whereT^A_(1,2,3),(4,5)= 101.4 1.4 5.35.2 1.η^24.η2.3^2 4.5 ×[5|K_14|η [2|K_14|η [3|K_14|η , T^B_(1,2,3),4,5=-101.4 1.5 2.3 [1|K_23|η^2 [5|K_23|η[4|K_23|η2.3 2.η^23.η^24.5 4.5and P_3 denotes summation over the three cyclic permutations of legs 1,2 and 3. P_6 denotes the three permutations of P_3 together withinterchange of legs 4 and 5. The λ_η independence of M^(1)_5(1^+,2^+,3^+,4^+,5^+) is not manifest. The factorisations of the five-point single minus amplitudes are more varied as shown on fig. <ref>.Using the BCFW shift on (λ̅_1,λ_2) we obtain the amplitude M^(1)_5 ( 1^-, 2^+,3^+,4^+,5^+)= 101.2^2 (∏_i,j=2,3,4,5,i<ji.j)( 1.5 1.52.5^3 3.4+1.3 1.32.3^3 4.5+1.4 1.42.4^3 5.3 ) + 1.2^23.43.54.5 ∏_i=3,4,51.i(2.3^54.51.3^5 2.3 +2.4^55.31.4^5 2.4 +2.5^53.51.5^5 2.5 ) + 11.2^2 3.43.54.5 (2.3 4.5^5 1.5^3 1.4^32.3 + 2.4 5.3^5 1.3^3 1.5^32.4 + 2.5 3.4^5 1.4^3 1.3^32.5 ) . The five-point MHV amplitude is non-zero.The non-zero factorisations of the amplitude are shown in fig. <ref>. This amplitudecan be obtained using a BCFW shift of either the two negative helicity legs or of a negative-positive pair.Shifting the two negative legs generates the expression (using only the second factorisation of fig. <ref>),M_5^(1) (1^-,2^-,3^+,4^+,5^+)= -s_341.51.53.4^23.5^34.5^31.2^2 2.32.4 -s_451.31.34.5^24.3^35.3^31.2^2 2.42.5-s_531.41.45.3^25.4^33.4^31.2^2 2.52.3. This completes the set of five-point amplitudes.We can continue in this way generating the tree-level S-matrix. We have made available M_n^(1) for n ≤ 7 in Mathematica format at http://pyweb.swan.ac.uk/ dunbar/Smatrix.html. The amplitudes have been generated up to n=8 and have the correct symmetries, are η-independent and have the correct leading soft-limits.We have evaluated amplitudes in a R+α R^3 theory.In ref. <cit.> amplitudes in Yang-Mills theory extended by F^3 terms were studied. Then using double copy techniques and the KLT relations <cit.> graviton scattering amplitudes were derived upto n=6. As noted in <cit.> these correspond to amplitudes in a R+α R^3+√(α)R^2ϕ theory. The four-pointamplitudes in the two theories are proportional <cit.> butbeyond four-point the two sets of amplitudes are functionally different.The all-plus amplitude in the two theories remain proportional for n>4withM^(1),R^3+R^2ϕ_n(1^+,2^+,⋯ n^+) = 5/2 M^(1),R^3_n(1^+,2^+,⋯ n^+) and we confirm this for n≤ 7.§ SOFT LIMITS Graviton scattering amplitudes are singular as a leg becomes soft. Weinberg <cit.> many years ago presented the leading soft limit.If we parametrise the momentum of the n-th leg as k_n^μ = t × k_s^μ then in the limit t⟶ 0 the singularity in the n-point amplitude isM_n ⟶1 t× S^(0)× M_n-1 +O(t^0)where M_n-1 is the n–1-point amplitude.The soft-factor S^(0) is universal andWeinberg showed that (<ref>) does not receive corrections in loop amplitudes.#1.#2.#3 ⟨#1^-|#2 |#3^-⟩ #1.#2.#3 ⟨#1^-|#2 |#3^+⟩ #1.#2.#3 ⟨#1^+|#2 |#3^+⟩ #1.#2.#3 ⟨#1^-|#2 |#3^-⟩ Recently it has also been proposed <cit.> that the sub-leading and sub-sub-leading terms are also universal. This can be best exposed, when a positive helicity leg becomes soft,by settingλ_n = t ×λ_s,λ̅_n =λ̅_s.In the t⟶ 0 limit the amplitude has t^-3 singularities. At tree level the amplitudessatisfy soft-theorems <cit.> whereby their behaviour as t⟶ 0 is_n= S_t_n-1 +O(t^0) =(1 t^3 S^(0) +1 t^2 S^(1) +1 tS^(2)) _n-1 +O(t^0)where, for a positive helicity-leg becoming soft <cit.> S^(0) = -∑_i=1^n-1s. ii.αi.βs. i s.αs.β , S^(1) = -1/2∑_i=1^n-1s.i s.i ( i.αs.α+i.βs.β) λ̅_s^ȧ∂∂λ̅_i^ȧ , S^(2) = 1/2∑_i=1^n-1i.s i.s λ̅_s^ȧλ̅_s^ḃ∂∂λ̅_i^ȧ∂∂λ̅_i^ḃ . The proof of the soft theorems follows from Ward identities of extended Bondi, van der Burg, Metzner and Sachs (BMS) symmetry <cit.>.Although exact for tree level amplitudesthese receive loop corrections <cit.>. Whether the soft theorems extend beyond Einstein gravity has been examined before. In particular the leadingsoft behaviour can often be used as a check upon amplitudes such, e.g. in <cit.>.The leading and sub-leading limits were shown to hold for a R^3 insertion in <cit.>.Here we examine the amplitudes and, in particular, test the sub-sub-leading soft behaviour.We can summarise the behaviour of the leading amplitudes, M^(1)_n, simplyby stating:All the amplitudes calculated satisfy the soft limits of(<ref>) up to and including the sub-sub-leading term.We have verified this for all helicity amplitudes up to n=8.Note: to check (<ref>) one must implement momentum conservation consistently between the n-point amplitudes and the n-1-point amplitudes which in essence specifies how the point t=0 is approached.These are several ways to do this. We have followed the prescription of <cit.> but alternative implementations are possible <cit.>. In principle we could have found a behaviour of the formM_n^(1)⟶ S_tM_n-1^(1) +S_t^α M_n-1^(0)+R_nwhere S_t^α would be an α correction to the soft functions and R_n is a non-factorising term.In terms of this we find S_t^α=R_n=0.Since the theory we are considering is higher derivative it is not surprising that the leading and sub-leading parts of S_t^αvanish however it is interesting that the vanishing continues for the sub-sub-leading - unlike the loop corrections to Einstein gravity.Incidentally as a consequence of eq.(<ref>) the amplitude M^(1),R^3+R^2ϕ_n(1^+,2^+,⋯ n^+) also satisfies the soft theorems to sub-sub leading level.§ OTHER THEORIESWe have chosen to extend gravity using a three-point vertex and use a diagrammar approach whereby we only consider the on-shell amplitudes. There is, of course, complementarity between this approach and that of Lagrangian based field theory.The single choice of three-point amplitude corresponds to the single R^3 field density thataffects on-shell amplitudes.This makes the extended S-matrix simply depend upon the single parameter α. If we were to deform Einstein gravity by an additional four-point amplitude then there are more choices consistent with symmetry andspinor weight, e.g. we could haveM_4 (1^+,2^+,3^+,4^+) =α_1 ( 1.2^43.4^4+1.3^42.4^4+1.4^42.3^4 ) + α_2 (1.22.33.44.1 +permutations )^2 +⋯From a field theory perspective this freedom corresponds to the observation that there are multiple R^4 tensors that contribute to on-shell amplitudes <cit.>.The same issue arises when we consider the further expansion in α.If we considerM_4^(2)(1^-,2^-,3^+,4^+) there is a single factorisation as shown in fig. <ref>.The amplitudeM^(2)_4 (1^-,2^-,3^+,4^+) =1.2^43.4^4 (tu+β s^2s )has the correct factorisation for any choice of β.This ambiguity means we also have to specify the four-point amplitude todetermine the S-matrix.In the diagrammar approach this ambiguity arises due to the existence of a polynomial function with the correct symmetries and spinor and momentum weight. From a field theory perspective, additional counterterms can contribute to this amplitude. Specifically,we could deform the theory viaR ⟶ R + C_αR^3+ C_β D^2 R^4and the four-point amplitude is only specified once C_α and C_β are determined. § CONCLUSION We have constructed the (leading part) of the S-matrix of an extended theory of gravity starting from three-point amplitudes and only demanding factorisation. The theory is extended by theaddition of amplitudes whichare polynomial in momentum, thus implicitly imposing locality andunitarity on the S-matrix. We also require the amplitudes to have the correct spinor helicity as appropriate for massless particles.The S-matrix is then generated entirely from on-shell amplitudes by demanding factorisation. Specifically, we have extended the theory by the additionof three-point amplitudes which, from a field theory perspective, corresponds to introducing R^3 terms.This S-matrix differs from that obtained byapplying double copy or KLT techniques to a F^3 extension of Yang-Mills.Beyond the leading part, polynomial amplitudes exist at higher point and these must be specified to fully determine the S-matrix.Consistency of this approach and a field theoretic approach beyond leading order requires a correspondence between these polynomial amplitudes andthe counter terms contributing to on-shell amplitudes.We find that these amplitudes satisfy the same soft theorems as the tree amplitudes of Einstein gravity up to and including the sub-sub leading terms. 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http://arxiv.org/abs/1702.08273v2

mymainaddress]Andrea Abrardo, Mauro Barni, Kassem Kallas and Benedetta Tondi [mymainaddress]Department of Information Engineering and Mathematics, Via Roma 56, Siena, Italy We consider a simple, yet widely studied, set-up in which a Fusion Center (FC) is asked to make a binary decision about a sequence of system states by relying on the possibly corrupted decisions provided by byzantine nodes, i.e. nodes which deliberately alter the result of the local decision to induce an error at the fusion center. When independent states are considered, the optimum fusion rule over a batch of observations has already been derived, however its complexity prevents its use in conjunction with large observation windows.In this paper, we propose a near-optimal algorithm based on message passing that greatly reduces the computational burden of the optimum fusion rule. In addition, the proposed algorithm retains very good performance also in the case of dependent system states. By first focusing on the case of small observation windows, we use numerical simulations to show that the proposed scheme introduces a negligible increase of the decision error probability compared to the optimum fusion rule. We then analyse the performance of the new scheme when the FC make its decision by relying on long observation windows. We do so by considering both the case of independent and Markovian system states and show that the obtained performance are superior to those obtained with prior suboptimal schemes. As an additional result, we confirm the previous finding that, in some cases, it is preferable for the byzantine nodes to minimise the mutual information between the sequence system states and the reports submitted to the FC, rather than always flipping the local decision. Adversarial signal processing, Decision fusion in adversarial setting, Decision fusion in the presence of Byzantines, Message passing algorithm, Factor graph.§ INTRODUCTION Decision fusion for distributeddetection has received an increasing attention for its importance in several applications, includingwireless networks, cognitive radio, multimedia forensics and many others. One of the most common scenarios is the parallel distributed fusion model.According to this model, the n nodes of a multi-sensor network gather information about a system and make a local decision aboutthe system status. Then the nodes send the local decisions to a Fusion Center (FC), which is in charge of making a final decision about the state of the system. <cit.>In this paper, we focus on an adversarial version of the above problem,in which a number of malicious nodes, often referred to as Byzantines <cit.>, aims at inducing a decision error at the FC <cit.>. This is a recurrent problem in many situations wherein the nodes may make a profit from a decision error. As an example, consider a cognitive radio system <cit.> in which secondary users cooperate in sensing the frequency spectrum to decide about its occupancy and the possibility to use the available spectrum to transmit their own data. While cooperation among secondary users allows to make a better decision, it is possible that one or more users deliberately alter their measurements to let the system think that the spectrum is busy, when in fact it is not, in order to gain an exclusive opportunity to use the spectrum. Online reputation systems offer another example <cit.>. Here a fusion center must make a final decision about the reputation of an item like a good or a service by relying on user's feedback. Even in this case, it is possible that malevolent users provide a fake feedback to alter the reputation of the item under inspection. Similar examples are found in many other applications, including wireless sensor networks <cit.>, <cit.>, distributed detection <cit.>, <cit.>, multimedia forensics <cit.> and adversarial signal processing <cit.>.In this paper we focus on a binary version of the fusion problem, wherein the system can assume only two states. Specifically, the nodes observe the system over m time instants and make a local decision about the sequence of system states. Local decisions are not error-free and hence they may be wrong with a certain error probability. Honest nodes send their decision to the fusion center, while byzantine nodes try to induce a decision error and hence flip the local decision with probability P_mal before sending it to the FC. The fusion center knows that some of the nodes are Byzantines with a certain probability distribution, but it does not know their position. §.§ Prior Work In a simplified version of the problem, the FC makes its decision on the status of the system at instant j by relying only on the corresponding reports, and ignoring the node reports relative to different instants. In this case, and in the absence of Byzantines, the Bayesian optimal fusion rule has been derived in <cit.>,<cit.> and it is known as Chair-Varshney rule. If local error probabilities are symmetric and equal across the network, Chair-Varshney rule boils down to simple majority-based decision. In the presence of Byzantines, Chair-Varshney rule requires the knowledge of Byzantines' positions along with the flipping probability P_mal. Since this information is rarely available, the FC may resort to a suboptimal fusion strategy. In <cit.>, by adopting a Neyman-Pearson setup and assuming that the byzantine nodes know the true state of the system, the asymptotic performance obtainable by the FC are analysed as a function of the percentage of Byzantines in the network. By formalising the attack problem as the minimisation of the Kullback-Leibler distance between the reports received by the FC under the two hypotheses, the blinding percentage, that is, the percentage of Byzantines irremediably compromising the possibility of making a correct decision, is determined.In order to improve the estimation of the sequence of system states, the FC can gather a number of reports provided by the nodes before making a global decision (multiple observation fusion). In cooperative spectrum sensing, for instance, this corresponds to collectively decide about the white holes over a time window, or, more realistically, at different frequency slots. The advantage of deciding over a sequence of states rather than on each single state separately, is that in such a way it is possible for the FC to understand which are the byzantine nodes and discard the corresponding observations (such an operation is usually referred to as Byzantine isolation). Such a scenario has also been studied in <cit.>, showing that - at least asymptotically - the blinding percentage is always equal to 50%. In <cit.>, the analysis of <cit.> is extended to a situation in which the Byzantines do not know the true state of the system. Byzantine isolation is achieved by counting the mismatches between the reports received from each node and the global decision made by the FC. The performance of the proposed scheme are evaluated in a cognitive-radio scenario for finite values of n. In order to cope with the lack of knowledge about the strategy adopted by the attacker, the decision fusion problem is casted into a game-theoretic formulation, where each party makes the best choice without knowing the strategy adopted by the other party.A slightly different approach is adopted in <cit.>. By assuming that the FC is able to derive the statistics of the reports submitted by honest nodes, Byzantine isolation is carried out whenever the reports received from a node deviate from the expected statistics. In this way, a correct decision can be made also when the percentage of Byzantines exceeds 50%. The limit of the approach proposed in <cit.>, is that it does not work when the reports sent by the Byzantines have the same statistics of those transmitted by the honest nodes. This is the case, for instance, in a perfectly symmetric setup with equiprobable system states, symmetric local error probabilities, and an attack strategy consisting of simple decision flipping.A soft isolation scheme is proposed in <cit.>, where the reports from suspect byzantine nodes are given a lower importance rather being immediately discarded. Even in <cit.>, the lack of knowledge at the FC about the strategy adopted by the attacker (and viceversa) is coped with by adopting a game-theoretic formulation.A rather different approach is adopted in <cit.>, where a tolerant scheme that mitigates the impact of Byzantines on the global decision is used rather that removing the reports submitted by suspect nodes from the fusion procedure.When the value of P_mal and the probability that a node is Byzantine are known, the optimum fusion rule under multiple observation can be derived <cit.>. Since P_mal is usually not known to the FC, in <cit.> the value of P_mal used to define the optimum fusion rule and the value actually used by the Byzantines are strategically chosen in a game-theoretic context. Different priors about the distribution of Byzantines in the network are considered ranging from an extreme case in which the exact number of Byzantines in the network is known to a maximum entropy case. One of the main results in <cit.> is that the best option for the Byzantines is not to always flip the local decision (corresponding to P_mal = 1), since this would ease the isolation of malicious nodes. In fact, for certain combinations of the distribution of Byzantines within the network and the length of the observation window, it is better for the Byzantines to minimise the mutual information between the reports submitted to the FC and the system states.§.§ Contribution The main problem of the optimum decision fusion scheme proposed in <cit.> is its computational complexity, which grows exponentially with the length of the observation window. Such a complexity prevents the adoption of the optimum decision fusion rule in many practical situations. Also the results regarding the optimum strategies of the Byzantines and the FC derived in <cit.> refer only to the case of small observation windows.In the attempt to diminish the computational complexity while minimising the loss of performance with respect to the optimum fusion rule, we propose a new, nearly-optimum, fusion scheme based on message passing and factor graphs. Message passing algorithms, based on the so called Generalised Distributive Law (GLD, <cit.>,<cit.>), have been widely applied to solve a large range of optimisation problems, including decoding of Low Density Parity Check (LDPC) codes <cit.> and BCJR codes <cit.>, dynamic programming <cit.>, solution of probabilistic inference problems on Bayesian networks <cit.> (in this case message passing algorithms are known as belief propagation). Here we use message passing to introduce a near-optimal solution of the decision fusion problem with multiple observation whose complexity grows only linearly with the size of the observation window, thus marking a dramatic improvement with respect to the exponential complexity of the optimal scheme proposed in <cit.>.Using numerical simulations and by first focusing on the case of small observation windows, for which the optimum solution can still be applied, we prove that the new scheme gives near-optimal performance at a much lower complexity than the optimum scheme. We then use numerical simulations to evaluate the performance of the proposed method for long observation windows. As a result, we show that, even in this case, the proposed solution maintains the performance improvement over the simple majority rule, the hard isolation scheme in <cit.> and the soft isolation scheme in <cit.>.As opposed to previous works, we do not limit our analysis to the case of independent system states, but we extend it to a more realistic scenario where the sequence of states obey a Markovian distribution <cit.> as depicted in Figure <ref>. The Markovian model is rather common in the case of cognitive radio networks <cit.> where the primary user occupancy of the spectrum is often modelledas a Hidden Markov Model (HMM).The Markovian case is found to be more favourable for the FC with respect to the case of independent states, due the additional a-priori information available to the FC in this case.Last but not the least, we confirm that the dual optimum behaviour of the Byzantines observed in <cit.> is also present in the case of large observation windows, even if in the Markovian case, the Byzantines may continue using the maximum attack power (P_mal = 1) for larger observation windows.The rest of this paper is organised as follows. In Section <ref>, we introduce the notation used in the paper and give a precise formulation of the addressed problem. In Section <ref>, we describe the new message passing decision rule based on factor graph. In Section <ref>, we first discuss the complexity of the proposed solution compared to the optimal solution. Then, by considering both independent and Markovian system states, we compare the performance of the message passing algorithm to the majority rule, the hard isolation scheme <cit.>, the soft isolation scheme described in <cit.> and the optimal fusion rule. In addition, we discuss the impact that the length of the observation window has on the optimal behaviour of the Byzantines. We conclude the paper in Section <ref> with some final remarks.§ NOTATION AND PROBLEM FORMULATIONThe problem faced with in this paper, is depicted in Figure <ref>. We let 𝐬 = {s_1,s_2,…,s_m} with s_i ∈{0,1} indicate the sequence of system states over an observation window of length m. The nodes collect information about the system through the vectors x_1,x_2 … x_n, with x_j indicating the observations available at node j. Based on such observations, a node j makes a local decision u_i,j about system state s_i. We assume that the local error probability, hereafter indicated as ε, does not depend on either i or j. The state of the nodes in the network is given by the vector 𝐡 = {h_1,h_2,…,h_n} with h_j = 1/0 indicating that node j is honest or Byzantine, respectively. Finally, the matrix 𝐑 = {r_i,j}, i = 1,…,m, j = 1,…,n contains all the reports received by the FC. Specifically, r_i,j is the report sent by node j relative to s_i. As stated before, for honest nodes we have u_i,j= r_i,j while, for Byzantines we have p (u_i,j r_i,j ) = P_mal. The Byzantines corrupt the local decisions independently of each other. By assuming that the transmission between nodes and fusion center takes place over error-free channels, the report is equal to the local decision with probability 1 for honest nodes and with probability 1-P_mal for Byzantines. Hence, according to the local decision error model, we can derive the probabilities of the reports for honest nodes: p(r_i,j | s_i, h_j = 1) = (1-ε)δ(r_i,j-s_i)+ε (1-δ(r_i,j-s_i)),where δ(a) is defined as:δ(a)=1,if a=0 0,otherwise. On the other hand, by introducing η = ε(1-P_mal) + (1-ε)P_mal, i.e., the probability that the fusion center receives a wrong report from a byzantine node, we have:p(r_i,j | s_i, h_j = 0) = (1-η)δ(r_i,j-s_i)+η (1-δ(r_i,j-s_i))As for the number of Byzantines, we consider a situation in which the states of the nodes are independent of each other and the state of each node is described by a Bernoulli random variable with parameter α, that is p(h_j=0)=α , ∀ j. In this way, the number of byzantine nodes in the network is a random variable following a binomial distribution, corresponding to the maximum entropy case <cit.> withp(𝐡) = ∏_j p(h_j), where p(h_j) = α(1-h_j) + (1-α)h_j.Regarding the sequence of states 𝐬, we assume a Markov model as shown in Figure <ref> , i.e., p(𝐬) = ∏_i p(s_i|s_i-1). The transition probabilities are given by p(s_i|s_i-1) = 1-ρ if s_i = s_i-1 and p(s_i|s_i-1) = ρ when s_is_i-1, whereas for i = 1 we have p(s_1|s_0) = p(s_1) = 0.5.In this paper we look for the the bitwise Maximum A Posteriori Probability (MAP) estimation of the system states {s_i} which reads as follows: [ŝ_i= max_s_i ∈{0,1}  p(s_i | 𝐑) ; =max_s_i ∈{0,1}∑_{𝐬,𝐡}\ s_i p( 𝐬, 𝐡 | 𝐑) (law of total probability); = max_s_i ∈{0,1}∑_{𝐬,𝐡}\ s_i p(𝐑 | 𝐬, 𝐡) p (𝐬) p (𝐡)(Bayes); = max_s_i ∈{0,1}∑_{𝐬,𝐡}\ s_i∏_i,jp(r_i,j | s_i, h_j )∏_i p(s_i|s_i-1) ∏_j p(h_j)]where the notation ∑_\ denotes a summation over all the possible combinations of values that the variables contained in the expression within the summation may assume by keeping the parameter listed after the operator \ fixed. For a given 𝐡, the matrix of the observations 𝐑 at the FC follows a HMM <cit.>.The optimisation problem in (<ref>) has been solved in <cit.> for the case of independent system states. Even in such a simple case, however, the complexity of the optimum decision rule is exceedingly large, thus limiting the use of the optimum decision only in the case of small observation windows (typically m not larger than 10). In the next section we introduce a sub-optimum solution of (<ref>) based on message passing, which greatly reduces the computational complexity at the price of a negligible loss of accuracy. § A DECISION FUSION ALGORITHM BASED ON MESSAGE PASSING§.§ Introduction to Sum-product message passing In this section we provide a brief introduction to the message passing (MP) algorithm for marginalization of sum-product problems. Let us start by considering N binary variables 𝐳 = {z_1,z_2,…,z_N}, z_i ∈{0,1}. Then, consider the function f(𝐳) with factorization:f(𝐳) = ∏_kf_k(𝒵_k)where f_k, k = 1,,M are functions of a subset 𝒵_k of the whole set of variables. We are interested in computing the marginal of f with respect to a general variable z_i, defined as the sum of f over all possible values of 𝐳, i.e.:μ(z_i) = ∑_𝐳\ z_i∏_kf_k(𝒵_k)where notation ∑_𝐳\ z_i denotes a sum over all possible combinations of values of the variables in 𝐳 by keeping z_i fixed. Note that marginalization problem occurs when we want to compute any arbitrary probability from joint probabilities by summing out variables that we are not interested in. In this general setting, determining the marginals by exhaustive search requires 2^N operations. However, in many situations it is possible to exploit the distributive law of multiplication to get a substantial reduction in complexity.To elaborate, let associate with problem (<ref>) a bipartite factor graph, in which for each variable we draw a variable node (circle) and for each function we draw a factor node (square). A variable node is connected to a factor node k by an edge if and only if the corresponding variable belongs to 𝒵_k. This means that the set of vertices is partitioned into two groups (the set of nodes corresponding to variables and the set of nodes corresponding to factors) and that an edge always connects a variable node to a factor node. Let now assume that the factor graph is a single tree, i.e., a connected graph where there is an unique path to connect two nodes. In this case, it is straightforward to derive an algorithm which allows to solve the marginalization problem with reduced complexity. The algorithm is the MP algorithm, which has been broadly used in the last years in channel coding applications <cit.>, <cit.>.To describe how the MP algorithm works, let us first define messages as 2-dimensional vectors, denoted by 𝐦 = {m(0),m(1)}. Such messages are exchanged between variable nodes and function nodes and viceversa, according to the following rules. Let us first consider variable-to-function messages (𝐦_vf), and take the portion of factor graph depicted in Fig. <ref> as an illustrative example. In this graph, the variable node z_i is connected to L factor nodes, namely f_1,f_2,…,f_L. For the MP algorithm to work properly, node z_i must deliver the messages 𝐦^(l)_vf, l = 1,…,L to all its adjacent nodes. Without loss of generality, let us focus on message 𝐦^(1)_vf. Such a message can be evaluated and delivered upon receiving messages 𝐦^(l)_fv, l = 2,…,L, i.e., upon receiving messages from all function nodes except f_1. In particular, 𝐦^(1)_vf may be straightforwardly evaluated by calculating the element-wise product of the incoming messages, i.e.:m^(1)_vf(q) = ∏_j=2^Lm^(j)_fv(q)for q=0,1.Let us now consider factor-to-variable messages, and refer to the factor graph of Fig. <ref> where P variable nodes are connected to the factor node f_k, i.e., according to the previous notation, 𝒵_k = {z_1,…,z_P}. In this case, the node f_k must deliver the messages 𝐦^(l)_fv, l = 1,…,P to all its adjacent nodes. Let us consider again 𝐦^(1)_fv: upon receiving the messages 𝐦^(l)_vf, l = 2,…,P, f_k may evaluate the message 𝐦^(1)_fv as:m^(1)_fv(q) = ∑_z_2,…,z_P[ f_k(q,z_2,…,z_P) ∏_p=2^Pm^(p)_vf(z_p)]for q=0,1. Given the message passing rules at each node, it is now possible to derive the MP algorithm which allows to compute the marginals in (<ref>). The process starts at the leaf nodes, i.e., those nodes which have only one connecting edge. In particular, each variable leaf node passes an all-ones message to its adjacent factor node, whilst each factor leaf node, say f_k(z_i) passes the message m^(k)_fv(q) = f_k(z_i = q) to its adjacent node z_i. After initialization at leaf nodes, for every edge we can compute the outgoing message as soon as all incoming messages from all other edges connected to the same node are received (according to the message passing rules (<ref>) and (<ref>)). When a message has been sent in both directions along every edge the algorithm stops. This situation is depicted in Fig. <ref>: upon receiving messages from all its adjacent factor nodes, node z_i can evaluate the exact marginal as: μ(z_i) = ∏_k=1,…,Lm^(k)_fv(z_i). With regard to complexity, factors to variables message passing can be accomplished with 2^P operations, P being the number of variables in f_k. On the other hand, variables to nodes message passing's complexity can be neglected, and, hence, the MP algorithm allows to noticeably reduce the complexity of the problem provided that the numerosity of 𝒵_k is much lower than N. With regard to the optimization, Equation (<ref>) evaluates the marginal for both z_i = 0 and z_i = 1, which represent the approximated computation of the sum-product for both hypotheses. Hence, the optimization is obtained by choosing the value of z_i which maximizes it. §.§ Nearly-optimal data fusion by means of message passing The objective function of the optimal fusion rule expressed in (<ref>) can be seen as a marginalization of a sum product of functions of binary variables, and, as such, it falls within the MP framework described in the previous Section. More specifically, in our problem, the variables are the system states s_i and the status of the nodes h_j, while the functions are the probabilities of the reports shown in equations (<ref>) and (<ref>), the conditional probabilities p(s_i|s_i-1), and the a-priori probabilities p(h_j). The resulting bipartite graph is shown in Figure <ref>.It is worth noting that the graph is a loopy graph, i.e., it contains cycles, and as such it is not a tree. However, although it was originally designed for acyclic graphical models, it was found that the MP algorithm can be used for general graphs, e.g., in channel decoding problems <cit.>. In general, when the marginalization problem is associated to a loopy graph, the implementation of MP requires to establish a scheduling policy to initiate the procedure, so that variable nodes may receive messages from all the connected factors, thus evaluating the marginals. In this case, a single run of the MP algorithm may not be sufficient to achieve a good approximation of the exact marginals, and progressive refinements must be obtained through successive iterations. However, in the presence of loopy graphs, there is no guarantee of either convergence or optimality of the final solution. In many cases, the performance of the message-passing algorithms is closely related to the structure of the graph, in general, and its cycles, in particular. Many previous works in the field of channel coding, e.g., see <cit.>, reached the conclusion that, for good performance, the factor graph should not contain short cycles. In our case, it is possible to see from Figure <ref> that the shortest cycles have order 6, i.e., a message before returning to the sender must cross at least six different nodes. We speculate that such a minimum cycles length is sufficient to provide good performance for the problem at hand. We will prove through simulations that such a conjecture is true.To elaborate further, based on the graph of Figure <ref> and on the general MP rules reported in the previous Section, we are now capable of deriving the messages for the scenario at hand. In Figure <ref>, we display all the exchanged messages for the graph in Figure <ref> that are exchanged to estimate in parallel each of the states s_i, i ∈{0,1} in the vector 𝐬 = {s_1,s_2,…,s_m}. Specifically, we have: [τ_i^(l)(s_i) =φ_i^(l)(s_i) ∏_j = 1^nν_i,j^(u)(s_i) i = 1,…,m;τ_i^(r)(s_i) =φ_i^(r)(s_i) ∏_j = 1^nν_i,j^(u)(s_i) i = 1,…,m;φ_i^(l)(s_i) =∑_s_i+1= 0,1 p(s_i+1 | s_i) τ_i+1^(l)(s_i+1) i = 1,…,m-1;φ_i^(r)(s_i) =∑_s_i-1 = 0,1 p(s_i | s_i-1)τ_i-1^(r)(s_i-1) i = 2,…,m;φ_1^(r)(s_1) =p(s_1);ν_i,j^(u)(s_i) = ∑_h_j= 0,1 p(r_i,j| s_i,h_j. ) λ_j,i^(u)(h_j) i = 1,…,m,    j = 1,…,n;ν_i,j^(d)(s_i) = φ_i^(r)(s_i)φ_i^(l)(s_i)∏_k = 1kj^nν_i,k^(u)(s_i) i = 1,…,m-1,    j = 1,…,n;ν_m,j^(d)(s_m) = φ_i^(r)(s_m)∏_k = 1kj^nν_m,k^(u)(s_m) j = 1,…,n;λ_j,i^(d)(h_j) =∑_s_i= 0,1 p(r_i,j| s_i,h_j. )ν_i,j^(d)(s_i) i = 1,…,m,    j = 1,…,n;λ_j,i^(u)(h_j) =ω_j^(u)(h_j) ∏_q = 1qi^mλ_j,q^(d)(h_j) i = 1,…,m,    j = 1,…,n;ω_j^(d)(h_j) = ∏_ i = 1 ^mλ_j,i^(d)(h_j) j = 1,…,n;ω_j^(u)(h_j) =p(h_j) j = 1,…,n; ] < g r a p h i c s >Factor graph for the problem at hand with the illustration of all the exchanged messages.As for the scheduling policy, we initiate the MP procedure by sending the messages λ_j,i^(u)(h_j) = ω_j^(u)(h_j) to all p(r_i,j| s_i,h_j. ) factor nodes, and by sending the message p(s_1) to the variable node s_1. Hence, the MP proceeds according to the general message passing rules, until all variable nodes are able to compute the respective marginals. When this happens, the first iteration is concluded. Then, successive iterations are carried out by starting from leaf nodes and by taking into account the messages received at the previous iteration for the evaluation of new messages. Hence, the algorithm is stopped upon achieving convergence of messages, or after a maximum number of iterations.The MP scheme described above can be simplified by observing that messages can be normalized without affecting the normalized marginals. Henceforward, let us consider as normalization factors the sum of the elements of the messages, i.e., if we consider for example τ_i^(l)(s_i), the normalization factor is τ_i^(l)(0)+τ_i^(l)(1). In this case, the normalized messages, say τ̅_i^(l)(s_i) can be conveniently represented as scalar terms in the interval (0,1), e.g., we can consider τ̅_i^(l)(0) only since τ̅_i^(l)(1) = 1 - τ̅_i^(l)(0). Accordingly, the normalized messages can be evaluated as:[τ̅_i^(l) = φ̅_i^(l)∏_j = 1^nν̅_i,j^(u)/φ̅_i^(l)∏_j = 1^nν̅_i,j^(u) + (1-φ̅_i^(l)) ∏_j = 1^n (1-ν̅_i,j^(u)) i = 1,…,m;τ̅_i^(r) = φ̅_i^(r)∏_j = 1^nν̅_i,j^(u)/φ̅_i^(r)∏_j = 1^nν̅_i,j^(u) + (1-φ̅_i^(r) ) ∏_j = 1^n (1-ν̅_i,j^(u) ) i = 1,…,m;φ̅_i^(l) =ρτ̅_i+1^(l) + (1-ρ) (1-τ̅_i+1^(l)) i = 1,…,m-1;φ̅_i^(r) =ρτ̅_i-1^(r) + (1-ρ) (1-τ̅_i-1^(r)) i = 2,…,m;φ̅_1^(r) =p(s_1 = 0);ν̅_i,j^(u) =p(r_i,j| 0,0. )λ̅_j,i^(u) + p(r_i,j| 0,1. ) (1-λ̅_j,i^(u))/p(r_i,j| 0,0. )λ̅_j,i^(u) + p(r_i,j| 0,1. ) (1-λ̅_j,i^(u)) + p(r_i,j| 1,0. )λ̅_j,i^(u) + p(r_i,j| 1,1. ) (1-λ̅_j,i^(u)) i = 1,…,m,    j = 1,…,n;ν̅_i,j^(d) = φ̅_i^(r)φ̅_i^(l)∏_k = 1kj^nν̅_i,k^(u)/φ̅_i^(r)φ̅_i^(l)∏_k = 1kj^nν̅_i,k^(u) + (1-φ̅_i^(r)) (1-φ̅_i^(l))∏_k = 1kj^n (1-ν̅_i,k^(u)) i = 1,…,m-1,    j = 1,…,n;ν̅_m,j^(d) = φ̅_m^(r)∏_k = 1kj^nν̅_m,k^(u)/φ̅_m^(r)∏_k = 1kj^nν̅_m,k^(u) + (1-φ̅_m^(r)) ∏_k = 1kj^n (1-ν̅_m,k^(u)) j = 1,…,n;λ̅_j,i^(d) = p(r_i,j| 0,0. )ν̅_i,j^(d) + p(r_i,j| 1,0. )(1-ν̅_i,j^(d))/p(r_i,j| 0,0. )ν̅_i,j^(d) + p(r_i,j| 1,0. )(1-ν̅_i,j^(d)) + p(r_i,j| 0,1. )ν̅_i,j^(d) + p(r_i,j| 1,1. )(1-ν̅_i,j^(d)) i = 1,…,m,    j = 1,…,n;λ̅_j,i^(u) =ω̅_j^(u)∏_q = 1qi^mλ̅_j,q^(d)/ω̅_j^(u)∏_q = 1qi^mλ̅_j,q^(d)+ (1-ω̅_j^(u)) ∏_q = 1qi^m (1-λ̅_j,q^(d)) i = 1,…,m,    j = 1,…,n;ω̅_j^(d) = ∏_ i = 1 ^mλ̅_j,i^(d)/∏_ i = 1 ^mλ̅_j,i^(d)+ ∏_ i = 1 ^m (1-λ̅_j,i^(d)) j = 1,…,n;ω̅_j^(u) =p(h_j = 0) j = 1,…,n; ]§ SIMULATION RESULTS AND DISCUSSIONSIn this section, we analyze the performance of the MP decision fusion algorithm. We first consider the computational complexity, then we pass to evaluate the performance in terms of error probability. In particular, we compare the performance of the MP-based scheme to those of the optimum fusion rule <cit.> (whenever possible), the soft isolation scheme presented in<cit.>, the hard isolation scheme described in <cit.> and the simple majority rule. In our comparison, we consider both independent and Markovian system states, for both small and large observation window m. §.§ Complexity DiscussionIn order to evaluate the complexity of the message passing algorithm and compare it to that of the optimum fusion scheme, we consider both the number of operations and the running time. By number of operations we mean the number of additions, substractions, multiplications and divisions performed by the algorithm to estimate the vector of system states 𝐬.By looking at equation (<ref>), we see that running the message passing algorithm requires the following number of operations:* 3n+5 operations for each of τ̅_i^(l) and τ̅_i^(r). * 3 operations for each of φ̅_i^(l) and φ̅_i^(r). * 11 operations for ν̅_i,j^(u).* 3n+5 operations for ν̅_i,j^(d).* 3n+2 operations for ν̅_m,j^(d).* 11 operations for λ̅_j,i^(d).* 3m+2 operations for each of λ̅_j,i^(u) and ω̅_j^(d).summing up to 12n+6m+49 operations for each iteration over the factor graph. On the other hand, in the case of independent node states, the optimal scheme in <cit.> requires 2^m(m+n) operations. Therefore, the MP algorithm is much less computationally expensive since it passes from an exponential to a linear complexity in m. An example of the difference in computational complexity between the optimum and the MP algorithms is depicted in Figure <ref>.With regard to time complexity, Table <ref> reports the running time of the MP and the optimal schemes. For n=20, the optimal scheme running time is 17.547 times larger than that of the message passing algorithm. On the other hand, for the case of n=100, the optimal scheme needs around 4.258 times more than the message passing scheme. The tests have been conducted using Matlab 2014b running on a machine with 64-bit windows 7 OS with 16,0 GB of installed RAM and Intel Core i7-2600 CPU @ 3.40GHz. §.§ Performance Evaluation In this section, we use numerical simulations to evaluate the performance of the message passing algorithm and compare them to the state of the art schemes. The results are divided into four parts. The first two parts consider, respectively, simulations performed with small and large observation windows m. Then, in the third part, we investigate the optimum behaviour of the Byzantines over a range of observation windows size. Finally, in the last part, we compare the case of independent and Markovian system states.The simulations were carried out according to the following setup. We considered a network with n = 20 nodes, ε = 0.15, ρ = {0.95, 0.5} corresponding to Markovian and independent sequence of system states, respectively. The probability α that a node is Byzantine is in the range [0,0.45] corresponding to a number of Byzantines between 0 and 9. As to P_mal we set it to either0.5 or 1[It is know from <cit.> that for the Byzantines the optimum choice of P_mal is either 0.5 or 1 depending on the considered setup.]. The number of message passing iterations is 5. For each setting, we estimated the error probability over 10^5 trials.§.§.§ Small m To start with, we considered a small observation window, namelym=10. With such a small value of m, in fact, it is possible to compare the performance of the message passing algorithm to that of the optimum decision fusion rule. The results we obtained are reported in Figure <ref>. Upon inspection of the figure, the superior performance of the message passing algorithm over the Majority, Soft and Hard isolation schemes is confirmed. More interestingly, the message passing algorithm gives nearly optimal performance, with only a negligible performance loss with respect to the optimum scheme.Figure <ref> confirms the results shown in Figure <ref> for Markovian system states (ρ = 0.95). §.§.§ Large m Having shown the near optimality of the message passing scheme for small values of m; we now leverage on the small computational complexity of such a scheme to evaluate its performance for large values of m (m = 30). As shown in Figure <ref>, by increasing the observation window all the schemes give better performance, with the message passing algorithm always providing the best performance. Interestingly, in this case, when the attacker uses P_mal=1.0, the message passing algorithm permits to almost nullify the attack of the Byzantines for all the values ofα. Concerning the residual error probability, it is due to the fact that, even when there are no Byzantines in the network (α=0), there is still an error floor caused by the local errors at the nodes ε. For the case of independent states, such an error floor is around 10^-4. In Figure <ref> and <ref>, this error floor decreases to about 10^-5 because of the additional a-priori information available in the Markovian case.§.§.§ Optimal choice of P_mal for the Byzantines One of the main results proven in <cit.>, is that setting P_mal = 1 is not necessarily the optimal choice for the Byzantines. In fact, when the FC manages to identify which are the malicious nodes, it can exploit the fact the malicious nodes always flip the result of the local decision to get useful information about the system state. In such cases, it is preferable for the Byzantines to use P_mal = 0.5 since in this way the reports send to the FC does not convey any information about the status of the system. However, in <cit.>, it was not possible to derive exactly the limits determining the two different behaviours for the Byzantines due to the impossibility of applying the optimum algorithm in conjunction with large observation windows. By exploiting the low complexity of the message passing scheme, we are now able to overcome the limits of the analysis carried out in <cit.>.Specifically, we carried out an additional set of experiments by fixing α = 0.45 and varying the observation window in the interval [5,20]. The results we obtained confirm the general behaviour observed in <cit.>. For instance, in Figure <ref>, P_mal = 1.0 remains the Byzantines' optimal choice up to m=13, while for m > 13, it is preferable for them to use P_mal=0.5. Similar results are obtained for independent system states as shown in Figure<ref>. §.§.§ Comparison between independent and Markovian System StatesIn this subsection, we provide a comparison between the cases of Markovian and independent system states.By looking at Figure <ref> and <ref>, we see that the Byzantinesswitch their strategy from P_mal=1 to P_mal = 0.5 for a smaller observation window (m=10) in the case of independent states (the switching value for the Markovian case is m = 13). We can explain this behaviour by observing that in the case of Markovian states, using P_mal = 0.5 results in a strong deviation from the Markovianity assumption of the reports sent to the FC thus making it easier the isolation of byzantine nodes. This is not the case with P_mal = 1, since, due to the symmetry of the adopted Markov model, such a value does not alter the expected statistics of the reports.As a last result, in Figure <ref>, we compare the error probability for the case of independent and Markov sources. Since we are interested in comparing the achievable performance for the two cases, we consider only the performance obtained by the optimum and the message passing algorithms. Upon inspection of the figure, it turns out that the case of independent states is more favourable to the Byzantines than the Markov case. The reason is that the FC may exploit the additional a-priori information available in the Markov case to identify the Byzantines and hence make a better decision. Such effect disappears when α approaches 0.5, since in this case the Byzantines tend to dominate the network. In that case, the Byzantines' reports prevail the pool of reports at the FC and hence, the FC becomes nearly blind so that even the additional a-priori information about the Markov model does not offer a great help.§ CONCLUSIONS In this paper, we proposed a near-optimal message passing algorithm based on factor graph for decision fusion in multi-sensor networks in the presence of Byzantines. The effectiveness of the proposed scheme is evaluated by means of extensive numerical simulations both for the case of independent and Markov sequence of states.Experiments showed that, when compared to the optimum fusion scheme, the proposed scheme permits to achieve near-optimal performance at a much lower computational cost: specifically, by adopting the new algorithm based on message passing we were able to reduce the complexity from exponential to linear. Such reduction of the complexity permits to deal with large observation windows, thus further improving the performance of the decision. Results on large observation windows confirmed the dual behavior in the attacking strategy of the Byzantines, looking for a trade-off between pushing the FC to make a wrong decision on one hand and reducing the mutual information between the reports and the system state on the other hand.In addition, the experiments showed that the case of independent states is more favorable to Byzantines than the Markovian case, due to the additional a-priori information available at the FC in the Markovian case.As future work, we plan to focus on a scenario more favorable to the Byzantines, by giving them the possibility to access the observation vectors. In this way, they can focus their attack on the most profitable cases and avoid to flip the local decision when it is very likely that their action will have no effect on the FC decision.Considering the case where the nodes can send to the FC more extensive reports (multi-bit case) <cit.> is another interesting extension. § REFERENCES

http://arxiv.org/abs/1702.08357v1

apsrev4-1_our_style

http://arxiv.org/abs/1702.08427v2

[ [ Received: date / Accepted: date ===================================In recent years prominent intellectuals have raised ethical concerns about the consequences of artificial intelligence. One concern is that an autonomous agent might modify itself to become "superintelligent" and, in supremely effective pursuit of poorly specified goals, destroy all of humanity. This paper considers and rejects the possibility of this outcome. We argue that this scenario depends on an agent's ability torapidly improve its ability to predict its environment through self-modification. Using a Bayesian model of a reasoning agent, we show that there are important limitations to how an agent may improve its predictive ability through self-modification alone. We conclude that concern about this artificial intelligence outcome is misplaced and better directed at policy questions around data access and storage.The appetite of the public and prominent intellectuals for the study of the ethical implications of artificial intelligence has increased in recent years. One captivating possibility is that artificial intelligence research might result in a `superintelligence' that puts humanity at risk. <cit.> has called for AI researchers to consider this possibility seriously because, however unlikely, its mere possibility is grave. <cit.> argues for the importance of considering the risks of artificial intelligence as a research agenda. For Bostrom, the potential risks of artificial intelligence are not just at the scale of industrial mishaps or weapons of mass destruction. Rather, Bostrom argues that artificial intelligence has the potential to threaten humanity as a whole and determine the fate of the universe. We approach this grand thesis with a measure of skepticism. Nevertheless, we hope that by elucidating the argument and considering potential objections in good faith, we can get a better grip on the realistic ethical implications of artificial intelligence.This paper is in that spirit. We consider the argument for this AI doomsday scenario proposed by Bostrom <cit.>. Section 1 summarizes Bostrom's argument and motivates the work of the rest of the paper. In focuses on the conditions of an “intelligence explosion” that would lead to a dominant machine intelligence averse to humanity. Section 2 argues that rather than speculating broadly about general artificial intelligence, we can predict outcomes of artificial intelligence by considering more narrowly a few tasks that are essential to instrumental reasoning. Section 3 considers recalcitrance, the resistance of a system to improvements to its own intelligence, and the ways it can limit intelligence explosion. Section 4 contains an analysis of the recalcitrance of prediction, using a Bayesian model of a predictive agent. We conclude that prediction is not something an agent can easily improve upon autonomously. Section 5 discusses the implication of these findings for further investigation into AI risk.§ BOSTROM'S CORE ARGUMENT AND DEFINITIONS Bostrom makes a number of claims in the course of his argument which I will outline here as distinct propositions.theoremTheorem proposition[theorem]Proposition A system with sufficient intelligence relative to other intelligent systems will have a `decisive strategic advantage' and will determine the fate of the world and universe. Concretely, Bostrom accepts human beings, governments, emulated brains, and computers as potential intelligent systems. In his implicit model of the world, these agents are in contest with each other. By Bostrom's definition, a `decisive strategic advantage' is the amount of technological advantage sufficient “to achieve complete world domination". It is beyond the scope of this paper to explore the nuances of Proposition 1. We will provisionally accept it and focus on the probability that a sufficiently intelligent system will arise. An intelligent system is likely to attain a decisive strategic advantage if it undergoes an `intelligence explosion', a rapidly accelerating rate of intelligence increase. Bostrom never offers a definition of intelligence that is amenable to quantification. He does leverage quantitative intuitions in the course of his argument when he proposes the following model of intelligence change. The rate of change in intelligence is equal to optimization power divided by recalcitrance.dI/dt = O/ROptimization power refers to the effort of improving the intelligence of the system. Recalcitrance refers to the resistance of the system to being improved.If an intelligent system works to improve its own intelligence, then optimization power will increase with the system's intelligence, leading to rapidly accelerating intelligence increase if recalcitrance is sufficiently low. We will consider more precise versions Proposition 2, 3, and 4 latter in this paper.Bostrom maintains that an intelligent system will attempt to recursively improve its own intelligence under very general conditions. Any intelligent system will have or develop increasing its own intelligence as an instrumental goal towards other goals. Proposition 5 is a consequence of Bostrom's instrumental convergence thesis,<cit.> Several instrumental values can be identified which are convergent in the sense that their attainment would increase the chances of the agent's goal being realized fora wide range of final goals and a wide range of situations, implying that theseinstrumental values are likely to be pursued by a broad spectrum of situated intelligent agents. This thesis is important for Bostrom's line of argument because the threat of AI comes from its predictably rapid takeoff as a `superintelligence' combined with the unpredictability of its goals. A machine intelligence is unlikely to have goals that are aligned with the values of humans. The developing field of value learning in artificial intelligence (cite) has been motivated in part by concerns akin to Proposition 6. In Bostrom's work, the problems of machine value misalignment are illustrated by many dystopian scenarios which we will not go into here. Recalcitrance is likely to be lower for machine intelligence than for human intelligence because of the physical properties of computers. The overall picture is a compelling narrative for many. A machine intelligence research project achieves the ability to modify itself to make itself more intelligent. It does so in service of some goal its programmers originally provided (Proposition 5). Since recalcitrance for improvements to machine intelligence is low (Proposition 7), it undergoes and intelligence explosion (Proposition 4), gets a decisive strategic advantage (Proposition 2) and determines the fate of humanity (Proposition 1). Since the machine's goals are likely misaligned with humanity's (Proposition 6), artificial intelligence poses a great risk.Bostrom provides a wide survey of the possibilities surrounding greater-than-human intelligence. We have outlined the logic of the argument that we believe provides most of the motivational force behind the book. In doing so, we have made it easier to verify the logical validity of the argument. We will continue to analyze this argument with a focus on the role of instrumental goals and recalcitrance in predicting artificial intelligence related risk. We will focus on Propositions 2, 3, 4, and 5, leaving other aspects of the argument to future work.§ INTELLIGENCE AND INSTRUMENTAL TASKS The use of the term “intelligence" in the preceding section has been vague. This is unfortunate and a consequence of some of the vagueness in discussion of artificial intelligence ethics and risk in Bostrom and elsewhere. Some of the discourse around the ethics of artificial intelligence anticipates qualitatively new risks associated with what has been called “Strong AI" (cite). One contribution of this paper is to narrow the discussion by showing that these risks can be understood in terms of well-understood “narrow" AI tasks. We anticipate that this narrower framing of the problems of AI risk will be more tractable.Bostrom leads with the provocative but fuzzy definition of superintelligence as “any intellect that greatly exceeds the cognitive performance of humans in virtually all domains of interest.” The logic of the argument shows that the “domains of interest" necessary and sufficient for intelligence explosion are limited to those that concern intelligence augmentation itself.Bostrom writes about these domains in two ways. In one section he discusses the “cognitive superpowers", domains that would quicken a superintelligence takeoff. These “superpowers" include: Intelligence amplification, Strategizing, Social manipulation, Hacking, Technology research, and Economic productivity. In another section he discusses “convergent instrumental values", values that agents with a broad variety of goals would converge on as important to their pursuit of final goals.. These values include: Self-preservation, Goal-content integrity, Cognitive enhancement, Technological perfection, Resource acquisition.There are striking parallels between the “superpowers" that would hasten takeoff and instrumental values. “Intelligence amplification" is a superpower, whereas "cognitive enhancement" is an instrumental value. “Technology research" is a superpower, "technological perfection" is a value. The danger of intelligence explosion is the danger that an intelligent system will confuse its power with its motives, in particular when its power and motive are both its own intelligence in a narrow instrumental sense. We have captured this aspect of Bostrom's argument in Proposition 5, above.We note that the motivation of a system to increase its own instrumental intelligence is necessary but not sufficient for an intelligence explosion. In addition to being motivated, an intelligent system must be capable of rapidly increasing its intelligence. By Proposition 4, this capability will be a function not only of the system's optimization power, but also its recalcitrance.The possibility of an intelligence explosion will be restricted specifically by the recalcitrance of the kinds of tasks that comprise instrumental intelligence. Narrowing our focus on specific tasks will make the problem of assessing AI risk more tractable because performance on more narrowly defined tasks is better specified. As a result, our judgements about the recalcitrance of improvement on those tasks can be better grounded in statistical and computer science theory, as opposed to being speculative.In pursuit of this more narrow and grounded understanding of AI risk, in the next section we will explore Bostrom's model of intelligence growth in more depth.§ RECALCITRANCE CONSIDERED Bostrom's model of intelligence change depends on two variables, optimization power and recalcitrance. These are presented as components in a qualitative model. Optimization power is the effort put into improving the intelligence of the system. Recalcitrance is the resistance of the system to improvement. While it's desirable to have units in which intelligence, optimization power, and recalcitrance could be measured, none have been provided by Bostrom. Nonetheless this model is a useful one for explicating intuitions about self-modifying intelligence. Bostrom's initial formulation of this model is: dI/dt = O(I)/R Bostrom's claim is that for instrumental reasons an intelligent system is likely to invest some portion of its intelligence back into improving its intelligence. He introduces a linear model of self-improvement that we will adapt here.By assumption we can model O(I) = α I + β for some parameters α and β, where α and β are positive and represent the contribution of optimization power by the system itself and external forces (such as a team of researchers), respectively. If recalcitrance is constant, e.g R = k, then we can compute: dI/dt = α I + β/k Under these conditions, I will be exponentially increasing in time t. This is the "intelligence explosion" that gives Bostrom's argument so much momentum. The explosion only gets worse if recalcitrance is below a constant. Implicitly, Bostrom appears committed to the following additional proposition: A system whose intelligent is growing exponentially is undergoing an intelligence explosion that will lead to a decisive strategic advantage. We provisionally accept this proposition. However, it's important to remember that recalcitrance may also be a function of intelligence. Bostrom does not mention the possibility of recalcitrance increasing in intelligence. Consider the following model where recalcitrance is, like optimization power, linearly increasing in intelligence. dI/dt = α_o I + β_o/α_r I + β_r Now there are four parameters instead of three. Note this model is identical to the one above it when α_r = 0. Assuming all these parameters are positive, as I increases the rate of intelligence growth approaches α_o / α_r from below. This is linear, not exponential, growth. In this circumstance, there would be no intelligence explosion and therefore much less catastrophic AI risk.There are many plausible reasons why recalcitrance might increase with intelligence levels. For example, if intelligence improvements vary considerably in the search cost of discovering them, then a system might first collect the “low hanging fruit" and then have to resort to searching for harder and harder to reach discoveries.This is not a decisive argument against intelligence explosion and thepossibility of a decisively strategic intelligent system. It is an argument for why considering recalcitrance seriously is important for assessing the likelihood of such an outcome. A firmer grip on the problem of predicting future AI risk can be gained by looking at the recalcitrance of specific instrumental reasoning tasks. In the next section, we consider specifically the recalcitrance of the general task of prediction. § RECALCITRANCE OF PREDICTION Prediction is a very well-studied problem in artificial intelligence and statistics. Many more specific intelligence tasks can be analyzed as special cases of prediction. For example, some of Bostrom's “cognitive superpowers", such as Hacking and Social Manipulation, are analysable partly as a matter of prediction in the domains of computer networks and interpersonal interaction. One reason why prediction is so well-studied is that it is so important instrumentally: skill at prediction is valuable in pursuit of a wide range of other goals.Prediction is such a critically important part of intelligence that we propose the following conjecture as an addendum to Bostrom's intelligence explosion argument: Part of what it means for an intelligent system to improve its own intelligence in a domain (including the domain of improving its own intelligence) is for it to improve its ability to make predictions in that domain. It follows that if the task of prediction is highly recalcitrant, then there will be no autonomous intelligence explosion.A benefit of looking at a particular intelligent task is that it allows us to think more concretely about what it would mean to become more intelligent. For prediction, we can consider intelligence to be the ability to make good predictions about the world based on valid inference from data.We will represent a predicting agent using the Bayesian formulation of statistical inference: P(H|D) = P(D|H) P(H)/P(D) Here, P(H|D) is the posterior probability of a hypothesis H given observed data D. If one is following statistically optimal procedure, one can compute this value by taking the prior probability of the hypothesis P(H), multiplying it by the likelihood of the data given the hypothesis P(D|H), and then normalizing this result by dividing by the probability of the data over all models, P(D) = ∑_iP(D|H_i)P(H_i).Statisticians will justifiably argue whether this is the best formulation of prediction. And depending on the specifics of the task, the target value may well be some function of posterior (such as the hypothesis with maximum likelihood) and the overall distribution may be secondary. These are valid objections that we would like to put to one side in order to get across the intuition of an argument.To the extent that the Bayesian formulation is an accurate representation of the general problem of prediction, we can analyze its recalcitrance. We start by enumerating the ways in which an agent might improve its performance on the prediction task, which is validly computing P(H|D) in such a way that best approximates the truth. * Computational accuracy. A system can improve its ability to compute the mathematical function of the Bayesian update. Many widely used statistical inference algorithms use numerical approximation rather and so it is possible for a system to improve its algorithm's faithfulness to the mathematical formula that defines its goal.* Computational speed. There are faster and slower ways to compute theinference formula. An intelligent system could come up with a way to make itself compute its answer faster. This might be independent of the accuracy of its answer.* Prior. The success of inference depends crucially on the priorprobability assigned to hypotheses or models. A prior is better when it assignshigher probability to the true process that generates observable data, or modelsthat are `close' to that true process. * Data. Assuming accurate Bayesian computation, performance at prediction will depend on the quality of the data used in the inference. Note that "better data" is not necessarilythe same as “more data". If the data that the system learns from is from a biasedsample of the phenomenon in question, then a successful Bayesian update couldmake its predictions worse, not better. Better data is data that is informativewith respect to the true process that generated the data. Now that we have enumerate the ways in which an intelligent system may improve its powerof prediction, we can ask: how recalcitrant are these factors to self-improvementby an intelligent system?* Recalcitrance of accuracy. It may be possible for a system to inspect itself and determine ways to modify its own algorithm to make it more accurate at computing a Bayesian update. However there is a hard limit to improvements of this kind. No system can compute a Bayesian update more accurately than computing it perfectly accurately. Therefore, in the limit, recalcitrance of computational accuracy is infinite.* Recalcitrance of speed. It is possible for a system to experiment with novel algorithms and select those that are provably faster than its current ones, or which perform better on benchmark tests. However, once again there is a hard limit to the speed of computation: the maximum speed of the hardware system on which the system is implemented. Without the ability to increase hardware resources available, an intelligent system will reach infinite speed recalcitrance in the limit.* Recalcitrance of the prior. An intelligent system could modify the parameters of its expectations independently of the data that it learns from. But it is essential to the abstraction of the Bayesian agent that the prior encodes whatever bias the agent has that is not learned from external data. So strictly speaking, there is no way for an intelligent agent to modify its own prior intelligently. Intelligence in prediction is a matter of using data intelligently, not being accidentally gifted with the correct prior beliefs. So the recalcitrance of improving the prior is infinite.* Recalcitrance of data. Better data improves performance on prediction. But data collection is not something an intelligent system can do purely autonomously, since it has to interact withthe phenomenon of interest to get more data. We cannot make assumptions about the recalcitrance of data collection without modeling the environment the agent is in. Contrary to the conditions of Bostrom's intelligence explosion scenario, we have identified ways in which the recalcitrance of prediction, an important instrumental reasoning task, is prohibitively high. Purely algorithmic self-improvement is particularly limited. If we allow a system to improve its own hardware, that allows the system to improve its speed. Overall performance depends critically on data collection. Neither hardware expansion nor data collection is a feature of the intelligent system alone, but rather the possibility of these depends on the context in which the system operates. If, for example, there are increasing search costs for the intelligent system as it seeks out new data and hardware improvements, that would imply an increase in recalcitrance as a function of intelligence. As we have seen in the previous section, this sort of dependence of recalcitrance on intelligence can mean that the probability of an intelligence explosion is negligible. § DISCUSSION AND DIRECTIONS FOR FUTURE WORK We have explicated the logic of one argument for concern about risk from artificial intelligence. This argument concerns the possibility that an autonomous intelligent system modifies itself, undergoes an intelligence explosion, and takes over the world in a way that is adverse to human interests.In our analysis, we discover that at the core of the argument are several claims that are much more narrow and tractable than appear on the surface. In particular, we can get a grip on the problem of predicting the behavior of self-modifying intelligent systems by focusing on instrumental reasoning tasks and their susceptibility to autonomous self-improvement. If we can show that recalcitrance on these tasks is predictably high, we can dismiss the probability of an intelligence explosion as being negligible.To demonstrate how such an analysis could work, we analyzed the recalcitrance of prediction, using a Bayesian model of a predictive agent. We found that the barriers to recursive self-improvement through algorithmic changes is prohibitively high for an intelligence explosion. Rather, an intelligent system attempting to improve its own abilities of prediction would need foremost to acquire faster hardware and better data.The recalcitrance of acquiring faster hardware and better data depend not just on the intelligence of the system, but also on the environment. If an environment imposes variable search and acquisition costs for hardware and data, we would expect recalcitrance of these improvements to increase with intelligence, which would curtail an intelligence explosion.While not a decisive argument against the possibility of an intelligence explosion, the preceding arguments do suggest that those concerned with the ethical implications of the future of AI should put their attention elsewhere. If intelligent systems engage in intelligence-expanding activities but in a non-explosive way, that suggests that the probably outcomes for the future of AI will be best modelled as multi-agent systems competing for cognitive resources rather than as a single, decisively controlling agent.If intelligence growth is limited by data and hardware, not by advancement in artificial intelligence algorithms, that also suggests that AI researchers may not be in the best position to mitigate the risks of artificial intelligence. Rather, regulators controlling the use of generic computing hardware and data storage may be more important to determining the future of artificial intelligence than those that design algorithms.aaai

http://arxiv.org/abs/1702.08495v2

2pt2pt

http://arxiv.org/abs/1702.08470v2

Supported by Strong Chain Rules for Min-Entropy under Few Bits Spoiledsupported by Maciej Skórski IST AustriaEmail: mskorski@ist.ac.at Accepted .Received ; in original form========================================================================= It is well established that the notion of min-entropy fails to satisfy the chain rule of the formH(X,Y) = H(X|Y)+H(Y), known for Shannon Entropy.The lack of a chain rule causes a lot of technical difficulties, particularly in cryptographywhere the chain rule would be a natural way to analyze how min-entropy is split among smaller blocks. Such problems arise for example when constructing extractors and dispersers.We show that any sequence of variables exhibits a very strong strong block-source structure (conditional distributions of blocks are nearly flat) when we spoil few correlated bits. This implies, conditioned on the spoiled bits, that splitting-recombination properties hold. In particular, we have many nice properties that min-entropy doesn't obey in general, for example strong chain rules, “information can't hurt” inequalities, equivalences of average and worst-case conditional entropy definitionsand others. Quantitatively, for any sequence X_1,…,X_t of random variables over an alphabet 𝒳 we prove that, when conditioned on m = t· O( loglog|𝒳| + loglog(1/ϵ) + log t) bits of auxiliary information, all conditional distributions of the form X_i|X_<i are ϵ-close to be nearly flat (only a constant factor away). The argument is combinatorial (based on simplex coverings).This result may be used as a generic tool for exhibiting block-source structures. We demonstrate this by reproving the fundamental converter due to Nisan and Zuckermann (J. Computer and System Sciences, 1996),which shows that sampling blocks from a min-entropy source roughly preserves the entropy rate. Our bound implies, only by straightforward chain rules, an additive loss of o(1) (for sufficiently many samples), which qualitatively meetsthe first tighter analysis of this problem due to Vadhan (CRYPTO'03), obtained by large deviation techniques. chain rule, min-entropy, spoiling knowledge, block sources, local extractors§ INTRODUCTION §.§ Strong vs Weak Entropy Chain RulesOne of the most useful properties of Shannon entropy is the chain rule,showing how entropy splits between distributionsH_1(X|Y) = H(X,Y) - H(Y).The notion of min-entropy, very important for cryptography <cit.>, fails to satisfy this property <cit.>.In the lack of a chain rule, a much weaker one-sided bound (e.g. of the form H_∞(X|Y) ⩾ H_∞(X)-H_0(Y), where H̃_∞ is an appropriate extension of min-entropy to conditional distributions) is sometimes used <cit.>, which we address as a weak chain rule[In leakage-resilient cryptography such bounds are simply called chain rules. In this paper we discuss chain rules in a strong sense.]. §.§ Need for Strong Chain Rules While the weak chain rule suffices for many applications related to bounded leakage  <cit.> (where Y is leakage much shorter than the amount of min-entropy in X), it is insufficient where one needs to estimate how entropy is distributed among blocks. In these settings, one would like to argue that (roughly) either X or Y|X has high min-entropy if the joint min-entropy (of (X,Y)) is high.Examples of such problems are randomness extraction in the bounded storage model <cit.>, constructions of dispersers <cit.>,or oblivious transfer protocols <cit.>.§.§ Our Contribution and Related Works Although the chain rule fails in general, we show that it is true conditioned on few spoiled bits. We actually show more, that (locally, conditioned on auxiliary bits) a very strong block-source structure exists. Namely, each block is nearly flat given previous blocks. Informally, the theorem reads as follows [Informal: exhibiting flat block-source structures] For any sequence X=X_1,…,X_t of correlated random variables each over 𝒳 and any ϵ there exists auxiliary information S of length m bits such that* S is short: m=t· O(loglog|𝒳|+loglog(1/ϵ) + log t) * Conditioned on S, conditional block distributions[X_i|X_i=x_i,X_i-1=x_i-1,…,X_1=x_1] are nearly flat (ϵ-close to a probability distribution whose values differ by a constant factor). The formal statement is given in <Ref>. For cryptographic applications, loglog(ϵ^-1) is pretty much a small constant (typically ϵ = 2^-100).Also, for sources with super-logarithmic entropy per block, that is when H_∞(X) ≫ t loglog|𝒳|), and the number of blocks t growing not too fast, e.g. t = log^O(1)(|𝒳|),the error term is of a smaller order than the entropy. Under these mild assumptions, conditioned on the partition generated by auxiliary bits, we conclude many nice properties that fail in general. Examples are chain rules, “conditioning only decreases entropy” properties, equivalences of conditional entropy defined in different ways and others.§.§.§ Our ToolsSpoiling Knowledge The spoiling knowledge technique is essentially about finding auxiliary information that increases entropy, and was introduced in <cit.>. We use the same idea to force block distributions to be nearly flat.Covering techniques (combinatorial geometry) In order to construct a good “spoiling”, we consider the logarithm of the chaining identityp_X_1… X_t(·) = ∏_j=1^tp_X_i|X_i-1… X_1(·) whichrepresents the “surprise” of the total distribution as a sum of “next-block surprises” of the form r_i(·) = -log p_X_i|X_i-1… X_1(·). It follows that the vector of all r_i (for i=1,…,t) lie in a (t-1)-dimensional simplex of edge roughly O(tlog|𝒳|). Our partition is obtained from coverings, asfor all x in the same part r_i(·) is roughly constant (when the radius is small enough). This approach can be likely optimized (we use a crude bound on the covering number). Also the lower bounds on the necessary number of spoiled buts are possible, by considering packings instead of coverings. We defer this discussion to the full version.§.§ Related worksAs far as we know, the presented result on spoiling min-entropy is knew. The Nisan-Zuckerman lemma, discussed in this paper as an application, was analyzed by Vadhan <cit.> and recently by Bellare <cit.>.These results study min-entropy present in random chunks of a larger source, and don't offer tools for splitting entropy in any source deterministically (particularly for a small number of blocks), as we do. §.§ Applications The important result due to Nisan and Zuckerman <cit.>, improved later by Vadhan <cit.>states that sampling from a given source of high min-entropy rate α yields a source of a comparable entropy rate β. This fact is a crucial step in constructions of so called local extractors, that extract randomness parsing only a part of input.For a while, only a lossy bound β≈α/log(1/α) was known. The reason was precisely the lack of a chain rule for min-entropy. As observed by Cachin <cit.> the proof for Shannon entropy (a less interesting case) is straightforward and follows basically by a splitting-recombining argument, which uses a chain rule in both directions. We demonstrate by our technique that (surprisingly) a very effective splitting-recombining approach actually works, and achieves β = α- o(1) in a very straightforward way. This matches the bound due to Vadhan. Concretely, if the original source is a sequence of t blocks over an alphabet 𝒳 and we take ℓ samples, then β = α - err_𝖲𝗉𝗈𝗂𝗅-err_𝖲𝖺𝗆𝗉 where the losses due to chain rules and sampling equal, respectively err_𝖲𝗉𝗈𝗂𝗅 = O(loglog|𝒳|+loglog(1/ϵ)+log t)/log|𝒳|err_𝖲𝖺𝗆𝗉 = O(√(ℓ^-1log(1/ϵ))).In particular β converges to α when the block length is log|𝒳| = log^ω(1)(1/ϵ) andlog |𝒳| = ω(1)· t. For more details see <Ref> in <Ref>. For the discussed result our bounds converge slightly slower than Vadhan's bounds derived by large deviation techniques. However, our spoiling technique can be used also for small number of samples.§.§ Organization In <Ref> we explain necessary notions and notations. Auxiliary facts that will be needed are discussed in <Ref>. In <Ref> we prove the main result. Applications to the bounded storage model are discussed in <Ref>.We conclude the work in <Ref>.§ PRELIMINARIES§.§ Basic NotationFor any random variables X_1,X_2 by p_X_1|X_2 we denote the distribution of X_1 conditioned on X_2, that isp_X_1|X_2(x_1,x_2) = [X_1=x_1|X_2=x_2]. Throughout this paper, all logarithms are taken to base 2.For any sequence of random variables X = X_1,…,X_n we denote X_<i=X_1,…,X_i-1,X_⩽ i=X_1,…,X_i and more generally for any subset I⊂{1,…,n} we put X_I = X_i_1X_i_2… X_i_m-1X_i_m where i_1<i_2<… <i_m are all elements of I. §.§ Distances, EntropiesIn the definitions below 𝒳 is an arbitrary finite set.For two random variables X,Y on 𝒳 by the statistical distance (total variation) we meand_TV(X;Y) = 1/2∑_x∈𝒳| [X=x] - [Y=x]| The Shannon entropy of a random variable X on 𝒳 equalsH_1(X) = -∑_x[X=x]log[X=x]. The min-entropy of a random variable X on 𝒳 equalsH_∞(X) = -logmax_x∈𝒳[X=x]. Let X,Y be random variables over 𝒳 and 𝒴 respectively. The worst-case min-entropy of X conditioned on Y equalsH_∞(X|Y) = min_y∈𝒴 H_∞(X|Y=y).The average min-entropy of X conditioned on Y equalsH_∞(X|Y) = -log(𝔼_y Y 2^-H_∞(X|Y=y)).The averaged notion is slightly weaker, but has better properties and actually better suits applications <cit.>.The notion of smooth entropy is more accurate than min-entropy because quantifies entropy up to small perturbations in the probability mass.The ϵ-smooth min-entropy of a random variable X on 𝒳 is defined asH_∞^ϵ(X)= max_X': d_TV(X,X')⩽ϵ H_∞(X')where the maximum is over all random variables X' on 𝒳. In other words, X has at least k bits of smooth min-entropy if there is X' of min-entropy at least k and ϵ-close to X.§.§ Randomness Extractors Below we recall the definition of seeded extractorsWe say that a function 𝖤𝗑𝗍:{0,1}^n×{0,1}^d →{0,1}^m is a (k,ϵ)-extractor if and only ifd_TV(𝖤𝗑𝗍(X,U_d),U_d ; U_m,U_d) ⩽ϵfor any X on {0,1}^n with min-entropy at least k.§.§ Samplers Averaging samplers are procedures which sample points, within a given domain, that are distributed enough random toapproximate every function. It turns out that there exist good averaging samples, using much less auxiliary randomness than necessary to generate independent points. For our applications we consider samplers that approximate averages from below.A function 𝖲𝖺𝗆𝗉:{0,1}^d → [t]^ℓ is a (μ,θ,γ)-averaging sampler if for every function f:[t]→ [0,1] with average value 1/ℓ∑_i=1^ℓf(i) ⩾μ it holds that_i_1,…,i_ℓ𝖲𝖺𝗆𝗉(U_r)[ 1/ℓ∑_i=1^ℓf(i) ⩾μ - θ] ⩽ 1-γ. For the setting in <Ref> there is an averaging sampler which for any ℓ' such that Ω(μθ^-2log(1/γ) ) ⩽ℓ' ⩽ℓ produces ℓ' distinct samples and uses d = log(t/ℓ') + log(1/γ) + 2log(μ/θ) + loglog(1/μ)+O(1) random bits. § AUXILIARY LEMMAS The following lemma is essentially the information can't hurt principle, well known for Shannon entropy, stated for the notion of min-entropy.It says that extra information can only reduce the uncertainty, and comes up often in cryptography.For any X,Y,Z we have H_∞(X|Y,Z) ⩽H_∞(X|Y).It is well known (see, e.g., <cit.>) that when each block has certain min-entropy entropy conditioned on previous blocks, the total entropy grows linearly with the number of blocks (as expected). The lemma is stated without a proof (it follows directly from definitions). We stress, however, that using the worst-case notion of conditional entropy is crucial. For any X,Y,Zwe haveH_∞(X,Y|Z) ⩾H_∞(X|Y,Z)+H_∞(Y|Z). Another lemma, well known in folklore, quantifies the intuition that conditioning on “large” events (not too surprising information) doesn't decrease entropy too much.For any random variable X and any events A,B we have H_∞(X|B,A) ⩾ H_∞(X|B)-log(1/(A|B)).The first part of the lemma follows because we have [X=x|A,B] = [X=x,A∩ B]/[A∩ B] ⩽[X=x,B]/([B]·[A|B]) for every x.§ MAIN RESULT§.§ Chain Rule Let 𝒳 be a fixed alphabet and X=(X_1,…,X_t) be a sequence of (possibly correlated) random variables each over 𝒳. Then for any 1>ϵ>0 and δ>0 there exists a collection ℬ of disjoint sets on 𝒳^t such that* ℬ can be indexed by a small number of bits, namelylog |ℬ| = t· O(loglog|𝒳| +loglog(ϵ^-1)+log (t/δ)) * ℬ almost covers the domain∑_B∈ℬp_X(B) ⩾ 1-ϵ * Conditioned on members of ℬ, block distributions X_i|X_<i are nearly flat.∀ x,x'∈ B: 2^-O(δ)⩽p_X(x_i|x_<i)/p_X(x'_i|x'_<i)⩽ 2^O(δ).for every B∈ℬ and i=1,…,t. In some applications (see for example <Ref>) it is convenient to work with parts that are not too small.By substituting ϵ:=ϵ/2, and deleting from ℬ all members of smaller than ϵ/2|ℬ| (their mass is at most ϵ/2), we obtainIn <Ref>, we may assume p_X(B) = Ω(ϵ· |ℬ|^-1) for every B∈ℬ.Under the assumptions of <Ref>, for every B∈ℬ for every index i and for everyset I⊂{1,…,i-1} we have* The chain rule for min-entropy H_∞(X_i|X_I,B) = H_∞(X_i,X_I|B) - H_∞(X_I|B)± O(δ). * The average and worst-case min-entropy almost matchH_∞(X_i|X_I,B) = H_∞(X_i|X_I,B)± O(δ). Fix any subset I⊂{1,…,i-1} of size m. By <Ref> for every x and B we have[X_⩽ i = x_⩽ i|B ] =2^± O(δ)[X_< i = x_< i |B ].Let J = {1,…,i-1}∖ I. Taking the sum of <Ref> over x such that x_I = x'_I and x_i = x” are fixed but x_J varies we obtain[X_i=x”,X_I = x'_I|B ] =2^± O(δ)[X_I = x'_I |B ].This impliesH_∞(X_i|X_I,B) = H_∞(X_i,X_I,B) - H_∞(X_I,B)± O(δ)H_∞(X_i|X_I,B) = H_∞(X_i|X_I,B) ± O(δ)which finishes the proof.Let p be the joint distribution of X_1,…,X_n. For any i denote p_i(x_i|x_i-1,…,x_1) = p_X_i|X_<i(x_i,…,x_1).and let the “surprise” of the bit x_i ber^i(x) = -log p_i(x_i,…,x_1).Note that p(x_1,…,x_n) = ∏_i=1^n p_i(x_i,x_i-1,…,x_1) and therefore, denoting x=(x_1,…,x_n), we obtain [Decomposing surprises] We have∑_i=1^tr^i(x) = -log p(x). The next claim follows by a simple Markov-type argument. [Significant probabilities] There exists a set A⊂𝒳^t of probability 1-ϵ such that p(x) ⩾ϵ/|𝒳|^tfor all x∈ A. Denoting x=x_1,…,x_n, we have∀ x∈ A: -log p(x_1,…,x_n) ⩽ tlog|𝒳| + log(1/ϵ).The claim below follows from <Ref> and <Ref> [Surprises live in a simplex] We have r^i(x) ⩾ 0 for i=1,…, t and ∑_i=1^t r^i(x) ⩽ tlog|𝒳| + log(1/ϵ), for all points x∈ A.[Simplex coverings imply a chain rule] If the simplex with side length tlog|𝒳| + log(1/ϵ) can be coveredby N balls of radius R in the ℓ_∞ norm, then the theorem holds with |ℬ| =N and δ =R. Let C⊂ℝ^t, |C|=N, be the set of the centers of the covering balls. Let S be the function which assigns to every point x∈ A (where A is defined in <Ref>) the point z∈ Cclosest to the vector (r_1(x),…,r_t(x)) in the ℓ_∞-norm. Fix any z and let B_z = {x: S(x)=z}. By the properties of the covering, for i=1,…,t we obtain∀ x∈ B: |r^i(x)-z_i| ⩽ RIn particular, the surprises for any two points in B are close∀ x,x'∈ B: |r^i(x) -r^i(x')| ⩽ 2R,i=1,…,t.Let p_B be the conditional probability of p given B. Denote byr^i_Bthe surprise of the i-th bit given previous bits and conditioned on B, that isr^i_B(x)= -log[X_i=x_i|X_i-1=x_i-1,…,X_1=x_1,B]= -log p_B(x_i | x_i-1,…,x_1)Note that p_B(x) = p(x)/p(B) for x∈ B, and hence r^i_B(x) = r^i(x) + log (p(B)^-1). Now <Ref> implies∀ x,x'∈ B: |r^i_B(x) -r^i_B(x')| ⩽ 2R,i=1,…,t.which finishes the proof.It remains to observe that the covering number for our case is log N = log N_0 + loglog(|𝒳|)+loglog(1/ϵ))+log (t/δ) (see for example <cit.>) which finishes the proof.§ APPLICATIONSLet 𝒳 be a fixed finite alphabet, and let X_1,…,X_t be a sequence of correlated random variables each over 𝒳.Let i_1,…,i_ℓ∈ [1,t], where ℓ < t, be chosen from the set {1,…,t} by an averaging (μ,θ,γ)-sampler. Then there is a random variable ℬtaking m = t· O(loglog|𝒳|+loglog(1/ϵ)+log t) bits,such that1/ℓlog|𝒳|H^ϵ_∞(X_i_ℓX_i_ℓ-1X_i_ℓ-2…,X_i_1|ℬ) ⩾ 1/tlog|𝒳|H_∞(X_i_ℓX_i_ℓ-1X_i_ℓ-2…,X_i_1)-err_𝖲𝗉𝗈𝗂𝗅-err_𝖲𝖺𝗆𝗉𝗅𝖾𝗋where the errors due to spoiling and sampling equalerr_𝖲𝗉𝗈𝗂𝗅 = O(loglog|𝒳|+loglog(1/ϵ)+log t)/log|𝒳| err_𝖲𝖺𝗆𝗉 = O(√(ℓ^-1log(1/ϵ))).Composing this with an extractor over 𝒳^ℓ one obtains a local extractor, whichreads only a small fraction (specifically ℓ/t) of input bits. We refer to <cit.> for a general discussion. We will argue that the sequence X_i_1,…,X_i_ℓ, for ℓ sufficiently big, likely has the same entropy rate (entropy per block) as the original sequence X_1,…,X_ℓ.Let ℬ be the family guaranteed by <Ref>. By part (a) of <Ref> applied t times (starting from i=t downto i=1) we have for every B∈ℬ∑_i=1^tH_∞(X_i|X_<i,B) ⩾ H_∞(X|B) - O(tδ)and now by part (b) applied to each summand∑_i=1^tH_∞(X_i|X_<i,B) ⩾ H_∞(X|B) - O(tδ).Note that𝔼_i_1,…,i_ℓ∑_j=1^ℓH_∞(X_i_j|X_<i_j,B) = 1/t∑_i=1^tH_∞(X_i|X_<i,B)In particular, with high probability over (i_j)_j=1,…,ℓ 1/ℓ∑_j=1^ℓH_∞(X_i_j|X_<i_j,B) ≳1/t∑_i=1^tH_∞(X_i|X_<i,B).For the sake of clarity, we comment later on the exact error in accuracy and probability in <Ref>. Observe that by part (b) of <Ref> we obtain1/ℓ∑_j=1^ℓH_∞(X_i_j|X_i_jX_i_j-1…,X_i_1|B)⩾ 1/ℓ∑_j=1^ℓH_∞(X_i_j|X_<i_j|B) -O(δ)(which is the conditioning reduces entropy property). Again, by applying part (b) of <Ref> to the sum on the right-hand side of <Ref> we get1/ℓ∑_j=1^ℓH_∞(X_i_j|X_i_jX_i_j-1…,X_i_1|B)⩾ 1/ℓ∑_j=1^ℓH_∞(X_i_j|X_i_jX_i_j-1…,X_i_1|B)- O(δ)By <Ref> from <Ref> we get for every B'∈ℬ'1/ℓH_∞(X_i_ℓX_i_ℓ-1X_i_ℓ-2…,X_i_1|B) ⩾ 1/ℓ∑_j=1^ℓH_∞(X_i_j|X_i_jX_i_j-1…,X_i_1|B)- O(δ).Combining this with <Ref>,<Ref> and <Ref> we finally obtain (with high probability)1/ℓH_∞(X_i_ℓX_i_ℓ-1X_i_ℓ-2…,X_i_1|B) ≳ 1/tH_∞(X|B) - O(δ).Note that this holds for every B. Recall that by <Ref> we can assume [B] = Ω(|ℬ|^-1ϵ). Now by <Ref> we have H_∞(X|B) > H_∞(X)-O(log|ℬ| + log(1/ϵ)) and thus1/ℓH_∞(X_i_ℓX_i_ℓ-1X_i_ℓ-2…,X_i_1|B) ≳ 1/tH_∞(X|B) -O(log|ℬ| + log(1/ϵ))/t.We can do slightly better. Namely, from <Ref> 1/ℓH_∞(X_i_ℓX_i_ℓ-1X_i_ℓ-2…,X_i_1|ℬ) ≳ 1/tH_∞(X|ℬ)- O(δ).where ℬ is a random variable that assigns to every point x the corresponding set B covering x, conditioned in addition on the map being defined (it fails when x is not covered by any B which happens w.p. at most ϵ) Indeed, we have2^-H_∞(X_i_ℓX_i_ℓ-1X_i_ℓ-2…,X_i_1|B)⩽ ( 2^-H_∞(X_i_tX_i_t-1X_i_ℓ-2…,X_i_1))^ℓ/t.and <Ref> follows by averaging over B and the Jensen Inequality (note that ℓ < t implies that the corresponding mapping is concave). By <Ref> applied to X|ℬ=B for all possible B to outcomes of ℬ we obtain1/ℓH_∞(X_i_ℓX_i_ℓ-1X_i_ℓ-2…,X_i_1|ℬ) ≳ 1/tH_∞(X) -log |ℬ|/t - O(δ)It remains to use an explicit bound on |ℬ| from <Ref>, set the sampler to γ = ϵ and compute θ from ϵ and ℓ. § CONCLUSION By a simple combinatorial argument combined with the spoiling knowledge technique we showed how to exhibits strong block-source structures in any min-entropy source. This approach may be applied to locally enforce chain rules (or other desired properties) for min-entropy.

http://arxiv.org/abs/1702.08476v1

On-the-Fly Computation of Bisimilarity Distances]On-the-Fly Computation of Bisimilarity Distances* G. Bacci]Giorgio Bacci Dept. of Computer Science, Aalborg University, Denmark {grbacci, giovbacci, kgl, mardare}@cs.aau.dk Work supported bySapere Aude: DFF-Young Researchers Grant 10-085054 of the Danish Council for Independent Research,by the VKR Center of Excellence MT-LAB and by the Sino-Danish Basic Research Center IDEA4CPS. G. Bacci]Giovanni Bacci K. G. Larsen]Kim G. Larsen R. Mardare]Radu Mardare G.3,I.1.4,I.6.4*An earlier version of this paper appeared as <cit.>. The present version extends <cit.> by considering the case of CTMCs and improves the linear programming approach of <cit.>. We propose a distance between continuous-time Markov chains (CTMCs) and study the problem of computing it by comparing three different algorithmic methodologies: iterative, linear program, and on-the-fly.In a work presented at FoSSaCS'12, Chen et al. characterized the bisimilarity distance of Desharnais et al. between discrete-time Markov chains as an optimal solution of a linear program that can be solved by using the ellipsoid method. Inspired by their result, we propose a novel linear program characterization to compute the distance in the continuous-time setting. Differently from previous proposals, ours has a number of constraints that is bounded by a polynomial in the size of the CTMC. This, in particular, proves that the distance we propose can be computed in polynomial time.Despite its theoretical importance, the proposed linear program characterization turns out to be inefficient in practice. Nevertheless, driven by the encouraging results of our previous work presented at TACAS'13, we propose an efficient on-the-fly algorithm, which, unlike the other mentioned solutions, computes the distances between two given states avoiding an exhaustive exploration of the state space.This technique works by successively refining over-approximations of the target distances using a greedy strategy, which ensures that the state space is further explored only when the current approximations are improved.Tests performed on a consistent set of (pseudo)randomly generated CTMCs show that our algorithm improves, on average, the efficiency of the corresponding iterative and linear program methods with orders of magnitude. [ [ December 30, 2023 ===================== § INTRODUCTIONContinuous-time Markov chains (CTMCs) are one of the most prominent models in performance and dependability analysis. They are exploited in a broad range of applications, and constitute the underlying semantics of many modeling formalisms for real-time probabilistic systems such as Markovian queuing networks, stochastic process algebras, and calculi for systems biology. An example of CTMC is presented in Figure <ref>(left). Here, state s_1 goes to state s_3 and s_4 with probability 1/3 and 2/3, respectively. Each state has an associated exit-rate representing the rate of an exponentially distributed random variable that characterizes the residence-time in the state [Note that the only residence time distributions that ensure the Markov property (a.k.a., memoryless transition probability) are exponential distributions.]. For example, the probability to move from s_1 to any other state within time t ≥ 0 is given by ∫_0^t 3e^-3x dx = 1 - e^-3t. A state with no outgoing transitions (as s_3 in Figure <ref>) is called absorbing, and represents a terminating state of the system.A key concept for reasoning about the equivalence of probabilistic systems is Larsen and Skou'sprobabilistic bisimulation for discrete-time Markov chains (MCs). This notion has been extended to several types of probabilistic systems, including CTMCs. In Figure <ref>(left) s_4 and s_5 are bisimilar. Moreover, although s_1 and s_2 move with different probabilities to states s_4 and s_5, their probabilities to reach any bisimilarity class is the same, so that, also s_1 and s_2 are bisimilar. However, when the numerical values of probabilities are based on statistical sampling or subject to error estimates, any behavioral analysis based on a notion of equivalence is too fragile, as it only relates processes with identical behaviors.This issue is illustrated in Figure <ref>(right), where the states t_1 and t_2 (i.e., the counterparts of s_1 and s_2, respectively, after a perturbation of the transition probabilities) are not bisimilar. A similar situation occurs considering perturbations on the exit-rates or on associated labels, if one assumes they are taken from a metric space.This is a common issue in applications such as, systems biology <cit.>, planning <cit.>, games <cit.>, or security <cit.>, where one is interested in knowing whether two processes that may differ by a small amount in the real-valued parameters (probabilities, rates, etc.) have “sufficiently” similar behaviors. This motivated the development of the metric theory for probabilistic systems, initiated by Desharnais et al. <cit.> and greatly developed and explored by De Alfaro, van Breugel, Worrell, and others <cit.>. It consists in proposing a bisimilarity distance (pseudometric), which measures the behavioral similarity of two models.These pseudometrics, e.g., the one proposed by Desharnais et al., are parametric in a discount factor that controls the significance of the future in the measurements.Since van Breugel et al. have presented a fixed point characterization of the aforementioned pseudometric in <cit.>, several iterative algorithms have been developed in order to compute its approximation up to any degree of accuracy <cit.>.Recently, Chen et al. <cit.> proved that, for finite MCs with rational transition function, the bisimilarity pseudometrics can be computed exactly in polynomial time. The proof consists in describing the pseudometric as the solution of a linear program that can be solved using the ellipsoid method. Although the ellipsoid method is theoretically efficient, “computational experiments with the method are very discouraging and it is in practice by no means a competitor of the, theoretically inefficient, simplex method”, as stated in <cit.>. Unfortunately, in this case the simplex method cannot be used to speed up performances in practice, since the linear program to be solved may have an exponential number of constraints in the number of states of the MC.In this paper, we introduce a bisimilarity pseudometric over CTMCs that extends that of Desharnais et al. over MCs, and we consider the problem of computing it both from a theoretical and practical point of view. We show that the proposed distance can be computed in polynomial time in the size of the CTMC. This is obtained by reducing the problem of computing the distance to that of finding an optimal solution of a linear program that can be solved using the ellipsoid method. Notably, differently from the proposal in <cit.>, our linear program characterization has a number of constraints that is bounded by a polynomial in the size of the CTMC. This, in particular, allows one to avoid the use of the ellipsoid algorithm in favor of the simplex or the interior-point methods.However, also in this case, the linear program characterization turns out to be inefficient in practice, even for small CTMCs. Nevertheless, supported by the encouraging results in our previous work <cit.>, we propose to follow an on-the-fly approach for computing the distance.This is inspired by an alternative characterization of the bisimilarity pseudometric based on the notion of coupling structure for a CTMC. Each coupling structure is associated with a discrepancy function that represents an over-approximation of the distance. The problem of computing the pseudometric is then reduced to that of searching for an optimal coupling structure whose associated discrepancy coincides with the distance. The exploration of the coupling structures is based on a greedy strategy that, given a coupling structure, moves to a new one by ensuring an actual improvement of the current discrepancy function. This strategy will eventually find an optimal coupling structure. The method is sound independently from the initial starting coupling structure. Notably, the moving strategy is based on a local update of the current coupling structure. Since the update is local, when the goal is to compute the distance only between certain pairs of states, the construction of the coupling structures can be done on-the-fly, delimiting the exploration only on those states that are demanded during the computation.The efficiency of our algorithm has been evaluated on a significant set of randomly generated CTMCs. The results show that our algorithm performs orders of magnitude better than the corresponding iterative and linear program implementations. Moreover, we provide empirical evidence that our algorithm enjoys good execution running times.One of the main practical advantages of our approach consists in that one can focus on computing only the distances between states that are of particular interest. This is useful in practice, for instance when large systems are considered and visiting the entire state space is computationally expensive. A similar issue has been considered by Comanici et al., in <cit.> in the case of Markov decision processes with rewards, who noticed that for computing the approximated pseudometric one does not need to update the current value for all the pairs at each iteration, but it is sufficient only to focus on the pairs where changes are happening rapidly. In our approach, the termination condition is checked locally, still ensuring that the local optimum corresponds to the global one.Our methods can also be used in combination with approximation techniques as, for instance, to provide a least over-approximation of the behavioral distance given over-estimates of some particular distances. Synopsis:The paper is organized as follows. In Section <ref>, we recall the basic preliminaries on continuous-time Markov chains and define the bisimilarity pseudometric. Section <ref> is devoted to the analysis of the complexity of the problem of computing such a distance. Here, two approaches are considered: an approximate iterative method and a linear program characterization. In Section <ref>, we provide an alternative characterization of the distance based on the notion of coupling structure. This is the basis for the development of an on-the-fly algorithm (Section <ref>) for the computation of the pseudometric, whose correctness and termination is proven in Section <ref>. The efficiency of this algorithm is supported by experimental results, shown in Section <ref>. Final remarks and conclusions are in Section <ref>.§ CONTINUOUS-TIME MARKOV CHAINS AND BISIMILARITY PSEUDOMETRICSWe recall the definitions of (finite) L-labelled continuous-time Markov chains (CTMCs) for a nonempty set of labels L, and stochastic bisimilarity over them. Then, we introduce a behavioral pseudometric over CTMCs to be considered as a quantitative generalization of stochastic bisimilarity.Given a finite set X, a discrete probability distribution over it is a function μ X → [0,1] such that μ(X) =1, where μ(E) = ∑_x ∈ Eμ(x), for E ⊆ X. We denote the set of finitely supported discrete probability distributions over X by (X).[Continuous-time Markov chain]An L-labelled continuous-time Markov chain is a tuple = (S, A, τ, ρ, ℓ) consisting of a nonempty finite set S of states, a set A ⊆ S of absorbing states,a transition probability function τ S ∖ A →(S), an exit rate function ρ S ∖ A →[>0], and a labeling function ℓ S → L. The labels in L represent properties of interest that hold in a particular state according to the labeling function ℓ S → L.If s ∈ S is the current state of the system and E ⊆ S is a subset of states, τ(s)(E) ∈ [0,1] corresponds to the probability that a transition from s to arbitrary s' ∈ E is taken, and ρ(s) ∈[>0] represents the rate of an exponentially distributed random variable that characterizes the residence time in the state s before any transition is taken. Therefore, the probability to make a transition from state s to any s' ∈ E within time unit t ∈ is given by τ(s)(E) ·[ρ(s)]([0,t)), where [r](B) = ∫_B r e^-rx dx, for any Borel subset B ⊆ and r > 0. Absorbing states in A ⊆ S are used to represent termination or deadlock states. An example of a CTMC is shown in Figure <ref>.For discrete-time Markov chains, the standard notion of behavioral equivalence is probabilistic bisimulation of Larsen and Skou <cit.>. The following definition extends it to CTMCs. To ease the notation, for E ⊆ S, we introduce the relation ≡_E⊆ S × S defined by s ≡_E s' if either s,s' ∈ E or s,s' ∉ E.[Stochastic Bisimulation]Let = (S, A, τ, ρ, ℓ) be a CTMC. An equivalence relation R ⊆ S × S is a stochastic bisimulation onif whenever s R t, then * s ≡_A t, ℓ(s) = ℓ(t), and* if s,t ∉A, then ρ(s) = ρ(t) and, for all C ∈ S/_R, τ(s)(C) = τ(t)(C).Two states s, t ∈ S are bisimilar with respect to , written s ∼_ t, if they are related by some probabilistic bisimulation on .Intuitively, two states are bisimilar if they have the same labels, they agree on being absorbing or not, and, in the case they are non-absorbing, their residence-time distributions and probability of movingby a single transition to any given class of bisimilar states is always the same.As an example of two stochastic bisimilar states, consider s_1 and s_2 in the CTMC depicted on the left hand side of Figure <ref>. A bisimulation relation that relates them is the equivalence relation with equivalence classes given by s_1,s_2, s_3, and s_4, s_5. §.§ Bisimilarity Pseudometrics on CTMCs In this section, we introduce a family of pseudometrics on CTMCs parametric in a discount factor λ∈ (0,1). Following the approach of <cit.>, given a CTMC = (S, A, τ, ρ, ℓ) we define a (1-bounded) pseudometric on S as the least fixed point of an operator on the setof functions from S × S to [0,1]. This pseudometric is then shown to be adequate with respect to stochastic bisimilarity: we prove that two states are stochastic bisimilar if and only if they have distance zero.Recall that dX × X → is a pseudometric on a set X if d(x,x) = 0 (reflexivity), d(x,y) = d(y,x) (symmetry)and d(x,y) + d(y,z) ≥ d(x,z) (triangular inequality), for arbitrary x,y,z ∈ X; it is a metric if, in addition, d(x,y) = 0 iff x = y. A pair (X,d) where d is a (pseudo)metric on X is called a (pseudo)metric space.Hereafter we will assume L to be equipped with a 1-bounded metric [Since the set S of states is assumed to be finite, one may assume the set of labels to be so as well. Thus, the metric d_L on labels can be bounded without loss of generality.] d_LL × L → [0,1].The operator we are going to introduce will use three key ingredients: the distance d_L between labels, a distance between residence-time distributions, and a distance between transition distributions. The first is meant to measure the static differences with respect to the labels associated with the states, the last two are meant to capture the differences in the dynamics, respectively, with respect to the continuous and discrete probabilistic choices. To this end, we consider two distances over probability distributions. The first one is the total variation metric, defined for arbitrary Borel probability measures μ, ν overasμν = sup_E |μ(E) - ν(E)|,where the supremum is taken over the Borel measurable sets of . The second one is the Kantorovich distance, which is based on the notion of coupling of probability measures, which we introduce next in the case of probability distributions over finite sets. [Coupling] Let S be a finite set, and let μ, ν∈(S). A probability distribution ω∈(S × S) is said a coupling for (μ, ν) if, for arbitrary u,v ∈ S ∑_v ∈ Sω(u, v) = μ(u)and∑_u ∈ Sω(u, v) = ν(v).In other words, ω is a joint probability distribution with left and right marginal, respectively, given by μ and ν. We denote the set of couplings for (μ, ν) by μν. For a finite set S and a 1-bounded distance dS × S → [0,1] over it, the Kantorovich distance is defined, for arbitrary distributions μ, ν∈(S) as follows [The minimum can be used in place of an infimum thanks to the Fenchel-Rockafeller duality theorem (see <cit.>).]𝒦_d(μ, ν) = min∑_u,v ∈ S d(u,v) ·ω(u,v) ω∈μν .Intuitively, 𝒦_d lifts a (1-bounded) distance over S to a (1-bounded) distance over its probability distributions. One can show that 𝒦_d is a (pseudo)metric if d is a (pseudo)metric.Now, consider the following functional operator. Let = (S, A, τ, ρ, ℓ) be CTMC and λ∈ (0,1) a discount factor. The function Δ^_λ→ is defined as follows, for dS × S → [0,1] and s, t ∈ SΔ^_λ(d)(s,t) =1if s ≢_A t st if s,t ∈ A maxst, λ·st if s,t ∉ Awhere []→ and ,S × S → [0,1] are respectively defined byst = st+ (1 - st) ·𝒦_d(τ(s),τ(t)), st = d_L(ℓ(s),ℓ(t)), andst = [ρ(s)][ρ(t)] . The functional Δ^_λ measures the difference of two states with respect to: their labels (by means of the pseudometric ), their residence-time distributions (by means of the pseudometric ), and their discrete probabilities to move to the next state (by means of the Kantorovich distance). If two states disagree on being absorbing (or not) they are considered incomparable, and their distance is set to 1. If both states are absorbing, they express no dynamic behavior, hence they are compared statically, and their distance corresponds to that occurring between their respective labels. Finally, if the states are non-absorbing, then they are compared with respect to both their static and dynamic features, namely, taking the maximum among their respective associated distances.Specifically, the value st corresponds to the least probability that two transitions are taken independently from the states s and t at different moments in time.This value is used by the functional [] to measure the overall differences that might occur in the dynamics of the two states in combination with the Kantorovich distance between their transition probability distributions (see Remark <ref> for more details).The setis endowed with the partial order ⊑ defined as d ⊑ d' iff d(s,t) ≤ d'(s,t) for all s,t ∈ S and it forms a complete lattice. The bottom element 0 is the constant 0 function, while the top element is the constant 1 function. For any subset D ⊆, the least upper bound D, and greatest lower bound D are, respectively, given by ( D)(s,t) = sup_d ∈ D d(s,t) and ( D)(s,t) = inf_d ∈ D d(s,t), for all s,t ∈ S.It is easy to check that, for anyand λ∈ (0,1), Δ^_λ is monotone (i.e., whenever d ⊑ d', then Δ^_λ(d) ⊑Δ^_λ(d')), thus, since (, ⊑) is a complete lattice, by Tarski's fixed point theorem Δ^_λ admits least and greatest fixed points.[Bisimilarity distance]Letbe a CTMC and λ∈ (0,1). The λ-disc­oun­ted bisimilarity pseudometric on ℳ, denoted by ^, is the least fixed point of Δ^ℳ_λ. The rest of the section is devoted to show that the least fixed point ^ is indeed a pseudometric and, moreover, is adequate with respect to stochastic bisimilarity (Theorem <ref>). This justifies the definition above. To this end we need some technical lemmas. In particular, we prove that Δ^_λ preserves pseudometrics (Lemma <ref>) and it is Lipschitz continuous (Lemma <ref>). Hereafter, unless mentioned otherwise, we fix a CTMC = (S, A, τ, ρ, ℓ) and a discount factor λ∈ (0,1). To ease the notation Δ^_λ, ^, and ∼_ will be denoted simply by Δ_λ, , and ∼, respectively, wheneveris clear from the context.The operator Δ_λ preserves pseudometrics. Let dS × S → [0,1] be a pseudometric. We want to prove that Δ_λ(d) is a pseudometric. Recall that ,S × S → [0,1] are pseudometrics. Thus, since the point wise maximum of pseudometrics is a pseudometric, it suffices to prove that [] preserves pseudometrics. Recall that, 𝒦_d (S) ×(S) → [0,1] is a pseudometric, since d is so. Thus, reflexivity and symmetry are immediate. The only nontrivial case is the triangular inequality. Let s,t,u ∈ S ∖ A, we want to prove st≤su + ut. First note that, for 0 ≤β≤ 1 and α' ≥α, the following holds:α + (1- α) β= β + (1 - β) αdistributivity≤β + (1 - β) α' α≤α' and 0 ≤β≤ 1= α' + (1 - α') β . distributivityThus, sinceis a pseudometric, by triangular inequality and the above we have st = st+ (1 - st) ·𝒦_d(τ(s),τ(t))def. []≤su + ut + (1 - (su + ut) ) ·𝒦_d(τ(s),τ(t)) * If we show that the last summand in (<ref>) is less than or equal to the sum of (1 - su) ·𝒦_d(τ(s),τ(u)) and (1 - ut) ·𝒦_d(τ(u),τ(t)), we get the following, and we are done≤su+ (1 - su) ·𝒦_d(τ(s),τ(u)) +ut+ (1 - ut) ·𝒦_d(τ(u),τ(t)) = su + utdef. [] .To this end, consider two cases. If su + ut > 1 then the inequality holds trivially, since 1 - (su + ut) < 0, so that the last summand in (<ref>) is negative and 1-su≥ 0.Instead, if su + ut≤ 1 then 1 - (su + ut) ≥ 0, so we have(1 - (su + ut) ) ·𝒦_d(τ(s),τ(t)) ≤(1 - (su + ut) ) ·(𝒦_d(τ(s),τ(u)) + 𝒦_d(τ(u),τ(t)) )triang. 𝒦_d=(1 - (su + ut) ) ·𝒦_d(τ(s),τ(u)) + (1 - (su + ut) ) ·𝒦_d(τ(u),τ(t)) ≤(1 - su) ·𝒦_d(τ(s),τ(u)) + (1 - ut) ·𝒦_d(τ(u),τ(t))and we are done. The setcan be turned into a metric space by means of the supremum norm dd' = sup_s,t ∈ S |d(s,t) - d'(s,t)|. Next we show that the λ-discounted functional operator Δ_λ is λ-Lipschitz continuous, that is Δ_λ(d')Δ_λ(d)≤λ·d'd, for any d,d' ∈. The operator Δ_λ is λ-Lipschitz continuous.By <cit.>, to prove that Δ_λ is λ-Lipschitz continuous it suffices to show that, whenever d ⊑ d' then, for all s,t ∈ S, Δ_λ(d')(s, t) - Δ_λ(d)(s, t) ≤λ·d'd. If s ≢_A t, then Δ_λ(d')(s, t) - Δ_λ(d)(s, t) = 1 - 1 = 0, so that, the inequality is satisfied. If s,t ∉ A, Δ_λ(d')(s, t) - Δ_λ(d)(s, t) = st - st = 0, and again the inequality holds trivially. The same happens when st≥λ·[d']st. Indeed, by monotonicity of [], st≥λ·st holds, so that, by definition of Δ_λ we haveΔ_λ(d')(s,t) - Δ_λ(d)(s,t) = λ·(st - st) = 0 ≤λ·d'd .It remains the case when s,t ∉ A and λ·st≤st < λ·[d']st.Assume that 𝒦_d(τ(s),τ(t)) = ∑_u,v ∈ S d(u,v) ·ω(u,v), for some ω∈τ(s)τ(t), then we haveΔ_λ(d')(s,t) - Δ_λ(d)(s,t)≤λ·( [d']st - [d]st) ≤λ·( 𝒦_d'(τ(s),τ(t)) - 𝒦_d(τ(s),τ(t)) ) ≤λ·( ∑_u,v ∈ S d'(u,v) ·ω(u,v) - ∑_u,v ∈ S d(u,v) ·ω(u,v) ) = λ·( ∑_u,v ∈ S (d'(u,v) - d(u,v)) ·ω(u,v) ) ≤λ·( ∑_u,v ∈ Sd'd·ω(u,v) ) = λ·(d'd·∑_u,v ∈ Sω(u,v) ) = λ·d'd . to 221 pt It is standard thatwith the supremum norm forms a complete metric space (i.e, every Cauchy sequence converges). Therefore, since λ∈ (0,1), a direct consequence of Lemma <ref> and Banach's fixed point theorem is the following. For any λ∈ (0,1),is the unique fixed point of Δ_λ. Moreover, for any n ∈, and dS × S → [0,1] we have Δ^n_λ(d)≤λ^n/1-λΔ_λ(d)d. Now we are ready to state the main theorem of this section.is a pseudometric. Moreover, for any s,t ∈ S, s ∼ t if and only if (s,t) = 0 We first prove thatis a pseudometric. By Lemma <ref> and Banach's fixed point theorem, = _n ∈Δ^n_λ(0). Clearly, 0 is a pseudometric.Thus by Lemma <ref>, a simple induction on n shows that, for all n ∈, Δ^n_λ(0) is a pseudometric. Since the least upper bound with respect to ⊑ preserves pseudometrics, we have thatis so.Now we are left to prove that, for any s,t ∈ S, s ∼ t iff (s,t) = 0.0ex* (⇐) We prove that R =(s,t) (s,t) = 0 is a stochastic bisimulation. Clearly, R is an equivalence. Assume (s,t) ∈ R, then, by definition of Δ_λ, one of the following holds: *s,t ∈ A and st = 0; *s,t ∉ A, st = 0, and []st = 0.If (<ref>) holds, by st = 0 we get that ℓ(s) = ℓ(t). If (<ref>) holds, we have st = 0 and 𝒦_(τ(s),τ(t)) = 0. By st = 0 we get [ρ(s)] = [ρ(t)] and hence ρ(s) = ρ(t). By <cit.>, 𝒦_(τ(s),τ(t)) = 0 implies that, for all C ∈ S/_R, τ(s)(C) = τ(t)(C). Therefore R is a bisimulation. * (⇒)Let R ⊆ S × S be a stochastic bisimulation on , and define d_RS × S → [0,1] by d_R(s,t) = 0 if (s,t) ∈ R and d_R(s,t) = 1 if (s,t) ∉ R. We show that Δ_λ(d_R) ⊑ d_R. If (s,t) ∉ R, then d_R(s,t) = 1 ≥Δ_λ(d_R)(s,t). If(s,t) ∈ R, then ℓ(s) = ℓ(t) and one of the following holds: *s,t ∈ A; *s,t ∉ A, ρ(s) = ρ(t) and, ∀ C ∈ S/_R.τ(s)(C) = τ(t)(C).If (<ref>) holds, Δ_λ(d_R)(s,t) = st = 0 = d_R(s,t).If (<ref>) holds, by <cit.> and the fact that, for all C ∈ S/_R, τ(s)(C) = τ(t)(C), we have 𝒦_d_R(τ(s),τ(t)) = 0. Moreover st = 0. This givesthat Δ_λ(d_R)(s,t) = 0 = d_R(s,t).Since ∼ is a stochastic bisimulation, Δ_λ(d_∼) ⊑ d_∼, so that, by Tarski's fixed point theorem, ⊑ d_∼. By definition of d_∼ and⊑ d_∼, s ∼ t implies (s,t) = 0. Observe that Theorem <ref> holds for alternative definitions of the functional Δ_λ.An example is given when the functional [] in the definition of Δ_λ is given as [d]st = max1/M· |ρ(s) - ρ(t)|, 𝒦_d(τ(s),τ(t)) ,where M = max_s ∈ Sρ(s) is used to rescale the symmetric difference |ρ(s) - ρ(t)| to a value within [0,1]. Another example is obtained by replacing the maximum above by a convex combination of the two values as below,for some α∈ (0,1),[d]st = α·1/M |ρ(s) - ρ(t)| + (1- α) ·𝒦_d(τ(s),τ(t)), So, what does it make a proposal for Δ_λ preferable to another? Although Theorem <ref> is an important property for a behavioral pseudometric, it does not say much about the states that have distance different from zero. In this sense, a good behavioral metric should relate the distance with a concrete problem. Our definition of Δ_λ, for example, is motivated by a result in <cit.> that states that the total variation distance of CTMCs (more generally, on semi-Markov chains) is logically characterized as the maximal difference w.r.t. the likelihood for two states to satisfy the same Metric Temporal Logic (MTL) formula <cit.>. It turns out that when d_L is the discrete metric over L (i.e., d_L(a,b) = 0 if a = b, and 1 otherwise),bounds from above the total variation distance. This relatesto the probabilistic model checking problem of MTL-formulas against CTMCs. The above alternative proposals for a distance do not enjoy this property. § COMPLEXITY AND LINEAR PROGRAMMING REPRESENTATIONIn this section, we study the problem of computing the bisimilarity distance by considering two different approaches. The former is an iterative method that approximatesfrom below (resp. above) successively applying the operator Δ_λ starting from the least (resp. greatest) element in . The latter is based on a linear program characterization ofthat is based on the Kantorovich duality <cit.>. In contrast to an analogous proposal in <cit.>, our linear program has a number of constraints that is polynomially bounded in the size of the CTMC. As a consequence, the bisimilarity distancecan be computed in polynomial time in the size of the CTMC. §.§ Iterative method By Theorem <ref>, for any ϵ > 0, it follows that to get ϵ-close to , it suffices to iterate the application of the fixed point operator ⌈log_λϵ⌉ times. For any ϵ > 0 and dS × S → [0,1], Δ^⌈log_λϵ⌉_λ(d)≤ϵ. By Theorem <ref> we have Δ^n_λ(d)≤λ^n/1-λΔ_λ(d)d and by Δ_λ(d)d≤ 1, we have Δ^n_λ(d)≤λ^n/1-λ. For n = log_λ (ϵ - ϵλ), we have ϵ = λ^n/1-λ. Therefore, by Lemma <ref> and ⌈log_λϵ⌉≥log_λ (ϵ - ϵλ), we have Δ^⌈log_λϵ⌉_λ(d)≤ϵ. By the above result, we obtain a simple method for approximating . If the starting point is 0 we obtain an under-approximation, whereas starting from 1 we get an over-approximation. Both the approximations can be taken arbitrary close to the exact value.However, as shown in the following example, the exact distance value cannot be reached in general. This holds for any discount factor. [<cit.>] Consider the {red, blue}-labeled CTMC below.[label1/.style=circle, draw=red!50,fill=red!20, thin, circle split, inner sep=0.6mm,label2/.style=circle,draw=blue!50,fill=blue!20, thin, circle split, inner sep=0.6mm] 2.5cm (0,0) node[label1] (s) s lower 1 ((s) +(right:)) node[label1] (t) t lower 1 ((t) +(right:)) node[label2] (u) u lower 1;[-latex, font=] (s) edge[loop above] node[above] 1 (s) (t) edge[loop above] node[above] λ (t)edge node[above] 1-λ (u) (u) edge[loop above] node[above] 1 (u);Let d_LL × L → [0,1] be the discrete metric over L, defined as d_L(l,l') = 0 if l = l' and 1 otherwise. One can check that (s,t) = λ - λ^2/1- λ^2 and, for all n ∈, Δ^n_λ(0)(s,t) ≤λ - λ^2n+1/1+λ. Since, for all n ∈, λ - λ^2n+1/1+λ < λ - λ^2/1- λ^2, we have that the fixed point cannot be reached in a finite number of iterations.In <cit.> is shown that the bisimilarity distance of Desharnais et al. <cit.> can be computed exactly by iterating the fixed point operator up to a precision that allows one to use the continued fraction algorithm to yield the exact value of the fixed point. This method can be applied provided that the pseudometric has rational values. In their case, this is ensured assuming that the transition probabilities are rational. Unfortunately, in our case this cannot be ensured under the same conditions. Indeed, the total variation distance between exponential distributions with rates r, r' > 0 is analyticallysolved as follows[r][r'] = 0if r = r' | ( r'/r)^r/r - r' -( r'/r)^r'/r - r'|otherwisethus, even restricting to rational exit-rates and probabilities, the distance may assume irrational values.As a consequence, we cannot assume to compute, in general, the exact distance values. §.§ Linear Program Characterization Our linear program characterization leverages on two key results. The first one is the uniqueness of the fixed point of Δ_λ (Theorem <ref>). The second one is a dual linear program characterization of the Kantorovich distance.For S finite, dS × S → [0,1], and μ, ν∈(S), the value 𝒦_d(μ,ν) coincides with the optimal value of the following linear program𝒦_d(μ,ν)=[t] min_ω∑_u,v ∈ S d(u,v) ·ω_u,v∑_vω_u,v = μ(u) ∀ u ∈ S ∑_uω_u,v = ν(v) ∀ v ∈ Sω_u,v≥ 0 ∀ u,v ∈ S . By a standard argument in linear optimization, the above can be alternatively represented by the following dual linear program𝒦_d(μ,ν)=[t] max_y∑_u ∈ S (μ(u) - ν(u) )· y_uy_u - y_v≤ d(u,v) ∀ u,v ∈ S .This alternative characterization is a special case of a more general result commonly known as the Kantorovich duality and extensively studied in linear optimization theory (see <cit.>).Let n = |S| and h = n - |A|. Consider the linear program in Figure <ref>, hereafter denoted by D, with variables d ∈[]^n^2, y ∈[]^h^2 + n andk,m ∈[]^h^2. The constraints of D are easily seen to be bounded and feasible. Moreover, the objective function of D attains its optimal value when the vectors k and m are maximized in each component[Indeed, for arbitrary x,y ∈^n and n ∈, if x_i ≤ y_i for all i = 1..n, then ∑_i =1^n x_i ≤∑_i =1^n y_i.]. Therefore, according to (<ref>), an optimal solution (d^*,y^*,k^*,m^*) ∈ D satisfies the following equalities∀ s,t ∉A. m^*_s,t = min{st, λ( st + (1 - st) 𝒦_d^*(τ(s), τ(t)) ) } , ∀ s,t ∉A.k^*_s,t = 𝒦_d^*(τ(s), τ(t)).By the above equalities and the constraints of D, it follows that d^* is a fixed point of Δ_λ. Since the distance is the unique fixed point of Δ_λ, d^* =. Let (d^*,y^*,k^*,m^*) be a solution of D. Then, for all s,t ∈ S, d^*_s,t = (s,t). Consider d^*∈[]^|S|^2 and the linear program above, hereafter denoted by D(d^*)*arg max_y,k,m∑_s,t ∉A k_s,t + m_s,t m_s,t≤st∀ s,t ∉Am_s,t≤λ( st + (1 - st) k_s,t)∀ s,t ∉Ak_s,t = ∑_u ∈ S (τ(s)(u) - τ(t)(u))· y_u^s,t∀ s,t ∉A y_u^s,t - y_v^s,t≤ d^*_u,v∀ s,t ∉A, ∀ u,v ∈ S Any feasible solution of D(d^*) can be improved by increasing any value of k_s,t or m_s,t for some s,t ∉ A. On the one hand, the value of k_s,t (for some s,t ∉ A) can be increased independently from that of (m_s,t)_s,t ∉ A since it only depends on the values of (y^s,t_u)_u ∈ S. For this reason, if we denote by k^*_s,t an optimal value for k_s,t we have the following equalityk^*_s,t = max_y ∑_u ∈ S (τ(s)(u) - τ(t)(u) )· y_u y_u - y_v≤ d^*_u,v∀ u,v ∈ SThus, by Equation (<ref>), we have that k^*_s,t = 𝒦_d^*(τ(s), τ(t)) for all s,t ∉A. On the other hand, the value of m_s,t (for some s,t ∉ A) is bounded from above by the constant st and the λ( st + (1 - st) k_s,t). The value of λ( st + (1 - st) k_s,t) increases with that of k_s,t, hence, if if we denote by m^*_s,t the optimal value for m_s,t, we have that∀ s,t ∉A. m^*_s,t = min{st, λ( st + (1 - st) k^*_s,t) }∀ s,t ∉A.k^*_s,t = 𝒦_d^*(τ(s), τ(t)) .Let (d',y',k',m') be an optimal solution of D and (y”,k”,m”) be an optimal solution of D(d').Notice that the optimal value ∑_s,t ∉A k”_s,t + m”_s,t of D(d') (i.e., ∑_s,t ∉A k”_s,t + m”_s,t) is greater than or equal to the optimal value of D (i.e., ∑_s,t ∉A k'_s,t + m'_s,t), since the constraints of D(d') are a subset of those of D. Consider now two possible cases. (Case 1:k' = k” and m' = m”) For s,t ∉ A we have that the following equalities holdd'_s,t = st + λ( st + (1 - st) k'_s,t) - m'_s,tby (d',y',k',m') feasible for D= st + λ( st + (1 - st) k”_s,t) - m”_s,tby k' = k” and m' = m”= max{st ,λ( st + (1 - st) k”_s,t) }by (<ref>)= max{st, λ( st + (1 - st)𝒦_d'(τ(s),τ(t)) ) }by (<ref>)= max{st, λ𝒯(d')(s,t) }by def. 𝒯= Δ(d')(s,t) by def. Δ_λFrom the above equality and feasibility of (d',y',k',m') for D, we have d'_s,t = Δ_λ(d')(s,t) for all s,t ∈ S. Thus, by Theorem <ref> we obtain d'_s,t = (s,t) for all s,t ∈ S. (Case 2: k' ≠ k” or m' ≠ m”) By the previous case we have that the optimal value of D is equal to ∑_s,t ∉A k”_s,t + m”_s,t. This is a sum of non negative values, indeed by (s,t) ≥ 0 for all s,t ∈ S and Equations (<ref>) and (<ref>), we have that k”_s,t≥ 0 and m”_s,t≥ 0 for all s,t ∉ S. Thus, if k' ≠ k” or m' ≠ m” we have ∑_s,t ∉A k”_s,t + m”_s,t > ∑_s,t ∉A k'_s,t + m'_s,t. This contradicts the assumption that (d',y',k',m') is optimal for D.This proves that if (d',y',k',m') is an optimal solution of D then d' =.In <cit.> it has been shown that the bisimilarity distance of Desharnais et al. on discrete-time Markov chains can be computed in polynomial time as the solution of a linear program that can be solved by using the ellipsoid method. However, in their proposal the number of constraints may be exponential in the size of the model. This is due to the fact that the Kantorovich distance is resolved listing all the couplings that correspond to vertices of the transportation polytopes involved in the definition of the distance <cit.>. In contrast, our proposal has a number of constraints and unknowns[Actually, the variables k are used only for ease the presentation but they can be removed by substitution.] bounded by 3 |S|^2 + |S|^4 and |S|^3 + 2 |S|^2, respectively. This allows one to use general algorithms for solving LP problems (such as the simplex and the interior-point methods) that, in practice, are more efficient than the ellipsoid method. Our LP characterization (Theorem <ref>) exploits the fact that, for λ∈ (0,1), the functional operator Δ_λ has unique fixed point (Theorem <ref>). In the so called undiscounted case, i.e., when λ = 1, the fixed point is not unique anymore and the LP characterization does not work directly. To obtain a similar LP characterization with a number of constraints polynomial in the size of the CTMC, by following <cit.> one may modify the operator Δ_1 to an operator Δ' that forces the distance to be 0 if the states are stochastic bisimilar, otherwise it behaves as Δ_1. The operator Δ' has unique fixed point when d_L is the discrete metric over L (i.e., d_L(a,b) = 0 if a = b, and 1 otherwise), but for other choices of d_L the uniqueness may not hold. Moreover, this allows us to state the following complexity result.can be computed in polynomial-time in the size of .By Theorem <ref>,can be computed within the time it takes to construct and solve D. D has a number of constraints and unknowns that is bounded by a polynomial in the size of , therefore its construction can be performed in polynomial time. For the same reason, D admits a polynomial time separation algorithm: whenever a solution is given, its feasibility is checked by scanning each inequality in D; otherwise the first encountered inequality that is not satisfied is returned as a separation hyperplane. Therefore, the thesis follows by solving D using the ellipsoid method together with the naïve separation algorithm described above.In Theorem <ref> the term “computed” should be replaced by “approximated”. We already discussed about the impossibility of obtaining an exact value of(the total variations distance between exponential distributions may assume irrational values!). In fact, for the construction of the linear program D one has to use an approximated (rational) version of the coefficients st, for all s,t ∉ A, say e_s,t. To ensure that the optimal solution d^* of D is at most ε apart from(i.e., |(s,t) - d^*_s,t | ≤ε, for all s,t ∈ S) one only needs that |st - e_s,t | ≤ε/2, for all s,t ∉ A. This can be done using the Newton-Raphson's iteration algorithm for approximating the n-square root of a number ζ∈ up to a precision ϵ > 0. This method is known to be polynomially computable in the size of the representation of ζ, n and ϵ <cit.>. Hencecan be approximated up to any precision ε > 0 in polynomial-time in the size of .§ ALTERNATIVE CHARACTERIZATION OF THE PSEUDOMETRICIn the following, we propose an alternative characterization of the bisimilarity distance , based on the notion of coupling structure. Our result generalizes the one proposed in <cit.> for MCs, to the continuous-time settings.[Coupling Structure]Let ℳ = (S, A, π, ℓ) be a CTMC. A coupling structure foris a function (S∖ A) × (S∖ A) →(S × S) such that, for all s,t ∉ A, (s,t) ∈τ(s)τ(t).Intuitively, a coupling structure forcan be thought of as an S × S-indexed collection of joint probability distributions, each having left/right marginals equal to τ.The following definition adapts the definition of the operator Δ_λ (see Definition <ref>) with respect to the notion of coupling structure for a CTMC.Let = (S, A, τ, ρ, ℓ) be CTMC,a coupling structure for , and λ∈ (0,1) a discount factor. The function Γ^_λ→ is defined as follows, for dS × S → [0,1] and s, t ∈ SΓ^_λ(d)(s,t) =1if s ≢_A t st if s,t ∈ A maxst, λ·st if s,t ∉ Awhere ,S × S → [0,1] are as in Definition <ref> and []→ is given byst = st+ (1 - st) ·∑_u,v ∈ S d(u,v) ·(s,t)(u,v) . Recall that the Kantorovich distance between two distributions μ and ν is defined as 𝒦_d(μ,ν) = min_ω∑_u,v ∈ S d(u,v) ·ω(u,v), where the minimum is taken over all the possible couplings ω∈μν. Thus, the operator Γ^_λ can intuitively be thought of as a possible instance of Δ^_λ with respect to a fixed choice of the couplings given by . One can easily check that Γ^_λ is monotone, thus, by Tarski's fixed point theorem, it admits least and greatest fixed points. The least fixed point, in particular, will be denoted by C and referred to as the λ-discrepancy of 𝒞.= minC𝒞 coupling structure for ℳ. We first prove that δ_λ⊑C, for any coupling structurefor . By Tarski's fixed point theorem, it suffices to prove that, for any dS × S → [0,1], Δ_λ(d) ⊑Γ^_λ(d). The only nontrivial case is when s,t ∉ A, which follows by definition of 𝒦_d, by noticing that [d]st≤[d]st and that the maximum is order preserving. It remains to prove that the minimum is attained. To this end, define a coupling structure ^* as ^*(s,t) = ω_s,t, for s,t ∉ A, where ω_s,t∈τ(s)τ(t) is such that 𝒦_(τ(s),τ(t)) = ∑_u,v(u,v) ·ω_s,t(u,v). By construction, = Γ^^*_λ(), hence C^*⊑. Since ^* is a coupling structure for , by what we have shown above we also have ⊑C^*. Therefore, = C^*. § GREEDY COMPUTATION OF THE BISIMILARITY DISTANCEInspired by the characterization given in Theorem <ref>, we propose a procedure to compute the bisimilarity pseudometric that is alternative to those previously described. The set of coupling structures forcan be endowed with the preorderdefined as ' iff C⊑C'.Theorem <ref> suggests to look at all the coupling structuresforin order to find an optimal one, i.e., minimal w.r.t. . However, it is clear that the enumeration of all the couplings is unfeasible, therefore it is crucial to provide an efficient search strategy which allows one to find an optimal coupling by exploring only a finite amount of them. Moreover we also need an efficient method for computing the λ-discrepancy associated with a coupling structure. §.§ Computing the Lambda-Discrepancy In this section we consider the problem of computing the λ-discrepancy associated with a coupling structure. By Tarski's fixed point theorem, C corresponds to the least pre-fixed point of Γ^_λ, that is C = { d ∈ [0,1]^S × S|Γ^_λ(d) ⊑ d }. This allows us to compute the λ-discrepancy associated withas the optimal solution of the following linear program, denoted by 𝐷𝑖𝑠𝑐𝑟_λ().*arg min_d∑_s,t ∈ S d_s,t d_s,t≥ 1 if s ≢_A t d_s,t≥stif s ≡_A t d_s,t≥λ( st+ (1 - st) ·∑_u,v ∈ S d_u,v·(s,t)(u,v) ) if s,t ∉ A𝐷𝑖𝑠𝑐𝑟_λ() has a number of inequalities that is bounded by 2 |S|^2 and |S|^2 unknowns, thus, it can be efficiently solved using the interior-point method. If one is interested in computing the λ-discrepancy for a particular pair of states (s,t), the method above can be applied on the least independent set of inequalities containing the variable d_s,t. Moreover, assuming that for some pairs the values associated to d are known, the set of constraints can be further decreased by substitution. §.§ Greedy Strategy for Optimal Coupling Structures In this section, we propose a greedy strategy that moves toward an optimal coupling structure starting from any given one. Then, we provide sufficient and necessary conditions for a coupling structure, to ensure that its associated λ-discrepancy coincides with .Hereafter we fix a CTMC =(S, A, τ, ρ, ℓ) and a coupling structurefor it.The greedy strategy takes a coupling structure and locally updates it at a given pair of states in such a way that it decreases it with respect to .For s,t ∉ A and ω∈τ(s)τ(t), we denote by [(s,t)/ω] the update ofat (s,t) with ω, defined as [(s,t)/ω](u,v) = (u,v), for all (u,v) ≠ (s,t), and [(s,t)/ω](s,t) = ω; it is worth noting that, by construction, [(s,t)/ω] is a coupling structure of .The next lemma gives a sufficient condition for an update to be effective for the strategy. Let s, t ∉ A and ω∈τ(s)τ(t). Then, for ℋ = [(s,t)/ω] and any λ∈ (0,1), if Γ^ℋ_λ(C)(s,t) < C(s,t) then H⊏C.It suffices to show that Γ^ℋ_λ (C) ⊏C, i.e., that C is a strict post-fixed point of Γ^ℋ_λ. Then, the thesis follows by Tarski's fixed point theorem. Let u,v ∈ S. If u ≢_A v, then Γ^ℋ_λ(C)(u,v) = 1 =Γ^_λ(C)(u,v) =C(u,v). If u,v ∈ A, then Γ^ℋ_λ(C)(u,v) = uv = Γ^_λ(C)(u,v) =C(u,v). If u,v ∉ A and (u,v) ≠ (s,t), by definition of ℋ, we have that (u,v) = ℋ(u,v), hence Γ^_λ(C)(u,v) = Γ^ℋ_λ(C)(u,v). The remaining case, i.e., (u,v) = (s,t), holds by hypothesis. This proves Γ^ℋ_λ (C) ⊏C. Lemma <ref> states that 𝒞 can be improved w.r.t.by updating it at (s,t), if s,t ∉ A and there exists a coupling ω∈τ(s)τ(t) such that the following holds∑_u,v ∈ SC(u,v) ·ω(u,v) < ∑_u,v ∈ SC(u,v) ·(s,t)(u,v).A coupling that enjoys the above condition is ω∈ TP(C, τ(s), τ(t)) where, for arbitrary μ,ν∈(S) and cS × S → [0,1]TP(c, μ, ν) = *arg min_ω∑_s,t ∈ S c(u,v) ·ω_u,v∑_vω_u,v = μ(u) ∀ u ∈ S ∑_uω_u,v = ν(v) ∀ v ∈ S ω_u,v≥ 0 ∀ u,v ∈ S .The above problem is usually referred to as the (homogeneous) transportation problem with μ and ν as left and right marginals, respectively, and transportation costs c. This problem has been extensively studied and comes with (several) efficient polynomial algorithmic solutions <cit.>.This gives us an efficient solution to update any coupling structure, that, together with Lemma <ref> represents a strategy for moving towardby successive improvements on the coupling structures.Now we proceed giving a sufficient and necessary condition for termination. If C≠, then there exist s,t ∉ A and a coupling structure ℋ = 𝒞[(s,t)/ω] for ℳ such that Γ^ℋ_λ(C)(s,t) < C(s,t). We proceed by contraposition. If for all s,t ∉ A and ω∈τ(s)τ(t), Γ^ℋ_λ(C)(s,t) ≥C(s,t), then C = Δ_λ(C). Since, by Theorem <ref>, Δ_λ has a unique fixed point, C =. The above result ensures that, unless 𝒞 is optimal w.r.t , the hypothesis of Lemma <ref> issatisfied, so that, we can further improve 𝒞 as aforesaid.The next statement proves that this search strategy is correct.= C iff there is no coupling ℋ for ℳ such that Γ^ℋ_λ(C) ⊏C. We prove: ≠C iffthere exists ℋ such that Γ^ℋ_λ(C) ⊏C. (⇒) Assume ≠C.By Lemma <ref>, there exist a pair of states s,t ∈ S and a coupling ω∈τ(s)τ(t) such that λ·∑_u,v ∈ SC(u,v) ·ω(u,v) < C(s,t). As in the proof of Lemma <ref>, we have that ℋ = 𝒞[(s,t)/ω]satisfies Γ^ℋ_λ (C) ⊏C. (⇐) Let ℋ be such thatΓ^ℋ_λ(C) ⊏C.By Tarski's fixed point theorem D⊏C.By Theorem <ref>, ⊑D⊏C. Note that, in general there could be an infinite number of coupling structures for a given CTMC. However, for each fixed d ∈, the linear function mapping ω to ∑_u,v ∈ S d(u,v) ·ω(u,v) achieves its minimum at some vertex in the transportation polytope τ(s)τ(t).Since the number of such vertices is finite, the termination of the search strategy is ensured by updating the coupling structure using optimal vertices. This does not introduce further complications in the algorithm since there are methods for solving the transportation problem (e.g., Dantzig's primal simplex method <cit.>) which provide optimal transportation schedules that are vertices. § THE ON-THE-FLY ALGORITHM In this section we describe an on-the-fly technique for computing the bisimilarity distancefully exploiting the greedy strategy of Section <ref>.Let Q ⊆ S × S and consider the problem of computing (s,t) for all (s,t) ∈ Q. Recall that the strategy proposed in Section <ref> consists in a traversal _0 ▹_λ_1 ▹_λ⋯▹_λ_n of the set of coupling structures forthat starts from an arbitrary coupling structure _0 and leads to an optimal one _n.We observe that, for any i < n * the improvement of each coupling structure _i is obtained by a local update at some pair of states u,v ∉ A, namely _i+1 = _i[(u,v)/ω] for some ω∈ TP(_i,τ(u),τ(v));* the pair (u,v) is chosen according to an optimality check that is performed locally among the couplings in τ(u)τ(v), i.e., _i(u,v) ∉ TP(_i,τ(u),τ(v));* whenever a coupling structure _i is considered, its associated λ-discrepancy _i can be computed by solving the linear program 𝐷𝑖𝑠𝑐𝑟_λ(_i) described in Section <ref>.Among the observations above, only the last one requires to look at the coupling structure _i. However, as noticed in Remark <ref>, the value _i(s,t) can be computed without considering the entire set of constraints of 𝐷𝑖𝑠𝑐𝑟_λ(_i), but only the least independent set of inequalities that contains the variable d_s,t.Moreover, provided that for some pairs of states E ⊆ S × S the value of the distance is known, the linear program 𝐷𝑖𝑠𝑐𝑟_λ(_i) can be further reduced by substituting the occurrences of the unknown d_u,v by the constant (u,v), for each (u,v) ∈ E.This suggests that we do not need to store the entire coupling structures, but they can be constructed on-the-fly during the calculation.Specifically, the couplings that are demanded to compute _i(s,t) are only those _i(u,v) such that , where __i,E^* is the reflexive and transitive closure of __i,E⊆ S^2 × S^2, defined by(s',t') __i,E (u',v')iff _i(s',t')(u',v') > 0and(u',v') ∉ E. The computation of the bisimilarity pseudometric is implemented byAlgorithm <ref>. It takes as input a finite CTMC= (S, A, τ, ρ, ℓ), a discount factor λ∈ (0,1), and a query set Q ⊆ S × S. We assume the following global variables to store:* 𝒞: the current (partial) coupling structure;* d: the λ-discrepancy associated with 𝒞;* ToCompute: the pairs of states for which the distance has to be computed; * Exact: the set pairs of states (s,t) such that d(s,t) = (s,t), i.e., those pairs which do not need to be further improved[Actually, the set Exact contains those pairs such that |d(s,t) - (s,t)| ≤ϵ where ϵ corresponds to the precision of the machine. In our implementation ϵ = 10^-9.];* Visited: the set of pairs of states that have been visited so far.Moreover, s,t will denote the set { (u,v) | (s,t) _,Exact^* (u,v) }.At the beginning(line <ref>–<ref>) both the coupling structure 𝒞 and thediscrepancy d are empty, there are no visited states, no exact computed distances, and the pairs to be computed are those in the input query.While there are still pairs left to be computed (line <ref>), we pick one (line <ref>), say (s,t). According to the definition of , if s ≢_A t then (s,t)=1; if s = t then (s,t) = 0 and if s,t ∈ A then (s,t) = st, so that, d(s,t) is set accordingly, and (s,t) is added to 𝐸𝑥𝑎𝑐𝑡 (lines <ref>–<ref>). Otherwise, if (s,t) was not previously visited, a coupling ω∈τ(s)τ(t) is guessed, and the routine 𝑆𝑒𝑡𝑃𝑎𝑖𝑟 updates the coupling structureat (s,t) with ω (line <ref>), then the routine 𝐷𝑖𝑠𝑐𝑟𝑒𝑝𝑎𝑛𝑐𝑦 updates d with the λ-discrepancy associated with(line <ref>). According to the greedy strategy,is successively improved and d is consequently updated, until no further improvements are possible (lines <ref>–<ref>). Each improvement is obtained by replacing a sub-optimal coupling C(u,v), for some (u,v) ∈s,t, by one taken from TP(d, τ(u), τ(v)) (line <ref>). Note that, each improvement actually affects the current value of d(s,t), since the update is performed on a pair in s,t. It is worth to note thatand 𝐸𝑥𝑎𝑐𝑡 are constantly updated, hence s,t may differ from one iteration to another.When line <ref> is reached, for each (u,v) ∈s,t, we are guaranteed that d(u,v) = (s,t), therefore s,t is added to 𝐸𝑥𝑎𝑐𝑡 and, for these pairs, d will no longer be updated. At this point (line <ref>), the couplings associated with the pairs in 𝐸𝑥𝑎𝑐𝑡 can be removed from . In line <ref>, the exact pairs computed so far are removed from ToCompute.Finally, if no more pairs need be considered, the exact distance on Q is returned (line <ref>).Algorithm <ref> calls the subroutines 𝑆𝑒𝑡𝑃𝑎𝑖𝑟 and 𝐷𝑖𝑠𝑐𝑟𝑒𝑝𝑎𝑛𝑐𝑦. The former is used to construct and update the coupling structure 𝒞, the latter to update the current over-approxi­ma­tion d during the computation. Next, we explain how they work. 𝑆𝑒𝑡𝑃𝑎𝑖𝑟 (Algorithm <ref>) takes as input a CTMC = (S, A, τ, ρ, ℓ), a pair of states s,t ∈ S, and a coupling ω∈τ(s)τ(t). In lines <ref>–<ref>, the coupling structureis set to ω at (s,t), then (s,t) is added to 𝑉𝑖𝑠𝑖𝑡𝑒𝑑. The on-the-fly construction of the coupling structure is recursively propagated to the demanded successor pairs of (s,t) according to the information accumulated so far.During this construction, if some states with trivial distances are encountered, d and Exact are updated accordingly (lines <ref>–<ref>).𝐷𝑖𝑠𝑐𝑟𝑒𝑝𝑎𝑛𝑐𝑦 (Algorithm <ref>) takes as input a discount factor λ∈ (0,1) and a pair of states s,t ∉ A. It constructs the least linear program obtained from 𝐷𝑖𝑠𝑐𝑟_λ(), that can compute (s,t) using the information accumulated so far (line <ref>). In lines <ref>–<ref> the current λ-discrepancy is updated accordingly; and those pairs (u,v) ∈s,t for which the current λ-discrepancy coincide with the distance are added to 𝐸𝑥𝑎𝑐𝑡.Next, we present a simple example of Algorithm <ref>,showing the main features of our method: (1) the on-the-fly construction of the (partial) coupling, and (2) the restriction only to those variableswhich are demanded for the solution of the system of linear equations.[On-the-fly computation]Consider the CTMC in Figure <ref>, and assume we want to compute the λ-discounted bisimilarity distance between states s_1 and s_4, for λ = 1/2.Algorithm <ref> starts by guessing an initial coupling structure 𝒞_0. This is done by considering only the pairs of states which are really needed in the computation. Starting from the pair (s_1,s_4) a coupling in ω_1,4∈τ(s_1)τ(s_4) is guessed as in Figure <ref> and assigned to _0(s_1,s_4).This demands for the exploration of the pairs(s_2,s_3), (s_2,s_4), (s_1,s_2) and the guess of three new couplings ω_2,3∈τ(s_2)τ(s_3), ω_2,4∈τ(s_2)τ(s_4), and ω_1,2∈τ(s_1)τ(s_2), to be associated in _0 with their corresponding pairs. Since no other pairs are demanded, the construction of _0 terminates as shown in Figure <ref>. The λ-discrepancy associated with _0 for the pair (s_1, s_4) is obtained as the solution of the following reduced linear program*arg min_d ( d_1,4 + d_2,3 + d_2,4 + d_1,2 ) d_1,4≥1/6d_1,4≥α/2 + (1-α)/2·( 4/9· d_2,3 + 17/63· d_2,4 + 11/63·d_4,4^=0) d_2,3≥1/2d_2,3≥α/2 + (1-α)/2·( 1/4· d_1,2 + 1/4·d_2,2^=0 + 1/2· d_2,4) d_2,4≥2/3d_2,4≥1/2·( 1/9· d_1,2 + 5/36· d_1,4 + 4/9· d_2,3 + 11/36· d_2,4) d_1,2≥1/2d_1,2≥α/2 + (1-α)/2·( 5/7·d_2,2^=0 + 1/4· d_1,4 + 1/28· d_2,4)where α = [15][9] = √(3/5)/25 (by Equation (<ref>)). Note that, the bisimilarity distance for the pairs (s_2,s_2) and (s_4,s_4) is always 0, thus d_2,2 and d_4,4 are substituted accordingly. The solution of the above linear program isd^_0(s_1,s_4) = α/2 + 5(1-α)/21, d^_0(s_2,s_3) = 1/2, d^_0(s_2,s_4) = 2/3, and d^_0(s_1,s_2) = 1/2.Since, the λ-discrepancy for (s_2,s_3), (s_2,s_4), and (s_1,s_2) equals the distancebetween their labels, it coincides with the bisimilarity distance, hence it cannot be further decreased. Consequently, the pairs of states are added to the set 𝐸𝑥𝑎𝑐𝑡 and their associated couplings are removed from _0. Note that, these pairs will no longer be considered in the construction of a coupling structure.In order to decrease the λ-discrepancy of (s_1,s_4), Algorithm <ref> constructs a new coupling structure _1. According to our greedy strategy, _1 is obtained from _0 updating _0(s_1,s_4) (i.e., the only coupling left) by the coupling ω'_1,4∈τ(s_1)τ(s_4) (shown in Figure <ref>) that is obtained as the solution of a transportation problem with marginals τ(s_1) and τ(s_4), where the current λ-discrepancy is taken as cost function. The resulting coupling does not demand for the exploration of new pairs in the CTMC, hence the construction of _1 terminates. The reduced linear program associated with _1 is given by *arg min_d d_1,4d_1,4≥1/6d_1,4≥α/2 + (1-α)/2·( 1/9·d_2,2^=0+ 4/9·d_2,3^=1/2+ 10/63·d_2,4^=2/3+ 2/7·d_4,4^=0)whose solution is d^_1(s_1,s_4) = α/2 + 31(1-α)/189.Solving again a new transportation problem with the improved current λ-discrepancy as cost function, we discover that the coupling structure _1 cannot be further improved, hence we stop the computation, returning (s_1,s_4) = d^_1(s_1,s_4) = α/2 + 31(1-α)/189. Algorithm <ref> can also be used for computing over-appro­xi­mateddistances. Indeed, assuming over-estimates for some particular distances are already known, they can be taken as inputs and used in our algorithm simply storing them in the variable d and treated as “exact” values. In this way our method will return the least over-approximation of the distance agreeing with the given over-estimates. This modification of the algorithm can be used to further decrease the exploration of the CTMC. Moreover, it can be employed in combination with approximated algorithms, having the advantage of an on-the-fly state space exploration.§ EXPERIMENTAL RESULTSIn this section, we evaluate the performance of the on-the-fly algorithm on acollection of randomly generated CTMCs[ The tests have been performed on a prototype implementation coded in Wolfram Mathematica^9 (available at<http://people.cs.aau.dk/ giovbacci/tools.html>) running on an Intel Core-i7 3.4 GHz processor with 12GB of RAM.].First, we compare the execution times of the on-the-fly algorithm with those of the iterative method proposed in Section <ref>.Since the iterative method only allows for the computation of the distance for all state pairs at once, the comparison is (in fairness) made with respect to runs of our on-the-fly algorithm with inputquery being the set of all state pairs. For each input instance, the comparison involves the following steps:

http://arxiv.org/abs/1702.08306v2

The role of the observer in goal-directed behaviorInés Samengo10pt 0.25in AbstractIn goal-directed behavior, a large number of possible initial states end up in the pursued goal. The accompanying information loss implies that goal-oriented behavior is in one-to-one correspondence with an open subsystem whose entropy decreases in time. Yet ultimately, the laws of physics are reversible, so entropy variations are necessarily a consequence of the way a system is described. In order to reconcile different levels of description, systems capable of yielding goal-directed behavior must transfer the information about initial conditions to other degrees of freedom outside the boundaries of the agent. To operate steadily, they must consume ordered degrees of freedom provided as input, and be dispensed of disordered outputs that act as wastes from the point of view of the aimed objective. Broadly speaking, hence, goal-oriented behavior requires metabolism, even if conducted by non-living agents. Here I argue that a physical system may or may not display goal-directed behavior depending on what exactly is defined as the agent. The borders of the agent must be carefully tailored so as to entail the appropriate information balance sheet. In this game, observers play the role of tailors: They design agents by setting the limits of the system of interest. Their computation may be iterated to produce a hierarchy of ever more complex agents, aiming at increasingly sophisticated goals, as observed in darwinian evolution. Brain-guided subjects perform this creative observation task naturally, implying that the observation of goal-oriented behavior is a goal-oriented behavior in itself. Minds evolved to cut out pieces of reality and endow them with intentionality, because ascribing intentionality is an efficient way of modeling the world, and making predictions. One most remarkable agent of whom we have indisputable evidence of its goal-pursuing attitude is the self. Notably, this agent is simultaneously the subject and the object of observation.A bunch of nucleic acids swim among many other organic compounds forming a cytoplasmatic soup, and somehow, manage to arrange themselves into precisely the sequence required for DNA replication. Carbon dioxide molecules steadily stick to one another materializing a solid tree trunk out of a tiny seed. Owls eat the young bats with poor navigation ability, thereby improving the eco-location proficiency of the species. The neurons in a dog's brain fire precisely in the required sequence to have the dog bury its bone, hiding it from other dogs. The wheels, breaks, and clutch of a self-driving car coordinate their actions in order to reach the parking area of a soccer field, no matter the initial location of the car, nor the traffic along the way. The limbs of the Argentine soccer players display a complex pattern of movements that carry the ball, through kicks and headers, at Messi's feet in front of the keeper... kick... goal!This essay is about goals. In all the above examples, a collection of basic elements, following local and apparently purpose-less laws, manage to steer the value of certain variables into some desired regime. The initial state is rather arbitrary, and yet, the agents manage to adaptively select, out of many possible actions, the maneuvers that are suited to conduct the system to the desired goal. Throughout these seemingly intelligent choices, order appears to raise from disorder. Scattered nucleotides become DNA. Air and dust become trees. Owl hunger becomes sophisticated eco-location organs. Neural activity becomes a buried bone. A car anywhere in the city becomes a car at a specific location. A football anywhere in the stadium becomes a football in the goal. How do the components of each system know what to do, and what not to do, in order to reach the goal? This is the question that will entertain us here.L0.5< g r a p h i c s > In DNA replication, many initial states are mapped onto a single final state. The entropy is therefore high at the beginning and low at the end.One important characteristic of goal-directed agents is that they are flexible: They reach the goal from multiple initial conditions, and are typically able to circumvent obstacles. For example, in DNA replication, the initial state is one out of many configurations in which nucleotides can be spatially distributed in a solution of organic compounds. The final state, the goal, is the precise spatial arrangement of those same nucleotides within the newly constructed DNA strand. In the soccer stadium, the ball may be initially in any location, the final state is the ball at the goal. Multiple initial states are hence mapped onto a single final state, as in Fig. <ref>.In physical terms, the non-injective nature of this mapping implies a reduction in entropy.Admittedly, the final state need not be strictly unique. In DNA replication, permutations of equal nucleotides are still allowed in the final state, and occasionally, there might also be a few errors in the replication process. Dogs may consider more than a single location for the concealed bounty, and Messi may choose to shoot the ball anywhere inside the 24 ft wide by 8 ft high of the goal. Such restricted amounts of freedom, or even the occasional failures to reach the final state (shooting an own goal, for example), by no means compensate the abrupt reduction in entropy that takes place throughout the process.In fact, were entropy not to decrease, the system would not exhibit goal-oriented behavior. We are used to associating entropy increments with information losses, and entropy reductions with information gains. Here I am taking the opposite view: Entropy reductions are associated with information losses. The two views are not incompatible, they simply refer to different things. The first case deals with a closed system and information about macroscopic variables. The second, with an open system and microscopic variables. When a closed macroscopic system evolves in the direction that maximizes the entropy of all compatible microscopic states (the usual case in closed thermodynamical systems), the final macro-state does not allow us to deduce the initial macro-state, since the mapping between them is non-injective. Were we to know the detailed final micro-state, however, we would be able to deduce the initial micro-state. When a goal-oriented system evolves in the direction of decreasing entropy, the final micro-state does not allow us to deduce the initial micro-state, but for a different reason: goal-oriented systems are open, and they interact with degrees of freedom we are not keeping track of. In this essay, the distinction between micro and macro-states is not emphasized, because the phenomena we deal with are not always divisible into separate scales.The notion of goal-oriented behavior that is used here always brings about an entropy reduction. I now want to demonstrate the reciprocal statement: If a system reduces its entropy, a goal can be ascribed to the process. Therefore, entropy reduction and goal-oriented behavior are in a one-to-one correspondence.The goal in question can always be defined by the restricted set of values that the variables acquire in the final state: the target DNA sequence, the buried bone, the ball at the goal. Of course, the reduction in entropy must first be verified: a broad set of initial states must evolve into a small final set. A car that in a single trial travels from one location to another is not guaranteed to be a self-driving car. Only if the initial location has proven to be arbitrary, and the traffic conditions variable, can goal-directed behavior be arrogated. The notion of entropy is subtle, since it not only characterizes a physical system, but also, the way it is described. When the universe is described at its utmost basic level (assuming there is one such level), all we have is a collection of fundamental particles evolving from some initial state. If the state of all particles is specified, the total entropy of the universe vanishes. Time reversibility of the laws of physics dictates entropy to remain zero for all past and future times. Therefore, there is no way to attain neither an increase nor a reduction in entropy. Energy dissipation and goal-directed behavior, hence, are absent from the complete description. We need to blur our point of view to give them a chance, either by restricting the description to macroscopic variables, or to subsystems. In fact, the main conclusion of this essay is that an observer with a very special point of view is required for agency to exist. If the information about the initial conditions is apparently lost in goal-oriented behavior, then such information must be somehow concealed in degrees of freedom we are not keeping track of.They may have been moved into too microscopic states to be monitored, or into fluid degrees of freedom that, by the time the goal is reached, have already exited the subsystem under study. What we track, and what we ignore, hence, plays a crucial role in agency.L0.5< g r a p h i c s > A demon controls the sliding door, allowing particles to pass from right to left, but not the other way round. The initial state of every molecule that the demon has already acted on (letting it pass or not) is recorded in its memory, and depicted in blue. To be consistent with the second law of thermodynamics, processes where entropy decreases are only possible in open systems that somehow interact with the external world. Originally, they were supposed to require an energy influx. This is, however, not a necessary condition: Sometimes, the sole exchange of information suffices. A good example is Maxwell's Demon <cit.>. Suppose we have a gas enclosed in two adjacent chambers communicated by a small hole in the wall between them (Fig. <ref>). The hole may or may not be covered by a sliding door controlled by a demon. Initially, both chambers have equal pressure and density. The demon then opens or closes the hole selectively, depending on whether a molecule approaches from one side, or the other. Molecules coming from the right are allowed to pass into the left chamber, but not the other way round. As time goes by, molecules accumulate on the left side, eventually leaving the right side empty. The collection of all gas molecules can be interpreted as performing goal-directed behavior: No matter the initial state, gas is gradually compressed into the left chamber. This final state can be conceived as a goal, and it comprises a reduction in entropy: initially each particle can be anywhere in the two chambers, and in the final state, they are all in the left side. Arrogating purpose, in this case, is to assume that the gas—who takes the role of the agent—wants to shrink. Other verbs may be used (tends to, is inclined to, etc.), but the phrasing is irrelevant. As uncanny as it may seem, arrogating purpose to the gas is a rather accurate description of the gas' phenomenology.The gas + demon is a toy model of a closed system, so no interaction with the outside world is allowed. To perform the task, the demon needs to acquire information on the location of each molecule approaching the hole, to then decide whether to let it pass or not. In a slightly modified version of this system, Bennett <cit.> demonstrated that the storage of information in the demon's memory can be done with no energy expenditure, as long as the memory is initially blank, and there is plenty of storage capacity. The work required to move and stop the door, as well as the energy needed to measure the position of particles and to maintain the demon alive, can also be made as small as desired, simply diminishing mechanical friction, and moving slowly. The demon is however not allowed to delete the acquired information, because information erasure requires energy consumption, at a minimal cost of k_ BT per erased bit <cit.>. Therefore, as time goes by, the information of the initial location of each gas molecule is erased from the gas, and copied onto the demon's memory.The gas gradually reduces its entropy only if we are careful to exclude the demon from what we define as the system. If we include the demon (and its memory), entropy simply remains constant, since all the details of the initial state are still stored. Depending on the observer's choices, then, entropy may or may not decrease, meaning that arrogating agency may or may not be possible. A subsystem can only decrease its entropy if it somehow gets rid of initial conditions. In DNA replication, after the addition of each new nucleotide to the developing strand, the initial location of the free nucleotide determines the final configuration of the mediating enzimes, thereby transferring the information of the initial state to a change in the 3-dimensional configuration of nearby proteins. If enzimes are not restored to their functional state, the process cannot be iterated. So enzimes, in turn, must pass the information on somewhere else. This transfer is actually the important point in the emergence of goal-directed behavior. Energy consumption is only helpful if energy is degraded in the process: ordered energy sources must be transformed into disordered products. In animal cells, order arrives as glucose and oxygen molecules. Disorder exits as carbon dioxide, water and faster molecular motion (heat). The input degrees of freedom, specifically in the case of glucose, are conformed of atoms tidily organized into large molecules. The output degrees of freedomare transported by smaller molecules, amenable to be arranged in many more configurations. The laws of physics are ultimately reversible, so initial conditions cannot be truly erased, they can only be shuffled around. As an example, Edward Fredkin studied how non-dissipative systems, such as our universe, may perform the usual logical computations (AND, OR, etc), which are themselves non-invertible <cit.>. We know that 0 AND 1 = 0. However, knowing that the result of the operation is 0 does not suffice to identify the two input variables. If the computation is performed by an ultimately non-dissipative system, the information of the initial input variables must be somehow moved into some other variable, albeit perhaps not in a manner that is easily accessible. Fredkin's solution was to prove thatcomputing required some extra input variables, not needed for the computation per se, but mandatory for the information balance. When performing a single logical operation (say, for example, AND), the additional variables are in a well defined state (no uncertainty), and throughout the computation, they acquire so-called garbage values (garbage because they are not required to perform the computation), that represent those input degrees of freedom that cannot be deduced from the output. Copying part of the input into garbage variables ensures that no information is lost, and the computation becomes feasible in a non-dissipative physical substrate.L0.5< g r a p h i c s > Goal-directed systems (blue ball) eat up ordered degrees of freedom, and produce disordered degrees of freedom.Ascribing agency is all about ignoring who really did the job (the Universe, to put it grandly), and arrogating intentionality to an entropy-reducing subsystem. The task of the observer is to design the borders of the subsystem so as to allow ordered degrees of freedom to be progressively incorporated, and/or disordered ones to be eliminated. If the goal is to be achieved repeatedly, a steady flow of order is required, as well as a regular garbage collection service. Purposeful agents, hence, only emerge from sub systems that eat up order (Fig. <ref>). Broadly speaking, they can be said to breathe, or to be endowed with metabolism,even if they need not be alive in the biological sense.Maxwell's demon hid the initial conditions of the gas in its memory. The dog, the self-driving car, and the soccer players, all hide their own initial state and that of the environment inside their memories. Memories can of course be erased, but erasures consume energy, and they are ultimately no more than flushing initial conditions into the high-entropy products of energy degradation.L0.5< g r a p h i c s > Observers, just as photographers, selectively focus on an aspect of reality, to satisfy their cognitive appetite <cit.>. Potograph kindly supplied by Lucía Samengo.For a long time, scientists failed to include memories as part of the systems under study, so goal-directed behavior sometimes appeared paradoxical. Here I argue that observers attribute agency by disregarding initial conditions. Of course, observers are free to delineate the borders of the subsystem under study as they wish. They can always shape the limits of what they define as the agent in such a way as to have it do all sorts of wonderful things, as achieve goals, and reduce entropy. The agent must be fed with order, and the mess must be cleaned up, but still, it can be done. The natural question is therefore: What is interesting in goal-directed behavior if the observer is allowed to engineer the very definition of the agent, in order to get the desired result? Plants grow because what we define as a plant is the stuff that grows every spring, and not the dirt left on the ground every autumn. Species improve because we restrict the definition of a species to the material that a posteriori is seen as successful, and exclude the corpses left behind of those who failed. Cell division seems to be a productive business because the waste products are not defined to be part of cells. Returning to the question posed above “How do the components of the system know what to do, and what not to do, in order to reach the goal?”, we can now provide an answer. Components know nothing, observers do. Just as photographers select an arbitrary plane in the visual world where to focus an image and engender a sharp object (Fig. <ref>), so do observers choose which variables compose the system, and which do not, so that a goal emerges.Should we be amazed that the world we live in allows observers to create agents? Could we not live in a universe where assigning agency were downright impossible? I would be very much surprised if it were so. The impossibility to define goal-directed behavior would mean that no subsystems exist where entropy decreases. The global entropy growth that takes place in the whole universe should develop uniformly and monotonously all throughout space and time, allowing for no local oscillations. That would be ordered indeed! I do not expect disorder to arise in such an orderly manner. Within this picture, all the interesting events seem to take place in the observer's creative act. Any local decrement of entropy, no matter how trivial, appears to suffice for an observer to be able to ascribe agency. We are demanding little of the world, and a lot of the observer. But does the evolving world not have organizational merits of its own? If we look at the history of events taking place in our planet, as time goes by, agents seem to become increasingly sophisticated. Compare the strategy of a replicating DNA molecule in a bacterium with the one of Menelaus of Sparta to recover Helen of Troy, and thereby, ultimately manage bisexual reproduction. All the complexity of the bacterium's strategy is present in Menelaus', but not the other way round. Evolution seems to be striving towards what appears to be a runaway escalation of sophistication and design <cit.>. Is the development of refined agents something that only depends on the observer's creativity, or is it something actually taking place independently of observers? Merits are shared, I believe. Observers produce agents. In the absence of agents, no subsystems are cut out of the wholeness of the cosmos, and complexity cannot be measured. Once observers are in play, even if they might have never intended it, it turns out that the computation they perform is liable to iteration. Agency allocation implies that many equal final states are produced from many different initial states. The final states are similar to one another, and similarity is a form of order. Such final states can therefore be used as the ordered degrees of freedom that a higher-level process may use as fuel. This fuel needs to enter into a noisy system for higher-level agents to emerge (Fig. <ref>).L0.5< g r a p h i c s > The goal achieved by low-level agents can become the ordered fuel of higher-level agents.Noise is typically instantiated by a changing environment, often in combination with the occasional mistakes that may have happened in the lower-level process, as mutations in DNA replication. By iterating the algorithm, profuse RNA replication in free solutions can be observed to give rise to prokaryote cells, who in turn evolve into eukaryotes, from which multi-cellular organisms appear, all the way up to the ever growing branches of the tree of life. In the way, conscious humans, civilization, and artificial intelligence emerge. As well as a lot of garbage, as environmentalists wisely remind us. The tree of life develops as a continuous process, irrespectively of whether observers interpret it or not. Observers are required to dissect it into agents, and to evaluate their sophistication. As we climb the ladder of evolutionary design, the fuel degrees of freedom diminish in number, and increase in complexity, the latter defined as the number of bits required to describe their inner structure. If the resources at the bottom level are finite, the process cannot be iterated indefinitely, since eventually, too few and too complex degrees of freedom may not be identifiable as multiple instances of one single ordered pattern.Brain-guided observers are continuously ascribing agency. They do so because the role of a brain is to model the world around its carrier, so that effective survival strategies can be implemented. They have evolved to do so. Mental models must capture the regularities of the world, and discard the noise. Here, noise is defined as the degrees of freedom that are irrelevant to predicting those features of the environment that affect the observer's fitness. It would be a waste of resources, if not impossible, for us to represent in our brains all what happens in a dog's brain. Much more efficient is to ascribe agency, and conclude that the dog wants to bury the bone. We cannot follow the evolution of all the bats that were eaten by owls, we therefore conclude that the predation of owls sharpens the eco-location capacity of bats. We do not care for the details with which self-driving cars are programmed, we just think of them as goal-directed. We need an economic description, so we assign agency. Observers do not assign agency to all the entropy-reducing systems they meet. Purpose is only arrogated to subsystems for which there is no evident source of order, or for sources that are too costly to represent. The cost of a representation is judged in terms of its contribution to prediction accuracy. If we only look at the gas controlled by Maxwell's demon, ignoring the demon itself, we conclude that the gas wants to shrink. If, however, the demon takes weekends off, the purposeful model of the gas loses accuracy. A more sophisticated representation discerning between week days and weekends is needed. Assigning agency may or may not be a convenient strategy, depending on the trade-off between the economy of the representation and the prediction errors it induces. Arrogating agency in excess, for example by believing that all what happens is maneuvered by some obscure intentionality, yields a poor prediction strategy.Observation is the result of development: Observers learn how to observe, and they do so within the framework of learning theory <cit.>. They are first exposed to multiple examples of the process, that act as the training set. Before learning, the final state can only be predicted from the initial one if all degrees of freedom are tracked - a representation capacity that observers typically lack. Making the best use of their resources, observers explore the power set of the system (the set of all subsets of the system) and search for some entropy-reducing subset from which an agent and a goal can be defined. They then discard the superfluous degrees of freedom, thereby compressing information. Yet, if the subsystem does indeed reduce entropy, they are still able to make predictions. With successful predictions the world begins to makes sense, so ascribing agency is in a way equivalent to constructing knowledge. In fact, the construction of knowledge can be argued to be the essence of a mind.In the last paragraphs, we have been observing observers. In doing so, we have placed ourselves one step above the hierarchical ladder of observation. We have concluded that purposeful agents do not exist per se, they are a mental construct of observers. This view may be easily accepted when regarding the agents (let us call them “zombies”) around us, but is more problematic when it applies to ourselves. In the end, we experience our own purposes in a most irrefutable manner. In the present context, a self with purposes is no more than a compressed representation of an observer. Within a physicalist's point of view, the observer of the self cannot be situated in any other place than in the same brain where the self emerges. Whether the observer coincides with the self, whether it only partially overlaps with it, whether it contains it, or is contained by it, I do not dare to assert. I conjecture, however, that brains create selves following the same principles with which they ascribe agency to external factors, the difference being that the creation of the self involves a vastly larger number of degrees of freedom. Those degrees of freedom, moreover, are typically only accessible to the local subsystem. They include the mental processes of which we have conscious access, encompassing external sensory input, and the detailed state of our body. The latter has been proposed as the base for emotion, and the higher-level neural patterns triggered by such a state, the base of feeling<cit.>.Within the self, the subject and the object of observation seem to coincide, forming a strange loop. Douglas Hofstadter <cit.> has suggested that the circular nature of the mind observing itself is essential to the self. I am not sure, however, whether this recursive hypothesis constitutes an actual explanation of the self, or simply a way to bind the two loose ends together and worry no more. It could also be the case that what we perceive as a unitary self is in fact a whole collection of disperse mental processes, inside which multiple observers coexist, although separately unaccessible. In the end, consciousness has been equated with complex and indivisible information processing <cit.>, so accessing subprocesses may not be possible. Dennet however very strongly argues that if such mental subprocesses can be considered multiple observers <cit.>, there is no such thing as a hierarchy, and even less, an ultimate observer. I am afraid I am unable to provide a finished picture of agency when going all the way up to the self. I hope, however, to have built a sensible image of other less intimate agents. The main conclusion of this essay is that the interesting part of agency is the observer. Physics does not make sense, observers make sense of it. Life does not have a meaning, we give it a meaning. Life may not even be fundamentally different from non-life, it may just be a collection of subsystems that appear to have goals. Goal-directed behavior does not exist if we do not define our variables in such a way as to bring goals into existence. Bringing goals into existence is a task that brains perform naturally, because they have evolved to model and predict the future. One fundamental agent that has emerged inside each one of us is the self. The mechanisms behind this process remain unclear, but its evolutionary utility is undisputed. Were we not able to produce meaning, we would not manage to distinguish ourselves as a special part of the cosmos. We would not have a sense of identity, a sense of self-preservation, nor the ability to think. The self is requiredto enunciate even the most basic statements, all the way up from cogito ergo sum. 99maxwell1908 Maxwell J C (1908). Theory of heat. London, New York: Longmans, Green.bennett1982 Bennett C H (1982). The thermodynamics of computation—a review. Int. J.Theoret. Phys. 21 (12): 905–940.landauer1961 Landauer RW (1961). Irreversibility and Heat Generation in the Computing Process IBM J. Res. Dev. 5(3): 183 - 191.fredkin1982 Fredkin E, Toffoli T (1982). Conservative Logic. Int. J. Theoret. Phys. 21 (3-4): 219–253.dennet1995 Dennet DC (1995). Darwin's dangerous idea: Evolution and the meaning of life. New York: Simon & Schuster.norretranders1999 Norretranders T (1999). The User Illusion: Cutting Consciousness Down to Size. New York: Viking.mackay2003 MacKay D (2003). Information theory, inference and learning algorithms. New York: Cambridge University Press.damasio1999 Damasio AR (1999). The Feeling of What Happens: Body and Emotion in the Making of Consciousness. San Diego: Harcourt Brace & Co.hofstadter2007 Hofstadter D (2007). I am a strange loop. New York: Basic Books.tononi2008 Tononi G (2008). Consciousness as Integrated Information: a Provisional Manifesto. Biol. Bull. 215: 216 – 242.dennet1991 Dennet DC (1991) Consciousness explained. Boston: Little, Brown and Co.

http://arxiv.org/abs/1702.08421v1

nolistsep UbbmmndefiDefinition definition[1]An Algebraic Treatment of Recursion R.J. van GlabbeekRob van Glabbeek Data61, CSIRO, Sydney, Australia School of Computer Science and Engineering, University of New South Wales, Sydney, Australia December 30, 2023 ====================================================================================================================================================Jan Bergstra has put his mark on theoretical computer science by a consistent stream of original ideas, controversial opinions, and novel approaches. He sometimes reorganised the arena, enabling others to follow. I, for one, might never have entered computer science if it wasn't for Jan's support and encouragement, and will never forget the team spirit in the early days of process algebra in his group at CWI. This paper is dedicated to Jan, at the occasion of his 65^th birthday and retirement.I review the three principal methods to assign meaning to recursion in process algebra: the denotational, the operational and the algebraic approach, and I extend the latter to unguarded recursion.13.6 pt 13.6 pt 13.6 pt§ PROCESS ALGEBRA In process algebra, processes are often modelled as closed terms of single-sorted specification languages.signatures Letbe a set of variables. A signature is a set of pairs (f,n) of a function symbol f∉ and an arity n ∈. The set (Σ) of terms over a signature Σ is generated by: * ⊆(Σ),* if (f,n) ∈Σ and t_1,…,t_n ∈(Σ) then f(t_1,…,t_n) ∈(Σ),* If V_⊆,  :V_→(Σ) and X∈ V_, then X|∈(Σ).A functionas appears in the last clause is called a recursive specification.A recursive specificationis often displayed as {X=_X| X ∈ V_}. An occurrence of a variable y in a term t is free if it does not occur[3] in a subterm of the form X| with y ∈ V_.A term is closed if it contains no free occurrences of variables.The semantics of such a language is a function _:(Σ) → (^→). It assigns to every term t∈(Σ) its meaning t∈^→. The meaning of a closed term is a value chosen from a class of values , called a domain. The meaning of an open term is a -ary operator on : a function of type ^→. It associates a value t(ρ)∈ to t that depends on the choice of a valuation ρ:→.Sometimes, only a subset of (Σ) is given a semantics, for instance by restricting to terms satisfying a syntactic criterion of guardedness.Another approach lacks the recursion construct itself, but declares a single recursive specification :V_→(Σ) for the entire language <cit.>. A term t in such a language can be seen as a the term t|, obtained from t by substituting, for each Y∈ V_, Y| for each occurrence of Y. Conversely, each term in the general language of signatures can be converted into the form t| with t andrecursion-free.§ DENOTATIONAL, OPERATIONAL AND ALGEBRAIC SEMANTICS The standard (denotational) semantics assigns to each function (f,n)∈Σ an n-ary operator f_n^:^n→. The semantics of a recursion-free expression t is then given by * X(ρ) = ρ(X)for X ∈, and* f(t_1,…,t_n)(ρ)=f_n^(t_1(ρ),…,t_n(ρ)) for (f,n)∈Σ. [3]Three approaches appear in the literature to give semantics to recursion.The denotational approach <cit.> recognisesas having type ^∖ V_→ (^V_→^V_) and defines X|(ρ) for ρ∈^∖ V_ to be the X-component of the least fixed point of (ρ). For this least fixed point to exists, either , equipped with a suitable preorder ⊑, needs to be a complete lattice, with the operators f^ monotonic, or (,⊑) be a c.p.o., with the f^ continuous, orbe a complete metric space, with the f^ contracting (or some variation on this theme).The operational approach <cit.> is based on a set of inference rules that derive a collection of (labelled) transitions betweenclosed terms. The semantic domain is now the collectionof process graphs (S,T,I), with S a set of states, T a set of transitions between states, and I∈ S an initial state, possibly subject to some cardinality restrictions. The operational semantics P of a closed term P takes S to be the set of closed terms, I=P, and T the derivable transitions. The semantics of open terms can be dealt with by encoding the process graphs ρ(X) for X∈ as constants in an appropriate extension of the process algebra. This approach covers the meaning of recursion constructs too.Let guardedness be a criterion on recursive specifications, such that ifis guarded then is has a unique solution, meaning that if ρ_i for i=1,2 are valuations with ρ_1(Z)=ρ_2(Z) for all Z∈∖ V_, andρ_i(X) = _X(ρ_i) for all X∈ V_, then ρ_1(X)=ρ_2(X) for all X∈ V_. The algebraic approach <cit.> yields a semantics for terms with guarded recursion only, where X|(ρ) for ρ∈^∖ V_ is the X-component of the unique solution of (ρ). § EXTENDING THE ALGEBRAIC APPROACH TO UNGUARDED RECURSION In <cit.> I proposed an extension of the algebraic approach to unguarded recursion. An expression X| is seen as a kind of variable, only ranging over the solutions of . Taking for example ACP <cit.>, interpreted in a domain of process graphs modulo strong bisimilarity <cit.>, then X|X=aX is a case of guarded recursion and denotes a specific process, namely an a-loop. On the other hand, X|X=X is an unguarded recursion, and seen a variable ranging over all processes, just like X itself. In between, X|X=X+aX is a case of unguarded recursion, and seen as a variable ranging over all processes of the form a^*P.To avoid ambiguity in deciding when two, almost identical, processes X| denote the same variable or different ones, here I formalise this approach only for terms t| where no further recursion occurs in t or , thus following the second approach of pa.A valuation ρ:→ is compatible with a recursive specificationiff ρ(Y) = _Y(ρ) for all Y∈ V_. The meaning t of a recursion-free term t in the context of a global recursive specificationis now a function intofrom the set of compatible valuations only. It is obtained from the semantics of t from den by restricting (t) to the compatible valuations.In particular, an equation t=u holds under this semantics iff t(ρ) = u(ρ) for all valuations ρ compatible with . Hence it is equivalent to the conditional equation (⋀_X∈ V_ X = _X )⇒ t=u.The laws of process algebra remain valid in this approach, including the congruence property for recursion: if _X(ρ)='_X(ρ) for all valuations ρ, and all X ∈ V_=V_' then t|=t|'.eptcsini

http://arxiv.org/abs/1702.07838v1

mohamed.fadlallah@fsc.bu.edu.eg Physics Department, Faculty of Science, Benha University, Benha, Egypt ulrich.eckern@physik.uni-augsburg.de Institute of Physics, University of Augsburg, 86135 Augsburg, Germany The structural, electronic, and optical properties of metal (Si, Ge, Sn, and Pb) mono- and co-doped anatase TiO_2nanotubes are investigated, in order to elucidate their potential for photocatalytic applications. It is found that Sidoped TiO_2 nanotubes are more stable than those doped with Ge, Sn, or Pb. All dopants lower the band gap, except the (Ge, Sn) co-doped structure, the decrease depending on the concentration and the type of dopant.Correspondingly, a redshift in the optical properties for all kinds of dopings is obtained. Even though a Pb mono- and co-doped TiO_2 nanotube has the lowest band gap, these systems are not suitable for water splitting, due to the location of the conduction band edges, in contrast to Si, Ge, and Sn mono-doped TiO_2 nanotubes. On the other hand, co-doping of TiO_2 does not improve its photocatalytic properties. Our findings are consistent with recent experiments which show an enhancement of light absorption for Si and Sn dopedTiO_2 nanotubes.Cation mono- and co-doped anatase TiO_2 nanotubes: An ab initio investigation of electronic and optical properties Ulrich Eckern December 30, 2023 ==================================================================================================================§ INTRODUCTION Titanium dioxide (TiO_2), also known as titania, has been widely studied as a promising material for many applications because of its low production cost, chemical stability, and non-toxicity <cit.>.Titania is useful for, in particular, solar cells <cit.>, batteries <cit.>, photochemical <cit.> and photocatalytic <cit.> applications, gas sensing <cit.>, and hydrogen storage <cit.>. However, TiO_2 can only be activated by ultraviolet light due to its large band gap (3.0 eV for the rutile,and 3.2 eV for the anatase phase). Therefore, engineering the band gap[The term “band gap engineering”, introducedmore than 30 years ago, generally refers to all attempts at modifying the band gap, e.g., by heterostructuring, combiningsuitable materials, and doping.] <cit.> of titania in order to increase its photosensitivity for visible light is a major target in photocatalyst studies. In recent years, various low-dimensional TiO_2 nanostructures have been prepared, such as thin films <cit.>,nanoparticles <cit.>, nanowires <cit.>, andnanotubes <cit.>. TiO_2 nanotube (TNT) arrays are most interesting for applications due to their large internal surface and highly ordered geometry <cit.>. The structural properties, stability and electronic structure of different TNT structures(anatase and lepidocrocite) have been discussed, e.g., in <cit.>. All anatase nanotubes are semiconductors withdirect band gaps while the lepidocrocite nanotubes are semiconductors with indirect gaps. In addition,anatase nanotubes were found to be most stable; their stability increases with increasing diameter <cit.>.The rolling of an anatase (101) sheet along the [101] and [010] directions has been usedto build (n,0) and (0,n) TNTs, respectively <cit.>. Further details of the geometrical properties of TNTs, in particular, about the folding procedure and the anatase layer basic translation vectors, can be found in Refs. ANE,AMF1,DSz,RAE,AVB.The experimental results show that the predominant peaks of anatase and rutile nanotubes are (101) and (110) <cit.>. Recently, several mono- and co-doped TNTs have been synthesized, e.g., C <cit.>,P <cit.>, Co <cit.>, Si <cit.>, and Sn <cit.> mono-doped, as well as (C/N, F) co-doped <cit.> TNTs. On the other hand, doped TNTs have been studied theoretically only occasionally, e.g., N and B doping <cit.>, C, N, S, and Fe doping <cit.>, (N, S) co-doping <cit.>, and nonmetal and halogen doping <cit.>.In the context of the present study, we note that an improvement of the photocatalytic properties of bulk TiO_2 has been observed experimentally <cit.> and calculated theoretically <cit.> for Si doping. Other dopings (Ge, Sn, Pb) are also known to reduce the band gap in the rutile bulk system, while Sn and Pb doping slightly broadens the band gap in anatase TiO_2 <cit.>. Experimentally an improvement of photocatalytic properties was found for Sn doped bulk systems synthesized by the hydrothermal method <cit.>. TiO_2 thin films doped with Si <cit.>, Sn <cit.>, Pb <cit.>, and Ge <cit.> have been prepared and investigated, generally showing an improvement of photocatalytic activity upon doping. With respect to TiO_2 nanotubes, a suitable doping with Si also improves the light absorption <cit.>.Similar results have been found for Sn doping where, however, also a transformation from anatase to rutile is observed <cit.>. Thus, in the light of these previous experimental and theoretical studies, and in view of their potential high relevance for photocatalytic applications, we perform a systematic study of doping anatase TNTwith group-IV elements, as a function of dopant concentrations; in addition, co-doping effects are alsoinvestigated.Our focus in the present work is on the anatase nanotube, because the TiO_2 anatase (101) surface is known to be a quite effective surface for solar cell applications <cit.>.In the following, after describing the methodology (section II), we investigate the effect of the mono-dopants (Si, Ge, Sn, Pb)on the structure and stability of anatase phase (8,0) titania nanotubes (section III). Then we study the electronic structure of doped TNTs (section IV), followed by a discussion of the optical properties (section V).An application of this study is the splitting of water (section VI). We close our work with a brief summary (section VII). § METHODOLOGY We apply density functional theory (DFT) employing the generalized gradient approximation (GGA) <cit.> and the Perdew-Burke-Ernzerhof functional <cit.> as implemented in the SIESTA package <cit.>.[Generallyspeaking, the accuracy of DFT-GGA calculations – which notoriously underestimate the band gap – isalways an issue. In this context, we mention that this question is thoroughly discussed in a recent paper<cit.>, with the conclusion that DFT-GGA is “an empirical, yet practical” approach.See also <cit.> in relation to the “scissors operation”.]The wave functions are expanded using a local atomic orbitals basis set; the energy cutoff is 300 Ry,and the Monkhorst-Pack k-meshes contain 1 × 1 × 12 points.Structural relaxation is carried out with the conjugate gradient method until the net force on every atom is smallerthan 0.04 eV/Å. As we are interested in the properties of nanotubes, a rectangular supercell, 20 × 20 × L Å^3,is used,where L is the length of the nanotube along the z axis.The distance between two neighboring TNTs, in x and y directions, is thus 20 Å,which is sufficient to avoid any image interaction.Test calculations, changing the size of the supercell and the number of k-points, show the convergence of our results.In particular, in order to check for the spurious dipole-dipole interaction between image supercells, the cell has been increased to 30 × 30 × L Å^3; however, no effect was observed. (This aspect is discussed in detail, e.g., in <cit.> and <cit.>.)Spin polarized calculations have also been performed for selected systems, but no modifications were found. § OPTIMIZED STRUCTURE AND STABILITY The total number of the atoms in the unit cell of a TNT is related to the number of atoms in one unit cell (48 atoms) in the surface layer.We use a supercell including two TNT unit cells to study the effect of changing the doping concentration.Figure <ref> shows the structure oftwo unit cells of (8,0) TNT. The fundamental periodic of the TNT nanotube(along z direction) is found to be 10.49 Å, which is only slightly larger than the value obtained in a previous study, 10.13 Å <cit.>. In that paper, only one unit cell was studied, whose length was determined to be 5.065 Å, hence we compare with twice this value. No symmetry was preserved during structure optimization. Concerning other geometric parameters, we find the inner diameter of the nanotube, cf. the lower part of Fig. <ref>, i.e., between an O3 and its opposite counterpart, to be given by 7.07 Å, while the distance between an O1 and its opposite counterpart is 12.05 Å; the diameter with respect to the Ti atoms is 9.51 Å, and the O1–O3 distance is 2.49 Å. The bond length Ti1–O1 (equal to Ti1–O3) is 1.84 Å, while the bond Ti2–O2 is slightly longer, 1.95 Å, in good agreement with previous works <cit.>.Cation doping of the TNT is introduced by replacing Ti atoms by the dopants. Replacing one Ti by a metal dopant correspondsto ∼ 1.0% dopant concentration.If two atoms are substituted, the doping concentrations will double, and so on. These dopant concentrations are comparable to those reported experimentally <cit.>. Though there are several possible dopant locations for 2% and 3% doping, we have opted in this work for configurations inwhich the dopants are as far apart as possible, namely9.8 Å for 2%, and 9.8 Å, 9.3 Å, 7.4 Å for 3% concentrations,thereby avoiding as muchas possible any dopant-dopant interaction. With this choice, we also avoid major distortions of the nanotube’s structure.Naturally, we thus exclude the possibility of dopant cluster formation (which could be an interesting question in itself <cit.>, but is beyond the scope of the present study).The optimized average bond lengths around the dopant atoms are listed in Tab. <ref>.The bond length between the dopant atom and the O atom increases as the ionic radius of the dopant increases:Si, Ge, Sn, and Pb, with radii 0.40, 0.53, 0.69, and 1.19 Å, respectively. In comparison, the ionicradius of Ti^4+ is 0.61 Å. The charge deficiency on the metal, estimated as the difference between electronic charge densities obtained with theMulliken population analysis, is also given in Tab. <ref>. The table shows that the charge transfer from the dopant atom to the surrounding O atom is rather high forSi and Sn, as compared to Ge and Pb. The formation energy of doped TNTs is used to investigate the stability of thestructures. The formation energy (E_form) of the dopant atoms can be calculated as follows <cit.>:E_form = E_M-TiO_2+μ_Ti-(E_TiO_2+μ_M),where E_M-TiO_2 and E_TiO_2 are the total energies of the metal-doped TiO_2 and the pristineTiO_2 nanotube, respectively, while μ_Ti and μ_M denote the chemical potentials for Ti and the dopant;the latter are assumed to be equal to the energy of one atom in its corresponding bulk structure.The formation energy depends on the growth conditions, which can be Ti-rich or O-rich <cit.>. For the Ti-rich condition, thermodynamic equilibrium is assumed for the Ti bulk solid phase, thus its chemical potentialis fixed at μ_Ti, while the chemical potential of O is fixed by the growth conditions.Under the O-rich condition, O is assumed to be in equilibrium with O_2 molecules, thus the chemical potential of Ois μ_O= μ_O_2/2.We present the formation energy under the O-rich condition, which is lower than for the Ti-rich condition.The stability of nanotubes with dopants is in the following order: Si, Ge, Sn, Pb.The behavior of the formation energies can be understood, to a large extent, in terms of the dopant'selectronegativity (see also section VI) given by1.90 (Si), 2.01 (Ge), 1.96 (Sn), and 2.33 (Pb) (Pauling scale). On the one hand,one notes that the formationenergy of Si is smaller than that of the other dopants, corresponding to the fact that Si has the smallest electronegativity.On the other hand, the Pb formation energy is the largest, and so is its electronegativity. From thispoint of view, Ge and Sn doped TNTs are “out of order”, which can be related to the effect of electronegativity on theionic radius, implying that the formation of Sn–O bonds is more favorable than Ge–O bonds.This behavior of formation energies and bond lengths is very similar to the behavior of the corresponding dopantin bulk TiO_2 <cit.>. § ELECTRONIC STRUCTUREIn this section, we discuss the density of states (DOS) and the partial density of states (PDOS) for the doped TNTs under consideration, in particular, the behavior of the valence bands (VBs) and the conduction bands (CBs) upon doping, with focus on the modifications of the energy gap. In order to present the results in a concise and systematic fashion, we have chosen to measure the energy in the DOS and PDOS plots, Figs. <ref>–<ref>, relative to the top of the valence band energy, E_TVB, of the pristine TNT. For water splitting applications, on the other hand, the absolute energies are required, see section VI, in particular, table <ref>.Before going into detail, we emphasize that we have carefully checked the dependence of theresults on the dopant positions. For example, for a concentration of 1%, we find that the total energy for different dopant positions varies only by less than 0.01 eV, and no change in the DOS is obtained. For 2% and 3% concentrations,we find that the stability increases upon increasing the distance between dopants. Hence our calculations have been doneat the largest possible distance(s) between dopant atoms.Figure <ref>(a) shows the density of states (DOS) for pristine titania nanotubes (8,0).The calculated band gap is 2.20 eV, which – as usual in DFT-GGA – is lower than the corresponding experimentalgap of the TiO_2 nanotube (3.18 – 3.23 eV <cit.>).The Ti (3d) states dominate in the unoccupied states, while the O (2p) states contribute mostly tothe occupied states witha minor contribution to the unoccupied states, see Fig. <ref>(b). The DOS and PDOS arevery similar to the results obtained in <cit.>. §.§ Mono-doped TNTs Si-doping. The effect of Si doping at different concentrations on the electronic structure of TNT is shown in Fig. <ref>(b-d).The band gap is 1.80 eV for 1% doping, less by 0.40 eV than that of the pristine TNT. The corresponding total DOS is similarto the pristine DOS, however, with a smaller band gap, see inset of Fig. <ref>(b) as compared to Fig. <ref>(a). When the concentration increases to 2% and 3%,we find that the Si–Si distance decreases to 9.4 Å after optimization for 2% concentration,and to 9.4 Å, 9.0 Å, and 7.3 Å between different pairs of Si atoms for 3%. These values have to be compared with the original Ti–Ti distance of 9.8 Å for 2%, and 9.8 Å, 9.3 Å, and 7.4 Å for3% concentrations. The band gap remains at 1.80 eV for 2%, and increases to 1.86 eV for 3%.The computed band gap reduction for the corresponding doped bulk system is found to be slightly smaller, 0.20 eV <cit.>. (In that paper, only 2% doping was studied.)Concerning the detailed behavior, we note that on the scale of the figure an almost rigid, concentration independent downshift of about 1.3 eV of the VB is observed, accompanied by a slight “smearing” of the oscillations which are visible below -0.5 eV in the pristine PDOS. The PDOSshows that the dopant states start contributing above 1 eV, with a distinct maximum at about 2.2 eV. The dopant contribution is rather small, but increases continuously with increasing concentration. Comparing with the Ti PDOS, Fig. <ref>(a), we realize that while the onset of Si states is clearly lower than the onset of the pristine Ti states, the latter coincides with the maximum of the Si PDOS. The evolution of the maximum of the Si PDOS can be seen more clearly in Fig. <ref>(b-d). It is located near 2.2 eV for 1% and 2% concentration, Fig. <ref>(b,c), then shifts downwards to about 1.8 eV for 3% concentration, see Fig. <ref>(d). As the location of dopant states shifts closer to the CB edge, the band gap increases. The decrease of the band gap – as compared to the pristine TNT – is consistent with the observed increase of optical absorption of TNT upon Si doping <cit.>.Ge-doping. The optimized Ge–Ge distances are slightly larger than the Si–Si values, consistent with the increase in ionic radius, namely 9.7 Å for 2% concentration (and hence only 0.1 Å smaller than the original Ti–Ti distance), and 9.7 Å, 9.2 Å, and 7.5 Å for 3% concentration. Figure <ref>(e,g) shows that the band gap is 1.86 eV,at any concentration, which is less than the pristine band gap but larger than that for Si doping TNT at 1% and 2% concentration,because the location of Ge states is closer to CB edge than the Si states at these concentrations. The band gap does not depend on the concentration because the dopant states peak position (1.86 eV) is rigid (Fig. <ref>(d)).The peak in the Ge PDOS can be attributed to the fact that the Ge ionic radius and electronegativity are only slightly different in comparison to Ti. The shifts of the VB and the CB are quite similar to the case of Si doping, even though the Ge states give a strongercontribution in the PDOS, Fig. <ref>(d). Accordingly, the DOS of the Ge doped system at different concentrationsis very similar to the DOS of the Si doped structures, see inset of Fig. <ref>(b)).Again, the bulk gap reduction was reported to be slightly smaller, only 0.15 eV <cit.> compared tothe present 0.34 eV. Sn-doping. For Sn doped TNTs, we find that for 2% concentration the Sn–Sn distance is 9.8 Å which is the same as the original distance of Ti atoms. Regarding 3% concentration, the distances are 9.8 Å, 9.6 Å, and 7.6 Åwhich are larger than original distances. Figure <ref>(b) shows that the contribution of Sn states at 1% concentrationis similar to Ge doping structure at 2% in the energy range -2.0 eV to 2.5 eV.The distinct peak appears at 1.8 eV which is the same as for Si at 3% and Ge at any concentration.Hence the band gap also is 1.86 eV. When the Sn concentration increases, the distinct peak sightly shifts tohigher energy (Fig. <ref>(b,c)), which is in the opposite direction compared to Si doping with increasing concentration.The distinct peak is located at 2.0 eV, and we find the gap to be 1.83 eV. Due to the similarity of the Ti PDOS of dopedSn structures with the corresponding one of Si doped structures, the general behavior of the DOS for Sn doping is similar to the DOS for Si doping. The computed reduction of the band gap through Sn doping also isin good agreement with the corresponding light absorption experiment <cit.>.Pb-doping. Within the mono-doped series, we finally consider Pb. The optimized Pb–Pb distances are 10.0 Å for 2% concentration,and 10.0 Å, 9.6 Å, and 7.7 Å for 3% concentration. These values are larger than the original distances of thehost atoms. The distinct peak of Pb states is not only located at a lower energy (1.2 eV) as compared to the peak of thepreviously discussed dopants, but also clearly lower than the Ti CB states, see Fig. <ref>(e-g), such that a separate dopant peak appears in the corresponding DOS. These states decrease the band gap to 1.56 eV, which is the lowest band gapin comparison to the other dopants at any concentration. On the other hand, the shift of the VB and CB edges is 0.5 eV and 0.3 eV,respectively, downwards in energy which is less than the corresponding values for the other systems. For the resulting DOS see inset of Fig. <ref>(e), as compared to the inset of Fig. <ref>(b). Increasing the concentration of Pb, the band gap slightly decreases to 1.50 eV and 1.44 eV, for 2% and 3%, respectively.The PDOS shows that the majority of the additional states derive from the Pb states, see Fig. <ref>(f-g).The peak energy of the Pb states hardly changes with increasing doping.In order to obtain a better understanding of the systematics of the above results, and of those presented in the following subsection, we emphasize that the relevant aspect is the energetic location of cation dopant states relative to the conduction band, in which the Ti 3d states dominate. In particular, we have been able to relate the characteristic concentration and dopant dependent shifts of the dopant PDOS, especially the distinct peak, to the behavior of the energy gap. In comparison to the doped bulk system <cit.>, we note first of all, that the gap reduction for the doped TNT generally is stronger than in the bulk, where, in fact, a gap enhancement was found for 2% Sn or Pb doping. However, the stronger effect of doping – compared to the bulk system – appears reasonable since the disturbances created by dopants are expected to have a stronger influence in a reduced-dimensionality system like a nanotube. With respect to the location of dopant states, we note that their respective peak energies obey the following inequality: E_Pb(6s) < E_Ti(3d) < E_Sn(5s) < E_Ge(4s) < E_Si(3s). In particular, the Si, Ge, and Sn states are well within the conduction band, such they are not able to form distinct dopoant states below the CB. Instead, they “only” reduce the energy gap. However, there is no obvious trend – except for the relation to the distinct dopant PDOS peak, see above – when the dopantconcentration is increased, see table <ref>: When the Si concentration is increased, the gap slightly increases, which likely can be related to the fact that a rather large geometric disturbance is created by Si which has the smallest ionic radius; and that this disturbance is reduced upon doping, at least for 3%. While the gap for Ge doping is concentration independent, it is found to slightly decrease for Sn doping, which is reasonable since the ionic radius of Sn is larger than the ionic radius of Ti. However, these are rather subtle effects, and we believe it is hardly possible to identify a single “cause” for the systematics.The only clear-cut case in the considered series is Pb, where the dopant states are strong and located in energy clearly below the conduction band. Thus a separate peak in the DOS is formed, whose amplitude increases with dopant concentration. In addition, a strong reduction of the energy gap is found.§.§ Co-doped TNTs Turning finally to co-doped TNTs, the doping concentration is 2% for two different substitutional atoms, and 3% fortwo atoms from the same kind plus one doping atom from another kind. We first study the effect of co-doping at 2%. Starting with Si–Ge, the optimized distance is 9.5 Å. This is approximately the average of the Si–Si and Ge–Ge distances at 2% concentration.Figure <ref>(b) shows the effect of (Si, Ge) co-doping on the electronic structure. The overlap between Si, Ge, and Ti statesappears near the CB band, therefore the bands shift to lower energy. This shift is less than the corresponding one in the case of 1% and 2% mono-doping with Si and Ge by 0.6 eV. The (Si, Ge) co-doped TNT has a band gap of 1.92 eV, larger than the gap for Si and Ge mono-doping at any concentration.As we go down the group of dopants in the periodic table (4A), the dopant-dopant distance increases to 9.6 Å forSi–Sn and 9.7 Å for Si–Pb. The overlap between the dopant and the host atom states also increases slightly nearthe CB edge, so the band gap of (Si, Sn) is 1.98 eV, see Fig. <ref>(c). Figure <ref>(d) shows the PDOS ofthe (Si, Pb) co-doped system, which is very similar to the DOS of the Pb mono-doped TNT. It is characterized byPb dopant states below the conduction band. The band gap for this co-doping is 1.50 eV, smaller than the gap of the 1% and equal to the 2% mono-doped Pb system. As compared to the co-doped structures discussed above, the bands of (Ge, Sn) shift to higher energy, and the band gap increases to 2.3 eV, Fig. <ref>(e), clearly larger than the gap of pristine TNT. The increase in the band gap can be attributedto the strong interaction (bonding) between the dopant (such as Sn) and the Ti CB states in the energy range 1.8–2.5 eV,see Fig. <ref>(e).The distance between Ge–Sn is similar to the Si–Sn distance (9.6 Å). If Sn is replaced by Pb, the distance ofGe–Pb increases to 9.8 Å, which is less than the Pb–Pb distance at 2%. Due to the interaction between Ge and Ti statesnear the CB edge, the Pb states slightly move towards higher energy, so the band gap slightly increases to 1.56 eV ascompared to (Si, Pb), see Fig. <ref>(f). The last 2% co-doped system is (Sn, Pb), Fig. <ref>(g),with 9.9 Å Sn–Pb distance. The distinct peak of the Sn dopantis located at the same position as for Ge (2.4 eV), Fig. <ref>(c), and the Pb midgap statesremain in their place, thus the band gap does not change.The band gap of co-doped systems at 2% concentration is larger than the band gap of the individual corresponding mono-doped structures because of a good co-dopant states interactions near the CB edge,except for Pb doping. Table <ref> summarizes the band gap values of all structures considered.Last we study co-doped TNTs at high concentration, i.e., 3%. The distances in the (2Si, Ge) co-doped structureare 9.1 Å, 7.3 Å for Si–Ge, and 9.4 Å for Si–Si, and for (Si, 2Ge) are 9.1 Å, 7.4 Å for Ge–Geand 9.6 Å for Ge–Si.Figure  <ref>(b) shows the PDOS of (2Si, Ge) which is practically identical to the (Si, Ge) case.Also there is no change when another configuration, (Si, 2Ge), is considered. The electronic structures of(Si, Ge)/(2Si, Ge)/(Si, 2Ge) co-dopants do not depend on the concentration of the individual dopantsbecause all configurations have a similar effect at the same energy. Regarding (2Si, Sn), we find, see Fig. <ref>(c),that the overlap between states in the CB reduces the band gap as compared to (Si, Sn) by ∼ 0.3 eV,and the gap becomes 1.86 eV, less than the band gap of the same co-doped system at 2% concentration.The DOS of the (Si, 2Sn) system is practically the same as the (2Si, Sn) DOS, even though the distances differ slightly: 9.1 Å, 7.5 Å for Si–Sn, and 9.8 Å for Si–Si, for the former, and9.2 Å, 7.4 Å for Si–Sn, and 9.4 Å for Si–Si for the latter case.For (2Si, Pb) co-doping, Fig. <ref>(d), the PDOSs show that the CB and Pb midgap states shift towards lower energy by 0.6 eV and 0.2 eV, respectively, as compared to the same co-doped system at low concentration,Fig. <ref>(d). This relatively strong shift in the CB is due to the shift of the corresponding Si states. The overlap between CB and Pb states is most pronounced at 1.4 eV, which results in a small shoulderin the DOS (see inset of figure).This reduces the band gap to 1.44 eV, less than the corresponding one for 2% co-doping but equalto the band gap of 3% Pb mono-doping. The PDOSs of (Si, 2Pb) show the same gap and DOS shape as (Si, Pb) due to the dominant effect of the Pb states.For (2Ge, Sn)/(Ge, 2Sn) co-doping, Fig. <ref>(e), the PDOSs are very similar to the case of (2Si, Sn), Fig. <ref>(c),with a small shift of bands to higher energy. The band gap is 1.86 eV, the same as for Ge mono-doping at any concentration.The PDOSs of (2Ge, Pb)/(Ge, 2Pb) are the same as for (Ge, Pb), with the same band gap.The last 3% co-doped structure is (2Sn, Pb): as compared to the (2Si, Pb) system, the PDOS is very similar with respect to the Pb contribution, but there is a shift in energy due to the Sn states (in comparison with the Si states), consistent with what we observed for the case of Sn versus Si mono-doping. As compared to (Sn, Pb) co-doping, the CB and Pb states for (2Sn, Pb) are lower in energy due to the higherconcentration of Sn; the band gap is 1.50 eV. For (Sn, 2Pb), due to the high concentration of Pb, the PDOS is similarto (Sn, Pb). The band gaps of the 3% co-doped structures are also presented in table <ref>.We note that for a given co-doped system, say, (2X, Y), there are different possibilities to position the X and Y atoms.We have considered such different cases, and we have confirmed that the DOSs are not affected.§ OPTICAL PROPERTIES The optical properties of a semiconductor photocatalyst are closely related to its electronic structure. The decrease of the band gap for all mono-dopants as compared to pristine TNT, see Fig. <ref>, leads to a redshift of the optical absorption edge. This redshift depends on the kind of dopant and the concentration.Clearly, several factors are relevant for the differences between doped bulk TiO_2 and doped TNTs, namely the geometry,electronic structure, and the interaction between dopant and neighboring Ti and O atoms, with the general tendencyof reducing the optical gap. As is apparent from Fig. <ref>, this leads to a shift of the absorption edge towards higherwavelengths, most pronounced for Pb mono- and (2Pb, Ge) co-doping. In contrast, a reduction of the optical gap upondoping in the bulk system is only found for Si and Ge doping <cit.>. Our results agree qualitatively with the recentlyobserved gap reduction for Sn doped TNTs <cit.>.The optical absorption is related to the complex dielectric function ε(ω)=ε_1(ω)+iε_2(ω),with ω the frequency. The imaginary part is calculated from the momentum matrix elementsbetween the occupied and unoccupied states, and the real part subsequently from the Kramers-Kronig relation.The absorption coefficient then is given by <cit.>α(ω) = √(2)ω√(√(ε_1^2(ω)+ε_2^2(ω))-ε_1(ω)).A “scissors operation” <cit.> of 1.0 eV, which corresponds to the difference between the calculatedand the experimental gap (3.2 eV) for pristine TNT, is also used for the doped system.As a side-remark, we wish to add that this “operation” – adding ad hoc a correction Δ to the conduction band energies, such that the calculated energy gap plus Δ (here 1.0 eV) equals the experimental gap, is pretty much standard. It relates to the well known problem of density functional theory that the gap calculated from the Kohn-Sham orbitals almost always is by far too small <cit.>. Another way out of this problem is to extend DFT and include so-called GW corrections, see <cit.> and references therein; but this approach is computationally quite costly, and hence not practical for systematic studies of doped systems.A pristine TiO_2 nanotube can only absorb the narrow UV light (370 nm), but shows no absorption for visible light,see Fig. <ref>. The calculated optical absorption spectra for all mono-doped TNTs show absorption in the visible-light region, namely in the range 380-410 nm. Also, a redshift isapparent for all mono-doped TNTs, consistent with the earlier discussion. § APPLICATION: WATER SPLITTINGThe improvement of the visible light activity of TiO_2 is very important for water splitting (H production) <cit.>.In this context, it is important to note that the absolute values of the conduction and valence band edges, E_CBE and E_VBE, are required. This issue has been discussed extensively in the literature; see, e.g., Ref. Gratzel2001, or, more recently, Ref. Wang2016. In short <cit.>, the conduction band edge is computed from the empirical relation E_CBE = X̅ - 0.5 E_gap - 4.5 eV, where X̅ denotes the geometric mean of theelectronegativities of the constituents (e.g., X̅ =(χ_Tiχ_O^2)^1/3 = 5.80 eV for the pristine case, using experimental values <cit.>), and E_gap is the scissors-corrected energy gap.Then E_VBE = E_CBE + E_gap.Calculations of the conduction band edge (CBE) and the valence band edge (VBE) have shown that theCBE of anatase TiO_2 is located at -0.29 eV, and the VBE at 2.91 eV <cit.>. Note that these band edges are measured with respect to the normal hydrogen electrode (NHE) potential of the reduction andoxidation levels of water: the reduction level (H^+/H_2) is located at 0 eV, and the oxidationlevel (H_2O/O_2) at 1.23 eV, respectively. Thus the CBE is “above” the water reduction (H^+/H_2) level, and the VBE “below” the water oxidation (H_2O/O_2) level, in the standard representation <cit.>.Table <ref> shows that Si, Ge, and Sn mono-doping of TNTs improves the photocatalytic properties, at any concentration.However, the CBE value is too high compared to the reduction level of water, hence Pb doped TNTs are useful for hydrogen production despite the fact that they have the lowest band gapsamong the mono-dopants. The low-concentration Si and Ge doped structures show a better efficiency than for high concentration.In contrast, for bulk TiO_2 anatase only Ge doping improves the photocatalytic properties <cit.>. We do not present the co-doping results here because all of them have CBEs around -2 eV, which is higher than the reduction level of water,and the VBEs are higher than the oxidation level.§ SUMMARYDensity functional theory has been employed to study the structural, electronic, and op- tical properties of cation mono- and co-doped titania nanotubes (TNTs) at different doping concentrations. All mono-/co-dopants, except (Sn, Ge) co-doping, decrease the band gap of the TNT, similar to previous results [40]. For mono-dopants, Pb doped TNTs have the lowest band gap at the studied concentrations (1% to 3%)due to the presence of distinct Pb states below the conduction band. The contribution of the dopant states in theconduction band increases as we move down the 4A group in the periodic table, i.e., from Si to Ge, Sn, and Pb. The decrease in the band gaps of mono-doped TNTs is accompanied by shifts in the band edges towards lower energy for Si, Ge, and Sn. The band gaps of 2% co-doped TNTs, except for (Ge, Sn), are smaller than those of Si, Ge, and Sn mono-doped TNTs at any concentration. The (Pb, 2X; X = Si, Pb) co-doped TNTs have the lowest band gap ofall mono- and co-doped TNTs. However (Ge, Sn) 2% co-doped TNT has the largest band gap not only of all mono- and co-dopedTNTs but also compared to the pristine nanotube. The influence of co-dopants can be understood, to a large extent, in terms of a superposition of individual mono-dopant effects. The study of optical properties illustrates that mono- and co-doped TNTs can absorb a wide range of visible light, in contrast to pristine TNT. This observation, consistent with recent experimental results, is related to the decrease of the band gap. The Si, Ge, and Sn mono-doped TNTs at low concentration (1%) have a high ability to produce hydrogen in the water splitting process, their performance being clearly better than for pristine TNT. 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http://arxiv.org/abs/1702.07919v3

WU-HEP-17-03 KIAS-P17014 Brane-localized masses in magnetic compactifications Makoto Ishida,^1[ncanis3@fuji.waseda.jp]Kenji Nishiwaki,^2[nishiken@kias.re.kr] and Yoshiyuki Tatsuta,^1[y_tatsuta@akane.waseda.jp]^1Department of Physics, Waseda University, Tokyo 169-8555, Japan^2School of Physics, Korea Institute for Advanced Study (KIAS), Seoul 02455, Republic of KoreaAbstract 0.9 We discuss effects of the brane-localized mass terms on the fixed points of the toroidal orbifold T^2/Z_2 under the presence of background magnetic fluxes, where multiple lowest and higher-level Kaluza–Klein (KK) modes are realized before introducing the localized masses in general. Through the knowledge of linear algebra, we find that, in each KK level, one of or more than one of the degenerate KK modes are almost inevitably perturbed, when single or multiple brane-localized mass terms are introduced. When the typical scale of the compactification is far above the electroweak scale or the TeV scale, we apply this mechanism for uplifting unwanted massless or light modes which are prone to appear in models on magnetized orbifolds. ===========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================empty§ INTRODUCTION The standard model (SM) of particle physics has been verified by the discovery of the final puzzle piece—i.e., the Higgs boson—in 2012 <cit.>. It is well known that the SM can explain almost all of the phenomena around the electroweak scale (∼ 100 GeV) with great accuracy. However, extensions beyond the SM are required, due to several theoretical difficulties which appear inevitably in the SM, e.g., the flavor puzzle, the lack of a dark matter candidate, the gauge hierarchy problem, and so forth.Among such extensions beyond the SM, extra dimensions have been studied from a phenomenological point of view. Indeed, the geometry of the compactified hidden directions determines phenomenological properties in the four-dimensional (4D) low-energy effective theory (LEET). For example, it is known that some extra-dimensional properties such as Kaluza-Klein (KK) wave functions reflect information on the extra-dimensional topologies. In particular, localization of the lowest wave function(s) among the KK-decomposed modes significantly affects the LEET obtained after dimensional reduction. Indeed, many phenomenological models accompanying localization of the KK-expanded modes have been proposed and investigated. For example, overlap integrations of KK wave functions are used to realize a huge hierarchy in Yukawa coupling constants <cit.>, where differences in the degrees of overlapping lead to the hierarchy in the eigenvalues of the Yukawa matrix. Similarly, such an overlapping of KK wave functions can provide the Froggatt-Nielsen mass matrix textures <cit.>, their Gaussian-extended version <cit.>, and so on. For such a reason, model builders interested in extra dimensions are intrigued by the localization of particle profiles in extra directions as a way to realize a huge hierarchy in a natural way.On the other hand, when we address orbifold compactifications, an interesting feature is found, which is useful for concrete model constructions: the existence of orbifold fixed point(s). Model builders have added desirable terms on the orbifold fixed points to derive necessary structures and/or to conquer problems that are more difficult in the bulk part of extra dimensions. For example, the authors of Ref. <cit.> pointed out that the Yukawa couplings can be introduced on the S_1/Z_2 orbifold fixed points, although Yukawa interactions are prohibited on the bulk of S_1/Z_2. There are other uses of fixed points.In some papers, model builders have introduced brane-localized mass term(s) on the fixed points in order to uplift dangerous massless (or very light) particles for consistent model constructions.In Refs. <cit.>, brane-localized mass terms on the fixed points of the toroidal orbifold T^2/Z_2 were investigated and it was declared that a massless zero mode can become massive via effects of the brane-localized mass. Recently, several groups have eagerly studied systems on magnetized backgrounds based on toroidal extra dimensions and their orbifolded versions <cit.>.[See also Refs. <cit.>.] This is because magnetic fluxes play important roles in constructing phenomenological models <cit.>. Indeed, the presence of magnetic fluxes leads to the multiplicity of the lowest KK modes, where such an emergence of the multiple lowest modes is considered as a spontaneous generation of a family replication in LEET. Also, specific configurations of the magnetic fluxes penetrating extra dimensions break supersymmetry, (see e.g., Ref. <cit.>).In this paper, we examine the situation where the two phenomenology fascinating ideas—namely, a magnetized background and a mass term localized on an orbifold fixed point—are taken into account simultaneously. Our major motivation for focusing on this configuration is as follows. A possible problem in realizing family structures when using the magnetic fluxes is that extra massless modes emerge in some concrete models. Introducing brane-localized mass terms may help such situations by making some of the light particles decoupled, which can be expected. Also, after the perturbation by the insertion of mass terms on fixed points, particle profiles are changed through the rediagonalization of a perturbed KK mass matrix, where some part of the mass spectrum may be unchanged.This paper is organized as follows. In Sec. <ref>, we briefly review the KK decompositions for the six-dimensional (6D) scalar and spinor fields on a magnetized two-dimensional torus, and show that the KK-expanding wave functions are described by the Jacobi theta function and the Hermite polynomials on the basic magnetic background. Then, we focus on the Z_2-orbifolded situation of a two-dimensional torus with magnetic fluxes, where important properties of the Z_2 eigenmodes are shown. In Sec. <ref>, we investigate effects on the KK mass spectra after taking care of effects from brane-localized mass(es), where the forms of eigenvalues and eigenvectors after the perturbation are investigated theoretically. Subsequently, in Sec. <ref> we directly explore deformations of the profile of KK particles (in the correct mass eigenbases under the brane-localized mass terms) through numerical calculations.In Sec. <ref>, we comment on the cutoff dependence of the KK mass eigenvalues. Section <ref> is devoted to a conclusion and discussion. In Appendix <ref>, we summarize our notation for 6D gamma matrices. In Appendix <ref>, we provide a discussion on the situation with multiple brane-localized mass terms.§ BRIEF REVIEW OF BULK WAVE FUNCTIONS In this section, we briefly review the wave functions of KK modes on a magnetized background, based on Refs. <cit.>. §.§ Flux backgroundWe consider the two actions of the 6D gauge theory on the 4D Minkowski spacetime times two-dimensional torus T^2 with a 6D Weyl spinor (Ψ) or a 6D complex scalar (Φ), e.g.,S_ Weyl = ∫ d^4 x ∫_T^2 d^2z{ i Ψ̅Γ^M D_M Ψ}, S_ scalar = ∫ d^4 x ∫_T^2 d^2z{ - |D_M Φ|^2 },where the index M runs over μ(= 0, 1, 2, 3), 5, 6 and Γ^M denotes the 6D gamma matrices (see Appendix <ref> for details of our notation). D_M ≡∂_M - iq A_M denotes a covariant derivative. Here, q denotes a U(1) charge.In the above action, we define a (dimensionless) complex coordinate z ≡ (y_5 + τ y_6)/2π R with two Cartesian coordinates y_5 and y_6 and τ∈ℂ to express two extra space directions. Also, R denotes a compactification radius of T^2 and is associated with a compactification scale M_C, i.e.,M_C ∼ 1/R. In the complex coordinate, the toroidal periodic condition is expressed as z ∼ z + 1 ∼ z + τ.In the six-dimensional action, we assume that the vector potential A_M possesses a (classical) nontrivial flux background b = ∫_T^2 F with field strength F = (ib/2Im τ) dz ∧ d z̅:A^(b) (z) = b/2Im τ Im z̅ dz.The consistency condition under contractible loops, e.g., z → z+1 → z+1+τ→ z+τ→ z, provides the Dirac charge quantization,qb/2π = M ∈ℤ.Indeed, the magnetic flux plays an important role in the context of higher-dimensional gauge theory. For example, it was found in Ref. <cit.> that the flux background can provide the multiplicity of KK-expanding wave functions and their localized profiles, as we will see below. §.§ KK modes on T^2 with magnetic fluxes Next, we briefly review KK-mode wave functions of 6D Weyl spinor and scalar fields, denoted by Ψ and Φ, on a magnetized T^2, based on Refs. <cit.>. First, we decompose them asΨ (x^μ, z)= ∑_n χ_n (x^μ) ⊗ψ_n (z),Φ (x^μ, z)= ∑_n φ _n(x^μ) ⊗ϕ_n (z),where the integer n = 0,1,2,⋯ discriminates values of KK levels. For later convenience, we adopt the above notation where a two-dimensional(2D) spinor ψ_n carries a 2D chirality distinguished by ± as ψ_n = (ψ_+, n,ψ_-, n)^T.The KK modes of the spinor in Eq. (<ref>) are designated as eigenstates of the covariant derivative D ≡[ ∂_z̅ + π M z/(2 Im τ) ]/(π R) with ∂_z̅ = π R (∂_y_5 + τ̅∂_y_6) as [ D^† D 0; 0DD^† ][ ψ_+, n; ψ_-, n ] = m_n^2[ ψ_+, n; ψ_-, n ] ,while those of the scalar in Eq. (<ref>) are eigenstates of the Laplace operator Δ≡{D^†, D}/2 asΔϕ_n = m_n^2 ϕ_n.For simplicity, we choose a simple complex structure parameter, i.e., τ=i. Also, we focus on the case of positive magnetic fluxes M>0. Because it is straightforward to apply the following discussions to nontrivial values of τ and negative fluxes, we will not address such a case on this paper.The form of the eigenstates of the KK modes is shown analytically by the Jacobi theta function and the Hermite polynomials. First, we focus on (massless) zero-mode wave functions for ψ_+, n. The zero-mode wave functions are multiply degenerate and given asψ^j_+, 0(z) =N e^π i MzImz ϑ[ j/M; 0 ] (Mz, Mi),where the Jacobi theta function is defined byϑ[ a; b ] (ν, τ) = ∑_ℓ∈ℤ e^π i (a+ ℓ)^2 τ + 2π i (a + ℓ) (ν +b),where a and b are real parameters, and ν and τ take complex values with Im τ> 0. In the above expression, the number of degenerate zero modes is determined by the magnitude of the magnetic fluxes, i.e., j=0, 1,..., M-1 (with modulo M). The normalization constant is calculated as N = (2M/ A^2)^1/4, where the area of the torus A = (2π R)^2, which is independent of j on T^2. On the other hand, massive KK-mode wave functions are given asψ_+, n^j(z) =N/√(2^n n!) e^π i M zImz∑_ℓ∈ℤ e^ -π M (j/M + ℓ )^2 + 2π i Mz (j/M + ℓ ) H_n ( √(2π M)(jM + ℓ +Imz ) ),with the Hermite polynomialsH_n(x) = (-1)^n e^x^2d^n/dx^n e^-x^2. We note that the form in Eq. (<ref>) is a specific case (n=0) of that in Eq. (<ref>). As shown in Refs. <cit.>, the squared KK mass eigenvalue is given asm_n^2= 4 π M/ A n,for n = 0,1,2,⋯, which is independent of the index j. Next, we address the case of ψ_-,n. Indeed, since nonvanishing magnetic fluxes cause a chirality projection for massless zero modes of ψ_n, the zero modes ψ_-, 0 are not normalizable for M>0. For n ≥ 1, wave functions of the KK modes are similarly written as ψ^j_-, n = D ψ^j_+, n/m_n.Note that the multiplicity of ψ^j_-, n is the same as that of ψ^j_+,n when n ≥ 1.The case of the 6D scalar field is treated similarly to the case of the 6D spinor which we discussed.The set of wave functions of the scalar field is exactly the same as that of the spinor, i.e.,ϕ_n^j(z) =N/√(2^n n!) e^π i M zImz∑_ℓ∈ℤ e^ -π M (j/M + ℓ )^2 + 2π i Mz (j/M + ℓ ) H_n ( √(2π M)(jM + ℓ +Imz ) ),where n ≥ 0 and j = 0,1,⋯,M-1 (with modulo M). An important difference between the scalar and spinor fields is found in their mass eigenvalues. The KK mass spectrum of the scalar is given asm_n^2 = 2π M/ A (2n+1),which implies that the lowest KK modes of the scalar are massive. [If one tries to embed the toroidal compactification with fluxes into the superstring/supergravity theories, it is plausible that the above charged (fundamental) scalar field may consist of some higher-dimensional gauge fields as a possibility for the UV completion. Although a derivation of the scalar is often difficult, in this paper we analyze the scalar spectrum from the field-theoretical point of view.] §.§ KK modes on T^2/Z_2 with magnetic fluxes Now, we are ready to address the wave functions on T^2/Z_2 with fluxes. In addition to toroidal conditions on the fields, we introduce an additional identification in the 2D space.In general, for N=2, 3, 4, and 6, the T^2/Z_N orbifold is defined by identifications under the twist,z ∼ e^2 π i /N z.It was concretely pointed out in Refs. <cit.> that the magnetic fluxes can coexist with the twist identification, and also that some parts of the KK-expanded modes are projected out. Accordingly, the multiplicity of the KK modes is changed and hence magnetized toroidal orbifolds can be an interesting framework for phenomenological model building.[ Another motivation for considering magnetized orbifolds is to realize the CP violation in the quark sector LEET via higher-dimensional supersymmetric Yang-Mills theories (see Ref. <cit.>). ]In this paper, we restrict ourselves to the Z_2 twisted orbifold as an illustration.Also, we assume that (discretized) Wilson line and Scherk-Schwarz twisting phases are all vanishing. Since an extension to the cases with such nonvanishing twisting phases can be done straightforwardly by means of the operator formalism <cit.>, we do not address such generalized situations. Under the above twist identification, we construct the Z_2 eigenstates of the KK modes which should obey the boundary conditions around z=0:ψ_T^2/Z_2±, n(-z)= ±ηψ_T^2/Z_2±, n(z),where η denotes the Z_2 parity η = ± 1.It was pointed out in Refs. <cit.> that the physical eigenstates ψ_T^2/Z_2±, n on T^2/Z_2 are easily obtained asψ^j_T^2/Z_2 +, 0(z)= 1/√(2) (ψ^j_+, 0 (z) + ηψ^j_+, 0(-z)) = 1/√(2) (ψ^j_+, 0 (z) + ηψ^M-j_+, 0(z)),where we used the important property ψ^j_+, 0(-z) = ψ^M-j_+, 0(z).This expression is just a formal solution of the zero-mode equations in Eqs. (<ref>) and (<ref>). For an arbitrary number of quantized fluxes, the number of independent zero-mode wave functions is counted as shown in Table <ref>.Next, normalizable wave functions of the excited KK modes (n ≥ 1) are similarly written asψ^j_T^2/Z_2±, n(z) = 1/√(2) (ψ^j_±, n (z) ±η (-1)^n ψ^M-j_±, n(z)),where we use a similar formula for the KK modes on T^2: ψ^j_±, n(-z) = (-1)^n ψ^M-j_±, n(z).The eigen–wave functions in Eqs. (<ref>) and (<ref>) keep the same mass spectrum as those on T^2, i.e.,[ D^† D 0; 0DD^† ][ ψ_T^2/Z_2 +, n; ψ_T^2/Z_2 -, n ] =m_n^2 [ ψ_T^2/Z_2 +, n; ψ_T^2/Z_2 -, n ],m_n^2 = 4π M/ A n,except for ψ_-, 0, which has no consistent solution when M>0 on T^2. These expressions are also formal solutions and the number of independent physical modes is calculated.In Tabs. <ref> and <ref>, the number of independent KK wave functions is shown.Here, we mention the ranges of the index j after the Z_2 orbifolding. The index j starts from zero or one in a Z_2-even or Z_2-odd case, respectively, since the j=0 component apparently vanishes in the latter case. Also, to avoid double counting, when the number of independent physical modes is n_mode, the first n_mode values of j are taken as individual degrees of freedom. Before closing this section, it is important to discuss the orthonormal condition for the physical eigenstates of the KK modes on T^2/Z_2. Let us consider an overlap integral of physical states on T^2/Z_2,∫_T^2 d^2zψ^j_T^2/Z_2±, n( ψ^j'_T^2/Z_2±, n')^† = δ_n, n'( δ_j, j'+ η (-1)^n δ_j+j', M),where the Kronecker delta appearing in this relation should be interpreted as that with modulo M. It is easy to see that the second term on the right-hand side provides a nonzero contribution only for j+j'=M, which is rephrased as j = j' = M/2    ( modM/2) for η = +1.Thus, by redefining a normalization constant asN = (2M/ A^2)^1/4 N_j = (2M/ A^2)^1/4/√(1+δ_j, M/2),the wave functions are normalized and orthogonal to each other, such as∫_T^2 d^2zψ^j_T^2/Z_2±, n( ψ^j'_T^2/Z_2±, n')^† = δ_n, n'δ_j, j'.Now, the normalization constant 𝒩_j becomes dependent on j.Being similar to the wave functions on T^2, the results for the spinor are applied to the scalar case asϕ^j_T^2/Z_2, n(z) = 1/√(2) (ϕ^j_n (z) + η (-1)^n ϕ^M-j_n(z)).Here, we immediately confirm the corresponding relations for the mass eigenvalues,Δϕ^j_T^2/Z_2, n(z) = m_n^2 ϕ^j_T^2/Z_2, n(z),m_n^2 = 2π M/ A (2n+1).The multiplicity of the wave finctions is also the same as that of the spinor.§ BRANE-LOCALIZED MASSES ON A MAGNETIZED ORBIFOLD BACKGROUNDBefore introducing brane-localized masses in magnetized extra dimensions, let us explain fixed points on toroidal orbifolds. As explained already, toroidal orbifolds T^2/Z_N for N=2,3,4,6 are obtained by an identification of two-dimensional extra dimensions under the toroidal periodicities and the Z_N rotation,z ∼ z + 1 ∼ z + τ∼ e^2π i/N z,where we keep the complex structure parameter in the general form. An important factor in extra-dimensional model constructions is the presence of orbifold fixed points z_ fixed. For T^2/Z_2, there exist four fixed points z_ fixed = 0, 1/2, τ/2, (1+τ)/2, as shown in Fig. <ref>.For the other orbifolds, the fixed points are located at z_ fixed = 0, (2+τ)/3, (1+2τ)/3 for T^2/Z_3, z_ fixed = 0, (1+τ)/2 for T^2/Z_4 and z_ fixed = 0 for T^2/Z_6, respectively.Except for T^2/Z_2, the complex structure parameter should be discretized as τ = e^2π i /N due to consistency conditions of crystallography <cit.>.For later convenience, the fixed points of T^2/Z_2 are labeled asz_1 = 0, z_2 = 1/2,z_3 = i/2,z_4 = 1+i/2.We recall that all of the actual calculations are done with the simple choice of τ = i.§.§ ScalarWe introduce a brane-localized mass at a fixed point corresponding to the origin of two extra directions, i.e., z = z_1 (= 0). The Lagrangian for the complex 6D scalar field under consideration is given asL = - | D_M Φ (x^μ, z) |^2 - h |Φ(x^μ, z)|^2 δ^2(z - z_1).Here, the real dimensionless variable h denotes the scalar mass localized at the fixed point, where the mass scale is provided by the radius R. Also, this Lagrangian straightforwardly provides the six-dimensional equation of motion,D_M D^M Φ - hΦδ^2(z - z_1) = 0.By substituting the KK-expanded scalar (<ref>) for Eq. (<ref>), the effective Lagrangian after dimensional reduction is calculated asL_ eff = - ∑_n, j (∂_μφ_n^j )^† (∂^μφ_n^j) = - ∑_n, n'∑_j, j' (φ_n^j)^†( ∫ d^2z (D ϕ^j_T^2/Z_2, n(z))^†(D ϕ^j'_T^2/Z_2, n'(z) ) + h (ϕ^j_T^2/Z_2, n(z_1))^†ϕ^j'_T^2/Z_2, n'(z_1) )φ_n'^j'≡ L_ kin - ∑_n, n'∑_j, j' (φ_n^j)^† M^2_(n, j), (n',j')φ_n'^j',where ℒ_kin represents the kinetic terms of KK scalar particles, which are canonically normalized. The explicit form of the KK mass matrix after the perturbation is given asM^2_(n, j), (n',j') = 2π M/ A (2n+1) δ_n, n'δ_j, j' + h (ϕ^j_T^2/Z_2, n(z_1))^† (ϕ^j'_T^2/Z_2, n'(z_1))= m_0^2(2n+1) δ_n, n'δ_j, j' + 2h (v_n, j)^† v_n', j'with m_0^2 = 2π M/ A, where we adopt the shorthand notationv_n, j ≡1/√(2)ϕ^j_T^2/Z_2, n(z_1) = ϕ^j_n(z_1) =N_j/√(2^n n!)∑_ℓ∈ℤ e^-π M (j/M + ℓ)^2 H_n ( √(2π M) (jM + ℓ) ).Here, we mention the sign of the parameter h. When h is positive/negative, no/possible tachyonic modes appear in the spectrum.An extension to the localized mass term at the other fixed points is straightforward; all we need to do is change z_1 to z_i(i=2,3,4) in v_n,j. Here, we comment on the fourth fixed point z_4 and effects from the localized mass on it. Via direct calculations, we obtain ϕ^j_T^2/Z_2, n(-z_k) = (-1)^M δ_k, 4 η ϕ^j_T^2/Z_2, n (z_k) for k=1,2,3, 4. This implies that when η=-1 there is no effect, except for k=4 with odd M.[ In other words, when η = +1, a brane-localized mass term manifestly vanishes only in the case of k=4 with odd M (see also Table <ref>). ] In this paper, we restrict ourselves to the case of η = +1. Such brane-localized mass terms of the scalar field possibly affect physics at low energies. Hence, we are interested in the eigenvalues and eigenvectors of the perturbed mass matrix in Eq. (<ref>). We note that the above relation is also derived from the pseudoperiodic boundary conditions on both T^2 and T^2/Z_2 (see, e.g., Ref. <cit.>),ϕ^j_T^2 orT^2/Z_2, n(z+1) = e^i q χ_1(z)ϕ^j_T^2orT^2/Z_2, n(z), ϕ^j_T^2 orT^2/Z_2, n(z+i) = e^i q χ_i(z)ϕ^j_T^2orT^2/Z_2, n(z), χ_1(z) =b/2Im(z),χ_i(z) = -b/2Im(i z),by setting z as 0, -1/2, -i/2, or -(1+i)/2. We will diagonalize this KK mass matrix in the following. First of all, we comment on the cutoff scale. Unfortunately, extra-dimensional models are nonrenormalizable, and hence they should be considered as a kind of LEET of a more fundamental theory, e.g., string theory at a scale below the cutoff scale Λ. In the following discussion, we define the cutoff scale as a certain level of the KK masses, i.e.,m_0^2 < m_1^2 (= 3m_0^2) < ... < m^2_n_max≡Λ^2 < m^2_n_max+1 < ⋯,where Λ is related to the size of the perturbed KK mass matrix by turning on the brane-localized mass term.Also, a few comments about the cutoff of theories in extra dimensions are in order. When the reference energy in the renormalization group evolution crosses the mass of KK particles, beta functions take contributions from the states. Cumulative spectra of KK particles, which are a generic structure of compact extra dimensions, lead to a rapid increase and decrease of effective 4D couplings immediately once the reference scale passes the lowest KK state (see, e.g., Ref. <cit.>). Therefore, in general the cutoff scale should be close to the typical size of KK particles. When the cutoff scale is not very far away from the electroweak scale, corrections via higher-dimensional operators are not suppressed. On the other hand, when the KK mass scale is far away from the electroweak scale, such contributions are subdued (though the cutoff scale is close to the typical scale of KK states). Hence, for an extra-dimensional theory with a sufficiently heavier KK mass scale compared with the scale of electroweak physics, the fact that the cutoff scale should be near the typical scale of KK particles does not seem to be problematic.It is convenient to relabel the indices (n^('),j^(')) in the KK mass matrix in Eq. (<ref>) in terms of a new label α. We describe the degeneracy of the wave functions for even n (odd n) by d_e (d_o), which gives us the useful relation M = d_e + d_o. We can define a one-to-one labeling as shown in Table <ref>.[ When η = +1 and n is odd, mode functions vanish in j=0. Therefore, j starts from one (not zero) in the category n=1 in Table <ref>. ] For example, when M=3, an explicit correspondence between α and (n, j) (up to the ninth mode) is shown in Table <ref>. In terms of this labeling, the KK mass matrix (“wavefunction vector”) is expressed as M^2_(n,j), (n',j')→ M^2_α, β (v_n,j→ v_α), respectively.Also, by use of the information in Tables <ref> and <ref>, the size of the mass matrix is easily estimated as α_max =M × (k-1) + d_eforn_max = 2(k-1), M ×k forn_max = 2k-1, ( k = 1,2,3,⋯),Now, we are ready to analyze the perturbed KK mass matrix. As a first illustration, let us consider M=3 and Λ = m_1, where d_e=2, d_o=1, and α runs over 1, 2, 3. The corresponding 3 × 3 KK mass matrix is given asM^2_α, β = [ m_0^2 + 2 h |v_1|^22h v_1^† v_22h v_1^† v_3;2h v_2^† v_1 m_0^2 + 2 h |v_2|^22h v_2^† v_3;2h v_3^† v_12h v_3^† v_2m_1^2 + 2h |v_3|^2 ],and three eigenvalues can be analytically solved asm_0^2,2 m_0^2 + h(|v_1|^2 + |v_2|^2 + |v_3|^2) ±√((m_0^2 + h (|v_1|^2 + |v_2|^2 + |v_3|^2))^2 -4h m_0^2 (|v_1|^2 + |v_2|^2)),where we used the relation m_1^2 = 3 m_0^2. Here, the eigenvalue is unperturbed as m_0^2. By focusing the property |v_α R|^2 ≲ 1, which is recognized by the correct normalization of the mode functions, the other eigenvalues are roughly estimated as m_0^2(1 +O(h)) and m_1^2(1 +O(h)). Thus, we find that one of the original lowest KK masses, i.e., m_0^2 appears after turning on the brane-localized mass. When the cutoff scale is chosen as Λ = m_3, where α=1,2, ..., 6, the corresponding eigenvalues are calculated in a similar manner as m_0^2, m_0^2(1 +O(h)), m_1^2(1 +O(h)), m_2^2, m_2^2(1 +O(h)), and m_3^2(1 +O(h)). We find that one of the second excited states is also unperturbed.The above discussion can be extended to the generic magnitude of the magnetic flux and an arbitrary cutoff scale. The KK mass matrix with a brane-localized mass can be symbolically expressed asM^2 =diag(m_0^2, m_0^2, ..., m_0^2_d_e,m_1^2,m_1^2,...,m_1^2_d_o,m_2^2, m_2^2, ..., m_2^2_d_e,⋯ ) + v^†⊗ v,where (v)_α = (v_1, v_2, ...) denotes an α_max component complex vector. For d_e ≥ 2, there always exists an eigenvector for the lowest mode (n=0),(u_n=0)_α = (u_1, u_2,..., u_d_e, 0, 0, ⋯),which satisfies v · u_n=0 =0. In the above example (M=3), there exist d_e-1 linearly independent eigenvectors that satisfy v · u_n=0 =0 as (u^(1)_n=0)_α = (-v_2, v_1, 0, 0, ..., 0_d_e, 0, 0, ⋯),(u^(2)_n=0)_α = (0, -v_3, v_2, 0, ..., 0_d_e, 0, 0, ⋯), ⋮ (u^(d_e -1)_n=0)_α = (0,0, ..., 0, -v_d_e, v_d_e-1_d_e, 0, 0, ⋯), where we cannot take another eigenvector that is linearly independent of all of u^(1)_n=0, u^(2)_n=0, ..., u^(d_e -1)_n=0. This fact suggests that one of the lowest modes is uplifted by the perturbation after turning on the brane-localized mass. For any level of the degenerate mass eigenvalues before the perturbation, we find that such vectors provide the corresponding eigenvalue asM^2 u_n = m_n^2 u_n + v^† (v · u_n) = m^2_n u_n.We would like to comment on the effects from multiple localized mass terms. For example, we turn on two localized masses h_1|Φ|^2 and h_2|Φ|^2 at z=z_1 and z=z_2, respectively. Here, the corresponding KK mass matrix is( M^2)_α, β = (2n+1)m_0^2 δ_α, β + 2h_1 v^(1)†_α v^(1)_β + 2h_2 v^(2)†_α v^(2)_β,where we definev^(1)_α ≡ v^(1)_n,j = ϕ^j_n(z_1),v^(2)_α ≡ v^(2)_n,j = ϕ^j_n(z_2). The degeneracies of the KK states via magnetic fluxes are degraded one by one as we place the brane-localized mass. We provide a detailed discussion on such cases in Appendix <ref>. Before closing this section, it is important to mention a 6D vector field, which is decomposed into a (4D) vector component (A_μ) and two scalar components (A_5, 6) from the four-dimensional point of view.The KK mass spectra of a vector field which feels magnetic fluxes on a flux background were analyzed in Ref. <cit.>, which provided KK eigenvalues of the two corresponding 4D scalars of ϕ_n, z≡ (A_5 + i A_6)/√(2) asm_n^2 = 2π M/ A (2n-1),and of ϕ_n, z̅≡ (A_5 - i A_6)/√(2) asm_n^2 = 2π M/ A (2 (n+1) +1),where the spectrum of the vector field is equivalent to ϕ_n, z̅ through suitable gauge fixing. Here, we would pay attention to two points. One is that the equation of motion of the 6D scalar Φ takes a different form than those of ϕ_n, z and ϕ_n, z̅, which leads to the difference in the mass spectra. The other is that in the present Abelian case, the 6D vector field does not feel any magnetic flux. The situation where the 6D vector field couples to magnetic flux is realized in a non-Abelian gauge theory, which is a reasonable playground for unified theories.From Eq. (<ref>), we recognize a pathology of the emergence of the tachyonic state in ϕ_n=0, z, which is a critical obstacle for constructing reasonable models. An ordinary remedy for conquering the difficulty is to address supersymmetrized theories, where if 4D 𝒩 = 1 supersymmetry remains in the action (before taking into account the connection to the supersymmetry-breaking sector), such tachyonic states are stabilized (see, e.g., Refs. <cit.>). On the other hand, issues discussed in this manuscript would provide another clue to circumventing the obstacle by uplifting the tachyonic mode via a brane-localized mass term for ϕ_n, z. When localized mass terms, e.g., h A_5^2 (and h A_6^2) are induced after non-Abelian gauge symmetry breaking (i.e., introducing fluxes and/or Wilson lines)[ Actually in the present situation, an Abelian gauge field (and Cartan parts of a non-Abelian gauge field) cannot feel magnetic fluxes and thus the lowest mode remains massless, while non-Cartan parts of a non-Abelian gauge field can detect such fluxes and can be massive. Also, we mention that a gauge-invariant mass term via (classical) flux configurations cannot be confined within a 4D world, which means that other extra spacial direction(s) should be required in addition to the present two directions to realize the localized mass term of ϕ_n,z and ϕ_n,z̅ in a gauge-invariant way.]the above discussion is relevant for analyzing the KK mass matrix of such kinds of scalars, in spite of the difference in the pattern of the KK masses.§.§ SpinorWe express the 6D Weyl spinor as Ψ = (λ_+, λ_-)^T in terms of four-component Weyl spinors λ_+ and λ_-. Then, its KK decomposition is given asλ_+ (x^μ, z) = ∑_n,jχ_+, n^j (x^μ) ⊗ψ_T^2/Z_2 +, n^j(z),λ_- (x^μ, z) = ∑_n,jχ_-, n^j (x^μ) ⊗ψ_T^2/Z_2 -, n^j(z). The Lagrangian for the 6D Weyl spinor is given asL = i Ψ̅Γ^M D_M Ψ + (i ν̅γ^μ∂_μν -g (ν̅λ_+ + H.c.)) δ^2(z-z_k),where g is a massless parameter associated with the localized mass of the spinor. Note that the 6D Weyl spinor cannot possess a Dirac mass term such as Ψ̅Ψ <cit.>. Then, we add a four-dimensionally localized Weyl spinor field ν(x^μ) and introduce a localized mass term at a fixed point z_k. In the following, we place the localized mass at a fixed point of T^2/Z_2 [see Eq. (<ref>)]. It is easily found that λ_- (x^μ, z_k)=0 (λ_+(x^μ, z_k)=0) [k=1,2,3] for η=+1 (η=-1), respectively. Hence, we cannot introduce such a mass for λ_- at the three fixed points. Therefore, we focus on the case of η=+1, as in the previous section. It is straightforward to expect that we can similarly analyze the case of η=-1. Hereafter, we choose the 6D chirality of Ψ as -1, which means that left-handed chiral modes are realized as zero modes of λ_+ (see Appendix <ref>). The 4D chirality of ν is automatically determined to be right-handed. The spinor Lagrangian in Eq. (<ref>) provides the six-dimensional equations of motion,i γ^μ∂_μλ_+ + D^†λ_- - g νδ^2(z - z_k)=0,i γ^μ∂_μλ_- - D λ_+= 0, ( i γ^μ∂_μν - g λ_+)δ^2(z - z_k)=0.Removing λ_- and ν from these equations leads to the (six-dimensional) equation of motion for λ_+,( ∂_μ∂^μ + D^† D )λ_+ - g^2λ_+ δ^2(z - z_k) =0,where the operator D^† D is equal to Δ - 2 π M/𝒜, which indicates the mass difference in the fermion case and scalar case as shown in Eqs. (<ref>) and (<ref>) when g=0.By plugging the KK decompositions (<ref>) and (<ref>) into Eq. (<ref>), the effective Lagrangian under the cutoff scale is calculated asL_ eff =L_ kin - ∑_n, j(n ≠ 0)(m_n χ̅^j_+, n_L χ^j_-, n_R+ g ν̅_R χ_+, 0^j_L ψ^j_T^2/Z_2 +, 0(z_k) + g ν̅_R χ^j_+,n_L ψ^j_T^2/Z_2 +, n(z_k) + H.c.) ≡ L_ kin - ∑_n, j(n ≠ 0) (ν̅_R,χ̅_-,n^j_R)M[ χ^j_+,0_L; χ^j_+,n_L ] + H.c.,where ℒ_kin contains kinetic terms and the corresponding 4D chiralities are explicitly shown for clarity.The perturbed KK mass matrix ℳ under the brane-localized spinor mass term is symbolically expressed asM =[ √(2) gψ^j_+,0(z_k) √(2) gψ^j_+,n(z_k);0m_n ].Here, note that the index n takes nonzero positive integer values in the above expressions, and also that we use the relation ψ^j_T^2/Z_2 +, n(z_k) = √(2)ψ^j_+,0(z_k).Here, the size of ℳ is (1 + N_KK) × (d_e + N_KK), where N_KK represents the number of excited KK modes that appear up to the level designated by n_max. The relation is easily understood among N_KK and α_max as defined in Eq. (<ref>),N_KK = α_max - d_e,because d_e represents the number of zero modes. Since the matrix ℳ is asymmetric, it is convenient to consider the following two forms of products of the matrix( MM^†)_α, β = (Π^2_R)_α,β + √(2) g (V^†_R Π_R)_α δ_0, β + √(2) g (Π_R V_R)_β δ_0, α+ 2 g^2[ ∑_j=0^d_e -1 |v_0,j|^2 + ∑_ρ = 1^N_KK |(V_R)_ρ|^2 ] δ_0, αδ_0, β (α, β = 0,1,⋯,N_KK), ( M^† M)_α, β = (Π^2_L)_α, β + √(2) g (V^†_R)_α (V_R)_β (α, β = 1,2,⋯,α_max=N_KK+d_e),for the right- and left-handed Weyl spinors (ν_R, χ_-,n^j_R) and (χ^j_+,0_L, χ^j_+,n_L), respectively. Here, we see that α=0 corresponds to the 4D localized field ν_R in Eq. (<ref>). The sizes of the matrices MM^† and M^† M are (1 + N_KK)×(1 + N_KK) and (d_e + N_KK)×(d_e + N_KK), respectively. Also, we define the following symbols(Π_R)_α, β ≡ diag (0,m_1, m_1, ..., m_1_d_o,m_2, m_2, ..., m_2_d_e, ⋯), (V_R)_α ≡ (0,v_1, 1, v_1, 2, ..., v_1, d_o_d_o,v_2, 0, v_2, 1, ..., v_2, d_e-1_d_e,⋯)^T, (Π_L)_α, β ≡ diag (0, 0, ..., 0_d_e,m_1, m_1, ..., m_1_d_o,m_2, m_2, ..., m_2_d_e, ⋯), (V_L)_α ≡ (v_0, 0, v_0, 1, ..., v_0, d_e-1_d_e,v_1, 1, v_1, 2, ..., v_1, d_o_d_o,v_2, 0, v_2, 1, ..., v_2, d_e-1_d_e,⋯)^T,with v_n, j≡ψ^j_+, n(z_k).In the spinor case, we recognize that the mass spectrum of the left-handed modes χ_+, 0^j_L and χ_+, n^j_Lis equivalent to that of the scalar because the mass matrix squared in Eq. (<ref>) takes the same form as that in the scalar case.On the other hand, we analyze the mass spectra of the 4D brane-localized field ν_R and the right-handed modes χ_-,n^j_R. For a vector (u)_α (α = 0,1,⋯,N_KK), we calculate the product of ( MM^†)_α, β and u_α as ∑_β=0^N_KK ( MM^†)_α, β (u)_β = √(2) g∑_β=1^N_KK (Π_R V_R)_β (u)_β + 2 g^2[ ∑_j=0^d_e -1 |v_0,j|^2 + ∑_ρ=1^N_KK |(V_R)_ρ|^2 ] (u)_0 (α=0),∑_β=0^N_KK (Π^2_R)_α, β (u)_β + √(2) g (V^†_R Π_R)_α (u)_0 (α≥ 1).Equation (<ref>) implies that ν_R is perturbed by the presence of the localized mass since the right-hand side for α=0 may be nonvanishing in almost all cases.This is understood as follows. Since the right-handed spinors χ^j_-, n_R are originally massive around the compactification scale (∼ 1/√( A)), we can determine whether ν_R is massless or massive only by investigating the determinant of MM^†. This is because the determinant of a matrix is equal to the product of its eigenvalues. If the determinant is nonzero, we can conclude that ν_R becomes massive.Here, let us focus on a simple example for M=3 and Λ=m_2. For the right-handed fields (ν_R, χ_-,1^0_R, χ_-,1^1_R, χ_-,2^1_R), the 4 × 4 KK mass matrix squared MM^† is symbolically written with the symbols a, b, c and d as MM^† = [ 2 g^2 d √(2) g am_1 √(2) g bm_1 √(2) g cm_2; √(2) g a^†m_1 m_1^2 0 0; √(2) g b^†m_1 0 m_1^2 0; √(2) g c^†m_2 0 0 m_2^2 ].Here, g is a dimensionless coefficient in the localized mass term of the spinor field. The determinant of this matrix is calculated as( MM^†) = 2 g^2 m_1^4 m_2^2 ( d - |a|^2 - |b|^2 - |c|^2 ),where mass dimensions of the four symbols are two (for d) and one (for a, b, c). For a nonzero coefficient (g ≠ 0), a miraculous cancellation should occur among the symbols a, b, c and d to realize MM^† =0, which suggests that ν_R is still massless after the localized-mass perturbation. Although we note that we could not find a concrete example where the cancellation happens, our conclusion is that ν_R gets a mass via the localized mass in almost any case. When M and/or Λ is arbitrary, configurations of nonzero components in the mass matrix squared ℳℳ^† look similar. Therefore, this kind of discussion is still valid.The matrices MM^† and ℳ^†ℳ contain N_KK + 1 and N_KK + d_e numbers of squared mass eigenvalues, respectively, where N_KK values are common in both of the matrices. These modes mainly originate from the KK mass terms χ^j_-,nRχ^j_+,nL + H.c. which exist even before the perturbation. § DEFORMATIONS OF THE KK WAVE FUNCTIONSAs discussed in Sec. 3, we found that at least one of the lowest KK masses remains after introducing one brane-localized mass term if the number of the lowest modes is greater than or equal to two. In this section, we investigate what happens on the corresponding KK wave function after the perturbation.A physical sense of such deformations is a possible modulation of three-point effective interactions, i.e., Yukawa couplings in phenomenological models, which are characterized by the overlap integrals of three types of KK wavefunctions.[For example, see Refs. <cit.> and also Ref. <cit.>.] The Yukawa couplings on T^2/Z_2 are expressed asy_αβγ∼∫_T^2 d^2zψ_T^2/Z_2^α(z) ψ_T^2/Z_2^β (z) (ϕ_T^2/Z_2^γ(z))^†,where α, β and γ discriminate the physical eigenstates on magnetized T^2/Z_2 before the perturbation by turning on single or multiple brane-localized mass terms. For example, when we introduce a localized mass term for the 6D scalar field, the profiles of KK mode functions describing the lowest states would be changed as ϕ_T^2/Z_2^α(z)(α = 1,2, ..., d_e) →ϕ^i_mass_T^2/Z_2(z)(i_mass =1,2, ..., d_e -1). Accordingly, the Yukawa couplings are expected to be changed as ∫_T^2 d^2zψ_T^2/Z_2^α(z)ψ_T^2/Z_2^β (z) (ϕ_T^2/Z_2^γ(z))^†→∫_T^2 d^2zψ_T^2/Z_2^α(z)ψ_T^2/Z_2^β (z) (ϕ_T^2/Z_2^i_mass (z))^†.It is also expected that the same holds for localized mass terms of the spinor. Although we do not analyze such Yukawa couplings in the presence of brane-localized masses in this paper, it is important to investigate the change of the lowest KK mode functions. As an illustrative example, we consider a simple example in the scalar case for M=3 and Λ=m_1=18π/ A, and then α=1, 2, 3 [corresponding to (n,j)=(0,0), (0,1), (1,1)]. Then, the KK mass matrix under consideration is the same as the expression in Eq. (<ref>). For h=0.5, the eigenvalues (divided by m_0) of Eq. (<ref>) are numerically obtained as(m/m_0)^2 = 1.00000,1.08008,3.05828,and also the corresponding eigenvectors are given asu^(1) = (0.459701, -0.888074, 0), u^(2)= (-0.88755, -0.45943, 0.0343449),u^(3)= (0.0305008, 0.0157884, 0.99941).The corresponding wave functions unaffected/affected by the brane-localized mass are obtained by internal products of the above eigenvectors and the KK wave functions before the perturbation ϕ_α≡ (ϕ^0_T^2/Z_2, 0, ϕ^1_T^2/Z_2, 0, ϕ^1_T^2/Z_2, 1), i.e.,ϕ^i_ mass(z) ≡∑_α=1^3 (u^(i_ mass))_αϕ_α (z) ( i_mass = 1, 2, 3 ).We show the probability densities |ϕ^i_ mass(z)|^2 of the KK wave functions with mass eigenvalues that are unaffected/perturbed by the brane-localized mass at z = z_1 for M=3 in Fig. <ref>, where a red cross denotes a position with the brane-localized mass. Figure <ref> tells that an unaffected mode (i_ mass=1) avoids the position with the localized mass, and also that the other affected modes (i_ mass=2, 3) are localized at the position with the localized mass.The trend that unaffected modes avoid the position with the localized mass is also found in situations with multiple localized masses (see Fig. <ref>).We provide another example with M=4 and Λ=m_1 (α=1,2,3,4), and two brane-localized mass terms at the two fixed points z=z_1 and z=z_2. The sketches of wave function localizations are shown in Fig. <ref>. Here, we can see that two of the three lowest modes before the perturbation (see Table <ref>) are uplifted.§ CUTOFF DEPENDENCE OF MASS EIGENVALUESOnce we specify a cutoff scale Λ, we can write down the KK mass spectra with the brane-localized mass below Λ. As concretely addressed in the previous section, when η = +1 and M ≥ 2, a part of the lowest modes of the KK mass spectrum in the scalar is unaffected by the presence of a single localized mass term, where the values of such unperturbed mass eigenstates are independent of the cutoff scale Λ. On the other hand, some mass eigenvalues are perturbed and get heavier by the effect of the localized mass term, where the degree of such deformations would depend on the cutoff scale. We suppose that Λ = m_n_ max for a certain n_ max. Hereafter, we consider n_ max instead of Λ. As before, we focus on M=3. The two lowest modes without localized masses are doubly degenerate and their KK masses are m_0^2 = 6π/ A. After introducing a localized mass at z=z_1, one of the two lowest modes gets heavier and the perturbed mass is roughly estimated as m^2 ∼ m_0^2 (1 +O(h)), assuming that |v_α R|^2 ∼ O(1). For the maximum of the KK levels n_ max, the ratio (m/m_0)^2 can be interpolated as(m/m_0)^2≃α_1 + α_2 ln n_ max + α_3 n_ max,where α_i(i=1,2,3) are real constants. The cutoff scale dependence of the (squared) mass ratio (m/m_0)^2 is shown in Fig. <ref> (Fig. <ref>), where we set h=0.5 (h=3.0), respectively. Figures <ref> and <ref> tell us that the choic of cutoff scale does not drastically affect the mass ratio, and we can conclude that such a cutoff dependence is irrelevant from a model-building point of view [except for h being as huge as ≳𝒪(10) and/or the compactification scale ∼ R^-1 being as small as ≲𝒪(1) TeV]. We also find that the cutoff dependence looks similar and would be irrelevant for greater magnitudes of the fluxes.[ Here, we comment on approaches to treat brane-localized mass terms. The simplest method adopted in our analysis—where an effective mass matrix with infinite numbers of columns and lows is derived through KK expansions, and we (numerically) diagonalize an approximated form with a truncation of higher modes holding heavier KK masses than a cutoff scale Λ—is enough when results are not sensitive to values of the cutoff. For more precise discussions, the techniques with the theta functions argued in Refs. <cit.> would be useful. See also, e.g., Refs. <cit.> (and references therein) for discussions on the bulk-boundary interplay of higher-dimensional fields. ] Also, we would like to comment on the testability of the mass correction via localized masses through high-energy experiments. When the coefficient of a localized mass h is small, the cutoff dependence is negligible (as already explained) and it seems difficult to probe the effect by discovering several KK modes and measuring the mass differences among them. Thus, the mechanism has difficulties from the testability point of view. However, the main motivation of this research is to reveal the relationship between the presence of localized masses and the number of fluxes (corresponding to the number of unperturbed matter generations).When we consider a high-scale extra-dimensional theory, the correction in KK states (before the mass perturbation) seems to be less important. Nevertheless, we can claim that the stability of the values of heavier states (compared with the electroweak scale) against the perturbation is a good feature of the present scenario.§ CONCLUSION In this paper, we have considered effects on the KK mass spectra via the presence of brane-localized masses at fixed points of a toroidal orbifold T^2/Z_2 with magnetic fluxes. Under the presence of the magnetic fluxes on the toroidal and orbifold compactifications, the magnetic fluxes are quantized and become topological indices, and then the multiplicity of KK-decomposing wave functions appears in the low-energy effective theory.We have added single or multiple brane-localized masses at the fixed points of T^2/Z_2 where some parts of the KK spectrum on T^2 are projected out by the orbifolding, while multiple degenerated modes still remain (if M ≥ 2). By analyzing the effects of the localized masses using linear algebra, we have found that, at each KK level, one or more of the degenerate KK masses are perturbed, when single or multiple brane-localized mass terms are introduced. This discussion is valid for both of the six-dimensional scalar and spinor fields. In addition, we have also investigated deformation in wave functions through the localized mass terms and the cutoff dependence of the magnitude of modulations in the mass eigenvalues.The mechanism which we have investigated in this paper is useful for phenomenologies on magnetized orbifolds, especially in constructing unified theories with a much heavier KK scale compared with the electroweak scale, for decoupling some light exotic particles away from the physics around the electroweak and TeV scales. An important point in the mechanism in 6D is that particle spectra do not seriously depend on the magnitude of the coefficients of brane-localized mass terms when the KK scale is sufficiently higher than the electroweak scale, while the number of fixed points where brane-localized mass terms are injected plays a significant role. Therefore, we can conclude that our mechanism is useful for removing unwanted exotic light modes from low-energy effective theories without relying on the details of theories in extra dimensions. Applications to more generalized situations with nontrivial Scherk-Schwarz and Wilson line phases as well as general choices in the complex structure parameter τ look fruitful. § ACKNOWLEDGMENTSM.I. and Y.T. would like to thank Hiroyuki Abe for helpful discussions. K.N. is grateful to Hiroyuki Abe and the particle physics group at Waseda University for the kind hospitality during the final stage of this work. We thank the PRD referee for giving us various fruitful comments. Y.T. is supported in part by Grants-in-Aid for Scientific Research No. 16J04612 from the Ministry of Education, Culture, Sports, Science and Technology of Japan.§ NOTATIONIn this appendix, we review our notation for 6D gamma matrices, which obey the Clifford algebra{Γ^M, Γ^N } = -2 g^MN, g^MN = diag(-1,+1,+1,+1,+1,+1).Our choice for the set of 6D gamma matrices is as follows:Γ^μ = [ γ^μ 0; 0 γ^μ ], Γ^5 = [0 iγ_5; iγ_50 ], Γ^6 = [0γ_5; -γ_50 ],where γ_5 describes the 4D chirality, which is defined as γ_5 = i γ^0 γ^1 γ^2 γ^3. The matrix denotes the 6D chirality and can be decomposed asΓ^7 = - Γ^0 Γ^1 Γ^2 Γ^3 Γ^5 Γ^6 = [γ_50;0 -γ_5 ] = Γ_4D chiral Γ_internalwithΓ_4D chiral = [ γ_5 0; 0 γ_5 ],Γ_internal≡ i Γ^5Γ^6 = [I_40;0 -I_4 ].The matrix Γ_internal describes eigenvalues of the internal chirality. The 6D chirality is calculated as the simple product of the 4D chirality and the internal chirality.§ GENERIC DISCUSSION IN THE SCALAR CASEIn the main sections, we mainly discussed the effects arising from the presence of a single brane-localized mass. In this appendix, we extend the discussion to multiple brane-localized mass terms and their effects on KK mass eigenvalues. The scalar case is addressed in the following discussion, where the method to treat the fermion case is straightforwardly recognized through the result for the scalar. The reason is that ℳ^2 for the scalar and ℳ^†ℳ for the fermion take the same form, as we pointed out in Sec. <ref>. For example, we add another brane-localized mass to Eq. (<ref>), and then obtainM^2 =diag(m_0^2, m_0^2, ..., m_0^2_d_e,m_1^2, m_1^2, ..., m_1^2_d_o,m_2^2, m_2^2, ..., m_2^2_d_e, ⋯ ) +(v^(1))^†⊗ v^(1) + (v^(2))^†⊗ v^(2).Here, we do not specify two distinct positions with the localized masses and symbolically express (v^(1))_α = ϕ_α(z_i) and (v^(2))_α = ϕ_α(z_j) for z_i ≠ z_j. In order to keep one of the original lowest eigenvalues as m_0^2 after diagonalizing the KK mass matrix, an eigenvector (u)_α which contains(u)_α = (u_1, u_2, ..., u_d_e, 0,0,⋯)should simultaneously satisfyv^(1)· u = 0v^(1)_1 u_1 + v^(1)_2 u_2 + ... + v^(1)_d_e u_d_e = 0, v^(2)· u = 0v^(2)_1 u_1 + v^(2)_2 u_2 + ... + v^(2)_d_e u_d_e = 0.The relations in Eqs. (<ref>) and (<ref>) ensure that (u)_α is an eigenvector with the eigenvalue m_0^2,ℳ^2 u = m_0^2 u + (v^(1))^†⊗( v^(1)· u ) + (v^(2))^†⊗( v^(2)· u ) = m_0^2 u.When we recognize that whether an eigenvector is normalized or not does not affect the number of linearly independent eigenvectors, we find that at least three nonzero components are required in u as,(u^(1))_α = (u_1^(1),u_2^(1),u_3^(1),0,0,⋯),which obeys the simplified constraints,v^(1)_1 u^(1)_1 + v^(1)_2 u^(1)_2 + v^(1)_3 u^(1)_3 = 0, v^(2)_1 u^(1)_1 + v^(2)_2 u^(1)_2 + v^(2)_3 u^(1)_3 = 0.When we take u^(1)_1 = 1, which is just a scaling, the corresponding values of u^(1)_2 and u^(1)_3 are fixed asu^(1)_2 = v^(1)_3 v^(2)_1 - v^(1)_1 v^(2)_3/-v^(1)_3 v^(2)_2 + v^(1)_2 v^(2)_3, u^(1)_3 = v^(1)_2 v^(2)_1 - v^(1)_1 v^(2)_2/ v^(1)_3 v^(2)_2 - v^(1)_2 v^(2)_3.Apparently, similar procedures can continue, e.g., for (u^(2))_α = (0,u_2^(2),u_3^(2),u_4^(2),0,0,⋯). Now, we can conclude that the number of linearly independent eigenvectors under two brane-localized mass terms is d_e - 2 (for even n) and d_o -2 (for odd n), respectively, unless anomalous situations arise, e.g., -v^(1)_3 v^(2)_2 + v^(1)_2 v^(2)_3 = 0. In such exceptionally special cases, the number of linearly independent eigenvectors does not obey the above criterion. Situations with brane-localized mass terms at three (four) fixed points are scrutinized in the same way, where d_e - 3 (d_e - 4) [for even n] and d_o -3 (d_o -4) [for odd n] independent physical modes remain unperturbed in the case without accidental cancellation in corresponding conditions.utphys

http://arxiv.org/abs/1702.08226v2

Symmetric flows for compressible heat-conducting fluids with temperature dependent viscosity coefficientsLing WanSchool of Mathematics and Statistics, Wuhan University, Wuhan 430072, ChinaTao WangCorresponding author. Email addresses: ling.wan@whu.edu.cn (L. Wan), tao.wang@whu.edu.cn (T. Wang).School of Mathematics and Statistics, Wuhan University, Wuhan 430072, ChinaDICATAM, Mathematical Division, University of Brescia, Via Valotti 9, 25133 Brescia, Italy ================================================================================================================================================================================================================================================================================================================================================================================= We consider the Navier–Stokes equations for compressible heat-conducting ideal polytropic gases in a bounded annular domain when the viscosity and thermal conductivity coefficients are general smooth functions of temperature. A global-in-time, spherically or cylindrically symmetric, classical solution to the initial boundary value problem is shown to exist uniquely and converge exponentially to the constant state as the time tends to infinity under certain assumptions on the initial data and the adiabatic exponent γ. The initial data can be large if γ is sufficiently close to 1. These results are of Nishida–Smoller type and extend the work [Liu et al., SIAM J. Math. Anal. 46 (2014), 2185–2228] restricted to the one-dimensional flows. Keywords: Compressible Navier–Stokes equations; Temperature dependent viscosity coefficients; Global symmetric solutions;Nishida–Smoller type result; Exponential stabilityMathematics Subject Classification: 35Q35 (35B40, 76N10)§INTRODUCTION The motion of a compressible viscous and heat-conducting fluid in Ω⊂ℝ^d can be described by the Navier–Stokes equations in Eulerian coordinates:∂_tρ+div(ρu)=0,∂_t(ρu)+div(ρu⊗u)+∇P=div 𝕊,∂_t(ρE)+div(ρuE+uP)=div(κ∇θ+𝕊·u).Herethe primary dependent variables are the density ρ, the velocity field u∈ℝ^d, and the temperature θ. The symbol E=e+12|u|^2 is the specific total energy. For ideal polytropic gases, the pressure P and the specific internal energy e are related with ρ and θ by equations of state:P=Rρθ, e=c_vθ,where R>0,c_v=R/(γ-1), and γ>1 are the specific gas constant, the specific heat at constant volume, and the adiabatic exponent, respectively.The fluid is assumed to be Newtonian so that the viscous stress tensor 𝕊 is of form𝕊=μ(∇u+(∇u)^𝖳) +λdiv u 𝕀,where(∇u)^𝖳 is the transpose matrix of ∇u and 𝕀 is the d× d identity matrix. The viscosity coefficients μ, λ, and the thermal conductivity coefficient κ are prescribed through constitutive relations as functions of the density and temperature satisfying μ>0, κ>0, and 2μ+dλ>0 (see <cit.>). In this paper we establish the existence and exponential decay rate of global-in-time,spherically or cylindrically symmetric, classical solutions to (<ref>)–(<ref>) in the bounded annular domain Ω={x=(x_1,⋯,x_d)∈ℝ^d: a<r<b} for 0<a<b<∞ with large initial data. Here we set d∈ℕ_+ andr=|x| in the spherically symmetric case, while d=3 and r=√(x^2_1+x^2_2) in the cylindrically symmetric case.The system (<ref>)–(<ref>) is supplemented withthe spherically or cylindrically symmetric initial data:(ρ,u,θ)(0,x) =(ρ_0,u_0,θ_0)(x)forx∈Ω,and the boundary conditions:u=0,∂θ/∂n=0 on∂Ω,where n denotes the unit outward normal vector to ∂Ω.We are interested in thecase where the transport coefficients μ, λ, andκ are smooth functions of the temperature. More specifically, we suppose thatμ,λ,κ∈ C^3(0,∞),μ(θ)>0,κ(θ)>0, 2μ(θ)+dλ(θ)>0for allθ>0.Our main motivation is provided by the kinetic theory of gases. By virtue of the Chapman–Enskog expansion, the compressible Navier–Stokes system (<ref>) is the first order approximation of the Boltzmann equation, and the transport coefficients μ, λ, and κ depend solely on the temperature (see <cit.> and <cit.>).In particular, if the intermolecular potential varies as r^-a with r being the molecule distance, then μ, λ, and κ satisfyμ=μ̅θ^a+4/2a,λ=λ̅θ^a+4/2a,κ=κ̅θ^a+4/2a fora> 0,where μ̅, λ̅, and κ̅are constants. In spite of this obvious physical relevance, there is no global existence result currently available beyond the small data <cit.> for the Navier–Stokes system (<ref>)–(<ref>)withgeneral adiabatic exponent γ and transport coefficients (<ref>). Let us mention some related results about the global existence and large-time behavior for the full compressible Navier–Stokes equations (<ref>) with large data.The global existence and uniqueness of smooth solutions to (<ref>)–(<ref>) in one-dimensional bounded domains are proved in the seminal work by <cit.> for constant transport coefficients and large initial data. The crucial step in <cit.> is to obtain the positive upper and lower bounds of the specific volume τ (i.e. the reciprocal of density ρ) and temperature θ, which is achieved by means of a decent representation forτ and the maximum principle.The results in <cit.> have been generalized to cover the spherically and cylindrically symmetric flows. In the case of spherical symmetry, <cit.> (resp. <cit.>) showed theexistence of global-in-time (generalized) solutions in bounded annular domains (resp. in exterior domains), while <cit.> investigated the flows between a static solid core and a free boundary connected to a surrounding vacuum state. In the cylindrically symmetric case, <cit.> established the global solvability with large data in a bounded annular domain. Later, <cit.> proved the global existence of a spherically or cylindrically symmetric weak solution with large discontinuous data in a ball. As for the large-time behavior of global solutions, see <cit.> (resp. <cit.>) for spherically symmetric flows in bounded domains (resp. in exterior domains), and <cit.> for cylindrically symmetric flows in bounded domains. In all of these works the transport coefficients μ, λ, and κ are supposed to be constants.The argument in <cit.> can be applied to the case of constant viscosity and temperature dependent thermal conductivity;see <cit.> for one-dimensional flows and <cit.> for cylindrically symmetric flows.Under certain assumptions ontemperature dependent thermal conductivity κ,pressure P,and internal energy e, <cit.> show the global existence of spherically or cylindrically symmetric, classical and strong solutions for (<ref>) with constant viscosity coefficients μ and λ, which generalizes the work by <cit.> for one-dimensional flows. The assumption of constant viscosity coefficients μ and λ is essential in<cit.>. Temperature dependence of the viscosity coefficients μ and λ turns out to have a strong influence on the solution and lead to difficulty in mathematical analysis for global solvability with large data.<cit.> recently obtain the first result on global well-posedness of smooth solutions to the one-dimensional Navier–Stokes system (<ref>)–(<ref>) with general adiabatic exponent γ,temperature dependent viscosity, and large initial data. Unfortunately, this result do not cover the models satisfying (<ref>). As far as we are aware, the only global solvability result currently available for (<ref>)–(<ref>) with constitutive relations (<ref>) and large data is due to<cit.> for one-dimensional flows. In <cit.> the initial density and velocity can be large, but γ-1 and the H^3-norm of θ_0-1 with θ_0 being the initial temperature have to be small.In other words, this is a Nishida–Smoller type global solvability result with large data. The original Nishida–Smoller type global solvability result is about the one-dimensional ideal polytropic isentropic compressible Euler system in <cit.> (see also <cit.> for thenonisentropic case). It is anatural and interesting problem to establish a Nishida–Smoller type global existence result for the spherically or cylindrically symmetric solutions of (<ref>)–(<ref>) with temperature dependent transport coefficients (<ref>). Our main objective is to study this problemand to show the exponential stability of solutions toward the constant equilibrium states as well. For this purpose, we reduce the initial boundary value problem (<ref>)–(<ref>) to the corresponding problem in either spherical or cylindrical coordinates.We let (u, v, w) be the velocity components in either spherical or cylindrical coordinates. Thusu(t,x)=ũ(t,r)/rx,v=w≡ 0 in the spherically symmetric case, whileu(t,x) =ũ(t,r)/r(x_1,x_2,0) +ṽ(t,r)/r(-x_2,x_1,0)+w̃(t,r)(0,0,1)in the cylindrically symmetric case.In both cases, the density and temperature depend only on the time t and the radius r, i.e. (ρ,θ)(t,x)=(ρ̃,θ̃)(t,r). The system for (ρ̃,ũ, ṽ,w̃,θ̃) then takes the form ρ̃_t+(r^m ρ̃ũ)_r/r^m=0, ρ̃(ũ_t+ũũ_r)-ρ̃ṽ^2/r+P̃_r= [ν̃(r^m ũ)_r/r^m]_r-2m ũμ̃_r/r,ρ̃(ṽ_t+ũṽ_r)+ρ̃ũṽ/r= (μ̃ṽ_r)_r+2μ̃ ṽ_r/r^m-m(μ̃r^m-1ṽ)_r/r^m-μ̃ṽ/r^2m, ρ̃(w̃_t+ũw̃_r)= (μ̃w̃_r)_r+mμ̃ w̃_r/r, ρ̃(ẽ_t+ũẽ_r)+P̃(r^m ũ)_r/r^m= (κ̃r^mθ̃_r)_r/r^m+𝒬̃, where μ̃=μ(θ̃), ν̃=2μ(θ̃)+λ(θ̃), P̃=Rρ̃θ̃, ẽ=c_vθ̃, and𝒬̃= ν̃(r^m ũ)_r^2/r^2m-2mμ̃(r^m-1ũ^2)_r/r^m+μ̃w̃_r^2+μ̃[ṽ_r-ṽ/r^m]^2.In spherically symmetric case, d∈ℕ_+, m=d-1, and ṽ=w̃=0, whereas in cylindrically symmetric case, d=3 and m=1.The initial and boundary conditions (<ref>) and (<ref>) are reduced to (ρ̃,ũ,ṽ,w̃,θ̃)(0,r) = (ρ̃_0,ũ_0,ṽ_0,w̃_0,θ̃_0)(r),a≤r≤b, (ũ,ṽ,w̃,θ̃_r)(t,a) =(ũ,ṽ,w̃,θ̃_r)(t,b)=0, t≥0.These boundary conditions are supposed to be compatible with the initial data.To establish the global existence, it is convenient to transform the initial boundary value problem (<ref>)–(<ref>) into that in Lagrangian coordinates. We introduce the Lagrangian coordinates (t,x) and denote(ρ,u,v,w,θ)(t,x) =(ρ̃,ũ,ṽ,w̃,θ̃)(t,r), wherer=r(t,x)=r_0(x)+∫_0^t ũ(s,r(s,x))ds,and r_0(x):=h^-1(x),h(r):=∫_a^r z^m ρ̃_0(z)dz.Notice thatthe function h is invertible on [a,b] provided that ρ̃_0(z)>0 for each z∈[a,b] (which will be assumed in Theorem <ref>).Here, without loss of generality, we set that h(b)=1. Due to (<ref>), (<ref>), and (<ref>), we see ∂/∂ t∫_a^r(t,x) z^m ρ̃(t,z)dz = ∂/∂ t∫_b^r(t,x) z^m ρ̃(t,z)dz=0.Then it is easy to check that∫_a^r(t,x) z^m ρ̃(t,z)dz=h(r_0(x))=xand∫_b^r(t,1) z^m ρ̃(t,z)dz=0.Hence r(t,0)=a, r(t,1)=b for t≥ 0, and the region {(t,r): t≥ 0, a≤ r≤ b} under consideration is transformed into {(t,x):t≥ 0,0≤ x≤ 1}. The identities (<ref>) and (<ref>) implyr_t(t,x)=u(t,x),r_x(t,x)=r^-mτ(t,x),where τ:=1/ρ is the specific volume. By virtue of (<ref>),the system (<ref>) is reformulated to that for (τ,u,v,w,θ)(t,x) as τ_t=(r^m u)_x,u_t-v^2/r+r^mP_x= r^m[ν(r^m u)_x/τ]_x-2mr^m-1uμ_x, v_t+uv/r= r^m[μr^m v_x/τ]_x+2μv_x-m( μr^m-1v)_x-μτv/r^2m,w_t= r^m[μr^m w_x/τ]_x+mμr^m-1 w_x,e_t+P(r^m u)_x= [κr^2m θ_x/τ]_x+𝒬,where t>0, x∈ℐ:=(0,1), andP =R θ/τ,e=c_v θ,c_v=R/γ-1, 𝒬 = ν(r^m u)_x^2/τ-2mμ(r^m-1 u^2)_x+μ r^2m w_x^2/τ +μτ[r^m v_x/τ-v/r^m]^2.The initial and boundary conditions are(τ,u,v,w,θ)(0,x) = (τ_0,u_0,v_0,w_0,θ_0)(x),x∈ℐ, (u,v,w,θ_x)(t,0) =(u,v,w,θ_x)(t,1)=0, t≥0,where (τ_0,u_0,v_0,w_0,θ_0):= (1/ρ̃_0,ũ_0,ṽ_0,w̃_0,θ̃_0)∘ r_0, the symbol ∘ denotes composition, and r_0 is defined by (<ref>).We now state our main results in the following theorem. Suppose that the transport coefficients μ, λ, and κ satisfy (<ref>). Let the initial data (τ_0,u_0,v_0,w_0,θ_0) be compatible with the boundary conditions (<ref>) and satisfy(τ_0,u_0,v_0,w_0)_H^3(ℐ)+(√(c_v)(θ_0-1),θ_0xx)_H^1(ℐ)≤Π_0,V_0^-1≤τ_0(x)≤ V_0,θ_0(x)≥ V_0^-1 for all x∈ℐ,whereΠ_0and V_0 are positive constants independent of γ-1. Then there exist constants ϵ_0>0 and C_1>0, which depend only on Π_0and V_0, such that if γ-1≤ϵ_0, then the initial boundary value problem (<ref>)–(<ref>) has a unique global solution(τ,u,v,w,θ)∈ C([0,∞),H^3(ℐ)) satisfying C_1^-1≤τ(t,x)≤ C_1,12≤θ(t,x)≤ 2for all (t,x)∈[0,∞)×ℐ,and the exponential decay rate(τ-τ̅,u,v,w,θ-θ̅)(t)_H^1(ℐ)+r(t)-r̅_H^2(ℐ)≤ C_γe^-c_γt for all t∈[0,∞), where C_γ and c_γ are positive constants depending on γ, andτ̅=∫_ℐτ_0dx,θ̅=∫_ℐ[θ_0+1/2c_v(u_0^2+v_0^2+w_0^2)]dx,r̅=[a^m+1+(m+1)τ̅x]^1/m+1. The techniques in this paper can be applied to obtain analogous results for the initial boundary value problem (<ref>)–(<ref>) with the following boundary conditions (u,v,w)|_x=0,1=0,θ|_x=0,1=1, t≥ 0. We deduce from (<ref>) and (<ref>) that no vacuum will be developed if the initial data do not contain a vacuum.It follows from Sobolev's imbedding theorem that the unique solution constructed in Theorem <ref> is a globally smooth non-vacuum solution with large initial data. Moreover, this result in Lagrangian coordinates can easily be converted to an equivalent statement for the corresponding problem in Eulerian coordinates.Now we outline the main ideas to deduce Theorem <ref>. As shown in <cit.>, the crucial step to construct the global solutions for the initial boundary value problem (<ref>)–(<ref>) with large initial data is to obtain the positive upper and lower bounds of the specific volume τ and the temperature θ. In the case of constant viscosity coefficients μ and λ, the pointwise bounds for the specific volume τ and the upper bound for the temperature θ can be obtained by modifying the argument in <cit.>.The positive lower bound for the temperature then follows from the standard maximum principle. However, we find that this methodology does not work in the case of density/temperature dependent viscosity. For ideal polytropic gases (<ref>)–(<ref>), the temperature θ satisfiesc_vθ_t+Rθ(r^m u)_x/τ= [κ r^2mθ_x/τ]_x+𝒬.It can be expected to get the uniform-in-time bounds for √(c_v)(θ-1,θ_t)(t)_H^1(ℐ) by very careful energy estimates even when the viscosity and thermal conductivity coefficients are functions of the temperature. Recalling that c_v=R/(γ-1), under the a priori assumption that γ-1 is sufficiently small, we can use the smallness of (θ-1,θ_t)(t)_H^1(ℐ) to handle the possible growth of solutions induced by the temperature dependence of the viscosity.The bounds for the specific volume τ from below and above can be established by developing the argument by <cit.> (see Lemma <ref>).There are three main differences between our results for the symmetric flows (<ref>) and the work in <cit.> restricted to the one-dimensional case. 1. First, the system (<ref>) under consideration here is more complicated than the one-dimensional compressible Navier–Stokes system (i.e. the system (<ref>) with m=v=w=0) which makes the form and treatment of equations simpler.2. The second difference is that the analysis in <cit.> takes place in the whole space without boundary. Thus the energy estimates for the two-order and three-order derivatives of the solutions (u_xx(t)_L^2(ℝ) etc.) can be deduced directly from integrating by parts (see, for instance, <cit.> for the estimate of u_xx(t)_L^2(ℝ)). Due to the presence of the boundary conditions (<ref>) in our case, we cannot use the methods in <cit.> to estimate the two-order and three-order derivatives of the solutions (τ,u,v,w,θ).To overcome such a difficulty, we make the estimates for (u_t,v_t,w_t,θ_t) and (u_xt,v_xt,w_xt,θ_xt), which yield thebounds for (τ_xx,u_xx,v_xx,w_xx,θ_xx)(t)_H^1(ℐ).3. A third difference concerns the assumptions on the initial data.In <cit.>, √(c_v)(θ_0-1)_H^3(ℝ) is required to be bounded by some (γ-1)-independent positive constant.According to this assumption, the H^1-norm of θ_0xx has to be small because of the smallness of γ-1.In Theorem <ref>, θ_0xx_H^1(ℐ) is assumed in (<ref>) to be bounded by some (γ-1)-independent positive constant. Thus θ_0xx_H^1(ℐ) can be large even when the adiabatic exponent γ goes to 1.The rest of this paper is organized as follows.First, in Section <ref>, we derive a number of desired a priori estimates. More specifically, the basic energy estimate is obtained in Subsection <ref>; the uniform-in-time pointwise bounds of the specific volume τ is shown in Subsection <ref> by applying the argument developed by Kanel'; the estimates on first-order, second-order, and third-order derivatives of the solution (τ,u,v,w,θ) will be deduced in Subsections <ref>, <ref>, and <ref>, respectively. Finally, in Section 3, by combining the a priori estimates and the continuation argument, we prove the existence, uniqueness and exponential decay rate of global-in-time solutions for the problem (<ref>)–(<ref>).Notations. Throughout this paper, we use ℐ:=(0,1). For1≤ q≤∞ and k∈ℕ, we denote by L^q(ℐ) the usual Lebesgue space on ℐ equipped with the norm ·_L^q(ℐ) and by H^k(ℐ) the standard Sobolev space in the L^2 sense equipped with the norm ·_H^k(ℐ).For notational simplicity, we shall use ·:=·_L^2(ℐ),·_k:=·_H^k(ℐ),·_L^q:=·_L^q(ℐ). If I⊂ℝ and X is a Banach space, thenwe denote by C(I; X) the space of continuous functions on I with values in X, by L^q(I; X) the space of L^q-functions on I with values in X, and by ·_L^q(I; X)the norm of the space L^q(I; X). To simplify the presentation, we employ C, c, and C_i (i∈ℕ) to denote various positive constants, depending only onΠ_0and V_0, whereΠ_0and V_0 are determined by (<ref>)–(<ref>).Hence C, c, and C_i are independent of γ-1 and t. The symbol A≲ B (or B≳ A) means that A≤ C B holds uniformly for some (γ-1)-independent constant C. We also use C_γ, c_γ, and C_i(γ) (i∈ℕ_+) to denote positive constants depending on γ. § A PRIORI ESTIMATES This section is devoted to deriving certain a priori estimates on the solutions (τ, u, v, w, θ)∈ X(0,T;M,N) to the initial boundary value problem (<ref>)–(<ref>) with temperature dependent transport coefficients (<ref>) for T>0, M≥ 1, andN≥ 1. Here we define the setX(t_1,t_2 ;M,N):={ (τ,u,v,w,θ)∈ C([t_1,t_2];H^3(ℐ)):τ_x∈ L^2(t_1,t_2;H^2(ℐ)),τ_t∈ C([t_1,t_2];H^2(ℐ))∩ L^2(t_1,t_2;H^2(ℐ)), (u_t,v_t,w_t,θ_t)∈ C([t_1,t_2];H^1(ℐ))∩ L^2(t_1,t_2;H^2(ℐ)),ℰ_t_1(t_2)≤ N^2, (u_x,v_x,w_x,θ_x)∈ L^2(t_1,t_2;H^3(ℐ)),τ(t,x)≥ M^-1 ∀ (t,x)∈[t_1,t_2 ]×ℐ},for constants M, N, t_1, and t_2 (t_1≤ t_2),whereℰ_t_1(t_2):=sup_t∈[t_1,t_2]{(u,√(c_v)(θ-1),√(c_v)θ_t)(t)_1^2 +θ_xx(t)^2} +∫_t_1^t_2√(c_v)θ_t(s)_1^2ds,withθ_t|_t=t_1 :=.1/c_v[-P(r^m u)_x+[κ r^2mθ_x/τ]_x+𝒬]|_t=t_1, θ_xt|_t=t_1 :=.1/c_v[-P(r^m u)_x+[κ r^2mθ_x/τ]_x+𝒬]_x|_t=t_1. We set without loss of generality that R=1 and hence c_v=1/(γ-1). For the sake of simplicity, we will use the following abbreviation:·:=·_L^∞([0,T];L^∞( ℐ)). Since (τ, u, v, w, θ)∈ X(0,T;M,N), it follows fromSobolev's inequality thatM^-1≤τ(t,x)≲ N,(θ-1,θ_t)(t)_1≲(γ-1)^1/2N∀ (t,x)∈[0,T]×ℐ, θ_x ≲ (γ-1)^1/4N, (θ-1,θ_t) ≲ (γ-1)^1/2N,∫_0^Tθ_t(t)_1^2dt≲ (γ-1)N^2. We shall make repeated use of the following estimate: a≤ r(t,x)≤ b∀ (t,x)∈[0,T]×ℐ, which follows from (<ref>) and (<ref>). §.§ Basic energy estimate In the following lemma we show the basic energy estimate and the pointwise bounds for θ.Assume that the conditions listed in Theorem <ref> hold.Then there exists a positive constant ϵ_1 depending only onΠ_0and V_0, such that if (γ-1)^1/4 M^2 N^2≤ϵ_1, then12≤θ(t,x)≤ 2∀ (t,x)∈[0,T]×ℐ,sup_t∈[0,T](√(ϕ(τ/τ̅)),u,v,w,√(c_v)(θ-1))(t)^2≲ 1,∫_0^T∫_ℐ[mτ u^2+τ v^2+τ_t^2+u_x^2+v_x^2+w_x^2+θ_x^2/τ]dxd t ≲ 1, where ϕ(z):=z-ln z -1 and τ̅ is given by (<ref>). The estimate (<ref>) follows immediately from (<ref>).In order to prove (<ref>) and (<ref>), we deduce an entropy-type energy estimate for the initial boundary value problem (<ref>)–(<ref>).Let θ̂>0 be an arbitrary but fixed constant.Multiplying (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>) by θ̂(τ̅^-1-τ^-1), u, v, w, and (1-θ̂θ^-1), respectively, we have η_θ̂(τ,u,v,w,θ)_t+θ̂κ r^2mθ_x^2/τθ^2+θ̂𝒬/θ=ℛ_x, whereη_θ̂(τ,u,v,w,θ):=θ̂ϕ(τ/τ̅)+12(u^2+v^2+w^2)+c_vθ̂ϕ(θ/θ̂) is the relative entropy (see, for instance, <cit.>), 𝒬 is given by (<ref>), and ℛ:=  ν r^m u(r^m u)_x/τ+[1-θ̂/θ]κ r^2mθ_x/τ+ r^m u[θ̂/τ̅-θ/τ]-2mμ r^m-1 u^2+μ r^2m (vv_x+ww_x)/τ-m μ r^m-1 v^2. Thanks to the boundary conditions (<ref>), we integrate (<ref>) over ℐ to obtaind/dt∫_ℐη_θ̂(τ,u,v,w,θ) +∫_ℐθ̂[κ r^2mθ_x^2/τθ^2 +𝒬/θ]=0.By virtue of (<ref>), we infer that, for σ>0, ν(r^m u)_x^2- 2mμτ(r^m-1 u^2)_x =ν r^2m u_x^2+(2mμ+ m^2 λ)τ^2 u^2/r^2+2m λ r^m-1τ u u_x =[mλ r^m u_x/√(σ)+ √(σ)τ u/r]^2+ [ν-m^2λ^2/σ]r^2mu_x^2+ [2mμ+m^2λ -σ]τ^2u^2/r^2≥[ν-m^2λ^2/σ]r^2mu_x^2 +[2mμ+m^2λ -σ]τ^2u^2/r^2.In light of (<ref>) and (<ref>), we can choose σ =m(mλ^2+2μ^2+(m+1)μλ)/ν>0 so thatν(r^m u)_x^2-2mμτ(r^m-1 u^2)_x/τθ≥β/τ[r^2mu_x^2+m τ^2 u^2/r^2],where β is given by β:=2μ+(m+1)λ/θmin{μν/mλ^2+2μ^2+(m+1)μλ,μ/ν}.Using (<ref>), (<ref>), (<ref>), and (<ref>), we integrate by parts to find2∫_ℐμ(θ)/θvv_x =-∫_0^T∫_ℐd/dθ[μ(θ)/θ]θ_xv^2 ≲ (γ-1)^1/4NM∫_ℐτ v^2, which combined with (<ref>) and (<ref>) implies∫_ℐμτ/θ[r^m v_x/τ-v/r^m]^2≳∫_ℐv_x^2/τ+(1-ϵ_1)∫_ℐτ v^2.Plug (<ref>) and (<ref>) into (<ref>),and use (<ref>) to derive that if (<ref>) holds for some sufficiently small ϵ_1>0, thend/dt∫_ℐη_θ̂(τ,u,v,w,θ) +c∫_ℐθ̂[mτ u^2+τ v^2+ u_x^2+v_x^2+w_x^2+θ_x^2/τ] ≤ 0. It follows from (<ref>), (<ref>), and (<ref>) that ϕ(θ)≳ (θ-1)^2 andτ_t^2/τ≲ mτ u^2+u_x^2/τ.If we take θ̂=1, by virtue of the conditions assumed in Theorem <ref>, we infer∫_ℐη_1(τ_0,u_0,v_0,w_0,θ_0)≲ 1.Integrating (<ref>) with θ̂=1 over (0,t) yields the estimates (<ref>) and (<ref>). §.§ Pointwise bounds for the specific volume In this subsection we employ the argument developed by Kanel' <cit.> to obtain the uniform bounds for the specific volume τ. To this end, we first make the estimate for τ_x/τin the following lemma. If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then sup_t∈[0,T]τ_x/τ(t)^2 +∫_0^T∫_ℐτ_x^2/τ^3≲ 1+lnτ.According to the chain rule, we have[ντ_x/τ]_t= [ντ_t/τ]_x+ ν'(θ)/τ(τ_xθ_t-τ_tθ_x), which combined with (<ref>) and (<ref>) implies[ντ_x/τ]_t-u_t/r^m-P_x=-v^2/r^m+1+2m uμ_x/r+ ν'(θ)/τ(τ_xθ_t-τ_tθ_x).Multiply (<ref>) by ντ_x/τ and use (<ref>), (<ref>) to deduce[1/2(ντ_x/τ)^2- u/r^mντ_x/τ]_t +νθτ_x^2/τ^3 +[u/r^mντ_t/τ]_x-(u/r^m)_x ντ_t/τ -ντ_xθ_x/τ^2 =ντ_x/τm u^2-v^2/r^m+1 +2m uμ_x/rντ_x/τ +ν'(θ)/τ^2(ντ_x-r^-muτ) (τ_xθ_t-τ_tθ_x).Integrating the last identity, we getd/dt∫_ℐ[1/2(ντ_x/τ)^2- u/r^mντ_x/τ] +∫_ℐνθτ_x^2/τ^3 =∑_q=1^7𝒦_q,where each term 𝒦_q in the decomposition will be defined and estimated below. First we consider the term𝒦_1:= ∫_ℐ(u/r^m)_xντ_t/τ.According to (<ref>) and (<ref>), we infer𝒦_1= ∫_ℐντ_t/τ(u_x/r^m-mτ u/r^2m+1)≲∫_ℐ[u_x^2/τ+ τ_t^2/τ+mτ u^2].The second term is defined and estimated as𝒦_2:=∫_ℐντ_xθ_x/τ^2≤ϵ∫_ℐτ_x^2/τ^3 +C(ϵ) ∫_ℐθ_x^2/τ.For the term 𝒦_3:= ∫_ℐντ_x/τm u^2-v^2/r^m+1,by virtue of the boundary conditions (<ref>), we integrate by parts and use (<ref>), (<ref>) to derive𝒦_3=-∫_ℐlnτ[ν/r^m+1(m u^2-v^2)]_x≲lnτ∫_ℐ|(mθ_x u^2,θ_x v^2, mτ u^2,τ v^2,mu u_x,vv_x)|.In view of (<ref>) and (<ref>), we have𝒦_3≲lnτ[1+ (γ-1)^1/4NM] ∫_ℐ[mτ u^2+τ v^2 +u_x^2+v_x^2/τ].The term 𝒦_4= ∫_ℐν'(θ)/τr^-mu τ_tθ_xcan be treated by using (<ref>) as𝒦_4 ≲u^1/2u_x^1/2τ_t/√(τ)θ_x/√(τ) ≲τ_t/√(τ)^2+ (γ-1)^1/2N^4M^2 u_x/√(τ)θ_x/√(τ)≲[1+ (γ-1)^1/2N^4M^2] ∫_ℐτ_t^2+u_x^2+θ_x^2/τ.We have from (<ref>) and (<ref>) that𝒦_5:=∫_ℐν'(θ)ν/τ^2θ_tτ_x^2≲Nθ_t∫_ℐτ_x^2/τ^3≲ϵ_1 ∫_ℐτ_x^2/τ^3.Using (<ref>), we get𝒦_6:= ∫_ℐτ_x/τθ_x (2m r^-1uμ'(θ)ν-νν'(θ)τ^-1τ_t) ≲ ϵ∫_ℐτ_x^2/τ^3 +C(ϵ)(γ-1)^1/2N^2M^2∫_ℐ[ mτ u^2+τ_t^2/τ].For the last term𝒦_7:=-∫_ℐν'(θ)/τ r^-mu τ_xθ_t ,we have from (<ref>) that𝒦_7 ≲ϵ∫_ℐτ_x^2/τ^3 +C(ϵ) u^2 Nθ_t^2 ≲ϵ∫_ℐτ_x^2/τ^3 +C(ϵ) N^3θ_t^2.Plugging (<ref>)–(<ref>) into (<ref>) and using (<ref>), (<ref>) yieldd/dt∫_ℐ[1/2(ντ_x/τ)^2- u/r^mντ_x/τ] +∫_ℐτ_x^2/τ^3≲ N^3θ_t^2+ [1+ lnτ] ∫_ℐ[mτ u^2+v^2+ τ_t^2+u_x^2+v_x^2+θ_x^2/τ].We integrate (<ref>) over (0,t), apply Cauchy's inequality, and use (<ref>), (<ref>), (<ref>) to conclude (<ref>). Now we establish the uniform bounds for the specific volume, which are essential for the proof of the main theorem. If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then C_1^-1≤τ(t,x)≤ C_1 for all (t,x)∈[0,T]×ℐ. DefineΦ(τ):=∫_1^τ/τ̅√(ϕ(z))/zdz,where τ̅ is defined by (<ref>). We infer that for suitably large constant C and τ≥ C,Φ(τ) ≳∫_C^τ/τ̅√(ϕ(z))/zdz≳∫_C^τ/τ̅ z^-1/2dz.Hence τ^1/2≲ 1+|Φ(v)|∀ τ∈(0,∞).Similarly, it follows that|lnτ|^3/2≲ 1+|Φ(τ)|∀ τ∈(0,∞).Thus, we haveτ^1/2+lnτ^3/2≲1+sup_(t,x)∈[0,T]×ℐ|Φ(τ(t,x))|.By virtue of (<ref>) and (<ref>), we obtain∫_ℐτ(t,x)dx=τ̅∀ t∈[0,T],which implies that for each t∈[0,T], there exists y(t)∈ℐ such that τ(t,y(t))=τ̅. Hence|Φ(τ)(t,x)|= |∫_y(t)^x∂/∂ xΦ(τ(t,y))dy|≤∫_ℐ√(ϕ(τ/τ̅(t,y)))|τ_x/τ(t,y)|dy ≤√(ϕ(τ/τ̅))(t)τ_x/τ(t),which combined with Lemmas <ref>–<ref>yields|Φ(τ)(t,x)| ≲ 1+lnτ^1/2∀ (t,x)∈[0,T]×ℐ.Combine (<ref>) with (<ref>) to deduce (<ref>). We plug (<ref>) into (<ref>), (<ref>), and (<ref>) to obtain the following corollary. If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then sup_t∈[0,T](τ,τ_x,u,v,w,√(c_v)(θ-1))(t)^2 +∫_0^T(τ_x,τ_t,√(m) u, u_x,v, v_x, w_x,θ_x )(t)^2d t ≤ C_2.§.§ Estimates on first-order derivatives This part is devoted to deducing the bounds for(u_x,v_x,w_x,√(c_v)θ_x)(t) uniformly in time t. We first make the estimate for w_x in the next lemma. If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then sup_t∈[0,T]w_x(t)^2 +∫_0^Tw_xx(t)^2dt≤ C_3.Multiply (<ref>) by w_xx, integrate the resulting identity over ℐ, and usew_t(t,0)=w_t(t,1)=0to findd/dtw_x(t)^2 +2∫_ℐμ r^2mw_xx^2/τ =-2∫_ℐw_xxr^mw_x[μ r^m/τ]_x -2m∫_ℐw_xxμr^m-1w_x.In view of (<ref>), (<ref>) and (<ref>), we have|[μ r^m/τ]_x|≲|(1,θ_x,τ_x)|.By virtue of (<ref>), (<ref>) and (<ref>),we havesup_t∈[0,T](θ-1,θ_t)(t)_1+(θ_x,θ_t)+ ∫_0^Tθ_t(t)_1^2dt≲ 1.Then using Cauchy's and Sobolev's inequalities yieldsd/dtw_x(t)^2 + w_xx(t)^2≲∫_ℐ w_x^2(1+θ_x^2+τ_x^2)≲[1+θ_x^2]w_x(t)^2 +sup_s∈[0,T]τ_x(s)^2w_x(t)_L^∞^2≲[1+C(ϵ)sup_s∈[0,T]τ_x(s)^4]w_x(t)^2+ϵw_xx(t)^2,which combined with (<ref>) impliesd/dtw_x(t)^2+ w_xx(t)^2≲w_x(t)^2.We conclude (<ref>) by integrating (<ref>) over (0,t).In the following lemma we obtain the bounds for u_x(t) and v_x(t) uniformly in time t. If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then sup_t∈[0,T](u_x,v_x)(t)^2 +∫_0^T(u_xx,v_xx)(t)^2dt≤ C_4. Multiply (<ref>) and (<ref>) by u_xx and v_xx, respectively, and then add the resulting identities to get (u_x^2+v_x^2)_t-2(u_xu_t+v_xv_t)_x +2r^2m/τ(ν u_xx^2+μ v_xx^2) -2u_xxr^m P_x  =2[-u_xxv^2/r+v_xxuv/r] -2u_xx r^m[(ν (r^m u)_x/τ)_x -ν r^m u_xx/τ]+4m r^m-1u μ_x u_xx -2v_xx[r^m v_x (μ r^m/τ)_x+2μ r^m-1 v_x-m(μ r^m-1 v)_x-μτ v /r^2m]. Integrating the last identity and using the boundary conditions u_t|_∂ℐ=v_t|_∂ℐ=0, we have d/dt(u_x,v_x)(t)^2+2∫_ℐr^2m/τ(ν u_xx^2+μ v_xx^2)=∑_q=8^12𝒦_q, where each term 𝒦_q in the decomposition will be defined and estimated below.First, we get 𝒦_8:=2∫_ℐ u_xxr^m P_x≤ϵu_xx(t)^2+C(ϵ) (θ_x,τ_x)(t)^2. For the term 𝒦_9:=2∫_ℐ[-u_xxv^2/r+v_xxuv/r], we have from Sobolev's inequality and (<ref>) that 𝒦_9 ≤ϵ(u_xx,v_xx)(t)^2+C(ϵ)|(u,v)(t)_L^∞^2v(t)^2≤ϵ(u_xx,v_xx)(t)^2+C(ϵ)(u_x,v_x)(t)v(t)^2.To estimate the term 𝒦_10:=-2∫_ℐu_xx r^m[(ν (r^m u)_x/τ)_x-ν r^m u_xx/τ],we first compute from (<ref>), (<ref>) and (<ref>) that|(ν (r^m u)_x/τ)_x-ν r^m u_xx/τ| ≲|(τ_x,θ_x)||(u_x,mu)|+|(u_x,muτ_x,mu)|.It follows from (<ref>), (<ref>), and (<ref>) that 𝒦_10 ≤ϵu_xx(t)^2+C(ϵ) ∫_ℐ[τ_x^2 u_x^2+(1+θ_x^2)|(u_x,muτ_x,mu)|^2]≤ϵu_xx(t)^2+C(ϵ) [1+θ_x^2][ (u_x,mu)(t)^2+τ_x(t)^2(u_x,mu)(t)_L^∞^2]≤ϵu_xx(t)^2+C(ϵ) [(u_x,mu)(t)^2+(u_x,mu)(t)(u_xx,u_x)(t)] ≤ 2ϵu_xx(t)^2+C(ϵ)(u_x,mu)(t)^2.The term𝒦_11 := 4m∫_ℐr^m-1u μ_x u_xxcan be easily estimated as𝒦_11≤ϵu_xx(t)^2+C(ϵ)θ_x^2 mu(t)^2 ≤ϵu_xx^2+C(ϵ)mu(t)^2.The last term is 𝒦_12:= -2∫_ℐv_xx[r^m v_x (μ r^m/τ)_x+2μ r^m-1 v_x -m(μ r^m-1 v)_x-μτ v /r^2m].Similar to the derivation of (<ref>), we can get𝒦_12 ≤ϵv_xx(t)^2 +C(ϵ)∫_ℐ[ v_x^2(1+θ_x^2+τ_x^2)+v^2]≤ 2ϵv_xx(t)^2+C(ϵ)(v_x,v)(t)^2.Plugging (<ref>)–(<ref>) into(<ref>) and taking ϵ sufficiently small yieldd/dt(u_x,v_x)(t)^2+( u_xx,v_xx)(t)^2 ≲(θ_x,τ_x,u_x,mu,v_x,v)(t)^2+(u_x,v_x)(t)v(t)^2.Integrating (<ref>) over (0,t), we deduce from (<ref>) that(u_x,v_x)(t)^2+∫_0^t( u_xx,v_xx)(s)^2ds≲ 1+sup_s∈[0,t](u_x,v_x)(s), from which we can conclude (<ref>). The following lemma is the estimate on √(c_v)θ_x(t).If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then sup_t∈[0,T]√(c_v)θ_x(t)^2 +∫_0^Tθ_xx(t)^2dt≤ C_5. Multiply (<ref>) by θ_xx, use the boundary condition for θ, and integrate the resulting identity to get d/dt√(c_v)θ_x(t)^2+2∫_ℐκ r^2mθ_xx^2/τ=∑_q=13^15𝒦_q, where the term 𝒦_q will be given below.For the term 𝒦_13:=∫_ℐ2θ_xx[θ(r^m u)_x/τ-ν(r^m u)_x^2/τ+2mμ(r^m-1 u^2)_x],it follows from (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) that𝒦_13 ≤ϵθ_xx(t)^2 +C(ϵ)∫_ℐ|(u_x,mu,1)|^2|(u_x,mu)|^2≤ϵθ_xx(t)^2 +C(ϵ)(u_x,mu,1)(t)_L^∞^2(u_x,mu)^2≤ϵθ_xx(t)^2 +C(ϵ)(u_x,mu,1)(t)^2(u_x,mu,u_xx)(t)^2 ≤ϵθ_xx(t)^2+C(ϵ)(u_x,mu,u_xx)(t)^2.Defining𝒦_14:=∫_ℐ2θ_xx[ -μ r^2m w_x^2/τ -μτ(r^m v_x/τ-v/r^m)^2], we have from (<ref>), (<ref>) and (<ref>) that𝒦_14 ≤ϵθ_xx^2 +C(ϵ)∫_ℐ|(w_x,v_x,v)|^4≤ϵθ_xx^2 +C(ϵ)(w_x,v_x,v)(t)_L^∞^2 (w_x,v_x,v)(t)^2≤ϵθ_xx^2+C(ϵ)(w_xx,v_xx,w_x,v_x,v)(t)^2The term𝒦_15:=∫_ℐ2 θ_xx[κ r^2mθ_xx/τ-(κ r^2mθ_x/τ)_x]can be bounded by using (<ref>) as𝒦_15 ≤ϵθ_xx(t)^2 +C(ϵ)∫_ℐθ_x^2[1+θ_x^2+τ_x^2]≤ϵθ_xx(t)^2+C(ϵ)[1+θ_x^2] (θ_x,τ_x)(t)^2≤ϵθ_xx(t)^2+C(ϵ)(θ_x,τ_x)(t)^2.We insert (<ref>)–(<ref>) into (<ref>) to obtaind/dt√(c_v)θ_x(t)^2+θ_xx(t)^2≲(mu,v,τ_x,u_x,v_x,w_x,θ_x,u_xx,v_xx,w_xx)(t)^2,which combined with (<ref>), (<ref>), (<ref>) and (<ref>) implies (<ref>). Using the system (<ref>), we can deduce the next lemma from Corollary <ref>, Lemmas <ref>–<ref>. If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, thensup_t∈[0,T]τ_t(t)^2 +∫_0^T(τ_xt,u_t,v_t,w_t,c_vθ_t)(t)^2dt≲ 1. We have from (<ref>), (<ref>), (<ref>), and (<ref>) that |τ_t|≲ |(mu,u_x)|,|τ_xt|≲ |(u_xx,u_x,mu,muτ_x)|, which combined with (<ref>) and (<ref>) yieldssup_t∈[0,T]τ_t(t)^2 +∫_0^Tτ_xt(t)^2dt≲ 1. By virtue of (<ref>), (<ref>) and (<ref>), we can deduce from (<ref>)that |c_vθ_t|≲|(mu,u_x,θ_xx,θ_x^2,τ_xθ_x,θ_x,mu^2,u_x^2,w_x^2,v_x^2,v^2)|, which combined with (<ref>), (<ref>), (<ref>), and (<ref>) implies c_vθ_t(t)^2 ≲(mu,u_x,θ_xx,θ_x,τ_x)(t)^2+∫_ℐ|(mu,u_x,w_x,v_x,v)|^4≲(mu,u_x,θ_xx,θ_x,τ_x)(t)^2+(mu,u_x,w_x,v_x,v)^2(mu,u_x,w_x,v_x,v)_1^2≲(mu,u_x,θ_xx,θ_x,τ_x,u_xx,w_x,w_xx,v_xx,v_x,v)(t)^2. Then it follows from Corollary <ref>, Lemmas <ref>–<ref> that∫_0^Tc_vθ_t(t)^2dt≲ 1. By virtue of (<ref>) and (<ref>), we have c_vθ_t(0)≤ C,for some (γ-1)-independent positive constant C. The other estimates in (<ref>) can be proved by a similar computation. §.§ Estimates on second-order derivatives In this subsection, we aim to derive the uniform bounds for (τ_xx,u_xx,v_xx,w_xx,θ_xx)(t). For this purpose, we make the estimates for (u_t,v_t,w_t) in the next lemma.If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then sup_t∈[0,T](u_t,v_t,w_t)(t)^2 +∫_0^T(u_xt,τ_tt,v_xt,w_xt)(t)^2dt≲ 1.The proof is divided into three steps. Step 1. We let ∂_t act on (<ref>), and multiply the resulting identity by u_t to yield [1/2u_t^2]_t+ν r^2mu_xt^2/τ+[u_tr^m P_t-u_t r^m (ν (r^m u)_x/τ)_t]_x=J_1+J_2+J_3, withJ_1 := -(r^m)_xu_t[ν(r^m u)_x/τ]_t, J_2 :=r^m u_xt[P_t-(ν/τ)_t(r^m u)_x-ν/τ((r^m u)_xt-r^m u_xt)],J_3 :=u_t[(v^2/r)_t-(r^m)_tP_x+(r^m)_xP_t+ (r^m)_t(ν (r^mu)_x/τ)_x-2m(r^m-1u μ_x)_t]. It followsfrom (<ref>), (<ref>) and (<ref>) that (u,v,w)≲ 1.To estimate J_1 and J_2, we first get from (<ref>) that|(r^m u)_xt-r^m u_xt|≲|((r^m)_txu,(r^m)_xu_t,(r^m)_tu_x)| ≲ |(mu^2,u_t,uu_x)|, |(r^m u)_xt|≲|(mu^2,u_t,uu_x,u_xt)|.Then we have from (<ref>) and (<ref>) thatJ_1≲|u_t| |(θ_tτ_t,τ_t^2,mu,u_t,u_x,u_xt)| ≲ϵ u_xt^2+C(ϵ) |(τ_t,τ_t^2,mu,u_t,u_x)|^2, andJ_2≲|u_xt| |(θ_t,τ_t,θ_tτ_t,τ_t^2,mu^2,u_t,uu_x)| ≲ϵ u_xt^2+C(ϵ) |(θ_t,τ_t,τ_t^2,mu,u_t,u_x)|^2.According to (<ref>), we have from (<ref>),(<ref>), (<ref>), and (<ref>) that |[ν (r^m u)_x/τ]_x|≲|(u_t,v^2,θ_t,τ_t,θ_x)|,andJ_3≲ |u_t||(v_t,v,θ_x,τ_x,θ_t,τ_t,u_t,θ_xt)|≲|(u_t,v_t,v,θ_x,τ_x,θ_t,τ_t,θ_xt)|^2. Integrating (<ref>) over (0,t)×ℐ, using (<ref>)–(<ref>), (<ref>)–(<ref>),(<ref>), and (<ref>) yield u_t(t)^2+∫_0^tu_xt^2≲ 1+∫_0^tθ_xt^2+ ∫_0^tτ_t_L^∞^2τ_t^2 ≲ 1+∫_0^tτ_t_1^2≲ 1. By virtue of (<ref>), τ_tt=(r^m u)_xt, and hence |τ_tt|≲ |u_xt|+|(u_t,uu_x)|+|mu||(u_x,1)|. Combining this with (<ref>),(<ref>), (<ref>),and (<ref>) implies∫_0^Tτ_tt(t)^2dt≲ 1. Step 2. Letting ∂_t act on (<ref>), andmultiplying the resulting identity by v_tyield [1/2v_t^2]_t+μ r^2mv_xt^2/τ -[v_t r^m (μ r^m v_x/τ)_t]_x =J_4+J_5, withJ_4 :=v_t[-(uv/r)_t+(r^m)_t(μ r^m v_x/τ)_x -(μτ v/r^2m)_t], J_5 := v_t [-(r^m)_x(μ r^m v_x/τ)_t+2(μ r^m-1v_x)_t-m(μ r^m-1v)_xt]-r^m v_xtv_x[μ r^m /τ]_t. By virtue of (<ref>), (<ref>) and (<ref>),we derive from (<ref>) that |[μ r^m v_x/τ]_x|≲|(v_t,v,θ_x,v_x)|,and J_4≲ |v_t||(u_t,v_t,v,v_x,θ_x,τ_t,θ_t)|.It follows from (<ref>) and (<ref>) that J_5 ≲|v_t||(v_xt,τ_t v_x,v_x,v_t,θ_xt,θ_x,θ_t,u_x,mu)|+|v_xtv_x||(θ_t,τ_t,u)|≲ϵ v_xt^2+C(ϵ)|(v_t, τ_t v_x,v_x,θ_xt,θ_x,θ_t,u_x,mu)|^2.Integrating (<ref>) over (0,t)×ℐ, using (<ref>)–(<ref>), (<ref>)–(<ref>), (<ref>), and (<ref>) yield v_t(t)^2+∫_0^tv_xt^2≲ 1+∫_0^tθ_xt^2+∫_0^tτ_t_L^∞^2v_x^2≲ 1+∫_0^tτ_t_1^2≲ 1. Step 3. Let ∂_t act on (<ref>), and multiply the resulting identity by w_t to get [1/2w_t^2]_t+μ r^2mw_xt^2/τ-[w_t r^m (μ r^m w_x/τ)_t]_x=J_6+J_7,withJ_6 :=w_t(r^m)_t[μ r^m w_x/τ]_x,J_7 : =-r^m w_x w_xt[μ r^m/τ]_t - w_t[ (r^m)_x(μ r^m w_x/τ)_t -m(μ r^m-1w_x)_t].According to (<ref>) and (<ref>), we inferJ_6 ≲ |w_t||(w_t,w_x)|≲ |(w_t,w_x)|^2,J_7 ≲|w_xt||(θ_tw_x,τ_tw_x,w_x,w_t)| +|w_t||(θ_tw_x,τ_tw_x,w_x)|≲ϵ w_xt^2+C(ϵ) |(w_t,θ_tw_x,w_x,τ_t w_x)|^2. Integrating (<ref>) over (0,t)×ℐ,using (<ref>)–(<ref>),(<ref>)–(<ref>), (<ref>), and (<ref>) yield w_t(t)^2+∫_0^tw_xt^2≲ 1+∫_0^tτ_x_L^∞^2w_x^2≲ 1+∫_0^tτ_t_1^2≲ 1.Combine the estimates (<ref>), (<ref>), (<ref>) and (<ref>) to derive(<ref>). Noting that the equations for (u,v,w) are parabolic, we deduce the uniformbounds for the L_x^2-norms of (u_xx,v_xx,w_xx) in the following lemma.If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then sup_t∈[0,T](u_xx,v_xx,w_xx,τ_xt)(t)≤ C_6.It follows from (<ref>), (<ref>), and (<ref>) that|(r^m)_x|≲ m, |(r^m)_xx|≲ |(τ_x,m)|, |(r^m)_xxx|≲ |(τ_xx,τ_x,m)|,and hence|(r^m u)_xx-r^m u_xx|≲ |(τ_x u,mu,u_x)|.According to (<ref>), we getν r^2m u_xx/τ = u_t-v^2/r+r^mP_x+2mr^m-1uμ_x -r^m[(ν(r^m u)_x/τ)_x- ν r^m u_xx/τ].Using (<ref>),(<ref>) and (<ref>), we obtain|u_xx|≲|(u_t,v,θ_x,τ_x)|+ |(θ_xτ_t,τ_xτ_t)|+ |(τ_x u,mu,u_x)|,which combined with Corollary <ref>, Lemmas <ref>–<ref> yieldssup_t∈[0,T]u_xx(t)≲ 1.Since the proof of the other estimates in (<ref>) can be shown in a similar way, we omit it.The following lemma concerns the bounds forthe L^2_x-norms of c_vθ_t and θ_xx. If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then sup_t∈[0,T](c_vθ_t,θ_xx)(t)^2 +∫_0^T√(c_v)θ_xt(t)^2dt≤ C_7.Let ∂_t act on (<ref>) and multiply the resulting identity with c_vθ_t to discover [(c_vθ_t)^2/2]_t+c_v κ r^2mθ_xt^2/τ-[c_vθ_t(κ r^2mθ_x/τ)_t]_x=c_vθ_t𝒬_t-c_vθ_t(P(r^m u)_x)_t-c_vθ_xθ_xt[κ r^2m/τ]_t.By virtue of the boundary conditions (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>), we integrate the last identity to getc_vθ_t(t)^2+∫_0^t√(c_v)θ_xt^2≲c_vθ_t(0)^2+∫_0^t(c_vθ_t,𝒬_t, (P(r^m u)_x)_t)^2 +∫_0^t∫_ℐc_vθ_x^2|(θ_t,τ_t,mu)|^2≲ 1+ ∫_0^t(𝒬_t,(P(r^m u)_x)_t)^2 +sup_s∈[0,T]√(c_v)θ_x(s)^2∫_0^t(θ_t,τ_t,mu)_1^2≲ 1+∫_0^t(𝒬_t, θ_tτ_t,τ_t^2,τ_tt)^2.Using (<ref>), (<ref>) and (<ref>), after some elementary calculations, we have|𝒬_t|≲ |(θ_t,τ_t)|τ_t^2+|τ_tτ_tt|+ |θ_t||(u_x,mu)|+|(u_x,mu,u_t,u_xt,u_xu_t)|+|w_xw_xt|+w_x^2|(θ_t,τ_t,mu)| +|(θ_t,τ_t)||(v_x,v)|^2 +|(v_x,v)||(v_xt,v_t,v_x,v_xτ_t,v)|,and𝒬_t^2≲ |(θ_t,τ_t,mu,w_x,v_x,v)|^6+τ_t^2τ_tt^2+w_x^2w_xt^2 +v_x^2v_xt^2+ |(θ_t,u_x,τ_t,u_t,v_x,v_t)|^4+ |(θ_t,u_x,mu,u_t,u_xt,v_xt,v_t,v_x,v)|^2.Employing Sobolev's inequality, (<ref>), Corollary <ref>, and Lemmas <ref>–<ref> yields∫_0^t∫_ℐ |(θ_t,τ_t,mu,w_x,v_x,v)|^6≲sup_[0,t](θ_t,τ_t,mu,w_x,v_x,v)^4 ∫_0^t(θ_t,τ_t,mu,w_x,v_x,v)_1^2 ≲ 1,∫_0^t∫_ℐ|(θ_t,θ_x,u_x,τ_t,u_t,v_x,v_t)|^4≲sup_[0,t](θ_t,θ_x,u_x,τ_t,u_t,v_x,v_t)^2 ∫_0^t(θ_t,θ_x,u_x,τ_t,u_t,v_x,v_t)_1^2 ≲ 1,and ∫_0^t∫_ℐ[τ_t^2τ_tt^2+w_x^2w_xt^2 +v_x^2v_xt^2]≲sup_[0,t](τ_t,w_x,v_x)_1^2∫_0^t(τ_tt,w_xt,v_xt)^2 ≲ 1.Combining (<ref>)–(<ref>) implies sup_t∈[0,T]c_vθ_t(t)^2 +∫_0^T√(c_v)θ_xt(t)^2dt≲ 1. By virtue of the equation (<ref>), we haveκ r^2mθ_xx/τ =c_vθ_t+Pτ_t -(κ r^2m/τ)_x θ_x-𝒬,which implies|θ_xx|≲ |(c_vθ_t,τ_t)|+|θ_x||(θ_x,τ_x,m)| +|(τ_t^2,u_x,mu,w_x^2,v_x^2,v^2)|.We use this estimate and (<ref>), (<ref>) to have sup_t∈[0,T]θ_xx(t)^2 ≲ 1.The estimate (<ref>) then follows from (<ref>) and (<ref>). In the following lemma we make an estimate for ((u_xxx,v_xxx,w_xxx,θ_xxx) in L^2(0,T;L^2(ℐ)).If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then ∫_0^T(u_xxx,v_xxx,w_xxx,θ_xxx/√(c_v))(t)^2dt≲ 1+∫_0^Tτ_xx(t)^2dt.Differentiating (<ref>) with respect to x, we have r^2mν u_xxx/τ+(r^m)_x[ντ_t/τ]_x+r^m[(ν/τ)_xτ_t]_x+r^m(ν/τ)_x(r^m u)_xx=u_xt-[v^2/r]_x+(r^m P_x)_x+2m(r^m-1u μ_x)_x-r^m ν/τ[(r^m u)_xxx-r^m u_xxx]. By a direct computation, applying (<ref>), (<ref>), (<ref>), and (<ref>) gives|u_xxx| ≲  |u_xt|+|(v_x,v,θ_x,τ_x,θ_xx,τ_xx,τ_x^2,u_x)| +|u||(τ_xx,τ_x,m)|+|u_x||(τ_x,1)|+|u_xx|+ |(τ_tx,θ_xτ_t,τ_xτ_t)| +|(θ_xxτ_t,τ_xxτ_t,τ_t,τ_x^2τ_t,θ_xτ_xt,τ_xτ_xt)| +|τ_tx||(θ_x,τ_x)|. Applying (<ref>), Corollary <ref> and Lemmas <ref>–<ref>, we infer∫_0^Tu_xxx^2 ≲ 1+∫_0^Tτ_xx^2 +∫_0^T∫_ℐ[τ_x^4+θ_xx^2τ_t^2+ τ_xx^2τ_t^2+τ_x^4τ_t^2+τ_x^2τ_xt^2].The last term on the right can be estimated by employing the Sobolev's inequality as∫_0^T∫_ℐ[θ_xx^2τ_t^2+ τ_xx^2τ_t^2+τ_x^4τ_t^2+τ_x^4+τ_x^2τ_xt^2]≲sup_t∈[0,T]τ_t_L^∞^2∫_0^T[(θ_xx,τ_xx)^2+τ_xxτ_x^3] +∫_0^Tτ_xτ_xx(τ_x,τ_tx)^2≲ 1+∫_0^Tτ_xx^2.Here we have used (<ref>), (<ref>) and (<ref>). Hence we get∫_0^Tu_xxx^2 ≲ 1+∫_0^Tτ_xx^2. The estimate for (v,w) in (<ref>) can be obtained similarly.We next show the estimate of θ_xxx in (<ref>). Differentiate (<ref>) with respect to x to getκ r^2mθ_xxx/τ =c_vθ_xt+(Pτ_t)_x- [κ r^2m/τ]_xxθ_x -2[κ r^2m/τ]_xθ_xx-𝒬_x,which combined with (<ref>) yields|θ_xxx|/√(c_v) ≲|√(c_v)θ_xt| +|(θ_xτ_t,τ_xτ_t,τ_xt)| +|(θ_xx,τ_xx,τ_x^2,θ_x,τ_x)| +|(θ_xx,τ_xθ_xx,𝒬_x)|, |√(c_v)θ_xt| ≲|θ_xxx|/√(c_v) +|(θ_xτ_t,τ_xτ_t,τ_xt)| +|(θ_xx,τ_xx,τ_x^2,θ_x,τ_x)| +|(θ_xx,τ_xθ_xx,𝒬_x)|.Applying (<ref>), Corollary <ref> and Lemmas <ref>–<ref>, we infer∫_0^Tθ_xxx/√(c_v)^2≲ 1+∫_0^Tτ_xx^2 +∫_0^T∫_ℐ[θ_xx^2τ_x^2+𝒬_x^2]≲ 1+∫_0^Tτ_xx^2 +sup_[0,T]θ_xx^2∫_0^Tτ_xxτ_x +∫_0^T𝒬_x^2≲ 1+∫_0^Tτ_xx^2 +∫_0^T𝒬_x^2.To conclude (<ref>), it remains to prove∫_0^T𝒬_x^2≲ 1.According to the definition of 𝒬, we can compute that|𝒬_x|≲  τ_t^2|(θ_x,τ_x)|+|τ_tτ_xt|+ |θ_x||(mu,u_x)|+|(mu,τ_x,u_x,u_xx,u_x^2)|+w_x^2|(θ_x,τ_x,1)|+|w_xw_xx|+ |(θ_x,τ_x)||(v_x,v)|^2+|(v_x,v)||(v_xx,v_x)|,which yields𝒬_x^2≲ |(θ_x,τ_x,τ_t,w_x,v,v_x)|^6 +|(θ_x,mu,u_x,w_x,v,v_x)|^4+|(mu,τ_x,u_x)|^2+τ_t^2τ_xt^2+w_x^2w_xx^2+v_x^2v_xx^2+u_xx^2+v_xx^2.Applying (<ref>)–(<ref>),Corollary <ref>, and Lemmas <ref>–<ref>, we can deduce (<ref>) and therefore complete the proof of this lemma.If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then sup_t∈[0,T]τ_xx(t)^2 +∫_0^T(τ_xx,u_xxx,τ_xxt,v_xxx,w_xxx,θ_xxx/√(c_v))(t)^2 dt≤ C_8. Let ∂_x act on (<ref>) and multiply the resulting identityby (ντ_x/τ)_x to find[1/2(ντ_x/τ)_x^2]_t +θ/ντ[ντ_x/τ]_x^2 -[ντ_x/τ]_x [ν'(θ)/τ(τ_xθ_t-τ_tθ_x)]_x=-[ντ_x/τ]_x [-θ_xx/τ-2θτ_x^2/τ^3 +2θ_xτ_x/τ^2-θτ_x/ντ(ν/τ)_x] +[ντ_x/τ]_x[ u_t/r^m-v^2/r^m+1 +2m uμ_x/r]_x.Integrating the last identity and using Cauchy's inequality yield(ντ_x/τ)_x(t)^2 +∫_0^t(ντ_x/τ)_x^2≲.(ντ_x/τ)_x|_t=0^2+∫_0^t∫_ℐ[|(θ_xx,τ_x^2,θ_xτ_x)|^2+ |(u_xt,u_t,v_x,v,u_xθ_x,θ_xx,θ_x)|^2] + ∫_0^t∫_ℐ[|(τ_xθ_tx,τ_tθ_xx)|^2+ |(τ_xxθ_t,τ_txθ_x)|^2+ |(τ_xθ_t,τ_tθ_x)|^2 |(τ_x,θ_x)|^2].We only show the estimates for the last three terms on the right-hand side of (<ref>), since the other terms can be easily treated by using Corollary <ref> and Lemmas <ref>–<ref>. The term∫_0^t∫_ℐ|(τ_xθ_t,τ_tθ_x)|^2 |(τ_x,θ_x)|^2can be estimated by using (<ref>) and (<ref>) as∫_0^t∫_ℐ|(τ_xθ_t,τ_tθ_x)|^2 |(τ_x,θ_x)|^2 ≲∫_0^t∫_ℐ |(τ_x,τ_t,θ_x)|^4 ≲ϵ∫_0^tτ_xx^2+C(ϵ).It follows from (<ref>), (<ref>), (<ref>), and (<ref>) that∫_0^t∫_ℐ|(τ_xθ_tx,τ_tθ_xx)|^2 ≲∫_0^t(τ_x,τ_t)(τ_xx,τ_xt)(θ_xt,θ_xx)^2≲ϵ∫_0^t(τ_xx,τ_xt)^2+C(ϵ) ∫_0^t(θ_xt,θ_xx)^2 ≲ϵ∫_0^tτ_xx^2+C(ϵ).In view of (<ref>) and (<ref>), we have∫_0^t∫_ℐ|(τ_xxθ_t,τ_txθ_x)|^2≲ (θ_t,θ_x)^2 ∫_0^t(τ_xx,τ_xt)^2≲ 1+ (γ-1)^1/2N^2∫_0^tτ_xx^2.Thanks to (<ref>)–(<ref>) and (<ref>), we derive(ντ_x/τ)_x(t)^2 +∫_0^t(ντ_x/τ)_x^2 ≲ C(ϵ) +(ϵ+ ϵ_1^2)∫_0^tτ_xx^2.Noting that|τ_xx| ≲|(ντ_x/τ)_x| +|(θ_xτ_x,τ_x^2)|, taking ϵ and ϵ_1 sufficiently small, we getτ_xx(t)^2+∫_0^tτ_xx^2≲ 1+∫_ℐτ_x^2(θ_x^2+τ_x^2)+ ∫_0^t∫_ℐτ_x^2(θ_x^2+τ_x^2)≲ 1+sup_s∈[0,t]τ_x(s)^2_L^∞≲ 1+sup_s∈[0,t]τ_xx(s).Apply Cauchy's inequality to (<ref>) to getsup_t∈[0,T]τ_xx(t)^2 +∫_0^Tτ_xx^2≲ 1,which combined with (<ref>) gives (<ref>).§.§ Estimates on the third-order derivatives This subsection is devoted to deriving the estimates for (τ_xxx,u_xxx,v_xxx,w_xxx,θ_xxx). To do this, we first make the estimates for (u_xt,v_xt,w_xt) in the following lemma.If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then sup_t∈[0,T](u_xt,v_xt,w_xt)(t)^2 +∫_0^T(u_xxt,v_xxt,w_xxt)(t)^2dt≲ 1,sup_t∈[0,T](u_xxx,τ_tt,τ_xxt,v_xxx,w_xxx)(t)≤ C_9. Letting ∂_t act on (<ref>) and multiplying the resulting identity by u_xxt yields [1/2u_xt^2]_t-(u_xtu_tt)_x+ν r^2mu_xxt^2/τ=u_xxtR_u withR_u:=  -(v^2/r)_t+(r^mP_x)_t+2m(r^m-1uμ_x)_t-(r^m)_t[ν (r^m u)_x/τ]_x-r^m[(ν/τ)_x(r^mu)_x]_t-r^m(ν/τ)_t(r^mu)_xx-r^mν/τ[(r^mu)_xxt-r^m u_xxt]. Thanks to (<ref>), we integrate (<ref>) and use Cauchy's inequality to have u_xt(t)^2+∫_0^tu_xxt^2≲u_xt|_t=0^2+∫_0^t∫_ℐR_u^2. By virtue of the chain rule, (<ref>), and (<ref>), for a general smooth function f(τ,θ), we have f(τ,θ)_xt≲  |(τ_xt,θ_xt)|+|(τ_x,τ_t,θ_x,θ_t)|^2, |(r^mu)_xxt-r^m u_xxt| ≲  |((r^m)_xxtu,(r^m)_xtu_x,(r^m)_xxu_t,(r^m)_xu_xt, (r^m)_tu_xx)| ≲  |(u_xx,u_x,mu,τ_x)| +|(u_x^2,u_x)|+|(τ_x u_t,u_t)|+|(u_xt,u_xx)|, and |R_u| ≲ |(v_t,v)|+|(θ_x,τ_x)|+|(τ_xt,θ_xt)|+|(τ_x,τ_t,θ_x,θ_t)|^2+|u_tθ_x|+ |(u_t,v,θ_x,τ_x)|+|(θ_x,τ_x)τ_tt|+|(τ_xt,θ_xt)||τ_t|+|(τ_x,τ_t,θ_x,θ_t)|^2|τ_t|+|(θ_t,τ_t)τ_xt|+|(u_xx,u_x,mu,τ_x,u_x^2,τ_x u_t,u_t,u_xt)|. Applying Sobolev's inequality, we get from(<ref>), (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>) that (τ_t,τ_x,u_x,v_x,w_x)≲ 1.Using (<ref>), (<ref>), Corollary <ref>, and Lemmas <ref>–<ref>, we derive∫_0^t∫_ℐR_u^2≲ 1.Insert the last estimate into (<ref>) to getu_xt(t)^2+∫_0^tu_xxt^2≲ 1.The estimates for v and w in (<ref>) can be obtained in a similar way.We next show the estimate for u_xxx in (<ref>).In light of (<ref>), we have u_xxx(t)^2≲ 1+θ_xxτ_t^2+τ_xxτ_t^2+τ_txτ_x^2≲1.The other estimates in (<ref>) can be proved similarly by using the system (<ref>). The proof of this lemma is completed. In the following lemma, we deduce the L^∞(0,T;L^2(ℐ))-norm for √(c_v)θ_xt, and we can have the bound for θ_xxx/√(c_v) in L^∞(0,T;L^2(ℐ)) due to the equation (<ref>).If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then sup_t∈[0,T](√(c_v)θ_xt,θ_xxx/√(c_v))(t)^2 +∫_0^Tθ_xxt(t)^2dt≤ C_10. Let ∂_t act on (<ref>) and multiply the resulting identity by θ_xxt to get [c_v/2θ_xt^2]_t-(c_vθ_ttθ_xt)_x+κ r^2mθ_xxt^2/τ= θ_xxtR_θ, withR_θ:=[P(r^m u)_x]_t-[(κ r^2mθ_x/τ)_xt-κ r^2mθ_xxt/τ]-𝒬_t. Integrating the last identity and using the boundary condition for θ yield √(c_v)θ_xt(t)^2+∫_0^tθ_xxt(s)^2ds≲√(c_v)θ_xt|_t=0^2+∫_0^t∫_ℐR_θ^2.By virtue of (<ref>) and (<ref>), we have√(c_v)θ_xt|_t=0^2≲ 1.To estimate the last term on the right, we compute|(P(r^m u)_x)_t|≲ |τ_t(θ_t,τ_t)|+|τ_tt|,|𝒬_t|≲ |(θ_t,τ_t)τ_t^2|+|τ_tτ_tt|+ |θ_t(r^m-1u^2)_x|+|(r^m-1u^2)_xt|+|w_xw_xt|+ |(μ r^2m/τ)_tw_x^2| +|(θ_t,τ_t)||(v_x,v)|^2+|(v_x,v)||(v_xt,v_x,v_t,v)|,and |[(κ r^2mθ_x/τ)_xt-κ r^2mθ_xxt/τ]| ≲|(θ_x,τ_x,1)θ_xt|+ |(θ_t,τ_t,1)θ_xx|+|θ_x||(mu,u_x)| +|θ_x|[|(θ_x,τ_x)|+|(θ_t,τ_t)|+|(θ_xt,τ_xt)|+|(θ_x,θ_t,τ_x,τ_t)|^2]. Plug (<ref>)–(<ref>) into (<ref>) and use (<ref>), (<ref>), (<ref>), Corollary <ref>,and Lemmas <ref>–<ref> to conclude sup_t∈[0,T]√(c_v)θ_xt(t) ^2 +∫_0^Tθ_xxt(t)^2dt≲ 1.The estimate (<ref>) follows by using (<ref>), (<ref>) and (<ref>). By using the system (<ref>), we can get the following estimates for (u_xxxx,v_xxxx,w_xxxx,θ_xxxx). The proof is similar to that of Lemma <ref> and hence we omit the details for brevity.If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then ∫_0^T(u_xxxx,v_xxxx,w_xxxx,θ_xxxx/c_v)(t)^2dt≲ 1+∫_0^Tτ_xxx(t)^2dt.The following lemma is to establish the bound for L_x^2-norm of τ_xx(t,x). We note that, unlike the above estimates, the bound in (<ref>) depends on γ-1. If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, thensup_t∈[0,T]τ_xxx(t)^2 + ∫_0^T(τ_xxx,u_xxxx,v_xxxx,w_xxxx, θ_xxxx/√(c_v))(t)^2 dt≲ c_v.Let ∂_x^2 act on (<ref>) and multiply the resulting identity by (ντ_x/τ)_xx to get [1/2(ντ_x/τ)_xx^2]_t +θ/ντ[ντ_x/τ]_xx^2 =[ντ_x/τ]_xxR_τ,withR_τ=[u_t/r^m-v^2/r^m+1 +2m uμ_x/r+ν'(θ)/τ(τ_xθ_t-τ_tθ_x)+(θ/τ)_x]_xx+θ/ντ(ντ_x/τ)_xx.Integrating (<ref>) and using Cauchy's inequality yield(ντ_x/τ)_xx(t)^2 +∫_0^t(ντ_x/τ)_xx(s)^2ds ≲ 1+∫_0^t∫_ℐR_τ^2. From (<ref>) and (<ref>), we have(τ_xt,u_xx,v_xx,w_xx)≲ 1.Using (<ref>) and (<ref>),after a direct calculation, we haveR_τ^2≲ |(u_t,u_xt,u_xxt)|^2+|(v,v_x,v_xx)|^2+|(θ_x,θ_xx,θ_xxx)|^2+|(τ_x,θ_x,τ_xx,θ_xx,θ_xxx)|^2 +|(τ_xxxθ_t,τ_xxt,τ_xxθ_xt,θ_xxt)|^2.Plug (<ref>) into (<ref>) and useCorollary <ref>, Lemmas <ref>–<ref> to get(ντ_x/τ)_xx(t)^2 +∫_0^t(ντ_x/τ)_xx(s)^2ds≲.(ντ_x/τ)_xx|_t=0^2 +∫_0^tθ_xxx^2+ θ_t^2∫_0^tτ_xxx^2+ ∫_0^t∫_ℐτ_xx^2θ_xt^2.By virtue of (<ref>), (<ref>) and (<ref>), we have(ντ_x/τ)_xx(t)^2 +∫_0^t(ντ_x/τ)_xx(s)^2ds ≲ c_v+ ϵ_1[∫_0^tτ_xxx^2+sup_s∈[0,t]τ_xx(s)^2].Employ (<ref>) and (<ref>) to have|ντ_xxx/τ| ≲|(ντ_x/τ)_xx|+ |(τ_xx,θ_xx,τ_x,θ_x)|,which combined with (<ref>) and (<ref>) implies (<ref>) if ϵ_1 is sufficiently small. § PROOF OF THEOREM <REF> In this section we complete the proof of Theorem <ref> by combining the a priori bounds obtained in Section <ref>, the continuation argument and Poincaré's inequality.We divide this section into two parts.The first is devoted to proving the existence and uniqueness of global solutions to the problem (<ref>)–(<ref>), and the second oneaims at showing the convergence decay rate of the solutions toward constant states. §.§ Global solvabilityWe first present the local solvability result to the initial boundary value problem (<ref>)–(<ref>) in the following proposition, which can be proved by the standard iteration method (see, for instance, <cit.>).Let the initial data (τ_0,u_0,v_0,w_0,θ_0)∈ H^3(ℐ) satisfy thatτ_0(x)≥λ_1^-1,θ_0(x)≥λ_2^-1∀ x∈ℐ,u_0_1+√(c_v)(θ_0-1,θ_t|_t=0)_1+θ_0xx≤Λ,(τ_0,u_0,v_0,w_0)_3+√(c_v)θ_0-1_1+(θ_0xx,√(γ-1)θ_0xxx)≤Π,for some positive constantsλ_1, λ_2, Λ, and Π, where θ_t|_t=0 and θ_xt|_t=0 are defined by (<ref>) and (<ref>), respectively. Then there exists a positive constant T_0=T_0(λ_1,λ_2,Λ,Π), which depends only on λ_1, λ_2, Λ, and Π, such that the initial boundary value problem (<ref>)–(<ref>) has a unique solution (τ, u, v, w, θ) ∈ X(0,T_0;2λ_1,2Λ). According to (<ref>) and (<ref>), there exists a (γ-1)-independent positive constant C_0, such that τ_0(x)≥ V_0^-1,θ_0(x)≥ V_0^-1∀ x∈ℐ,u_0_1+√(c_v)(θ_0-1,θ_t|_t=0)_1+θ_0xx≤ C_0,(τ_0,u_0,v_0,w_0)_3+√(c_v)θ_0-1_1+(θ_0xx,√(γ-1)θ_0xxx)≤Π_0.Let ϵ_1, C_j (j=1,⋯,10) be chosen in Section <ref>. We assume that γ-1≤ϵ_0 withϵ_0:=min{δ_1,δ_2},whereδ_1:=[ϵ_1/(2C_0)^2(2V_0)^2]^4,δ_2:=[ϵ_1/5C_11(2C_1)^2 ]^4, C_11:=∑_j=1^10C_j. ApplyingProposition <ref>, we can find a positive constant t_1=T_0(V_0,V_0,C_0,Π_0) such that there exists a unique solution(τ, u, v, w, θ) ∈ X(0,t_1;2V_0,2C_0) to the initial boundary value problem (<ref>)–(<ref>). Since γ-1≤δ_1, we can applyCorollary <ref> and Lemmas <ref>–<ref> with T=t_1 to deduce that there exists a positive constant C_1(γ), depending only on V_0, Π_0, and γ, such that for each t∈[0,t_1],the local solution(τ, u, v, w, θ) satisfies ℰ_0(t)≤ C_11,τ(t,x)≥ C_1^-1,θ(t,x)≥12∀ x∈ℐ,(τ,u,v,w)(t)_3+√(c_v)(θ_0-1)(t)_1+(θ_xx,√(γ-1)θ_xxx)(t)≤ C_1(γ). If we take(τ, u, v, w, θ) (t_1,·) as the initial data and apply Proposition <ref> again, we can extend the local solution(τ, u, v, w, θ) to the time interval [0,t_1+t_2] with t_2=T_0(C_1,2,C_11,C_1(γ)) such that (τ, u, v, w, θ)∈ X(t_1,t_1+t_2;2C_1,2√(C_11)). Hence the local solution (τ, u, v, w, θ)∈ X(0,t_1+t_2;2C_1,√(5C_11)).Noting that γ-1≤δ_2, we can applyCorollary <ref> and Lemmas <ref>–<ref> with T=t_1+t_2 to deduce that (<ref>) and (<ref>) hold for each t∈[0,t_1+t_2].Repeating the above procedure, we can then extend the solution (τ, u, v, w, θ) step by step to a global one provided thatγ-1≤ϵ_0 with ϵ_0 given by (<ref>). Furthermore,(τ,u,v,w,θ)(t)_3^2+∫_0^∞[(u_x,v_x,w_x,θ_x)_3^2+τ_x_2^2]dt≤ C_2(γ) ∀ t∈[0,∞),where C_2(γ) is some positive constant depending on γ, Π_0, and V_0.§.§ Convergence decay rateIt follows from (<ref>) and Poincaré's inequality that(τ-τ̅)(t)^2= ∫_ℐ|τ(t,x)-∫_ℐτ(t,y)dy|^2dx≲τ_x(t)^2.Thanks to the boundary conditions for (u,v,w), we apply Poincaré's inequality to get(u,v,w)(t)≲(u_x,v_x,w_x)(t).Multiplying (<ref>), (<ref>), (<ref>) by u, v, w, respectively,adding them with (<ref>), and integrating the resulting identity over [0,t]×ℐ yield∫_ℐ[c_vθ+1/2(u^2+v^2+w^2)](t,x)dx=c_vθ̅∀ t∈[0,∞),where θ̅ is given by (<ref>). We use Poincaré's inequality and(<ref>) to obtain(θ-θ̅)(t)^2≤∫_ℐ|θ(t,x)-∫_ℐθ(t,y)dy|^2dx+(u,v,w)(t)^2 ≲(u_x,v_x,w_x,θ_x)(t)^2.We have from(<ref>) that for all (t,x)∈[0,∞)×ℐ,[r^m+1(t,x)]_x=(m+1)τ(t,x), r^m+1(t,x)=a^m+1+(m+1)∫_0^xτ(t,y)dy.Then |r^m+1(t,x)-r̅^m+1| = (m+1)|∫_0^x(τ(t,y)-τ̅)dy|, [r^m+1(t,x)-r̅^m+1]_x =(m+1)[τ(t,x)-τ̅],which combined with (<ref>) implyr(t)-r̅_2≲r^m+1(t)-r̅^m+1_2 ≲τ(t)-τ̅_1. To derive the exponential stability of the solution (τ,u,v,w,θ), we define the following energy functionals:ℋ_η(t) :=∫_ℐη_θ̅(τ,u,v,w,θ)(t,x)dx, ℋ_τ(t) :=∫_ℐτ_x^2(t,x)dx,ℋ_θ(t):=∫_ℐθ_x^2(t,x)dx, ℋ_U(t) :=∫_ℐ[u_x^2+v_x^2+w_x^2](t,x)dx, ℋ_0(t) :=∫_ℐ[1/2(ντ_x/τ)^2- u/r^mντ_x/τ](t,x)dx,where η_θ̅(τ,u,v,w,θ) is defined by (<ref>).Then we have from (<ref>) thatℋ_τ(t)-Cu(t)^2≲ℋ_0(t)≲ℋ_τ(t)+u(t)^2 ≲ℋ_τ(t)+ ℋ_U(t). It follows from (<ref>), (<ref>), and (<ref>) thatd/dtℋ_U(t)+c(u_xx,v_xx,w_xx)(t)^2≲ℋ_τ(t)+ℋ_U(t)+ℋ_θ(t).In light of (<ref>) and (<ref>), we havec_vd/dtℋ_θ(t)+cθ_xx(t)^2≲ℋ_τ(t)+ℋ_U(t)+ℋ_θ(t)+(u_xx,v_xx,w_xx)(t)^2.Multiplying (<ref>) by a sufficiently small σ_1>0 and adding this with (<ref>) yieldd/dt[ℋ_U(t)+c_vσ_1ℋ_θ(t)]+cσ_1(u_xx,v_xx,w_xx,θ_xx)(t)^2 ≲ℋ_τ(t)+ℋ_U(t)+ℋ_θ(t).By virtue of (<ref>), (<ref>), and (<ref>),we getd/dtℋ_0(t)+cℋ_τ(t) ≲  [1+(γ-1)^2][ℋ_U(t)+ℋ_θ(t)]+(γ-1)^2[ ℋ_τ(t)+(u_xx,v_xx,w_xx,θ_xx)(t)^2].Since c_v^-1=|γ-1|≤ϵ_0, we multiply (<ref>) by a sufficiently small σ_2 and add this into (<ref>) to findd/dt[ℋ_0(t)+σ_2ℋ_U(t)+c_vσ_2σ_1ℋ_θ(t)]+cℋ_τ(t)+cσ_2σ_1(u_xx,v_xx,w_xx,θ_xx)(t)^2 ≲[1+ϵ_0^2][ℋ_U(t)+ℋ_θ(t)]+ϵ_0^2[ ℋ_τ(t)+(u_xx,v_xx,w_xx,θ_xx)(t)^2].Take ϵ_0>0 suitably small to inferd/dt[ℋ_0(t)+σ_2ℋ_U(t)+c_vσ_2σ_1ℋ_θ(t)]+cℋ_τ(t) ≲ℋ_U(t)+ℋ_θ(t).It follows from (<ref>) and thatd/dtℋ_η(t)+c[ℋ_U(t)+ℋ_θ(t)]≤ 0.Multiplying (<ref>) by a sufficiently small σ_3 and adding this into (<ref>) implyd/dtℋ(t)+c[ℋ_τ(t)+ℋ_U(t)+ℋ_θ(t)] ≤ 0,where ℋ(t):=ℋ_η(t)+σ_3ℋ_0(t)+σ_3σ_2ℋ_U(t)+c_vσ_3σ_2σ_1ℋ_θ(t).Using (<ref>), (<ref>), (<ref>), and (<ref>), we have[1/2-σ_3C]u(t)^2+cσ_3[ℋ_τ(t)+ℋ_U(t)+ℋ_θ(t)]≤ℋ(t)≲ℋ_τ(t)+ℋ_U(t)+ℋ_θ(t).Take σ_3>0 with Cσ_3≤ 1/4 so thatcσ_3[ℋ_τ(t)+ℋ_U(t)+ℋ_θ(t)]≤ℋ(t)≲ℋ_τ(t)+ℋ_U(t)+ℋ_θ(t), which combined with (<ref>) yieldsℋ_τ(t)+ℋ_U(t)+ℋ_θ(t)≲ℋ(t)≲e^-ct.We combine (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>) to conclude the exponential decay estimate (<ref>). Therefore, the proof of Theorem <ref> has been completed. Acknowledgement Research of the authors was supported by the Fundamental Research Funds for the Central Universities, the Project funded by China Postdoctoral Science Foundation, and the grants from National Natural Science Foundation of China under contracts 11601398 and 11671309. The authors express much gratitude to Huijiang Zhao for his support and advice. Tao Wang would like to warmly thank Paolo Secchi, Alessandro Morando, and Paola Trebeschi for support and hospitality during his postdoctoral stay at University of Brescia.abbrvnat

http://arxiv.org/abs/1702.07896v1

Trace of the Twisted Heisenberg Category Michael Reeks December 30, 2023 ======================================== We consider the distribution of free path lengths, or the distance between consecutive bounces of random particles, in an n-dimensional rectangular box.If each particle travels a distance R, then, as R →∞ the free path lengths coincides with the distribution of the length of the intersection of a random line with the box (for a natural ensemble of random lines) and we give an explicit formula (piecewise real analytic) for the probability density function in dimension two and three.In dimension two we also consider a closely related model where each particle is allowed to bounce N times, as N →∞, and give an explicit (again piecewise real analytic) formula for its probability density function.Further, in both models we can recover the side lengths of the box from the location of the discontinuities of the probability density functions. § INTRODUCTIONWe consider billiard dynamics on a rectangular domain, i.e., point shaped “balls” moving with linear motion with specular reflections at the boundary, and similarly for rectangular box shaped domains in three dimensions.We wish to determine the distribution of free path lengths of ensembles of trajectories defined by selecting a starting point and direction at random. The question seems quite natural and interesting on its own, but we mention that it originated from the study of electromagnetic fields in “reverberation chambers” under the assumption of highly directional antennas <cit.>.Briefly, the connection is as follows (we refer to the forthcoming paper <cit.> for more details): given an ideal highly directional antenna and a highly transient signal, then the wave pulse dynamics is essentially the same as a point shaped billiard ball traveling inside a chamber, with specular reflection at the boundary.Signal loss is dominated by (linear) “spreading” of the electromagnetic field and by absorption occurring at each interaction (“bounce”) with the walls. The first simple model we use in this paper neglects absorption effects, and models signal loss from spreading by simply terminating the motion of the ball after it has travelled a certain large distance.The second model only takes into account signal loss from absorption, and completely neglects spreading; here the motion is terminated after the ball has bounced a certain number of times.We remark that the distribution of free path lengths is very well studied in the context of the Lorentz gas — here a point particle interacts with hard spherical obstacles, either placed randomly, or regularly on Euclidean lattices; recently quasicrystal configurations have also been studied (cf. <cit.>.) Let R>0 be large and let a rectangular n-dimensional box K⊆^n be given, where n≥ 2. We send off a large number M>0 of particles, each with a random initial position p^(i)∈ K chosen with respect to a given probability measure μ on K, and each with a uniformly random initial direction v^(i)∈𝕊^n-1={x∈^n:x=1}, i=1,…,M, for a total distance R each.Each particle travels along straight lines, changing direction precisely when it hits the boundary of the box, where it reflects specularly. We record the distancetravelled between each pair of consecutive bounces for each particle. (Note in particular that we obtain more bounce lengths from some particles than from others.)Let X_M,R be the uniformly distributed random variable on this finite set of bounce lengths of all the particles. More precisely, a random sample of X_M,R is obtained as follows: first take a random i.i.d.sample of points (with respect to the measure μ) p^(1), …, p^(M)∈ K, and a random sample of directions v^(1),…,v^(M)∈𝕊^n-1 (with respect to the uniform measure).Each pair (p^(i),v^(i)) then defines a trajectory T^i of length R, and each such trajectory gives rise to a finite multiset B^i of lengths between consecutive bounces.Finally, with B = _i=1^M B^i denoting the (multiset) union of bounce length multisets B^1,…,B^M, we select an element of B with the uniform distribution.(That is, with 1_B denoting the integer valued set indicator function for B, and B' = { x : 1_B(x) ≥ 1 } we select the element b ∈ B' with probability 1_B(b)/∑_x ∈ B' 1_B(x).) We are interested in the distribution of X_M,R for large M and R, and this turns out to be closely related to a model arising from integral geometry. Namely, let ℓ denote the unique (up to a constant) translation- and rotation-invariant measure on the set of directed lines ℓ in ^n, and consider the restriction of this measure to the set of directed lines ℓ intersecting K, normalized such that it becomes a probability measure. Denote by X the random variable Xlength(ℓ∩ K) where ℓ is chosen at random using this measure.For any dimension n≥ 2, and for any distribution μ on the starting points, the random variable X_M,R converges in distribution to the random variable X, as we take R→∞ followed by taking M→∞, or vice versa. The mean free path length has a quite simple geometric interpretation. We haveX = 2π|𝕊^n-1||𝕊^n|Vol(K)Area(K)= 2√(π)·Γ(n+1/2)Γ(n/2)Vol(K)Area(K)where Area(K) is the (n-1)-dimensional surface area of the box K, Vol(K) is the volume of the box K, Γ is the gamma function, and where |𝕊^n-1|=2π^n/2/Γ(n/2) is the (n-1)-dimensional surface area of the sphere 𝕊^n-1⊆^n. The formula in (<ref>) has been proven in a more general setting earlier (see formula (2.4) in <cit.>); for further details, see Section <ref>. For the convenience of the reader we give a short proof of formula (<ref>) in our setting in Section <ref>. Throughout the paper, we will write pdf_Z and cdf_Z for the probability density function and the cumulative distribution function of Z, respectively, for random variables Z.We next give explicit formulas for the probability density function of X in dimensions two and three.For a box of dimension n=2 with side-lengths ≤, the probability density function of X is given bypdf_X(t) = 1/+· 1, if t<, ^2t^2√(t^2-^2), if <t<-1 + 1t^2( ^2√(t^2-^2) + ^2√(t^2-^2)), if ,<t.for 0<t<√(^2+^2). We note that the probability density function in Theorem <ref> is analytic on all open subintervals of (0, √(^2+^2)) not containingor . Moreover, it is constant on the interval (0,min(,)) and has singularities of type (t-)^-1/2 and (t-)^-1/2 just to the right ofand , respectively. See Figure <ref> for more details.For an explanation of these singularities, see Remark <ref>.For a box of dimension n=3 with side-lengths ,,, the probability density function of X is given bypdf_X(t) = F(,,,t) + F(,,,t) + F(,,,t)3 πt^3 (++)where F is the piecewise-defined function given byF(,,,t) = t^3(8-3t)for 0<t<, and byF(,,,t) = (6t^4-^4+6 π^2 )-4(+)√(t^2-^2)(^2+2t^2)for <t<√(^2+^2), and byF(,,,t) = 6 π^2+^4-3 t^4-6 ^2 ^2 +√(t^2-^2-^2) 4( ^2 + ^2 +2 t^2)++4√(t^2-^2)(^2+2t^2)-12 ^2 ·arctan(√(t^2-^2-^2))+-4√(t^2-^2)(^2+2t^2)-12 ^2 ·arctan(√( t^2-^2-^2)/)for √(^2+^2)<t<√(^2+^2+^2).We note that the probability density function in Theorem <ref> is analytic on all open subintervals of (0, √(^2+^2+^2)) not containing any of the points,,,√(^2+^2), √(^2+^2), √(^2+^2).Moreover, it is linear on the interval (0,min(,,)) and has positive jump discontinuities at the points ,,. At the points {√(^2+^2), √(^2+^2), √(^2+^2)}∖{,,}, it is continuous and differentiable. Note that the probability distribution X_M,R gives a larger “weight” to some particles than others, since some particles get more bounces than others for the same distance R. One could also consider a similar problem where we send off each particle for a certain number N>0 of bounces, and then consider the limit as M→∞ followed by taking the limit N→∞, where M is the number of particles. This would give each particle the same “weight”.Denote the finite version of this distribution by Y_M,N and its limit distribution as M→∞ and then N→∞ by Y.With regard to the previous discussion about signal loss, we call the limit distribution X of X_M,R the spreading model and we call the limit distribution of Y_M,N the absorption model. Determining the probability density function of the absorption model appears to be the more difficult problem, and we give a formula only in dimension two:For a box of dimension n=2 with side-lengths ≤, the random variable Y_M,N converges in distribution to the random variable Y, as we take M→∞ followed by taking N→∞, where the probability density function pdf_Y(t) is given by2/π( 2 (+)/(^2+^2)-2 /(^2+^2)^3/2(tanh^-1(/√(^2+^2)) + tanh^-1(/√(^2+^2))) )for 0<t<,, and by2/π( (- √(t^2-^2)) /t(+√(t^2-^2)) √(t^2-^2) + 2+2 t - 2√(t^2-^2)/t (^2+^2)+ 2 (-tanh^-1(t/√(^2+^2))+tanh ^-1(√(t^2-^2)√(^2+^2)/t ) - tanh^-1(/√(^2+^2)))/(^2+^2)^3/2)for <t<, and by2/π( (- √(t^2-^2))/t(+√(t^2-^2)) √(t^2-^2)+ (- √(t^2-^2))/t(+√(t^2-^2)) √(t^2-^2)+ 22- √(t^2-^2) - √(t^2-^2)/t (^2+^2)+2 (-2tanh^-1(t/√(^2+^2))+tanh ^-1(√(t^2-^2)√(^2+^2)/t ) + tanh ^-1(√(t^2-^2)√(^2+^2)/t ) )/(^2+^2)^3/2)for ,<t<√(^2+^2). See Figure <ref> for a comparison between the probability density functions for the two different models in dimension 2.It is not a priori obvious that the two limit distributions should differ, and it is natural to ask how much, if at all, they differ. We start by remarking that the expression for pdf_Y(t) does not simplify into the expression for pdf_X(t); indeed, for (,)=(1,2) we have pdf_X(t)=1/3 but pdf_Y(t)≈ 0.32553 on the interval (0,1).For very skew boxes, with =1 and →∞, it is straightforward to show thatpdf_Y(/2)/pdf_X(/2)→∞as →∞.§.§ DiscussionGiven a closed convex subset C ⊂^n with nonempty interior it is possible to define a natural probability measure on the set of lines in ^n that have nonempty intersection with C.The expected length of the intersection of a random line is then, up to a constant that only depends on n,given by Vol(C)/Area(C); this is known as Santalo's formulain the integral geometryand geometric probability literature (cf. <cit.>). A billiard flow on a manifold M with boundary ∂ M gives rise to a billiard map (roughly speaking, the phase space Ω is then the collection of inward facing unit vectors v at each point x ∈∂ M).Given (x,v) ∈Ω we define the associated free path as the distance the billiard particle, starting at x in the direction v, covers before colliding with ∂ M again.As the billiard map carries a natural probability measure ν we can view the free path as a random variable, and the mean free path is then just its expected value.Remarkably, the mean free path (again up to a constant that only depends on the dimension) is then given by Vol(M)/Vol(∂ M) — even for non-convex billiards.This was deduced in the seventies at the Moscow seminar on dynamical systems directed by Sinai and Alekseev but was never published and hence rederived by a number of researchers.For further details and an interesting historical survey, see Chernov's paper <cit.>.In spirit our methods are closely related to the ones used by Barra-Gaspard <cit.> in their study of the level spacing distribution for quantumgraphs, and this turns out to be given by the distribution of return times to a hypersurface of section of a linear flow on a torus.In particular, for graphs with a finite number of disconnected bonds of incommensurable lengths, the hypersurface of section is the “walls” of the torus, and the level spacings of the quantum graph is exactly the same same as the free path length distribution in our setting when all particles have the same starting velocity.(In particular, compare the numerator in (<ref>) for v fixed with <cit.>.)In <cit.>, Marklof and Strömbergsson used the results by Barra-Gaspard to determine the gap distribution of the sequence of fractional parts of {log_b n }_n ∈^+.The gap distribution depends on whether b is trancendental, rational or algebraic;quite remarkably the density function P(s) for these gaps share a number of qualitative features with the density function pdf_X(s) for free paths in our setting.Namely, the density functions both have compact support and are smooth apart from a finite number of jump discontinuities.Further, in some cases the density function is constant for s small; compare Figure <ref> (here d=2) with <cit.> (here b=√(10)).However, there are some important differences: for P(s), left and right limits exist at the jump discontinuities, whereas for d=2, the right limit of pdf_X(s) is +∞ at the jumps (cf. Figure <ref>.) Further, despite appearences, P(s) is notlinear near s=0 (cf. <cit.> corresponding to b=e) whereas for d=3, pdf_X(s) is indeed linear near s=0 (cf. Figure <ref>). §.§ AcknowledgementsWe would like to thank Z. Rudnick for some very helpful discussions, especially for suggesting the connection with integral geometry.We also thank J. Marklof for bringing references <cit.> to our attention.S.H. was partially supported by a grant from the Swedish Research Council (621-2011-5498).P.K. was partially supported by grants from the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine, and the Swedish Research Council (621-2011-5498).§ PROOF OF THEOREM <REF>In this section, we prove Theorem <ref>.For notational simplicity, we give the proof in dimension three; the general proof for n≥ 2 dimensions is analogous.Given a particle with initial position p and initial direction v, let N_R,p,v be the number of bounce lengths we get from that particle as it has travelled a total distance R>0, and let N_R,p,v(t) be the number of such bounce lengths of length at most t≥ 0. The uniform probability distribution on the set of bounce lengths of M particles with initial positions p^(1),…,p^(M) and initial directions v^(1),…,v^(M) has the cumulative distribution functioncdf_X_M,R(t)=∑_i=1^M N_R,p^(i),v^(i)(t)∑_i=1^M N_R,p^(i),v^(i)= 1/M∑_i=1^M N_R,p^(i),v^(i)R N_R,p^(i),v^(i)(t)N_R,p^(i),v^(i)1/M∑_i=1^MN_R,p^(i),v^(i)R .(Note that the denominator is uniformly bounded from below, which follows from equation (<ref>) below.) By the strong law of large numbers, the function (<ref>) converges almost surely to∫_K∫_𝕊^2N_R,p,vR N_R,p,v(t)N_R,p,v S(v)μ p∫_K∫_𝕊^2N_R,p,vRS(v)μ pas M→∞, where dμ is the probability measure with which we choose the starting points, and S is the surface area measure on the sphere 𝕊^2.By symmetry, we may restrict the inner integrals to 𝕊^2_+{(,,)∈𝕊^2: ,,> 0}.We now look at the limit of (<ref>) as R→∞, and we note that since the integrands are uniformly bounded, we may move the limit inside the integrals by the Lebesgue dominated convergence theorem. Fix one of the integrands, and denote it by f(R,p,v,t). We will show that its limit g(p,v,t)lim_R→∞ f(R,p,v,t) exists for all t and all directions v∈𝕊^2.Moreover, if p^(i) and v^(i) denote random variables corresponding to an initial position and an initial direction, respectively, as above, thenh(p^(i),v^(i),t) := lim_R →∞N_R,p^(i),v^(i)R N_R,p^(i),v^(i)(t)N_R,p^(i),v^(i)is a random variable with finite variance (and similarly for the terms in the denominator of (<ref>); in particular recall it is uniformly bounded from below),and thus the strong law of large numbers gives that the limit of (<ref>) as R→∞, and then M →∞ almost surely equals (<ref>). This shows thatlim_M→∞lim_R→∞cdf_X_M,R(t) exists almost surely and is equal tolim_R→∞lim_M→∞cdf_X_M,R(t).Consider a particle with initial position p and initial direction v=(,,)∈𝕊^2_+. By “unfolding” its motion with specular reflections on the walls of the box to the motion along a straight line in ^n — see Figure <ref> for a 2D illustration — we see that the particle's set of bounce lengths is identical to the set of path lengths between consecutive intersections of the straight line segment {p+tv: 0≤ t≤ R} with any of the planes x=n, y=n, z=n, n∈. Thus we see thatN_R,p,v = R/+R/+R/+(1)for large R, and thereforeN_R,p,vR→/+/+/as R→∞. Now project the line {p+tv:0≤ t≤ R} to the torus ^3/Λ where Λ={(n_1,n_2,n_3): n_1,n_2,n_3∈} and let us identify the torus with the box K; see Figure <ref>. Each bounce length corresponds to a line segment which starts in one of the three planes x=0, y=0 or z=0 and runs in the direction v to one of the three planes x=, y= or z=. There are R/+(1) line segments which start from the plane z=0, and thus the probability that a line segment starts from the plane z=0 is//+/+/as R→∞.By the ergodicity of the linear flow on tori (for almost all directions), the starting points of these line segments become uniformly distributed on the rectangle [0,]×[0,]×{0} for almost all v∈𝕊^2_+ as R→∞; from here we will assume that v is such a direction, and we will ignore the measure zero set of directions for which we do not have ergodicity.Consider one of these line segments and denote its length by T and its starting point by (x_0,y_0,0). For an arbitrary parameter t≥ 0, we have T≤ t if and only if t≥-x_0 or t≥-y_0 or t≥; the starting points (x_0,y_0)∈[0,]×[0,] which satisfy this are precisely those outside the rectangle [0,-t]×[0,-t] assuming that t≤ and otherwise it is the whole rectangle [0,]×[0,]. The area of that region is-(-t)(-t)if ≥ t, ≥ t, ≥ t and otherwise it is . Since the starting points (x_0,y_0) are uniformly distributed in the rectangle [0,]×[0,] as R→∞, it follows that the probability that T≤ t is1-(-t)(-t)χ(≥ t, ≥ t, ≥ t),where χ(P) is the indicator function which is 1 whenever the condition P is true, and 0 otherwise. We get analogous expressions for the case when a line segment starts in the plane x=0 or y=0 instead. Thus the proportionof all linesegments with length at most t as R→∞ islim_R→∞N_R,p,v(t)N_R,p,v = //+/+/(1-(-t)(-t)χ(≥ t, ≥ t, ≥ t))+=.//+/+/(1-(-t)(-t)χ(≥ t, ≥ t, ≥ t))+=.//+/+/(1-(-t)(-t)χ(≥ t, ≥ t, ≥ t)) which can be written1-χ(≥ t, ≥ t, ≥ t) (/+/+/)× ×( (-t)(-t)+ (-t)(-t)+ (-t)(-t) ).Recognizing that both integrands (<ref>) and (<ref>) are independent of the position p, we see that the limit of (<ref>) as R→∞ may be written as lim_R→∞lim_M→∞cdf_X_M,R(t) = 1- 1 ∫_𝕊^2_+ (++) S(v) × ×∫_ v∈𝕊^2_+ ≤/t≤/t≤/t ( (++) -2t(++) +3t^2 ) S(v)for all t>0. The corresponding formula in n dimensions is given bylim_R→∞lim_M→∞cdf_X_M,R(t) = 1- ∫_ v∈𝕊^n-1_+v_i ≤_i/t for i=1,…,n ( ∑_i=1^n v_i ∏_j≠ i(a_i-tv_j) )S(v) (∏_i=1^n _i) ∫_𝕊^n-1_+( ∑_i=1^n v_i/_i)S(v)for all t>0, where the side-lengths of the box K are _1,…,_n and S is the surface area measure on 𝕊^n-1_+∩ [0,∞)^n. (The denominator can be given explicitly by using Lemma <ref> below.) We have thus proved that the random variable X_M,R converges in distribution to a random variable with probability density function given by (<ref>) as we take M→∞ followed by taking R→∞, or alternatively, first taking R→∞ followed by taking M→∞. It remains to prove that this distribution agrees with the distribution of the random variable X defined in the introduction.§.§ Integral geometryWe start by recalling some standard facts from integral geometry (cf. <cit.>.)The set of directed straight lines ℓ in ^3 can be parametrized by pairs (v,q) where v∈𝕊^2 is a unit vector pointing in the same direction as ℓ and q∈ v^ is the unique point in ℓ which intersects the plane through the origin which is orthogonal to v. The unique translation- and rotation-invariant measure (up to a constant) on the set of directed straight lines in ^3 is ℓ A(q)S(v) where A is the surface measure on the plane through the origin orthogonal to v∈𝕊^2, and S is the surface area measure on 𝕊^2. Consider the set L_,, of directed straight lines in ^3 which intersect the box K. Now, since ++ is the area of the projectionof the box K onto the plane v^ for v∈𝕊^2_+, it follows that the total measure of L_a,b,c with respect to ℓ isC_,, 8∫_𝕊^2_+(++) S(v) = 2π (++)where we used symmetry, and the integral may be evaluated by switching to spherical coordinates. It follows that ℓ / C_,, is a probability measure on the set of directed lines intersecting the box L_a,b,c. Let ℓ be a random directed line with respect to this measure, and define the random variable Xlength(ℓ∩ K), as in the introduction. Let us determine the probability that X≤ t for an arbitrary parameter t≥ 0. By symmetry it suffices to consider only directed lines with v∈𝕊^2_+. The set of all intersection points between the rectangle [0,]×[0,]×{0} and the lines ℓ with X≤ t and direction v∈𝕊^2_+ has area -(-t)(-t)χ(≥ t,≥ t,≥ t), as in (<ref>), and its projection onto the plane v^ has area [-(-t)(-t)χ(≥ t,≥ t,≥ t)].By symmetry it follows that the area of the set of directed lines ℓ∈ L_,, with X≤ t and direction v∈𝕊^2_+ projected down to v^ is U(v,t)[-(-t)(-t)χ(≥ t,≥ t,≥ t)] + .[-(-t)(-t)χ(≥ t,≥ t,≥ t)] + .[-(-t)(-t)χ(≥ t,≥ t,≥ t)],and it follows thatProb[X≤ t] = 1C_,,∫_X≤ tℓ = 8C_,,∫_𝕊^2_+ U(v,t) S(v),which we see is identical to (<ref>), and we have thus proved that X_M,R converges in distribution to X as we take M→∞ and then R→∞. This concludes the proof of Theorem <ref>. Computing the mean value We will determine the mean value (<ref>) of X; to do this we exploit the integral geometry interpretation of the random variable X.By symmetry it suffices to restrict to directed lines ℓ with v∈𝕊^2_+.For fixed v∈𝕊^2_+, denote by Q(v)=(K+span(v))∩ v^ the set of q∈ v^ such that the directed line ℓ parametrized by (v,q) intersects K.We note that X A(q) is a volume element of the box K for any fixed v∈𝕊^2_+, and thus integrating X A(q) over all q yields the volume of the box. Hence the mean value is X= 8C_,,∫_𝕊^2_+∫_Q(v)X A(q) S(v) = 8C_,,∫_𝕊^2_+ S(v) = 2++.In n dimensions we get a normalizing factor Area(K)/2· 2^n∫_𝕊^n-1_+ v_n S(v), so with the aid of the Lemma <ref> in the Appendix, it follows that the mean value in n dimensions is X = 12^n 1π|𝕊^n|2^nArea(K)/2 2^nVol(K)|𝕊^n-1|2^n = 2π|𝕊^n-1||𝕊^n|Vol(K)Area(K)where Area(K) is the (n-1)-dimensional surface area of the box K, and Vol(K) is the volume of the box K. § PROOF OF THEOREM <REF>Using formula (<ref>) in dimension n=2, we getcdf_X(t) = 1- ∫_ v∈𝕊^1_+ ≤/t≤/t((-t) +(-t) )S(v) ∫_𝕊^1_+( /+/)S(v) . We use polar coordinates =cosθ, =sinθ so that S(v)=θ. Then the above becomes 1- ∫_cos (min(a/t, 1))^ sin (min(b/t, 1)) ( cosθ+sinθ-2tsinθcosθ) θ∫_0^π/2( cosθ + sinθ) θ = 1- 1/+[ sinθ-cosθ+tcos^2θ] _cos (min(a/t, 1))^ sin (min(b/t, 1)) .The numerator of the second term may be writtenχ(< t)( ·/t-√(1-^2/t^2)+t(1-^2/t^2) ) +χ(≥ t)(- · 0 + t· 0) + -χ(< t)( √(1-^2/t^2) - ·/t + t·^2/t^2) -χ(≥ t)( · 0 - +t )which can be simplified to χ(< t)( t--√(1-^2/t^2)) +χ(< t)( t - -√(1-^2/t^2)) +(+-t ).Inserting this into (<ref>) and differentiating yields Theorem <ref>.§ PROOF OF THEOREM <REF>We will evaluate the cumulative distribution function (<ref>) and then differentiate. The denominator of the second term of (<ref>) is∫_𝕊^2_+(++) S(v) = π4 (++),as may be evaluated by switching to spherical coordinates. Definef(,,)S(v), g(,,)-2t S(v), h(,,)3t^2S(v).By symmetry, we havef(,,) = S(v) = S(v), f(,,) = S(v) = S(v), g(,,) = -2t S(v) = -2t S(v), g(,,) = -2t S(v) = -2t S(v),and thus we can write the numerator in the second term of (<ref>) asf(,,)+ f(,,)+ f(,,)+ g(,,)+ g(,,)+ g(,,)+ h(,,).Exploiting the symmetries, it suffices to evaluate h(,,), g(,,) and f(,,) (note the order of the arguments to f).We will evaluate these integrals by switching to spherical coordinates, but first we need to parametrize the part of the sphere inside the box 0≤≤/t, 0≤≤/t, 0≤≤/t. Fix t∈(0,√(^2+^2+^2)). We haveF(,,)S(v) =( ∫_^θ_∫_0^π/2 + ∫_θ_^∫_φ_^π/2 - ∫_θ_^∫_φ_^π/2) F̃(θ,φ) sinθφθfor any integrable function F: 𝕊^2_+→, where F̃(θ,φ)F(sinθcosφ, sinθsinφ, cosθ), where /t,θ_ max(, /t),θ_ max(, /t), √(^2+^2)t,φ_ /tsinθ(whenever ≤ tsinθ), φ_ /tsinθ(whenever ≤ tsinθ).and where we have used the shorthand umin(u,1).We will parametrize the set of points v=(,,) on the sphere 𝕊^2 such that 0< ≤/t, 0< ≤/t, 0< ≤/t.Switch to spherical coordinates =sinθcosφ, =sinθsinφ, =cosθ. The non-negativity conditions of (<ref>) are equivalent to the condition θ,φ∈(0,π/2). For such angles, the condition ≤/t is equivalent to/t≤θ,and the conditions ≤/t, ≤/t are equivalent totsinθ≤φ≤tsinθ.The interval (<ref>) is non-empty for precisely those θ∈(0,π/2) such that θ≤ since1 ≤tsinθ^2 + tsinθ^21 ≤(tsinθ)^2 + (tsinθ)^2sinθ≤√(^2+^2)tθ≤√(^2+^2)t.Thus we may restrict θ to the interval given by the inequalities≤θ≤.Note that we have ≤ for all t≤√(^2+^2+^2) since≤ 1 ≤/t^2 + √(^2+^2)/t^2 1 ≤(/t)^2 + (√(^2+^2)/t)^2t^2 ≤^2+^2+^2.We conclude that we can writeF(,,)S(v) = ∫_^∫_/tsinθ^/tsinθF̃(θ,φ) sinθφθ. For θ∈(0,π/2), note that /tsinθ is definedprecisely when /t≤θ and that /tsinθ is defined precisely when /t≤θ. We have < θ_ if and only if t< √(^2+^2), and we have <θ_ if and only if t<√(^2+^2). Moreover we note that we always have θ_, θ_∈ [, ].Let us rewrite the integration limits in the right-hand side of (<ref>) in terms of φ_ and φ_. A priori, we need to distinguish between the two cases θ_≤θ_ and θ_< θ_. If θ_≤θ_ then we get (∫_^∫_x/tsinθ^y/tsinθ) = ( ∫_^θ_∫_0^π/2 + ∫_θ_^θ_∫_φ_^π/2 + ∫_θ_^∫_φ_^φ_) =( ∫_^θ_∫_0^π/2 + ∫_θ_^∫_φ_^π/2 - ∫_θ_^∫_φ_^π/2 + ∫_θ_^∫_φ_^π/2 - ∫_θ_^∫_φ_^π/2) =( ∫_^θ_∫_0^π/2 + ∫_θ_^∫_φ_^π/2 - ∫_θ_^∫_φ_^π/2).If on the other hand θ_<θ_ then (∫_^∫_x/tsinθ^y/tsinθ) = ( ∫_^θ_∫_0^π/2 + ∫_θ_^θ_∫_0^φ_ + ∫_θ_^∫_φ_^φ_) =( ∫_^θ_∫_0^π/2 + ∫_θ_^θ_∫_0^π/2 - ∫_θ_^θ_∫_φ_^π/2 + ∫_θ_^∫_φ_^π/2 - ∫_θ_^∫_φ_^π/2)which we see is identical to (<ref>). Combining (<ref>) and (<ref>) we get the conclusion of the lemma. Applying Lemma <ref>we geth(,,) = 3t^2S(v) = 3t^2 ( ∫_^θ_∫_0^π/2 + ∫_θ_^∫_φ_^π/2 - ∫_θ_^∫_φ_^π/2) (sin^2θcosθcosφsinφ) sinθφθ.An antiderivative of the integrand cosφsinφ·sin^3θcosθ with respect to φ is -1/2cos ^2φsin ^3θcosθ, and thus the above is3t^2 ( ∫_^θ_. cos ^2φ|_φ=0 + ∫_θ_^. cos ^2φ|_φ=φ_ - ∫_θ_^. cos ^2φ|_φ=φ_) 1/2sin ^3θcosθθ = 3t^2 ( ∫_^θ_ 1 + ∫_θ_^a^2t^2sin^2θ + ∫_θ_^(b^2t^2sin^2θ-1) ) 1/2sin ^3θcosθθ =3/2( ∫_^θ_ t^2sin ^3θcosθθ + ∫_θ_^ a^2 sinθcosθθ + ∫_θ_^(b^2sinθ - t^2sin^3θ) cosθθ) =3/2( [ t^21/4sin ^4θ]_^θ_ + [ a^2 1/2sin^2 θ]_θ_^+ [ b^21/2sin^2θ-t^21/4sin^4θ]_θ_^). Next considerg(,,) = -2t S(v) = -2t( ∫_^θ_∫_0^π/2 + ∫_θ_^∫_φ_^π/2 - ∫_θ_^∫_φ_^π/2) (sin^2θcosφsinφ) sinθφθ.An antiderivative of the integrand cosφsinφ·sin^3θ with respect to φ is -1/2cos ^2φsin ^3θ, and thus the above isg(,,) = -2t S(v) = -t( ∫_^θ_.cos ^2φ|_φ=0 + ∫_θ_^.cos ^2φ|_φ=φ_ - ∫_θ_^.cos ^2φ|_φ= φ_) sin ^3θθ = -t( ∫_^θ_ 1 + ∫_θ_^a^2t^2sin^2θ + ∫_θ_^(b^2t^2sin^2θ-1) ) sin ^3θθ = -t( ∫_^θ_sin ^3θθ + ∫_θ_^a^2sinθt^2θ + ∫_θ_^(b^2sinθt^2-sin ^3θ)θ) = -t( [cos^3θ/3-cosθ]_^θ_+ a^2t^2[-cosθ]_θ_^+ [-b^2cosθt^2-cos^3θ/3+cosθ]_θ_^).We obtain g(,,) and g(,,) by switching the roles of ,, in (<ref>). We remark that trying to obtain g(,,) and g(,,) directly, by integratingand , respectively, by first integrating with respect to φ, taking the limits φ→φ_a and φ→φ_b, and then finding an antiderivative with respect to θ, seem to result in much more complicated expressions.Finally considerf(,,) = S(v) =( ∫_^θ_∫_0^π/2 + ∫_θ_^∫_φ_^π/2 - ∫_θ_^∫_φ_^π/2) (sinθsinφ) sinθφθ.An antiderivative of the integrand sinφ·sin^2θ with respect to φ is -cosφ·sin^2θ, and thus the above is( ∫_^θ_. cosφ|_φ=0 + ∫_θ_^. cosφ|_φ=φ_ - ∫_θ_^. cosφ|_φ=φ_) sin ^2θθ =( ∫_^θ_ 1 + ∫_θ_^atsinθ - ∫_θ_^√(1-b^2t^2sin^2θ)) sin ^2θθ =( ∫_^θ_sin^2θθ + ∫_θ_^asinθtθ - ∫_θ_^√(sin^2θ-b^2t^2)sinθθ) =( 1/2[ θ-sinθcosθ]_^θ_ + [ -acosθt]_θ_^ - ∫_θ_^√(1-b^2t^2 - cos^2θ)sinθθ)where the last integral inside the parentheses may be written as[ -1/2(cosθ√(1-b^2t^2-cos^2θ)+(1-b^2t^2) (cosθ/√(1-b^2t^2-cos^2θ))) ]_θ_^ =[ -1/2(cosθ√(sin^2θ-b^2t^2)+(1-b^2t^2) (cosθ/√(sin^2θ-b^2t^2))) ]_θ_^whenever θ_<π/2, by using the fact that 1/2(x √(c-x^2)+c (x/√(c-x^2))) is an antiderivative of √(c-x^2) with respect to x when c is a constant.We obtain f(,,) and f(,,) by switching the roles of ,, in (<ref>).It remains to insert the limits ,θ_,θ_, into the antiderivatives (<ref>), (<ref>) and (<ref>) above. Noting that ,θ_,θ_, are expressed in terms of piecewise-defined functions, the following manipulations will be useful. For any function ψ, we haveψ() =ψ(cos/t)χ_ + ψ(cos 1)(1-χ_) =(ψ(cos/t)-ψ(0))χ_ + ψ(0)where χ_χ(t>). Similarly,ψ() =(ψ(sin√(^2+^2)/t)-ψ(π/2))χ_, +ψ(π/2)where χ_,χ(√(^2+^2)>t), andψ(θ_)= (1-χ_)ψ(π/2) + (χ_-χ_,) ψ(sin/t) + χ_,ψ(cos/t) = χ_,·( ψ(cos/t)-ψ(sin/t) )+ χ_·( ψ(sin/t)-ψ(π/2) )+ ψ(π/2)and similarly, ψ(θ_) can be written asχ_,·( ψ(cos/t)-ψ(sin/t) )+ χ_·( ψ(sin/t)-ψ(π/2) )+ ψ(π/2). With this we can evaluate [ψ]_^θ_, [ψ]_θ_^, [ψ]_θ_^. But since we know that we will get a function symmetric with respect to the values ,,, it suffices to keep only those terms with χ_ and χ_,, say, and then the other terms may be evaluated by just switching the order of ,,.Upon inserting the limits and differentiating, one obtains (after tedious calculations) that pdf_X(t) = F(,,,t) + F(,,,t) + F(,,,t)3 πt^3 (++)whereF(,,,t) (8 t^3-3t^4) +χ(t≥)((6t^4-^4+6 π^2 )- (8t^3-3t^4)-4(+)√(t^2-^2)(^2+2t^2))+ χ(t≥√(^2+^2))[ ^4+^4-9 t^4-6 ^2 ^2+√(t^2-^2-^2) 4( ^2 + ^2 +2 t^2)+ 4√(t^2-^2)(^2+2t^2)-12 ^2 ·arctan(√(t^2-^2-^2))+ 4√(t^2-^2)(^2+2t^2)-12 ^2 ·arctan(√( t^2-^2-^2)/) ].Rewriting F as a piecewise function, we get Theorem (<ref>).§ PROOF OF THEOREM <REF>Consider the distribution of the random variable Y_M,N.Since we record the same number of bounces for each choice of angle φ we may replace the M-particle system with a one particle system Y_N as follows: randomly select, with uniform distribution, the angle φ and generate N bounce lengths and randomly select one of these bounce lengths (with uniform distribution); by the strong law of large numbers, Y_M,N converges in distributionto Y_N as M→∞.We now determine the limit distribution of Y_N.As before, we first unfold the motion, and replace motion in a box with specular reflections on the walls with motion in ^2; see Figure <ref>.The path lengths between bounces is then the same as the lengths between the intersections with horizontal or vertical grid lines.To understand the spatial distribution, we project the dynamics to the torus ^2/Λ where Λ is the latticeΛ = {(n_1,n_2) : n_1,n_2 ∈},and we may identify the torus with the rectangle [0,]×[0,]. Let us first consider the motion of a single particle with an arbitrary initial position, and direction of motion given by an angle φ. Taking symmetries into account, we may assume that φ∈ [0,π/2].(Note that d φ/π/2 gives a probability measure on these angles.)If the particle travels a large distance R>0, the number of intersections with horizontal, respectively vertical, grid lines is R sinφ/ +O(1), respectively R cosφ/ +O(1).Thus, in the limit R →∞, the probability of a line segment beginning at a horizontal (respectively vertical) grid line is given by P_h, respectively P_v (here we suppress the dependence on φ) whereP_hsinφ//sinφ/+cosφ/,P_vcosφ//sinφ/+cosφ/.The unfolded flow on the torus is ergodic for almost all φ, and thus the starting points of the line segments becomes uniformly distributed as R→∞ for almost all φ.Let T = T(φ) /cosφ.Since sinφ = √(T^2-^2)/T, we obtain thatP_h = √(T^2-^2)/+√(T^2-^2), P_v = /+√(T^2-^2).Let θ = arctan/ denote the angle of the diagonal in the box, and assume that 0 ≤φ≤θ.We then observe the following regarding the line segment lengths.First, if the segment begins at a horizontal line, it must end at a vertical line, and the possible lengths of these segment lie between 0 and T.We find that these lengths are uniformly distributed in [0,T] since the starting points of the segments are uniformly distributed.On the other hand, if the line segment begins at a vertical line, it can either end at a vertical or horizontal line.Since the starting points are uniformly distributed, the former happens with probabilitytanφ/ =√(T^2-^2)//=√(T^2-^2)/and the length of the segment is again uniformly distributed in [0,T], whereas the latter happens with probability - tanφ/ =1- √(T^2-^2)/in which case the segment is always of length T.Now, φ∈ [0,θ] implies that T ∈ [,√(^2+^2)], and noting that d φ/ T = /T √(T^2-^2)we find that the probability of observing a line segment of length t is the sum of a “singular part” (the segment begins and ends on vertical lines; note that all such segments have the same lengths) and a “smooth part” (the segment does not begin and end on vertical lines).Moreover, the smooth part contribution equals1/π/2∫_max(,t)^√(^2+^2)1/T( P_h + P_vtanφ/) d φ/ T Twhich, on inserting (<ref>), equals1/π/2∫_max(,t)^√(^2+^2)1/T·( √(T^2-^2)/+√(T^2-^2) +/+√(T^2-^2)tanφ/) ·/T √(T^2-^2) T = 1/π/2∫_max(,t)^√(^2+^2)1/T·( √(T^2-^2)/+√(T^2-^2) +/+√(T^2-^2)√(T^2-^2)/) ·/T √(T^2-^2) T = 1/π/2∫_max(,t)^√(^2+^2)2/+√(T^2-^2)· T/T^2 .On the other hand, the “singular part contribution”, provided t ≥, to the probability of a segment having length t equalsP_v/π/2·- tanφ/·d φ/dt= 1/π/2·/+√(t^2-^2)·(1- √(t^2-^2)/) ·/t √(t^2-^2) = 1/π/2·/t(+√(t^2-^2)) √(t^2-^2)·( - √(t^2-^2)). In case θ≤φ≤π/2, a similar argument (we simple reverse the roles ofand ) shows that the smooth contribution equals1/π/2∫_max(,t)^√(^2+^2)2/+√(T^2-^2)· T/T^2and that the singular contribution (if t ≥) equals1/π/2·/t(+√(t^2-^2)) √(t^2-^2)·(- √(t^2-^2)) . Thus, if we let P_sing(t) denote the “singular contribution” to the probability density function we find the following: if t <, thenP_sing(t) = 0if t ∈ [,], thenP_sing(t) =1/π/2·(- √(t^2-^2)) /t(+√(t^2-^2)) √(t^2-^2)and if t ∈ [, √(^2+^2)], thenP_sing(t)=1/π/2·( (- √(t^2-^2))/t(+√(t^2-^2)) √(t^2-^2)+ (- √(t^2-^2))/t(+√(t^2-^2)) √(t^2-^2)).Note that P_sing has a singularity of type (t-)^-1/2 just to the right of t= (and similarly just to the right of t=). In a sense this singularity arises from the singularity in the change of variables φ↦ T since d φ/dT = /T √(T^2-^2).The reason for the singularities in the spreading model for n=2 is similar, as the spreading model can be obtained from the absorption model by a smooth change of the angular measure.Similarly, the “smooth part” of the contribution is (for t ∈ [0,√(^2+^2)]) given byP_smooth(t) = 1/π/2( ∫_max(,t)^√(^2+^2)2/+√(T^2-^2)· T/T^2+ ∫_max(,t)^√(^2+^2)2/+√(T^2-^2)· T/T^2)Hence the probability density function of the distribution of the segment length t is given by pdf_Y(t) = P_sing(t) +P_smooth(t).We will now evaluate P_smooth(t). An antiderivative of 2/+√(T^2-^2)·1/T^2 with respect to T for T∈(,√(^2+^2)) is2 ( √(T^2-^2)- )/T (^2+^2)+2 (tanh^-1(T/√(^2+^2))-tanh ^-1(√(T^2-^2)√(^2+^2)/T ))/(^2+^2)^3/2where tanh(z)=1/2log1+z/1-z for z<1. (A quick calculation shows that √(T^2-^2)√(^2+^2)/T< 1 whenever <T<√(^2+^2).) We can rewrite (<ref>) as2 ( √(T^2-^2)- )/T (^2+^2)+log((√(^2+^2)+T) (T -√(T^2-^2)√(^2+^2))/(√(^2+^2)-T) (T+ √(T^2-^2)√(^2+^2))) /(^2+^2)^3/2By l'Hôpital's rule we havelim_T→√(^2+^2)+T -√(T^2-^2)√(^2+^2)/√(^2+^2)-T=lim_T→√(^2+^2)+-T/√(T^2-^2)√(^2+^2)/-1=^2/so the limit of (<ref>) as T→√(^2+^2)+ islog( (^2/)·(√(^2+^2)+√(^2+^2)) /(√(^2+^2)+√(^2+^2))) /(^2+^2)^3/2 = 2log( /) /(^2+^2)^3/2.The limit of (<ref>) as T→+ is-2 /(^2+^2)+2 tanh^-1(/√(^2+^2))/(^2+^2)^3/2.Thus, assuming <, we can write π/2P_smooth(t) as2 (+)/(^2+^2)-2 /(^2+^2)^3/2(tanh^-1(/√(^2+^2)) + tanh^-1(/√(^2+^2)))if t<,, or as2+2 t - 2√(t^2-^2)/t (^2+^2)+ 2 (-tanh^-1(t/√(^2+^2))+tanh ^-1(√(t^2-^2)√(^2+^2)/t ) - tanh^-1(/√(^2+^2)))/(^2+^2)^3/2if <t< or as22- √(t^2-^2) - √(t^2-^2)/t (^2+^2)+2 (-2tanh^-1(t/√(^2+^2))+tanh ^-1(√(t^2-^2)√(^2+^2)/t ) + tanh ^-1(√(t^2-^2)√(^2+^2)/t ) )/(^2+^2)^3/2if ,<t. Adding P_sing(t) to this, we get Theorem <ref>. § CALCULATION OF AN INTEGRALWrite |𝕊^n-1| for the (n-1)-dimensional surface area of the sphere 𝕊^n-1⊆^n. Then we have ∫_𝕊^n-1_+ v_nS(v) = 1π|𝕊^n|2^n.where 𝕊^n-1_+ 𝕊^n-1∩ (0,∞)^n is the part of the sphere 𝕊^n-1 with positive coordinates. We may parametrize v=(v_1,…,v_n)∈𝕊^n-1_+ withv_1= cosθ_1 v_2= sinθ_1 cosθ_2 v_3= sinθ_1 sinθ_2 cosθ_3 ⋮v_n-1 = sinθ_1⋯sinθ_n-2cosθ_n-1v_n= sinθ_1⋯sinθ_n-2sinθ_n-1for θ_1,…,θ_n-1∈(0,π/2). We have the spherical area elementS(v) = sin^n-2θ_1 sin^n-3θ_2 ⋯sinθ_n-2θ_1⋯θ_n-1.Thus we get∫_𝕊^n-1_+ v_nS(v) =∏_i=1^n-1∫_0^π/2sin^n-1-iθ_i θ_i.Introducing an additional integration variable θ_n, we recognize the integrand as the spherical area element in n+1 dimensions, and thus the above is1∫_0^π/2θ_n∏_i=1^n∫_0^π/2sin^n-1-iθ_i θ_i = 1π/2|𝕊^n|2^n+1.since ∫_𝕊^n_+ S(v) = |𝕊^n|/2^n+1. 99barra-gaspard-level-spacings-quantum-graphs F. Barra and P. Gaspard. On the level spacing distribution in quantum graphs. J. Statist. 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On the distribution of free path lengths for the periodic Lorentz gas. II. M2AN Math. Model. Numer. Anal., 34(6):1151–1163, 2000.klain-rota-integral-geometry-book D. A. Klain and G.-C. Rota. Introduction to geometric probability. Lezioni Lincee. [Lincei Lectures]. Cambridge University Press, Cambridge, 1997. mansson-pers-comm D. Månsson, personal communication.marklof-strombergsson-annals-2010 J. Marklof and A. Strömbergsson. The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems. Ann. of Math. (2), 172(3):1949–2033, 2010.marklof-strombergsson-boltzman-grad-annals2011 J. Marklof and A. Strömbergsson. The Boltzmann-Grad limit of the periodic Lorentz gas. Ann. of Math. (2), 174(1):225–298, 2011.marklof-strombergsson-gaps-between-logs J. Marklof and A. Strömbergsson. Gaps between logs. Bull. Lond. Math. Soc., 45(6):1267–1280, 2013.marklof-strombergsson-CMP-quasicrystal J. Marklof and A. Strömbergsson. Free path lengths in quasicrystals. Comm. 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http://arxiv.org/abs/1702.08096v1

School of Engineering and Applied Sciences, Harvard University, Cambridge,MA 02138, USA. Department of Physics, Harvard University, Cambridge,MA 02138, USA. Physical Chemistry and Soft Matter, Wageningen University, Stippeneng 4, 6708 WE, Wageningen, the NetherlandsDepartment of Chemical Engineering and Biotechnology, University of Cambridge, New Museums Site, Pembroke Street, CB2 3RA Cambridge, UK. School of Engineering and Applied Sciences, Harvard University, Cambridge,MA 02138, USA. Van der Waals-Zeeman Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands School of Engineering and Applied Sciences, Harvard University, Cambridge,MA 02138, USA. Department of Physics, Harvard University, Cambridge,MA 02138, USA.[Corresponding Author: ]weitz@seas.harvard.edu Crystals with low latent heat are predicted to melt from an entropically-stabilised body-centered cubic (BCC) symmetry. At this weakly first-order transition, strongly correlated fluctuations are expected to emerge, which could change the nature of the transition. Here we show how large fluctuations stabilise BCC crystals formed from charged colloids, giving rise to strongly power-law correlated heterogeneous dynamics. Moreover, we find that significant non-affine particle displacements lead to a vanishing of the non-affine shear modulus at the transition. We interpret these observations by reformulating the Born-Huang theory to account for non-affinity; illustrating a scenario ofordered solids reaching a state where classical lattice dynamics fail.64.70.dj, 82.70.Dd, 61.50.AhDirect observation of entropic-stabilisation of BCC crystals near melting David A. Weitz=========================================================================Many common metals, including lithium, sodium and most transition metals of group IV and V, transform from a close-packed structure to a body-centered cubic (BCC) phase at high temperatures. These BCC phases are remarkable as they derive their stability, with respect to other lattices, from a large vibrational entropy<cit.>. Strongly correlated fluctuations of the lattice are predicted to emerge on approach to the solid-liquid boundary, and, for some BCC crystals, the melting transition becomes only weakly first order<cit.>. Similar fluctuations are also expected for melting in the absence of interfaces or dislocations where liquid nucleation is normally initiated<cit.>. This is a different form of melting, first described by Born<cit.>. These correlated fluctuations could change the very nature of the solid itself. However, these have never been experimentally observed. Here, we explore BCC crystals formed by charged colloids and experimentally probe the nature of these stabilizing fluctuations, and their behavior upon approach to melting. We show that the crystals are stabilized at low densities by correlated lattice vibrations in direct analogy to their atomic counterparts. We identify “hot” particles that have the largest fluctuations away from their average lattice positions; they exhibit strong spatial correlation, forming extended connected clusters. These clusters exhibit fractal configurations on the lattice, whose size diverges critically as the solid-liquid transition is approached. This gives the phase transition a pronounced second-order character. However, the affine shear modulus remains non-zero right up to the transition, and thus the transition is strictly first-order. We show that these large fluctuations cause the system to respond non-affinely, and the non-affine shear modulus vanishes continuously. These results demonstrate that thermal excitation of the BCC crystal lead to a new state, where the average structure is crystalline but, surprisingly, long-ranged correlated fluctuations govern the behavior. We study charged colloids of poly(methyl methacrylate) of d = 2.1 μm, suspended in a density-matching mixture of decahydronaphtalene and tetrachloroethylene. The addition of 15 mM AOT, acting as a charge-control agent, leads to the charging of the colloidal particles<cit.> with a screening length 1/κ = 1 μm. As a result of the strong electrostatic repulsion, these colloids form solids at very low densities<cit.>, which we are known as colloidal Wigner crystals <cit.> by analogy to the crystals formed by electrons on a semiconductor surface <cit.>. The main control parameter in our experiments is the particle volume fraction ϕ.We image each particle in a 90x90x15 μm volume at 1 Hz with confocal fluorescence microscopy. To determine the short-time dynamics, we complement these data with horizontal two-dimensional slices at 28 Hz.The phase behaviour of this system as a function of ϕ is shown in Figure 1. For ϕ> 0.20, the charged colloidal particles form face-centered cubic (FCC) crystals (Fig. 1d); as ϕ is decreased the solid adopts a BCC symmetry with few defects <cit.> (Fig. 1c). At even lower ϕ the sample melts into a liquid (Fig.1a). Themelting of crystalline solids from a superheated state is accompanied by complex dynamics in the form of migrating topological defects <cit.>, particle-exhange loops <cit.> or premelting at defects <cit.>. In our experiments however, we study the approach to melting of samples at a fixed value of ϕ; to the best of our knowledge, the samples are in equilibrium and thus representative of the thermodynamic phase boundaries of the system. We also note that the measurement of absolute volume fractions in these systems is challenging and may have some error; in the analysis below we present our data as a function of relative volume fractions, such that these errors cancel and do not alter the validity of our conclusions. Previous work has shown a quantitative agreement between the experimental phase behavior of this system and that predicted by theory <cit.>, further supporting the assumption of equilibrium.We observe liquid-crystal coexistence in a narrow rangebetween 0.060<ϕ<0.066 (Fig. 1b); this coexistence region is irrefutable evidence of a first-order transition. The long-range order of the crystals is reflected by Bragg peaks in the structure factor (bottom row Fig.1c&d). By contrast, the liquid sample exhibits only isotropic short-range order (bottom row Fig.1a). For the sample in coexistence, we calculate S(q_x,q_y) for each region separately, and find distinct Bragg peaks for the crystalline region, and isotropic scattering for the liquid region (Fig. 1b).Thesolid-solid transformation from a FCC to a lower density BCC structure which melts into the liquid is in direct analogy to a wide variety of metals, which exhibit a similar transition upon increasing the temperature up to the melting point. Moreover, it is also in accord with the theoretical predictions for crystals with a low latent heat of melting<cit.>. The system of charged colloids, which can be analyzed at the single-particle scale, is an ideal model to explore the general scenario of entropically-stabilised BCC phases and their weak first-order transition. We explore the dynamics of this system by determining the mean-square displacement < Δ r^2(t) >. We observe two distinct behaviors; a purely diffusive behaviour in the liquid and a time-independent plateau of height δ^2 in the solid (inset Fig. <ref>a). The time-independent plateau in the solid reflects the amplitude of particle fluctuations around their mean lattice positions. Normalising the amplitude of particle fluctuations with the lattice constant a gives the Lindemann parameter δ_L = δ/a . We observe a sharp rise to δ_L = 0.25 in the solid phase at melting, where the crystal and liquid coexist; beyond this, δ_L can no longer be defined (Fig.<ref>a). A defining feature of all solids is a finite shear modulus. The value of the affine shear elastic constant C_44 is determined by the symmetry of the lattice and the strength of interparticle bonds <cit.>, assuming that all particles are displaced proportionally to the external deformation. From our confocal microscopy data we can directly measure C_44, as detailed elsewhere<cit.>. For volume fractions right up to ϕ_m, C_44 remains non-zero (squares in Fig.<ref>a), decaying as k_BT/a^3, consistent with the predictions of affine theory. Once the sample becomes liquid and a lattice can no longer be defined, C_44 jumps discontinuously to zero. The anisotropy in crystal elasticity for these systems also persists up to the melting transition <cit.>. Combined with the distinct Bragg peaks for a crystal coexisting with the liquid (Fig.<ref>b), these data are in full accord with a transition that is strictly first-order.Our microscopy data provide a means to investigate the nature of the solid close to this transition, where it exhibits large fluctuations. Indeed, inspection of the images (Fig.<ref>) and movies(see SI) suggests that large fluctuations create pronounced deformations of the lattice, especially at low ϕ. To ascertain the nature of these deformations, we identify "hot" particles as those that display instantaneous displacement amplitudes larger than δ_L = 0.25, the ensemble-averaged value at melting <cit.>. The fraction of "hot" particles in the crystal n_L rises steeply to 0.5 just below the melting transition, whereupon it jumps to a value of 1 in the liquid (Fig. <ref>b). We find no detectable differences in the local surroundings of "hot" particles and all others (inset Fig.<ref>b).The "hot" particles are not homogeneously distributed, butform connected clusters. We color-code these for several volume fractions in Fig.<ref>c-e.Both the size and spatial extent of the clusters increase as the sample approaches the melting transition, where they percolate the field of view (Fig.<ref>e). These extended and transient clusters are the first observation of the correlated fluctuations implied by the Alexander-McTague theory for any BCC lattice where the difference in free energy between the liquid and solid is small<cit.>. We determine their size distribution P(n), with n the number of particles within a connected cluster. We find a distinct power-law correlation with an exponential cut-off (Fig. <ref>a), whose power-law exponent of 1.75 is independent of volume fraction. We confirm that the clustering of "hot" particles is statistically significant by comparing an experimental P(n) with a simulated distribution for samples in which "hot" particles have been randomly placed on the lattice (Fig.<ref>b). We also calculate the scaling between cluster size n and their radius of gyration R_g and find that they are fractal in nature, with a characteristic fractal dimension d_f =1.7 that is universal for all ϕ (Fig.<ref>c).Remarkably, these data exhibit hallmarks of a second-order transition. To demonstrate this we calculate the average cluster size⟨ n ⟩; the mean cluster size diverges upon approaching a critical volume fraction ϕ_c with a characteristic power-law exponent of -3/4 (Fig.<ref>d). As a confirmation of this critical behavior we also compute the effective cluster volume fraction that also diverges at ϕ_c (see SI). Thus, not only do we find heterogeneous dynamics within an on-average perfect crystal, which exhibit fractal correlations in space, but the size and volume fraction of these clusters diverge critically. The critical volume fraction ϕ_c for this divergence appears to coincide with the symmetry change at the melting point, which we have shown to be a strictly first-order transition. This provide new insight into the nature of the weak first-order transition described by Alexander and McTague<cit.>, but also raises the intriguing question how hallmarks of a continuous transition can coincide with a phase transition which is clearly first-order in nature.The key to understanding the origin of this behavior lies with the elasticity of the solid. The affine elasticity measured by C_44 assumes that every particle is displaced exactly proportionally to an applied strain. However, the dynamic disorder caused by the large thermal excitations at low ϕ breaks the local BCC symmetry and causes the local coordination number to differ substantially from the value prescribed by a perfect lattice. These transient violations of centre-of-inversion symmetry must result in net non-zero forces on every particle in its affine position (Fig.<ref>b), which can only be relaxed upon allowing non-affine displacements<cit.>. Such non-affine displacements remove elastic energy from the lattice and thus reduce the overal crystal elasticity. To verify this ansatz, we calculate the amplitude of non-affine displacements<cit.> < D^2 >; we indeed find that < D^2 > grows upon approaching the melting transition, where the deviations of perfect lattice order are largest (inset <ref>a). Clearly, any consideration of the crystal rigidity under these conditions must take non-affinity into account. We measure the non-affine shear modulus G_0from the correlations in motion between pairs of particles, averaged isotropically and over all pairs in the field-of-view<cit.>. Strikingly, the non-affine rigidity G_0 vanishes at ϕ_c (circles in Fig.<ref>a), where the affine modulus C_44 remains non-zero. This confirms the importance of non-affinity due to thermal fluctuations as the crystal approaches melting. We note that the non-affine modes and a vanishing non-affine modulus are crucial features of several approaches to describe disordered solids <cit.>, yet they have remained largely unexplored for crystals to date.A remarkable paradox emerges from these observations; whereas the phase transition is strictlyfirst-order, we measure a continuous vanishing of the non-affine shear rigidity and divergence of collective length scales, akin to a second-order transition. The affine elastic constants are based on the symmetry of a perfect lattice; thus C_44 must be discontinous as the symmetry changes discontinuously at ϕ_m. By contrast, the non-affine modulus G_0 does not require assumption of a specific symmetry and is thus very sensitive to thermal disorder. In effect, the non-affine modulus provides a local probe of a more random random environment with a symmetry much closer to the liquid; this results in a transition which is continuous to within experimental resolution.For an ordered network of springs deforming affinely, the affine elastic modulus can be estimated as<cit.> kT/a^3. By contrast, our experiments show a non-affine modulus which is order-of-magnitude below this limit. Naively, we can set the typical length scale of the clusters of "hot" particles R_g as the relevant scale governing the non-affine mechanics in proximity to ϕ_c as G_0 ∝ kT/R_g^3, which can be related to the cluster statistics as R_g^3 ∝⟨ n ⟩^3/d_f. From our experiments we know that ⟨ n ⟩∝ (ϕ-ϕ_c)^-3/4; thus G_0 ∝ (ϕ - ϕ_c)^9/4d_f. Remarkably, this simple argument explains two key experimental observations: the vanishing of the non-affine modulus at ϕ_c and the convergence of G_0 and the affine C_44 at high volume fractions where R_g ≈ a.To quantitatively explain our results, we must explicitly account for the non-affinity by extending the original Born-Huang theory<cit.>. Non-affine displacements lower the free energy of deformation and hence reduce the nonaffine shear modulus G_0 with respect to its affine counterpart C_44: G_0=C_44-G_NA, where G_NA is the nonaffine correction<cit.>. To find G_NA, we adopt a framework for the rigidity of networks with central force bonds. The mechanical stability is governed by the distance from isostaticity Z - Z_c, where Z is the average number of bonds at each node and its critical value Z_c =6 defines the isostatic point<cit.>. The coordination number Z represents the number of stress-bearing, permanent, bonds with neighbors; in this case these are the bonds not part of the clusters of "hot" particles. The stability parameter Z-Z_c can be directly related to experimentally measurable properties of the clusters, such as d_f and their pair correlation function.<cit.> Within this approach, the full derivation of which is given in the SI, we predict the shear modulus to vanish as G_0=G_A-G_NA=K ϕ(ϕ-ϕ_c )^0.56/a ∝ϕ(ϕ - ϕ_c)^0.64, in which a is the lattice spacing and K a proportionality constant, which is the only adjustable parameter in our model. Remarkably, this theoretical prediction is in quantitative agreement with the experiments as shown bythe solid line in Figure <ref>a. These results give rise to an unexpected picture of the behaviour of entropically-stabilised BCC crystals, which by nature are subject to strong thermal excitations. Due to the inherently low coordination number in the BCC phase, softening of the crystal triggers the emergence of strongly correlated heterogeneous dynamics on the lattice, while the average structure of the crystal lattice remains perfect. The correlated fluctuations and associated non-affine mechanics exhibit the hallmarks of a continuous, critical, transition, that paradoxically coincides with the strictly first-order melting point of the crystal. Such large collective fluctuations increase the entropy of the solid<cit.>, which extend the crystal stability to lower densities and lead to a very small jump in enthalpy at the first-order solid-liquid transition. They are also observed for various atomic crystals such as those formed by sodium or lithium. Moreover, these correlated fluctuations provide the mechanism for the elastic collapse that causes melting of a superheated crystal that was first anticipated by Born.Our observation of strongly correlated fluctuations are unique to the BCC phase, in contrast to, for example, the more common FCC structure in colloidal crystals. More than other structures, the BCC structure is stabilized by entropy with respect to the liquid <cit.>. As a result, its first-order melting transition can become sufficiently weak that the effects of non-affinity become significant. We also find the high density FCC phase in our experiments (Fig.<ref>), which is stable down to ϕ = 0.2. Even at these low densities, and at δ_L ≈ 0.1, where the hard-sphere FCC would melt <cit.>, the FCC crystals do not show any non-affinity. This is corroborated by experiments on FCC crystals formed from colloidal hard spheres, in which no deviations from continuum lattice dynamics were observed <cit.>. Finally, we notice that the affine and non-affine moduli converge upon approaching the FCC phase (Fig.<ref>a). Non-affine displacements are therefor not of significance for the FCC symmetry, but that they are a particular feature of the high-temperature, or low density, BCC phase. This is in full agreement with the predictions of Alexander-McTague<cit.>.The collective fluctuations we observe increase the entropy of the solid<cit.> and lead to a very small jump in enthalpy at the first-order solid-liquid transition, also observed for various atomic crystals such as those formed by sodium or lithium. Our results for colloidal crystals may thus help in understandingweak BCC phases near melting in a much wider variety of systems. For example, over 40 elements in the periodic table exhibit a high-temperature BCC phase close to their melting line. Moreover, low-density BCC crystals of charged particles areof interest in astrophysics, as they are expected to be an important state of matter in neutron stars and pulsars <cit.>. Our results illustrate a scenario in which large thermal fluctuations may bring perfectly ordered solids to a state where classical theories for lattice mechanics break down, and new, richer physics emerges.§ ACKNOWLEDGEMENTSThis work was supported by the National Science Foundation (DMR-1310266, DMR-1206765), the Harvard Materials Research Science and Engineering Center (DMR-1420570) and NASA (NNX13AQ48G). The authors thank Peter J. 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http://arxiv.org/abs/1702.07873v1

jrice@physics.unlv.edu zhang@physics.unlv.eduDepartment of Physics and Astronomy, University of Nevada, Las Vegas, NV 89154, USA The cosmological evolution of primordial black holes (PBHs) is considered. A comprehensive view of the accretion and evaporation histories of PBHs across the entire cosmic history is presented, with focus on the critical mass holes. The critical mass of a PBH for current era evaporation is M_cr∼ 5.1×10^14 g. Across cosmic time such a black hole will not accrete radiation or matter in sufficient quantity to hasten the inevitable evaporation, if the black hole remains within an average volume of the universe. The accretion rate onto PBHs is most sensitive to the mass of the hole, the sound speed in the cosmological fluid, and the energy density of the accreted components. It is not easy for a PBH to accrete the average cosmological fluid to reach 30M_⊙ by z∼0.1, the approximate mass and redshift of the merging BHs that were the sources of the gravitational wave events GW150914 and GW151226. A PBH located in an overdense region can undergo enhanced accretion leading to the possibility of growing by many orders of magnitude across cosmic history. Thus, two merging PBHs are a plausible source for the observed gravitational wave events. However, it is difficult for isolated PBHs to grow to supermassive black holes (SMBHs) at high redshift with masses large enough to fit observational constraints.§INTRODUCTIONPrimordial black holes (PBHs) are among the most intriguing ghosts in the universe. A singular PBH of sufficient mass can navigate the history of the universe without detectable clues to its existence; a true cosmic ghost. Low mass PBHs evaporate before the current epoch and the radiation signature of an isolated high mass PBH is too weak to detect. The last moments of a PBH evaporation reveal the hole through a burst of high-energy radiation that is distinguishable from that of short gamma-ray bursts (GRBs) <cit.>.The upper limits on the number density of PBHs across a wide range of masses is discussed extensively in <cit.>. To date there are no confirmed PBH burst signals, but these compelling ghosts are ripe cosmological messengers that will enhance our understanding of the universe if observed. The PBHs evolving through cosmic history could be used as a proxy for understanding the conditions in the early universe. PBHs of significant mass may gain a dark matter (DM) halo, e.g. <cit.>. Since the PBH evaporation rate depends only on the mass of the hole and the assumed particle physics model <cit.>, PBHs in similar astrophysical environments should produce similar radiation signatures; the ultimate “standard candles.”This study explores the evolution of PBHs through accretion and evaporation across the entire cosmic history. Special attention is paid to the changes in the density, temperature, and sound speed in the cosmological fluid because of their influence on the accretion rate of that fluid onto the PBHs. In <ref> the concordance cosmological model of ΛCDM is discussed. In <ref> the PBH accretion and evaporation models are discussed and formulae are given for the accretion rates in the various cosmological eras. In <ref> the results of the study are discussed. Finally in <ref> the conclusions are presented and a discussion of astrophysical implications is made. §COSMOLOGICAL MODELThe concordance cosmology assumed throughout this study is the six parameter ΛCDM model, implementing the most recent Planck Collaboration results <cit.>. The model consists of the homogeneous and isotropic Friedmann-Robertson-Walker (FRW) geometry dynamically evolving according to the Einstein field equations. The Einstein equations, also called the Friedmann equations <cit.> in this case, describe the evolution of the curvature and energy content of the universe as( ȧ/a)^2 +kc^2/a^2 = 8π G/3ρ,ä/a = -4π G/3(ρ+3P/c^2),where the scale factor is a≡ a_0(1+z)^-1, with the scale factor today a_0≡1 and z the cosmological redshift, k=0,±1 indicating zero, positive, or negative spatial curvature respectively, G is the universal gravitation constant, and c is the speed of light. The term ρ is the sum of the proper inertial mass densities of the cosmological fluid and the contribution from spatial curvature, and P is the pressure contribution from matter, radiation, and vacuum energy (or cosmological constant Λ).The equation of state of each cosmological fluid can be expressed P_i = w_iρ_i c^2 (no sum over i) with equation of state parameter w_i. The equation of state parameters for matter, radiation, cosmological constant, and spatial curvature are 0, 1/3, -1, and -1/3 respectively. Note that the equation of state of a baryonic gas P∝ρ^γ, where γ is the adiabatic index, is relevant when calculating accretion rates onto a compact object. The approximation w_i=b≈0 for a baryonic gas holds on cosmological scales.The Hubble parameter H is a measure of the temporal (extrinsic) curvature of the FRW geometry and is defined(ȧ/a)^2≡ H^2 = H_0^2ℰ^2,where subscript-0 implies evaluation of a quantity today. The Hubble constant is H_0 = 100h km s^-1Mpc^-1 where the dimensionless Hubble parameter is h=0.6774 from Planck <cit.>. Before defining ℰ^2, it is convenient to introduce the dimensionless density parameters todayΩ_i,0 ≡ρ_i,0/ρ_cr,0 = 8π G/3H_0^2ρ_i,0,where i indicates baryonic matter, dark matter, radiation, and Λ. Dividing Eq. (<ref>) by H^2 and evaluating the quantities today gives an expression for the `effective' dimensionless density parameter for spatial curvatureΩ_k,0 = 1 - Ω_0,where Ω_0 ≡Ω_r,0+Ω_m,0+Ω_Λ,0. The combined Planck and baryon acoustic oscillation data <cit.> are consistent with Ω_k,0 = 0.000±0.005, i.e. the universe has zero spatial curvature to within 0.5% accuracy. The term ℰ^2 in Eq. (<ref>) is a function of the dimensionless density parameters with their redshift dependenciesℰ^2≡∑_iΩ_i,0(1+z)^3(1+w_i)=Ω_r,0(1+z)^4 + Ω_m,0(1+z)^3==+(1-Ω_0)(1+z)^2 + Ω_Λ,0.In the FRW geometry, proper time is related to redshift through the differential ż = -H(z)(1+z). Therefore the time Δ t≡ t_2-t_1 elapsed between any two redshifts z_1 and z_2 is given by the integralΔ t≡∫_t_1^t_2 dt = H_0^-1∫_z_2^z_1dz/(1+z)ℰ(z),which has no tractable analytic solution ordinarily, but may be calculated analytically in simple cases or numerically in general. The age of the universe calculated numerically from Eq. (<ref>) is t_0=13.8 Gyr, which was reported in the 2015 Planck results <cit.>.The spatially-averaged inertial mass densities of the various components of the cosmological fluid decrease as power-laws with decreasing redshift according to their equation of state, i.e.ρ_i∝ (1+z)^3(1+w_i)for matter, radiation, curvature, and the cosmological constant. The average matter density in the universe evolves as ρ_m ∝ (1+z)^3. The effective mass density of radiation evolves as ρ_r ∝ (1+z)^4. In the early universe the redshift dependence of this term is more complicated due to the presence of radiation in the form of neutrinos and other relativistic Standard Model (SM) particles in addition to photons. Thus ρ_revolves <cit.> according to the expressionρ_r(T)= π^2/30g_⋆(T)k_B^4T^4/c^5ħ^3,where g_⋆(T) is the effective number of relativistic degrees of freedom, k_B is Boltzmann's constant, and T is the temperature. Thus the effective radiation mass density evolves with redshift as ρ_r∝ g_⋆(1+z)^4. A list of the important particle mass and energy thresholds is given in Table <ref>. Shown in Fig. <ref> is the corresponding plot of g_⋆ as a function of temperature.The factor g_⋆ increases by up to a factor 106.75/3.38∼31.6 at high redshift when all SM particles are relativistic. The factor g_⋆(T) for all relativistic particle species in thermal equilibrium can be calculated as the sum <cit.>g_⋆(T)= ∑_i=bg_i(T_i/T)^4 + 7/8∑_i=fg_i(T_i/T)^4,where the first term is the sum over all bosons and the second term is the sum over all fermions.Each component of the cosmological fluid has an associated temperature whose value depends on redshift. At high redshift after inflation the universe is dominated by radiation and the components of the cosmological fluid are in equilibrium. The temperature of radiation evolves simply as T_r = T_0(1+z) where T_r is the radiation temperature at any redshift lower than the neutrino decoupling redshift z_dec,ν and the temperature today is T_0 = 2.72548 K <cit.>. The baryonic matter is coupled to the radiation through Compton scattering prior to the thermalization redshift <cit.>1 + z_th ∼ 800(Ω_b,0h^2)^2/5∼ 174,and in this redshift regime the baryonic matter temperature evolves as T_b = T_0(1+z). After z_th, the baryonic matter temperature evolves adiabatically until reionization asT_b = T_0(1+z)^2/1+z_th,so that at thermalization, T_b = T_r(z=z_th).It is assumed that the dark matter is mostly weakly interacting massive particles (WIMPs) denoted χ. For simplicity other dark matter models are not considered in this study; see <cit.> for more details on all the dark matter models. For a discussion on the possibility of the PBHs themselves being the dark matter see <cit.>. If the dark matter is composed of WIMPs the DM temperature decouples from the radiation temperature at high redshift when thermal freeze out of the dark matter particles occurs <cit.>. The model forces the DM thermal freeze out to occur when k_BT_fr∼ m_χ c^2/20. If m_χ c^2 = 100 GeV the freeze out redshift is 1+z_fr = T_fr/T_0∼2×10^13.The root-mean-square velocity v_rms is approximated by comparing the relativistic kinetic energy of the dark matter particles to their thermal energy:(γ_χ-1)m_χ c^2= 3/2k_BT_χ,where the Lorentz factor associated with β_χ≡ v_rms/c of the dark matter particles is given byγ_χ ≡ (1-β_χ^2)^-1/2.It is useful to define the dimensionless quantityΘ_χ ≡3k_BT_χ/2m_χ c^2such that Eq. (<ref>) becomes γ_χ = 1+Θ_χ and thereforeβ_χ^2= Θ_χ^2+2Θ_χ/(1+Θ_χ)^2,which is friendly to numerical evaluations for all possible physical values of Θ_χ. The value of β_χ becomes close to unity quite rapidly and for the assumed freeze-out temperature, β_χ≃ 0.367, i.e. the dark matter particles are mildly relativistic at freeze-out. §PRIMORDIAL BLACK HOLE ACCRETION AND EVAPORATIONThe history of the universe may be divided into redshift regimes to simplify analysis. The relevant physical processes in the very early universe are distinct from those acting in the current era and thus it is important to summarize the physics in each regime.Though important to the dynamics of the universe in general, the history of the very early universe (prior to inflation) is not considered in detail in this study. The number density of a pre-inflation cosmological relic, e.g. any pre-inflation PBHs, is negligible after the inflationary epoch. The number density of a pre-inflation relic depends on the amount of inflation. The amount of inflation is calculated by finding the number of e-foldings during the inflationary epochN≡∫_a_i^a_fdln a = ∫_t_i^t_f H dt,where t_i denotes the start of inflation, t_f denotes the end of inflation, and H is the Hubble parameter given in Eq. (<ref>). A successful inflation model requires the number of e-foldings to be at least N_min∼50 in order to solve the horizon problem <cit.>. The number density of a relic which formed prior to inflation will thus decrease by a factor e^3N_min∼ e^150∼ 10^65. Thus any PBHs which formed prior to inflation are unlikely to be located in the observable universe today.After inflation, PBHs may form through a variety of mechanisms including collapse of primordial inhomogeneities, phase transitions, and cosmic string or domain wall collisions <cit.>. If the energy density fluctuations have a strength δρ/ρ∼ 1 in a particular spacetime volume, the region will likely collapse to a black hole. In this study it is assumed that the collapse to a black hole occurs on a time scale much shorter than the Hubble time so that the expansion is irrelevant to PBH formation. A black hole forming at a time t after the Big Bang will have a mass less than or equal to the Hubble mass at that time, i.e.M_H = c^3t/G∼ (4.0×10^14 g)t_-24,using the useful notation f = 10^nf_n. At t=1.0 s after the Big Bang, the Hubble mass is M_H ∼ 2.0×10^5M_⊙. A derivation of Eq. (<ref>) can be found in <ref>. Models for the mass function of PBHs are discussed in <cit.> with emphasis on the behavior of the mass function near the critical mass regime.Particles with spin s between energy E and E+dE are emitted near the horizon of a Schwarzschild black hole of mass M at a rate <cit.>dṄ = Γ_sdE/2πħ[exp(8π GME/ħ c^3)-(-1)^2s]^-1,where Γ_s is the absorption probability for a mode with spin s <cit.>. This is the so-called Hawking radiation which resembles emission from a blackbody with radius R_s = 2GM/c^2 of temperaturek_BT_BH = ħ c^3/8π GM = (10.6 MeV)M_15^-1.The BH can only radiate when the temperature of the hole is greater than that of the radiation bath of the early universe. The temperature of radiation in the early universe evolves as T_r∝ 1+z and is less than 10.6 MeV when z<4.5×10^10, i.e. when t>0.01 s. This will have a negligible effect on the evolution of PBHs near 10^15 g. The absorption probability asymptotes to the geometric optics limitΓ_s= 27G^2M^2E^2/ħ^2c^6when the particle energy isE ≫ k_BT_BH. The functional form of Γ_s is much more complicated for lower energy E∼ k_BT_BH interactions as discussed in <cit.>.The mass loss rate due to the Hawking emission from a Schwarzschild black hole of mass M requires a sum over all particle types and an integration over the particle energies leading to the simple equation dM_15/dt = (-5.34×10^-5 gs^-1)f(M)M_15^-2,where f(M) is a function <cit.> allowing for the emission of particles other than photons and f(M)=1 for M≫10^17 g. The function f(M) increases when the mass of the PBH crosses a particle mass threshold (see Table <ref>), after which the PBH may emit that particle. A good approximation is f(M) ∼ f(M_i) because for the majority of its lifetime the mass of a PBH remains near its formation mass M_i <cit.>.For supermassive black holes or stellar mass black holes the evaporation rate in Eq. (<ref>) is negligibly small. The evaporation rate becomes important on cosmological time scales for black holes with mass M∼10^15 g. This is seen by integrating Eq. (<ref>) to get the evaporation timescalet_evap = (6.24× 10^18 s)f(M_i)^-1M_i,15^3.Assuming t_evap = t_0 = 13.8 Gyr, the critical formation mass for evaporation today isM_cr = 5.1× 10^14 g,where the parameter f for the critical mass is f(M_cr)∼1.9 as assumed in <cit.>.In every cosmic era the Hawking evaporation of a near-critical mass PBH will compete with accretion of the cosmological fluid onto the hole. For the PBHs of M∼ M_cr the accretion turns out to be irrelevant if the hole accretes the cosmological fluid at spatially-averaged densities. For PBHs much smaller than M_cr accretion is completely unimportant. For PBHs larger than M_cr the accretion becomes ever more important and the evaporation rate becomes ever smaller. Thus it is important to quantify the various accretion rates at the relevant cosmic epochs. The accretion rates are dependent on the physical parameters of the cosmological fluid, which change dramatically with redshift. The full equation to be solved is the first-order nonlinear ordinary differential equation in MdM/dt = Ṁ_evap(f,M;z) + Ṁ_acc(ρ,c_s,M;z),where Ṁ_evap is given by Eq. (<ref>) and the mass accretion term Ṁ_acc will be calculated in the following sections. The equation is integrated from the formation time t_i to any desired final time (or the evaporation time for small holes) using Eq. (<ref>), with the concordance cosmological model accounted for at all times. The Hawking evaporation term has an explicit dependence on f(M) and the mass M of the hole and an implicit dependence on z (or t). The accretion term has explicit dependence on ρ, c_s, and M and an implicit dependence on z due to the evolution of those quantities across cosmic time.The mass accretion term in Eq. (<ref>) is split into its component partsṀ_acc(ρ,c_s,M;z)= Ṁ_r + Ṁ_b + Ṁ_χ,where r indicates radiation (γ, ν, and other SM particles), b indicates baryonic matter, and χ the dark matter particles. When the universe is cool enough (i.e. T_r < 0.511 MeV), the radiation term consists only of photons and neutrinos. At higher redshift, the other SM particles become relativistic and can be accreted. When the baryonic matter is coupled to the radiation the two accretion rates become coupled and are written Ṁ_b+r. In the sections following, the mass accretion term is calculated explicitly for the different cosmic eras.§.§Late universe accretionIn the late universe at z ≲ z_th the relevant cosmic scales are set by the formation and evolution of structure, i.e. the distribution of dark and baryonic matter in the cosmic web. The details of cosmic structure formation are ripe with rich and complicated physics and are not included in this study; see <cit.>.To set a bound on late universe accretion, all accretion terms here are set by the spatially-averaged fluid quantities. The PBHs in our universe will likely form and evolve within overdense regions, so the use of spatially-averaged quantities gives a good idea what to expect with relatively isolated holes. The accretion of radiation in this redshift regime is unimportant for ∼ M_cr PBHs because it is a horizon-limited growth given in the approximate formṀ_r= 4π R_S^2cρ_r= 16π G^2/c^3ρ_r,0(1+z)^4M^2= (6.5×10^-48 gs^-1)(1+z)^4M_15^2,where ρ_r is the equivalent mass density in radiation. The accretion rate in Eq. (<ref>) is comparable to the magnitude of the Hawking evaporation rate when the mass of the PBH isM= (6.3×10^25 g)(1+z)^-1.Thus the accretion of background radiation in this redshift regime is unimportant to critical mass holes. A PBH of the mass given in Eq. (<ref>) will not evaporate until long after the current era.The accretion of baryonic matter is more complicated as it is governed by gas dynamics in the vicinity of the PBH. If the PBH is in an `average' region of the universe, i.e. of average baryonic matter density and temperature, the accretion of baryons will be a competition between Bondi and Eddington-limited accretion <cit.>. The accretion rate found in this manner will inform a lower bound for any relevant PBH accretion activity. Below z∼ 30 the details of cosmic structure will change this simplified picture, but it is useful to set a first approximation. A complete picture of the baryonic accretion has not been properly solved and is the subject of intense study from both theoretical and observational perspectives. With the complicated gas dynamics removed from the analysis in this simplified calculation, the accretion rate can be expressed asṀ_b= min(Ṁ_b,B,Ṁ_b,E)= min(4πλ_sG^2/c_s,b^3ρ_b,0(1+z)^3M^2,4π Gm_p/σ_TcM).where λ_s = 1/4 for a γ = 5/3 baryonic gas <cit.>, m_p is the proton mass, and σ_T is the Thomson scattering cross section for electrons. It is clear that the Eddington limit is redshift-independent and is equal toṀ_E= (7.03×10^-2 gs^-1)M_15.The Bondi rate is also redshift-independent in this redshift regime. The temperature of the baryonic gas in this regime isT_b= T_r(1+z)/1+z_th = T_0(1+z)^2/1+z_th,and therefore the sound speed in the baryonic gas (assuming it is entirely hydrogen) isc_s,b = (5k_BT_b/3m_p)^1/2= (1+z)(5k_BT_0/3m_p(1+z_th))^1/2= (1.5×10^3 cms^-1)(1+z).Thus the redshift dependence of the Bondi rate goes away and Eq. (<ref>) becomesṀ_b= min[π G^2(5k_BT_0/3m_p(1+z_th))^-3/2ρ_b,0M^2,4π Gm_p/σ_TcM]=(1.9×10^-24 gs^-1)M_15^2, M<M_cr,1(7.0×10^-2 gs^-1)M_15, M>M_cr,1,where M_cr,1 = 3.8×10^37 g is the mass of a PBH that gives an equivalence in the Bondi and Eddington rates in this redshift regime. Comparing Eq. (<ref>) to Eq. (<ref>) it is clear that an isolated near-critical mass PBH cannot accrete sufficiently to beat the Hawking evaporation rate. The Bondi rate is comparable to the magnitude of the Hawking evaporation rate when the PBH has the characteristic massM_ch,1 = 7.3×10^19 g.Any relevant growth of a near-critical mass PBH in this redshift regime will have to come from enhanced accretion if the hole is located within a significant density perturbation such as an individual galaxy or galaxy cluster.§.§ Post-recombination accretionIn the post-recombination universe (z_th < z < z_rec) the matter temperature is coupled to the radiation temperature via Compton scattering, i.e. T_b = T_0(1+z). The recombination redshift is listed in <cit.> as z_rec = 1089.90. Starting 381,000 yr after the Big Bang and until thermal decoupling, the sound speed in the baryonic gas can be expressed asc_s,b = (5k_BT_b/3m_p)^1/2= (1+z)^1/2(5k_BT_0/3m_p)^1/2= (1.9×10^4 cms^-1)(1+z)^1/2.In this redshift regime, the accretion of radiation is still horizon limited and given by Eq. (<ref>). The accretion of baryonic matter is the Bondi accretion rate at lower mass and is Eddington-limited growth if the mass is large enough. Using the same arguments as beforeṀ_b= min(Ṁ_b,B,Ṁ_b,E)= min(π G^2/c_s,b^3ρ_b,0(1+z)^3M^2,4π Gm_p/σ_TcM)=(8.1×10^-28 gs^-1)(1+z)^3/2M_15^2, M<M_cr,2(7.0×10^-2 gs^-1)M_15, M>M_cr,2,where M_cr,2 = (8.7×10^40 g)(1+z)^-3/2 is the PBH mass that gives an equivalent Bondi and Eddington rate. The Bondi accretion rate in Eq. (<ref>) is comparable to the magnitude of the Hawking evaporation rate when the PBH has a characteristic massM_ch,2 = (5.1×10^20 g)(1+z)^-3/8.Thus in the post-recombination era until thermal decoupling, the relevant process for near-critical mass PBHs is Hawking evaporation.§.§Pre-recombination accretionIn the pre-recombination era (z_rec < z < z_mr) after matter-radiation equality the baryonic matter and radiation are fully coupled and cannot accrete independently. Thus the assumptions present in the Bondi accretion formula fail <cit.> and the accretion of the coupled fluid is horizon-limited. The temperature of the baryonic gas is coupled to the radiation temperature and the sound speed in the fluid can be written (see <ref>)c_s^2= c^2/34ρ_r/4ρ_r+3ρ_b.At the matter-radiation equality the sound speed in Eq. (<ref>) is a few percent below the asymptotic value c/√(3).The accretion rate of this coupled fluid onto a PBH is the horizon-limited rateṀ_b+r = 4π R_S^2c_s(ρ_r+ρ_b)= 16π G^2/√(3)c^3[4ρ_r,0(1+z)^4/4ρ_r,0(1+z)^4+3ρ_b,0(1+z)^3]^1/2==×[ρ_r,0(1+z)^4+ρ_b,0(1+z)^3]M^2.This rate has a complicated dependence on redshift so it is useful to expand the right hand side of Eq. (<ref>) near the boundaries of this redshift regime. Defining the intermediary terms ρ_r'≡4ρ_r,0 and ρ_b'≡3ρ_b,0 near z_rec the rate takes the formṀ_b+r = (6.8×10^-36 gs^-1)[0.26ρ_r'(1+z/1+z_rec)^4==+0.17ρ_b'(1+z/1+z_rec)^3]M_15^2.The redshift dependence of the sound speed in Eq. (<ref>) is included in the expansion above and in the expansion that follows. The rate in Eq. (<ref>) becomes comparable in magnitude to the Hawking evaporation rate when the PBH has a characteristic massM_ch,3a ≃ (6.2×10^22 g)[0.26ρ_r'(1+z/1+z_rec)^4==+0.17ρ_b'(1+z/1+z_rec)^3]^-1/4.So again the Hawking evaporation is most important for critical mass, i.e. Eq. (<ref>), PBHs. Closer to z_mr the rate in Eq. (<ref>) takes the formṀ_b+r = (5.5×10^-34 gs^-1)[0.25ρ_r'(1+z/1+z_mr)^4==+0.21ρ_b'(1+z/1+z_mr)^3]M_15^2,so the accretion of the baryonic radiation fluid occurs slowly for near critical PBHs. This rate becomes comparable in magnitude to the Hawking evaporation rate when the hole is of characteristic massM_ch,3b ≃ (2.1×10^22 g)[0.25ρ_r'(1+z/1+z_mr)^4==+0.21ρ_b'(1+z/1+z_mr)^3]^-1/4.The accretion of dark matter onto a PBH will be horizon-limited and should be quite small if the spatially-averaged cosmological value for ρ_χ,0 is assumed. The dark matter accretion rate isṀ_χ = 4π R_S^2cβ_χρ_χ,where β_χ is defined in Eq. (<ref>). The mass density of dark matter evolves according to ρ_χ = ρ_χ,0(1+z)^3 such that Eq. (<ref>) becomesṀ_χ = 16π G^2/c^3(Θ_χ^2+2Θ_χ)^1/2/1+Θ_χρ_χ,0(1+z)^3M^2.If there is an enhancement of the DM density term ρ_χ due to the formation of a DM halo there will be an appropriate enhancement of the DM accretion rate. Thus Eq. (<ref>) represents a lower limit on the DM accretion rate. For a treatment of accretion from an enhanced DM halo see <cit.>. Since the temperature T_χ of dark matter decoupled from the radiation temperature at z_fr∼2.1×10^13, the dimensionless quantity Θ_χ in this redshift regime is quite small. The expansion of β_χ for Θ_χ≪ 1 is β_χ≃ (2Θ_χ)^1/2. Thus Eq. (<ref>) becomesṀ_χ = 16π G^2/c^3[3k_BT_0/m_χ c^2(1+z_fr)]^1/2ρ_χ,0(1+z)^4M^2= (3.4×10^-58 gs^-1)(1+z)^4M_15^2,which is about ten orders of magnitude smaller than the accretion rate due to the baryon-radiation coupled fluid. In this regime the accretion rate of dark matter onto a PBH becomes similar to the Hawking evaporation rate whenM_ch,3c = (2.3×10^28 g)(1+z)^-1.The constraints on the accretion rates further strengthens the argument that accretion onto a critical mass PBH is unimportant and most if not all of the lifetime of such a PBH is dominated by the Hawking evaporation.§.§ Post-DM freeze-out accretionIn the post-DM freeze-out (z_mr<z<z_fr) era the universe is dominated by radiation. The dark matter, if it comprised of WIMPs, will be non-relativistic until redshifts higher than z_fr <cit.> and will accrete at a horizon-limited rate. The accretion of baryonic matter and radiation is horizon-limited as before. It is convenient to apply ρ_b≪ρ_r and therefore ignore the baryonic matter terms and allow c_s∼ c/√(3). Also in this redshift regime, the effective number of relativistic degrees of freedom g_⋆ begins to increase at higher redshift so it is important to express the radiation term as in Eq. (<ref>). The accretion rate is thereforeṀ_b+r = 4π R_S^2c_s(ρ_r+ρ_b)≃8π^3 G^2k_B^4T_0^4/15√(3)c^8ħ^3g_⋆(1+z)^4M^2= (2.0×10^7 gs^-1)(g_⋆/86.25)(1+z/1+z_fr)^4M_15^2.For a near critical mass PBH this is a large accretion rate compared to the magnitude of the Hawking rate. Thus the mass of a PBH in this redshift regime where these two rates balance isM_ch,4 = (1.5×10^12 g)(g_⋆/86.25)^-1/4(1+z/1+z_fr)^-1.The period of enhanced accretion in the early universe is quite short due to the strong redshift dependence, i.e. Ṁ∝ (1+z)^4 so no significant accretion is expected for critical mass PBHs. This is consistent with the findings from previous studies on PBH accretion, i.e. <cit.>. At high redshift a critical mass PBH will not accrete significantly, but massive PBHs can grow by about an order of magnitude by z_mr.In this redshift regime the accretion of DM onto the PBH is small. It is increasingly important at higher redshift but is never larger than the radiation accretion rate in Eq. (<ref>). At the DM freeze-out redshift the DM particles are somewhat relativistic, i.e. Θ_χ∼ 0.075, such that the accretion rate is of the same form as Eq. (<ref>) to a good approximation. In this regime Eq. (<ref>) also remains valid.§.§ Pre-DM freeze-out accretionIn this redshift regime the universe undergoes many changes as g_⋆ increases and all particles become relativistic. At high enough redshifts all particles have the same temperature and follow T=T_0(1+z). The accretion rate at these high redshifts is therefore the same as Eq. (<ref>). The PBH will not accrete radiation in the early universe if T_BH>T_r, which corresponds to z<4.5×10^10 if M = 10^15 g. The radiation accretion at these high redshifts is highly dependent on the particle physics model. This study employs the Standard Model with all the latest particle masses from <cit.>. The equivalent mass density in radiation changes dramatically in the early universe because of the change in g_⋆ as shown in Table <ref>.Table <ref> summarizes the relevant properties of the universe with reference to the equations they are first noted. Table <ref> summarizes the relevant evaporation and accretion rates of PBHs in the relevant redshift regimes with reference to the equations or sections they are first noted. §RESULTSFrom the evaporation and accretion expressions in <ref> it is possible to construct a rough accretion or evaporation history for any PBH with mass M_i forming at redshift z_i. The critical mass holes with M_i = M_cr∼ 5.1×10^14 g will suffer no significant accretion in their entire lifetime if located in a suitably `average' volume of the universe. They will assume the evaporation timescale in Eq. (<ref>) and evaporate according to Fig. <ref>. If the same PBHs of Fig. <ref> happened to form later, say at redshift z_i = 10^8, it would not affect their history due to the small timescales in the early universe. The accretion rate of the cosmological fluid onto larger PBHs at high redshift will be more important.The analysis in <ref> can be summarized in a look-up plot of M_f against M_i. The regime important for M_i∼ 10^-4M_⊙ holes is shown in Fig. <ref> and the entire mass regime is shown in Fig. <ref>. Note the agreement of Fig. <ref> in the near-critical mass regime to Figure 2 of <cit.>. The holes evaporating at higher redshift must have initial masses slightly lower than M_cr. Note that no significant accretion occurs across the intermediate mass regime between M_cr and ∼10^36 g due to the low accretion rates for BHs of this mass. Isolated PBHs in this mass regime accreting the spatially-averaged cosmological fluid do not grow much. This does not account for enhancement of the accretion rates due to structure formation and thus represents a first approximation. If the accretion rate is enhanced via Ṁ_b,B→ f_bṀ_b,B where f_b = ρ_enh/ρ_b is an enhancement factor and ρ_enh is the enhanced baryonic matter density, then a PBH of given initial mass can reach a higher mass for a given final redshift. This is reflected in the dotted lines of Fig. <ref>, which show the final mass of a PBH growing from 1.0 s after the big bang to z=0.1 given an enhancement factor f_b=10^1,10^2,10^3. Even a small enhancement of the baryonic matter density leads to a large increase in the possible final mass of the accreting PBH. Since the Bondi accretion rate is proportional to M^2, higher mass PBHs will accrete more than lower mass PBHs and this increase in the accretion rate is indicated by the increasing M_f in Fig. <ref> around M∼10^36 g.In the first few seconds of the universe (z≳10^9), PBHs approaching the formation mass limit around 10^38 g have a large accretion rate (see Eq. <ref>). This large accretion rate, though short-lived, can increase the mass of the PBH by about an order of magnitude by z=10^9. This effect is absent in lower mass PBHs and thus is visible in Fig. <ref> as a small increase beginning above M_i∼10^38 g. §CONCLUSIONS AND DISCUSSIONA comprehensive view of the evolution of PBHs throughout cosmic history was presented. The accretion and evaporation histories of PBHs with masses in the approximate range 10^14 g < M < 2×10^38 g were calculated. PBHs with lower masses will have evaporated prior to the current era and are not considered and PBHs with higher masses are not allowed due to the Hubble mass constraint of Eq. (<ref>). The accreted fluids were assumed to have spatially averaged cosmological densities and the details of structure formation were not included. The important quantities for accretion are the mass densities of the various cosmological fluids, the sound speed in those fluids, and the details of their behavior at all relevant redshifts. All of these details were calculated precisely for the ΛCDM concordance cosmology.The important findings of this study are the following: * A PBH with initial mass near M_cr=5.1×10^14 g will not accrete radiation or matter in any significant quantity and will thus evaporate according to the timescale given in Eq. (<ref>). A PBH with initial mass less than M_cr will evaporate prior to the current era.* A PBH with initial mass in the approximate range 10^15 g < M_i < 10^35 g neither evaporates nor accretes significantly over a Hubble time. Such a PBH would have to grow by other means, i.e. merging with other BHs or accreting while in an overdense region of the universe. Since the Hawking evaporation rate is so small for PBHs in this mass regime, the lower limit on the final (observed) mass of such PBHs is thus simply M_f = M_i.* A PBH with initial mass M < 10^38 g will not grow significantly in the early universe, i.e. within the first few minutes after the Big Bang. This finding is consistent with other PBH accretion studies, e.g. <cit.>. The small increase for BHs with M_i∼10^38 g seen in Fig. <ref> results from the large accretion rate for high-mass holes in Eq. (<ref>). It represents a growth of approximately one order of magnitude in the early universe, consistent with previous studies. There is negligible growth of critical mass PBHs in the radiation-dominated era.* A PBH with initial mass in the approximate range 10^35 g < M_i < 10^37 g can accrete significantly during its lifetime. In the redshift regime z_th<z<z_rec, a PBH with M < (8.7×10^40 g)(1+z)^-3/2 accretes at the Bondi rate and is Eddington-limited above that. In the redshift regime z≤ z_th, a PBH with M < 3.8×10^37 g accretes at the Bondi rate and is Eddington-limited above that. A PBH with such a mass that grows at the Bondi rate for its whole lifetime can thus grow by one or two orders of magnitude.* When a PBH grows enough to have its baryonic matter accrete at an Eddington-limited rate, the hole can increase in mass by many orders of magnitude if evolving into the late universe z_f∼ 0. Since the PBH will grow by accreting the spatially averaged cosmological gas, this growth represents how an `average' PBH accretes at the Eddington limit. The true accretion history of course will be complicated by feedback effects which were not modeled here. The curves in Fig. <ref> thus represent an `average' growth. A true astrophysical hole of this mass may grow at either a higher or a lower rate.* A PBH with initial mass in the approximate range 4×10^37 g < M_i < 10^38 g will accrete at an Eddington limited rate after z_rec and the final mass of such a hole depends on its observed redshift. At z_f=30, the hole can only grow to M_f=1.01M_i. The hole can grow to M_f=7.7M_i if z_f=6. The hole can grow to M_f=6.4×10^11M_i if z_f=0.1 and to M_f=1.1×10^13M_i if z_f=10^-4. See Fig. <ref> for more details.The PBH mass histories discussed in this study represent a first approximation of their cosmic behavior. Several astrophysical applications may be discussed in the context of the above results: * It is impossible to explain the large BHs with M∼10^10M_⊙ observed <cit.> at z>6 via PBHs with Eddington-limited accretion of the `average' baryonic gas, even with M_i∼ 10^5M_⊙. These holes must be explained through multiple massive PBH mergers, mergers with BH seeds from the first generation of stars, or PBHs in overdense regions accreting at super-Eddington rates.* PBHs do not easily grow to 30M_⊙ by z∼0.1 through Bondi accretion of the `average' cosmological fluid. These PBHs cannot easily explain the binary BH mergers observed by LIGO as the gravitational wave events GW150914 <cit.> and GW151226 <cit.> unless they experience an enhancement of the Bondi rate through various channels. One such channel is a baryonic matter density enhancement leading to Ṁ_b,B→ f_bṀ_b,B as discussed in 4. Small enhancement factors allow a lower mass PBH to reach 30M_⊙ compared to those PBHs accreting the average cosmological baryonic matter. Another possibility is the LIGO BHs were PBHs that formed with an initial mass M_i=M_f, where M_f is their mass at the merger time. According to <cit.>, the event rate for PBH mergers would be high enough to explain the GW events if the PBHs constitute a large enough fraction of the dark matter. However, PBHs in the appropriate mass range to explain these LIGO events are unlikely to be a large enough fraction of the DM as constrained from CMB measurements discussed in <cit.>, <cit.>, and <cit.>. Either LIGO has chanced upon two relatively rare PBH mergers or there is a common stellar evolution channel that produces BHs of these masses. Both explanations are interesting and more data are needed to distinguish these two possibilities.* Searches for PBH bursts <cit.> are ongoing. Although there are candidates for such events, no confirmed PBH burst event has been detected. The spectral properties of such bursts should be distinguishable from the `normal' GRBs. The non-detection of such an event has a few explanations. First, the fraction of PBHs that make up the dark matter must be quite low for PBHs of the relevant mass scale (see Fig. 9 of <cit.>). Thus it is plausible that not enough of these PBHs exist to subsequently evaporate and trigger gamma-ray detectors. Second, it might be possible for the critical mass holes to accrete enough to no longer evaporate in the current era. However, the accretion rate is too small and this would not explain the non-detection of PBH bursts. Even if the accretion rate onto small PBHs happened to be large enough, there would be smaller PBHs that would accrete enough to reach M_cr anyway, filling the void of critical mass holes.§ ACKNOWLEDGMENTSThis work was supported by the Nevada NASA Space Grant College and Fellowship Training Program Cooperative Agreement #NNX10AN23H. We thank the anonymous referee for multiple suggestions that significantly improved the quality of the manuscript. J.R.R. thanks the UNLV High Energy Astrophysics Group for fruitful discussions. We extend our gratitude to Dr. Qing-Guo Huang of the Institute of Theoretical Physics, Chinese Academy of Sciences who alerted us to an error in an equation that is now fixed.§MAXIMUM PBH FORMATION MASSAs discussed in <ref> the maximum formation mass of a PBH will be the Hubble mass, i.e. the mass contained within the Hubble volume at a given time. The Hubble volume isV_H= 4/3π R_H^3,where R_H = 2ct is the Hubble radius in the early radiation dominated universe. The critical density is the Hubble mass in a Hubble volume and thusM_H(t)= ρ_crV_H= 3H^2/8π G·4/3π(2ct)^3= 4H^2c^3t^3/G,but in a radiation dominated universe H=1/(2t) so thatM_H= c^3t/G,which recovers Eq. (<ref>). See <cit.> for more detailed discussions of the PBH mass function.§SOUND SPEED IN THE COSMOLOGICAL FLUIDIn the late universe at redshifts lower than z_th, the temperature of the baryonic matter decouples from the CMB photon temperature. Thus the sound speed in the baryonic fluid is given by Eq. (<ref>)c_s,b = (1.5×10^3 cms^-1)(1+z),where the increase due to reionization around z∼9 is not taken into account. In the redshift regime z_th<z<z_rec the redshift dependence changes due to the temperature coupling between the baryonic matter and the CMB radiation. Thus the sound speed evolves as Eq. (<ref>)c_s,b = (1.9×10^4 cms^-1)(1+z)^1/2.In the above equations it is assumed that the baryonic matter is composed entirely of hydrogen; corrections due to the helium and metal content of the baryonic matter need to be made for a more realistic calculation.In the early universe at redshifts higher than the recombination redshift z_rec∼1090, the baryonic matter is coupled to the CMB radiation. The sound speed in such a coupled fluid can be found by calculatingc_s^2= (∂ P/∂ρ)_s,where the subscript s on the right hand side indicates taking the derivative at constant entropy. The dominant pressure term is the radiation pressure and the density is a sum of radiation and baryonic terms ρ = ρ_r + ρ_b. The dark matter does not contribute to the pressure or density terms but has an early influence when it is relativistic at redshifts greater than z_fr∼2.1×10^13.Rewriting the partial derivatives of Eq. (<ref>) in terms of temperature givesc_s^2= (∂ P_r/∂ T)_s/(∂ρ_r/∂ T)_s+(∂ρ_b/∂ T)_s.Recalling Eq. (<ref>) and P_r = ρ_r c^2/3 the numerator of Eq. (<ref>) is(∂ P_r/∂ T)_s= 4π^2/90g_⋆(T)k_B^4T^3/c^3ħ^3 = 4ρ_rc^2/3T, ignoring the small ∂ g_⋆/∂ T terms. Similarly, the first term in the denominator of Eq. (<ref>) is(∂ρ_r/∂ T)_s= 4π^2/30g_⋆(T)k_B^4T^3/c^5ħ^3 = 4ρ_r/T.Recalling at high redshift the radiation and baryonic gas temperatures are coupled, i.e. T_r = T_b = T and using T = T_0(1+z), the second term in the denominator of Eq. (<ref>) is(∂ρ_b/∂ T)_s= ∂/∂ T(ρ_b,0T^3/T_0^3) = 3ρ_b/T.Combining Eq. (<ref>–<ref>) into Eq. (<ref>) givesc_s^2= c^2/34ρ_r/4ρ_r+3ρ_b.It is clear that at redshifts higher than z_mr∼3400 the sound speed calculated using Eq. (<ref>) asymptotes to c_s∼ c/√(3)∼ 0.577c. The behavior of the sound speed across all relevant redshifts is plotted in Fig. <ref>. § REFERENCES

http://arxiv.org/abs/1702.08069v2

D.V. Skobeltsyn Institute of Nuclear Physics, M.V. Lomonosov Moscow State University, Moscow 119991, RussiaD.V. Skobeltsyn Institute of Nuclear Physics, M.V. Lomonosov Moscow State University, Moscow 119991, RussiaInstitute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary Prospect 7a, Moscow 117312, RussiaD.V. Skobeltsyn Institute of Nuclear Physics, M.V. Lomonosov Moscow State University, Moscow 119991, RussiaInstitute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary Prospect 7a, Moscow 117312, RussiaInstitute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary Prospect 7a, Moscow 117312, RussiaD.V. Skobeltsyn Institute of Nuclear Physics, M.V. Lomonosov Moscow State University, Moscow 119991, Russiast@ms2.inr.ac.ru Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary Prospect 7a, Moscow 117312, RussiaResults of the search for ∼ (10^16 - 10^17.5) eV primary cosmic-ray photons with the data of the Moscow State University (MSU) Extensive Air Shower (EAS) array are reported. The full-scale reanalysis of the data with modern simulations of the installation does not confirm previous indications of the excess of gamma-ray candidate events. Upper limits on the corresponding gamma-ray flux are presented. The limits are the most stringent published ones at energies ∼ 10^17 eV.Constraints on the flux of ∼ (10^16 - 10^17.5) eV cosmic photons from the EAS–MSU muon data S. V. Troitsky December 30, 2023 ===========================================================================================§ INTRODUCTION Searches for astrophysical gamma rays with energies 10^15 eV≲ E_γ≲ 10^20 eV attracted considerable attention for years <cit.>. At these energies, photons interact with the atmosphere and produce extensive air showers (EAS) which may be detected by installations studying cosmic rays. The primary motivation for the studies includes multimessenger astronomy. Photons work as a diagnostic tool to distinguish between various models of the origin of energetic cosmic rays and neutrinos. In particular, because of the pair production on cosmic background radiation, the photon flux from extragalactic sources is strongly suppressed at sub-PeV to sub-EeV energies, and non-observation of gamma rays in this energy band would strongly favor the extragalactic origin of IceCube-detected astrophysical neutrinos over the Galactic models <cit.>. At higher energies, non-observation of cosmogenic photons <cit.> would strongly constrain models with proton composition of cosmic rays with E ≳ 10^19.5 eV, see e.g. Refs. <cit.>. Contrary, observation of high-energy photons may be a smoking-gun signal of new physics, including superheavy dark matter <cit.> (see also Ref. <cit.> and, for a recent reanalysis, Ref. <cit.>), axion-like particles <cit.>, or an ultimate test of Lorentz-invariance <cit.>.For air-shower experiments, the main problem in the photon search is to separate gamma-ray induced events from the usual, hadron-induced air showers. One of the best discriminating variables is the muon number of a EAS, which is much lower in photon-induced events, compared to the bulk of showers. Indeed, a photon-induced shower develops by means of electromagnetic interactions mostly, and the only source of muons is provided by photonuclear reactions, which have a relatively small cross section. Contrary, in a hadron-induced shower, lots of muons are produced in decays of π mesons, born in hadronic interactions. Unfortunately, muon detectors are small or missing in many modern cosmic-ray experiments. Here, we take advantage of having a large muon detector in the EAS-MSU experiment and use its data to search for air showers poor in muons.Previous preliminary studies indicated some excess of gamma-ray candidates in the EAS-MSU data with respect to the expected background <cit.>. These results invited for a detailed study of the data with state of art methods. This study has been performed: a full Monte-Carlo model of the data and the detector was constructed <cit.> which gives an excellent description of the surface-detector results <cit.> as well as of the bulk of muon-detector data <cit.>. Here, we present the ultimate results of this study and consider muonless photon-candidate events. We will see that, within the present approach, no excess of these photon-like events is seen in the data. This allows us to put upper limits on the primary gamma-ray flux which are world-best at energies E_γ∼ 10^17 eV.The rest of the paper is organized as follows. In Sec. <ref>, we briefly describe the installation, the data and the Monte-Carlo simulations used in this work; references to more detailed descriptions are given. We derive the main result of the paper, the limits on the gamma-ray flux, in Sec. <ref>. In Sec. <ref>, we estimate systematic uncertainties of the result and compare it with other studies, including previous photon searches in the EAS-MSU experiment. We briefly conclude in Sec. <ref>.§ DATA AND SIMULATIONS The EAS-MSU experiment and the Monte-Carlo model are described in detail in the previous papers <cit.>. The installation consists of 76 charged-particle detector stations, in which multiple Geiger-Mueller counters are located which determine the number of charged particles (N_e) in a shower. The total number of Geiger-Mueller counters in the surface part of the installation is about 10,000. The total area of all Geiger-Mueller counters is ∼ 250 m^2; the total area of the installation is ∼ 0.5 km^2. In this work, we use the data of the large muon detector located at the center of the array at a depth of 40 meters of water equivalent underground corresponding to the threshold energy of 10 GeV for vertical muons. The muon detector had the total area of 36.4 m^2 and consisted of Geiger-Mueller counters with area of 0.033 m^2 each.For the present study, we select events with the following cuts:* the event passes the reconstruction procedure and the reconstruction quality criteria are satisfied <cit.>; * the age parameter of the EAS is in the range 0.3<S<1.8; * the reconstructed zenith angle θ is below 30^∘;* the distance between the reconstructed shower axis and the array center where the muon detector is located is R<240 m;* the reconstructed shower size is N_e>10^7;* out of 32 sections of the muon detector, at least 28 were operational.The only difference with the cuts used in Ref. <cit.> is the lower N_e threshold which is chosen to cover a wider range of energies of potential photons. The extension of the N_e range requires a new comparison of data to Monte-Carlo to validate the simulations. The corresponding distributions of N_e, shower age S, core distance of the array center R, zenith angle θ and muon density at 100 meters ρ_μ(100) are shown in Appendix.With the account of the muon detector operation cut, we are left with 1204 days of data taking within 1984–1990. The data set contains 3148 events.Following the previous studies, we consider the following criteria to select photon-candidate events: the muon detector is not triggered by the shower. However, low-energy protons do not always produce a sufficient number of muons in EAS to activate the muon detector, in particular, at large distances between the detector and the shower axis. To evaluate the background of muonless events from hadronic primaries, we make use of the full Monte-Carlo (MC) model of the EAS-MSU array described in Ref. <cit.>. It includes simulations of artificial air showers by the CORSIKA7.4001 <cit.> package with theQGSJET-II-04 <cit.> high-energy hadronicinteraction model, FLUKA 2011.2c <cit.> low-energy hadronic interaction model and EGS4 <cit.> electromagnetic model. Artificial EAS are recorded and processed identically to the experimental data. The Monte-Carlo simulations have been performed for proton and iron primaries, see Ref. <cit.> for details; a realistic composition was assumed to be a mixture of the two. The muon component of EAS is highly dependent on the primary composition, therefore the number of muonless background events depends on the assumed proton fraction. For this study, the primary composition has been determinedin Ref. <cit.> by fitting the observed distribution of the muon densities. The assumed fraction of protons is 46 ± 6 %. We note that, as discussed in Ref. <cit.>, it agrees well with the composition obtained from the EAS-MSU surface-detector data <cit.>. More details on the MC set with hadronic primaries may be found in Ref. <cit.>.We also need a MC simulation with gamma-ray primaries which is used to determine the efficiency of the installation for the gamma-ray detection, to relate the reconstructed N_e to the primary-photon energy E_γ and to account for the amount of photon showers which do not produce photon-candidate (that is, muonless) events. The simulation and reconstruction of the artificial photon showers was performed in a way similar to that for hadron-induced events <cit.>. The total number of simulated independent gamma-induced showers is 300, their thrown energies follow the E_γ^-1 spectrum with lower bound of MC 10^16<E_γ≤ 10^17.5 eV, and they are selected at the reconstruction stage to reproduce the primary spectrum ∼ E_γ^-2, as is customary in high-energy photon searches. The dependence of the resulting limits on the assumed spectrum is through the efficiency only and is weak. The total number of MC realizations of gamma-induced events is 27310 (see Ref. <cit.> for description of the sampling). Of them, 3898 passed all cuts. § THE GAMMA-RAY FLUX LIMIT The total number of muonless events in the set is 86, while the expected number of background muonless events from primary hadrons is 80.1. The muon detector core distance distribution of the observed and expected muonless events is shown in Fig. <ref>. To study various energy ranges, we consider certain subsamples of the data. The quantity reconstructed for each shower is N_e, not the primary energy E; the relation between the two quantities may be obtained from simulations. The (E-N_e) relations are different for photons and for hadrons, and it is important to keep track of this difference for gamma-ray searches, see e.g. Refs. <cit.>. The mean gamma-ray energy E_γ is related to N_e asN_e(E_γ)= 4.1 × 10^-10E_γ/eV,see Fig. <ref>for the MC simulated points and the fit. The condition used to select the data for the search of photons with E_γ> E_γ^ min is defined asN_e>max{ 10^7, aN_e(E_γ^ min) },where the coefficient a=0.56 was chosen in such a way that at least 90% of MC photon-induced showers with E_γ>E_γ^ min are reconstructed with these N_e.For each energy cut E_γ^ min, we determine the number of observed muonless events, n_ obs, in the sample, as well as the expected number of muonless events from the background of hadron-induced showers, n_ b. No excess of muonless events is seen, and we estimate the maximum number of photon-induced events in the sample, n_γ^ FC, by means of the standard Feldman–Cousins method for the Poisson distribution <cit.>, for the 90% and 95% confidence levels (CL). The upper limits on the photon flux are then estimated as a ratio of n_γ^ FC and the effective exposure of the experiment to photons of the given energy range. The effective exposure accounts for the fraction of photon events lost in the reconstruction and, also, of reconstructed photon events which are not muonless. We estimate the effective exposure as follows.The geometrical exposure for the conditions we use (R ≤ R_ max=240 m, θ≤θ_ max=30^∘) is given by A_ geom=Ω× S × T, where Ω=2π(1-cosθ_ max) is the solid angle, S=π R_ max^2 is the area and T=14060.7 hours is the on-time of the installation corresponding to the data set used. Note that the R cut is defined in the plane orthogonal to the shower axis and therefore Ω is calculated differently from the conventional case when exposure is determined by the area of the array.MC photon-induced showers are thrown in a square with area S_ MC=(280  m)^2=0.3136 km^2 and with zenith angles up to θ_ MC=35^∘. The corresponding MC geometrical exposure is then A_ MC=Ω_ MC× S_ MC× T, where Ω_ MC=πsin^2θ_ MC and θ_ MC=35^∘. We calculate the number n_ pass, 0μ of events from the MC set which passed all cuts (that is, in particular, were reconstructed with geometrical properties corresponding to A_ geom), satisfy the criterion (<ref>) and are muonless and divide it by the number n_ MC of thrown MC events (corresponding to A_ MC). The effective exposure is then given byA_ eff=n_ pass, 0μ/n_ MC A_ MC . The flux limit is then obtained asI_γ=n_γ^ FC/A_ eff,where n_γ^ FC corresponds to the required confidence level. Next, we define the exposure correction as a ratio of the effective exposure to geometrical one:ξ = A_ eff/A_ geom . Note that the exposure correction factor may exceed unity because Monte-Carlo events are thrown to the area larger than the installation. Figure <ref>presents the exposure correction factor ξ as a function of energy. For comparison, the reconstruction efficiency for primary photons and protons is also shown (it is determined in a similar way as ξ but without the muonlessness condition and with the criterion N_e>10^7 instead of Eq. (<ref>)). Decline of ξ at higher energies reflects the fact that the probability for a primary photon to produce a EAS which is not muonless grows with energy.Our limits on the integral gamma-ray flux, which represent the main result of this work, are presented in Table <ref>. § DISCUSSION§.§ Systematic uncertainties The main part of systematic uncertainties in the study comes from the simulation of the background of muonless events from hadronic showers. Indeed, it is known that hadronic-interaction models used in the air-shower simulations are not perfect, in particular in the part related to the description of the muon content of EAS. While our previous study indicates <cit.> that the bulk of E>10 GeV muon data of the EAS-MSU experiment is well described by the QGSJET-II-04 simulations, assuming the primary chemical composition implied by the surface-detector studies, this is not directly tested for muon-poor showers. The uncertainties of the hadronic model, in principle, may reveal itself in the incorrect estimation of the background. Indeed, we note (cf. Table <ref>) that for certain energy ranges, the number of observed muonless events in our sample is smaller than expected under the background-only hypothesis. Therefore, in addition to the standard statistical estimate of the upper limit on the gamma-ray flux, we calculated also the “expected” flux upper limits which would be obtained if the number of observed muonless events followed the simulations under the background-only hypothesis. Alternatively, one may estimate the flux limits which might be obtained under assumption that the MC model doesn't provide a reliable prediction of the background. To this end, we use the “data-driven background”, that relies on the assumption that the correct background is equal to the number of the observed muonless events. These “expected” and “data-driven-background” limits on the gamma-ray flux are presented in Table <ref> and compared in Fig.<ref>.We also estimate the systematic errors associated with the uncertainty of the chemical composition. The change of the proton fraction within it's error ± 6% results in the energy dependent correction of flux limits. The variation of limits is 21% for the minimum energy and 4% for maximum energy. A careful look at the relation between n_ b and n_ obs reveals one peculiarity which is most probably related to the modelling of hadronic interactions but, in principle, might also be explained in terms of the presence of a certain amount of primary photons. The ratio n_ obs/n_ b remains constant, ∼ 0.4, for energies E_γ≳ 8 × 10^16 eV, but quickly raises to ∼ 1 below this energy. If, due to some systematics in the modelling of the background, the real n_ b is indeed ∼ 0.4 of the MC one, then we have a certain excess of muonless events at 2× 10^16 eV≲ E ≲ 8 × 10^16 eV, which may correspond to an excess of primary photons expected, for instance, in the heavy dark-matter decay scenario. §.§ Comparison with previous EAS-MSU results Previous analyses of the EAS-MSU data in the same energy range suggested an excess of muonless events which might be explained by the presence of a certain amount of photons in the primary cosmic-ray flux at E∼ 10^17 eV <cit.>. The present study does not confirm that claim and puts strong upper limits on the gamma-ray flux, see Table <ref>. It is therefore important to understand the differences between this analysis and the previous ones.With respect to the previous preliminary analysis, this final study has several important advantages. First, it is based on the new full Monte-Carlo description of the air showers and of the installation <cit.>. This uses modern simulation tools and an analysis technique with the real and simulated events processed by one and the same reconstruction program. In such a way, we take into account all possible biases introduced at the reconstruction step, as well as keep track of rare fluctuations in the EAS development and registration. Second, the reconstruction program has been slightly revised for this study. The main overall effect of the reconstruction update is that, while the muonless events remain in the data sample, their reconstructed energies become, for most of them, lower than before. For these lower energies, the background of muonless hadron-induced showers is higher, and the same amount of muonless events does no longer represent an excess. Third, an additional check of the quality of muon data was performed: at least 28 out of 32 sections of the muon detector are required to be operational. To be specific, let us consider 48 muonless events with N_e≥ 2× 10^7 studied in Ref. <cit.>. Of them, 28 events have N_e< 2× 10^7 in the new analysis; 4 events arrived before 1984 (not included in the present data set); 10 events arrived at the days excluded from the present analysis because of stricter criteria on the quality of muon data; 6 muonless events remained in the data set. In addition, 3 new muonless events joined the data set in the new reconstruction (they did not pass the cuts in the old one), so the total number of muonless events with N_e≥ 2× 10^7 is now 9. This reduced number of photon candidates is below the MC background expectation of 18.9 events, so no excess is present in the updated data set. Our new results are compared to previous studies in Fig.<ref>. Finally, the efficiency of the muonless detection of photons is estimated in the present work with the full photon Monte-Carlo. The numerical values of limits became apparently weaker due to account of the fact that only about 40% of the photon-induced showers are registered as muonless. §.§ Comparison with other results and possible applications Many experiments searched for primary photons with the EAS technique and none has yet found any. Our flux limits are compared to others in Fig. <ref>.We see that our limits are similar to those of the KASCADE-Grande experiment, being world-best for certain energies. At the same time, one can see that the energy range (10^16-10^18) eV discussed here is one of the least studied bands. While future studies to improve the limits are important, already the present data may be used to constrain astrophysical models with Galactic sources of PeV neutrinos <cit.>or decaying dark matter with appropriate mass and sufficiently hard spectrum (see <cit.> for a review). § CONCLUSIONS This work presents the results of the search for primary photons in the EAS-MSU data. The photon candidate events are defined as ones givingno signal in the muon detector of the installation. We made use of the full Monte-Carlo simulation of the installation and of the updated reconstruction of EAS parameters. Contrary to the previous analysis of the same data, no evidence was found for an excess of photon-candidate events, and this fact allowed us to put upper limits on the diffuse flux of primary gamma rays at energies above ∼ (10^16-10^17) eV. For certain energies, the limits are world-best ones. The difference with the previous study is, mainly, due to the change of reconstruction: the energies of muonless events moved downwards, while the background of muonless showers from primary hadrons is higher at lower energies. The limits obtained in this work may be used in multimessenger astrophysics as well as for constraining exotic particle-physics models.§ ACKNOWLEDGEMENTSST thanks J.-M. Frère for a discussion on the data-driven background and the Service de Physique Théorique at Université Libre de Bruxelles for hospitality at the final stages of the work. Monte-Carlo simulations have been performed at the computer cluster of the Theoretical Physics Department, Institute for Nuclear Research of the Russian Academy of Sciences. The experimental work of the EAS-MSU group is supported by the Government of the Russian Federation (agreement 14.B25.31.0010) and the Russian Foundation for Basic Research (project 14-02-00372). Development of the analysis methods and application of them to the EAS-MSU data is supported by the Russian Science Foundation (grant 14-12-01340). 381975 G.B. Khristiansen, G.V. Kulikov, Yu.A. Fomin,“Ultra-high-energy cosmic radiation”, Moscow, Atomizdat, 1975.Risse M. Risse and P. Homola, “Search for ultrahigh energy photons using air showers,” Mod. Phys. Lett. A 22 (2007) 749 [astro-ph/0702632].AhlersM. Ahlers and K. Murase,“Probing the Galactic Origin of the IceCube Excess with Gamma-Rays,” Phys. Rev. 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DISTRIBUTION OF EAS PARAMETERS FOR DATA AND MONTE-CARLO In this appendix, we provide the data to Monte-Carlo comparison of the EAS parameter distribution for the extended cut N_e>10^7 used is the present work. The primary composition is the same as determined in Ref. <cit.> by fitting the observed distribution of the muon densities: 46% protons and 54% iron. The distributions of S, N_e, R, θ and muon density at 100 meters ρ_μ(100) are shown in Figures <ref>-<ref>.The validity of the N_e cut extension is verified by reasonable agreement of the data and Monte-Carlo.

http://arxiv.org/abs/1702.08024v1

german.lugones@ufabc.edu.br Centro de Ciências Naturais e Humanas, Universidade Federal do ABC, Avenida dos Estados 5001, 09210-580 Santo André, São Paulo, Brazil.jose.arbanil@upn.pe Departamento de Física, Instituto Tecnológico de Aeronáutica, Centro Técnico Aeroespacial, 12228-900 SãoJosédos Campos, São Paulo, Brazil. Departamento de Ciencias, Universidad Privada del Norte, Av. Alfredo Mendiola 6062 Urb. Los Olivos, Lima, Lima, Perú. We study the properties of compact stars in the Randall-Sundrum II type braneworld model. To this end,we solve the braneworld generalization of the stellar structure equations for a static fluid distribution with spherical symmetry considering that the spacetime outside the star is described by a Schwarzschild metric. First, the stellar structure equations are integrated employing the so called causal limit equation of state (EOS), whichis constructedusing a well established EOS at densities below a fiducial density, and the causal EOS P= ρ above it. It is a standard procedure in general relativistic stellar structure calculations to use such EOS for obtaining a limit in the mass radius diagram, known as causal limit, above which no stellar configurations are possible if the EOS fulfills that the sound velocity is smaller than the speed of light. We find that the equilibrium solutions in the braneworld model can violate the general relativistic causal limitand, for sufficiently large mass they approach asymptotically to the Schwarzschild limit M = 2 R.Then, we investigate the properties of hadronic and strange quark stars using two typicalEOSs: a nonlinear relativistic mean-field model for hadronic matter andthe MIT bag model for quark matter.For masses below ∼ 1.5 - 2 M_⊙,the mass versus radius curves show the typical behavior found within the frame of General Relativity. However, we also find a new branch of stellar configurations that can violate the general relativistic causal limit and that in principle may have an arbitrarily large mass.The stars belonging to this new branch are supported against collapse by the nonlocal effects of the bulk on the brane. We also show that these stars are always stable under small radial perturbations.These results support the idea that traces of extra-dimensions might be found in astrophysics, specifically through the analysis of masses and radii of compact objects.04.50.-h, 11.25.Wx, 97.60.Jd, 26.60.-cCompact stars in the braneworld: a new branch of stellar configurations with arbitrarily large mass José D. V. Arbañil Received: date / Accepted: date =================================================================================================== § INTRODUCTIONBraneworld models represent the universe as a three-dimensional brane where elementary particles live embedded in a higher-dimensional spacetime called the bulk, only accessed by gravity <cit.>. Within thisframework, astrophysical and cosmological models can be constructed where the gravitational effect of extra-dimensions can be assessed. Two well known examples of braneworld models are Randall-Sundrum (RS) <cit.>and Dvali-Gabadadze-Porrati (DGP) <cit.> models. In RS models, ultraviolet modifications to General Relativity are introduced. Significant deviations from Einstein's theory occur at very high energies, e.g. in the early universe, in gravitational collapse and in compact objects. DGP models present infrared modifications with respect to General Relativity.In the present work we focus on the RS type II braneworld model <cit.>, whichhas attracted much attention because it includes nontrivial gravitational dynamics despite a simple construction. In the RS model, our universe is a brane embedded in one extra dimension (the bulk)whichis a portion of a 5D anti-de Sitter spacetime (AdS_5);i.e. the extra dimension is curved or warped rather than flat. Significant deviations from Einstein's theory occur at very high energies. At low energies, gravityhasan exponentially suppressed tail into the extra dimension due to a negative bulk cosmological constant, Λ_5 = - 6/ℓ^2 where ℓ is the curvature radius of AdS_5. The brane gravitates with self-gravity in the form of a brane tension λ, where λ = 3 M_p^2 / (4 πℓ^2) and M_p^2 = M_5^3 ℓ.On the brane, the negative Λ_5 is counterbalanced by the positive brane tension λ. The Einstein's field equationtakes the conventional form but with an effective energy-momentum tensor T^ eff_μν, i.e., it reads: G_μν=8π G T^ eff_μν, where G_μν is the usual Einstein field tensor, and we consider c=1. The effective energy-momentum tensor has the form<cit.> T_μν^eff=T_μν + 6/λ S_μν-1/8π G ℰ_μν , where the firstterm contains the standard energy momentum tensor; e.g. for a perfect fluid we haveT_μν = ρ u_μu_ν+ph_μν, wherep is the pressure of the fluid,ρ is its energy density,u^μ is the four-velocity and h_μν=g_μν+u_μu_ν is the projection orthogonal to u^μ. The secondand third terms include modifications with respect to the standard 4D Einstein's field equation. The bulk correction includes a local termand a nonlocal one (second and third terms respectively) <cit.>. For a perfect fluid, the local contribution reads S_μν = 1/12ρ^2 u_μ u_ν + 1/12ρ(ρ+2p) h_μν. The nonlocal contribution for static spherical symmetry reads: ℰ_μν=-6/8π Gλ [ 𝒰u_μu_ν+ 𝒫 r_μ r_ν+ (𝒰-𝒫)/3h_μν], where 𝒰 and 𝒫 are respectively the nonlocal energy density and nonlocal pressure on the brane and r_μ is a unit radial vector. Notice that,as λ→∞, the bulk corrections vanish and General Relativity is recovered. Some consequences for compact star physicshave been explored within the above described braneworld model. In their pioneering work,Germani and Maartens <cit.> showed thatthe Schwarzschild solution is no longer the unique asymptotically flat vacuum exterior. In general, the exterior carries an imprint of nonlocal bulk graviton stresses andknowledge of the 5D Weyl tensor is needed as a minimum condition for uniqueness. They also found an exact uniform-density stellar solution on the brane, and showed that the existence of neutron stars leads to a constraint on the brane tension that is stronger than the big bang nucleosynthesis constraint, but weaker than the Newton-law experimental constraint.After this work, some other studies of spherical static stars considering a Schwarzschild exterior metric in the braneworld model were done in order to determine how local and nonlocal corrections affect the stellar structure.In Refs. <cit.>the role of the local and nonlocal corrections is examined in nonuniform and uniform stars. In these works, to overcome all problems associated with the system of equations,the time metric component is prescribed. This approach helps to find an exact solution of theEinstein's equations on the brane and an analytic form of the Weyl curvature terms. More recently, someproperties of compact stars in the braneworld model were analyzedusing neutron star equations of state and assuming thatthe Weyl terms obey the simplest relation 𝒫 = w 𝒰 <cit.>. However we must notice that in this work the boundary condition for the nonlocal energy density 𝒰 was set at the stellar center and not at the surface of the star as it should be.In this work we investigate several aspects of compact stars within braneworld models. In Sec. <ref> we review the stellar structure equations on the brane, the boundary conditions and explain the shooting method used for the numerical integration of the equations. In Sec. <ref> we describe the causal limit equation of state (EOS) employed in the literature to obtain the causal limit above which compact stars are not expected in General Relativity. Thereafter, we employ the causal limit EOS to obtain such limit in braneworld models, finding that the general relativistic limit can be violated. In Sec. <ref> we study the properties of hadronic and quark stars using typical EOSs for hadronic and quark matter and find striking features that differentiate their structure with respect to the general relativistic case. Since only stellar configurationsin stable equilibrium are acceptable from the astrophysical point of view, we analyze in Sec. <ref> the stability of the compact stars under small radial perturbations using a static method that allows to determine the precise number of unstable normal radial modes. Finally, inSec. <ref> wesummarize and discuss our results.§ STELLAR STRUCTURE ON THE BRANE §.§ Stellar structure equations and boundary conditionsGermani and Maartens <cit.>solved the Einstein's equations on the brane and derived the braneworld generalization of the stellar structure equations for a static fluid distribution with spherical symmetry dm/dr = 4π r^2ρ_eff, dp/dr = -(ρ+p)dϕ/dr, dϕ/dr = Gm+4π Gr^3(p_eff+4𝒫/(8π G)^2λ)/r(r-2Gm), d𝒰/dr = - (4𝒰+2𝒫 )dϕ/dr-2(4 π G)^2 (ρ+p) d ρ/dr-2d𝒫/dr-6/r𝒫 , where ρ^ eff=ρ(1+ρ/2λ)+6U/(8π G)^2λ, p^ eff=p + ρ/2 λ(ρ + 2 p)+2U/(8π G)^2λ. To solve Eqs. (<ref>)-(<ref>) we need an equation of state ρ = ρ(p) and a relation of the form 𝒫 = 𝒫(𝒰) relating the nonlocal components (“dark” equation of state).Two of the boundary conditions of the stellar structure equations on the brane are the same as for the standard General Relativistic equations. Specifically, at the center of the star (r=0) the enclosed mass is zero:m(r=0) = 0 , and at the surface of theobject the pressure vanishes: p(R) = 0 . The remaining boundary condition is determined by the Israel-Darmois matchingcondition [G_μνr^ν]_Σ=0at the surface of the object Σ, where [f]_Σ≡ f(R^+)-f(R^-) (in the following we use the superscripts - and + to indicate quantities inside and outside the star respectively). By the brane field equation (<ref>), this implies [T^ eff_μν r^ν]_Σ=0, leading to [p^ eff + 4P / ((8π G)^2 λ) ]_Σ=0.Since at the surface of the object we have p(R)=0, we have: (4π G)^2ρ^2(R)+ U^-(R) +2P^-(R)=U^+(R) + 2P^+(R) , whichholds for any static spherical star with vanishing pressure at the surface.In BW models, the Schwarzschild solution is no longer the unique asymptotically flat vacuum exterior. In general, the exterior carries an imprint of nonlocal bulk graviton stresses. Knowledge of the 5D Weyl tensor is needed as a minimum condition for uniqueness.If there are no Weyl stresses in the interior(U^- =P ^- = 0), and if the energy density is non-vanishing at the surface, ρ(R)≠ 0, then there must be Weyl stresses in the exterior, i.e. the exterior solution cannot be the Schwarzschild one <cit.>. Equivalently, if we assume a Schwarzschild exterior solution (U^+ =P^+ = 0) and the energy density is nonzero at the surface, then the interior solution must havenonvanishingnonlocal Weyl stresses.On the other hand, despite previous no-go results,brane-world compact objects with a Schwarzschild exterior are obtained in Ref. <cit.>.This result is arises at the price of having negative pressure inside a narrow shell at the star surface,which effectively acts as a solid crust separating the inner fluid from the vacuum exterior <cit.>. Such crust has a negligible thickness, falling below any physically sensible length scale for astrophysical sources, and the discontinuities in 𝒰 and 𝒫 at r=R are negligibly small. Therefore, the crust can be neglected in the calculation of the mass-radius diagram andother global stellar properties. However, we emphasize that in the present work, the physicality of the Schwarzschild exterior is not necessarily guaranteed. Nonetheless, in order to simplify the analysis and to facilitate the comparison with GR, we focus here on a class of models that satisfy the following properties:* we consider a Schwarzschild exterior solution (𝒰^+ = 𝒫^+ = 0); * we assume 𝒫^- = 0, which is consistent with the isotropy of the physical pressure in the star. As a consequence, the interior must have nonvanishing nonlocal Weyl stresses (𝒰^- ≠ 0).Therefore, the boundary condition for 𝒰at r=R simplifies to: (4π G)^2ρ^2(R)+ U^-(R)=0.In summary, the full set of equations to be solved is:dm/dr= 4πρ^ effr^2 , dp/dr= -(p+ρ)[4π Gp^ effr+mG/r^2]/[1-2mG/r], d U^-/dr=4 U^-/p+ρdp/dr -2(4π G)^2(ρ+p)dρ/dr, with the boundary conditions m(r=0) = 0, p(R) = 0 and (4π G)^2ρ^2(R)+ U^-(R)=0. An equation of state ρ = ρ(p) must be supplied to close the system. In thelimit λ→∞, we have ρ^ eff→ρ and p^ eff→ p, and the General Relativistic stellar structure equations are recovered.§.§ Numerical integration of the structure equationsFor a given EOS of the form ρ = ρ(p) and a given value of the brane tension λ, Eqs. (<ref>)-(<ref>) can be integrated simultaneously with a Runge-Kutta method from the center towards the surface of the object. However, since the boundary conditionforU^-(r) is given at the star's surface, a shooting method is used in order to match Eq. (<ref>).The integration of Eqs. (<ref>), (<ref>) and (<ref>) begins with the values m(0) = 0,p(0) = p_c,U^-(0) =U^-_c, trial , where p_c is a given value for the central pressure, and U^-_c, trial isa trial value of U^- at r=0. The integration proceeds outwards until the pressure vanishes in order to verify Eq. (<ref>). However,after such integration Eq. (<ref>) is not necessarily fulfilled. Therefore,the trial value of U_c^- is corrected through a Newton-Raphson iteration scheme in order to improve the matching of Eq. (<ref>) in the next integration.The integration from r=0 is repeated successively until Eq. (<ref>) is satisfiedwith the desired precision. Once such precision is attained,the point at which the pressure of the fluid vanishes determines the star's radius R and the star's mass M= m(R).It is worth mentioning that for some simple EOSs, Eq. (<ref>) can be integrated analytically.In Appendix <ref> we derive the explicit solution for a linear EOS of the form ρ= p / c_s^2 + b, where c_s^2 and b are arbitrary constants.This EOS is very useful because it includes as special cases the causal EOS ρ=p,the ultra-relativistic EOS ρ=3 p, and the MIT bag model EOS for massless quarks ρ = 3 p + 4 B, that we will use below. § UPPER BOUND ON THE MAXIMUM MASS OF COMPACT STARS IN THE BRANEWORLD MODEL: THE CAUSAL LIMIT §.§ The causal limit EOS A complete knowledge of the equation of state of neutron star matter is still a challenge at present. The EOS can be reliably determined up to ∼ 2ρ_sat, being ρ_sat≈ 151MeV / fm^3 the nuclear saturation density. However, for larger densities, the determination of a well-founded EOS strongly depends on the knowledge of strong interactions in a regime that cannot be reached experimentally.As a consequence, there is a large amount of high-density EOSsin the literature that incorporate several aspects that may play a crucial role at the inner core of the star, such as three-body forces, bosonic condensates, hyperonic degrees of freedom and quarkmatter <cit.>.An important aspect of neutron stars within the frame of General Relativity, is that there exists a maximum gravitational mass above which there are no stable stellar configurations. The maximum mass exists no matter what the EOS, but its determination depends on a deep comprehension of the EOS up to several times ρ_sat. However,using the so called causal limit EOS, it is possible to circumvent the uncertainties related to the properties of high-density matter and obtain upper bounds to the maximum allowedmass of a neutron star <cit.>. The causal limit EOS can be constructed by using a detailed EOS at density ranges where they can be safely regarded as accurate and imposing generic constraints at densities exceeding some fiducial density, e.g., subluminal sound velocity and thermodynamic stability (see e.g. <cit.>).In this work, we adopt thewell established Baym, Pethick, and Sutherland (BPS) EOS<cit.> at densities below a fiducial density ρ_t, and a causal equation of state (i.e. sound velocity = speed of light) p = ρ - aabove ρ_t <cit.>. Since both EOSsare matched at an energy density ρ_t and a pressure p_t,the constant a in the high density EOS is given by a = ρ_t - p_t, where ρ_t and p_t also fulfill the BPS EOS (see Table <ref>). §.§ The causal limit in General RelativityUsing the causal limit EOS, it is possible to obtain a curve, know as causal limit, that represents an upper bound in the mass-radius diagram for compact stars. The procedure to find the causal limitwithin the frame of General Relativity has been explained in several textbooks (see e.g. <cit.>). For completeness, we present it here and in the next subsection we discuss it within the braneworld model.For a given value of a, the stellar structure equations can be integrated, and a maximum stellar mass can be determined together with the corresponding stellar radius. For example, using ρ_t=260.1MeV/fm^3 and p_t=3.809MeV/fm^3), the sequence has a maximum mass object with: M_max=3.131M_⊙ ,R = 13.35 km. Repeating the calculations for many different values of a, it can be shown that the maxima fall on a straight line given byM =0.345 R (see Fig. <ref>). Therefore, the region excluded by causality in the M-R diagram is given by <cit.>: M ≳0.345 R, or, equivalently: ( M/M_⊙) ≳ 0.234 ( R/km) . §.§ The causal limit in the braneworld model In order to determine the causal limit in the braneworld model, we integrate the stellar structure equations on the brane using the causal limit equation of state presented in Sec. <ref>. For small masses (≲ 1.5 - 2 M_⊙),the curves show the typical behavior found within the frame of General Relativity. Specifically, very small mass stars have very large radii, and as the mass increases above a few tenths of solar masses the radii fall within a range of few kilometers around ∼ 10 km. Nevertheless, for large mass objects,local high-energy effects as well as nonlocal corrections lead to significant deviations with respect to General Relativity. At around 1.5 - 2 M_⊙ the M(R) curves bend anticlockwise as in the general relativistic case. However, instead of reaching a maximum mass as in General Relativity, the curves bend once more (clockwise) for larger masses and thereafter they increase roughly linearly (see Fig. <ref>).A striking feature of this behavioris that once the M - R curves bend clockwise they mayfall above the causal limit obtained within General Relativity (c.f. Eqs. (<ref>) - (<ref>)). It can also be checked thatas the masses and radii increase, the curves tend asymptotically to the Schwarzschild limit M = 2 R. The asymptotic approach depends on the value of the brane tension λ: when λ is small the curves go close to the line M = 2 R at relatively small masses; but, for large λthe approaching occurs at higher masses. Since the M - R curves for the causal EOS approach asymptotically the line M = 2 R, but do not go beyond it, the Schwarzschild limit M = 2 R is a good representation of the causal limit in the braneworld model  [In a more general way,this is also discussedin <cit.>. They focus onthe field theoretical description of a generic theory of gravity flowing to Einstein General Relativity in IR and prove that, if ghost-free, in the weakly coupled regime such a theory can never become weaker than General Relativity.]. In other words, the equilibrium solutions found in the braneworld can violate the limit of causality for General Relativity (Eqs. (<ref>) - (<ref>)) and, for sufficiently large mass, can occupy the region between the straight lines shown in Fig. <ref>.§ MODELS FOR HADRONIC AND STRANGE QUARK STARS In this section, we investigate the properties of hadronic and strange quark stars using typical models for the equations of state. As mentioned in Sec. <ref> there is a large amount of high density EOS that fulfill present experimental and astrophysical constrains. However, our purpose is not making an exhaustive survey of all the available EOSs, but rather to explore the qualitative properties of hadronic and strange quark stars using two models that have been extensively employed in the literature: a nonlinear relativistic mean-field model for hadronic matter andthe MIT bag model for quark matter.In Sec. <ref> we briefly summarize the EOSs and in Sec. <ref> we study the structural properties of the resulting compact objects.§.§ Equations of stateFor the hadronic phase we use a non-linear Walecka model <cit.> including the whole baryon octet, electrons and the corresponding antiparticles. The Lagrangian is given by L= L_B+ L_M+ L_L, where the indices B, M and L refer to baryons, mesons and leptons respectively. For the baryons we have L_B= ∑_B ψ̅_B [γ^μ (i∂_μ - g_ω B ω_μ- g_ρ B τ⃗·ρ⃗_μ) -(m_B-g_σ B σ)]ψ_B,with B extending over nucleons n, p and the following hyperons Λ, Σ^+, Σ^0, Σ^-, Ξ^-, and Ξ^0. The contribution of the mesons σ, ω and ρ is given by L_M = 1/2 (∂_μσ ∂^μσ -m_σ^2σ^2) - b/3m_N(g_σσ)^3 -c/4 (g_σσ)^4-1/4 ω_μν ω^μν +1/2m_ω^2ω_μ ω^μ-1/4 ρ⃗_μν·ρ⃗^μν+ 1/2 m_ρ^2 ρ⃗_μ·ρ⃗^μ, and the coupling constants are g_σ B=x_σ B  g_σ,  g_ω B=x_ω B  g_ω,  g_ρ B=x_ρ B  g_ρ. Electrons are included as a free Fermi gas, L_L=∑_l ψ̅_l (i /∂ - m_l)ψ_l, in chemical equilibrium with all other particles.The constants in the model are determined by the properties of nuclear matter and hyperon potential depths known from hypernuclear experiments. In the present work we use the GM1 parametrization for which we have (g_σ/m_σ)^2 = 11.79 fm^-2, (g_ω/m_ω)^2 = 7.149 fm^-2, (g_ρ/m_ρ)^2 = 4.411 fm^-2, b = 0.002947 and c = 0.001070<cit.>. For the hyperon coupling constants we adopt x_σ i = x_ρ i = 0.6 and x_ω i = 0.653<cit.>.At low densities we use the Baym, Pethick and Sutherland (BPS) model <cit.>. For details on the explicit form of the equation of state derived from this Lagrangian the reader is referred to Refs. <cit.> and references therein.We describe quark matter through the MIT bag model. For simplicity we assume a zero strong coupling constant and consider massless quarks. If such effects were taken into account, the equation of state would be qualitatively the same but we would find nonanalytic expressions. In practice, only u, d, and s quarks appear in quark matter because other quark flavors have masses much larger that the chemical potentials involved (roughly 300 MeV). Since these quarks are assumed to be massless, leptons are not necessary to electrically neutralize the phase, and thus, they are not present in the system <cit.>. In such a case, the equation of state adopts the simple form ρ = 3 p + 4 B, where B is the bag constant. Witten <cit.> conjectured that, at zero pressure and temperature, three flavor quark matter may have an energy per baryon smaller than ordinary nuclei. This would make strange quark matter the true ground state of strongly interacting matter and would lead to the existence of strange quark stars i.e. stellar objects completelycomposed by strange quark matter <cit.>.Within the MIT bag model for massless quarks and zero strong coupling constant, the Witten hypothesis is verified if the bag constant is in the range 57 MeV/fm^3≲ B ≲ 94 MeV/fm^3. In this paper we adopt B = 60MeV/fm^3. §.§ Structural properties of hadronic and strange quark stars In the following we present our results for hadronic and strange quark stars using the equations of state presented in the previous subsection. In Fig. <ref> we show the mass-radius relationship for some values of the brane tension λ.At the top panel we display the results for strange quark matter and at the bottom panel for hadronic matter. We also include the causal limit found before for General Relativity and the Schwarzschild limit M = 2 R.For small masses (≲ 1.5 - 2 M_⊙),the curves show the typical behavior found within the frame of General Relativity, i.e. very small mass hadronic stars have very large radii,while strange stars follow roughly M(R)∝ R^3. For large mass objects, braneworld effects lead to the deviations with respect to General Relativity that were explained in the case of the causal EOS of previous section.At around 1.5 - 2 M_⊙ the M(R) curves for hadronic and quark starsbend anticlockwise as in the general relativistic case. But then, the curves bend once more (clockwise) for larger masses and thereafter they increase roughly linearly and approach asymptotically to the Schwarzschild limit.In summary, the main characteristics of the mass-radius relationship already found for the causal EOS are confirmed for both the hadronic and the strange quark mater EOSs: * The M(R) curves violate the general relativistic causal limit for large enough masses; instead, they can occupy the region between thegeneral relativistic causal limit andthe Schwarzschild limit. * Static stellar configurations do not have a maximum mass as in thegeneral relativistic case, i.e. objects of any mass are possible in principle.Notice that, differently from the causal EOS, we find now that in some cases there is a local maximum in the M(R) curves at M ∼ 2 M_⊙. Nevertheless, after bending clockwise, the behavior of all the M(R) curves is qualitatively the same. Fig. <ref> shows the dependence of the masswith the central energy densityρ_c for some values of the brane tension λ. For a given value of ρ_c, the mass of a star is larger in the braneworld modelthan in General Relativitydue to local and nonlocal extra-dimensional modifications to the structure equations on the brane. As expected, these corrections are small for low central energy densities but they become significant with increasing central energy density, specially for the smaller values of the brane tension λ. A remarkable feature of the M(ρ_c) curves is that there is a value of ρ_c for which the stellar mass diverges.This means that for large enough masses the nonlocal energy density U^- supports the star against gravitational collapse. The maximum value of ρ_c increases with the brane tension λ as can be seen in Fig. <ref>. In particular, as we approach to the general relativistic case (λ→∞) the maximum density is shifted to infinity.In Fig. <ref> we showthe nonlocal energy densityU^-as a function of the radial coordinate r for a central energy densityρ_c =2500MeV/fm^3 and five values of the brane tension λ. For both,strange quark stars and hadronic stars,the nonlocal energy density startsat a large negative value at the center of the star and growsmonotonically towards the stellar surface. The more negative values of U^- are found for the lower values of λ.A star with a more negative nonlocal energy densityadmits more mass, because U^-acts as an effective negative pressure helping against the collapse. § STELLAR STABILITYIn the previous sections we found that in braneworld models there is a new branch of stellar configurations that is not present within General Relativity. Since only stellar configurationsin stable equilibrium are acceptable from the astrophysical point of view,we should check the stability of the previously obtained stellar models. A well known static criterionthat is widely used in the literature states that a necessarycondition for a model to be stable is that its mass M increases with growing central density, i.e.d M/d ρ_c > 0 . The latter is a necessary but not sufficient condition. The opposite inequality dM/d ρ_c < 0 always implies instability of stellar models; i.e configurations lying on the segments with dM/d ρ_c < 0 are always unstable with respect to small perturbations. In Figs. <ref> and <ref>there are two qualitatively different types of M(R) andM(ρ_c) curves. One typepresentsone local maximum and one local minimum in both theM(R) and the M(ρ_c) curves. The other one has no critical points. These two types are represented separately in Fig. <ref>, where we showthe M(R) andM(ρ_c) curves forstrange quark stars for two different values of λ (for simplicity we do not show hadronic stars because the stability analysis is completely equivalent, as we shall see below). For λ=(551 MeV)^4 (upper panels) the stellar massis always an increasing function of the central density ρ_cand the M(R) curve has no local maxima or minima. Thus, the above necessary stability condition is always verified in this case.Forλ=(724 MeV)^4(lower panels)the part of the M(ρ_c) curve between the points M1 and M2 verifies dM/d ρ_c < 0, i.e. those configurations are unstable.M1 indicates the local maximumand M2 indicates the local minimum of the mass in both plots.Thenecessary stability condition is verified for the branch to the left of M1 and to the right of M2 in the M(ρ_c) curve and in the corresponding branches of the M(R) curve. Therefore, the branches thatapproach asymptotically to the Schwarzschild limitverify the necessary conditiondM/dρ_c > 0 for any λ, but, as stated before, this is not a sufficient condition for stability. A more detailed study of the stability of non-rotating spherically symmetric equilibrium models against small perturbations should be carried out through the analysis of their radial oscillations. However, this is left for future work because it is necessary to derive and solve the pulsation equations on the brane. Instead, we employ here a sufficient criterion which enables one to determine the precise number of unstable normal radial modes using the M(R) curve <cit.>. According to such criterion, at each critical point of the M(R) curve (local maxima or minima) one and only one normal radial mode changes its stability, from stable to unstable or vice versa. There are no changes of stability associated with radial pulsations at other points of the M(R) curves. Moreover, one mode becomes unstable (stable) if and only if the M (R) curve bends counterclockwise (clockwise) at the critical point. In order to analyze the stellar stability using the above criterion,we assume that the low mass branch (up to ≲ 1.5 - 2 M_⊙) of the M(R) curves is stable for all radial modes,as it is in the general relativistic case.For the curves with two critical points,the M(R) curve bends counterclockwise at the local maximum and the fundamental oscillation mode becomes unstable. However, at the local minimum the fundamental mode becomes stable again because the curve bends clockwise there. Beyond the local minimum there are no more critical points and all the radial modes remain stable. In the case without critical points, the whole sequence remains stable for all radial modes provided that the low mass configurations are stable. Thus, we can conclude thatthe branches thatapproach asymptotically to the Schwarzschild limit are always stable under small radial perturbations. As a consequence, stellar configurations of arbitrarily large mass are allowed within braneworld models. § SUMMARY AND CONCLUSIONS In this work we have studied the structure of compact stars in a Randall-Sundrum type II braneword model. To this end,we employed the braneworld generalization of the stellar structure equations for a static fluid distribution with spherical symmetry. We considered that the spacetime outside the star is described by a Schwarzschild metric, i.e. the nonlocal pressure and energy density vanish outside the star, and assumed that the nonlocal pressure is zero in the stellar interior.As a consequence, the interior must have nonvanishing nonlocal Weyl stresses (𝒰^- ≠ 0).In order to obtain an upper bound to the maximum mass of compact stars in the braneworld model, we integrated the stellar structure equations employing the causal limit EOS, which is obtained adopting the well established Baym, Pethick, and SutherlandEOSat densities below a fiducial density, and an EOS with the sound velocity equal to the speed of light above it. Assuming the causal limit EOS,it can be shown thatthe region above the causality limit depicted in Fig. <ref> is forbidden within General Relativity. However, the equilibrium solutions found in the braneworld model can violate the limit of causality for General Relativity and, for sufficiently large mass they approach asymptotically to the Schwarzschild limit M = 2 R. Hence, there is a region in the M-R plane that is forbidden in General Relativity for causal equations of state but that can be accessed within braneworld models; i.e.the region between the straight lines shown in Fig. <ref>.Then, we investigated the properties of hadronic and strange quark stars using two typicalEOSs that have been extensively employed in the literature: a nonlinear relativistic mean-field model for hadronic matter andthe MIT bag model for quark matter.The main characteristics of the mass-radius relationshipfound using the causal limit EOS are confirmed for bothhadronic andstrange quark stars.For small masses (≲ 1.5 - 2 M_⊙),the curves show the typical behavior found within the frame of General Relativity, i.e. very small mass hadronic stars have very large radii,while strange stars follow roughly M(R)∝ R^3.Moreover,the M(R) curves for hadronic and quark starsbend anticlockwiseat around 1.5 - 2 M_⊙ as in the general relativistic case. However, the curves bend once more (clockwise) for larger masses and thereafter they increase roughly linearly and approach asymptotically to the Schwarzschild limit. Again, two remarkable features are thatthe M(R) curves violate the general relativistic causal limit, andthatstatic stellar configurations do not have a maximum mass as in thegeneral relativistic case, i.e. objects of any mass are possible in principle (see Fig. <ref>).These large mass stars are supported against collapse by the nonlocal effects of the bulk on the brane.Finally, we studied the stability under small perturbations of the stellar configurations in the braneworld. We used a static criterion which enables one to determine the precise number of unstable normal radial oscillation modes analyzing the bending of the mass-radius curves at the critical points (see Fig. <ref>). We assumed that the low mass branch (up to ≲ 1.5 - 2 M_⊙) of the M(R) curves is stable for all radial modes,as it is in the general relativistic case. For the models without critical points, there are no changes of stability associated with radial oscillation modes, and therefore all configurations are stable. For the models with two critical points,the M(R) curve bends counterclockwise at the local maximum and the fundamental oscillation mode becomes unstable there. However, the fundamental mode regains stability at thelocal minimum because the curve bends clockwise there. Beyond that minimum there are no more critical points and all the radial modes remain stable. In summary, within braneworld models we obtain the low mass branch of compact star configurations already known from general relativistic calculations, but we also find a new branch that approaches asymptotically to the Schwarzschild limit which is always stable under small radial perturbations. This new branch contains stellar configurations of arbitrarily large mass,supported against collapse by the nonlocal effects of the bulk on the brane. It is worth emphasizing that black holes are still possible within the here studied braneworld models. Moreover, the stellar configurations that asymptotically approach to the Schwarzschild limit are expected to be stable under small perturbations, but not necessarily under large ones. Therefore, a very large mass braneworld compact star could collapse into a black hole if strongly perturbed in a catastrophic astrophysical event, e.g. in a binary stellar merging. Finally, we remark that although a complete 5D analysis would be necessary to fully understand the properties of the new branch, these results serve as a proof of principle that traces of extra-dimensions might be found in astrophysics, specifically through the analysis ofmasses and radii of compact objects.GL acknowledges the Brazilian agencies CNPqand FAPESP for financial support.JDVA thanks the Brazilian agencies CAPES and FAPESP (Project 13/26258-4). § ANALYTIC SOLUTION FOR THE NONLOCAL ENERGY DENSITY U^- FOR A LINEAR EQUATION OF STATE For an arbitrary equation of state, the nonlocal energy density inside the star U^-(r) must be obtained through the numerical integration of Eq. (<ref>) together with Eqs. (<ref>) and (<ref>). However, as we show below, an analytic solution for U^-(r) can be obtained in the case of a linear EOS. First, we rewrite Eq. (<ref>) in the form d U^-(r)/dr+ U^-(r)g(r)= f(r), where g(r)=-4/p(r)+ρ(r)dp(r)/dr, f(r)=-2(4π G)^2(ρ(r)+p(r))dρ(r)/dr. Multiplying by the integrating factor e^∫ g(r)dr we have d/dr( U^-(r)e^∫ g(r)dr)= f(r)e^∫ g(r)dr, which gives: U^-(r)=e^-∫ g(r)dr[∫ f(r)e^∫ g(r)drdr+C_1],where C_1 is an integration constant. Now, we consider a linear EOS of the formρ=p/c^2_s+b, where c^2_sand b are constants (c_s is the speed of sound). Notice that this EOS includes as special cases the causal EOS ρ=p,the ultra-relativistic EOS ρ=3 p, and the MIT bag model EOS for massless quarks ρ = 3 p + 4 B. For such EOS,Eq. (<ref>) reads U^-(r)= [ (1 + c^2_s) p(r) + b c^2_s]^4c^2_s/(1+c^2_s)[-2(4π G)^2/c^4_s×. .∫[ (1+c^2_s)p(r) + b c^2_s]^1-3 c^2_s/1+c^2_s dp + C_1],with k_1 being an integration constant that comes from the integral ∫ g(r)dr.Integrating Eq. (<ref>) we find U^-(r)={ [2p(r)+b]^2[-(4π G)^2ln(2p(r)+b)..+k_2],c_s=1;[(1+c^2_s)p(r) + b c^2_s ]^4 c^2_s/(1+c^2_s)[-(4π G)^2/(1-c^2_s)×..[ (1+c^2_s) p(r)+ b c^2_s]^2(1-c^2_s)/(1+c^2_s) + k_2],c_s<1; . where k_2=C_2+k_1 C_1, being C_2 another integration constant.To determine k_2 we use the boundary condition: (4π G)^2ρ^2(R)+ U^-(R)=0. The value U^-(R) is obtained evaluating Eq. (<ref>) at the surface of the star, and from the EOS we obtain ρ(R) = b because p(R) =0. Therefore, Eq. (<ref>) reads: k_2={(4π G)^2[ln(b)-1],c_s=1; (4π G)^2 b(b c_s^2)^(1- 3 c^2_s)/(1+c^2_s)/(1-c^2_s),c_s<1.. Replacing k_2 fromEq. (<ref>)into Eq. (<ref>)we find U^-(r)={-(4π G)^2[2p(r)+b]^2[ln(2p(r)+b)..-ln(b)+1],c_s=1;-(4π c_s G)^2/(1-c^2_s)[1/c^2_s[(1+c^2_s)/c^2_sp(r)+b]^2..-b^2(1-c^2_s)/(1+c^2_s)[(1+c^2_s)/c^2_sp(r)+b]^4c^2_s/(1+c^2_s)],c_s<1;. which gives U^-(r) as a function of p(r), where p(r) is to be obtained from the integration of Eqs. (<ref>) and (<ref>). An equivalent expression ofU^-(r) as a function of ρ(r) can be obtained using the EOS: U^-(r)={-(4π G)^2[2ρ(r)-b]^2[ln(2ρ(r)-b)..-ln(b)+1],c_s=1;-(4π c_sG)^2/(1-c^2_s)[c^2_s[(1+c^2_s)/c^2_sρ(r)-b]^2..-b^2(1-c^2_s)/(1+c^2_s)[(1+c^2_s)ρ(r)-bc^2_s]^4c^2_s/(1+c^2_s)], c_s<1.. In particular, we can evaluate the latter expression at the center of the star in order to obtain a relationship between the central nonlocal energy density U^-_c≡ U^-(0) and the central mass-energy density ρ_c ≡ρ(0) U^-_c={-(4π G)^2[2ρ_c-b]^2[ln(2ρ_c-b)..-ln(b)+1],c_s=1;-(4π c_s G)^2/(1-c^2_s)[c^2_s[(1+c^2_s)/c^2_sρ_c-b]^2..-b^2(1-c^2_s)/(1+c^2_s)[(1+c^2_s)ρ_c-bc^2_s]^4c^2_s/(1+c^2_s)],c_s<1.. 99maartens2010 R. Maartens and K. Koyama,Living Rev. Relativity 13, 5 (2010).Randall1999 L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999).dvali2000 G. Dvali, G. Gabadadze, and M. Porrati, Phys. Lett. B 485,208 (2000). Shiromizu2000 T. Shiromizu, K. -I. Maeda and M. Sasaki, Phys. Rev.D 62, 024012 (2000). maartens2001 C. Germani and R. Maartens, Phys. Rev. D 64, 124010 (2001).ovalle2008 J. Ovalle,Modern Physics Letters A 38, 3247 (2008).ovalle2009 J. Ovalle, Int. J. Mod. Phys. D 18, 837 (2009).ovalle2010 J. Ovalle, Modern Physics Letters A 25, 3323 (2010).castro2014 L. B. Castro, M. D. Alloy, and D. P. Menezes, J. Cosmol. Astropart. Phys. 08, 047 (2014).ovalle2015 J. Ovalle, L. Á. Gergely, and R. Casadio, Class. Quantum Grav. 32, 045015 (2015).haensel P. Haensel, A. Y. Potekhin, and D. G. Yakovlev Neutron Stars 1: Equation of Stateand Structure Springer Verlag, New York, (2007).glen N. K. Glendenning, Compact Stars: Nuclear Physics, Particle Physics and General Relativity 2nd ed. Springer Verlag, New York, (2000).rhoades1972 C. E. Rhoades and R. Ruffini, Phys. Rev. Lett. 32, 324 (1974).bps G. Baym, C. Pethick, and P. Sutherland,Astrophys. J. 299, 170 (1971). lattimer2012 J. M. Lattimer,Annu. Rev. Nucl. Part. Sci 62, 485 (2012).lattimer2004 J. M. Lattimer and M. Prakash, Science 304, 536 (2004). Dvali2011 G. Dvali, S. Folkerts and C. Germani,Phys. Rev.D 84024039 (2011).witten1984 E. Witten,Phys. Rev. D 30, 272 (1984).Walecka J. D. Walecka, Ann. Phys. 83,491 (1974).Walecka2B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16,1 (1986).glendenning1991N. K. Glendenning and S. A. Moszkowski, Phys. Rev. Lett. 67, 2414 (1991).lugones_eos1 G. Lugones, T. A. S. do Carmo, A. G. Grunfeld, and N. N. Scoccola, Phys. Rev. D 81, 85012 (2010).lugones_eos2T. A. S. do Carmo, G. Lugones and A. G. Grunfeld,J. Phys. G: Nucl. Part. Phys. 40, 035201 (2013).farhi1984E. Farhi and R. L. Jaffe,Phys. Rev. D 30, 2379 (1984).Alcock1986 C. Alcock, E. Farhi, and A. Olinto, Astrophys. J. 310, 261 (1986).htww B. K. Harrison, K. S. Thorne, M. Wakano, and J. A. Wheeler, Gravitation Theory and Gravitational Collapse, University of Chicago Press, Chicago, (1965).

http://arxiv.org/abs/1702.07824v1

Unravelling the Dodecahedral Spaces Jonathan Spreer and Stephan Tillmann December 30, 2023 ========================================In this paper, we study the properties of Carmichael numbers, false positives to several primality tests. We provide a classification for Carmichael numbers with a proportion of Fermat witnesses of less than 50%, based on if the smallest prime factor is greater than a determined lower bound. In addition, we conduct a Monte Carlo simulation as part of a probabilistic algorithm to detect if a given composite number is Carmichael. We modify this highly accurate algorithm with a deterministic primality test to create a novel, more efficient algorithm that differentiates between Carmichael numbers and prime numbers.§ INTRODUCTION In recent years, cybersecurity has been an issue because of insecure cryptosystems. Primality testing is an important step in the implementation of the RSA cryptosystem. In the search for time-efficient primality tests, composite numbers have been inadvertently selected for key generation, rendering the system fatally vulnerable <cit.>. Carmichael numbers are false positives to several primality tests, including the Fermat test and the Miller-Rabin test <cit.>. This paper provides both a classification of Carmichael numbers and a novel, highly accurate algorithm that detects Carmichael numbers. Section 2 of this paper provides the necessary background for studying the proportion of Fermat witnesses for Carmichael numbers. Furthermore, Section 2 concludes with the observation that many Carmichael numbers have a proportion of Fermat witnesses of less than 50%. The results pertaining to the classification of Carmichael numbers with a proportion of Fermat witnesses of less than 50% are detailed in Section 3.1. This classification provides a lower bound for the smallest prime factor of certain Carmichael numbers with a proportion of Fermat witnesses of less than 50% using both inequalities from the initial observation and Newton's method for approximating the root of a function.The observation made in Section 2.4 served as the motivation for creating an algorithm that differentiates between Carmichael numbers and other composite numbers. Section 3.2 discusses this algorithm, which uses a Monte Carlo simulation to check if a composite number is Carmichael with a certain high probability. The proof of this algorithm and its probability of correctness are detailed in Sections 3.3 and 3.4, respectively. In addition, Section 3.6 provides a modified version of this algorithm that allows for the detection of Carmichael numbers among both composite numbers and prime numbers. The proof of this modified algorithm and its probability of correctness are detailed in Sections 3.7 and 3.8, respectively.Sections 3.5 and 3.9 analyze the efficiencies of the first algorithm and the modified version. The first algorithm has a run-time of O(t(log n)^3), where n is the number that is tested and t is the sample size of the number of integers selected in the random sample. The run-time of the second algorithm is O(nt(log n)^3 + (nlog n+C(n))· x ), where x is the run-time of the deterministic primality test that is combined with the original algorithm. Detailed analyses of these efficiencies are provided in Sections 3.5 and 3.9. § BACKGROUND §.§ Primality Testing The RSA algorithm requires two large prime numbers, p and q, from which the keys are generated. To determine if a randomly generated large number n is prime, deterministic primality tests (tests with 100% accuracy) may seem to be the primary option. However, even the fastest known deterministic tests, such as the Agrawal-Kayal-Saxena primality test (or the AKS test), have a run-time of O((log n)^6), where n is the number that is tested for primality <cit.>. Thus, more efficient primality testing algorithms that maintain a high accuracy are needed. Many practical primality tests for larger numbers are probabilistic. In probabilistic primality tests, either (1) a positive integer n is determined to be composite (with 100% accuracy) or (2) the integer n is determined to be prime with a certain probability. To maximize the probability that the primality test works correctly, one must conduct a Monte Carlo simulation so that the chance that n is incorrectly shown to be prime is strictly less than a predetermined value.§.§ Fermat Test The Fermat test is a probabilistic primality test that utilizes notions from Fermat’s little theorem <cit.>. In the Fermat test, a random number a is chosen from (ℤ/n ℤ)\{0}. The test then checks if a^n-1≡ 1n. If a^n-1≢1n, then n is not a prime number. Otherwise, if a^n-1≡ 1 n, then n is said to be prime with a certain probability. In particular, there are some composite numbers n for which there exists an a∈(ℤ/nℤ)\{0} such that a^n-1≡ 1 n; one such composite number is n=561=3·11·17. In this case, if a=2, a^n-1≡2^560≡ 1561. After randomly selecting an element of (ℤ/nℤ)\{0} and calculating a^n-1n, n=561 turns out to be a false positive for the Fermat test. One large class of such false positives is Carmichael numbers, which have the property that for all a∈ (ℤ/nℤ)^×, a^n-1≡ 1 n.Consider the set of all a in {1, 2, 3,…, n-1} for which a^n-1≢1 n. Such values for a are called Fermat witnesses for the Fermat primality test because these values of a show that n is not a prime number. Table <ref> shows a^n-1n for all a∈ (ℤ/nℤ)\{0} in the case when n=21. In Table <ref>, the values of a that are Fermat witnesses are colored in blue, and for those values, a^n-1≡ a^20≢1 21. In the 20-element set {1, 2, 3,…, 20}, 16 elements are Fermat witnesses. In other words, for n=21, the proportion of Fermat witnesses is 80%. A number a is defined to be a non-trivial Fermat witness if (a,n)=1 and a^n-1≢1 n. Note that a would be considered a trivial Fermat witness if (a,n)>1 because a would not be an element of (ℤ/nℤ)^×, which implies that a^n-1≢1n. It has been shown that for n∈ℕ, if there exists a non-trivial Fermat witness, then the proportion of Fermat witnesses is greater than 50% (see Theorem 3.5.4 of <cit.>). The proof of this claim uses the idea of three disjoint subsets (A, B, and C) that categorize all integers in the set {1, 2, 3,…, n-1}: * A = {1 ≤ a ≤ n-1: a^n-1≡ 1n}* B={1≤ a≤ n-1: (a,n)=1anda^n-1≢1n}* C={1≤ a≤ n-1: (a,n)>1}Composite numbers with no non-trivial Fermat witnesses (equivalently, |B|=0) are called Carmichael numbers, which are further detailed in Section 2.3. §.§ Carmichael numbersCarmichael numbers are composite numbers n with the property that for all a∈ℕ such that (a,n)=1, a^n-1≡ 1n. The Fermat test is vulnerable because there are infinitely many Carmichael numbers <cit.>. Carmichael numbers obey Korselt's criterion, which is the equivalent condition to a composite number n being Carmichael <cit.>. Korselt's criterion states that a composite number n is Carmichael if and only if the following are true:(i) the number n does not have a square factor greater than 1(ii) for all prime factors p of n, (p-1)| (n-1). Suppose n=(6m+1)(12m+1)(18m+1), where (6m+1), (12m+1), and (18m+1) are prime numbers. It is not difficult to show that 6m| (n-1), 12m| (n-1), and 18m| (n-1) because n-1=(6m+1)(12m+1)(18m+1)-1=1296m^3+396m^2+36m+1-1=1296m^3+396m^2+36m <cit.>. Thus, by Korselt's criterion, such n is Carmichael.Carmichael numbers are important to study and classify because of their significant role in primality tests. By understanding the importance of Carmichael numbers, cryptographers and number theorists can modify primality tests in a way that Carmichael numbers can be easily identified. §.§ Fermat Witnesses for Carmichael Numbers Let a be an element of (ℤ/nℤ)\{0}. Recall that a is a Fermat witness for a Carmichael number n if and only if (a,n)>1. The proportion of Fermat witnesses for Carmichael numbers is an important subject for investigation because it determines the probability that Carmichael numbers will be correctly determined to be composite numbers. Because ϕ(n)=|{a∈ (ℤ/nℤ)|(a,n)=1 }|, the proportion of Fermat witnesses for Carmichael number is given by 1-ϕ(n)n-1.It is important to consider a few small examples of the proportion of Fermat witnesses for Carmichael numbers. For the Carmichael number n=561, the proportion of Fermat witnesses is equal to 1-ϕ(n)n-1=1-320560≈0.4286. For the Carmichael number n=1105, the proportion of Fermat witnesses is equal to 1-ϕ(n)n-1=1-7681104≈0.3043. For the Carmichael number n=1729, the proportion of Fermat witnesses is equal to 1-ϕ(n)n-1=1-12961728≈0.2504.The examples above seem to suggest that the rate of Fermat witnesses is less than 50% for all Carmichael numbers. However, this conjecture is not correct; Table <ref> lists all Carmichael numbers less than 10^21 with the property that 1-ϕ(n)n-1 is greater than 50% <cit.>. Although the rate of Fermat witnesses for Carmichael numbers is not bounded above by 50%, the observations pertaining to the rate of Fermat witnesses for Carmichael numbers are essential to the creation of the algorithms detailed in this paper. § RESULTS The properties of Carmichael numbers were used to examine the proportion of Fermat witnesses to find a classification of Carmichael numbers n with the property that the proportion of Fermat witnesses, 1-ϕ(n)n-1 (approximated as 1-ϕ(n)n for larger values of n in this paper), is less than 50%. Furthermore, this paper provides a novel algorithm thatdetects if a given composite number n is Carmichael using observations made about the proportion of Fermat witnesses for Carmichael numbers. In addition, a scheme that combines this highly accurate test with a deterministic primality test is provided to determine if a given number is Carmichael. §.§ Classification of Carmichael Numbers n with 1-ϕ(n)n-1<50% Let n be a Carmichael number such that n=p_1p_2⋯ p_r and p_i are all distinct prime factors of n (it is possible to express a Carmichael number as the product of distinct prime factors by the definition provided in Section 2.3). Let a≤ p_1<p_2<⋯<p_r. This section focuses on bounding the value of a for which n is guaranteed to be a Carmichael number with 1-ϕ(n)n-1<50%. Because there are r prime factors of n, a^r ≤ n. Using this inequality yields the following:rloga≤logn r≤log_an. So, it follows that:1a≥1p_1 (1-1a)^log_an≤(1-1a)^r≤ϕ(n)n.The last inequality results from the fact that a is less than every prime factor of n, which has r prime factors. Note that ϕ(n)=n·(1-1p_1)·(1-1p_2)⋯(1-1p_r)≥(1-1a)^r. Furthermore, (1-1a)^log_an≤ϕ(n)n. It was observed that many Carmichael numbers have proportions of Fermat witnesses of less than 50%. To characterize some Carmichael numbers that exhibit this property, it must now be checked when the following occurs:12≤(1-1a)^log_an≤(1-1a)^r≤ϕ(n)n 12≤(1-1a)^log_an 12≤(a-1a)^log_an.Note that log_a n=log_(a-1)/anlog_(a-1)/aa,which implies that:12≤(a-1a)^(log_(a-1)/an)/(log_(a-1)/a a)=n^log_aa-1/a. Taking the log of both sides results in:log12≤(log_aa-1a)·(log n) log_n 12≤log_aa-1a.Let k=log_n12. Note that k≤log_aa-1a, which implies that a^k≤a-1a. Multiplying both sides by a yields a^k+1≤ a-1. Thus, a^k+1-a+1≤ 0. Now, it remains to find the values of a for which a^k+1-a+1≤ 0.Let f(a)=a^k+1-a+1. Figure <ref> shows f(a) for the case when n=1729. To find the values of a for which f(a)≤0, the zero of f(a) must be calculated. Theorem <ref> focuses on this calculation, which results in a classification of Carmichael numbers with a proportion of Fermat witnesses of less than 50%. If the smallest prime factor p_1 of a Carmichael number n satisfies the following:1+log_2n-((1+log_2n)^log_nn/2-log_2n(log_nn/2)· (1+log_2n)^log_n1/2-1) ≤ p_1,then the proportion of numbers from 1 to n-1 that are Fermat witnesses is less than 50%.To find a bound for the zero of f(a)=a^k+1-a+1, it suffices to use Newton's method to approximate a lower bound for the smallest prime factor p_1 of n. This method begins with a function f(x) defined over the real numbers such that the derivative of f(x) exists and is defined over all reals. An initial guess x_0 is made to approximate the root of the function. A new approximation x_1 is made using the following equation:x_1=x_0 -f(x_0)f'(x_0).This process of approximating the roots of the function f(x) continues with:x_n+1=x_n - f(x_n)f'(x_n). Note that the tangents to the function f(a)=a^k+1-a+1 have x-intercepts that are greater than the zero of f(a) because f(a) is a concave function. Thus, if the approximation of the zero of f(a) is less than p_1, then the zero of f(a) is less than p_1, which implies that the proportion of Fermat witness is less than 50% for the Carmichael number.To first approximate the zero of f(a), let x_0=1. Note that f'(a)=(k+1)· a^k-1, which means that f'(1)=(k+1)· 1 -1=k. Also, note that f(1)=1^k+1-1+1=1. Thus,x_1=1-f(1)f'(1)=1-1k=1+1log_n2=1+log_2n.Now, consider the second iteration of Newton's method. Note that:x_2=x_1-f(x_1)f'(x_1)=(1+log_2n)-f(1+log_2n)f'(1+log_2n). Furthermore, the numerator of f(1+log_2n)f'(1+log_2n) can be rewritten as:f(1+log_2n)=(1+log_2n)^log_n(n/2)-(1+log_2n)+1=(1+log_2n)^log_n(n/2)-log_2n.The derivative of f(a) evaluated at a=1+log_2n is given by:f'(1+log_2n)=(1+log_n(1/2))· (1+log_2n)^log_n(1/2)-1.Thus, if the following is true:a<1+log_2n-((1+log_2n)^log_n(n/2)-log_2n(log_n(n/2))· (1+log_2n)^log_n(1/2)-1)≤ p_1,then the proportion of Fermat witnesses for the Carmichael number n is less than 50%, as desired. □ Theorem <ref> exploits an interesting observation about Carmichael numbers: the proportion of Fermat witnesses for many Carmichael numbers is less than 50%. This property is quite fascinating because every composite number with non-trivial Fermat witnesses has a proportion of Fermat witnesses of greater than 50%. This key observation can be further utilized to create an algorithm that distinguishes between Carmichael numbers and other composite numbers. §.§ Algorithm that Distinguishes Carmichael Numbers and Other Composite Numbers This section provides the details for the probabilistic algorithm that determines if a composite number is Carmichael. The algorithm works as follows. Consider a composite number n. Conduct a Monte Carlo simulation by first randomly selecting t numbers from the set {1, 2,…, n-1}, where t=⌊(ln n)^2⌋. Note that ⌊(ln n)^2⌋ is the sample size temporarily because ⌊ln n⌋ is quite small for larger values of n and the variation would be quite significant with a smaller sample size. For larger numbers, a sample size of ⌊(ln n)^2⌋ yields more accurate results. Now, check for each such a from the randomly sample if a^n-1≡ 1n. Next, calculate the proportion of values of a for which a^n-1≢1n from the random sample. If the proportion of such numbers is less than 45% [Note that there are other composite numbers for which the proportions of Fermat witnesses are close to 50%. Such numbers would be incorrectly determined to be Carmichael because of sampling variations.], then the composite number n is “probably” Carmichael. Otherwise, check every instance in which a^n-1≢1n and check if (a,n)=1. If (a,n)=1, then the number n is declared as an “other composite number.” If there are no such a relatively prime to n, then the number n is Carmichael with a high accuracy. The pseudocode for this algorithm is detailed in Algorithm <ref>. §.§ Proof of Correctness for Algorithm 1 If a number n is Carmichael, then n must have no non-trivial Fermat witnesses. Thus, a Carmichael number n will be accurately determined as Carmichael. Otherwise, other composite numbers, which must have non-trivial Fermat witnesses, will be correctly determined as “other composite numbers” with a certain high probability, as described in Section 3.4. §.§ Justification of Algorithm 1 To show that Algorithm <ref> works with high accuracy, one must consider the probability that a number is Carmichael given that the number is composite and has no non-trivial Fermat witnesses for a random sample of t=⌊(ln n)^2⌋ integers from 1 to n-1. The proof of this algorithm requires Bayes' rule in conditional probability.Let X be the random variable for the event that a 1024-bit integer n is Carmichael. Let Y_t be the random variable for the event that either the proportion of Fermat witnesses is less than 45% for the random sample of size t or no non-trivial Fermat witnesses are found after checking if a^n-1≡ 1n for each element of the random sample. Also, let Z be the event that a 1024-bit integer n is composite. The desired probability is equivalent to Pr(X|(Y_t ∩ Z)).Recall that Bayes' rule states that:Pr(X|(Y_t ∩ Z)) =Pr((Y_t ∩ Z)|X)· Pr(X)Pr((Y_t∩ Z)|X)· Pr(X) +Pr((Y_t∩ Z)|X')· Pr(X').Note that X' refers to the event that n is not Carmichael. First, consider the numerator of the probability described above. Note that Pr((Y_t∩ Z)|X)=1 because if a number is Carmichael, then Z must be true because all Carmichael numbers are composite numbers and Y_t must be true because Carmichael numbers have no non-trivial Fermat witnesses. Thus, the numerator is equal to Pr(X). Finding the probability that a given 1024-bit integer (a common size of the prime numbers chosen for the RSA cryptosystem) is Carmichael is equivalent to finding the proportion of 1024-bit integers that are Carmichael numbers. The probability Pr(X) can also be expressed as C(2^1024)-C(2^1023)2^1023, where C(n) is a function of n that denotes the number of Carmichael numbers less than a number n. It has been found thatC(n) = n·exp(-k(n)·log n logloglog nloglog n)for some function k(n) defined over ℝ <cit.>. Note that C(n)n has been shown to be approximately n^0.34n for larger values of n. Thus, the numerator can be expressed as Pr(X)=(2^1024)^0.34-(2^1023)^0.342^1023.Consider the denominator of the probability of accuracy for Algorithm <ref>:Pr((Y_t∩ Z)|X)· Pr(X) + Pr((Y_t∩ Z)|X')· Pr(X').Note that Pr((Y_t∩ Z)|X)· Pr(X) is equal to the numerator, which is simply Pr(X). Now, considerthe term Pr((Y_t∩ Z)|X')· Pr(X'). Recall that Pr(X') denotes the probability that n is not Carmichael. Because X' is the random variable for the event that n is not Carmichael, Pr((Y_t∩ Z)|X') is the probability that either the proportion of Fermat witnesses is less than 45% for the random sample or no non-trivial Fermat witnesses are found and n is composite, given that the number is not Carmichael. Note that:Pr((Y_t∩ Z)|X')=p + (1-p)·[(1-|B|n)^t - (|A|n)^t],where p is the probability that less than 45% of the random sample are Fermat witnesses given that n is not Carmichael. Also, recall that B denotes the set of non-trivial Fermat witness and A denotes the set of all Fermat non-witnesses. The expression for Pr((Y_t∩ Z)|X') provided in the previous paragraph can be explained by the intuition behind Algorithm <ref>. In this algorithm, Carmichael numbers are first detected based on whether or not the proportion of Fermat witnesses is less than 45%. If the proportion of Fermat witnesses is greater than or equal to 45%, then the algorithm checks if there are any non-trivial Fermat witnesses. Similarly, in calculating the probability Pr((Y_t∩ Z)|X'), one must first account for the event that the proportion of Fermat witnesses is less than 45% for the sample. This first part is denoted by p, as defined earlier. Otherwise, if the proportion of Fermat witnesses for the sample is greater than or equal to 45%, then the probability is given by (1-p)·[(1-|B|n)^t - (|A|n)^t]. This is because the probability that the proportion of Fermat witnesses for the sample is greater than or equal to 45% is (1-p) and the probability that there are no non-trivial Fermat witnesses but there are some trivial Fermat witnesses found in the sample is [(1-|B|n)^t - (|A|n)^t] (in the case that there are no Fermat witnesses found, the number n could be prime, which would violate the event X).The equivalent expression for Pr((Y_t∩ Z)|X') described earlier can be evaluated by first approximating the value of p. The distribution of proportions of Fermat witnesses for the random samples is a binomial distribution with an average value of 1-|A|n because A denotes the set of all Fermat non-witnesses. Because the proportions of Fermat witnesses from random samples follow a binomial distribution, the standard deviation is given by σ =√(1t(|A|n)·(1-|A|n)). Since the RSA cryptosystem selects two large prime factors (of about 300 digits), the binomial distribution can be approximated by the probability density function, which describes a normal model (see Figure <ref>).gauss2 1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2)) tick label style=color=white, label style=font=, legend style=font=It suffices to find an approximate value of p, which may be found by approximating the value of σ and finding the value of x for which the lower values of x represent the event that the proportion of Fermat witnesses is less than 45% for the random sample found by a Monte Carlo simulation. Recall that the proportion of Fermat witnesses for all composite numbers with non-trivial Fermat witnesses is greater than 50%. In other words, |A|n<12, which implies that (1-|A|n)>12, because there exists at least one non-trivial Fermat witness when determining Pr((Y_t∪ Z)|X). To prove that Algorithm <ref> works for approximately 100% of the time, it suffices to show that when the 1-|A|n=12 this accuracy still holds.[Note that it follows from Lagrange's theorem that the group of all Fermat non-witnesses divides the order of the group (ℤ/nℤ)^×. The least proportion of Fermat witnesses for a number n with non-trivial Fermat witnesses is 1-ϕ(n)2n because of numbers such as 91 that can be expressed as q· (2q-1), where q and 2q-1 are prime. For 91, q=7.] To find the probability p, one must calculate the number of standard deviations x=0.45 is from the mean of 12 (this value is also referred to as a z-score or standard score):z=0.45-12√(1t(12)·(12)). Recall that p is the probability that less than 45% of the random sample are Fermat witnesses given that the number n is not Carmichael. The value of p is also equal to the area under the probability density function from -∞ to z. This area can be calculated using the cumulative distribution function, F(x):F(z)=1√(2π)∫_-∞^ze^-t^2/2dt,where z is the standard score.For the calculation of the value of p using the cumulative distribution function, a program in Mathematica can be used to approximate the value of z, which can then be used to evaluate F(z). To calculate the probability that Algorithm <ref> works correctly, one may use the approximate size (≈ 10^300) of the prime numbers used in the RSA cryptosystem to approximate the value of n. In particular, the calculation of the probability depends only on the size of the number n and not on actual prime factors of the number n. The calculated probability p yields a probability of approximately 100%.§.§ Efficiency of Algorithm 1 Using the Algorithm <ref> implementation and the Algorithm <ref> pseudocode, it can be calculated that Algorithm <ref> has a time complexity of O(t(log n)^3), where t is the sample size. The (log n)^3 represents time needed for determining the greatest common divisor of an element of the sample and n using the Euclidean algorithm. Although this algorithm maintains both high efficiency and high accuracy, Algorithm <ref> may not be compared to previous primality testing algorithms or previous Carmichael detecting algorithms because it relies on the fact that the number n is composite. Thus, to compare this algorithm with existing algorithms, modifications must be made in a way that Carmichael numbers are detected among not just composite numbers but all numbers. §.§ Algorithm 1 Modifications: Detecting Carmichael Numbers In Section 3.2, a novel algorithm for distinguishing Carmichael numbers and other composite numbers was described. This algorithm combined the properties of the Fermat witnesses for Carmichael numbers and other fundamental properties. This section exploits the aforementioned scheme to show a new algorithm that allows for the detection of Carmichael numbers and not just the separation between Carmichael numbers and other composite numbers.Instead of differentiating between Carmichael numbers and other composite numbers, one may modify the algorithm so that it could differentiate between the set of both Carmichael numbers and prime numbers and the set of all other composite numbers. This modification allows for a deterministic (or almost deterministic) primality test to check all of the numbers in the set of all Carmichael numbers and prime numbers, which is much smaller to check than the set of all integers.§.§ Proof of Correctness for Modified Algorithm If a number n is Carmichael or prime, then n must have no non-trivial Fermat witnesses. Otherwise, other composite numbers, which must have non-trivial Fermat witnesses, will be correctly determined to be “other composite numbers” with a certain high probability, as described in Section 3.8. Furthermore, a highly accurate primality test that has been proven for correctness will correctly distinguish between Carmichael numbers and prime numbers. §.§ Justification of the Modified Algorithm This section provides a proof for the high accuracy of the modified algorithm. The proof detailed in this section uses similar notions as those used in Section 3.4. However, the random variable for the event that the number is composite will not be of use in this proof that justifies the distinction of Carmichael numbers among all other integers. Let X be the random variable for the event that a 1024-bit integer is either Carmichael or prime. Also, let Y_t represent the random variable for the event that after random sampling t=⌊(ln n)^2⌋ times, either the proportion of Fermat witnesses for a number n is less than 45% or there are no non-trivial Fermat witnesses. The probability that must be calculated is as follows:Pr(X|Y_t)=Pr(Y_t|X)· Pr(X)Pr(Y_t|X)· Pr(X)+Pr(Y_t|X')· Pr(X'). Note that Pr(Y_t|X)=1 because Carmichael numbers and prime numbers have no non-trivial Fermat witnesses. So, the numerator is equal to Pr(X), which is the probability that a randomly chosen number is Carmichael or composite. Calculating this probability is the same as calculating the proportion of numbers less than a number n that are Carmichael or prime. As detailed in Section 3.4, the proportion of numbers that are Carmichael is approximately n^0.34n. The proportion of numbers less than n that are prime is approximately 1ln n, which is a result of the prime number theorem. Thus, accounting for the size of the prime numbers used in the RSA cryptosystem, it may be calculated that Pr(X)=(2^1024)^0.34-(2^1023)^0.342^1023+2^1024ln 2^1024-2^1023ln 2^10232^1023.For the denominator, Pr(Y_t|X)· Pr(X) is the same as the numerator. Now, consider the term Pr(Y_t|X')· Pr(X'). The left term, Pr(Y_t|X'), represents the probability that a number that is neither Carmichael nor prime has either a proportion of Fermat witnesses that is less than 45% or no non-trivial Fermat witnesses. This probability is exactly the same as Pr(Y_t∩ Z|X') = p + (1-p)·[(1-|B|n)^t - (|A|n)^t], as shown in Section 3.4. Thus, the probability Pr(X|Y_t) is equal to:Pr(X|Y_t)=Pr(X)Pr(X) + (1-Pr(X))·[p + (1-p)·[(1-|B|n)^t - (|A|n)^t]],which can be evaluated using Mathematica to approximate the probability using large numbers for n. Thus, the probability of accuracy of the modified algorithm is approximately 100%. §.§ Efficiency of the Modified Algorithm The modified algorithm is useful for finding a list of Carmichael numbers less than or equal to n. Suppose that the algorithm runs for the first n numbers. Then, the time complexity of the modified algorithm is O(nt(log n)^3 + (nlog n+C(n))· x ), where x is the run-time of a deterministic primality test that is combined with Algorithm <ref>. Note that (nlog n+C(n))· x represents the time needed for the modified part of the algorithm. Although this modified algorithm is efficient for the purposes of determining a list of Carmichael numbers, the efficiency could be optimized by finding a value of t for which the accuracy is still maintained.§ CONCLUSIONS AND FUTURE EXTENSIONS This paper determined both a classification of Carmichael numbers and a method for detecting Carmichael numbers, pseudoprimes to several primality tests. To further the research in this paper, one may examine the proportion of Fermat witnesses to find the percentage of Carmichael numbers with a proportion of Fermat witnesses of less than 50%. These findings may be used to modify the upper bound for which the proportion of Fermat witnesses is checked in Algorithm <ref>. Furthermore, the algorithm may be modified with an efficient deterministic primality test. Moreover, the value of t must be modified to improve the efficiency of the algorithm. To extend the idea of detecting pseudoprimes, one may examine either the proportion of witnesses for false positives of other primality tests that have many false positives. § ACKNOWLEDGMENTS The author wishes to thank his mentor, Hyun Jong Kim, for his guidance throughout this project. The author would also like to thank Dr. Tanya Khovanova for helping to edit this paper and the MIT PRIMES program for making this research possible. apacite

http://arxiv.org/abs/1702.08066v1

empty 1]Rui M. Castro 2]Ervin Tánczos [1]Technische Universiteit Eindhoven[2]University of Wisconsin - MadisonAre there needles in a moving haystack?Adaptive sensing for detection of dynamically evolving signals [======================================================================================================= In this paper we investigate the problem of detecting dynamically evolving signals.We model the signal as an n dimensional vector that is either zero or has s non-zero components.At each time step t∈ℕ the non-zero components change their location independently with probability p.The statistical problem is to decide whether the signal is a zero vector or in fact it has non-zero components.This decision is based on m noisy observations of individual signal components collected at times t=1,…,m.We consider two different sensing paradigms, namely adaptive and non-adaptive sensing.For non-adaptive sensing the choice of components to measure has to be decided before the data collection process started, while for adaptive sensing one can adjust the sensing process based on observations collected earlier.We characterize the difficulty of this detection problem in both sensing paradigms in terms of the aforementioned parameters, with special interest to the speed of change of the active components.In addition we provide an adaptive sensing algorithm for this problem and contrast its performance to that of non-adaptive detection algorithms. § INTRODUCTIONDetection of sparse signals is a problem that has been studied with great attention in the past.The usual setting of this problem involves a (potentially) very large number of items, of which a (typically) much smaller number may be exhibiting anomalous behavior.A natural question one can ask if it is possible to reliably detect if there are indeed some items showing anomalous behavior?Questions like this are encountered in a number of research fields.Some examples include epidemiology where one wishes to quickly detect an outbreak or the environmental risk factors of a disease <cit.>, identifying changes between multiple images <cit.>, and microarray data studies <cit.> to name a few.A common point in the examples above is that even though it is not known which items are anomalous, their identity remains fixed throughout the sampling/measurement process.However, in certain situations the identity of these items may change over time.Consider for instance a signal intelligence setting where one wishes to detect covert communications.Suppose that our task is to survey a signal spectrum, a small fraction of which may be used for communication, meaning that some frequencies would exhibit increased power.On one hand we do not know beforehand which frequencies are used, but also the other parties may change the frequencies they communicate through over time.This means we will be chasing a moving target.This introduces a further hindrance in our ability to detect whether someone is using the surveyed signal spectrum for covert communications.Other motivating examples for such a problem include spectrum scanning in a cognitive radio system <cit.>, detection of hot spots of a rapidly spreading disease <cit.>, detection of momentary astronomical events <cit.> or intrusions into computer systems <cit.>.The main question that we aim to answer in this paper is how the dynamical aspects of the signal affect the difficulty of the detection problem.In the more classical framework of the signal detection problem, inference is based on observations that are collected non-adaptively.However, dealing with time-dependent signals naturally leads to a setting where measurements can be obtained in a sequential and adaptive manner, using information gleaned in the past to guide subsequent sensing actions.Furthermore, in certain situations it is impossible to monitor the entire system at once, but instead one can only partially observe the system at any given time.It is known that, in certain situations, adaptive sensing procedures can very significantly outperform non-adaptive ones in signal detection tasks <cit.>.Hence our goal is to understand the differences between adaptive and non-adaptive sensing procedures when used for detecting dynamically evolving signals, in situations where the system can only be partially monitored. Contributions: In this paper we introduce a simple framework for studying the detection problem of time-evolving signals.Our signal of interest is an n-dimensional vector x_t ∈^n, where t∈ denotes the time index.We take a hypothesis testing point of view.Under the null the signal is static and equal to the zero vector for all t, while under the alternative the signal is a time-evolving s-sparse vector.At each time step t∈ we flip a biased coin independently for each non-zero signal component to decide if these will “move” to a different location.Thus, the coin bias p encodes the speed of change of the signal support in some sense.At each time step we are allowed to select one component of the signal to observe through additive standard normal noise, and we are allowed to collect up to m measurements.Our goal is to decide whether the signal is zero or not, based on the collected observations.We present an adaptive sensing algorithm that addresses the above problem, and show it is near-optimal by deriving the fundamental performance limits of any sensing and detection procedure.We do this in both the adaptive sensing and non-adaptive sensing settings for a range of parameter values p and s.It is easy to see that the above problem can not be solved reliably unless we are allowed to collect on the order of n/s measurements.When the number of measurements is of this order, we can reliably detect the presence of the signal when the smallest non-zero component scales roughly like √(p log (n/s)) in the adaptive sensing setting (Theorems <ref> and <ref>).In the non-adaptive sensing setting detection is possible only when the smallest non-zero component scales like √(log (n/s)) (Theorem <ref>).Hence, under the adaptive sensing paradigm the speed of change influences the difficulty of the detection problem, with slowly changing signals being easier to detect.Contrasting this, in the non-adaptive sensing setting the speed of change appears to have no strong effect in the problem difficulty when m is of the order n/s.When the number of measurements m is significantly larger than n/s the picture changes quite a bit, and a theoretical analysis of that case is beyond the contribution of this paper.Nevertheless we provide some simulation results indicating that, in the non-adaptive sensing setting, the signal dynamics will then influence the detection ability.Despite its simplicity, the setting introduced in this paper provides a good starting point to understand the problem of detecting dynamically evolving signals.Although we provide several answers in this setting many questions remain (both technical and conceptual).We hope that this work opens the door for many interesting and exciting extensions and developments, some of which are highlighted in Section <ref>. Related work: The setting where the identity of the anomalous items is fixed over time has been widely studied in the literature.Classically this problem has been addressed when each entry of the vector is observed exactly once.In this context both the fundamental limits of the detection problem and the optimal tests are well understood (see <cit.> and references therein).The same problem has been investigated in the adaptive sensing setting as well.In <cit.> the authors provide an efficient adaptive sensing algorithm for identifying a few anomalous items among a large number of items.These results were generalized in <cit.> to cope with a wide variety of distributions.The algorithms outlined in these works can in principle also be used to solve the detection problem, that is where only the presence or absence of anomalous items needs to be decided.In <cit.> and <cit.> bounds on the fundamental difficulty of the estimation problem were derived, whereas in <cit.> bounds for the detection problems were provided as well.Our work here has a similar flavor to all the above, but tackling the problem when the anomalous items may change positions while the measurement process is taking place.This brings a new temporal dimension to the signal detection problems referenced above.Statistical inference problems pertaining time-dependent signals have been investigated in various settings in the past.However, the papers referenced below only have varying degrees of connection to the problem we are considering, as despite our best efforts, we were only able to find a few instances that resemble our setting.A setting that has some degree of temporal dependence is the monitoring of multi-channel systems.This problem was introduced in <cit.> and later revisited in <cit.> and <cit.>.In this setting each channel of a multi-channel system contains a Wiener process, a few of which are anomalous and have a deterministic drift.The observer is allowed to monitor one channel at a time with the goal to localize the anomalous channels as quickly as possible.Although there is a clear temporal aspect to these problems, the anomalous channels identity is unchanged during the process.Another prototypical example of inference concerning temporal data is change-point detection in a system involving multiple processes.In this problem we have multiple sensors observing stochastic processes.After some unknown time a change occurs in the statistical behavior of some of the processes, and our goal is to detect when such a change occurs as quickly as possible.This setting has been studied in <cit.>, a Bayesian version of the problem was investigated in <cit.>, while the authors of <cit.> deal with a version of the above problem where only one of the sensors is compromised.This setting shares similarities to ours, but there are some key differences.In the change-point detection setting, once a process becomes anomalous it remains so indefinitely.Since some processes are bound to exhibit anomalous behavior, the goal is to minimize the detection delay.Contrasting this, in the setting we consider an anomalous process can revert back to the nominal state, and there is a possibility that none of the processes are anomalous at any time.Hence our goal is to decide between the presence or absence of any anomalous processes over the measurement horizon.A set of more closely related work is concerned with the spectrum scanning of multichannel cognitive radio systems.Here the aim is to quickly and accurately determine the availability of each spectrum band of a multi-band system where the occupancy status changes over time.Alternatively one might only aim to quickly find a single band that is available.This problem has been studied in <cit.> and <cit.>, in which the authors provide efficient algorithms for the problem at hand.A very similar problem was investigated in <cit.>, where one observes multiple ON/OFF processes and wishes to catch one in the ON state.Although the underlying models of these problems come very close to the one we consider, these works are also change-point detection problems in spirit.Hence a similar comment applies here as well, namely that the goal of the algorithms of <cit.> and <cit.> is to detect a change-point while minimizing some notion of regret (such as detection delay or sampling cost), which is somewhat different to the problem we are aiming to tackle. Organization: Section <ref> introduces the problem setup, including the signal and observation models and the inference goals.In Section <ref> we introduce an adaptive sensing algorithm and analyze its performance.Section <ref> is dedicated to the characterization of the difficulty of the detection of dynamically evolving signals.In particular we show that the algorithm presented in Section <ref> is near-optimal, and examine the difference between adaptive and non-adaptive sensing procedures.In Section <ref> we present numerical evidence supporting a conjecture on the non-adaptive sensing performance limit in the regime when m is of the order n/s.Concluding remarks and avenues for future research are provided in Section <ref>. § PROBLEM SETUPFor notational convenience let [k]={ 1,… ,k } where k∈.In our setting the underlying (unobserved) signal at time t is a n-dimensional vector, where time t∈ is discrete.Let μ>0 and denote the unknown signal at time t∈ by x⃗^(t)≡(x_1^(t),…,x_n^(t))∈^n, wherex^(t)_i = {[ μif i ∈ S^(t); 0 ifi ∉ S^(t) ]. ,and S^(t)⊂ [n] is the support of the signal at time t.We refer to the components of x⃗^(t) corresponding to the support S^(t) as the active components of the signal at time t.In Section <ref> we model the signal as a random process with the property that, at any time, the number of active components is much smaller than n.In this idealized model the active components of x⃗^(t) have all same value, which might seem restrictive at first.However, when the active components have different signs and magnitudes, the arguments of all the proofs hold throughout the paper with μ playing the role of the minimum absolute value of the active components.Although a more refined analysis is likely possible, where the minimum is replaced by a suitable function of the magnitudes of active components, we choose to sacrifice generality for the sake of clarity (see also Remark <ref> below).The signal is only observable through m noisy coordinate-wise measurements of the formY_t = x^(t)_A_t + W_t , t∈ [m] ,where A_t ∈ [n] is the index of the entry of the signal measured at time t and W_t are independent and identically distributed (i.i.d.) standard normal random variables.In the general adaptive sensing setting A_t is a (possibly random) measurable function of { Y_j ,A_j }_j∈ [t-1] and W_t is independent of {x⃗^(j),A_j }_j∈ [t] and {Y_j}_j∈[t-1].This means the choice of signal component to be measured can depend on the past observations.A more restrictive setting is that of non-adaptive sensing, where the choice of components to be measured has to be made before any data is collected.Formally A_t is independent from {Y_j ,A_j }_j∈ [t-1] for all t∈[m]. This measurement model is very similar to that of <cit.>, <cit.> and <cit.>, where measurements are of the formY_t = x_A_t + Γ_t^-1 W_t , t=1,2,… ,when x is a (time-independent) signal, A_t are as above, and Γ_t ∈ represent the precision of the measurements (that can be also chosen adaptively).In those papers the authors impose a restriction on the total precision used (and not on the number of measurements).However, since often the precision is related to the amount of time we have for an observation it is somewhat more appealing to consider fixed precision measurements instead.See also Remark <ref> for an alternative model closer in spirit to that of the above papers.Recently <cit.> considered an extension of the classical sparse signal detection problem in which the measurements are heteroscedastic, and derived the asymptotic constants of the detection boundary.In principle, a model similar in spirit to the one presented in that work could also be considered here as well, by assuming that measurements on active components not only have elevated means, but also variance different to 1.The ideas of <cit.> can be used to modify our detection procedure (in particular the Sequential Thresholding Test – see Algorithm <ref>) to craft a procedure that can deal with measurements of different variances.However, the question of heteroscedasticity for dynamically evolving signals is too rich to be dealt with in the present work. §.§ Signal dynamics We consider what might be the simplest non-trivial stochastic model for the evolution of the signal.Our goal is to model situations where the signal support S^(t) changes “slowly” over time.For concreteness consider first a particular situation, where we assume that at any time t there is a single active component (so |S^(t)|=1 for all t∈).We model the support evolution as a Markov process: the support S^(1) is chosen uniformly at random over the set [n] (that is, the active component is equally likely to be any of the [n] components); for t≥ 1 we flip a biased coin with heads probability p∈[0,1] independent of all the past, and if the outcome is heads then S^(t+1) is chosen uniformly at random over [n], otherwise S^(t+1)=S^(t). In words, at each time instant the active component stays in place with probability 1-p and “jumps” to another location with probability p.Thus when p=1 the signal has a new support drawn uniformly at random at each time t∈, whereas in case p=0 the support is chosen randomly at the beginning and stays the same over time.In general, the parameter p can be interpreted as the speed of change of the support, with larger values corresponding to a faster rate of change.This basic model of signal dynamics can be easily generalized to multiple active components model as follows.Let s∈[n] be the sparsity of our signal.We enforce that |S^(t)|=s for t∈, meaning the signal sparsity does not change over time.For t=1, S^(t) is chosen uniformly at random from the set {S⊆ [n] : |S|=s}.For time t ≥ 1, we flip s independent biased coins, each corresponding to an active component, to decide which components move and which stay in the same place.Formally take p∈[0,1] and let θ^(t)_i ∼ (p) be independent for every i ∈ [s], t∈.Consider an enumeration of S^(t) as S^(t)≡{S_i^(t)}_i∈ [s].If θ_i^(t)=0 component S_i^(t) will also be included in S^(t+1), otherwise it will move.The support set S^(t+1) is chosen uniformly at random from the set{ S ⊂ [n]: |S|=s, S∩ S^(t) = { S^(t)_i: θ^(t)_i =0 }} .For illustration purposes we provide some simulated results in Figure <ref> (n is chosen quite small for visual clarity only). Although we consider time to be discrete, continuous-time counterparts of this model are certainly possible (e.g., by taking the transition times to be generated by a Poisson process).A realistic measurement model in this case would require the variance of the observation noise to be inversely proportional to the time between consecutive measurements, effectively playing a similar role to the precision parameter as in <cit.>.§.§ Testing if a signal is present In the setting described one can envision several inference goals.One might try to “track” the active components of the signal, attempting to minimize the total number of errors over time.A somewhat different and in a sense statistically easier goal is to detect the presence of a signal, attempting to answer the question: are there any needles in this moving haystack?This is the question we pursue in this paper, and it can be naturally formulated as a binary hypothesis test.Under the null hypothesis there is no signal present, that is S^(t) = ∅ for every t∈.Under the alternative hypothesis there is a signal support evolving according to the model described above, for some s∈[n] and p∈[0,1].Ultimately, after we collected m observations we have to decide whether or not to reject the null hypothesis.Formally, let Ψ : { A_t,Y_t }_t∈ [m]→{ 0,1 } be a test function where the outcome 1 indicates the null hypothesis should be rejected.We evaluate the performance of any test Ψ≡Ψ({ A_t,Y_t }_t∈ [m]) in terms of the maximum of the type I and type II error probabilities, which we call the risk of a test R(Ψ).Namely we requireR(Ψ)≡max_i=0,1_i (Ψ≠ i) ≤ε ,with some fixed ε∈ (0,1/2), where _0 and _1 denote the probability measure of the observations and the null and alternative hypothesis, respectively.Later on we also use the notation _i, i∈{0,1} to denote the expectation operator under the null and alternative hypothesis respectively.Note that both the null and alternative hypothesis are simple in the current setup (as we assume p and μ to be known).In particular, the density of the observations = (y_1,… ,y_m) under the alternative can be written as the following mixture:_1 ( ) = [ ∏_t∈ [m] g ( A_t | { y_j ,A_j }_j∈ [t-1]) ( { A_t ∈ S^(t)} f_μ (y_t) + { A_t ∉ S^(t)} f_0 (y_t) ) ],where f_μ is the density of a normal distribution with mean μ and variance 1, { S^(t)}_t∈ [m] are the supports evolving as defined in Section <ref>, and g(A_t | { y_j ,A_j }_j∈ [t-1] ) is the density of the sensing action at time t.Note, however, that our detection procedures in Section <ref> do not require knowledge of μ or p.The main goal of this work is to understand how large the signal strength μ needs to be, as a function of n,m,s,p and ε to ensure (<ref>) is satisfied.To this end we first propose a specific adaptive sensing algorithm and evaluate its performance in Section <ref>.Furthermore in Section <ref> we prove that, in several settings, this algorithm is essentially optimal, by showing lower bounds on μ that are necessary for detection by any sensing and testing strategy.In the subsequent sections we will see that there is a complex interplay between the parameters n,m,s and p in how they affect the minimum signal strength required for reliable detection.It is noteworthy to stress that even when we restrict ourselves to the case p=1 the nature of the optimal test changes radically depending on the interplay between the remaining parameters n,m and s.In this case, the signal support is reset at every time t∈, which means that regardless of the sampling strategy (the choice of A_t) we are in the situation akin to a so-called sparse mixture model.These models are now well understood (see <cit.>, <cit.>, <cit.> and references therein).We know that in the case of mixture models, for very sparse signals a type of scan test (which is essentially a generalized likelihood-ratio test) performs optimally, whereas for less sparse signals a global test based on the sum of all the observations is optimal.In our case the interplay between the parameters n,s and m determines the level of sparsity of the sample under the alternative.This in turn means that when p=1 the optimal test and the scaling required for μ, depends on the relation between m and s/n.The above phenomenon becomes even more complex when p<1.Note, however, that unless m is at least of the order of n/s reliable detection is impossible (regardless of the value of p).The reason behind this is that no sampling strategy will sample an active component under the alternative in fewer measurements with sufficiently large probability.To see this consider the case p=0 and suppose there is no observation noise.Let the sampling strategy be arbitrary and let Ω denote the event that the algorithm does not sample an active component.When m≤ n/s we have_1 (Ω )≥n-sm/nm = (n-s)(n-s-1)… (n-s-m+1)/n(n-1)… (n-m+1)≥( 1-s/n-m)^m ≥( 1-2s/n)^n/s .The expression on the right is bounded away from zero when n/s is large enough.Hence regardless of the sampling strategy, there is a strictly positive probability that no active components are sampled under the alternative, which shows that (<ref>) can not hold for ε smaller than ( 1-2s/n)^n/s.When p> 0, sampling an active component becomes even harder, hence the same rationale holds.In this paper we focus primarily on the regime where the number of measurements m is only slightly larger than n/s (what might be deemed to be the “small sample” regime).If we are interested in scenarios where one needs a detection outcome as soon as possible this is the interesting regime to consider.Interestingly, when m is significantly larger than n/s the optimal sensing and testing strategies, as well as the fundamental difficulty of the problem appears to be quite different than that of the small sample regime, and is an interesting and likely fruitful direction for future work.In Section <ref> we conducted a small numerical experiment illustrating how the fundamental performance behavior changes in that regime. The results in this paper can be very naturally generalized for signals with different signs and magnitudes, by considering the class of signals characterized by the minimum signal magnitude.In the regime where m is of the order of n/s this is essentially the most natural characterization, since only a very small number of active components will actually be observed (so a very low magnitude component will hinder the performance of any method).When m is significantly larger the picture changes quite significantly and pursuing these results is an interesting avenue for future research beyond the scope of this paper. § A DETECTION PROCEDUREIn this section we present an adaptive sensing detection algorithm for the setting in Section <ref> and analyze its performance.To devise such a procedure we use a similar approach as taken by <cit.> — first devise a sensible procedure that works when there is no observation noise (i.e., when W_t≡ 0), and then make it robust to noise by using sequential testing ideas.Consider a setting where there is no measurement noise, that is, when measuring a component of x⃗^(t) we know for sure whether that component is zero or not.In such a setting if we find an active component we can immediately stop and deem Ψ =1.Note that it is wasteful to make more than one measurement per component, and that, before hitting an active component, we have absolutely no prior knowledge on the location of active components.Therefore an optimal adaptive sensing design is random component sampling without replacement.If we look at a large enough number of randomly chosen components and only observe zeros, it becomes reasonable to conclude that there are no active components and so we deem Ψ=0.Bear in mind though that in case we did not observe any active components we might have simply been unlucky, and missed them even though they are present.Hence, there is always a possibility for a false negative decision regardless of how many components we observe, unless p=0 and m≥ n-s.The procedure that we propose is a “robustified” version of the one explained above, so that it can deal with measurement noise.This is done by performing a simple sequential test to gauge the identity of the component that we are observing.A natural candidate for this is the Sequential Likelihood-Ratio Test (SLRT), introduced in <cit.>. However, the dynamical nature of the signal causes some difficulties.In particular the identity/activity of the component that we are observing might change while performing the test, creating many analytic hinderances in the study of the SLRT performance.We instead use a simplified testing/stopping criteria that is easier to analyze in such a scenario.The basic detection algorithm, presented in Algorithm <ref>, queries components uniformly at random one after another and tests their identity (whether they are active or not during the subsequent time period) using the sequential test to be described later.Once a component is deemed to have been active we set Ψ =1 and stop collecting data.If after examining T components or exhausting our measurement budget no components are deemed active we set Ψ = 0.Formally, let { Q_j }_j∈ [T] denote the components queried by Algorithm <ref>.We choose Q_j, j∈ [T] to be independent ([n]) random variables.[In principle one could ensure these are sampled without replacement from [n], but this would only unnecessarily complicate the analysis without yielding significant performance gains.] The appropriate number of queries T≤ m will be chosen later.For each Q_j we run a sequential test to determine the identity of that component.We refer to our sequential test as Sequential Thresholding Test (STT).To gauge the identity of Q_j, j∈ [T], the STT algorithm makes multiple measurements at that coordinate.The exact number of measurements depends on the observed values (in a way we describe in detail later), and hence it is random.We denote the number of observations collected by STT at coordinate Q_j by N_j.Formally, this means that A_t = Q_j for t∈[ 1+ ∑_i=1^j-1 N_i , ∑_i=1^j N_i ].At the end of the jth run of STT (j=1,2,… ,T), the STT returns either that an active component was present at coordinate Q_j, or that no active component was present at that location.In the former case there is no need to collect any more samples: Algorithm <ref> stops and declares Ψ =1.Otherwise we continue with applying STT to coordinate Q_j+1.If all T runs of STT found no signal, or we exhaust our measurement budget, Algorithm <ref> stops and returns Ψ =0.The sequential test that we use to examine the identity of a queried component is based on the ideas of distilled sensing introduced and analyzed in <cit.> and the Sequential Thresholding procedure of <cit.>.The distilled sensing algorithm is designed to recover the support of a sparse signal (whose active components remain the same during the sampling process).The main idea there is to use the fact that the signal is sparse and try to measure active components as often as possible, while not wasting too many measurements on components that are not part of the support.Our aim here is somewhat similar: on one hand we wish to quickly identify when the component that we are sampling is non-active so that we can move on to probe a different location of the signal.On the other hand in case we are sampling an active component we wish to keep sampling it as long as it is active to collect as much evidence as possible.However, unlike in the original setting of distilled sensing, we need to be able to quickly detect that we are sampling an active component, as it will eventually move away because of the dynamics.To address the last point the STT algorithm in Algorithm <ref> uses an evolving threshold for detection depending on the number of observations collected.We present STT in a way that emphasizes that it is a stand-alone routine plugged into the detection algorithm above, and not necessarily specific to the problem at hand.Hence, when discussing STT, the observations the STT makes are denoted by X^(1),X^(2),….In the context of Algorithm <ref>, for the jth call of STT we have X^(1),X^(2),… to be independent normal random variables with variance one and means respectively x^(T_j)_Q_j x^(T_j +1)_Q_j,…, where T_j=1+∑_i=1^j-1 N_i.In words, STT collects at most k measurements sequentially and keeps track of the running average until one of the stopping conditions is met.The first stopping condition says that once the running average drops below the threshold t_k we stop and declare that there is no signal present.The second says that if the running average at step j exceeds a threshold t_j, we stop and conclude that a signal component is present.Note that after each measurement the upper threshold decreases, eventually reaching t_k, hence the procedure necessarily terminates after at most k measurements.Key to the performance of the STT is a good choice of k and { t_j }_j∈ [k], which is informed by the following heuristic argument:the sample collected by the detection algorithm consists of T blocks of measurements, where each block corresponds to an application of STT.Let the block lengths be denoted by { N_j }_j∈ [T].Suppose for a moment that blocks entirely consist of either zero mean or non-zero mean measurements.In this case we can simply think of each block j as a single measurement with mean multiplied by √(N_j) for all j∈ [T].This would reduce the problem to a detection problem in a T-dimensional vector, each component being normally distributed and having unit variance.This is a well-understood setting, and we know that in this case the signal strength needs to scale as √(log T) when there are not too many active components (see for instance <cit.> and the references therein).Recall that we are concerned with the case where the number of measurements we are allowed to make is of the order n/s.Hence we do not expect to encounter active components too many times.This heuristic shows that we should calibrate STT in a way that when it encounters j consecutive measurements with elevated mean, it should be able to detect it when μ≈√(1jlog T)[In this informal discussion, the notations ≈ and ≳ hide constant factors and/or log (1/ε) terms.].Furthermore, considering the tail properties of the Gaussian distribution, it is easy to see that we also need μ≳√(log1ε) for reliable detection.Recalling that j≤ k, this shows that choosing k greater than log T does not buy us anything.Informed by the above heuristic argument we choose the parameters of STT so that the following result holds. Let ε∈ (0,1) and define the parameters of STT ask = ⌊log (T/2) ⌋ , t_j = √(c(2ε /T)/jlog2T/ε) , j∈ [k],wherec(x)= 2( 1+ loglog (1/x)/log (1/x)) .Denote the observations available to the STT by X^(1),… ,X^(k) (note that the STT may terminate without observing all the variables).Then the following holds: (i)If X^(i)∼^i.i.d.𝒩(0,1) for i∈ [k], then STT declares “Signal" with probability at most ε /T.(ii)For any j∈[k], if the X^(i)∼^i.i.d.𝒩(μ ,1) for i∈ [j] withμ≥√(c(2ε /T )/jlog2T/ε) + √(2 log4/ε) ,then STT declares “No Signal" with probability at most ε/3.Note that, for (ii) it suffices for the first j observations to have elevated mean to guarantee the good performance of the STT. For the first part suppose note that the STT declares “Signal" if at any time step j∈ [k] the running average X_j exceeds the threshold t_j.( ∃ j ∈ [k]: X^(j)≥ t_j )≤∑_j=1^k (X^(j)≥ t_j) ≤∑_j=1^k1/2exp( - j t_j^2/2)= ∑_j=1^⌊log (T/2) ⌋1/2exp( -c(2ε /T)/2logT/2ε) ≤1/2log (T/2) ·( 2ε/T)^c(2ε /T)/2 ,where the first inequality follows by a union bound, and the second inequality is follows by a tail bound on Gaussian random variables noting that X_j∼𝒩(0,1/j).The last expression above is at most ε /T, which can be checked by taking the logarithm:log( 1/2log (T/2) ·( 2ε/T)^c(2ε /T)/2)= loglog (T/2) + ( 1- loglog (T/(2ε))/log (2ε/T)) log (2ε /T) -log 2 = loglog (T/2) + log (2ε/T) - loglog (T/(2ε)) -log 2≤logε/T . For the second part assume the conditions in (ii) hold for μ as given in the lemma.Define the eventΩ = {∃ i∈ [j-1]: X^(i)≤ t_k } .Note that if this event happens, we stop and declare “No signal" in one of the first j-1 steps.(Declare “No signal") =(Ω ) +(Declare “No signal")|Ω) ( Ω ) ≤ (Ω ) +( X^(j)≤ t_j|Ω) ( Ω ) ≤ (Ω ) + (X^(j)≤ t_j ).Using a union bound and the same Gaussian tail bound as before, the last expression can be upper bounded by∑_i=1^j-11/2exp( -i (μ -t_k)^2/2) + 1/2exp( - j(μ -t_j)^2/2). Considering the first term above, note thatμ - t_k ≥ t_j + √(2 log4/ε) -t_k ≥√(2 log4/ε) ,since t_j ≥ t_k (recall that j≤ k).Hence the first term can be upper bounded as∑_i=1^j-11/2exp( -i (μ -t_k)^2/2) ≤1/2∑_i=1^j-1 (ε /4)^i ≤ε/21/4-ε≤ε /6.On the other hand, when μ satisfies the inequality above, the second term is simply upper bounded by (ε /4)^j, and so the left-hand-side of (<ref>) is less than ε/6+ε/8<ε/3. Using Lemma <ref>, we can establish a performance guarantee for our detection algorithm.Though it is possible to derive a result for fixed n and s it is more transparent to state a result for large n instead, better highlighting the impact of parameter p.Keeping this comment in mind, note that 2 ≤ c(x) ≤ 2(1+1/e)≤ 2√(2) and c(x)→ 2 as x → 0.Thus, keeping ε fixed and letting T→∞, we see that if there exists a τ >1 for whichμ≥τ√(2/jlog T) + √(2 log4/ε) ,then for T large enough the condition on μ in Lemma <ref> is satisfied.Furthermore, recall that our main interest is how the algorithm performs when the time horizon (number of measurements) is only slightly larger than n/s.Fix ε∈ (0,1/3) and assume s≡ s_n=o(n/(log n)^2) as n→∞.The parameter p≡ p_n is also allowed to depend on n.Set T= 9n2slog_2 3ε and the parameters of STT according to Lemma <ref>.If the measurement budget is m≥ 2T the detection algorithm satisfiesR(Ψ)=max_i=0,1_i (Ψ≠ i) ≤ε ,wheneverμ≥τ√(2 max{ 2p,1log (n/s)}log (n/s)) + √(2 log4/ε) ,for n large enough and τ>1 fixed (but arbitrary).Before we move on to the proof of this result, let us discuss its message.First note that the detection algorithm is agnostic about the speed of change p and the signal strength μ, though it does require knowledge of the sparsity s to set the parameter T. The number of measurements that we require is a multiple of n/s, which is the minimum amount necessary to be able to solve the problem (see Section <ref>).Furthermore, when p < 1/(2log (n/s)) the signal strength needs to scale as √(log (1/ε )), and when p≥ 2/log (n/s) it needs to scale as √(p log (n/s)).This matches the intuition that the speed of change p affects the problem difficulty in a monotonic fashion.We will show in Section <ref> that in the regime m ≈ n/s this scaling of μ is necessary to reliably solve this detection problem.In Figure <ref> we present an illustration of the above detection algorithm.We can clearly see the “random” exploration (in red) and the “tracking” of active components (in green).Note that in this case the algorithm missed that an active component was hit at time 8, so more exploration was needed. As we have mentioned in Section <ref>, for now we are interested in the case where the number of observations we can make is roughly n/s.Note that Theorem <ref> claims the same performance guarantee for every m that is at least of order n/s.In fact, it is not hard to see that the performance of this algorithm does not improve as m increases, hinting that it is suboptimal for large m.Actually this algorithm completely ignores the fact that a component might have multiple periods of activity over time, and that activity evidence from multiple components might be combined for detection, in a more global fashion.Consider the following simple algorithm: sample components uniformly at random in each step t∈ [m].Then in each step we hit an active component with probability s/n.We then roughly have ms/n active components in our sample under the alternative.Consider the standardized sum of our observations.Under the null this follows a standard normal distribution, whereas under the alternative it is distributed as N(√(m) s μ/n,1).Thus reliable detection using this simple global algorithm is possible when μ is of the order n/(√(m) s).Hence this algorithm clearly outperforms the one above when m is large enough (compared to n/s).This phenomena is not unlike that present in sparse mixture detection problems (e.g.as in <cit.>) where depending on the sparsity a global test might be optimal.In light of Lemma <ref>, the type I error probability is at most ε by a union bound.Hence we are left with studying the alternative.There are two ways that our algorithm can make a type II error.Either the measurement budget is exhausted, or we fail to identify an active component in T runs of STT.We bound the probability of the first event by ε/3, and of the second event by 2ε/3 ensuring that under the alternative the probability of error is bounded by ε.We start with upper bounding the probability of exhausting our measurement budget.Let N_j denote the number of measurements that STT makes when called for the jth time, for j∈ [T].Note that these random variables are independent and identically distributed, because the components to query are selected uniformly at random independently from the past, the dynamic evolution of the model is memoryless, and the observation noise is independent.First we upper bound _1 (N_1).Note that 1 ≤ N_1 ≤ k, where k=⌊log (T/2) ⌋ by Lemma <ref>.Let Ω denote the event that a non-zero mean observation appears at location A_1 in any of the first k steps.By the law of total expectation we have_1 (N_1) ≤ k _1 (Ω ) + _1 (N_1 |Ω ).Note that_1 (Ω ) = _1 (∃ t∈[k]: A_1 ∈ S^(t) ) ≤∑_t=1^k _1 (A_1 ∈ S^(t) ) ≤s/n + (k-1)s/n-s≤ks/n-s ,since the choice of A_1 (and S^(1)) is random, and in each subsequent step the probability that a signal component moves to location A_1 is at most s/(n-s) regardless of p.On the other hand, recalling that t_k = √(c(2ε /T)klogT2ε)≥√(2) is the lower stopping boundary of STT,_1 (N_1|Ω) = 1+∑_t=2^k _0 (N_1 ≥ t)≤ 1+∑_t=2^k _0 (X_t-1 > t_k) ≤ 1+∑_t=2^k _0 (X_t-1 > √(2)) ≤ 1+ 1/2∑_t=1^k-1 e^-t≤ 1+ 1/2(e-1) < 3/2.Hence_1 (N_1) ≤ 1+ 1/2(e-1) + k^2s/n-s < 3/2,for large enough n, since the last term can be made arbitrarily small by the definition of T, and the assumption on s.Since N_1 is also a bounded random variable, an easy (but crude) way of proceeding is to use Hoeffding's inequality to get_1 ( ∑_j=1^T N_j > m ) = _1 ( ∑_j=1^T N_j - _1 ( ∑_j=1^T N_j ) > m - _1 ( ∑_j=1^T N_j ) ) ≤_1 ( ∑_i=1^T N_i - _1 ( ∑_i=1^T N_i ) > T/2 ) ≤exp( - T/2k^2)=exp( - T/2⌊log(T/2)⌋^2)≤ε/3,provided T is large enough, which is the case if n is large enough.This shows that the probability that the measurement budget is exhausted is bounded by ε/3.The final step in the proof is to guarantee that the algorithm identifies an active component in one of the T tests with high probability.To show this, we first guarantee that there will be an instance in the repeated application of STT where the first 1/(2p) observations that the procedure has access to have elevated mean (when p=0 we only need that the STT probes an active component at least once).Then we can apply Lemma <ref> together with a union bound to conclude the proof.Let T_j = 1+∑_i=1^j-1 N_i denote the time when STT starts for the jth time.Let N = ∑_j=1^T { Q_j ∈ S^(T_j)} denote the number of times an active component is sampled at the start of an STT.Note that N ∼ (T,s/n).In these situations the STT has access to a sequence of active measurements (of random length).Denote the number of consecutive active observations these STTs have access to by {η_i }_i∈ [N], and for now assume p>0.Note that η_i ∼ (p) and {η_i }_i∈ [N] are independent.We have(∀ i ∈ [N]: η_i < 1/(2p) )≤( ∀ i∈ [N]: η_i <1/(2p) | N≥log_2 3ε)+ ( N < log_2 3ε).On one hand, note that the median of η_i is ⌈ -1/log_2 (1-p) ⌉ which is greater than 1/(2p).This can be easily checked by considering the cases p≥ 1/2 and p<1/2 separately.Hence the first term above can be upper bounded as( ∀ i∈ [N]: η_i <⌈ -1/log_2 (1-p) ⌉ | N≥log3ε) ≤ 2^-log_2 3ε = ε /3. On the other hand, N ∼ (T,s/n) and so by Bernstein's inequality,( N < (1-δ ) Ts/n) ≤exp( -3 δ^2/8Ts/n),for any δ∈ (0,1).However, note that plugging in the value of T together with δ = 2/3 yields( N<log_2 3ε) = ( N < (1-δ)Ts/n)≤exp( -49/48log_2 3ε) < ε /3,since log_2 x > log x for x>1.So we conclude that the probability that there is no block (out of T) with the first 1/(2p) observations active is bounded by 2ε/3.When p=0, we only need to control (N=0), for which we can simply use the inequality above since log_2 3ε >0.Finally, if such a block is present the probability STT will not detect it is bounded by ε/3 via part (ii) of Lemma <ref>, providedμ≥√(c(2ε/T)/min{ 1/(2p),⌊log (T/2) ⌋}log(T/2ε))+√(2log4/ε) ,where one should note that the blocks sampled by the STT are never larger than ⌊log (T/2) ⌋.It is easily checked that the above condition is met for the choices in the theorem, provided n is large enough, concluding the proof. § LOWER BOUNDSIn this section we identify conditions for the signal strength that are necessary for the existence of a sensing procedure to have small risk, namelyR(Ψ)=max_i=0,1_i (Ψ≠ i) ≤ε .We consider first the non-adaptive sensing setting.This is done both for comparison purposes (to highlight the gains of sensing adaptivity) but also illustrates some of the interesting features of this problem.In this case the sensing procedure is simply the choice of when and where to measure a component, before any data is collected.Then we consider the adaptive sensing setting to show the near-optimality of the algorithm proposed in Section <ref>.In both cases our primary interests in on the regime m ≈ n/s, as highlighted in Section <ref>. §.§ Non-adaptive sensingIn the non-adaptive sensing setting, the sampling strategy { A_t }_t∈ [m] needs to be specified before any observations are made.Note that this does not exclude the possibility of having a random design of the sensing actions.Common sense tells us that supports that are changing fast are harder to detect than those that are changing slowly, provided all other parameters are fixed.In other words, the problem difficulty should be increasing in the parameter p, meaning the signal magnitude μ needed to ensure (<ref>) should grow monotonically in p.Formalizing this heuristic in general turns out to be technically challenging with the methodologies we are aware of.Because of this we focus on the two extreme cases: when the signal is static (p=0), and when the entire signal resets at each time instance (p=1). Note that in the case s=1 it is relatively easy to formalize that the problem difficulty is non-decreasing in p.Suppose there exists an algorithm (denoted by Alg) that performs accurate detection for some p>0, and suppose we need to perform the detection task of a static signal.The idea is to transform the signal into one that has the same distribution as if it were generated according to the model of Section <ref> with parameter p, and apply Alg to the modified signal.If such a transformation is possible than the existence of Alg implies the existence of an accurate detection procedure – in other words, the problem difficulty is non-decreasing in p.Such a transformation is easy to construct for s=1, in fact one can almost follow the description of the signal model of Section <ref> word-by-word.Let {θ_t }_t∈ [m-1] be i.i.d.(p) variables and w.l.o.g.θ_m =1 — these represent the coin flips in the description of Section <ref>.Let N=∑_t∈ [m]{θ_t =1 } be the number of times the coin came up heads and τ_0 =0 and τ_j = inf{ t> τ_j-1: θ_t =1 }, j∈ [N] be the instances when the coin came up heads.Finally, let {π_i }_i∈ [N] be permutations of [n] drawn independently and uniformly at random (from the set of possible permutations).It is clear that a static support that is permuted by π_i on the time intervals [τ_i-1+1,τ_i ] will "look" like a support sequence evolving with parameter p.Formally, one can show that if ≡{ S^(t)}_t∈ [m] is a static support sequence (chosen uniformly at random) then ≡{S̃^(t)}_t∈ [m] defined asS̃^(t) = ∑_i∈ [N]{ t∈ [τ_i-1+1 ,τ_i ] π_i (S^(t))is distributed as a support sequence generated according to the model described in Section <ref> with parameter p.Hence for s=1 the problem difficulty is indeed non-decreasing in p.Nonetheless the authors did not find an obvious way to extend this argument to general sparsities, because the signal components change their locations at possibly different times.We note at this point that if one considered a more restrictive model where the entire support of the signal would reset simultaneously (a setting perhaps not vastly different to the one we are considering) would enable an argument similar to the above. We have the following result for these two extreme cases, which we prove at the end of the section.Note that these are not asymptotic, and hold for any n,m and s satisfying the assumptions in the statement. Let n,s,m ∈ be fixed (with s≤ n), consider a setup described in Section <ref>, and suppose there is a non-adaptive sensing design and a test Ψ satisfyingR(Ψ)=max_i=0,1_i (Ψ≠ i) ≤ε .(i)If p=0, s≤ n/2, n/s≤ m and ε≤ 1/(2e) then necessarilyμ≥√(n/2mslog( 2 n/s^2log( 1e-4ε)+1 )) .(ii)If p=1 and ε <1/2 then necessarilyμ≥√(log( n^2/s^2 mlog( 4(1-2ε )^2 +1 ) +1 )) .Considering the case p=1, the result above tells us that when m scales like n/s, the signal strength needs to scale as √(log (n/s)) for detection to be possible.This is the same scaling that is guaranteed by Theorem <ref>.This should come as no surprise, since when p=1 we have { A_t ∈ S^(t)}∼ (s/n) independently for every t∈ [m], regardless of the choice of A_t.Hence the resulting measurements { Y_t }_t∈ [m] follow the same mixture distribution under the alternative, no matter what sampling strategy we use.Although settings like these have been studied extensively (see <cit.> and references therein), those works consider asymptotic results.As such we find it useful to prove a non-asymptotic result for our particular problem, though we point out that this can be simply established by following the steps of the referenced proofs.Contrasting with this one has the (arguably) more interesting case when the signal is static (p=0).Although the problem of detecting static signals have been the focus of much work (see for instance <cit.>), a key difference in our setting is that the sensing actions of the experimenter are not fixed, but are free to choose.This results in a qualitatively different statement, as the following remark attests. In particular, the first part of the theorem above is interesting in its own right.It tells us that, for static signals, if the experimenter is free to choose the sensing actions, the signal magnitude needs to scale at least as √(nsmlogns^2) for detection to be possible.It is easy to see that this rate can (almost) be achieved using a sub-sampling scheme: select roughly n/s components at random and collect an equal number of samples of each.Average the observations for each component separately, and declare a signal if any of these averages is above the threshold √(nsmlogns).Basic calculations show that this procedure has low probability of error.Contrasting this, the lower bounds of <cit.>, which pertain the situation where we measure each component of the vector exactly once, scale as √(logns^2).Hence, the additional flexibility of where to sample buys us a multiplicative factor of √(nsm), even though no feedback from the observations is used.If we can use this feedback, we can also get rid of the log-factor, as shown in <cit.>.In light of the previous remark, the authors suspect the lower bound in part (i) of the Theorem is slightly loose.Namely, the term s^2 appears to be due to slack in the second moment method in Equation <ref>, and it might be possible to replace it by s via a more sophisticated truncation argument. The result above tells us that in the regime m ≈ n/s, the signal strength needs to scale as √(log (n/s^2)) for detection to be possible — approximately the same magnitude as required for p=1.On the other hand Theorem <ref> guarantees the existence of an adaptive sensing procedure that reliably detects static signals of constant magnitude (in terms of the parameters n and s) using roughly n/s measurements.This shows that adaptive sensing gains over non-adaptive sensing become more pronounced as the speed of change decreases.Finally we point out once more that the requirements for the signal strength of Theorem <ref> are essentially the same for p=0 and p=1.Although we did not succeed in proving a result that holds for any value of p due to technical difficulties, we conjecture that the lower bound or general values of p should interpolate between these two extremes.In other words, we suspect that the problem difficulty is essentially independent of p in the non-adaptive case when m is of the order (or slightly larger than) n/s.This conjecture is further supported by numerical simulations of testing error probability presented in Section <ref>. (i): To prove the claim above for p=0 we use the truncated second moment method, an approach suggested by <cit.> to address problems in the regular second moment method when the distribution of the likelihood ratio under the null has tails that are too heavy (and therefore too large of a second moment).First, note thatmax_i=0,1_i (Ψ≠ i) ≥1/2∑_i=0^1 _i (Ψ≠ i) = 1/2( 1-1/2_0 ( |L(Y⃗)-1|) ),where L(Y⃗) denotes the likelihood-ratio of the observations Y⃗=(Y_1,…,Y_m), and _0 is the expectation taken with respect to the distribution of the observations Y⃗ under the null.The second equality is well known (see for instance <cit.>), and can be easily checked using simple algebraic manipulations.A common way to proceed is to use either Cauchy-Schwarz's or Jensen's inequality to get_0 ( |L(Y⃗)-1|) ≤√(_0 ( (L(Y⃗)-1)^2 )) = √(_0 (L(Y⃗))) .Therefore, to get a lower bound on the risk we need to get a good upper bound on the variance of the likelihood ratio.This is often referred to as the second moment method.However, in some cases there is a lot of slack in the bound and the variance is too large to yield interesting results — so a modification of the above argument is needed.Let 𝒴 denote the sample space, and let L̃(y⃗): 𝒴→ be an arbitrary function.Instead of using the Cauchy-Schwarz inequality right away, let us continue the first chain of inequalities as_0 ( |L(Y⃗)-1|) = _0 ( |L(Y⃗) -L̃(Y⃗) + L̃(Y⃗) -1| ) ≤_0 ( |L̃(Y⃗) -1| ) + _0 ( |L(Y⃗) -L̃(Y⃗)| ) ≤√(_0 ( L̃(Y⃗)^2 ) -2_0 ( L̃(Y⃗)) +1) + _0 ( |L(Y⃗) -L̃(Y⃗)| ).Furthermore, if L̃(y⃗)≤ L(y⃗) for every y⃗∈𝒴, then we have_0 ( |L(Y⃗)-1|) ≤√(_0 ( L̃(Y⃗)^2 ) -2_0 ( L̃(Y⃗)) +1) +1 -_0 ( L̃(Y⃗) ).In order to proceed, we need to lower bound _0 (L̃(Y⃗)) and upper bound _0 (L̃(Y⃗)^2).To get a sharp lower bound with this method, we need a good choice for L̃(y⃗).This is often achieved by truncating the original likelihood-ratio by multiplying with the indicator of a well chosen event.In our setting the likelihood-ratio can be expressed in a convenient way.Note that under the null the observations are independent standard normal regardless of the sensing actions, hence_0 (y⃗) = ∏_t∈ [m] f_0(y_t),where f_μ (· ) is the density of a normal random variable with mean μ and variance 1.Under the alternative, the density of the observations is a mixture.Recall that we are considering the case p=0 therefore the signal support S^(t) does not change over time, namely S^(t)=S for all t∈[m].The conditional density of the observations given the sensing actions A=(A_1 ,… ,A_m ) and the support S can be written as_1 (y⃗|A,S) = ∏_t∈ [m]( { A_t ∈ S} f_μ (y_t) + { A_t ∉ S} f_0 (y_t) ).Hence the likelihood-ratio can be expressed asL(y⃗) = ( exp( ∑_t∈ [m]{ A_t ∈ S}logf_μ (y_t)/f_0 (y_t)) ). Using the second moment method without truncation, one would need to upper bound the second moment of the likelihood ratio above.Unfortunately, this yields a loose bound on μ.The reason is that the second moment will be extremely large when the signal is sampled often, even if this event is relatively rare.In other words, if ∑_t∈ [m]{ A_t ∈ S} is large one will face problems.Note that, since the support is chosen uniformly at random,( ∑_t∈ [m]{ A_t ∈ S }) = ms/n.However, for certain choices of design ∑_t∈ [m]{ A_t ∈ S} can be very far from the mean (e.g., if A_1=⋯=A_m then ∑_t∈ [m]{ A_t ∈ S} is equal to m with probability s/n and zero otherwise).This causes the second moment of the likelihood ratio to be extremely large.To resolve this issue we truncate the likelihood-ratio to exclude these somewhat troublesome instances.Begin by defining the setsA_ big = { i: ∑_t∈ [m]{ A_t =i } > 2ms/n } andA_ small = [n] ∖ A_ big .In words, for a given sensing design the signal components are divided in two disjoint subsets: one subset contains signal components that are sampled often, whereas the other contains the remaining components.A simple pigeon hole principle shows that |A_ big|≤ n/(2s).Now defineL̃(Y⃗) = [.{S⊆ A_ small}exp( ∑_t∈ [m]{ A_t ∈ S}logf_μ (Y_t)/f_0 (Y_t))| Y⃗].Clearly L̃(y⃗) ≤ L(y⃗) for all y⃗∈𝒴, and so we can apply (<ref>) by controlling the first and second moments of L̃(Y⃗).First note that, since the event S⊆ A_ small does not involve the observations Y⃗ we can easily conclude that_0 ( L̃(Y⃗) ) =(S⊆ A_ small) = ( (S⊆ A_ small|A⃗) ),where A⃗≡ (A_1 ,… ,A_m ).The conditional probability on the right can be lower bounded as(S⊆ A_ small|A⃗) = |A_ small|s/ns = |A_ small| (|A_ small|-1)… (|A_ small|-s+1)/n(n-1)… (n-s+1)≥( |A_ small|-s+1/n-s+1)^s ≥( n(1-12s)-s+1/n-s+1)^s= ( 1-n/2s(n-s+1))^s ≥( 1-1/s)^s≥1/e ,where we used |A_ small|≥ n(1-1/2s) and 1≤ s≤ n/2.We are left with upper bounding the second moment of L̃(Y⃗).First, note that in the non-adaptive sensing setting A=(A_1,… ,A_m) and S are independent.The proof proceeds by careful conditioning on these random quantities.We use Jensen's inequality to write_0 (L̃(Y)^2) = _0 [ ([{S⊆ A_ small}exp( ∑_t∈ [m]{ A_t ∈ S}logf_μ (Y_t)/f_0 (Y_t))| Y⃗] )^2] ≤_0[[([{S⊆ A_ small}exp( ∑_t∈ [m]{ A_t ∈ S}logf_μ (Y_t)/f_0 (Y_t))_h(S,Y⃗,A⃗)| Y⃗,A⃗])^2|Y⃗]] .At this point it is convenient to introduce an extra random variable S', independent from S and identically distributed.Then([.h(S,Y⃗,A⃗)| Y⃗,A⃗])^2= [.h(S,Y⃗,A⃗)| Y⃗,A⃗] [.h(S',Y⃗,A⃗)| Y⃗,A⃗]= [.h(S,Y⃗,A⃗)h(S',Y⃗,A⃗)| Y⃗,A⃗] .Therefore we conclude that_0[L̃(Y⃗)^2]≤_0 [{S⊆ A_ small}{S'⊆ A_ small}exp( ∑_t∈ [m]({ A_t ∈ S}+{ A_t ∈ S'}) logf_μ (Y_t)/f_0 (Y_t))]= [ _0[.{S,S'⊆ A_ small}exp( ∑_t∈ [m]({ A_t ∈ S}+{ A_t ∈ S'}) logf_μ (Y_t)/f_0 (Y_t))|A⃗,S,S']]= [{S,S'⊆ A_ small}∏_t∈ [m]_0[.exp(({ A_t ∈ S}+{ A_t ∈ S'}) logf_μ (Y_t)/f_0 (Y_t))|A⃗,S,S']]= [{S,S'⊆ A_ small}exp(μ^2 ∑_t∈ [m]{A_t∈ S∩ S'})] .We are now in a good position to finish the bound.Note that, when S,S'⊆ A_ small we have ∑_t∈[m]{A_t=i}≤ 2ms/n.It follows that_0[L̃(Y⃗)^2]≤[[.{S,S'⊆ A_ small}exp(μ^2 ∑_i∈[n]{i∈ S∩ S'}∑_t∈[m]{A_t=i}) |A⃗]]≤[[.exp(2msμ^2/n∑_i∈[n]{i∈ S∩ S'}) |A⃗]]= [ exp( λ∑_i∈[n]{i∈ S∩ S'}) ] ,where λ=2msμ^2/n.The beauty of the last expression is that it no longer involves the sensing actions or the observations, and depends only on the support.Using the negative association property of { i∈ S∩ S'} as introduced in <cit.> we can finally bound the second moment of the truncated likelihood as_0 (L̃(Y⃗)^2)≤[exp(λ∑_i∈[n]{i∈ S∩ S'}])] = [∏_i∈[n] e^λ{i∈ S∩ S'}] ≤∏_i∈[n][ e^λ{i∈ S∩ S'}] = ( 1+ s^2/n^2( e^λ -1 ) )^n = ( 1+ s^2/n^2( e^2μ^2 ms/n -1 ) )^n.We have now all the ingredients needed to complete the proof.Note that, on one hand, if max_i=0,1_i (Ψ≠ i) ≤ε then necessarily _0[|L(Y⃗)-1|]≥ 2-4ε.On the other hand, from (<ref>) we know that_0[|L(Y⃗)-1|]≤√(_0 ( L̃(Y⃗)^2 ) -2_0 ( L̃(Y⃗)) +1) +1 -_0 ( L̃(Y⃗) )< (1+ s^2/n^2( e^2μ^2 ms/n -1) )^n/2 +1/2 .This means thats^2/n^2(e^2μ^2 ms/n -1) > (3/2-4ε)^2/n-1≥2/nlog(3/2-4ε) ,where the last inequality uses the fact that x-1 ≥log x.The final result ensues by simple algebraic manipulation.(ii): Proving the claim for p=1 requires considerably less technical effort.In particular we can use the original second moment method, without truncation.Therefore, we simply need to upper bound the second moment of the likelihood-ratio.Using essentially the same calculations as before, we get_0 ( L(Y)^2 ) = [ exp( μ^2 ∑_t∈ [m]{ A_t ∈ S^(t)∩ S'^(t)}) ].When p=1 we have that { A_t ∈ S^(t)∩ S'^(t)}∼ (s^2 /n^2 ) and these random variables are independent, so we can simply evaluate the above expression and get _0 ( L(Y)^2 ) = ( 1+ s^2/n^2( e^μ^2 -1 ) )^m. Plugging this into the inequalities above (not using the truncation), we getμ≥√(log( n^2/s^2( √(4(1-2ε )^2)-1 ) +1 )) .The desired result follows by using x-1 ≥log x. §.§ Adaptive sensingIn the adaptive sensing setting, the decision where to sample at time t can depend on information gleaned up to that point.For the static case (p=0) the fundamental limits of the detection problem using adaptive sensing have been studied in <cit.>.Those lower bounds are derived for a slightly more general setting than the one considered here, in that the total precision of the measurements is constrained, but not the total number of measurements.Nevertheless, this bound is still valid in our setting, and states that for any adaptive sensing and testing procedure Ψ ifmax{_0 (Ψ≠ 0 ), _1 (Ψ≠ 1) }≤εthen necessarilyμ≥√(2 n/smlog1/2ε) . In the regime m ≈ n/s the bound states that the signal strength needs scale as √(log (1/ε )).This coincides (up to constants) with the bound of Theorem <ref> when p≤ 2/log (n/s).This tells us that when the signal changes slowly enough, the problem is essentially non-dynamic in nature.On the other extreme end of the spectrum is the case p=1.We have seen previously that in this case the non-adaptive and adaptive sensing settings are identical, by virtue of the fact that { A_t∈ S^(t)}∼ (s/n) for every t∈ [m] and independent, regardless of the choice of A_t.What remains to be understood are the fundamental limits for the intermediate regime. §.§.§ Non-extreme dynamics (p∈ (0,1)) For general values of p we start by considering the case s=1, which we call the 1-sparse case.This case is considerably simpler to analyze than the general s-sparse setting, as now whenever the active component changes the entire signal resets.This effectively creates a number of independent static signals on the time horizon. Consider the setup in Section <ref> and suppose there exists a test Ψ such thatmax_i=0,1_i (Ψ≠ i) ≤ε .(i)The signal strength must satisfyμ≥√(2n/smlog1/4ε) .(ii) When s=1 and p≥ 8/m, then necessarilyμ≥√(p/2clog( log( ( 54 -4ε)^2 +12) p^2 n^2/4 c^2 m +1 )) ,with c=6+3log 2.We provide the proof of Theorem <ref> at the end of the section.Part (i) holds regardless of the values of p and s, so it is necessarily loose when p is large.On the other hand part (ii) already captures the role of the rate of change p, and it is the main contribution in this result.Let us compare the above bound on μ with the guarantees for Algorithm <ref> proved in Theorem <ref>.Note that c and ε are constants.Thus the bound on the signal strength in the above result scales as √(p log(p^2 n^2/m)).Recall that we are interested in the regime m≈ n/s and that s=1, as we are considering the 1-sparse case.In that setting the bound above scales as √(p log (p^2 n)).Also note that the scaling of the performance guarantee of Theorem <ref> matches that of the lower bound from <cit.> when p<1/log n.Hence we only need to assess the result of Theorem <ref> when p≥ 1/log n.In this case, the scaling of that bound is at least as big as √(p(log n - 2loglog n))≈√(p log n).This shows near-optimality of the algorithm proposed in Section <ref>, in terms of its scaling in the parameters n and p.Due to technical reasons we were unable to generalize the result for signals of sparsity greater than one.As noted above, a key feature of the 1-sparse case is that the signal decouples into independent static signals over time.This key property is lost when we consider signals with sparsity greater than one, and this proves to be a major obstacle to obtain a rigorous formal proof.However, we conjecture that a similar result to the one above holds for s-sparse signals, with n replaced by n/s.The heuristic behind this is that a general s-sparse signal of dimension n should behave very much like an s-fold concatenation of an 1-sparse signal of dimension n/s, when viewed through the lens of one measurement per time-index (one expects this to actually be a statistical reduction, and this problem should be statistically easier than the original one).For such a signal the result above would follow directly with the signal dimension n replaced by n/s. When p≥ 8/m, if the risk of an adaptive sensing and test procedure is less or equal to ε then necessarilyμ≥√(p/2clog( log( ( 54 -4ε)^2 +12) p^2 n^2/4 c^2 s^2 m +1 )) ,with c=6+3log 2.We prove the two parts of the statement separately.(i): The proof is very similar to that of Theorem 3.1 in <cit.>, with small modifications to be able to deal with dynamically evolving signals (which actually simplify the argument).By Theorem 2.2 of <cit.> we haveinf_Ψmax_i=0,1_i (Ψ≠ i) ≥1/4 e^-(_0 _1 ) ,where (_0 _1 ) denotes the Kullback-Leibler divergence between the distribution of the dataunder the null and alternative respectively.This divergence can be simply upper bounded using Jensen's inequality as(_0 _1 ) = _0 [ -log L( ) ] ≤_0 [ [ .-∑_t∈ [m]{ A_t ∈ S^(t)}logf_μ (Y_t)/f_0 (Y_t)| ] ]. Changing the order of integration and expanding the densities f_μ (· ) and f_0 (· ) we get(_0 _1 ) ≤μ^2/2[ ∑_t∈ [m]{ A_t ∈ S^(t)}] = μ^2/2sm/n ,where the last step follows from the symmetry of the supports.In particular note that [ { A_t ∈ S^(t)} | A_t ] = s/n for every t∈ [m].Plugging this bound into the right side of (<ref>), using that the left side of (<ref>) is at most ε due to our assumption, and rearranging concludes the proof of the first claim.(ii): We use the truncated second moment method, as in the proof of Theorem <ref>.Recall that, from (<ref>) and (<ref>) , we have2 max_i=0,1_i (Ψ≠ i) ≥ 1- 1/2( √(_0 ( L̃(Y)^2 ) -2_0 ( L̃(Y)) +1) +1 -_0 ( L̃(Y) ) ),for any function L̃(· ) satisfying L̃( ) ≤ L( ), ∀∈𝒴, where L(· ) is the likelihood function.To aid the presentation we begin by introducing some convenient notation, illustrated in Figure <ref>.Recall that the variables θ^(t)_i∼^i.i.d.(p), t∈ [m], i∈ [s] identify the change points of the signal.Since now we are dealing with the 1-sparse case we have one variable per time index, so in what follows we drop the subscript from the previous notation.Furthermore, note that our time horizon is m, so we enforce θ^(m) =1 as this does not change the model and it is convenient for the presentation.Let the total number of change points over the time horizon be N = ∑_t∈ [m]{θ^(t) =1}.Note that N-1 ∼ (m-1,p).Let τ_0 = 0 and for j∈ N let τ_j = min{ t>τ_j-1: θ^(t) =1 } denote the time instances when the signal changes (so τ_N =m), as illustrated in Figure <ref>.Note that on the time intervals [ τ_j +1,τ_j+1 ] the signal is static.Let l_j=τ_j+1-τ_j denote the length of these intervals, and S_j, j∈ [N] denote the correspoding signal support.Finally, for any t∈ [m] let the number of change points up to time t be N(t) = max{ j: τ_j ≤ t }.It is important to note that the random variables θ^(t) completely determine the variables τ_j, N(t) and N.Let us first explicitly write the likelihood of the observations in the model under consideration.We use the shorthand notation = { y_t }_t∈ [m],={ A_t }_t∈ [m],={ S^(t)}_t∈[m],={θ_t }_t∈ [m].As before, the density ofunder the alternative is a mixture.In particular, denoting the density of N(μ ,1) by f_μ, the conditional density ofcan be written as_1 ( | , ) = ∏_t∈ [m]( { A_t ∈ S^(t)} f_μ (y_t) + { A_t ∉ S^(t)} f_0 (y_t) )= ∏_j∈ [N]∏_t= τ_j-1+1^τ_j( { A_t ∈ S_j } f_μ (y_t) + { A_t ∉ S_j } f_0 (y_t) ).Hence, the likelihood ratio isL( ) = [ [ .exp( ∑_j∈ [N]∑_t=τ_j-1+1^τ_j{ A_t∈ S_j }logf_μ (y_t)/f_0 (y_t)) |,] ],where conditioning onandis done in order to conveniently define L̃().Consider the eventΩ_c = {∀ j: l_j ≤ 2c/p } ,with some fixed c>0.This event says that the signal is never static for a time longer than 2c/p.Note that this event is determined exclusively by the variables {θ_t }_t∈ [m].We define the truncated likelihood asL̃( ) = [ {Ω_c}[ .exp( ∑_j∈ [N]∑_t=τ_j-1+1^τ_j{ A_t∈ S_j }logf_μ (y_t)/f_0 (y_t)) |,] ]. As in the proof of Theorem <ref>, we need to upper bound _0 ( L̃()^2 ) and lower bound _0 ( L̃()).We start with the latter.Since the event Ω_c only involves the variables , we have_0 ( L̃()) =(Ω_c ).We have the following result, the proof of which is presented in the Appendix. Consider the eventΩ_c = {∀ j: l_j ≤ 2c/p } .In the model described above (Ω_c ) >1/4 whenever c≥ 6 + 3 log 2 and p ≥ 8/m. According to Lemma <ref>, we have an appropriate bound for _0 ( L̃(Y)) when c≥ 6 + 3 log 2.All that remains is to derive an upper bound on the truncated second moment.This can be done much the same way as in the proof of Theorem <ref>.Using Jensen's inequality, we have_0 [ L̃( )^2 ] ≤_0 [ [ {Ω_c}[ .exp( ∑_j∈ [N]∑_t=τ_j-1+1^τ_j{ A_t∈ S_j }logf_μ (y_t)/f_0 (y_t)) |,]^2 ] ].Note that given , the S_j ∼ ([n]) and independent for j∈ [N].Let { S'_j }_j∈ [N] be an independent copy of { S_j }_j∈ [N].Following the same reasoning as in Theorem <ref> we can write the square of the conditional expectation above as the product of two expectations using the random variables {S_j, S'_j }_j∈ [N], and change the order of the expectations to get_0 [ L̃( )^2 ] ≤[ {Ω_c}[ .exp( μ^2 ∑_j∈ [N]∑_t=τ_j-1+1^τ_j{A_t ∈ S_j ∩ S'_j }) |,] ]. So far we have not taken into account the fact that we are allowed an adaptive design.This is captured by the crude bound below.∑_t=τ_j-1+1^τ_j{A_t ∈ S_j ∩ S'_j }≤ l_j {∃ t∈ [τ_j-1+1,τ_j ]: A_t ∈ S_j ∩ S'_j } .Informally this means that, if the used design “hits” the signal at any place in the interval [τ_j-1+1,τ_j ] it is assumed the design hit the signal in the entire interval (capturing more information).Furthermore(.∃ t∈ [τ_j-1+1,τ_j ]: A_t ∈ S_j ∩ S'_j|,) = (.S_j ∈{ A_t : t∈ [τ_j-1+1,τ_j ] }|,)^2.However, |{ A_t: τ_j-1 +1 ≤ t ≤τ_j }| ≤τ_j-τ_j-1 := l_j thus the probability above is bounded from above by l_j^2 /n^2.Using all this yields_0 [ L̃( )^2 ]≤[ {Ω_c}∏_j∈ [N][ .exp( l_j μ^2 {∃ t∈ [τ_j-1+1,τ_j ]: A_t ∈ S_j ∩ S'_j }μ^2 ) |,] ] ≤[ {Ω_c}∏_j∈ [N]( 1+ l_j^2/n^2( e^l_j μ^2 -1 ) ) ]. The last expression is readily upper bounded by the fact that N≤ m.Although this is a crude bound[In principle one can recall that N-1∼(m-1,p) and proceed from there, although it will overcomplicate the derivation.In any case, this will at most allow us to replace the term p^2 by p inside the logarithm in the statement of the theorem, which is not very relevant.] it is enough for our purposes.Also, on the event Ω_c we have the upper bound l_j ≤ 2c/p for every j∈ [N].We conclude that_0 ( L̃( )^2 ) ≤( 1+ 4c^2/p^2 n^2( e^2cμ^2 /p -1 ) )^m.Combining our results yields that if there exists a test for which max_i=0,1 (Ψ≠ i) ≤ε, we must have√(( 1+ 4c^2/p^2 n^2( e^2cμ^2 /p-1 ) )^m -1/2) +3/4≥ 2-4ε .Rearranging gives4c^2/p^2 n^2( e^2cμ^2 /p-1 ) ≥√(( 5/4 - 4ε)^2 + 1/2)-1.Using the inequality log x ≤ x-1 on the right hand side, and rearranging concludes the proof.§ NUMERICAL EVALUATION OF THE NON-ADAPTIVE LOWER BOUND Although the lower bound in Theorem <ref> only deals with the extreme cases p∈{ 0,1 }, we conjecture that in the regime m≈ n/s the same scaling of μ is necessary for reliable detection, regardless of the value of p.To corroborate this conjecture we provide a brief section of numerical experiments.We numerically estimate the right hand side of (<ref>), which is a lower bound on the maximal probability of error.We do so for several values of p∈ [0,1], and for each p we plot the value of the lower bound as a function of μ.Note that the sampling strategy has a large impact on the value in question.We know that when p=0 a sub-sampling scheme is near-optimal (see Remark <ref>), and so it should also be reasonable for small values of p.On the other hand, the sampling strategy is irrelevant for p=1, and probably essentially irrelevant for large p.This motivates using a sub-sampling scheme in all the experiments.Furthermore, note that unless we sample c· n/s different components, the probability _1 (∀ t∈ [m]: A_t ∉ S^(t)) can not be small.To ensure an upper bound of ε on the previous probability, we need to choose c ≡ c(ε ) = log (1/ε ).Considering all the above, we set up our experiment as follows.We set n=5000, s=⌈ n^1/4⌉ =9 and m=c(ε )n/s with ε =0.05.In this case, sub-sampling reduces to measuring m randomly selected components (one measurement each).We note that we experimented using multiple values of s across a wide range of sparsity levels, but found qualitatively the same result in all cases.Based on previous work concerning the sparse-mixture model (e.g.<cit.>) we expect the lower bound to reach the value ε when μ≈√(2log (n/s)).Hence, we set μ_t ≈ t·√(2log (n/s)), and plot the r.h.s.of (<ref>) as a function of t.The left panel of Figure <ref> seems to support our conjecture that the problem difficulty is independent of p in the regime m≈ n/s, as all the curves are on top of each other.Furthermore, since there is always a non-negligible chance of not sampling a signal component, the lower bound is bounded away from zero, even as μ_t grows large.To contrast this, we present another simulation with the same setup, except that the number of measurements m ≫ n/s.In particular, we set m=n, but otherwise use the same parameters.Note that in this case, sub-sampling amounts to sampling c(ε ) n/s randomly chosen components, but now we sample each of these m/(c(ε ) n/s) consecutive times.To keep the two plots on the same horizontal scale, we set μ_t = t·√( (2c(ε )n /sm) log (n/s)) in the right panel of Figure <ref>.It seems that in this case, the curves are no longer on top of each other, suggesting that the value of p has an impact on the problem difficulty.Surprisingly, the curve corresponding to p=1 is the one that descends the fastest, though the difference is only marginal.Though the cause of this is unclear, a possible reason might be that for faster signals the chance of not sampling active components at all is diminished, an effect that is more pronounced when m is large.In any case, this shows that in the regime m≫ n/s the speed of change might have a non-trivial effect on the problem difficulty.Exploring this is out of the scope of this work, but might be an interesting topic of future research. § FINAL REMARKSIn this paper we studied the problem of the detection of signals that evolve dynamically over time.We introduced a simple model for the evolution of the signal that allowed us to explicitly characterize the difficulty of the problem with a special regard to the effect of the speed of change.We also showed the potential advantages that adaptively collecting the observations bring to the table and showed that these are more and more pronounced as the speed of change decreases, which is in line with previous results dealing with signal detection using adaptive sensing.The lower bounds derived in this paper provide a clear picture of the role of the rate of change parameter p, but unfortunately still do not span the entire range of problems we would like to consider (e.g.Theorem <ref> applies only to p=0,1 and part (ii) of Theorem <ref> applies only to s=1).The latter difficulties appear to be mostly technical and the authors suspect these might be possible to address with carefully chosen reductions.Our contributions merely scratch the surface of this interesting problem, and below we highlight a few interesting directions for future work in this regard. Large vs.small sample regimes: in this work we focus primarily on the case m≈ n/s, which may be deemed as the small sample regime.When the number of measurements m is significantly larger the type of tests and performance tradeoffs will likely be different, even under the non-adaptive sensing paradigm.For instance, we expect the signal dynamics to have an effect on performance, meaning that it is easier to detect signals non-adaptively when p is smaller.Other interesting questions arise in that setting as well — what is the optimal non-adaptive sensing design?These questions become even more intriguing when one considers adaptive sensing. Restricted dynamics: in the model considered in this paper when signal components change they can move to any unoccupied location in the signal vector.This assumption simplifies the setup, but in some applications might be too unrestrictive.For instance, if signal components can only move to adjacent locations at each time step the effect of the speed of change will likely be less pronounced in the difficulty of detection (at least for adaptive sensing).Understanding the effect of such restrictions could prove valuable in certain applications, such as detection of a disease outbreak in a network, besides being interesting from a theoretical point of view. Structures: in certain situations the signal support can be assumed to have structure to it, for instance all anomalous items might be consecutive or have some other pattern.In some cases the structure of the support has a huge effect on the difficulty of the problems of detection and recovery (see for instance <cit.>).How structural restrictions affect these tasks for dynamically evolving signals could be a fruitful avenue of research. Support recovery: another common question in such settings is how well can we estimate the support of a signal.That is, instead of deciding only if there are anomalous items or not, we need to determine which of the items are anomalous.This is also an interesting problem to study for dynamically evolving signals, although a precise formulation of the objective and performance metric for such estimators is less immediate than for static signals. § ACKNOWLEDGEMENTS This work was partially supported by a grant from the Nederlandse organisatie voor Wetenschappelijk Onderzoek (NWO 613.001.114). We are very grateful for the comments of the two anonymous referees, which helped improving the presentation.§ APPENDIXWe write(Ω_c )≥( { N-1 >mp/2}∩{∀ j: l_j ≤ cm/N })= [ { N-1 > mp/2 }[ {∀ j: l_j≤ cm/N } | N ] ], We first lower bound the inner conditional probability.Note that if N≤ c this probability is one (since cm/N≥ m and l_j ≤ m by definition).When N>c, we will upper bound the probability of the complementary event.Note that given N the distribution ofis uniform from the set of 0-1 sequences of length m containing exactly N ones, and for which also θ_m =1.Hence, to upper bound (∃ j: l_j > cm/N), we simply need to count the number of sequences described above for which we have a long block.We can get an upper bound on this count in the following way.First note that since the last element of the sequence is always one, we can simply think of sequences of length [m-1] containing N-1 ones.Consider an interval of length cm/N in the set [m-1].Now consider the sequences containing N-1 ones, and for which there are no ones in the aforementioned interval.Note that for all such sequences the existence of at least one long interval is guaranteed.We can simply count how many 0-1 sequences can be generated like this.This number is an upper bound on the number of 0-1 sequences that have N ones, the last element of the sequence is one and for which ∃ j: l_j > cm/N.We thus have(∃ j: l_j> cm/N |N)≤ (m-cm/N) m-cm/NN-1/m-1N-1 = (m-cm/N) (m-cm/N)(m-cm/N -1)… (m-cm/N -N+2)/(m-1)(m-2)… (m-N+1)≤m/m-1(1-c/N) ( m-cm/N/m-2)^N-2 < ( m-cm/N/m-2)^N-2 .Now consider the logarithm of the expression above.Using log (1+x) ≤ x, we getlog (∃ j: l_j> cm/N |N) < (N-2) ( logm/m-2 + log (1-c/N) ) ≤ (N-2) ( 2/m-2 - c/N) ≤ -log2 ,whenever c≥ 6+3log 2, using the fact that 3≤ c≤ N≤ m.Hence (Ω_c ) ≥ (N-1 >mp/2)/2.All that remains is to use the fact that N-1 ∼ (m-1,p).For instance Chebyshev's inequality yields(N-1 ≤ mp/2) ≤4(m-1)p(1-p)/(mp)^2≤ 1/2,when p≥ 8/m and so the claim is proved. chicago

http://arxiv.org/abs/1702.07899v2

[]johannes.aberl@jku.at Institute of Semiconductor and Solid State Physics, Johannes Kepler University Linz, Altenbergerstraße 69, A-4040 Linz, Austria []klenovsky@physics.muni.cz Department of Condensed Matter Physics, Masaryk University, Kotlářská, CZ-61137 Brno, Czech Republic Central European Institute of Technology, Masaryk University, Kamenice 753/5, CZ-62500 Brno, Czech Republic Institute of Semiconductor and Solid State Physics, Johannes Kepler University Linz, Altenbergerstraße 69, A-4040 Linz, Austria Institute of Semiconductor and Solid State Physics, Johannes Kepler University Linz, Altenbergerstraße 69, A-4040 Linz, Austria Institute of Semiconductor and Solid State Physics, Johannes Kepler University Linz, Altenbergerstraße 69, A-4040 Linz, Austria Institute for Integrative Nanosciences, IFW Dresden, Helmholtzstraße 20, D-01069 Dresden, Germany Paul-Drude-Institut für Festkörperelektronik, Hausvogteilplatz 5-7, 10117 Berlin, Germany Department of Condensed Matter Physics, Masaryk University, Kotlářská, CZ-61137 Brno, Czech Republic Central European Institute of Technology, Masaryk University, Kamenice 753/5, CZ-62500 Brno, Czech Republic Institute of Semiconductor and Solid State Physics, Johannes Kepler University Linz, Altenbergerstraße 69, A-4040 Linz, Austria []rinaldo.trotta@jku.at Institute of Semiconductor and Solid State Physics, Johannes Kepler University Linz, Altenbergerstraße 69, A-4040 Linz, Austria We show that anisotropic biaxial stress can be used to tune the built-in dipole moment of excitons confined in In(Ga)As quantum dots up to complete erasure of its magnitude and inversion of its sign. We demonstrate that this phenomenon is due to piezoelectricity. We present a model to calculate the applied stress, taking advantage of the so-called piezotronic effect, which produces significant changes in the current-voltage characteristics of the strained diode-membranes containing the quantum dots. Finally, self-consistent 𝐤·𝐩 calculations reveal that the experimental findings can be only accounted for by the nonlinear piezoelectric effect, whose importance in quantum dot physics has been theoretically recognized although it has proven difficult to single out experimentally. 78.67.Hc, 73.21.La, 85.35.Be, 77.65.LyInversion of the exciton built-in dipole moment in In(Ga)As quantum dots via nonlinear piezoelectric effect Rinaldo Trotta December 30, 2023 ===========================================================================================================§ INTRODUCTION Semiconductor quantum dots (QDs) are currently emerging as one of the most promising sources of nonclassical light on which to base future quantum technologies <cit.>. This success is in large part due to the outstanding experimental and theoretical work on QD physics that has been carried out over the last decades. These studies have not only enabled a detailed understanding of the fundamental properties of these “artificial atoms", but have also offered the means to tailor their interactions with the environment, which is the key to make them suitable for envisioned applications. In spite of these accomplishments, however, the extreme sensitivity of the electronic properties of QDs to tiny variations of their shape, size, composition, built-in strain fields, as well as to external perturbations <cit.> very often makes it difficult to single out the physical effects which are responsible for particular experimental observations. This is especially true for statistical studies performed on dissimilar QDs that aim at grasping general trends applicable to all of them. To explain this point further, we focus on the effect of piezoelectric fields, whose importance in theoretical semiconductor physics is well documented <cit.>. Seminal works have demonstrated that in conventional III-V QDs grown on (100) substrates, first- and second-order contributions to the piezoelectric field tend to oppose each other so that its total effect on the QD properties is found to be small <cit.>. This is clearly not the case for GaN QDs <cit.> and CdSe nanocrystals <cit.>, but for In(Ga)As QDs, which are usually grown on nonpolar GaAs(001) substrates and are of interest for quantum optics, piezoelectricity is very often neglected <cit.>. From the experimental side it is not straightforward to recognize its importance. Despite the weak effect on the energy of the states, piezoelectricity is expected to have a strong influence on the position and shape of the electron and hole wave functions <cit.> and, in turn, on the sign and magnitude of the exciton (X) built-in dipole moment. Early experiments <cit.> showed that the sign of the built-in dipole moment is inverted with respect to the predicted sign <cit.>, a fact which was mainly explained by the shape and composition of the investigated QDs. In contrast, subsequent studies performed with In(Ga)As QDs grown on high-index substrates with polar orientation <cit.> demonstrated that the observed electron-hole alignment is not a general feature, and that piezoelectricity has to be taken into account for a correct interpretation of the experimental results, as confirmed very recently <cit.>. Related experiments have demonstrated that the sign of the X dipole can even vary for different QDs within the same sample <cit.>. It is therefore rather evident that it is experimentally challenging to single out the role of piezoelectricity in QDs. In this paper, we demonstrate that externally induced anisotropic strain fields can be used for wave function engineering and to even force an inversion of the exciton built-in dipole moment in the very same In(Ga)As QD. We achieve this result by integrating light-emitting-diode (LED) nanomembranes onto a piezoelectric actuator [see Fig. <ref>(a)] capable of delivering variable strain fields <cit.>. Differently from previous results <cit.> revealing a strain-dependence of the X dipole moment, we demonstrate that its tuning (and inversion) is driven by the piezoelectric effect. The induced piezoelectric field was estimated by exploiting a piezotronic-like phenomenon in the LED used <cit.>, i.e., a sizable strain-induced shift of its current-voltage (I-V) characteristics. Finally, self-consistent 𝐤·𝐩 calculations reveal that the inversion of the dipole moment is dominated by the nonlinear terms of the piezoelectric field.§ EXPERIMENTAL METHODS The microphotoluminescence (μ-PL) measurements were performed at low temperatures (8 K) by using a helium flow cryostat. A femtosecond Ti-sapphire-laser (operated at 850 nm) was focused by a microscope objective (0.42 numerical aperture) to address single QDs. The μ-PL spectra were recorded via a spectrometer connected to a liquid-nitrogen-cooled charge-coupled device. Polarization-resolved measurements were performed using a combination of a rotatable λ/2 wave plate and a fixed linear polarizer placed in front of the spectrometer in order to identify the origin of the transitions in the μ-PL spectra [see Fig. <ref>(b)] and to estimate the exciton fine-structure splitting (FSS). The p-i-n diode nanomembrane containing In(Ga)As QDs was grown by molecular beam epitaxy. The nanomembrane was transferred via a flip-chip process and bonded onto a [Pb[Mg_1/3Nb_2/3]O_3]_0.72-[PbTiO_3]_0.28 (PMN-PT) piezoelectric actuator by gold thermo-compression bonding (for further details on the sample structure and device fabrication see Refs. Trotta:12,Trotta:12_2). The emission properties of the QDs can be varied using two “tuning knobs": stress and electric fields, which are applied via voltages V_p and V_d, respectively. A positive (negative) voltage V_p applied to the PMN-PT induces a compressive (tensile) anisotropic biaxial stress in the nanomembrane. The corresponding electric field across the PMN-PT is given by F_p=V_p/d_p, where d_p≈ 300 μ m is its thickness. The electric field across the LED is instead given by F_d=-(V_d+V_bi)/d_d, where -1.9 V<V_d<1.9 V is the applied voltage, V_bi the diode's built-in voltage (positive) and d_d≈ 150 nm the thickness of the intrinsic region. For operation in reverse bias, F_d is oriented parallel to the QD's growth direction, i.e. from top to bottom in Fig. <ref>(a). Increasing the magnitude of F_d leads to a redshift of all transitions, as shown in Fig. <ref>(b).§ TUNING OF THE X ELECTRIC DIPOLE MOMENT We now show how strain can be used to invert the X built-in dipole moment. We start out by fitting the measured Stark shift of the X transition via E=E_0-p F_d+β F_d^2 where E_0 is the transition energy at F_d=0, p=ez the built-in dipole moment (e elementary charge, z is the carrier separation along the growth direction), and β the polarizability <cit.>. Repeating this procedure for each value of F_p allows us to extract the stress dependence of E_0, p and β, as illustrated in Fig. <ref>(a). Obviously, the determination of these parameters is tightly connected to the use of correct values of F_d and therefore of V_bi, which can be estimated from the I-V trace of the diode [see Fig. <ref>(c)]. Herein it turned out that there is a substantial shift of V_bi with applied stress of up to Δ V_bi= 108 mV (for Δ V_p=900 V) which is about an order of magnitude larger than the expected shift produced by the strain-induced changes of the energy band gap (∼10 meV). This effect, which we attribute to piezoelectricity, is of great importance not only for the data evaluation, but also for the theoretical model discussed below. In Fig. <ref>(b) we report the strain-dependence of p/e for the X confined in one of the nine measured QDs. The dipole moment shifts almost linearly with F_p, i.e. with applied stress, with an average tuning range of ⟨Δ p|/⟩e=(0.071± 0.007) nm and an average slope of ⟨dp/dF_p|=⟩(0.26± 0.04) μ eV kV^-2 cm^2. Most importantly, the applied stress is sufficient to suppress the electric dipole and invert its sign, i.e. to swap the position of electron and hole wave functions inside the QD. The inversion of p has been observed in four out of the nine measured QDs, and it is mainly determined by the value of p for F_p=0, see Fig. <ref>(c). This is found to be always negative, i.e. at F_p=0 the hole (electron) tends to be located closer to the QD apex (base). The measured (linear) strain-induced shift of the zero-field transition energy E_0 [⟨dE_0/dF_p|=⟩(0.21± 0.02) μ eV kV^-1 cm] is in good agreement with previous works <cit.>. The full set of data including E_0, p and β for all transitions can be found in the Supplemental Material <cit.>.§ DETERMINATION OF STRESS CONFIGURATION To explain the physics underlying the inversion of the X built-in dipole moment, it is fundamental to gain knowledge of the type of in-plane stress delivered by the PMN-PT actuator. Any in-plane stress configuration can be described by the three independent components of the stress tensor (s_xx, s_yy and s_xy) or, equivalently, by two principal stresses S_1, S_2 applied at an angle α with respect to a crystal axis ([100] in our case). The two sets of three independent parameters are related to each other via s_xx,yy=S_1+S_2/2±S_1-S_2/2 cos(2α) and s_xy=S_1-S_2/2 sin(2α). Therefore, an arbitrary in-plane stress can be fully characterized by S_1-S_2, S_1+S_2, and α. This requires the knowledge of several observables as F_p is varied. In our experiment we monitor (i) the shift of the X transition energy Δ E_0 [see inset of Fig. <ref>(a)], (ii) the changes of the magnitude of the FSS along with the corresponding X polarization angle, and (iii) the shift of the I-V trace of the diode. Point (i) is related to the hydrostatic part of the stress given by Δ E_0=a(S_1+S_2), where a is a parameter related to the elastic constants renormalized by the deformation potentials. Since a is known <cit.>, we can estimate S_1+S_2. Next, we use (ii) to estimate the direction α of the applied stress by using a recently developed model for the X Hamiltonian <cit.>. For the QDs investigated in this work we estimate α=55^∘ (note that α=45^∘ correspond to [110] direction), as discussed in the Supplemental Material <cit.>.Finally, we exploit (iii) to estimate s_xy and, since α is known, S_1-S_2 (see later in the text). As the shift of the I-V trace onset is a quite peculiar phenomenon, we discuss its origin below. As mentioned, the measured shift of the I-V trace cannot be accounted for by the strain-induced change of the i-Al_0.4Ga_0.6As band gap. In highly doped n-type GaAs the Fermi level (E_F) lies inside the conduction band <cit.>, and there is a potential difference at the interface between a semiconductor and a metal stemming from their different work functions <cit.>. Although there are several possibilities of the arrangement <cit.> we restrict ourselves (motivated by a typical scenario for n-type GaAs/Au interface <cit.>) to the case when a Schottky barrier ϕ_Sb is present at the Au contacts to the p-i-n diode <cit.>. Under this condition, the observed I-V shift stems from a strain-induced modification of ϕ_Sb. More specifically, for V_p≠ 0 piezoelectric charges are generated at the edges of our structure as well as at every interface between two different materials. As illustrated via the band scheme in Fig. <ref>(a) these charges produce an additional potential θ at the Au contacts, which effectively changes the current onset in the I-V trace <cit.>. Furthermore, an additional (net) electric field F_qd is created in the inner i-GaAs layer which acts on the QDs hosted therein. The magnitude and direction of F_qd depends on the applied stress (T or C) as well as on the arrangement of different materials <cit.>. The surface (piezoelectric) charge density σ_p at the n-GaAs/Au interface is linked to the shear stress via σ_p=e^n_14S_44s_xy, where e^n_14 is the piezoelectric constant of n-GaAs and S_44 is one of the elastic compliance constants. By obtaining a relation between σ_p and θ we can therefore exploit the measured shift of the I-V trace onset to calculate s_xy (and therefore S_1-S_2). The obtained dependence of s_xy on θ as well as of the additional electric field in the QD layer F_qd that is related to piezoelectric charges is shown in Fig. <ref>(b). The corresponding relations and their derivation are provided in the Supplemental Material <cit.>. The sensitivity of s_xy and F_qd on θ is evident, especially for the small values of ϕ_Sb typical for our structure at low temperature (we estimated 0.1 V for the doping concentration of 5· 10^18 cm^-3 present in the n-GaAs<cit.>). The knowledge of the shear part of the stress allows us to calculate S_1-S_2 for each value of F_p and, finally, magnitude, direction and anisotropy of the in-plane stress delivered by the PMN-PT. This is found to be highly anisotropic (with a ratio |S_1|/|S_2|≈ 3.16), applied at 55^∘ with respect to the [100] direction and with magnitudes (S_1+S_2) as high as -180 MPa (see Supplemental Material <cit.>). While this stress anisotropy is not expected for the [001] piezo cut used in this work, it is a common feature reported in the literature <cit.>. § ORIGIN OF THE DIPOLE INVERSION On this basis we performed calculations of the electronic structure of In(Ga)As QDs and investigate their dependence on the externally applied stress. The single-particle electronic levels were obtained by using the envelope function approximation based on the eight-band 𝐤·𝐩 method for electrons and holes employing the NEXTNANO^3 software package <cit.>. The simulated QD was assumed to be of truncated cone shape with a radius of 20 nm (10 nm) at its base (apex) and a height of 3.5 nm. The QD was embedded in a GaAs host and consisted of an In_xGa_1-xAs alloy with the In content linearly increasing from base (x=0.45) to apex (x=0.8) <cit.>. The externally applied stress was simulated by changing the corresponding elements of the Bir-Pikus Hamiltonian <cit.> and the calculations were performed in self-consistent Poisson-Schrödinger equation loops. The calculations furthermore account for the effect(s) induced by piezoelectric fields. Following the common definition, the piezoelectric response of a material is given in terms of the created polarization 𝐏 which can be expanded as P_i=∑_j=1^6e_ijϵ_j+1/2∑_jk=1^6B_ijkϵ_jϵ_k+… where ϵ represents the independent components of the strain tensor in Voight notation (i.e. ϵ_1=ϵ_xx, ϵ_2=ϵ_yy, ϵ_3=ϵ_zz, ϵ_4=2ϵ_yz, ϵ_5=2ϵ_xz and ϵ_6=2ϵ_xy) and e and B are the linearand nonlinear piezoelectric coefficients, respectively <cit.>. In zinc-blende crystals only four independent coefficients e_14, B_114, B_124 and B_156 are nonzero due to symmetry considerations <cit.>. The importance of the different-order contributions to the piezoelectric polarization depends essentially on the magnitude of strain and its particular configuration in the considered material and/or structure. Differently from the diode model presented above, where the relatively small applied stresses (or equivalently strains) justify the use of the linear contribution only, the strain field around and inside the QD is at least an order of magnitude larger. Therefore the nonlinear response is expected to be strongly magnified and can even dominate <cit.>. It is consequently reasonable to include at least the second-order contributions whereby the values for linear and quadratic piezoelectric coefficients were taken from Ref. Bester:06 (cubic terms are expected to be negligible <cit.>). In Fig. <ref> we present the dependence of p/e on S_1+S_2 for the stress configuration we estimated above with and without taking into account the second-order term of the piezoelectric field. Furthermore, we show the results obtained for stress fields with the same anisotropy and magnitude but with S_1 aligned along the [110] direction (i.e., the direction which maximizes the piezoelectric effect), the [100] direction (no piezoelectricity present), as well as for the case of purely biaxial stress (no piezoelectricity present). A rich scenario can be observed. First, we notice that no appreciable variation of p/e can be observed for s_xy=0, that is, in the absence of the piezoelectric effect (see the brown and the orange lines in Fig. <ref>). In strong contrast, for the stress configuration estimated above (blue line) and for a similar one in which the piezoelectric field is maximized (green line), we observe a variation of the dipole moment comparable to the experimental observations. The discrepancy between the experimental and theoretical values of p (and absolute value of E_0) is probably due to the specific QD shape considered for the theoretical calculations, which probably differs from the experimental one. It is also likely that the strain configuration we estimated is not exactly the one experienced by the specific QD due to strain inhomogeneities across the membranes.Nonetheless, our calculations clearly show that the change in the dipole moment observed in the experiment can be only explained by the anisotropic strain configurations which switch on the piezoelectric effect. Moreover, the large tuning rates achieved in the experiment can only be reproduced by including the second-order contribution to the piezoelectric field as apparent from the red line in Fig. <ref>. Thus, the inversion of the exciton dipole moment we report in this work constitutes clear and rare experimental evidence of the importance of the nonlinear terms of the piezoelectric field in III-V QD systems. It is worth emphasizing that, while in the calculations of Fig. <ref> we used the values of the nonlinear piezoelectric coefficients reported in Ref. Bester:06 there are several works <cit.> that have reported different values for these coefficients. In the Supplemental Material <cit.> we discuss this point in detail and we show that the results of our calculations and our findings are preserved upon exchange of the values of the piezoelectric coefficients reported in Refs. Bester:06,Wakata:11,Tse:13,Caro:15. We would also like to mention that we approximated the (experimental) absolute value of p/e at F_p=0 in the performed calculations by theoretically considering a shear pre-stress of s^pre_xy=200 MPa. This pre-stress, which is already present in our device at F_p=0, is a common feature and is attributed to the bonding and poling process <cit.>. However, it has no qualitative influence on the presented behavior of p/e vs S_1+S_2, which, obviously, purely depends on the applied stress (which is estimated from the experimental data). § CONCLUSION In conclusion, we have demonstrated experimentally and theoretically that piezoelectric fields can be used to engineer the wave function of excitons confined in In(Ga)As QDs and that, in this phenomenon, the nonlinear terms of the piezoelectric field dominate over the linear term. Our results are relevant not only for fundamental physics, because the effect of the piezoelectric field on the few-particle states in QDs can be now pinpointed from the experiments, but also for future applications. In fact, piezoelectricity can be used to modify the radiative lifetime, similar to vertical electric fields <cit.>. Moreover, the dipole moment can be engineered to limit the interaction of excitons with charges in the vicinity of the QD <cit.> or to modify the response of QDs to electric fields <cit.>. In this context, it is worth noting that the tunability of the exciton dipole moment offered by the piezoelectric field is at least an order of magnitude larger than what can be obtained using magnetic fields <cit.> and, in addition, can be achieved using a compact and scalable approach. This work was financially supported by the European Research council (ERC) under the European Union's Horizon 2020 Research and Innovation Programme (SPQRel, Grant agreement No. 679183) and by the European Union Seventh Framework Programme 209 (FP7/2007-2013) under Grant Agreement No. 601126 210 (HANAS). P. K. and J. H. have been financially supported by the Ministry of Education, Youth and Sports of the Czech Republic under the project CEITEC 2020 (LQ1601). Furthermore, we acknowledge E. Magerl, R. Singh, G. Bester, P. Atkinson and O. G. Schmidt for fruitful discussions and support.

http://arxiv.org/abs/1702.08314v3

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http://arxiv.org/abs/1702.07860v2

[ Variational Inference using Implicit Distributionsequal* Ferenc Huszártwtr twtrTwitter, London, U.KFerenc Huszárfhuszar@twitter.comvariational inference, adversarial training, density ratio estimation0.3in ]Generative adversarial networks (GANs) have given us a great tool to fit implicit generative models to data. Implicit distributions are ones we can sample from easily, and take derivatives of samples with respect to model parameters. These models are highly expressive and we argue they can prove just as useful for variational inference (VI) as they are for generative modelling. Several papers have proposed GAN-like algorithms for inference, however, connections to the theory of VI are not always well understood. This paper provides a unifying review of existing algorithms establishing connections between variational autoencoders, adversarially learned inference, operator VI, GAN-based image reconstruction, and more. Secondly, the paper provides a framework for building new algorithms: depending on the way the variational bound is expressed we introduce prior-contrastive and joint-contrastive methods, and show practical inference algorithms based on either density ratio estimation or denoising. § INTRODUCTIONImplicit distributions are probability models whose probability density function may be intractable, but there is a way to * sample from them exactly and/or calculate and approximate expectations under them, and* calculate or estimate gradients of such expectations with respect to model parameters. A popular example of implicit models are stochastic generative networks: samples from a simple distribution - such as uniform or Gaussian - are transformed nonlinearly and non-invertably by a deep neural network. Such networks can flexibly parametrise a wide range of probability distributions, including even degenerate ones which may not even have a continuous density.Implicit models have been successfully applied to generative modelling in generative adversarial networks <cit.> and subsequent work <cit.>. They work particularly well for visual data, partly because they can exploit the inductive biases of convolutional neural networks, and partly because they can flexibly model potentially degenerate, manifold-like distributions which natural images are assumed to follow.This note is about using implicit distributions in another important probabilistic machine learning problem: approximate inference in latent variable models. Unlike in the first applications of GANs to generative modelling where an implicit model directly models the distribution of observed data, in approximate inference we are interested in modelling the posterior distribution of latent variables given observed data. Direct generative modelling and approximate inference are very different problems indeed: in the former we are provided with samples x_i from the distribution to be modelled, in the latter we are given a joint distribution of latents z and observations x, and a set of observed samples x_i, but no samples from the posterior itself.In this note we focus on variational inference (VI) which works by minimising a divergence between the approximate and real posteriors. More precisely we follow the usual KL-divergence formulation, but other, more general variational methods exist <cit.>. VI also provides a lower bound to the marginal likelihood or model evidence - the evidence lower bound or ELBO - which can be maximised with respect to parameters of the latent variable model to approximate maximum likelihood learning. It's important to keep in mind that despite several algorithms in this note look and feel like adversarial training procedures, the way the model is fit to observed data is more akin to variational auto-encoders <cit.> than to GANs.There are several reasons to explore implicit distributions in the context of variational inference. Firstly, explicit VIis often limited to exponential family distributions or other distributions with tractable densities <cit.> which may not be expressive enough to capture non-trivial dependencies that the real posterior exhibits. A flexible, implicit distribution may provide a better approximation to the posterior and a sharper lower bound. Secondly, It may be desirable to use an implicit likelihood as the resulting latent variable model may fit the data better. For example, the likelihood or forward model might be described as a probabilistic program <cit.> whose density is intractable or unknown. Finally, sometimes we may want to use implicit priors over latent variables. For example in a deep hierarchical latent variable model the prior for a layer may be a complicated probabilistic model with an intractable density. Or, when solving inference problems in computational photography, the prior may be the empirical distribution of natural images as in <cit.>. In summary, any or all of the prior, the likelihood and the approximate posterior may have to be modelled implicitly, and we need VIprocedures that are ready to tackle these situations.In this note we present two sets of tools to handle implicit distributions in variational inference: GAN-like adversarial algorithms which rely on density ratio estimation, and denoising-based algorithms which build a representation of the gradients of each implicit distribution's log-density and use these gradient estimates directly in a stochastic gradient descent (SGD) algorithm. We further classify algorithms as prior-contrastive and joint-contrastive depending on which form of the variational bound they use. Prior-contrastive methods only deal with implicit distributions over latent variables (the prior or approximate posterior), while joint-contrastive methods can handle fully implicit models where none of the distributions involved has a tractable density. This classification gives rise to a range of algorithms listed in Table <ref>, alongside related algorithms from prior work. All of the algorithms presented here can perform variational approximate inference, which is the main focus of this note, but not all of them can perform learning unless the likelihood is explicitly defined.§.§ Overview of prior work Several of the algorithms proposed here have been discovered in some form before. However, their connections to variational inference is rarely made explicit. In this section we review algorithms for inference and feature learning which use implicit distributions or adversarial training. As we will see, several of these admit a variational interpretation or can be rather straightforwardly modified to fit the variational framework.GANs have been used rather successfully to solve inverse problems in computer vision. These inverse problems can be cast as a special case of approximate inference. <cit.> used GANs to reconstruct and generate images from non-linear feature representations. As pointed out later by <cit.>, this method, DeePSiM, can be interpreted as a special case of amortised maximum a posteriori (MAP) or variational inference with a Gaussian observation model. GANs have also been used for inference in image super-resolution <cit.>. Connections between GANs and VIin this context were first pointed out in <cit.>. <cit.> also introduced a modified objective function for the GAN generator which ensures that the algorithm minimises the Kullback-Leibler divergence as opposed to the Jensen-Shannon divergence, an essential step to using GANs for VI. The AffGAN algorithm presented there is highly application specific, thus, it does not solve VI in general.In more recent, parallel work, <cit.> proposes adversarial variational Bayes(AVB), perhaps the best description of the use of GANs for variational inference. AVB is a general algorithm that allows for implicit variational distributions and is in fact equivalent to the prior-contrastive adversarial algorithm (, Algorithm <ref>) described in Section <ref>. Operator variational inference <cit.> formulates a general class of variational lower bounds based on operator divergences, resulting in a practical algorithm for training implicit inference networks without a tractable density. As is shown in the paper, the KL-divergence-based variational bound used here and in <cit.> is a special case of OPVI. Adversarial autoencoders <cit.> are similar to variational autoencoders where the KL-divergence term is replaced by an adversarial objective. However, AAEs do not use the KL-divergence formulation of the adversarial loss and their discriminator is independent of the encoder's input, thus they are not a true variational method. Finally, <cit.> proposed variational message passing, in which adversaries are employed to minimise local Jensen-Shannon divergences in an algorithm more akin to expectation propagation <cit.> than to variational inference.Another line of research extends GANs to latent variable models by training the discriminator on the joint distribution of latent and observed variables. This technique has been independently discovered as bi-directional GAN <cit.> and adversarially learned inference <cit.>. These algorithms are closely related to the joint-contrastive adversarial algorithm (, Algorithm <ref>). ALI and BiGAN use the Jensen-Shannon formulation of GANs rather than the KL-divergence ones used here. On the one hand, this means that the Jensen-Shannon variants aren't technically VIalgorithms. On the other hand, the symmetry of the Jensen-Shannon divergence makes ALI and BiGAN completely symmetric, enabling not only approximate inference but also learning in the same algorithm. Unfortunately, this is no longer true when KL divergences are used:is an algorithm for inference only.The algorithms mentioned so far are examples of adversarial techniques which rely on density ratio estimation as the primary tool for dealing with implicit distributions <cit.>. <cit.> and <cit.> demonstrated an alternative or complementary technique based on denoising autoencoders. As shown by <cit.> the optimal denoising function learns to represent gradients of the log data density - which in turn can be used in an inference method. <cit.> used this insight to build a denoiser-based inference algorithm for image super-resolution and connected it to amortised maximum a posteriori (MAP) inference. The extension from MAP to variational inference is straightforward and this method is closely related to the prior-contrastive denoising VI (, Algorithm <ref>) algorithm presented here.§ VARIATIONAL INFERENCE: TWO FORMS In this section, we give a lightweight overview of amortised variational inference (VI) in a latent variable model, in a model similar to variational autoencoders <cit.>. We observe an sequence of N observed data 𝒟 = {x_n, n=1… N}. For each data point there exists an associated latent variable z_n, n=1… N. We specify a prior p_θ(z) over latent variables and a forward model p_θ(x| y) which describes how the observations are related to latents. In such model we are interested in maximum likelihood learning, which maximises the marginal likelihood or model evidence ∑_n=1^Nlog p_θ(x_n) with respect to parameters θ, and inference which involves calculating the posterior p_θ(z| x). We assume that neither the marginal likelihood or the posterior are tractable.In amortized VI we introduce an auxiliary probability distribution q(z| x_n; ψ), known as the recognition model, inference network or approximate posterior. Using q_ψ we define the evidence lower bound (ELBO) as follows: ℒ(θ, ψ) = ∑_n=1^N{log p_θ(x_n) - q_ψ(z| x_n)p_θ(z| x_n)} As the name suggests, ELBO is a lower bound to the model evidence p_θ(𝒟) and it is exact when q_ψ matches the true posterior p_θ(z| x) exactly. Maximising ELBO with respect to ψ is known as variational inference. This minimises the KL divergence [q_ψ| p_θ(z| x)] thus moving the q_ψ closer to the posterior. Conversely, maximising ELBO with respect to θ is known as variational learning which approximates maximum likelihood learning.The ELBO can be calculated exactly for many combinations of p_θ and q_ψ, whose densities are tractable. VAEs use a re-parametrisation trick to construct a low variance estimator to ELBO, but still require tractable densities for both the model p_θ and recognition model q_ψ. If p_θ and/or q_ψ are implicit the ELBO needs to be approximated differently. As we will see in the next sections, it is useful to formulate ELBO in terms of density ratios.There are two main forms considered here. Firstly, the prior-contrastive form used also by VAEs <cit.>: ℒ = ∑_n=1^Nz∼ q_ψ(z| x_n)[ log p_θ(x_n| z) - _θ, ψ(z,x_n) ]= ∑_n=1^N[z∼ q_ψ(z| x_n)log p_θ(x_n| z) - q_ψ(z| x_n)p_θ(z)], where we introduced notation for the logarithmic density ratio _θ, ψ = logq_ψ(z| x_n)/p_θ(z).We call Eqn. (<ref>) the prior-contrastive expression as the KL term contrasts the approximate posterior q_ψ with the prior p_θ(z).Alternatively, we can write ELBO in a joint-contrastive form as follows: ℒ(θ, ψ) = -N·(q_ψ(z| x)(x)p_θ(x, z) - ℍ[])= - ∑_n=1^Nz∼ q_ψ(z| x_n)_θ, ψ(x_n, z) - N·ℍ[],where we introduced notationto denote the real data distribution and ℍ[] denotes its entropy[In practice,is an empirical distribution of samples, so technically it does not have a continuous density or differential entropy ℍ[]. We still use this notation liberally to avoid unnecessarily complicating the derivations.]. ℍ[] can be ignored as it is constant with respect to both θ and ψ. We also introduced notation _θ, ψ to denote the logarithmic density ratio _θ, ψ(x_n, z) = logq_ψ(z| x_n)(x)/p_θ(x_n, z). Note that while _θ, ψ was a log-ratio between densities over z, _θ, ψ is the ratio of joint densities over the tuple (x,z). As this form contrasts joint distributions, we call Eqn. (<ref>) the joint-contrastive expression.When using implicit models the density ratios _θ, ψ and _θ, ψ cannot be computed analytically. Indeed, even if all distributions involved are explicitly defined,is only available as an empirical distribution, thus _θ, ψ cannot be calculated even if the densities of other distributions are tractable. In this note we rely on techniques for estimating _θ, ψ or _θ, ψ, or their gradients, directly from samples. For this to work we need to deal with a final difficulty: that _θ, ψ or _θ, ψ themselves implicitly depend on the parameter ψ which we would like to optimise. §.§ Dependence of _θ, ψ and _θ, ψ on ψ The KL-divergences in equationsanddepend on ψ in two ways: first, an expectation is taken with respect to q_ψ - this is fine as we assumed expectations under implicit distributions and their gradients can be approximated easily. Secondly, the ratios _θ, ψ and _θ, ψ themselves depend on ψ, which may cause difficulties. If one optimised ELBO na ively via gradient descent, one should back-propagate through both of these dependencies. Fortunately, the second dependence can be ignored:∂/∂ψ.z ∼ q_ψ_θ, ψ(z) |_ψ = ψ_0 = . ∂/∂ψz ∼ q_ψ_θ, ψ_0(z) |_ψ = ψ_0 The only difference between the LHS and RHS of the equation is in the subscripts _θ, ψ v. s. _θ, ψ_0. As _θ, ψ_0 is a constant with respect to ψ, Eqn. (<ref>) reduces to the gradient of an expectation with respect to q_ψ, which we assumed we can approximate if q_ψ is an implicit distribution. The detailed proof of Eqn. (<ref>) is in Appendix <ref>, the key idea of which is the observation that for any ψ_0 z ∼ q_ψ_θ, ψ(z) = x, z ∼ q_ψ_0_θ, ψ(x, z) + q_ψq_ψ_0 A similar equation analogously holds for _θ, ψ in Eqn. (<ref>), or indeed, any other KL divergence as well. §.§ Approximate SGD Algorithms for VI In the following sections we outline algorithms for VI which allow for implicit distributions. These algorithms can generally described as two nested loops of the following nature:* the outer loop performs stochastic gradient descent (SGD) on an approximation to ELBO with respect to ϕ, using gradient estimates obtained by the inner loop* in each iteration of outer loop, with ψ=ψ_t fixed, the inner loop constructs an estimate to _θ, ψ_t, θ, ψ_t, or more generally to the gradient in Eqn. (<ref>) As long as the gradient estimates provided by the inner loop has no memory between subsequent iterations of the outer loop, and the gradient estimates provided by the inner loop on average constitute a conservative vector field, the algorithms can be seen as instances of SGD, and as such, should have the convergence properties of SGD.§ DIRECT DENSITY RATIO ESTIMATION Direct density ratio estimation, also known as direct importance estimation, is the task of estimating the ratio between the densities of two probability distribution given only i. i. d. samples from each of the distributions <cit.>. This task is relevant in many machine learning applications, such as dealing with covariate shift or domain adaptation. A range of methods have been introduced to learn density ratios from samples, here we focus on adversarial techniques which employ a discriminator trained via logistic regression. We note that other methods such as KLIEP <cit.> or LSIF <cit.> could be used just as well. §.§ Adversarial approach using discriminators <cit.> proposed estimating density ratios by training a logistic regression classifier between samples from the two distributions. Assuming the classifier is close to a unique Bayes-optimum, it can then be used directly to provide an estimate of the logarithmic density ratio. This approach has found great application in generative adversarial networks <cit.>, which work particularly well for generative modelling of images <cit.>.Let us use this to construct an approximation _ϕ to the logarithmic density ratio _ψ,θ from Eqn. (<ref>). We can do this by minimising the following objective function, typically via SGD: ℓ^_ψ,θ(ϕ)= ∑_n=1^Nz∼ p_θ(_ϕ(x_n,z)) - ∑_n=1^Nz∼ q_ψ(z| x_n)(_ϕ(x_n,z)), where (t) = log(1+exp^t) and (t) = t - (t) are the softplus and softminus functions, respectively. Once the approximate log ratio _ϕ is found, we can use it to take a gradient descent step along the approximate negative ELBO: -ℒ_ϕ(ψ, θ)= ∑_n=1^Nϵ∼𝒩(0,I)_ϕ(x_n,g_ψ(x_n,ϵ))- ∑_n=1^Nϵ∼𝒩(0,I)log p_θ(x_n| g_ψ(x_n,ϵ)), where we re-parametrised sampling from q_ψ in terms of a generator function g_ψ and noise ϵ. When q_ψ is explicitly defined, this re-parametrisation is the same as the re-parametrisation in VAEs. When q_ψ is an implicit distribution, it often already defined in terms of a non-linear function g_ψ and a noise variable ϵ which it transforms.Equationsandare analogous to the discriminator and generator losses in generative adversarial networks, with 1/(1+exp(-_ϕ)) taking the role of the discriminator. Optimising the two losses in tandem gives rise to the prior-contrastive adversarial algorithm (, Algorithm <ref>) for variational inference. This algorithm is equivalent to the independently developed adversarial variational Bayes <cit.>.As the likelihood p_θ(x| z) appears in Eqn. (<ref>), in Algorithm <ref> the forward model has to be explicitly defined, but the prior p_θ(z) and approximate posterior q_ψ can be implicit. Algorithm <ref> only describes variational inference - finding ψ given θ - but the approximate ELBO in Eqn.can be used for variational learning of θ as well, with the exception for parameters of the prior p_θ(z).Learning prior parameters involves minimising the KL-divergence ∑_n=1^Nq_ψ(z| x_n)p_θ(z) which is akin to fitting p_θ to samples from the aggregate posterior ∑_n=1^Nq_ψ(z| x_n) via maximum likelihood. If the prior has a tractable density, this may be an easy task to do. A more interesting case is though when the prior p_θ(z) itself is a latent variable model, in which case we can lower bound the said KL divergence with another ELBO, thereby stacking multiple models on top of each other in a hierarchical fashion <cit.>.A similar adversarial algorithm (, Algorithm <ref>) can be constructed to target _θ,ψ in the joint-contrastive formulation of ELBO (Eqn. <ref>).is very similar to ALI <cit.> and BiGAN <cit.> in that it learns to discriminate between the joint distributions (x)q_ψ(z| x) and p_θ(x,z). Unlike these methods, however,uses the correct loss functions so it maximises an approximation to the ELBO. Unlike in , which required a tractable likelihood p_θ(x| z),also works with completely implicitly defined models. As a downside,provides no direct way for variational learning of θ. ALI and BiGAN exploit the symmetry of the Jensen-Shannon divergence to optimise for θ, but asuses the asymmetric KL-divergence, this is not an option. Section <ref> explores an idea for fixing this drawback of .§ DENOISER-GUIDED LEARNING Although most versions of GAN use an adversarial discriminator based on logistic regression, there are other ways one can tackle learning and inference with implicit distributions. One interesting tool that has emerged in recent papers <cit.> is the use of denoising autoencoders <cit.> or reconstruction-contractive autoencoders <cit.>.The key observation for using denoising is that the Bayes-optimal denoiser function captures gradients of the log density of the data generating distribution: v^*(x,z)= _v x,z∼ q(x,z), η∼𝒩_σ_nv(x,z+η) - z^2 ≈ z + σ^2_n ∂log q(z| x)/∂ z as σ_n→ 0 This allows us to construct an estimator to the score of a distribution by fitting a DAE to samples. We note that it is possible to obtain a more precise analytical expression for for the optimal denoising function <cit.>.Let's see how one can use this in the prior-contrastive scenario to deal with an implicit q_θ. First, we fit a denoising function _ϕ by minimising the following loss: ℓ^_ψ, σ_n(ϕ) = ∑_n=1^N z∼ q_θ(z| x_n), η∼𝒩_σ_n v_ϕ(z + η,x_n) - z ^2, which can then be used to approximate the gradient of ELBO (Eqn. (<ref>)) with respect to ψ as follows: ∂ℒ(θ,ψ)/∂ψ≈∑_n=1^Nϵ∼𝒩∂ g_ψ(x_n,ϵ))/∂ψ.∂log p_θ(x_n, z)/∂ z|_z=g_ψ(x_n,ϵ)+ ∑_n=1^Nϵ∼𝒩∂ g_ψ(x_n,ϵ))/∂ψg_ψ(x_n,ϵ) - _ϕ(g_ψ(x_n,ϵ))/σ^2_n.Several SGD methods only require gradients of the objective function as input, this gradient estimate can be readily used to optimise an approximate ELBO. The resulting iterative algorithm, prior-contrastive denoising VI (, Algorithm <ref>) updates the denoiser and the variational distribution in tandem. Following similar derivation one can construct several other variants of the algorithm. The denoiser approach is more flexible than the adversarial approach as one can pick and choose which individual distributions are modelled explitly, and which ones are implicit. For example, when the prior p_θ is implicit, we can train a denoiser to represent its score function. Or, one can start from the joint-contrastive formulation of the ELBO and train a DAE over joint distribution of x and z, giving rise to the joint-contrastive denoising VI (). In the interest of space the detailed description of these variants is omitted here.As the denoising criterion estimates the gradients of ELBO but not the value itself, the denoising approach does not provide a direct way to learn model parameters θ. The denoising method may be best utilised in conjunction with an adversarial algorithm such as a combination ofand . The denoising method works better early on in the training when q_ψ and p_θ are very different, and therefore the discrimination task is too easy. Conversely, as q_ψ approaches p_θ, the discriminator can focus its efforts on modelling the residual differences between them rather than trying to model everything about q_θ in isolation as the denoiser in Algorithm <ref> does. <cit.> already used such combination of adversarial training with a denoising criterion for generative modelling. However, the additional nonlinear transformation before denoising introduced in that work breaks the mathematical connections to KL divergence minimisation. § SUMMARY To summarise, we have presented two main ways to formulate ELBO in terms of logarithmic density ratiosand . We called these prior-contrastive (PC) and joint-contrastive (JC). We have then described two techniques by which these density ratios, or their gradients, can be estimated if the distributions involved are implicit: adversarial methods ( and , Algorithms <ref>&<ref>) directly estimate density ratios via logistic regression, denoising methods ( and , Algorithm <ref>) estimate gradients of the log densities via denoising autoencoders. We have mentioned that these methods can be combined, and that such combination may improve convergence.While all of these algorithms can perform variational inference - fitting the variational parameters ψ - not all of them can perform full variational learning of model parameters θ if the model itself is implicitly defined. In Section <ref> we outline an idea based on reverse mode differentiation (RMD) idea by <cit.> which, giving rise to an algorithm we refer to as , which can in theory perform fully variational inference and learning in a model where all distributions involved are implicit.The capabilities of algorithms presented here and in related work are summarised in Table <ref>. The adversarial variational Bayes <cit.> is equivalent to PC-Adv, while ALI <cit.> and BiGAN <cit.> are closely related to JC-Adv. <cit.> and <cit.> are closely related to PC-Adv, although the former solves a limited special case and the latter uses the Jensen-Shannon formulation and hence is not fully variational. § EXPERIMENTS Several related papers have already demonstrated the success of methods surveyed here on real world datasets, see for example <cit.> for , <cit.> forand <cit.> for denoiser-based techniques. Experiments in these papers typically focus on the models' ability to learn θ, and the quality of samples from the learnt generative model p_θ(x).As the focus here is on inference rather than learning, the goal of this section is to validate the algorithms' ability to perform inference. To this end, we have devised a simple toy problem loosely based on the “sprinkle” example which exhibits explaining away <cit.>. In our “continuous sprinkler” model, two independent scalar hidden variables z_1 and z_2 are combined nonlinearly to produce a univariate observation x: (z_1,z_2)∼𝒩(0, σ^2 I_2×2)x∼EXP(3 + max(0,z_1)^3 + max(0,z_2)^3) Although z_1 and z_2 are a priori independent, the likelihood introduces dependence between the two variables once conditioned on data: either latent variable taking a large value can explain a large observed x. This is an example of explaining away which is an important phenomenon in latent variable models that is known to be hard to model with simple, unimodal distributions.Column A in Figure <ref> illustrates the joint posterior density of z_1 and z_2 for various values of x. The subsequent columns show the posterior approximations by ,and , respectively. q_θ is implemented as a stochastic generative network where the observation x and Gaussian noise variables ϵ are fed as input to a multilayer perceptron g_ψ. The discriminators and denoisers were implemented as multilayer perceptrons as well. Columns C and D illustrate the limiting behaviour of the discriminator in thealgorithm: as q_ψ converges to the true posterior, _ψ is expected resemble the likelihood p_θ(x| z) up to an additive constant. Inthe discriminator eventually converges to the flat _ϕ=0 solution. § DISCUSSION AND FUTURE WORK Are adversaries really needed? When using adversarial techniques for VI, we model the distribution of latent variables rather than observations. The distributions we encounter in VI are usually thought of as simpler than the distribution of observed data, so the question arises whether the flexibility of the adversarial framework is really needed.Is the prior-contrastive too much like noise-contrastive? In thealgorithm, the discriminator compares samples from the approximate posterior to the prior, and the prior is often high-dimensional Gaussian noise. Even at convergence, the two distributions the discriminator sees never overlap, and this may slow down training. This can be remedied by observing that as q_ψ converges to the true posterior, the discriminator will converge to the log-likelihood plus constant _ϕ(x,z)≈log p(x| y) + c. Hence, the task of the discriminator can be made easier by forming an ensemble between a neural network and the actual log-likelihood.Aren't denoising methods imprecise? The main criticism of denoiser-based methods is that the gradient estimates are imprecise. As <cit.> pointed out, the optimal denoising function represents the gradients of the noise-corrupted distribution rather than the original, and in practical cases the noise level σ_n may not be small enough for this effect to be negligible. <cit.> observed that denoiser-based methods can not produce results as sharp as adversarial counterparts. Finally, for the outer loop SGD to work consistently, the gradient estimates provided by the inner loop have to form a conservative vector field. While the Bayes-optimal denoiser function satisfies this, it is unclear to what degree this property is preserved when using suboptimal denoisers <cit.>. We believe that an alternative approach based on score matching <cit.> - a task intimately related to denoising <cit.> - might overcome both of these issues.How to learn θ? The focus of this note is on variational inference, which is finding ψ. However, it is equally important to think about learning θ. Unfortunately, none of the algorithms presented here allow for fully variational learning of model parameters θ when p_θ(x| z) is implicit. ALI and BiGAN do provide an algorithm, but as we mentioned, they are not fully variational. We close by highlighting one possible avenue for future work to enable this: differentiating the inner loop of thealgorithm via reverse mode differentiation <cit.>. To learn θ via SGD, one only needs an estimate of the gradient ∂_ψ,θ/∂θ. We can't compute _ψ,θ, only an approximation _ϕ which is reached via SGD. Each step of SGD depends implicitly on θ. Following <cit.> we can algorithmically differentiate the SGD algorithm in a memory-efficient way to obtain an estimate of the gradient we need for learning θ. We have not validated this approach experimentally, but included it asin Table <ref>.icml2017§ IGNORING IMPLICIT DEPENDENCE ON Ψ Proof of Eqn. (<ref>): . ∂/∂ψq_ψp_θ|_ψ = ψ_0 = ∂/∂ψ.z ∼ q_ψ_θ, ψ(z) |_ψ = ψ_0= . ∂/∂ψz ∼ q_ψlogq_ψ_0(z)/p(z)|_ψ = ψ_0 + . ∂/∂ψq_ψq_ψ_0|_ψ = ψ_0= . ∂/∂ψz ∼ q_ψlogq_ψ_0(z)/p(z)|_ψ = ψ_0= . ∂/∂ψz ∼ q_ψ_θ, ψ_0(z) |_ψ = ψ_0, where the third line is obtained by noting that ψ_0 is a local minimum of q_ψq_ψ_0, hence the second term in the second line is 0.

http://arxiv.org/abs/1702.08235v1

Elliptic Dynamical Quantum Groups and Equivariant Elliptic CohomologyElliptic Dynamical Quantum Groupsand Equivariant Elliptic Cohomology[This paper is a contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications. The full collection is available at https://www.emis.de/journals/SIGMA/EHF2017.htmlhttps://www.emis.de/journals/SIGMA/EHF2017.html]Giovanni FELDER ^†, Richárd RIMÁNYI ^ and Alexander VARCHENKO ^G. Felder, R. Rimányi and A. Varchenko^† Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland mailto:felder@math.ethz.chfelder@math.ethz.ch^ Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA mailto:rimanyi@email.unc.edurimanyi@email.unc.edu, mailto:anv@email.unc.eduanv@email.unc.eduReceived April 30, 2018, in final form December 12, 2018; Published online December 21, 2018We define an elliptic version of the stable envelope of Maulik and Okounkov for the equivariant elliptic cohomology of cotangent bundles of Grassmannians. It is a version of the construction proposed by Aganagic and Okounkov and is based on weight functions and shuffle products. We construct an action of the dynamical elliptic quantum group associated with 𝔤𝔩_2 on the equivariant elliptic cohomology of the union of cotangent bundles of Grassmannians. The generators of the elliptic quantum groups act as difference operators on sections of admissible bundles, a notion introduced in this paper.elliptic cohomology; elliptic quantum group; elliptic stable envelope17B37; 55N34; 32C35; 55R40 § INTRODUCTIONMaulik and Okounkov <cit.> have set up a program to realize representation theory of quantum groups of various kinds on torus equivariant (generalized) cohomology of Nakajima varieties. A central role is played by the stable envelopes, which are maps from the equivariant cohomology of the fixed point set of the torus action to the equivariant cohomology of the variety. Stable envelopes depend on the choice of a chamber (a connected component of the complement of an arrangement of real hyperplanes) and different chambers are related by R-matrices of the corresponding quantum groups. The basic example of a Nakajima variety is the cotangent bundle of the Grassmannian Gr(k,n) of k-planes in ℂ^n. The torus is T=U(1)^n× U(1), with U(1)^n acting by diagonal matrices on ℂ^n and U(1) acting by multiplication on the cotangent spaces. Then the Yangian Y(𝔤𝔩_2) acts on H_T(⊔_k=0^nT^*Gr(k,n)) and the action of generators is described geometrically by correspondences. It turns out that this representation is isomorphic to the tensor products of n evaluation vector representations with the equivariant parameters of U(1)^n as evaluation points and the equivariant parameter of U(1) as the deformation parameter of the quantum group. The choice of a chamber is the same as the choice of an ordering of the factors in the tensor product. The same holds for the affine quantum universal enveloping algebra U_q(𝔤𝔩_2) if we replace equivariant cohomology by equivariant K-theory. As was shown in <cit.>, the stable envelopes, which realize the isomorphisms, are given by the weight functions, which originally appeared in the theory of integral representations of solutions of the Knizhnik–Zamolodchikov equation, see <cit.>. Their special values form transition matrices from the tensor basis to a basis of eigenvectors for the Gelfand–Zetlin commutative subalgebra.The recent preprint <cit.> of Aganagic and Okounkov suggests that the same picture should hold for equivariant elliptic cohomology and elliptic dynamical quantum groups and this is the subject of this paper. The authors of <cit.> define an elliptic version of the stable envelopes and show, in the example of the cotangent bundle of a projective space, stable envelopes corresponding to different orderings are related to the fundamental elliptic dynamical R-matrices of the elliptic dynamical quantum group E_τ,y(𝔤𝔩_2). Our paper is an attempt to understand the elliptic stable envelope in the case of cotangent bundles of Grassmannians. In particular we give a precise description of the space in which the stable envelope takes its values. Our construction of stable envelopes is based on elliptic weight functions. In Appendix <ref> we also give a geometric characterization, in terms of pull-backs to the cohomology of fixed points, in the spirit of <cit.>.While our work is inspired by <cit.>, we do not know whether the two constructions are equivalent or not. The interesting project of understanding the exact relation between our construction and the construction of Aganagic–Okounkov requires more work.Compared to equivariant cohomology and K-theory, two new features arise in the elliptic case. The first new feature is the occurrence of an additional variable, the dynamical parameter, in the elliptic quantum group. It also appears in <cit.>, under the name of Kähler parameter, in an extended version of the elliptic cohomology of Nakajima varieties. The second is a general feature of elliptic cohomology: while T-equivariant cohomology and K-theory are contravariant functors from T-spaces to supercommutative algebras, and can thus be thought of as covariant functors to affine superschemes,[The reader may safely ignore the super prefixes, as we only consider spaces with trivial odd cohomology, for which one has strictly commutative algebras.] in the elliptic case only the description as covariant functor to (typically non-affine) superschemes generalizes straightforwardly.Our main result is a construction of an action of the elliptic quantum group associated with 𝔤𝔩_2 on the extended equivariant elliptic cohomology scheme Ê_T(X_n) of the union X_n=⊔_k=0^n X_k,n of cotangent bundles X_k,n=T^*Gr(k,n) of Grassmannians. The meaning of this is that we define a representation of the operator algebra of the quantum group by difference operators acting on sections of a class of line bundles on the extended elliptic cohomology scheme, which we call admissible bundles: up to a twist by a fixed line bundle, admissible bundles on Ê_T(X_k,n) are pull-backs of bundles on Ê_U(n)× U(1)(pt) (by functoriality there is a map corresponding to the map to a point and the inclusion of the Cartan subalgebra T→ U(n)× U(1)). The claim is that there is a representation of the elliptic quantum group by operators mapping sections of admissible bundles to sections of admissible bundles.This paper may be considered as an elliptic version of the paper <cit.> where analogous constructions are developed for the rational dynamical quantum group E_y(𝔤𝔩_2). §.§ NotationFor a positive integer n, we set [n]={1,…,n}. It K is a subset of [n] we denote by |K| its cardinality and by K̅ its complement. Throughout the paper we fix τ in the upper half plane and consider the complex elliptic curve E=ℂ/(ℤ+τℤ). The odd Jacobi theta functionθ(z)=sinπ z/π∏_j=1^∞(1-q^j e^2π iz)(1-q^j e^-2π iz)/(1-q^j)^2, q= e^2π iτ,is normalized to have derivative 1 at 0. It is an entire odd function with simple zeros at ℤ+τℤ, obeying θ(z+1)=-θ(z) andθ(z+τ)=- e^-π iτ-2π izθ(z). § DYNAMICAL R-MATRICES AND ELLIPTIC QUANTUM GROUPS §.§ Dynamical Yang–Baxter equationLet 𝔥 be a complex abelian Lie algebra and V an 𝔥-module with a weight decomposition V=⊕_μ∈𝔥^*V_μ and finite dimensional weight spaces V_μ. A dynamical R-matrix with values in End_𝔥(V⊗ V) is a meromorphic function (z,y,λ)↦ R(z,y,λ)∈End_𝔥(V⊗ V) of the spectral parameter z∈ℂ, the deformation parameter y∈ℂ and the dynamical parameter λ∈𝔥^*, obeying the dynamical Yang–Baxter equationR(z,y,λ-y h^(3))^(12)R(z+w,y,λ)^(13) R(w,y,λ-y h^(1))^(23) = R(w,y,λ)^(23)R(z+w,y,λ-y h^(2))^(13) R(z,y,λ-y h^(3))^(12)in End(V⊗ V⊗ V) and the inversion relationR(z,y,λ)^(12)R(-z,y,λ)^(21)=Idin End(V⊗ V). The superscripts indicate the factors in the tensor product on which the endomorphisms act non-trivially and h is the element in 𝔥^*⊗End(V) defined by the action of 𝔥: for example R(z,y,λ-y h^(3))^(12) acts as R(z,y,λ-yμ_3)⊗Id on V_μ_1⊗ V_μ_2⊗ V_μ_3. [<cit.>] Let 𝔥≃ℂ^N be the Cartan subalgebra of diagonal matrices in 𝔤𝔩_N(ℂ). Let V=⊕_i=1^N V_ϵ_i the vector representation with weights ϵ_i(x)=x_i, x∈𝔥 and one-dimensional weight spaces. Let E_ij be the N× N matrix with entry 1 at (i,j) and 0 elsewhere. The elliptic dynamical R-matrixfor 𝔤𝔩_N is[We use the convention of <cit.>. This R-matrix is obtained from the one introduced in <cit.> by substituting y=-2η and replacing z by -z.] R(z,y,λ) =∑_i=1^NE_ii⊗ E_ii+∑_i≠ jα(z,y,λ_i-λ_j) E_ii⊗ E_jj +∑_i≠ jβ(z,y,λ_i-λ_j) E_ij⊗ E_ji, where α(z,y,λ)= θ(z)θ(λ+y)/θ(z-y)θ(λ), β(z,y,λ)= -θ(z+λ)θ(y)/θ(z-y)θ(λ). It is a deformation of the trivial R-matrix R(z,0,λ)=id_V⊗ V.A dynamical R-matrix defines a representation of the symmetric group S_n on the space of meromorphic functions of (z_1,…,z_n,y,λ)∈ℂ^n×ℂ×𝔥^* with values in V^⊗ n. The transposition s_i=(i,i+1), i=1,…,n-1, acts asf↦ S_i(z,y,λ)s_i^*f, S_i(z,y,λ)=R(z_i-z_i+1,y,λ-y∑_j=i+2^nh^(j))^(i,i+1) P^(i,i+1),where P∈End(V⊗ V) is the flip u⊗ v↦ v⊗ u and s_i^* acts on functions by permutation of z_i with z_i+1.To a dynamical R-matrix R there corresponds a category “of representations of the dynamical quantum group associated with R”. Fix y∈ℂ and let 𝕂 be the field of meromorphic functions of λ∈𝔥^* and for μ∈𝔥^* let τ_μ^*∈Aut(𝕂) be the automorphism τ_μ^*f(λ)=f(λ+yμ). An object of this category is a 𝕂-vector space W=⊕_μ∈𝔥^*W_μ, which is a semisimple module over 𝔥, with finite dimensional weight spaces W_μ, together with an endomorphisms L(w)∈End_𝔥(V⊗ W), depending on w∈ U⊂ℂ for some open dense set U, such that =0pt(i) L(w)u⊗ fv=(id⊗τ^*_-μf) L(w)u⊗ v, f∈𝕂, u∈ V_μ,v∈ W.(ii) L obeys the RLL relations: R(w_1-w_2,y,λ-y h^(3))^(12)L(w_1)^(13) L(w_2)^(23) = L(w_2)^(23)L(w_1)^(13) R(w_1-w_2,y,λ)^(12). Morphisms (W_1,L_W_1)→ (W_2,L_W_2) are 𝕂-linear maps φ W_1→ W_2 of 𝔥-modules, commuting with the action of the generators, in the sense that L_W_2(w) id_V⊗φ=id_V⊗φL_W_1(w) for all w in the domain of definition. The dynamical quantum group itself may be defined as generated by Laurent coefficients of matrix elements of L(w) subject to the RLL relations, see <cit.> for a recent approach in the case of elliptic dynamical quantum groups and for the relations with other definitions of elliptic quantum groups.The basic example of a representation is the vector evaluation representation V(z) with evaluation point z∈ℂ. The vector representation has W=V⊗_ℂ𝕂 andL(w)v⊗ u=R(w-z,y,λ)v⊗τ_-μ^* u, v∈ V_μ, u∈ W.Here τ^*_-μ (v⊗ f)=v⊗τ_-μ^*f for v∈ V and f∈𝕂, and R acts as a multiplication operator.More generally we have the tensor product of evaluation representations V(z_1)⊗⋯⊗ V(z_n) with W=V^⊗ n⊗𝕂, and, by numbering the factors of V⊗ V^⊗ n by 0,1,…,n,L(w)v⊗ u =R(w-z_1,y,λ-y∑_i=2^nh^(i))^(01) R(w-z_2,y,λ-y ∑_i=3^nh^(i))^(02)⋯ × R(w-z_n,y,λ)^(0,n) v⊗τ_-μ^* u,v∈ V_μ, u∈ W.For generic z_1,…, z_n the tensor products does not essentially depend on the ordering of the factors: the operators S_i defined above are isomorphisms of representationsV(z_1)⊗⋯⊗ V(z_i)⊗ V(z_i+1)⊗⋯⊗ V(z_n) → V(z_1)⊗⋯ ⊗ V(z_i+1)⊗ V(z_i)⊗⋯⊗ V(z_n). It is convenient to consider L-operators L(w), such as (<ref>), which are meromorphic functions of w and are thus only defined for w in an open dense set. But one may prefer to consider only representations with L(w) defined for all w∈ℂ. This may be obtained for the representation given by (<ref>) by replacing L(w) by the product of L(w) with ∏_a=1^nθ(w-z_a+y). §.§ Duality and gauge transformationsSuppose that R(z,y,λ) is a dynamical R-matrix with 𝔥-module V. Let V^∨=⊕_μ (V^∨)_μ with weight space (V^∨)_μ the dual space to V_μ. Then R^∨(z,y,λ)=(R(z,y,λ)^-1)^*, the dual map to R(z,y,λ)^-1, is a dynamical R-matrix with values in End_𝔥(V^∨⊗ V^∨). It is called the dual R-matrix to R.Another way to get new R-matrices out of old is by a gauge transformation. Let ψ_V(λ) be a meromorphic function on ℂ×𝔥^* with values in Aut_𝔥(V). Let ψ_V⊗ V(λ)=ψ_V(λ-y h^(2))^(1)ψ_V(λ)^(2)∈End_𝔥(V⊗ V). ThenR_ψ(z,y,λ)=ψ_V⊗ V(λ)^-1R(z,y,λ)ψ_V⊗ V(λ)^(21)is another dynamical R-matrix. The corresponding representations of the symmetric group are related by the isomorphismψ_V^⊗ n(λ)=∏_i=1^nψ_V(λ-y∑_j=i+1^nh^(j))^(i).§.§ The elliptic dynamical quantum group E_τ,y(𝔤𝔩_2) In this paper, we focus on the dynamical quantum group E_τ,y(𝔤𝔩_2). The corresponding R-matrix is the case N=2 of Example <ref>. With respect to the basis v_1⊗ v_1, v_1⊗ v_2, v_2⊗ v_1, v_2⊗ v_2,R(z,y,λ)= (1 0 0 0 0α(z,y,λ)β(z,y,λ) 0 0β(z,y,-λ)α(z,y,-λ) 0 0 0 01),where λ=λ_1-λ_2. Since R depends only on the difference λ_1-λ_2 it is convenient to replace 𝔥 by the 1-dimensional subspace ℂ spanned by h=diag(1,-1). Then, under the identification 𝔥≅ℂ via the basis h, v_1 has weight 1 and v_2 has weight -1. Let (W,L) be a representation of E_τ,y(𝔤𝔩_2) and write L(w)=∑_i,j=1^2 E_ij⊗ L_ij(w). Then L_ij(w) maps W_μ to W_μ+2(i-j) and for f(λ)∈𝕂, L_i2(w)f(λ) =f(λ+y)L_i2(w) and L_i1(w)f(λ) =f(λ-y)L_i1(w). [the vector representation V(z)] Let V=ℂ^2 with basis v_1, v_2, then L_11(w)v_1=v_1, L_22(w)v_2=v_2, L_11(w)v_2= θ(w-z)θ(λ+y)/θ(w-z-y)θ(λ) v_2,L_22(w)v_1= θ(w-z)θ(λ-y)/θ(w-z-y)θ(λ) v_2, L_12(w)v_1 =-θ(λ+w-z)θ(y)/θ(w-z-y)θ(λ) v_2,L_21(w)v_2=-θ(λ-w+z)θ(y)/θ(w-z-y)θ(λ) v_1,and the action on other basis vectors is 0.§.§ The Gelfand–Zetlin subalgebraLet W be a representation of the elliptic dynamical quantum group E_τ,y(𝔤𝔩_2). Then L_22(w), w∈ℂ and the quantum determinant <cit.>Δ(w)=θ(λ-y h)/θ(λ) (L_11(w+y)L_22(w)-L_21(w+y)L_12(w))generate a commutative subalgebra of End_𝔥(W). It is called the Gelfand–Zetlin subalgebra.§ SHUFFLE PRODUCTS AND WEIGHT FUNCTIONSWeight functions are special bases of spaces of sections of line bundles on symmetric powers of elliptic curves. They appear in the theory of hypergeometric integral representation of Knizhnik–Zamolodchikov equations. In <cit.> they were characterized as tensor product bases of a space of function for a suitable notion of tensor products. In this approach the R-matrices for highest weight representations of elliptic quantum groups arise as matrices relating bases obtained from taking different orderings of factors in the tensor product. We review and extend the construction of <cit.> in the special case of products of vector representations. §.§ Spaces of theta functions Let z∈ℂ^n,y∈ℂ,λ∈ℂ and define Θ^-_k(z,y,λ) to be the space of entire holomorphic functions f(t_1,…,t_k) of k variables such that=0pt * For all permutations σ∈ S_k, f(t_σ(1),…,t_σ(k))=f(t_1,…,t_k). * For all r,s∈ℤ, the meromorphic function g(t_1,…,t_k)=f(t_1,…,t_k)/∏_j=1^k∏_a=1^n θ(t_j-z_a) obeys g(t_1,…, t_i+r+sτ,…,t_k)= e^2π is(λ-ky)g(t_1,…,t_i,…,t_k).Let z∈ℂ^n, y∈ℂ, λ∈ℂ and define Θ^+_k(z,y,λ) to be the space of entire holomorphic functions f(t_1,…,t_k) of k variables such that=0pt * For all permutations σ∈ S_k, f(t_σ(1),…,t_σ(k))=f(t_1,…,t_k). * For all r,s∈ℤ, the meromorphic function g(t_1,…,t_k)=f(t_1,…,t_k)/∏_j=1^k∏_a=1^n θ(t_j-z_a+y), obeys g(t_1,…, t_i+r+sτ,…,t_k)= e^-2π is(λ-ky)g(t_1,…,t_i,…,t_k).These spaces are spaces of symmetric theta functions of degree n in k variables and have dimension n+k-1k. Actually Θ^- depends on the parameters only through the combination ∑_a=1^n z_a+λ-ky and Θ^+ through the combination ∑_a=1^nz_a-λ-(n-k)y. For z∈ℂ and all k=0,1,2,…, Θ^-_k(z,y,λ) is a one-dimensional vector spacespanned by ω^-_k(t;z,y,λ)=∏_j=1^kθ(λ-t_j+z-ky), Θ^+_k(z,y,λ) is a one-dimensional vector space spanned by ω^+_k(t;z,y,λ) =∏_j=1^kθ(λ+t_j-z+(1-k)y). For z∈ℂ^n, y,λ∈ℂ, Θ^-_k(z,y,λ)=Θ^+_k(z,y,-λ-(n-2k)y) and ω^+_k(t;z,y,λ)=(-1)^kω^-_k(t;z,y,-λ-(1-2k)y). It is however better to keep the two spaces distinct as they will be given a different structure.§.§ Shuffle productsLet Sym denote the map sending a function f(t_1,…,t_k) of k variables to the symmetric function ∑_σ∈ S_nf(t_σ(1),…,t_σ(k)).Let n=n'+n”, k=k'+k” be non-negative integers, z∈ℂ^n, z'=(z_1,…,z_n'), z”=(z_n'+1,…, z_n). Then the shuffle product * Θ^±_k'(z',y,λ+y(n”-2k”))⊗Θ^±_k”(z”,y,λ)→Θ^±_k(z,y,λ), sending f⊗ g to f*g(t)=1/k'!k”!Sym(f(t_1,…,t_k') g(t_k'+1,…,t_k)φ^±(t,z,y)), with φ^-(t,z,y) =∏_j=1^k'∏_l=k'+1^k θ(t_l-t_j+y)/θ(t_l-t_j)∏_l=k'+1^k∏_a=1^n'θ(t_l-z_a) ∏_j=1^k'∏_b=n'+1^nθ(t_j-z_b+y),φ^+(t,z,y) =∏_j=1^k'∏_l=k'+1^k θ(t_j-t_l+y)/θ(t_j-t_l)∏_l=k'+1^k∏_a=1^n'θ(t_l-z_a+y) ∏_j=1^k'∏_b=n'+1^nθ(t_j-z_b), is well-defined and associative, in the sense that (f*g)*h=f*(g*h), whenever defined. In the formula for f*g in Proposition <ref> we can omit the factor 1/k'!k”! and replace the sum over permutations defining Sym by the sum over (k',k”)-shuffles, namely permutations σ∈ S_k such that σ(1)<…<σ(k') and σ(k'+1)<⋯<σ(k). This is essentially the first part of Proposition 3 of <cit.> in the special case of weights Λ_i=1. The proof is straightforward: the apparent poles at t_j=t_l are cancelled after the symmetrization since θ(t_j-t_l) is odd under interchange of t_j with t_l. Thus f*g is a symmetric entire function. One then checks that every term in the sum over permutations has the correct transformation property under lattice shifts. The maps * of Proposition <ref> define isomorphisms ⊕_k'=0^k Θ^±_k'(z',y,λ+y(n”-2k”)) ⊗Θ^±_k-k'(z”,y,λ) →Θ^±_k(z,y,λ) for generic z, y, λ.We prove this Proposition in <ref> below. §.§ Vanishing conditionThe shuffle product * preserves subspaces defined by a vanishing condition. It is the case of the fundamental weight of a condition introduced in <cit.> for general integral dominant weights.Let (z,y,λ)∈ℂ^n×ℂ×ℂ. We define a subspace Θ̅^±_k(z,y,λ)⊂Θ^±_k(z,y,λ) by a vanishing condition:Θ̅^±_k(z,y,λ) = Θ^±_k(z,y,λ) if k=0,1, {ff(t_1,…,t_k-2,z_a,z_a-y)=0, 1≤ a≤ n,t_i∈ℂ} if k≥2. For n=1, Θ̅^±_k(z,y,λ) =Θ^±_k(z,y,λ)≅ℂ, k=0,1, 0, k≥2. Indeed, the condition is vacuous if k≤ 1 and if k≥2 then ω^±_k(z;z-y,t_3,…)=θ(λ-ky)θ(λ+(1-k)y) times a nonzero function. For k=1, n≥1, Θ̅^±_1(z,y,λ) = Θ^±_1(z,y,λ). For k=2, n=2, Θ̅^±_2(z_1,z_2,y,λ) is a one-dimensional subspace of the three-dimensional space Θ^±_2(z,y,λ). The shuffle product restricts to a map⊕_k'=0^k Θ̅^±_k' (z',y,λ+y(n”-2k”)) ⊗Θ̅^±_k-k' (z”,y,λ) →Θ̅^±_k(z,y,λ), which is an isomorphism for generic values of the parameters.The proof is postponed to Section <ref> below. By iteration we obtain shuffle multiplication mapsΦ̅_k^±(z,y,λ)⊕_Σ k_a=k⊗_a=1^n Θ̅^±_k_a(z_a,y,λ-y∑_b=a+1^n(2k_a-1)) →Θ̅^±_k(z_1,…,z_n,y,λ),defined for (z,y,λ)∈ℂ^n×ℂ×ℂ and k=0,1,2,…. The direct sum is over the nk n-tuples (k_1,…,k_n) with sum k and k_a∈{0,1}, a=1,…,n.The maps Φ̅_k^±(z,y,λ) are isomorphisms for generic (z,y,λ)∈ℂ^n×ℂ×ℂ. Thus, for generic z,y,λ∈ℂ^n×ℂ×ℂ, Θ̅^±_k(z,y,λ) has dimension nk and is zero if k>n. §.§ Duality The identificationϱ Θ^-_k(z,y,λ)→Θ^+_k(z,y,-λ-(n-2k)y)of Remark <ref> (the identity map) restricts to an isomorphismΘ̅^-_k(z,y,λ)→Θ̅^+_k(z,y,-λ-(n-2k)y),also denoted by ϱ. For f∈Θ^-_k' and g∈Θ^-_k” as in Proposition <ref>, the shuffle product ϱ(g)*ϱ(f) is well-defined and obeysϱ(f*g)=ϱ(g)*ϱ(f).It is clear that the vanishing condition is preserved. The last claim follows from the identity φ^-(t,z,y)= φ^+(t_k'+1,…,t_k,t_1,…,t_k',z_n'+1,…,z_n,z_1,…,z_n',y) for the functions appearing in the definition of the shuffle product.For n=1 we have ϱ(ω^-_k)=(-1)^kω^+_k, see Example <ref>. §.§ Weight functionsFor (z,y,λ)∈ℂ^n×ℂ×ℂ, letΘ̅^±(z,y,λ)=⊕_k=0^n Θ̅^±_k(z,y,λ).It is an 𝔥-module with Θ̅^±_k of weight -n+2k. Let v_1, v_2 be the standard basis of ℂ^2. If n=1, we identify Θ̅^±(z,y,λ) with ℂ^2 via the map ω^±_1↦ v_1, ω^±_0↦ v_2. Then Φ̅^±(z,y,λ)=⊕_kΦ̅^±_k(z,y,λ) is a linear map(ℂ^2)^⊗ n→Θ̅^±(z,y,λ).It is a homomorphism of 𝔥-modules. Then a basis of (ℂ^2)^⊗ n is labeled by subsets I of [n]={1,…,n}: v_I=v_j(1)⊗⋯⊗ v_j(n) with j(a)=2 if a∈ I and j(a)=1 if a∈I̅, the complement of I.The weight functions ω^±_I(t;z,y,λ) are the functions ω^±_I(·;z,y,λ)=Φ̅^±(z,y,λ)v_I∈Θ̅^±(z,y,λ). In particular, for n=1, ω^±_∅=ω^±_0, ω^±_{1}=ω_1^±. Corollary <ref> implies:Let (z,y,λ) be generic. The weight functions ω^±_I(·; z,y,λ) with I⊂ [n], |I|=k form a basis of the space Θ̅^±_k(z,y,λ) of theta functions obeying the vanishing condition. For k=1 and n=1,2…, z∈ℂ^n, y∈ℂ, λ∈ℂ, a=1,…,n, ω^-_{a}(t;z,y,λ) =θ(λ-t+z_a+y(n-a-1)) ∏_b=1^a-1θ(t-z_b) ∏_b=a+1^n θ(t-z_b+y), ω^+_{a}(t;z,y,λ) =θ(λ+t-z_a+y(n-a)) ∏_b=1^a-1θ(t-z_b+y) ∏_b=a+1^n θ(t-z_b).§.§ R-matricesNote that while Θ̅^±_k(z,y,λ) is independent of the ordering of z_1,…,z_n the map Φ̅_k^± does depend on it and different orderings are related by R-matrices, as we now describe. We define R-matrices R^±(z,y,λ)∈End_𝔥(ℂ^2⊗ℂ^2) byR^±(z_1-z_2,y,λ) =Φ̅^±(z_1,z_2,y,λ)^-1Φ̅^±(z_2,z_1,y,λ)P,where Pu⊗ v=v⊗ u is the flip of factors. Up to duality and gauge transformation, these R-matrices coincide with the elliptic R-matrix of Section <ref>: =0pt (i) Let s_i∈ S_n be the transposition (i,i+1). Then Φ̅^±(s_iz,y, λ) =Φ̅^±(z,y,λ) R^±(z_i-z_i+1,y,λ -y∑_j=i+2^nh^(j))^(i,i+1)P^(i,i+1).(ii) The R-matrices R^± obey the dynamical Yang–Baxter equation (<ref>) and the inversion relation (<ref>). (iii) With respect to the basis v_1⊗ v_1, v_1⊗ v_2, v_2⊗ v_1, v_2⊗ v_2 of ℂ^2⊗ℂ^2, R^-(z,y,λ)= (1 0 0 0 0α(-z,y,-λ)β(-z,y,λ) 0 0β(-z,y,-λ)α(-z,y,λ) 0 0 0 01)=R^∨(z,y,λ) is the dual R-matrix, see Section <ref>, with the standard identification of ℂ^2 with (ℂ^2)^* and R^+(z,y,λ)= (1 0 0 0 0α(z,y,-λ)β(z,y,λ) 0 0β(z,y,-λ)α(z,y,λ) 0 0 0 01)= R_ψ(z,y,λ) is the gauge transformed R-matrix with ψ(λ)=(θ(λ)θ(λ-y) 00 1). Let (z,y,λ)∈ℂ^n×ℂ×ℂ be generic and set S_i(z,y,λ)= R(z_i-z_i+1,y,λ-y∑_j=i+2^nh^(j))^(i,i+1)P^(i,i+1)∈End_𝔥((ℂ^2)^⊗ n), i=1,…,n-1, cf. (<ref>).=0pt (i) For t∈ℂ^k, let ω^-(t;z,y,λ)=∑_I⊂[n], |I|=kω^-_I(t;z,y,λ)v_I. Then ω^-(t;z,y,λ)=S_i(z,y,λ) ω^-(t;s_iz,y,λ).(ii) Let ψ_V^⊗ n(λ)=∏_i=1^nψ(λ-y ∑_j>ih^(j))^(i), cf. Section <ref>. Then Φ̅^+(s_iz,y,λ)ψ_V^⊗ n(λ)^-1 =Φ̅^+(z,y,λ)ψ_V^⊗ n(λ)^-1 S_i(z,y,λ). §.§ A geometric representationLet z_1,…,z_n,y,λ be generic and w∈ℂ. Recall that we identify Θ̅^+(w,y,λ) with V=ℂ^2 via the basis ω^+_1, ω^+_0. Consider the shuffle products[The compressed notation we are using might be confusing: the map p_+ is actually defined on ⊕_k Θ̅^+(w,y,λ+(n-2k)y)⊗Θ̅^+_k(z,y,λ). The identification of the first factor with V depends on k through the λ-dependence of the basis vectors ω^+_i.]p_+ V ⊗Θ̅^+(z_1,…,z_n,y,λ)→Θ̅^+(w,z_1,…,z_n,y,λ),p_- Θ̅^+(z_1,…,z_n,y,λ-yh^(2))⊗ V →Θ̅^+(w,z_1,…,z_n,y,λ).Then varying w and denoting P the flip of tensor factors, we get a homomorphismℓ(w,y,λ)=p_+^-1∘ p_-∘ P∈Hom(V⊗Θ̅^+(z,y,λ-y h^(1)),V⊗Θ̅^+(z,y,λ)).By construction it obeys the dynamical Yang–Baxter equationR^+(w_1-w_2,y,λ-y h^(3))^(12)ℓ(w_1,y,λ)^(13)ℓ(w_2,y,λ-y h^(1))^(23)= ℓ(w_1,y,λ)^(23)ℓ(w_2,y,λ-y h^(2))^(13) R^+(w_1-w_2,y,λ-y h^(3))^(12)in Hom(V⊗ V⊗Θ̅^+(z,y,λ-y(h^(1)+h^(2))), V⊗ V⊗Θ̅^+(z,y,λ)). By varying λ we obtain a representation of the elliptic dynamical quantum group as follows. Let (z,y)∈ℂ^n×ℂ be generic and consider the space Θ̅^+_k(z,y)_reg of holomorphic functions f(t,λ) on ℂ^k×ℂ such that for each fixed λ, t↦ f(t,λ) belongs to Θ̅^+(z,y,λ). It is a module over the ring 𝒪(ℂ) of holomorphic functions of λ. We setΘ̅^+_k(z,y)=Θ̅^+_k(z,y)_reg⊗_𝒪(ℂ)𝕂.It is a finite dimensional vector space over 𝕂, and for generic z, y it has a basis given by weight functions ω^+_I, |I|=k. Let z_1,…,z_n,y be generic complex numbers. Then Θ̅^+(z_1,…,z_n,y)=⊕_k=0^nΘ̅^+_k(z_1,…,z_n,y). is a representation of the elliptic quantum group E_τ,y(𝔤𝔩_2) with the L-operator L(w)(v⊗ u)=ψ(λ-yh^(2))^(1)ℓ(w,y,λ)(ψ(λ)^-1)^(1)(v⊗τ_-μ^*u), v∈ V_μ. Here ψ is the gauge transformation of Proposition <ref>.The homomorphisms ℓ obey the RLL-type relations (<ref>) with R-matrix R^+ which, according to Proposition <ref>, is obtained from R by the gauge transformation ψ. It is easy to check thatℓ̂(w,y,λ)= ψ(λ-yh^(2))^(1)ℓ(w,y,λ)(ψ(λ)^-1)^(1)obeys the same relations but with R^+ replaced by R. It follows that the corresponding difference operators define a representation of the elliptic dynamical quantum group.It follows from the previous section that this representation is isomorphic to the tensor product V(z_σ(1))⊗⋯⊗ V(z_σ(n)) for any permutation σ∈ S_n. However this identification with a tensor product of evaluation vector representations depends on a choice of ordering of the z_1,…,z_n, while Θ̅^+(z_1,…,z_n,y,λ) depends as a representation only on the set {z_1,…, z_n}.§.§ PairingWe define a pairing of Θ^-_k with Θ^+_k, taken essentially from <cit.>. Note that the product of a function in Θ^-_k(z,y,λ) and a function in Θ^+_k(z,y,λ) divided by the products of Jacobi theta functions in part (2) of Definitions <ref> and <ref>, is a function which is doubly periodic in each variable t_i with poles at t_i=z_a and at t_i=z_a-y, a=1,…,n. It can thus be viewed as a meromorphic function on the Cartesian power E^k of the elliptic curve E=ℂ/(ℤ+τℤ).Let z_1…,z_n,y∈ E such that z_a≠ z_b+jy for all 1≤ a,b≤ n, 1≤ j≤ n-1, and let γ∈ H_1(E∖{z_1,…,z_n}) be the sum of small circles around z_a, a=1,…,n, oriented in counterclockwise direction. Let D⊂ E^k be the effective divisor D=∪_a=1^n∪_i=1^k({t∈ E^k t_i=z_a}∪{t∈ E^k t_i=z_a-y}). The symmetric group S_k acts by permutations on the sections of the sheaf 𝒪(D) of functions on E^k with divisor of poles bounded by D. Let ⟨ ⟩Γ(E^k,𝒪(D))^S_k→ℂ be the linear form f→⟨ f⟩= θ(y)^k/(2π i)^kk!∫_γ^k f(t_1,…,t_k)∏_1≤ i≠ j≤ kθ(t_i-t_j)/θ(t_i-t_j+y) dt_1⋯ dt_k.For k=0 we define ⟨ ⟩ℂ→ℂ to be the identity map. Let f∈Γ(E^k,𝒪(D))^S_k. Then ⟨ f⟩=θ(y)^k∑_1≤ i_1<⋯<i_k≤ nres_t_1=z_i_1⋯res_t_k=z_i_k(f(t_1,…,t_k)∏_i≠ jθ(t_i-t_j)/θ(t_i-t_j+y)).By the residue theorem, ⟨ f⟩ is a sum of iterated residues at t_i=z_a(i) labeled by maps a [k]→[n]. Since θ(t_i-t_j) vanishes for t_i=t_j, only injective maps a contribute non-trivially. Moreover, since the integrand is symmetric under permutations of the variables t_i, maps a differing by a permutation of {1,…,k} give the same contribution. Thus we can restrict the sum to strictly increasing maps a and cancel the factorial k! appearing in the definition. Denote Q=∏_i=1^k∏_a=1^nθ(t_i-z_a)θ(t_i-z_a+y) and let ⟨ ,⟩ Θ^-_k(z,y,λ)⊗Θ^+_k(z,y,λ)→ℂ be the bilinear pairing ⟨ f,g⟩= ⟨ fg/Q⟩, defined for generic z∈ℂ^n, y∈ℂ. Note that fg/Q is an elliptic function of t_i for all i.Here is the explicit formula for the pairing:⟨ f,g⟩=θ(y)^k/(2π i)^kk!∫_γ^kf(t_1,…,t_k)g(t_1,…,t_k)/∏_i,aθ(t_i-z_a)θ(t_i-z_a+y)∏_i≠ jθ(t_i-t_j)/θ(t_i-t_j+y) dt_1⋯ dt_k.Let n=1. Then ⟨ω^-_0,ω^+_0⟩=1, ⟨ω^-_1,ω^+_1⟩=θ(λ-y) θ(λ), and ⟨ω^-_k,ω^+_k⟩=0 for k>1. The first claim holds by definition. We have ⟨ω^-_1,ω^+_1⟩ =θ(y) res_t=zθ(λ-t+z-y) θ(λ+t-z)/θ(t-z)θ(t-z+y) dt =θ(λ-y) θ(λ). For k≥ 2, the residue at t_1=z is regular at t_i=z for i≥2 and thus the iterated residue vanishes.=0pt (i) The pairing restricts to a non-degenerate pairing Θ̅^-_k(z,y,λ) ⊗Θ̅^+_k(z,y,λ) →ℂ for generic z_1,…,z_n,y,λ. (ii) In the notation of Proposition <ref>, suppose f_i∈Θ̅^-_k'_i(z',y,λ+y(n”-2k_i”)), g_i∈Θ̅^+_k”_i(z”,y,λ), i=1,2 and k_1'+k_2'=k_1”+k_2”. Then ⟨ f_1*f_2,g_1*g_2⟩= ⟨ f_1,g_1⟩⟨ f_2,g_2⟩,if k'_1=k”_1 and k'_2=k”_2,0, otherwise.It is sufficient to prove (ii), since with Lemma <ref> it implies that, with a proper normalization, weight functions form dual bases with respect to the pairing.We use Lemma <ref> to compute ⟨ f_1*f_2,g_1*g_2⟩. Let us focus on the summand in Lemma <ref> labeled by i_1<⋯<i_n and suppose i_s≤ n'<i_s+1. Due to the factor θ(t_l-z_a) in φ^-, see Proposition <ref>, the only terms in the sum over shuffles having nonzero first s residues res_t_1=z_i_1, …, res_t_s=z_i_s are those for which t_1,…, t_s are arguments of f_1. In particular the summand vanishes unless s≤ k_1'. Similarly the factors θ(t_j-z_b) in φ^+ restrict the sum over shuffles to those terms for which t_s+1,…,t_k are arguments of g_2, so that the summand vanishes unless s≥ k-k_2”=k_2'. It follows that if k_1'<k_2' then ⟨ f_1*f_2,g_1*g_2⟩ vanishes and that if k_1'=k_1”, the pairing can be computed explicitly as sum over i_1<⋯<i_s≤ n'<i_s+1<⋯<i_k, with s=k_1', of terms involving f_1g_1(z_i_1,…,z_i_k_1')f_2g_2(z_i_k_1'+1,…,z_i_k). The coefficients combine to give ⟨ f_1,g_1⟩⟨ f_2,g_2⟩.There remains to prove that the pairing vanishes also if k_1'>k_2'. Here is where the vanishing condition comes in. We first consider the case where k_1'-k_2'=1 and then reduce the general case to this case.As above the presence of the vanishing factors in φ^± imply that the non vanishing residues in Lemma <ref> are those labeled by i_1<⋯<i_k such that at least k_1” indices are ≥ n' and the corresponding variables t_i are arguments of g_1 and at least k_2' indices are ≤ n' and the corresponding variables are arguments of f_2. If k_1'-k_2'=1 there is one variable left and we can write the pairing as a sum of one-dimensional integrals over this variable: I_A,B=∫_γf_1(z_A,t)g_1(z_B)f_2(z_A)g_2(t,z_B)/h(z_1,…,z_n,y,t) dt.Here z_A=z_a_1,…,z_a_k_2' with a_i≤ n' and z_B=z_b_1,…,z_b_k_1” with b_i>n'. The point is that in h(z,t) several factor cancel and one obtainsh(z,y,t)=C(z,y)∏_c∈A∪ Bθ(t-z_c) ∏_c∈ A∪ Bθ(t-z_c+y),for some t-independent function C(z,y). Because of the vanishing condition, the integrand in (<ref>) is actually regular at t=z_c-y and the only poles are at t=z_c, c∈A∪ B. By the residue theorem I_A,B=0.Finally, let us reduce the general case to the case where k_1'-k_2'=1. We use induction on n. By Lemma <ref> the pairing vanishes unless k=1,0 so there is nothing to prove in this case. Assume that the claim is proved for n-1. By Proposition <ref>, we can write g_1=h_1*m_1 and g_2=h_2*m_2 with m_i∈Θ̅^-_r_i(z_n,y,λ). By Lemma <ref> we can assume that r_i∈{0,1}. By the associativity of the shuffle product we can use the result for k_1'-k_2'=1 to obtain that the pairing vanishes unless r_1=r_2 and⟨ f_1*g_1,f_2*g_2⟩=⟨ f_1*h_1,f_2*h_2⟩⟨ m_1,m_2⟩.By the induction hypothesis, this vanishes unless k_1'=k_2'. We obtain orthogonality relations for weight functions. To formulate them we introduce some notation. For I⊂ [n] and 1≤ j≤ n we setn(j,I)=|{l∈[n] | l∈ I, l>j}|, w(j,I)=n(j,I)-n(j,I̅).Thus -w(j,I) the sum of the weights of the tensor factors to the right of the j-th factor in v_I.[cf. <cit.>] ⟨ω^-_I,ω^+_J⟩ = δ_I,Jψ_I(y,λ), where ψ_I(y,λ)= ∏_j∈ Iθ(λ-(w(j,I)+1)y)θ(λ-w(j,I)y).§.§ Normalized weight functions By construction the weight functions ω^±_I are entire functions of all variables and obey the vanishing conditions ω^±_I(z_a,z_a-y,t_3,…,t_k;z,y,λ)=0,a=1,…,n. This motivates the following definition.The normalized weight functions w_I^± are the functions w_I^-(t;z,y,λ) =ω^-_I(t;z,y,λ)/∏_1≤ j≠ l≤ kθ(t_j-t_l+y), w_I^+(t;z,y,λ) = ω^+_I(t;z,y,λ)/ψ_I(y,λ) ∏_1≤ j≠ l≤ kθ(t_j-t_l+y).The factor 1/ψ_I, defined in Corollary <ref>, simplifies the orthogonality relations and the action of the permutations of the z_i at the cost of introducing poles at λ+yℤ moduloℤ+τℤ.Let I={i_1,…,i_k}⊂[n] and f(t_1,…,t_k) a symmetric function of k variables. We write f(z_I) for f(z_i_1,…,z_i_k). For each I,J⊂[n] such that |I|=|J|, the weight functions w_I^-(z_J;z,y,λ) and ψ_I(y,λ) w_I^+(z_J;z,y,λ) are entire functions of z, y, λ. The vanishing condition implies that ω_I^±(z_a,z_b,t_3,…) is divisible by θ(z_b-z_a+y) so that the quotient by θ(t_2-t_1+y) is regular at z_b=z_a-y after substitution t_1=z_a, t_2=z_b. Since ω_I^± is a symmetric function, the same holds for any other pair t_j, t_l.The orthogonality relations become:[cf. <cit.>] Let I,J⊂[n], |I|=|J|=k. The normalized weight functions obey the orthogonality relations ∑_Kw^-_I(z_K,z,y,λ)w^+_J(z_K,z,y,λ)/∏_a∈ K∏_b∈K̅θ(z_a-z_b)θ(z_a-z_b+y) =δ_I,J. The summation is over subsets K⊂[n] of cardinality |K|=k. This is a rewriting of Corollary <ref> by using Lemma <ref>. We will also use the orthogonality relations in the following equivalent form. Let I,K⊂ [n], |I|=|K|=k. We have∑_J w^-_J(z_I,z,y,λ) w_J^+(z_K,z,y,λ)= ∏_a∈ I, b∈I̅θ(z_a-z_b)θ(z_a-z_b+y), I=K,0, otherwise. Letx_IK =w^-_I(z_K,z,y,λ)/∏_a∈ K, b∈K̅θ(z_a-z_b),y_KJ =w^+_J(z_K,z,y,λ)/∏_a∈ K, b∈K̅θ(z_a-z_b+y).Proposition <ref> claims that the matrix (x_IK)_I,K is the left inverse of the matrix (y_KJ)_K,J. This implies, however, that the matrix (x_IK)_I,K is also a right inverse of (y_KJ)_K,J, which is equivalent to the statement of the corollary.Weight functions have a triangularity property. Introduce a partial ordering on the subsets of [n] of fixed cardinality k: if I={i_1<…<i_k} and J={j_1<…<j_k}, then I≤ J if and only if i_1≤ j_1, …, i_k≤ j_k.Let ϵ[n]^2→{0,1} be such that ϵ(a,b)=1, if a>b, 0, if a<b. Then=0pt (i) w_I^-(z_J;z,y,λ) vanishes unless J≤ I and w_I^-(z_I;z,y,λ)=∏_a∈ Iθ(λ-(w(a,I)+1)y)∏_a∈ I,b∈I̅θ(z_a-z_b+ϵ(b,a)y).(ii) w_I^+(z_J,z,y,λ) vanishes unless I≤ J and w_I^+(z_I;z,y,λ)=∏_a∈ I, b∈I̅θ(z_a-z_b+ϵ(a,b)y)/∏_a∈ Iθ(λ-(w(a,I)+1)y).§.§ Eigenvectors of the Gelfand–Zetlin algebraThe normalized weight functions w^-_I evaluated at z_J provide the (triangular) transition matrix between the standard basis of (ℂ^2)^⊗ n and a basis of eigenvectors of the Gelfand–Zetlin algebra. The Gelfand–Zetlin algebra is generated by the determinant Δ(w), see (<ref>), and L_22(w). The determinant acts by multiplication by∏_i=1^nθ(w-z_i+y)/θ(w-z_i). We thus need to diagonalize L_22(w).Let 0≤ k≤ n, [k]={1,…,k}. Then ξ_[k]= ∏_i=1^kθ(λ+(n-k-i)y)v_[k]∈ V(z_1)⊗⋯⊗ V(z_n) is an eigenvector of L_22(w) with eigenvalue ∏_a=1^kθ(w-z_a)/θ(w-z_a-y).(See <cit.>.) Since L_21(w)v_1=0=L_12(w)v_2, the action of L_22(w) on v_1^⊗ k⊗ v_2^⊗ n-k is simply the product of the action on all factors, with the appropriate shift of λ. Since L_22(w) acts diagonally in the basis v_1, v_2 one gets the result by straightforward calculation. For I⊂[n], |I|=k, defineξ_I=ξ_I(z,y,λ)= ∑_|J|=kw^-_J(z_I,z,y,λ)/∏_a∈ I,b∈I̅θ(z_a-z_b+y) v_J.By Lemma <ref> this definition is consistent with the one for ξ_[k] above. [cf. <cit.>] The vectors ξ_I, I⊂[n], |I|=k form a basis of eigenvectors of the operators of the Gelfand–Zetlin algebra on V(z_1)⊗⋯⊗ V(z_n): Δ(w)ξ_I=∏_a=1^nθ(w-z_a+y)/θ(w-z_a)ξ_I,L_22(w)ξ_I= ∏_a∈ Iθ(w-z_a)/θ(w-z_a-y)ξ_I. By Corollary <ref>(i), we have that ξ_I(z,y,λ) =S_i(z,y,λ)ξ_s_i· I(s_iz,y,λ).Thus ξ_I(z,y,λ) is related to ξ_s_i· I(s_iz,y,λ) by a morphism of representations of the elliptic dynamical quantum group. If |I|=k then there is a permutation σ such that σ· I=[k] and thus ξ_I(z,y,λ)=ρ(σ) ξ_[k](σ· z,y,λ) for some morphism ρ(σ). Since ξ_[k](z,y,λ) is an eigenvector of L_22(w) with eigenvalue μ_[k](w;z,y), see Lemma <ref>, we deduce that ξ_I(z,y,λ) is an eigenvector with eigenvalue μ_I(w;z,y)=μ_[k](w;σ· z,y). §.§ An explicit formula for weight functionsLet I={i_1<i_2<⋯<i_k}⊂ [n] and recall the definition (<ref>) of w(i,I) and of ψ_I in Corollary <ref>. We setl^+_I(r,a;t,z,y,λ) =θ(t_r-z_a+y), if a<i_r,θ(λ+t_r-z_a-w(i_r,I)y),if a=i_r,θ(t_r-z_a),if a>i_r,andl^-_I(r,a;t,z,y,λ) =θ(t_r-z_a), if a<i_r,θ(λ-t_r+z_a-(w(i_r,I)+1)y),if a=i_r,θ(t_r-z_a+y),if a>i_r.Thenw^+_I(t;z,y,λ)=1/ψ_I(y,λ)Sym( ∏_r=1^k∏_a=1^nl^+_I(r,a;t,z,y,λ) /∏_1≤ i<j≤ kθ(t_i-t_j)θ(t_j-t_i+y)),andw^-_I(t;z,y,λ)=Sym( ∏_r=1^k∏_a=1^nl^-_I(r,a;t,z,y,λ) /∏_1≤ i<j≤ kθ(t_j-t_i)θ(t_i-t_j+y)). §.§ Dual bases and resonancesHere we prove Propositions <ref> and <ref>. They are corollaries of the following more precise statement:Let Λ=ℤ+τℤ and fix (z,y,λ)∈ℂ^n×ℂ×ℂ. Assume that z_a-z_b-sy∉Λ, for all a≠ b, s=0,…,k, λ-sy∉Λ for all s∈ S⊂ℤ for some finite S depending on k and n.=0pt (i) Let ω^±_k_a=ω_k_a^±(t;z_a,λ+y∑_b=a+1^n(1-2k_b)), see Example <ref>. Then the family ω^±_k_1*…*ω^±_k_n,k∈ℤ^k_≥0,∑_a=1^n k_a=k is a basis of Θ^±_k(z,y,λ). (ii) The subfamily indexed by (k_a)_a=1^n such that k_a∈{0,1} for all a is a basis of Θ̅^±_k(z,y,λ).Part (i) and a special case of (ii) are proved in <cit.>. The proof relies on the following construction of linear forms whose evaluations on the members of the family form a non-degenerate triangular matrix. For a symmetric function f(t_1,…,t_k) and w∈ℂ, define ev_wf to be the symmetric function of k-1 variablesev_wf(t_1,…,t_k-1)=f(t_1,…,t_k-1,w).It is easy to check that ev_w maps Θ^±_k(z,y,λ) to Θ^±_k-1(z,y,λ± y). For ℓ∈ℤ_≥0 and f a symmetric function of k variables setev_w,ℓ(f)=f if ℓ=0, ev_w-ℓ y∘⋯∘ev_w-y∘ev_w(f)if 1≤ℓ≤ k, 0, otherwise.Finally, for ℓ∈ℤ_≥0^n, ∑_a=1^nℓ_a=k we introduce linear forms ϵ_ℓ_1,…,ℓ_n∈Θ^±(z,y,λ)^*:ϵ_ℓ_1,…,ℓ_n= ev_z_n,ℓ_n∘⋯∘ev_z_1,ℓ_1.The following result is a special case of Proposition 30 from <cit.> (adapted to the conventions of this paper). It can be checked by induction using the fact that the evaluation points are such that at most one shuffle in the definition of the shuffle products contributes nontrivially.Let (z,y,λ)∈ℂ^n×ℂ×ℂ, k,ℓ∈ℤ_≥0^n with ∑_a=1^nℓ_a= ∑_a=1^n k_a. Then=0pt (i) Let f_a∈Θ^-_k_a(z_a,y,λ-y∑_b=a+1^n(2k_b-1)), a=1,…,n. Then ϵ_ℓ_1,…,ℓ_n(f_1*⋯*f_n)=0, unless ℓ_1+⋯+ℓ_p≤ k_1+⋯+k_p for all p=1,…,n, and ϵ_k_1,…,k_n (f_1*⋯*f_n) =∏_a=1^nev_z_a,k_a(f_a) ∏_a<b(∏_s=0^k_b-1θ(z_b-z_a-ys) ∏_s=0^k_a-1θ(z_a-z_b+y(1-s))).(ii) Let ω^-_k be the basis of Θ^-_k(z,y,λ), z∈ℂ, defined in Example <ref>. Then ev_z,kω^-_k=∏_s=1^kθ(λ-sy).Setting f_a=ω^-_k_a, a=1,…, n gives a proof of Proposition <ref> (i) in the case of Θ^-. The case of Θ^+ is reduced to this case by Proposition <ref>.We turn to the proof of Proposition <ref>(ii). In the notation we have introduced here, Θ̅^±_k(z,y,λ) is the intersection of the kernels of ev_z_a,2 for a=1,…,n.Let (z,y,λ)∈ℂ^n×ℂ×ℂ and 1≤ c≤ n. In the following proposition we describe the interaction of ev_z_c and ev_z_c-y with the shuffle product. By using the identifications of Remark <ref>, we view these maps as maps between the following spaces:ev_z_c Θ^±_k(z,y,λ)→Θ^±_k-1(z_1,…,z_c-y,…,z_n,y,λ), ev_z_c-y Θ^±_k(z,y,λ)→Θ^±_k-1(z_1,…,z_c+y,…,z_n,y,λ± 2y). In the notation of Proposition <ref>, let f∈Θ^-_k'(z',y,λ+y(n”-2k”)), g∈Θ^-_k”(z”,y,λ). We have =0pt(i) ev_z_c(f*g)= ev_z_c(f)*g ∏_b=n'+1^nθ(z_c-z_b+y), 1≤ c≤ n',(ii) ev_z_c-y(f*g)= f*ev_z_c-y(g) ∏_a=1^n'θ(z_c-z_a-y), n'<c≤ n.Similarly, let f∈Θ^+_k'(z',y,λ+y(n”-2k”)), g∈Θ^+_k”(z”,y,λ). We have =0pt(iii) ev_z_c-y(f*g)= ev_z_c-y(f)*g ∏_b=n'+1^nθ(z_c-z_b+y), 1≤ c≤ n',(iv) ev_z_c(f*g)= f*ev_z_c(g) ∏_a=1^n'θ(z_c-z_a-y), n'<c≤ n.(i) Due to the factor ∏_l=k'+1^kθ(t_l-z_c) in the definition of φ^-, see Proposition <ref>, the only terms in the sum over permutations contributing nontrivially to ev_z_c(f*g) are such that t_k is an argument of f. Thus ev_z_c(f*g)=ev_z_c(f)*g times a factor that is computed explicitly. The proof of (ii)–(iv) is similar. By iterating, we obtain:Let f∈Θ^-_k'(z',y,λ+y(n”-2k”)), g∈Θ^-_k”(z”,y,λ). We have ev_z_c,2(f*g)= ev_z_c,2(f)*g ∏_b=n'+1^nθ(z_c-z_b+y)θ(z_c-z_b),1≤ c≤ n',ev_z_c,2(f*g) = f*ev_z_c,2(g) ∏_a=1^n'θ(z_c-z_a-y)θ(z_c-z_a),n'<c≤ n, and similarly for Θ^+.In particular, if f and g satisfy the vanishing condition thenalso f*g does.We give the proof for Θ̅^-_k(z,y,λ). The proof for Θ̅^+_k(z,y,λ) is similar or can be deduced using the duality map of Proposition <ref>. It follows from Corollary <ref> that the indicated subfamily does belong to Θ̅^-_k(z,y,λ). It remains to show that it is a spanning set. By Proposition <ref>, (i), we know that any element of Θ̅^-_k(z,y,λ) can be written as linear combinations∑_k_1+⋯+k_n=kλ_k_1,…,k_nω_k_1*⋯*ω_k_n.On the other hand, the linear form ϵ_k_1,…,k_n vanishes on Θ̅^-_k(z,y,λ) if k_a≥2 for some a, since it involves the evaluation at z_a,z_a-y. By Lemma <ref>, the coefficients λ_k_1,…,k_n must thus vanish if at least one k_a is ≥2 which is what we had to prove. § EQUIVARIANT ELLIPTIC COHOMOLOGY OF GRASSMANNIANS Let E be an elliptic curve and G a compact group. Equivariant elliptic cohomology was postulated by Ginzburg, Kapranov and Vasserot in <cit.> as a functor E_G from pairs of finite G-CW complexes to superschemes satisfying a set of axioms, generalizing those satisfied by equivariant cohomology and equivariant K-theory. One of them being that for a point pt, E_G(pt) is a suitable moduli scheme of G-bundles on the dual elliptic curve. For example E_U(n)(pt)=E^(n)=E^n/S_n and for an abelian group A with group of characters X(A)=Hom(A,U(1)), E_A(pt)=Hom(X(A),E). By functoriality, the scheme E_G(M) comes with a structure mapp_G E_G(M)→ E_G(pt).For a complex elliptic curve, the case we consider here, a construction of equivariant elliptic cohomology was given by Grojnowski <cit.>. It has the property that for a connected Lie group G with maximal torus A and Weyl group W then W acts on E_A(M) and E_G(M)=E_A(M)/W. §.§ Tautological bundles and Chern classesLet E be a complex elliptic curve. The unitary group U(n) and its maximal torus A≅ U(1)^n act on the Grassmannian Gr(k,n) of k-dimensional subspaces of ℂ^n. The A-equivariant cohomo­logy of Gr(k,n) was described in <cit.>, and is analogous to the classical description of ordinary equivariant cohomology in terms of Chern classes of tautological bundles. The Grassmannian has two tautological equivariant vector bundles of rank k and n-k, respectively. They give rise to a characteristic map <cit.>χ E_A(Gr(k,n))→ E_U(k)(pt)× E_U(n-k)(pt)=E^(k)× E^(n-k).Here E^(k)=E^k/S_k denotes the symmetric power of the elliptic curve, which is the U(k)-equivariant cohomology of a point. Together with the structure map to E_A(pt) we have a description of the equivariant elliptic cohomology as the fiber product of E^(k)× E^(n-k) and E^n over E^(n), namely, we have the Cartesian square:[ E_A(Gr(k,n))⟶ E^(k)× E^(n-k);↓ ↓;E^n⟶ E^(n). ]The left vertical arrow is the structure map to E_A(pt); the maps E^n→ E^(n) and E^(k)× E^(n-k)→ E^(n) are the canonical projections. Thus E_A(Gr(k,n))=(E^(k)× E^(n-k))×_E^(n)E^n.The symmetric group S_n (the Weyl group of U(n)) acts on the diagram above (with trivial action on the right column) and the U(n)-equivariant cohomology is the quotient by this action:E_U(n)(Gr(k,n))=E_A(Gr(k,n))/S_n=E^(k)× E^(n-k).§.§ Moment graph descriptionAn alternative useful description of the equivariant elliptic cohomology is via the localization theorem, proved by Goresky, Kottwitz and MacPherson <cit.> for equivariant cohomology and generalized to elliptic cohomology by Knutson and Roşu <cit.>. For partial flag varieties such as Grassmannians it is described explicitly in <cit.>. The action of A on the Grassmannian Gr(k,n) has isolated fixed points labeled by subsets of [n]={1,…,n} with k elements. The fixed point x_I labeled by I⊂[n] is the k-plane spanned by the coordinate axes indexed by I. The inclusion of the fixed points x_I induces a map ι_I E_A(pt)→ E_A(Gr(k,n)) and it turns out that E_A(Gr(k,n)) is the union of the ι_IE_A(pt)≃ E^n where I runs over the subsets of [n] with k elements.Let Γ be the graph with vertex set Γ_0 the set of subsets I⊂ [n] with |I|=k elements and an edge connecting I with I' for each pair of vertices such that |I∩ I'|=k-1. In this case I=J∪{a} and I'=J∪{b} with |J|=k-1 and we set Δ_I,I'={z∈ E^n, z_a=z_b}. We then have inclusion maps Δ_I,I'→ι_IE_A(pt)=E^n, Δ_I,I'→ι_I'E_A(pt)=E^n.We have the coequalizer diagram ⊔_|I∩ I'|=k-1Δ_I,I'⇉⊔_I⊂ [n],|I|=kE^n→ E_A(Gr(k,n))). In other words, E_A(Gr(k,n)) is the union of copies of E^n labeled by subsets I⊂ [n] of size k, glued along the diagonals Δ_I,I'. The structure map E_A(Gr(k,n))→ E^n is the identity on each copy. The isomorphism between the two descriptions of E_A(Gr(k,n)) is induced by the map⊔_I⊂ [n],|I|=kE^n→(E^(k)× E^(n-k))×_E^(n)E^n,whose restriction to the copy E^n labeled by I isz↦ (z,z_I,z_I̅), z_I=(z_i)_i∈ I,z_I̅=(z_j)_j∈I̅.It is easy to check directly Proposition <ref> using the fiber product as a definition of the equivariant elliptic cohomology.§.§ Cotangent bundles and dynamical parameterThe action of U(n) on the Grassmannian induces an action on its cotangent bundle X_k,n=T^*Gr(k,n). Additionally we have an action of U(1) on the cotangent bundle by scalar multiplication on the fibers, so we get an action ofG=U(n)× U(1)and its Cartan torusT=A× U(1)≅ U(1)^n+1.Since the cotangent bundle is equivariantly homotopy equivalent to its zero section, the equivariant elliptic cohomology is simplyE_T(X_k,n)=E_A(Gr(k,n))× E,a scheme over E_T(pt)=E^n× E, andE_G(X_k,n)=E_U(n)(Gr(k,n))× E,a scheme over E_G(pt)=E^(n)× E.We will consider, as in <cit.>, an extended version of elliptic cohomology to accommodate for dynamical variables in quantum group theory, namelyÊ_T(X_k,n):=E_T(X_k,n)× (Pic(X_k,n)⊗_ℤE)≅ E_T(X_k,n)× E,a scheme over Ê_T(pt)=E^n× E× E (the Picard group of the Grassmannian is infinite cyclic generated by the top exterior power of the tautological bundle). Similarly, we setÊ_G(X_k,n)=E_G(X_k,n)× E,which is a scheme over Ê_G(pt)=E^(n)× E× E.The fixed points x_K for the A-action on the Grassmannian are also isolated fixed points in the cotangent bundle of the Grassmannian for the T-action and we have maps ι_K=Ê_T(i_K)Ê_T(pt) →Ê_T(X_k,n) induced by the inclusion i_Kpt↦ x_K. Then Ê_T(X_k,n) consists of the components ι_KÊ_T(pt), where K runs over the subsets of [n] with k elements. By Section <ref>, we have a description of Ê_T(X_k,n) as a fiber product:Ê_T(X_k,n)≅(E^(k)× E^(n-k))×_E^(n)E^n× E× E.In particular we have the characteristic embeddingc Ê_T(X_k,n)→ E^(k)× E^(n-k)× E^n× E× Eof the extended T-equivariant elliptic cohomology scheme into a non-singular projective variety.§ ADMISSIBLE LINE BUNDLES ON Ê_T(X_K,N) §.§ Line bundles on E^pLine bundles on complex tori are classified by the Appel–Humbert theorem, see <cit.>. In the special case of powers of generic elliptic curves this reduces to the following explicit description: let E=ℂ/Λ with Λ=ℤ+τℤ so that E^p=ℂ^p/Λ^p. For each pair (N,v) consisting of a symmetric integral p× p matrix N and v∈(ℂ/ℤ)^p let ℒ(N,v) be the line bundlewith actionλ·(x,u)=(x+λ,e_λ(x)u),λ∈Λ^p,x∈ℂ^p, u∈ℂ,and cocyclee_n+mτ(x)=(-1)^n^tNn(- e^ iπτ)^m^tNm e^2π i m^t(Nx+v),n,m∈ℤ^p.=0pt (i) ℒ(N,v) is isomorphic to ℒ(N',v') if and only if N=N' and v≡ v'Λ^p. (ii) For generic E, every holomorphic line bundle on E^p is isomorphic to ℒ(N,v) for some (N,v). (iii) ℒ(N_1,v_1)⊗ℒ(N_2,v_2)≅ℒ(N_1+N_2,v_1+v_2). (iv) Let σ∈ S_p act by permutations on E^p and ℂ^p. Denote also by σ the corresponding p× p permutation matrix. Then σ^*ℒ(N,v)=ℒ(σ^t Nσ,σ^t v).To an integral symmetric p× p matrix N and a vector v∈ℂ^p we associate the integral quadratic form N(x)=x^tNx and the linear form v(x)=x^tv on the universal covering ℂ^p of E^p and we call them the quadratic form and the linear form of the line bundle ℒ(N,v). The linear form is defined up to addition of an integral linear form. Exceptions to (ii) are elliptic curves with complex multiplication, in which case there are additional line bundles that are not isomorphic to those of the form ℒ(N,v). The map E^p→Pic(E^p) sending v to ℒ(0,v) is an isomorphism onto the subgroup Pic^0(E^p) of classes of line bundles of degree 0. If E is a generic elliptic curve, the Néron–Severi group NS(E^p)=Pic(E^p)/Pic^0(E^p) is a free abelian group of rank n(n+1)/2 identified with the group of integral symmetric matrices via N↦ℒ(N,0).Sections of ℒ(N,v) are the same as functions f on ℂ^p such that f(x+λ)=e_λ(x)^-1f(x) for all λ∈Λ^p, x∈ℂ^p. Explicitly, a function on ℂ^p defines a section of ℒ(N,v) if and only if f(x_1,…,x_j+1,…,x_p) = (-1)^N_jjf(x),f(x_1,…,x_j+τ,…,x_p) = (-1)^N_jj e^-2π i(∑_k N_jkx_k+v_j)-π iτ N_jj f(x),for all x∈ℂ^p, j=1,…,p. The factors of -1 in the cocycle can be removed by going to an equivalent cocycle. With the present convention the line bundles ℒ(N,0) correspond to divisors whose irreducible components are subgroups. Let θ(z) be the odd Jacobi theta function in one variable, see (<ref>). Then, for any r∈ℤ^p and z∈ℂ, θ(r^tx+z)=θ(r_1x_1+⋯+r_px_p+z) is a holomorphic section of ℒ(N,v) with quadratic form N(x)=(∑_i=1^pr_ix_i)^2, and linear form v(x)=z∑_i=1^p r_ix_i.If z=0 this section vanishes precisely on the subtorus Ker(ϕ_r), the kernel of the group homomorphism ϕ_r E^p→ E, x↦∑_ir_ix_i. Since an integral quadratic form is an integral linear combination of squares of integral linear forms, ℒ(N,0) has a meromorphic section which is a ratio of products of theta functions θ(r^tx) with r∈ℤ^p. §.§ Admissible line bundlesThe elliptic dynamical quantum group acts on sections of admissible line bundles, which are, up to a twist by a fixed line bundle, those coming from the base scheme Ê_T(pt). Let p_T be the structure mapp_T Ê_T(X_k,n)→Ê_T(pt),and χ̂=χ×id×id the characteristic mapχ̂ Ê_T(X_k,n)=E_A(Gr(k,n))× E× E→ E^(k)× E^(n-k)× E× E.Let t_1,…,t_k,s_1,…,s_n-k,y,λ be coordinates on the universal covering of E^k× E^n-k× E× E and N the quadratic formN_k,n(t,s,y,λ)=2∑_i=1^k t_i(λ+(n-k)y)+∑_i=1^k∑_j=1^n-k(t_i-s_j)^2.Clearly N_k,n is symmetric under permutation of the coordinates t_i and of the coordinates s_j and thus ℒ(N_k,n,0) can be considered as a bundle on E^(k)× E^(n-k)× E× E. The twisting line bundle on X_k,n is 𝒯_k,n=χ̂^*ℒ(N_k,n,0) An admissible line bundle on Ê_T(X_k,n) is a line bundle of the form p_T^*ℒ⊗𝒯_k,n, for some line bundle ℒ on Ê_T(pt). §.§ Holomorphic and meromorphic sectionsWe will consider meromorphic sections of line bundles on elliptic cohomology schemes. Since these schemes are singular, we need to be careful about the definition. Recall that Ê_T(X_k,n) has components Y_K=ι_KÊ_T(pt), corresponding to the inclusion of the fixed points x_K, labeled by subsets K⊂[n] of cardinality k. We say that a meromorphic section on a complex manifold restricts to a meromorphic section on a submanifold if it is defined at its generic point, i.e., if the divisor of poles does not contain a component of the submanifold.Let ℒ be a line bundle on Ê_T(X_k,n). A meromorphic section of ℒ is a collection of meromorphic sections s_I of ℒ|_Y_I, labeled by I⊂[n] with |I|=k and restricting to meromorphic sections on all intersections Y_I_1∩⋯∩ Y_I_s and such that s_I|_Y_I∩ Y_J=s_J|_Y_I∩ Y_J,for all I, J. A holomorphic section is a meromorphic section whose restriction to each Y_I is holomorphic. We denote by Γ(Ê_T(X_k,n),ℒ) the space of holomorphic sections of ℒ and by Γ_mer(Ê_T(X_k,n),ℒ) the space of meromorphic sections of ℒ.§.§ Weight functions and admissible line bundlesWith the description of line bundles of Section <ref>, the weight functions w^+_I(t_1,…,t_k,z_1,…,z_n, y,λ) can be viewed as sections of certain line bundles on E^(k)× E^(n-k)× E^n× E× E, namely as (s-independent) functions of the coordinates (t_1,…,t_k,s_1,…,s_n-k,z,y,λ) on the universal covering space, with proper multipliers under lattice translations. Their pull-back by the characteristic embeddingcÊ_T(X_k,n)↪ E^(k)× E^(n-k)× E^n× E× E,see (<ref>), is a section of the pull-back bundle and its restriction to ι_JÊ_T(pt) is the evaluation of w^+_I at t=z_I. Let I⊂[n], |I|=k. Then the restriction c ^*w^+_I of w^+_I to Ê_T(X_k,n) is a meromorphic section of the admissible bundle p_T^*ℒ_I⊗𝒯_k,n for some line bundle ℒ_I on Ê_T(pt). Moreover ψ_Ic ^*w^+_I is holomorphic. We need first to check that all terms in the sum over S_k defining the symmetrization map Sym in Section <ref> have the same transformation properties under shifts of the variables by the lattice so that they define sections of the same line bundle on E^k× E^n× E^2. The symmetrization map then produces a section symmetric under permutations of t_i, which is the same as a section of a line bundle onE^(k)× E^n× E^2. The transformation properties are encoded in the quadratic form: the argument of Sym is a section of the line bundle ℒ(M_I,0) with M_I(t,z,y,λ) =2∑_r=1^k t_r(λ+(n-k)y)+∑_r=1^k∑_a=1^n (t_r-z_a)^2-2∑_1≤ r<s≤ k(t_r-t_s)^2 + M̃_I(z,y,λ),where M̃_I is independent of t_1,…,t_k. Since M_I is symmetric under permutations of the variables t_i it defines an S_n-equivariant line bundle. All terms in the sum over permutations are sections of this line bundle and their sum is a symmetric section, i.e., the pull-back of a section on the quotient E^(k)× E^n× E^2, which we understand as a section on E^(k)× E^(n-k)× E^n× E^2, constant along E^(n-k). The restriction to the component ι_KÊ_T(pt) of Ê_T(X_k,n) is w^+_I(z_K,z,y,λ), the result of substituting the variables t_i by z_K=(z_i)_i∈ K. It is a section of the line bundle with quadratic form M_I(z_K,z,y,λ) = 2∑_i∈ Kz_i(λ+(n-k)y) +∑_i∈ K∑_a=1^n(z_i-z_a)^2 -∑_i,j∈ K(z_i-z_j)^2+M̃_I(z,y,λ)=2∑_i∈ Kz_i(λ+(n-k)y) +∑_i∈ K,j∈K̅(z_i-z_j)^2 +M̃_I(z,y,λ) = N_k,n(z_K,z_K̅,y,λ)+M̃_I(z,y,λ), cf. (<ref>). Thus the symmetrization is a section of the tensor product of the twisting bundle and the bundle with quadratic form M̃_I which is independent of K and thus comes from Ê_T(pt)=E^n× E^2. The section ψ_Ic ^*w^+_I is holomorphic because of Lemma <ref>. Thus c ^*w_I^+ is a meromorphic section of an admissible line bundle p_T^*ℒ_I⊗𝒯_k,n with poles on a finite set of hypertori with equation λ-jy=0, j∈ℤ, the divisors of zeros of ψ_I. The bundle ℒ_I can be calculated: ℒ_I=ℒ(N_I,0) withN_I=-2∑_a∈I̅n(a,I)z_ay-2∑_a∈ Iz_a(λ+n(a,I̅)y)+(k(n-k)-∑_a∈ In(a,I̅))y^2-∑_a∈ I(λ- (n(a,I)+1)y+n(a,I̅)y)^2,see (<ref>) for the definition of n(a,I). Let D_I be the divisor of zeros of the section ψ_I(y,λ) on Ê_T(pt), I⊂ [n], see Corollary <ref>. Then the normalized weight function can be understood as a holomorphic section of an admissible bundle:c ^*w^+_I∈Γ(X_k,n, p_T^*ℒ_I(D_I)⊗𝒯_k,n).Here the notation ℒ(D) means as usual the invertible sheaf of meromorphic sections of a sheaf L whose poles are bounded by the divisor D.Let n=1. Then w^+_∅(t,z,y,λ)=1 and c ^*w_∅=1 is a section of the trivial bundle (ℒ_∅ and 𝒯_0,1 are both trivial). For k=1,w^+_{1}(t,z,y,λ)=θ(λ+t-z)/θ(λ)θ(λ-y),and c ^*w^+_{1} is obtained by substituting t=z:c ^*w^+_{1}(z,y,λ)=1/θ(λ-y).This is a meromorphic section of the line bundle ℒ(-(λ-y)^2,0) with a simple pole at λ=y on Ê_T(X_1,1)=Ê_T(pt)≅ E^3. The quadratic form is composed from the quadratic forms 2zλ of 𝒯_1,1 and -2zλ-(λ-y)^2 of ℒ_{1}. §.§ Elliptic cohomology classes and stable envelopeHere we introduce an elliptic version of the Maulik–Okounkov stable envelope. It is constructed in terms of weight functions. In Appendix <ref> we give an axiomatic definition in the spirit of <cit.>. It would be interesting to understand the relation of our definition with the one sketched in <cit.>. Let ℒ∈Pic(Ê_T(pt)). A T-equivariant elliptic cohomology class on X_k,n of degree ℒ is a holomorphic section of the admissible bundle p_T^*ℒ⊗𝒯_k,n on Ê_T(X_k,n). We denote by H_T^ell(X_k,n)_ℒ the vector space of T-equivariant elliptic cohomology classes of degree ℒ on X_k,n. We denote by H_T^ell(X_n)_ℒ the 𝔥-module ⊕_k=0^n H_T^ell(X_k,n)_ℒ, with k-th direct summand of 𝔥-weight -n+2k. The stable envelope is the mapStab (ℂ^2)^⊗ n→⊕_k=0^n⊕_I⊂ [n], |I|=k H_T^ell(X_k,n)_ℒ_I(D_I),sending v_I to the cohomology class c ^*w_I^+. The basis vector v_I should be viewed as the generator of the space of elliptic cohomology classes of the fixed point x_I, see Section <ref> below. The class c ^*w_I^+ has analogs in equivariant cohomology and equivariant K-theory of X_k,n, see <cit.>. The analog of c ^*w_I^+ in equivariant cohomology is the equivariant Chern–Schwartz–MacPherson class of the open Schubert variety Ω_I, see <cit.>. Hence c ^*w_I^+ may be considered as an elliptic equivariant version of the Chern–Schwartz–MacPherson class.§.§ Sheaf of elliptic cohomology classes and theta functionsHere we realize elliptic cohomology classes as sections of coherent sheaves on Ê_T(pt) and relate their sections to the theta functions with vanishing condition of Section <ref>.Let k=0,…,n, μ=-n+2k and 𝒯_k,n be the twisting line bundle of Definition <ref>. The sheaf of elliptic cohomology classes of weight μ is the sheaf ℋ_T^ell(X_k,n)=p_T*𝒯_k,n on Ê_T(pt). Here p_T*=(p_T)_* denotes the direct image by the structure map p_TÊ_T(X_k,n)→Ê_T(pt).By the projection formula, ℒ⊗ p_T*𝒯_k,n≅ p_T*(p_T^*ℒ⊗𝒯_k,n) for any line bundle ℒ∈Pic(Ê_T(pt)). Thus a section of ℋ_T^ell(X_k,n)⊗ℒ on an open set U is a section of the admissible line bundle p_T^*ℒ⊗𝒯_k,n on p_T^-1(U). In particular,H^ell_T(X_k,n)_ℒ=Γ(Ê_T(pt),ℋ^ell_T(X_k,n)⊗ℒ). The space Θ^+_k(z,y,λ) of theta functions introduced in Section <ref> is the fiber of a vector bundle Θ^+_k,n on Ê_T(pt). In the language of Section <ref>, Θ^+_k,n=p_*ℒ(N^Θ_k,n,0) is the direct image by the projection p E^(k)×Ê_T(pt)→Ê_T(pt) onto the second factor of the line bundle associated with the quadratic formN^Θ_k,n =2∑_i=1^kt_i(λ+(n-k)y)+∑_i=1^k∑_a=1^n(t_i-z_a)^2+k(k-1)y^2.Here, as usual, the t_i are coordinates on the universal covering of E^k and z_a, y, λ are coordinates on the universal covering of Ê_T(pt). In fact only the terms involving t_i in N^Θ_k,n are determined by the transformation properties of the fibers Θ^+_k(z,y,λ). We choose the remaining terms to simplify the formulation of Theorem <ref> below.The space of theta functions Θ̅^+_k(z,y,λ) satisfying the vanishing condition of Section <ref> is the generic fiber of a coherent subsheaf Θ̅^+_k,n of Θ^+_k,n on Ê_T(pt) (it is the intersection of kernels of morphisms ev_z_a,2 of coherent sheaves). The sheaves Θ̅^+_k,n and ℋ_T^ell(X_k,n) are closely related: there is a morphismφ Θ̅^+_k,n→ℋ_T^ell(X_k,n),defined as follows. A section of Θ̅^+_k,n on an open set U is given by a function f(t;z,y,λ) on ℂ^k× U, which, as a function of t belongs to Θ^+_k(z,y,λ) and obeys the vanishing conditionsf(z_a,z_a-y,t_3,…,t_k;z,y,λ)=0,a=1,…,n.The morphism φ sends f to (φ_If)_I⊂ [n],|I|=k where φ_If is the restriction of φ f to ι_I Ê_T(pt)≅Ê_T(pt):φ_If(z,y,λ)= f(t;z,y,λ)/∏_i≠ jθ(t_i-t_j+y)|_t=z_I,cf. Section <ref>.Let D⊂Ê_T(pt) be the union of the hypertori z_a=z_b+y, 1≤ a≠ b≤ n and λ=jy, -n≤ j≤ n. The map φ f↦ (φ_If)_I⊂ [n], |I|=k given by formula (<ref>) is a well-defined injective morphism of 𝒪_Ê_T(pt)-modules φ Θ̅^+_k,n↪ℋ^ell_T(X_k,n), which is an isomorphism on Ê_T(pt)∖ D.We first prove that the morphism is well-defined. The function φ_If of z, y, λ defines a section of the line bundle ℒ(Q,0) with quadratic form Q=(N^Θ_k,n-∑_i≠ j(t_i-t_j+y)^2)|_t=z_I. An explicit calculation shows that Q=N_k,n|_t=z_I,s=z_I̅. It follows that φ f is a meromorphic section of p_T*𝒯_k,n. By Lemma <ref> (which applies to any symmetric theta function obeying the vanishing condition), φ f is actually holomorphic.To show that the morphism is injective, we use the fact that the weight functions ω_I^+ form a basis of Θ̅^+_k(z,y,λ) at the generic point of Ê_T(pt), see Proposition <ref>. Thus every local section of Θ̅^+_k,n can be written as linear combination of normalized weight functions with meromorphic coefficients. If this section is in the kernel of our morphism then its restriction to each component vanishes. By the triangularity property of weight functions of Lemma <ref> all coefficients must vanish and the kernel is trivial. We now construct the inverse map on the complement of D. A section s of ℋ^ell_T(X_k,n) on an open set U is a collection of sections s_I of 𝒯_k,n on the various components of p_T^-1(U) and agreeing on intersections. Then f=φ^-1s is f(t;z,y,λ)= ∑_I,Kψ_K(y,λ)^-1w_K^-(z_I,z,y,λ)s_I(z,y,λ)/∏_a∈ I, b∈I̅θ(z_a-z_b)θ(z_a-z_b+y)ω_K^+(t,z,y,λ). It is easy to check that this is a meromorphic section of Θ̅^+_k,n on U with poles at z_b=z_a+y, 1≤ a≠ b≤ n and at the zeros of ψ_K. It is regular at the apparent poles at z_a=z_b since the sections s_I agree on intersections of the components. Let us compute φ f: f(t;z,y,λ)/∏_i≠ jθ(t_i-t_j+y) =∑_I,Kw_K^-(z_I,z,y,λ)s_I(z,y,λ)/∏_a∈ I, b∈I̅θ(z_a-z_b)θ(z_a-z_b+y) w_K^+(t,z,y,λ). The orthogonality relations, see Corollary <ref>, imply φ_Jf =∑_I,Kw_K^-(z_I,z,y,λ) s_I(z,y,λ)/∏_a∈ I,b∈I̅θ(z_a-z_b) θ(z_a-z_b+y)w_K^+(z_J,z,y,λ) =s_J(z,y,λ).* §.§ Symmetric group and G-equivariant cohomology classes The symmetric group S_n on n letters acts on ℂ^n by permutation of coordinates. This action induces an action of S_n on the Grassmannians, their cotangent bundles X_k,n and on T so that the action map T× X_k,n→ X_k,n is S_n-equivariant. The induced action on the cohomology schemes Ê_T(X_k,n) can be easily described: on Ê_T(pt)=E^n× E^2, S_n acts by permutations of the first n factors and σ∈ S_n sends the component ι_KÊ_T(pt) of Ê_T(X_k,n) to ι_σ(K)Ê_T(pt) so that the diagram[ Ê_T(pt)σ⟶Ê_T(pt);ι_K↘↙ι_σ(K); Ê_T(X_k,n) ]commutes for any K⊂[n] with |K|=k elements. The structure map Ê_T(X_k,n)→Ê_T(pt) is S_n-equivariant and the quotient by the action of S_n is the G-equivariant elliptic cohomology scheme. The twisting bundle is S_n-equivariant, i.e., the S_n-action lifts to an S_n action on the bundle. This follows since the twisting bundle is the pull-back by an S_n-equivariant map of a bundle on E^(k)× E^(n-k) on which the action of the symmetric group is trivial. In particular for each σ∈ S_n and admissible line bundle ℳ, we have an admissible line bundle σ^*ℳ and a mapσ^* Γ_mer(Ê_T(X_k,n),ℳ)→Γ_mer(Ê_T(X_k,n),σ^*ℳ),and also a mapσ^* ℋ^ell_T(X_k,n)→σ^*ℋ^ell_T(X_k,n).Let πÊ_T(pt)→Ê_G(pt)=Ê_T(pt)/S_n. Then we obtain an action of the symmetric group on π_*ℋ^ell_T(X_k,n).Let G=U(n)× U(1). The sheaf of G-equivariant elliptic cohomology classes is ℋ^ell_G(X_k,n)=π_*ℋ^ell_T(X_k,n)^S_n,a coherent sheaf on Ê_G(pt)=E^(n)× E× E. Let ℒ∈Pic(Ê_G(pt)). The space of G-equivariant elliptic cohomology classes of degree ℒ on X_k,n is H^ell_G(X_k,n)_ℒ=Γ(Ê_G(pt),ℋ_G^ell(X_k,n)⊗ℒ). We set H^ell_G(X_n)_ℒ to be the 𝔥-module⊕_k=0^n H^ell_G(X_k,n)_ℒ, with the summandlabeled by k of 𝔥-weight -n+2k.§.§ Admissible difference operatorsRecall that Ê_T(X_k,n) has a factor E× E corresponding to the U(1)-action on the cotangent spaces and the dynamical parameter. For j∈ℤ, let τ_j=τ_1^j be the automorphism of E× E such thatτ_j(y,λ)=(y,λ+jy).Denote also by τ_j the automorphism id×τ_j of Ê_T(X_k,n)=E_A(Gr(k,n))× E× E. If ℒ is a line bundle on Ê_T(X_k,n) then τ_j lifts to a (tautological) bundle map ℒ→τ_j^*ℒ, also denoted by τ_j^*. It maps meromorphic sections to meromorphic sections and is thus a well-defined operatorτ^*_j Γ_mer(Ê_T(X_k,n),ℒ)→Γ_mer(Ê_T(X_k,n),τ^*_jℒ). Let k=0,…,n and ℒ be a line bundle on Ê_T(pt), μ∈ 2ℤ, ν∈ℤ. An admissible difference operator on meromorphic sections of an admissible line bundle ℳ_1=p_T^*ℒ_1⊗𝒯_k,n on Ê_T(X_k,n) of degree (ℒ,μ,ν) is a linear map φΓ_mer(Ê_T(X_k,n), ℳ_1)→Γ_mer(Ê_T(X_k+μ,n),ℳ_2)) such that=0pt (i) ℳ_2 is the admissible bundle p_T^*ℒ_2⊗𝒯_k+μ/2,n with ℒ_2=ℒ⊗τ_ν^*ℒ_1. (ii) For each section s of ℳ_1 and fixed pointx_K∈ X_k+μ,n, ι_K^*φ(s)=∑_K'φ_K,K'τ_ν^*ι_K'^*s for some sections φ_K,K'∈Γ_mer(Ê_T(pt),ι^*_Kℳ_2⊗ι^*_K'τ^*_νℳ_1^-1).By inserting the definition, we see that the line bundle of which φ_K,K' is a section isι_K^*ℳ_2⊗τ_ν^*ι_K'^*ℳ_1^-1= ℒ⊗ι^*_K𝒯_k+μ/2,n⊗τ_ν^*ι_K'^*𝒯_k,n^-1.This line bundle is independent of the admissible line bundle the operator acts on. It thus makes sense to let the same admissible difference operator act on sections of different admissible line bundles. We set𝒜_k,n(ℒ,μ,ν),k=0,…,n, 0≤ k+μ≤ n,to be the space of admissible difference operators of degree (ℒ,μ,ν).It is convenient to extend the above definitions to the case of varying k. We denote by X_n=⊔_k=0^n X_k,n the disjoint union of cotangent bundles to Grassmannians of subspaces of all dimensions in ℂ^n. The extended elliptic cohomology scheme is thenÊ_T(X_n)=⊔_k=0^n Ê_T(X_k,n).It comes with a map p_TÊ_T(X_n)→⊔_k=0^nÊ_T(pt). An admissible line bundle on Ê_T(X_n) is a line bundle whose restriction to each Ê_T(X_k,n) is admissible. Let ℒ=(ℒ^0,…,ℒ^n) be a line bundle on ⊔_k=0^nÊ_T(pt), μ,ν∈ℤ. An admissible difference operator of degree (ℒ,μ,ν) acting on sections of an admissible line bundle ℳ_1 is a linear map Γ_mer(Ê_T(X_n),ℳ_1)→Γ_mer(Ê_T(X_n),ℳ_2) restricting for each k=0,…,n such that k+μ∈{0,…,n} to an admissible difference operator Γ_mer(Ê_T(X_k,n),ℳ_1|_X_k,n)→Γ_mer(Ê_T(X_k+μ,n),ℳ_2|_X_k+μ,n), of degree (ℒ^k,μ,ν). We denote by 𝒜_n(ℒ,μ,ν)=⊕_0≤ k,k+μ≤ n A_k,n(ℒ^k,μ,ν) the space of admissible difference operators of degree (ℒ,μ,ν). We will not need to consider operators on components for k such that k+μ∉{0,…,n}. However to have a correct definition we may set Ê_T(X_k,n) to be the empty set if k∉{0,…,n} and declare the space of sections of any line bundle on the empty set to be the zero vector space.§.§ Left and right moment maps Examples of admissible difference operators are multiplication operators by sections of pull-backs of line bundles on Ê_T(pt). A subclass of these operators appear as coefficients in the defining relations of the quantum group: they are the entries of the R-matrix and are functions of the dynamical and deformation parameter, and appear in the relations in two different guises: with and without “dynamical shift”. We borrow the terminology of <cit.>, where these two appearances are called the left and right moment maps.Let ℒ be a line bundle on Ê_T(pt). We define two line bundles μ_ℓℒ, μ_rℒ on ⊔_k=0^nÊ_T(pt): =0pt* μ_rℒ is the line bundle (ℒ,…,ℒ);* μ_ℓℒ is the line bundle (τ_n^*ℒ,τ_n-2^*ℒ,…,τ_-n^*ℒ). The left moment map is the map μ_ℓ Γ_mer(Ê_T(pt),ℒ)→𝒜_n(μ_ℓℒ,0,0), sending a section s to the operator whose restriction to 𝒜_k,n is the multiplication by τ_n-2k^*p_T^*s.The right moment map is the map μ_rΓ_mer(Ê_T(pt),ℒ) →𝒜_n(μ_rℒ,0,0) sending s to the operator whose component in 𝒜_k,n is the multiplication by p_T^*s. §.§ Sections of admissible bundles as a module over multiplication operators Let ℒ∈Pic(Ê_T(pt)) and set 𝒜^0_n(ℒ)=(𝒜_n(ℒ,0,0)). Then the family 𝒜_n^0=(𝒜^0_n(ℒ))_ℒ∈Pic(Ê_T(pt)) is a commutative subalgebra graded by the Picard group of the base. It acts on meromorphic sections of admissible bundles by mapping Γ_mer(X_k,n,ℳ) to Γ_mer(X_k,n,ℳ⊗ p_T^*ℒ) for any admissible line bundle ℳ. Then the weight functions form a system of free generators of the module of sections of admissible line bundles over 𝒜^0_n in the following sense.Let ℒ∈Pic(Ê_T(pt)). Every section ω∈Γ_mer(Ê_T(X_k,n),p_T^*ℒ⊗𝒯_k,n) can be uniquely written as ω=∑_I⊂ [n], |I|=k a_I Stab(v_I), for some a_I∈𝒜^0_n(ℒ⊗ℒ_I^-1), where L_I is the line bundle of Proposition <ref>. Denote by Y_I=ι_IÊ_T(pt) the component labeled by I. Suppose that ω is a meromorphic section vanishing on Y_J for all J>I and such that ω|_Y_I≠ 0. By Lemma <ref>, we can subtract from ω a multiple of c ^*w_I^+ to get a section that vanishes on Y_J, J≥ I. By induction we may subtract from ω a suitable linear combination of weight functions to get 0.§.§ S_n-equivariant admissible difference operators An admissible difference operator is called S_n-equivariant if it commutes with the action of the symmetric group on sections.Let ℒ be an S_n-equivariant line bundle on Ê_T(pt). An admissible difference operator φ of degree (ℒ,μ,ν) is S_n-equivariant if and only if its matrix elements φ_K,K' obey σ^*φ_σ(K),σ(K')=φ_K,K'. §.§ Graded algebras, graded modulesLet Q be a group. Recall that an Q-graded algebra A over ℂ is a collection (A_γ)_γ∈ Q of complex vector spaces with associative linear multiplication maps A_γ⊗ A_γ'→𝒜_γγ', a⊗ b↦ a· b. Let P be a set with a left action of Q. A P-graded (left) module over A is a collection (M_p)_p∈ P of complex vector spaces indexed by P together with linear action maps A_γ⊗ M_p→ A_γ· p, a⊗ m↦ a· m, obeying (a· b)· m=a·(b· m). A unital Q-graded algebra is an Q-graded algebra with an identity element 1∈ A_e in the component indexed by the identity element e of Q. We require 1 to act as the identity on P-graded modules. §.§ The grading of admissible difference operatorsLet Q=Pic(Ê_T(pt))⋉ 2ℤ×ℤ be the product of the Picard group of Ê_T(pt)≅ E^n+2 by 2ℤ×ℤ with group law(ℒ,μ,ν)(ℒ',μ',ν') =(ℒ⊗τ^*_νℒ', μ+μ',ν+ν'). The collection (𝒜_n(ℒ,μ,ν))_(ℒ,μ,ν)∈ Q with the composition of operators is a unital Q-graded algebra. An admissible difference operator of degree (ℒ',μ',ν') sends a section of an admissible line bundle ℳ_1=p_T^*ℒ_1⊗𝒯_k,n to a section of ℳ_2=p_T^*ℒ_2⊗𝒯_k+μ'/2,n with ℒ_2=ℒ'⊗τ^*_ν'ℒ_1. An operator of degree (ℒ,μ,ν) sends this section to a section of p_T^*ℒ_3⊗ T_k+μ/2+μ'/2,n with ℒ_3=ℒ⊗τ^*_νℒ_2 =ℒ⊗τ_ν^*ℒ'⊗τ^*_ν+ν'ℒ_1.It is clear that the μ-components of the degree add, so the composition has degree (ℒ⊗τ_ν^*ℒ',μ+μ', ν+ν'), as required. The identity element is the multiplication by constant function 1, a section of the trivial bundle 𝒪.There is a slight abuse of notation, since 𝒜(ℒ,μ,ν) is defined for a line bundle ℒ and not for its equivalence class. The point is that 𝒜(ℒ,μ,ν) for equivalent bundles ℒ are canonically isomorphic: if φ is an admissible difference operator of degree (ℒ,μ,ν) and ψℒ→ℒ' is an isomorphism then φ'=ψ∘φ∘ψ^-1 is an difference operator of degree (ℒ',μ,ν). This establishes the isomorphism ψ̅ 𝒜(ℒ,μ,ν)→𝒜(ℒ',μ,ν), which we claim is independent of ψ. Indeed any two choices of ψ differ by the composition with an automorphism of ℒ. Since Aut(ℒ)=ℂ^×, ψ and ψ' differ by multiplication by a nonzero scalar which does not affect ψ̅.Let P be the set of pairs (ℒ,μ) with ℒ∈Pic(Ê_T(X_n)) and μ∈ℤ. Then Q acts on P via(ℒ,μ,ν)·(ℒ',μ') = (ℒ⊗τ_ν^*ℒ',μ+μ').Let ℳ=p_T^*ℒ_1⊗𝒯_k,n be an admissible bundle on X_k,n.Admissible difference operators map sections of admissible line bundles to sections of admissible line bundles. This is formalized as follows.The collection of vector spaces Γ_mer(Ê_T(X_k,n),p_T^*ℒ⊗ T_k,n), labeled by (ℒ,μ), with μ=-n+2k is a P-graded module over the Q-graded unital algebra 𝒜_n of admissible difference operators. This is an immediate consequence of the definition, see Definition <ref>(i). § ACTION OF THE ELLIPTIC DYNAMICAL QUANTUM GROUPIn this section we construct an action of the elliptic dynamical quantum group associated with 𝔤𝔩_2 on the extended equivariant elliptic cohomology Ê_T(X_n) of the union of cotangent bundles of the Grassmannians of planes in ℂ^n. The action is by S_n-equivariant admissible difference operators acting on admissible line bundles on the cohomology scheme. Thus each generator L_ij(w)of the elliptic dynamical quantum group acts on sections of any admissible line bundle by an admissible difference operator of some degree (ℒ_ij(w),μ_ij,ν_ij) which we give below. We also compute the action on T-equivariant elliptic cohomology classes and use the S_n-equivariance to show that the action descends to an action on G-equivariant classes, with G=U(n)× U(1).We construct the action in such a way that at the generic fibre of Ê_T(X_k,n)→ E_T(pt)=E^n× E (i.e., for fixed z_1,…,z_n,y) the map (<ref>) defines a morphism of representations from the tensor product of evaluation representations. In other words, suppose thatL_ij(w)v_I=∑_K L_ij(w,z,y,λ)_I^Kv_K,for some meromorphic coefficients L_ij(w,z,y,λ)_I^K. Then we want thatL_ij(w)Stab(v_I) =∑_K L_ij(w,z,y,λ)_I^KStab(v_K).The matrix coefficients L_ij(w,z,y,λ)_I^K are certain meromorphic functions of z,y,λ with theta function-like transformation properties and can thus be considered as meromorphic sections line bundles on Ê_T(pt). Therefore each summand on the right-hand side is a meromorphic section of an admissible line bundle. The content of the following theorem is that the sum defines uniquely an admissible difference operator. =0pt(i) The formula (<ref>) uniquely defines admissible difference operators L_ij(w), i,j∈{1,2}, of degree (ℒ_ij(w),2(i-j),ϵ(j)) with ϵ(1)=-1, ϵ(2)=1, for some S_n-equivariant line bundle ℒ_ij(w) on Ê_T(pt).(ii) These operators obey the RLL relations of the elliptic dynamical quantum group in the form μ_ℓ R(w_1-w_2,y,λ)^(12) L(w_1)^(13) L(w_2)^(23) = L(w_2)^(23)L(w_1)^(13)μ_rR(w_1-w_2,y,λ)^(12). Here the coefficients of the quadratic relations are in 𝒜^0_n and the action of μ_ℓ, μ_r is on each matrix element of R.The proof of this theorem is by explicit description of the action and is parallel to the case of Yangians and affine quantum enveloping algebras, <cit.>. We give the formulae for the action in Sections <ref>, <ref> and <ref>. The proof of Theorem <ref> is in Section <ref>.By Theorem <ref>, the generators L_ij(w) send meromorphic sections of admissible bundles to meromorphic sections of admissible bundles. The next result gives a more precise control on the poles of coefficients. We give the action on holomorphic sections, i.e., equivariant elliptic cohomology classes, both for the torus T=U(1)^n× U(1) and the group G=U(n)× U(1).=0pt(i) Let D be the divisor on Ê_T(pt) whose components are the hypersurfaces defined by equations z_a+y=w, for 1≤ a≤ n and λ+yj=0, for j=-n…,n-1,n. Then L_ij(w) maps H^ell_T(X_n)_ℒ to H^ell_T(X_n)_τ_ϵ(j)^*ℒ⊗ℒ_ij(w)(D) for any ℒ∈Pic(Ê_T(pt)).(ii) Let H^ℓ_G(X_n)_ℒ be the space of G-equivariant elliptic cohomology classes of degree ℒ for G=U(n)× U(1), see Definition <ref>. Let πÊ_T(pt)→Ê_G(pt) be the canonical projection. View line bundles on Ê_G(pt) as S_n-equivariant line bundles on Ê_T(pt). Then the opera­tors L_ij(w) induce well-defined operators from H^ell_G(X_n)_ℒ to H^ell_G(X_n)_τ_ϵ(j)^*ℒ⊗ℒ_ij(w)(π(D)) for each ℒ∈Pic(Ê_G(pt)). The proof of this theorem is contained in Section <ref>.Let qÊ_T(pt)=E_T(pt)× E→ E_T(pt) be the projection onto the first factor. Since the action of the generators L_ij(w) is by admissible difference operators it preserves the fiber of q_*ℋ_T^ell(X_n) at a generic point of the non-extended E_T(pt). If we realize this fiber as a certain space of functions of λ and tensor with all meromorphic functions of λ we get a representation of the quantum group in the sense of Section <ref>. By construction, it is isomorphic to the tensor product of evaluation representations. Thus we can think of the action of the quantum group on equivariant elliptic cohomology classes as a tensor product of evaluation representations with variable evaluation points and deformation parameter.§.§ Action of the Gelfand–Zetlin subalgebraThe Gelfand–Zetlin subalgebra is the commutative subalgebra generated by L_22(w) and the determinant Δ(w). As shown in Section <ref> these operators act diagonally in the basis ξ_I of V(z_1)⊗⋯⊗ V(z_n). It follows that the vectorsξ̂_I=∑_|J|=kw^-_J(z_I,z,y,λ)/∏_a∈ I, b∈I̅θ(z_a-z_b+y)Stab(v_J),(cf. (<ref>)), which by construction are sums of sections of certain admissible line bundles, are eigenvectors of the Gelfand–Zetlin subalgebra. It turns out that they are sections of admissible bundles with support on a single irreducible component of Ê_T(X_n):Let I,K⊂ [n], |I|=|K|=k. The restriction of ξ̂_I to the component of Ê(X_k,n) labeled by K isι_K^*ξ̂_I= ∏_a∈ I, b∈I̅θ(z_a-z_b), I=K,0,otherwise.From the definition of ξ̂_I and Stab(v_J) we haveι_K^*ξ̂_I =ι_K^*(∑_Jw_J^-(z_I,z,y,λ)/∏_a∈ I,b∈I̅θ(z_a-z_b+y) w^+_J(t,z,y,λ) ) =1/∏_a∈ I,b∈I̅θ(z_a-z_b+y)∑_J w_J^-(z_I,z,y,λ) w^+_J(z_K,z,y,λ),which, using Corollary <ref>, proves our statement. Thus we can write any section (s_I)_I⊂[n] of an admissible bundle as linear combination∑s_Iξ̂_I/∏_a∈ I, b∈I̅θ(z_a-z_b).Since the ξ̂_I are eigenvectors the action of the Gelfand–Zetlin algebra is given by admissible difference operator with diagonal matrices of coefficients.The action of the determinant Δ(w) is easiest to describe: it is given by an admissible difference operator of degree (ℒ_Δ(w),0,0), where ℒ_Δ(w)=ℒ(N_Δ,v_Δ(w)) is the line bundle associated with the dataN_Δ(z,y,λ)= ∑_a=1^n (2z_a+y)y,v_Δ(w;z,y,λ)=-wy.Since N_Δ and v_Δ are symmetric under permutations of the variables z_i, the corresponding bundle is naturally S_n-equivariant. The determinant acts on sections of any admissible line bundle ℒ_1 on Ê_T(X_k,n) it acts by multiplication by the section∏_i=1^nθ(w-z_i+y)/θ(w-z_i)of ℒ_Δ(w).The action of L_22(w) is by an operator of degree (ℒ_22(w),0,1). It is defined on the components byι^*_KL_22(w)s=∏_i∈ Kθ(w-z_i)/θ(w-z_i-y)τ_1^*ι_K^*s,and ℒ^k_22(w)=ℒ(2k∑_a=1^nz_ay,kwy), which is an S_n-equivariant line bundle.These formulae define S_n-equivariant admissible difference operators Δ(w)∈𝒜(ℒ_Δ(w),0,0) and L_22(w)∈𝒜(ℒ_22(w),0,1). Both operators are defined by diagonal matrices (φ_K,K') in the notation of Definition <ref>. It is straightforward to check that the diagonal matrix elements φ_K,K are sections of the correct line bundle. The equivariance property of Lemma <ref> is clearly satisfied. Moreover the divisor of poles does not contain any diagonal Δ_I={z∈ E^n+2 z_i=z_j, ∀i,j∈ I}, I⊂[n] so that the difference operator maps meromorphic sections to meromorophic sections. It remains to check that the sections on the different components coincide on their intersections, namely that for every a∈ K, b∈K̅, φ_K,K|_z_a=z_b=φ_K̃,K̃|_z_a=z_b, K̃=K∖{a}∪{b}. This can be checked directly but also follows from the equivariance condition for the permutation of a and b.§.§ Action of L_12 and L_21Let k=1,…,n and K⊂[n] with |K|=k-1, thenι^*_K L_12(w)s =(-1)^kθ(y)∑_a∈K̅θ(λ+w-z_a+(n-2k+1)y)/θ(w-z_a-y) ×∏_j∈ Kθ(w-z_j)/θ(w-z_j-y)∏_j∈ Kθ(z_a-z_j-y) /∏_j∈K̅∖{a}θ(z_a-z_j) ι_K∪{a}^*τ_1^*s. L_12(w) is an S_n-equivariant admissible difference operator of degree (ℒ_12(w),-2,1) with ℒ^k_12(w)=ℒ(-(λ+(n-2k)y)^2,-w(λ+(n-k+1)y)). Let k=0,…,n-1 and K⊂[n] with |K|=k+1, thenι^*_K L_21(w)s =(-1)^n-kθ(y)/θ(λ)θ(λ-y)∑_a∈ Kθ(λ-w+z_a)/θ(w-z_a-y) ×∏_j∈ K∖{a}θ(w-z_j)/θ(w-z_j-y)∏_j∈K̅θ(z_j-z_a-y) /∏_j∈ K∖{a}θ(z_j-z_a) ι_K∖{a}^*τ_-1^*s L_21(w) is an S_n-equivariant admissible difference operator of degree (ℒ_21(w),2,-1) with ℒ^k_21(w)=ℒ(-λ^2-(n-2k+2)y^2+2y(λ-∑_i=1^nz_i),w(λ-(k+1)y)).Lemmas <ref> and <ref> are proved the same way as Lemma <ref>. The only new feature is the appearance of simple poles on diagonals z_i=z_j and it is thus not a priori clear that these operators map meromorphic sections to meromorphic sections in the sense of Definition <ref>. The point is that when acting on meromorphic sections, these poles cancel by the equivariance conditions. For example let us consider the behaviour of φ=L_12(w)s in the vicinity of the diagonal z_a=z_b. The matrix element φ_K,K∪{a} has a simple pole there if b∈K̅∖{a} and so has the matrix element φ_K,K'∪{b} which by equivariance is obtained from φ_K,K∪{a} by the transposition σ_12 of a, b. In local coordinates and trivializations compatible with the S_n-action, by setting f_j=ι_K∪{j}τ^*_1s, j=a,b, the potentially singular term in ι_K^*L_12(w)s at z_a=z_b has the formg(z,y,λ)/z_a-z_bf_a(z,y,λ) +g(s_abz,y,λ)/z_b-z_af_b(σ_12z,y,λ).Since s_abz=z and f_a=f_b on the diagonal z_a=z_b, the poles cancel. The same argument works for L_21. §.§ Action of L_11Since L_22(w) is an invertible admissible difference operator, the action of L_11(w) can be obtain from the action of the Gelfand–Zetlin algebra and the action of L_12, L_21 via the formula for the determinantΔ(w)=μ_ℓ(θ(λ))/μ_r(θ(λ)) (L_11(w+y)L_22(w)-L_21(w+y)L_12(w)).Here θ(λ) is considered as a section of the bundle ℒ(N,0) on E^2 with N(y,λ)=λ^2. §.§ Proof of Theorems <ref> and <ref>Theorem <ref>(i) for L_22, L_12 and L_21 follows from Lemma <ref>, Lemma <ref> and Lemma <ref>, respectively. The operator L_11 can be expressed as composition of these admissible difference operators via the determinant and is thus also admissible. Part (ii) follows by construction. To prove Theorem <ref>(i) we need to check that in the matrix elements of L_ij(w) only simple poles at z_i=w+y and λ=jy, j∈ℤ appear. This is clear from the explicit formulae except for L_11(w). To prove it for this operator, we use two formulae for it: one is using the definition and the orthogonality relations, and the other expressing it in terms of the other L_ij and the determinant, as in Section <ref>.The first formula givesι_L^*L_11(w)s=∑_I,J,Kw^-_K(z_I)L_11(w)_K^Jw^+_J(z_L)/∏_a∈ I, b∈I̅θ(z_a-z_b) θ(z_a-z_b+y)τ_-1^*ι_I^*s.The matrix elements L_11(w)_K^J of L_11(w) in the tensor basis v_I of (ℂ^2)^⊗ n are sums of products of matrix elements of R-matrices and have at most simple poles at z_a=w+y and possible poles at λ=jy, j∈ℤ, see (<ref>). Thus the right-hand side of the (<ref>) has (among other poles) at most simple poles at z_a=w+y. The second formula is in terms of the determinant:L_11(w)s=(θ(λ)/θ(λ-yμ)Δ(w-y) +L_21(w-y)L_12(w))L_22(w+y)^-1s,for s∈ H_T^ell(X_k,n)_ℳ with μ=-n+2k. From this formula and the explicit expression of L_12, L_21, L_22 we see that only simple poles at λ=yμ, -n≤μ≤ n occur and that the remaining apparent poles at z_a-z_b=0, z_a-z_b+y=0 in (<ref>) are spurious.Finally Theorem <ref>(ii) holds since the bundles ℒ_ij(w) are S_n-equivariant (and can thus be viewed as line bundles on the quotient) and the action is given by S_n-equivariant difference operators.§ SHUFFLE PRODUCTS AND STABLE ENVELOPES FOR SUBGROUPSThe stable envelope of <cit.> is a map from the equivariant cohomology of the fixed point set for a torus action on a Nakajima variety to the equivariant cohomology of the variety. The goal of this section is to extend this interpretation of the stable envelope to the elliptic case for cotangent bundles of Grassmannians. In our construction the stable envelope is built out of weight functions, which in turn are obtained from shuffle products of elementary weight functions associated with the one-point spaces T^*Gr(0,1), T^*Gr(1,1). Thus the first step is to extend the fiber-by-fiber construction of the shuffle product of Section <ref> to a shuffle product defined on sections of the coherent sheaf Θ̅^+_k,n on Ê_T(pt). By using the isomorphism (outside the divisor D) of Theorem <ref>, we get a shuffle product on the sections of the sheaf of elliptic cohomology classes ℋ^ell_T(X_k,n). The n-fold shuffle product of classes in ℋ^ell_T(X_k,1), k=0,1 is then essentially the stable envelope.We propose to view shuffle products of factors of an arbitrary number of factors as stable envelopes corresponding to subgroups of T. Their geometric interpretation is that they correspond to maps from the cohomology of the fixed point set for the action of a subgroup of the torus T, cf. <cit.>.The basic case, which as we shall see corresponds to the shuffle product of two factors, is the subgroup B_m⊂ U(1)^nB=B_m={(z,…,z_m,1,…,1)∈ Az∈ U(1)},isomorphic to U(1).Fixed points for the action of this subgroup on Gr(k,n) are k-planes of the form V_1⊕ V_2, with V_1 in the span of the first m coordinate axes and V_2 in the span of the last n-m coordinate axes. Thus the fixed point set decomposes into connected components according to the dimension of V_1. Each of these components is a product of Grassmannians. Similarly, the B_m-invariant part of the cotangent space at a fixed point splits as a direct sum of cotangent spaces at the factors and we get an isomorphismX_k,n^B_m≅⊔_d=0^k X_d,m× X_k-d,n-m.As above we consider the action of A_n=U(1)^n on X_k,n. Then the embedding X_d,m× X_k-d,n-m↪ X_k,n is A_m× A_n-m= A_n-equivariant. The Künneth formula <cit.> predicts that this embedding induces a mapE_A_m(X_d,m)× E_A_n-m(X_k-d,n-m)→ E_A_n(X_k,n),In the description as a fiber product,E_A_m(X_d,m)=E^(d)× E^(m-d)×_E^(m)E^m,and the map is the obvious one: ((t',s',z'),(t”,s”,z”))↦ (p(t',t”),p(s',s”),z',z”). Here t'∈ E^(d), t”∈ E^(k-d), p E^(d)× E^(k-d)→ E^(k) is the canonical projection and similarly for the other factors.As in Section <ref>, we consider the extended equivariant elliptic cohomology Ê_T_n(X_k,n)=E_A_n(X_k,n)× E^2 for the torus T_n=A_n× U(1) where the additional U(1) factor acts by multiplication on each cotangent space. We then have the corresponding embeddingÊ_T_m(X_d,m)×_E^2Ê_T_n-m(X_k-d,n-m)→Ê_T_n(X_k,n),where the map to E^2 is the projection onto the second factor. Both are schemes over Ê_T_n(pt)=Ê_T_m(pt)×_E^2E_T_n-m(pt). The shuffle product of Proposition <ref> defines a map* τ^*_n”-2k”Θ̅^+_k',n'⊠Θ̅^+_k”,n”→Θ̅^+_k,n⊗ℒ_k',k”,n',n”of sheaves of 𝒪_Ê_T(pt)-modules, where k=k'+k”, n=n'+n” and ℒ_k',k”,n',n”= ℒ(k”y((n'-k')y-2∑_a=1^n'z_a),0)∈Pic(Ê_T_n(pt)). The sheaf Θ^+_k,n is defined by the quadratic form N^Θ_k,n(t,z,y,λ), see (<ref>). Let us write t=(t',t”), z=(z',z”), with t'=(t_1,…,t_k'), t”=(t_k'+1,…,t_k) and similarly for z. Then τ^*_n”-2k”Θ̅^+_k',n'⊠Θ̅^+_k”,n” is associated with the quadratic form M(t,z,y,λ)=N^Θ_k',n'(t',z',y,λ+y(n”-2k”)) +N^Θ_k”,n”(t”,z”,y,λ). The shuffle product maps a section of this bundle to a section of a bundle associated with the sum of this quadratic form and the quadratic forms of the theta functions in φ^+, see Proposition <ref>, namely M(t,z,y,λ) + ∑_j=1^k'∑_l=k'+1^k ((t_j-t_l+y)^2-(t_j-t_l)^2)+∑_l=k'+1^k ∑_a=1^n'(t_l-z_a+y)^2+∑_j=1^k'∑_b=n'+1^n (t_j-z_b)^2. It is straightforward to verify that this is equal to N^Θ_k,n(t,z,y,λ)+k”y((n'-k')y-2∑_a=1^n'z_a). This shows that the shuffle product takes values in Θ^+_k,n⊗ℒ_k',k”,n',n”. The fact that it actually takes values in the subsheaf defined by the vanishing condition follows from Proposition <ref>.Let k=k'+k”, n=n'+n”. The unnormalized stable envelope associated with the component X_k',n'× X_k”,n” of the fixed point set X_k,n^B_n' is the shuffle product map Stab τ_n”-2k”^*Θ̅^+_k',n'⊠Θ̅^+_k”,n”→Θ̅^+_k,n⊗ℒ_k',k”,n',n” of sheaves of 𝒪_Ê_T_n(pt)-modules.By using the isomorphism Θ̅_k,n≅ℋ_T_n^ell(X_k,n) on the complement of the divisor D of Theorem <ref>, we obtain a mapStab τ_n”-2k”^*ℋ^ell_T_n'(X_k',n') ⊠ℋ^ell_T_n”(X_k”,n”) →ℋ^ell_T_n(X_k,n)⊗ℒ_k',k”,n',n”,on Ê_T(pt)∖ D.More generally, we may consider subgroups B=B_n_1×⋯× B_n_r⊂ U(1)^n whose fixed point sets have components X_k_1,n_1×⋯× X_k_r,n_r and define stable envelopes given by r-fold shuffle products and thus by compositions of stable envelopes for two factors.Two special cases give the stable envelope of Section <ref> and the action of the elliptic dynamical quantum group.In the first case we take B=U(1)^n. The fixed points are isolated and labeled by I⊂[n]. We think of the fixed point labeled by I as a product X_k_1,1×⋯× X_k_n,1 with k_i=1 if i∈ I and k_i=0 otherwise. The unnormalized stable envelope on the component labeled by I⊂ [n] is thenStab ⊠_j=1^n τ^*_w(j,I)ℋ^ell_T_1(X_k_i,1) →ℋ^ell_T_1(X_k_i,1)⊗ℳ_I,(the factors are ordered from left to right) for some suitable line bundle ℳ_I∈Pic(Ê_T_n(pt)) obtained as tensor product of line bundles L_k',k”,n',n”. In this case the map is defined everywhere, not just on the complement of D, since Θ̅_k,1≅ℋ^ell_T_1(X_k,1) on Ê_T_1(pt).The stable envelope of Section <ref> is obtained by taking the tensor product with suitable line bundles τ^*_w(j,I)ℳ_k_j on Ê_T_n(pt) so that ⊕_k=0^1Γ(X_k,1,ℋ^ell_T_1(X_k,1)⊗ℳ_k) is identified with ℂ^2 via the basis ω_0^+,ω_1^+, passing to global sections and normalizing by dividing by ψ_I.In the second case we reproduce the construction of Section <ref> in the global setting. Thus we consider the stable envelope for two factors X_d,1× X_k-d,n⊂ X_k,n+1. We obtain two maps⊕_d=0^1τ^*_n-2(k-d)Θ̅^+_d,1⊠Θ̅^+_k-d,n⊗ℒ_d,k-d,1,n^-1→Θ̅^+_k,n+1 ⊕_d=0^1τ^*_1-2dΘ̅^+_k-d,n⊠Θ̅^+_d,1⊗ℒ_k-d,d,n,1^-1→Θ̅^+_k,n+1which are invertible at a generic point. Since Θ̅^+_k,1 is isomorphic to ℋ^ell_T_1(X_k,1) we get a map⊕_d=0^1τ^*_1-2dΘ̅^+_k-d,n⊠ℋ^ell_T_1(X_d,1) ⊗ℒ_k-d,d,n,1^-1 →⊕_d=0^1τ^*_n-2(k-d)ℋ^ell_T_1(X_d,1)⊠Θ̅^+_k-d,n⊗ℒ_d,k-d,1,n^-1,defined on some dense open set. This map contains the information of the action of the elliptic dynamical quantum group on the elliptic cohomology of Ê_T_n(X_n). The action of the generators is given as explained in Section <ref>: one needs as above to take the tensor product with a suitable line bundle to associate elliptic cohomology classes ω^+_1,ω^+_0 with the standard basis of ℂ^2. Then we are in the setting of Section <ref> and we obtain an action of the elliptic dynamical quantum group which is up to gauge transformation the one described in the previous section.§ AXIOMATIC DEFINITION OF ELLIPTIC STABLE ENVELOPES In this section we give an axiomatic definition of the elliptic stable envelopes in the spirit of Maulik–Okounkov <cit.>, see also <cit.>.Recall that c ^*w^+_I is a meromorphic section (with controlled denominators) of an appropriate line bundle over Ê_T(X_k,n). The scheme Ê_T(X_k,n) has components Y_J=ι_J Ê_T(pt), and the restriction of a section to Y_J is the result of substituting the variables t_i by z_J=(z_i)_i∈ J.A meromorphic section of an admissible line bundle p_T^*ℒ(N,0)⊗𝒯_k,n restricted to Y_J can be written as a meromorphic function F^n+2→ whose transformation properties with respect to the lattice ^n+2+τ^n+2 are determined by p_T^*ℒ(N,0)⊗𝒯_k,n, see Remark <ref>. Below we will consider special forms of such functions. For any I the section c ^*w^+_I satisfies the following properties. =0pt* It is a meromorphic section of an admissible line bundle p_T^*ℒ(N,0)⊗𝒯_k,n.* The restriction of c ^*w^+_I to Y_I, written as a function ^n+2→ with transformation pro­per­ties determined by p_T^*ℒ(N,0)⊗𝒯_k,n, is ∏_a∈ I,b∈I̅θ(z_a-z_b+ϵ(a,b)y)/∏_a∈ Iθ(λ-(w(a,I)+1)y),where ϵ(a,b) is defined in Lemma <ref> and w(a,I) is defined in (<ref>).* The restriction of c ^*w^+_I to any Y_J, written as a function ^n+2→ with transformation properties determined by p_T^*ℒ(N,0)⊗𝒯_k,n, is of the form1/ψ_I∏_a∈ J∏_b∈J̅,b<aθ(z_a-z_b+y) · F_I,J,where F_I,J is a holomorphic function.Moreover, these three properties uniquely determine c ^*w^+_I.From the second property one can calculate the quadratic formN_I =-2∑_a∈I̅n(a,I)z_ay-2∑_a∈ Iz_a(λ+n(a,I̅)y)+(k(n-k)-∑_a∈ In(a,I̅))y^2-∑_a∈ I(λ- (n(a,I)+1)y+n(a,I̅)y)^2,cf. (<ref>). Lemma <ref> (ii) implies the triangularity property =0pt* the restriction of c ^*w^+_I to Y_J is 0 unless J≤ I.According to Theorem <ref> this property is a consequence of the three properties listed. The third listed property is a local version of a support condition used in the axiomatic description of cohomological stable envelopes in <cit.>; see also the corresponding axiom in K-theory in <cit.>. The first two properties of c ^*w^+_I are claimed in Proposition <ref> and Lem­ma <ref>(ii). Inspecting the explicit formula for w^+_I in Section <ref> one finds that, after substitution t_i by z_a, a∈ J, all non-zero terms are of the form (<ref>), which proves the third property.Now we prove that the three properties uniquely determine c ^*w_I^+. Let a section satisfy the three listed properties, and let κ_I be the difference of that section and c ^*w_I^+. Assume that κ_I is not 0. Then there exists a J such that κ_I restricted to Y_J is not 0. For a total ordering ≺ refining the partial order < on the cardinality k subsets of [n] let us choose J to be the largest with the property κ_I|_Y_J≠0. We have J≠I because of the second property.We claim that κ_I|_Y_J, written as a function ^n+2→ with transformation properties determined by p_T^*ℒ(N_I,0)⊗𝒯_k,n, is of the form1/ψ_I∏_a∈ J∏_b∈J̅,b<aθ(z_a-z_b+y) ·∏_a∈ J∏_b∈J̅, b>aθ(z_b-z_a)· F_1,where F_1 is holomorphic. The fact that this function can be written in the form1/ψ_I∏_a∈ J∏_b∈J̅, b<aθ(z_a-z_b+y) · F_2,with F_2 holomorphic, is explicit from the third property. We need to prove that F_2 is the product of ∏_a∈ J∏_b∈J̅,b>aθ(z_b-z_a) and a holomorphic function. Let a∈ J, b∈J̅ and b>a. Denote J'=(J-{a})∪{b}. Observe that J<J' and hence J≺ J'. From the choice of J therefore it follows that κ_I restricted to Y_J' is 0. The diagonal Δ_J,J'={z_a=z_b} is included both in Y_J and Y_J', hence we obtain that the substitution of z_a=z_b into κ_I|_Y_J vanishes. It follows that the function F_2 vanishes on the hyperplane z_a=z_b and its translates by the lattice ^n+2+τ^n+2. The zeros of θ(z_b-z_a) are exactly these hyperplanes and are of first order, therefore F_2 can be written as a product of θ(z_b-z_a) times a holomorphic function. Iterating this argument for all (a,b) with a∈ J, b∈J̅, b>a we obtain that (<ref>) is in fact of the form (<ref>), what we claimed.Observe that the product of theta functions in (<ref>) is the numerator of c ^*w_J^+|_Y_J. Hence we obtain that (<ref>) further equals∏_a∈ Jθ(λ-(w(a,J)+1)y)/ψ_I· c ^*w_J^+|_Y_J· F_1.Since the transformation properties of κ_I|_Y_J are determined by p_T^*ℒ(N_I,0)⊗𝒯_k,n, and those of c ^* w_J^+|_Y_J are determined by p_T^*ℒ(N_J,0)⊗𝒯_k,n, we have that the transformation properties of∏_a∈ Jθ(λ-(w(a,J)+1)y)/ψ_I· F_1are determined by p_T^*ℒ(N_I-N_J,0) – the factor 𝒯_k,n canceled.Let a∈ J ∩I̅, and consider (<ref>) as a function of z_a, let us call it f(z_a). Since the first factor (the fraction) only depends on λ and y, f is a holomorphic function of z_a for generic y,λ. Comparing the z_a dependence of N_I and N_J we obtain thatf(z_a+τ)= e^-2π i (λ+sy) f(z_a),f(z_a+1)=f(z_a),for some integer s. Using the 1-periodicity, we expandf(z_a)=∑_m∈ℤ a_me^2π i m z_a,and using the first transformation property of (<ref>) we obtain∑_m a_me^2π i m z_a(e^2π i m τ -e^-2π i (λ+sy))=0,implying a_m=0 for all m∈ℤ. We obtained F_1=0, and in turn, κ_I|_Y_J=0. This is a contradiction proving that κ_I is 0 on all Y_J.§.§ AcknowledgmentsWe thank Nora Ganter and Mikhail Kapranov for explanations on equivariant elliptic co­ho­mo­logy. G.F. was supported in part by the National Centre of Competence in Research “SwissMAP – The Mathematics of Physics” of the Swiss National Science Foundation. R.R. was supported by the Simons Foundation grant #523882. A.V. was supported in part by NSF grant DMS-1362924 and Simons Foundation grant #336826. 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http://arxiv.org/abs/1702.08060v2

2 SDN as Active Measurement Infrastructure Erik Rye US Naval Academy rye@usna.eduRobert BeverlyNaval Postgraduate School rbeverly@nps.edu =================================================================================================================empty Active measurements are integral to the operation and management of networks, and invaluable to supporting empirical network research. Unfortunately, it is often cost-prohibitive and logistically difficult to widely deploy measurement nodes, especially in the core.In this work, we consider the feasibility of tightly integrating measurement within the infrastructure by using Software Defined Networks (SDNs). We introduce“SDN as Active Measurement Infrastructure” () to enable measurements to originate from any location where SDN is deployed, removing the need for dedicated measurement nodes and increasing vantage pointdiversity. We implement and using , as well as a proof-of-concept custom measurement protocol to demonstrate the power and ease of 's open framework.Via alarge-scale measurement campaign using SDN switches as vantage points, we show that is accurate, scalable, and extensible.§ INTRODUCTION Software Defined Networking (SDN) has emerged as a powerful architectural paradigm, enabling innovations in network virtualization, provisioning, verification, and security (e.g., <cit.>). Within the context of measurement, SDNs are commonly instrumented to monitor network utilization and can quickly modify their forwarding behavior in response to dynamic workloads <cit.>.In contrast to such passive measurements (e.g., packet or flow-level statistics, heavy-hitters, or anomaly detectors), this work considers the feasibility of performing active network measurement using SDNs.Intuitively, SDNs provide the basic building blocks for facilitating programmable active measurement vantage points.Via the standardized protocol <cit.>, SDN controllers can generate any arbitrary packet (measurement probes) and instructa switch to emit the packet out a specified interface.Similarly, controllers can instantiate fine-grained flow rules that match measurement probe responses.Controllers may then perform arbitrary computation over probe responses as they arrive encapsulated in an message from the switch. In this work, we introduce and advocate for an architecture using these primitives which we term “SDN as Active Measurement Infrastructure” ().While active measurements today are performed using end-hosts, we believe provides compelling advantages.First, tighter integration of active measurements within the network allows operators, administrators, and researchers to place measurements anywhere an SDN switch exists – without the traditional cost of configuring, certifying, securing, installing, powering, managing and maintaining a dedicated measurement host.Often, installing a measurement host within the core or edge of the network presents an insurmountable administrative or policy hurdle, or is not physically possible (due to space, availability and expense of consuming an interface on the router, etc).By leveraging existing production equipment, lowers deployment and vantage point diversity barriers.Second, placing active measurements within SDN more closely couples the ability to programatically enable action to be taken in response to measurement results.In a similar fashion to using passive measurements to drive the behavior of the SDN data plane, active measurements can provide actionable information for network reconfiguration and adaptability.Third, follows the philosophy of inexpensive commodity hardware, centralized control, and compositional network architecture.uses thestandardized API to drive commodity switches, thereby removing the traditional need to develop, configure, and deploy custom APIs for requesting measurement tasks and receiving measurement results.As we will show, using prevents over-specialization of the measurement platform, making it easy to extend to new and unanticipated measurement tasks.Toward the goal of converging on a standards-based strategy foractive measurement platforms for operators and researchers, weinvestigate the feasibility and limitations of . Our primary contributions include: -0.5 * The architectural vision to converge activemeasurement facilities into commodity hardware using standardized protocols. * A large-scale Internet measurement study using to sourceactive probes from commodity SDN switches, and acorresponding analysis of the resulting fidelity ofround-trip latency measurements. * Implementation of a new measurement protocol using the primitives as a demonstration of the ability to easily and rapidly innovate and deploy measurements within the framework.* A fundamentally different approach to Internet measurement, in which corenodes (in addition to edge systems) participate as vantage points for activemeasurements, potentially in reaction to events in the traffic streamsthey observe.The remainder of this paper is organized as follows.Section <ref> describes , while Section <ref> details results and accuracy from using inreal-world large-scale Internet measurement experiments.We discuss related work in Section <ref>, and conclude with a discussion of deployment scenarios and suggestions for future work in Section <ref>. § IMPLEMENTATIONActive measurement is widely used for network management and debugging.Canonical examples include the and utilities that provide reachability, round-trip time, and forward path information.Not only are such tools used for troubleshooting, they are integral to the operation of large providers, content distribution networks, and at-scale services.Myriad other active measurement techniques and tools have been developed, for instance host and service fingerprinting <cit.>, capacity and bandwidth estimation <cit.>, censorship detection <cit.>, network neutrality <cit.>, residential broadband performance <cit.>, and congestion localization <cit.>, to name only a few.It is well-known that the accuracy and generality of inferences from these active measurements can depend strongly on where in the network the measurements are performed.For instance, censorship may only affect nodes behind a certain middlebox, congestion may affect only a single network, or a BGP hijack event may affect reachability for only a subset of vantage points.To this end, the research community has developed several active measurement platforms consisting of varying (but relatively small) numbers of nodes distributed (in an ad-hoc fashion) across the Internet, e.g., Archipelago <cit.>, Bismark <cit.>, Dasu <cit.>, and Atlas <cit.> to name a few.Unfortunately, these platforms are frequently: i) designed for a specialized task; ii) under-powered (either in terms of compute, memory, or network abilities); and iii) lacking in network or geographic diversity.Rather than the current fractured environment of incompatible network measurement platforms, abilities, APIs, and output formats, we consider the feasibility of performing active measurementusing existing SDN standards and capabilities. Our vision that by using SDN, the measurement and operational community can utilize a standard and open API to avoid over-specialization, lower deployment barriers, and facilitate vantage points otherwise not possible.While significant prior work utilizes the passive network measurement capabilities afforded by SDNs, e.g., <cit.> and novel methods to gather measurements from SDNs, e.g., <cit.>, our work seeks to understand the feasibility of SDN for active measurement – i.e., where the SDN switch generates specialized measurement probes and gathers their responses.Figure <ref> illustrates the high-level architectural view of within an SDN.We envision co-existing with other applications on an SDN controller that is responsible for one or more SDN switches within a provider or enterprise network.Our implementation is based on the popular open-source Ryu <cit.> controller.In response to measurement tasks (<ref>) instructs the SDN controller to send various  <cit.> messages to the SDN switch(es) in order to induce active IP measurement probes.In particular, we utilize the message to instruct a switch to send a particular packet.Further, installs flow table rules such that it can receive and process measurement replies; we rely on messages from the switch that encapsulate data-plane probe packets matching particular criteria. In this fashion, acts as the abstraction layer between measurement tasks and the SDN.§.§ CalibrationWe note that many measurement tasks, including and , require accurate timing information, for round-trip time (RTT) latency estimation. must thus address two timing challenges related to using commodity switches and the protocol: approximating the time at which measurement packets are sent and received.With respect to probe transmission, both the delay in sending the instruction (the “”) from the controller to the switch, and the switch's processing delay in executing that instruction, contribute potentially variable latency.Further, while provides a mechanism to forward probe responses from the switch to the controller (the “”), there is nostandardized way to obtain the time when the packet was received at the switch.We must therefore estimate these values via an assessment of the RTT between the controller and the switch, and via empirical analysis of commodity SDN switch behavior.While it is possible to design specialized hardware and implement API changes to address these sources of inaccuracy, our goal in this work to determine the feasibility of current SDN hardware and software implementations to support .§.§.§ Switch Processing Delays Let R(X) be the time at which some packet X is received at a switch port and T(X) be the time at which some packet X is emitted from a switch port.* Assume a switch receives a probe within a at R(pktout).To determine the switch's delay is emitting the probe, we compute: T(probe) - R(pktout). * Further, we wish to determine whether any output reordering occurs, ∃ is.t.T(probe_i+1) < T(probe_i) and R(pktout_i) < R(pktout_i+1).* Assume a switch receives a response packet that matches a flow rule.The switch's delay in generating the message is: T(pktin) - R(response).To isolate sources of delay and calibrate our measurements, we create an isolated testbed using a Linux machine with two physical Ethernet interfaces as shown in Figure <ref>.OneEthernetdirectly connects to the Out-of-Band Management (OOBM) port on a commodity commercial SDN switch, while the other Ethernet is connected to one of the switch's data-plane interfaces. In our experiments, we use a commodity HP2920 commercial switch running 1.3.We then perform a packet capture from the Linux machine to time packets in relation to dataplane packets (either generated or received).Because the packet capture is performed on two interfaces of the same physical machine, time is synchronized. Since the host and the SDN switch are directly connected via a <1 meter cable, propagation delay is a negligible component of the measured delay.We send 100 messages to the switch, and measure the switch's delay in emitting the corresponding packet. Figure <ref> displays the cumulative fraction of delays T(probe_i) - R(pktout_i) over the 100 requests.We see that 97% have a delay between 1.5 and 2.0ms.Two messages require ≈ 23ms, while the switch took approximately 50ms before emitting one of the packets.Overall, the delay is both small and tightly bounded.Further, we observe no packet reordering.Similarly, we evaluate the switch's delay in emitting a message in response to receiving a packet that matches a flow rule, T(pktin_i) - R(response_i).We see a qualitatively similarly shaped distribution in Figure <ref>, however 95% of the messages are generated in 1.0ms or less. §.§.§ Bundled Messages We observe instances of multiple messages bundled into single TCP segments by the controller.Such effects are due to operating systems and their corresponding TCP stack implementations. Quantifying the effect that bundling has on time-sensitive measurements is therefore important for RTT estimation. Thus, we must assess the variation in time for packets to be emitted by an SDN switch, whether arriving at the switch as messages in separate TCP segments, or bundled with other messages in a single TCP segment.In order to measure this effect, we use messages to instruct the switch to emit ICMP packets.We calculate the difference in time between an ICMP Echo Request leaving the SDN switch and its corresponding message arriving at the switch, accounting for messages in individual segments and multiple messages contained in a single segment separately. We sent 15,000 probe requests with 5 probes apiece to a host running <cit.> acting as the SDN switch. Of the 75,000 probes generated, only 61 probes are sent asmessages contained in a TCP segment with other messages. The time difference between ICMP Echo Requests exiting the switch and the time the message entered the switch is characterized in Figure <ref>. On average, the overall switch delay is approximately 10% higher for probes sent in multiple--containing segments – a 206 μ s delay was incurred by multipart messages, whereas individually sent pings had only a 189 μ s mean delay.§.§.§ Controller to Switch RTT EstimationThe controller measures the RTT between controller and associated SDN switch by issuing and timestamping anmessage, a built-in message type that is used by default as a “heartbeat” between the controller and switch. When the correspondingreturns from the switch, the controller timestamps this reply, allowing for the calculation of the RTT between the controller and switch RTT_C-S. For our implementations of(Section <ref>) and(Section <ref>) we estimate RTT_C-S for each target separately before generating probes. In order to account for isolated, drastic changes in controller-to-switch latencies, we keep an exponential moving weighted average of RTT_C-S times for use in true RTT estimation calculations.Note that performing the controller to switch latency estimation within the protocol, as opposed to, e.g., using a simple ICMP echo, provides a more reliable approximation of the delay (and processing) incurred by messages.§.§ ConfigurationFor experiments in which probes must eventually return to the switch that emitted them (and ), these probes must use a source address that is routed to or through the switch that generated the probe[We discuss more complicated scenarios where a different switch in the network receives probe responses in <ref>.].Thus, must know what source IPaddress to use when generating messages and installing flow rules.While this requires specific addressing information to be known, such knowledge is integral to SDN controllers.In our testing, we chose an unused address on a subnet that is routed to a network on which the SDN switch is connected.Because it was not possible to assign an IP address to the commodity SDN switch port,deliberately generates gratuitous ARPs.This allows the router to which our SDN switch is connected to pre-populate its ARP cache and prevent any additional delay.We expect that generating these ARPs will not be necessary in other deployments where SDN switches act as layer-3 forwarders.Thus, for correct operation, must at a minimumbe configured with: i) the IP and MAC addresses to use when sourcing measurement probes; and ii) the IP and MAC address of the next hop such that the probe is properly forwarded.As is typical in SDN installations, the switches are configured with the IP address of the controller, and listens for incoming connections. §.§ Measurement APIFollowing our general philosophy of leveraging existing protocols, utilizes HTTP and JSON as its measurement API with which the consumers of measurement tasks interface.In this way, acts as the interface between high-level measurement tasks, and the marshaling of messages within the SDN plane.The controller therefore runs an HTTP server that listens for incoming measurement instructions.We leverage HTTP as a standard RPC-like mechanism with the ability to easily support encryption, integrity, and authentication.As a concrete example, we detail here the REST API calls necessary for our implementation described in Section <ref>. Many common measurement tasks may leverage or require the emission of ICMP Echo Request packets, such as RTT estimation or to elicit responses containing IP Identifier values. To enable these experiments, the controller has a defined “probe URL”, a URL that when requested with the required parameters included in a JSON array via an HTTP PUT, will emit Echo Requests destined for a particular target. For instance,the controller uses URLas the Echo Request-emitting resource locator (by default, Ryu listens for API calls on port 8080). Our implementation requiresa JSON array be sent with the HTTP PUT request containing three keywords extending the JSON schema: i) , the IP address of the target; ii) , the number of Echo Requests to be emitted; and iii) , an optional payload for inclusion in the Echo Requests emits from the SDN switch. Upon receipt of the PUT request, the controller parses the JSON array and passes the fields to the method responsible for creating the ICMP Echo Requests, encapsulating them in IP datagrams and Ethernet frames, and delivering messages to the SDN switch. Switch for packet emission and output port selection are specified in message data fields, ensuring the correct switch emits the packet from the proper output interface.The use of the REST API creates an extensible measurement framework – in our example, it is trivial to add an additional optional source IP address parameter for Echo Request generation or to define a default TTL for the emitted datagrams – that is easily automated using scripts (withor ). Furthermore, the API enables the experimenter to operate independently from the measurement platform, without needing to log into the controller to start or stop measurements.Experiment output is retrieved via REST API calls; we obtain state maintained by the controller again through the use of HTTP GET requests to predefined URLs. For example, an HTTP GET request toreturns a JSON array containing thepacket emission and arrival times associated with each ICMP ID and Sequence Numbers sent in our implementation, using the ICMP ID as the JSON schema keyword. ICMP Echo Requests that receive no corresponding Echo Reply simply contain a null entry in the response timestamp position in its corresponding field, allowing unresponsive destination hosts to be treated differently than responsive destination addresses. This raw data can then be manipulated by the experimenter according to their own needs. A practical application using this data (estimating RTTs) is demonstrated in Section <ref>.Figure <ref> is a high-level view of 's implementation.§.§ Common Measurement Tasks §.§.§ Pingimplements the network utility through an HTTP PUT message to the controller from a client.This PUT message includes a JSON array with target IP address, count and payload fields.In our implementation, the client is a trivial 12 line Python program, illustrating the ease with which it is possible to develop within the framework.Upon receipt of the request via the PUT, the controller first sends an ARP reply to the router containing the MAC and IP addresses of the controller's interface to the switch as described in <ref>.This gratuitous ARP reply prevents any ARP cache expiration;without this gratuitous ARP reply, we risk needing to reply to ARP requests and thereby negatively affecting time-sensitive RTT estimation.Next, initiates RTT estimation to the switch as outlined in Section <ref>. The controller then creates an ICMP Echo Request destined for the target IP address, encapsulates it in an Ethernet frame addressed to the next hop's MAC address, and emits an message to an SDN switch. Concurrently, the controller creates a map between the ICMP Identifier field value used and target IP address to maintain state of in-flight probes. initially sets the ICMP Sequence Number field to 0 for the first ICMP Echo Request. The payload of our Echo Request is empty, but can be specified by the requester in the JSON array sent with the HTTP PUT.timestamps the packet and delivers a message to the SDN switch, which emits the ICMP Echo Request out of the switch port specified as an argument to themethod. then increments the ICMP Sequence Number field value and repeats until the requested number of ICMP Echo Requests have been delivered to the switch via messages. For each subsequent ping target, increments the ICMP Identifier field value, establishing a linkage between ICMP Identifier values and ping destination IP addresses. Concurrently, is delivered ICMP Echo Replies for the probes it has transmitted by the switch via messages, due to the installation of a flow rule during initialization directing the switch to forward these to the controller. When a probe reply is received notes the time the ICMP Echo Replies were received by the controller and stores this value. For a given target Tgt, we can therefore calculate RTT_C-Tgt, the total time elapsed between transmission from to the switch and messages received by from the switch for each ICMP Sequence Number. RTT_C-Tgt - RTT_C-S then approximates the true RTT between the SDN switch and probe target. Targets that do not respond to probes have no ICMP Echo Reply timestamp and leave us unable to calculate the RTT; we therefore discard any targets with no probe replies. A REST API call retrieves the timestamp values stored by for analysis by the client. Because the ICMP ID field is 16 bits, this implementation allows for approximately 65 thousandtargets before 's state table is full. We work around this limitation by pulling the current state from via a REST API call; specifically a GET request to a specific URL returns a JSON array of 's state table to the experimenter. Another REST API call clears the state table and measurements can be resumed, thus allowing for a potentially unlimited number oftargets. An alternative implementation might leverage the payload field of the ICMP Echos to maintain the state oftargets; we abstain from this approach in order to maintain consistency of sent packet sizes for all probes when conducting large-scale measurements. §.§.§ TracerouteAnother common measurement task we implement in is, initiated by a specific REST API call[While there are several traceroute variants, we provide basic implementation here to demonstrate the ease with which measurements can be created.]. The keywords contained in the JSON array of the HTTP PUT message are the target, and number of probes per TTL.'s implementation operates by first determining RTT_C-S as described in Section <ref>. then creates ICMP Echo Request messages with TTL values beginning at 1, delivering these to the switch via messages. Initially, the ICMP Identifier value is set to 0, as in our implementation. maintains TTL and target information for each probe by ICMP identification and sequence numbers.When intermediate routers respond with ICMP Time Exceeded messages, correlates the response with its corresponding probe by parsing the ICMP quote, and thereby determines the IP address and RTT for each hop along the forward path.increments the IP TTL value after each user-specified number of probes at each TTL value, until one of two conditions occurs: i) the target is reached and responds with an ICMP Echo Reply, or ii) the TTL value reaches 30 without having reached the target. The controller then increments the ICMP Identifier value used for correlating messages bound for the same target together, as in our implementation in Section <ref>. Our implementation is asynchronous; each successive ICMP packet is sent to the switch in a message to be emitted as soon as it is created by the controller. is capable of handling these multiple packets in flight simultaneously due to the state table it maintains of ICMP ID/Sequence Number values. Like our implementation, probes for which intermediate routers or destination addresses are unresponsive simply contain a null value as the return timestamp. A REST API call returns the state to the experimenter, who can then reproduce the path from source to destination from the IP addresses. §.§ Custom MeasurementsIn addition to performing existing active measurement tasks from within SDN (and as previously described), facilitates the creation and execution of new and novel measurements in the spirit of network innovation.As a proof-of-concept, we partially implement the “router ID” primitive from our nascent work in developing and using within protocol measurements <cit.>.Today's method for active router discovery has remained essentially unchanged for almost 30 years: induces routers along the forward path to a destination to return ICMP messages with one of the router's interface IP addresses as the source.Unfortunately, IP addresses are a poor proxy for a router identifier.Our reliance on leads to the cumbersome and error-prone processes of: i) alias resolution, to determine the set of IP addresses belonging to the same physical router; and ii) router ownership inference.In the case of alias resolution, the existing techniques require intensive active probing, only work for a subset of addresses, and can both produce false aliases and miss true aliases <cit.>.Similarly, the method of using the autonomous system (AS) that originated the address space to which a router's interface belongs is often not indicative of the AS that owns or maintains the router, due to delegated and off-path addresses <cit.>.Our intent in implementing the router ID is not to perform exhaustive measurements, but rather to provide a concrete example of the types of measurement primitives that could be created with , and their immediate benefit to network management and diagnosis.For debugging and management purposes, we imagine that a provider wishes to extend the functionality of their core networking devices such that they respond to an identification query with a unique router identifier and the AS number to which the device belongs.Using , we instantiate a rule in the switch to encapsulate and forward to the controller any ICMP packets destined to the switch with type 200 and code 0[corresponding to a currently unused ICMP type].The controller parses this router ID query message and creates a response with the device's AS and identifier.This trivial example highlights several important characteristics of . First, while any database, the DNS, could conceivably provide an identical functionality, doing so requires keeping different portions of the namespace consistently updated – a significant practical difficulty.Instead, provides a much closer coupling where the control plane of the network device knows its AS and router identifier.Second, router ID demonstrates the ease with which a new protocol or protocol extension can be implemented to provide substantial measurement benefit.In this case, router ID effectively solves both the aliasing and ownership problems with an explicit mechanism, rather than the brittle and error-prone inferences the measurement community is currently forced to employ.§.§ Deployment ScenariosWe envision core network operators facilitating experiments by allowing researchers access to the application running on their production controllers. This immediately introduces several implementation challenges. First, how might service providers arbitrate access to their instance? Ryu supports a robust PKI-based authentication system, allowing network operators to permit access to authorized users and specific switches via public key authentication. Second, allowing users the ability to inject an arbitrary number of packets as quickly as the controller can generate them into their network is likely an unappealing prospect for service providers. But because is a controller application, policies limiting maximum packet rates, connection rates, and bitrates can be specified and tailored to individual operators' risk assessments. Third, the potential exists for measurement experiments to install flow rules on SDN switches that conflict with production dataplane forwarding rules.This potential conflict can be overcome by assigning a lower priority to user-generated flow rules, which would then be ignored if a conflicting flow rule installed by the network operator exists, thereby preventing unintended and potentially harmful consequences; already supports this type of flow rule prioritization. Finally, should support multiple experimenters concurrently; because the controller is simply a production server, partitioning of multiple users is a benefit inherited from the deployment environment itself. Conflicting flow rules generated by individual experimenters can be handled by assigning higher priority to certain experiments, in a first-come, first-served manner, or according to operator-specific instructions. Finally, to mitigate risk assumed by network administrators running instances, can be configured to only allow certain types of packets to be generated via its API. For instance, the ability to send ICMP-based or probes at a low datarate, or packets to TCP port 80 might be enabled, while the emission of more unusual packets (like those discussed in Section <ref>) could be restricted only to certain users or disabled entirely.§ RESULTS §.§ Ping In order to measure the accuracy of 's implementation, we capture traffic entering and leaving the SDN switch through a hub connected to the machine running with . In this manner, weobtain ground truth for the times our -generated ICMP Echo Requests and Replies exit and enter the switch. We account for varioushardware and software implementations, using both an HP 2920 SDN switch and a Linux machine running , as well as remote versus local controller scenarios. Both the HP switch and machine running are located in Monterey, California, while the remote controller is located in Boston, Massachusetts. Figure <ref> shows the CDF of absolute error (between -recorded RTT and actual RTT) using an HP 2920 series switch, as well as using the machine running as the switch itself with (labeled as “Local controller HP2920” and “Local Controller OVS”, respectively). Additionally, the software and hardware switches were controlled from a controller operating on the machine in Boston, shown in the “Remote Controller” curves. All trials probed the same 15,000 IP addresses and received ∼11,000 replies each.Our results show that the instance controlling the remote SDN switch has the least amount of absolute error, with nearly 99% of errors at 10 ms or less. The lower bound is the remote instance controlling the HP switch, with only ∼70% of errors at the 10 ms of less mark. Figure <ref> is a CDF of the absolute error as a fraction of the total RTT. In our implementation, the implementations consistently produce a lower error/RTT value, with the locally controlled trial obtaining an error of <10% of the total RTT for more than 95% of all probes. Both HP switch implementations produce the most error in RTT measurement, with the remote controller architecture achieving the same error percentage for approximately 20% fewer RTTs. One method that can be used to reduce the absolute and relative RTT errors is simply to send more Echo Requests to the targets. In Figure <ref>, we show the error relative to the actual RTT in two trials, one in which five Echo Requests were sent to the destination, and the other in which only one Echo Request was emitted. For the trial in which five Echo Requests were sent to the target, we record the mean RTT for all probes to each target. This strategy dramatically reduces the overall RTT error. § RELATED WORKZeng first posited the notion of SDN controllers instructing switches to send test packets in the context of their Automated Test Packet Generation system in <cit.>.We extend this idea to the variety of active measurements in use today, and implement such a system in real SDN hardware and software.Our work relies on RTT measurements of probes to various network targets generated and received by a controller from SDN switches; quantifying latencies in SDNs is therefore fundamental to our work. Rostos develop the OFLOPS platform to evaluate switch implementations in various use-cases <cit.>.OFLOPS examines processing delays in switches when performing specific actions such as forwarding and packet modification.Our work also examines processing delays in both hardware and software implementations of modern devices. In <cit.>, He measure inbound (switch to controller) and outbound (controller to switch) latencies induced by events and flow-modification rule insertions, deletions, and modifications, respectively. Our timing analysis is impacted by delays caused by events as well, which we similarly analyze on our devices using a passive tap. SLAM <cit.>, a tool used to monitor and estimate latencies in data centers, sends packets between SDN switches to estimate the latency along the path between them. SLAM generates notifications to a controller via notification messages sent by the first and last switches along a path when a probe is received. As in our work, in <cit.> the authors account for controller to switch latency by continuously monitoring it viamessages.The notion of using network devices to perform active measurements was first standardized in <cit.> and <cit.>, protocols for performing one and two-way active measurements of delay and loss. Our work is broader, generalizes these protocols, and provides a means for such schemes to be implemented in network switches without explicit vendor support. Most closely related to our own work is SDN traceroute <cit.>, a technique toreveal the sequence of switches and ports that actual data-plane packets traverse in an SDN network.SDN traceroute relies on injecting measurement probes via messages, and installing rules to match and retrieve tagged probe traffic via messages.While SDN traceroute and rely on the same primitives, is designed to provide a platform for existing measurements, IP-level ping and traceroute, using SDNs. § CONCLUSIONS AND FUTURE WORKWe introduce , an active measurement application for SDNs. enables active measurement experiments to be conducted wherever SDNs are deployed, bringing measurement capability to the core of the network as opposed to traditional edge-based platforms. We implement two common active network measurement utilities – , to determine RTTs from an SDN switch to network target, and an ICMP-based . To validate our RTT measurements, we conduct multiple experiments to identify sources of error and quantify their effects when using as a measurement platform. We find that these effects vary significantly according to the location of the controller and between hardware and software SDN switches, but can be mitigated by sending several packets to each target, and using the mean RTT from all probes. Finally, we demonstrate the ability to conduct custom measurements by designing a capability to perform active router discovery by querying routers for an imagined “router ID” value containing that device's ASN and an identifying string. By sending an ICMP packet with an unused ICMP type and code value to a router supporting this capability, obtains topological data about a device directly, rather than via current, error-pronemethods. §.§ Future WorkWith instances deployed at the network core, we aim to understand forwarding behavior not accessible to measurement experiments performed at the network edge. For example, do s initiated from the network edge follow the same terminal path to a destination as those initiated from a tier 1 AS? That is, do operator policies pertaining to source IP address affect the path selection in ways that are not discernible to edge-based vantage points? Additionally, as we note in Section <ref>, SDN controllers are capable of arbitrary packet creation. These packets need not adhere to the constraints placed upon ordinary hosts; for example, a controller may generate IP datagrams with any source IP address. If destined for another switch, these spoofed source address packets can reveal source address validation policies implemented by network operators unmeasurable by platforms at the edge of the Internet. Finally, 's unique contribution of offering measurement vantage points from the core of the network affords the opportunity to conduct certain estimates, like bandwidth usage, closer to the quantity being measured. We envision using , or a similar system, to “tag” portions of the traffic flowing through vantage points to better understand traffic characteristics as it is routed through network operator-administered infrastructure. Finally, while we focus on OpenFlow-based SDNs in this work due to the ubiquity of their real-world deployment, we note that the SDN space is rapidly evolving.For example, recent work on programmable network processors <cit.> promises a much richer set of abstractions and capabilities that can utilize if realized.§ ACKNOWLEDGMENTSWe thank Mark Allman, Steve Bauer, and Ethan Katz-Bassett forvaluable early feedback. Views and conclusions are those of the authors and should not be interpreted as representing the official policies or position of the U.S. government.abbrv

http://arxiv.org/abs/1702.07946v1

xian.chen@pku.edu.cn Astronomy Department, School of Physics, Peking University, 100871 Beijing, China Kavli Institute for Astronomy and Astrophysics at Peking University, 100871 Beijing, Chinapau@ice.cat Institut de Ciències de l'Espai (CSIC-IEEC) at Campus UAB, Carrer de Can Magrans s/n 08193 Barcelona, SpainKavli Institute for Astronomy and Astrophysics at Peking University, 100871 Beijing, ChinaInstitute of Applied Mathematics, Academy of Mathematics and Systems Science, CAS, Beijing 100190, ChinaZentrum für Astronomie und Astrophysik, TU Berlin, Hardenbergstraße 36, 10623 Berlin, Germany The formation of compact stellar-mass binaries is a difficult, but interesting problem in astrophysics.There are two main formation channels: In the field via binary star evolution, or in dense stellar systems via dynamical interactions.The Laser Interferometer Gravitational-Wave Observatory (LIGO) has detected black hole binaries (BHBs) via their gravitational radiation. These detections provide us with information about the physical parameters of the system. It has been claimed that when the Laser Interferometer Space Antenna (LISA) is operating, the joint observation of these binaries with LIGO will allow us to derive the channels that lead to their formation.However, we show thatfor BHBs in dense stellar systems dynamical interactions could lead to high eccentricities such that a fraction of the relativistic mergers are not audible to LISA.A non-detection by LISA puts a lower limit of about 0.005 on the eccentricity of a BHB entering the LIGO band. On the other hand, a deci-Hertz observatory, like DECIGO or Tian Qin, would significantly enhance the chances of a joint detection, and shed light on the formation channels of these binaries. Revealing the formation of stellar-mass black hole binaries: The need for deci-Hertz gravitational wave observatories Pau Amaro-Seoane December 30, 2023 =====================================================================================================================Introduction.–The first LIGO events, GW150914 and GW151226 <cit.>, are consistent with mergers of General-Relativity black holes (BHs). Data analysis reveal that the orbits started at a semi-major axis of a∼10 Schwarzschild radii (R_S) with an eccentricity of e<0.1. The BH masses are about M_1≃36 and M_2≃29 M_⊙ for GW150914 and M_1≃14 and M_2≃7.5 M_⊙ for GW151226.The detections can be used to infer new, more realistic event rates, of about 9-240  Gpc^-3 yr^-1 <cit.>. This rate agrees with two formation channels: (i) evolution of a binary of two stars in the field of the host galaxy, where stellar densities are very low (e.g <cit.>) or (ii) via exchange of energy and angular momentum in dense stellar systems, where the densities are high enough for stellar close encounters to be common (e.g. <cit.>).LIGO and other ground-based gravitational wave (GW) observatories, such as Virgo, are, however, blind with regarding the formation channels of BH binaries (BHBs). Both channels predict populations in the 10-10^3  Hz detector band with similar features, i.e. masses larger than the nominal 10 M_⊙, a mass ratio (q≡ M_2/M_1) of about 1, low spin, and nearly circular orbits <cit.>.It has been suggested that a joint detection with a space-borne observatory such as LISA <cit.> could allow us to study different moments in the evolution of BHBs on their way to coalescence:LISA can detect BHBs when the BHs are still 10^2-10^3 R_S apart, years to weeks before they enter the LIGO/Virgo band <cit.>.At such a separation, the orbital eccentricity bears the imprint of the formation channel because (i) BHBs in dense stellar systems form on systematically more eccentric orbits and (ii) the GW radiation at this stage is too weak to circularize the orbits <cit.>. Therefore, circular binaries typically form in the field, while eccentric ones through the dynamical channel.Recent studies further predict that those BHBs with an eccentricity of e>0.01 in the LISA band preferentially originate from the dynamical channel <cit.>.In this letter we prove that eccentric BHBs originating in dense stellar environments have a large chance to elude the LISA band.Inaudible black hole binaries–Non-circular BHBs have two distinct properties. (i) Eccentricity damps the characteristic amplitude (h_c) of each GW harmonic, as compared to a circular BHB. In Figure <ref> we depict two sources similar to GW150914 but originating from two distinct channels, i.e. with two different initial eccentricities. In the low-eccentricity case, the n=2 harmonic predominates and it is strong enough to be jointly detected by LISA and LIGO/Virgo.In the (very) eccentric case, however, the amplitudes of the harmonics are orders of magnitude below the noise level of LISA, so that a joint detection is ruled out. When the eccentricity has been significantly damped, about one hour before the merger, the dominant harmonic starts to converge to the n=2 one, and later, upon entering the LIGO band, becomes indistinguishable from that in the circular case. Therefore, the imprint about the formation channel is lost.(ii) Increasing the eccentricity shifts the peak of the relative power of the GW harmonics towards higher frequencies (see Fig. 3 of <cit.>). Hence, more eccentric orbits emit their maximum power at frequencies farther away from LISA. More precisely, when e=0, all the GW power is radiated through the n=2 harmonic, so that the GWs have a single frequency of 2/P, where P=2π(GM_12/a^3)^-1/2 is the orbital period and M_12=M_1+M_2.On the other hand, when e≃1, the n=2.16(1-e)^-3/2 harmonic becomes predominant <cit.>, so most GW power is radiated at a frequency of f_ peak=2.16(1-e)^-3/2P^-1.In Figure <ref> we display the a-(1-e) plane for a BHB.The boundaries of the stripes have been estimated by looking at the minimum and maximum frequencies audible by the detectors, f_1 and f_2, and letting f_1< f_ peak< f_2, with f_ peak defined before.If a BHB is evolving only due to GW emission, it will evolve parallel to the green lines. Thesetrack are parallel to the stripes because as long as e≃1, the pericenter distance, r_p=a (1-e), is almost constant during the evolution <cit.>, and a constant r_p corresponds to a constant f_ peak. Because of this parallelism, a BHB cannot evolve into the band of a GW detector if it initially lies below the detector stripe.Hence, we can see that some binaries will fully miss the LISA/TJ range. A good example is the eccentric BHB we chose for Figure <ref>. A detector operating at higher frequencies, such as TQ or DECIGO, can however cover the relevant part of the phase-space, so that a joint search is possible. These detectors could alert LIGO/Virgo decades to hours before an event is triggered, as one can read from the isochrones of Figure <ref>.Dense stellar environments.–BHBs such as the one we have used for our last example completely miss the LISA/TJ band. Eccentric binaries typically originate from dense stellar systems such as globular clusters (GCs) and nuclear star clusters (NSC), as shown by a number of authors in a number of publications <cit.>. In these systems, BHs diffuse towards the center via a process called mass segregation <cit.>. To model it, we adopt a Plummer model <cit.>, and we assume that the mean stellar density is ρ_*=5×10^5 M_⊙  pc^-3 and the one-dimensional velocity dispersion is σ_*=15  km s^-1. These values correspond to a typical GC with a final mass of M_ GC≈10^5 M_⊙ and a half-mass radius of R_h≈0.5 pc.We note, however, that the main conclusions derived in this work do not significantly change for a NSC.The two driving and competing mechanisms in the evolution of any BHB in the center of the cluster are (i) interaction with other stars, “interlopers”, which come in at a rate of Γ∼2π Gρ_*a(M_12/M_*)/σ_*, with M_*=10 M_⊙ the mean mass of the interlopers because the cluster has gone through mass segregation, and (ii) gravitational radiation, which shrinks the orbital semi-major axis at a rate of ȧ_ gw=-8 c R_S^3q (1+q)/5a^3(1-e^2)^7/2(1+73/24e^2+37/96e^4), <cit.>. We can readily separate the phase-space in two distinct regimes according to these two competing processes by equating their associated timescales: t_ int:=1/Γ and t_ gw:=(1/4) |a/ȧ_ gw|, which defines the threshold shown as the thick, black line in Figure <ref>. The reason for the 1/4 factor is given in <cit.>.Below the curve, BHBs will evolve due to GW emission. Above it, close encounters with interlopers are the main driving mechanism, so that BHBs can be scattered in both directions in angular momentum in a random-walk fashion. The scattering in energy is less significant but also present (see <cit.> and discussion in <cit.>).Possible ways of forming relativistic BHBs.– Different mechanisms have been proposed in the literature to form a BHB which eventually might end up emitting detectable GWs.(1) Primordial binaries: In stellar dynamics this term refers to binaries already present in the cluster which form via stellar evolution.Population synthesis models predict that these binaries populate the area of phase-space displayed as the grey thick-dashed box of Figure <ref> (see e.g. <cit.>). We note that only a small fraction of them are in the LISA/TJ band.(2) Dynamics: (2.1) Close encounters of multiple single, i.e. initially not bound, objects also form BHBs (see e.g. <cit.>). Their formation follows a thermal distribution in e (e.g. <cit.>), like primordial binaries, but the distribution of a is better constrained: When the binding energy of the binary, E_b=GM_1M_2/(2a) becomes smaller than the mean kinetic energy of the interlopers, E_*=3M_*σ_*^2/2, the binary ionizes <cit.>. The threshold condition E_b=E_* can be expressed in terms of a “hard radius”, a_h=GM_1M_2/(3M_*σ_*^2).These “hard” binaries heat up the system, meaning that they deliver energy to the rest of the stars interacting with them: Binaries with a<a_h impart on average an energy of Δ E≃ kGμ M_*/a to each interloper, where μ is the reduced mass of the binary and k is about 0.4 when M_1≃ M_2≃ M_* <cit.>.The interloper hence is re-ejected into the stellar system with a higher velocity because of the extra energy, v∼(3σ_*^2+2kGμ/a)^1/2, and the center-of-mass of the BHB recoils at a velocity of v_b∼ M_*v/(M_1+M_2).Occasionally, the BHB will leave the system if this velocity exceeds the escape velocity of the GC, v_ esc=√(2.6GM_ GC/R_h) <cit.>. The threshold for this to happen is defined by the condition v_b=v_ esc, i.e. the binary must have a semi-major axis smaller than the “ejection radius”, a_ ej. Therefore, all of these BHBs are confined in a_h<a<a_ ej of Figure <ref>. Because of their thermal distribution, we have that 95% of them have e<0.975. Therefore, they populate an even smaller area than those primordial binaries.(2.2) Binary-single interactions: Initially we have a hard BHB which interacts with a single object in a chaotic way. During the interaction the interloper might excite the eccentricity of the inner binary to such high values that the binary is on an almost head-on-collision orbit, to soon merge and emit a detectable burst of GWs <cit.>. This happens only if t_ gw is shorter than the period of the captured interloper P_ int. The event rate for BHBs has not been calculated for this scenario but earlier calculations for neutron-star binaries find it to be 1  Gpc^-3 yr^-1 <cit.>.We derive now the eccentricities of these BHBs: Suppose the semi-major axis of a BHB changes from a (with, of course, a_ ej< a< a_h) to a', and e to e' during the three-body interaction, and the final orbit of the interloper around the center-of-mass of the BHB has a semi-major axis of a_ int.Energy conservation results in the following relations,a'>a and a_ int≃ 2a/(1-a/a') (see <cit.>), where we neglect the initial energy of the interloper because the BHB is assumed to be hard.Then using a conservative criterion for a successful inspiral, t_ gw(a', e')=P_ int(a_ int), we derive e' for the BHB, which allows us to confine the range of eccentricities as the dashed, blue curve of Figure <ref>.(2.3) Hierarchical triple: This is similar to the previous configuration, but now we only consider 1<a'/a<1.5, because this requires that a_ int > 6 a, in which case theconfiguration is stable <cit.>.This leads to a secular evolution of the orbital eccentricity of the inner BHB which is known as the Lidov-Kozai resonance (see <cit.> and also <cit.>).The inner BHB will decouple via GW emission and merge at a critical eccentricity, and the merger rate has been estimated to be 0.3-6  Gpc^-3 yr^-1 <cit.>.We follow the scheme of <cit.> of isolated hierarchical triples but impose four additional requirements which are fundamental for a realistic estimation of the threshold eccentricity in our work: (a) The BHB has a_ ej< a< a_h.(b) The third body orbiting the BHB has a mass of M_ int=10 M_⊙ because of mass segregation, and an eccentricity of e_ int=2/3, which corresponds to the mean of a thermal distribution <cit.>.(c) The outer binary, i.e. the third object and the inner BHB, is also hard, so that a_ int<GM_12/(3σ_*^2). (d) The pericenter distance of the outer binary, a_ int(1- e_ int) should meet the criterion for a stable triple (Eq. 90 in Ref.<cit.>). These conditions delimit the range of eccentricities as shown by the dashed, orange lines in Figure <ref>.(3) Gravitational braking: There is a small probability that two single BHs come to such a close distance that GW radiation dissipates a significant amount of the orbital energy, leaving the two BHs gravitationally bound <cit.>.For GCs, and using optimistic assumptions, these binaries contribute an event rate of 0.06-20  Gpc^-3 yr^-1 in the LIGO band <cit.>, while in NSCs it has been estimated to range between 0.005-0.02  Gpc^-3 yr^-1 <cit.>. The boundaries in Figure <ref> for BHBs formed via this mechanism can be calculated using the formulae of <cit.>.For that, we choose an initial relative velocity v in the range σ_*<v<3σ_* and an initial impact parameter b in the range 0.3b_ max<b<0.99b_ max to account for the majority of the encounters, because the encounter probability is proportional to b^2, and b_ max is the maximum impact parameter that leads to a bound binary. The first LIGO detections, had they been originated via this mechanism, should originate from the red area above the green line. Discussions and conclusions.–A joint detection of BHBs with LIGO/Virgo and LISA/TJ would be desirable because of the science payback. In this paper we show that the actual number of BHBs to be coincidentally detected is very uncertain.As Figure <ref> shows, LISA/TJ is already deaf to mildly eccentric BHBs: For example, a BHB at milli-Hertz orbital frequencies starting at a ∼ 10^-3 AU and 0.7≲ e≲0.9 will also be missed by LISA/TJ, but later be detectable by LIGO/Virgo.BHBs can form via the five mechanisms which we discussed in the list of possible formations. This allows us to pinpoint the regions in phase-space which produce BHBs that eventually will merge via gravitational radiation. The total area of these five regions is a small subset of phase-space.It is an error to assume that all binaries born in this subset are jointly detectable by LIGO/Virgo and LISA/TJ.Only a subset of that subset of phase-space will lead to successful joint detections. This sub-subset depends on the masses of the BHBs.We can see this in Figures <ref> and <ref>. While in the first figure the hierarchical triple gets into the LISA/TJ band, it does not in the second one.On the other hand, up to 95% of primordial and dynamical binaries (1 and 2.1 in the list of possible formations) are produced in the box delimited by grey dashed lines. In that box, and in principle, the BHBs can lead to sources jointly detectable by LIGO/Virgo and LISA/TJ. However, exceptions might occur if a scatter results in a BHB jumping towards high eccentricities. This probability has not been fully addressed. It requires dedicated numerical scattering experiments with relativistic corrections (e.g.<cit.>), as well as a proper star-cluster model to screen out BHBs that can decouple from the stellar dynamics (e.g. our model as presented in Figures <ref> and <ref>).We have shown that mergers in GCs produced by the mechanisms (2.2), (2.3), and (3) are inaudible to LISA.The event rates corresponding to these mergers have been largely discussed in the literature, but are uncertain, due to questionable parameters, such as the cosmic density of GCs and the number of BHs in them.Nevertheless, it has been estimated that the rate could be as large as 20  Gpc^-3 yr^-1 <cit.>, while the current LIGO detections infer a total event rate of 9-240  Gpc^-3 yr^-1. Moreover, these mergers could also originate in NSCs, <cit.>, and the event rates there are higher, up to 10^2  Gpc^-3 yr^-1 <cit.>. Therefore, future multi-band GW astronomy should prepare for LIGO/Virgo BHBs that do not have LISA/TJ counterparts. A non-detection by LISA/TJ is also useful in constraining astrophysics: It puts a lower limit on the eccentricities of the LIGO/Virgo sources, which according to Figures <ref> and <ref> is about 0.005. A deci-Hz detector, by covering the gap in frequencies between LISA/TJ and LIGO/Virgo,would drastically enhance the number of jointly detectable binaries.Acknowledgement.–This work is supported partly by the Strategic Priority Research Program “Multi-wavelength gravitational wave universe” of the Chinese Academy of Sciences (No. XDB23040100) and by the CAS President's International Fellowship Initiative. PAS acknowledges support from the Ramón y Cajal Programme of the Ministry of Economy, Industry and Competitiveness of Spain.We thank Bence Kocsis and Fukun Liu for many fruitful discussions, and Eric Peng for a thorough reading of our manuscript. 74 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Abbott et al.(2016a)Abbott, Abbott, Abbott, Abernathy, Acernese, Ackley, Adams, Adams, Addesso, Adhikari, and et al.]ligo16a author author B. P. Abbott, author R. Abbott, author T. D. Abbott, author M. R. 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http://arxiv.org/abs/1702.08479v2

National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan Department of Physics, University of Florida, Gainesville, FL 32611, USA ben.wu@nao.ac.jpDepartment of Physics, University of Florida, Gainesville, FL 32611, USA Department of Astronomy, University of Florida, Gainesville, FL 32611, USADepartment of Astronomy, University of Florida, Gainesville, FL 32611, USANational Astronomical Observatory, Mitaka, Tokyo 181-8588, JapanSchool of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UKDepartment of Physics, Florida State University, Tallahassee, FL 32306-4350, USA We study giant molecular cloud (GMC) collisions and their ability to trigger star cluster formation. We further develop our three dimensional magnetized, turbulent, colliding GMC simulations by implementing star formation sub-grid models. Two such models are explored: (1) “Density-Regulated,” i.e., fixed efficiency per free-fall time above a set density threshold; (2) “Magnetically- Regulated,” i.e., fixed efficiency per free-fall time in regions that are magnetically supercritical. Variations of parameters associated with these models are also explored. In the non-colliding simulations, the overall level of star formation is sensitive to model parameter choices that relate to effective density thresholds. In the GMC collision simulations, the final star formation rates and efficiencies are relatively independent of these parameters. Between non-colliding and colliding cases, we compare the morphologies of the resulting star clusters, properties of star-forming gas, time evolution of the star formation rate (SFR), spatial clustering of the stars, and resulting kinematics of the stars in comparison to the natal gas. We find that typical collisions, by creating larger amounts of dense gas, trigger earlier and enhanced star formation, resulting in 10 times higher SFRs and efficiencies. The star clusters formed from GMC collisions show greater spatial sub-structure and more disturbed kinematics.§ INTRODUCTION Most stars are thought to form in clusters within giant molecular clouds (GMCs). GMCs have typical hydrogen number densities of n_ H=100cm^-3, diameters of ∼tens of parsecs, masses of up to 10^6 M_⊙, and average temperatures of ∼ 10-30K. Dense clumps within GMCs, potentially traced as, e.g., Infrared Dark Clouds (IRDCs), are recognized as being the likely precursors to star clusters <cit.>.IRDCs have such high mass surface densities (Σ≳ 0.1  g cm^-2) that they are dark at mid-IR (∼10 μ m) and even far-IR (∼70 μ m) <cit.>. Their low temperatures (10-20K) <cit.>, high volume densities (n_ H>10^5  cm^-3), relatively compact sizes (∼few pc), and masses (∼ 10^2-10^5 M_⊙) indicate that they have the potential to be the precursors to most of the observed mass range of star clusters known in the Galaxy. The initial and early stages of star cluster formation can also be traced by dust continuum emission<cit.> and by samples based on emission of dense gas tracers <cit.>. Surveys of young embedded stars also can probe the structure <cit.>, age distribution <cit.> and kinematics of young clusters<cit.>. Currently, the dominant processes that induce the collapse and fragmentation of GMCs into star-forming clumps are poorly understood.Various theoretical models includeregulation by turbulence <cit.>, regulation by magnetic fields <cit.>, triggering by stellar feedback <cit.>, triggering by converging atomic flows <cit.>, and triggering via converging molecular flows, i.e., GMC-GMC collisions <cit.>. Semi-analytic models <cit.> and numerical simulations <cit.> of global galactic disks have shown that GMCs collide relatively frequently due to the approximately 2D geometry of a thin disk and interaction rates driven by differential rotation of galactic orbits. The average timescale between GMC collisions was found to be about 20% of a local orbital period within a flat rotation curve disk <cit.> <cit.>. A growing number of numerical studies have also shown that collisions between molecular clouds can provide conditions favorable for massive star and star cluster formation <cit.>. We note that in general comparison of the results between the simulations of different groups is complicated by the use of different initial conditions, different numerical methods and different included physics.Our approach here is to systematically build-up realism for our GMC collision simulations by including additional physics step by step that allows an understanding of the relative importance of different input assumptions. <cit.>and <cit.> developed a numerical study of GMC-GMC collisions, focusing on understanding the physical mechanisms as well as using them to predict observational diagnostics. Comparing magnetized, supersonically turbulent GMCs in colliding and non-colliding cases over a wide parameter space and investigating a varied array of potential observational signatures, they found that a number of indicators suggest similarities between the colliding scenarios to observed GMCs and IRDCs. Further, dynamical virial analysis suggested that dense ^13CO-defined structures created through GMC collisions were more likely to collapse and form massive star clusters when compared with more quiescently evolving structures.The next stage in our work is the crucial transition from collapsing clumps into star clusters. Properties of the stars that form, along with their dynamical evolution shortly thereafter, may provide insight into the dominant star formation mechanisms.The goal of this study is to answer the question: do realistic models of GMC collisions create star clusters that closely match the properties of observed young star-forming regions?We approach this question by further building upon our previous numerical framework of GMCs through the development of star formation sub-grid models, one of which is a novel magnetically-regulated model. We combine our existing gas-focused observational diagnostic methods with additional information from the population of star particles. Thus, we hope to provide insight to the star formation process by analyzing the evolution of IRDC-type structures into young star clusters.Section <ref> describes our numerical setup and the various star formation models.We then present our results in Section <ref>, which include:gas and star cluster morphologies (<ref>),properties of star-forming gas (<ref>), global star formation rates (SFRs) and efficiencies(<ref>), spatial clustering (<ref>),and star particle kinematics (<ref>).In Section <ref> we discuss our conclusions. § NUMERICAL MODEL§.§ Initial Conditions We further develop the numerical framework described in Paper II and introduce two star formation routines.Our GMCs are identical to those initialized in Paper II, which are motivated by observed GMC properties. The clouds are self-gravitating, supersonically turbulent, and magnetized. They are initialized with identical densities and offset by an impact parameter. The clouds are embedded in an ambient medium of ten times lower density (i.e., an atomic cold neutral medium, CNM), which for the colliding case, is converging along with the GMCs. The initial simulation properties are summarized in Table <ref>.The simulation domain is (128pc)^3 and contains two neighboring GMCs. The GMCs are initially uniform spheres, with Hydrogen number densities of n_ H,GMC = 100cm^-3 and radii R_ GMC = 20.0 pc. This gives each GMC a mass M_ GMC = 9.3 × 10^4 M_⊙.The ambient gas represents the atomic cold neutral medium (CNM) and has a density of n_ H,0 = 10cm^-3. The centers of the GMCs are offset by 2R_ GMC in the collision axis (x), 0 in the y-axis, and b=0.5R_ GMC in the z-axis.To approximate the density and velocity structures observed in GMCs, our clouds are initialized with a supersonic turbulent velocity field which is random, purely solenoidal, and follows the v_k^2∝ k^-4 relation, where k=π/d is the wavenumber for an eddy diameter d. Conventionally, the “k-mode” is normalized to the simulation box length. The gas within the GMC is initialized with Mach number ℳ_s≡σ/ c_s= 23 (for T=15 K conditions), of order virial. We set our fiducial k-modes to be {2,...,20}, where each mode within this range is excited.This is representative of the large-scale turbulent velocities (small k) spanning from the GMC diameters down to a small enough minimum scale (large k), which is numerically resolved, but expected to cascade to smaller scales. We do not drive turbulence, instead letting it decay within a few dynamical times.Note also that turbulence is initialized only within the initial volume of the GMCs while we leave the ambient medium non-turbulent. Note also the GMC collision will also drive turbulence in the clouds in that case.A large-scale uniform magnetic field of strength 10 μ G is initialized throughout the box at an angle θ=60^∘ with respect to the collision (x-) axis. This choice of |B| is motivated by the Zeeman measurements of typical GMC field strengths, summarized by <cit.>.In the fiducial colliding case, the bulk flows (including both the ambient gas and the GMCs) have a relative velocity of v_ rel=10km s^-1. In the non-colliding case, there is no bulk velocity flow.The simulations are run for 5Myr to investigate the onset of star formation. Note that this is 1Myr longer than the simulations described in Paper II, which focused on gas properties of the pre-star-forming clump. Note also that the freefall time given the initial uniform density GMCs is t_ ff=(3π/[32Gρ])^1/2≃ 4.35 Myr. However, the values of t_ ff for the denser substructures created by turbulence and by the collision are much shorter. Star formation is expected to occur in both non-colliding andcolliding cases, with the detailed properties of resulting starclusters acting as the key point of our investigation. llllclc Initial Simulation Properties4 0ptGMCAmbientn_ H(cm^-3) 100 10R(pc)20 ...M(M_⊙) 9.3×10^4...T(K) 15150 t_ ff (Myr) 4.35 ...c_ s(km/s)0.230.72v_A(km/s)1.845.83v_ vir(km/s)4.9...σ (km/s)5.2...ℳ_ s... 23 ... ℳ_A ...2.82 ... k-mode (k_1,k_2) (2,20) ...v_ bulk (km/s)±5±5B(μ G) 1010λa...4.3 1.5 βb...0.015 0.015 anormalized mass-to-flux ratio: λ = (M/Φ)/(1/2π G^1/2) bthermal-to-magnetic pressure ratio: β=8π c_s^2 ρ_0 / B^2§.§ Numerical CodeOur models are run using [http://enzo-project.org (v2.4)], a magnetohydrodynamics (MHD) adaptive mesh refinement (AMR) code <cit.>. We use the Dedner-MHD method, which solves the solves the MHD equations using the Harten-Lax-van Leer with Discontinuities (HLLD) method and a piecewise linear reconstruction method (PLM). The time is evolved using the MUSCL 2nd-order Runge-Kutta method. The ∇·B=0 solenoidal constraint of the magnetic field is maintained via a hyperbolic divergence cleaning method <cit.>.The simulation domain is realized with a top level root grid of 128^3 with 3 additional levels of AMR. Our models thus have an effective resolution of 1024^3, with a minimum grid cell size of 0.125 pc. We refine solely on the local Jeans length, setting a necessary requirement of resolving by 8 cells, i.e., higher than the 4 cells typically used to avoid artificial fragmentation <cit.>). Our higher resolution leads to larger volumes of the GMCs being better resolved and thus generally better resolution of, e.g., shocks <cit.>. We note that the Jeans criterion assumes purely thermal support. The gas in our simulations also has some magnetic support, so its effective “magneto-Jeans length” will be significantly larger than the thermal Jeans length in directions perpendicular to the magnetic field.Due to the relatively high bulk velocities andpotentially strong magnetic fields, we require the use of the “dual energy formalism” <cit.> , which separately solves the internal energy equation as well as the total energy equation, ensuring accurate calculation of pressures and temperatures in these conditions. If the ratio of thermal to total energy is less than 0.001, then the temperature is calculated from the internal pressure. Otherwise, the total energy is used.Additionally, we employ the “Alfvén limiter” (described in Paper II) to avoid exceedingly small timesteps set by Alfvén waves. This acts by choosing a maximum Alfvén velocity, v_A, max=B/√(4πρ_ min)=1×10^7cms^-1, and setting a density floor that is determined by the magnetic field. This predominantly affects only small pockets of very low-density gas with which we are less interested, and thus the dynamical results are deemed unaffected by this limiter.§.§ Thermal ProcessesWe assume a constant mean particle mass (μ=2.33 m_ H) throughout the simulation domain for simplicity, as our focus is on the dense molecular gas of GMCs.We also choose a constant adiabatic index γ=5/3. Note that this essentially ignores certain excitation modes of H_2 that may be relevant (i.e., shocks), but it is still the most appropriate single-valued choice of γ, given our focus on the dynamics of cold H_2. Also, we assume n_ He=0.1n_ H, giving a mass per H of 2.34×10^-24g.The PDR-based heating and cooling functions developed in Paper I are again used in these simulations. The assumptions are: (1) FUV radiation field of G_0=4 (i.e., appropriate conditions for the inner Galaxy, e.g., at Galactocentric distances of ∼ 4 kpc) and (2) a background cosmic ray ionization rate of ζ = 1.0 × 10^-16s^-1.The heating/cooling functions trace the atomic to molecular transition and recreate a multi-phase ISM. They span density and temperature ranges of 10^-3≤ n_ H/ cm^-3≤ 10^6 (extended to 10^10  cm^-3 via extrapolation) and 2.7 ≤ T/ K≤ 10^7, respectively.We use theexternal chemistry and cooling library[https://grackle.readthedocs.org/] <cit.> to incorporate our heating/cooling functions in tabular form into , modifying the energy equation.In order to avoid numerical instabilities related to the heating/cooling processes, we limit the timestep on each AMR level to a factor of 0.2 the minimum cooling time. Additionally, we set a hard floor for the minimum cooling timestep of 625  yrs.§.§ Star FormationWe utilize the particle machinery ofto model star formation.Specifically, star particles (i.e., collisionless, point particles with mass m_⋆) form within a simulation cell if certain local criteria are met.Two star formation routines are developed: (1) density-regulated star formation; (2) magnetically-regulated star formation.§.§.§ Density-regulated star formationOur first star formation routine is a “density-regulated” model, based on that of <cit.> <cit.>. Stars are formed within a cell only if they have been refined to the finest level of resolution and the density exceeds a particular threshold value, n_ H,sf. The fiducial star formation density threshold is chosen to be n_ H,sf = 10^6  cm^-3, which is set partly based on observed densities of pre-stellar cores <cit.>. We will consider variation of this parameter by a factor of two to higher and lower values. The temperature in the cell is also required to be <3000 K, to avoid star formation in dense, shock-heated regions, but we will see that this constraint is not of practical concern for the simulations presented here. Note that there is no requirement for gravitational boundedness of gas in the cell. Nor is there a requirement for net convergence of gas flow to the cell. These choices are motivated by the fact such conditions are not well resolved on the local scales associated with an individual cell. In addition, we expect that processes such as turbulence and diffusion of magnetic flux that are occurring on sub-grid scales (or scales near the grid scale that are not well resolved) will regulate star formation, e.g., creating local conditions that are gravitationally unstable, perhaps via converging flows. With these points in mind, this star formation sub-grid model using a density threshold is thus designed to be as simple as possible, enabling us to gain a clear understanding of how the results depend on its input parameters.In cells meeting the above conditions, star particles are then produced so that the SFR is, on average, equal to that expected if there is a fixed star formation efficiency per local free-fall time, ϵ_ ff, where the local free-fall time, t_ ff, is expressed as t_ ff=(3π/32Gρ)^1/2 = 4.4×10^4 n_ H,6^-1/2yr,i.e., the value for collapse of a uniform density sphere, where n_ H,6≡ n_ H/10^6cm^-3. We adopt a fiducial choice of ϵ_ ff=0.02, motivated by observations of GMCs, their star-forming clumps and stellar populations in embedded clusters, which suggest fairly low and density-independent values of ϵ_ ff <cit.>.Thus the SFR isṁ_⋆=ϵ_ ffm_ gas/t_ ff = 2.9×10^-5(ϵ_ ff/0.02) (Δ x/0.125pc)^3 n_ H,6^3/2 M_⊙yr^-1,where we have normalized to the minimum cell size, Δ x, relevant for the simulations in this paper.The timesteps in the simulation are typically quite short, i.e., much less than the signal crossing time of a cell, i.e., ≪ 1.2×10^5yr for a signal speed of 1 km s^-1. Thus the average mass of stars that are expected to be created in a given cell in a given simulation timestep is often very small, i.e., <1 M_⊙.To enable both a practical computation that does not involve too many star particles, but also with the eventual aim of producing star particles with masses that are characteristic of observed stellar masses, the star formation sub-grid model also involves a parameter of a minimum star particle mass, m_⋆,min. For the density-regulated models we consider here, we set m_⋆,min=10 M_⊙. Thus in this case the star particles represent small (sub-)clusters of stars, since the mean stellar mass is ≲ 1 M_⊙ for realistic stellar initial mass functions <cit.> (see also discussion of star particle dynamics in <ref>). With this value of m_⋆,min we are almost always in a regime in which the mass of stars to be created in a given timestep is smaller than m_⋆,min and so the decision to form a star particle or not needs to be implemented probabilistically, i.e., the “stochastic star formation” regime. In this case, the star particle is formed with probability ṁ_⋆Δ t / m_⋆,min, where Δ t is the simulation timestep. If on the other hand ṁ_⋆Δ t > m_⋆,min (which can occur in certain circumstances), then the star particle is simply created with this mass.Another factor affecting the choice of m_⋆,min is the desire not to change the gas mass in a cell by too large a fraction when the star particle is created, i.e., to avoid too large changes in density, pressure, etc. In general we set an upper limit of this fraction of 0.5. In the fiducial case a cell of size 0.125 pc at the star formation threshold density contains a minimum gas mass of m_ gas,min=63 M_⊙, so this fraction is ≲ 0.17 for these models (≲ 0.34 for the lower threshold density case).Overall there are three density (“d”)-regulated runs (i.e., three choices of threshold density) for each of the noncolliding (“nocol”) and colliding (“col”) simulation set-ups. The parameters of these star formation models and simulations are listed in Table <ref>. §.§.§ Magnetically-regulated star formation We introduce a new “magnetically-regulated” star formation model that takes into account magnetic criticality, i.e., star formation is only allowed to proceed if a cell has a mass to flux ratio that is greater than a certain value. If the cell is magnetically “supercritical” by this criterion, then it forms stars at a fixed efficiency per local free-fall time, ϵ_ ff, where we will adopt the same value of 0.02 that was used in the density-regulated models. Thus this magnetic criticality condition acts to replace the density-threshold criterion of <ref>. However, as we discuss below, the choice of m_⋆,min also introduces an effective minimum density for star formation in this model also.To assess the mass-to-flux ratio criterion, as an approximation, we treat each grid cell individually and calculate the dimensionless mass-to-flux ratioμ_ cell = ρΔ x √(G)/B c_1,where ρ is the density within a cell of length Δ x, G is the gravitational constant, B is the strength of the magnetic field within the cell, and c_1 comes from defining the critical mass-to-flux ratio as( M/Φ)_ crit = c_1/√(G)and is ultimately dependent of the geometry of the system. For an infinite disk, the value is c_1=1/(2π) <cit.>; for an isolated cloud it is roughly 1/√(63)∼ 0.126 <cit.>. We will consider variations of c_1 of a factor of two to higher and lower values. These and other parameters of the magnetically (“B”)-regulated star formation models are also listed in Table <ref>. We note that although the true mass-to-flux ratio depends on the geometry of the entire flux tube and cannot be completely confined to a localized quantity, this only acts as a first-order correction.If a cell is magnetically subcritical (i.e., μ_ cell<1), the magnetic pressure is deemed strong enough to withstand gravitational contraction, preventing any stars from forming within that cell. For those cells that are magnetically supercritical (i.e., μ_ cell>1), then star formation may be allowed to occur. However the two other criteria introduced in <ref>, i.e., the cell is resolved at the finest refinement level and T<3000 K, also must be satisfied. In addition, we also limit the fraction of gas mass in a cell that is turned into stars at a single timestep to be <0.5. Given the minimum star particle mass, m_⋆,min, this imposes an “effective density threshold” for the magnetically-regulated star formation model. For this reason, we will investigate magnetically-regulated models with m_⋆,min = 10 M_⊙ (as in the density-regulated models), but also with m_⋆,min = 1 M_⊙. These choices correspond to effective minimum threshold densities of n_ H,sf= 3.55× 10^5cm^-3 for m_⋆,min=10 M_⊙ and n_ H,sf= 3.55× 10^4cm^-3 for m_⋆,min=1 M_⊙.If all the above conditions are satisfied, then the star formation process is allowed to occur at fixed efficiency per local free-fall time, as described in <ref>. We will see that the magnetically-regulated models with m_⋆,min=1 M_⊙ can form significant numbers of star particles out of the stochastic regime, but these masses should not be interpreted as being a realistic assessment of the stellar initial mass function, since their values depend on the size of the simulation timestep.Overall there are four magnetically (“B”)-regulated runs (i.e., three choices of mass-to-flux threshold for m_⋆,min=10 M_⊙ and one run with m_⋆,min=1 M_⊙ at the fiducial mass-to-flux threshold) for each of the noncolliding (“nocol”) and colliding (“col”) simulation set-ups. The parameters of these star formation models and simulations are also listed in Table <ref>.§.§.§ Star particle and star cluster dynamics Once the star formation criteria are met, mass is removed from the cell and placed into a point-like star particle. These evolve as a collisionless N-body system. However, these are not treated as accreting sink particles, so they do not gain additional mass from the gas, which we expect to be realistic due to the action of stellar winds from the young stars.The particles still interact with the gas gravitationally via a cloud-in-cell (CIC)algorithm which maps the particle positions onto the grid. This limits the closest distances between star particles to the grid resolution, ultimately resulting in softer mutual gravitational interactions.As a result, small scale, i.e., internal, star cluster dynamics is not expected to be well-modeled.However, the early stages and larger scales of the spatial and kinematic distribution of the stars, should be more accurately followed.Note also that our ability to follow the true internal dynamics of the formed star clusters is limited by the fact that we do not fully allow for the presence of a range of stellar masses, including both low-mass and high-mass stars, or the presence of binary or higher order multiple star systems. However, since our ability to accurately follow the dynamical evolution of the star cluster is mostly limited by the fact that gravitational forces are not well resolved below the grid scale of the simulation, our focus is mostly on the global distribution of stars in the simulations and the large scale spatial and kinematic distributions of the stars in the clusters, e.g., low order spatial mode asymmetries.The current modeling also does not include feedback from the formed star particles. A goal of a future paper is to include protostellar outflow momentum feedback in these models, but at the moment the star formation that results should be considered a baseline estimate in the limit of zero feedback.§ RESULTSWe perform analysis of each the simulations, comparing and contrasting star formation models as well as non-colliding vs. colliding cases. In particular, we discuss: morphology of the clouds and clusters (<ref>); properties of star-forming gas (<ref>); global star formation rates (<ref>); spatial clustering of stars (<ref>); and star vs. gas kinematics (<ref>). §.§ Cloud and Cluster Morphologies The morphology of the gas and the stars are shown in Figure <ref> for the non-colliding clouds and Figure <ref> for the colliding clouds. In the non-colliding cases, the gas evolution is essentially identical, where turbulent velocities and self-gravity create a network of relatively slowly growing filaments with increasing differentiation in mass surface density. Evolution is relatively passive and quiescent.In general, the onset of star formation takes place near 4.5Myr for the m_⋆,min=10 M_⊙ cases and 3.5Myr for the m_⋆,min=1 M_⊙ case.Both density-regulated and magnetically-regulated models result inpockets of localized star formation concentrated at density peakswithin filamentary structures. The slightly more populated network offilaments in the northeast region forms a higher number of(small) star clusters, but overall, star formation is scatteredsparsely throughout both GMCs and remains relatively spatiallyisolated.The magnetically-regulated models form clusters with ahigher degree of elongation, i.e., following the axes of the natalfilaments. There is also slightly more widespread star formationactivity compared with the density regulated case. By t=5.0  Myr,approximately 5-8 clusters have formed in the density-regulated cases,with the higher critical density models forming fewer clusters, whereas10-12 separate clusters have formed in the magnetically-regulated cases.Differences are more pronounced in the m_⋆,min=1 M_⊙ case, as stars form in elongated clusters along the filaments instead of the more localized spherical clusters as in the density regulated case or the slightly eccentric clusters as in the m_⋆,min=10 M_⊙ magnetically regulated cases. However, one needs to bear in mind that the magnetically-regulated models also involve an effective minimum density threshold as an additional requirement for star formation. This threshold density depends on the minimum star particle mass that is allowed in the model via the requirement that no more than 50% of the cell's gas mass can be converted to a star particle (see <ref>). These effective threshold densities are n_ H,⋆ min= 3.55× 10^5cm^-3 for m_⋆,min=10 M_⊙ and n_ H,⋆ min= 3.55× 10^4cm^-3 for m_⋆,min=1 M_⊙. Thus the variation in m_⋆,min is a way of investigating how varying this effective density threshold influences the resulting stellar population. Recall that for star-forming gas, the star formation activity in lower density regions is suppressed because the rate scales inversely with the local free-fall time, i.e., SFR =ϵ_ ff m_ cell/t_ ff∝ n_ H^3/2. Thus the overall SFR in these magnetically-regulated models will depend on both the probability distribution function (PDF) of densities of the gas above the effective threshold density that achieves the magnetic criticality condition. In the simple density-regulated models, the SFR will simply depend on the PDF of densities above the threshold density.Considering now the GMC collision cases (Figure <ref>), we see that they produce a much more active and dynamic environment that leads to creation of much denser gas structures (see also Paper II). At the interface of the colliding flows, a primary high-density filamentary structure is formed.This relatively compact, sheet-like structure lies predominantly in the plane perpendicular to the collision axis, with smaller filaments extending outward in various directions. Mass surface densities of ∼1  g cm^-3 are reached much sooner compared to the non-colliding case.This results in earlier and more rapid star formation, generally beginning near 3.0 to 3.5Myr for the m_⋆,min=10 M_⊙ cases and earlier than 3.0Myr for the m_⋆,min=1 M_⊙ case of magnetically-regulated star formation.In all such cases, the clusters form in the central colliding region, at the peaks of filaments located in the primary filamentary network. These sites often correspond with overdense clumps located at filament junctions, potentially pointing toward star formation triggered by filament-filament interactions on the smaller scale. By t=4.5  Myr, the individual star clusters have grown and merged into one dominant star cluster located near (x,y)=(5 pc,10 pc), while stars continue to form from dense clumps scattered throughout the post-shock colliding region. This large cluster appears to contain multiple populations of smaller star clusters that have merged together through a combination of gravitational attraction and initial velocity inherited from the natal gas of the collision.The spatially separated clusters from earlier times have grown in population and are moving toward the main cluster, while a few smaller clusters are continuing to form along the still-colliding dense filamentary gas. By t=5.0 Myr, the main cluster (which has grown to a few thousand stars in the m_⋆,min=10 M_⊙ case and a factor of 10 higher in the 1 M_⊙ case) is co-located with the majority of the dense gas, as more star clusters form in the vicinity. There exists a small population of individual stars that form in relative isolation and/or are dynamically ejected from the denser regions.The factor of two variations in n_ H,sf do not greatly alter the overall cluster morphology. However, there are small differences in total cluster number as well as cluster size corresponding to the density threshold, with increasing thresholds leading to reduced star and cluster formation.The magnetically-regulated models exhibit slightly earlier star formation, initializing just prior to t=3.0 Myr in each case, and a higher number of clusters formed, which culminates in a larger central cluster at late times compared with the density-regulated models. Within these models, increasing values of c_1 result in reduced star formation overall, though the locations where star formation is centered do not change.Themodel initiates star formation the earliest, with a primary central cluster and 5-8 smaller clusters already formed by t=3.0  Myr.Stars form in elongated structures directly corresponding to the dense gas filaments similar, on small scales, to that of themodel.By t=4.0  Myr, stars are present throughout the primary filament, still generally following the filamentary structure of the gas, with smaller clusters forming elsewhere throughout the colliding region. By t=5.0  Myr, the primary central cluster has grown directly as well as from gravitational interactions with the nearby clusters. Outlying clusters have continued to increase in size and number. Within dense filaments, the B-field is generally aligned perpendicular to the filament axis (see Paper II). Qualitatively, the mass-to-flux is expected to be highest at the density peaks, locally, but is expected to decrease when the entire flux tube is taken into account due to the lower-density environment surrounding the filaments.We note also that although these are ideal MHD simulations, some numerical diffusion of flux is expected to occur that may influence the star formation activity. The effects of modeling non-ideal MHD processes will be explored in a future paper in this series. Figure <ref> shows the combined 3D structure of the gasdensity, magnetic field geometry, and star particles. Theandmodels are compared at the same time, t=5.0  Myr,revealing the denser and more compact structure created in the GMC collision.These figures show the contrasting global morphology of the gas and stellar structure. Detailed analysis of various aspects of these properties is performed in subsequent sections.The different star formation modelscan be directly compared via |B| vs. n_ H phaseplots (Figure <ref>).The critical thresholds for each model are plotted over phaseplots inrespective non-colliding and colliding runs without star formation (i.e., fiducial runs from Paper II extended to t=5.0  Myr).In this manner, the total mass along with various properties of thegas affected by each star formation model can be estimated. The colliding case forms regions of overall higher density and magnetization, both enhanced by approximately an order of magnitude. At a given density, gas in the colliding case generally contains stronger field strengths due to the nature of the compressive flows. The star formation thresholds for both the density-regulated and magnetically-regulated star formation routines are overplotted as blue and red lines, respectively.The density-regulated star formation regime affects a greater total gas mass in the colliding case. As the critical density threshold decreases, the number of affected cells in both scenarios increases, leading to increased star formation regardless of magnetic field strength.As the threshold for mass-to-flux ratio is lowered, a similar pattern of increasing star formation occurs.Key differences from the density-regulated models become apparent as star formation is now allowed to occur in regimes of lowdensity, super-critical gas and is inhibited in high-density,sub-critical gas. Overall, the various models primarily create starsfrom the same gas, though narrow regimes exist in which stars formexclusively within certain routines. As discussed above, in these magnetically-regulated models, the effective minimum density thresholds for star formation provide an additional bound.In the m_⋆,min=10 M_⊙ models, much of the gas in the colliding cases–and even more so in the non-colliding cases–is limited by this effective density threshold.The 1 M_⊙ cases allow star formation from a larger amount of locally super-critical gas. However, we will see below that the overall mass of stars formed by 5 Myr in the colliding case depends only weakly on this choice. §.§ Properties of Star-forming Gas We examine the masses of young stars and properties of their progenitor gas cells, just before a star particle is created. Figure <ref> displays the cumulative histograms over 5 Myr of the stellar masses and key properties of the star-forming gas.For the non-colliding cases, the stars form strictly at their threshold masses, indicating purely stochastic star formation is occurring. The density-regulated models form approximately 100-250 stars each, with higher critical density thresholds resulting in fewer stars.The distributions of cell densities peak at the thresholds of 0.5, 1, and 2 × 10^6cm^-3 for the respective models and extend above the cutoffs by factors of a few. Gas temperatures range from 6 to 40 K, averaging approximately 10 to 20 K with higher density thresholds resulting in slightly lower temperatures. The local normalized mass-to-flux ratio of the star-forming cells in these models is supercritical by factors of few forto few tens for . The velocities of these cells are generally a few km s^-1, consistent with values expected from decaying turbulence in the self-gravitating GMCs.The magnetically-regulated non-colliding models exhibit slightly higher numbers of stars (few hundred for the m_⋆,min=10 M_⊙ models).Across these three models, distributions for density have peaks at the cutoff of ∼ 3×10^5cm^-3, temperatures primarily near 20 K (slightly above equilibrium), μ_ cell near 3-5 (slightly supercritical), and |v| near 2 km s^-1 (supersonic but consistent with decay of the initial turbulence). There exist slight trends of increasing density and decreasing temperature as the thresholds for μ_ cell increase between models. For the m_⋆,min=1 M_⊙ model, approximately 20 times more star particles are created.The distribution of cell densities also exhibit a cutoff at the effective minimum density of ∼ 3×10^4cm^-3 in this model, with the spread of densities reaching up to a factor of 20 higher. The temperature and velocity distributions exhibit similar peaks and spreads as their higher minimum stellar mass counterparts. However, the cells generally have a lower μ_ cell near 1 (magnetically critical) when stars are formed. This suggests that the condition for criticality is reached before the cell density grows to a point at which it can produce more massive stars in a given timestep and so stays in the stochastic limit.The colliding models with density-regulated star formation produce ∼ 6×10^3 stars, over 20 times the number formed from the non-colliding models over the same time period. The density distributions are similar to the non-colliding cases, peaking at the cutoffs, but exhibit an increased spread with cells reaching densities higher by factors of a few. The gas temperatures are also higher, averaging near 30-40 K but with a few cells reaching ∼ 90  K. Temperatures are generally lower for the higher-density cutoff models, but all peak at temperatures higher than equilibrium, likely due to shocks produced throughout the primary colliding region. The collision also produces high-density gas at a wide range of μ_ cell, ranging from a few times subcritical up to ∼ 20 times supercritical. The distributions peak near the value for magnetic criticality, with lower higher-density cutoff models corresponding with higher values of μ_ cell.Cell velocity distributions are nearly identical, peaking near 10-20 km s^-1. For the magnetically-regulated colliding models with m_⋆, min=10 M_⊙, a similar star particle count is seen, exceeding their respective non-colliding models by factors of 10 to 20. Themodel also forms some stars outside of the stochastic regime, as masses of ∼ 12-13M_⊙ are created. It is important to recall that the expected mass of the star particle to be created depends on the local SFR in the cell, i.e., on ϵ_ ff and the cell density, but also on the timestep of the simulation. Thus the presence and mass distribution of these higher mass star particles should not be over-interpreted. The presence of stars outside the stochastic regime simply indicates that some very high density, high SFR cells are present, and this is confirmed in the plots showing the density distributions, with some densities up to n_ H=10^7  cm^-3. We note that star-forming cells can also have higher temperatures near 30 to 40 K, perhaps indicating creation of the dense gas in shocks, but recall that the star formation sub-grid model does not assess the degree of gravitational instability in the gas. The star-forming cells show a concentration of gas at the minimum magnetic criticality cutoff. They also exhibit generally higher velocities indicating strongturbulence and/or bulk motion associated with the GMCs.Themodel has the greatest total number of stars formed, i.e., ∼ 5×10^4 and forms a range of stellar masses up to ∼ 7 M_⊙ (but again this mass function should not be expected to be compared to a real IMF, rather being simply the way the model ensures the total mass of stars formed is correct given the model parameters).Cell number densities range from ∼ 3×10^4 to ∼ 9×10^6cm^-3. Temperatures are near 40 K and μ_ cell reaches a few tens but increases in cell number towards the critical value cutoff of 1. Velocities exhibit a similar trend as the 10 M_⊙ models, showing high levels of turbulence, bulk motion,and/or infall to the primary cluster.§.§ Star Formation Rates and EfficienciesThe star formation rate (SFR) and overall star formation efficiency (SFE) of molecular clouds are important quantities that help determine the global galactic star formation process.The time evolution of these quantities for both non-colliding and colliding cases, for each star formation model, is shown in Figure <ref>.The SFR is calculated as the time derivative of the total mass of the star particles. The efficiencies are determined by normalizing the total stellar mass by the combined gas mass of the two original GMCs. The evolution of these quantities is measured in simulation time, as well as relative to the freefall time of the initial GMC density (t_ ff=4.35 Myr).In the non-colliding density-regulated cases, star formation initiates shortly after 1 t_ ff, i.e., at approximately t=4.4 Myr. The higher-cutoff density models form stars at slightly later times corresponding to when the critical density is achieved. At a given time, SFR and SFE vary by factors of a few between models. Over thecourse of the next 0.5 Myr (until simulation completion), the SFRsincrease to ∼ (3-6)×10^-3 M_⊙yr^-1 andthen generally level off. Total stellar masses of ∼ 9×10^2-3×10^3 M_⊙ are created, corresponding to ϵ≈ 1% by 1.15 t_ ff.For the magnetically-regulated cases with m_⋆,min=10 M_⊙, star formation also starts after 1 t_ ff, and evolves in a similar manner as the density-regulated models except with slightly higher SFRs and efficiencies. The differences between these models is also much smaller, as the three 10 M_⊙ magnetically-regulated models reach about 6×10^-3 M_⊙yr^-1 by 1.1 t_ ff and then level off. The SFEs also reach and slightly exceed 1% by the simulation end time. Thecase exhibits the most dissimilar behavior of the non-colliding models, initiating star formation approximately 1 Myr earlier and reaching 4×10^-3 M_⊙yr^-1 and ϵ=1% by 1 t_ ff. By 5 Myr, the SFR and SFR are approximately 2-3 times greater than the other magnetically-regulated models.The above trends are mostly likely caused by the fact that all these star formation models have effective density thresholds that need to be met to allow star formation to proceed, even the magnetically-regulated models (see <ref>). These thresholds decrease monotonically as we consider the density-regulated models, then the B-(0.5,1,2)-nocol models, and then the B-1-1M-nocol model. The simulations are in a regime in which the total SFR and eventual total SFE are set mostly by the fraction of gas in the GMCs that can meet these density threshold criteria. For the particular B-field strengths in these simulations (i.e., 10μ G), the choice of the magnetic threshold parameter does not play a significant role in setting the SFR.The colliding cases produce much higher SFRs and SFEs during their evolution. The density-regulated models begin forming starsat a rapid pace shortly after t=3.2 Myr, with higher densitythresholds slightly delaying the onset of star formation. There is some oscillation in the growth of the SFRs, but overall it increases from onset until t_ ff near 0.08 M_⊙yr^-1 andthen levels off through the culmination of the simulations. Star formation efficiencies reach 1% by 3.7 to 4 Myr and more than 20% by t_ ff. While the early behavior differs slightly in time between the density-regulated models, they appear to converge at later times.The m_⋆,min=10 M_⊙ magnetically-regulated colliding cases show very similar results to each other throughout the whole evolution, which indicates that the SFR is not limited by the mass-to-flux thresholds in this simulation set-up. Indeed, these models also converge with the density-regulated models by about 4 Myr. starts forming stars at the earliest times, but it also shows convergence in SFR by about 4 Myr.These results indicate that the SFR is in fact not limited by the density threshold criteria either. In the GMC-GMC collision the SFRs appear to be set by the creation of structures that can place gas at densities greater than any of the threshold densities, after which, even with ϵ_ ff=0.02, it is turned quite efficiently into stars. It is important to note that stellar feedback has not yet been included in our star formation models. Our current treatment may be a good approximation for initialSFRs, but mechanisms such as protostellar outflows that becomeimportant during formation, and ionization, winds and radiation pressurefrom massive stars soon after, will likely result in reduced SFRs.§.§ Spatial ClusteringWe investigate various quantitative metrics for spatial structure of the star clusters formed in our simulations. Global star and gasproperties of the primary clusters are measured and the angular dispersion parameter (ADP) and minimum spanning tree (MST) methods are used to analyze cluster substructure. The ADP is sensitive to angular substructure at chosen radii, whilethe MST determines the degree of overall centrally concentratedclustering.In order to define the primary cluster within a given model, we use the Density-based spatial clustering of applications with noise (DBSCAN) algorithm <cit.>. This density-based clustering algorithm is applied to our projected star particle data and the median particle position of the highest population cluster is used as our initial cluster center. A circular aperture with initial radius of 0.4 pc is centered at this point, and a new center isdetermined by finding the center of mass using stars included onlywithin this aperture. This process is repeated with aperture radiiiteratively decreasing by factors of two, down to length scales of 0.1 pc. The ADP is found for the primary cluster using each of these defined centers, while the MST is found for the entire domain. §.§.§ Global structure of primary clusterWe measure global structural properties of the primary star clusters created in the simulations. Figure <ref> shows results for these clusters that have formed by t=5  Myr in four models: , , , and .Due to relatively sparse particle density, the clusters in the non-colliding m_⋆,min=10 M_⊙ models were not included in this analysis. Also note that while elliptical annuli are displayed (see ADP discussion in Section <ref>), global properties for cluster structure are calculated using circular annuli.The top row of Figure <ref> shows the mass surface density of stars locally within each annulus, Σ_⋆, the enclosed average mass surface density of stars, Σ_⋆, and the enclosed average mass surface density of the gas, Σ_ gas.For the non-colliding case, Σ_⋆ ranges from about 10^3 to 5× 10^4  M_⊙ pc^-2, whereas the colliding cases form clusters with approximately 10^5 to 10^6  M_⊙ pc^-2. It can be seen that Σ_⋆ falls off quickly, reaching ∼10^2 and ∼10^4  M_⊙ pc^-2 by R=0.4  pc for non-colliding and colliding cases, respectively. We find best fit power law profiles of the form:Σ_*(R) = A ( R/0.2  pc)^-k_Σ_*,where A is a normalization factor, R is the distance from cluster center, and k_Σ_⋆ is the power law exponent. k_Σ_⋆ is found to be ∼2.3 to 2.6 for these primary clusters.We denote a half-mass radius, R_1/2, as the radius within which half of the total mass of the cluster out to R=0.4  pc is contained. Within our chosen clusters, R_1/2≃0.08-0.1 pc. Within R_1/2, the total enclosed stellar masses are 4.0× 10^2 M_⊙, 2.3× 10^4 M_⊙, 2.3× 10^4 M_⊙, and 2.2× 10^4 M_⊙ for models , , , and , respectively. This shows that the properties of the primary cluster in the colliding simulations are not much affected by the choice of star formation subgrid model. Within R_1/2, the averaged stellar mass surface density is Σ_*,1/2=2.1× 10^4M_⊙ pc^-2 (4.4g cm^-2) for the non-colliding case and 1.1× 10^6M_⊙ pc^-2 (2.28×10^2g cm^-2), 8.0× 10^5M_⊙ pc^-2 (1.7×10^2g cm^-2), and 1.06× 10^6  M_⊙ pc^-2 (2.22×10^2g cm^-2) for the respective colliding GMC models.The respective gas masses are 53 M_⊙, 5.7×10^2 M_⊙, 7.7×10^2 M_⊙, and 4.5×10^2 M_⊙.Note that the ∼0.1  pc scales are barely resolved in our simulation, so only the average enclosed gas masses are measured.At this stage, the cluster in the non-colliding case has Σ_⋆ < Σ_ gas. The colliding cases all have Σ_⋆ > Σ_ gas.We compare these cluster properties with those of observed young clusters <cit.>.The cluster formed in the non-colliding simulationthat has Σ_⋆≃ 2×10^4 M_⊙pc^-2 (4.4g cm^-2) is much denser than any known young cluster of comparable mass (i.e., with M_*,1/2≲1000 M_⊙).The colliding simulations produce more massive clusters and these are also seen to have much higher mass surface densities (by more than a factor of ten) at their half-mass scale than the densest known Galactic clusters, such as the Arches or Westerlund 1. We note that stellar feedback is not currently included within our simulations and expect the implementation of protostellar outflow feedback, planned in a future paper, to result in a reduction of Σ_⋆.The 1-D velocity dispersion, σ_z,1/2, is calculated for the stars seen to be within projected radii of R_1/2.The cluster formed in the non-colliding case has σ_z,1/2=1.57km s^-1, while those formed from GMC collisions have much higher values of 9.94, 9.83, and 8.81km s^-1, respectively for these clusters shown in Figure <ref> (left to right).More detailed kinematic analysis is performed in Section <ref>.The dynamical state of the clusters is investigated via calculation of the virial ratio,Q_i=-T_⋆/Ω=3σ^2R/2GM_⋆,where T_⋆ and Ω are the total kinetic and gravitationalpotential energies of the stars, respectively. For a given radius R,M_⋆ is the total enclosed stellar mass and σ is the 1-Dvelocity dispersion of the enclosed particles. Values of Q_i<1 indicate a bound cluster, while Q_i=0.5 represents a state of virial equilibrium. We find virial ratios at R_1/2 of 0.34, 0.24, 0.29, and 0.19 for the primary clusters from the simulations , , , and , respectively. These are all sub-virial, with collisions forming more tightly bound clusters at this stage in their evolution. However, as noted in <ref>, due to R_1/2 approaching the grid scale, gravitational forces are not well resolved and thus accurate dynamical evolution of the cluster is limited.When the virial ratio is calculated at better-resolved scales of R=0.4  pc, we find increased values of 0.44, 0.38, 0.36, and 0.33 for the same clusters, respectively. §.§.§ Angular Dispersion ParameterThe angular dispersion parameter (ADP), δ_ ADP,N(R) <cit.>, is a technique for quantifing the degree of substructure of a stellar distribution, especially designed for application to centrally-concentrated star clusters. It is similar to the azimuthal asymmetry parameter (AAP) developed by <cit.>. In its simplest form, this technique divides the distribution spatially into equal-area circular sectors and compares the dispersion of the number counts contained within each region. Further division using concentric annuli allows study of this substructure as a function of radius. In order to account for a global elongation or eccentricity of the cluster, best-fitted elliptical annuli can be used. We will adopt this as our fiducial method. To obtain the best-fit ellipse shape and orientation, a linear fit to the stars projected in the central 0.2 pc of the cluster is used to set the position angle, θ_e of the semi-major axis, a. Then the dispersion in position in directions parallel and perpendicular to θ_e are calculated to derive the eccentricity. For results in a given annulus of semi-major axis, a, we display them at a radius, R, for which the circular area would be equal to that of the ellipse.For a given annulus divided into a total of N equal sectors, each ith sector contains n_i stars. The ADP is defined as:δ_ ADP,N=√(1/(N-1)n∑_i=1^N(n_i-n)^2) = √(σ^2/σ^2_ Poisson),where σ is the standard deviation of the n_i values, n is the average of the number of stars per sector in the given annulus, and σ_ Poisson is the standard deviation expected from a Poisson distribution. Thus, values of δ_ ADP,N≃ 1 indicate nearly random distributions of sources, azimuthally. ADP analysis was performed using 20 equally-spaced concentric annuli out to a maximum radius of 0.4 pc with N=6 equally-divided sectors. δ_ ADP,6 is computed for twenty orientations of the sector pattern at every 3^∘ angular rotation, and the final value is averaged.Both circular and elliptical annuli are used to calculate δ_ ADP,6, using the same previously determined cluster center.The star cluster formed in the non-colliding model, which has a relatively low number of stars and thus larger Poisson errors, has δ_ ADP,6≃ 1.5 - 2.5 for circular annuli and ≃ 1 - 2 for elliptical annuli.In both cases, δ_ ADP,6 peaks near R=0.15  pc.The primary clusters from the colliding models have similar morphologies, especially the density-regulated and magnetically-regulated m_⋆,min=10 M_⊙ cases.In themodel, there exists a denser population of lower mass stars and the location of the subcluster R=0.4  pc is at a slightly different position. This slight deviation may be attributed to the earlier onset of star formation from lower density gas.The circular and elliptical δ_ ADP,6 values are similar, although again the latter is slightly smaller in size.For the clusters in theandsimulations, δ_ ADP,6≃2-3, while themodel has overall higher values of δ_ ADP,6≃6-7.The radial behavior of the clusters from the three colliding models is similar as well, in that the cluster has moderate values of δ_ ADP,6 out to R≈ 0.2  pc. These decrease at the outskirts of the defined cluster, then increase relatively sharply out to R≈ 0.4  pc upon the presence of the subcluster.<cit.> carried out a similar ADP analysis of the Orion Nebula Cluster (ONC). They found δ_ ADP,N with N=4,6,9 rises from below 1 in the very center of the ONC to reach fairly constant values of about 1.5 to 2.5 from 0.1 pc to about 1.3 pc. Accounting for ellipticity in the annuli brings these values of δ_ ADP,N down to about 1 to 1.5, with some variations near 2. Da Rio et al. concluded the projected spatial distribution of young, embedded stars in the ONC is relatively smooth, which may be evidence for dynamical processing if the cluster is older than a few orbit crossing times.The primary cluster formed in themodel returns similar values of δ_ ADP,6 as the ONC, while the m_⋆,min=10 M_⊙ colliding models return slightly higher values.Thecase has much higher δ_ ADP,6, which may be attributed to its much larger number of stars (∼10 times higher) resulting in smaller Poisson errors.As discussed in <ref>, we caution that the simulation code is not able to resolve small scale gravitational interactions between stars, potentially affecting the outcome of this metric. Still, we expect future application of this technique to better resolved clusters, including models where final stellar concentrations are reduced by including local feedback, will provide useful comparisons with observed young, embedded clusters to help test different formation scenarios.§.§.§ Minimum spanning tree Another method of studying the hierarchical structure of stellar distributions is through the use of the minimal spanning tree (MST). The MST <cit.> is a technique borrowed from graph theory in which all of the vertices of a connected, undirected graph are joined such that the total weighting for the graph edges is minimized. In the case of star clusters, the projected euclidean distances between the individual stars acts as the edge weight.To study the hierarchical structure of a collection of stars, <cit.> introduced a dimensionless parameter, Q, which can distinguish and quantify between smooth radial clustering (i.e., more centrally concentrated) vs. multi-scale type clustering (i.e., more substructure). Specifically,Q=s/m. The numerator is the normalized correlation lengths=d/R_ cluster,where d is the mean pairwise separation distance between the stars and R_ cluster is the overall cluster radius, calculated as the distance from the mean position of all stars to the farthest star.The denominator is the normalized mean edge lengthm=∑_i=1^N_⋆-1e_i/√((N_⋆A)(N_⋆-1)) ,where N_⋆-1 is the total number of edges, e_i is the length of each edge, and A=π R_ cluster^2 is the cluster area.lcccccr Parameters of Observed Clusters 0ptCluster Qs mTaurus 0.470.55 0.26IC2391 0.660.74 0.49Chameleon0.670.63 0.42ρ Ophiuchus 0.850.53 0.45IC3480.980.49 0.482<cit.>The threshold of Q_0=0.785 determines a quantitative threshold of either smooth radial clustering (Q>Q_0) or multi-scale clustering (Q<Q_0). Table <ref> lists Q, s, and m for various observed clusters <cit.>. We track the evolution of Q throughout our simulations (see Figure <ref>). In the m_⋆,min=10 M_⊙ non-colliding cases, clustering begins with Q≈ 0.5, at the low end of observed clusters (e.g., Taurus), and continues to decrease monotonically. For , an initially much higher Q is seen, but this quickly decreases to values even lower than the other cases. This general behavior can be understood as the formation of an initial cluster (more tightly concentrated in the m_⋆,min=1 M_⊙ case). However, the overall stellar distribution soon appears very dispersed as other independent clusters form throughout the GMCs.The colliding cases exhibit very different behavior. The density-regulated cases begin with similar Q parameter values and initially decrease from 0.5 to 0.25. However, beginning near 3.4 Myr, they experience a sharp increase in Q, reaching between 0.8 and 1.2, corresponding with the high end of observed clusters (e.g., ρ Ophiuchus and IC348). The higher-density cutoff models reach higher maximum Q values and peak at later times. After this peak, the Q values drop to fairly multi-scale-type clustering, but then rise again toward 0.5 to 0.7 in the final 0.5 Myr. The m_⋆,min=10 M_⊙ magnetically-regulated cases have similar qualitative behavior, with the Q peak occurring earlier in time, near 3.6 Myr and reaching very large values, surpassing centrally clustered observations. However, Q also drops down to ∼ 0.25 but again equalizes to values near 0.6. These can be understood as the initial formation of a moderately distributed star cluster which quickly becomes very centrally dominated as a result of new star formation in the compressed gas due to the collision that forms a primary cluster. However, the colliding region soon produces other clusters separated from the primary star cluster, thus decreasing Q. Then, beginning from t≈ 4.0 Myr, the gas and stars continue to coalesce, growing in size, number of stars, and central concentration. The primary cluster grows and accumulates more of the surrounding clusters, leading to the growth into a slightly multi-scale distribution overall. Themodel begins near Q=0.6 and experiences a lower peak near 0.8 before dropping off to 0.25. The final rise of Q is concurrent with the other colliding models, but instead of settling near 0.6, Q continues to rise until the end of the simulation t=5.0 Myr, reaching a very high central clustering value of 1.5.When comparing the results from our non-colliding vs. colliding simulations, clusters formed by GMC collisions spend a much greater fraction of the initial 5 Myr evolution with Q parameters within the range of observed clusters. While this result should not be overinterpreted, as clusters produced in the non-colliding cases may evolve into more centrally peaked distributions at beyond 5 Myr, a much stronger clustering of stars naturally arises from colliding gas, and this behavior is quantitatively realized in our simulations. §.§ Gas and Star Kinematics The relationship between the kinematics of young stars and their surrounding gas has been studied in order to gain insight into the formation and early evolution of young, embedded stellar populations. For example, using data from the INfrared Spectra of Young Nebulous Clusters (IN-SYNC) survey <cit.>, which achieves radial velocity accuracies of about 0.3 km s^-1, <cit.> have studied the kinematic properties of young stars in NGC 1333, while <cit.> have carried out a similar analysis of IC 348. <cit.> <cit.> analyzed similar data for the ONC and its extended southern filament, including comparison to gas tracers such as ^13CO.Our simulations allow a similar investigation of the kinematic properties of both ^13CO-defined gas and the young stars under various star formation scenarios.Our gas structures are defined using synthetic ^13CO(J=1-0) emission, based on the same observational assumptions as Paper II (i.e., GMCs are at a distance d=3 kpc, the optically thin limit applies, and we bin with a spectral resolution of 0.212  km s^-1). Figure <ref> shows position-velocity diagrams for non-colliding and colliding cases for density and magnetically-regulated star formation models<cit.>. The mean gas velocity, mean stellar velocity, and difference in these means as functions of position are also shown as profiles plotted below their respective colormaps.Mean values are taken using positional bins of 0.5 pc (i.e., 9.5× 10^-3  deg).The non-colliding cases show widely dispersed gas over the positional space, with clumpy morphology in ^13CO(J=1-0). The velocity v_z is fairly low, staying within ± 5  km s^-1. The gas velocity gradient is relatively shallow, following the general structure of the clouds. The gas and stellar kinematics in the density-regulated and magnetically-regulated star formation models are similar, with more star clusters present in themodel. The star clusters can be seen localized in positional space with a small scatter in velocity space. Generally, the stars are positioned in the vicinity of other high-intensity ^13CO clumps.The colliding cases show very different behavior in position-velocity space. The gas is much more localized spatially in x, the direction of the colliding flows. The structure is more concentrated in the y-direction as well due to the higher central gravitational potential formed. The average ^13CO-weighted velocity gradient is much steeper in x for the colliding cases, but relatively similar in magnitude to the non-colliding cases in the y line of sight.Larger clumps with higher intensities of ^13CO gas are seen in the colliding cases, with a dense network of filamentary structures present. Additionally, the gas velocity dispersion is much greater, with portions reaching velocity dispersions of ± 20  km s^-1. Gas and stellar kinematic morphologies are also similar among the different star formation models.The central star cluster is seen to have a very large velocity dispersion, with the primary clusters inandmodels found in  <ref> to have σ_z,1/2=9.94 and 9.83  km s^-1, respectively.Separate, smaller clusters can also be seen with their own stellar populations near high-intensity clumps of gas.From the position-velocity information, we calculate velocity gradients of the gas (dv_ los/ds), the velocity dispersion of the gas (σ_ gas), the velocity dispersion of the stars (σ_ *), the ^13CO-weighted average velocities of the gas (v_ gas), the mass-weighted average velocities of the stars (v_*), and the velocity offset between the two (Δv). Table <ref> summarizes these properties for the four models as viewed from x, y, and z lines of sight, as well as their RMS values.We note that the average σ_* values (9.86 and 9.53 km s^-1) in the whole domain are only slightly lower than those of the primary cluster (9.94 and 9.83 km s^-1). In these cases, the central cluster contains the majority of the stars and thus dominates the overall distribution.Within the non-colliding cases, both the gas and stellar kinematics agree fairly closely between different star formation models. The velocity gradients are larger in the x and y directions due to asymmetries from the impact parameter. A RMS value of 0.072 km s^-1 pc^-1 is recorded. The dispersion of the gas and stars are similar in both models, with σ_ gas, RMS≈ 1.76 km s^-1 and σ_ *, RMS≈ 1.68 km s^-1. The mean velocities of the gas and stars are also similar, with v_ gas, RMS≈ 0.91 km s^-1 and v_ *, RMS≈ 0.94 km s^-1. The stellar velocities in the density-regulated model have slightly lower dispersions, but higher mean values. Overall, the velocity offset between the gas and the stars for the non-colliding models is approximately 0.35 km s^-1.For the colliding cases, the density-regulated and magnetically-regulated star formation models exhibit similar kinematic properties of the gas and stars. On average, dv_ los/ds=0.26 km s^-1 pc^-1, σ_ gas, RMS=4.14 km s^-1, σ_ *, RMS=9.70 km s^-1, v_ gas, RMS=0.50 km s^-1, v_ *, RMS=1.16 km s^-1, and Δv=1.35 km s^-1. Relative to non-colliding clouds, the collision induces a much larger velocity gradient (∼3-4 times greater), a larger velocity dispersion in the gas (∼2 times greater), and a much larger stellar velocity dispersion (∼5 times greater).As functions of position, the mean velocities of gas, stars, and their offsets, are compared. Offsets exist in both non-colliding and colliding cases, becoming most apparent in close proximity to star clusters.For the non-colliding case, these differences are relatively small, at a few km s^-1. However, the colliding case contains regions in which the offsets exceed 5  km s^-1.Averaged over position space, colliding GMCs result in velocity offsets a factor of ∼4 times higher than non-colliding GMCs. This parameter may be an indicator for determining the dynamical formation history of young star clusters, as results from collisions show more disturbance kinematically.We make a first, simple comparison with the results of the IN-SYNC survey of the ONC and surroundings <cit.>. This survey found offsets between gas and star velocities of approximately Δ v_r∼ -0.5 km/s, but up to -1.0 to -1.5 km/s in some regions. The magnitude of such offsets are in general consistent with those seen in both the non-colliding and colliding models, especially considering variations associated with the particular line of sight. Given that the observational data for Orion is just a single example of a star-forming region, viewed on a particular sight line, it is difficult to draw definitive conclusions about whether the gas and star kinematics favor one scenario over another. Larger numbers of star-forming regions need to be studied with similar methods. In addition, other metrics, such as the comparison of low and high density gas tracers <cit.>, need to be examined, which on the simulation side requires extensionof the astrochemical modeling to include species such as N_2H^+.§ DISCUSSION AND CONCLUSIONSWe have implemented two classes of star formation sub-grid routines into the MHD code Enzo that we are using to study GMC collisions: a density-regulated model based on a threshold density and a new magnetically-regulated model based on a threshold mass-to-flux ratio. Varying key parameters for each star formation routine, we explored the large-scale morphology, properties of star-forming gas, global star formation rates and efficiencies over time, spatial clustering of the stars, and gas and stellar kinematics. For each model, we investigated scenarios of non-colliding and colliding GMCs.The non-colliding cases evolved in a relatively quiescent manner, driven by the initial turbulence and interplay of self-gravity and magnetic fields. Star clusters formed only in the very late stages of the simulations, from overdense clumps located within filaments and dispersed throughout the GMC complex. Generally, these clusters contained hundreds of solar masses each and grew at a relatively slow rate. Star clusters in the density-regulated star formation routines were smaller and more isolated. The clusters formed in the magnetically-regulated models exhibited slightly more elongated morphologies. For this simulation set up, the level of star formation activity appears to be regulated by the effective density threshold that is used in each of these models, with the mass-to-flux criterion not having a large influence.During collisions between GMCs, stars formed earlier and in larger clusters, from high-density gas produced in the primary filamentary colliding region.While star formation rates level off by the completion of thesimulations, extrapolation of future behavior is unclear.Nevertheless, by t=5  Myr, individual clusters have grown and merged to form one large, dominant cluster with a total stellar mass of 5× 10^4 M_⊙. For this particular set-up, the final overall level of star formation is relatively independent of all the explored star formation sub-grid models. Star formation appears to be limited by the ability of the collision to direct mass into high density regions, which then eventually form stars with high overall efficiency.Just prior to star formation, both density and magnetically regulated star formation result in fairly similar gas properties of parent cells. However, colliding cases experience relatively wider ranges of densities, temperatures, μ_ cell, and velocity magnitude. Higher mean values for density and temperature are found, while gas is more magnetically subcritical and turbulent.The primary star clusters formed in the various models were analyzed and found to have much higher surface densities at their half-mass scale than any observed cluster. We expect the future inclusion of stellar feedback will reduce these surface densities.The angular dispersion parameter (ADP) analysis was carried out on the primary clusters in the simulations. ADP values are generally greater than those see in the ONC, which may indicate the ONC is dynamically older than the simulated clusters. The MST Q parameter was also used to investigate the global spatial distribution properties of the star, with non-colliding cases resulting in overall highly multi-scaled clustering due to the scattered formation of independent clusters. Colliding GMCs produce clusters with Q parameters that vary between those expected of multi-scale and centrally clustered distributions.Kinematically, our colliding GMC cases produce velocity gradients 3-4 times greater than those of the non-colliding cases. The velocity dispersions also differ, with the gas in the colliding clouds having approximately twice the velocity dispersion. Stellar velocity dispersions in the simulations are dominated by the potentials of the primary clusters that form, with this leading to much greater dispersions in the colliding case. We find that the colliding cases produce typically 4 times larger offsets between the mean gas and mean star velocities compared to the non-colliding case.Finally, we remind the reader of several important caveats. The young stars do not inject feedback, especially protostellar outflow feedback, into the surrounding gas. Internal star cluster dynamics are not well followed because of gravitational softening at the grid scale (∼ 0.1 pc) and because the star particles lack realistic mass and multiplicity distributions. Still the conditions that are simulated here may provide boundary conditions for more detailed models that are able to follow full N-body evolution of the clusters <cit.>. Finally, in the context of GMC collisions, a wide range of cloud (e.g., degree of initial magnetization) and collision (e.g., velocities; impact parameters) parameters remain to be explored with these models. These items will be addressed in subsequent papers in this series.The authors would like to thank Nicola Da Rio and Juan Pablo Farias for useful discussions.Computations described in this work were performed using the publicly-availablecode (http://enzo-project.org), This research also made use of the yt-project (http://yt-project.org), a toolkit for analyzing and visualizing quantitative data <cit.>. Volume renderings were performed using(https://wci.llnl.gov/simulation/computer-codes/visit). 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http://arxiv.org/abs/1702.08117v2

rotrotAltPFTrtrSoc The Markoff Group of Transformationsin Prime and Composite Moduli Chen Meiri and Doron Puderwith an Appendix by Dan Carmon December 30, 2023 =================================================================== The Markoff group of transformations is a group Γ of affine integral morphisms, which is known to act transitively on the set of all positive integer solutions to the equation x^2+y^2+z^2=xyz. The fundamental strong approximation conjecture for the Markoff equation states that for every prime p, the group Γ acts transitively on the set X^*(p) of non-zero solutions to the same equation over ℤ/pℤ. Recently, Bourgain, Gamburd and Sarnak proved this conjecture for all primes outside a small exceptional set.In the current paper, we study a group of permutations obtained by the action of Γ on X^*(p), and show that for most primes, it is the full symmetric or alternating group. We use this result to deduce that Γ acts transitively also on the set of non-zero solutions in a big class of composite moduli. Our result is also related to a well-known theorem of Gilman and Evans, stating that for any finite non-abelian simple group G and r≥3, the group Aut(_r) acts on at least one “T_r-system” of G as the alternating or symmetric group. In this language, our main result translates to that for most primes p, the group Aut(_2) acts on a particular T_2-system of PSL(2,p) as the alternating or symmetric group.§ INTRODUCTION The Markoff surface 𝕏 is the affine surface in 𝔸^3 defined by the equation[Sometimes the Markoff equation is written as x^2+y^2+z^2=3xyz. However, these two equations are equivalent in the sense that their integer solutions are related bijectively by (x,y,z)⟷(3x,3y,3z). This bijection holds also for solutions in ℤ/pℤ for every prime p≠3.]x^2+y^2+z^2=xyz.The Markoff triples MM is the set of positive integer solutions to Equation (<ref>), such as (3,3,3). The Markoff group of automorphisms of 𝕏 is the group ΓΓ generated by permutations of the coordinates and the Vieta involutions R_1, R_2 and R_3R_i where R_3(x,y,z)=(x,y,xy-z) and R_1 and R_2 are defined analogously. It is easy to see that M is invariant under Γ and Markoff proved that Γ acts transitively on M <cit.>. Let Δ be the group generated by Γ and the involutions that replace two of the coordinates by their negatives. Then the set 𝕏(ℤ) of integer solutions to (<ref>) has two Δ-orbits: {(0,0,0)} and its complement X^*(ℤ)def=𝕏(ℤ)∖{(0,0,0)}.§.§ Prime Moduli If p is a prime number, then 𝕏(ℤ/pℤ) is the finite set of solutions to (<ref>) in ℤ/pℤ, and we denote X^*(p)=𝕏(ℤ/pℤ)∖{(0,0,0)}X^*(p). The strong approximation conjecture for the Markoff equation (<ref>) states that for every prime p, the reduction mod p of the set of Markoff triples M→ X^*(p) is onto. This is clearly equivalent to Γ acting transitively on X^*(p). Recently, Bourgain, Gamburd and Sarnak proved this conjecture for all primes outside of a small exceptional set:Let E be the set of primes for which Γ does not act transitively on X^*(p). For any ε>0, the number of primes p≤ T with p∈ E is at most T^ε, for T large enough.Moreover, for any ε>0, the largest Γ-orbit in X^*(p) is of size at least |X^*(p)|-p^ε, for p large enough (whereas |X^*(p)|∼ p^2).Let Γ_pΓ_p be the finite permutation group induced by the action of Γ on X^*(p). In the current work we study the nature of this group. The first step here is to notice that Γ_p preserves a block structure as follows: For (x,y,z)∈ X^*(p) denote by [x,y,z][x,y,z] the block of all solutions obtained from (x,y,z) by sign changes, so [x,y,z]def={(x,y,z),(x,-y,-z),(-x,y,-z),(-x,-y,z)} .Then Γ_p preserves this block structure. Let Y^*(p)Y^*(p)denote the set of blocks in X^*(p), and Q_pQ_p denote the permutation group induced by the action of Γ (or Γ_p) on Y^*(p). Simulations suggest the following conjecture:For every p≥5, the permutation group Q_p is the full alternating or symmetric group.This conjecture was also raised, independently, in <cit.>, where the authors also state precisely for which primes one can expect the alternating group (p≡316) and for which the full symmetric group (p≢316). If this conjecture holds, then roughly speaking (we give the precise formulation in Theorem <ref> below), Γ acts transitively on the solutions of (<ref>) modulo n, for every square free.Here we prove this conjecture for most primes. More particularly, we prove it for every p≡1(4) outside the exceptional set from Theorem <ref>, and for density-1 of the primes p≡3(4): If p≡1(4) and Q_p is transitive, then Q_p is the full alternating or symmetric group on Y^*(p). Namely, Q_p is the full alternating or symmetric group for all p≡1(4) outside the exceptional set from Theorem <ref>. In fact, our proof yields that for every p≡1(4), the group Γ acts as the full alternating or symmetric group on the large component described in Theorem <ref>. In the case p≡3(4), our proof is more involved and requires one further assumption:Let p be a prime. Assume that:* p≡3(4).* Q_p is transitive.* The order of 3+√(5)/2∈𝔽_p^2 is at least 32√(p+1).Then Q_p is the full alternating or symmetric group on Y^*(p).The number 3+√(5)/2 is related to the special solution [3,3,3]∈ Y^*(p): its order inside 𝔽_p^2 gives the length of the cycle of the transformation [x,y,z]↦[x,z,xz-y] containing the element [3,3,3]. For details see Sections <ref> and <ref>.As shown in Appendix <ref>, the condition regarding the order of 3+√(5)/2 is satisfied for density-1 of the primes[A set of primes A has density 1 if lim_n→∞| A∩ P_n|/| P_n|=1, where P_n={ 1<p≤ n | p is prime}. In fact, the set of primes for which 3+√(5)/2 has order at least 32√(p+1) satisfies something slightly stronger than density 1 – see Appendix <ref>.], henceFor density-1 of all primes p≡3(4), the group Q_p is the full alternating or symmetric group on Y^*(p).§.§ Composite Moduli Let n be a positive integer which is square-free, so n=p_1⋯ p_k where p_1,…,p_k are distinct primes. Let 𝕏(n)𝕏(n) denote the set of solutions to the Markoff equation (<ref>) in ℤ/nℤ. By the Chinese Remainder Theorem, 𝕏(n)=𝕏(p_1)×…×𝕏(p_k), and let X^*(n)=X^*(p_1)×…× X^*(p_k)X^*(n)be the set of solutions which are non-zero modulo any of the primes composing n. The action of Γ on 𝕏(n) is the diagonal action on the 𝕏(p_i), and the subset X^*(n) is invariant under this action. Denote the corresponding permutation group Γ_nΓ_n. Is the action on X^*(n) transitive? It turns out that this would follow from Conjecture <ref> and indeed holds true for the cases of that conjecture we establish:Let n=p_1⋯ p_k be a product of distinct primes. If for every j=1,…,k, Q_p_j≥Alt(Y^*(p_j)), then Γ acts transitively on X^*(n).In particular, if conjecture <ref> holds, then Γ acts transitively on X^*(n) for every square-free n. Letdenote the set of primes that satisfy the assumptions of Theorem <ref> or of Theorem <ref>. Then for every set of distinct primes p_1,…,p_k∈, Γ acts transitively on X^*(p_1⋯ p_k).Bourgain, Gamburd and Sarnak already proved Corollary <ref> for primes p≡1(4) for which Γ_p is transitive. This result should appear in the series announced in <cit.>. We stress that our proof is entirely different: while Bourgain, Gamburd and Sarnak improve their techniques from the proof of Theorem <ref> so that the argument work for several primes simultaneously, our proof is group-theoretic and uses Theorem <ref> as a black box. Both proofs rely on solutions containing the parabolic elements ±2 – see Figure <ref> and Section <ref>. For n=p_1⋯ p_k as above, we use the notation Y^*(n)=Y^*(p_1)×…× Y^*(p_k)Y^*(n) for the set of blocks in X^*(n) and Q_nQ_n for the permutation group induced by the action of Γ on Y^*(n). Note that these blocks are given by sign changes modulo every prime separately and are usually of size 4^k each (if all primes are odd). It is quite straight-forward to prove that under the assumptions of Theorem <ref>, Γ acts transitively on Y^*(n), using composition factors of Q_n. It requires some further argument to show that Γ acts transitively on the full set X^*(n). We elaborate in Section <ref>. At this point we would like to remark on the dependence of our results on the Classification of Finite Simple Groups (CFSG)CFSG. We use the classification only in the proof of Theorem <ref>: we first give an elementary proof that for a prime p satisfying the assumptions in the theorem, Q_p is a primitive permutation group[Recall that a permutation group G≤Sym(m) is called primitive if it does not preserve any non-trivial block-structure. In particular, if m≥3, G must be transitive.], and then rely on (results depending on) the CFSG to deduce that Q_p is the full alternating or symmetric group. If we rely on Theorem <ref> to deduce Corollary <ref>, the latter also becomes partly dependent on the CFSG. This can be avoided, however, and to this aim we also give a proof that Γ acts transitively on X^*(n) assuming only that Q_p_1,…,Q_p_k are primitive permutation groups, without using the CFSG (see Theorem <ref> below). To sum up, the only results depending on the CFSG are Theorem <ref>, Corollary <ref>, and the part of Theorem <ref> relating to primes p≡3(4). In contrast, Theorems <ref> and <ref> and Corollary <ref> do not depend on the CFSG. We illustrate this in Figure <ref>.Indeed, the following result does not depend on the CFSG:Let n=p_1⋯ p_k be a product of distinct primes. If Q_p_1,…,Q_p_k are primitive permutation groups, then Γ acts transitively on X^*(n).§.§ T_2-systems Let G be a finitely generated group and _r the free group on r generators. A normal subgroup N⊴_r is said to be G-defining if _r/N≅ G. Denote by Σ_r(G)Σ_r(G) the set of G-defining normal subgroups in _r. Consider the action of Aut(_r) (in fact, of Out(_r)) on Σ_r(G). The orbits of this action are called T_r-systems of G. The following theorem is due to Gilman (for r≥4) and Evans (who extended to r=3):<cit.> Let G be a finite non-abelian simple group and r≥3. Then Aut(_r) acts on at least one T_r-system of G as the alternating or symmetric group.In fact, Gilman and Evans provide more information about the special T_r-system on which Aut(_r) acts as the full alternating or symmetric group, and show it is especially large. Gilman also showed that for G=PSL(2,p) with p≥5 prime, there is only one T_r-system for r≥3. Namely, he proved that Aut(_r) acts transitively on Σ_r(G). Theorem <ref> says, of course, that the permutation group in this case is the alternating or symmetric group. For more details we refer the reader to the beautiful surveys <cit.>.When r=2, the action of Aut(_2) on Σ_2(G) is not transitive for any finite non-abelian simple group G. In fact, the number of T_2-systems tends to infinity as |G|→∞ <cit.>. The main reason for this phenomenon is that if { a,b} are a set of generators of _2, and φ_2↠ G an epimorphism, then the set of conjugacy classes of[Here, [a,b] denotes the commutator aba^-1b^-1.] φ([a,b]) and of φ([a,b])^-1 is a well-defined invariant of the G-defining subgroup N=φ, which is also invariant under Aut(_2). We elaborate more in Section <ref>.Our result sheds more light on the case of T_2-systems for G=PSL(2,p). If A,B∈SL(2,p) and we denote x=tr(A), y=tr(B) and z=tr(AB), then tr([A,B])=x^2+y^2+z^2-xyz-2.In Section <ref> it is explained why the map (A,B)↦(tr(A),tr(B),tr(AB)) yields a bijection between the elements in Σ_2(PSL(2,p)) with associated trace -2 and the elements of Y^*(p). In this language, the main result of <cit.> – Theorem <ref> above – says that outside the exceptional set of primes, these elements form a single T_2-system. See <cit.> for an extensive survey of the connection between the Markoff equation (<ref>) and T_2-systems of PSL(2,p). Through this connection, Theorems (<ref>) and (<ref>) translate to a result in the spirit of Theorem <ref>:Assume that the prime p satisfies the assumptions of Theorem <ref> or of Theorem <ref>. Then Aut(_2) acts on the trace-(-2) T_2-system of PSL(2,p) as the full alternating or symmetric group.The paper is organized as follows. Section <ref> gives some more notation and collects some results from <cit.> we use here. In the short Section <ref> and longer Section <ref> we prove Theorem <ref> for p≡1(4) and Theorem <ref> for p≡3(4), respectively. Section <ref> is dedicated to proving the transitivity of Γ in certain composite moduli: first assuming the groups Q_p contain the alternating group (in Section <ref>), and then assuming only that Q_p is primitive (Section <ref>). In Section <ref> we give some background on T-systems and prove Theorem <ref>. Finally, Appendix <ref>, by Dan Carmon, shows that the assumption in Theorem <ref> regarding the order of 3+√(5)/2∈𝔽_p^2 holds for most primes.§.§ Acknowledgments We are indebted to Peter Sarnak for his encouragement, and for stimulating discussions, enlightening suggestions and clever advice. We would also like to thank Zeev Rudnick and Pär Kurlberg for beneficial comments, and to Dan Carmon for writing the useful Appendix <ref>. We have benefited much from the mathematical open source community, and in particular from SageMath. Author Meiri was supported by BSF grant 2014099 and ISF grant 662/15. Author Puder was supported by the Rothschild fellowship, by the NSF under agreement No. DMS-1128155 and by the ISF grant 1071/16. Author Carmon was supported by the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n^o 320755. § PRELIMINARIES Before proving our main results, let us describe some further notation and collect further results from <cit.> that we use below. §.§ Further notation* We already introduced above the notation [x,y,z] for the block of the solution (x,y,z) in X^*(p), so [x,y,z]∈ Y^*(p). We also use this notation for a composite (square-free) modulo n: here [x,y,z] is the element (block) in Y^*(n) containing the solution (x,y,z).* Some elements in Γ are permutations of the three coordinates of solutions. We denote these elements by τ_(12)τ_(12),τ_(123) for the permutation exchanging the first and second coordinates, by τ_(123) for the cyclic permutation and so on. By abuse of notation, we use the same notation for the corresponding elements in Γ, Γ_p, Q_p, Γ_n and Q_n. * The analysis in <cit.>, as well as in the current work, relies heavily on three “rotation” elements _1,_2,_3∈Γ_i. They are defined by_jdef=R_j+2∘τ_(j+1  j+2)(the indices are taken modulo 3). For example, (x,y,z)_1↦(x,z,xz-y). The rotation _j fixes the j-th coordinate and its action on X^*(p) and on Y^*(p) is completely analyzed in <cit.> – see Lemmas <ref> and <ref> below. Again, by abuse of notation we write _i for the rotation element in the different groups Γ, Γ_p, Q_p, Γ_n and Q_n. * Following <cit.>, we denote the “conic sections” by C_j(a) C_j(a), j=1,2,3. These are defined asC_j(a)={(x_1,x_2,x_3)∈ X^*(p) | x_j=a} .When we write C_j(± a)C_j(± a), we mean the conic section in Y^*(p):C_j(± a)={[x_1,x_2,x_3]∈ Y^*(p) | x_j=a} . * For every prime p we let ii denote a square root of -1 (in 𝔽_p or in 𝔽_p^2).* For x∈ℤ/pℤ we use the standard Legendre symbol (x/p)(x/p) to denote the image of x under the character of order 2. Namely,(x/p)= 1 x is a square in 𝔽_p^*-1 x is a non-square in 𝔽_p^*0 x=0 . * The notation |x| is used to denote the order of the group element x∈ G in the group G.§.§ Rotation elements The action of _1 on the conic section C_1(x)⊆ X^*(p) is a linear map on the last two coordinates given by the matrix([01; -1x ]).The eigenvalues of this matrix are given by x±√(x^2-4)/2. This leads to the following definitions and lemmas from <cit.>: * An element x∈𝔽_p if called hyperbolichyperbolic if (x^2-4) is a square in 𝔽_p^*.* An element x∈𝔽_p if called ellipticelliptic if (x^2-4) is a non-square in 𝔽_p^*.* An element x∈𝔽_p if called parabolicparabolic if (x^2-4)=0 in 𝔽_p, namely, if x=±2. Notice that this categorization of the elements is invariant under sign change x↦-x. The following lemmas are based on Lemmas 3-5 of <cit.> which describe the action of _i on X^*(p). We adapt them below in order to describe the action of _i on Y^*(p) and add some further details, all follow easily from Section 2.1 in <cit.>. We state the lemmas for C_1(± x), but the same statements holds, evidently, for C_2(± x) and for C_3(± x).<cit.> Let p≡1(4) be prime. Then, * |Y^*(p)|=p(p+3)/4.* |C_1(±2)|=p; The permutation induced by _1 on C_1(±2) consists of a single p-cycle. * There are p-1/4 hyperbolic elements up to sign. For x hyperbolic, |C_1(± x)|=p-1/2. Let ω^±1∈𝔽_p be the eigenvalues of the matrix (<ref>), so x=ω+ω^-1. The permutation induced by _1 on C_1(± x) consists of p-1/2d cycles of length d each, where d=max(|ω|,|-ω|)/2 and |ω| is the order of ω in the multiplicative group 𝔽_p^*. The solutions in C_1(x) have the form (x,α+β,αω+βω^-1) for α,β∈𝔽_p^* with αβ=x^2/x^2-4, and (x,α+β,αω+βω^-1)_1↦(x,αω+βω^-1,αω^2+βω^-2). * There are p-1/4 elliptic elements up to sign. For x elliptic, |C_1(± x)|=p+1/2. Define ω as for hyperbolic elements by x=ω+ω^-1, only now ω∈𝔽_p^2∖𝔽_p. The permutation induced by _1 on C_1(± x) consists of p+1/2d cycles of length d each, where d=max(|ω|,|-ω|)/2 and |ω| is the order of ω in the multiplicative group 𝔽_p^2^ *. Moreover, ω^p+1=1, i.e. |ω| | (p+1). The solutions in C_1(x) have the form (x,A+A^p,Aω+A^pω^-1) with A∈𝔽_p^2^* and A^p+1=x^2/x^2-4, and (x,A+A^p,Aω+A^pω^-1)_1↦(x,Aω+A^pω^-1,Aω^2+A^pω^-2).We sum up the content of Lemma <ref> in Table <ref>.When p≡3(4), our results are somewhat weaker and the proofs more involved. The main reason for that is the lack of solutions with the parabolic elements ±2:<cit.> Let p≡3(4) be prime. Then, * |Y^*(p)|=p(p-3)/4* There are no solutions in Y^*(p) involving the parabolic elements ±2, nor the elliptic element 0.* There are p-3/4 hyperbolic elements up to sign. For x hyperbolic, the size and structure of C_1(± x) and the action of _1 on C_1(± x) have the same properties as for x hyperbolic when p≡1(4) (see Lemma <ref>).* There are p-3/4 non-zero elliptic elements up to sign. For x elliptic, the size and structure of C_1(± x) and the action of _1 on C_1(± x) have the same properties as for x elliptic when p≡1(4) (see Lemma <ref>). We sum up the content of Lemma <ref> in Table <ref>.For x∈𝔽_p, denote by d_p(± x)d_p(± x) the order of _1∈ Q_p in its action on C_1(± x). Namely, the solutions with first coordinate ± x in Y^*(p) belong to cycles of length d_p(± x).§ ALTERNATING GROUP FOR P≡1(4)This section contains the proof of Theorem <ref>, which states that if p≡1(4) and Q_p is transitive, then Q_p contains the entire alternating group Alt(Y^*(p)). As mentioned above, the existence of parabolic elements when p≡1(4) allows a rather short argument in this case.We use the following classical theorem of Jordan: Let G≤Sym(n) be a primitive permutation group containing a cycle of prime length p≤ n-3. Then G≥Alt(n).Assume p≡1(4), and let _1∈ Q_p be the rotation element defined on Page rotations. This element has one p-cycle, while all its other cycles have length coprime to p (see Table <ref>). Thus its power σ=_1^ |_1|/p∈ Q_p is a p-cycle. As |Y^*(p)|=p(p+3)/4≥ p+3, it is now sufficient to show, by Jordan's Theorem (Theorem <ref> above), that Q_p is primitive in Sym(Y^*(p)). We need to show that the group Q_p preserves no non-trivial block structure. Assume there is a block structure { B_1,…,B_m} preserved by Q_p. So ⋃ B_i=Y^*(p) and B_i∩ B_j=∅ for i j, and for every g∈ Q_p and every i, g(B_i)=B_j for some j.Consider C_1(±2)⊂ Y^*(p), the p elements contained in the cycle of size p in σ. The set C_1(±2) must be contained in a block, for otherwise it has to be the union of several equally-sized blocks, but p is prime. Say C_1(±2)⊆ B_1. So B_1 contains all solutions with ±2 in the first coordinate. In particular, it contains [2,2,2+2i] and [2,2+2i,2]. But the same argument with _2 and _3 shows that B_1 contains all solutions with ±2 in any coordinate. So B_1 is invariant under all three rotations and under all permutations of coordinates, and therefore invariant under the action of the whole group Q_p. By the transitivity of Q_p, B_1=Y^*(p). The proof of Theorem <ref> in <cit.> shows that for every prime p, the large component of X^*(p) contains all solutions with parabolic (±2) coordinates. Thus, our proof of Theorem <ref> applies to the general case: the group Γ acts on the large component of Y^*(p) as the alternating or symmetric group. § ALTERNATING GROUP FOR P≡3 (4) In the case where p≡3(4), there are no parabolic elements, and in Sections <ref> and <ref> we establish the primitivity of Q_p for density-1 of these primes rather than for all those outside the exceptional set from Theorem <ref>. We also rely on much deeper theorems, involving the classification of finite simple groups (CFSG), to conclude in Section <ref> that whenever Q_p is primitive, it contains Alt(Y^*(p)). Throughout this section, we assume that p≡3(4). §.§ Primitivity of Q_p when p≡3(4) In this subsection we prove that under the assumptions of Theorem <ref>, the permutation group Q_p is primitive. Namely,Let p be prime with p≡3(4). Assume that Q_p is transitive and that the order of 3+√(5)/2∈𝔽_p^2 is at least 32√(p+1). Then Q_p is primitive.To establish primitivity of Q_p, one needs to show there are no non-trivial blocks in the action of Q_p on Y^*(p): a block is a subset B⊆ Y^*(p), such that for every g∈ Q_p, either g.B=B or g.B∩ B=∅. As Q_p is assumed to be transitive, if B is proper (B⫋ Y^*(p)) and of size at least two, then the subsets { g.B | g∈ Q_p} constitute a partition of Y^*(p) which is a non-trivial block structure preserved under the action of Q_p. So proving Q_p is primitive is equivalent to showing that every proper block is a singleton.The proof of Theorem <ref> relies on the following two propositions which contain properties of blocks in Y^*(p). We defer the proofs of these two propositions to the next subsection, and complete the proof of Theorem <ref> in the current subsection, assuming the two propositions.We say that some coordinate j∈{ 1,2,3} is homogeneous in a block B⊆ Y^*(p) if the j-th coordinate of every solution in B is of the same type (either all hyperbolic or all elliptic). Let p≡3(4). Assume that Q_p acts transitively on Y^*(p), and let B⫋ Y^*(p) be a proper Q_p-block. Then at least two of the coordinates { 1,2,3} are homogeneous in B.The most technical ingredient of the proof of primitivity is the following. Recall that d_p(± x) denotes the length of the cycles of _1∈ Q_p containing elements of C_1(± x).Assume that Q_p is transitive and let x∈𝔽_p∖{ 0,±2} satisfy d_p(± x)≥16√(p+1). Then, for every j∈{ 1,2,3}, every proper Q_p-block B⫋ Y^*(p) contains at most one solution with j-th coordinate ± x.The idea of the proof of this proposition is the following: assume there are two solutions in the block B with first coordinate ± x. Say these are [x,y_0,y_1] and [x,z_0,z_1]. Then for every 1≤ m, the block _1^ m(B) contains the solutions [x,y_m,y_m+1] and [x,z_m,z_m+1] with y_m and z_m defined recursively by y_m+1=xy_m-y_m-1 and z_m+1=xz_m-z_m-1. By Proposition <ref>, at least one of the two coordinates 2,3 in every block is homogeneous, meaning that for every m, either y_m and z_m have the same type (hyperbolic or elliptic), or y_m+1 and z_m+1 have the same type. Using classical results in number theory, we show such “high correlation” between two cycles of _1 is impossible whenever these cycles are long enough.Section <ref> gives the details of the proof, and assuming it, we finish the proof of Theorem <ref>. We need the following corollary showing that elements of high order in the sense of Proposition <ref> appear in the same block and the same coordinate only with other elements of the same type and the same order:Assume that Q_p is transitive and that x∈𝔽_p∖{ 0,±2} satisfies d_p(± x)≥16√(p+1). If B⫋ Y^*(p) is a proper Q_p-block containing some solution with first coordinate ± x, and another solution with first coordinate ± x', then d_p(± x)=d_p(± x'). In particular, x and x' are of the same type (both hyperbolic or both elliptic). Note that _1^ d_p(± x)(B)=B. By Proposition <ref>, _1^ m(B) B for 1≤ m<d_p(± x). Hence, d_p(± x') is some multiple of d_p(± x). In particular, the assumption of Proposition <ref> holds for x', and by symmetry, d_p(± x) is a multiple of d_p(± x'). Hence d_p(± x')=d_p(± x). Assume that Q_p is transitive and ω=3+√(5)/2∈𝔽_p^2^ * has order at least 32√(p+1). We need to show that Q_p is primitive. We use the special symmetric solution [3,3,3]∈ Y^*(p). Whenever ω∈𝔽_p^2 has high order in the multiplicative group 𝔽_p^2^*, the cycle of _1∈ Q_p containing the solution [3,3,3] is long. More concretely, 3=ω+ω^-1, and by Lemma <ref> and Table <ref>, d_p(±3) is either |ω| or |ω|/2, where |ω| is the order of ω in the multiplicative group 𝔽_p^2^  *. So d_p(±3)≥16√(p+1).Assume that [a,b,c] and [3,3,3] are two distinct solutions lying in the same proper Q_p-block B⫋ Y^*(p). By Lemma <ref>, d_p(±3)≥16√(p+1), and by Corollary <ref>, d_p(± a)=d_p(± b)=d_p(± c)=d_p(±3). As [3,3,3] is the only solution of the form [x,x,x] or [x,x,-x], we can assume without loss of generality that {± b}{± c}. Since τ_(2 3) stabilizes [3,3,3], we have τ_(2 3)(B)=B, so the two distinct solutions [a,b,c] and [a,c,b] both belong to B. This contradicts Proposition <ref>: d_p(± a)=d_p(±3) is large and thus a cannot appear twice in the same coordinate in the same block.As mentioned in Section <ref>, the assumptions in Theorem <ref> hold for density-1 of the primes p≡3(4). Indeed, relying on strong results of Ford <cit.>, Dan Carmon proves in Proposition <ref> in Appendix <ref> that under some assumptions, the order of a quadratic integer modulo primes is high for density-1 of the primes. From Proposition <ref> we deduce:For density-1 of all primes, the element ω=3+√(5)/2∈𝔽_p^2 has order at least 32√(p+1) in the multiplicative group 𝔽_p^2^  *, in which case d_p(±3)≥16√(p+1).Combining Theorem <ref> with Corollary <ref> shows why the assumptions in Theorem <ref> hold for density-1 of all primes p≡3(4), hence:For density-1 of all primes p≡3(4), the group Q_p is primitive in its action on Y^*(p). It is conceivable that there is a stronger version of Proposition <ref> which states there cannot be correlation between two long cycles of _1∈ Q_p even with two different first coordinates. Were we able to prove this, we could omit the condition about the order of 3+√(5)/2 in the statements of Theorems <ref> and <ref> and assume only that Q_p is transitive to conclude that it is primitive and, moreover, contains Alt(Y^*(p)). (This would make Theorem <ref> completely parallel to Theorem <ref> dealing with p≡1(4).) In Remark <ref> below we explain the obstacle to proving this more general version of Proposition <ref>.§.§ Properties of blocks in the action of Q_p on Y^*(p) In the current subsection we prove the two propositions that were stated without proof in the previous subsection. Proposition <ref> is proved in Section <ref>, and Proposition <ref> proved in Sections <ref> (the hyperbolic case) and <ref> (the elliptic case).§.§.§ Homogeneity of coordinates in blocks The subgroup H=⟨_1,_2,_3⟩≤Γ has index at most 2 in Γ. By definition, Γ is generated by the three Vieta involutions and permutations of coordinates. Since R_3=_1·τ_(2 3) and likewise for R_1 and R_2, since τ_(1 3 2)=_3·_1 and since S_3=⟨(12),(132)⟩, we obtain that Γ=⟨_1,_2,_3,τ_(1 2)⟩ =⟨ H,τ_(1 2)⟩. It is easy to check that τ_(1 2)_jτ_(1 2)∈ H for j=1,2,3, so H⊴Γ and Γ=H·⟨τ_(1 2)⟩. This finishes the proof.Recall that Proposition <ref> says that if Q_p acts transitively on Y^*(p), and if B⫋ Y^*(p) is a proper block of the action of Q_p on Y^*(p), then at least two of the coordinates { 1,2,3} are homogeneous in B.Assume that some coordinate, say j=1, is not homogeneous in B. We need to show that the second and third coordinates are homogeneous. The element _1^(p-1)/2 fixes every solution with first coordinate hyperbolic, while _1^(p+1)/2 fixes every solution with first coordinate elliptic. Hence B is invariant under both elements, and thus by _1. By the same argument, if all three coordinates are not homogeneous, B is invariant under H_p=⟨_1,_2,_3⟩≤ Q_p. By Lemma <ref>, [Q_p:H_p]≤2, and transitivity implies there are at most two blocks in the action: B and B'=γ(B) for some γ∈ Q_p. But the block containing [3,3,3] is also invariant under τ_(1 2), hence is invariant under the whole of Q_p – a contradiction.Thus at least one coordinate – the second or the third – is homogeneous. Notice that _1, which stabilizes B, moves the third coordinate of the solutions to the second. Hence both the second and third coordinates must be homogeneous.In fact, the proof of the last lemma yields something slightly stronger. Denote the type of a solution in Y^*(p) by some triple in { h,e} ^3, depending on whether every coordinate is hyperbolic or elliptic. Then, every block B as above contains either only solutions of the same type (homogeneous in all coordinates), or only solutions of exactly two types: one type is (h,h,h) or (e,e,e), and the other differs from the first type in one coordinate (the sole non-homogeneous coordinate). §.§.§ No correlation between two long _1-cycles with the same first hyperbolic coordinate We now prove Proposition <ref> stating that if Q_p is transitive and d_p(± x)≥16√(p+1), then ± x cannot appear twice in the same coordinate in the same proper Q_p-block B⫋ Y^*(p). What we actually prove is the lack of correlation between two long enough cycles of _j with the same j-th coordinate (including the case of two different offsets of the same cycle). The proof of Proposition <ref> is split to the case where x is hyperbolic (in the current subsection) and the case it is elliptic (given in Section <ref>).We use the following classical number-theoretic result:Let f(x)∈𝔽_p[x] be a polynomial with m distinct roots in 𝔽_p which is not a square in 𝔽_p[x]. Then |∑_s∈𝔽_p(f(s)/p)|≤(m-1)√(p).Assume that x is hyperbolic with d_p(± x)≥16√(p+1), and that there are two elements in the proper Q_p-block B⫋ Y^*(p) with ± x in the first coordinate. The same arguments holds, evidently, for every coordinate j=1,2,3.Assume that [x,y_0,y_1] and [x,z_0,z_1] belong to B. By Lemma <ref>, x=ω+ω^-1 with ω∈𝔽_p^* and we can assume |ω|=2d≥32√(p-1): otherwise, replace x with -x and ω with -ω. Write y_0=α+β with α,β∈𝔽_p^* so that αβ=x^2/x^2-4 and y_1=αω+βω^-1 (see Lemma <ref>). The cycle of _1 containing [x,y_0,y_1] is [x,y_0,y_1]=[x,y_d,y_d+1],[x,y_1,y_2],…,[x,y_d-2,y_d-1],[x,y_d-1,y_d]withy_j=αω^j+βω^-j.The set {ω^j} _0≤ j≤2d-1 is the same as the set { s^m} _s∈𝔽_p^* where m=p-1/2d (with every element in {ω^j} covered by p-1/2d different values of s). So as sets,{ y_0,…,y_2d-1} ={αω^j+βω^-j} _0≤ j≤2d-1={ f_α,β(s)def=α s^m+β s^-m} _s∈𝔽_p^*.The same holds for the cycle of _1 containing [x,z_0,z_1] with γ,δ∈𝔽_p^* in the role of α,β, so that z_j=γω^j+δω^-j. We may assume that γ±α, for otherwise [x,y_0,y_1]=[x,z_0,z_1]. Moreover, if s^m=ω^j then f_α,β(s)=y_j and f_γ,δ(s)=z_j. Notice that y_j and z_j are of different types (one hyperbolic and the other elliptic) if and only if ((y_j^2-4)(z_j^2-4)/p)=-1.Since [x,y_j,y_j+1] and [x,z_j,z_j+1] both belong to the block _1^ j(B), we derive from Proposition <ref> that (<ref>) cannot hold for two consecutive values of j. In the parametrization given by s∈𝔽_p^*, this means that ((f_α,β(s)^2-4)(f_γ,δ(s)^2-4)/p)=((f_αω,βω^-1(s)^2-4)(f_γω,δω^-1(s)^2-4)/p)=-1cannot hold for any s∈𝔽_p^*.Write g_α,β(s)def=(s^m)^2(f_α,β(s)^2-4)=[(α s^2m+β)^2-4s^2m]∈𝔽_p[s],and k_1(s)=g_α,β(s)g_γ,δ(s) and k_2(s)=g_αω,βω^-1(s)g_γω,δω^-1(s). Now (<ref>) is equivalent to (k_1(s)/p)=(k_2(s)/p)=-1.Denote by N_(-1,-1) the number of s∈𝔽_p for which (<ref>) holds. Our goal is to show that N_(-1,-1)>0, whence (<ref>) has some solution s0, yielding a contradiction (note that s=0 is not a solution to (<ref>)). Note that k_1(s),k_2(s)0 for every s∈𝔽_p: indeed, g_α,β(0)=β^20, and if 0 s∈𝔽_p and g_α,β(s)=0 then f_α,β(s)=±2 is y_j for some j, but there are no solution in X^*(p) containing ±2 when p≡3(4). Therefore (k_1(s)/p),(k_2(s)/p)0 for s∈𝔽_p and N_(-1,-1)=1/4∑_s∈𝔽_p(1-(k_1(s)/p))(1-(k_2(s)/p)) =1/4[p-∑_p∈𝔽_p(k_1(s)/p)-∑_p∈𝔽_p(k_2(s)/p)+∑_p∈𝔽_p(k_1(s)k_2(s)/p)].For every ∅ B⊆{ 1,2}, define M_Bdef=∑_s∈𝔽_p(∏_j∈ Bk_j(s)/p).Then (<ref>) becomes N_(-1,-1)=1/4[p-M_{ 1}-M_{ 2}+M_{ 1,2}].We use Theorem <ref> to estimate the M_B's. First, we show that none of k_1,k_2 and k_1k_2 are squares in 𝔽_p[x]. The roots of g_α,β(s)=(α s^2m+β+2s^m)(α s^2m+β-2s^m)satisfy s^m=±2±√(4-4αβ)/2α=±1±√(1-x^2/x^2-4)/α=±1±√(-4/x^2-4)/α.As x is hyperbolic and p≡3(4), we have that -4/x^2-4 is not a square in 𝔽_p, so 1 and √(-4/x^2-4) are linearly independent over 𝔽_p, and ±1±√(-4/x^2-4)/α are four distinct values for S^m, different from zero. Moreover, the polynomial s^m-ξ is separable for 0ξ∈𝔽_p^2 because m=p-1/2d<p. So g_α,β(s), which is of degree 4m, has 4m distinct roots in 𝔽_p, and in particular is not a square in 𝔽_p[x]. This analysis shows that g_α,β and g_γ,δ have a common root if and only if α=±γ. Since α±γ by assumption, k_1=g_α,βg_γ,δ and k_2=g_αω,βω^-1g_γω,δω^-1 are both separable of degree 8m. Finally, k_1k_2, of degree 16m, is also not a square in 𝔽_p[x]: for α±αω and if α=±γω then αω±γ.Theorem <ref> yields that |M_{ 1}|,|M_{ 2}|≤(8m-1)√(p) and |M_{ 1,2}|≤(16m-1)√(p). From (<ref>) we now obtainN_(-1,-1) ≥ 1/4[p-2(8m-1)√(p)-(16m-1)√(p)] =1/4[p-32m√(p)+3√(p)]m=p-1/2d= 1/4[p-16(p-1)/d√(p)+3√(p)]d≥16√(p-1)≥ 1/4[p-√((p-1)p)+3√(p)]>3√(p)/4>0.§.§.§ No correlation between two long _1-cycles with the same first elliptic coordinate The general proof strategy for the elliptic case is the same as for the hyperbolic case, albeit with a few extra technical details. In the hyperbolic case, we used a parametrization of the elements of a cycle of _1 as a function over 𝔽_p^*, which allowed us to use Weil's bound (Theorem <ref> above). In the elliptic case, a similar approach requires that we go over the elements in the cyclic subgroup of size p+1 in 𝔽_p^2^ *. The following lemma allows us to parametrize this subgroup as a function over 𝔽_p:The multiplicative subgroupH H≤𝔽_p^2^* of order p+1 satisfiesH={θ+iη | θ,η∈𝔽_p,θ^2+η^2=1} ={2s/1+s^2+i1-s^2/1+s^2 | s∈𝔽_p}∪{ -i}(where i=√(-1)∈𝔽_p^2). Note that (θ+iη)^p=θ-iη (recall that p≡3(4) so i^p=i^4k+3=i^3=-i). So (θ+iη)^p+1=(θ+iη)(θ-iη)=θ^2+η^2. This gives the first equality in (<ref>). A straightforward computation yields the second equality.We assume that x is elliptic with d_p(± x)≥16√(p+1), and assume that there are two elements in the proper Q_p-block B⫋ Y^*(p) with ± x in the first coordinate. We use the notation H for the subgroup of order p+1 in 𝔽_p^2^ *, as in Lemma <ref>. Assume that [x,y_0,y_1] and [x,z_0,z_1] both belong to B. By Table <ref>, x=ω+ω^-1 with ω∈ H, and we can assume that |ω|=2d≥32√(p+1), for otherwise replace ω by -ω and x by -x. Let A∈𝔽_p^2 satisfy that A^p+1=x^2/x^2-4, that y_0=A+A^p and that y_1=Aω+A^pω^-1 (see Lemma <ref>). The cycle of _1 containing [x,y_0,y_1] is [x,y_0,y_1]=[x,y_d,y_d+1],[x,y_1,y_2],…,[x,y_d-2,y_d-1],[x,y_d-1,y_d]withy_j=Aω^j+A^pω^-j.The set {ω^j} _0≤ j≤2d-1 is the same as the set { h^m} _h∈ H where m=p+1/2d, with every element in {ω^j} covered by m different values of h. So as sets, { y_0,…,y_2d-1} ={ Aω^j+A^pω^-j} _0≤ j≤2d-1={ f_A(h)def=Ah^m+A^ph^-m} _h∈ H.The same holds for the cycle of _1 containing [x,z_0,z_1] with C∈𝔽_p^2 in the role of A, so that z_j=Cω^j+C^pω^-j. We may assume that C± A, for otherwise [x,y_0,y_1]=[x,z_0,z_1]. Moreover, if h^m=ω^j then f_A(h)=y_j and f_C(h)=z_j. As in the proof of the hyperbolic case, we derive from Proposition <ref> that ((f_A(h)^2-4)(f_C(h)^2-4)/p)=((f_Aω(h)^2-4)(f_Cω(h)^2-4)/p)=-1cannot hold for any h∈ H. To be able to use Theorem <ref>, we want to reparametrize (<ref>) as polynomials in s∈𝔽_p, using Lemma <ref>. Denoteg_A(s)def=(1+s^2)^2m[f_A(h(s))^2-4]where h(s)=2s+i(1-s^2)/1+s^2=-i(s+i)^2/1+s^2=-i(s+i)/(s-i).Let also k_1=g_Ag_C and k_2=g_Aωg_Cω. Then (<ref>) is equivalent to (k_1(s)/p)=(k_2(s)/p)=-1.As in the proof of the hyperbolic case, denote by N_(-1,-1) the number of s∈𝔽_p for which (<ref>) holds. Our goal is to get a contradiction by showing that N_(-1,-1)>0. Note that g_A(s)0 for s∈𝔽_p because g_A(s)=(1+s^2)^2m(y_j^2-4) for some y_j as above, and s± i and y_j±2. Thus k_i(s)0 neither, and (k_i(s)/p)∈{ 1,-1}. As in equations (<ref>)-<ref> in the hyperbolic case, we get that N_(-1,-1)=1/4[p-M_{ 1}-M_{ 2}+M_{ 1,2}],where for ∅ B∈{ 1,2}, we define M_Bdef=∑_s∈𝔽_p(∏_j∈ Bk_j(s)/p).We use Theorem <ref> to estimate the M_B's. First, we show that k_1,k_2∈𝔽_p[x]. Notice that h(s)^-1=(s-i)/-i(s+i)=i(s-i)/(s+i),so g_A(s) =(1+s^2)^2m[f_A(h(s))+2][f_A(h(s))-2] =(1+s^2)^2m(Ah(s)^m+A^ph(s)^-m+2)(Ah(s)^m+A^ph(s)^-m-2) =(A[-i(s+i)^2]^m+A^p[i(s-i)^2]^m+2[1+s^2]^m)· ·(A[-i(s+i)^2]^m+A^p[i(s-i)^2]^m-2[1+s^2]^m).The last expression shows that g_A(s)∈𝔽_p^2[s]. Its degree is 4m: indeed, the leading coefficient is (-1)^m(A^2+A^2p)+2A^p+1-4,and for m even this coefficient equals (A+A^p)^2-4=y_0^2-4 which is not zero since y_0±2 (see Lemma <ref>). For m odd, this coefficient is -(A+A^p)^2+4(A^p+1-1)=-y_0^2+4(A^p+1-1),which is not zero because A^p+1-1=4/x^2-4 is not a square in 𝔽_p when x is elliptic. As 𝔽_p^2=𝔽_p+i𝔽_p, we can write g_A=g'_A+ig”_A, where g'_A,g”_A∈𝔽_p[s]. By definition, for every s∈𝔽_p, we have h=h(s)∈ H, and g_A(s)=(1+s^2)^2m[f_A(h)^2-4]∈𝔽_pso g”_A(s)=0 for every s∈𝔽_p. Since (g”_A)≤4m<p, we conclude that g”_A is the zero polynomial, hence g_A(s)=g_A'(s)∈𝔽_p[s] and so k_1,k_2∈𝔽_p[x].Next, we wish to show that k_1, k_2 and k_1k_2 are not squares in 𝔽_p[x]. There is a one-to-one correspondence between the roots of g_A in 𝔽_p and the roots ofr_A(h)def=(Ah^2m+2h^m+A^p)(Ah^2m-2h^m+A^p),in 𝔽_p given by α ↦h(α)=-i(α+i)/(α-i) 1+ih/h+i=α(h)h,because ± i is never a root of g_A (recall that g_A(s)∈𝔽_p[s] has the form from (<ref>)) and -i never a root of r_A (because r_A(h)=h^2m(f_A(h)^2-4), -i∈ H and thus f_A(-i)=y_j for some y_j as above, and y_j±2). It is easier to analyze the roots of r_A than those of g_A: if h is a root of r_A thenh^m=±1±√(1-κ(x))/A,where κ(x)=A^p+1=x^2/x^2-4. Now note the following:* The four possible values of h^m are distinct and different from zero (this follows from κ(x)0,1).* Because (m,p)=1, the four polynomials h^m-±1±√(1-κ(x))/A are separable, so r_A has 4m distinct roots in 𝔽_p, and so does g_A.* If A± C, the 4m roots of r_A are distinct from the 4m roots of r_C: certainly 1+√(1-κ(x))/A±1+√(1-κ(x))/C, and if 1+√(1-κ(x))/A=±1-√(1-κ(x))/C we obtainC =± A·1-√(1-κ(x))/1+√(1-κ(x)) κ(x)=C^p+1= A^p+1(1-√(1-κ(x))/1+√(1-κ(x)))^p+1=κ(x)ξ^p+1with ξ=1-√(1-κ(x))/1+√(1-κ(x))∈𝔽_p because 1-κ(x)=-4/x^2-4 is a square in 𝔽_p. Then ξ=±1, that is, C=± A – a contradiction. Hence k_1=g_Ag_C and k_2=g_Aωg_Cω are separable of degree 8m each.* Finally, if C± A, the polynomial k_1k_2=g_Ag_Aωg_Cg_Cω is not a square in 𝔽_p[x]: it is separable unless A=± Cω or Aω=± C, but the two cannot hold simultaneously.We can now apply Theorem <ref> to obtain the same bounds on the M_B's as in the hyperbolic case, and from (<ref>) we now obtainN_(-1,-1) ≥ 1/4[p-2(8m-1)√(p)-(16m-1)√(p)] =1/4[p-32m√(p)+3√(p)]m=p+1/2d= 1/4[p-16(p+1)/d√(p)+3√(p)]d≥16√(p+1)≥ √(p)/4[√(p)-√((p+1))+3]>0. As we noted in Remark <ref> above, it is conceivable that a stronger version of Proposition <ref> holds. Let us point to the phase in the current argument that fails in this more general setting. The simplest case to consider if that of x,x'∈𝔽_p both hyperbolic of maximal order, so d_p(x)=d_p(x')=p-1/2. Assume that x=ω+ω^-1 and x'=ω'+ω'^-1, and that ω'=ω^r. Then, in the notation of Section <ref>, if y_j=α s+β s^-1, then y_j'=α's^r+β's^-r, and our goal is to show that (α s+β s^-1) and (α's^r+β's^-r) cannot be of the same type (hyperbolic/elliptic) for too many values of s∈𝔽_p^*. The problem is that r can be of any order, and is generically of order ≥√(p). For polynomials of such degree Weil's Theorem <ref> is useless. §.§ Deducing Alternating group from primitivity Finally, in this section, we show how to deduce that Q_p≥(Y^*(p)) whenever Q_p is primitive. Throughout this section we denote the symmetric group Sym(n) by S_n and Alt(n) by A_nS_n,A_n. Here we use the following result of Guralnick and Magaard, classifying primitive subgroups of S_n containing an element with at least n/2 fixed points. This theorem relies heavily on the CFSG. We adjust the statement of the theorem to our needs – the original statement in <cit.> is more detailed. In the statement we use the notation Soc(G) for the socle of the group G (see Section <ref> for details), and the standard notation G_1≀ G_2 for the wreath product of two groups. Let G≤S_n be a primitive group, and let x∈ G have at least n/2 fixed points. Then one of the following holds:* G=Aff(2,k) is the affine group acting on 𝔽_2^ k and x is a transvection[To be sure, x is a transvection when Aff(2,k) is embedded in GL(2,k+1) as the matrices with bottom row (0,…,0,1).] and is, in particular, an involution. In this case x has exactly n/2 fixed points.* There are r≥1, m≥5 and 1≤ k≤ m/4 such that n=mk^r, the group S_m acts on the set Δ of k-subsets of { 1,…,m} in the natural way, G≤ S_m≀ S_r acts on Δ^r and Soc(G)=A_m^ r.* For some r≥1, n=6^r, the group S_6 acts on Δ={ 1,…,6} by applying an outer automorphism[Namely, for some fixed φ∈Aut(S_6)∖Inn(S_6), the permutation σ∈ S_6 acts on Δ by σ.i=φ(σ)(i).], G≤ S_6≀ S_r acts on Δ^r and Soc(G)=A_6^ r.* The group G is some variant of an orthogonal group over the field of two elements acting on some collection of 1-spaces or hyperplanes, and the element x is an involution. The following lemma helps us rule out Case <ref> of the above theorem with r=1.Consider the embedding ι S_m↪ S_n given by the natural action of the symmetric group S_m on the set Δ of n=mk k-subsets of m, for some 2≤ k≤m/4. If, for some π∈ S_m, the image ι(π) has a cycle of size divisible by q and a cycle of size divisible by s for some distinct primes q and s, then ι(π) also has a cycle of size divisible by qs. Assume that { a_1,…,a_k}∈Δ belongs to a cycle α of length divisible by q in ι(π). Assume that in π, the elements a_1,…,a_k belong to t distinct cycles: the elements a_1,…,a_ℓ_1 belong to the cycle σ_1, the elements a_ℓ_1+1,…,a_ℓ_2 belong to the cycle σ_2, and so on (each σ_j may contain additional elements not from { a_1,…,a_k}). Let o_1 be the smallest power of σ_1 that maps { a_1,…,a_ℓ_1} to itself. Define o_2,…,o_t analogously. Then, q |lcm(o_1,…,o_t). In particular, q | o_i for some i, and so q||σ_i|. Without loss of generality, assume q||σ_1|, so that a_1 belongs to a cycle σ=σ_1 of π of size divisible by q. Likewise, assume that b_1 belongs to a cycle τ of π of size divisible by s.Denote A={ 1,…,m}∖(σ∪τ) (namely, A consists of the elements not belonging to the cycle σ nor to τ). Assume first that στ. If |A|≥ k-2, then a k-subset containing a_1, b_1 and k-2 elements from A belongs to a cycle of ι(π) of size divisible by qs. If |A|<k-2, then, as k≤m/4, at least one of σ or τ has more than k element. Assume without loss of generality it is σ. Consider the k-subset { b_1,a_1,π(a_1),π^2(a_1),…,π^k-2(a_1)}. This subset belongs to a cycle of ι(π) of size lcm(|τ|,|σ|), which, in particular, is a multiple of qs.Finally, assume σ=τ. Then qs||σ|. If the length of σ is at least k+1, the k-subset { a_1,π(a_1),π^2(a_1),…,π^k-2(a_1),π^k-1(a_1)} belongs to a cycle of ι(π) of size dividing qs. If |σ|≤ k then A contains more than k-1 elements, and the k-subset containing a_1 and k-1 elements from A belongs to a cycle of ι(π) of size dividing qs.Let p≡3(4) be prime. If Q_p is primitive, then Q_p≥(Y^*(p)). Consider _1∈ Q_p. Among the p(p-3)/4 elements in Y^*(p), (p-1)(p-3)/8 belong to cycles of length at least 3 and dividing p-1/2, and (p+1)(p-3)/8 belong to cycles of length at least 3 and dividing p+1/2 (see Table <ref>). Since gcd(p-1/2,p+1/2)=1, the permutation σ=_1^(p+1)/2 fixes exactly (p+1)(p-3)/8>|Y^*(p)|/2 elements of Y^*(p). Thus Q_p satisfies the assumptions in Theorem <ref>. We can now rule out all options except for Q_p=(Y^*(p)) or Q_p=Sym(Y^*(p)).Cases <ref> and <ref> are immediately ruled out because the permutation σ∈ Q_p is not an involution. Case <ref> with r≥2 and Case <ref> are immediately ruled out because |Y^*(p)|=p(p-3)/4 is not a proper power nor equal to six. It remains to consider Case <ref> with r=1. Let q be some prime factor of p-1/2, and let s be some prime factor of p+1/2. By Table <ref>, _1 contains cycles of size divisible by q (indeed, even of size q exactly), and of size divisible by s. However, it does not contain any cycle of size divisible by qs. Using Lemma <ref>, this rules out Case <ref> from Theorem <ref> with r=1 and k≥2. The remaining case, that of Case <ref> with r=k=1, is precisely the case that the group in question is either A_n or S_n. This finishes the proofs of Theorem <ref> and of Corollary <ref>: Theorem <ref> is now a consequence of Theorem <ref> and Proposition <ref>, while Corollary <ref> follows from Corollary <ref> and Proposition <ref>.§ STRONG APPROXIMATION FOR SQUARE FREE COMPOSITE MODULI In this section we derive our main application of the results on the groups Q_p and show that Γ acts transitively on X^*(n) for various square-free composite values n=p_1⋯ p_k. First, in Section <ref>, we prove that if Q_p_j≥Alt(Y^*(p_j)) for every j=1,…,k, then Γ acts transitively on Y^*(n). In Section <ref> we strengthen this result to showing that, moreover, Γ acts transitively on X^*(n), namely, that strong approximation for the Markoff equation holds in modulo n, thus proving Theorem <ref>.At this point, we are able to prove Theorem <ref> that Q_p≥Alt(Y^*(p)) for p≡3(4) satisfying the assumptions in the statement of Theorem <ref>, only while relying on the classification of finite simple groups (CFSG) – see Section <ref>. However, the CFSG is not necessary for establishing the transitivity of Γ on X^*(n) when n=p_1⋯ p_k and p_1,…,p_k are distinct primes satisfying the assumptions in Theorems <ref> or <ref> (this is Corollary <ref>). In Section <ref> we give an alternative proof for the transitivity of Γ on X^*(n), which uses only the primitivity of Q_p, as in Theorem <ref>, thus proving Theorem <ref>. The point is that we want to provide a proof of the transitivity on X^*(n) which can be potentially understood in full, from basic principles, by a motivated reader. This is practically impossible if one relies on the CFSG. §.§ Transitivity of Γ on Y^*(n) Here we prove the following lemma:Let n=p_1⋯ p_k be a product of distinct primes. If Q_p_j≥Alt(Y^*(p_j)) for j=1,…,k, then Γ acts transitively on Y^*(n).Moreover, Q_n, which is a subgroup of Sym(Y^*(p_1))×…×Sym(Y^*(p_k)), contains Alt(Y^*(p_1))×…×Alt(Y^*(p_k)). We prove the proposition by induction on k, the case k=1 being trivial. Assume k≥2. It is enough to show that for every j=1,…,k,Q_n≥1×…×1×Alt(Y^*(p_j))×1×…×1.Recall that Y^*(3)=∅, so we may assume 3∤ n. Without loss of generality we assume that j=k. We first prove (<ref>) assuming p_k≥5. Note that |Y^*(p_k)|≥5 (see Lemmas <ref> and <ref>), and so Alt(Y^*(p_k)) is simple. This group is never a composition (Jordan-Hölder) factor of Alt(Y^*(p_ℓ)) when p_k p_ℓ, because[For p odd the size of |Y^*(p)| is p(p±3)/4 as given in Section <ref>, and |Y^*(2)|=4.] |Y^*(p_k)||Y^*(p_ℓ)|. Now consider the normal seriesQ_n=Q_n∩[Sym(Y^*(p_1))×…×Sym(Y^*(p_k))]Q_n∩[1×…×1×Sym(Y^*(p_k))]@-[u]^⊴Q_n∩[1×…×1×Alt(Y^*(p_k))]@-[u]^⊴1@-[u]^⊴The group Q_p_k is a quotient of Q_n, and so Alt(Y^*(p_k)) a composition factor of Q_n, and thus a composition factor of one of the quotients in (<ref>). But the upper quotient is isomorphic to Q_p_1⋯ p_k-1, which by the induction hypothesis has composition factors Alt(Y^*(p_ℓ)) for ℓ k,p_ℓ2 and possibly some copies of ℤ/2ℤ coming from Sym(Y^*(p_ℓ))/Alt(Y^*(p_ℓ)) or copies of ℤ/2ℤ and ℤ/3ℤ coming from Sym(Y^*(2)). The middle quotient is either trivial or ℤ/2ℤ. Thus Alt(Y^*(p_k)) must be a composition factor of the bottom quotient, so 1×…×1×Alt(Y^*(p_k))≤ Q_n.Finally, if p_k=2, note that |Y^*(2)|=4. The composition factors of Alt(4) are one copy of ℤ/3ℤ and two copies of ℤ/2ℤ. By an argument as above, the factor ℤ/3ℤ must belong to the bottom quotient in (<ref>). Denote Hdef=Q_n∩[1×…×1×Alt(Y^*(2))]=1×…×1× H'.It is easy to check that H⊴ Q_n. For every g_k∈Alt(Y^*(2)) there are g_1,…,g_k-1 with g_j∈Sym(Y^*(p_j)) such that (g_1,…,g_k)∈ Q_n, thus H'⊴Alt(Y^*(2))≅Alt(4). But the only normal subgroup of Alt(4) containing the composition factor ℤ/3ℤ is Alt(4) itself. §.§ Transitivity of Γ on X^*(n) We now finish the proof of Theorem <ref> and prove that if n=p_1⋯ p_k is a product of distinct primes with Q_p_j≥Alt(Y^*(p_j)) for every 1≤ j≤ k, then Γ acts transitively on X^*(n).We want the proof of this section to work in a slightly greater generality than the assumption that Q_p_j≥Alt(Y^*(p_j)), so that it applies also for the next section, where we do not rely on the CFSG. This is part of the motivation for the following notation: Let n=p_1⋯ p_k be a product of distinct primes for which Q_p_j is primitive. We assume further that * The primes are ordered by the order of the rotations _i in the groups Q_p_j, which is |_1| in Q_p= 3 p=2 p(p^2-1)/4p≡1(4) p^2-1/4p≡3(4) .For instance, 7 comes before 5. We break potential ties by putting the larger prime first: for example, we put 11 before 5.* Without loss of generality, 2,5,7,11| n and so the first four primes are 2,7,11,5 (in that order). This assumption is possible because in these four cases, computer simulations indicate that Q_p=Sym(Y^*(p)) is the full symmetric group, so our assumptions always hold.Furthermore, for j=1,…,k, * Let M_j=p_1⋯ p_jM_j denote the product of the first j primes. * Let Ω_jΩ_j⊴Γ denote the kernel of the action of Γ on Y^*(M_j). Note than Ω_j+1⊴Ω_j.* Let Λ_jΛ_j⊴Γ denote the kernel of the action of Γ on X^*(M_j). Note that Λ_j+1⊴Λ_j⊴Ω_j.Finally, for every prime p, we let π_pΓ→ Q_pπ_p denote the projection. In Section <ref> we shall prove the following lemma without relying on the CFSG:Let n=p_1⋯ p_k with Q_p_j primitive for j=1,…,k as in Notation <ref>. Then, for every j=2,…,k, the image of Ω_j-1 in Q_p_j contains a subgroup H_j≤Sym(Y^*(p_j)) satisfying: * H_j is transitive on Y^*(p_j)* H_j is isomorphic to a direct product of non-abelian simple groups[Note that we assume j≥2. Indeed, this does not hold for p_1=2: there are no simple non-abelian subgroups inside Sym(Y^*(2))≅Sym(4).] T_1×…× T_m for some m=m(j)∈ℤ_≥1.In particular, Ω_j-1 acts transitively on Y^*(p_j) and Γ acts transitively on Y^*(n).Note that if we assume that Q_p_j≥Alt(Y^*(p_j)), the conclusion of Lemma <ref> follows immediately from Lemma <ref>: indeed, for p≥5, Alt(Y^*(p)) is indeed transitive on Y^*(p) and is a product of a single non-abelian simple group. So Lemma <ref> is already proven relying on the CFSG, or if one assumes that p_j≡1(4) for j=1,…,k. In the remaining part of this subsection we rely only on the conclusion of Lemma <ref>. We assume Notation <ref> throughout.For j=2,…,k, the group Λ_j-1 acts transitively on Y^*(p_j). Consider the normal series Γ Ω_j-1@-[u]^⊴ Λ_j-1@-[u]^⊴and its projection on Q_p_j via π_jΓ↠ Q_p_j. By Lemma <ref>, π_j(Ω_j-1)≥ H_j where H_j acts transitively on Y^*(p_j) and is a direct product of non-abelian simple groups. As Ω_j-1 fixes Y^*(M_j-1)=Y^*(p_1)×…× Y^*(p_j-1), its action on X^*(M_j-1) fixes every 4-block and only permutes elements inside the 4-blocks, hence the image of Ω_j-1 in Γ_M_j-1 is a subgroup of Sym(4)^|Y^*(p_1)|+…+|Y^*(p_j-1)|. Hence this image is solvable of order 2^α·3^β for some α,β∈ℤ_≥0, so all its composition factors are either ℤ/2ℤ or ℤ/3Z. We deduce that the quotient Ω_j-1/Λ_j-1 has only composition factors ℤ/2ℤ and\or ℤ/3ℤ. Let Ω_j-1=N_0⊵ N_1⊵…⊵ N_r=Λ_j-1be a normal series with quotients ℤ/2ℤ and\or ℤ/3ℤ. Note that the index [H_jπ_j(N_1)∩ H_j] is at most 3, but as H_j is a direct product of non-abelian simple groups, it has no proper subgroups of index[To be sure, the reason that H=T_1×…× T_m with T_1,…,T_m finite non-abelian simple groups has no subgroups of index 2 or 3 is that the normal subgroups of H are B_1×…× B_m where B_i∈{ 1,T_i} for every i (this is standard: if N⊴ H and N∩ T_11, then 1[N,T_1]⊴ T_1, and so [N,T_1]=T_1 and N≥ T_1). In particular, since the smallest non-abelian simple group is Alt(5), any proper normal subgroup of H is of index at least 60. If K≤ H has index 2 or 3, then its core, ∩_h∈ HhKh^-1, is proper normal subgroup of index at most 6, which is impossible.] ≤3, hence π_j(N_1)≥ H_j. By induction, the same argument shows that π_j(N_ℓ)≥ H_j for every ℓ, and, in particular, π_j(Λ_j-1)≥ H_j. For j=5,…,k (so p_j≥13), Λ_j-1 acts transitively on X^*(p_j). Our strategy is to find a triple (x,y,z)∈ X^*(p_j) and elements in Λ_j-1 mapping (x,y,z) to the other elements in its 4-block: (x,-y,-z), (-x,y,-z) and (-x,-y,z). Together with the transitivity of Λ_j-1 on Y^*(p_k) established in Lemma <ref>, this would complete the proof. As in other places in this paper, we deal separately with the case p_j≡1(4) and the case p_j≡3(4), the argument in the former case being simpler. Case 1: p=p_j≡1(4)Take some x∈𝔽_p hyperbolic of maximal order (namely, the _1-cycles in C_1(x) are of length p-1≥12 each). Since 0 has order 4, x0 and (0,x,ix)∈ X^*(p). Let (r,s,t)∈ X^*(p) be another solution with r elliptic. As all _1-cycles in C_1(0) have length 4 and (p+1≡24), we get that _1^ p+1 fixes all four elements in [r,s,t] while mapping (0,x,ix)↦(0,-x,-ix). By Lemma <ref>, there is some g∈Λ_j-1 mapping [0,x,ix]↦[r,s,t]. The element h_1=g^-1·_1^ -(p+1)· g·_1^ p+1 is in Λ_j-1 (as Λ_j-1⊴Γ) and maps (0,x,ix)↦(0,-x,-ix). Since x is maximal hyperbolic, its order is (p-1) which is divisible by 4. Hence -x is also maximal hyperbolic. Let now (r',s',t')∈ X^*(p) be a solution with s' elliptic. Note that (p^2-1/4≡0 p+1) while (p^2-1/4≡p-1/2 p-1). Thus _2^ (p^2-1)/4 fixes all four elements in [r',s',t'] while mapping (0,x,ix)↦(0,x,-ix) and (0,-x,-ix)↦(0,-x,ix). By Lemma <ref>, there is some g'∈Λ_j-1 mapping [0,x,ix]↦[r',s',t']. The element h_1=(g')^-1·_1^ -(p^2-1)/4· g'·_1^ (p^2-1)/4 is in Λ_j-1 and maps (0,x,ix)↦(0,x,-ix) and (0,-x,-ix)↦(0,-x,ix). Case 2: p=p_j≡3(4)In Proposition <ref> below, we prove there is a solution (x,y,z)∈ X^*(p) with both x and y elliptic of order divisible by 4. In this case, -x has the same order as x, say this order is 4m and note that 4m|(p+1). Let (r,s,t)∈ X^*(p) be another solution with r hyperbolic. As p-1≡2(4), there is a number q with (q≡2m4m) and (q≡0 p-1). We get that _1^ q fixes all four elements in [r,s,t] while mapping (x,y,z)↦(x,-y,-z) and (-x,-y,z)↦(-x,y,-z). By Lemma <ref>, there is some g∈Λ_j-1 mapping [x,y,z]↦[r,s,t]. The element h_1=g^-1·_1^ -q· g·_1^ q is in Λ_j-1 and maps (x,y,z)↦(x,-y,-z) and (-x,-y,z)↦(-x,y,-z). In the same fashion, we find an element of Λ_j-1 mapping (x,y,z)↦(-x,y,-z) and we are done. Modulo Proposition <ref> which we prove at the end of this subsection, we can now complete the proofs of Theorem <ref> and Corollary <ref>: We use Notation <ref>. We need to show that Γ acts transitively on X^*(n). We prove that Γ acts transitively on X^*(M_j) for j=1,…,k (recall that M_k=n). For j=4 we verified by computer that Γ is transitive on X^*(2·5·7·11). For j≥5, we use induction and assume that Γ acts transitively on X^*(M_j-1). From Lemma <ref> it follows that Γ is transitive on X^*(M_j).We complete the subsection with the proposition we use in the proof of case 2 in Lemma <ref>:For every prime p3,11 with p≡3(4), there is a solution (x,y,z)∈ X^*(p) with two coordinates elliptic of order divisible by 4.In the proof of Proposition <ref> we use notation as in Section <ref>. As 4|(p+1), if ω∈ H is not a square then 4||ω|. Thus, it is enough to find a solution (x,y,z)∈ X^*(p) with x,y elliptic and the corresponding ω_x,ω_y not squares in H.Assume y=ω+ω^-1 is elliptic (so ω∈ H). Then ω is a square in H if and only if y+2 is a square in 𝔽_p. Note that y+2=ω+ω^-1+2=(ω^1/2+ω^-1/2)^2. If ω^1/2∈ H then ω^1/2+ω^-1/2∈𝔽_p. On the other hand, if ω^1/2∉ H, then ω^(p+1)/2=-1 and so ω^1/2+ω^-1/2∉𝔽_p, because(ω^1/2+ω^-1/2)^p=ω^(p+1)/2ω^-1/2+ω^-(p+1)/2ω^1/2=-(ω^-1/2+ω^1/2)(ω^1/2+ω^-1/2)(the last inequality stems from (ω^1/2+ω^-1/2)^2=y+20).Fix x∈𝔽_p elliptic of maximal order (p+1). So 4||ω_x|=p+1. By Lemma <ref>, it is enough to find y,z∈𝔽_p such that (x,y,z)∈ X^*(p) is a solution, y is elliptic and y+2 is a non-square. Since y elliptic means that y^2-4=(y+2)(y-2) is not a square, we need to find y,z with (x,y,z)∈ X^*(p) and y+2 a non-square and y-2 a square.Imitating the notation from Section <ref>, assume x=ω+ω^-1 with ω∈ H, choose some A∈𝔽_p^2 for which A^p+1=x^2/x^2-4, and let f_A(h)=Ah+A^ph^-1 for h∈ H. Then, {(f_A(h),f_Aω(h)) | h∈ H} ={(y,z) | (x,y,z)∈ X^*(p)} .Recall the parametrization of H∖{ -i} by elements from 𝔽_p described in Lemma <ref>: h(s)=2s+i(1-s^2)/1+s^2=-i(s+i)/s-i. Define g_1,g_2∈𝔽_p[s] as follows:g_1(s)def= (1+s^2)^2[f_A(h(s))+2]=(1+s^2)[2s(A+A^p)+(1-s^2)i(A-A^p)+2(1+s^2)]g_2(s)def= (1+s^2)^2[f_A(h(s))-2]=(1+s^2)[2s(A+A^p)+(1-s^2)i(A-A^p)-2(1+s^2)].It is not hard to see that g_j(s)∈𝔽_p[s]: indeed, A+A^p,i(A-A^p)∈𝔽_p. We now show that for large enough p, there is some s∈𝔽_p for which (g_1(s)/p)=-1  and  (g_2(s)/p)=1.Denote by N_(-1,1) the number of s∈𝔽_p for which (<ref>) holds. Our goal is to show that for large enough p, N_(-1,1)>0. As in the proof of Proposition <ref>, g_1 and g_2 have no zeros inside 𝔽_p because there are no solutions in X^*(p) involving ±2. SoN_(-1,1)=1/4∑_s∈𝔽_p(1-(g_1(s)/p))(1+(g_2(s)/p)).For ∅ B⊆{ 1,2}, let M_Bdef=∑_s∈𝔽_p(∏_j∈ Bg_j(s)/p) and then (<ref>) becomesN_(-1,1)=1/4(p-M_{ 1}+M_{ 2}-M_{ 1,2}).Note that g_1(s)g_2(s)=(1+s^2)^4[f_A(h(s))^2-4]=(1+s^2)^2g_A(s)where g_A(s) is defined as in Equation (<ref>) in Section <ref> for m=1. As our analysis in Section <ref> shows, all roots of g_A, except for ± i, have multiplicity 1. Thus, none of g_1, g_2 or g_1g_2 is a square in 𝔽_p[x]. Now g_1 and g_2 have each at most 4 distinct roots and by Theorem <ref>, |M_{ 1}|,|M_⌊ 2⌋|≤3√(p). Their product g_1g_2 has at most 6 distinct roots, hence by Theorem <ref> |M_{ 1,2}|≤5√(p). From (<ref>) we getN_(-1,1) ≥ 1/4(p-2·3√(p)-5√(p))=p-11√(p)/4.So for p>11^2=121 we have N_(-1,1)>0 and we are done.For all primes p with p≡3(4), p≤121 and p3,11, we verified by a computer there is a solution (x,y,z)∈ X^*(p) with x,y elliptic and of order divisible by 4. For example, one can take (3,3,3)∈ X^*(7), (6,6,8)∈ X^*(19), (3,3,3)∈ X^*(23) and (4,4,9)∈ X^*(31).§.§ Transitivity without the classification In this section we prove Theorem <ref> concerning the transitivity of Γ in square free composite moduli without relying on the CFSG. We are going to use some strong results from the theory of permutation groups, mostly revolving around O'Nan-Scott theorem. While strong, the proofs of these results are completely contained in the book <cit.> and are not more than a few-page-long each. We stress that if all primes in the decomposition of n are 2 or (14), then already the proof in the previous sections does not rely on the CFSG. More concretely, let n=p_1⋯ p_k be a product of distinct primes, and we assume that Q_p_j is a primitive permutation group in its action on Y^*(p_j) for every j=1,…,k. Our goal is to show then that Γ acts transitively on X^*(n). It is enough to prove Lemma <ref> above, as we already showed in Section <ref> how it yields the conclusion we seek. Throughout this subsection we assume Notation <ref>.The CFSG-free proof of Lemma <ref> uses the important concept of the socle:A minimal normal subgroup of a non-trivial group G is a normal subgroup K1 of G which does not contain properly any other non-trivial normal subgroup of G. The socle of G, denoted (G)(G), is the subgroup generated by the set of all minimal normal subgroups of G. Note that (G) is generated by normal subgroups of G and thus (G) G.For example, if m≥5 then (Sym(m))=(Alt(m))=Alt(m). In contrast, (Sym(4))=(Alt(4))={ 1,(12)(34),(13)(24),(14)(23)}.Let G≤Sym(n) be a primitive subgroup. Then exactly one of the following holds: * For some prime p and some integer d, the group G is permutation isomorphic[Two permutation groups are permutation isomorphic if they are the same permutation groups except for, possibly, the labeling of the points in the sets they act on.] to a subgroup of the affine group Aff(p,d) acting on 𝔽_p^ d, so, in particular, n=p^d. In this case, (G) is a regular[A permutation group H≤Sym(n) is called regular if it is sharply transitive. Namely, it is transitive and free. In other words, it is transitive and of order n. The name originates from the observation that such subgroups are obtained as the (left or right) regular representation of order-n groups.] elementary abelian subgroup of order p^d.* (G)=K_1× K_2 where K_1,K_2 G are minimal normal subgroups of G, which are regular, non-abelian and permutation isomorphic to each other. Moreover[For G a group and K≤ G a subgroup, C_G(K)={ g∈ G | gk=kg ∀ k∈ K} is the centralizer of K in G.], C_G(K_1)=K_2 and C_G(K_2)=K_1. In addition, K_1≅ K_2≅ T^m for some finite simple non-abelian group T and some m∈ℤ_≥1.* (G) is a minimal normal subgroup of G. Moreover, C_G((G))=1 and (G)≅ T^m for some finite simple non-abelian group T and some m∈ℤ_≥1.If G≤Sym(n) is a primitive permutation group and 1 H G is a non-trivial normal subgroup, then H is transitive. If p≥5 is prime and Q_p is primitive, then (p)(p)def=(Q_p) acts transitively on Y^*(p) and is a direct product of non-abelian simple groups. Transitivity follows from Theorem <ref> and the fact that the socle is a normal subgroup. Case (1) of Theorem <ref> is ruled out because |Y^*(p)|=p(p±3)/4 is not a prime power (or, alternatively, because Aff(p,d) has no non-identity elements fixing more than half of the points, such as _1^p(p+1)/2∈ Q_p). So either Q_p falls into case (2) or it falls into case (3).We also use the following result giving strong limitations on primitive groups: Let GSym(n), GAlt(n), be a primitive permutation group.* If G is not 2-transitive then |G|<exp{ 4√(n)(ln n)^2}.* If n≥216 and G is 2-transitive and contains a section[A section of a group is some quotient of a subgroup.] isomorphic to Alt(k), then k<6ln n.Let p and q be distinct primes with Q_p and Q_q primitive, and such that p precedes q in the order defined in Notation <ref>. Then Q_pq≥1×(q) (sitting inside Sym(Y^*(p))×Sym(Y^*(q))). Recall that the primes are sorted by the order of rotation elements. So if o_p (o_q, respectively) is the order of _1 in Q_p (Q_q, respectively) then o_p≤ o_q.Case 1: o_p<o_qIf the inequality is strict, then the image of g=_1^o_p∈Γ in Q_p is the identity whereas its image g in Q_q is not. By Corollary <ref>, (q) falls under one of cases (2) or (3) from Theorem <ref>.Assume first that (q) falls under case (3). Since C_Q_p((q))=1, there is some h∈(q) not commuting with g∈ Q_q, so e[g,h]=ghg^-1h^-1∈(q)∩π_q((Γ↠ Q_p)). Since (q) is a minimal normal subgroup of Q_q, it is generated by the conjugates of [g,h] in Q_q, all of which also belong to π_q((Γ↠ Q_p)). Thus (q)≤π_q((Γ↠ Q_p)).Now assume that (q) falls under case (2). Since regular subgroups of Sym(n) are obtained as the (left or right) regular representation of a group of order n, every element of a regular permutation group has all its cycles with equal length. Since _1∈ Q_q contains cycles of coprime lengths, no non-trivial power of it can belong to a regular subgroup, so g=_1^o_p∉ K_1∪ K_2. So there are h_1∈ K_1 and h_2∈ K_2 not commuting with g. Consider h=h_1h_2∈ K_1× K_2=(q). Then [g,h]=([g,h_1],[g,h_1])∈ K_1× K_2=(q) belongs also to π_q((Γ↠ Q_p)) but not to K_1∪ K_2. The only normal subgroups of Q_p which are contained in K_1× K_2 are 1,K_1,K_2 and K_1× K_2. Hence K_1× K_2 is generated by the conjugates in Q_q of [g,h], all of which belong to π_q((Γ↠ Q_p)). Thus (q)≤π_p((Γ↠ Q_p)). Case 2: o_p=o_qWe are left with the rare case[In fact, the only such case with p<1,000,000 is p=5.] that o_p=o_q, as in p=11 and q=5. In this case p>q, p≡3(4), q≡1(4) and (p^2-1)=q(q^2-1). In particular, as Q_q is primitive, it contains the full alternating group Alt(Y^*(q)) by the CFSG-free Theorem <ref>. We claim that Q_p has no composition factor isomorphic to Alt(Y^*(q)). Using this, we can finish as in the proof of Lemma <ref>: indeed, consider the following normal series of Q_pq1⊴ Q_pq∩[1×Alt(Y^*(q))]⊴ Q_pq∩[1×Sym(Y^*(q))]⊴ Q_pq.Since Q_q is a quotient of Q_pq, Alt(Y^*(q)) is a composition factor of Q_pq, so it has to be a composition factor of one of the quotients in (<ref>). The rightmost quotient is Q_p which we show below has no composition factor isomorphic to Alt(Y^*(q)). The second quotient is ℤ/2ℤ or trivial. Thus, the leftmost quotient contains Alt(Y^*(q)) as a composition factor, namely, Q_pq≥1×Alt(Y^*(q)), and we are done as (q)=Alt(Y^*(q)).So we have left to show that Q_p has no composition factor isomorphic to Alt(Y^*(q)). This is certainly the case if Q_p≥Alt(Y^*(p)) (as in the case p=11,q=5). So assume Q_pAlt(Y^*(p)) and proceed using Theorem <ref>. First, assume that Q_p is not 2-transitive. Asymptotically, its order is smaller than that of Alt(Y^*(q)): indeed, if n=|Y^*(p)|=p(p-3)/4 then n≈ q^3, and so by Theorem <ref>,ln|Alt(Y^*(q))| =ln(1/2·(q(q+3)/4)!)≈ q^2ln q ln|Q_p|≤4√(n)(ln n)^2≈ q^1.5(ln q)^2.In fact, this asymptotic reasoning starts taking effect for q≥203,897, but for smaller values of q there are no cases for which o_p=o_q except for q=5 (this was easily verified by computer). Finally, assume that Q_p is 2-transitive. Then, not only does it not have a composition factor isomorphic to Alt(Y^*(q)), it does not even have a section isomorphic to it: since p(p-3)/4≥216, Theorem <ref> says that k=q(q+3)/4<6lnp(p-3)/4. This is impossible when q≥13. We can now finish our CFSG-free proof of Lemma <ref>. Assume n=p_1⋯ p_k is a product of distinct primes with Q_p_1,…,Q_p_k primitive and p_1,…,p_k ordered as in Notation <ref>. We need to show that for every j=2,…,k, the image of Ω_j-1 in Q_p_j, π_p_j(Ω_j-1) contains a subgroup H_j≤Sym(Y^*(p_j)) which is transitive and isomorphic to a direct product of non-abelian simple groups. We show that π_p_j(Ω_j-1)≥(p_j), which is enough by Corollary <ref>.Without loss of generality, it is enough to prove this when j=k. As (p_k)≅∏_i=1^mT_i with T_1,…,T_m non-abelian simple groups, each of them satisfies [T_i,T_i]=T_i. Hence for any t∈ℤ_≥1 there is a sequence of elements g_1,…,g_t∈(p_k) so that the nested commutator […[[g_1,g_2],g_3],…,g_t]has non-trivial projection in each of the T_i's. Choose such a sequence of length t=k-1. By Lemma <ref>, for every i=1,…,k-1, there is an element g_i∈Γ with π_p_i(g_i)=1 and π_p_k(g_i)=g_i. The element g=[…[[g_1,g_2],g_3],…,g_k-1]∈Γsatisfies then that π_p_i(g)=1 for all i=1,…,k-1, whereas π_p_k(g)∈(p_k) in not contained in any proper normal subgroup of (p_k). Hence every element of (p_k) is a product of conjugates of π_p_k(g), and we obtain that π_p_k(Ω_k-1)≥(p_k). § T_2-SYSTEMS This section explains why Theorem <ref> is equivalent to Theorems <ref> and <ref>. Namely, if we let Σ_2,-2(p) denote the set of PSL(2,p)-defining subgroups of _2 with associated trace -2, our goal here is to show: * A one-to-one correspondence between Y^*(p) and Σ_2,-2(p), and* An isomorphism between Q_p, the group of permutations induced by the action of Γ on Y^*(p), and the group of permutations induced by the action of Aut(_2) on Σ_2,-2(p).First, let us define Σ_2,-2(p) properly. For A,B∈PSL(2,p), define (A,B)def=( A, B, AB)∈𝔽_p^ 3/∼,where ∼ is the equivalence of changing the sign of two of the coordinates (each of A and B is a well-defined matrix in SL(2,p) up to a sign). Assume ⟨ A,B⟩ =PSL(2,p), and let φ_2↠PSL(2,p) be the epimorphism mapping the generators a and b of _2 to A and B, respectively. The kernel N=φ is a PSL(2,p)-defining subgroup of _2, and define(N)def=(A,B).Recall that Σ_2(G) denotes the set of G-defining subgroups of _2.The map Σ_2(PSL(2,p))→𝔽_p^ 3/∼ is well-defined. Let G=PSL(2,p). Given N∈Σ_2(G), all epimorphisms _2↠ G with kernel N are obtained one from the other by post-composition with some automorphism from Aut(G). But every automorphism of G is obtained by a conjugation by some element from PGL(2,p). Evidently, such conjugation does not effect the image ofon the images of the generators a and b of _2.Recall that ([A,B])=Q( A, B, AB) where Q(x,y,z)=x^2+y^2+z^2-xyz-2. Thus, for N∈Σ_2(PSL(2,p)), the elementQ(N)def=Q((N))∈𝔽_pis well-defined, and we denoteΣ_2,-2(p)Σ_2,-2(p)def=Q^-1(-2)⊆Σ_2(PSL(2,p)).Note that, by definition, for every N∈Σ_2,-2(p) the triple (N) is (an equivalence class up to sign changes of) a solution to the Markoff equation (<ref>) over ℤ/pℤ.The map |_Σ_2,-2(p) is a bijection from Σ_2,-2(p) to Y^*(p). Consider the map SL(2,p)×SL(2,p)→𝔽_p^ 3 defined as in (<ref>). By <cit.>, if (x,y,z)∈𝔽_p^ 3 is the image of some generating pair in SL(2,p), then every two pairs in ^-1((x,y,z)) are conjugated one to the other by an element g∈SL(2,𝔽_p). Since these pairs are generating, this conjugation by g is an automorphism of SL(2,p). As every automorphism of SL(2,p) is also an automorphism of PSL(2,p), we obtain that |_Σ_2,-2(p)Σ_2,-2(p)→𝔽_p^ 3/∼is injective.By <cit.>, the mapis surjective. The analysis in <cit.> shows that the only triple (x,y,z)∈𝔽_p^ 3 with Q(x,y,z)=-2 which does not correspond to generating pairs is[To see that (0,0,0) is not associated with a generating pair, note that if A∈PSL(2,p) has trace 0, then A is an involution. If both A and B are involutions, then ⟨ A,B⟩ is a dihedral group, which is a proper subgroup of PSL(2,p). ] (0,0,0). This completes the proof of the claim.We have left to show the isomorphism of Q_p and the permutation group induced by Aut(_2) on Y^*(p)≅Σ_2,-2(p). Recall that Q_p=⟨τ_(12),τ_(23),R_3⟩. For _2=(a,b), Aut(_2) is generated by the following Nielsen moves[We deliberately copy the notation for these Nielsen moves from <cit.>.]: r(a,b)↦(a^-1,b), s(a,b)↦(b,a) and t(a,b)↦(a^-1,ab). The induced action of these three automorphisms on Y^*(p) is easily seen to be the same action given by R_3, τ_(12) and τ_(23), respectively. § APPENDIX § ON THE ORDER OF A QUADRATIC INTEGER MODULO MOST PRIMESBY DAN CARMON Throughout this appendix, we use the notation f≪ g to mean that there exists an absolute constant C>0 for which f≤ Cg for all valid values of the implicit variables. The similar notation f≪_ag means there exists a function C=C(a)>0 for which f≤ Cg. The notation f≍ g is shorthand for “f≪ g and g≪ f”. §.§ The main claim Let a∈ℚ(√(D)) be a fixed quadratic integer with norm 1 and absolute value |a|>1 (e.g. a=3+√(5)/2). For primes p∤ D, consider the residue a̅=(a p), as an element of either 𝔽_p or 𝔽_p^2, depending on whether D is a quadratic residue modulo p. In both cases there are two possible choices for a̅, but its order o_p(a), which is the smallest positive integer satisfying a̅^o_p(a)=1∈𝔽_p^2 is well-defined. Let π(x)=#{p≤ x:p– prime}π(x) be the prime counting function. We prove the following claim: For any constant C≥1, #{p≤ x:o_p(a)≤ C√(x)}≪_aπ(x)/(log x)^δ(loglog x)^3/2-δ,where δ is the Erdős-Tenenbaum-Ford constant, δ=1-1+loglog2/log2=0.086071....In particular, the set of primes with o_p(a)>C√(p) has relative density 1.§.§ Proof outline Proposition <ref> follows from the combination of two sub-lemmas:Let α=α(x) tend to infinity arbitrarily slowly with x, and let y=√(x/α). Then #{p≤ x:o_p(a)≤ y}≪_aπ(x)/α.Let α,y be as in the previous Lemma. Define z=C√(x), and u_0=logα/log x. Suppose further that α∈(4/C^2,√(x)/C). Then #{p≤ x:∃ d∈(y,z], p≡±1 d}≪ u_0^δ(log2u_0)^-3/2π(x). Indeed, since a has norm 1, o_p(a) is always a factor of either p-1 when D is a quadratic residue modulo p, or of p+1 when D is a non-quadratic residue, i.e. p≡±1o_p(a) in either case. Thus o_p(a)≤ C√(x) implies that p is either included in the set of the first lemma if o_p(a)≤ y, or in the set of the second lemma if o_p(a)∈(y,z]. Choosing the optimal value α=(log x)^δ(loglog x)^3/2-δyields the claimed value in the right hand side of both lemmas. §.§ Proofs of the lemmasThe following proof is an adaptation of an argument from Erdős and Murty <cit.>, in which only integral values a and a specific choice of α were considered.For every k≥1 define A_k=a^k-a^-k/√(D). Note that A_k is always an integer, with |A_k|<|a|^k, and that o_p(a)=k implies p| A_k. Define B_y=∏_k=1^⌊ y⌋A_k,so that o_p(a)≤ y implies p| B_y. We now observe thatlog B_y=∑_k=1^⌊ y⌋log A_k≤∑_k=1^⌊ y⌋klog|a|≪_ay^2=x/α,and on the other hand log B_y ≥∑_p| B_ylog p≥∑_p : o_p(a)≤ ylog p≥∑_o_p(a)≤ y log√(x) =12log x·#{√(x)<p≤ x:o_p(a)≤ y},whence #{p≤ x:o_p(a)≤ y}≤π(√(x))+2log B_y/log x≪_a2/αx/log x≪π(x)/α.This lemma is a direct application of results due to Ford <cit.>. We cite the relevant definitions and theorems. Ford's main object of study is the function H(x,y,z)=#{n≤ x:∃ d∈(y,z],d| n}.We are particularly interested in the specialized function H(x,y,z;P_λ)=#{n≤ x:n∈ P_λ,∃ d∈(y,z],d| n},where P_λ={p+λ:p– prime} is a set of shifted primes, and more specifically only for λ=±1.In <cit.>, Ford estimates H(x,y,z) for all possible choices of y≤ z≤ x. The relevant case for our choice of y,z is the third subcase of case (v), wherein x,y,z are all large, y≤√(x), and z∈[2y,y^2], all of which are immediately validated for our values, due to the constraint on α. For this case, the theorem states H(x,y,z)/x≍ u^δ(log2u)^-3/2,where u is the number satisfying z=y^1+u, or equivalentlyu=log(z/y)/log y=log(C√(α))/log(√(x/α))=logα+2log C/log x-logα≍logα/log x=u_0.In <cit.>, Ford estimates H(x,y,z;P_λ), for any fixed non-zero λ. The behaviour of the function is determined by whether z is greater or lesser than y+(log y)^2/3. The constraint on α implies z≥2y, so we are certainly in the regime of z≥ y+(log y)^2/3, in which the theorem yieldsH(x,y,z;P_λ)≪_λH(x,y,z)/log x.Combining the estimates (<ref>),(<ref>),(<ref>) yields (<ref>), proving the lemma.alpha Chen Meiri,Department of Mathematics,Technion - Israel Institute of TechnologyHaifa 32000 Israelchenm@tx.technion.ac.il Doron Puder, School of Mathematical Sciences,Tel-Aviv University,Tel-Aviv 69978 Israeldoronpuder@gmail.com Dan Carmon, School of Mathematical Sciences,Tel-Aviv University,Tel-Aviv 69978 Israeldancarmo@post.tau.ac.il

http://arxiv.org/abs/1702.08358v2

^1 Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institut, Villigen PSI, Switzerlandartur.glavic@psi.ch Polarized neutrons are a powerful probe to investigate magnetism in condensed matter on length scales from single atomic distances to micrometers. With the ongoing advancement of neutron optics, that allow to transport beams with increased divergence, the demands on neutron polarizes and analyzers have grown as well. The situation becomes especially challenging for new instruments at pulsed sources, where a large wavelength band needs to be polarized to make efficient use of the time structure of the beam.Here we present a polarization analysis concept for highly focused neutron beams that is based on transmission supermirrors that are bend in the shape of equiangular spirals. The method allows polarizations above 95% and good transmission, without negative impact on other beam characteristics. An example of a compact polarizing device already tested on the AMOR reflectometer is presented as well as the concept for the next generation implementation of the technique that will be installed on the Estia instrument being build for the European Spallation Source. § INTRODUCTIONNeutron radiation has several properties desired in the investigation of many condensed matter systems. In addition to a suitable wavelength on thelength scale, high sensitivity to lighter elements and energies useful for spectroscopic studies, the main advantage the neutrons pose over x-ray photons is the sensitivity to magnetic induction inherent to the magnetic moment of the particle. The neutron is a spin 1/2 particle and thus has only two allowed values for the z-component of its magnetic moment, aligned parallel or anti-parallel to an external magnetic field.One speaks of spin-up/-down polarization, denoted by |+>/|->.Although interaction with magnetic moments in the investigated sample alone can reveal information about the magnetic structure, e.g. magnetic unit cell, symmetry and local moment from powder diffraction, the determination of the neutron polarization before and after the scattering process yields additional knowledge about the direction of magnetization<cit.>. There are basically two principles for the interaction of polarized neutrons with magnetic moments in the sample. Only the projection M⃗_⊥ of the magnetization vector M⃗ onto the plane perpendicular to the scattering vector Q⃗ is detected. Secondly, the relative orientation of this projection to the neutron polarization vector P⃗ determines the strength of the interaction and change of polarization state: non spin-flipscattering:M⃗_⊥|| P⃗ spin-flipscattering:M⃗_⊥⊥P⃗ In the first case the scattering potential is increased in the parallel and decreased in the anti-parallel case.Neutron beams can essentially be polarized using one of three methods:* Magnetic Bragg-scattering in special Heusler alloy crystals, where the nuclear and magnetic cross-sections balance out in the structure factor to only allow one spin state to get reflected<cit.>.The practical advantage of this technique is the simple implementation at instruments that already use crystal monochromators without the need for extra space or complicated arrangements. Polarization efficiency, however, is rather limited and the technique is not suitable for polychromatic beams. * The spin dependent absorption cross-section of ^3He can be used to polarize neutrons by generating a partially nuclear spin polarized gas volume where the neutron beam is transmitted through <cit.>. The much larger attenuation of the undesired spin-state results in a polarized neutron beam. Main advantages of this method are the possibility to specifically optimize the polarization efficiency to the desired application and the relatively low dependence of the device properties on the neutron trajectory. Especially the latter can be used for polarization analysis with large angular coverage. A main limitation of ^3He polarizers is the sensitivity to magnetic field perturbation that make it difficult to use them near high-field magnets. Secondly, the wavelength dependence of the absorption process leads to significant losses in transmission for longer wavelength, especially relevant for instruments with large wavelength bands. The transmission over the bandwidth of a time of flight instrument can vary by a factor of 5 or larger. * Reflection of an artificial multilayer or supermirror with different reflectivities for spin-up and spin-down neutrons<cit.>. These devices can use varying mirror geometries adapted to the specific application, but the physical principle is the same. While limited by the small reflection angle needed for each mirror, these solid state devices can reach very high neutron polarizations with only minimal losses to the intensity of the desired neutron polarization state. In environments with constrained space the necessary system size and/or the high magnetic field needed to saturate the magnetic coating can lead to interference with other instrument components. Here we present a system based on the latter method. The fundamental properties of reflecting polarization mirrors are:* Reflectivity R is a function of the momentum transfer normal to the reflecting surface q_z = 4π/λsinα with the neutron wavelength λ and the incident angle α. * The effective upper critical q_z for the two spin states are q_c^|+> := m  q_c^Ni andq_c^|->, where the critical q_z of natural Niq_c^Ni≈ 0.0022^-1 is used as a reference (m=1). * The polarization efficiency of the polarizer for the reflected beam isP_R= R^|+> - R^|->/R^|+> + R^|->≈1-R^|+>/R^|-> q_c^|-> < q_z < q_c^|+>.Here R^|->≪ R^|+> was used. Typical polarizers based on Fe/Si or FeCoV/Ti:N supermirrors (SM) reach P = 92%…99% with a single reflection. * For the transmitted beam the situation is different:The transmission T^|-> = 1-R^|-> is close to 100% for q_z > q_c^, but T^|+> = 1-R^|+> can reach up to 50% for high m.After a single transmission the polarizing efficiencyP_T= T^|-> - T^|+>/T^|+> + T^|->thus is in the range 30% to 90%.For a good polarization, several SM coatings must be used in sequence.Often the supporting substrate (a Si waver) is coated on both sides, and sometimes a stack of substrates is used. The advantage of using the transmitted beam is, that its trajectory is hardly affected by the polarizer.In addition such devices can be used to suppress long wavelength contamination from frame overlap in pulsed applications, as the transmission for both spin states below the critical edge of the substrate is 0. This manuscript deals essentially with a special geometry of a SM used in transmission that allows polarization of neutron beams with large divergence. We make use of the small focus size in modern, highly focused neutron instruments that provide the necessary boundary condition, namely a limited local angular distribution of neutron trajectories.§ BASIC PRINCIPLEFor a given angular distribution Δα and wavelength spread [λ_min, λ_max] the necessary m of the coating can be calculated withq_c^|-> < 4π/λ_maxsinα_min α_min > arcsinq^|->λ_max/4πandm q_c^Ni ≥ 4π/λ_minsin(α_min+Δα) ,This limits the application of SM polarizers to cases where either the α- or the λ-range are moderate or where a lot of space is available (due to the small angles of incidence α on the polarizer).The latter is the case if a (part) of the neutron beam guide is coated with a polarizing SM or if a polarizing cavity<cit.> is used. A transmission filter used for polarizing a beam or for frame-overlap suppression (i.e. used as a low-pass for λ) should ideally intersect the beam at the same optimized angle α̂ everywhere within the beam.For a parallel beam this is fulfilled by a flat surface, inclined relative to the beam by α̂. Although a highly beam divergence (Δθ) seems to be incompatible with this condition on the first glance, the knowledge of a point of origin/convergence is enough to generate a curved surface that intersects with the beam under the same angle over the whole divergence range. The corresponding surface has the shape of a logarithmic spiral, also called equiangular spiral.§ FEATURES OF AN EQUIANGULAR SPIRALThe general parametric representation of an equiangular spiral is x=a ^bθ cosθ y=a ^bθ sinθwhere a is a scaling factor and b = 1/tanα̂ relates to the intersection angle α̂ of the spiral with an arbitrary trajectory y=x tanθ through the pole at x=y=0. A feature of the equiangular spiral is that a central stretching, i.e. changing a is equivalent to a rotation θ→θ+θ' around the pole.For a small α̂ and a rather low divergence Δθ≪ 10^∘ thespiral can be approximated by the function f(x)= α̂xln[ x/x_min ]where x_min is a scaling factorgiven by the intersection of f(x) with the horizon f(x_min) = 0.§ SINGLE LAMELLA DEVICEThe simplest realization of a polarization- or wavelength filter using the presented concept is a single face with spiral bending in one direction(illustrated in figure <ref>) and no bending in the perpendicular direction.Such a device can in principle cover arbitrary divergences Δθ, where the length of the device growth exponentially with Δθ.The optimization of the device, i.e. the chioce of α̂ and m, has to take into account the final width of the source ±Δ y.This leads to angular errors at the polarizer of Δα= arcsinΔ y / x largest for the point closest to the pole x_min.Thus eqns. (<ref>) and (<ref>) are replaced byα̂ > arcsinq_c^ λ_min/4π+ arcsinΔ y/x_minandm> 4π/q_c^Ni λ_min( sinα̂ + Δ y/x_min) §.§ Demonstrator deviceA polarizer based on this principle was designed and build for the Selene demonstrator at the AMOR reflectometer of PSI<cit.>. The device was designed at the Laboratory for Neutron Scattering and manufactured by SwissNeutronics. The rather limited space of 40cm from source to guide entrance and the relatively large source with a height of 2Δ y = 1mm necessitate a split assembly, i.e. the device has a V-shape comparable to a polarizing V-cavity<cit.>. [ Despite the similar shape, this is no cavity. The polarization is based on exactly one interaction of the beam with the polarizing reflector, there are no outer (reflecting) walls.]Figure <ref> shows the first design sketch and the final device. The technical parameters of the latter are λ∈ [4, 16], x_min = 170mm and x_max = 395mm (point furthest from the pole). The divergence to be covered is Δθ = 1.8^∘.The angle of incidence is α̂= 1.5^∘ and the polarizing coating is a m=4.2 FeCoV / Ti:N supermirror, covered with Ni to simultaneously act as a low-pass for λ < 16.8, i.e.q_min = q_c^Ni in this case. This polarizer was characterized using the Selene guide to focus the divergent beam behind the source at the pole position, a 1mm wide slit behind the neutron guide, to an analyzer.The latter was a Si wafer coated with a polarizing m=6 SM used in reflecting geometry.The magnetization field was 2000Oe to ensure a high efficiency. The spin state of the neutron beam could be switched using an RF flipper located between polarizer and analyzer.In the following we make the conservative (for the calculated performance of the polarizer) assumption that the flipper is 100% efficient.The efficiency of the polarizer P_p is a function of θ and λ, while the efficiency of the analyzer P_a depends on the angle of incidence on its surface, ω, and on λ.The relative alignment of the analyzer ω_a determines the relation ω = θ+ω_a. This can be used to estimate a lower boundary for P_p by performing several measurements with varying ω_a.The measured spin asymmetry P_ω_a(λ,θ) for each pixel is a product of P_p(λ,θ) and P_a(λ,ω_a+θ). Using P_a≤ 1 this gives P_p(λ,θ) ≥ P_ω_a(λ,θ)∀ω_a. Figure <ref> shows an intensity map for the lower boundary for P_p obtained from measurements with 4 different ω_a positions.The gradual decay of P_a with q and its rapid drop at q_c^ leads to the visible intensity steps in the upper left corner of the map.The overall polarization is surprisingly good for only two polarizing transmission interfaces. Therefore the device is now used regularly for user experiments at AMOR. The darker line for θ≈ 0.22^∘ corresponds to trajectories traveling through an area of strongly reduced transmission of the Selene guide, which leads to worse signal to noise ratio for the spin-flip channel and thus reduction of the measured polarization.The brighter region at θ≈ 0^∘ to 0.3^∘ corresponds to trajectories intersecting both substrates, i.e. 3 or 4 transmissions through SM interfaces.The lower performance for λ≈ 4 and θ > 0.6^∘ or θ < -0.6^∘, respectively, means that the angle of incidence at the highest θ angles is larger than expected, which is the result of a too large distance of the polarizer from the source.A better alignment (tilt and distance to the source) is expected to deliverP_p≥93%λ > 4≥96%λ > 7over the full Δθ = 1.8^∘, andP_p≥97%λ > 4over the range at the device front, where the two mirrors overlap, θ = 0.15^∘. The latter value can also be seen as a lower boundary for the performance of a device with two lamella in sequence.The limitations of a single-lamella transmission polarizer are * Δ y/x_min must be small, typically below arcsin 0.35^∘ = 12* The coating is restricted to m < 7, where P_T drops considerably towards this limit, while the costs grow. This restricts α̂* The wavelength band to be polarized is restricted due to the too high transmission forneutrons at large q_z and total reflection from the substrate for low q_z.* For small α̂ and large λ a parallel offset of the beam is caused by refraction.The offset is in the order of a few ten microns and thus only relevant for very good focusing systems.§ MULTI LAMELLAE DEVICES PLANNED FOR ESTIAEstia will be a focusing reflectometer at the European Spallation Source (ESS) that is being build in Lund, Sweden. The instrument concept is based on the Selene neutron guide, but in contrast to the focusing option available at AMOR, the elliptical mirrors will form the complete beamline neutron transport system. Increasing the size of the focusing neutron optics lead to an increased space available for the polarizing optics.The goal is to achieve state of the art neutron polarization analysis while keeping the focusing capability and large intensity towards which the instrument is optimized. Therefore, larger devices with more than one lamella will be constructed for the beamline.Two separate mirrors for the polarizer will be placed close to the focus in the middle of the Selene guide and a double lamella device on the detector arm as an analyzer.§.§ polarizer Figure <ref> shows a conceptual sketch of the Estia polarization concept. Two equiangular spirals intersect the beam before and after the focal point in the middle between the ellipses of the Selene neutron guide, where a large free space is available to place these components. A large polarization (>99%) and decent transmission[The transmission of an unpolarized beam is >40% as >80% of spin-down neutrons are transmitted.] (>40%) can be expected from this geometry, as 4 SM interfaces (with m=5.0 Fe/Si) intersect with the neutron path.The change in intersection direction inherent to the symmetry of the mirror has two additional advantages:* Trajectories that do not pass through the focal point due to the finite size of the source will hit the second mirror with the inverse angular offset δα present at the fist mirror. Therefore a too large angle of incident that reduces polarization efficiency of one mirror leads to improved efficiency in the second mirror and vice versa. * The small linear offset due to refraction in the first silicon wafer will mostly be reversed in the second mirror as the angle of incidence has the same magnitude but opposite sign.With α̂=1.65^∘ and m=5 the polarization system can cover the whole accessible wavelength range from 4 to 25 and, at the same time, allow frame overlap suppression above 30 by using the silicon total reflection. Coatings will be made of Fe/Si in the case of Estia, as the large neutron flux would lead to high activation of the component when using Co containing supermirrors. The two devices will be independent, allowing the user to choose between the use of one or two polarizing devices if higher intensity is desired. §.§ analyzer Analysis on Estia can not follow the same approach as the polarization, as there is not focal point behind the sample. To still be able to reach good polarization efficiency two subsequent transmissions need to be performed. When the analyzer spirals are curved in the direction perpendicular to the reflection plane and the geometry (scaling factor a) is correctly chosen, it is possible to not only measure the transmitted beam with the position sensitive detector but also the neutrons reflected from the first analyzer mirror.For this reason the Estia analyzer, which will be placed together with the detector on a movable table, will consist out of two supermirror lamellae (fig. <ref>). The first will be a m=4 reflection polarizer with just one SM coated side to produce a well polarized spin-up state. After this a second, double sided SM with slightly larger α̂ and m-value will be placed, together with an absorber at the end of the device to capture neutrons only reflected on the second lamella. As a result the transmitted neutrons will have passed three reflective interfaces before reaching the detector, leading to a relatively good polarization. A great gain of detecting both spin-states at the same time can be achieved when analyzing reflectivity from an unpolarized neutron beam, allowing polarized measurements with only slightly reduced intensity (85%) compared to unpolarized measurements. The advantage for polarization analysis is often limited, as the spin-flip signal is much weaker than the non spin-flip reflectivity and dominates the measurement time. On the other hand, if two polarized beams, spin-up and spin-down, hit the sample surface at the same time, this method can be used for complete polarization analysis in time resolved measurements. A later upgrade to Estia, installing two separate Selene neutron guides which share a common focus at the sample position, will make this simultaneous 4-state measurement possible. § CONCLUSIONWe have presented a novel concept to use curved polarizing supermirrors in transmission geometry to cover large beam divergences for instruments with beam focusing. A demonstration device for the focusing option of the AMOR reflectometer was build and tested on the beamline, showing the expected polarization efficiencies. Following the same general concept, the Estia reflectometer at ESS will be build using two spiral shaped polarizers and a double lamella analyzer. With this setup the instrument will achieve a high polarization of 99% over a broad wavelength band with only minor impact on instrument intensity and allow simultaneous measurement of spin-up and spin-down reflectivity on the analyzer side. After full extension of the instrument with two separate Selene beam paths, the ability to measure all 4 spin-states at once will allow novel time resolved polarization analysis studies.As all other polarization devices, the equiangular spiral has its limitations imposed by the SM reflectivity that restricts the range of neutron wavelength and maximum focus size to which the system can be applied. With the ongoing advances in pulsed neutron sources and neutron optics, more instruments will have improved focusing capabilities to be able to measure tiny samples. In this scenario, current ^3He systems are unable to deliver sufficient polarization efficiencies or suffer from significant attenuation when applied to beams with typical wavelength bands used at pulsed sources. Other solid states devices as e.g. cavities, mirror fans or bender systems significantly distort the neutron trajectories and thus the focusing ability of the instrument (or detector resolution). The presented devices are optimized for the focusing geometry and therefore offer significant advantages over the established polarization components. § REFERENCES 10 url<#>1#1urlprefixURL Halpern1939 Halpern O and Johnson M H 1939 Phys. Rev. 55 898–Blume1963 Blume M 1963 Phys. Rev. 130 1670–Lovesey1986 Lovesey S W 1986 Theory of Neutron Scattering from Condensed Matter: Volume 2: Polarization Effects and Magnetic Scattering. 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http://arxiv.org/abs/1702.08315v1

This paper finds the bulk local limit of the swap process of uniformly random sorting networks. The limit object is defined through a deterministic procedure, a local version of the Edelman-Greene algorithm, applied to a two dimensional determinantal point process with explicit kernel. The latter describes the asymptotic joint law near 0 of the eigenvalues of the corners in the antisymmetric Gaussian Unitary Ensemble. In particular, the limiting law of the first time a given swap appears in a random sorting network is identified with the limiting distribution of the closest to 0 eigenvalue in the antisymmetric GUE. Moreover, the asymptotic gap, in the bulk, between appearances of a given swap is the Gaudin-Mehta law – the limiting universal distribution for gaps between eigenvalues of real symmetric random matrices.The proofs rely on the determinantal structure and a double contour integral representation for the kernel of random Poissonized Young tableaux of arbitrary shape. Symmetric flows for compressible heat-conducting fluids with temperature dependent viscosity coefficientsLing WanSchool of Mathematics and Statistics, Wuhan University, Wuhan 430072, ChinaTao WangCorresponding author. Email addresses: ling.wan@whu.edu.cn (L. Wan), tao.wang@whu.edu.cn (T. Wang).School of Mathematics and Statistics, Wuhan University, Wuhan 430072, ChinaDICATAM, Mathematical Division, University of Brescia, Via Valotti 9, 25133 Brescia, Italy =================================================================================================================================================================================================================================================================================================================================================================================§ INTRODUCTION §.§ Overview The main object of this article is the uniformly random sorting network, as introduced by Angel, Holroyd, Romik, and Virág in <cit.>. Let 𝔖_n denote the symmetric group and τ_i denote the transposition between i and i+1 for 1 ≤ i ≤ n-1. The τ_i are called adjacent swaps. Let rev = n, n-1, …, 1 denote the reverse permutation of 𝔖_n. A sorting network of 𝔖_n is a sequence of permutations σ_0 = id, σ_1, …, σ_N = rev of shortest length with the property that for every k,σ_k+1 = σ_k ∘τ_i for somei.In other words, the permutations change by swapping adjacent labels at each step and must go from the identity to the reverse in the shortest number of swaps. The number of adjacent swaps required in any sorting network of 𝔖_n is n2. See Figure <ref> for an example of a sorting network in the wiring diagram representation. We identify a sorting network of 𝔖_n by its sequence of swaps(s_1, …, s_n2),where s_i denotes the adjacent swap (s_i, s_i+1).A random sorting network of 𝔖_n is a sorting network of 𝔖_n chosen uniformly at random. Computer simulations were used to conjecture many beautiful asymptotic properties of random sorting networks. See <cit.> for an account of these statements and the first rigorous results, and also <cit.> for other asymptotic theorems. The proofs of the conjectures from <cit.> were recently announced in <cit.>.In many examples, random combinatorial structures built out of symmetric groups are known to exhibit the same asymptotic behavior as random matrices. The most famous result of this sort due to Baik–Deift–Johansson <cit.> identifies the fluctuations of longest increasing subsequences of random permutations with the fluctuations of largest eigenvalue of random Hermitian matrices. Its further upgrades, <cit.> link fluctuations of several first rows of the Young diagram distributed according to the Plancherel measure for symmetric groups to those of several largest eigenvalues. A connection also exists for “bulk” (i.e. not largest) rows and eigenvalues, see <cit.>.On the other hand, up to now no such connections were known for random sorting networks. In the present article we find such a connection. It exists for a sort of local limit of random sorting networks. Indeed, we find the bulk local limit of random sorting networks, by proving that it is given by a simple, local, deterministic algorithm (the local Edelman-Greene algorithm) applied to a specific random point process on ×_≥ 0. In turn, we describe this point process by showing that its correlation functions have determinantal form and provide explicit expressions for the corresponding correlation kernel. The very same point process appeared in the work of Forrester–Nordenstam <cit.> (see also Deffoseux <cit.>) as the hard edge limit of antisymmetric GUE–corners process; it describes the asymptotic distribution of the closest to 0 eigenvalues of the corners of large antisymmetric matrix with i.i.d. (modulo symmetry) Gaussian entries of mean 0.A corollary of our theorem is that the rescaled, asymptotic distribution of the first time when the swap between ⌊n(1+α)/2⌋ and ⌊n(1+α)/2⌋ + 1 appears, for α∈ (-1,1), is the same as the rescaled, asymptotic distribution of the closest to 0 eigenvalue of an antisymmetric-GUE random matrix. Another corollary is that within the bulk, the asymptotic gap between appearances of the aforementioned swap is described by the Gaudin–Mehta law — the asymptotic universal distribution of the gap between eigenvalues of real symmetric random matrices in the bulk. Complete statements are given in the next section.In an independent and parallel work, Angel, Dauvergne, Holroyd, and Virág <cit.> also study the bulk local limit of random sorting networks. Their approach is very different from ours. We deduce explicit formulas for the prelimit local structure of random sorting networks, and then analyze the asymptotic of these formulas in the spirit of Integrable Probability, see <cit.> and also <cit.>. On the other hand, <cit.> argue probabilistically, analyzing a Markov chain (whose transition probabilities are expressed through the hook formula for dimensions) for sampling random Young tableaux. The connection to random matrices remains invisible in the results of <cit.>. It would be interesting to match these two approaches, but it has not been done so far.§.§ Bulk limit of random sorting networks. We now describe our main result. Informally, we study the asymptotics of the point process (s_i,i), i=1,…,n2, in a window of finite height and order n width, so that the number of points in the window remains finite; see Figure <ref>. Here (s_1,…,s_n2) are swaps of a random sorting network of 𝔖_n.In <cit.> it is proven that the point process (s_i,i) is stationary with respect to the second coordinate. Therefore, it suffices to study windows adjacent to 0 in second coordinate, which we do.The limiting object S_local is a point process on ×_≥ 0 defined by a two–step procedure. First, we introduce an auxiliary point process _edge on ×_≥ 0 through its correlation functions._edge is the (unique) determinantal point process on ×_≥ 0 with correlation kernel K_edge(x_1,u_1;x_2,u_2) = 2/π∫_0^1 t^x_2-x_1cos( tu_1 + π/2 x_1) cos( tu_2 + π/2x_2) dt,if x_2 ≥ x_1; - 2/π∫_1^∞ t^x_2-x_1cos( tu_1 + π/2 x_1) cos( tu_2 + π/2x_2) dt,if x_2 < x_1. We refer to <cit.> and Section <ref> for more detailed discussions of determinantal point processes. We note that the particles of _ edge on adjacent lines {x}×_≥ 0 and {x+1}×_≥ 0 almost surely interlace, see Figure <ref>. The point process _edge has appeared in the random matrix literature before in <cit.>, <cit.>. In more details, let G be an infinite random matrix with rows and columns indexed by _> 0, and whose entries are independent and identically distributed, real-valued, standard Gaussians. Let A = G - G^T/√(2). The top–left m× m corner of A almost surely has 2 ⌊ m/2⌋ non-zero eigenvalues of the form±𝐢_1^m,±𝐢_2^m ,…,±𝐢_⌊ m/2 ⌋^m,where 0 < _1^m < _2^m < ⋯ < _⌊ m/2 ⌋^m. Forrester and Nordenstam prove that _ edge is the weak limit of the point process {(j,√(2M)_i^2M+j)}⊂ℤ×ℝ_≥ 0, i∈ℤ_> 0, j∈ℤ, as M →∞.Particle configurations of S_local are obtained from _edge by a deterministic procedure, which is a local version of the well-known Edelman–Greene bijection <cit.> between staircase shaped tableaux and sorting networks. In the following we describe this procedure. A rigorous definition of the procedure utilizes properties of _edge that are not immediate from Definition <ref>. We provide the rigorous construction in Section <ref> where the description is given in the language of Young tableau, which is the more standard setup for defining the Edelman-Greene bijection. Local Edelman-Greene algorithm Fix a configuration X of _ edge and suppose that we want to define the positions of all particles of S_local inside the rectangle [a,b]× [0,T] with a<0<b. Then almost surely there are two integers â<2a and b̂>2b such that X has no particles on the segments â× [0,T] and b̂× [0,T]. The particles of X outside [â,b̂]×[0,T] are further ignored.We now define a particle configuration Y – the restriction ofS_local onto [â,b̂]×[0,T] – through an iterative procedure. Start by declaring Y=∅, and setting X̂ to be the restriction of X onto [â,b̂]×[0,T]. Repeat the following until X̂ is empty: *Let (x,u) be an element of X̂ with smallest second coordinate.The parity of x will be even. Add (x/2,u) to Y, i.e., redefine Y:=Y∪{(x/2,u)}. * Define the sliding path (x_1,u_1), (x_2,u_2), … as a unique collection ofpoints in X̂ (of maximal length) such that* (x_1,u_1)=(x,u), * u_1<u_2<⋯<u_k and |x_i-x_i+1|=1 for i=1,…,k-1, * For each i=1,…,k-1, the only points of X̂ in the rectangle[x_i-1,x_i+1]× [u_i,u_i+1] are (x_i,u_i) and (x_i+1,u_i+1).In other words, (x_i+1,u_i+1) is the point in [x_i-1,x_i+1] × (u_i,T], which is closest to (x_i,u_i). See Figure <ref> for an illustration. * Remove the k points (x_1,u_1),…, (x_k,u_k) from X̂ andreplace them by k-1 points (x_1,u_2), (x_2,u_3), …, (x_k-1,u_k). * Go back to Step (<ref>), unless X̂ is empty.The first coordinates of the particles of Y will be integral; this follows from the interlacing property of the particles of X̂, which is preserved throughout the steps of the procedure.One immediate property of the just defined map _edge↦ S_local is that the position of the first particle of S_local in the ray {a}×_≥ 0 almost surely coincides with the position of the first particle of _edge in the ray {2a}×_≥ 0. Therefore, the joint law of the positions of the first particles of S_local in the rays {a_i}×_≥ 0, for i=1,…,k, can be explicitly evaluated as a Fredholm determinant. See Corollary <ref> for the case k=1 and <cit.> for general statements.We also show that S_local is invariant under translations and reflections of the first (–valued) coordinate, ergodic with respect to translations of the first coordinate, and stationary in the second (_≥ 0–valued) coordinate; see Proposition <ref>.We are ready to formulate the main result. Fix α∈ (-1,1), and let s_1,s_2,…,s_n 2 be swaps of a random sorting network of 𝔖_n. Define the point process S_α,n of rescaled swaps near the point (n(α+1)/2,0) throughS_α,n={(s_i-⌊n(α+1)/2⌋ , √(1-α^2)·2i/n)}_i=1^n 2.Then as n→∞, the point process S_α,n converges weakly to S_ local.It is proven in <cit.> that the global scaling limit of the space-time swap process of random sorting networks is the product of the semicircle law and Lebesgue measure. The √(1-α^2) scaling of Theorem <ref> is consistent with the semicircle result.We emphasize that Theorem <ref> states both that S_α,n converges and the limit is obtained by applying the localized Edelman-Greene algorithm to _edge. Theorem <ref> does not cover the case |α|=1, where the asymptotic behavior changes. It is plausible that the methods of the present article can be adapted to this remaining case, but we not address it here; see <cit.> for another approach to |α|=1 case.Theorem <ref> implies that the first swap times in random sorting networks converge to a one–dimensional marginal of S_ local; the distribution of the latter can be expressed as a Fredholm determinant. Figure <ref> shows the approximate sample distribution of the rescaled first swap time and (<ref>) shows the tail asymptotics. . Let _FS, α, n be the first time the swap interchanging ⌊n(α+1)/2⌋ with ⌊n(α+1)/2⌋+1 appears in a random sorting network of 𝔖_n. The following convergence in law holds:lim_n→∞ 2 √(1-α^2)/n _FS, α, n = _FS,where_FS > t = 1 + ∑_k=1^∞(-1)^k/k!∫_[0,t]^k[ K_edge(u_i,u_j)] du_1 ⋯ du_k,andK_edge(u_1,u_2)= sin(u_1-u_2)/π (u_1-u_2) + sin(u_1+u_2)/π (u_1+u_2) . Connection to the Gaudin-Mehta law A further consequence deals with the limiting law of the gap between swaps on the same horizontal line in random sorting networks. Fix β∈ (0,1). Given a random sorting network of 𝔖_n, let _+ be the distance between ⌊βn2⌋ and the closest to its right swap interchanging ⌊n(α+1)/2⌋ with ⌊n(α+1)/2⌋+1.Let _- be the analogous distance to the closest to its left swap.Due to stationarity of random sorting networks, the joint law _- and _+ is given by_- > a, _+ > b = _FS, α, n > a+ b.Indeed, due to stationarity, both sides of (<ref>) give the probability of the event that there are no swaps in the interval [-a,b] after the appropriate re-centerings. Equation <ref> shows that the law of (_+, _-), and hence, of the gap _- + _+, is determined by the law of the first swap time _FS, α, n. In particular, their limiting law after rescaling by √(1-α^2)/ n is uniquely determined from the distribution function (<ref>).This is connected to the scaling limit of the point process of eigenvalues of GOE random matrices in the bulk. The scaling limit of the eigenvalues of GOE random matrices near 0 is stationary. Let -Λ_- and Λ_+ be, respectively, the closest to 0 negative and closest to 0 positive point in the limit process. If the matrices are scaled so that the mean eigenvalue gap near 0 is 1, then (<ref>) is the distribution function of (π/2)Λ_+. In other words,(<ref>) is the asymptotic probability to see no eigenvalues in an interval of length (2/π)t for large GOE random matrices, normalized so that the mean eigenvalue gap around the interval is 1; see e.g. <cit.>, <cit.>, <cit.>. The gap between points, Λ_- + Λ_+, has its law determined from that of Λ_+ according to (<ref>). This is the celebrated Gaudin–Mehta law, originally put forward by Wigner as a model for the gap between energy levels in heavy nuclei and later found in numerous systems. We arrive at the following corollary.For α∈ (-1,1) and β∈ (0,1), let Gap_α,β,n be the distance in a random sorting network of 𝔖_n between the two swaps interchanging ⌊n(α+1)/2⌋ with ⌊n(α+1)/2⌋+1: the one closest from the left to time βn2 and the one closest from the right to βn2. Then, the distributional limitlim_n→∞ 4√(1-α^2)/π n Gap_α,β,nis the Gaudin–Mehta law, i.e. the asymptotic gap in the bulk between eigenvalues of real symmetric random matrices with mean gap one. The proof of Theorem <ref> builds upon two ideas. The first one (which is also used in most of the rigorous results on sorting networks) is to reduce the study of random sorting networks to uniformly random staircase shaped standard Young tableaux via the Edelman–Greene bijection <cit.> (see also <cit.>). Our observation is that if we Poissonize uniformly random standard Young tableaux (of arbitrary shape!), then the result can be described by a determinantal point process with an explicit correlation kernel written as a double contour integral. We further show that the Poissonization does not change the local statistics, and therefore, the limit theorem is reduced to the asymptotic analysis of the aforementioned double contour integral, which we perform.Our results on the correlations and limiting behavior of random standard Young tableaux might be of independent interest, and so we present them in the next section. §.§ Random Standard Young Tableaux A partition λ is a sequence of non-negative integers λ_1 ≥λ_2 ≥⋯≥ 0 such that |λ|:=∑_i=1^∞λ_i<∞. The length of λ, denoted ℓ(λ), is the number of positive _i and the size ofis ||.We identify a partition with a Young diagram (YD), which is the set of lattice points{ (i,j) ∈^2:i≥ 1,1 ≤ j ≤_i}.The points of the Young diagramare its cells and we say the Young diagram has shape . Given a pair of YDsand μ, we write ≼μ if the cells ofare contained within the cells of μ. If the containment is strict then ≺μ. If ≼μ then μ∖ denotes the cells of μ that are not in . A standard Young tableau (SYT) of shapeis an insertion of the numbers 1,2, …, || into the cells ofsuch that they strictly increase along the rows (from left to right) and also along the columns (from bottom to top). The numbers within a SYT are its entries. The set of SYTs of shapeis in bijection with the set of increasing sequences of YDs∅ = ^(0)≺^(1)≺^(2)≺⋯≺^(||) =such that the entry k is inserted into the singleton cell of ^(k)∖^(k-1).A staircase shaped SYT of length n-1 (or also n-1 rows) is a SYT of shape (n-1,n-2,…,2,1), which we denote . The Edelman–Greene bijection <cit.> gives a one-to-one correspondence between staircase shaped SYTs and sorting networks; see Section <ref> for the details. This is the reason for our interest in SYTs.A Poissonized Young tableau (PYT) of shape λ is an insertion of distinct real numbers from the interval (0,1) into the cells ofsuch that they strictly increase along the rows andalong the columns. Note that if we replace the entries of a PYT by their relative ranks then we get a SYT. The set of PYTs of shapeis in bijection with the set of increasing sequences of YDs indicating the times of jumps:∅ = ^(0)t_1≺^(1)t_2≺^(2)t_3≺⋯t_||≺^(||) =such that the entry t_k is inserted in the singleton cell of ^(k)∖^(k-1). These increasing sequences of Young diagrams with labels were discussed in <cit.> in the connection to the Young bouquet; see also <cit.>.We would like to identify a PYT with a collection of non-intersecting paths. For that we first map a Young diagram λ to a countable particle configuration {λ_i-i+1/2}_i=1,2,…⊂ℤ+1/2. This procedure can be viewed as projecting the boundary of the Young diagram in Russian notation onto a horizontal line, see Figure <ref>. The empty Young diagram ∅ corresponds to {-1/2,-3/2,-5/2,…}.Give a PYT, for each t consider the countable particle configuration corresponding to the Young diagram filled with the entries ≤ t in the PYT. The trajectories of particles then form a collection of paths, making jumps to the right at the times indexed by the entries t_k of the tableau (equivalently, labels in (<ref>)). Let us draw a cross at a point (x,t), x∈ℤ, 0<t<1, if a particle jumps from (x-1/2) to (x+1/2) at time t; see Figure <ref>. Although there are infinitely many particles, the only ones that move are the ℓ() particles that correspond to the rows ofwith positive size. Given a finite Young diagram λ, consider the point process _ of jumps of a uniformly random Poissonized Young tableau of shape λ. _ is a determinantal point process on × [0,1] with correlation kernel K_(x_1,t_1; x_2,t_2) as follows. For x_1,x_2 ∈ and t_1, t_2 ∈ [0,1],K_(x_1,t_1; x_2;t_2) = t_2 > t_1,x_1 > x_2 (t_1-t_2)^x_1-x_2-1/(x_1-x_2-1)! +[0,_1-x_2)[0,n+x_1)-w/z+1·G_(z+x_2)/G_(x_1-1-w)·(1-t_2)^z (1-t_1)^w/w+z+x_2-x_1+1,where G_(u) = u+1∏_i=1^∞u+i/u-λ_i+i= u+1+n/∏_i=1^n (u-_i +i),n≥ℓ(λ).The contours C_z[0,_1-x_2) and C_w[0,n+x_1) are as shown in Figure <ref>. Both are counter-clockwise, encloses only the integers in the respective half open intervals [0,_1-x_2) and [0,n+x_1), and arranged such that w+z+x_2-x_1+1 remains uniformly bounded away from 0 along the contours.When t_1 or t_2 equals 1, K_ is to be understood in the limit as t_1 or t_2 tends to 1. The contours C_z[0,_1-x_2) and C_w[0,n-x_2) may also be replaced by unbounded contours C_z[0,∞) and C_w[0,∞) with bounded imaginary parts, respectively. The proof of Theorem <ref> is through a limit transition in the correlation kernel of <cit.> (see also <cit.>) for the uniformly random Gelfand–Tsetlin patterns; the proof is in Section <ref>. Such a limit transition can be viewed as a degeneration of the combinatorial structures related to the representation theory of the unitary groups U(N) to those related to the symmetric groups 𝔖_n, see <cit.> for a discussion.Let us emphasize that the very same procedure can be used to identify a uniformly random SYT with a point process of jumps, however, the resulting process will not be determinantal – this is why we need to pass from SYTs to PYTs. §.§ From Poissonized tableaux to local statisticsWe close the introduction with an outline of the argument that takes us from Poissonized tableaux to local statistics of sorting networks. By the nature of the Edelman-Greene bijection, the swaps of a sorting network of 𝔖_n near time 0 are determined by the location of the largest entries of an SYT of shape . These entries reside within unit order distance of the edge of , which consist of the cells (i,n-i) for 1 ≤ i ≤ n-1. As a result, the first step to deriving local statistics of random sorting networks is to derive the statistics of a uniformly random PYT of shapenear its edge.Let _ denote a uniformly random PYT of shape . We are interested in the statistics of the entries of _ that lie within the following windows. A window is parameterized by a center α∈ (-1,1) (corresponding to the center (1+α)n/2 of the swaps of a sorting network), a length L, and an entry height u. The window then consist of entries _(i,j) that satisfy |i-(1+α)n/2| ≤ L and _(i,j) ≥ 1 - u/n. In other words, roughly the largest n u entries of _ with row indices in the interval [(1+α)n/2-L, (1+α)n/2+L].We study the statistics of _ in a window in terms of its associated process of jumps _, rescaled accordingly. For each integer n ≥ 1, let c_n be an integer having the same parity as n and such that |c_n - α n| = O(1) as n →∞. Consider the rescaled process of jumps_α, n = { (x,u) ∈×_≥ 0:(x + c_n, 1 - u/n √(1-α^2) ) ∈_}.Theorem <ref> and Proposition <ref> together imply that _α,n converges weakly to the point process _edge from Definition <ref>. It is the building block for the proof of Theorem <ref>.In Section <ref> we construct the local staircase shaped tableau _edge by using the local jump process _edge. We prove in Theorem <ref> that it provides the limiting statistics of _ in local windows. Using a de-Poissonization argument we conclude in Theorem <ref> that uniformly random staircase shaped SYTs also converge within local windows to _edge. Although Poissonization is not important for the local limit, it is important for the proof.In Section <ref> we prove Theorem <ref>. First, we give a proof of Corollary <ref> in Section <ref>. In Section <ref> we define the local version of the Edelman-Greene algorithm that maps _edge to S_local. In Section <ref> we complete the proof of Theorem <ref> and conclude with some statistical properties of S_local. §.§ AcknowledgementsWe would like to thank Balint Virág, who brought our attention to the problem of identifying the local limit of sorting networks. We are grateful to Alexei Borodin, Percy Deift, and Igor Krasovsky for valuable discussions and references. We would also like to thank an anonymous referee for an exceptionally careful reading of the paper and some helpful feedback.V. Gorin's research was partially supported by NSF grants DMS-1407562, DMS-1664619, by a Sloan Research Fellowship, by The Foundation Sciences Mathématiques de Paris, and by NEC Corporation Fund for Research in Computers and Communications. M. Rahman's research was partially supported by an NSERC PDF award.§ PRELIMINARIESThis section presents basic facts about Young tableaux, Poissonization and determinantal point processes. Some material from the Introduction is repeated for convenience. §.§ Gelfand-Tsetlin patternsA semi-standard Young tableau of shape = (_1, …, _M), where _1 ≥_2 ≥⋯≥_M ≥ 0 are integers, is an insertion of numbers from {1,…, M} into the cells of the YDsuch that the entries weakly increase along each row and strictly increase along every column. It is important to emphasize that while for the definitions of Young diagrams and standard Young tableaux the value of M is not important, here the object substantially depends on M. Semi-standard Young tableaux (SSYTs) are in bijection with interlacing particle systems, often known as Gelfand-Tsetlin patterns (or schemes).A Gelfand-Tsetlin pattern (GTP) with M rows is a triangular array of non-negative integers [a(i,j)] with row i containing i entries a(i,1), …, a(i,i). The array satisfies the following order and interlacing constraints.Order & Interlace: a(i,j) ≥ a(i-1,j) ≥ a(i,j+1) for everyiandj.Let a^(i) = (a(i,1), …, a(i,i)) denote the i-th row of the GTP. Each row corresponds to a YD due to the order constraints. The interlacing conditions ensure that a^(i-1)≼ a^(i), and in fact, a^(i)∖ a^(i-1) is a horizontal strip which means that the cells in any row of a^(i)∖ a^(i-1) are to the left of the cells in the previous row. Figure <ref> provides an example. The set of GTPs with a fixed top row a^(M) is in bijection with the set of SSYTs of shape = a^(M). Indeed, given a GTP [a(i,j)] with top row , such a tableaux is obtained by inserting the value i into the cells of a^(i)∖ a^(i-1) for every 1 ≤ i ≤ M (set a^(0) = ∅). If a^(i)∖ a^(i-1) is empty then i is not inserted. In the reverse direction, given a SSYT of shape , a GTP with top rowis obtained by setting a^(i) to be the YD consisting of the cells ofwith entries ≤ i and removing trailing zero rows to ensure that a^(i) has i entries.A GTP may also be represented as an interlacing particle system on ( +1/2)× as follows. Given a GTP [a(i,j)] with M rows, the particle system [ν(i,j)] associated to it has M rows of particles, with particles on row i being placed on the horizontal line {y=i} of the plane, and the position of the j-th particle on row i is(ν(i,j),i ) =(a(i,j) - j + 1/2, i )for every 1 ≤ j ≤ i.The transformation a(i,j) → a(i,j)-j + 1/2 makes the order constraints strict and the interlacing constraints semi-strict:ν(i,j) ≥ν(i-1,j) > ν(i,j+1).The jumps of an interlacing particle system ν with M rows is a set of points in ×{1, …, M-1}, defined inductively from the top row to the bottom as follows. Given two consecutive rows [ν(i,·)] and [ν(i-1,·)], the jumps on row i consist of particles at the positions(k,i-1) ∈^2 for every integerk ∈ [ν(i-1,j), ν(i,j)] and every 1 ≤ j ≤ i-1.In other words, the jumps of row i are placed on the horizontal line {y = i-1} and fill out integers in the intervals [ν(i-1,j),ν(i,j)] for every 1 ≤ j ≤ i-1. Note that ν may determined from its top row and set of jumps. §.§ Poissonized Young tableauxFor a YD , let [0,1]^ denote the set of functions from the cells ofinto [0,1]. Let PYT() denote the set of all functions T∈ [0,1]^ that satisfy the following tableau constraints.(1) T(i,j) ≤ T(i,j+1)for every(i,j)and(i,j+1) ∈, (2) T(i,j) ≤ T(i+1,j)for every(i,j)and(i+1,j) ∈.The Poissonized tableau (PYT) of shapeis an element of PYT(). The Poissonized staircase shaped tableau of size N = n2 is an element of PYT().Let _ denote a uniformly random element of PYT(). Then _ is related to a uniformly random SYT of shapein the following way. First, the entries of _ are distinct with probability 1. Given that, consider the random SYT _ obtained by inserting k into the cell that contains the k-th smallest element of _. Then _ is a uniformly random SYT of shape . In the other direction, _ can be generated by first sampling _, then independently sampling a uniformly random Y ∈ [0,1]^, and setting _(i,j) to be the _(i,j)-th smallest entry of Y.Throughout the paper, _ denotes a uniformly random element of PYT() and _ denotes a uniformly random SYT of shape . §.§ Jumps of Poissonized tableaux and local limitAny T ∈PYT() can be represented as an interlacing particles system with a fixed top row in the following manner. Consider 0 ≤ t ≤ 1 and letYD^(t) = { (i,j) ∈ : T(i,j) ≤ t}.The tableau constraints (<ref>) ensure that YD^(t) is a YD for every t. Recall that a YD can be made to have an infinite number of rows by appending rows of size 0 after the last positive row. Encodeas particle configuration on + 1/2 by placing a particle at positionν_j = _j - j + 1/2for j ≥ 1.This is an infinite particle configuration on + 1/2 such that ν_1 > ν_2 > ⋯ and ν_j - ν_j+1 = 1 for j > ℓ() (shown in Figure <ref>). Let ν^(t) be the particle configuration associated to YD^(t) via (<ref>) and let ν = (ν^(t); 0 ≤ t ≤ 1) be the particle system on (+ 1/2) × [0,1] with a particle at position (x,t) if and only if x ∈ν^(t).The particle system ν^(t) viewed in reverse time, i.e., from t=1 to t=0, can be interpreted as an ensemble of non-intersecting and non-increasing paths p(i,u), for 1 ≤ i ≤ℓ(). Let p(i,u) be the (+ 1/2)-valued path starting from p(i,0) = ν^(0)_i and decreasing an integer unit at the times 1-T(i,_i), 1- T(i,_i-1), …, 1-T(i,1). If some of the entries are equal then p(i,u) decreases by the number of consecutive equal entries. The paths should be left continuous so that the jumps occur immediately after the jump times. The path p(i,u) decreases by _i units with final position p(i,1) = -i + 1/2. Due to the columns of T being non-decreasing – condition (2) of (<ref>) – the paths are non-intersecting: p(i,u) > p(i+1,u) for every i and u. Figure <ref> shows the paths associated to a staircase shaped PYT.The jumps of p(i,u) consists of points (x,t) ∈× [0,1] such that * 1-t is a discontinuity point of p(i,u), i.e., t equals some entry of T on row i.* x in an integer in the interval [p(i,(1-t)+), p(i,1-t) ], where p(i,u+) = lim_s ↓ u p(i,s).The paths can be reconstructed from their jumps and initial positions. The jumps of ν, and also of T, is the (possibly) multiset of × [0,1] defined byX = { (x,t) : (x,t) is a jump of some path p(i,u)}.X may be a multiset because two adjacent paths may jump at the same time by amounts that causes some of their jumps to coincide. The coinciding jumps has be counted with multiplicity. However, if the entries of T are distinct then X is a simple set. The tableau can be reconstructed from its jumps and the initial position of the paths.Let _ denote the jumps of a uniformly random element _ of PYT(). _ is simple almost surely since _ has distinct entries almost surely. Theorem <ref> asserts that _ is a determinantal point process on × [0,1]. §.§ Determinantal point processesWe describe some basic notions about point processes; for a thorough introduction see <cit.>. Let S be a locally compact Polish space. A discrete subset X of S is a countable multiset of S with no accumulation points. By identifying X with the measure ∑_x ∈ Xδ_x, the space of discrete subsets can be given the topology of weak convergence of Borel measures on S. This means that X_n → X_∞ if for every compact subset C ⊂ S, lim sup_n# (C ∩ X_n) ≤#(C ∩ X_∞), where cardinality is taken with multiplicity. A discrete set is simple if every point in it has multiplicity one. A point process on S is a Borel-measurable random discrete set of S. All point processes considered in this paper will be simple almost surely.Throughout the paper we denote #_ to be counting measure onand (A) to be Lebesgue measure on a measurable subset A ⊂. Also, μ_1 ⊗μ_2 denotes the product of measures μ_1 and μ_2, and μ^⊗ k denotes the k-fold product of μ.A determinantal point processon S is a simple point process for which there is a correlation kernel K : S × S →, and a Radon measure μ on S, called the reference measure, with the following property. For every continuous f : S^k → of compact support,𝔼[∑_(x_1,…, x_k) ∈^k x_1, …, x_kdistinct f(x_1, …, x_k)] = ∫_S^k [ K(x_i,x_j) ] f(x_1,…,x_k)μ^⊗ k(dx_1,…,dx_k).Expectations of the form given by the l.h.s. of (<ref>) determine the law ofunder mild conditions on K <cit.>. This will be the case in this paper as the correlation kernels we consider will be continuous. If S is discrete then it is customary to take the reference measure to be counting measure. In this caseis determinantal if for every finite A ⊂ S,A ⊂ =[ K(x,y)]_x,y ∈ A. The correlation kernel of a determinantal point process is not unique. Ifis a determinantal point process with correlation kernel K then K may be replaced by g(x)/g(y)K(x,y), for any non-vanishing function g, without changing determinants on the r.h.s. of (<ref>). Thus the new kernel determines the same process. This observation will be used multiple times. The determinantal point processes that we consider will be on spaces of the form S = ×{1, …, M}, or S = × [0,1], or S = ×_≥ 0, with reference measures being, respectively, counting measure, #_⊗[0,1] and #_⊗(_≥ 0). The following lemma records some facts that will be used in deriving weak limits of determinantal point processes. We do not include the proof as it is rather standard; see <cit.>.I) Let _M be a determinantal point process on ×{1, …, M} with correlation kernel K_M. For x_1,x_2 ∈ and 0 ≤ t_1,t_2 ≤ 1, letk_M(x_1,t_1; x_2,t_2) = M K_M(x_1, ⌈ Mt_1 ⌉; x_2, ⌈ Mt_2 ⌉).Suppose that k_M → k uniformly on compact subsets of × (0,1). Then the point process_M^scaled = { (x, t/M): (x,t) ∈_M}restricted to × (0,1) converges weakly to a determinantal point processwhose reference measure is #_⊗(0,1) and whose correlation kernel is k. II) Let _n be a determinantal point process on × (0,1) with reference measure #_⊗(0,1) and correlation kernel K_n. For c_n ∈ and β > 0, define a point process on ×_> 0 by_n^scaled = {(x- c_n,β n(1-t) ): (x,t) ∈_n }.The correlation kernel of _n^scaled with reference measure #_⊗(_> 0) isk_n(x_1,u_1; x_2,u_2) = (β n)^-1 K_n(x_1+c_n, 1- u_1/β n;x_2+c_n, 1- u_2/β n).If k_n → k uniformly on compact subsets of ×_> 0 then _n^scaled converges weakly to a determinantal point processwith reference measure #_⊗(_> 0) and correlation kernel k.Extendto a point process on ×_≥ 0 without additional points. Then _n^scaled converges weakly toon ×_≥ 0 if the points of _n^scaled do not accumulate at the boundary in the sense that for every x ∈,lim_→ 0 lim sup_n →∞ _n^scaled∩ ({x}× [0,]) ≠∅ = 0.§ DETERMINANTAL REPRESENTATION OF POISSONIZED TABLEAUX §.§ Determinantal representation of discrete interlacing particle systemsIn order to prove Theorem <ref> we use a determinantal description of discrete interlacing particle systems due to Petrov. This is the main tool behind the proof.Let ν = (ν(M,1) + 1/2 > ⋯ > ν(M,M) + 1/2) be a fixed particle configuration on + 1/2. Here we abuse notation from Section <ref> to have the ν(M,j)s be integers instead of half-integers. Let ℙ_ν be the uniform measure on all interlacing particle systems or, equivalently, GTPs as described in Section <ref>, with fixed top row ν. Let _ν be the point process of jumps of an interlacing particle system sampled according to ℙ_ν, where the jumps are as described in Section <ref>.Petrov <cit.> proves that ℙ_ν is a determinantal point process on(+ 1/2) ×{1, …, M} with an explicit correlation kernel. According to the notation there, particles live onbut we have translated particle systems by 1/2 so that the jumps are integral. In particular, in the notation of <cit.>, one has N=M, x ^M,j = ν(M,j) and the variables x_1, x_2 take integer values. In <cit.> it is explained that the point process of jumps, _ν, is also determinantal on ×{1, …, M-1} and its correlation kernel is given in terms of the correlation kernel of ℙ_ν in <cit.> (up to the translation by 1/2).In particular, <cit.> proves that the correlation kernel of the jumps isK__ν(x_1,m_1;x_2,m_2) = (-1)^x_2-x_1 + m_2-m_1K_ℙ_ν(x_1-1,m_1+1;x_2,m_2),where K_ℙ_ν is the kernel presented in <cit.>. The discussion there is in terms of lozenge tilings of polygonal domains using three types of lozenges as depicted in Figure <ref>. It is proved that the positions of any of the three types of lozenges in such a uniformly random tiling is a determinantal point process. The jumps of an interlacing particle system are given by the positions of the lozenges of the rightmost type from Figure <ref>, where as the particles themselves are given by the positions of lozenges of the leftmost type. Jumps occur when a lozenge of the leftmost type is glued along its bottom diagonal to a lozenge of the rightmost type; see <cit.> for such a tiling. By Remark <ref>, (-1)^x_2-x_1 + m_2-m_1 K_ℙ_ν(x_1-1,m_1+1;x_2,m_2) defines the same determinantal point process as K_ℙ_ν(x_1-1,m_1+1;x_2,m_2), and we will use the latter kernel.Some notation is needed in order to express the kernel for the point process of jumps. For integers a and b, let C[a,b) denote a closed, counter-clockwise contour onthat encloses only the integers a, a+1, …, b-1 if a > b, and empty otherwise. Throughout the paper, all contours intersect the real line at points which have distance at least 1/10 from the integers. This ensures that the integrands of all contour integrals will be a uniform distance away from their poles.For z ∈ and an integer m ≥ 1, let(z)_m = z(z+1) ⋯ (z-m+1),(z)_0 = 1. The process of jumps, _ν, of a uniformly random interlacing particle system with fixed top row ν =(ν(M,1) + 1/2 > ν(M,2) + 1/2 > ⋯ > ν(M,M) + 1/2) is a determinantal point process on ×{1,…,M-1} with correlation kernel K as follows. For x_1, x_2 ∈ and 1 ≤ m_1, m_2 ≤ M-1,K(x_1,m_1; x_2,m_2) = - m_2 ≤ m_1,x_2 < x_1 (x_1-x_2)_m_1-m_2/(m_1-m_2)! + + [x_2,ν(M,1)+1)[x_1-M,ν(M,1)+1) [(z-x_2+1)_M-m_2-1/(w-x_1+1)_M-m_1·(M-m_1-1)!/(M-m_2-1)!××1/w-z·∏_j=1^M w - ν(M,j)/z - ν(M,j)].The contour C_w[-M,ν(M,1)+1) contains C_z[x_2,ν(M,1)+1) without intersecting it.§.§ Proof of Theorem <ref>Letbe a Young diagram with at most n rows of positive length, that is, ℓ() ≤ n. For M ≥ n, consider semi-standard Young tableaux of shape_M = (_1, …, _n, 0, …, 0_M-n zeroes).The effect of adding M-n zero rows is to allow the entries in the non-zero rows ofto be between 1 to M. The law of a uniformly random PYT of shapeis the weak limit of a uniformly random semi-standard Young tableau of shape _M, as M →∞, after the entries are rescaled onto the interval [0,1]. Indeed, the law of a uniformly random PYT of shapecan be approximated by the uniform distribution on points [T(i,j)] ∈ [0,1]^ that satisfy the tableau constraints (<ref>), with each T(i,j) = k/M for some 1 ≤ k ≤ M, and the column constraints being strict.The top row of particle systems associated to semi-standard Young tableaux of shape _M under the bijection described in Section <ref> isν^(M)_ =((_1-1) + 1/2, …, (_n-n)+1/2, -(n+1) + 1/2,-(n+2) + 1/2,…, -M + 1/2 ).Due to the approximation scheme above, and the bijection between semi-standard Young tableaux and interlacing particle systems discussed in Section <ref>, the process of jumps, _, of a uniformly random PYT of shapeis the weak limit of the process of jumps, __M, of a uniformly random interlacing particle system with top row ν^(M) after these jumps are rescaled onto ×{1/(M-1), …, 1}. Thus, we derive a determinantal description of the rescaled jumps of __M in the large M limit. Since _ almost surely contains no jumps on the boundary ×{0, 1}, it suffices to derive the determinantal description with _ restricted to × (0,1).Let K_M denote the kernel from Theorem <ref> for the process 𝒳__M. As explained in Remark <ref>, the kernel (M-1)^x_2-x_1K_M(x_1,m_1;x_2,m_2) determines the same determinantal point process. By Lemma <ref>, in order to prove the theorem it suffices to show that(M-1)^x_2-x_1+1 K_M(x_1, ⌈ (M-1)t_1 ⌉; x_2, ⌈ (M-1)t_2 ⌉)⟶K_,uniformly over compacts subsets of x_1,x_2 ∈ and t_1, t_2 ∈ (0,1).We begin by deforming the contours in the double contour integral from (<ref>) that defines K_M. This will simplify the representation of K_M for taking the large M limit. Deform the w-contour, C_w[x_1-M,ν__M(M,1)+1), by pushing it leftward past the z-contour, C_z[x_2,ν__M(M,1)+1), so that it encloses the consecutive integers min{x_1,x_2} -1, …, x_1-M. The deformation results in picking up residues at w = z and also at the consecutive integers w = x_1-1, …, x_2, if x_2 < x_1.Let J_M(w, x_1,m_1; z, x_2, m_2) denote the integrand of the double contour integral from (<ref>) but without the factor of 1/(w-z). Note that ν__M(M,1) +1= _1. Calculating the residues at w=z and leaving the remaining residues as a contour integral provides the following representation of K_M.K_M(x_1,m_1;x_2,m_2) = - m_2 ≤ m_1,x_2 < x_1 (x_1-x_2)_m_1-m_2/(m_1-m_2)!+ (Ia)+ 1/2 π𝐢∮_C_z[x_2, _1) dz (z-x_2+1)_M-m_2-1/(z-x_1+1)_M-m_1·(M-m_1-1)!/(M-m_2-1)! + (Ib)+ 1/(2 π𝐢)^2∮_C_z[x_2, _1) dz ∮_C_w[x_1-M, min{x_1,x_2}) ∪ C_w[x_2, x_1) dw J_M(w,x_1,m_1; z, x_2, m_2)/w-z(II). The following lemma simplifies (Ia) + (Ib).For 0 ≤ m_1, m_2 ≤ M-1,1/2 π i∮_C_z[x_2, _1) dz (z-x_2+1)_M-m_2-1/(z-x_1+1)_M-m_1·(M-m_1-1)!/(M-m_2-1)!- m_2 ≤ m_1,x_2 < x_1 (x_1-x_2)_m_1-m_2/(m_1-m_2)! = m_2 > m_1, x_1 > x_2(m_1-m_2+1)_x_1-x_2-1/(x_1-x_2-1)!. The integral (Ib) is evaluated in <cit.> by summing over residues at z = x_2, …, x_1-1 and evaluating the resulting sum in closed form via a hypergeometric identity. We have(Ib) = x_1 > x_2(m_1-m_2+1)_x_1-x_2-1/(x_1-x_2-1)!.Therefore,(Ia) + (Ib) = x_1 > x_2 [ (m_1-m_2+1)_x_1-x_2-1/(x_1-x_2-1)! - m_2 ≤ m_1 (x_1-x_2)_m_1-m_2/(m_1-m_2)!].If m_2 ≤ m_1 then the above is 0 because both terms in the difference are equal to (m_1-m_2+x_1-x_2-1)!/(m_1-m_2)!(x_1-x_2-1)!. Hence, (Ia) + (Ib) in non zero only if x_1 > x_2 and m_2 > m_1 and equals what is given in the statement of the lemma. Now consider the expression (II) from (<ref>). Observe that the zeroes of (w-x_1+1)_M-m_1 are at consecutive integers x_1-1, x_1-2, …, x_1-(M-m_1). On the other hand, the polynomial ∏_j (w-ν__M(M,j)) also has zeroes at ν__M(M,n+i) = -(n+i) for 1 ≤ i ≤ M-n. Therefore, the only poles of J_M in the w variable are at the integers x_1-1, x_1-2, …, -n so long as x_1+m_1 ≥ 0. For fixed x_1 and m_1 ≥√(M), say, the condition x_1+m_1 ≥ 0 is satisfied for all large M. Thus, the contour integral over C_w[x_1-M, min{x_1,x_2}) may be shortened to C_w[-n,min{x_1,x_2}) for all large M if x_1 remains fixed and m_1 ≥√(M).Having done so, (II) becomes the following integral after changing variables z ↦ z+x_2 and w ↦ -w + x_1-1:(II) = [0,_1-x_2)[0,n+x_1)(z+1)_M-m_2-1 (M-m_1-1)! ∏_j=1^M -w+x_1-1+ν__M(M,j)/z+x_2-ν__M(M,j)/(-w)_M-m_1 (M-m_2-1)! (w+z+x_2-x_1+1).By a slight abuse of notation, let J_M henceforth denote the integrand of (<ref>) without the factor 1/(w+z+x_2-x_1+1). The following lemma provides the asymptotic form of J_M.Fix integers x_1 and x_2. Suppose 0 < t_1,t_2 < 1 and w and z remain bounded and have distance at least 1/10 from the integers. Then for m_1 = ⌈ t_1(M-1)⌉ and m_2 = ⌈ t_2(M-1)⌉,(M-1)^x_1-x_2-1J_M(w,x_1,m_1;z,x_2,m_2) = (1-t_1)^w (1-t_2)^z -w G_(z+x_2)/z+1 G_(x_1-1-w)(1 + O(M^-1)).The big O term is uniform over z,w so long as the stated assumptions hold and t_1,t_2 remain in compact subsets of (0,1). The function G_ is as stated in Theorem <ref>.We will use the following identity:(y)_m = y+m/y,y ∉{0,-1,-2, …}.We will also use Stirling's approximation of the Gamma function in the following form:y+m/(m-1)! = m^y(1+O(m^-1)),m ≥ 1.The big O term is uniform in m so long as y is bounded and bounded away from negative integers. Using these two properties, if m_1 =⌈ t_1(M-1)⌉ and m_2 =⌈ t_2(M-1)⌉ then(z+1)_M-m_2-1/(-w)_M-m_1·(M-m_1-1)!/(M-m_2-1)! = (1-t_1)^w (1-t_2)^z -w/z+1(M-1)^z+w (1+ O(M^-1)) Now consider the term∏_j=1^M(w-ν__M(M,j) )= ∏_j=1^n (w-_j+j) · (w+n+1)_M-n= ∏_j=1^n (w - _j+j) ·w+M+1/w+n+1.Applying (<ref>) to w+M+1 and z+M+1 gives∏_jw-ν__M(M,j)/z-ν__M(M,j) = z+n+1/w+n+1·∏_j=1^n w-_j+j/z- _j+j· M^w-z(1+O(M^-1)).Substituting in z+x_2 and -w + x_1-1 then gives∏_j -w+x_1-1-ν__M(M,j)/z+x_2-ν__M(M,j) = G_(z+x_2)/G_(x_1-1-w)(M-1)^-(w+z) +x_1-x_2-1 (1+O(M^-1)).Combining (<ref>) with (<ref>) provides the desired conclusion of the lemma. We now prove that (M-1)^x_2-x_1+1K_M converges to K_. Suppose x_i are fixed and m_i = ⌈ t_i(M-1) ⌉ for i = 1,2 and t_i ∈ [δ, 1-δ] for some δ > 0. Recall that K_M is given in (<ref>). For all sufficiently large values of M, Lemma <ref> and then the identity (<ref>) followed by the estimate (<ref>) show that (Ia) + (Ib) of (<ref>) equals(Ia) + (Ib) = m_2 > m_1, x_1 > x_2(m_1-m_2+1)_x_1-x_2-1/(x_1-x_2-1)! = t_2 > t_1, x_1 < x_2(t_1-t_2)^x_1-x_2-1/(x_1-x_2-1)!· (M-1)^x_1-x_2-1 (1+ O(M^-1)). Now we consider (II) in the form given in (<ref>) following the change of variables. Lemma <ref> implies that as M →∞,(M-1)^x_2-x_1+1 J_M(w,x_1,t_1; z, x_2,t_2) → (1-t_1)^w (1-t_2)^z-w G_(z+x_2)/z+1 G_(x_1-1-w).The convergence is uniform over compact subsets of w and z so long as w and z are uniformly bounded away from the integers. For all large values of M the contours of integration of (II) become free of M, namely, z ∈ C_z[0,_1-x_2) and w ∈ C_w[0,n+x_1). The contours may also be arranged such that they remain bounded away from the integers and |w+z+x_2-x_1+1| ≥ 1/10 throughout, say. This implies that as M →∞, (M-1)^x_2-x_1+1· (II) converges to[0,_1-x_2)[0,n+x_1) (1-t_1)^w (1-t_2)^z -w G_(z+x_2)/z+1 G_(x_1-1-w) ×× 1/(w+z+x_2-x_1+1). We have thus concluded that (M-1)^x_2-x_1+1 K_M converges to the kernel K_ given in Theorem <ref>. Moreover, the estimates show that the converge is uniform over compact subsets of × (0,1). Indeed, so long at t_1,t_2 ∈ [δ,1-δ] and |x_1|, |x_2| ≤ B, the error term in the convergence is of order O_B,δ(M^-1), by Lemma <ref>, because the double contours eventually become free of M and the integrand converges uniformly over the contours. Part (I) of Lemma <ref> now implies that the rescaled process of jumps, ^scaled__M, converges weakly on × (0,1) to a determinantal point process with kernel as given in Theorem <ref>. § BULK LOCAL LIMIT OF THE JUMPS OF POISSONIZED STAIRCASE SHAPED TABLEAUXIn this section we prove that the point process X_α,n from (<ref>) converges weakly to the point process _edge from Definition <ref>. This is done in a two-step procedure. First, we prove in Theorem <ref> that the limit of X_α, n is a determinantal point process whose kernel is given in terms of a double contour integral. Second, we identifying this kernel with the one from Definition <ref> in Proposition <ref>.The point process _α, n from (<ref>) converges weakly to a limiting determinantal point process _edge on ×_≥ 0. The correlation kernel of _edge with respect to reference measure #_⊗(_≥ 0) is given as follows. For u_1,u_2 ∈_≥ 0 and x_1, x_2 ∈,K_edge(x_1,u_1;x_2,u_2) = u_2 < u_1, x_2 < x_1 (u_2-u_1)^x_1-x_2-1/(x_1-x_2-1)! +[0,∞)[0,∞)π/2·G(w; x_1,u_1) G(z; x_2,u_2)/w+z +x_2-x_1+1, where G(z; x,u) = u^z/z+1sin(π/2(z+x_2)).The contours C_z[0,∞) and C_w[0,∞) are unbounded, contain the non-negative integers but remain uniformly bounded away from them and are arranged such that w+z+x_2-x_1+1 remains uniformly bounded away from 0. They may also be arranged such that their imaginary parts remain bounded and C_z contains C_w.The value of K_edge when u_1 or u_2 equals 0 is to be understood in the sense of the limit as u_1 or u_2 tends to 0.The proof proceeds in two steps, each verifying the conditions of part (II) of Lemma <ref>. First, we will show that the correlation kernel of _α,n converges to K_edge uniformly on compact subsets of x_1,x_2 ∈ and u_1,u_2 ∈_> 0. Then we will argue that points of _α,n do not accumulate on the boundary ×{0} as n →∞.Let β = √(1-α^2). Part (II) of Lemma <ref> and Theorem <ref> imply that the correlation kernel of _α,n with reference measure #_⊗(_≥ 0) isK_n(x_1,u_1;x_2,u_2) = (β n)^-1 K_(x_1 + c_n, 1- u_1/β n; x_1 + c_n, 1- u_1/β n ).The kernel (β n)^x_1-x_2 K_n determines the same point process by Remark <ref>. Using part (II) of Lemma <ref> it suffices to show that (β n)^x_1-x_2 K_n converges uniformly over compact subsets of ×_> 0 to K_edge as n →∞ in order to deduce convergence of _α,n to _edge on ×_>0.Let G_n denote the function 2^n-1G_, where G_ is as in Theorem <ref>. ThenG_n(z) = 2^n-1z+n+1/∏_j=1^n-1 (z-n + 2j) = z+n+1/(z-n/2+1)_n-1 = z+n+1z-n+2/2/z+n/2.Substitute in x_i +c_n for the variables x_i and 1 - (β n)^-1u_i for the variables t_i in K_. Then,K_n(x_1,u_1;x_2,u_2) = u_2 < u_1, x_2 < x_1 (u_2-u_1)^x_1-x_2-1/(x_1-x_2-1)! (β n)^x_2-x_1 +[0,n-1-c_n-x_2)[0,n-1+c_n+x_1)-w G_n(z+x_2+c_n) u_1^w u_2^z (β n)^-w-z-1/z+1 G_n(x_1-1+c_n-w) (w+z+x_2-x_1+1).Using the formula for G_n from (<ref>) and applying the identity1-yy = π/sin(π y),y ∉{0,-1,-2,…},to y = n-c_n-z/2givesG_n(z+c_n)= z+n+c_n+1/n+c_n+z/2n-c_n-z/2·π/sin (π/2(n-c_n-z) ). The estimate for z from (<ref>) along with the observation that n ± c_n = (1 ±α)n+ O(1)→ +∞ implies the following asymptotic bevaviour as n →∞. The symbol ∼ denotes a multiplicative term 1+O(n^-1) where the big O error is uniform over z in compact subsets of ∖.n ±(c_n+z)/2 ∼n ± c_n/2·( n ± c_n/2)^±z/2,n+c_n+z+1 ∼n+c_n· (n+c_n)^z+1.Therefore,n+c_n+z+1/n+c_n+z/2n-c_n-z/2·n+c_n+w/2n-c_n-w/2/n+c_n+w+1 ∼(n+c_n/n-c_n)^w-z/2 (1+α)^z-wn^z-w∼( 1-α/1+α)^z-w/2 (1+α)^z-wn^z-w = (β n)^z-w.Substituting in z+x_2 for z and -w+x_1-1 for w in this estimate givesG_n(z+x_2+c_n)/G_n(-w+x_1-1+c_n)∼sin (π/2(-w+x_1-1-n-c_n) )/sin (π/2(z+x_2-n-c_n) ) (β n )^w+z + x_2-x_1+1.The error in the above estimate vanishes as n →∞ so long as w, z and the x_i lie in compact subsets of their respective domains. We also have thatsin( π/2(n-c_n-w) )/sin(π/2(n-c_n-w)) = sin(π/2 w )/sin(π/2 z )if n-c_n is even.Since n-c_n is assumed to be even we conclude from (<ref>) that so long w and z are bounded and remain uniformly bounded away from the integers thenlim_n →∞(β n)^x_1-x_2+1 G_n(z+x_2+c_n) (β n)^-w-z-1/G_n(x_1+c_n-1-w) = -sin (π/2 (w-x_1+1) )/sin (π/2 (z+x_2)). The above displays the pointwise limit of the part of the integrand from (<ref>) that depends on n. In order to interchange the pointwise limit with the contour integral we must show that the integrand is bounded uniformly over n by a function that is integrable over the contours z ∈ C_z[0,n-1-c_n-x_2) and w ∈ C_w[0,n-1+c_n+x_1), also uniformly over n. Then we may apply the dominated convergence theorem.Towards this end suppose z is such that (1) |(z)| ≤ 2n, (2) |(z)| is uniformly bounded over n, say by 100, and (3) z remains bounded away fromby distance at least 1/10. In this case Stirling approximation to the Gamma function implies that modulus of the ratio of the l.h.s. of (<ref>) to its r.h.s. is bounded above and below by exponential factors of uniform rate in |(z)|. The same holds for the ratio of the l.h.s. of (<ref>) to its r.h.s. That is, for some constant C,e^-C(|(z)| + 1)≤ | n ±(c_n+z)/2n ± c_n/2·( n ± c_n/2)^±z/2| ≤ e^C(|(z)| + 1),e^-C(|(z)| + 1)≤ | n+c_n+z+1n+c_n· (n+c_n)^z+1 |≤ e^C(|(z)| + 1). Throughout the following C denotes a constant that is free of n but its value may change from line to line. We combine the estimates above with the equation for G_n(z+c_n) from (<ref>) and observe that there is a C such that 1/C ≤ |sin(n-c_n-z)| ≤ C due to the assumptions on z. This in turn implies that there is a C such thate^-C (|(z)| +1)≤ | G_n(z+c_n)/(β n)^z| ≤ e^C (|(z)| + 1). The contours C_z[0,n-1-c_n-x_2) and C_w[0,n-1+c_n+x_1) can certainly be arranged such that for fixed x_1 and x_2 the variables z+x_2 and -w+x_1-1 satisfy the aforementioned assumptions (1)–(3) uniformly over n. Thus, we get the following uniform estimate over n with z ∈ C_z[0,n-1-c_n-x_2) and w ∈ C_w[0,n-1+c_n+x_1):| G_n(z+x_2 + c_n)/G_n(-w+x_1-1+c_n)| ≤ (β n)^(z+w) +x_2-x_1+1e^C(|(z)| + |(w)| +1).The contours may also be arranged such that |w+z+x_2-x_1+1| ≥ 1/10, say. Then the modulus of the integrand of the double contour integral from (<ref>) satisfies(β n)^x_1-x_2 | -w G_n(z+x_2+c_n) u_1^w u_2^z (β n)^-w-z-1/z+1 G_n(x_1-1+c_n-w)(w+z+x_2-x_1+1)| ≤|-w/z+1 | |u_1^w| |u_2^z| e^C (|(z)| + |(w)| +1). Stirling's approximation implies that if (z) ≥ 1/10 and |(z)| remains bounded then|u^z|/|z+1| = e^-(z) log(z) + (z)(log u + O(1)).Applying (<ref>) with y = -w also gives -w = - π [w+1sin(π w)]^-1. Note that |sin (π w)|^-1≤ C so long as w remains uniformly bounded away from the integers. Combining this with (<ref>) shows that the r.h.s. of (<ref>) is integrable over unbounded double contours C_z[0,∞) ∋ z and C_w[0,∞) ∋ w as long as the contours are arranged such that z,w remain uniformly bounded away from the integers, have uniformly bounded imaginary parts, and z+w+x_2-x_1+1 remains uniformly bounded away from 0. Thus, the limit (<ref>), upper bound (<ref>) and the dominated convergence theorem implies that as n →∞(β n)^x_1-x_2K_n(x_1,u_1; x_2,u_2) →u_2 < u_1, x_2 < x_1 (u_2-u_1)^x_1-x_2-1/(x_1-x_2-1)! + [0,∞)[0,∞) u_1^w u_2^z ·--wsin (π/2 (w-x_1+1) )/z+1sin (π/2 (z+x_2) )·1/w+z+x_2-x_1+1. Furthermore, our estimates show that the convergence is uniform over x_i in compact subsets ofand u_i in compact subsets of _> 0. (In fact, when some u_i → 0 the integral contributes only through residues at the origin. Lemma <ref> computes the limit as u_i → 0.) Comparing the limit integrand with the one presented in (<ref>) we observe that the proof of the kernel convergence will be complete once it is shown that- -wsin (π/2 (w-x_1+1) ) = (π/2)/w+1sin (π/2 (w+x_1) ) .From (<ref>), --w = π [w+1sin(π w)]^-1. Also, sin ( π/2(w-x+1)) = cos(π/2(w-x)). Finally, double angle trigonometric formulae imply sin(π w) = 2 sin( π/2(w+x)) cos( π/2(w-x)). Substituting these equations into the l.h.s. of the above verifies the equality with the r.h.s.To complete the proof of convergence of _α,n to _edge on ×_≥ 0 it is enough to show, using Lemma <ref>, that for every x ∈,lim_→ 0 lim sup_n →∞ # _α,n∩ ({x}× [0,]) = 0.In other words, points do not accumulate at the boundary in the limit. From relation (<ref>) for determinantal point processes and (<ref>) we get# _α,n∩ ({x}× [0,]) = ∫_0^ K_n(x,t;x,t)dt= [0,n-1-c_n-x)[0,n-1+c_n+x)-w G_n(z+x+c_n) (∫_0^ t^w+zdt ) (β n)^-w-z-1/z+1 G_n(x-1+c_n-w) (w+z+1)= [0,n-1-c_n-x)[0,n-1+c_n+x)-w G_n(z+x+c_n)^w+z+1(β n)^-w-z-1/z+1 G_n(x-1+c_n-w) (w+z+1)^2. The quantity above is of the form I_,n. Arguing exactly as in the derivation of the limit kernel, I_,n→ I_, where I_ is given by the double contour integral in the definition of K_edge(x,;x,) from (<ref>) but with an additional factor of w+z+1 in the denominator of the integrand. Indeed, I_ = ∫_0^ K_edge(x,t;x,t) dt, which is the expected number of points of _edge on {x}× [0,]. The quantity I_ remains uniformly bounded near = 0 since, as → 0, thecontribution to the integral that defines I_ comes from the residues at w,z = 0 and these residues do not depend on . (See Lemma <ref> where lim_→ 0 K_edge(x,;x,) is derived analogously.) Consequently, I_→ 0 as → 0 and thus the condition from (<ref>) holds.§.§ Integral representation of the edge kernel We begin with an auxiliary lemma.Let C_w [0,∞) and C_z [0,∞) be contours as in the statement of Theorem <ref> and let G(z;x,u) be as in (<ref>). For t > 0 letI(t) = [0,∞)[0,∞)π/2·G(w; x_1,tu_1) G(z; x_2,tu_2))/w+z +x_2-x_1+1.Then,lim_t → 0 I(t) = 2/π cos (π/2x_1) cos(π/2 x_2) /x_2-x_1+1 if x_2 ≠ x_1-1-x_1even if x_2 = x_1-1.Moreover, I is continuously differentiable on _> 0 and can be differentiated by interchanging differentiation with the contour integration.The integrand of I is continuously differentiable in t. Observe that the contours of integration contain no singularities of the integrand, and in fact, are arranged to be a positive distance from all zeroes of sin(π/2(z+x_2)) and sin(π/2(w+x_1)) in the denominator. The estimate for |u^z| / |z+1| from (<ref>) shows that the derivative of the integrand in the variable t is absolutely integrable over the contours as long as t lies in a compact subset of _≥ 0. Consequently, by the dominated convergence theorem, I is continuously differentiable and the derivative may be interchanged with integration.Let us now consider the limiting value of I(t) as t → 0. Decomposing C_w as C_w[0,1) ∪ C_w[1,∞), and similarly for C_z, gives∮_C_z∮_C_w = ∮_C_z[0]∮_C_w[0] + ∮_C_z[0]∮_C_w[1,∞]+ ∮_C_z[1,∞]∮_C_w[0] + ∮_C_z[1, ∞]∮_C_w[1,∞].These four contours may also be arranged such that (w+z) > 0 unless both w ∈ C_w[0] and z ∈ C_z[0]. Recall thatG(z; x,u) = u^z/z+1sin (π/2(z+x)).Thus, the integrand of I(t) converges to 0 as t → 0 so long as w ∉ C_w[0] and z ∉ C_z[0]. So each of the double contour integrals above except for the first has a limit value of 0 as t → 0 (the limit operation may be interchanged with integration as argued above). To complete the proof it suffices to calculatelim_t → 0 [0][0]π/2·G(w; x_1,tu_1) G(z; x_2,tu_2))/w+z +x_2-x_1+1. The integral above is evaluated via residues at w=0 and z=0. If x_2 ≠ x_1-1 then (<ref>) equalsRes_z=0( Res_w=0 ( (π/2)G(w;x_1,tu_1) G(z;x_2,tu_2)/w+z+x_2-x_1+1) ) = 2/πcos (π/2x_2)cos (π/2x_1)/x_2-x_1+1.This is the limit value of I(t) in the statement of the lemma for x_2 ≠ x_1-1.Now consider the limit (<ref>) in the case x_2 = x_1-1. As the contour C_w[0] can be arranged to be contained inside C_z[0], the integral in w equals the residue of the integrand at the only possible pole at w = 0. This equalsRes_w=0 ((π/2) G(w;x_1,tu_1) G(z;x_2,tu_2)/w+z) = cos(π/2 x_1) (tu_2)^z/zz+1sin(π/2(z+x_2)).If x_1 is odd then the above equals 0. Otherwise, cos( (π/2)x_1) = (-1)^x_1/2 and the integral of the above over C_z[0] is given by its residue at the pole z=0 (note that sin(π/2 x_2) ≠ 0 since x_2 = x_1-1 is odd). The residue equalsRes_z=0 (cos(π/2 x_1) (tu_2)^z/zz+1sin(π/2(z+x_2))) = cos(π/2x_1)/sin(π/2x_2) = -1.Thus, if x_2 = x_2-1 then (<ref>) equals - x_1even and this completes the proof. The kernel K_edge has the following form.K_edge(x_1,u_1;x_2,u_2) = 2/π∫_1^0 t^x_2-x_1cos( tu_1 + π/2 x_1) cos( tu_2 + π/2x_2) dt,if x_2 ≥ x_1; - 2/π∫_1^∞ t^x_2-x_1cos( tu_1 + π/2 x_1) cos( tu_2 + π/2x_2) dt,if x_2 < x_1. For t > 0 letf(t) = K_edge (x_1, tu_1; x_2, tu_2) = t^x_1-x_2-1u_1 > u_2, x_1 > x_2(u_2-u_1)^x_1-x_2-1/(x_1-x_2-1)! + I(t),where I(t) is as defined in Lemma <ref>. By Lemma <ref>, f is continuous differentiable on _>0 and the function t^x_2-x_1+1f(t) may be differentiated by interchanging differentiation with integration. Differentiating t^x_2-x_1+1f(t) and clearing common powers of t gives(x_2-x_1+1)f + tf'= [0,∞)[0,∞)π/2· G(w; x_1,tu_1) G(z; x_2,tu_2)) = π/2·1/2 π𝐢∮_C_w[0,∞) dwG(w; x_1,tu_1) ·1/2 π𝐢∮_C_z[0,∞) dzG(z; x_2,tu_2)). The contour integrals can be evaluated by summing over residues of G. Inside the contour C_z[0,∞], the function G(z,x,u) has simple poles at integers z such that sin (π(z+x)/2) = 0. In other words, z has to have the same parity as x. Let x̅ =xodd. The residues of the integral come from integers z = 2k + x̅ for k ≥ 0. Since Res_y=2k(sin(π y/2)^-1) = (2/π)(-1)^k, summing over the residues gives1/2 π𝐢∮_C_z[0,∞) dw (tu)^z/z+1sin (π/2(z+x)) = 2(-1)^x+x̅/2/π∑_k ≥ 0(-1)^k (tu)^2k + x̅/(2k + x̅)! = 2(-1)^x+x̅/2/πcos( ut -π/2x̅) = 2/πcos( ut +π/2 x).Consequently,(x_2-x_1+1)f(t) + tf'(t) = 2/πcos( u_1t +π/2x_1 ) cos ( u_2t + π/2x_2 ).Multiplying (<ref>) by t^x_2-x_1 then implies that[t^x_2-x_1+1f]' = 2/π t^x_2-x_1cos( u_1t +π/2x_1) cos ( u_2t + π/2x_2 ). For x_2 ≥ x_1, the r.h.s. of (<ref>) is integrable over t in [0,1]. Moreover, t^x_2-x_1+1f(t) → 0 as t → 0 because lim_t → 0 f(t) = lim_t → 0 I(t), and the latter limit is finite whereas t^x_2-x_1+1→ 0 as t → 0. Therefore, (<ref>) impliesf(1) = 2/π∫_0^1 t^x_2-x_1cos( u_1t +π/2x_1 ) cos ( u_2t + π/2x_2 )dt. Next, consider the case x_2 < x_1-1. Now the relation from (<ref>) should be integrated from 1 to ∞, which is convergent since x_2-x_1 ≤ -2. The formula follows so long as lim_t →∞ t^x_2-x_1+1f(t) = 0. Rather than derive this limit we take a slightly indirect approach by considering the limit of f(t) near t=0. For t > 0 defineg(t)= - 2/π∫_1^∞ s^x_2-x_1cos ( u_1st +π/2x_1) cos ( u_2st + π/2x_2)ds = - 2/πt^x_1-x_2-1∫_t^∞ s^x_2-x_1cos ( u_1s +π/2x_1) cos ( u_2s + π/2x_2)ds.Upon differentiating g it follows readily that g satisfies the same differential equation as f displayed in (<ref>). Therefore, f(t) = g(t) + C for some constant C. In order to identify C as zero it suffices to show that lim_t → 0 f(t)-g(t) = 0. Since t^x_1-x_2-1→ 0 as t → 0, due to x_2 < x_1-1, both f(t) and I(t) have the same limit as t → 0. Thus, utilizing Lemma <ref>, showing C=0 amounts to provinglim_t → 0 g(t) = 2/π cos (π/2x_1) cos(π/2 x_2) /x_2-x_1+1. The limit of g(t) can be found using L'Hôspital's rule, which shows thatlim_t → 0 g(t)= lim_t → 0- 2/π∫_t^∞ s^x_2-x_1cos ( u_1s +π/2x_1) cos ( u_2s + π/2x_2) ds/t^x_2-x_1+1= lim_t → 0 2/πt^x_2-x_1cos ( u_1t +π/2x_1) cos ( u_2t + π/2x_2)/(x_2-x_1+1) t^x_2-x_1 = 2/π cos(π/2x_1) cos(π/2x_2) /x_2-x_1+1.We conclude that g(t) = f(t) for t ∈_> 0, and in particular that f(1) = g(1), as required.Finally, consider the case x_2 = x_1-1. The r.h.s. of (<ref>) is continuous for t in [0,1] because one of cos ( u_1t +π/2x_1) or cos ( u_2t + π/2x_2) has a zero at t=0 depending upon the parity of x_1. From Lemma <ref>, lim_t → 0 f(t) = u_1 > u_2 -x_1even. Therefore, (<ref>) implies thatf(1) = u_1 > u_2 -x_1even + 2/π∫_0^1 t^-1cos ( u_1t +π/2x_1) cos ( u_2t + π/2x_2) dt. We now express (<ref>) as an integral over t ∈ [1,∞) as given in the proposition. First, note K_edge may be modified on the measure zero set consisting of (x_1,u_1;x_2,u_2) such that u_1=u_2 without changing determinants in (<ref>), and thus, this does not affect the law of _edge. We will modify the kernel on this zero set after the following calculations to get the form given in the proposition.Observe that cos(u_2t + π/2 x_2) = sin(u_2t + π/2x_1) if x_2=x_1-1. Using trigonometric formulae the integrand of (<ref>) becomes2 cos ( u_1t +π/2x_1) sin ( u_2t + π/2x_1)/π t = sin((u_1+u_2)t) + sin((u_2-u_1)t)/π t, x_1even -sin((u_1+u_2)t) + sin((u_2-u_1)t)/π t, x_1odd.Using the fact that ∫_0^∞sin t/t = π/2, we get that for a ∈,∫_0^1 sin (at)/π t dt = sgn(a) ∫_0^|a|sin t/π t dt= sgn(a)/2 - ∫_1^∞sin(at)/π t dt. Using (<ref>) and the representation of the integrand in (<ref>) we infer that if x_1 is even thenu_1 > u_2 -x_1even + 2/π∫_0^1 t^-1cos ( u_1t +π/2x_1) sin ( u_2t + π/2x_1) dt = u_1 > u_2 - 1 + sgn(u_1+u_2) + sgn(u_2-u_1)/2 - ∫_1^∞sin((u_1+u_2)t) + sin((u_2-u_1)t)/π t dt =- u_1=u_2 (1 + u_1=0/2) - 2/π∫_1^∞ t^-1cos ( u_1t +π/2x_1) sin ( u_2t + π/2x_1) dt.This shows that (<ref>) equals the expression given in the statement of the proposition for x_2 = x_1-1 and x_1 even except for the additive term - u_1=u_2 (1 + u_1=0/2). By modifying K_edge on the zero set {u_1=u_2, x_2=x_1-1, x_1even} we may ignore this term.For x_1 being odd we argue in the same manner as above to infer that (<ref>) equalsu_1 > u_2 -x_1even + 2/π∫_0^1 t^-1cos ( u_1t +π/2x_1) sin ( u_2t + π/2x_1) dt = u_1 > u_2- sgn(u_1+u_2) + sgn(u_1-u_2)/2 + ∫_1^∞sin((u_1+u_2)t) + sin((u_1-u_2)t)/π t dt== -u_1=u_2 > 0/2 - 2/π∫_1^∞ t^-1cos ( u_1t +π/2x_1) sin ( u_2t + π/2x_1) dt.Once again, we modify K_edge on the zero set {u_1 = u_2, x_2=x_1-1, x_1odd} to ignore the additive term -1/2u_1=u_2 > 0 and get the expression of the kernel given in the proposition. §.§ Statistical properties of _edgeThis section derives certain properties of _edge, namely, Proposition <ref>, Proposition <ref>, Lemma <ref> and Lemma <ref>, that will be used to derive the local limit of staircase shaped tableaux and of sorting networks.The process _edge has the following statistical properties.I) Translation and reflection invariance: For any integer h the translated process_edge + (2h,0) = { (x+2h,u) : (x,u) ∈_edge}and the reflected process(-1,1) * _edge = {(-x,u): (x,u) ∈_edge}have the same law as _edge.II) One dimensional marginals: For any x ∈ and u_1, u_2 ∈_≥ 0,K_edge(x,u_1; x,u_2) = sin(u_1-u_2)/π(u_1-u_2) + (-1)^xsin(u_1+u_2)/π(u_1+u_2).Therefore, _edge∩ ({x}×_≥ 0) is a determinantal point process with reference measure (_≥ 0) and correlation kernel (<ref>).Part I) From Lemma <ref> the correlation kernel of _edge + (2h,0) equals K_edge(x_1-2h,u_1; x_2-2h,u_2). The integral representation of K_edge in Proposition <ref> implies thatK_edge(x_1-2h,u_1; x_2-2h,u_2) = K_edge(x_1,u_1; x_2,u_2)upon observing that cos (x + π h) = (-1)^h cos(x), which implies that the integrands do not change after the kernel is transformed. Consequently, the translated point process has the same law as the original. Similarly, the correlation kernel for the reflected process is K_edge(-x_1,u_1;-x_2;u_2) = (-1)^x_1-x_2 K_edge(x_2,u_2;x_1,u_1). The latter kernel defines the same determinantal point process as _edge in law. 0.1in Part II) Proposition <ref> gives thatK_edge(x,u_1;x,u_2)= 2/π∫_0^1 cos(tu_1 + π/2x) cos(tu_2 + π/2x) dt = 1/π(sin(tu_1-tu_2)/u_1-u_2 + sin(tu_1 + tu_2 + π x)/u_1+u_2 )|_t=0^t=1= 1/π ( sin(u_1-u_2)/u_1-u_2 + (-1)^x sin(u_1+u_2)/u_1+u_2).The fact hat _edge∩ ({x}×_≥ 0) is determinantal with kernel as stipulated follows from the relation (<ref>) for determinantal point processes. There is a universal constant C such that for x_1,x_2 ∈ and u_1, u_2 ∈_≥ 0,| K_edge(x_1,u_1;x_2,u_2) | ≤C/max{ |x_1-x_2|, |u_1-u_2| } + 1. Throughout this argument C denotes a universal constant whose value may change from line to line. We begin with the case x_2 ≠ x_1-1. From the integral representation of K_edge we see that if x_2 ≥ x_1 then|K_edge(x_1,u_1;x_2,u_2)|= 2/π | ∫_0^1 t^x_2-x_1cos(tu_1+π/2x_1)cos(tu_2+π/2x_2) dt| ≤2/π∫_0^1 t^x_2-x_1 dt ≤C/|x_2-x_1|+1.Similarly, if x_2 < x_1 then x_2 ≤ x_1-2 and|K_edge(x_1,u_1;x_2,u_2)|= 2/π | ∫_1^∞ t^x_2-x_1cos(tu_1+ π/2x_1)cos(tu_2+π/2x_2) dt| ≤C/|x_2-x_1|+1.Combining these bounds we deduce that if x_2 ≠ x_1 -1 then| K_edge(x_1,u_1;x_2,u_2)| ≤C/|x_1-x_2|+1. Now we consider decay in the u-variables, assuming that x_2 ≠ x_1-1. Define v(t) asv(t) = sin (tu_1-tu_2 + π/2 (x_1-x_2))/π(u_1-u_2) + sin (tu_1+tu_2 + π/2 (x_1+x_2))/π(u_1+u_2) ifu_1 ≠ u_2 t cos(π/2(x_1-x_2))/π + sin (tu_1+tu_2 + π/2 (x_1+x_2))/π(u_1+u_2) ifu_1 = u_2.Then v'(t) = 2/πcos(tu_1 + π/2x_1)cos(tu_2 + π/2x_2). Using that |sin(y)/y| ≤ 1, and the formula for v(t), we observe that there is a C such that|v(t)| ≤C/|u_1-u_2|+1if |u_1-u_2| ≥ 1. Applying integration by parts to the integral form of K_edge gives, for x_2 ≥ x_1,K_edge(x_1,u_1;x_2,u_2) = v(1) -v(0)x_1=x_2 - ∫_0^1 (x_2-x_1)t^x_2-x_1-1 v(t) dt.Now the triangle inequality and (<ref>) imply that if |u_1-u_2| ≥ 1 then|K_edge(x_1,u_1;x_2,u_2)|≤C/|u_1-u_2|+1 + C |x_2-x_1|/|u_1-u_2|+1 ∫_0^1 t^x_2-x_1-1dt ≤2C/|u_1-u_2|+1.If |u_1-u_2| < 1, then we use the bound (<ref>) to reach the same conclusion as above. An entirely analogous bound holds when x_2 < x_1 because then x_2 ≤ x_1 -2, and t^x_2-x_1 is integrable over t ∈ [1,∞). Therefore, for x_2 ≠ x_1-1,| K_edge(x_1,u_1;x_2,u_2)| ≤C/|u_1-u_2|+1.Combining (<ref>) with (<ref>) implies the required inequality (<ref>) for x_2 ≠ x_1-1.The case x_2 = x_1-1 requires some care. The representation (<ref>) for the integrand of K_edge(x_1,u_1;x_1-1,u_2) givesK_edge(x_1,u_1;x_1-1,u_2)= - 2/π∫_1^∞ t^-1cos( tu_1 + π/2 x_1) sin( tu_2 + π/2x_1)dt= - ∫_1^∞(-1)^x_1sin((u_1+u_2)t) + sin((u_2-u_1)t)/π tdt. Integration by parts and the triangle inequality imply that for a ≥ 1,| ∫_1^∞ dt sin(a t)/t | =| cos(a)/a - ∫_1^∞ dt cos(at)/at^2| ≤C/a .For 0 ≤ a ≤ 1, we have| ∫_1^∞ dt sin(a t)/t | =| π/2 - ∫_0^1 dt sin(at)/t | ≤ C + ∫_0^1 dta ≤ C.Together, these bounds imply that for a ∈,| ∫_1^∞ dt sin(a t)/t | ≤C/|a|+1. Separating (<ref>) naturally into two integrals and applying (<ref>) implies that|K_edge(x_1,u_1;x_1-1,u_2)| ≤C/|u_1-u_2|+1.This establishes (<ref>) for the case x_2 = x_1-1 and completes the proof.Spatial ergodicity of _edge For h ∈, denote by τ^h the translation that maps (x,u) ↦ (x+2h,u) for (x,u) ∈×_≥ 0. So _edge is invariant under the action of every τ^h by Proposition <ref>. An event E associated to _edge is invariant if for every h ∈, E = τ^h E, where τ^hE = {τ^h(ω): ω∈ E} and τ^h(ω) is the action of τ^h on a sample outcome ω of _edge. The invariant sigma-algebra of _edge is the sigma-algebra _inv consisting of all the invariant events._edge is ergodic w.r.t. spatial translations in that if E ∈_inv then E∈{0,1}.For A ⊂×_≥ 0, let (A) = σ (_edge∩ A) be the sigma-algebra generated by the points of _edge restricted to A.For A, B ⊂×_≥ 0, letdist(A,B) = inf {max{|x-y|, |u-v|}: (x,u) ∈ A, (y,v) ∈ B }.For k ≥ 1, suppose f: (×_≥ 0)^k → is continuous and compactly supported. LetN(f) = ∑_(x_1,u_1), …, (x_k,u_k) ∈_edge(x_i,u_i) all distinct f(x_1,u_1;⋯; x_k,u_k).Now suppose f, g : (×_≥ 0)^k → are continuous and compactly supported such that there are disjoint subsets A, B ⊂×_≥ 0 with support(f) ⊂ A^k and support(g) ⊂ B^k. This implies that if (x_1,u_1;⋯; x_k,u_k) ∈support(f) and (x_k+1,u_k+1;⋯; x_2k, u_2k) ∈support(g), then (x_i,u_i) ≠ (x_k+j,u_k+j) for every 1 ≤ i,j ≤ k. We first show that in this case| N(f)N(g) - N(f)N(g) | ≤(2k)! C^2k/dist(A,B)^2 +1||f||_1 ||g||_1,where C is the universal constant from Lemma <ref> and ||f||_1 is the L^1-norm of f with respect to (#_⊗(_≥ 0))^⊗ k.Indeed, the assumption on the supports of f and g imply from (<ref>) thatN(f)N(g) = ∫_(×_≥ 0)^2k[ K_edge(x_i,u_i;x_j,u_j)]_1 ≤ i,j ≤ 2kf(x_1,u_1;⋯; x_k,u_k) × g(x_k+1,u_k+1;⋯; x_2k,u_2k)d(#_⊗(_≥ 0) )^⊗ 2k.Let us expand the determinant of the (2k) × (2k) matrix above as a sum over all permutations. We break up the permutations into two types: permutations that map the subsets {1,…, k} and {k+1,…, 2k} into themselves and those that do not. When summed over permutations of the first type the integral above equals N(f)N(g). For a permutation σ of the second type, observe that there are two indices i and j, with i ≤ k and j > k, such that σ(i) > k and σ(j) < k. Then for ℓ∈{i,j}, Lemma <ref> gives|K_edge(x_ℓ,u_ℓ; x_σ(ℓ), u_σ(ℓ)) | ≤C/max{ |x_ℓ - x_σ(ℓ)|, |u_ℓ-u_σ(ℓ)|} + 1≤C/1 + dist(A,B).For all other indices ℓ we have |K_edge(x_ℓ,u_ℓ; x_σ(ℓ), u_σ(ℓ))| ≤ C. Consequently, the term involving σ contributes at most C^2k-2 (1 + dist(A,B))^-2 in absolute value to the determinant above for every (x_1,u_1; ⋯; x_k,u_k) ∈support(f) and (x_k+1,u_k+1;⋯;x_2k, u_2k) ∈support(g). Since there are (2k)! -(k!)^2 such permutations σ, we conclude that| N(f)N(g) - N(f)N(g) |≤(2k)!C^2k/dist(A,B)^2+1∫_(×_≥ 0)^2k |f g| d(#_⊗(_≥ 0) )^⊗ 2k = (2k)!C^2k/dist(A,B)^2+1||f||_1 ||g||_1. Let ^k(A) be the sigma-algebra generated by the random variables N(f), where f : A^k → is continuous and compactly supported. The bound (<ref>) implies that if A and B are disjoint, X is ^k(A)-measurable and Y is ^k(B)-measurable, then,| XY - XY| ≤(2k)! C^2k/dist(A,B)^2 + 1 |X||Y| . The bound in (<ref>) implies ergodicity of _edge as follows. Let E ∈_inv. Given 0 << 1, we may choose an event E' ∈^k([-n,n] ×_≥ 0), for some k and n, such that E Δ E' <. Since _edge is invariant under τ^h, we have that τ^h E Δτ^h E' = E Δ E' for every h. Therefore by the triangle inequality,| E' ∩τ^h E' - E ∩τ^h E| ≤E' Δ E + τ^h E Δτ^h E'≤ 2 .Due to invariance of E this implies that |E' ∩τ^h E' - E| ≤ 2.Set h = n + m for an integer m ≥ 1. Then τ^h E' ∈^k([n+2m,3n+2m] ×_≥ 0). We now apply (<ref>) with A = [-n,n] ×_≥ 0 and B = [n+2m,3n+2m] ×_≥ 0, observing that dist(A,B) = 2m. Since τ^h E' = E' by translation invaraince, we infer that| E' ∩τ^h E' - E'^2| ≤(2k)!C^2k/4m^2.Since | E' - E| ≤, we conclude that|E - E^2| ≤(2k)!C^2k/4m^2 + 5 .Letting m →∞ followed by → 0 shows that E = E^2, as required. The proof above may be used to deduce that _edge is in fact space-time mixing. Almost surely, _edge has an unbounded collection of points on every line {x}×_≥ 0. In fact, the following holds. Let N_x(t) = # (_edge∩ ({x}× [0,t]) ). For every x, the sequence N_x(t)/t → 1/π in probability as t →∞.Fix x ∈. Using part (II) of Proposition <ref> we see that for any interval [a,b] ⊂_≥ 0,N_x([a,b]) = ∫_a^b K_edge(x,u,x,u) du = b-a/π + (-1)^x/2 π∫_2a^2b du sin u/u .Observe from (<ref>) that N_x(t)/t = 1/π + O(1/t) as t →∞.From (<ref>) we see that K_edge(x,u_1;x,u_2) is symmetric in the variables u_1 and u_2. Thus,ρ(u_1,u_2) := K_edge(x,u_1;x,u_2) K_edge(x,u_2;x,u_1) = K_edge(x,u_1;x,u_2)^2 ≥ 0.From the relation (<ref>) for determinantal point processes we have thatN_x(t) · (N_x(t)-1) = ∫_0^t ∫_0^t[ K_edge(x,u_i;x,u_j)]_i,j=1,2du_1 du_2=(∫_0^t K_edge(x,u;x,u)du )^2 - ∫_0^t ∫_0^t ρ(u_1,u_2) du_1 du_2 ≤N_x(t)^2.This inequality implies that Var(N_x(t)) ≤N_x(t). Since N_x(t) = (t/π) + O(1), Chebyshev's inequality implies that for any > 0,| N_x(t)/t - 1/π | >=O(1)/^2 t.This provides the claimed convergence in probability.Convergence in probability implies that there is a sequence of times t_k →∞ such that N_x(t_k)/t_k → 1/π almost surely as k →∞. This in turn implies that there is an unbounded collection of points of _edge on {x}×_≥ 0 almost surely. An union bound over x provides the claim in the lemma. The following event occurs almost surely. For every t > 0 there exists a doubly infinite sequence of integers x_i, i ∈, such that _edge contains no points on each of the segments {2x_i}× [0,t].By monotocity and an union (or rather intersection) bound over rational values of t, it suffices to show that the event occurs almost surely for every fixed t > 0. Given a fixed t, let X_i be the indicator of the event that _edge has no points on {2i}× [0,t]. It suffices to show that almost surely infinitely many of the X_is equal 1 for i ≥ 0. Then, reflection invariance of _edge and another union bound imply that almost surely a doubly infinite collection of the X_is are equal to 1, as required.Due to translation invariance of _edge the sequence X_i, i = 0,1,2,…, is stationary in that (X_0,X_1, …) has the same law as (X_1,X_2, …). It is also ergodic by Proposition <ref>. Therefore, by the Ergodic Theorem,lim_n →∞ 1/n∑_i=0^n-1 X_i = X_0 = 1,almost surely.The probability that X_0 = 1 is the probability that _edge has no points in {0}× [0,t]. This is strictly positive by (<ref>) below. As a result, an infinite number of the X_is equal 1 whenever the limit in the above holds.§ THE LOCAL STAIRCASE SHAPED TABLEAUThe local staircase shaped tableau, henceforth, local tableau, is a random function on= { (x,y) ∈^2: y ≥ 0, x ≡ y(mod2) }.Figure <ref> provides an illustration. The rows and columns ofare given by the diagonal linesrow2x = {(2x-k,k): k ≥ 0}, column2x = {(2x+k,k): k ≥ 0}. In order to define tableaux onand their convergence, we first explain the topology on [0,∞] since tableau entries will take values there (we allow the value ∞). The topology on [0,∞] is the usual topology on _≥ 0 extended in the natural way by stipulating that a sequence converges to ∞ if its values diverge to ∞, possibly stabilizing to the value ∞. In this case we will say that the sequence grows to ∞.For example, 1,2,3,4, … grows to ∞, as does 1,∞,2,∞, …, as well as 1,∞,∞,∞,….A tableau is a function T: → [0,∞]such that it satisfies the tableau constraintsI)T(x,y) ≤min{ T(x-1,y+1), T(x+1,y+1) }for every (x,y) ∈.II)Along every row and column of Tthe entries grow to ∞. The YDembeds intovia (i,j) ∈↦ (j-i - nodd, n-i-j) ∈. This is a rotation that puts row r ofon row 2(⌊ n/2 ⌋ - r) of ; see Figure <ref>. In this manner any PYT T of shapeembeds as a tableau F_T : → [0,∞] by settingF_T(x,y) =n (1-T( ⌊n/2⌋ - x+y/2, ⌊n/2⌋ + x-y/2) ),if (⌊n/2⌋ - x+y/2, ⌊n/2⌋ + x-y/2) ∈;∞,otherwise.By an abuse of notation we denote F_T by T.We say that a sequence of tableaux T_n converges if there is a tableau T such that, in the aforementioned topology on [0,∞], T_n(x,y) → T(x,y) for every (x,y) ∈Δ_∞. Note we stipulate that a limit of tableaux remain a tableau.A random tableau 𝐓 : → [0,∞] is a Borel probability measure on tableaux with respect to the topology above. Convergence of a sequence of random tableaux means weak convergence with respect to this topology. §.§ Bulk local limit of staircase shaped tableauxSection <ref> describes how PYTs of a given shape are in bijection with ensembles of non-increasing and non-intersecting paths whose initial positions are given in terms of the shape. We describe the bijection explicitly for tableaux defined onas it will be useful in the proof of the local limit theorem.Consider an ensemble of paths { p(2x,u) }, for x ∈ and u ∈_≥ 0, that satisfy the following.I) p(x,·) : _≥ 0→ + 1/2is left continuous, non-increasing with p(x,0)=2x + 1/2.II) p(x,·) are non-intersecting: p(x,u) > p(x-1,u) for every x ∈, u ∈_≥ 0.III)The jumps of the paths as defined by(<ref>)is a discrete subset of×_≥ 0. For a tableau T : → [0,∞], paths satisfying (<ref>) are obtained by setting p(x,u) = 2x + 1/2 for 0 ≤ u ≤ T(2x,0), and for k ≥ 1,p(x,u) = p(x,0) - k if T(2x-k+1,k-1) < u ≤ T(2x-k,k).In other words, p(x,·) is left continuous and decreases by integer units at times indexed by row 2x of T. The paths are non-intersecting due to the columns of T being non-decreasing. Indeed, p(x,u) - p(x-1,u) = 2 +N_x-1(u) - N_x(u), where N_x(u) is the number of entries of T on row 2x with value at most u. Due to the columns being non-decreasing, N_x-1(u) ≥ N_x(u) -1, and thus, p(x,u) - p(x-1,u) ≥ 1. The jumps of the paths form a discrete set due to the rows and columns of T growing to ∞. When a row entry equals ∞ then the corresponding path jumps only a finite number of times.Let X denote the jumps for the ensemble of paths associated to a tableau T on . The jumps can be read off from T in the following manner. For every x ∈, the jumps on the line {x}×_≥ 0 are the entries of T whose cells have first coordinate x in . More precisely, if u is the k-th smallest point of X on {x}×_≥ 0 thenu =T (x, 2k-1-xeven).If there are less than k points on {x}×_≥ 0, there is no such u and the tableau entry above equals ∞.To see this, observe that u is the time when the path starting at initial position (x+ 1/2)+2k-1 - x even jumps for the (2k-x even)-th time. Indeed, this is the k-th path starting at or to the right of position x+ 1/2 and it hits position x - 1/2 after jump number 2k-x even. The first k jumps on {x}×_≥ 0 are the times when the first k paths starting at or to the right of position x + 1/2 hits position x - 1/2. Also, when there is no such u it means that the path starting from (x+ 1/2)+2k-1 - x even has exhausted its jumps and it does not get to position x - 1/2.Let M_T → X denote the map from tableaux defined onto jumps of paths satisfying (<ref>). This map is invertible with the inverse given by the relation (<ref>). Namely, T(x,y) is the [(y+1+x even)/2 ]–th smallest jump of X on {x}×_≥ 0 with the convention that T(x,y) = ∞ if no such jump exists. Let M_X → T denote the inverse map.The set of jumps of paths satisfying (<ref>) is closed in the topology on discrete subsets of ×_≥ 0. The map M_T → X is a homeomorphism from the set of tableaux onto the set of jumps of paths satisfying (<ref>), with inverse given by M_X → T.The maps M_T → X and M_X → T are inverses by design. We must show that they are continuous. We begin with continuity of M_T → X. Let T_n be a sequence of tableaux such that T_n converges to a tableau T_∞. Let X_n = M_T → X(T_n) and X_∞ = M_T → X(T_∞). Recall from Section <ref> that convergence of X_n to X_∞ requires that for every x ∈ and k ≥ 1, the k-th smallest point of X_n on {x}×_≥ 0 must converge to the corresponding point of X_∞, while accounting for the case that there may be less than k points.Let y = 2k-1-xeven. Then, by (<ref>), T_n(x,y) is the k-th smallest point of X_n on {x}×_≥ 0 and similarly for T_∞(x,y). There are two cases: T_∞(x,y) < ∞ or T_∞(x,y) = ∞. In the former case, the k-th smallest point of X_n on {x}×_≥ 0 is eventually finite and the same for X_∞. Moreover, we have convergence of these points since T_n(x,y) → T_∞(x,y). In the latter case, given any bounded subset of {x}×_≥ 0, the k-th smallest point of X_n eventually escapes the set or is non-existent due to T_n(x,y) →∞. This is as required since X_∞ has no k-th smallest point on {x}×_≥ 0. This proves that M_T → X is continuous.Now we show that the set of jumps of paths satisfying (<ref>) is a closed set in the space of discrete subsets of ×_≥ 0, as well as that that M_X → T is continuous. Suppose X_n is a sequence of such jumps sets and that it converges to a discrete subset X_∞.Let T_n = M_X → T(X_n). First, we show that T_n(x,y) converges for every (x,y) ∈. Indeed, T_n(x,y) is the k-th smallest point of X_n on {x}×_≥ 0 for k = (y+1+x even)/2. Therefore, convergence of X_n to X_∞ implies that T_n must converge to some function T_∞: → [0,∞]. Note that T_∞(x,y) = ∞ if and only if X_∞ has less than k points on {x}×_≥ 0.The function T_∞ is a tableau because the tableau inequalities from (<ref>) continue to hold in the entry-wise limit, and the rows and columns will grow to ∞ due to X_∞ being a discrete set. Thus, consider X̂_∞ = M_T → X(T_∞). By the first part of the proof, X_n →X̂_∞. But then, X̂_∞ = X_∞ because limits of discrete subsets of ×_≥ 0 are unique. This shows both the closure property of sets of jumps for paths satisfying (<ref>) and the continuity of M_X → T. We now state the local limit theorem for Poissonized staircase shaped tableaux in the bulk. For α∈ (-1,1), let c_n = 2 ( ⌊n/2⌋ - ⌊(1+α)n/2⌋). For a PYT T having shape , embed it as a tableau onaccording to (<ref>) and consider the rescaled tableau T_α,n : → [0,∞] defined byT_α,n(x,y) = √(1-α^2)T(x+c_n, y).Let _ be a uniformly random PYT of shapeand denote by _α,n the random tableau associated to _ by (<ref>).The sequence of random Poissonized tableaux _α,n converges weakly to a random tableau _edge. Moreover, the law of _edge is M_X → T( _edge).Observe that |c_n - α n| ≤ 2 for every n. With this choice of c_n, the jump process associated to _α,n has law _α,n from (<ref>) because these jumps are simply the jumps of _ rescaled onto ×_≥ 0 as in (<ref>). Theorem <ref> asserts that _α,n converges weakly to _edge. Due to being a weak limit of the jumps of ensembles of paths satisfying (<ref>), _edge is also almost surely the jumps of such an ensemble of paths by the closure property given in Lemma <ref>. The continuity of M_X → T then implies that M_X → T(_α,n) converges weakly to M_X → T(_edge). Therefore, _α,n converges weakly to a random tableau _edge having the law of M_X → T(_edge).Bulk local limit theorem for random staircase shaped SYT Recall _ denotes a uniformly random SYT of shapeand N = n2. Consider the rescaled tableau^rsc_(i,j) = 𝐓_(i,j)/N+1. The random tableau ^rsc_ converges to _edge in the bulk local limit, that is, under the embedding and rescaling from (<ref>).Consider the following coupling between _ and _. Given _, independently sample P_(1) < P_(2) < ⋯ < P_(N) according to the order statistics of N i.i.d. random variables distributed uniformly on [0,1]. Insert the entry P_(k) into the cell ofthat contains entry k of _. The resulting tableau has the law of _. Using this coupling, and due to the manner the scaling from (<ref>) is defined, it suffices to show the following in order to conclude that ^rsc_ converges to _edge by way of Theorem <ref>.Fix an L > 0 and consider any (x,y) ∈ such that (x,y) ∈ [-L,L] × [0,L]. Let P_(k) be the entry of _ inside (x,y) under the embedding from (<ref>). Then as n →∞, we need to show thatn | P_(k) - k/N+1 | ⟶0 in probability.The number k is random, its distribution depends onas well as n and α.In order to establish (<ref>) we will use the following auxiliary fact, which is a byproduct of <cit.>. There is a number δ_n of order o(N) as n →∞, such that with probability tending to 1 as n →∞, every entry of _ within the cells of ∩ [-L,L] × [0,L] under the embedding (<ref>) has value at least N - δ_n. As a consequence, k ≥ N - δ_n with probability tending to 1. We writen| P_(k) - k/N+1| > ≤ n^2/^2 | P_(k)- k/N+1|^2 |k ≥ N- δ_n + k < N-δ_n. For a fixed deterministic j,P_(j) has a Beta distribution with parameters j and N+1-j, which has mean j / (N+1) and variance| P_(j) - j/N+1|^2= j (N+1-j)/(N+1)^2(N+2).Since the P_(j)s are independent of , employing the bound above for j ≥ N- δ_n and summing over the probabilities of k give| P_(k)- k/N+1|^2 |k ≥ N- δ_n≤δ_n/N^2.The latter quantity is of order o(1)/N as n →∞. Since N= n2, we conclude that both terms on the right hand side of (<ref>) tend to 0 as n →∞.Statistical properties of the local staircase shaped tableau The setcan be made into a directed graph by putting directed edges from each vertex (x,y) ∈ to the vertices (x-1,y+1) and (x+1,y+1). The automorphisms of this graph consists of translations ϕ_h, for h ∈, given by ϕ_h(x,y) = (x+2h,y), as well as a reflection ϕ_- given by ϕ_-(x,y) = (-x,y). Tableaux are preserved by these automorphisms.A random tableau 𝐓 is translation invariant if 𝐓∘ϕ_h has the same law as 𝐓 for every translation ϕ_h. The random tableau is reflection invariant if 𝐓∘ϕ_- has the same law as 𝐓. The translation invariant sigma-algebra of 𝐓 is the sigma-algebra of events that remain invariant under every translation:_inv = {EventsEassociated to 𝐓 s.t. ϕ_h E = E for everyh ∈. }.(Recall that ϕ_h E = {ω∘ϕ_h: ω∈ E}.) We say 𝐓 is ergodic under translations if _inv is the trivial sigma-algebra.The local tableau _edge has the following statistical properties. * Almost surely, _edge(x,y) is finite for every (x,y) ∈ and the entries of _edge are all distinct. * The law of _edge is both translation and reflection invariant. * _edge is ergodic under translations. * Almost surely, for every t > 0 there are infinitely many positive and negative x ∈ such that T_edge(2x,0) > t. Almost surely, _edge has an infinite and unbounded collection of points on every line {x}×_≥ 0 by Lemma <ref>. Also, almost surely, _edge does not contain two points of the form (x,u) and (y,u) with x ≠ y. To see this, observe from the relation (<ref>) for determinantal point processes that the expected number of such pairs of points in _edge is 0 due to the set of such pairs having measure zero with respect to the measure (#⊗(_≥ 0))^⊗ 2. When both these properties hold, _edge satisfies (1).The law of _edge is invariant under translations because for every translation ϕ_h, the tableau _edge∘ϕ_h is constructed from the jump process _edge + (2h,0), which has the same law of _edge by Proposition <ref>. Similarly, reflection invariance of _edge follows from reflection invariance of _edge. This establishes (2).The ergodicty of _edge under translations follows from the ergodicity of _edge under translations (Proposition <ref>). This is because a translation invariant event for _edge is the image of a translation invariant event for _ edge under the map M_X → T. Finally, (4) is the statement of Lemma <ref>. § RANDOM SORTING NETWORKS §.§ Sorting networks, Young tableaux and Edelman-Greene bijectionStanley <cit.> enumerated the number of sorting networks of 𝔖_n, which equalsn2!/∏_j=1^n-1 (2n-1-2j)^j.Following Stanley, Edelman and Greene <cit.> provided an explicit bijection between sorting networks and staircase shaped SYT. An account of further combinatorial developments may be found in <cit.>. We describe the part of the Edelman-Greene bijection that maps staircase shaped tableaux to sorting networks. The inverse map is a modification of the RSK algorithm; we do not describe it here since it is not used in the paper. See <cit.> or <cit.> for a full description of the bijection.Recall that a sorting network of 𝔖_n is identified by its sequence of adjacent swaps (s_1,…,s_N), where N = n2. For the rest of the paper we will use N to denote n2. For T ∈SYT(), we adopt the convention that T(i,j) = - ∞ if (i,j) ∉.The Schützenberger operator Let (i_max(T),j_max(T)) denote the cell containing the maximum entry of aSYT T. The Schützenberger operator Φ : SYT() →SYT() is a bijection defined as follows. Given T ∈SYT(), construct the sliding path of cells c_0, c_1,…, c_d-1∈ iteratively in the following manner. Set c_0 = (i_max(T), j_max(T)) and c_d = (1,1). Then setc_r+1 = argmax { T(c_r - (1,0)), T(c_r - (0,1)) }.Let Φ(T) = [T̂(i,j)] where T̂(c_r) = T(c_r+1) + 1 for 0 ≤ r ≤ d-1, T̂(c_d) = 1, and T̂(i,j) = T(i,j)+1 for all other cells (i,j) ∈∖{c_0,…, c_d}. Figure <ref> provides an illustration.The Edelman-Greene map EG : SYT() ↦{sorting networks of 𝔖_n} is defined byEG(T) =( j_max(Φ^k(T)))_0 ≤ k ≤ N-1 ,where Φ^k is the k-th iterate of Φ. Edelman and Greene <cit.> proved that EG indeed maps to sorting networks and that is has an inverse. §.§ First swap times of random sorting networks: proof of Corollary <ref>Let T_FS(s) be the first time the adjacent swap (s,s+1) appears in a sorting network ω of 𝔖_n. According to the Edelman-Greene bijection, this time is recorded in the entry (n-s,s) of EG(ω). Thus,T_FS(s) = N+1 - EG(ω)(n-s,s).In terms of the rescaled tableau _^rsc from (<ref>) we have that_FS(s)law=(N+1) · (1 - _^rsc(n-s,s)).This implies the following for _FS, α,n – the first time an adjacent swap between ⌊n(1+α)/2⌋ and⌊n(1+α)/2⌋+1 appears in a random sorting network of 𝔖_n:2 √(1-α^2)/n _FS, α,n law= 2N+2/n^2 _α, ^rsc(0,0).Here, _α, ^rsc is the tableau _^rsc rescaled and embedded intoaccording to (<ref>). Theorem <ref> implies that _α, ^rsc(0,0) converges weakly to _edge(0,0). Since (2N+2)/n^2 → 1, we conclude that the rescaled _FS, α,n from above converges weakly to _edge(0,0). Thus, _FS has the law of _edge(0,0).Now we explain how to get the distribution function of _FS given in (<ref>). Observe that the event {_edge(0,0) > t} is the event {_edge∩ ({0}× [0,t]) = ∅}. The probability of the latter (often known as “gap probability") has the representation given by (<ref>), which is the Fredholm determinant of K_edge over L^2({0}× [0,t]). This is a well-known property of determinantal point processes under the condition that the kernel be of trace class <cit.>. The kernel K_edge is of trace class on L^2({0}× [0,t]) simply because |K_edge(0,u_1;0,u_2)| ≤ 2/π.The asymptotic behaviour of the distribution function of _FS is well-known:log_FS > t = - 1/4t^2 - 1/2t - 1/8log t +7/24log 2+ 3/2ζ'(-1) + o(1)ast →∞.The formula (<ref>) has a history. In theoretical physics literature, the leading term in (<ref>) was first studied in <cit.>, while the full expansion was given in <cit.>. The complete mathematical treatment was developed in <cit.>; the present form of (<ref>) is given in the last reference.We will only need the simple corollary of (<ref>) that _FS> t > 0 for every t. §.§ Edelman-Greene algorithm on the local tableauThe procedure described here is the same as the one given in the Introduction except that it is in the language of tableaux instead of their jumps. In order to define the Edelman-Greene algorithm on the local tableau we first introduce some concepts that allow us to define Edelman-Greene algorithm on tableaux defined on .A directed path from (x,y) ∈ to (x',y') ∈ is a sequence of cells c_0 = (x,y), c_1, …, c_k = (x',y') ofsuch that c_i+1-c_i ∈{ (-1,1), (1,1)} for every i. The cells ofcan be partially ordered as follows: (x,y) ≤ (x',y') if there is a directed path from (x,y) to (x',y'). Recall thatis a directed graph with edges from (x,y) to (x± 1, y+1). It can also be thought of as an undirected graph by forgetting the direction of the edges. A connected subset ofis a connected subgraph ofin the undirected sense.A Young diagram (YD) ofis a connected subsetthat is downward closed in the partial order, that is, if (x,y) ∈ and (x',y') ≤ (x,y) then (x',y') ∈. For example,is a YD of . The boundary of , ∂, consists of cells (x,y) ∉ such that there is a directed edge from some cell (x',y') ∈ to (x,y). The peaks ofconsists of the maximal cells ofin the partial order.Let T: → [0,∞] be a tableau as in (<ref>). A sub-tableau is the restriction of T to a YD ; we sayis the support of the sub-tableau. Let T^finite = { T(x,y): T(x,y) ≠∞}. We take the support of T to be the support of T^finite. Observe that T^finite is a countable disjoint union of sub-tableaux of T, say T_1, T_2, …. Indeed, the support of the T_is are the connected components of the subgraph spanned by cells (x,y) such that T(x,y) ≠∞. We will call the T_is the clusters of T. The tableau T is EG-admissible if all the entries of T^finite are distinct and every cluster T_i is supported on a YD of finite size.Edelman-Greene algorithm on a finite tableauLetbe a YD ofof finite size and T: →_≥ 0 a tableau such that all its entries are distinct. The Edelman-Greene map EG takes as input T and outputs a triple (x,t,T̂), where x ∈, t ∈_≥ 0 and T̂ is a sub-tableau.The sliding path of T is a directed path c_0, c_1, …, c_k defined by * c_0 = argmin {T(x,y): (x,y) ∈}.* c_i+1 = argmin {T(c_i + (-1,1)), T(c_i + (1,1))}.* c_k = peak ofobtained when both c_k + (± 1,1) belong to ∂.Let = ∖{c_k} and define T̂ : →_≥ 0 byT̂(x,y) =T(x,y),if (x,y) ∈∖{c_0, …, c_k-1}; T(c_i+1),if (x,y) = c_ifor some 0 ≤ i ≤ k-1.The cell c_0 must be on the bottom level ofand has the form (2x,0) for some x ∈. Set t = T(c_0). The output is EG(T) = (x,t,T̂), and empty if T is the empty tableau.The Edelman-Greene algorithm on T outputs a discrete subset S(T) ⊂×_≥ 0, denoted the swaps of T. Let (x_j, t_j, T̂_j), for 1 ≤ j ≤ ||, be defined iteratively by (x_1,t_1,T̂_1) = EG(T) and (x_j,t_j,T̂_j) = EG(T̂_j-1) for 2 ≤ j ≤ ||. Then,S(T) = { (x_j,t_j): 1 ≤ j ≤ || }.If the cell (x,y) ∈ contains the k-th smallest entry of T then its entry is removed during the k-th iteration of the algorithm. We will say that the entry at (x,y) exits at time t_k from row x_k. We will also say that (x_k, t_k) originates from cell (x,y).Edelman-Greene algorithm on an admissible tableau Let T_1, T_2, … be the clusters of an EG-admissible tableau T. Observe that for i ≠ j, the swaps of T_i and T_j exit from mutually disjoint rows. Thus, the swap sets S(T_1), S(T_2), … are row-wise mutually disjoint. The swaps of T are defined asS(T) = ⋃_iS(T_i). The local tableau _edge is not EG-admissible. In order to define swaps for the local tableau we cut off large entries so that it becomes EG-admissible, and then process the tableau in a graded manner. For this to be successful, the EG algorithm ought to be consistent in the sense that running it on a tableau, and then restricting to swaps that originate from a sub-tableau, must produce the same outcome as the algorithm applied to the sub-tableau. This is not always the case and the following explains when it may be so.Given two tableaux T_small and T_big, we say T_small≤ T_big if the following criteria hold. * T_small(x,y) = T_big(x,y) for every (x,y) ∈support(T_small).* For every (x,y) ∈support(T_small), and (x',y') ∈support(T_big) ∖support(T_small), if (x,y) belongs to the same cluster of T_big as (x',y') then T_small(x,y) < T_big(x',y'). Let T_small≤ T_big, and suppose that T_big is EG-admissible. Then, applying the EG algorithm to T_big and restricting to the swaps that originate from the cells of T_small produces the same outcome as applying the EG algorithm to T_small. In particular, S(T_small) ⊂ S(T_big).Observe that the clusters of T_small are contained within the clusters of T_big. The EG algorithm acts independently on each cluster of T_big is a row-wise disjoint manner. Fix a particular cluster T of T_big, and suppose that the clusters of T_small that are contained inside T are T_1, …, T_k. It suffices to prove that the EG algorithm applied to T, and then restricted to the swaps that originate from T_1, …, T_k, produces the same outcome as the algorithm applied to each individual T_i.Let = support(T) and _i = support(T_i). The assumption is that each entry of ∖ (∪_i _i) is larger than every entry of ∪_i _i. Therefore, the EG algorithm applied to T will process every entry of ∪_i _i before it ever processes an entry from the complement. When some entry from ∖ (∪_i _i) enters a cell of some _i during the first ∑_i |_i| steps, the algorithm treats that entry as if it were ∞. Since T_small agrees with T_big on ∪_i _i, the EG algorithm will output the swaps of T_1, …, T_k during the first ∑_i |_i| steps, and then output the remaining swaps of T ∖ (∪_i T_i). This is what was claimed. A tableau T is graded EG-admissible if all of its finite-valued entries are distinct and, if for every t > 0, the sub-tableauT^≤ t = { T(x,y): T(x,y) ≤ t}is EG-admissible.Observe that T^≤ t_1≤ T^≤ t_2 whenever t_1 ≤ t_2. Lemma <ref> thus implies that S(T^≤ t_1) ⊂ S(T^≤ t_2). Therefore, for a graded EG-admissible tableau T, we may defineS(T) = ⋃_t ≥ 0S(T^≤ t). Suppose a sequence of tableaux T_n → T_∞, and also that every T_n and T_∞ are graded EG-admissible. Then for every integer x and t ≥ 0, there is a finite YDthat contains the cluster of (2x,0) in T_n^≤ t for every n.This follows from a diagonalization argument, more precisely, König's infinity lemma, which states that every infinite connected graph with finite vertex degrees contains an infinite path.Suppose for the sake of a contradiction that the conclusion of the lemma fails. Let T_n,x,t denote the cluster of (2x,0) in T_n^≤ t. Call a cell (x',y') ∈ bad if there is a undirected path infrom (2x,0) to (x',y') that is contained in infinitely many of the clusters T_n,x,t. Consider the connected component of (2x,0) inthat is spanned by the subgraph of bad vertices. If the component is finite then there is a finite YDthat contains the component. This implies that for all sufficiently large n, every cell of ∂ lies outside T_n,x,t because any path from (2x,0) to a cell outsidemust pass through ∂. Therefore, T_n,x,t⊂ for all large n. Since every T_n is graded EG-admissible, this means that there is a finite YD that contains every T_n,x,t, which is a contradiction.Therefore, the connected component of (2x,0) spanned by the bad vertices is infinite. Since every vertex ofhas degree at most 4, König's lemma provides an infinite path of (distinct) bad vertices (x_0,y_0), (x_1,y_1), … starting from (x_0,y_0) = (2x,0). By definition of being bad, for every m, there is a path from (2x,0) to (x_m,y_m) that is contained in some infinite subsequence of the clusters T_n^m_i,x,t with n^m_i →∞ as i →∞. Let ℓ_m be the length of this path. Observe that ℓ_m →∞ with m because the distance from (2x,0) to (x_m,y_m) inmust tend to infinity due to every vertex having degree at most 4.The YD _m formed by the cells ofthat are at or below the cells on the path from (2x,0) to (x_m,y_m) must be contained in every cluster T_n^m_i,x,t. Since T_n converges to T_∞, this implies that _m ⊂ T_∞,x,t for every m. Since |_m| ≥ℓ_m →∞, we deduce that T_∞,x,t is infinite. However, this is a contradiction to T_∞ being graded EG-admissible. Suppose a sequence of tableaux T_n → T_∞, and that every T_n as well as T_∞ is graded EG-admissible. Then S(T_n) → S(T_∞) as discrete subsets of ×_≥ 0.A compact subset of ×_≥ 0 is a finite, disjoint union of sets of the from {x}× C for x ∈ and compact C ⊂_≥ 0. Therefore, we must show that for every such x and C,lim sup_n#[S(T_n) ∩ ({x}× C)]≤# [ S(T_∞) ∩ ({x}× C) ].Fix a t > 0 such that C ⊂ [0,t].Suppose T is a graded EG-admissible tableau. The swaps of T on {x}× [0,t] are the entries of T^≤ t that exit from row x. Let T_x,t denote the cluster of (2x,0) in T^≤ t. By Lemma <ref>, the swaps of T on {x}× [0,t] are completely determined by running the EG algorithm on T_x,t. We deduce from Lemma <ref> that there is a finite YDsuch thatsupport(T_n,x,t) ⊂for every nandsupport(T_∞,x,t) ⊂. Since sup_(x',y') ∈ | T_n(x',y') - T_∞(x',y') | → 0, we conclude that the following must occur for all sufficiently large n. * The order of the entries of T_n onstabilizes to the order of the entries of T_∞ on .* For every (x',y') ∈, if T_∞(x',y') ∉ C then T_n(x',y') ∉ C. Once condition (1) holds then, due to T_n,x,t⊂, a swap from S(T_n) lies on {x}× C if and only if there is a cell (x',y') ∈ such that T_n(x',y') ∈ C and, when the EG algorithm is applied to T_∞ restricted to , the entry at cell (x',y') exits from row x. The same conclusion holds for swaps of S(T_∞) on {x}× C. This property along with condition (2) implies thatS(T_n) ∩ ({x}× C) ⊂ S(T_∞) ∩ ({x}× C) for all large n.This completes the proof. §.§ Completing the proof of Theorem <ref> Theorem <ref> will follow from Theorem <ref> once we prove that the local tableau _edge is graded EG-admissible almost surely. To this end, first observe that the entries of _edge are finite and distinct by part (1) of Proposition <ref>. We must show that, almost surely, the clusters of _edge^≤ t are finite for every t.By part (4) of Proposition <ref>, the local tableau satisfies the following almost surely: for every t and x, there are integers a,b ≥ 0 such that _edge(2x-2a,0) > t and _edge(2x+2b,0) > t. When this property holds the tableau constraints imply that the cluster of _edge^≤ t containing (2x,0) must be contained within cells whose row and column indices are both between 2x-2a and 2x+2b. The set of such cells is finite, and so the cluster of every bottom level cell in _edge^≤ t is finite. Now if _edge(2x-k,k) ≤ t then cell (2x-k,k) belongs to the same cluster as (2x,0) in _edge^≤ t since the row entries are non-decreasing. This implies that, almost surely, _edge^≤ t is EG-admissible for every t, as required.Finally, we complete the proof. The law of S_α,n is that of the Edelman-Greene algorithm applied to the rescaled uniformly random staircase shaped tableau ^rsc_ from (<ref>). Theorem <ref> asserts that ^rsc_ converges weakly to _edge as a tableau embedded in . By Skorokhod's representation theorem, there exists random tableaux _n and _∞ defined on a common probability space such that _n has the law of ^rsc_, _∞ has the law of _edge, and _n →_∞ almost surely.The tableaux _n and _∞ are graded EG-admissible almost surely. Theorem <ref> then implies that S(_n) converges to S(_∞) almost surely. This means that S_α,n, which has the law of S(_n), converges weakly to S(_edge), which is the law of S(_∞). We conclude with some statistical properties of the local swap process. The process S_local has the following properties. * S_local is invariant under translations and reflection of the -coordinate.* S_local is stationary in time in that for every t ≥ 0, the process S_local∩ (×_≥ t) has the same law as (shifted) S_local.* S_local is ergodic under translations of the -coordinate in that the sigma-algebra_inv = {Events ofS_local that are invariant under every translation}is trivial . We believe that S_local is also ergodic in the time coordinate. However, the proof of this is more challenging and, therefore, we leave it as a conjecture. We have that S_local = S(_edge) in law. Applying a -automorphism to S_local is the same as first applying its analogue to _edge (the maps ϕ_h and ϕ_-), and then applying the EG algorithm to the resulting tableau. Thus, the invariance of S_local under -automorphisms follows from the corresponding invariance of _edge stated in Proposition <ref>.Time stationarity of S_local is a consequence of the stationarity of finite random sorting networks <cit.>, as we explain. If (s_1,…, s_N) is the sequence of swaps of a random sorting network of 𝔖_n, then (s_1,…, s_N-1) has the same law as (s_2, …, s_N).The ergodicity of S_local under -translations is a consequence of the ergodicty of _edge under translations (part 3 of Proposition <ref>). Indeed, a translation invariant event for S_local is the image of a translation invariant event of _edge under the EG algorithm. ADHV[AGH]AGH O. Angel, V. Gorin, A. E. Holroyd, A pattern theorem for random sorting networks. 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http://arxiv.org/abs/1702.07895v4

[ Abraham Smith Paul Bendich John Harer Alex Pieloch Jay Hineman October 24, 2016 ==================================================================§ INTRODUCTION The relativelyrecent discovery of the present acceleration of our Universe is strictly proved by Type Ia Supernovae, Cosmic Microwave Background Radiation, and Sloan Digital Sky Survey <cit.>-<cit.>.This fact initiated a great number of theoretical hypothesis, and inspired researchers to a diversity of different explanationsof such an unusual behavior of the Universe. Nevertheless, mostly two distinct approaches to explain the phenomenon of accelerated expansion of the Universe have been well developed during the recent years. One of them deals with the so-called dark energy (DE) with negative pressure in General Relativity, another one acts within the frameworks of some acceptable modifications of the gravity theory on the long distance.According to several estimations, the dark energy occupies up to 72 % of the total energy of the universe, while dark matter gives about 24 %,and the usual baryonic matter consists of about 4 % of the total energy. In spite of all attempts to explain the late acceleration of our Universe, DE is still an open problem to the theoretical physicists because its nature is unknown so far. As known, there are numerous models available in the literature devoted to different descriptions of the nature of DE but none of them are fully definitive and reliable.The simplest and the most natural candidate among them is the cosmological constant <cit.> with the equation of state (EoS) parameter w_Λ =-1. Although this model meets the observational data but it suffers at least from two serious problems such as fine tuning and cosmic coincidence.Currently, there are numerous DE and modified gravity models to explain the present acceleration of the universe.Alternatively, to explain the decay of the density, the different forms of dynamically changing DE with an effective equation of state (EoS) w_de = p_de/ρ_de < -1/3, were proposed instead of a constant vacuum energy density. Other possible forms of DE include quintessence -1/3> w_q > -1, phantom (w_ph < -1)etc.In the last few decades there has been considerable interest in alternative theories of gravitation. One of the most interesting among them is the scalar-tensor theory proposed by G. Lyra <cit.>, and essentially developed by Sen and Dunn <cit.>. They have constructedan analogue of the Einstein field equation based on Lyra's geometry. According to Halford <cit.> the scalar-tensor treatment based on Lyra's geometry predicts the same effects within observational limits as in Einstein's theory.Moreover, as it pointed out by Soleng <cit.>, the constant displacement field in Lyra's geometry will either include a creation field and be equal to Hoyle's creation field cosmology or contain a special vacuum field, which together with the gauge vector form may be considered as a cosmological term. Subsequent investigations were done by several authors in cosmologywithin the framework of Lyra's geometry (see, e.g., <cit.>-<cit.> and references therein).Involving of the exotic forms of matter and fields for the explanation of the phenomenon of accelerated expansion in the context of Lyra's geometry is reasonable in the same extent as it is in Einstein's theory of gravity. Thereby the models on Lyra manifold with the quintessence, phantom field, tachyons, Chaplygin gas and some other forms of matter were studied by several authors. Nevertheless, the problem which has to be investigated in detail still remains unsolved, namely, whether the resource of Lyra's geometry in the form of the displacement field is quite enough to describe the acceleration of the universe filled by the ordinary matter and fields.As known, the electromagnetic fields could be one of these natural sources of gravity . A number of studies in the framework of Lyra's geometry involved the electromagnetic fields as the sources of gravity in the cosmological models (see, e.g., <cit.>-<cit.> and references therein). As far as we know, the models in Lyra's geometrywhich include the Yang-Mills fields <cit.> have notbeen studied yet. At the same time, a largenumber of works investigated the influence and role of these fields on the behavior of the Universe in the framework of Einstein's theory (see, e.g. <cit.>-<cit.> to mention just a few).Motivated by the above mentioned investigations, we have considered the homogeneouscosmological model with the Yang-Mills field in Lyra geometry. The aim of this paper is to search for the exact solutions for such a model, that is to find explicit expressions for the scale factor and the displacement field of this model.As consequence, through the time dependence of the YM content of the Universe, solutions are found for the energy density and the pressure. Some geometrical and physical aspects of the model also havebeen studied. For instance, exact solutions for the scalar field and its potential have been obtained in two specific cases regarding the effective equation of state and the exponential law of expansion. § THE MODEL EQUATIONSThe Einstein's gravitational equations in Lyra's geometryin normal gauge (see, e.g. <cit.> and <cit.>) can be written asR_ik- 1/2 g_ik R - Λ g_ik +3/2ϕ_i ϕ_k - 3/4g_ikϕ^j ϕ_j = T^(tot)_ik,where ϕ_i is a displacement vector, andΛ is a time-varying cosmological term. For simplicity, we assumethe gravitational constant 8π G=1. All other symbols have their usual meanings in the Riemannian geometry, and T^(tot)_ik stands for the total energy-momentum tensor (EMT) of all kinds of matter in the Universe. The Friedmann-Robertson-Walker (FRW) line element can be represented byds^2 = d t^2- a^2 (t)[d r^2+ξ^2(r)d Ω^2],where a(t) is a scale factor of the Universe, andξ_(r)={[sin(r), k =+1,; r,k =0,; sinh(r), k =-1.;].in accordance with thesign of the spatialcurvature k.In the case of minimal coupling of the displacement field with matter, the energy-momentum tensor of matter can be derived in a usual manner from the Lagrangian of matter. Therefore, the EMT of the YM fieldsis followed from the Lagrangian density of SO_3 YM field <cit.> asL_YM=-1/16πF_ik^aF^a ik,where F_ik^a=∂_iW_k^a-∂_kW_i^a+ eε_abcW_i^bW_k^c is the stress tensor of YM fields W^a_i. Varying this Lagrangian with respect to metric, one can easily get the following EMT of YM fieldsT_i^k=-1/4πF_ij^a F^akj+1/16πδ_i^k F_mn^a F^amn.At the same time, the variation of Lagrangian (<ref>) with respect to YM fields W^a_i yields the following YM equationD_i(√(-g)F^aik)=0,where D_i stands for the covariant derivative.Let us suppose further that the remaining content of the universe can be described as a perfect fluid with the EMT of the following form𝒯_ik= (ρ_m +p_m)u_i u_k -p_mg_ik,whereu_i = (1,0,0,0) is4-velocity of the co-moving observer, satisfying u_i u^i = 1, ρ_m and p_m are the energy density and pressure of matter, consequently. Hence, the total EMT equals T^(tot)_ik=T_ik+𝒯_ik.§ EXACT SOLUTION FOR THE YM EQUATION As well-known, the generalized Wu-Yang ansatz for the SO_3 YM fields can be written as <cit.>W^a_μ =ε_μ abx^bK(r,t)-1/gr^2+(δ^a_μ-x^ax_μ/r^2)S(r,t)/gr, W^a_0 = x^a W(r,t)/gr,where μ=1,2,3. The example of exact solution for the YM equation (<ref>) in metric (<ref>) has been obtained in <cit.> with the help of the following substitutionW = d α/d t,   K = P(r)cosα(t),   S = P(r)sinα(t),where α is an arbitrary function of time.As a result of simple calculation on the base of (<ref>) and (<ref>), we have the following expressions for the YM tensor components{[ F_12=g^-1 P'(r)( mcosα +l sinα),; F_13=g^-1 P'(r)sinθ( msinα- l cosα),; F_23=g^-1sinθ(P^2(r)-1) n,;F_01= F_02= F_03=0,;].wheren= (sinθcosϕ, sinθsinϕ, cosθ),        l = (cosθcosϕ, cosθsinϕ, -sinθ),     m= (-sinϕ, cosϕ, 0)are the orthonormalized isoframe vectors, and the prime means a derivative with respect to r. As it is noted in <cit.>, the YM fields (<ref>) have only magnetic components. Here and below an overdot denotes the partial derivative with respect to time t, and a prime denotes the partial derivative with respect to r. Taking into account the Wu-Yang ansatz (<ref>), the energy-momentum tensor (<ref>) can be written down in the formT_0^0 = 1/8π g^2 a^4(a^2 W'^2+a^2 Z + X + Y),                T_1^1 =1/8π g^2 a^4(a^2 W'^2-a^2 Z - X + Y),                T_2^2 = T_3^3 = - 1/8π g^2 a^4(a^2 W'^2+ Y), T_0^1=1/2π g^2 a^2J,whereX=2(K'^2+S'^2)/ξ^2(r), Y=(K^2-1+S^2)^2/ξ^4(r),Z=2[(K̇+WS)^2+(Ṡ-WK)^2]/ξ^2(r),         J=K'(K̇+WS)+S'(Ṡ-WK)/ξ^2(r). The set of equations for the YM fields (<ref>) can be written down with the help of ansatz (<ref>) as follows(ξ ^2 W')' +2[(Ṡ-WK )K-(K̇+WS )S]=0,a∂/∂ t[a(Ṡ-WK)]- S”                                +[(K^2-1+S^2 )S/ξ^2 -(K̇+WS )W]=0,a∂/∂ t[a(K̇+WS)]- K”                                +[(K^2-1+S^2 )K/ξ^2 +(Ṡ-WK )W]=0.One can readily verify that the substitution(<ref>) into equations (<ref>) yields the only equation for unknown function P(r), namelyP”-(P^2 -1)P/ξ^2= 0.At the same time, inserting (<ref>) into(<ref>), one can obtain the following components of EMT for the YM fieldT_0^0 = 1/8π g^2 a^4 ξ^2( 2 P'^2+(P^2 -1)^2/ξ^2),       T_1^1 =1/8π g^2 a^4 ξ^2(- 2 P'^2+(P^2 -1)^2/ξ^2),    T_2^2 = T_3^3 = - 1/8π g^2 a^4(P^2 -1)^2/ξ^4,and T_0^1=0. Assumingthe displacement field ϕ_i(t) is a homogeneousvector field, and considering the homogeneity and isotropy of the line element (<ref>), it is necessary to require the independence of the EMT components (<ref>) on the radial coordinate r. Thus, we get{[ 2 P'^2+(P^2 -1)^2/ξ^2=A ξ^2,; ; 2 P'^2-(P^2 -1)^2/ξ^2=B ξ^2,; ; (P^2 -1)^2/ξ^2=Cξ^2,;].where A, B and C are some constants. A simple analysis of equations (<ref>) and (<ref>) allows to find out that A=3C and B=C. Therefore, the set of equations (<ref>) is reduced to the following two equationsP'^2=C ξ^2,      P^2=1+ϵ√(C)ξ^2,where ϵ = ± 1. The direct consequence of (<ref>) is as followsξ'^2=1+ϵ√(C)ξ^2.Taking into account (<ref>) and (<ref>), we can get C=|k|, ϵ=-k andP(r)=±√(1- k ξ^2).Hence, due to (<ref>), the latter leads to the nontrivial solution for equation (<ref>) given byP(r)=±ξ'(r)={[±cos r, k =+1,; ± 1,k =0,; ±cosh r, k =-1.;]. Interpreting EMT of the YM gauge field as the EMT of a perfect fluid, that isT_ik= (ρ_g +p_g)u_i u_k -p_gg_ik,we can get from (<ref>) and (<ref>) thatT_0^0=ρ_g=3|k|/8π g^2 a^4, T_1^1=T_2^2=T_3^3=-p_g=- |k|/8π g^2 a^4,Thus, the effective EoS of the given YM field is the same one asfor the pure radiation, that is p_g=ρ_g/3, when k=±1.It is useful to mention that the conservation equation for YM field, T^k_i;k=0, for EMT of YM field in form (<ref>), namelyρ̇_g+3H(ρ_g+p_g)=0,where H=ȧ/a is the Hubble parameter, is satisfied identically by our solution (<ref>). As can be seen, the reason of such satisfaction of the continuity equation in the case k=0 is nothing more than the equality of ρ_g and p_g to zero according to (<ref>). That is why we will consider further only two types of the non-flat models with k = ± 1. § COSMOLOGICAL EQUATIONSLet ϕ_i be a time-like displacement vector field,ϕ_i = (2/√(3) β,0,0,0),where β = β(t) is a function of time only, and the factor 2/√(3) is used in order to simplify the writing of all the following equations. Given the line element (<ref>) andtaking the total EMT as a sum of (<ref>) and (<ref>), one can reduce the main equations of the model (<ref>) to the following set of equations:3H^2 + 3 k/a^2 - β^2 = ρ_m + ρ_g +Λ,            2 Ḣ + 3H^2 + k/a^2 + β^2 = -p_m -p_g +Λ.The continuity equation follows from Eqs. (<ref>) and (<ref>) asρ̇_m + ρ̇_g + Λ̇+ 2 ββ̇+ 3 H (ρ_m+ρ_g + p_m +p_g+ 2β^2 )=0. Obviously, the displacement vector field can be treated through the EMT in the same way as in the most studies on the Lyra geometry in cosmology. In such approach, the conservation of the energy-momentum should be accounted for the sum of EMTs <cit.>.As emphasized in <cit.>, the displacement vector field can give rise to an effective cosmological term Λ_eff. It means that Λ(t) is not an independent dynamical parameter of the model, and it should be removed from the system of equations (<ref>) and (<ref>). Moreover,we are going to preserve the continuity equation for matter in its standard form,ρ̇_m + 3 H (ρ_m + p_m )=0,which follows from the conservation equation, 𝒯^k_i;k=0, and (<ref>). Taking into account equations (<ref>) and (<ref>), the continuity equation(<ref>) becomesΛ̇+ 2 ββ̇+ 6 H β^2 =0.Since Λ =constant, equation (<ref>) supposes the displacement field to be the so called stiff fluid. The assumption of a non-vanishing and time-varying Λ term gives us some new possibilities <cit.>. Therefore, we assume thatΛ≠ constant. Moreover, we suppose that(<ref>) can be satisfied by some Λ(t), which can be found as the result of formal integration of equation (<ref>),Λ(t) =λ_0 -β^2-6∫ H β^2 d t,where λ_0 is a constant of integration. Substituting this expression along with ρ_g, p_g from (<ref>) into equations (<ref>) and (<ref>), one can get the basic equations of the model as follows3H^2 + 3k/a^2 = 3/8π g^2 a^4+ρ_eff,             2 Ḣ -2k/a^2 = -1/2π g^2 a^4 - (ρ_eff + p_eff),where we have introduced the effective cosmological termΛ_eff(t)=λ_0 - 6 ∫ H β^2 d t,and the effective energy density and pressureρ_eff=ρ_m +Λ_eff,     p_eff=p_m -Λ_eff - Λ̇_eff/3 H. One can readily verify that due toequations (<ref>) and (<ref>), and the continuity equation (<ref>), the effective energy density and pressure (<ref>) also satisfy the continuity equation in its usual form:ρ̇_eff + 3 H (ρ_eff + p_eff )=0. At the same time, the deceleration parameter in Lyra's geometry is defined just as in the standard cosmology, that isq = -a^2 ä/ȧ^2 = -1-Ḣ/H^2.It should be noted that the set of dynamical equations (<ref>), (<ref>) and (<ref>) consists of two independent equations, and fully determines the dynamics of our model. However, to determine three parameters, say a(t), ρ_eff and p_eff,one more condition should be set. For example,an effective EoS w_eff=p_eff/ρ_eff can play the role of suchadditional equation. Assuming the matter also obeysa barotropic EoSp_m = w_m ρ_m, we also can obtain the full setof equations but only if we make some assumption about Λ_eff(t) (see, e.g. <cit.>). Let us now consider one interesting example of exact solution for our model.§ SOLUTION FOR THE EFFECTIVE VACUUM MODELIn the following, we provide the exact solution for our model in one simple case. Let us supposep_eff=-ρ_eff,that is the effective vacuum EoS w_eff=-1. From equation (<ref>), we getρ_eff = Λ_0=constant>0.Hence, the only independent equation, which has to be solved, follows from (<ref>) as3H^2 + 3k/a^2 = 3/8π g^2 a^4+Λ_0.Equation (<ref>) can be readily solved for all possible relations between its constant parameters (see, e.g., <cit.> and references therein). In particular, for a special choice of the constants of integration, these solutions can be given by a=a_k,δ(t),a={[ (√(A)sinh[2√(Λ_0/3)(t+t_ks)]+3k/2Λ_0)^1/2, δ=+1,; ; a_0 exp[√(Λ_0/3)t],                           δ=0,; ; (√(A)cosh[2√(Λ_0/3)(t+t_kc)]+3k/2Λ_0)^1/2, δ=-1,;].whereA=3/4Λ_0|1/2π g^2-3/Λ_0|,  δ=sgn(1/2π g^2-3/Λ_0),and the constants of integrationare supposed to bet_ks=(1-k)/4√(3/Λ_0)arsinh(3/2√(A)Λ_0)andt_kc=(1-k)/4√(3/Λ_0)arcosh(3/2√(A)Λ_0). Therefore, we can obtain from (<ref>) the following expression for the Hubble parameter H(t)=H_k,δ(t),H_k,δ(t)={[ H_0cosh[2H_0(t+t_ks)]/sinh[2H_0(t+t_ks)]+B_k,  δ=+1,; ; H_0sinh[2H_0(t+t_ks)]/cosh[2H_0(t+t_ks)]+B_k,  δ=-1,;] .where H_0=√(Λ_0/3), B_k=k/2√(A)H_0^2, and H(t)=H_0 for the case of δ=0. As one can see, at any case, H_0 = lim_t →∞H(t).Taking into account these equations and (<ref>), one can get the deceleration parameter q=q(t) as followsq={[ B_k^2+1-(sinh[2H_0(t+t_ks)]+ B_k)^2/cosh^2[2H_0(t+t_ks)], δ=+1,;; B_k^2-1-(cosh[2H_0(t+t_ks)]+ B_k)^2/sinh^2[2H_0(t+t_ks)], δ=-1.; ] . The behavior of theHubble parameter (<ref>) and the deceleration parameter (<ref>) with time for k=+1 and different values of δ is shown in Fig. 1. To be specific, we have chosen A=1. Then, one can see the black lines for H_0=1 and δ=+1, the blue lines for H_0=1 and δ=-1, the red lines for H_0=1.25 and δ=+1, and the green lines for H_0=1.25 and δ=-1. Applying (<ref>) for (<ref>) and (<ref>), we getp_m+ρ_m=Λ̇_eff/3 H,and, taking into account (<ref>) and (<ref>),ρ_m=Λ_0-Λ_eff,    p_m=-Λ_0+Λ_eff +Λ̇_eff/3 H.The last means that the matter EoS isw_m=p_m/ρ_m=-1+Λ̇_eff/3 H(Λ_0-Λ_eff). SinceΛ̇_eff= - 6 H β^2in accordance to (<ref>), than equation (<ref>) leads top_m+ρ_m=-2β^2.Let us suppose hat the matter in our model exists in the form of a scalar field ϕ(t) with a potential V(ϕ), that isρ_m=1/2ϵϕ̇^2 + V(ϕ),     p_m=1/2ϵϕ̇^2 - V(ϕ),where ϵ=±1 for the quintessential and phantom fields, respectively. Due to the real character of the displacement vector and non-negativity of the matter density,we obtainw_m ≤ -1. It means that the matter should be of the phantom nature. For example, it can be a phantom scalar field ϕ(t), for which ϵ=-1, that isρ_m+p_m=-ϕ̇^2,   ρ_m-p_m= 2 V(ϕ)From these expressions and equations (<ref>) - (<ref>), we haveϕ̇^2=2β^2,andV =Λ_0+β^2-Λ_eff.The further investigation of this model involves the imposition of some additional conditions. For example, we can supplement the model by the following simplest assumption regarding the displacement fieldβ=β_0=constant.which is often discussed in the literature. Taking into account equations (<ref>) and (<ref>), we obtainϕ(t)=√(2)β_0 t+ϕ_0,where ϕ_0 is a constant of integration. In view of (<ref>), the integration of equation (<ref>) yields the following effective cosmological termΛ_eff= λ_0- 6 β_0^2ln[a(t)/a_0],where λ_0 is a constant of integration. Substituting (<ref>) into (<ref>) alon with (<ref>) and (<ref>), we get the following potential of the scalar fieldV(ϕ)= V_0 + 6 β_0^2ln|a(ϕ-ϕ_0/√(2)β_0)|,where V_0 = Λ_0+β_0^2-λ_0- 6 β_0^2ln a_0. Substituting equation (<ref>) for different δ and k=±1 in (<ref>), we can obtain the scalar potential V(ϕ)=V_k,δ(ϕ) as followsV_k,+1(ϕ)= V_0 + 3 β_0^2ln|sinh(1/β_0√(2Λ_0/3) ϕ)+3k/2√(A)Λ_0|,for δ=+1 and ϕ_0=√(2)β_0 t_ks,V_k,-1(ϕ)= V_0 + 3 β_0^2ln|cosh(1/β_0√(2Λ_0/3) ϕ)+3k/2√(A)Λ_0|,for δ=-1 and ϕ_0=√(2)β_0 t_kc, andV(ϕ)= V_0 + √(6Λ_0) β_0 ϕ,when δ=+1. Here V_0 is a constant.For the specific values of constants, the scalar potentials (<ref>) and (<ref>) are shown in Fig.2.Substituting (<ref>) in equation (<ref>), we get the following equation for the scalar field EoSw_m(t)=-1-1/R+3ln[a(t)/a_0],where R=(Λ_0-λ_0)(2β_0^2)^-1, and a(t) is defined by equation (<ref>). As can be seen from equation (<ref>), the EoS of matterw_m could remain less than -1 all the time but tends to -1 as t→∞.§ COSMOLOGICAL MODEL WITH A CONSTANT HUBBLE PARAMETERNow we consider the case of inflationary model, when H=H_0 = constant, and hencea(t)=a_0 exp(H_0 t).Then from equation (<ref>), one can obtainΛ_eff(t)=λ_0 - 6H_0∫β^2(t) d t . According to equations (<ref>), (<ref>) and (<ref>), it is easy to get the following energy density and pressure of matter ρ_m=3k/a^2-3/8π g^2a^4-λ_0+3H_0^2+6H_0∫β^2 d t,      p_m =-k/a^2-1/8π g^2a^4+λ_0-3H_0^2-6H_0∫β^2 d t -2 β^2.By summarizing these equations, one can get thatρ_m + p_m = 2k/a^2-1/2π g^2a^4-2β^2. Once again, the further investigation of this model requires to apply some extra conditions. For example, we can supplement the model by the following simplest assumption regarding the displacement fieldβ(t)= β_0/a^2(t)=β_0/a_0^2exp(-2H_0 t),where β_0 is a constant. Taking into account (<ref>), (<ref>) and(<ref>), one can get the following equation for the EoS of matter w_m=p_m/ρ_mw_m =-k/a_0^2 e^-2H_0 t+1/2a_0^4(1/4π g^2+β_0^2) e^-4H_0 t+ρ_0/3k/a_0^2 e^-2H_0 t-3/2a_0^4(1/4π g^2+β_0^2) e^-4H_0 t+ρ_0,where and further ρ_0=3H_0^2-λ_0 is the asymptotic (at t→∞) value of the energy density of matter. It is interesting to note that this equation asymptotically leads to the constant quintessential EoSw_m=-1/3 when and only when ρ_0=0, that is λ_0=3H_0^2, for any sign of the curvature, and for any numeric values of other parameters of the model.In general, the detailed behavior of w_m(t) depends on the numeric values of parameters k, a_0, β_0 and g in equation (<ref>), and also depends on the relation between them. If we introduce the following notationC_1=2/a_0^2,   C_2=2/a_0^4(1/4π g^2+β_0^2),we can find that the EoS parameter (<ref>) behaves itselfindifferent manners depending to the relative relation between C_1 and C_2. The dependence of EoS w_m(τ)on the dimensionless time τ=H_0 t forC_1=0.8, S_2=1.6 and k=+1 (in blue), k=-1 (in red) is shown in Fig. 3 for several values of ρ_0.Let us assume again that the matter exists in the form of a scalar field ϕ(t) with a potential V(ϕ). Therefore, it follows from equation (<ref>) thatρ_m+p_m=ϵϕ̇^2,   ρ_m-p_m= 2 V(ϕ),From these equations and (<ref>)-(<ref>),we can obtainϵ H_0^2 (ϕ')^2 =kC_1-C_2x^2,andV(x) =kC_1x^2-1/4C_2x^4 + ρ_0,where x=exp(-H_0 t)∈ (0,1] and ϕ'=dϕ/dx. The exact solutions for equation (<ref>) can be readily obtained as followsϕ(x) =±1/2H_0[x√(C_1-C_2x^2)                    +C_1/√(C_2)arcsin(√(C_2/C_1)x)]+ϕ_0,for k=+1, C_1≥ C_2, ϵ=+1, orC_1<C_2, x∈ [0,x_0] where x_0=√(C_1/C_2),ϕ(x) =±1/2H_0[x√(C_2x^2-C_1)                    -C_1/√(C_2)arcosh(√(C_2/C_1)x)]+ϕ_0,for k=+1, C_1<C_2, ϵ=-1, x∈ [x_0,1], andϕ(x) =±1/2H_0[x√(C_1+C_2x^2)                    +C_1/√(C_2)arsinh(√(C_2/C_1)x)]+ϕ_0,for k=-1, ϵ=-1.Notably that almost the same result could be obtained from the assumption of the other rate of changing for the displacement field, namelyβ(t)= β_0/a(t)=β_0/a_0exp(-H_0 t).In this case, equation (<ref>) for the EoS of matter is rewritten as followsw_m =-1/a_0^2(k-β_0^2) e^-2H_0 t+1/8π g^2a_0^4 e^-4H_0 t+ρ_0/3/a_0^2(k-β_0^2) e^-2H_0 t-1/8π g^2a_0^4 e^-4H_0 t+ρ_0,and both equations (<ref>) and (<ref>) are valid again but with re-defined constants C_1, C_2 byC_1=2(1-kβ_0^2)/a_0^2,   C_2=1/2π g^2a_0^4.Therefore, we can obtain the same solutions (<ref>)-(<ref>) with new constants (<ref>).Putting aside a detailed discussion of a generalization of this model in the spirit of Ref.<cit.>, let us note that the consequencesof such generalization for the cosmic acceleration could be more radical than those obtained here. § CONCLUSION In this paper, we have investigated some cosmological non-flat FRW models with YM fields on Lyra manifold. By using the exact solution of the YM equations on the FRW cosmological background, the exact solutions in Lyra's geometry have been obtained withthe certain propositions regarding the effective equation of state and the rate of expansion.As a result, we have obtained the explicit expressions for the EoS of matter. The scalar field potentialhas been reconstructed assuming that the matterconsists of the quintessential or phantom scalar field. The results of this paper may also be extended without much effort to the case of coupling the YM fields with the ordinary matter. In that case the accelerated regime of expansion could be reached by means of energy exchange between different forms of matter.Finally, we can conclude that these models still contain a great potential for the study of cosmological expansion within Lyra geometry based on the exact solutions with participation of the Yang-Mills fields. 99 Riess A. G. 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Samaroo, "Generalized Exact Cosmologies with Interacting Yang-Mills and Nonlinear Scalar Fields", General Relativity and Gravitation, 36(7) (2004),1661. http://dx.doi.org/10.1023/B:GERG.0000032158.16161.1bMatyjasek3 J. Matyjasek, "Cosmolagical models with a time-dependent Λ term", Physical Review D, 51(8) (1995), 4154. https://doi.org/10.1103/PhysRevD.51.4154Stephani H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Herlt. Exact Solutions of Einsteins Field Equations, 2nd Edition, Cambridge University Press, 2003.

http://arxiv.org/abs/1702.07940v2

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http://arxiv.org/abs/1702.08029v2

0000-0002-4528-7637]S. E. Lake Physics and Astronomy Department, University of California, Los Angeles, CA 90095-15470000-0001-5058-1593]E. L. Wright Physics and Astronomy Department, University of California, Los Angeles, CA 90095-15470000-0002-9508-3667]R. J. Assef Núcleo de Astronomía de la Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Av. Ejército Libertador 441, Santiago, Chile Astronomy Department University of Cape Town Private Bag X3 Rondebosch 7701 Republic of South Africa0000-0003-0624-3276]S. Petty NorthWest Research Associates 4118 148th Ave NE Redmond, WA 98052-5164 Institute of Geophysics and Planetary Physics, Lawrence Livermore National Laboratory, Livermore CA 94551 Department of Physics, University of California, Davis, CA 95616Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 911090000-0002-9390-9672]C.-W. Tsai Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109 Physics and Astronomy Department, University of California, Los Angeles, CA 90095-1547 S. E. Lake lake@physics.ucla.edu Thesatellite surveyed the entire sky multiple times in four infrared wavelengths <cit.>.The unprecedented combination of coverage area and depth gives us the opportunity to measure the luminosity function of galaxies, one of the fundamental quantities in the study of them, at 2.4 to an unparalleled level of formal statistical accuracy in the near infrared.The big advantage of measuring luminosity functions at wavelengths in the window ≈ 2 to 3.5 is that it correlates more closely to the total stellar mass in galaxies than others.In this paper we report on the parameters for the 2.4 luminosity function of galaxies obtained from applying the spectroluminosity functional based methods defined in <cit.> to the data sets described in <cit.> using the mean and covariance of 2.4 normalized spectral energy distributions (SEDs) from <cit.>.In terms of single Schechter function parameters evaluated at the present epoch, the combined result is: ϕ_⋆ = 5.8 ± [0.3_stat,0.4_sys] × 10^-3Mpc^-3, L_⋆ = 6.4 ± [0.1_stat,0.3_sys] × 10^10L_2.4 ⊙ (M_⋆ = -21.67 ± [0.02_stat,0.05_sys]AB mag), and α = -1.050 ± [0.004_stat,0.04_sys].The high statistical accuracy comes from combining public redshift surveys with the wide coverage from , and the unevenness in statistical accuracy is a result of our efforts to work around biases of uncertain origin that affect resolved and marginally resolved galaxies.With further refinements, the techniques applied in this work promise to advance the study of the spectral energy distribution of the universe. § INTRODUCTIONThe luminosity function (LF) is one of the most basic statistical properties measured for any class of objects in astronomy.The fundamental nature of the LF means that it has been measured for galaxies many times, in many different bandpasses <cit.>. The release of the AllWISE catalog generated from the data gathered by thesatellite, described in <cit.> and <cit.>, marks the availability of 3.4 (W1) and 4.6 (W2) photometric data that is better than 95% complete over the vast majority of the sky down to 44 and 88 (19.79 and 19.04 AB mag), respectively.This new data set presents the opportunity to utilize the large number of public redshift surveys, in tandem with a small -selected survey of our own, to measure the near-IR luminosity function of galaxies at 2.4 to unprecedented accuracy.The advantage of measuring the luminosity function in this range of wavelengths is that fluxes suffer from minimal dust extinction in both the target galaxy and the Milky Way, according to dust extinction models like the one from <cit.>.Further, near infrared light traces the target galaxy's stellar mass in evolved stars more faithfully than optical wavelengths <cit.>, as long as the contribution of thermally pulsing asymptotic giant branch (TP-AGB) stars can be correctly accounted for in the population synthesis models <cit.>.We measure the luminosity function at 2.4, in particular, because it is the wavelength directly observed by W1 for galaxies at the median redshift, z=0.38, of galaxies with F_W1 > 80 in <cit.>.This new opportunity also presents new challenges. First, even limiting the redshift surveys to those that are publicly available and that are, primarily, selected by flux at a single wavelength meant that there were a lot of details that needed to be addressed in the characterization and selection process for the six surveys used here.Because of this, the primary characterization is done separately in a companion paper <cit.>.Second, we limited the surveys to be well above the sensitivity limits of the AllWISE data set, a minimum flux of 80 in W1 (19.14 AB mag), in order to minimize the additional incompleteness from the cross-match and to match the properties of <cit.> as closely as possible.This means that the surveys have flux limits at two wavelengths.The two flux limits, combined with the wide range of redshifts included in the analysis, 0.01 < z ≤ 1.0, meant that the existing luminosity function measurement tools were not adequate to the challenge of analyzing the collected data set.For this reason, a new estimator based on analyzing the likelihood of a galaxy's entire spectral energy distribution (SED) is derived in a companion paper <cit.>.The likelihood estimate used in this work is based on the mean and covariance of SEDs as measured in <cit.>. The structure of this paper is as follows: Section <ref> summarizes properties of the data used to measure the 2.4 luminosity function described in LW18II, Section <ref> summarizes the estimators used in this paper to perform the measurements, Section <ref> contains the results of the analysis (including comparisons to other measured LFs), Section <ref> places the results of this work in context and outlines possible improvements to the methods used, and Section <ref> contains the conclusions drawn from the analysis.The cosmology used in this paper is based on the WMAP 9 year ΛCDM cosmology <cit.>[<http://lambda.gsfc.nasa.gov/product/map/dr5/params/lcdm_wmap9.cfm>], with flatness imposed, yielding: Ω_M = 0.2793, Ω_Λ = 1 - Ω_M, redshift of recombination z_recom = 1088.16, and H_0 = 70 ^-1^-1 (giving Hubble time t_H = H_0^-1 = 13.97Gyr, andHubble distance D_H =c t_H = 4.283 Gpc). All magnitudes are in the AB magnitude system, unless otherwise specified.In cases where the source data was in Vega magnitudes and a magnitude zero point was provided in the documentation, they were used for conversion to AB (2MASS[<http://www.ipac.caltech.edu/2mass/releases/allsky/faq.html#jansky>] and AllWISE[<http://wise2.ipac.caltech.edu/docs/release/allsky/expsup/sec4_4h.html#WISEZMA>]).For the surveys without obviously documented zero points (NDWFS[<http://www.noao.edu/noao/noaodeep/>], SDWFS[<http://irsa.ipac.caltech.edu/data/SPITZER/SDWFS/>]) we converted Vega magnitudes to AB magnitudes using those provided in <cit.>.When computing bandpass solar luminosities we utilized the 2000 ASTM Standard Extraterrestrial Spectrum Reference E-490-00[<http://rredc.nrel.gov/solar/spectra/am0/>].For our standard bandpass, W1 at z=0.38, we get an absolute magnitude of M_2.4 ⊙ = 5.337AB mag, L_2.4 ⊙ = 3.344 × 10^-8^2.§ DATA SUMMARYThe full description of the data sets used in this work can be found in LW18II.Table 1 of LW18II summarizes the properties of the spectroscopic surveys used here, and we refer interested readers there.As described in LW17I, the parametric model used here requires models for the relationship between noise and flux.Because the flux selection cuts are all relatively bright a single power law of the form(σ_F/F)^2 = (F/A)^Bwas adequate.The model also requires an estimate of the mean, ⟨τ⟩, and standard deviation, σ_τ, of the dust obscuration for each selection filter in the field of the redshift survey.Table <ref> contains the parameters measured for each survey, with separate values for the optical selection filter and W1.The calculation of the incompleteness due to flux variety also uses the parameters in the table to calculate the selection function in redshift-luminosity space, S(L,z).l|cccccccc Noise Models Used for Parametric Fits and Completeness Survey A_opt B_opt A_W1 B_W1 ⟨τ_opt⟩ σ_τ opt ⟨τ_W1⟩ σ_τ W1 10^2 10^2 10^3 10^3 6dFGS 43410 -0.9068 5.469× 10^-6 -0.2774 2.559 3.142 12.69 15.58 SDSS 203.7 -1.174 0.9552 -0.5334 0 0 6.173 6.107 GAMA 66.15 -1.419 6.793 -0.6317 0 0 5.914 1.801 AGES 37.12 -1.472 1.570 -0.5651 2.002 0.4561 2.015 0.4591 /DEIMOS 89.52 -0.457 675.6 -1.016 23.91 23.89 15.8 15.79 zCOSMOS 102.1 -1.379 139.1 -0.8576 3.134 0.2279 3.141 0.2283 Noise model parameters fit to each survey separately. The `opt'parameters pertain to the optical filter used for selection in the survey, and the `W1' parameters are the parameters for the W1 filter. The A and B parameters relate to a power law fit of signal-to-noise ratio to flux (see Equation <ref>), and ⟨τ⟩ and σ_τ are the mean and standard deviation of the optical depth from foreground dust obscuration, averaged over targets in the survey. When the dust obscuration parameters are zero, target selection was performed on fluxes after extinction correction instead of before. All quantities given to four significant figures, regardless of the statistical or systematic uncertainties in the quantities. There is one additional plot set not in LW18II that is necessary for understanding the choices made about how to analyze the data, found in Figure <ref>.It shows that w1rchi2, the reduced χ^2 of fitting a point spread function to the source in the W1 images <cit.>[<http://wise2.ipac.caltech.edu/docs/release/allwise/expsup/sec5_3bii.html#review_stationary_model>],behaves about as one would expect for galaxies with z<1 where, even if galaxies in the past were the same size as today, the increasing angular diameter distance gives them, overall, a smaller radius on the sky.The vertical line at z=0.2 is placed there after trial and error as a dividing line between samples that are significantly contaminated by resolved and marginally resolved objects (z ≤ 0.2), and those that are sufficiently point-like to render resolution concerns moot (z > 0.2).In this work we describe the set of all galaxies that come from below z=0.2 as the low z sample, and those from above the line as the high z sample. As a check on the impact the new analysis techniques from LW17I had on the results, we have also analyzed high and low redshift subsamples that had completeness above 98% of the maximum value for the survey, as defined by Equation 21 of LW17I, and as shown by the light blue contours on the luminosity-redshift plots in LW18II.These samples, called the `trim' simples, have significantly lower numbers of galaxies with the accompanying increase in statistical uncertainty, but also reduced systematic uncertainty from the constancy of the selection function. The number of sources each survey makes to the combined samples, as well as their overall sizes, can be found in Table <ref>.l|rrrrrr|r Combined Samples Sizes Sample N_6dFGS N_SDSS N_GAMA N_AGES N_WDa N_zCb N_tot Low z 27,071 450,731 28,619 2,000 36 133 508,590 High z 0 26,556 15,872 3,741 171 1,091 47,431 Low z Trim 15,891 106,003 9,513 1,212 25 116 132,760 High z Trim 0 21 125 452 17 647 1,262 Number of sources contributed by each survey to the combined samples named in the first column. The last column contains the total number of sources in each combined sample. a bzCOSMOS § ANALYSIS METHODS AND MODELS SUMMARYThe analyses in this work are centered around maximum likelihood estimates of the quantities related to the LF.As is customary in most works on the LF, we analyze the data using both binned and parametric estimators.The full description and derivation of the estimators used here is in LW17I.Two standard binned estimators were adapted for this work: 1/V_max from <cit.> as modified by <cit.>; and N_obs/N_mdl from <cit.>.The adaptation was to handle situations where the selection function, S(L,z), varies significantly over the bin in luminosity-redshift space. Note that the selection function, S, is here not regarded as a particular function but as an assessment of the probability that a galaxy with the properties specified in the arguments would be selected for inclusion in the data set. For the 1/V_max estimator, the varying completeness corrected version of the estimator, with log-spaced bins in luminosity, is given by:Φ(L_i,z_i) = 1/S(L_i,z_i) L_i Δln L_i∑_j1/Δ V_j,where the redshift-luminosity bin is labeled by the index i, Φ(L_i,z_i) is the constant estimate of the luminosity function for that bin, the sum is over sources that fall into the bin, and Δ V_j is the volume available to the source to still be in both the bin and the selection criteria of the survey. The varying completeness corrected version of N_obs/N_mdl is, formally, much closer to the form of the original estimator from <cit.>:Φ(L_i,z_i) = Φ^mdl(L_i,z_i) ·N_i/⟨ N^mdl_i⟩ = Φ^mdl(L_i,z_i)/∫_binΦ^mdl(L,z)S(L,z)ḶṾ· N_i,where Φ^mdl(L,z) is an approximate (`model') luminosity function that brings the estimator closer to evaluating the luminosity function at the center of the bin, (z_i, L_i), as long as Φ^mdl(L,z) is closer to the true luminosity function than the implicitly assumed Φ^mdl(L,z) = constant of an uncorrected estimator.For these purposes, this work assumes Φ^mdl(L,z) is a Schechter function that has a faint end slope α = -1 and M_⋆ = -22 (near the peak of the luminosity histograms in LW18II).The parametric estimator used in this work is based on the spectro-luminosity functional, Ψ.The final estimator of the likelihood of the data is a little complicated, requiring a few nested equations to express.The outermost equation is given by:ln(ℒ) = ∑_galaxiesln (S(F_sel, F_0, x⃗) (F_sel, F_0|L_0) Φ(L_0, z))- ∫ S(F_sel, F_0, x⃗) (F_sel, F_0|L_0) Φ(L_0, z) F̣_selF̣_0 Ḷ_0 Ṿ_c,where the selection function, S, calculates the probability that a source with given observed fluxes used for primary and secondary target selection, F_sel and F_0 (respectively), and at spatial position x⃗ is selected for inclusion in the survey.Here F_sel differs by survey (see the band column of Table 1 in LWII), and F_0 is always the W1 flux because the source needs to be well detected there to use F_W1 in the calculation of L_2.4. This flux selection function is assumed to take only two values: 0 for excluded regions, and and s for sources in selected regions, where s (0 < s < 1) is the overall completeness of the survey.Basically, its primary purpose is to set the limits of the integration.is the likelihood for a galaxy to have a particular SED given it has 2.4 luminosity L_0, projected down from the full function space to the fluxes F_sel and F_0.In this work we approximate the fullas a Gaussian and incorporate an uncertainty model for the fluxes and dust extinction to get:(F_sel, F_0|L_0) = ^τ_1 + τ_2/√((2π)^2 det(σ))·(4π D_L(z)^2/(1+z)L_0)^2 ·exp(-1/2∑_i,j=2^2 [ℓ_i - μ_i][σ^-1]_ij·[ℓ_j - μ_j]), and ℓ_i= F_i 4π D_L(z)^2 ^τ_i/(1+z)L_0.τ_i is the optical depth due to dust in the Milky Way present for flux in filter i, when the selection fluxes weren't extinction corrected, 0 when they were.D_L(z) is the luminosity distance to the source. The F_i are one of F_sel or F_0 and are the measured fluxes.μ_i is the mean value ℓ_i takes, as predicted from the mean SED.σ is a covariance matrix that takes the following form (no summation):σ_ij = Σ_ij + (δ_ij A_i F_i^B_i + σ_τiσ_τj) μ_i μ_j,where A and B are model parameters fit using ordinary least squares in log-space of σ_F^2 F^-2 to F, σ_τi is the standard deviation of the optical depth present for fluxes in channel i over the survey targets, and Σ_ij is the covariance of SEDs projected down to apply to the space spanned by F_sel and F_0, similar to the calculations described in <cit.>.The optical depths are calculated from the dust extinction model of <cit.> using the E(B-V) dust maps from <cit.>.Φ(L_0,z) is a Schechter luminosity function, originally defined in <cit.>,Φ(L, z) = ϕ_⋆/L_⋆(L/L_⋆)^α^-L/L_⋆.The parameterization for evolution in ϕ_⋆ and L_⋆ are:ϕ_⋆= ϕ_0 ^-R_ϕ t_L(z), and L_⋆= L_0 ^-R_L t_L(z)(1 - t_L(z)/t_0)^n_0,where t_L(z) is the lookback time of a source at redshift z, R_ϕ is the specific evolution rate in ϕ and is assumed to be constant, R_L is the luminosity evolution rate, t_0 is the time of first light (set here to t_0 = t_L(z_recom) since our data cannot meaningfully constrain it),and n_0 is the power law index for the early time increase in L_⋆.As explained in LW17I, the actual fitting was done using derived parameters that proved to be more statistically orthogonal than the ones given above.In terms of the parameters from Equation <ref>, they are:R_n = R_ϕ - min(1+α, 0) [R_L + n_0/t_0], and κ_⋆= ϕ_0 L_0^3/2/Ω_sky 4π^1/2Γ(α + 5/2),which are named the specific rate of change in galaxy number density at z=0, and the normalization to the source flux counts distribution.If the luminosity function were a static Schechter function in a static Euclidean universe of infinite radius then Ṇ/F̣Ω̣ = κ_⋆ F^-5/2. §.§ Error Analysis DetailsThe statistical uncertainty in the binned estimators is assumed to be fundamentally Poisson combined with propagation of errors that assumes all factors other than the number of galaxies in a bin are constants, as described in LW17I.For the parametric estimator in Equation <ref>, its complicated form makes performing a second order expansion about the maximum likelihood parameters, as is done in frequentist statistics, to find the uncertainty in those parameters tedious and error prone.Further, when there is any region of the parameter space where the likelihood becomes unusually flat, as happens frequently when selection functions are involved, the Taylor expansion must be carried out to higher order to get an estimate in the parameters' uncertainties.These conditions make the work done here an ideal case for the application of a Bayesian analysis using Markov Chain Monte Carlo (MCMC) to estimate the uncertainties in the parameters.The software tool used to perform this error analysis is the Python package known as emcee version 2.1.0[<http://dan.iel.fm/emcee/current/>], described in <cit.>, an implementation of the stretch-move algorithm proposed in <cit.>.When doing any MCMC analysis the algorithm needs to run for a number of steps before it starts to provide an accurate and uncorrelated sample of the posterior.This process is called `burn in.'If the initial point is far from the mode of the posterior, then this first step is dominated by a pseudo-random walk toward that mode, effectively making it an inefficient optimization algorithm.This process is even less computationally efficient for emcee because it uses an entire ensemble of independent walkers.To short circuit this process, we started the walkers in a ball around an extremum found using the optimize package of SciPy.When fitting models as complex as the ones used here, there is the added downside that there are frequently multiple local extrema.This factor causes a tradeoff in the usage of emcee and how spread out the initial positions of the walkers are: if the spread is large then the odds of finding a better minimum than the current guess goes up, but walkers will also get stuck in local minima that are uninteresting outliers; if the spread is small, then all of the walkers are characterizing the minimum of interest, but it takes much longer to find any possible lower minima.In principle, it would be possible to design an algorithm that was mostly emcee, but that periodically trimmed outliers and started new walkers at large distances to search for possible new minima.In practice, it is easier to break the process up into different steps.In minimum finding mode, the initial spread of the walkers is large, centered on the minimum found by a comparatively efficient algorithm, and it restarts any time a new minimum is found that is lower than the initial one.The software then switches to minimum characterizing mode where the initial spread is small, and after a number of burn in steps that are discarded the final sample is produced.If a new minimum is found during minimum characterizing mode, then the minimum characterizing process begins again from the beginning, but the switch is not made back to minimum finding mode under the assumption that the improvement from doing so is marginal.With any Bayesian analysis the prior must be described.Table <ref> contains the explicit descriptions of the ranges of the parameters, and the priors assumed on the parameters, for the LF.In most cases the priors are flat in the given parameter, and the majority of the remainder are flat in the logarithm of the parameters.The exceptions to this are the faint end slopes, α, and the initial luminosity index, n_0.Because α=-2 gives an unphysical infinite background radiation, we chose to impose a mildly informative prior with a beta distribution shape that excluded the end points of the allowed interval.With n_0 the buildup of L_⋆ should neither be discontinuously fast, so n_0 > 0, nor should it be much slower than the integral of a linear accretion rate, so n_0 < 10.Further, the data used here does not constrain the buildup of L_⋆ directly, so an informative prior that constrains the value of n_0 is used.Explicitly, it is expected that their initial luminosity is proportional to the gas accretion rate, and should therefore be reasonably near linear, so we impose n_0 = 1 ± 0.2 with a log-normal distribution.cccccc 0.5 LF Parameter Priors Param Min Max Units Prior Notes κ_⋆ 10^-3 10^2 Jy^3/2^-1 κ_⋆^-1 a R_n0 -50 50 t_H^-1 flat b L_0 10^9 10^13 L_2.4 ⊙ L_0^-1 c R_L -50 50 t_H^-1 flat α -2 0 — -α(2+α) n_0 10^-15 10 — n_0^-1e^-25 ln(n_0)^2 / 2 d Parameter (Param) ranges (from Min to Max) and priors used in fitting the luminosity function. Priors are given in unnormalized form. a Normalization to the static Euclidean number counts. Its relationship to standard Schechter function parameters can be found in Equation <ref>. b Specific rate of change of the galaxy number density at z=0. Its relationship to standard Schechter function parameters can be found in Equation <ref>. c L_2.4 ⊙ = 33.44 ^2 ↔ M_2.4 ⊙ = 5.337 AB absolute. d This prior has mean 1 and standard deviation 1/5. § LUMINOSITY FUNCTIONSRunning emcee produces a sequence of model parameters that are distributed as though they were samples taken from the posterior distribution.The sequence of parameters is, therefore, the result of the analysis from which all other results are derived.The chains are published with this paper and at <www.figshare.com> under the digital object identifier (DOI) https://dx.doi.org/10.6084/m9.figshare.410962510.6084/m9.figshare.4109625 in https://www.gnu.org/software/gzip/gzipped http://irsa.ipac.caltech.edu/applications/DDGEN/Doc/ipac_tbl.htmlIPAC table format.An few example lines from one of the chains can be found in Table <ref>.rrcccccc 0.97 Example Lines from MCMC Chains StepNum WalkerNum ln_KappaStar R_nln_LstarR_L alpha ln_n0 — — ln(^3/2^-1)^-1 ln(^2) ^-1 — — 0 02.0414 -0.0562898.7190 -0.13463 -1.5071 -0.30514 0 11.8404-0.0136868.2423 -0.19569 -1.2721 -0.03764 0 22.13090.2246667.6358-0.29812-0.9931-0.01672 0 32.55180.0950798.6992-0.22876-1.4027 0.41558 0 41.8430-0.0072508.2185-0.21643 -1.3473 0.03394 Example lines from one of the chains produced by emcee in the tables under DOI https://dx.doi.org/10.6084/m9.figshare.410962510.6084/m9.figshare.4109625. Floating point values truncated here for brevity, but not in the downloadable tables. StepNum is the zero indexed step number that the ensemble was at in the chain, and WalkerNum is the number of the walker which was at the position defined by the row for that step. ln_KappaStar is the natural logarithm of κ_⋆ in ^3/2^-1 (see Equation <ref>). R_n is the specific rate of change of the galaxy number density evaluated at the present time in ^-1 (see Equation <ref>). ln_Lstar is the natural logarithm of L_⋆ in ^2 evaluated at z=0 ( M_⋆ = -2.5/ln 10ln[L_⋆/3631 4π 10^-10^2 ]≈ -1.086ln[L_⋆/^2 ] -13.352for AB absolute mag). R_L is the long term decay constant in L_⋆ in ^-1. alpha is the faint end slope of the luminosity function. ln_n0 is the natural logarithm of the early time power law index of the evolution of L_⋆ (see Equation <ref>) The content of that file set is as follows: there is an individual chain for each individual spectroscopic survey (with file names matching the names of the survey), one for each merged subsample described in Table <ref> with matching names, and two chains, labeled `High z Prior' and `High z Trim Prior', where the High z samples were analyzed with the mean and standard deviation in α from the matching Low z samples used as additional Gaussian priors in the analysis.The reason for only using the faint end slope as a prior is discussed in Section <ref>.Of all of the analysis chains produced, High z Prior is the canonical one for this work based on a subjective evaluation of statistical accuracy and bias.A comparison of the N_obs / N_mdl binned estimator using all of the data to the LF with the mean parameters from the High z Prior chain can be found in Figure <ref>.The solid red lines, of varying opacity and brightness, are the result of evaluating the High z Prior LF model at five equally spaced redshifts from 0.01 to 1 (0.01, 0.258, 0.505, 0.743, 1).The three 3.6 LFs from <cit.>, color corrected to 2.4 using the mean SED from <cit.>, are plotted at z=0.38 using black dashed (`all'), red dotted (`early'), and blue dash-dotted (`late') lines.Two features stick out most prominently in Figure <ref>.First, the parametric estimator places the value of ϕ_⋆ a factor of about 1.7 higher than the binned estimator, though this is within the statistical uncertainty in ϕ_⋆ for Hi z Prior.Second, the falloff at the bright end appears to be better described as a power law than exponential.The former is likely caused by the fact that the data content of Figure <ref> is drawn from all the samples, without restriction on redshift, while the plotted LF is of the mean parameters from the High z Prior chain.The cause of the latter is uncertain.Averaging the evolving fit LF over different redshifts using the observed redshift distribution did not produce the observed power law shape.Further, limiting the plot to data in the High z sample does not alter this feature significantly, either, so it is unlikely to be an artifact related to photometry of resolved sources.The explanation that, qualitatively, seems most likely to cause the feature is the presence of active galactic nuclei (AGN) in the sample, which are observed to have a power law falloff to the LF on the bright end <cit.>.While this means that the fit LF doesn't match all of the details of the real LF, this was an expected consequence of using a single Schechter function for the LF and not separate LFs for different galaxy types.The global properties of the LF, like predictions of galaxy count and luminosity density, should still correspond with the observable values in the same way that a Gaussian fit to a data set will reproduce the mean and standard deviation, even if the data are not Gaussian distributed.Detailed analyses of the different posterior chains is done in Subsection <ref>, including an examination of the necessity of splitting the combined samples by redshift, and evidence for luminosity uncertainty contributing to the softening of the high luminosity falloff.Comparisons with the results of other measurements of the luminosity function is done in Subsection <ref>. §.§ Internal ComparisonsBecause the parametric estimator used in this work is new, it is important to analyze multiple data set that all have different selection criteria and compare the results to see if the systematic biases have been correctly managed.The mean parameters from each of the posterior chains can be found in Table <ref>, and parameters derived from those mean parameters are found in Table <ref>. All of the uncertainties given in the tables are purely statistical uncertainties derived from the Bayesian posterior of the data.They do not include sources of error that are, for the purposes of this work, systematic, including: the accuracy of the cosmological parameters (≈ 3%), the peculiar velocity the Milky Way , the accuracy of the completeness assessments of the different surveys , selection effects not modeled, the accuracy of the AllWISE W1 photometric zero point <cit.>, and the accuracy of the numerical integration algorithms used .Cosmic variance, the additional variability of the data set induced by the tendency of galaxies to cluster more than random chance, was estimated to be about 5% for SDSS in <cit.>, and <cit.> found GAMA to be 15% under-dense with respect to SDSS. Based on this, we assign cosmic variances of: 15%, 5%, 15%, 20%, 20%, and 20% to the 6dFGS, SDSS, GAMA, AGES, , and zCOSMOS surveys, respectively, with 4% for the combined analyses. The combination of these effects implies 5.3% systematic uncertainty in the determination of L_⋆, 5.2% in κ_⋆, 7.3% in ϕ_⋆, an assumed -fold per Hubble time in the evolution rate parameters, and an assumed 4% in α, for the combined samples.All of the parameters on a given line in Tables <ref> and <ref> are correlated, to greater or lesser degrees.Including tables or plots of the correlation among the parameters would take up a prohibitive amount of room, so this work only contains a single example of the covariance matrix among the primary parameters constructed from the High z Prior chain in Table <ref>.It should be noted that the full correlation among parameters is not necessarily encapsulated by a covariance matrix, particularly the correlation between ln n_0 and R_n.In other words, an examination of the pairwise distribution of parameters in each Markov chain does not always show the elliptical structure that would suggest they are well characterized by a Gaussian distribution, and hence by only a mean vector and covariance matrix, particularly when the sample size is small. The biggest trend in Table <ref> is in R_n, with the lower depth surveys consistent with the number density of galaxies currently declining and the higher depth surveys with the opposite.While it is possible that this is a real feature of the data, whether from a turnover in the comoving number density of galaxies or cosmic variance from the Milky way existing in an low density region, selection biases must first be ruled out.A preliminary analysis of the SDSS subset with a W1 flux maximum of 2(15.65AB mag) suggested that this effect was being driven by the galaxies with high apparent flux, and not the faint galaxies that dominate number counts for samples with minimum luminosity significantly lower than L_⋆.Because faint galaxies dominate the number density of galaxies, it is unlikely that a trend driven by the presence of bright objects in the sample is a real phenomenon, neither cosmic trend nor cosmic variance.As the error bars on R_n show, this parameter is poorly constrained by the data, so a small bias in a correlated parameter, even a weekly correlated one, can drive a big change in R_n.Since bright galaxies are also more likely to be resolved or marginally resolved, we decided to work around the problem by dividing the combined samples at z=0.2 where the number of galaxies with high w1rchi2 fell to a level low enough to be negligible (see Figure <ref>).The advantage of a redshift split instead of a flux cut is that it does not increase the impact of the systematic uncertainties inherent in the the final estimator used.The only parameter from the untrimmed low redshift sample analysis that is plausibly not affected by a bias affecting resolved sources is the faint end slope.Given the luminosity range covered by the Low z samples makes it the strongest constraint on α available, we use the mean and standard deviation of α from the corresponding Low z samples as Gaussian priors on the High z Prior samples. In order to estimate the possible systematic impact the Trim samples were constructed.The Trim samples are identical to their corresponding combined samples, but each survey is limited to the region in luminoisty-redshift space where the completeness is at least 98% of its maximum value; the regions enclosed by the faint blue lines in the L-z plots of LW18II.This substantially reduces the size and depth of the sample, so it is more vulnerable to cosmic and statistical variance.The Low z and Low z Trim chains agree within the statistical uncertainties, with the exception of L_⋆.Even though the effects on the High z samples was more dramatic, this is to be expected given the reduced effective depth and loss of low luminosity sources, especially in the AGES andsamples.When the power law part of the LF is not directly sampled, where the LF is linear in a log-log plot, the information about α is encoded in the higher order moments of what was observed, increasing sensitivity to statistical and cosmic variance fluctuations. lcccccc Luminosity Function Bayesian Mean Parameters Surveyκ_⋆ R_naL_⋆aR_L α n_0 —^3/2^-1 t_H^-1 10^10 L_2.4 ⊙ h^-2 t_H^-1 — — 6dFGS 2.94 ± 0.07 -4.9 ± 0.8 3.15 ± 0.07 -3.4 ± 0.3 -0.91 ± 0.02 0.9 ± 0.2 SDSS 4.06 ± 0.03 -3.55 ± 0.09 3.34 ± 0.03 -2.4 ± 0.2 -0.957 ± 0.005 1.0 ± 0.2 GAMA 4.0 ± 0.1 0.2 ± 0.2 2.67 ± 0.09 -12.7 ± 0.8 -0.95 ± 0.02 8.3 ± 0.6 AGES 5.4 ± 0.3 1.8 ± 0.3 2.2 ± 0.2 -5.0 ± 0.7 -0.57 ± 0.06 1.8 ± 0.4 /DEIMOS 8 ± 2 1 ± 2 5 ± 2 -3 ± 1 -1.1 ± 0.3 1.0 ± 0.2 zCOSMOS 4.4 ± 0.5 1.8 ± 0.8 2.6 ± 0.5 -4.5 ± 0.5 -0.9 ± 0.1 0.9 ± 0.2 Low z 3.11 ± 0.02 -7.47 ± 0.09 3.41 ± 0.02 -1.5 ± 0.2 -1.059 ± 0.004 0.8 ± 0.2 High z 5.0 ± 0.2 0.6 ± 0.2 2.66 ± 0.05 -2.8 ± 0.1 -0.68 ± 0.03 0.56 ± 0.08 High z Prior 5.7 ± 0.2 0.1 ± 0.2 3.12 ± 0.05 -2.6 ± 0.1 -1.050 ± 0.004 0.50 ± 0.07 Low z Trim 3.24 ± 0.03 -7.7 ± 0.2 3.62 ± 0.04 -0.4 ± 0.2 -0.972 ± 0.008 0.9 ± 0.2 High z Trim 9 ± 1 -0.8 ± 0.7 3.9 ± 0.4 -4.8 ± 0.5 -1.93 ± 0.04 1.0 ± 0.2 Hi z Trim Prior 5.0 ± 0.6 1.9 ± 0.7 2.9 ± 0.2 -4.3 ± 0.4 -0.935 ± 0.008 1.0 ± 0.2 Mean parameters from the Bayesian posterior functions. The top half of the table is broken down by survey, and the bottom half is one of the combined analyses of all data sets. κ_⋆ is the Euclidean flux counts normalization (see Equation <ref>). R_n is the specific rate of change of the numeric density of galaxies (see Equation <ref>). R_L is the long time decay constant in L_⋆, and n_0 is the initial luminosity index (see Equation <ref>). For κ_⋆, L_⋆, and n_0 the means are geometric means, in keeping with the values published in the posterior chains. a Parameter evaluated at z=0. lccccccc Luminosity Function Derived Parameters Surveyϕ_⋆a R_ϕ M_⋆a - 5 log_10 h z_⋆ ρ_L2.4a R_ρa z_ρ —10^-2h^3 ^-3 t_H^-1 AB mag — 10^8 L_2.4 ⊙^-3t_H^-1 — 6dFGS 0.86 ± 0.03 -5.2 ± 0.7 -20.91 ± 0.03 1.7 ± 0.3 1.81 ± 0.04 -7.6 ± 0.6 4.1 ± 0.6 SDSS 1.09 ± 0.01 -3.61 ± 0.09 -20.973 ± 0.008 1.0 ± 0.1 2.49 ± 0.01 -4.93 ± 0.07 2.8 ± 0.4 GAMA 1.51 ± 0.07 0.0 ± 0.2 -20.73 ± 0.04 0.43 ± 0.01 2.76 ± 0.05 -4.2 ± 0.2 0.43 ± 0.02 AGES 2.4 ± 0.2 0.4 ± 0.2 -20.53 ± 0.09 1.3 ± 0.2 3.4 ± 0.2 -2.8 ± 0.4 1.1 ± 0.1 /DEIMOS 1.2 ± 0.8 1 ± 2 -21.4 ± 0.5 1.3 ± 0.6 5 ± 1 -1 ± 1 0.5 ± 0.7 zCOSMOS 1.8 ± 0.5 1.5 ± 0.5 -20.7 ± 0.2 2.3 ± 0.3 3.1 ± 0.3 -2.0 ± 0.4 1.5 ± 0.3 Low z 0.814 ± 0.009 -7.43 ± 0.09 -20.994 ± 0.008 0.65 ± 0.09 2.013 ± 0.009 -8.09 ± 0.06 4.8 ± 0.6 High z 1.81 ± 0.08 -0.1 ± 0.2 -20.73 ± 0.02 2.4 ± 0.2 3.0 ± 0.1 -2.4 ± 0.2 2.5 ± 0.3 High z Prior 1.69 ± 0.08 0.2 ± 0.2 -20.90 ± 0.02 2.5 ± 0.3 3.8 ± 0.1 -1.9 ± 0.2 2.3 ± 0.3 Low z Trim 0.77 ± 0.01 -7.7 ± 0.2 -21.06 ± 0.01 -0.7 ± 0.2 1.92 ± 0.02 -7.1 ± 0.2 4.2 ± 0.6 High z Trim 1.1 ± 0.3 3 ± 1 -21.2 ± 0.1 2.4 ± 0.4 50 ± 40 -1.1 ± 0.6 0.8 ± 0.5 Hi z Trim Prior 1.7 ± 0.3 1.7 ± 0.7 -20.80 ± 0.08 2.1 ± 0.3 3.3 ± 0.4 -1.6 ± 0.5 1.2 ± 0.3 Bayesian mean values related to the luminosity function calculated from the posterior chains separately from the parameters in Table <ref>, with h=0.7. The means of ϕ_⋆ and ρ_L2.4 are geometric means. R_ϕ is the specific rate of change of ϕ_⋆ (assumed constant for all redshifts, see Equations 31 of LW17I). z_⋆ is the redshift at which the model predicts L_⋆ will peak (Equation 32 of LW17I). ρ_2.4 is the present day 2.4 luminosity density of galaxies (see Equation 41 of LW17I), R_ρ is its specific rate of change (Equation 42 of LW17I), and z_ρ is the redshift at which the model predicts j to have peaked (Equation 45 of LW17I). a Parameter evaluated at z=0. ll|llllll 0.95 Hi z Sample with Low z Prior Luminosity Function Bayesian Parameter Posterior Covariance Parameterσ δlnκ_⋆δ R_n t_Hδln L_⋆(0) δ R_L t_H δα δln n_0 δlnκ_⋆ 0.03029 1.000 0.9023 -0.5915 -0.1510 -0.04423 -0.2330 δ R_n t_H 0.2040 0.9023 1.000 -0.8257 -0.4389 0.06990 -0.02677 δln L_⋆(0) 0.01524 -0.5915 -0.8257 1.000 0.7285 -0.1243 -0.2852 δ R_L t_H 0.1210 -0.1510 -0.4389 0.7285 1.000 -0.009374 -0.8508 δα 0.004266 -0.04423 0.06990 -0.1243 -0.009374 1.000 -0.009308 δln n_0 0.1431 -0.2330 -0.02677 -0.2852 -0.8508 -0.009308 1.0000 The σ column contains the standard deviation of the parameters, and the remaining rows and columns comprise the matrix of correlation coefficients between the parameters, considered pairwise. All values given to four significant figures. The numerical comparisons discussed above provide a nice overview of the behavior of the different sets, but no work is complete without graphical comparisons of the fit models to binned estimators for the data.The comparisons of the unbinned evolving model ML fits to the entire samples with two binned estimators, 1/V_max and N^obs / N^mdl, can be found in Figures <ref>, <ref>, and <ref>.The first notable feature of the grid of plots in Figure <ref> is that the N^obs/N^mdl agrees better with the fit luminosity functions than the 1/V_max estimator.This is to be expected since N^obs/N^mdl is closer to a maximum likelihood estimator, and the fits were done using unbinned maximum likelihood.In particular, the N^obs/N^mdl outperforms 1/V_max where the approximation that the selection function is constant with redshift is a bad one.The difference is particularly stark for panel b, SDSS, where the shallow optical selection makes SED variability particularly relevant.The next obvious feature is the disagreement of the fitted faint end slope with the binned ones in panel d, AGES.There is a local maximum in the likelihood with a faint end slope closer to α = -1, but it is not a global maximum.There are two reasons this fails to be a global maximum: the low faint end completeness of AGES giving the less luminous galaxy bins bigger error bars, and a fluctuation in the data.These facts are more apparent in an examination of the first row of Figure <ref>, where the apparent disagreement vanishes.The final feature of note in the luminosity function plots is the upturns at the faint ends of panels b (SDSS) and c (GAMA).It is likely the same feature that caused <cit.> and <cit.> to use a double luminosity function to fit the data.While a double LF would provide better agreement to the data, it is unclear without a deeper examination of the data the extent to which the additional LF is modeling a fundamental feature of the universe (for example, the split between red and blue galaxies) or a cosmic variance fluctuation in the data.One example of an even bigger fluctuation can be seen in panel c of Figure 12 of LW18II.There is a significant over-density in the Sloan data near its peak at around z=0.75.The over-density that causes that bump goes by the name the Sloan Great Wall, discovered in <cit.>, and it is a good example of how large cosmic variance can get.It is also profitable to compare the observed redshift histograms against the predictions based on the luminosity function and selection function; plots containing such comparisons can be found in Figure <ref>.The power of this comparison is that, unlike the binned/unbinned LF comparisons in the earlier figures, the data in this figure need not be limited to sources with measuredfluxes.Thus, the comparison between the black histogram and darker lines is not entirely one of a fit with the data it was fit to, but the extrapolation of the LF and selection function with new data.The first noteworthy feature of the plots in Figure <ref> is that the High z Prior LF (blue dashed lines) provide more accurate extrapolations, overall, than the individual survey fits (solid red lines). That said, there are some features of the extrapolation in Figure <ref> that need explaining. First, the model over-predicts the number of galaxies observed at low z in Panel a, 6dFGS. The likely dominant culprit there is the bias against unresolved galaxies induced by the use of the 2MASS Extended Source Catalog (2MASS XSC) to produce the target list for 6dFGS, as evidenced by the dearth of sources with w1rchi2 < 3 relative to other surveys in Figure <ref>. Next, the high redshift tails of of Panels a–c also disagree with the model. Confusingly, the model under-predicts the number of galaxies in the high z tail of a, and over-predicts the tails of b and c. Because the high-z tail of the histograms is controlled primarily by the way the selection function limits sources to those with L>L_⋆, this is the same as saying that: one, there are more L>L_⋆ galaxies than the model predicts; and two, that those galaxies are redder than average in the optical for the r selection of SDSS and GAMA to remove them. This appears consistent with the textbook level knowledge that the galaxies found in clusters are redder and larger than the smaller and bluer field galaxies. Put simply, this is probably a limitation of the single SED single LF model used in this paper. We have not performed any quantitative investigations into whether this explanation is sufficient because to do so would be to use multiple luminosity functions with their own mean SEDs to build a more accurate spectro-luminosity functional and then fit that to the data, which is beyond the scope of this work.We have, however, investigated a number of factors that should contribute to the disagreement, but all of them are either the wrong magnitude or incapable of explaining both of the high z tails:* anyrelated selection effects (the extrapolated graphs remove all -based selection criteria),* contamination of the aperture photometry in the target selection catalogs causing sources that are too faint to be included (number of sources is too small, and makes the under-predictions worse), and* the spread of predictions consistent with the uncertainty in the model parameters (the 1-σ band is thinner than the lines in the high z regions).§.§ External ComparisonsIn order to make the Schechter parameters in other papers comparable to the ones measured in this one it was necessary to use the mean SED from this work to color correct their values of L_⋆ and, wherever possible, use the evolution measured in the other papers to bring them all to redshifts 0, 0.38, and 1.5.The literature parameters can be found in Table <ref> alongside the mean High z Prior parameters from this work.Further, the external works often have measurements of the LF in multiple filters; when that is the case, the observation filter with wavelength closest to the W1's 3.4 was used for the primary parameter comparisons.On the whole, this work's estimate of L_⋆ is lower than the literature and its estimate for ϕ_⋆ is higher, but not radically so, especially compared to the spread among the literature values.The values for L_⋆ at high redshift (z=1.5) have a much larger spread, making this epoch ripe for studies based on deeper imaging surveys.The spread in measured specific evolution rates, shown in Table <ref>, is considerably larger than the primary parameters.The uncertainties are not included in the table, but they're generally more than 0.1 t_H^-1 and less than 1 t_H^-1.The most directly comparable values, the ones at ^0.1z, [3.6], and 2.4, are all largely consistent.Most importantly, the specific rate of change in the density of galaxies, R_n, should be the same for all the different surveys in all bandpasses.That the spread is so large is likely attributable to a combination of selection effects, the inadequacy of a single Schechter function to describe all types of galaxies in all bandpasses, and the fact that R_n is one of the least well constrained parameters by the data.The final comparison is a graphical one of the models for the evolution of L_⋆ and ρ_L2.4 from this work to an empirical models from <cit.> and <cit.> in this work's Figure <ref>.The model from the review in <cit.> is for the evolution of the cosmic star formation rate density, ψ (Equation 15 there), and the the model from <cit.> is a 1-σ variability band in luminosity density evolution for theK filter (λ≈ 2.2) scaled using the mean SED (L_K / L_ν(2.4) = 1.199).The data used to produce the empirical model from <cit.> was, essentially, scaled luminosity densities measured in the far ultraviolet and far infrared.Most importantly, the measurements used to produce the empirical model for ψ spans the range of redshifts from 0 to 8, so their estimate would place the peak of the luminosity density somewhere between redshifts 1.3 and 2.5 based on data.The data on ρ_K evolution from <cit.> is primarily below redshift 2, and based on statistically interpolating measured values from the literature, and constructing the band from an ensemble of such interpolations.Considering that all of the data used in this work has a redshift below 1, and most of that less than 0.5, the crude evolution model used here does surprisingly well at locating the epoch where L_⋆ and ρ_L2.4 peak.Figure <ref> also shows that the peaks in L_⋆ and ρ_L are, possibly, too broad, but this is not surprising given the simplicity of the model and limits of the data used.lccc|ccc|ccc Comparison with Other IR Schechter LF Measurements Paper Obs Band α 10^-8 ρ_L2.4(0) ϕ_⋆(0) ϕ_⋆(0.38) ϕ_⋆(1.5) L_⋆(0) L_⋆(0.38) L_⋆(1.5) — — — h L_2.4 ⊙^-310^-2h^3 ^-310^10 L_2.4 ⊙ h^-2 <cit.> K -1.2±0.2 7±6 1.2±0.8 — — 5±2 — — <cit.> K_s -1.09±0.06 7.5±0.9 1.2±0.1 — — 4.5±0.2 — — <cit.>b K_s -1.2 ± 0.1 2±3 0.1±0.1 0.2 0.4 19±9 17 14 <cit.> K_s -0.86 ± 0.04 16±1 4.2±0.2 — — 4.1±0.2 — — <cit.> c1a -0.92 ± 0.04 5± 5 1±1 2 2 4.4±0.4 0.28 0.25 <cit.>b K -0.99±0.04 5.0±0.8 0.65±0.09 0.80 0.63 7.8±0.7 7.8 10.0 <cit.>b K -1.1±0.2 12±3 1.5±0.6 1.1 0.64 8±3 8 11 <cit.> c1a -1.12 ± 0.16 3.9±0.9 1.08±0.03 — — 3.3±0.6 5.1 18 <cit.> K -0.81±0.04 5.7±0.3 1.66±0.08 — — 3.7±0.1 5.3 14.9 <cit.> z -1.07±0.02 5.1±0.8 1.3±0.2 1.1 0.63 3.9±0.2 7.0 41 <cit.> K -1.16±0.04 7±2 0.7±0.1— — 9±2 — — This work W1 -1.050± 0.004 5.4 ± 0.2 1.69 ± 0.08 1.6 1.5 3.12 ± 0.05 5.6 9.9 Paper is the work from which the measurements came, in chronological order. `Obs Band' is the observation band used to calculate the luminosity function. ρ_L2.4(0) is the 2.4 luminosity density at z=0, ϕ_⋆(z) is the LF normalization at redshift z, and L_⋆(z) is the value of L_⋆ at redshift z. Uncertainties from this work do not include systematic uncertainties, and so are underestimated. The conversion factors used are:and aSpitzer/IRAC channel 1, λ=3.6. bParameters here are based on linear interpolation/extrapolation. lcc|cc|cc 0.68 Evolution Rate Comparisons Paper z_0Band ∂ln L_⋆ / ∂ t R_ϕ R_n R_ρ t_H^-1 t_H^-1 t_H^-1 t_H^-1 <cit.> 0.1 ^0.1u -3.4 -2.6 -2.3 -5.9 ^0.1g -1.6 -0.3 -0.1 -1.9 ^0.1r -1.3 -0.1 -0.2 -1.4 ^0.1i -1.3 -0.5 -0.5 -1.8 ^0.1z -0.6 -1.8 -1.9 -2.4 <cit.> 0.25 [3.6] -0.8 0a -0.1 -0.8 [4.5] -0.7 0a 0.0 -0.7 <cit.> 0.3 ^0.1r -1.0 0.4 0.4 -0.5 <cit.> 0.13 ^0.1u -4.8 6.5 6.1 1.8 ^0.1g -2.2 1.2 0.9 -1.1 ^0.1r -0.5 -1.4 -1.5 -1.9 ^0.1i -1.1 0.0 -0.1 -1.2 ^0.1z -1.3 0.4 0.3 -0.9 <cit.> 0.2 ^0.1r -0.7 -0.7 -0.9 -1.4 This work 0.38 2.4 -1.8 0.2 0.1 -1.7 Paper is the work from which the measurements came, in chronological order. z_0 is the mean or median redshift of the data in the work, and is the redshift at which the parameters were evaluated to make this table. Band is the passband in which the luminosity function was measured. ∂ln L_⋆ / ∂ t is the specific rate of change of L_⋆, measured in units of inverse Hubble times. R_ϕ is the specific rate of change of ϕ_⋆ (see Equations 31 of LW17I), R_n is the specific rate of change in the number density of galaxies (see Equation <ref>), and R_ρ is the specific rate of change in the 2.4 luminosity density (see Equation 42 of LW17I) aThis parameter was set to this value, not measured. § DISCUSSION The consistency of the results in this work with the literature shows that, while the goal of increasing the statistical accuracy of the LF measurements has been met, that increased accuracy has not, yet, uncovered any new facets of galaxy evolution. The internal comparisons show that there is still room for improvement in the techniques used here.In particular, closer attention paid to ensuring that the full flux of resolved galaxies is measured without contamination from foreground stars will permit a further large jump in sample size.Improvement in the performance of numerical integration of arbitrary high dimensional Gaussian functions over rectangular regions would make spectro-luminosity functional, Ψ, based techniques able to analyze samples constructed using the sort of complicated color selection done for DEEP2 <cit.>, the SDSS luminous red galaxy sample <cit.>, and the SDSS quasar sample <cit.>. Improvements in the form of Ψ that could allow it to fit the data more closely need a little more thought to the overall approach of the analysis before implementation.The primary approach used in the measurement of LFs is to classify galaxies into types and measure separate LFs for each type.It is not obvious, but the improvements suggested in LW17I, where the total Ψ is written as a sum over components, falls into the classification category.The way that the sum over components with Ψ becomes a classification scheme, effectively, is because the fluxes measured for each source will place it closest to one mean SED.If we define the square distance to the SED as the exponent inthen the majority of sources will most strongly affect the the LF parameters that correspond to the SED to which they are closest by that distance measure – with a few in a boundary region dividing their influence among the terms.Thus, especially if the mean SEDs are fixed, the model effectively classifies the sources by which term in the total spectro-luminosity functional the source has the greatest impact on.The more fluxes per source brought to bear in the analysis, the sharper that divide between source types is. The less common approach would be to, instead of classifying galaxies, analyzing their composition.Think of it as the difference between deciding whether a galaxy `is a' versus how much the galaxy `has a'.The advantage of the composition approach is that it addresses an ambiguity not dealt with directly by the model for estimating Ψ developed here: how many galaxies does each source represent?We know from images of local galaxies that the moderate to large galaxies have smaller galaxies in their halos (for example: the Large and Small Magellanic Clouds).As distance increases, the light from any satellite galaxies must, inevitably, be merged into the light from their primaries.The hallmark of a compositional approach is that it does not just assign probabilities to a source being in different classes, it divides the source's luminosity among them.So, for example, it could be possible to talk about dividing the Ψ into massive stars (say O, B, and A), intermediate stars (F and G), light stars (K, M, and lighter), stellar and supernova remnants, nebular emission, and AGN.The primary challenge would be to figure out what form the base luminosity function, Φ, of each of these should take.While it would seem that an increased richness of photometric data would also be required in order to analyze the composition of each galaxy, it isn't necessary to get a detailed analysis of every galaxy to get an accurate picture of the average composition of galaxies that is encoded in Ψ.One question that is answerable by improving the techniques developed here, and adding data for galaxies in the redshift range of 1 to 3 is: which peaked first, L_⋆ or ρ_L?The reason this question is of interest is because it encodes information about the comparative rate of star formation versus galaxy mergers; the former boosts both quantities, the latter only boosts L_⋆.We would, for example, expect L_⋆ to peak first in a universe where star formation continued in small galaxies after it and collisions slowed down for large ones.The converse would be the case if star formation was a relatively short epoch that cut off in most galaxies and the majority of individual galaxy luminosity growth was through accretion.The dataand simple evolution models used here cannot answer this question, though they hint that L_⋆ peaked first.Another question that will require even greater care to answer is: when did (or will) the number density of galaxies peak?The question is equivalent to asking when R_n=0.Though that is an equation that can, in principle, be solved using the parameters in this work, the spread in values produced are so large as to make the answer meaningless.It would take an evolutionary model with greater physical fidelity combined with more data to provide an answer worth examining.§ CONCLUSIONThe combination of the six different redshift surveys described in LW18II, made possible by the analysis techniques derived in LW17I, has produced measurements of the Schechter LF parameters that are comparable to the literature in terms of statistical precision for ϕ_⋆, and a marked improvement for L_⋆ and α.The parameters describing the evolution of L_⋆ and ϕ_⋆, R_L and R_ϕ, are less well constrained, but still comparable to the literature.Improving the constraints will require refinements in the process from end to end.The photometry of bright, resolved and marginally resolved, galaxies requires improvements that bring them in line with the quality of our photometry for point source.In the ideal case, a photometric survey with sufficient sensitivity to resolve the wings of the Airy profile produced by even the largest galaxies, and the software tools needed to remove foreground contamination from stars, would guarantee that close enough to all of the light in the galaxy has been directly observed to measure accurate luminosities and colors for each whole galaxy. The accuracy of the spectro-luminosity functional, Ψ[L_ν](z), can be improved in the straightforward way: writing it as a sum of spectro-luminosity functionals, each with its own LF and Gaussian .It may also be possible to improve the performance of Ψ by a rethinking how the LF is defined – instead of classifying galaxies into mutually exclusive categories, split into constituent parts with their own luminosity.The upside of a such an approach is that it naturally handles cases where multiple unresolved galaxies are contained in the same object.Perhaps more important than increasing the fidelity of Ψ is adding a model of the effective radius of galaxies so that surface brightness limits on galaxy selection can be modeled.Likewise, re-deriving an approximation of the estimator for the likelihood of observing an entire catalog so that it includes the effect of galaxy environment.Adding the effect of environment, for example using the two point function (ξ(r)), in the likelihood of the catalog is the most natural way to introduce cosmic variance to the process.Finally, measuring fluxes and redshifts for even fainter galaxies using instruments like the Multi-Object Spectrometer for Infra-Red Exploration (MOSFIRE) on the Keck II telescope, and the James Webb Space Telescope (JWST), will allow for explorations of the evolution of the faint end slope of galaxies, better constrain the evolution of ϕ_⋆ and L_⋆ with more high redshift data, and even, potentially, find a downturn in the LF at faint luminosity where galaxies and star clusters overlap as gas accreting gravitationally bound systems.We would like to thank theteam.This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, and NEOWISE, which is a project of the Jet Propulsion Laboratory/California Institute of Technology. WISE and NEOWISE are funded by the National Aeronautics and Space Administration.We would like to thank the SDSS team.Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/.SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.We would like to thank the GAMA team.GAMA is a joint European-Australasian project based around a spectroscopic campaign using the Anglo-Australian Telescope. The GAMA input catalogue is based on data taken from the Sloan Digital Sky Survey and the UKIRT Infrared Deep Sky Survey. Complementary imaging of the GAMA regions is being obtained by a number of independent survey programs including GALEX MIS, VST KiDS, VISTA VIKING, WISE, Herschel-ATLAS, GMRT and ASKAP providing UV to radio coverage. GAMA is funded by the STFC (UK), the ARC (Australia), the AAO, and the participating institutions. The GAMA website is http://www.gama-survey.org/ .Based on observations made with ESO Telescopes at the La Silla or Paranal Observatories under programme ID 175.A-0839.We would like to thank the 2MASS team.This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.We would like to thank the MAST team.Some/all of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX13AC07G and by other grants and contracts.We would like to thank the NDWFS team.This work made use of images and/or data products provided by the NOAO Deep Wide-Field Survey (Jannuzi and Dey 1999; Jannuzi et al. 2005; Dey et al. 2005), which is supported by the National Optical Astronomy Observatory (NOAO). NOAO is operated by AURA, Inc., under a cooperative agreement with the National Science Foundation.We would like to thank the IPAC team.This research has made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.We would like to thank the GALEX team.Based on observations made with the NASA Galaxy Evolution Explorer.GALEX is operated for NASA by the California Institute of Technology under NASA contract NAS5-98034.We would also like to thank the teams behind 6dFGS, AGES, zCOSMOS, SDWFS, and COSMOS.RJA was supported by FONDECYT grant number 1151408.

http://arxiv.org/abs/1702.07829v2

CDER]Supervised Learning of Labeled Pointcloud Differences via Cover-Tree Entropy Reduction A. SmithMathematics, Statistics, and Computer ScienceUniversity of Wisconsin-StoutMenomonie, WI, 54751, USAGeometric Data Analytics, Inc. Durham, NC, 27707, USA smithabr@uwstout.edu P. Bendich Mathematics Duke University Durham, NC, 27708, USA Geometric Data Analytics, Inc. Durham, NC, 27707, USA United States bendich@math.duke.edu J. Harer Mathematics, Electrical and Computer EngineeringDuke UniversityDurham, NC, 27707, USA Geometric Data Analytics, Inc. Durham, NC, 27707, USA harer@math.duke.edu A. PielochMathematicsColumbia UniversityNew York, NY, 10027, USA pieloch@math.columbia.edu J. Hineman Geometric Data Analytics, Inc.Durham, North Carolina, 27707, USA jay.hineman@geomdata.comWe introduce a new algorithm, called CDER, for supervised machine learning that merges the multi-scale geometric properties of Cover Trees with the information-theoretic properties of entropy.CDER applies to a training set oflabeled pointclouds embedded in a common Euclidean space.If typical pointclouds corresponding to distinct labels tend to differ at any scale in any sub-region, CDER can identify these differences in linear time, creating a set of distributional coordinates which act as a feature extraction mechanism for supervised learning. We describe theoretical properties and implementation details of CDER, and illustrate its benefits on several synthetic examples.MSC 62H30 MSC 60G55 [ Abraham Smith Paul Bendich John Harer Alex Pieloch Jay Hineman October 24, 2016 ================================================================== § ACKNOWLEDGMENTS All authors partially supported by OSD contract number N00024-13-D6400 via JHU-APL subcontract number 131753.(Distribution Statement A - Approved for public release; distribution is unlimited.) Bendich and Harer were also partially supported by NSF award BIGDATA 1444791.Harer was also partially supported by the DARPA MoDyL program, number HR0011-16-2-0033. Pieloch was supported by the National Science Foundation Graduate Student Fellowship Program through grant DGE 16-44869. We thank Christopher J. Tralie for many helpful insights about cover trees, and we thank David Porter and Michael Williams at JHU-APL for many motivational discussions.Our code is available at <https://github.com/geomdata/gda-public/> with documentation at <https://geomdata.github.io/gda-public/>§ OVERVIEW AND ASSUMPTIONSWe propose a new supervised machine-learning method for classification, where each object to be classified is a weighted pointcloud, and the classification task is to learn which of a finite set of labels should be applied to the pointcloud. The method is fast, data-driven, multi-scale, and requires no tuning.Additionally, its details are transparently geometric; it does not suffer from the “black box” interpretation difficulties that arise in many machine-learning methods.We call it Cover-tree Differencing via Entropy Reduction [CDER, pronounced “cedar”].A high-level sketch of CDER is as follows. We construct a partial cover tree <cit.> on the union of all labeled pointclouds that are given in some training set, and we search the cover tree for convex regions that are likely to be local minima of entropy <cit.>.For each such region, we build distributional coordinates from the dominant labels' data.An ordered sequence of these distributional coordinates allows us to determine the likelihood of each label for an unlabeled test pointcloud.Section <ref> explains the notion of distributional coordinates defined on pointclouds as a front-end for supervised learning. Section <ref> details the cover-tree algorithm and our enhancements to it.Section <ref> explains our simple approach to entropy minimization in the labeled pointcloud context. Section <ref> gives a formal description of CDER. Section <ref> illustrates CDER with several synthetic examples.The remainder of this section establishes context and notation for this problem. Fix a Euclidean[See the Discussion section for further thoughts on this requirement.] space ℝ^D.A pointcloud is a finite[In this document, |X| indicates the cardinality of a finite set X, whereas v indicates a norm.] set X = { x_1, x_2, …, x_|X|}⊂ℝ^D.We allow weighted pointclouds, where each x ∈ X has a positive weight w(x); the utility of weights is explained in Section <ref>.A cloud collection is a finite set of pointclouds𝒳 = { X_1, X_2, …, X_N},with N=|𝒳|.Let 𝒳= ⋃_i=1^NX_i ⊂ M denote the union of all pointclouds in a cloud collection, which is itself a pointcloud with weights and labels of each point x ∈𝒳 inherited from the respective X_ix.For our supervised learning task, the training data is a cloud collection 𝒳, where each pointcloud is labeled from a finite set of labels, Λ.For the sake of discussion and visualization, we usually interpret Λ as a set of colors, but for mathematical purposes, Λ = { 1, …, L} where L = |Λ|.Let λ: 𝒳→Λ denote the label function, and we also use λ to indicate a generic label in Λ.It is acceptable that the pointclouds X_i have unequal sizes across X_i ∈𝒳.It is also acceptable that the labeled sub-collections 𝒳|_λ = { X_i  : λ(X_i) = λ} = λ^-1(λ) have unequal sizes across labels λ∈Λ. Pointwise weights can be assigned to compensate for these sizes, as in Section <ref>. Our only structural hypothesis is that, for each labeled sub-collection, the pointclouds are sampled from an underlying density function—or from several density functions as chosen by a random process—on ℝ^D.For each label λ, let f_λ denote this density function.We aim to efficiently find regions in ℝ^D where one label has greater density than the other labels.That is, we seek convex regions Ω⊂ℝ^D that are “characteristic” of a particular label λ because f_λ is unusually prominent there, while ignoring regions where the various density functions are similar.See Figure <ref>.We analyze the relative density of labels over regions Ω using the information-theoretic notion of entropy (see Section <ref>), locating subregions of low entropy. Because of the use of cover trees, our method emphasizes regions Ω that are very large, very dense, or small-yet-remote (see Section <ref>). For each of these regions Ω, we construct a function g that approximates f_λ near Ω; such a g is an example of what we call a distributional coordinate (see Section <ref>).§ WEIGHTSTypically, pointclouds are experimental samples that do not come with pre-assigned weights, and we assign weights as follows: We have the prior assumption that each label λ∈Λ is equally likely among the test data.[See the discussion section for more general thoughts about this assumption.] We give the training set total weight 1.That is, each label is alloted total weight 1/L, where L = |Λ|.Moreover, we make the assumption that each sample X_i is equally representative of its underlying density function f_λ_i, regardless of |X_i|.That is, each pointcloud X_i is alloted total weight 1/L N_i, where N_i is the cardinality of X|_λ_i, the number of training pointclouds with label λ_i.Finally, we assume all points x ∈ X_i have equal weight, so w(x) = 1/L N_i |X_i|.Let 𝒳 denote the labeled and weighted pointcloud given by the union of all X_i with these labels and weights.For example, suppose that the training set consists of X_1, X_2, X_3 where |X_1| = 4, |X_2|=10, and |X_3|=3 with labels red, blue, and red, respectively.Suppose we want 𝒳 to have unit weight. Then each x ∈ X_1 has weight w(x) = 1/2·2·4 = 1/16, each x ∈ X_2 has weight w(x)= 1/2·1·10 = 1/20, and each x ∈ X_3 has weight w(x) = 1/2·2·3= 1/12.These weights are chosen to model relative likelihoods in the following way: If each label λ has an underlying density function f_λ, and each X_i with label λ is a sample from f_λ, then ∫_Ω f_λ should be proportional to ∑_X_i ∈𝒳|_λ∑_x ∈ X_i∩Ω w(x) for any measurable set Ω. § DISTRIBUTIONAL COORDINATES Given a cloud collection 𝒳 as above, we would like to map each pointcloud X to a vector v(X) ∈ℝ^k for some fixed dimension k. While this is an interesting topic in its own right, we are mainly concerned with viewing this as a feature extraction method: that is, the vectors v_i = v(X_i) ∈ℝ^k and corresponding labels λ_i = λ(X_i) will be used as input for any supervised learning algorithm that requires collections of vectors in a common Euclidean space, along with class labels.One could imagine a supervised learning approach to labeled pointclouds that does not perform feature extraction; for example, by first defining a (non-Euclidean) metric between pointclouds, and then applying a nearest-neighbor or clustering method.However, most such metrics are unstable to noise (such as Hausdorff distance) or are slower-than-linear to compute (such as various Kullback-Leibler Divergence <cit.> or the Wasserstein metric <cit.>).This section frames the problem of feature extraction in the common language of distributional coordinates. We start with a definition of this intuitive concept, and then describe the advantages and disadvantages of some standard examples. The approach we advocate comes in the next section.Let g be an integrable function on Euclidean space.For any pointcloud X = {x_1, …, x_|X|}, with pointwise weights w_i = w(x_i), define∫_X g = w_1 g(x_1) + … + w_|X|g(x_|X|).We refer to g as a distributional coordinate.Any ordered set (g_1, …, g_k) of distributional coordinates transforms X into a vector 𝐯_X = (∫_X g_1, …, ∫_X g_k) ∈ℝ^k. It is computationally important to keep the dimension k—or at least the intrinsic dimension of {𝐯_X :  X ∈𝒳}—as small as possible, as we hope to minimize the negative effect of the curse of dimensionality on the supervised learning problem.We use the cloud collection 𝒳 shown on the bottom of Figure <ref> as a notional working example in the Euclidean plane. There are two class labels, “magenta” and “green.” We imagine that each magenta pointcloud (a typical example on the top-left of the figure) contains a large number of points sampled from a large central blob and a few points sampled from much smaller blobs along the horizontal axis. Each green pointcloud (a typical example on the top-right of the figure) contains a large number of points sampled from the same large central blob and a few points sampled from smaller, different blobs along the horizontal axis.In a typical supervised learning context, 𝒳 would be split (say 80:20) into training and testing sets, and we would build a set of distributional coordinates from the labeled training set which could then inferthe correct color labels on the testing set. §.§ BinningA simple approach would be to let each g_i be the indicator function on some region R_i; that is, g_i takes the value 1 on some region R_i and 0 elsewhere. Then ∫_X g_i counts the total weight of the points X ∩ R_i. Assuming the regions R_1, … R_k cover the domain with overlap only on boundaries (for example, the R_i could be the squares shown in Figure <ref>) this is what is commonly referred to as binning, and it has been used (for example, <cit.>) successfully for supervised-learning on labeled pointclouds. On the other hand, binning is obviously unstable: consider a point x ∈ X that lies very close to the boundary between two adjacent bins. It also can suffer from the curse of dimensionality: a larger number of bins than are shown in Figure <ref> would be needed to distinguish the class labels.§.§ Gaussian MixturesTo address instability, one could define each g_i to be a Gaussian with mean μ_i and covariance matrix Σ_i. The obvious questions are: how many Gaussians (modes) should be used, what are their means, and what are their covariances?A data-driven approach should be used to fit a Gaussian Mixture Model (<cit.>, VI.8) to 𝒳.It is apparent to the human eye that nine Gaussians (1 large green and magenta, 4 small magenta, and 4 small green) model the example in Figure <ref>.That is, we could “by hand” define a good set of distributional coordinates to establish a feature vector 𝐯 = (g_1, …, g_9) : 𝒳→ℝ^9.Moreover, for the purposes of classification, the central “large” distributional coordinate g_1 is useless.And perhaps the two innermost “small” Gaussians are less useful, as the remaining six distributional coordinates are sufficient for accurate classification.Our goal is to replicate this observation: we construct Gaussian modes (g_1, …, g_k) in a data-driven manner from 𝒳 that are appropriate for use as labeled distributional coordinates for the label classification problem.We now proceed to the computational details. § COVER TREESCover trees were originally conceived <cit.> as a way to accelerate the nearest-neighbors problem.More recently, they have been used in dictionary learning <cit.> and for speeding up the analysis of the topology of pointclouds <cit.>, to pick just a few applications. This section gives basic definitions for cover trees as well as a fast algorithm for their construction; our account is far more detailed than usually appears in the literature and is meant to accompany our publicly available code. Sections <ref> and <ref> use cover trees to define a set of label-driven distributional coordinates on a cloud collection with class labels.For any finite pointcloud[For the moment we forget that 𝒳 is a union of pointclouds and simply treat it as a single pointcloud to be studied.] 𝒳⊂ M, consider a filtrationA_0 ⊂⋯⊂ A_ℓ⊂ A_ℓ+1⊂⋯⊂𝒳.An element a ∈ A_ℓ is called an adult[To reduce confusion (hopefully) by maintaining a consistent metaphor, we diverge from prior works' notation in two places.Our adult is called a center elsewhere.Our guardian is usually called a parent.We avoid the word parent because there are many distinct tree relations embedded in this algorithm.] at level ℓ.The set A_ℓ∖ A_ℓ-1 is the cohort at ℓ.A cover tree builds a filtration by covering 𝒳 with balls of smaller and smaller radius centered at the points in A_ℓ. See Figure <ref>.Specifically, a cover tree is a filtration of 𝒳 with the following additional properties: * |A_0|=1.Let A_0 = {a_0};* There is a radius r_0 such that 𝒳⊂ B(a_0, r_0);* There is a real number 0< θ < 1, called the shrinkage ratio such that, for every ℓ, 𝒳⊂∪_a_i ∈ A_ℓ B(a_i, r_ℓ) where r_ℓ = θ^ℓ r_0.* For each ℓ, if a_i, a_j ∈ A_ℓ, then d(a_i, a_j) > r_ℓ.* For each ℓ, each point x ∈𝒳 is assigned to a guardian a_i ∈ A_ℓ such that x lies in the ball B(a_i, r_ℓ); we say x is a child of a_i.Each a_i ∈ A_ℓ is its own guardian and its own child.* There is a tree structure on the (level, adult) pairs of the filtration (ℓ, a_i), where there is a tree relation (ℓ, a_i) → (ℓ+1, a_k) if a_k was a child of a_i at level ℓ.We say a_k is a successor of a_i, and a_i is a predecessor of a_k. Note that (ℓ, a_i) → (ℓ+1, a_i) for all a_i ∈ A_ℓ.§.§ VocabularyCover trees can be generated using a fast algorithm relying on the notion of friends, first introduced in <cit.>.The rough algorithm is familiar to many practitioners, but is not widely published.We enhance this algorithm by giving each point a weight and a label. We describe our method here because the details of the construction are crucial to the supervised-learning task of this article.Extending the maturation/reproduction metaphor of adults, children, and guardians above, a child x with guardian a_i at level ℓ is called a teen if 1/2 r_ℓ < d(a_i, x), and it is called a youngin if d(a_i, x) ≤1/2 r_ℓ.See Figure <ref>. When changing from level ℓ to level ℓ+1, the radius of each ball shrinks to r_ℓ+1 = θ r_ℓ.Children farther than r_ℓ+1 from their guardians become orphans.We must decide whether these orphans should be adopted by other adults at level ℓ+1, or if the orphans should be emancipated as new adults at level ℓ+1. That is, the newly emancipated adults at level ℓ+1 comprise the cohort at level ℓ+1. Our algorithm uses several notions of friends, set by the following bounds at level ℓ:T_1(ℓ)= (2 + θ)r_ℓ,T_2(ℓ)= (2 + 2θ)r_ℓ,T_3(ℓ)= 2( 1 + θ + θ^2 + ⋯ ) r_ℓ = 2/1-θr_ℓ.For each k=1,2,3, we say that adults a_i and a_j at level ℓ are type-k friends if d (a_i, a_j) ≤ T_k(ℓ).It is easy to verify the following recursion relations:T_1(ℓ) ≤ T_2(ℓ) ≤ T_3(ℓ),T_1(ℓ+1) ≤ T_1(ℓ),T_2(ℓ+1) ≤ T_2(ℓ),T_3(ℓ+1) ≤ T_3(ℓ).Here are the essential facts about friends, which are immediate consequences of the triangle inequality:T1 If x is a child of a_i at level ℓ that is orphaned at level ℓ+1, then x can be adopted only by either the type-1 friends of a_i at level ℓ or the newly-emancipated children at level ℓ+1 of those type-1 friends.See Figure <ref>. T2 If we choose to re-assign children to the nearest adult, then teen x with guardian a_k can be adopted by adult a_h at level ℓ+1, where the level-ℓ predecessors of a_k and a_h are type-2 friends. See Figure <ref>. T3 If a_k and a_h are type-3 friends at level ℓ+1, then their predecessors a_i and a_j are type-3 friends at level ℓ. See Figure <ref>.Suppose a_k is an adult at level ℓ+1 with predecessor a_i. If x has guardian a_k at level ℓ+1 and guardian of a_j at level ℓ, then a_j is a type-1 friend of a_i, and d(a_k, a_j) ≤ r_ℓ + r_ℓ+1= 1/2 T_2(ℓ).See Figure <ref>. Based on this scenario, we say a_j ∈ A_ℓ is an elder of a_k ∈ A_ℓ+1 if d(a_k, a_j) ≤1/2T_2(ℓ). Moreover, if a_i = a_k is its own predecessor, then a_j = a_i = a_k is its only elder, so the notion of elder is nontrivial only if a_k belongs to cohort ℓ. For the purpose of computing elders, there are three particularly good choices of θ which require no additional computation. If θ = 1/2(√(5)-1), then 1/2 T_2(ℓ) = T_1(ℓ+1). If θ = 1/2, then 1/2 T_2(ℓ) = T_2(ℓ+1). If θ = √(2)-1, then 1/2 T_2(ℓ) = T_3(ℓ+1). In these cases, the elders of a_k are the friends (of the respective type) a_j that have cohort ℓ or less.See Figure <ref>.These facts allow the following algorithm. §.§ AlgorithmAt level 0, the cover tree has a single adult, a_0.All points are its children.Its only friend is itself.Every x ∈𝒳 has some weight and some label, which we assume as given input.For each label λ, compute the total weight and weighted mean of all children of a_0 with that label.Suppose at level ℓ, the friends, children, per-label weights, and per-label means are known for all adults a_i ∈ A_ℓ.We compute these for level ℓ+1 as follows:* Advance. Shrink the radius by a factor of θ to r_ℓ+1 = r_ℓθ. Every adult a_i ∈ A_ℓ remains an adult a_i ∈ A_ℓ+1.Mark a_i as its own elder. * Orphan. For each a_i ∈ A_ℓ, find its orphaned children: those x such that d(a_i, x) > r_ℓ+1.Sort the orphans so that the first orphan is the one nearest the per-label mean for the label of greatest weight, the second orphan is the one nearest the per-label mean of the label of second-greatest weight, and so on, via a two-dimensional argsort without repetition.The reason for this sorting will become apparent in Section <ref>. * Adopt or emancipate. For each a_i ∈ A_ℓ, consider its orphans in order.For each x orphaned by a_i, check whether x can be adopted by another adult using the type-1 friends of a_i: * If so, the adoptive guardian will be a adult a_h such that d(a_h, x) ≤ r_ℓ+1.The predecessor a_j of a_h has d(a_j, a_i) ≤ T_1(l). * If not, then x is a new adult at level ℓ+1.Its only child is itself.Its friends are initialized to be itself and its predecessor a_i (but these will be updated in the Befriending stage).Note the set A_ℓ+1 is ordered by cohort, and within a cohort is ordered by predecessor, and within a predecessor block is ordered by proximity to the predecessor's per-label means.* Exchange Teens.[This step is optional for general cover tree, but it is particularly useful for the computation of elders, which are needed in Section <ref>.] If x has guardian a_i, then check whether it is closer to some other guardian using type-2 friends.That is, find a_h = _A_ℓ+1 d(x, ·). The predecessor a_j of a_h has d(a_j, a_i) ≤ T_2(ℓ).Note that a_h ≠ a_i implies d(a_h, x) > 1/2 r_ℓ+1, meaning that only teens (not youngins) can be exchanged. * Befriend: Now, all (ℓ+1)-adults and their children are known.We need to find new friends using the recursion relation and the predecessors. * A pair of ℓ+1 adults, a_k and a_h, are type-3 friends if their predecessors a_i and a_j are type-3 friends at level ℓ and d(a_k, a_h) ≤ T_3(ℓ+1). * A pair of ℓ+1 adults, a_k and a_h, are type-2 friends if they are type-3 friends and d(a_k, a_h) ≤ T_2(ℓ+1). * A pair of ℓ+1 adults, a_k and a_h, are type-1 friends if they are type-2 friends and d(a_k, a_h) ≤ T_1(ℓ+1). * For each newly emancipated adult a_k of cohort ℓ+1, find its elders by searching its friends for elements of cohort ℓ or less. * Weigh: For each adult a_k at level ℓ+1, compute the per-label weights and per-label weighted means for its children. Sort the labels by their weights from greatest to least.At level ℓ+1, the friends, children, per-label weights, and per-label means are all known. §.§ Algorithmic EfficiencySuppose n = |𝒳|, and the cover-tree filtration stops at level ℓ_max.We are interested in the storage and computational complexity of this algorithm in terms of n, for a fixed ambient dimension D.The desired output is the successor/predecessor tree and the children/guardian relationships at each level.Thus, storage of friend lists is the only issue that might lead us to another algorithm. The main question is: given an adult a_i at level ℓ, how many other adults are within T_3(ℓ) = 2/(1-θ)r_ℓ of a_i?Since no two adults at level ℓ are within r_ℓ of one another, this is related to the idea of “doubling dimension.” Define 𝔻(D, α) to be the maximum number of points in ℝ^D that can be placed in the ball of radius α, including the center, so that no two points have distance ≤ 1.This number grows with the ambient dimension D.In Euclidean space, it is independent of scale; that is, the same number works for a ball of radius rα using points of distance at least r.The number can be bounded by the number of simplices with edge length at least 1 that can be put into a ball of radius α.Since these simplices have a minimum volume √(D+1)/D!2^D/2, we get a bound like:𝔻≤(B_α)D! 2^D/2/√(D+1)∼D! (α)^D (2 π)^D/2/(D/2)!√(D+1)∼ O(D! 2^D α^D) .Setting α = 2/1-θ, we get an upper bound on the expected size of the friends list, which is a constant independent of the level ℓ. Thus, the total storage size is O(n); in particular, it is at most 3 𝔻n, where the 3 accounts for the lists of type-1, type-2, and type-3 friends.Setting α = 1/θ, we get an upper bound on the expected number of successors of an adult from level ℓ to ℓ+1, which is a constant independent of the level ℓ.The size of A_0 is |A_0|=1. The maximum expected size of A_1 is 𝔻.The maximum expected size of A_2 is 𝔻^2.Generally, the maximum expected size of A_ℓ is 𝔻^ℓ.Moreover, the larger the size of each cohort A_ℓ­ A_ℓ-1, the shallower the cover tree.That is, ℓ_max is expected to be of the order log_𝔻(n).For each adult a_i ∈ A_ℓ, we must: * Sum and sort the weights of its children, by label.The expected number of children is n/𝔻^D.Sorting the labels is O(L log(L)) for L labels. * Compute the distance (an O(D) operation) to each of its roughly θ^ℓ n children. These distances yield the orphans and teens.The expected number of orphans is (1-θ^D)n/𝔻^ℓ, and the expected number of teens is ( 1 - (1/2)^D)n/𝔻^ℓ. * For each orphan, determine whether it should be adopted or liberated.Compute the distance (an O(D) operation) to the T_1 friends or successors of the T_1 friends of a_i.This set has expected size 𝔻^2, which is constant in n. * For each teen, determine whether it should be exchanged.Compute the distance (an O(D) operation) to the T_2 friends or successors of the T_2 friends of a_i.This set has expected size 𝔻^2, which is constant in n. * For each successor—a list of size 𝔻—update its friends by computing the distance (an O(D) operation) to the successors of the T_3 friends of a_i.This is expected to be 𝔻^2 distance computations. For each adult at level ℓ, this is roughly n/𝔻^ℓ operations, albeit with an additional penalty from D.Level ℓ has at most 𝔻^ℓ adults, in which case the total number of computations is roughly∑_ℓ = 0^ℓ_max𝔻^ℓn/𝔻^ℓ O(𝔻^2) ∼ O(n log_𝔻n)O(𝔻^2). It is interesting that the rapid growth from the ambient dimension D is counteracted by the logarithm term. In our experience with applications, this is effectively linear in n, as the sum terminates earlier than this estimate. This is because the estimate of ℓ_max by log_𝔻(n) is typically irrelevant; the maximum level ℓ_max of the cover tree is actually controlled by the ratio of the maximum to the minimum distances in the pointcloud.In applications, these are typically within several orders of magnitude, so with θ = 1/2, we expect ℓ_max∼ 14. Moreover, in our supervised-learning application in Section <ref>, we abort the cover tree much earlier, once the low-entropy regions have been found.§ ENTROPY REDUCTIONFor each label λ, let 𝒳|_λ denote the (weighted) points in 𝒳 that have label λ.For any convex, compact Ω, let Ω_λ = Ω∩𝒳|_λ.The total weight in Ω of each label is w_λ = ∑{ w(x)  :  x ∈Ω_λ}, which by Section <ref> corresponds to the relative likelihood of a point with label λ being sampled from Ω. Let W = ∑_λ w_λ.To quantify the relative prominence of the various labels, we use the information-theoretic notion of entropy. Entropy is characterized by several nice properties, detailed in <cit.>. The formula isS(Ω)= - ∑_λw_λ/Wlog_L( w_λ/W)= -1/W∑_λ w_λlog_L( w_λ) + log_L(W). If all weights are roughly equal, then entropy is 1.If one label has high weight and the others have weight 0, then the entropy is 0.Generally, entropy decreases as a smaller subset of labels becomes more likely.Thus, low entropy is a good indicator that the region Ω is particularly prominent for a small subset of labels. The question remains: how can we locate low-entropy regions?Our search for low-entropy regions is motivated by the following elementary theorems; however, we do not insist that these strict regularity hypotheses hold for any given application. Fix a function f:ℝ^D →ℝ that is continuous on a compact, convex set Ω. Fix a radius r>0.Let μ_r(x) denote the centroid of f on the compact, convex set Ω∩ B_r(x) (the closed ball). There exists x ∈Ω such that μ_r(x) = x.Proof: Note that μ_r(x) varies continuously with x, and apply Brouwer's fixed-point theorem (<cit.>, II.22). Suppose that f:ℝ^D→ℝ is an analytic function with local max at x. For any ϵ >0, there exists a radius r such that the centroid μ_r(x) of f on B_r(x) has d(μ_r(x) ,x ) < ϵ.Proof: Expand f as a power series around x, and approximate f by truncating at the second-order term.The level sets are concentric ellipsoids about x, as given by the positive-definite symmetric matrix f”. Suppose f and f̃ are positive, continuous functions defined on a compact, convex region Ω⊂ℝ^D. Suppose that f achieves a local max f(x) = M on Ω and that f̃(x) ≤ε on Ω, with ε < M.Then, there is a radius r such that the average value of f on Ω∩ B_s(x) is greater than the average value of f̃ on Ω∩ B_s(x) for all s ≤ r.Proof: The mean value theorem for integrals.Putting these together, we try to find a region Ω and a ball of radius r_ℓ in that region whose center is near the mean of a particular label. We ask whether the entropy of that ball is non-decreasing with radius;that is, does the entropy become lower (or remain the same) when the ball shrinks? If true, this is consistent with the hypothesis that a local maximum of a particular label occurs near the center of that ball, while other labels remain roughly constant.If false, then we subdivide the region and search around the boundary of the original ball for other smaller regions of low entropy.Specifically, the compact, convex regions corresponds to the children of an adult a_i ∈ A_ℓ in the cover tree, and the sub-balls correspond to the children of successors of a_i at level ℓ+1 in the cover tree. This is the heart of the CDER algorithm in Section <ref>. § COVER-TREE DIFFERENCING VIA ENTROPY REDUCTIONThe input data is a ratio θ and a training set of pointclouds X_1, X_2, …, X_N with labels λ_1, λ_2, …, λ_N, respectively.Set weights as in Section <ref>.§.§ Region SelectionHere we describe the heart of the CDER algorithm, wherein we search the adults of the cover tree as a data-driven method to locate regions on which we want to build distributional coordinates.The construction and weighting of those distributional coordinates occurs in Section <ref>.For each level of the cover tree, define a subset Â_ℓ⊂ A_ℓ of adults that are still potential candidates for distributional coordinates. Set Â_0 = A_0 = { a_0}.Once Â_ℓ = {}, we .The cover tree of 𝒳 is constructed level-by-level as in Section <ref>.At each level ℓ of the cover tree (until ), we perform the following computation:For each adult a_i ∈ A_ℓ, let β_ℓ(i) denote the set of children of a_i at level ℓ. Let α_ℓ(i) = β_ℓ+1(i) denote the set of children of a_i at level ℓ+1.Let γ_ℓ(i) denote the union of the children (at level ℓ-1) of the elders of a_i. (If a_i belongs to cohort ℓ-1 or less, then γ_ℓ(i) = β_ℓ-1(i).) Note that α_ℓ(i) ⊂β_ℓ(i) ⊂γ_ℓ(i); see Figure <ref>.Then, we decide what to do with a_i:* if Â_ℓ = ∅, then .Nothing more is needed.* for each adult a_i ∈Â_ℓ: * if a_i has only one child (itself) at level ℓ+1, then .This region is useless.* elif a_i has only one child (itself) at level ℓ, then .This region is useless.* elif S(α) ≤ S(β) ≤ S(γ) < 1, thena distributional coordinate using the children of a_i of each of the dominant labels.[A label is dominant if its points represent more than 1/L of the total weight of the children.For a two-label task, this is simply the majority label.]* elif S(γ) ≤ S(α) ≤ S(β) < 1, thena_i to Â_ℓ+1.The smaller ball B(a_i, r_ℓ+1) may contain a lower-entropy region.* elif S(β) ≤ S(α) ≤ S(γ) < 1, thenthe successors of a_i—other than itself!—to Â_ℓ+1.The annulus B(a_i, r_ℓ) ∖ B(a_i, r_ℓ+1) may contain a lower-entropy region.* elif S(β) ≤ S(γ) ≤ S(α) < 1, thenthe successors of a_i—other than itself!—to Â_ℓ+1. The annulus B(a_i, r_ℓ) ∖ B(a_i, r_ℓ+1) may contain a lower-entropy region.* elif S(γ) ≤ S(α) ≤ S(β) < 1, then .The region B(a_i, r_ℓ) is useless.* elif if S(γ) ≤ S(β) ≤ S(α) < 1, then . The region B(a_i, r_ℓ) is useless.* else (meaning max{ S(α), S(β), S(γ)} = 1)the successors of a_i to Â_ℓ+1.There is too little information to decide whether B(a_i, r_ℓ) contains a low-entropy region, so re-analyze everything at the next level.Recall that our method in Section <ref> produces the list A_ℓ which is ordered by cohort, and within a cohort is ordered by predecessor, and within a predecessor block is ordered by proximity to the predecessor's per-label means.Therefore, if a_i < a_j in this ordering, then a_i tends to be nearer the mean of the most prominent label of a larger region than a_j.Hence, the distributional coordinates are sorted by “granularity,” from coarse to fine.In some scenarios, detailed analysis of clusters is more important than efficient classification.Therefore, one can implement a “non-parsimonious” version by replacing alloperations with .For machine learning applications, the “parsimonious” version is far faster, as the cover tree can be stopped much earlier than the limit suggested in Section <ref>. §.§ Build Distributional CoordinatesFor each a_i ∈ A_ℓ selected to , we construct a distributional coordinate g_i in the following way.For each of the dominant labels λ among the children of a_i, use PCA/SVD to build a Gaussian model g_i,λ of mass 1. Let Δ S denote the entropy difference on β_ℓ caused by erasing the non-dominant labels. We amplify (or attenuate) each Gaussian model by the coefficientm_i = w_λ(1 - Δ S) (r_ℓ)^D.The distributional coordinate is m_i g_i,λ(x).One can experiment with different weighting systems; we have found these to be generally effective, and we justify this choice for the following geometric reasons: * The weight represents the relative likelihood that a point of the given label was selected from the region on which this coordinate was detected. Perhaps, by chance in our selection process, a single region was detected at a later level in the form of several smaller regions.The weight of the smaller regions will sum to the weight of the larger region. * The entropy term is the “certainty” with which this region was selected.The entropy Δ S is the information that is lost if the other labels are removed from the region, so 1-Δ S is the information that remains. In other words, this term penalizes regions according to the impurity of their labels. * The (r_ℓ)^D = (r_0 θ^ℓ)^D term accounts for the relative remoteness of the sample region.All else being equal, remote regions are particularly distinctive. Because of the cover-tree construction, remoteness can be measured by the size of the cover-tree ball. For example,suppose that two different regions are detected by CDER, each with the same entropy, weight, and configuration of points; the Gaussian coordinates are translates of one-another.Suppose that the first region is very distant from the rest of the pointcloud, so it is detected at an early level ℓ.Suppose that the second region is surrounded closely by other data, so it is not detected by the cover tree until a later level ℓ' > ℓ. The first region's relative volume is θ^(ℓ-ℓ')D.§ EXAMPLESThis section illustrates the promise of CDER on several synthetic examples of varying levels of complexity that are designed to stress certain features of the algorithm. §.§ BlobsThis example shows that CDER ignores high-density regions unless they have low entropy.Consider two labels[Labels are indicated by colors, and we have intentionally chosen colors that seem to show up non-catastrophically in black-and-white printouts. Nonetheless, we recommend printing this paper in color or at least reading this section on the computer screen!]: 0/magenta and 1/green. The cloud collection has 25 pointclouds of each label. Each magenta pointcloud consists of: * 100 points sampled from a standard normal distribution* 2 points sampled from a normal distribution with σ = 0.2 and mean (4,0)* 2 points sampled from a normal distribution with σ = 0.2 and mean (5,0)* 2 points sampled from a normal distribution with σ = 0.2 and mean (-3,0)* 2 points sampled from a normal distribution with σ = 0.2 and mean (-6,0)Each green pointcloud consists of * 100 points sampled from a standard normal distribution* 2 points sampled from a normal distribution with σ = 0.2 and mean (-4,0)* 2 points sampled from a normal distribution with σ = 0.2 and mean (-5,0)* 2 points sampled from a normal distribution with σ = 0.2 and mean (3,0)* 2 points sampled from a normal distribution with σ = 0.2 and mean (6,0)This produces the cloud collection in Figure <ref>. The output of CDER appears in Figure <ref> In a typical run, the algorithm terminates at cover-tree level 8, even though the complete cover tree of 𝒳 ends at level 14.A total of 26 distributional coordinates are produced, and their weights vary by about 3 orders of magnitude. Table <ref> shows how conservative the entropy-reduction process is at selecting regions to generate distributional coordinates.To judge accuracy on testing data in a supervised-learning scenario, we need a method to label a test pointcloud using the distributional coordinates.Each test pointcloud X_test is mapped to a point in ℝ^26. Many sophisticated methods are possible, but for simplicity, we simply ask: For a given pointcloud X_test, which is bigger:the Euclidean norm of the magenta Gaussian coordinates evaluated on X_test,√(∑_g magenta(∫_X_test g)^2)or the Euclidean norm of the green Gaussian coordinates evaluated on X_test,√(∑_g green(∫_X_test g)^2)? With this simple comparison, the algorithm achieves 100% accuracy in a 5-fold cross-validation of this cloud collection with a 80/20 training/testing split. More precisely, for each run of the cross-validation, we take 80 percent of the point clouds, turn these into a cloud collection, build distributional coordinates using this cloud collection, and then test the results on the remaining 20 percent of the point clouds. This entire procedure is then repeated five times.Moreover, the relative masses of the 28 distributional coordinates vary over four orders of magnitude, so for this sort of comparison, one could dispose of many of them in the interest of speed while preserving accuracy. Note that these distributions are contrived to have the same mass (0-moment) and mean (1-moment) and variance.Elementary statistical tests would not distinguish them; 2-moments or skewness tests would be necessary. §.§ BlocksThis example shows that CDER is fairly robust against background noise that prevents strict separation.It also demonstrates that smoothness of the underlying distributions is not necessary for good results.Consider two labels: 0/magenta and 1/green. The cloud collection consists of 100 magenta pointclouds and 100 green pointclouds. Each magenta pointcloud in generated by sampling 30 points uniformly in the unit square, as well as 2 extra points sampled uniformly in a 0.1× 0.1 square at (1/4, 1/4), as well as 2 extra points sampled uniformly in a 0.1× 0.1 square at (1/2, 1/2). Each green pointcloud in generated by sampling 30 points uniformly in the unit square, as well as 2 extra points sampled uniformly in a 0.1× 0.1 square at (3/4, 3/4), as well as 2 extra points sampled uniformly in a 0.1× 0.1 square at (1/2, 1/2). See Figures <ref> and <ref>.Using the same simple comparison as in Section <ref>, the algorithm achieves 88% accuracy despite the high background noise. §.§ Deep FieldIn this example, we demonstrate that Gaussian mixtures are (nearly) “fixed points” of CDER.It also demonstrates that the algorithm can handle unequal sample sizes via the weighting system in Section <ref>. Consider two labels: 0/magenta and 1/green. The underlying distribution is build from 50 Gaussian distributions. For each of these, we choose a random label.We choose a mean point, uniformly on the 10 × 10 square.We choose a pair of orthonormal covariances uniformly from 0 to 0.5.We choose a random rotation angle.We choose an amplification factor (that is, a sample size) uniformly between 50 and 5000. For a particular random seed, we arrive at some combined density function f_0 and f_1, as in Figure <ref>Then, for each label, we produce between 20 and 40 pointclouds, each with between 50 and 500 points.For a particular random seed, we arrive at the cloud collection in Figure <ref>.The CDER algorithm builds a Gaussian mixture model focused on regions of low entropy (Figure <ref>), so it should not be surprising that it builds Gaussians near the original Gaussians, as in Figure <ref>.§.§ Three LabelsFor simplicity, the previous examples have involved only two labels.In a two-label system, a low-entropy region has exactly one dominant label. However, the algorithm is sensible for any number of labels, and it may be that the low-entropy regions are dominated by multiple labels.Hence, an ensemble of regions may be necessary to distinguish pointclouds. Consider three labels:0/magenta, 1/green, and 2/orange. Let f_0 be a triple of standard normal distributions: one each at (0,0), (4,0), and (-2, 2√(3)). Let f_1 be a triple of standard normal distributions: one each at (0,0),(-2,2√(3)), and (-2, -2√(3)). (-2,2√(3)), and (-2, -2√(3)). Let f_2 be a triple of standard normal distributions: one each at (0,0), (-2,-2√(3)), and (4, 0).See Figure <ref>.The CDER algorithm detects these shared regions perfectly, achieving 100% in a 5-fold cross-validation of this cloud collection with a 80/20 training/testing split, using the same simple comparison method as in Section <ref>. § STABILITYIdeally, one would like to prove that CDER is stable or backward-stable, in the sense of <cit.>. To formulate stability or backward-stability, CDER must be expressed as a numerical approximation of a formal function between normed vector spaces.Let C denote the vector space of piecewise continuous functions on our compact domain.Define a norm on C × C by (f_1,f_2) = √(f_1_2^2 + f_2_2^2), where each term is the L_2 norm on our (compact) domain. Given two “input ”functions (f_1, f_2) ∈ V we consider them as density functions for each of two labels.Consider the pointwise entropy,H(x) = -f_1(x)/f_1(x)+f_2(x)log_2(f_1(x)/f_1(x)+f_2(x)) -f_2(x)/f_1(x)+f_2(x)log_2(f_2(x)/f_1(x)+f_2(x)). Let T denote the formal function T: C × C → C × C by T(f_1, f_2) = (g_1, g_2) whereg_1(x)=H(x)f_1(x), if f_1(x) > f_2(x),0, otherwise.andg_2(x)=H(x)f_2(x), if f_2(x) > f_1(x),0, otherwise.A numerical algorithm T̃ for T can be obtained roughly as “sample equally from f_1 and f_2, and apply CDER to generate weighted distributional coordinates.The weighted sum of the distributional coordinates for label 1 is g_1, and the sum of the distributional coordinates for label 2 is g_2.”A stability result such as this is a subject of future work. We have not yet succeeded in proving either stability statement for CDER, but the high cross-validation of the examples above is promising. § DISCUSSIONThis paper introduced CDER, a data-driven, label-driven feature extraction method for collections of labeled pointclouds. CDER is fast, both in terms of theoretical complexity and in initial tests on examples. It does not require the user to choose any tuning parameters, and the geometric meaning of its output features are transparently clear. This section outlines some future directions and makes a few generalizing remarks.The attentive reader of Section <ref> may notice that cover trees can be defined in an arbitrary metric space, and indeed this is how they were originally defined <cit.>. We use them to construct distributional coordinates, and our algorithm for doing so (Section <ref>) demands that we be able to quickly compute means of sets of points. While the Fréchet Mean (<cit.>, IX) can be defined in any metric space, there are not always fast algorithms for its computation (it also need not exist in all cases). And so while more generality could be achieved in the definition of CDER, we only make our complexity statements for cloud collections in a common Euclidean space.All examples in Section <ref> were artificial, and simply intended to help visualize the novel and fairly-technical CDER algorithm and emphasize its key properties. Future work will involve applications of CDER to real data from a variety of real fields, including large administrative datasets of interest to social scientists and to vehicle tracking <cit.>.We also hope that CDER will prove useful as a feature-extraction method in topological data analysis <cit.>, since persistence diagrams can be thought of as point clouds in the plane. Future work will compare the performance of CDER against other such feature-extraction methods (for example, <cit.> and <cit.>).Finally, we recall the weighting discussion in Section <ref>, where we used the simplifying assumptions that each color/label was equally likely and that each point within a single point cloud should be given equal weight. We note that CDER can be easily adapted to accommodate other prior assumptions about relative likelihoods of labels or even prior assumptions about outlier status of certain points in a cloud, say as part of a Bayesian learning process. plain

http://arxiv.org/abs/1702.07959v3

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http://arxiv.org/abs/1702.08035v2

empty ^1 Department of Physics, St. Petersburg State University, 7/9 Universitetskaya Emb., 199034 St. Petersburg, Russia ^2 NRC “Kurchatov Institute” - ITEP,Bolshaya Cheremushkinskaya 25, 117218 Moscow, Russia ^3 Institut für Theoretische Physik, Technische Universität Dresden, Mommsenstraße 13, D-01062 Dresden, GermanyA new relativistic method based on the Dirac equation for calculating fully differential cross sections for ionization in ion-atom collisions is developed. The method is applied to ionization of the atomic hydrogen by antiproton impact, as a non-relativistic benchmark.The fully differential, as well as various doubly and singly differential cross sections for ionization are presented. The role of the interaction between the projectile and the target nucleus is discussed. Several discrepancies in available theoretical predictions are resolved. The relativistic effects are studied for ionization of hydrogenlike xenon ion under the impact of carbon nuclei. 34.10.+x, 34.50.Fa Relativistic calculations of differential ionization cross sections: Application to antiproton-hydrogen collisions A. I. Bondarev^1,2, Y. S. Kozhedub^1, I. I. Tupitsyn^1, V. M. Shabaev^1, and G. Plunien^3 December 30, 2023 =================================================================================================================== § INTRODUCTION Ionization in ion-atom collisions is of fundamental importance for atomic physics. Within the modern “Reaction Microscope” techniques <cit.>, it is possible to experimentally investigate ionization dynamics at the differential level. The measured fully differential cross sections (FDCS) for ionization being a stringent test of theory stimulate theoretical studies of collisions of ions with atoms and molecules. Up to date the FDCS for ionization have been successfully measured in collisions involving light targets such as helium <cit.>, lithium <cit.>, and molecular hydrogen <cit.>. For these targets, non-relativistic theoretical treatmentis sufficient. However, for heavy targets relativistic effects cannot be neglected, and appropriate target description is required. We note that the relativistic effects induced by fast nuclear motion were investigated for FDCS in the 100 MeV/u C^6+-He collision in Refs. <cit.>. The Facility for Antiproton and Ion Research (FAIR) being constructed in Darmstadt (Germany) <cit.> will be able to provide heavy ions up to bare uranium and has an extensive scientific program aimed at the research of heavy-ion collision dynamics. At the same time, we would like to point out promising plans for antiproton research at the FAIR <cit.> and the European Organization for Nuclear Research (CERN) <cit.>. Experimental and theoretical studies of ionization processes in collisions of antiprotons with atoms and molecules have been recently reviewed <cit.>. Despite of the lack of the experimental FDCS, the essentially non-relativistic antiproton-hydrogen collision serves as an ideal benchmark to various theories. This is due to the absence of the charged-transfer channel, in contrast to the collisions with protons, and absence of the electron correlations, in contrast to the electron-impact or multi-electron-target ionization.The perturbative calculations of FDCS for ionization in antiproton-hydrogen collision have been performed in Refs. <cit.>. Recently these FDCS have also been studied by several non-perturbative methods <cit.>. Firstly, McGovern et al. <cit.> developed a method for extracting the FDCS from an impact-parameter treatment of the collision within a coupled pseudostate (CP) formalism. Later, Abdurakhmanov et al. <cit.> worked out the fully quantal time-independent convergent-close-coupling (QM-CCC) approach to differential ionization studies in ion-atom collisions. Recently, Ciappina et al. <cit.> applied the time-dependent close-coupling (TDCC) technique to investigate the role of the nucleus-nucleus interaction in the FDCS. Afterwards, Abdurakhmanov et al. <cit.> used the semiclassical wave-packet convergent-close-coupling (WP-CCC) method to examine the FDCS. We also would like to mention the recent paper by Sarkadi and Gulyás <cit.>, where the FDCS were investigated using the classical-trajectory Monte Carlo method.In this contribution, we present a relativistic single-center semiclassical coupled-channel approach based on the Dirac equation to calculation FDCS for ionization in ion-atom collisions. The basis of target pseudostates is used for the scattering wave function expansion. These pseudostates representing bound states as well as discretized positive- and negative-energy Dirac continua are obtained by diagonalization of the target Hamiltonian utilizing B-splines. B-splines were introduced in atomic physics calculations in 1970s and are broadly used in various problems (see, e.g., reviews <cit.>). In particular, the B-spline expansion has already been applied to calculate the total ionization cross sections in antiproton-hydrogen collisions in Refs. <cit.>.We report the results of the developed method application to the problem of antiproton-impact ionization of atomic hydrogen, where noticeable disagreements in available theoretical predictions exist. We also report the total ionization probabilities of hydrogenlike xenon ion under the impact of carbon nuclei to demonstrate importance of the relativistic effects.The paper is organized as follows. In Sec. <ref> the relativistic method is described. Details of the calculations are given in Sec. <ref>. The results for the -H and C^6+-Xe^53+ collisions are presented in Secs. <ref> and <ref>, respectively. In Sec. <ref> we give the conclusions. Atomic units (a.u.) ħ = e = m_e = 1 are used throughout the paper unless otherwise stated. § THEORY §.§ Time-dependent Dirac equation in a finite basis setWe consider the collision of a one-active-electron target with a bare projectile. Within the semiclassical approximation, we treat the nuclei as sources of an external time-dependent potential.Thus the many-particle problem is reduced to the motion of the relativistic electron in a two-center time-dependent potential. The electron dynamics is described by the time-dependent Dirac equation,i∂Ψ(r⃗,t,R⃗)/∂ t = H(t)Ψ(r⃗,t,R⃗), where the total Hamiltonian of the system is the sum of the free relativistic Dirac Hamiltonian and the interactions between the active electron with the target atom and the projectile, and is given by H(t) =H_0+V_P(t), H_0 = c(α⃗·p⃗)+(β-1) c^2+V_ T, where α⃗ and β are the Dirac matrices.Let us assume that the target is located at the origin, while the projectile moves along a straight-line trajectory R⃗ = b⃗+v⃗t with the constant velocity v⃗ and at the impact parameter b⃗, so that b⃗·v⃗ = 0. Then the total two-center potential V(r⃗,t) is written asV(r⃗,t) = V_ T(r)+V_ P(|r⃗-R⃗(t)|).We note that the potential V(r⃗,t) does not include the interaction between the target nucleus and the projectile. This nucleus-nucleus (NN) interaction does not affect cross sections, which are not differential in the scattered projectile variables. For cross sections, which are differential in these variables, it can be taken into account by a phase transformation in Eq. (<ref>).The target potential V_ T consists of the Coulomb potential of the nucleus V_ nucl and the screening potential of the passive electrons V_ scr, V_ T = V_ nucl+V_ scr. The finite nuclear size effects are incorporated in V_ nucl using an appropriate nuclear charge distribution. The local screening potential of the passive electrons V_ scr can be obtained using various approximate methods.To solve Eq. (<ref>), we expand the time-dependent wave function Ψ(r⃗,t,R⃗) over a finite basis set, Ψ(r⃗,t, R⃗) =∑_aC_a(t,b⃗) e^-i_at_a(r⃗), where the basis functions _a are orthonormal and obtained by diagonalization of the stationary atomic Hamiltonian H_0 employing B-splines <cit.>, _a | H_0 | _a= _a, _a | _b= δ_ab. Since the target potential V_ T(r) possesses the spherical symmetry, the basis function _a(r⃗) may be represented as the bispinor _n_a _a μ_a(r⃗) with a given principal quantum number n_a, angular momentum-parity quantum number _a = (-1)^l_a+j_a+1/2(j_a+1/2), and angular momentum projection on the z-axis μ_a, _a(r⃗) ≡_n_a _a μ_a(r⃗) = 1/r[ G_n_a _a(r) χ__a μ_a(r̂⃗̂); iF_n_a _a(r) χ_-_a μ_a(r̂⃗̂) ],where G_n_a_a(r) and F_n_a_a(r) are the large and small radial components, respectively, and χ__aμ_a(r̂⃗̂) are the spherical spinors, and r̂⃗̂ = r⃗/r <cit.>. In the following, we assume that the z-axis is directed along the vector v⃗. The basis functions _a represent bound states, positive-energy, as well as negative-energy Dirac continuum. Moreover, for low-lying bound states they are very close to the exact ones. Their quality and overall number depends on the size of the B-spline basis set.We note that due to using the dual-kinetic-balance approach <cit.>, the basis setis free from the so-called spurious states, which may arise in a finite-basis-set representation of the Dirac equation <cit.>. Substituting Eq. (<ref>) into Eq. (<ref>), one derives the set of coupled-channel equations for the expansion coefficients, idC_a(t,b⃗)/dt = ∑_b C_b(t,b⃗) e^i(_a-_b)t _a|V_P|_bwith the initial conditions corresponding to the initial active electron state i, C_a(t →-∞,b⃗) = δ_ai. It should be noted that the atomic-like basis set centered at the target does not allow for the explicit description of charge transfer processes. So the method is reliable, if the charge transfer processes are minor compared to the direct ionization ones. This condition is met for fast projectiles, relatively (compared to the target) light projectiles, and projectiles without electron bound states. From the properties of the matrix element V_ab(R⃗) ≡⟨_a|V_P|_b⟩ under rotation around the z-axis, it follows that V_ab(R⃗) = V_ab(t,b)e^i(μ_b-μ_a)ϕ_b, where ϕ_b is the azimuthal angle of b⃗.Then the dependence of the expansion coefficient C_a(t,b⃗) on ϕ_b can also be factorized, C_a(t,b⃗) = C_a(t,b)e^i(μ_i-μ_a)ϕ_b, where C_a(t,b) satisfies the system idC_a(t,b)/dt = ∑_bC_b(t,b) e^i(_a-_b)tV_ab(t,b)with the initial conditionsC_a(t → -∞,b) = δ_ai. To evaluate the matrix elements, it is convenient to reexpand the potential of the projectile to the target position, where the basis functions are centered. If the finite nuclear size effect for the projectile is neglected, the reexpansion of its Coulomb potential can be done analytically <cit.>,-Z_ P/|r⃗-R⃗| = -Z_ P/r_>∑_l=0^∞(r_</r_>)^l ∑_m=-l^lC^l_m(r̂⃗̂)C_m^l*(R̂⃗̂), where r_< andr_> are the minimum and maximum values of (r,R), respectively, and C^l_m denotes the spherical tensor, which is related to the spherical harmonic Y_lm as C^l_m(r̂⃗̂) = √(4π/2l+1)Y_lm(r̂⃗̂). Thus the matrix element V_ab may be represented in the following form:V_ab(t,b) ≡V_n_a_aμ_a n_b_bμ_b(t,b) = ∑_lm R^l_n_a_a n_b_b(t,b)A^lm__a μ_a _b μ_b C^l*_m( vt/b,0), where the radial part is given by R^l_n_a_a n_b_b = -Z_ P∫_0^∞ dr 1/r_>(r_</r_>)^l [G_n_a_a(r) G_n_b_b(r) + F_n_a_a(r) F_n_b_b(r)], and the angular part is the so-called relativistic Gaunt coefficient, A^lm__a μ_a _b μ_b = χ__a μ_a|C^l_m| χ__b μ_b = g^lm(j_a μ_a; j_b μ_b). It may be expressed through the 3j-symbols as g^lm(j_a μ_a; j_b μ_b) =(-1)^1/2+μ_a √((2j_a+1)(2j_b+1))[j_alj_b;1/20 -1/2;] [j_alj_b; -μ_amμ_b ], where l_a+l_b+l should be even number, otherwiseg^lm(j_a μ_a; j_b μ_b) = 0.The commonly used non-relativistic Gaunt coefficient is proportional to the well-known integral of three spherical harmonics <cit.>. We note that here the matrix elements are calculated in the laboratory reference frame. From the computational point of view, this is not the most efficient way. There are two alternative possibilities. One may calculate them in the local reference frame, where z-axis is parallel to the internuclear vector R⃗ at each time moment. Then one should either rotate these matrix elements from the local to the laboratory reference frame using the Wigner D-functions (see, e.g., Ref. <cit.>), or rewrite the time-dependent equation (<ref>) in this local rotating reference frame. Since the rotating reference frame is non-inertial, an additional term arises in the Hamiltonian (<ref>) (see, e.g., Ref. <cit.>). We also would like to mention the symmetry properties of the matrix elements, which can be used for their calculation and storage: V_n_a_a-μ_a n_b_b-μ_b = (-1)^(j_b+l_b+μ_b-j_a-l_a-μ_a)V_n_a_aμ_a n_b_bμ_b.The system of equations (<ref>) may be rewritten in the matrix form, idC⃗dt = MC⃗, M_ab = e^i(_a-_b)tV_ab,where C⃗ is the vector incorporating the expansion coefficients C_a.To solve Eq. (<ref>), we use the short iterative Lanczos propagator <cit.>. It is an exponential-type propagator, where the matrix exponential is approximated in the Krylov subspace <cit.>. The Lanczos propagation is a standard procedure widely used in various chemical and physical calculations <cit.>.§.§ Cross sectionsThe total ionization probability is calculated as the following sum over the positive-energy basis states: P_ ion(b⃗) = P_ ion(b) = ∑_a^(_a > 0) |C_a(t→∞,b)|^2. An alternative method used in Refs. <cit.>, where the summation runs over all basis states and for each of them the overlap with the positive-energy continuum is taken into account, gives almost the same results in a sufficiently large basis set. The total ionization cross section follows from σ_ ion = ∫ db⃗P_ ion(b⃗) = 2π∫_0^∞ dbbP_ ion(b). Using the Stieltjes technique for every symmetry _a, we are also able to calculate partial transition probabilities differential in the energy of the electron <cit.>, dP^_a_ tr/d(_n_a+1^_a+_n_a^_a/2,b) =1/2P_n_a+1^_a(b)+P_n_a^_a(b)/_n_a+1^_a-_n_a^_a,P_n_a^_a(b) = ∑_μ_a|C_n_a_aμ_a(t→∞,b)|^2. After interpolation of the partial probabilities on a common energy grid, summation over the symmetries and integration over the impact parameter, one obtains the single differential cross section for the transition, dσ_ tr/d = 2π∫_0^∞ dbb ∑__adP^_a_ tr(b)/d. We note that the energies _a ≡_n_a^_a are obtained by diagonalization of the stationary atomic Hamiltonian H_0 in the finite B-spline set (see Eq. (<ref>)) and can not be chosen arbitrary. Moreover, basis functions _a with energy _a near the ionization threshold have a similar behavior for positive and negative values of energy _a. Thus Eq. (<ref>) can be used for _a < 0 as well, giving in this case the excitation probability into an energy interval, in contrast to the differential ionization probability for _a > 0. We proceed with evaluation of the probability of the electron ejection in a given direction. The spherical-wave decomposition of the outgoing continuum electron wave function Ψ^(-)_μ_s(r⃗) with a given asymptotic momentum p⃗ and spin projection at the z-axis μ_s is <cit.> Ψ^(-)_μ_s(r⃗) = ∑_μ m i^le^-iΔ_ C^jμ_lm,1/2μ_s Y^*_lm() ψ_μ(r⃗), where C^jμ_lm,1/2μ_s is the Clebsch-Gordan coefficient, ψ_μ is the Dirac partial wave, and Δ_ is the difference between the asymptotic large-distance phase of the Dirac-Coulomb solution and the free Dirac solution <cit.>. The Dirac partial wave ψ_μ with a given energy , angular momentum-parity quantum number , and angular momentum projection μ is represented by ψ_μ(r⃗) = 1/r[ G_(r) χ_μ(r̂⃗̂); iF_(r) χ_-μ(r̂⃗̂) ], and normalized on the energy scale, ψ_μ | ψ_'μ = δ(-'). The radial components G_ and F_ of the wave function ψ_μ and the phase shift Δ_are obtained using the RADIAL package <cit.>. In contrast to the energies _a used in Eq. (<ref>),may be chosen arbitrary.Note that since we quantize the spin of the ejected electron in the direction of the z-axis, the summation over μ_s = ± 1/2 is required in final expressions for observables. Alternatively, one may quantize the spin of the ejected electron in the direction of its propagation. Then the components with different projections (helicities) can be obtained and, in principle, measured. The transition amplitude T^μ_s(, θ_e, ϕ_e, b, ϕ_b) is obtained projecting the wave function Ψ(r⃗,t,R⃗) on the wave function Ψ^(-)_μ_s(r⃗) at the asymptotic time, T^μ_s(, θ_e, ϕ_e, b, ϕ_b) = Ψ^(-)_μ_se^-i t | Ψ,t →∞, where the angles θ_e and ϕ_e correspond to the directionof the ejected electron. Projecting Ψ^(-)_μ_s onto the basis states _a and using their orthonormality (see Ref. <cit.> for details) we come to the following expression for the transition amplitude: T^μ_s(, θ_e, ϕ_e, b, ϕ_b) = ∑_(-i)^l e^iΔ_∑_μ m C^jμ_lm1/2μ_s Y_lm(θ_e,ϕ_e) e^i(μ_i-μ)ϕ_b∑_n I^_ nC_nμ(t→∞,b), where μ_i is the angular momentum projection of the initial state and the radial overlapping integral I_ n^ is given by I^_ n = ∫_0^∞ dr [G_(r)G_n(r)+F_(r)F_n(r)]. Then the fully differential ionization probability as a function of the impact parameter b⃗, the electron ejection energy , and the electron ejection angles θ_e and ϕ_e is given by d^3P(b⃗)/dd(cosθ_e)dϕ_e = ∑_μ_s = ±1/2|T^μ_s(, θ_e, ϕ_e, b, ϕ_b)|^2. We note that in the non-relativistic limit, the electron spin projection at any axis is conserved and, as a result, one term in Eq. (<ref>) vanishes. For comparison with an experiment, it is usually more convenient to express the differential probabilities in terms of the transverse(perpendicular to v⃗)component η⃗ of the projectile momentum transfer q⃗ rather than the impact parameter b⃗. The projectile momentum transfer is the difference between the initial (k⃗_⃗i⃗) and final (k⃗_⃗f⃗) projectile momenta q⃗ = k⃗_⃗i⃗-k⃗_⃗f⃗. Transition amplitudes in the b- and η-representations are related by a two-dimensional Fourier transform <cit.>, T^μ_s(,θ_e,ϕ_e,η,ϕ_η) = 1/2π∫ db⃗e^i·e^iδ(b) T^μ_s(,θ_e,ϕ_e,b,ϕ_b), where δ(b) is the additional phase due to the NN interaction omitted in Eq. (<ref>). This phase depends on the explicit form of the NN interaction, which may include the Coulomb interaction between the projectile and the target nucleus, the projectile and the passive target electrons, as well as polarization effects. In the simple approximation, where the presence of the passive target electrons is accounted for by changing the target charge Z_ T to some screened value Z_ eff, V_NN(R) = Z_ effZ_ P/R. In this case, the phase factor δ(b) reads as δ(b) = 2Z_effZ_P/vlnvb. Some useful remarks on the derivation and applicability of this expression can be found in Ref. <cit.>. Moreover, in the present calculations, we explicitly checked that inclusion of the NN interaction (<ref>) directly in Eq. (<ref>) or as the phase factor (<ref>) in Eq. (<ref>) gives indistinguishable results. Using the Jacobi-Anger expansion <cit.>, we express the Fourier transform of the amplitude T^μ_s(,θ_e,ϕ_e,b,ϕ_b) as T^μ_s(,θ_e,ϕ_e,η,ϕ_η) = 1/2π∫_0^2π dϕ_b ∫_0^∞ bdb ∑_n i^ne^in(ϕ_b-ϕ_η) J_n(η b)e^iδ(b) T^μ_s(,θ_e,ϕ_e,b,ϕ_b). Here J_n(η b) is the n-th order Bessel function of the first kind and ϕ_η is the azimuthal angle of the transverse component of the momentum transfer η⃗.The integration over ϕ_b givesT^μ_s(,θ_e,ϕ_e,η,ϕ_η) = ∑_(-i)^l e^iΔ_∑_μ m C^jμ_lm1/2μ_s Y_lm(θ_e,ϕ_e)i^(μ-μ_i)e^i(μ_i-μ)ϕ_η∑_n I^_ n B^μ-μ_i_nμ(η), where B^m_nμ(η) = ∫_0^∞ bdbJ_m(η b)e^iδ(b) C_nμ(t→∞,b). Then the fully differential ionization probability as a function of the transverse component of the momentum transfer η⃗, the electron ejection energy , and the electron ejection angles θ_e and ϕ_eis calculated as d^3P(η⃗)/dd(cosθ_e)dϕ_e = ∑_μ_s = ±1/2|T^μ_s(,θ_e,ϕ_e,η,ϕ_η)|^2. The (fully) triply differential cross section (TDCS) may be expressed as d^3σ/ddΩ_edΩ_ P = k_i k_f d^3P(η⃗)/dd(cosθ_e)dϕ_e. This is the cross section for the electron being ejected with the energy in the range fromto +d into the solid angle dΩ_e, while the projectile is scattered into the solid angle dΩ_ P. It depends on the reference frame through the initial and final projectile momenta, since the solid angle dΩ_P is different in the laboratory and center of mass reference frames.Integrating the TDCS over corresponding variables, one can obtain various doubly differential cross sections (DDCS), singly differential cross sections (SDCS), and, finally, the total ionization cross section.From the sets of DDCS and SDCS, we focus here only on those, in which significant disagreements with the previously published results have been found. These are the DDCS d^2σ/ddη and SDCS dσ/d. The former is defined byd^2σ/ddη = η/k_i k_f∫_0^2πd^2σ/ddΩ_ P dϕ_ P, where d^2σ/ddΩ_ P =∫d^3σ/ddΩ_edΩ_ P dΩ_e and ϕ_ P is the azimuthal angle of the scattered projectile. In our approach, it can be calculated as d^2σ/ddη = η∫_0^2π dϕ_η ∫_-1^1 d(cosθ_e) ∫_0^2π dϕ_e d^3P(η⃗)/dd(cosθ_e)dϕ_e. The latter is defined as dσ/d = ∫ dΩ_e ∫ dΩ_ P d^3σ/ddΩ_edΩ_ P and can be easier calculated in the b- rather than in η-representation as dσ/d =∫_0^∞ bdb ∫_0^2π dϕ_b ∫_-1^1d(cosθ_e) ∫_0^2π dϕ_e d^3P(b⃗)/dd(cosθ_e)dϕ_e. We note that due to the symmetry, the fully differential ionization probability d^3P(b⃗)/dd(cosθ_e)dϕ_e depends on ϕ_b and ϕ_e only through their difference, and the integration in Eq. (<ref>) can be simplified. The similar arguments for the d^3P(η⃗)/dd(cosθ_e)dϕ_eallow us to simplify the integration in Eq. (<ref>).The SDCS calculated by means of Eqs. (<ref>) and (<ref>) for positive energies _a should be the same. This criterion can serve for checking of the calculations involving the wave function of the ejected electron defined by Eq. (<ref>). A useful check for the convergence over the basis set size is to obtain the first-order perturbative solution of the coupled-channel equations (<ref>): C^ B1_a(t,b⃗) = δ_ai-i∫_-∞^t dt' e^i(_a-_i)t'_a | V_ P | _i . Cross sections calculated using this perturbative solution should then be compared with the corresponding cross sections in the first Born approximation (FBA). We note that in the FBA, the NN interaction does not contribute to the cross sections due to orthogonality of the wave functions in Eq. (<ref>).§ RESULTS§.§ Details of calculations We used the theory described above to calculate cross sections for ionization in the antiproton-hydrogen collision. In the present calculation for the antiproton-hydrogen collision, we did not include in the expansion (<ref>) negative-energy continuum states, which result from the target Hamiltonian diagonalization. Furthermore, we omitted high-energy states with _k > 10 a.u. With these restrictions, the basis set consisted of 45 radial functions for each angular symmetry. Thestates with the angular momentum-parity quantum number = ±1,…,±8,-9, which corresponds tol = 0,…,8, were included in the basis set. The coupled-channel equations (<ref>) were solved from z_ min = -60 a.u. to z_ max = 60 a.u., where z=vt is the z-component of the projectile position. §.§ Antiproton-impact ionization of atomic hydrogenLet us start with presenting the total ionization cross sections. In Table <ref>, the present results of the full coupled-channel (cc) as well as corresponding FBA mode calculations, obtained by Eq. (<ref>) are compared with the results of the non-perturbative approaches of Refs. <cit.> and the analytical FBA results (see, e.g., Refs. <cit.>).Comparing the second and third columns of the table, one can see that the present FBA mode results are in excellent agreement with the analytical ones at all antiproton impact energies under consideration. The results of the full calculation are also in good agreement with the previous studies of Refs. <cit.>. However, the results of Ciappina et al. <cit.> at 30 keV impact are noticeably larger than the present ones and the results of Ref. <cit.>. Briefly discussed the total ionization cross sections, we turn to the triply differential cross sections. Following Abdurakhmanov et al. <cit.> and Ciappina et al. <cit.>, we adopt their conventions. So the direction of the scattered projectile is fixed by the value of the momentum transfer q or by the projectile deviation angle θ_ P. The polar angle θ_e of the ejected electron runs from -180 to 180 relative to the direction of the momentum transfer. The electron emission is considered in the scattering plane only. Fig. <ref> shows the TDCS for ionization of atomic hydrogen by impact of 200-keV antiprotons with a scattering angle of 0.2 mrad and for an ejected electron energy of 4 eV.The results of the non-perturbative QM-CCC <cit.> and TDCC <cit.> approaches along with the FBA analytical results and the present FBA mode results (see Eq. (<ref>)) are also shown. In the figure, one can see perfect agreement between the FBA mode results and the analytical FBA results, which in the following will be labeled as FBA without indicating the type. This agreement verifies the convergence of our results in the FBA mode as well as in the full calculation. All displayed curves demonstrate the two-peak structure with the binary peak in the direction of the momentum transfer and the recoil peak in the opposite direction. Note that the FBA TDCS are always symmetric with respect to the momentum transfer direction. Comparing to the FBA, all presented non-perturbative theories predict the reduced binary and enhanced recoil peaks both rotated away from the direction of the scattered antiproton. For both peaks the expected positions agree with each other, however, there is a noticeable discrepancy in the magnitude. The present results being in good agreement with the QM-CCC results lie significantly above the TDCC results. Ciappina et al. <cit.> assumed that it is the non-perturbative treatment of the higher-order electron-projectile terms of close-coupling formalisms rather than the NN interaction effect, as it was proposed by Abdurakhmanov et al. <cit.>, which leads to the shift of the binary and recoil peaks relatively to the FBA results. In our semiclassical calculations the NN interaction is treated as the phase factor in Eq. (<ref>), i.e. in the same way as in the TDCC calculations of Ref. <cit.>. Thus we are also able to examine the role of the NN interaction by taking it into account or ignoring in the performed calculations. In Fig. <ref>, we display the TDCS for the same parameters as in Fig. <ref>, together with the results of the calculation neglecting the NN interaction (δ(b) ≡ 0 in Eq. (<ref>)). The corresponding results of Ref. <cit.> and the FBA results are also shown. Here we indeed see that inclusion of the NN interaction does not affect the position of the binary peak, in accordance with the suggestion of Ref. <cit.>. Moreover, in our calculation the inclusion of the NN interaction also significantly reduces the TDCS. However, the peak value of the present TDCS obtained in the calculation ignoring the NN interaction is about 10% smaller than the FBA result, whereas the peak value of the TDCC TDCS <cit.> is only about 50% of the FBA result. In Figs. <ref> and <ref>, the TDCS for higher electron ejection energies of 7 and 10 eV, respectively, are presented. For every curve in these figures, the overall form is the same as for a lower energy ejection of 4 eV (see Fig. <ref>) and the positions of the binary and recoil peaks are nearly unchanged. One again can see good agreement between the present and QM-CCC results of Ref. <cit.>, which are almost indistinguishable except for the binary-peak maximum at about 13 and the minimum at about 86. The small differences at these regions increase with increasing the energy of the ejected electron. The binary peak positions of the TDCC TDCS of Ref. <cit.> agree with the present for both energies, however, there is again the inconsistency in the magnitude. Moreover, the TDCC TDCS increase with increasing the energy of the ejected electron in contradiction with the other theories. Next, following Refs. <cit.>, we investigate the TDCS for various projectile scattering angles. The results are shown in Figs. <ref> and <ref>.For all presented non-perturbative theories the magnitude of the binary peak decreases with increasing the projectile scattering angle, in accordance with the FBA. The position of the binary peak shifts to its FBA position with increasing the projectile scattering angle. Here we again see the mismatch between the present and QM-CCC results from the one hand and the TDCC results from the other hand. This mismatch grows with increasing the projectile scattering angle. The TDCS at an antiproton incident energy of 500 keV are shown in Fig. <ref>. Note that at such a high impact energy, the FBA TDCS still differs from the non-perturbative ones, while the total ionization cross sections predicted by all approaches agree much better with each other (see Table <ref>). Here the results of the present approach, QM-CCC and TDCC agree in magnitude. However, in contrast to the previously discussed examples for the 200 keV impact, the binary peak of TDCC TDCS is slightly shifted to the right compared to the present results and QM-CCC data. This may be caused by the inconsistency in the main text and the caption of Fig. 3 in Ref. <cit.>. In the caption, it is stated that TDCS is plotted for the value of the total momentum transfer q = 0.25 a.u., while in the main text, that for the value of the transverse component of the momentum transfer q_⊥≡η = 0.25 a.u., which corresponds to the antiproton scattering angle θ_ P = 0.061 mrad indicated there. The angle θ_f between the direction of the final projectile momentum k⃗_⃗f⃗ and the direction of the momentum transfer q⃗ equals to 52.3 and 58.6 for q = 0.25 a.u. and η = 0.25 a.u., respectively. We would like to point out that the TDCS for this kinematical regime has been first calculated within the CP method by McGovern et al. <cit.>. However, it is almost indistinguishable from the QM-CCC results of Ref. <cit.>. The DDCS in energy of the ejected electron and transverse component of the projectile momentum transfer d^2σ/ddηfor various energies of the ejected electron as a function of the transverse component of the projectile momentum transfer η at an incident antiproton energy of 200 keV is presented in Fig. <ref>. The results of the TDCC approach of Ref. <cit.> are also shown.It is clearly seen that for all energies of the ejected electron the present results disagree with the TDCC results both in the magnitude and shape. For small values of η, the present results for low energies of the ejected electron are larger than those for high ejection energies. For large values of η, the picture is inverted in accordance with the FBA, which is not shown here. It means that for large values of the momentum transfer the maximum of the DDCS is shifted from zero emission energy. For example, for η = 1.75 a.u. this maximum is located about = 25 eV in the FBA. In contrast, the TDCC results for high energy of the ejected electron are larger than those for low energy in the whole range of the momentum transfer. The DDCS d^2σ/ddη being integrated over η gives the SDCS dσ/d, which in this case unexpectedly increases with increasing the energy of the ejected electron. The TDCC DDCS indicate also pronounced structures in the variation of η, which are not observed in our results. At a higher antiproton incident energy of 500 keV, Ciappina et al. <cit.> found similar patterns as shown in Fig. <ref> for the 200 keV impact. However, these patterns are still too far from ours, which are very close to the FBA results and are not shown here. In order to explore the role of the NN interaction, it is more useful to consider DDCS at lower projectile incidentenergies. In Fig. <ref>, we display the DDCS d^2σ/ddη as a function of transverse momentum transfer η for an ejected electron energy of 5 eV at an antiproton incident energy of 30 keV.It is obvious that the DDCS is strongly influenced by the NN interaction in the whole range of the momentum transfer. However, no oscillatory structures in the variation of η are observed again.The reason of the strong contradiction between the present and TDCC results for DDCS is unclear to us. Fig. <ref> shows the SDCS in energy of the ejected electron dσ/d at an incident antiproton energy of 30 keV together with the results of the non-perturbative approaches of Refs. <cit.>. The results of all approaches are in good agreement for the electron ejection energies larger than 7 eV. However, the low-energy behavior is different. The WP-CCC SDCS of Abdurakhmanov et al. <cit.> has a maximum away from the zero emission energy, contrary to the other results. The present SDCS calculated using Eq. (<ref>) is in excellent agreement with the CP results of McGovern et al. <cit.>, and monotonically increases with decreasing the electron ejected energy. In order to verify this behavior, we also calculated SDCS using Eq. (<ref>), which is also valid for negative energies, where it describes the excitation rather than the ionization process. For positive energies, the results obtained by Eq. (<ref>) are in perfect agreement with the results obtained by Eq. (<ref>), and smoothly increase with decreasing the energy below the ionization threshold. This smooth transition between the excitation to high-energy bound states and the ionization to low-energy continuum states is quite reasonable from a general point of view.The WP-CCC method recently developed by Abdurakhmanov et al. <cit.> is formulated in the framework of the single-center semiclassical convergent close coupling approach. The key feature of the method is using stationary wave packets for discretization of the continuous spectrum of the target. Such continuum discretization allows one to generate pseudostates with arbitrary energies and distribution. The reason of the low-energy fall of the WP-CCC results might be a poor implementation of the wave packets describing low-energy states. By construction, the wave packets form an orthonormal basis for positive-energy states. However, a low-energy wave packet of a fine width has a huge size in the coordinate space. This requires the upper limit of the integration over the radial variable in the calculation of the matrix elements to be very large, which is hard to achieve. Furthermore, the results of the FBA mode calculation are determined by the matrix element involving the initial rather localized ground state, and thus are insensitive to the shape of the final state wave packet at large distances. This might be a reason of the good agreement of the WP-CCC results in the FBA mode with the analytical FBA predictions (see Fig. 9 in Ref. <cit.>). §.§ C^6+-impact ionization of hydrogenlike xenon ionFinally, in order to examine relativistic effects, we have considered the 100 MeV/u C^6+-Xe^53+ collision. The 100 MeV/u carbon nuclei have already been used to study the fully differential cross sections for single ionization of helium atom <cit.>. The impact-parameter dependencies of the total ionization probabilities from the K- and L-shells have been calculated. In order to explore the relativistic effects induced by a large target charge, we also carried out the calculation in the non-relativistic limit,where the standard value of the speed of light c was multiplied by a factor of 1000. The comparison of the results of both calculations is shown in Fig. <ref>.From the figure, one can see that the relativistic effects enhance the total ionization probability at small impact parameters and reduce it at large ones for all considered states. Also noticeable is the dominance of the ionization from the 1s state at small impact parameters. In contrast to the ionization from the 1s and 2s states, the total ionization probability from the 2p states, averaged over the values of total angular momentum and its projections, is convex upwards at small impact parameters. It is also worth to consider the impact-parameter dependence of the total ionization probability from the 2p states with various quantum numbers j and μ. These results together with the results of the corresponding non-relativistic calculation are shown in Fig. <ref>.The total ionization probability does not depend on the sign of the projections μ and m. However, for both calculations, it significantly depends on its absolute value. In the relativistic calculation, the total ionization probability depends also on the total angular momentum j of the initial 2p state. § CONCLUSION In this study, we have presented the relativistic semiclassical approach based on the Dirac equation to calculation of differential ionization cross sections in ion-atom collisions. B-splines are used to discretize the Dirac continua of the target. As the first test, the method has been applied to calculation of various differential cross sections for antiproton-impact ionization of atomic hydrogen. Several discrepancies in available results of non-perturbative approaches based on the Schrödinger equation have been resolved. We may assume that the TDCC calculations performed by Ciappina et al. <cit.> have an issue at the stage of the Fourier transform from the b- to η-representation of the ionization amplitude. We also suppose that the low-energy behavior of the WP-CCC SDCS found by Abdurakhmanov et al. <cit.> arises from the lack of normalization of thin wave packets with a small energy. The method has also been applied to explore the relativistic effects on the total ionization probability from the K- and L-shells of hydrogenlike xenon ion under the impact of carbon nuclei. The approach is also suitable for investigation of more complicated many-electron targets. In future, we plan to apply the developed approach to study ionization processes at the differential level in collisions involving heavy targets, where the relativistic effects are extremely important.§ ACKNOWLEDGEMENTSWe thank Alisher Kadyrov and Igor Bray for valuable discussions. This work was supported by RFBR (Grants No. 15-03-07644, No. 16-02-00233, and No. 17-52-53136), by SPSU (Grants No. 11.38.237.2015 and 11.65.41.2017), and by the President of the Russian Federation (Grant No. MK-6970.2015.2). A.I.B. acknowledges the support from the German-Russian Interdisciplinary Science Center (G-RISC) funded by the German Federal Foreign Office via the German Academic Exchange Service (DAAD) and the FAIR-Russia Research Center. h-physrev

http://arxiv.org/abs/1702.08285v1

-0.1in 6.in 8.5in 0in 0in From kinetic Cucker-Smale to pressureless Euler]A rigorous derivation from the kinetic Cucker-Smale model to the pressureless Euler systemwith nonlocal alignmentFigalli]Alessio Figalli [Alessio Figalli]ETH Zürich, Department of Mathematics, Rämistrasse 101, 8092 Zürich, Switzerland. alessio.figalli@math.ethz.ch Kang]Moon-Jin Kang [Moon-Jin Kang]Sookmyung Women's University, Department of Mathematics & Research Institute of Natural Sciences, 100, Cheongpa-ro 47-gil, Yongsan-gu, Seoul, 04310, Koreamoonjinkang@sookmyung.ac.krtheoremTheorem[section] proposition[theorem]Proposition lemma[theorem]Lemma corollary[theorem]Corollary claim[theorem]Claim definition remark[theorem]Remark example[theorem]Example definition[theorem]Definition conjecture[theorem]Conjecture question[theorem]Question equationsection figuresection

http://arxiv.org/abs/1702.08087v2

𝐫𝐒𝐐Wilczek Quantum Center, School of Physics and Astronomy and T. D. Lee Institute,Shanghai Jiao Tong University, Shanghai 200240, ChinaDepartment of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USADepartment of Physics and Astronomy, George Mason University, Fairfax, Virginia 22030, USAWilczek Quantum Center, School of Physics and Astronomy and T. D. Lee Institute,Shanghai Jiao Tong University, Shanghai 200240, ChinaDepartment of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260Center for Cold Atom Physics, Chinese Academy of Sciences, Wuhan 430071, China Motivated by the experimental realization of quantum spin models of polar molecule KRb in optical lattices, we analyze the spin 1/2 dipolar Heisenberg model with competing anisotropic, long-range exchange interactions. We show that, by tilting the orientation of dipoles using an external electric field, the dipolar spin system on square latticecomes close to a maximally frustrated region similar, but not identical, to that of the J_1-J_2 model. This provides a simple yet powerful route to potentially realize a quantum spin liquid without the need for a triangular or kagome lattice.The ground state phase diagrams obtained from Schwinger-boson and spin-wave theories consistently show a spin disordered region between the Néel, stripe, and spiral phase.The existence of a finite quantum paramagnetic region is further confirmed by an unbiased variational ansatz based on tensor network states and a tensor renormalization group. Frustrated Magnetism of Dipolar Molecules on a Square Optical Lattice: Prediction of a Quantum Paramagnetic Ground State W. Vincent Liu Received: date / Accepted: date ========================================================================================================================Understanding highly entangled quantum matter remains a challenging goal of condensed matter physics <cit.>. One paradigmatic example is quantum spin liquids in frustrated spin systems which defy any conventional long range order characterized by broken symmetry at zero temperature <cit.>. Instead, the ground state features long-range entanglement and nonlocal excitations.Spin liquids are also fertile ground for studying quantum phases described by gauge field theories and topological order <cit.>.While the existence of spin liquids has been firmly established in a number of exactly solvable models, e.g., the toric code <cit.> or the honeycomb Kitaev model <cit.>, the nature of the ground states for many frustrated spin models, e.g., the Heisenberg model on kagomelattices or the J_1-J_2 model on square lattices, still remains controversial despite the great theoretical progress in recent years <cit.>.An unambiguous experimental identification of quantum spin liquids in solid state materials also seems elusive <cit.>.It is, then, important to explore new physical systems that can cleanly realize well-defined spin models which have potential spin liquid ground states.Recent breakthrough experiments on magnetic atoms <cit.> and polar molecules  <cit.> confined in deep optical lattices introduceda new class of lattice spin models with competing exchange interactions that are long-ranged and anisotropic.The resulting spin Hamiltonians, such as the dipolar XXZ and dipolar Heisenberg models, are highly tunable by the external fields that couple to the magnetic and electric dipoles <cit.>. Here, we show that these models on square lattices feature strong exchange (not geometric) frustration and a quantum paramagnetic ground state for intermediate dipole tilting angles. This claim is consistently supported by physical arguments, two independent semiclassical analytical methods, and full numerical calculation based on tensor network ansatz <cit.>.Our key insight is that spin liquids may arise naturally from the system of tilted, interacting dipoles on square lattices, without the requirement of peculiar (e.g., triangular or kagome) lattices or exotic (e.g., Kitaev or ring-exchange) interactions.The dipolar XXZ and Heisenberg model.—First, we define the dipolar XXZ model on a square optical lattice, H_XXZ=J/2∑_i≠ j f(𝐫_i-𝐫_j) (S^x_i S^x_j+S^y_i S^y_j+η S^z_i S^z_j).Here i and j label the lattice sites, 𝐒_i=(S^x_i,S^y_i,S^z_i) are the spin (or pseudospin) operators at site i, and η is the exchange anisotropy. The key new feature here is that the coupling between the two spins depends on their relative position 𝐫=𝐫_i-𝐫_j and the external field (dipole) direction d̂ f(𝐫)=[1-3(r̂·d̂)^2](a/r)^3,with a the lattice constant [Fig. 1(a)]. This geometric factor, characteristic of the dipole-dipole interaction, dictates that spin interactions are long-ranged and anisotropic. For the special case of η=1, H_XXZ reduces to the dipolar Heisenberg modelH_d=J/2∑_i≠ j f(𝐫_i-𝐫_j) 𝐒_i ·𝐒_j,and for η=0, it reduces to the dipolar XY model, H_XY.Spin models of the form of H_XXZ have been realized experimentally in two settings. In Ref. <cit.>, the spin dynamics of a gas of ^52Cr atoms in optical lattices was observed. Each Cr atom carries a magnetic moment of 7μ_B and hyperfine spin S=3. An external magnetic field is used to align the magnetic dipoles in the direction of d̂. Such a dipolar gas of Cr in a deep lattice is shown to be described by H_XXZ with J=-μ_0(gμ_B)^2/4π a^3<0 and η=-2<cit.>. Note that J induced by the dipolar interaction is, contrary to the superexchange, independent of the tunneling, and it can be set as the unit of energy. Polar molecules such as ^40K^87Rb confined in optical lattices with negligible tunneling provide another way to realize H_XXZ with S=1/2 and tunable J and η<cit.>. Each molecule carries an electric dipole moment 𝐝 and undergoes rotation with angular momentum 𝐉 [see Fig. 1(a)]. Here, the pseudospin 1/2 refers to two rotational states of the molecule labeled by |j,m⟩, where j is the quantum number of the rotational angular momentum 𝐉 and m is its projection onto the quantization axis, chosen as the direction of the external electric field E. More details can be found in Ref. <cit.>.The dipole-dipole interaction projected onto the sub-Hilbert space of the pseudospins then takes the form of a spin Hamiltonian, where the spin flips correspond to transitions between the rotational states.For example, by choosing |j,m⟩=|0,0⟩ and |1,0⟩ as the pseudospin down and up respectively, Refs. <cit.> showed that the system is described by the effective Hamiltonian H_XXZ with J=D_t ^2/2πϵ_0 a^3>0 and η=(D_1-D_0)^2/2D_t^2>0.Here the dipole matrix element D_t=⟨1,0|d^0|0,0⟩, D_1=⟨1,0|d^0|1,0⟩, D_0=⟨0,0|d^0|0,0⟩, and d^0 together with d^± form the vector dipole operator in the spherical basis <cit.>.The anisotropy η increases monotonically with E. As shown in Ref. <cit.>,when E≃ 1.7B/|𝐝| with B the energy splitting of the two pseudospin states, η=1, and one arrives at the dipolar Heisenberg model H_d. In the KRb experiment <cit.> carried out at zero field and cubic lattice, η→ 0, the dipolar XY model H_XY was realized withJ on the order of 100 Hz. Despite the low filling factor and high entropy, coherent spin dynamics was observed via Ramsey spectroscopy <cit.> and modeled theoretically in Ref. <cit.>. Recently Yaoet al. <cit.> considered general η and worked out the phase diagram of H_XXZ on the Kagome and triangular lattice using Density Matrix Renormalization Group (DMRG).. For both lattices, they found evidence for quantum spin liquid centering around the Heisenberg limit, η=1 and θ=0, in which θ is defined by d̂·x̂=sinθcosϕ with x̂ representing a base vector of the square lattice. Thus the physics is connected to a geometrically frustrated Heisenberg model on both lattices, with additional longer range interactions and anisotropy η. In this Letter, we study the phases of H_d on a square lattice as the dipoles are tilted towards the lattice plane [see Fig. 1(a)] for S=1/2 and J>0. We show that strong frustration occurs at intermediate dipole tilting angle θ,leading to a quantum paramagnetic ground state. We emphasize that, here, the frustration is not imposed by the lattice geometry, but instead, is due to the competition between the exchange interactions, analogous to the J_1-J_2 model. Relatedly, the quantum paramagnetic phase appears at intermediate θ values (not around θ=0 as in Ref. <cit.>) between the Néel and the stripe orders. Thus, it differs qualitatively from the spin liquids studied in Ref. <cit.>. We will also employ different methods to solve the dipolar quantum spin models. Competing exchanges for tilted dipoles.—To appreciate the possible phases of H_d as d̂ is tuned as well as its connection to frustrated quantum spin models <cit.>, let us consider the leading exchange couplings between the nearest neighbors, J_x=Jf(ax̂) and J_y=Jf(aŷ), and the next nearest neighbors, J_d=Jf(ax̂+aŷ) and J'_d=Jf(ax̂-aŷ) [Fig. 1(b)]. Their relative magnitudes and signs depend sensitively on the dipole tilting angle θ and ϕ. One example is shown in Fig. 1(b) for fixed ϕ=25^∘. At small θ, J_x∼ J_y dominates because it is about three times that of J_d∼ J'_d. The situation is reminiscent of the J_1-J_2 model in the regime of the Néel order.As θ is increased, J_d and J'_d grow relative to J_x and J_y. The system becomes more frustrated due to the increased competition of the exchanges. This is the most interesting parameter region. Around θ≃ 40^∘, J_x and J_d vanish while J'_d∼ 0.4 J_y. The model can be viewed as coupled Heisenberg chains. For even larger θ, J_x and J_d switch signs to become ferromagnetic, and the stripe order is expected. Clearly, the physics of H_d is much richer than the J_1-J_2 model.In fact, the two models only overlap at one single point, θ=ϕ=0, where J_2/J_1=1/2√(2)≈ 0.35 and the system is Néel ordered. The degree of frustration can be measured by the “spin gap" Δ, the energy difference between the ground and the first excited state, from exact diagonalization of H_d for a 4× 4 lattice <cit.>. For example, we observe a pronounced peak in Δ around θ∼ 28^∘ for ϕ=25^∘, which indicates strong frustration and points to a gapped, spin disordered ground state <cit.>. For fixed ϕ=35^∘, the spin structure factor shows a clear peak at (π,π) for θ∼ 15^∘ for the Néel order, a peak at (0,π) for θ∼ 50^∘ for the stripe order, but no well defined peaks around θ∼ 35^∘, consistent with the argument above.Spin-wave and Schwinger-boson theory.—First, we obtain a coarse phase diagram of H_d on the (θ,ϕ) plane using two widely adopted analytical methods in frustrated quantum magnetism. This will help identify the interesting regions for the more expensive tensor network calculations to focus on. The starting point is the classical solution of H_d by the Luttinger-Tisza method <cit.>. H_d is of the form ∑_ijJ_ij𝐒_i·𝐒_j with hard spin constraint 𝐒_i=S and J_ij only depends on 𝐫_i-𝐫_j. A theorem states that the classical ground state is a planar spin spiral, _/S=x̂cos(·)+ŷsin(·) with an ordering wave vector = (Q_x,Q_y)<cit.>. The classical phase diagram <cit.> consists of three phases. The first is the Néel order corresponding to =(π,π) for small θ. The second is the stripe phase with =(0,π) for large θ but not too large ϕ. These two spin orders are collinear. The third, spiral phase fills the rest of the phase diagram, for large θ and ϕ, wherevaries continuously and, in general, is incommensurate with the lattice.Beyond the classical limit, quantum fluctuations will suppress the magnetic order and shift the phase boundary. These effects can be described qualitatively by modified spin wave theory  <cit.>.In the Holstein-Primakoff representation, we expand H_d in a series of 1/S and keep up to the quartic order of bosonic operators, i.e., we take into account the interactions between the linear spin waves. The bosonic Hamiltonian is solved by self-consistent mean field theory <cit.>. The result is summarized in Fig. 2(a). We find that the phase boundary of the Néel (stripe) phase moves towards smaller (larger) θ values, opening up an intermediate region in between where the magnetization vanishes. The spiral phase also recedes to higher ϕ values. We label this quantum paramagnetic region with QP. This is precisely the region where the various exchanges compete and the system is most frustrated. Alternatively, we can take into account quantum fluctuations by the rotationally invariant Schwinger boson mean field theory which is nonperturbative in S<cit.>. It is a well tested method capable of describing both magnetically ordered and spin liquid states of frustrated spin models <cit.>.The resulting phase diagram is shown in Fig. 2(b). Here, each magnetic order corresponds to condensation of bosons at a certain wave vector . Within a finite strip region labeled by QP between the Néel and stripe phase, the condensation fraction vanishes and the spin excitations are gapped, corresponding to a quantum paramagnetic phase. The fact that two different approximations agree on the existence of QP indicates that it must be a robust feature of the model H_d.Phase diagram from a tensor network ansatz.—A variational ansatz based on tensor network states <cit.> has recently emerged as an accurate and unbiased algorithm for solving two dimensional frustrated quantum spin models <cit.>. In this approach, the ground state many-body wave function |Ψ⟩ is constructed from a network of tensors T_i defined on lattice site i: |Ψ⟩=∏_i T_i, wherestands for contraction of neighboring tensors. Each tensor T_i has four virtual legs (indices), each with bond dimension D designed to build up the quantum entanglement between lattice sites, and one physical leg representing the spin. We choose a L× L cluster as the unit cell with periodic boundary conditions. The algorithm starts with L^2 random tensors, and imaginary time evolution is used to update the local tensors, |ψ'⟩=exp(-τ H)|ψ⟩, until convergence is achieved. We adopt the simple update scheme <cit.> based on singular value decomposition. By using the Trotter-Suzuki formula exp(-τ H)≈∏_i=1^4exp(-τ H_i)+O(τ^2), each iteration of projection for one plaquette can be done using exp(-τ H_i)(i=1,2,3,4) in four separate steps, in which each step evolves three sites (a right triangle) in one plaquette with H_i contains only three terms of the Hamiltonian. For example, H_1,2 contains J_x, J_y, and J_d terms and H_3,4 contains J_x, J_y and J'_d terms (See Refs. <cit.>).The expectation value of a local operator O_j at site j, ⟨ O_j⟩=⟨Ψ|O_j|Ψ⟩/⟨Ψ|Ψ⟩, can be computed by tensor contraction, ⟨ O_j⟩= (𝒪_j∏_i≠ j𝒯_i)/∏_i 𝒯_i where 𝒯_i=T_i^† T_i and 𝒪_j=T_j^† O_j T_j. We evaluate it using an iterative, real space coarse-graining procedure known as the tensor renormalization group which enables one to reach the thermodynamic limit <cit.>. In this way, we calculate the order parameters such as magnetization M=√(⟨ S_x⟩^2+⟨ S_y⟩^2+⟨ S_z⟩^2) <cit.>.With increasing D, quantum fluctuations beyond spin wave or Schwinger boson analysis are taken into account. The suppression of M is illustrated in Fig. 3(a) for different D values at fixed ϕ=15^∘. By extrapolating the results to infinite D, we can determine the phase boundary of the Néel and stripe phases. Repeating the procedure for different ϕ values, we obtain the phase diagram Fig. 3(b). It firmly establishes the existence of a finite quantum paramagnetic region (in red), about one degree wide in θ and persisting from ϕ=0 up to ϕ=20^∘, where the magnetization is completely suppressed to zero. The paramagnetic phase is narrower than the prediction of the Schwinger boson mean field theory which tends to overestimate the spin disordered region. Inside the Néel phase, there is a sudden drop of M.Note that the spiral phase, in general, is incompatible with the L× L cluster choice, even for large L. So we refrain from carrying out the tensor network ansatz beyond ϕ=20^∘. On the other hand, our numerics indicates that the phase boundary presented in Fig. 3(b) is not expected to depend sensitively on L as it varies <cit.>. Finally, we point out that the quantum paramagnetic phase is a robust feature of the dipolar XXZ model. It persists when η is tuned away from the Heisenberg limit, e.g., down to η=0.5 <cit.>. It is challenging to pin down the precise nature of the paramagnetic phase found here in the dipolar Heisenberg model. Similar difficulties also arise for the J_1-J_2 model where the latest DMRG result <cit.> suggests that the paramagnetic region may consist of a subregion with a plaquette valence bond solid (VBS) order and a second, spin liquid or quantum critical region. Possible spin liquid states for the J_1-J_2 model on square lattices have been classified within the framework of the Schwinger boson mean field theory <cit.>. Yet it remains unclear which one is realized in the ground state. It is possible that the QP region of H_d may contain some VBS order. Unlike the J_1-J_2 model, the C_4 rotation symmetry is broken in H_d as soon as the dipoles are tilted, which may disfavor the plaquette VBS. Because of the limitation of the cluster size, we could not accurately compute the dimer correlation functions. Future numerical work with larger L and D is required to shed light on this open issue. The new formulation of symmetric tensor networks <cit.> and Lanczos iteration <cit.> seems promising to detect the possible topological order and accessing the excitation spectrum.In summary, we presented consistent evidence that a quantum paramagnetic phase emerges from the simple physical system of interacting, tilted dipoles confined on square optical lattices. Our analysis of the dipolar Heisenberg model for general (θ,ϕ) adds a new dimension to frustrated quantum magnetism. It allows the exploration of potential spin liquids beyond the J_1-J_2 model which has not been realized cleanly so far. For KRb, J is about 100 Hz, or 5 nK, similar to the superexchange scale t^2/U of the Fermi Hubbard model recently studied using quantum gas microscope <cit.>. Thus, it seems possible to probe the spin order or spin correlations of H_d and related models in future experiments.We thank Ying-Jer Kao, Bo Liu, Jaime Merino, and Ling Wang for helpful discussions. This work is supported by the U.S. AFOSR Grant No. FA9550-16-1-0006 (H.Z., E.Z., and W.V.L.), NSF PHY-1205504 (E.Z.), and ARO Grant No. W911NF-11-1-0230, the Chinese National Science Foundation through the Overseas Scholar Collaborative Program (Grant No. 11429402) sponsored by Peking University and another grant (No.11227803), and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB21010100) (W.V.L.). 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The set of wavevectorsminimizing the classical energy will be denoted as {}.We compare the energies of the incommensurate spiral {_I}, the stripe _s = (0, π) and the Néel _n = (π, π) order. The result is the classical phase diagram shown in Fig. <ref>. Upon crossing the phase boundary, e.g., from the Neel (or the stripe) phase to the incommensurate spiral phase, the wavevector Q varies continuously. For example, in the special case of ϕ=45^∘ (see the inset of Fig. <ref>), Q_x=Q_y≡ Q, where Q changes continuously from π on the phase boundary between Neel and spiral phase to π/4 at the upper right corner of the (θ, ϕ) diagram. § EXACT DIAGONALIZATIONWe calculate the “spin gap" Δ, the energy defference between the ground and the first excited state from exact diagonalization of the dipolar Heisenberg Hamiltonian for a 4× 4 lattice. Fig. <ref> shows Δ as functions of θ for different ϕ. For ϕ≤ 30^∘, the pronounced peak at each line indicates strong frustration. For ϕ=35^∘, The appearance of the second peak corresponds to the transition from the stripe phase to the incommensurate spiral phase as θ increases. For ϕ≥ 40^∘, the disappearance of the first peak indicates the transition from the Neel phase to the spiral phase.§ SCHWINGER BOSON THEORYWe outline the Schwinger boson mean field theory (SBMF) of the dipolar Heisenberg model. The starting point is the bosonic representation of the spin operatorsS^+=a^† b,S^-=b^† a,S_z=1/2(a^† a- b^† b).with the constraint 1/2(a^† a + b^† b)=S.For the square lattice, introduce the antiferromagnetic (A) and ferromagnetic (B) bond operatorsA_ij=1/2(a_i b_j - b_i a_j),B_ij=1/2(a^†_i a_j + b^†_i b_j). In terms of the bond operators, the spin exchange term becomes _i ·_j=:B^†_ij B_ij:-A^†_ij A_ij,where :: means normal order of bosonic operators. Note that B and A are related by operator identity :B^†_ij B_ij:+A^†_ij A_ij=S^2.We adopt the rotational invariant formulation of SBMF and perform mean field decoupling for both A and B, _i ·_j≃ [β_ij B_ij-α_ijA_ij + h.c. ]-|β_ij|^2+|α_ij|^2,whereα_ij=⟨ A^†_ij⟩, β_ij=⟨ B^†_ij⟩, and ⟨ ... ⟩ denotes the ground state expectation value. This is known to perform better in describing the phases of frustrated spin systems compared to antatz that only keep either A or B. Also, within SBMF, the constraint Eq. (<ref>) is only enforced on average by introducing the Lagrange multiplier λ. The mean field Hamiltonian then takes the formℋ_MF = 1/2∑_i≠ jJ_ij [β_ij B_ij-α_ijA_ij+ h.c.- |β_ij|^2+|α_ij|^2 ] + λ∑_i[a^†_i a_i + b^†_i b_i-2S]. Long range magnetic order corresponds to condensation of the a and/or b bosons. To treat the condensate fraction, we decompose each operator intoa_i = ã_i + x_i, x_i=⟨ a_i ⟩ ,b_i = b̃_i +y_i, y_i=⟨ b_i ⟩ ,where x_i and y_i are c-numbers describing the condensate, while operators ã_i and b̃_i annihilate excitations over the condensate. We assume α_ij=α_δ with δ=_j-_i and similarly for β_ij. Namely they only depends on δ and not on _i. For our model, it is sufficient to keep δ=±x̂, ±ŷ, ±x̂±ŷ, i.e. the nn and nnn couplings. Fourier transform to 𝐤 space, e.g. ã_i→ã_k,ℋ_MF becomes a quadratic form of operators ã_k, b̃_k and c-numbers x_k, y_k. In accordance with the classical analysis, we assume x_k and y_k are nonzero only at a pair of wave vector ±𝐐/2. It is then diagonalized by a standard Bogoliubov transformation,ℋ_MF = ∑_k [c^†_k c_k + d^†_k d_k +1]ω_k +N/2∑_δ J_δ (|α_δ|^2-|β_δ|^2) + ∑_𝐤=±𝐐/2 [β_k (|x_k|^2+|y_k|^2) +(ix_ky_-kα^*_k + h.c.)]- Nλ (2S +1) +λ∑_𝐤=±𝐐/2 (|x_k|^2+|y_k|^2). Here N is the number of lattice sites, c_k and d_k are the eigenmodes of spin excitations with dispersion ω_k=√((λ+β_k)^2-|α_k|^2).andα_k=1/2∑_δ J_δsin (𝐤·δ) α_δ, β_k=1/2∑_δ J_δcos (𝐤·δ) β_δ.We adopt the sprial ansatz x_i=√(2m)cos(𝐐/2·_i), y_i=√(2m)sin(𝐐/2·_i). Then x_Q/2=√(Nm/2), y_Q/2=-ix_Q/2. Minimizing the SBMF ground energy with respect to the variational parameters {λ, α_δ, β_δ,x_𝐐/2,y_𝐐/2} leads to the self-consistency equations,S+1/2=1/2N∑_k λ+β_k/ω_k+m,α_δ=1/2N∑_k α_k/ω_ksin (𝐤·δ)+msin (𝐐/2·δ), β_δ=1/2N∑_k λ+β_k/ω_kcos (𝐤·δ)+mcos (𝐐/2·δ), λ+β_𝐐/2 = α_𝐐/2.The last equation is equivalent to the requirement that 𝐐 is chosen to be the minimum of ω_k. And the SBMF ground state energy simplifies toE_MF=∑_k ω_k -Nλ (2S +1) -N/2∑_δ J_δ (|β_δ|^2-|α_δ|^2).In the large S limit, we have m≃ S, α_δ= m sin (𝐐/2·δ), β_δ= m cos (𝐐/2·δ), λ=-∑_δ J_δcos ( 𝐐·δ), and E_cl=1/2NS^2∑ _δ J_δcos ( 𝐐·δ),which agrees with the classical result as expected. § MODIFIED SPIN WAVE THEORY We represent the spin operator using Holstein-Primakoff (HP) bosons,S^- = a^†√(2S-a^† a),S^+ = √(2S-a^† a) a,S_z = S-a^† a.Proper number of boson operators are introduced for the two-sublattice case (Neel phase) and the four-sublattice case (stripe phase). Take the two-sublattice for example, a_i (or b_j) are Bose annihilation operators on the A (or B) sublattice.The dipolar Heisenberg Hamiltonian can then be expanded in series of boson operators,ℋ_SW=ℋ_cl+ℋ^(2)+ℋ^(4),where the classical part ℋ_cl is given previously in Eq. <ref>, the quadratic part ℋ^(2) is ℋ^(2)/S = J_x∑_nn_x(a^†_i a_i+b^†_j b_j-a^†_i b^†_j-a_i b_j) + J_y∑_nn_y(a^†_i a_i+b^†_j b_j-a^†_i b^†_j-a_i b_j) + J_d'∑_nnn_1(a^†_i a_i'+b^†_j b_j'-a^†_i a_i-b^†_j b_j) + J_d∑_nnn_2(a^†_i a_i'+b^†_j b_j'-a^†_i a_i-b^†_j b_j),and the quartic part ℋ^(4) is 4ℋ^(4) = J_x∑_nn_x(a^†_i a_ia_ib_j+a_jb^†_j b_jb_j-2a^†_ia_ib^†_jb_j) + J_y∑_nn_y(a^†_i a_ia_ib_j+a_jb^†_j b_jb_j-2a^†_ia_ib^†_jb_j) + J_d'∑_nnn_1(a^†_ia^†_i'a_ia_i'-a^†_i a^†_i'a_i'a_i') + J_d'∑_nnn_1(b^†_jb^†_j'b_jb_j'-b^†_j b^†_j'b_j'b_j') + J_d∑_nnn_2(a^†_ia^†_i'a_ia_i'-a^†_i a^†_i'a_i'a_i') + J_d∑_nnn_2(b^†_jb^†_j'b_jb_j'-b^†_j b^†_j'b_j'b_j')+H.C.. For the Neel phase, the expectation values of many operator pairs vanish, e.g.,⟨ a_ia_i⟩=⟨ a_ia_i'⟩=⟨ a^†_ib_j⟩=0.We define the following nonzero averages of boson operators describing the quantum fluctuations of spinsf_0≡⟨ a^†_ia_i⟩, g_1≡⟨ a_ib_j⟩, f_2≡⟨ a^†_ia_j⟩,and apply self-consistent mean field decoupling of the quartic terms in ℋ^(4) a^†_ia_ia_ib_j = f_0 a_ib_j+g_1 a^†_ia_i-f_0 g_1, a^†_ia^†_i'a_i'a_i' = f_0 a^†_ia_i'+f_2 a^†_i'a_i'-f_0 f_2, a^†_ia_ib^†_jb_j = (1-λ_1)(f_0 b^†_jb_j+f_0 a^†_ia_i-f_0^2) + λ_1(g_1a_ib_j+g_1a^†_ib^†_j-g_1^2), a^†_ia_ia^†_i'a_i' = (1-λ_2)(f_0 a^†_i'a_i'+f_0 a^†_ia_i-f_0^2) + λ_2(f_2a^†_ia_i'+f_2a^†_i'a_i-f_2^2),where 0≤λ_1,λ_2≤ 1 are parameters determined by minimizing the ground state energy. The magnitudes of λ_1 or λ_2 describe the competition between the diagonal and off-diagonal terms of spin deviation operators. After Bogoliubov transformation to diagonalize the resulting Hamiltonian, the self-consistent equations can be solved by minimizing the ground state energy E_0 with respect to the variational parameters {λ_1, λ_2}.The energy and the staggered magnetization are given byE_0 = ℋ_cl+E_1+∑_𝐤ϵ_k,⟨ S_z⟩ = S-f_0,where E_1= J_x[(1-λ_1)(f_0^2-f_0)+λ_1g_1^2+(1/2-f_0)g_1+S] + J_y[(1-λ_1)(f_0^2-f_0)+λ_1g_1^2+(1/2-f_0)g_1+S] - J_d'[(1-λ_2)(f_0^2-f_0)+λ_2f_2^2+(1/2-f_0)f_2+S] - J_d[(1-λ_2)(f_0^2-f_0)+λ_2f_2^2+(1/2-f_0)f_2+S], and ϵ_k=√(h_k^2-Δ_k^2),with h_k = J_x[S-(1-λ_1)f_0+1/2g_1] + J_y[S-(1-λ_1)f_0+1/2g_1] - J_d'[S-(1-λ_2)f_0+1/2f_2)] - J_d[S-(1-λ_2)f_0+1/2f_2)] + J_d'[cos(k_x+k_y)(S-f_0/2+λ_2f_2)] + J_d[cos(k_x-k_y)(S-f_0/2+λ_2f_2)],and Δ_k = J_xcos(k_x)(f_0/2-S-λ g_1) + J_ycos(k_y)(f_0/2-S-λ g_1). The self-consistency equations aref_0 = 1/N∑_𝐤(h_k/ϵ_k-1), g_1 = -1/N∑_𝐤Δ_k/2ϵ_k(cos k_x+cos k_y), f_2 = 1/N∑_𝐤1/ϵ_kcos k_xcos k_y.The criterion for Neel order is a finite ⟨ S_z⟩_Neel.For the stripe case, a similar procedure can be applied except that four types of boson operators should be introduced. Correspondingly, three variational parameters λ_i are needed due to the difference between x and y directions. Using a similar self-consistent mean-field approximation, the boundary of stripe phase can be determined. The criterion for stripe phase is a finite ⟨ S_z⟩_stripe and real, positive-definite spin deviation operators.The mean field phase diagrams in Fig. 2 of the main text obtained by two different methods give us the same qualitatively picture butdifferent areas of the spin disordered region. This is not surprising, since different spin representations and mean field decoupling schemes are used. For example, in Eq. <ref>, the expectation values of bond operators are used in SBMF while the quartic terms in Eq. <ref> are described by the variational parameters of λ_1, λ_2 for the modified spin wave theory.§ TENSOR NETWORK ANSATZ§.§ Simple UpdateWe choose a L× L unit cell (i.e. L× L local tensors) with different virtual bond dimension D=2,4,6 to form the initial tensor network state |Ψ⟩ and set the time interval τ=0.005J^-1 for imaginary time evolution iterations for local tensors |ψ⟩,|ψ'⟩=exp(-τ H)|ψ⟩,until convergence is achieved.Taking L=2 as an example and using the Trotter-Suzuki formula <cit.>, we can express the projection operator as exp(-τ H)≈∏_i=1^4exp(-τ H_i)+O(τ^2),whereH_1 = J_x𝐒_A ·𝐒_B+J_y𝐒_B ·𝐒_C+2J_d𝐒_A ·𝐒_C,H_2 = J_x𝐒_C ·𝐒_D+J_y𝐒_A ·𝐒_D+2J_d𝐒_A ·𝐒_C,H_3 = J_x𝐒_ C·𝐒_D+J_y𝐒_B ·𝐒_C+2J'_d𝐒_B ·𝐒_D,H_4 = J_x𝐒_A ·𝐒_B+J_y𝐒_A ·𝐒_D+2J'_d𝐒_B ·𝐒_D.This means that each iteration of projection can be done using exp(-τ H_i)(i=1,2,3,4) in four separate steps for one plaquette. While in each step three out of four tensors are evolved (Fig. <ref>).§.§ Tensor Renormalization GroupStarting from the converged local tensors T_i obtained from the simple update, one can construct new two-dimensional local tensors (Fig. <ref>(a)), 𝒯_i = T_i^† T_i, 𝒪_i = T_i^† O_i T_i,where O_i is an operator.The expectation value ofO_i, ⟨ O_i⟩=⟨Ψ|O_i|Ψ⟩/⟨Ψ|Ψ⟩, can then be obtained by⟨ O_i⟩= (𝒪_i∏_j≠ i𝒯_j)/∏_j 𝒯_j,in the thermodynamic limit by using Tensor Renormalization Group (TRG) method <cit.>, where tr stands for contraction of neighboring tensors. Taking the denominator ∏_i 𝒯_i as an example (the numerator can be coarse-grained with the same procedure because the local operator O has the same structure with 𝒯). As shown in Fig. <ref>(b), for each step, one can decompose each 𝒯 to two 𝒮 via singular value decomposition,𝒯_i_1,a,k_1,b≈∑_ĩ𝒮_i_1,k_1,ĩ𝒮_a,b,ĩ.The truncation bond dimension of coarse-graining (dimension of the third leg of 𝒮) is set as χ. The new local tensor 𝒯̃ with the same structure as 𝒯 can be constructed from contracting the inner legs of four 𝒮, 𝒯̃_ĩ,j̃,k̃,l̃=∑_a,b,c,d𝒮_a,b,ĩ𝒮_d,c,j̃𝒮_a,c,k̃𝒮_d,b,l̃.Using 𝒯̃ as the starting tensors, these steps are repeated until ∏_i 𝒯_i is converged. In our TRG calculation, the truncation bond dimension is fixed as χ = 8 to make D the only tuning parameter of the whole procedure.§.§ Comparison of Different Unit Cell Sizes The phase boundary between the quantum paramagnetic phase and other long-range orders can be inferred from the disappearance of magnetic order parameters. It is crucial to determine whether the phase boundary depends sensitively on L, the size of the unit cell. To address this question, we calculate the magnetization M and the average energy E with different unit cell size (L=2,4,8) at fixed D. Fig. <ref> shows examples with D=4, from which we conclude that increasing L does not increase the accuracy significantly. Thus the scaling of L to larger value gives essentially the same result as L=2 and we can use L=2 to obtain the phase diagram in the main text.§.§ Average Energy and Extrapolation of M The simple update and coarse-graining TRG steps are repeated until the average energy E (Fig. <ref>) is converged for given D. To obtain the phase boundary, we apply the finite-size extrapolations of M using second-order polynomial fit in 1/D to infinite D <cit.>. One example at ϕ=15^∘ is shown in Fig. <ref>. Suppression of the magnetization to zero as D→∞ suggests a quantum paramagnetic region. §.§ Results for Finite Anisotropy We apply the same extrapolations of M for different anisotropy η (Fig. <ref>), which shows that the quantum paramagnetic region persists away from the Heisenberg limit η=1. Specifically, for η< 1, the quantum paramagnetic region remains robust for a large region, e.g., down to η=0.5. While for η>1, long range order is preferred when η is increased to η = 1.1. This seems to suggest that the Heisenberg limit is close to the upper limit of the quantum paramagnetic region. 7 fxundefined [1]ifx#1 fnum [1] #1firstoftwosecondoftwo fx [1] #1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0] ` 12 `$12 `&12 `#12 `1̂2 `_12 `%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty [Luttinger and Tisza(1946)]LuttingerTisza2 authorauthorJ. M. Luttinger and authorL. Tisza,10.1103/PhysRev.70.954journaljournalPhys. Rev. volume70,pages954 (year1946)NoStop[Jiang et al.(2008)Jiang, Weng, and Xiang]simpleup2 authorauthorH. C. Jiang, authorZ. Y. Weng, and authorT. Xiang,10.1103/PhysRevLett.101.090603journaljournalPhys. Rev. Lett. volume101, pages090603 (year2008)NoStop[Corboz et al.(2010)Corboz, Jordan, and Vidal]nnnsimple2 authorauthorP. Corboz, authorJ. Jordan, and authorG. Vidal,10.1103/PhysRevB.82.245119journaljournalPhys. Rev. B volume82, pages245119 (year2010)NoStop[Levin and Nave(2007)]LevinTRG2 authorauthorM. Levin and authorC. P. Nave,10.1103/PhysRevLett.99.120601journaljournalPhys. Rev. Lett. volume99, pages120601 (year2007)NoStop[Xie et al.(2012)Xie, Chen, Qin, Zhu, Yang, and Xiang]HOTRG12 authorauthorZ. Y. Xie, authorJ. Chen, authorM. P. Qin, authorJ. W. Zhu, authorL. P. Yang,and authorT. Xiang,10.1103/PhysRevB.86.045139journaljournalPhys. Rev. B volume86, pages045139 (year2012)NoStop[Jiang et al.(2012)Jiang, Yao, and Balents]J1J2Jiang2 authorauthorH.-C. Jiang, authorH. Yao,andauthorL. Balents,10.1103/PhysRevB.86.024424journaljournalPhys. Rev. B volume86, pages024424 (year2012)NoStop[Wang et al.(2016)Wang, Gu, Verstraete, and Wen]tnj1j22 authorauthorL. Wang, authorZ.-C. Gu, authorF. Verstraete,andauthorX.-G. Wen,10.1103/PhysRevB.94.075143journaljournalPhys. Rev. B volume94, pages075143 (year2016)NoStop

http://arxiv.org/abs/1702.08517v3

Cosmological evolution of primordial black holes Bing Zhang December 30, 2023 ================================================We consider using a battery storage system simultaneously for peak shaving and frequency regulation through a joint optimization framework which captures battery degradation, operational constraints and uncertainties in customer load and regulation signals. Under this framework, using real data we show the electricity bill of users can be reduced by up to 12%. Furthermore, we demonstrate that the saving from joint optimization is often larger than the sum of the optimal savings when the battery is used for the two individual applications. A simple threshold real-time algorithm is proposed and achieves this superlinear gain. Compared to prior works that focused on using battery storage systems for single applications, our results suggest that batteries can achieve much larger economic benefits than previously thought if they jointly provide multiple services. Battery management system, frequency regulation service, power system economics, data centers § INTRODUCTIONBattery energy storage systems are becoming increasingly important in power system operations. As the penetration of uncertain and intermittent renewable resources increase, storage systems are critical to the robustness, resiliency, and efficiency of energy systems. For example, studies suggest that 22 GW of energy storage would be needed in California by 2050 <cit.> and the entire United States could require 152 GW of storage <cit.>. Much of these capacities are expected to be achieved by distributed storage systems owned by individual consumers <cit.>.Currently, the two most prominent types of consumers that own significant levels of storage are information technology companies and operators of large buildings. Companies such as Microsoft and Google use battery storages extensively in their data centers as failover to onsite local generation <cit.>. These systems are sized to the capacity of the data center: a 10 MW data center will have a storage system with the power rating of 10 MW with several minutes of energy capacity. In commercial buildings, batteries are used to smooth their load and provide backup services <cit.>. These batteries tend to be slightly smaller, but are still in the 100's of kW/kWh range.Today, despite their potential to grid services, these battery storage systems are not integrated with the power system. To a storage owner, whether a battery taking part in grid services is predominantly determined by the economic benefits of these services. For example, a data center replaces its batteries every four years or so under normal conditions <cit.>. If the battery participates in the electricity market, batteries may degrade faster and require more frequent replacements. Do the gains from the market justify the additional operational and capital costs?The question of optimally operating a battery to maximize its economic benefit is a central one and has spurred a substantial body of research. The problems include energy arbitrage, peak shaving, frequency regulation, demand response and others (e.g. see <cit.> and the references within). In the past several years, it has been recognized that because of the high capital cost of batteries <cit.>, serving a single application is often difficult to justify their investments <cit.>. In addition, picking a single application does not consider the possibility of multiple revenue streams and may leave “money on the table”. Consequently, a recent line of research has started to analyze the co-optimization of batteries for both energy arbitrage and regulation services <cit.>. In this paper, we consider the joint optimization of using a battery storage system for both peak shaving and frequency regulation for a commercial customer. Peak shaving can be used to reduce the peak demand charge for these customers and the (fast) frequency regulation is an ideal service to provide for batteries because of their near instantaneous response time. The challenge in combining these two applications lies in their vastly different timescales: peak demand charge is calculated every month on a smoothed power consumption profile (e.g. 15-minute averages), while fast frequency regulation requires a decision every 2 to 4 seconds. The key observation in our work is that serving different applications over different timescales is economically beneficial to the battery: by exploring the diversity in different applications, we can obtain a so-called superlinear gain. An example of the superlinear gain is presented in Fig. 1. It gives the annual electricity bill savings for a 1MW data center under three scenarios, using batteries for frequency regulation service, peak shaving and joint optimization. For joint optimization, we use a simple online threshold algorithm given in Section IV. While for peak shaving and regulation service, the solutions are offline optimal. The super-linear gain arises for reasons that would be explored in depth in the rest of the paper, but briefly speaking, the randomness of frequency regulation signal could contribute to more efficient peak shaving. By exploring the diversity and mutual benefit in different applications, we have this non-linear behavior.It should be noted that the key function of the existing batteries in data centers is to provide backup capacity. The proposed joint optimization framework is deploying only part of the battery energy capacity while a large portion of the battery energy has been reserved for backup purpose. A more detailed discussion on the division of battery for grid service and backup is provided in Section V-A. §.§ Literature ReviewThe line of literatures consider co-optimization of storage starts from <cit.>. In <cit.>, the authors analyze the economics of using storage device for both energy arbitrage and frequency regulation service. The work in <cit.> extended this “dual-use” idea by considering plug-in electric vehicles as grid storage resource for peak shaving and frequency regulation. Both works showed that dual-use of storage often leads to higher profits than single applications. However, the aforementioned works mainly rely on heuristic analysis under different price and user patterns without directly using optimization models. The work of <cit.> bridged the methodology gap by proposing a systematic co-optimization framework, which could be applied for evaluating different application combinations at different timescales. This framework assumes that all future information is known, so it cannot be extended directly to deal with potential uncertainties from energy and ancillary service markets (e.g., price, frequency regulation signal, etc.).To deal with uncertainties, <cit.> formulate the battery co-optimization problem as a stochastic program. In <cit.>, stochastic programming was solved to obtain hourly optimal decisions. The work in <cit.> included applications of different time-scales in its optimization and tackled computational challenges by taking advantage of the problem's nested structure. §.§ Our ContributionsOur work is close in spirit to <cit.>, which captures both the future market uncertainties and timescale difference of multiple applications. However, compared with <cit.>, our work contributes in two significant ways: * We propose a joint optimization framework for batteries to perform peak shaving and provide frequency regulation services. This framework accounts for battery degradation, operational constraints, and the uncertainties in both the customer load and regulation signals. All of the previous works, to our knowledge, do not include the operational cost of batteries in their optimization models, which can potentially lead to aggressive charging/discharging responses and severely suboptimal operations <cit.>. Since batteries cycle multiple times a day when used for frequency regulation and peak shaving, the degradation effect plays an important role in determining their operations.* We show that there is a superlinear gain: where the revenue from joint optimization is larger than the sum of performing the individual applications. We quantify this gain using real world data from two large commercial users: a Microsoft data center and the University of Washington EE & CSE building. Figure <ref> gives an example daily load profile for both cases. The superlinear gain is fundamentally different from previous observations in <cit.> which only compared the revenue from co-optimization with one single application rather than the sum of the applications. The results in <cit.> hinted at the relationship between co-optimization revenue and the sum of multiple revenue streams, while mainly focusing on the trade-off between different applications and their “subadditivity”. The key observation in our work is that batteries can achieve much larger economic benefits than previously thought if they jointly provide multiple services by exploring the diversity of different applications. The rest of the paper is organized as follows. Section <ref> provides some background on electricity market rules and battery cell degradation. Section <ref> gives the joint optimization formulation. In Section <ref>, we propose a simple threshold control algorithm which achieves the superlinear gain in real time. Section <ref> analyzes the superlinear benefits using both real data and synthetic data. Finally, Section <ref> concludes the paper and outlines directions for future work. § PROBLEM FORMULATION This section provides some basic definitions and the detailed model setup.We consider a finite time horizon partitioned into T discrete intervals, indexed by t ∈{1,2,...,T}.This section sets up the overall optimization problem in three steps. First, we explain how the electricity bill is calculated for a large commercial user. Then we focus on two potential applications of using batteries: peak shaving and frequency regulation. Then we introduce the battery degradation model considered in this paper. §.§ Electricity Bill of Commercial UsersWe consider commercial consumers whose electricity bill consists of two parts: energy charge and peak demand charge. Let s(t) be the power consumption at at t and t_s be the size of a time step. Then the energy charge is given by:J^elec = λ_elec∑_t=1^T s(t) · t_s,where λ_elec is the price of energy with a unit of $/MWh. The peak demand charge is based on the maximum power consumption. In practice, this charge is calculated from a running average of power consumption over 15 or 30 minutes. Let s̅(t) denote the smoothed demand and then the peak demand charge can be written as,J^peak = λ_peakmax_t = 1,2,...,T [s̅(t)].For the rest of the paper, the time step size t_s is absorbed into the price coefficients for simplicity. Hence, the total electricity bill for a commercial user over time T is,J=λ_elec∑_t=1^T s(t)+ λ_peakmax_t = 1,2,...,T [s̅(t))] ,This cost function is convex in s(t) since it is a linear combination of linear functions and a piece-wise max function. In this paper, we investigate how to reduce the total cost (<ref>) by using battery energy storage (BES). Specifically, we consider two applications, peak shaving and frequency regulation.§.§ Peak ShavingThe peak demand charge of commercial users could be as large as their energy cost. Therefore smoothing or flattening peak demand represents an important method of reducing their electrical bills. A myriad of methods for peak shaving have been proposed in the literature, e.g., using energy storage <cit.>, load shifting and balancing <cit.>. Here we focus on using batteries. Batteries can discharge energy when demand is high and charge in other times to smooth user's consumption profiles. Let b(t) denote the power injected by the battery, with the convention that b(t)>0 represents discharging and b(t)<0 represents charging. Then s(t)-b(t) is the actual power draw from the grid. Let b=[b(1)…b(T)] be the vector of battery actions. The total electricity bill becomesJ^P = λ_elec∑_t=1^T[s(t)-b(t)] + f( b)+ λ_peakt=1...Tmax[(s̅(t)-b̅(t))]whereb̅(t) is the averaged power injection of the battery and f( b) models the degradation effect of using the battery. We note that battery degradation costs maybe somewhat complicated functions of the entire profile. More details about the battery cost model will be introduced in Section <ref>. §.§ Frequency Regulation ServiceBesides doing peak shaving, commercial users could earn revenue by providing grid services. In this paper, we consider using batteries owned by these users to participate in the frequency regulation market. In particular, we adopt a simplified version of the PJM frequency regulation market <cit.>. Fig. <ref> gives an example of the PJM fast frequency regulation (RegD) signal for 2 hours. Compared with traditional frequency regulation signals, it has a much faster ramping rate and is designed to have a zero-mean within a certain time interval, which is well aligned with the characteristics of batteries. For providing frequency regulation service, the grid operator pays a per-MW option fee λ_c to a resource with stand-by power capacity C for each hour. While during the frequency regulation procurement period, the resource is subjected to a per-MWh regulation mismatch penalty (λ_mis) for the absolute error between the instructed dispatch and the resource's actual response. Let r(t) be the normalized RegD signal, and the revenue from providing frequency regulation service over time T is:R = λ_c C · T - λ_mis∑_t=1^T|b(t)-Cr(t)| - f( b),where f( b) is again the operating cost of the battery. §.§ Battery Cell Degradation A key factor in the operational planning of battery energy storage (BES) is its operating cost, a majority of which stems from the degradation of battery cells subjected to repeated charge/discharge cycles. Different batteries exhibit different degradation behaviors, and their understanding and characterization is a major area of study (see <cit.> and references within). In this paper, we focus on lithium-ion batteries, which are one of the most popular batteries used in practice today.Modeling battery degradation is a challenging task and no single model can be used for all types of chemistry. We do not attempt to propose a single detailed model in this paper; instead, some general features of battery degradation are captured and included in our optimization framework.For instance, the authors in <cit.> showed that the capacity of the Lithium Manganese Oxide (LMO) batteries are sensitive to both the number of cycles as well as the cycle depth of discharge (DoD). The data in Fig. <ref> comes from a LMO cycle life test data in <cit.>. It shows that, if we limit the battery operation within certain DoD region to avoid the overcharge and over-discharge effects, there is a constant marginal cost for the cycle depth increase. In addition, cycle DoD is a linear function of the amount of battery charging/discharging,DoD = ∑_t∈ T_i b(t)t_s,where T_i denotes a set of timesteps belongs to the same battery degradation cycle. Thus, we could assign a constant marginal cost for the amount of battery charging/discharging:f( b) ∝λ_b |b(t)|, where λ_b is the linearized battery degradation cost co-efficient. To get λ_b, we normalize battery lifetime into the amount of energy a battery cell can process before reaching end-of-life, and prorate the battery cell cost into a per-MWh cost with respect to the charged and discharged energy.λ_b = λ_cell· 10^6/2N · (SoC_max - SoC_min)In the above formula, λ_cell is the battery cell price ($/Wh), N is the number of cycles that the battery could be operated within SoC limit [SoC_min, SoC_max].For Lithium-ion batteries based on other chemistries such as lithium iron phosphate (LiFePO_4), the loss of life per cycle is interpreted as a function of cycle numbers rather than the amount of energy charged or discharged <cit.>. This type of degradation model works well for long timescale applications considered in <cit.>. However, we consider battery for fast frequency regulation service, where the direction of b(t) changes fairly quickly. If one interprets the degradation cost as a function of the charging/discharging direction change times, the model would be too aggressive. In fact, this model suggests that a battery may die in a matter of days. Therefore, we adopt a cost that is proportional to the battery power. We limit the battery operation within certain DoD range (70%), and assign a constant marginal cost for battery energy charging and discharging. § JOINT OPTIMIZATION FRAMEWORK§.§ The Joint Optimization ModelIn this paper, we consider using a battery to provide frequency regulation service and peak shaving simultaneously, thus to boost the economic benefits. The stochastic joint optimization problem is given in (<ref>), which captures both the uncertainty of future demand s(t) and the uncertainty of future frequency regulation signals r(t). J^joint =min_C, b^ch(t), b^dc(t), y(t) λ_elec∑_t=1^TE_ s[s(t)- b(t)] + λ_peakt=1...Tmax E_ s[s̅(̅t̅)̅-b̅(̅t̅)̅]+∑_t=1^Tf(b(t))-E_ r,s[λ_c T · C - λ_mis∑_t=1^T|-s(t)+ b(t)+y(t)-Cr(t)|]s.t. b(t) = b^dc(t)-b^ch(t) ,C ≥ 0 , SoC_min≤SoC_ini+ ∑_τ=1^t[b^ch(τ)η_c-b^dc(τ)/η_d]t_s/E≤SoC_max 0 ≤ b^ch(t) ≤ P^max , 0 ≤ b^dc(t) ≤ P^max .The objective function (<ref>) minimizes the total electricity cost of a commercial user for the next day, including the energy cost, peak demand charge, battery degradation cost and frequency regulation service revenue. The optimization variables are frequency regulation capacity C, battery charging/discharging power b^ch(t), b^dc(t) and frequency regulation load baseline y(t). Participants in frequency regulation market should report a baseline y(t) to the grid operator ahead of their service time <cit.>. For a commercial user, the baseline y(t) is its load forecasting. Constraint (<ref>) guarantees a non-negative frequency regulation capacity bidding. (<ref>), (<ref>) and (<ref>) represent the battery SoC limit and power limits. §.§ BenchmarkTo show the gain of joint optimization, we describe two benchmark problems: the offline (deterministic) peak shaving problem and the offline (deterministic) frequency regulation service problem. In these benchmarks, we assume complete knowledge of the future. In essence, the benchmarks here represent the best possible performance of any algorithms that solve these problems individually.The offline peak shaving problem is: J^p = min_b(t)λ_elec∑_t=1^T[s(t)-b(t)] + λ_peakt=1...Tmax[s̅(̅t̅)̅-b̅(̅t̅)̅] +∑_t=1^T f(b(t))s.t. b(t) = b^dc(t)-b^ch(t) ,SoC_min≤SoC_ini+ ∑_τ=1^t[b^ch(τ)η_c-b^dc(τ)/η_d]t_s/E≤SoC_max 0 ≤ b^ch(t) ≤ P^max , 0 ≤ b^dc(t) ≤ P^max . The above problem is convex in terms of b. We solve it and denote the optimal bill value as J^p. The offline frequency regulation problem is: R^* = max_C, b(t)λ_c T · C - λ_mis∑_t=1^T|b(t)-Cr(t)| - ∑_t=1^T f(b(t))s.t. b(t) = b^dc(t)-b^ch(t) ,C ≥ 0,SoC_min≤SoC_ini+ ∑_τ=1^t[b^ch(τ)η_c-b^dc(τ)/η_d]t_s/E≤SoC_max 0 ≤ b^ch(t) ≤ P^max , 0 ≤ b^dc(t) ≤ P^max .The above regulation revenue maximization problem does not consider the effect of providing frequency regulation service on electricity bills. Recall that frequency regulation is a service managed by grid operators, while as an end consumer, the commercial user's electricity supply contracts with the utility is unchanged, thus the user still subjects to the energy and peak demand charge. Therefore, the overall electricity bill J^r is,J^r = λ_elec∑_t=1^T[s(t)-b^r(t)] + λ_peakt=1...Tmax[s̅(̅t̅)̅-b̅^r(t)] -R^* ,where b^r(t) is the optimal battery responce for frequency regulation service and R^* the optimal service revenue.Both of the benchmark problems are convex because all of the constraints are linear and objectives are an addition of convex functions (pointwise maximum is a convex function). To solve these problems, we use the CVX package for Matlab <cit.>, a generic package for solving convex problems. We used a 2.5 GHz Intel Core i7 Macbook with 16 GB memory.The problem size can be fairly large, since the time resolution is 4s. But even for an 8-hour horizon, the problem can be solved in about 10 minutes. §.§ The Superlinear Gain Our results highlight that a superlinear gain can often be obtained: the saving from the stochastic joint optimization can be larger than the sum of two benchmark optima. In mathematical form, superlinear gain denotes the following phenomenon, which often holds in practice,J-J^joint > (J-J^r)+(J-J^p) ,where the left side of Eq. (<ref>) is the saving from joint optimization, and the right side represents the sum of savings from two benchmark problems. The key observation in such case is the “super-additivity”. The revenue from co-optimization of multiple applications is not only higher than the revenue from any single application (which may be obvious), but also higher than the sum of revenues from all individual applications.We provide an example to demonstrate the superlinear gain. Table <ref> gives the daily electricity bill under four scenarios for a 1MW data center: the original bill (batteries are left idle), using battery only for frequency regulation service, using battery only for peak shaving, and using battery for both services (detailed algorithm discussed in Section <ref>). The bill savings are highlighted, from which we observed that saving from joint optimization is larger than the sum of each individual applications. The load curve and frequency regulation signals for that day are given in Fig. 2a and Fig. <ref>. § ONLINE BATTERY CONTROLThe previous section demonstrates that battery joint optimization could have superlinear gains. In this section, we propose an online battery control algorithm for joint optimization. The challenge in combining peak shaving and frequency regulation service together lies in their vastly different timescales. To deal with the timescale difference, we divided the optimization problem into two stages, 1) day-ahead decision on peak shaving threshold and frequency regulation capacity bidding; 2)real-time control of battery charging/discharging. Fig. <ref> summarizes the workflow of the overall control algorithm. In this section, we first introduce the load prediction and scenario reduction method for solving the day-ahead optimization problem, and then a real-time battery operation algorithm is presented.§.§ Load Prediction and Scenario Generation We use a multiple linear regression (MLR) model <cit.> for day-ahead load prediction, which is simple, easy to implement in commercial user's site, yet achieves high prediction accuracy. Details of the load prediction algorithm are given in Appendix A. We used the 10-fold cross validation method to evaluate the MLR load prediction model, and the resulting mean absolute percentage error (MAPE) is 3.7% for Microsoft data center load and 2.3% for University of Washington EE & CSE building. MAPE is a measure of prediction accuracy, which is calculated by averaging the absolute deviation between real value and prediction divided by the actual value.To deal with the uncertainty of future regulation signal, a scenario-based method is implemented. Here, we use one-year historical data to empirically model the distribution of regulation signals. Each daily realization of the regulation signal is called a “scenario”, and thus we obtain 365 scenarios. We applied the forward scenario reduction algorithm in <cit.> to select the best subset of scenarios. We set the number of selected scenarios as 10, which strives for a balance between performance and computational complexity by simulation results. Therefore, we have a set of 10 scenarios for frequency regulation signals denoted as Ω and each scenario associated with a realization probability ω_i, which in total compose the uncertainty set of frequency regulation signal.We solve the stochastic problem in (<ref>) using the load prediction ŝ(t) and contructed regulation signal uncertainty sets Ω. Define the optimal battery response and frequency regulation capacity as b^*(t) and C^*, then the optimal peak shaving threshold U^* is,U^* = t=1...Tmax[ŝ(t)-b̅^*(t)] , §.§ Real-time Control for Battery Charging/Discharging Section <ref> describes how to make day-ahead decisions on capacity bidding and peak threshold. Here we introduce a simple real-time battery control algorithm for joint optimization. It is computationally efficient, which only requires the measurement of battery's real-time state of charge (SoC) and achieves near-optimal performance compared with the offline optima with perfect foresight. Although more sophisticated methods such as model predictive control <cit.> or dynamic programming <cit.> have been proposed in the previous literature, they are not needed in this case given the near-optimal performance and high computational efficiency of the proposed online control method. The intuition for the real-time joint optimization control algorithm comes from the optimal battery control algorithm for frequency regulation service. Recall the benchmark frequency regulation problem with objective (<ref>): under linear battery cost model, given a fixed capacity bidding C, we have a simple yet optimal real-time battery control method. Theorem <ref> describes the optimal control algorithm for batteries providing frequency regulation service. Assume λ_b < λ_mis.[Otherwise the battery would not be used at all.] If the marginal battery charging/discharging cost is constant within the operation region, that is, f(b(t)) = λ_b |b(t)|. For a given capacity C, the optimal battery response b^*(t) for providing frequency regulation service is: * min{Cr(t), P^max, η_d [soc(t)-SoC_min]E/t_s}, if r(t)≥ 0 * max{Cr(t),-P^max ,[SoC(t)-SoC_max] E/η_c t_s},if r(t) < 0where SoC(t) of the battery state of charge at the beginning of time step t. The proof of Theorem <ref> is given in the Appendix C. As the theorem shows, when the marginal operation cost for battery charging/discharging is constant, the optimal battery control policy is a simple threshold policy. Following Theorem <ref>, we propose a real-time control algorithm for joint optimization in Algorithm <ref>.Table <ref> gives a comprehensive comparison between the simple online control algorithm and the offline optima with perfect foresight based on half year of simulation results of Microsoft data center and one-year data of UW EE & CSE building. Both results show that by implementing the simple threshold algorithm, we can achieve near-optimal performance compared with the perfect foresight case.§ SIMULATION RESULTSWe provide a case study using half year power consumption data from Microsoft data center and one year data from University of Washington EE & CSE building. The frequency regulation signal is from PJM fast frequency regulation market<cit.>, where the considered Microsoft data center locates. We implement the simple threshold control algorithm in Section <ref> for battery joint optimization. Simulation results demonstrate that over 80% of time, we will have the superlinear benefits by joint optimization.§.§ Parameter SetupAssume that the battery optimization horizon is 1 day and the time granularity of t is 4s, so that T=4320. The electricity price is 47 $/MWh and peak demand charge is 12 $/kW per month. For frequency regulation service, suppose the capacity payment is 50$/MWh and set mismatch penalty to guarantee at least 80% performance score <cit.>. The BES for optimization is Lithium Manganese Oxide (LMO) battery, with high power capacity and low energy capacity. Within the SoC operation region SoC_min = 0.2 and SoC_max = 0.8, LMO battery has a constant marginal degradation cost with regard to how much energy is charged and discharged.In this work, we consider using existing batteries in commercial users, e.g., the backup batteries in data centers, to participate in power market and reduce users' electricity bills. The key function of these batteries for users is to provide backup capabilities and the proposed joint optimization framework is deploying only part of the battery energy capacity. We assume the overall battery has a 1MW power capacity and 15 minutes energy capacity, which is a typical size of an industrial-scale grid-tied battery. Then, different portions of the total energy capacity are considered for grid service, 3 minutes, 5 minutes and 10 minutes respectively. The results are presented in Table <ref>, where the metrics for comparison under different scenarios are the annual bill savings and battery life expectancy.An aggressive user may try to replace their battery in a yearly basis for the largest bill reduction. More likely, for a building or a data center, a 3 year cycle is preferred. In fact, most data centers already replace their batteries every 3 to 4 years for reliability reasons <cit.>, so using 3 minutes of the battery capacity for grid services would lead to considerable gains without any additional burdens. Of course, the remaining portion of battery energy storage is reserved for emergency backup.Therefore we assume for joint optimization usage, the battery power rating P^max is 1MW, energy capacity E is 3 minute,and battery cell price is 0.5 $/Wh. Accoring to Fig. <ref>, the LMO battery can be operated for N = 10,000 cycles when the average cycle DoD is 60%. Using Eq. (<ref>), we calculate the battery degradation cost as 83$/MWh.In order to evaluate the performance of the proposed battery joint optimization algorithm, we compare the savings from joint optimization with the sum of savings from benchmark peak shaving and frequency regulation service. A criteria q (joint optimization saving ratio) is defined as below,q = (J-J^joint)-[(J-J^r)+(J-J^p)]/J ,which describes the percentile of superlinear saving compared to the original bill.§.§ Results for Synthetic Load: Peak Shape and Superlinear GainIn the previous sections (Figure 1 and Table II), we observed that by doing battery joint optimization, we have the superlinear gain. One natural question may come up, why we have the superlinear gain? Before we dive into more simulations on real data, we pick a simple rectangle peak where the base load is 0.5 MW, peak load is 1MW in this section for analysis, in order to better understand the conditions that lead to superlinear gains. We change the duration of peak from 3 minutes (a sharp peak) to 15 minutes (a flat peak) in order to study the effect of peak shape on the probability of superlinear gain.Intuitively, the superlinear gain is related to the shape of demand curve. Consider two different peaks, a narrow peak (Fig. <ref>) and a wide peak (Fig. <ref>), we find that the main difference lies in the peak shaving part. For a 3 minute short-time peak, the battery could shave a large portion of the peak before hitting the SoC bound (Fig.<ref> (b)). Thus, we save a lot from only doing peak shaving and the two applications do not interact much , and there is no superlinear saving. However, when the peak duration is long, it takes more battery energy to shave the same height off the peak. As seen from Fig.<ref> (b), the battery doesn't respond much in the peak shaving only case because the cost of using battery gets close (or even exceeds) the saving from reduced peak demand charge. This argument is verified by Fig.<ref> (d), where we find only doing peak shaving does not reduce the bill much. But if we consider joint optimization, the randomness of regulation signal helps break down the one flat peak into several short-time peaks, and we could save more from doing peak shaving on top of providing regulation service. This is where the superlinear saving comes from. §.§ Results for Real-life Data: Microsoft Data Center and UW EE & CSE BuildingThis section conducts simulations based on real-life data from Microsoft data center and UW EE & CSE building. Table <ref> and <ref> summarize the simulation results. We consider using a 1MW, 3 minutes battery for grid service, and the reported numerical results are achieved by the implementing the proposed simple online control algorithm. For a 1MW data center with $488,370 annual electricity bill, the cost saving by joint optimization is around $52,282 (10.72%), with $13,234 extra saving compared with the sum of benchmark optima. For UW EE & CSE building, 362 out of the 365 days, we have the superlinear gain. The annual electricity bill for UW EE & CSE building is around $359,634, from which we save $44,420 (12.35%) by implementing battery joint optimization. The superlinear gain is $14,061 per year. To link the analysis of synthetic load in the previous section to the real-life cases of data center and UW EE & CSE building, we preform a statistical analysis of peak durations. We plot the Cumulative Distribution Function (CDF) of peak duration for the data center and UW EE & CSE building in Fig. <ref>, where the average peak duration for data center is 0.75h (about 45 minutes) and 8.33h for the building.According to the observations in Figures 6 and 7, the proposed battery joint optimization has a larger gain for flat peaks compared to sharp peaks. Since the randomness of regulation signal helps break down one flat peak into several short-time peaks, we could save more from doing peak shaving on top of providing frequency regulation service. The exact definition of “long” and “short” peaks depend on the size of the battery. For a 3 minute battery, if the peak is shorter than 3 minutes, then performing joint optimization is not critical and we do not have a superlinear gain. On the other hand, for a peak that is longer than 3 minutes, it is important to use the regulation signal to break it up into smaller peaks. Therefore, both case studies have high superlinear gain probability, which is greater than 80%. The superlinear gain ratio of UW building (99%) is higher than the ratio of the data center (82.5%) since there are virtually no peaks shorter than the battery capacity in the former and a still a few short peaks in the latter. §.§ Sensitivity AnalysisHere, we preform sensitivity analysis about how different price settings, including different demand charge prices λ_peak, battery degradation costs λ_b and regulation payments λ_c, influence the superlinear gain ratio. In order to quantatively evaluate the conditions when superlinear gain will happen and generalize the analysis to all potential scenarios, we pick a simple load curve with a rectangle peak (base load 0.5 MW, peak load is 1 MW, peak duration 15 minutes) and a truncated Gaussian signal as frequency regulation signal, with μ = 0, σ^2 = 0.12 (variance of the PJM RegD signal) and range [-1,1]. The simple rectangle peak and synthetic frequency regulation signal are given in Fig. <ref>. Fig. 10a shows how the chance of having superlinear gain changes with regard to battery cell price and peak demand charge. The probability of superlinear gain increases as the battery cell price goes down, or as the peak demand charge goes up. The “physical origin” of the superlinear gain is the positive interaction between peak shaving and frequency regulation service. Since the randomness in the frequency regulation signal breaks the flat peak into several smaller peaks, more savings are obtained by performing peak shaving on the top of frequency regulation. Therefore, as the peak demand price goes up, it yields more economic benefits to jointly optimize the two applications. Similarly, the as the battery prices decreases, it can be used more aggressively for both applications. As battery prices continue to decrease in the future, the benefits of joint optimization will increase.Fig. 10b demonstrates how the probability of having superlinear relates to battery cell price and regulation capacity payment. The chance of having superlinear gain is the highest when both the battery cell price and regulation capacity payment are low. When the capacity payment is high enough, it yields much more economic benefits to provide frequency regulation service than peak shaving. In such condition, theprobability of having superlinear gain decreases.§ CONCLUSION This paper addresses using battery storage in large commercial users to reduce their electricity bills. We consider two sources of cost savings: reducing the peak demand charge and gaining revenue from participating in frequency regulation market. We formulate a framework that jointly optimizes battery usage for both of these applications. Surprisingly, we observe that a superlinear gain can often be obtained: the savings from joint optimization can be larger than the sum of the individual savings from devoting the battery to one of the applications. We also developed an online control algorithm which achieves the superlinear gain. The battery degradation model presented in this paper is in simplified linear form, which applies to certain battery operation range. Incorporating a more general and accurate battery degradation model, such as cycle-based degradation model <cit.> into the joint optimization framework might be an interesting direction for future work. § COMMERTIAL USER'S LOAD PREDICTION Solving the stochastic joint optimization problem in (<ref>) requires accurate short-term load forecasting (STLF) for the next 24 hours. A lot of research has been done in the area of STLF. There are two major factors determine the quality of load prediction, input features and prediction model. On the one hand, selecting features or a group of features which affect the future load most is important. The input features mainly include the effect of nature (eg. temperature) and the effects of human activities (calendar variables, e.g., business hours), and the interaction of above two factors. On the other hand, deciding which kind of models to forecast future load is also crucial. People have been adopting or developing various techniques for day-ahead load forecasting, including regression, time series analysis, neural networks, support vector machine and a combination of the above methods (see <cit.> and references within).In this paper, we used a multiple linear regression (MLR) model that takes 𝒳={trend, temperature forecasting (TMP), month, Hour × TMP, month × TMP, day × Hour, adjacency day's load, weekend and holiday effect, recent similar days' average} as input, and use the following MLR model to predict the power demand for next 1 day. Fig <ref> presents the day-ahead load prediction result for a data center. Y = β_0 + β_1 × Trend + β_2 × TMP + β_3 × Month+β_4 × Hour × TMP + β_5 × month × TMP+ β_6 × day × Hour + β_7 × Load(day-1)+ β_8 × weekend +β_9 × holiday + β_10×L̅o̅a̅d̅(day-1) § FREQUENCY REGULATION SIGNAL SCENARIOS REDUCTION In order to solve the stochastic joint optimization problem in (<ref>), we also need to model the uncertainty of future regulation signals. In this paper, we use one-year historical data to empirically model the distributions of regulation signals. Each daily realization of the regulation signal is called a “scenario”, and thus we obtain 365 scenarios.Because a large number of scenarios will reduce the computational tractability of the joint optimization problem, it is useful to choose a smaller subset of scenarios that can well approximate the original entire scenario set. We applied the forward scenario reduction algorithm in <cit.> to select the best subset of scenarios, and assign new probabilities to the selected scenarios. The key idea of scenario reduction is to pick a subset of scenarios which preserve as much information as the original set. We set the number of selected scenarios as 10, which strives for a balance between performance and computational complexity by simulation. For visualization clarity, we plot 4 out of the 10 selected scenarios in Fig. <ref>.§ PROOF OF THEOREM 1 Here we provide a detailed proof of Theorem 1 in <ref>. Under linear battery cost model, it is obvious that b^*(t) and r(t) always have the same sign. Or equivalent saying,b(t) and r(t) are always both positive or both negative. Based on the relative sizes of coefficients and sign, there are 5 cases to be considered: λ_b < λ_mis r(t) ≥ 0 b(t) ≥ Cr(t)(i) b(t) < Cr(t)(ii) r(t) <0 b(t) ≥ Cr(t)(iii) b(t) < Cr(t)(iv) λ_b ≥λ_mis (v) (i). λ_b < λ_mis, r(t) ≥ 0 and b(t) ≥ Cr(t) In this case, b(t) ≥ 0, battery is discharging. And the objective function (<ref>) becomes, C,b(t)maximizeλ_c C + 1/TE {-(λ_mis+λ_b)∑_t=1^Tb(t)+λ_mis∑_t=1^T Cr(t)}, Notice that the coefficient in front of b(t) is negative. In order to maximize the objective, we need to minimize b(t) under the following constraints: b(t) ≥ 0 b(t) ≥ Cr(t) b(t) ≤ P^max b(t) ≤η_d [soc(t)-SoC_min]E/t_s * If either Cr(t) > P^max or Cr(t) > η_d [soc(t)-SoC_min]E/t_s, there is no feasible solution for b(t). * If Cr(t) ≤ P^max and Cr(t) ≤η_d [soc(t)-SoC_min]E/t_s, we have the optimal b^*(t)=Cr(t). (ii). λ_b < λ_mis, r(t) ≥ 0 and b(t) < Cr(t) In this case, b(t) ≥ 0, battery is discharging, and the objective function (<ref>) becomes, C,b(t)maximizeλ_c C + 1/TE {(λ_mis-λ_b)∑_t=1^Tb(t)-λ_mis∑_t=1^T Cr(t)}, Notice the coefficient in front of b(t) is (λ_mis-λ_b), which is positive in this case. So in order to maximize the objective, we need to maximize b(t) under the following constraints: b(t) ≥ 0 b(t) < Cr(t) b(t) ≤ P^max b(t) ≤η_d [soc(t)-SoC_min]E/t_s So b^*(t) = min{Cr(t),P^max, η_d [soc(t)-SoC_min]E/t_s}. Summarizing case (i, ii), we get if λ_b < λ_mis, r(t) ≥ 0 b^*(t) = min{Cr(t),P^max, η_d [soc(t)-SoC_min]E/t_s} . (iii). λ_b < λ_mis, r(t) < 0 and b(t) ≥ Cr(t) In this case, b(t) < 0, battery is charging, and the objective function (<ref>) becomes, C,b(t)maximizeλ_c C + 1/TE {(-λ_mis+λ_b)∑_t=1^Tb(t)+λ_mis∑_t=1^T Cr(t)}, The coefficient in front of b(t) is (-λ_mis+λ_b), which is negative. So in order to maximize the objective, we need to minimize b(t) under the following constraints: b(t) ≤ 0 b(t) ≥ Cr(t) b(t) ≥ -P^max b(t) ≥[SoC(t)-SoC_max] E/η_c t_s The minimal b(t) is optimal, b^*(t) = max{Cr(t),-P^max, [SoC(t)-SoC_max] E/η_c t_s}. (iv). λ_b < λ_mis, r(t) < 0 and b(t) < Cr(t) In this case, b(t) < 0, battery is charging, and the objective function (<ref>) becomes, C,b(t)maximizeλ_c C + 1/TE {(λ_mis+λ_b)∑_t=1^Tb(t)-λ_mis∑_t=1^T Cr(t)}, The coefficient in front of b(t) is positive, so in order to maximize the objective, we need to maximize b(t) under the following constraints: b(t) < 0 b(t) < Cr(t) b(t) ≥ -P^max b(t) ≥[SoC(t)-SoC_max] E/η_c t_s * If either Cr(t) <-P^max or Cr(t) < [SoC(t)-SoC_max] E/η_c t_s, there is no feasible solution for b(t). * If Cr(t) ≥ -P^max and Cr(t) ≥[SoC(t)-SoC_max] E/η_c t_s, we have the optimal b^*(t)=Cr(t). Summarizing case (iii, iv), we get if λ_b < λ_mis, r(t) < 0, b^*(t) = max{Cr(t),-P^max, [SoC(t)-SoC_max] E/η_c t_s} . (v). λ_b ≥λ_mis We know that b(t) and r(t) always have the same sign, so the objective function (<ref>) could be expressed as, λ_c · C - 1/T E{∑_t=1^Tλ_mis ||b(t)|-C|r(t)|| + ∑_t=1^Tλ_b |b(t)|} , For each time step t, we take derivative of the objective function w.r.t. |b(t)|, δ J/δ |b(t)|= - (λ_mis·1⃗_|b(t)>C|r(t)|| -λ_mis·1⃗_|b(t)<C|r(t)|| + λ_b) = - λ_mis·1⃗_|b(t)>C|r(t)|| +λ_mis·1⃗_|b(t)<C|r(t)|| - λ_b≤λ_mis·1⃗_|b(t)<C|r(t)|| - λ_b≤λ_mis - λ_b ≤ 0 Since δ J/δ |b(t)|≤ 0, so in order to maximize J (the regulation service benefits), |b|=0. Therefore, when λ_b ≥λ_mis, b^*(t)=0 is optimal for ∀ C ≥ 0.ieeetr

http://arxiv.org/abs/1702.08065v3

FERMILAB-PUB-17-061-A 0.2inORCID: http://orcid.org/0000-0001-8837-4127 dhooper@fnal.govicholis1@jhu.eduORCID: http://orcid.org/0000-0002-3805-6478 ORCID: http://orcid.org/0000-0001-9888-0971 linden.70@osu.edu ORCID: http://orcid.org/0000-0002-5387-8138 kefang@umd.edu[a]Fermi National Accelerator Laboratory, Center for Particle Astrophysics, Batavia, IL 60510 [b]University of Chicago, Department of Astronomy and Astrophysics, Chicago, IL 60637 [c]University of Chicago, Kavli Institute for Cosmological Physics, Chicago, IL 60637 [d]Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland, 21218 [e]Ohio State University, Center for Cosmology and AstroParticle Physics (CCAPP), Columbus, OH43210 [f]University of Maryland, Department of Astronomy, College Park, MD, 20742 [g]Joint Space-Science Institute, College Park, MD, 20742 Recent measurements of the Geminga and B0656+14 pulsars by the gamma-ray telescope HAWC (along with earlier measurements by Milagro) indicate that these objects generate significant fluxes of very high-energy electrons. In this paper, we use the very high-energy gamma-ray intensity and spectrum of these pulsars to calculate and constrain their expected contributions to the local cosmic-ray positron spectrum.Among models that are capable of reproducing the observed characteristics of the gamma-ray emission, we find that pulsars invariably produce a flux of high-energy positrons that is similar in spectrum and magnitude to the positron fraction measured by PAMELA and AMS-02. In light of this result, we conclude that it is very likely that pulsars provide the dominant contribution to the long perplexing cosmic-ray positron excess.HAWC Observations Strongly Favor Pulsar Interpretations of the Cosmic-Ray Positron Excess and Ke Fang^f,g========================================================================================= § INTRODUCTIONMeasurements of the cosmic-ray positron fraction by the PAMELA <cit.> and AMS-02 <cit.> experiments (as well as HEAT <cit.>, AMS-01 <cit.> and Fermi <cit.>) have identified an excess of high-energy positrons relative to the standard predictions for secondary production in the interstellar medium (ISM). This result indicates that significant quantities of ∼0.01-1 TeV positrons are produced within the local volume (the surrounding ∼ kpc) of the Milky Way. The origin of these particles has been debated, and possible sources include nearby and young pulsars <cit.>, as well as annihilating dark matter particles <cit.>. This excess could also plausibly arise in nearby supernova remnants, which may be able to produce and accelerate secondary positrons before they escape into the ISM <cit.>. From the catalog of known pulsars, Geminga (PSR J0633+1746) and B0656+14 (PSR J0659+1414, thought to be associated with the Monogem supernova remnant) are particularly attractive candidates to account for the observed positron excess. These pulsars are each relatively young (370,000 and 110,000 years, respectively) and are located within a few hundred parsecs of the solar system (250^+230_-80 and 280^+30_-30 pc, respectively <cit.>). Much of what we know about these and other pulsars is based on gamma-ray observations. Geminga, in particular, is an extremely bright source of GeV-scale emission, thought to originate from the pulsar itself <cit.>. In contrast, observations of Geminga at very high energies reveal a significantly extended morphology. In particular, the Milagro Collaboration has reported the detection of gamma-ray emission at ∼ 35 TeV with a flux of (3.77 ± 1.07) × 10^-16 TeV^-1 cm^-2 s^-1 and an extended profile with a full-width at half-maximum of 2.6^+0.7_-0.9 degrees <cit.>.Very recently, the HAWC Collaboration has presented their measurements of the Geminga pulsar, confirming its angular extension (∼ 2^∘ radius) and reporting a flux of (4.87 ± 0.68) × 10^-14 TeV^-1 cm^-2 s^-1 at 7 TeV, with a local spectral index of -2.23 ± 0.08 <cit.> (see also Refs. <cit.>). HAWC has also reported the detection of very high-energy emission from B0656+14, with a similar degree of spatial extension and with a flux and spectral index at 7 TeV of (2.30 ± 0.73) × 10^-14 TeV^-1 cm^-2 s^-1 and -2.03 ± 0.14, respectively <cit.>.These observations by HAWC and Milagro allow us to conclude that these pulsars, in fact, deposit a significant fraction of their total spindown power into high-energy leptons. Furthermore, we will show in this paper that the flux of leptons required to explain these observations is roughly equal to that required for the Geminga and B0656+14 pulsars to produce an order one fraction of the positron excess observed by PAMELA and AMS-02. The spectrum, morphology and intensity of the gamma-ray emission measured by HAWC and Milagro leave us with little choice but to conclude that nearby pulsars are likely to be the dominant source of the observed cosmic-ray positrons.§ INVERSE COMPTON SCATTERING OF VERY HIGH-ENERGY ELECTRONS AND POSITRONS NEAR PULSARS The gamma-ray emission observed from Geminga and B0656+14 by HAWC and Milagro is almost certainly generated through the inverse Compton scattering of very high-energy leptons. The angular extension of this signal rules out other scenarios, with the possible exception of pion production. A pion production origin, however, would require an unrealistically large quantity ( 10^46 erg) of 𝒪(10^2) TeV protons to be confined to the region surrounding Geminga for10^5 years. Such a scenario could also be constrained to some degree by the lack of TeV neutrinos detected by the IceCube experiment <cit.>. To study the diffusion and energy losses of electrons and positrons produced in nearby pulsars, we utilize the standard propagation equation: ∂/∂tdn_e/dE_e(E_e,r,t)= ▽⃗·[D(E_e) ▽⃗dn_e/dE_e(E_e,r,t) - v⃗_cdn_e/dE_e(E_e,r,t) ] + ∂/∂ E_e[dE_e/dt(r)dn_e/dE_e(E_e,r,t)]+ δ(r) Q(E_e,t), where dn_e/dE_e is the differential number density of electrons/positrons at a distance r from the pulsar, D is the diffusion coefficient, v⃗_c is the convection velocity, and the source term Q describes the spectrum and time profile of electrons/positrons injected into the ISM. Energy losses are dominated by a combination of inverse Compton and synchrotron losses, and are given by <cit.>: -dE_e/dt(r)= ∑_i 4/3σ_T ρ_i(r) S_i(E_e) (E_e/m_e)^2 + 4/3σ_T ρ_ mag(r) (E_e/m_e)^2≡b(E_e,r)(E_e/ GeV)^2,where σ_T is the Thomson cross section and b(r) ≈1.02 × 10^-16GeV/ s ×[ ∑_i ρ_i(r)/ eV/ cm^3S_i(E_e) + 0.224(B(r)/3 μG)^2 ]. The sum in this expression is carried out over the various components of the radiation backgrounds, consisting of the cosmic microwave background (CMB), infrared emission (IR), starlight (star), and ultraviolet emission (UV). Throughout our analysis, we adopt the following parameters: ρ_ CMB=0.260 eV/cm^3, ρ_ IR=0.60 eV/cm^3, ρ_ star=0.60 eV/cm^3, ρ_ UV=0.10 eV/cm^3, ρ_ mag=0.224 eV/cm^3 (corresponding to B=3 μG), and T_ CMB =2.7 K, T_ IR =20 K, T_ star =5000 K and T_ UV =20,000 K. For low to moderate electron energies, these parameters correspond to a value of b ≃ 1.8 × 10^-16 GeV/s. At very high energies (E_em^2_e/2T), however, inverse Compton scattering is further suppressed by the following factor: S_i (E_e) ≈45 m^2_e/64 π^2 T^2_i/(45 m^2_e/64 π^2 T^2_i)+(E^2_e/m^2_e). To solve Eq. <ref>, we calculate the distribution of the electrons and positrons that were emitted a time t ago, and then sum the contributions produced over different periods of time. Considering an injected spectrum of the form Q(E_e,t) =δ(t) Q_0 E^-αexp(-E_e/E_c), the solution to Eq. <ref> (neglecting convection) is given by: dn_e/dE_e(E_e,r,t) =Q_0 E^2-α_0/8π^3/2E_e^2L^3_ dif(E_e,t) exp[-E_0/E_c]exp[-r^2/4L^2_ dif(E_e,t)], where E_0 ≡ E_e/(1-E_e b t) is the initial energy of an electron that has an energy of E_e after a time t, and the diffusion length scale is given by: L_ dif(E_e,t)≡ [ ∫^E_e_E_0D(E')/-dE_e/dt(E') dE' ]^1/2, = [D_0/b (E_e/ GeV)^1-δ (1-δ)(1-(1-E_e b t)^1-δ)]^1/2. In the last step we had adopted a parameterization of D(E_e) = D_0 E_e^δ for the diffusion coefficient. Note that for E_e b t > 1, there are no electrons/positrons of energy E_e and the contribution to dn_e/dE_e is set to zero. To account for the time profile of the electrons and positrons injected from a given pulsar, we adopt a function proportional to the spin-down power (the rate at which the pulsar loses rotational kinetic energy through magnetic dipole braking) <cit.>: Ė =-8π^4 B^2 R^6/3 c^3 P(t)^4≈1.0 × 10^35erg/ s×(B/1.6 × 10^12G)^2(R/15 km)^6(0.23 s/P(t))^4, where B is the strength of the magnetic field at the surface of the neutron star, R is the radius of the neutron star, and the rotational period evolves as follows: P(t) = P__0 (1+t/τ)^1/2, where P__0 is the initial period, and τ is the spindown timescale: τ = 3c^3 I P__0^2/4π^2B^2 R^6≈9.1 × 10^3 years (1.6× 10^12G/B)^2 (M/1.4 M_⊙)(15 km/R)^4(P__0/0.040 sec)^2. In Eqs. <ref> and <ref>, we had adopted benchmark values for the neutron star's magnetic field, radius and mass, chosen to match Geminga's observed period and its rate of change. At energies within the range measured by MILAGRO and HAWC, inverse Compton scattering yields photons with energies not very far below that of the incident electrons and positrons, E_γ∼ E_e. Adopting this approximation, the angular profile of gamma rays generated through Inverse Compton scattering is given by: Φ_γ(E_γ=E_e,ψ)∝ ∫Ėdt∫_ losdn_e/dE_e(E_e,r,t) ρ_ rad(r) dl ∝ ∫τ^2/(t+τ)^2dt ∫_ lose^-r^2/4L^2_ dif(E_e,t) ρ_ rad(r) dl, where ψ is the angle observed away from the pulsar, and r^2=l^2+d^2-2ld cosψ, where d is the distance between the pulsar and the observer. If we adopt a uniform distribution of radiation in the vicinity of the pulsar, this reduces to a profile of the form Φ_γ(ψ) ∝exp[-d^2 sin^2 ψ/4L_ dif^2(E_e,t)]. Observations of Geminga by both Miligro and HAWC indicate that the very high energy gamma-ray emission from this source is extended over a region of a few degrees across the sky. This in turn requires a diffusion length given by L_ dif(E_e) ≃ (250 pc)sin (0.5× 2.6^∘)/2(ln(2))^1/2≃ 2.6 pc. In contrast, adopting parameters appropriate for the ISM (D_0≃ 2× 10^28 cm^2/s, δ≃ 0.4, b=1.8× 10^-16 GeV/s), we find L_ dif(E_e,t) ≃ 200 pc(35TeV/E_e)^0.3 (1-(1-E_e b t)^0.6)^1/2. Assuming conditions for cosmic ray transport that are similar to those found in the ISM, this calculation shows that we should have expected the Inverse Compton emission observed at very high-energies to be extended over a scale of ∼ 60^∘, dramatically more than the ∼ 2^∘ extension reported by both Milagro and HAWC (see Fig. <ref>).To resolve this puzzle, one might be tempted to consider the possibility that the pulsar is surrounded by a dense radiation field, which intensifies the resulting inverse Compton emission from the surrounding parsecs. The problem with this scenario, however, is that there is not nearly enough power available to generate the required density of radiation. More quantitatively, in order for a r^-2 profile of radiation to exceed the energy density of the CMB at a distance of 1 parsec from the pulsar would require an amount of energy equivalent to more than ten times the total spin-down power of Geminga. Adopting an extreme benchmark in which 100% of Geminga's energy budget is transferred into radiation, the profile of Inverse Compton emission is altered only very modestly; by less than 10% at ψ = 1^∘ (see Fig. <ref>). Based on these considerations, it does not appear that local concentrations of radiation play a significant role in explaining the angular extent of the gamma-ray emission observed from Geminga or B0656+14. A more likely solution is that the conditions that dictate cosmic-ray diffusion around Geminga and B0656+14 are very different from those found elsewhere in the ISM, leading energetic leptons to escape from the surrounding regions much slower. In order to accommodate the observed extension in this way, we require that these particles only diffuse a distance of a few parsecs before losing most of their energy. For an energy loss time of 5000 years (corresponding to E_e=35 TeV and b=1.8× 10^-16 GeV/s), this requires a diffusion coefficient of D ∼10^27 cm^2/s (see Fig. <ref>). Although this is significantly smaller than that found in the bulk of the ISM, we note that it is similar to that predicted for standard Bohmian diffusion, D_ Bohm = r_L c/3 ≈ 1.2 × 10^27 cm^2/s ×(E_e/35 TeV)(μ G/B). If pulsars such as Geminga are typically surrounded by a region with inefficient diffusion (D ≪ D_ ISM), the volume of such regions must be fairly small to avoid conflicting with secondary-to-primary ratios in the cosmic-ray spectrum as measured at Earth. In particular, if such regions have a typical radius of r_ region, then such regions will occupy roughly the following fraction of the volume of the Milky Way's disk: f∼ N_ region×4π/3 r^3_ region/π R^2_ MW× 2 z_ MW∼0.25 ×(r_ region/100 pc)^3(Ṅ_ SN/0.03 yr^-1)(τ_ region/10^6 yr)(20 kpc/R_ MW)^2(200 pc/z_ MW), where Ṅ_ SN is the rate at which new pulsars appear in the Galaxy, τ_ region is the length of time that such regions persist, and N_ region = Ṅ_ SN×τ_ region is the number of such regions present at a given time. The quantities R_ MW and z_ MW denote the radius and half-width of the Galaxy's cylindrical disk. Combined with Milagro and HAWC observations of Geminga and B0656+14, these considerations suggest 5pc r_ region 50pc, for which there will be little impact on the observed secondary-to-primary ratios (other than the positron fraction).In Fig. <ref>, we plot the gamma-ray spectrum from the region surrounding Geminga, compared to the measurements by HAWC and Milagro (at 7 and 35 TeV, respectively). In performing this calculation, we utilized the full differential cross section for inverse Compton scattering <cit.>. In the left (right) frame, we have adopted a spectrum of injected electrons with an index of α=1.5 (1.9), and in each case selected a value of E_c that provides the best-fit to the combination of these two measurements. We have also allowed for the possibility that convective winds play a role in cosmic-ray transport <cit.>, moving particles away from the pulsar at a velocity given by either v_c = 55.4 km/s × (r_ region/10 pc) (blue dashed) or v_c = 554 km/s × (r_ region/10 pc) (green solid). In these four cases, the best-fits were found for E_c=44 TeV (α=1.5, low convection), E_c=35 TeV (α=1.5, high convection), E_c=67 TeV (α=1.9, low convection), and E_c=49 TeV (α=1.9, high convection). In each case, convection dominates over diffusion in transporting cosmic rays out of the region surrounding the pulsar.In addition to their flux measurement, the HAWC Collaboration has also reported a value of -2.23 ± 0.08 for Geminga's spectral slope at 7 TeV. Among the models shown in Fig <ref>, those with a low convection velocity (v_c = 55.4 km/s × (r_ region/10 pc)) predict spectral slopes at 7 TeV of -2.47 (α=1.5) or -2.59 (α=1.9). Such values are highly inconsistent with that reported by HAWC. In contrast, for those models with a higher degree of convection (v_c = 554 km/s × (r_ region/10 pc)), we instead predict a spectral slope of -2.23 (α=1.5) or -2.32 (α=1.9), in excellent agreement with HAWC's measurement. This favors scenarios in which convection plays a very important role in the transport of high-energy leptons, especially among those with energies below ∼10 TeV. We lastly note that among this range of models, between 7.2% and 29% of Geminga's total current spindown power is being deposited into electron-positron pairs with E_e > 10 GeV. § IMPLICATIONS FOR THE COSMIC-RAY POSITRON EXCESS Although the angular extent of the emission observed from Geminga and B0656+14 by HAWC and Milagro indicates that very high-energy (E_e10 TeV) leptons are effectively trapped within a few parsecs of these sources, the same fate need not be experienced by lower energy electrons and positrons. In particular, even a modest degree of convection (i.e. the streaming of particles away from the source at a constant velocity) can remove sub-TeV leptons from the region before they lose a substantial fraction of their energy, while higher energy leptons lose the vast majority of their energy to inverse Compton scattering before escaping the same region <cit.>. To address this more quantitatively, we can compare the timescale for energy losses, τ_ loss = 1/E_e b, to that for the escape from the region, τ_ escape = r_ region/v_c. For convective winds with a velocity of v_c, particles with E_e ≪ v_c /b r_ region are left largely unaffected, while those with E_e ≫ v_c /b r_ region lose the majority of their energy before escaping. In the left frame of Fig. <ref>, we plot the fraction of energy that an electron loses before escaping a region of radius 10 parsecs for several values of v_c. It is expected that future observations by the Cherenkov Telescope Array (CTA) will provide an important test of this transition by measuring the intensity and angular extent of the ∼ 0.1-10 TeV emission from Geminga and B0656+14. Although we have focused here on a scenario in which convection is responsible for expelling sub-TeV leptons from the regions surrounding Geminga and B0656+14, other means may also be possible. For example, if we simply introduce a diffusion coefficient with no energy dependance (δ=0), we effectively mimic the effects of convection (see the right frame of Fig. <ref>). In Fig. <ref>, we plot the cosmic-ray positron fraction as measured by AMS-02, compared to the predictions from the Geminga pulsar, using the same choices of parameters as adopted in Fig. <ref>. In each frame, the solid black curve denotes the contribution from standard secondary production in the ISM, while the other curves include both this contribution and that from Geminga. We remind the reader that those models with only weak convection (dotted blue curves) do not lead to a spectral index compatible with the measurement of HAWC, and thus should be viewed as a poor fit to the data.The positron fraction presented in Fig. <ref> includes a distinctive feature at 400-500 GeV, which is a result of energy losses. More specifically, a positron with an infinite energy will be reduced over a time t to an energy of E_e=(bt)^-1, which for t=370,000 years (the age of Geminga) yields a final energy of 476 GeV. Any positrons from Geminga above this energy were injected at later times and thus have not cooled to the same extent. The main lesson from the results shown in Fig. <ref> is that when the spectral shape and overall normalization of Geminga are fixed to reproduce the results of HAWC and Milagro, this pulsar is found to generate a non-negligible portion of the observed positron fraction. That being said, the overall size of this contribution to the cosmic-ray positron flux can vary by a factor of order unity depending on the precise values of the convection velocity, v_c, and spindown timescale, τ (see Eq. <ref>) that are adopted. The impact of the convection velocity is clearly evident in Fig. <ref>. The resulting positron flux scales approximately as τ^-1 (we have adopted a value of τ=9.1× 10^3 years). Furthermore, the time profile of a pulsar's emission could plausibly depart to some extent from that predicted from standard magnetic dipole braking <cit.>, potentially altering the normalization of the positron flux predicted here, as well as the inferred age of the pulsar. Finally, in Fig. <ref>, we plot the contributions to the positron fraction from the Geminga and B0656+14 pulsars, as well as the average contribution from those pulsars located more than 500 parsecs away from the Solar System. For each source, we have adopted α=1.9, E_c=49 TeV, v_c=554 km/s × (r_ region/10 pc), and normalized their contributions with τ=4.3× 10^3 years, and adopting a total birth rate of two pulsars per century in the Milky Way.[We produce nearly identical results if we instead adopt our default value for τ≃ 9.1× 10^3 and a somewhat higher value for the convection velocity, v_c ≃1160 km/s × (r_ region/10 pc).] For other details regarding the calculation of the contribution from distant pulsars, we direct the reader to Ref. <cit.>. In reality, we expect many of these parameters to vary from pulsar-to-pulsar, making a detailed prediction of this kind difficult and possibly unreliable. That being said, this exercise clearly demonstrates that in light of the measurements by HAWC and Milagro, it appears very likely that a sizable fraction of the observed positron excess originates from this class of sources. In addition, we note that it was recently shown that the stochastic acceleration of cosmic-ray secondaries in supernova remnants is also likely to contribute to the local positron flux <cit.>.§ SUMMARY AND CONCLUSIONS In this paper, we have made use of measurements by the very high-energy gamma-ray telescopes HAWC and Milagro to better understand and constrain the injection of high energy electrons and positrons from the nearby pulsars Geminga and B0656+14. The angular extension of the  TeV gamma-ray emission observed from these pulsars indicates that very high-energy leptons are effectively trapped within the surrounding several parsecs around these sources. Furthermore, their very high-energy gamma-ray spectra indicate that lower energy leptons are able to escape more easily, suggesting the presence of strong convective winds, with velocities of several hundred kilometers per second. In models that are able to reproduce the characteristics of the gamma-ray emission reported by HAWC and Milagro, these pulsars invariably provide a significant contribution to the local cosmic-ray positron spectrum, and thus to the measured positron fraction. Although it is not yet possible to precisely predict the normalization of the positron flux from these sources, these results show that Geminga and B0656+14 are expected to generate a significant fraction of the observed high-energy positron flux. If we make the entirely reasonable assumption that other pulsars in the Milky Way behave in a similar fashion to Geminga and B0656+14, we find that it is very likely that pulsars are responsible for much, if not the entirety, of the reported positron excess. An important test of this conclusion will come from future Imaging Atmospheric Cherenkov Telescopes (IACTs), such as the Cherenkov Telescope Array (CTA). Although existing IACTs have not yet reported any significant detection of TeV-scale emission from Geminga or B0656+14 <cit.>, next generation telescopes will be far better suited to detect emission that is extended over the angular scales reported by Milagro and HAWC. Such a measurement is expected to be able to confirm the transition to convection-dominated transport at energies below several TeV, and enable us to produce a more detailed determination of the spectrum of electrons and positrons that are injected from this class of sources. Acknowledgments. DH is supported by the US Department of Energy under contract DE-FG02-13ER41958. Fermilab is operated by Fermi Research Alliance, LLC, under contract DE- AC02-07CH11359 with the US Department of Energy. IC acknowledges support from NASA Grant NNX15AB18G and from the Simons Foundation. TL acknowledges support from NSF Grant PHY-1404311.JHEP

http://arxiv.org/abs/1702.08436v1

†]Erik Burman ⋆]Peter Hansbo [†]Department of Mathematics, University College London, London, UK–WC1E6BT, United Kingdom [⋆]Department of Mechanical Engineering, Jönköping University, SE-55111 Jönköping, SwedenDeriving robust unfitted finite element methods from augmented Lagrangian formulations [ December 30, 2023 ====================================================================================== In this paper we will discuss different coupling methods suitable for use in the framework of the recently introduced CutFEM paradigm, cf. Burman et al. <cit.>. In particular we will consider mortaring using Lagrange multipliers on the one hand and Nitsche's method on the other. For simplicity we will first discuss these method in the setting of uncut meshes, and end with some comments on the extension to CutFEM. We will, for comparison, discuss some different types of problems such as high contrast problems and problems with stiff coupling or adhesive contact. We will review some of the existing methods for these problems and propose some alternative methods resulting from crossovers from the Lagrange multiplier framework to Nitsche's method and vice versa. § INTRODUCTION Recently there has been increased interest in unfitted finite element for the imposition of boundary conditions or more generally for the coupling of physical systems over an interface. The unfitted discretization is then made as independent as possible of the geometric description of interfaces and domain boundaries in order to minimize the complexity of mesh generation.One such method is the Cut Finite Element Method (CutFEM) <cit.>, the goal of which is to retain the accuracy and robustness of a standard finite element method. To reach this aim, stabilization techniques are applied to make both the accuracy of the approximation and the system condition number independent of the mesh/boundary intersection and physical parameters. Thanks to this robustness of the discretization,powerful linear algebra techniques developed for finite element methods are made available for solving the linear systems obtained by the CutFEM discretization. In the CutFEM approach the boundary of—or interfaces in—a given domain is represented on a backgroundgrid, e.g., using a level set function, and the background grid isalso used to discretize the governing partial differentialequations. CutFEM builds on a general finite element formulation for the approximationof partial differential equations, in the bulk and on surfaces (interfaces or boundaries), that can handle elementsof complex shape and where boundary and interface conditions are built into the discrete formulation. CutFEM requires only a low-quality, even non-conforming, surface mesh representations of the computational geometry, thus mitigating mesh generation work.Unfitted methods typically use either Lagrange multipliers or Nitsche's method for the mortaring on interfaces or boundaries and it has been shown that Nitsche's method can be derived from a stabilized Lagrange muliplier method, due to Barbosa and Hughes <cit.> after static condensation of the multiplier <cit.>, cf. Section <ref>. Another, fluctuation based, stabilization of the Lagrange multiplier was proposed in <cit.> and further developed in the works <cit.> and also <cit.> of this collection. For this type of methods the multiplier typically may not be eliminated, which can be a disadvantage; however, in some situations it is desirable to define the multiplier on a different space anyway. Regardless of the relative virtues of the two approaches it is interesting to compare them and see how developments for one type of methods can be exploited in the context of the other. Similarly as in <cit.> we will here be interested in deriving methods using Lagrange multipliers and then recover the associated Nitsche method by formal elimination of the multiplier. However in our case we will base the discussion on the concept of augmented Lagrangian methods, which has recentlybeen successfully applied in the context of contact problems using Nitsche's method <cit.> and Lagrange multipliers <cit.>. The idea behind the augmented Lagrangian is to add a least squares term on the constraint to the Lagrangian functional of the constrained optimization problem on the discrete level. Provided this modification is not too strong it is expected to improve the conditioning as well as improving the control of the constraint compared to the case where only the multiplier is used to enforce the constraint. For early work on augmented Lagrangian methods in computational methods for partial differential equations we refer to Glowinski and le Tallec <cit.> or Fortin and Glowinski <cit.>. However, in cases where the problem depends on physical parameters that have some singular behavior, the augmented Lagrangian can lead to a severely ill-conditioned problem. This is typically the case for problems with high contrast in the diffusivity or strongly bonded adhesive problems. The ill-conditioning results from the fact that in the singular limit the least squares term on the constraint blows up, leading to ill-conditioning and also locking, unless the mesh-size is small enough to resolve the singularity. In many problems such a resolution is unfeasible and it is then useful to relax the size of the least squaresterm. This idea has been implicitly used in a number of works, starting with the paper <cit.> on debonding problems using Nitsche's method on unfitted meshes and then further developed for free flow porous media coupling in <cit.> and boundary conditions in <cit.>.A recent inventive use of this frameworkwas proposed in <cit.> where it wasapplied to coupling schemes in fluid-structure interaction.All of these works are concerned with Nitsche type formulations.Nevertheless there seems to have been no attempts at exploring these ideas directly in the framework of augmented Lagrangian methods. Our main objective in this paper is to study some model problems, exhibiting the typical parameter dependent behaviour, in the framework of augmented Lagrangian methods.The idea is to first revisit the imposition of boundary conditions with Lagrange multipliers and Nitsche's method. Then we consider three different situations of domain decomposition. Here we assume that the problem is set on a domain Ω with two systems, defined in the subdomains Ω_1 and Ω_2, that are coupled over a smooth boundary Γ.For simplicity we consider Poisson type problems and do not discretize the boundary Γ, that is, we consider the semi-discretized setting. The discussion can easily be extended to for instance compressible elasticity. We will consider the following three model problems: * general boundary conditions;* Poisson's equation with high contrast in the diffusion coefficient;* debonding and adhesive contact.First, in Section <ref>, we will recall, for an uncut mesh with Dirichlet boundary conditions, how least squares stabilized Lagrange multiplier methods lead toNitsche's method following <cit.>, and show how, alternatively, the augmented Lagrangian approach leads to the same formulation. Then, in Sections <ref>–<ref> we consider the three different model problems in the augmented Lagrangian framework and derive robust Nitsche methods as well as robust Lagrange multiplier methods, still formulated on uncut meshes. Some of the proposed methods appear to be new, whereas others are known in the literature and we will discuss existing results for the methods and without going into technical details we will speculate on what results are likely to carry over to the cases considered herein. In Section <ref>, we round off with some remarks considering the extension to cut finite element meshes, in particular with respect to stabilization of the discrete system, and, finally, in Section <ref>, we give a numerical example for one of the model problems.§ DERIVATIONS OF NITSCHE'S METHOD FROM LAGRANGE MULTIPLIERS §.§ Model problemLet us first consider the typical Poisson model problem of finding u such that-Δ u= f  in Ω , u=g  on Γ:=∂Ω,where Ω is a bounded domain in two or three space dimensions, with outward pointing normal n, and f and g are given functions. For simplicity, we shall assume that Ω is polyhedral (polygonal). The typical way of prescribing u=g on the boundary is to pose the problem (<ref>) as a minimization problem with side conditions and seek stationary points to the functionalℒ(v,μ) := 1/2 a(v,v)- μv-g- fv ,where fv := ∫_Ωf v dΩ, a(u,v) := ∫_Ω∇ u·∇ vdΩ,μv-g := ∫_Γμ (v-g) ds .The stationary points are given by finding (u,λ)∈ H^1(Ω)× H^-1/2(Γ) such thata(u,v)-λv = (f,v)∀ v∈ H^1(Ω), μu = μg∀μ∈ H^-1/2(Γ) .As is well known, the discretization of this problem requires balancing of the discrete spaces for the multiplier λ and the primal solution u in order for the method to be stable, for examples cf. <cit.>. In the following we shall not consider balanced (inf-sup stable) discrete methods but instead focus on stabilized methods. §.§ Nitsche's method as a stabilized multiplier methodFormally, the Lagrange multiplier in (<ref>) is given by λ = ∂_n u, where ∂_nv:= n·∇ v, and a well known stabilization method <cit.> for the discretization of (<ref>)–(<ref>) to is to add a term penalizing the difference between the discrete multiplier and the discrete normal derivative of the primal solution.To this end, we assume that 𝒯_h is a conforming shape regular meshe on Ω, consisting of triangles T anddefine V_h as the space of H^1–conforming piecewise polynomial functionson 𝒯,V_h := {v_h ∈ H^1(Ω): v_h|_T ∈ℙ_k(T),∀ T ∈𝒯}, k ≥ 1.As discrete space for the multiplier, we define the trace mesh on Γ as the set ℱ_h of element faces F on Γ and setΛ_h := {q_h ∈ L_2(Γ): q_h|_F ∈ℙ_l(F),∀ F ∈ℱ}, l ≥ 0.Thenwe seek (u_h,λ_h)∈ V_h×Λ_h such thata(u_h,v)-λ_hv-μu_h-1/γ_0h(λ_h-∂_n u_h)μ-∂_n v = (f,v)-μgfor all (v,μ)∈ V_h×Λ_h. Here h is the meshsize of the trace mesh on Γ, interpreted as a piecewise constant function along Γ and γ_0 is a number to be chosen sufficiently large to obtain a stable method. Following Stenberg <cit.> we now let P_h:L_2(Γ)→Λ_h denote the L_2–projection, and considering Λ_h to be a space of discontinuous discrete multipliers, continuous inside each face F of the Lagrange multiplier mesh on Γ, we can eliminate the multiplier from (<ref>):λ_h|_E = P_h∂_n u_h|_E -γ_0 h^-1P_h (u_h-g)|_E∀ E.Now considering the limiting case of Λ_h → L_2(Γ) we see thatλ_h →∂_n u_h -γ_0 h^-1 (u_h-g)and we can reintroduce this multiplier into our stabilized method, replacing also μ by ∂_n v - γ_0 h^-1 v_h, to obtain the problem of finding u_h∈ V_h such thata(u_h,v)-∂_n u_hv-∂_n v_hu_h+γ_0h^-1 u_hv = L(v)∀ v∈ V_h,whereL(v) :=(f,v)-∂_nvg+γ_0 h^-1vgwhich is Nitsche's method <cit.>. §.§ Nitsche's method as an augmented Lagrangian methodThe other approach to deriving Nitsche's method from Lagrange multipliers is more in the vein of Nitsche' original paper <cit.> where the method was derived from a discrete minimization problem without multipliers. TheLagrangian in (<ref>) on the discrete spaces is augmented by a penalty term mutliplied by γ∈R^+ so that we seek stationary points toℒ(v,μ) := 1/2 a(v,v)- μv-g+1/2γ^1/2 (v-g)^2_Γ - fv ,leading to the problem of finding (u,λ)∈ H^1(Ω)× H^-1/2(Γ) such thata(u,v)- λv+γuv -μu =fv+γgv-μgfor all (v,μ)∈ H^1(Ω)× H^-1/2(Γ). Choosing now in the discrete case λ_h := ∂_n u_h, μ = ∂_n v, and γ = γ_0h^-1 we recover (<ref>). It should be noted that augmented Lagrangian methods are not in general inf-sup stable; typically an unstable method remains unstable and the augmentation rather serves the purpose of strengthening a method where the side condition is too weakly enforced. It is therefore rather remarkable that in the particular case where the discrete multiplier is replaced by the discrete normal derivative, the augmentation always works as a stabilization mechanism, see also the discussion in <cit.>. An important feature of the augmented Lagrangian approach is that it directly carries over to the case of inequality constraints, as first shown by Chouly and Hild in the context of elastic contact <cit.>. In our model problem we replace the constraint u=g on Γ by an inequality constraint u -g ≤ 0 on Γ. We then have the following Kuhn–Tucker conditions on the multiplier and side condition:u -g ≤ 0,λ≤ 0, λ (u-g)=0.The key to incorporating these conditions into the augmented Lagrangian scheme, as pioneered by Alart and Curnier <cit.>, is to make the observation that (<ref>) is equivalent to λ = -γ [u-g-γ^-1 λ]_+u - g = [u-g-γ^-1 λ]_-where γ∈ℝ^+, [x]_±=±max (± x,0), cf. <cit.>.Denoting P_γ(u,λ) := γ (u-g)-λ we see that λ = -[P_γ(u,λ)]_+ = [P_γ(u,λ)]_- -P_γ(u,λ).We then formally write the augmented Lagrangian, similar to (<ref>), but using the second relation in (<ref>) for the contact constraint,ℒ(v,μ) := 1/2 a(v,v)- μv-g- [v-g-γ μ]_-+1/2γv-g- [v-g-γ^-1 λ]_-^2_Γ - fv.Observe that this naive formulation is not differentiable, so of little practical use. Our aim is now to propose a modified Lagrangian that is consistent with (<ref>), but smoother. We use the relations (<ref>) for the modifications, in order not to perturb the stationary points. The modifications must also respect the saddle point structure of the system. First observe that by the two relations of (<ref>) the stationary point must satisfy μ [v-g-γ μ]_- = 0, using this in the second term in the right hand side of (<ref>) and developing the square of the third term we obtainℒ(v,μ) := 1/2 a(v,v)- μv-g+1/2γv-g^2+ 1/2γ^-1 [P_γ(u,μ) ]_-^2_Γ -(v-g) [P_γ(u,μ )]_-- fv.We then use that the stationary point must satisfy the second relation of (<ref>) implying that(v-g) [P_γ(u,μ )]_- = γ^-1 [P_γ(u,μ) ]_-^2_Γresulting in ℒ(v,μ) := 1/2 a(v,v)- μv-g+1/2γv-g^2- 1/2γ^-1 [P_γ(u,μ) ]_-^2_Γ - fv.This formulation is differentiable and the corresponding Euler-Lagrange equations reada(u,v)- λv+γuv -μu-γ^-1[γ (u-g)-∂_n u]_-γ v-μ=fv+γgv-μg.Choosing now in the discrete case λ_h := ∂_n u_h and μ = ∂_n v results in a formulation that equals (<ref>) up to a nonlinear perturbation: we seek u_h∈ V_h such thata(u_h,v)-∂_n u_hv-∂_n v_hu_h+γ u_hv-γ^-1 [γ (u_h-g)- ∂_n u_h]_-γ v-∂_n v = L(v)∀ v∈ V_h,whereL(v) :=(f,v)-∂_nvg+γvgfor all v∈V_h.To see the equivalence of this formulation with that introduced by Chouly and Hild <cit.> we once again use the relation (<ref>) to write-γ^-1 [γ (u_h-g)- ∂_n u_h]_-γ v-∂_n v = -[P_γ(u,∂_n u) ]_- -P_γ(u,∂_n u)v-γ^-1∂_n v- P_γ(u,∂_n u)v-γ^-1∂_n v= [P_γ(u,∂_n u) ]_+ v-γ^-1∂_n v- P_γ(u,∂_n u)v-γ^-1∂_n v. Applying this relation in the last term of the left hand side of (<ref>) and simplifying results inwe seek u_h∈ V_h such thata(u_h,v)+γ [u_h-g-γ^-1 ∂_n u_h]_+v-γ^-1 ∂_n v - γ^-1 ∂_n u_h∂_n v= (f,v),for all v∈V_h. With the choice γ = γ_0/h (<ref>) precisely coincides with the nonlinear Nitsche method proposed by Chouly and Hild <cit.>. This variational problem corresponds to stationarity of the Lagran­gianℒ(v,μ) := 1/2 a(v,v)+ 1/2‖γ^1/2[v-g-γ^-1μ]_+‖_Γ^2 -γ^-1μμ - fv ,cf. <cit.>.In the following we shall explore how the augmented Lagrangian approach can be used in different situations.§ POISSON'S EQUATION WITH GENERAL BOUNDARY CONDITIONSWe first consider the simple case of imposing general boundary conditions in a finite element method with the mesh fitted to the boundary. Following <cit.> we consider the model problem-∇·(ε∇ u) = f Ω ε∂_n u = ^-1 (u_0 - u)+g Γ=∂Ω ,where ε, u_0, andare given functions of sufficient regularity. The problem can be equivalently formulated as the stationary points of the Lagrangianℒ(v,μ) := 1/2 a(v,v)- μv- 1/2μ^2_Γ- fv +(u_0 +g)μ.where a(u,v) := (ε∇ u, ∇ v)_Ω.The augmented Lagrangian is obtained by adding a scaled least squares term on the constraint. This leads toℒ(v,μ) :=1/2 a(v,v) - μv- 1/2μ^2_Γ + 1/2μ+ v - u_0 -g_Γ^2- fv +(u_0 +g)μ.Developing the square of the least squares contribution leads to the elimination of the multiplier and the formℒ(v) := a(v,v) - 1/2 v^2_Γ - fv - (^-1u_0 + g)v.The corresponding optimality system is the standard weak formulation of the Robin problem: find u ∈ H^1(Ω) such thata(u,v) + ^-1uv = fv + (^-1 u_0 + g)v, ∀ v ∈ H^1(Ω).Restricting the formulation to finite dimensional spaces results inthe finite element formulation: find u_h ∈ V_h such thata(u_h,v_h) + ^-1u_hv_h = fv_h + (^-1 u_0 + g)v_h.We see that as the “compliance” → 0, u|_Γ→ u_0, implying a Dirichlet condition in the limit and as →∞ we recover the weak formulation for the Neumann problem with ε∂_n u|_Γ = g. As was pointed out in <cit.> the corresponding linear system becomes ill-posed if << h. As a remedy for this we will temper the coefficient in front of the least squares term in the augmented Lagrangian. Indeed if we introduce S_h = ( + h/γ_)^-1,where γ_ is a free parameter, and use this as coefficient, we obtain the augmented Lagrangianℒ(v,μ) :=1/2 a(v,v) - μv- 1/2μ^2_Γ + S_h/2μ+ v - u_0 -g_Γ^2- fv - (u_0 +g)μ.The cancellations that led to the previous simple optimality system will not take place this time as the moderated parameter S_h will prevent the system from becoming too stiff. Deriving we find the following optimality systema(u,v) - (1-S_h) λv-(1-S_h) μu- (1-S_h ) λμ +S_h uv =fv+ (u_0 +g)(1- S_h )μ+S_h v.Using this formulation in a finite element method with a pair V_h ×Λ_h that satisfies the inf-sup condition leads to a robust and accurate method.Formally replacing λ by ε∂_n u and μ by ε∂_n v and writing the form on the finite space V_h yields the method: find u_h ∈ V_h such thata(u_h,v_h) - (1-S_h) ε∂_n u_hv_h-(1-S_h)ε∂_n v_hu_h- (1-S_h ) ε∂_n u_hε∂_n v_h +S_hu_hv_h= fv_h+ (u_0 + g)(1-S_h )ε∂_n v_h+S_h v_h.We identify the Nitsche method proposed in <cit.> and conclude that their analysis is valid for u_h if the parameter γ_ is chosen large enough. Observe that the Lagrange multiplier formulation (<ref>), which appears to be new, also varies robustly between a Neumann and a Dirichlet condition in the two limits, without succumbing to the ill-conditioning in the limits of high or low. We note in passing that sometimes the multiplier method can have advantages compared to Nitsche's method, in particular if fields defined on different meshes must be coupled, see for instance <cit.>. § FEM FOR POISSON'S PROBLEM WITH DISCONTINUOUS COEFFICIENTS We are interested in the following problem: find u_i:Ω_i ↦ℝ, i=1,2, such that-∇·ε_i ∇ u_i = f Ω_i, i=1,2u_i =0 ∂Ω∩Ω_i i=1,2 ε∂_n u =gΓ :=∂Ω_1∩∂Ω_2 u =0 Γ.We use the notation u=(u_1,u_2)∈ V_1× V_2 with the continuous spacesV_i={v_i∈ H^1(Ω_i): ∂ v_i/∂ n_i∈ L_2(Γ),v_i|_∂Ω∩∂Ω_i=0} , i=1,2.Then u denotes the jump of u over Γ defined asu := lim_ϵ→ 0^+ u(x-ϵ n) - u(x+ϵ n)for x ∈Γ and n denoting the normal on Γ pointing from Ω_1 to Ω_2. The diffusion coefficients ε_i, i=1,2, are assumed to be constant functions. We will also use the weighted averagesu := lim_ϵ→ 0^+ ( w_1 u(x-ϵ n)+ w_2 u(x+ϵ n))andu := lim_ϵ→ 0^+ ( w_2 u(x-ϵ n)+ w_1 u(x+ϵ n))where w_1,w_2 ≥ 0 are positive weights such that w_1 + w_2 = 0. This problem can be shown to be equivalent to finding (u,λ) ∈ H^1(Ω_1∪Ω_2) × H^-1/2(Γ), the saddle point of the constrained minimization problem defined by the Lagrangianℒ(v,μ) := 1/2 a(v,v) + <μ,v>_-1/2,1/2,Γ- fv -gv.where nowa(u,v) := (ε∇ u,∇ v)_Ω_1 ∪Ω_2.Thus, (u,λ) fulfillsℒ(u,λ)=inf_v ∈ Vsup_μ∈Λℒ(v,μ).We also know that for the exact solution there holds λ = -ε∇ u · n, for any admissible weights w_1,w_2. To formulate a discrete method, we suppose that we have regular finite element partitionings 𝒯_h^i of the subdomains Ω_i into shape regular simplexes. These two meshes have correspondingtrace meshes on the interface and for simplicity we assume that the meshes match across the interface so that the trace meshes are equivalent and we may writeℱ_h={ F: F=T∩Γ,T∈𝒯_h^i,i=1 or 2 }.We seek the approximation u_h=(u_1,h,u_2,h) in the space V^h = V_1^h× V_2^h, whereV^h_i={ v_i∈ V_i: v_i|_T∈ℙ_k(T),∀ T ∈𝒯_h },fork ≥ 1,and for λ_hin Λ_h := {q_h ∈ L_2(Γ): q_h|_F ∈ℙ_l(F),∀ F ∈ℱ_h }, for l ≥ 0 .If we now restrict the infimum of the supremum in (<ref>) to our finite dimensional subspaces V_h and Λ_h chosen such that the discrete spaces satisfy the inf-sup condition, we immediately obtain the standard Lagrange multiplier domain decomposition method <cit.>, with the Lagrangian given byℒ(v_h,μ_h) := 1/2a(v_h,v_h) + μ_hv_h - fv_h -gv_h.The augmented Lagrangian is obtained by adding a least squares penalty on the constraint:ℒ(v_h,μ_h) :=1/2 a(v_h,v_h)+μ_hv_h+γ/2v_h^2_Γ -fv_h -gv_h.This allows us to instead look for stationary points of the following augmented Lagrangian since we are working in discrete spaces and all pairings are L^2 scalar products: ℒ(v_h,μ_h) := 1/2 a(v_h,v_h)+1/2 γμ_h+γv_h^2_Γ -1/2γ^-1μ_hμ_h -fv_h -gv_h. The Euler–Lagrange equations characterising the saddle point of the system (if it exists) takes the form: find (u_h,λ_h) ∈ V_h ×Λ_h such thata(u_h,v_h) + γ^-1λ_h + γu_hμ_h+γv_h - γ^-1λ_hμ_h =fv_h + gv_hfor all v_h,μ_h ∈ V_h ×λ_h. Developing the second term of the left hand side we see that this is equivalent toa(u_h,v_h) + λ_h + γu_hv_h = fv_h + gv_h u_hμ_h= 0.This shows that the effect of the augmented Lagrangian compared to the standard Lagrange multiplier method is simply the addition of a penalty term on the constraint which, as mentioned above, gives us stronger control of the constraints than would otherwise be possible. Indeed for the standard Lagrange multiplier method only π_Λu_h is controlled, where π_Λu_hμ_h =u_hμ_h for all μ_h ∈Λ_h. This formulation is however stable only for well balanced choices of V_h and Λ_h. In case the spaces do not satisfy the inf-sup condition one may add a stabilizing term j(λ_h,λ_h) satisfyingh^1/2 (λ_h - π_Γλ_h)_Γ≲ j(λ_h,λ_h)^1/2where π_Γ satisfies π_Γμ_hv_h =μ_hv_h for all v_h in the trace mesh of 𝒯_1 (or 𝒯_2). For instance if V_h is the space of piecewise affine, continuous functions in each subdomain and Λ_h is the space of piecewise constant functions defined on the elements cut by Γ we may choosej(λ_h,λ_h) := ∑_F ∈ℱhλ_h_∂ F∖∂Γ^2and follow the analysis of <cit.> to prove error estimates forthe formulation: find (u_h,λ_h) ∈ V_h ×Λ_h such that a(u_h,v_h) + λ_h + γu_hv_h = fv_h + gv_h u_hμ_h - j(λ_h,μ_h)= 0for all v_h,μ_h ∈ V_h ×λ_h. On the other hand, if we formally replace λ_h by -μ∂_n u_h and μ_h by -μ∂_n v_h we obtain the formulation: find u_h∈ V_h such thata(u_h,v_h) -μ∂_n u_hv_h - u_hμ∂_n v_h+ γu_hv_h = fv_h + gv_hfor all v_h∈ V_h, and we recognise Nitsche's formulation from <cit.>.§ DEBONDING AND ADHESIVE CONTACTA robust discretization of the debonding problem was proposed in <cit.>. We will revisit their arguments in the context of Lagrange multipliers as an augmented Lagrangian formulation . The linear model problem in this case takes the form-∇·ε∇ u = f Ω_1 ∪Ω_2u =0 ∂Ω ε∂_n u =0Γ u =-ε∂_n uΓ.Here we note that by the continuity of the fluxes the formulation is independent of the choice of the weights w_i, i=1,2. This time the physical solution is discontinuous over the boundary and u ∈ H^1(Ω_1∪Ω_2).The critical case is whenbecomes large and a naive coupling strategy leads to an ill-conditioned system or even locking on the interface. We therefore follow <cit.>, but contrary to the discussion in that paper we herein use the augmented Lagrangian formulation to arrive at the method. In the intermediate step we obtain a robust Lagrange multiplier method for the debonding problem. This time we start from the following Lagrangian, the saddle points of which coincides with the solution of the debonding problem,ℒ(v,μ) := 1/2 a(v,v) - <μ,v>_-1/2,1/2,Γ- 1/2μ^2_Γ- fv.The augmented Lagrangian is obtained adding a least squares term on the constraintℒ(v,μ) := 1/2 a(v,v) - <μ,v>_-1/2,1/2,Γ- 1/2μ^2_Γ + 1/2v +μ_Γ^2- fv.Developing the square we see that the multiplier is eliminated and we obtainℒ(v) := 1/2 a(v,v) + 1/2v_Γ^2- fv.Studying the corresponding optimality system leads to: find u_h ∈ V_h such thata(u_h,v_h) + S u_hv = fv,∀ v_h ∈ V_h.where S=^-1. As for the problem in Section <ref> this formulation becomes ill-conditioned forsmall since the two terms of the left hand side will have sizes of different orders of magnitude. A possible remedy is to replaceby max(h,) in equation (<ref>), but this results in a nonconsistent perturbation of the system and reduced accuracy, in the regime where < h.If we instead modify the size of the least squares contribution in the augmented Lagrangian we can moderate the strength of the imposition of the constraint in a consistent manner. Introducing the parameter S_h =(h_k/γ_ + )^-1,similarly as for the method in Section <ref>, we may writeℒ(v,μ) := 1/2ε^1/2∇ v_Ω^2 - <μ,v>_-1/2,1/2,Γ- 1/2μ^2_Γ + 1/2 S_h v +μ_Γ^2- fv.Observe that the saddle point to this system is a weak solution to (<ref>)–(<ref>), but the size of the weight in front of the least squares term can never be larger than γ_/h, effectively bounding the stiffness of the system. The corresponding optimality system now reads:a(u,v) - (1-S_h) λv-(1-S_h) μu- (1-S_h ) λμ +S_h uv = fv.Formally replacing λ by ε∂_n u and μ by ε∂_n v and writing the form on the finite space V_h yields: find u_h ∈ V_h such thata(u_h,v_h) - (1-S_h) ε∂_n u_hv_h-(1-S_h) ε∂_n v_hu_h- (1-S_h ) ε∂_n u_hε∂_n v_h +S_h u_hv_h= fv_h.We recognise the same Nitsche type method as proposed in <cit.>, but this time with the weights chosen as in the previous section to also be robust with respect to the contrast in the diffusivity. §.§ Adhesive contactIn the previous linear model only the adhesive forces are accounted for, which implies that penetration is possible. A more physically realistic model excludes penetration by formulating the problem as a variational inequality. Our last model problem concerns this nonlinear model and we will combine the arguments developed above with those of <cit.>. Here for simplicity we assume that both Ω_1 and Ω_2 intersects the boundary ∂Ω. -∇·ε∇ u = f Ω_1 ∪Ω_2 u =0 ∂Ω ε∂_n u =0Γ u ≤ 0Γ ^-1u+ε∂_n u ≤ 0Γ u( ^-1u+ε∂_n u) =0 ΓTo cast this problem on the form of an augmented Lagrangian method westart out with the functional (<ref>) with an additional constraint on v: ℒ(v,μ) := 1/2ε^1/2∇ v_Ω^2 -<μ,v>_-1/2,1/2,Γ+ 1/2v_Γ^2- fv.The Euler–Lagrange equationsare to find (u,λ) such thata(u,v)-<λ,v>_-1/2,1/2,Γ + 1/uv = fv∀ v∈H^1(Ω_1∪Ω_2) , < μ ,u>_-1/2,1/2,Γ = 0 ∀μ∈ H^-1/2(Γ) ,and we note from (<ref>) that, formally, the multiplier is given byλ = ϵ∂_n u + ^-1u but of course (<ref>) enforces u=0 weakly. In order to create a one-sided contact condition we now consider the Kuhn–Tucker conditions (<ref>)–(<ref>) as λ≤ 0,  u≤ 0,  and λu = 0on Γ.These conditions can equivalently be formulated asλ = -γ [u -γ^-1λ]_+ ,where [x]_+=max (x,0), cf. <cit.>. To introduce this condition in the Lagrangian (<ref>) we note that, if λ,μ∈ L^2(Γ) (or if γ^-1:H^-1/2(Γ)↦ H^1/2(Γ), with suitable properties), in (<ref>) we have-λv = -λv-γ^-1μ - λγ^-1μ.Using now (<ref>) in the first term of the right hand side we havea(u,v)+<γ[u -γ^-1λ]_+,v -γ^-1μ)>_Γ + 1/uv = fv.Moving over to discrete spaces, where the assumption on λ_h makes sense, we write the corresponding Lagrangian on augmented form asℒ_a(v_h,μ_h) := a(v_h,v_h) +γ/2‖ [v_h-γ^-1μ_h]_+‖_Γ^2 - 1/2γ‖μ_h‖_Γ^2+1/2 ‖v_h‖_Γ^2 - fv_hleading to the problem of finding (u_h,λ_h)∈ V_h×Λ_h such thata(u_h,v_h)+γ<[u_h -γ^-1λ]_+,v_h -γ^-1μ_h)>_Γ -γ^-1< λ_h, μ_h >_Γ+ 1/< u_h, v_h>_Γ =fv_hfor all (v_h,μ_h)∈ V_h×Λ_h.Inserting now the definition (<ref>) of the multiplier and choosing μ = ϵ∂_n v + ^-1v we seek u_h∈ V_h such thata(u_h,v_h) + 1/< u_h, v_h>_Γ+γ<[(1-(γ)^-1)u_h -γ^-1ϵ∂_n u_h]_+,(1-(γ)^-1)v -γ^-1ϵ∂_n v)>_Γ -γ^-1<ϵ∂_n u_h + ^-1u_h,ϵ∂_n v_h + ^-1v_h>_Γ= fv_h∀ v_h∈ V_h . It is instructive to consider the two limiting cases of full contact and of no contact: at contact, (<ref>) givesa(u_h,v_h) -< u_h,ϵ∂_n v_h>_Γ -< v_h,ϵ∂_n u_h>_Γ+ (γ-1/)<u_h,v_h>_Γ= fv_h∀ v_h∈ V_h.With the particular choice γ := γ_/h + 1/,we obtain the following discrete problem: find u_h∈ V_h such thata(u_h,v_h)-<ϵ∂_n u_hv_h>-<ϵ∂_n v_hu_h> + γ_/h< u_h, v_h>_Γ = fv_hfor all v_h∈ V_h, which is the standard Nitsche method (<ref>) for the adhesion free problem (with γ_ =γ_0). In the case of no contact observe that by using γ^-1 = γ^-1-+,1/< u_h, v_h>_Γ - γ^-1<ϵ∂_n u_h + ^-1u_h,ϵ∂_n v_h + ^-1v_h>_Γ= -<ϵ∂_n u_h,v_h+ϵ∂_n v_h>_Γ-<u_h+ϵ∂_n u_h,ϵ∂_n v_h>_Γ -<ϵ∂_n u_h,ϵ∂_n v_h>_Γ(-γ^-1) <ϵ∂_n u_h + ^-1u_h,ϵ∂_n v_h + ^-1v_h>_Γ .The finite element formulation then takes the form: we seek u_h ∈ V_h such thatfv_h = a(u_h,v_h)-<ϵ∂_n u_h,v_h+ϵ∂_n v_h>_Γ-<u_h+ϵ∂_n u_h,ϵ∂_n v_h>_Γ -<ϵ∂_n u_h,ϵ∂_n v_h>_Γ + 1/γ_1< u_h+ϵ∂_n u_h, v_h +ϵ∂_n v_h>_Γ , ∀ v_h∈ V_h,where γ_1 := ^2/-γ^-1 = + h/γ_,which coincides with the form (<ref>). § STABILIZATION FOR THE EXTENSION TO CUTFEM Contrary to the methods discussed above, where the domains are meshed in the usual way, the CutFEM approach instead represents the boundary of a given domainon a backgroundgrid, for instance using a level set function. The background grid isthen also used to represent the approximate solution of the governing partial differentialequations. Consequently, CutFEM eases the burden of mesh generation by requiring only a low-quality surface mesh representation of the computational geometry.Cutting the mesh will, however, result in boundary elements with very small intersection with the physical domain. This may lead to a poorly conditioned system matrix or failure of stability of the discrete scheme. A remedy to this problem is to add a penalty term in the cut element zone that extends the coercivity to the whole mesh domain, i.e., in the O(h) zone of the mesh domain (of each subdomain for interface problems) that does not intersect the associated physical domain. This penalty term, termed ghost penalty due to it acting partly outside of the domain of interest, must be carefully designed to add sufficient stability, while remaining weakly consistent for smooth solutions. The basic methods described above are then applied to the cut meshes and the only additional term is the ghost penalization.To illustrate this idea, we consider the CutFEM method for the Poisson problem (<ref>). We observe that by taking v=u_h in the bilinear form a(u_h,v), we have the coercivity∇ u_h_L^2(Ω)^2 ≤ a(u_h,u_h).However, to obtain coercivity of the form a_h(u_h,v) using this stability and the boundary penalty term, the penalty parameter will depend on how the elements are cut, since, denoting the set of elements that are cut by Γ by𝒢_h := {K ∈𝒯_h: K ∩Γ≠∅} we have, with a_h(u_h,v) :=a(u_h,v)-∂_n u_hv-∂_nv_hu_h+γ u_hv thata_h(u_h,u_h) ≥ ∇ u_h^2_L^2(Ω) + γ^1/2u_h^2_L^2(Γ) - 2 ∑_K ∈𝒢_h∇ u_h_L^2(Γ∩ K) u_h_L^2(Γ∩ K). Using the following well known trace inequality: underreasonable mesh assumptions there exists aconstant C_T, depending on Γ but independent of the mesh, such thatw_L_2(Γ_K)^2≤ C_T ( h_K^-1 w_L_2(K)^2 + h_K ∇ w_L_2(K)^2), ∀ w∈ H^1(K) ,we have, ∇ u_h_L^2(Γ∩ K)≤C_Γ(| K∩Γ|/| K ∩Ω|)^1/2∇ u_h_L^2(K ∩Ω),where |·| denotes the measure of the indicated quantity. It follows that in principle we obtain coercivity by choosing γ|_K > 2 C_γ^2(| K∩Γ|/| K ∩Ω|),since by an arithmetic-geometric inequality, we havea_h(u_h,u_h) ≥ ∇ u_h^2_L^2(Ω) + γ^1/2 u_h^2_L^2(Γ) - 1/2∇ u_h^2_L^2(Ω) - ∑_K ∈𝒢_h C_Γ^2 (| K∩Γ|| K ∩Ω|) u_h^2_L^2(K ∩Γ) ≥ 1/2∇ u_h^2_L^2(Ω) + (γ - 2 C_Γ^2 (| K∩Γ|| K ∩Ω|))^1/2 u_h^2_L^2(Γ).Unfortunately this makes γ strongly dependent on the cut, since for | K∩Γ| = O(h_K), the volume measure | K ∩Ω| can be arbitrarily small, resulting in problems both with conditioning and accuracy. A solution to this problem is to add a stabilizing term g_h(u_h,v) to the form a_h(·,·). The role of this term is to extend the coercivity from the physical domain Ω to the mesh domain Ω_𝒯:= Ω∪𝒢_h. In order to have this effect, the stabilization term should have the following properties. * It should give a bound on the energy norm on the mesh domain in the sense thatc_G ∇ u_h^2_Ω_𝒯≤∇ u_h^2_Ω + g_h(u_h,u_h),where c_G>0 is bounded away from zero independent of the mesh/boundary intersection for positive ghost penalty stabilization parameter γ_g.* For an interpolant of the extension of u, i_h u := i_h 𝔼 u we must have the weak consistencyg_h(i_h u,i_h u) ≤ C h^ku_H^k+1(Ω).where the constant C is independent of the mesh/boundary intersection.One example of such a term is the ghost penalty stabilizationg_h(u_h,v) :=∑_F ∈ℱ_G (γ_g h ∂_n_F u_h,∂_n_F v)_F,valid for piecewise affine approximation.Here, we introduced the set of element faces ℱ_G associated with 𝒢_h, defined as follows: for each face F ∈ℱ_G there exists two simplices K and K' such that F=K ∩ K' and at least one of the two is a member of 𝒢_h.This means in particular that the boundary faces of the mesh 𝒯_h are excluded from ℱ_G. We also used ∂_n_F to denote the derivative in the direction of the normal to F. Coercivity now follows from (<ref>) and (<ref>) as followsa_h(u_h,u_h) ≥ ∇ u_h^2_L^2(Ω) + γh^-1/2 u_h^2_L^2(Γ) - 2 C_T ∇ u_h_L^2(𝒢_h)h^-1/2 u_h_L^2(Γ) + g_h(u_h,u_h)≥c_G ∇ u_h^2_Ω_𝒯 + γh^-1/2 u_h^2_L^2(Γ)- 2 C_T ∇ u_h_L^2(Ω_𝒯)h^-1/2 u_h_L^2(Γ) ≥ c_G/2∇ u_h^2_Ω_𝒯 + (γ - 2 C_T^2 c_G^-1) h^-1/2 u_h^2_L^2(Γ).Here C_T is the constant of the trace inequality (<ref>) and c_G is the coercivity constant of the stability estimate (<ref>). We conclude by choosing γ_0 > 2 C_T^2 c_G^-1, where the lower bound is independent of the mesh/boundary intersection, but not of the penalty parameter γ_g in g_h(·,·). Error estimates now follow in a similar fashion as for the standard Nitsche's method, using (<ref>) and the consistency of the penalty term. One may also show that the conditioning of the system matrix is bounded independently of the mesh/boundary intersection. For further details see <cit.>.Extension to the other model problems is straightforward. We considerthe problem of interface coupling using (<ref>). In the original paper on cut finite elements <cit.> the method for meshed subdomains was carried over to the cut element case using piecewise affine elements and weightsw_1 = K∩Ω_2/|K|, w_2 = K∩Ω_1/|K| .However, for problems with large contrast ε_max/ε_min it is known that thise choice is not stable for arbitrary cuts if the mesh size is not small enough to resolve the contrast. Indeed too large contrast can lead to a phenomenon reminiscent of locking for unfortunate cuts (i.e. if no H^1-conforming subspace with approximation exists). In case robustness is necessarywe instead choose the weights to be w_1 = ϵ_2/(ϵ_1+ϵ_2) and w_2 = ϵ_1/(ϵ_1+ϵ_2) and γ^-1 = γ_0 h^-1ω(ϵ) with ω(ϵ):=2 (ϵ_1 ϵ_2)/(ϵ_1+ϵ_2) we identify this method as the Nitsche method discussed in <cit.> which was shown to be stable on unfitted meshes provided a ghost penalty term is added. This additional stabilization term should here have the properties analogous to those for the cut fictitious domain method discussed above, and a typical example is the modification of (<ref>) now acting across the faces of the cut elements on the interface. The resulting method takes the forma(u_h,v_h) -μ∂_n u_hv_h - u_hμ∂_n v_h+ γu_hv_h+g_h(u_h,v_h) = fv_h + gv_h.In particular it was proven in <cit.> that for piecewise affine elements and smooth enough Γ there holds∑_i=1^2 ε_i ∇ (u_i - u_i,h)_Ω_i≤ C h f_Ωwhere the constant C is independent of ε and h. This result is possible to obtain thanks to the fact that the weights shift the interface term to the side where ε_i is the smallest. Observe now the resemblance between the formulation (<ref>) and (<ref>). The latter formulation however is not in general robust for large contrast.Indeed regardless of the contrast the constraint will be satisfied equally strongly and we know from the experience of <cit.> that the trick to obtaining robustness is to relax the control obtained by the multiplier by redefining λ_h. Instead of identifying λ_h = ε∂_n u_h we use that ε∂_n u_h = ω(ε) {∂_n u_h} where {·} denotes the standard arithmetic average. If we then instead identify λ_h= {∂_n u_h} and introduce the factor ω(ε) in the formulation (<ref>) we obtain a(u_h,v_h) + ω(ε) λ_h + γu_hv_h = fv_h + gv_h u_hω(ε) μ_h -j(λ_h,μ_h)= 0where once again γ = γ_0 h^-1ω(ϵ) and the stabilization operator j(·,·) also must scale as ω(ε). This formulation will relax the jump in a similar fashion as (<ref>), but control of λ_h is sacrificed as ω(ε) becomes small.§ A NUMERICAL EXAMPLE We give an example of how the method works in the case of adhesion, with and without contact. To exemplify how the different aspect of adhesion come into play we considera domain (0,1)×(0,1) with u=0 at x=0 and at y=0 and with ∂_n u=0 on other boundaries. The domain is cut by a half circle with radius r=0.74; ε =2 on the domain Ω_1 containing the origin and ε =1/2 on Ω_2. The right-hand side is given byf= {[ 1if y≤ 1/2,; -7/2if y > 1/2. ].We set γ_0=100, used no jump stabilization, and set ω(ε) = 1 but used geometric averages of the type (<ref>). In Fig.<ref> we show the solution using the standard Nitsche method (<ref>), in Fig. <ref> we show the solution with a cohesive interface with =1/2 using (<ref>), and in Fig. <ref> we show the solution using a one sided contact condition as in (<ref>). § ACKNOWLEDGEMENTSThe contribution of the first author was supported in part by the EPSRC grants EP/J002313/2 and EP/P01576X/1, the contribution of the second author was supported in part by the Swedish Foundation for Strategic Research Grant No. AM13-0029 and the Swedish Research Council Grant No. 2011-4992. 10AC91 Pierre Alart and Alain Curnier. A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput. Methods Appl. Mech. Engrg., 92(3):353–375, 1991.BaHu91 Helio J. C. Barbosa and Thomas J. R. Hughes. The finite element method with Lagrange multipliers on the boundary: circumventing the Babuška-Brezzi condition. Comput. Methods Appl. Mech. Engrg., 85(1):109–128, 1991.BC12 Gabriel R. 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Splitting schemes for incompressible fluid/thin-walled structure interaction with unfitted meshes. C. R. Math. Acad. Sci. Paris, 353(7):647–652, 2015.FG83 Michel Fortin and Roland Glowinski. Augmented Lagrangian methods, volume 15 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam, 1983. Applications to the numerical solution of boundary value problems, Translated from the French by B. Hunt and D. C. Spicer.GT82 R. Glowinski and P. Le Tallec. Numerical solution of problems in incompressible finite elasticity by augmented Lagrangian methods. I. Two-dimensional and axisymmetric problems. SIAM J. Appl. Math., 42(2):400–429, 1982.HaHa02 Anita Hansbo and Peter Hansbo. An unfitted finite element method, based on Nitsche's method, for elliptic interface problems. Comput. Methods Appl. Mech. Engrg., 191(47-48):5537–5552, 2002.HaHa04 Anita Hansbo and Peter Hansbo. 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http://arxiv.org/abs/1702.08340v1

http://arxiv.org/abs/1702.08284v3

firstpage–lastpageEmergence of Multi-Scaling in a Random-Force Stirred Fluid. Diego DonzisDecember 30, 2023 =============================================================Properties of the cold interstellar medium of low-metallicity galaxies are not well-known due to the faintness and extremely small scale on which emission is expected. We present deep ALMA band 6 (230 GHz) observations of the nearby, low-metallicity (12+log(O/H)=7.25) blue compact dwarf galaxy at an unprecedented resolution of 0.2 arcsec (52 pc).The ^12CO J=2→1 line is not detected and we report a 3-σ upper limit of L_ CO(2-1)=3.6×10^4 . Assuming that molecular gas is converted into stars with a given depletion time, ranging from 0.02 to 2 Gyr, we find lower limits on the CO-to-H2 conversion factor in the range 10^2 - 10^4 .The continuum emission is detected and resolved over the two main super star clusters. Re-analysis of the IR-radio spectral energy distribution suggests that the mm-fluxes are not only free-free emission but are most likely also associated with a cold dust component coincident with the position of the brightest cluster. With standard dust properties, we estimate its mass to be as large as 10^5 .Both line and continuum results suggest the presence of a large cold gas reservoir unseen in CO even with ALMA. galaxies: dwarf – galaxies: star formation – galaxies: individual:–submillimeter: ISM§ INTRODUCTIONStar-formation conditions in low-metallicity environments are not well known due to the lack of observational constraints on the cold dust and gas.Recent efforts with ground-based telescopes have tried to detect the cold gas reservoirs through carbon monoxide (CO) in nearby low-metallicity galaxies <cit.>. CO emission is found to be weak in those galaxies while active, bursty star formation is often evident. This would imply significantly different star-formation efficiencies unless CO no longer traces the bulk of the star-forming gas. Star formation taking place in the atomic gas <cit.> as well as changes in the standard CO-to-conversion factor between high- and low-metallicity galaxies have been suggested <cit.> though the amplitude of those changes is unknown. A complementary probe of the ISM in galaxies comes from the continuum emission. Recent, sensitive observations in the far-IR/millimeter (mm) with <cit.> have allowed us to constrain the spectral energy distributions (SEDs) of many nearby low-metallicity dwarf galaxies and to derive more robust dust masses <cit.>. However, most galaxies with metallicities ≤1/5 Z_ remain undetected in the submm/mm regime, implying that characterizations of a possible cold (T_ dust<20 K), massive dust reservoir are still uncertain. In an effort to characterize the very low-metallicity regime, we present new ALMA observations of the dwarf galaxy .§.§ General properties ofAmong nearby low-metallicity galaxies, the blue compactdwarf at 54 Mpc is one of the best laboratories to study vigorous star formation in quasi-pristine conditions.It is composed of a main body (Fig. <ref>) that we refer to as . Its general properties are listed in Table <ref>.is well known as one of the lowest metallicity galaxies of the nearby Universe, with 12+log(O/H)∼7.25 <cit.>, and exhibits exceptionally active star formation, with a total star-formation rate (SFR) of 0.7 M_ yr^-1 <cit.> and Σ_ SFR as high as 20 M_ yr^-1 kpc^-2 <cit.>.Star formation in is located in 6 compact (<60 pc)super star clusters (SSCs; ).The bulk of its stellar population has an age of less than 500 Myr.The two brightest SSCs, located to the south (S1 and S2 on Fig. <ref>), are the youngest, with ages ≤3 Myr and stellar masses of ∼10^6 <cit.> each.ISO and observations reveal strong mid-IR continuum from warm dust, no polycyclic aromatic hydrocarbons, and hard radiation fields <cit.>. While optical studies measure an average visual extinction A_V of∼0.7 mag <cit.>, with values similar in S1 and S2, dust modeling of the unresolved IR emission yieldsA_V≥10 mag <cit.>.This discrepancy suggests the presence of a separate componentwith high IR optical depth and probably hidden star formation. Although this galaxy remains undetected in CO <cit.>, warm molecular gas is present, since near-IR lines are detected in the southern clusters <cit.>. In this letter, we present new ALMA band 6 observations of at a resolution of 0.2 arcsec (52 pc). Section <ref> describes the observations. The molecular gas reservoir and spectral energy distribution are analysed in Sections <ref> and <ref>. We discuss and summarize our results in Sections <ref> and <ref>.§ DATA§.§ ALMA band 6 data Observations of were carried out in 5 blocks between August, 30 2015 and September, 23 2015 as part of the Cycle 2 program 2013.1.00916.S (PI Cormier). We targeted the continuum and ^12CO J=2→1 line in band 6 at a spatial resolution of 0.2 arcsec. Observations were made in extended configuration with 35 antennas. The field-of-view is 15 arcsec. One spectral window was centered at a rest frequency of 230.5 GHz with resolution 1.3 and the 3 other spectral windows were centered at 232.5, 245.5 and 247.5 GHz. Each window has bandwidth of 1.875 GHz. All windows were combined to make the continuum map (total bandwidth of 7.5 GHz centered around 239 GHz).We reprocessed and recalibrated the raw data from the archive using the Common Astronomy Software Applications version 4.6.0 <cit.>. Data from each execution block were first processed through a set of standard calibration steps to correct phase and amplitude variations versus time and versus channel and to flux calibrate the data. Data from antennas or channels which have irregular phases or amplitudes are flagged during the calibration. The final images were then created using the clean algorithm with natural weighting. The source was calibrated using quasars where the estimated flux densities have uncertainties of <∼15 per cent. The reconstructed Gaussian beam size in the final images is 0.23 arcsec×0.17 arcsec with a position angle of 65^∘, which matches the full-width at half maximum of the axes of the beam before deconvolution as well as the axis orientation. The continuum is detected at a signal-to-noise of 10 towards the peak of emission and resolved into two compact knots (S1 and S2, see Fig. <ref>). We achieve an rms of 0.011 mJy/beam in the continuum. We extract fluxes for the two SSCs by fitting 2D ellipses constrained by the shape of the ALMA beam. We fit each SSC at a time, masking out the other SSC. Emission from S1 is marginally resolved (fitted with an ellipse of size 1.15× the beam). Over the region S, we measure a total flux of 0.32 mJy (Table <ref>) with a circular aperture of radius 0.5 arcsec covering S1 and S2. The sum of the Gaussians agrees with the total within errors, indicating no significant extended emission. Uncertainties are given as the rms per beam times √(N_ beams).The CO J=2→1 line is not detected. We report an rms of 0.41 mJy/beam, for a spectral resolution of 5 .The maximum recoverable scale of our observations is ∼5 arcsec (∼1.3 kpc), so we are not missing extended emission in our analysis. §.§ Ancillary data and star-formation rates We compare our ALMA observations to ISM data of as well as radio data. We use photometry at a resolution >5 arcsec from <cit.>. We complement those with ALMA band 7 (346 GHz) data from <cit.> at ∼0.6 arcsec resolution. Additionally, we use radio continuum data from the VLA (1.3 to 6 cm) at ∼0.5 arcsec resolution and SSC-extracted fluxes from <cit.>.was observed at 1.5–22 arcsec resolution with the VLA (1.3 to 21 cm) by <cit.>. Although is not resolved in the data or in the VLA data from <cit.>, there is evidence that S1 and S2 dominate the global emission at other wavelengths. Hence we consider emission from region S as total emission.The total SFR of the galaxy, measured by combining the and total infrared emission, is ∼0.7 M_ yr^-1 <cit.>. From resolved studies, 1.3 cm emission yield SFRs of 0.38 and 0.27 M_ yr^-1 for S1 and S2 <cit.>. § STAR FORMATION AND MOLECULAR RESERVOIR The new ALMA observations put stringent constraints on the amount of CO-emitting molecular gas in . Assuming a broad line width of 50 , as measured from <cit.>, we derive a conservative 3-σ L_ CO(2-1) limit of 3.6×10^4 . Our solid angle is 0.044 arcsec^2. Taking a CO(2-1)/CO(1-0) ratio of 0.8 <cit.>, this corresponds to a L_ CO(1-0) limit of 4.5×10^4 .Previous CO J=1→0 and CO J=3→2 measurements by <cit.> and <cit.>, with solid angles of 23 arcsec^2 and 1 arcsec^2 and a CO(3-2)/CO(1-0) ratio of 0.6, put L_ CO(1-0) limits of 8.9×10^6 and 1.8×10^5 , respectively.Figure <ref> shows the CO J=1→0 luminosity as a function of SFR for and several nearby galaxies which measurements are compiled in <cit.>. Our ALMA data provide a new limit on the CO luminosity in . clearly stands out in being so CO faint for a star-forming galaxy.The CO-to-conversion factor () being unconstrained at low metallicity, we do not convert this luminosity into a limit on the mass of . Instead, we hypothesize that star formation proceeds with a given depletion time-scale, τ_ dep=M()/SFR. We assume three different molecular gas depletion times: 2 Gyr - a common value for normal, disc-type galaxies <cit.>; 0.2 Gyr - a common value for starburst galaxies <cit.>; and 20 Myr - a common value for local molecular clouds <cit.>. Significantly reduced depletion time-scales - similar to starburst values -have been found by <cit.> for another dwarf galaxy, II Zw 40. From the CO luminosity and SSC-derived SFR values, we can constrain . The limits that we find for S1 and S2 are shown in Figure <ref> and are around 10^2 for cloud depletion times (20 Myr), and 10^3 - 10^4 , amongst the highest, for whole-galaxy depletion times (0.2 - 2 Gyr).The most surprising finding is that, even at our 50 pc resolution, CO is not detected. <cit.> analyse variations of the factor as a function of visual extinction in numerical simulations of low-metallicity clouds. Following their study, our results are compatible with mean visual extinctions of 0.1-1 mag. This is within values found from optical data <cit.> and much lower than expected from IR data. The filling factor of the shielded and IR-bright component in must be very small. <cit.> estimate a clumpy dust covering factor of 60 per cent. With our 1-σ limit of W_ CO(1-0)≤ 5 K at a resolution of 50 pc, and a typical CO clump brightness of 15 K <cit.>, we estimate a beam filling factor for the CO-emitting clumps of less than 33 per cent. § SPECTRAL ENERGY DISTRIBUTION Figure <ref> shows the total FIR-radio SED of . The photometry consists of , ALMA, and VLA observations. <cit.> model the ALMA 345 GHz and VLA data with free-free emission and the data with warm dust (T_ dust=59 K). Our ALMA observation (red point) falls slightly below their free-free model.Figure <ref> shows a zoom on the radio SED, focusing on the high-resolution data. <cit.> model the emission from S1, S2, and region S with three distinct free-free components. For region S, their model passes in between the ALMA band 6 and band 7 data but disagrees at a >2-σ level from the two observations. Moreover, the ALMA fluxes cannot be reproduced simultaneously with free-free emission only. In particular, the slope between the ALMA bands 7 and 6 is incompatible with gas emission and is rather indicative of thermal dust emission.For S2, the ALMA band 6 data match well the free-free model of <cit.>. For S1, our flux at 230 GHz is 1.3 times higher than their model (10-σ significance) and we cannot fit both the VLA and ALMA data within errors with a single free-free model. Therefore in the following we consider an additional dust component that is most visible in S1 and not S2. We investigate the significance of a dust excess by focusing on S1. Fitting the SED of S1 with an additional dust component results in a better χ^2 than without it, and an F-test indicates that the probability of this dust component is high (>98 per cent). This is valid when employing modified black-body spectra as well as more realistic dust optical properties and a distribution of temperatures <cit.>.To model the dust emission present in the ALMA data of region S, we subtract the free-free emission predicted by the <cit.> models for the clusters S1 and S2. Note that we do not subtract their free-free model for region S (shown for reference in the top panel of Fig. <ref>) because there are discrepancies in the VLA observations/models of <cit.> and <cit.> for region S and the whole galaxy, respectively, whose origin is not clear to us. Then, we fit a dust component to the ALMA excess of region S (open circles in Fig. <ref>).We find that the PACS data and ALMA excess cannot be reproduced simultaneously with a single modified black-body (MBB). This remains true even if the PACS data encompass extended emission not belonging to region S. In addition to the MBB fitted to the PACS observations (59 K, β=2), we require a second dust component, for which we assume common values of κ_160=1.6 m^2 kg^-1 and β=1.7 <cit.>. The PACS 160 measurement puts an upper limit on the dust temperature of T_ dust≤ 14 K, which gives a massM_ dust≃1.5×10^5 (grey lines in Figs. <ref> and <ref>). With a temperature of 10 K, the required mass is ∼3×10^5 .Those dust masses are about 4 times larger than the dust mass of 3.8×10^4 found by <cit.> and more than 10 times larger than previous dust estimates based on data only <cit.>. We note that this cold dust mass is an estimate only. In particular, alternative dustproperties or mantle coating, resulting in a larger overall emissivity would reduce the mass <cit.>. It is also possible that the grains could exhibit an intrinsic excess at longer wavelength <cit.>. Further constraints between 200 and 600 would be extremely valuable to confirm the cold dust hypothesis.§ DISCUSSIONis a very intriguing galaxy in the sense that it is actively star-forming, metal-poor, with a large reservoir and extremely faint, still undetected CO emission. It potentially harbours a large dust/hidden molecular reservoir. Here we discuss the implications of such massive cold reservoirs on the galaxy properties and theory of star-formation (varying efficiencies, role of in metal-poor environments). Concerning the gas reservoir, with the SFR and τ_ dep values used in Sect. <ref>, we find molecular gas masses of 5×10^6-8×10^8 . This is the mass of gas within S1 and S2 required to explain the current star formation. If the depletion time is long (∼2 Gyr), the inferred molecular mass is larger than the mass within region S, estimated to be around 4×10^6-10^7 by <cit.> and <cit.>, and it is on the order of the total mass of 4×10^8 <cit.>. If the depletion time is short (∼20 Myr), the inferred molecular mass is of the order of the mass in region S and lower than the total mass.Recent numerical simulations at low metallicity demonstrate that star formation can happen in a primarily atomic-dominated ISM <cit.>. In the case of very short depletion times, no hidden molecular gas is required in to explain the SFR, while in the case of longer depletion times, a massive amount of hidden molecular gas would be required. Concerning the dust, the cold material found in Sect. <ref> could be responsible for the large extinction (A_V≃20 mag) needed in the mid-IR SED modeling. <cit.> estimate the dust surface density to be 1.5 M_⊙ pc^-2. Since the ALMA emission in S1 is barely resolved, we assume that the cold dust spatial extent corresponds to our beam size. Following the calculations of <cit.> and assuming a uniform dust distribution, the cold dust mass needed to account for the mid-IR extinction is ∼5×10^3 , i.e. 30 times lower than our new estimate. This estimate will go up if the dust is distributed in clumps, as suggested by <cit.>.It is noteworthy that the gas mass that is implied by our new dust estimate, assuming full condensation of the metals into dust, is of the same order of the gas mass estimated assuming τ_ dep of 2 Gyr. This cold gas mass is a few times larger than the warm mass of ∼10^8 (based on modeling of the NIR lines; ), as found in nearby metal-rich galaxies <cit.>. Those facts further support our interpretation of an extreme value needed to explain the non-detection of CO emission even with ALMA.§ CONCLUSIONWe present ALMA band 6 (230 GHz) observations in the blue compact dwarf galaxy at an unprecedented resolution of 0.2 arcsec (52 pc).The CO J=2→1 line is not detected. We report a very deep 3-σ L_ CO(2-1) limit of 3.6×10^4 . This corresponds to a lower limit on the conversion factor of 10^2 when assuming cloud depletion times and of 10^3 - 10^4 for whole-galaxy depletion times. The faintness of CO emission implies, for a whole-galaxy star-formation depletion time, a dominant amount of CO-dark gas.The ALMA continuum emission is detected and resolved over the two main SSCs. Re-analysis of the IR-radio SED suggests that the mm-fluxes are not only free-free emission but that there is a cold dust component coincident with the position of S1. Assuming standard dust properties, the dust mass could be as large as 10^5 . This mass of cold dust corroborates the CO-dark gas interpretation. § ACKNOWLEDGEMENTS We acknowledge support from the SYMPATICO grant (ANR-11-BS56-0023) of the French Agence Nationale de la Recherche; the DAAD/PROCOPE projects 57210883/35265PE; the DFG for the programmes BI 1546/1-1 and HO 5475/2-1. and for funding in SFB 881 'The Milky Way System' (subprojects B1, B2, and B8) and in SPP 1573 'Physics of the ISM' (grants KL 1358/18.1, KL 1358/19.2); the ERC via the Advanced Grant 'STARLIGHT' (project number 339177).This paper makes use of the following ALMA data: ADS/JAO.ALMA#2013.1.00916.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.mnras

http://arxiv.org/abs/1702.08467v2

APS/123-QED holgerflechsig@hiroshima-u.ac.jp Department of Mathematical and Life Sciences, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan Allosteric effects are often underlying the activity of proteins and elucidating generic design aspects and functional principles which are unique to allosteric phenomena represents a major challenge. Here an approach which consists in the in silico design of synthetic structures which, as the principal element of allostery, encode dynamical long-range coupling among two sites is presented. The structures are represented by elastic networks, similar to coarse-grained models of real proteins. A strategy of evolutionary optimization was implemented to iteratively improveallosteric coupling. In the designed structures allosteric interactions were analyzed in terms of strain propagation and simple pathways whichemerged during evolution were identified as signatures through which long-range communication was established. Moreover, robustness of allosteric performance with respect to mutations was demonstrated. As it turned out, the designed prototype structures reveal dynamical properties resembling those found in real allosteric proteins. Hence, they may serve as toy models of complex allosteric systems, such as proteins. Design of elastic networks with evolutionary optimised long-range communication as mechanical models of allosteric proteins. Holger Flechsig February 24, 2017 ============================================================================================================================§ INTRODUCTIONThe functional activity of proteins and its precise regulation often relies on allosteric coupling between different functional regions within the macromolecular structure. According to the mechanical perspective allosteric communication originates from structural changes mediated by a network of physically interacting residues <cit.>. Much resembling the occurrence of a proteinquake, local conformational motions initiated, e.g., upon ligand binding to one specific site propagate across the protein structure to spatially remote regions eventually generating a functional change in the conformation of another site.Experiments have indeed evidenced the existence of communication networks in proteins, which are formed by only a set of amino acids and constitute allosteric pathways, physically linking remote binding sites <cit.>. Various computational strategies aimed at predicting such pathways have been developed, including structure-based network analysis <cit.>, stochastic Markov modelling <cit.>, and sequence-based statistical methods <cit.>. Understanding of allosteric communication at the molecular level has also been widely addressed at atomistic resolution in molecular dynamics (MD) simulations <cit.>. To circumvent the heavy computational burden present in MD simulations, collective conformational motions and allosteric transitions in proteins have been, to a vast extend, investigated at the coarse-grained level using elastic-network models and the analysis of normal modes <cit.>. Despite their approximate nature such simplified descriptions have significantly contributed in understanding the mechanistic underpinnings of allostery in proteins <cit.>.While in the majority of cases the important question is considered to be how allostery works for a particular protein given its specific structure, there may also be more fundamental and general questions to be addressed such as what are generic design and functional principles, requisite to make allostery work, and, which dynamical properties are unique to allostery. One possibility to approach such aspects would consist in a systematic screening of the biophysical properties among allosteric proteins with available structural data, which has already been sporadically attempted earlier <cit.>.Here, in an attempt to address this subject, we present an alternative approach which is motivated by the idea to engineer, in silico, artificial spatial structures with dynamical properties resembling those found in real proteins. In previous works such an approach was employed to design structures of elastic networks which can operate as a model molecular machine <cit.> or swimmer <cit.>. Moreover, we recently used this machine to construct a model motor which in its function mimics the myosin protein motor <cit.>.In this paper we aim to establish a generalised structural model of an allosteric system. To this end we design through evolutionary optimisation elastic-network structures which, as the principal element of allostery, encode long-range coupling among two spatially remote local regions. We first explain how the strategy of iterative evolution was developed and applied to stepwise improve, starting from a random elastic network, the allosteric response in the emerging structures. In the designed structures allosteric communication was then analysed in terms of the propagation of strain and its spatial distribution was used to identify pathways through which remote interactions are established. Moreover, the effect of mutations was demonstrated and robustness of allosteric performance in the designed structures examined. Finally we discuss the relevance of our model system in the light of actual allosteric proteins.§ RESULTS§.§ Random elastic network and Evolutionary optimisation Our structural model of an allosteric system was based on evolutionary optimisation starting from a random elastic network. The initial random network was constructed as a two-domain structure. It was obtained by randomly folding two polymer chains independently, each consisting of 100 elastically linked identical beads and representing one domain, and then merging them such that they form a common domain interface. In the model physical interactions between all beads, beyond those acting between neighbouring beads in each chain, were introduced by connecting those two beads by an elastic link which have a separation smaller than a prescribed interaction radius. Thus the complete elastic network was obtained. Details of the construction are described in the Methods section.When elastic networks are constructed based on the structures of real proteins, the beads would corresponds to atoms or typically represent entire amino acid residues, and elastic links between them empirically mimic effective interactions between them <cit.>.Here, we proceed with the constructed random network of two compactly folded coupled domains shown in Fig. 1. In each network domain a pocket site, for simplicity formed by two beads only, was chosen such that a sufficient separation of the two sites was ensured (see Methods). One site represented the allosteric pocket whereas the other mimicked the regulated pocket, respectively. As shown in Fig. 1, the pockets were located opposite to each other near the surface of the respective domain.With this setup, the ability of the elastic network to conduct allosteric communication between the pockets was examined using a simple force-probe scheme, in which conformational dynamics in the allosteric site was initiated through local binding of an additional ligand bead, and the subsequent response generated in the regulated site has been detected. In particular, attractive pair forces which were acting on the pocket beads of the allosteric site have been applied to mimic binding of the fictitious substrate bead to its centre. The dynamics of protein elastic-networks consists in processes of over-damped relaxation motions (see Methods). Hence, as a result of such forces, all beads underwent coupled relaxation motions bringing the elastic network from its original conformation (without the forces) to a deformed steady state of the network, having the ligand bead tightly bound to the allosteric site (i.e, with the forces applied for a long time). During this process the additional forces tend to close this site, i.e. its two beads move towards each other, and first the elastic links localised in their vicinity became deformed. Eventually deformations propagated through the entire network structure until a final steady network conformation was reached. In the simulations the conformational motions inside the network were followed by numerically integrating the equations of motion for all network beads (see Methods). The response generated inside the regulated pocket in the second domain was quantified in terms of distance changes between its two corresponding beads.As we found, the random network structure did not reveal any allosteric coupling between the two sites. While the allosteric site closed upon substrate binding, with the distance between pocket beads changing by approx. -4Å, no response was detected at the regulated site with a change of <10^-4Å (for the choice of lengths units, see Methods). Proceeding with this observation, an algorithm of evolutionary optimisation aimed to design networks with pronounced allosteric communication was established. The optimisation process consisted in sequences of mutations followed by selection, which were applied iteratively starting from the initial random network. In each evolution step the following sequence was carried out: i) mutation; a single structural mutation was performed by randomly selecting one network bead (excluding one of the four pocket beads) and changing its equilibrium position, which corresponded to an alteration of elastic connections in the vicinity of the mutation spot. ii) probing of allostery; the force-probe scheme was applied in a simulation of the new mutant network structure and its conformation with the ligand bead bound to the allosteric site was obtained. iii) selection; the allosteric response in the regulated pocket of the network before and after the mutation was compared and the mutation was scored favourable and was accepted if the mutant network showed improved allosteric coupling between the two sites; otherwise it was rejected. The optimisation procedure was iteratively applied and terminated, once a network structure with a prescribed level of sufficient allosteric coupling was obtained. Details of the implementation are described in the Methods section.In this work two prototype examples of elastic network structures with allosteric coupling were designed and analysed.In two independent simulations of evolutionary optimisation, starting both times with the same initial random network, two different network structures with optimised allostery have been designed. The first one was designed under the side condition that allosteric coupling between the remote sites had the type of symmetric cooperativity, i.e., binding of the substrate bead to the allosteric site and the concomitant closing of that pocket resulted in consequent closing of the regulated site. In the second case, the requirement was vice versa; allosteric communication was optimised under the premise that substrate induced closing of the allosteric pocket triggers opening in the regulated site, i.e. asymmetric cooperative coupling was realised. In both cases, as a result of several hundreds of mutations which were needed during evolution to improve allosteric coupling, the networks underwent significant structural remodelling and concomitant rewiring of elastic interactions between their beads. In Fig. 2 snapshots of network architectures along the processes of evolution are displayed. For both designed networks it could be observed that besides mutation-induced changes taking place inside the individual domains, in particular their common interface became significantly remodelled. As evolution progressed the allosteric response generated inside the regulated pocket upon ligand-binding to the allosteric pocket was gradually enhanced during both design processes, as can be seen by comparing corresponding snapshots of network conformations in Fig. 2. In the next paragraphs the two prototype designed network structures are presented and their dynamical properties analysed. §.§ Designed prototype networks The designed networks are shown in Fig. 2 (bottom panel). Both networks retained the initial two-domain architecture after the evolution, their structures, however, were clearly different from the initial random network. In both designed networks a pronounced domain interface with many inter-domain links had emerged; such interface was rather sparse in the initial random network (see Fig. 2). Remodelling of that region under the process of evolution apparently points towards the importance of the pattern of elastic connections in the domain contact region for allosteric communication in our model systems. Those aspects will be further discussed in the next section.In Fig. 3 (A,a) the two designed networks are shown each in their equilibrium conformation and in the respective steady conformations with the ligand bound to the allosteric pocket and the allosteric response having been triggered in the remote regulated sites. Traces showing changes in the size of both pockets upon binding of the ligand bead to the allosteric pockets are also provided (see Fig. 3B,b). In both designed networks the allosteric pocket closed rapidly when the substrate bead bound there, with the size changing by -3.8Å in both cases. In striking contrast, motions inside the regulated pocket were much slower, when changes in its size only gradually set in and the pocket smoothly approached its closed conformation in the network with symmetric cooperativity (change by -2.3Å), or open conformation in the network withasymmetric cooperativity, respectively (change by +2.0Å). Movies visualising conformational motions during the allosteric transition in the designed networks are provided as Supplementary Videos V2 and V3.Comparing the ligand-bound network conformation with the initial structure reveals that in both designed networks the conformational changes underlying allosteric dynamics did not involve any large-scale structural rearrangements but was rather governed by small-amplitude subtle motions. In fact, initial and substrate-bound structures compared by RMSD's of only 0.6Åin the symmetric case and 0.46Å in the asymmetric case, respectively. While local motions in the two pockets were indeed pronounced, the communication between them must therefore have been resulted from cascades of small-amplitude displacements of network beads located in the region connecting both pockets.By the application of evolutionary optimisation to the random network we could successfully design special networks whose two-domain structure encodes enhanced allosteric interactions between two remote pockets. All three network structures are provided as Supporting Data. Next, to unravel signatures which may underly allosteric communication in the two prototype networks we have analysed the conformational dynamics in terms of mechanical strain propagating through the network structures. §.§ Propagation of strain and communication pathways Apparently, allosteric interactions in our model systems result from conformational changes propagating from the allosteric site across the domain interface to the regulated site. Therefore, in both designed structures the elastic strain inside the networks was computed during the entire simulation following ligand binding to the allosteric site until the steady state of the respective network was reached. The strain of a network was determined in terms of the deformation of elastic links connecting the beads, i.e. deviations from their natural lengths in the initial ligand-free network. Details of the computation are found in the Methods section and SI text. To conveniently visualise strain propagation, the network links are displayed bicoloured, to distinguish stretching or compression, and their width is proportional to the magnitude of the respective deformation. In order to compare to the designed networks the strain propagation was also followed in the initial random network which was allosterically inactive. Snapshots of the strain distribution in all three networks taken at different time moments during the simulation are shown in Fig. 4. Movies visualising conformational motions and the propagation of strain are provided as Supplementary Movies V1-V3.In all three networks the strain was first localised in the vicinity of the allosteric pocket where the ligand was bound. After that, rapidly interactions between other beads also set in and the majority of links in domain 1 became deformed. In the random network the distribution of strain in domain 1 was rather homogeneous; most links were stretched or compressed at similar levels and there was no distinctive feature present. In this network any significant deformations of links located in the central domain interface region could be detected and therefore spreading of strain into the second domain during the simulation was practically absent. The structure of this domain is apparently very stiff preventing any internal conformational motions to occur. Hence, the random elastic network cannot conduct allosteric communication. The situation in both designed networks was completely different. There, first, after ligand binding to the allosteric site conformational motions spread across the domain interface and generated deformations in the second domain and in the vicinity of the regulated site. Second, not all elastic pair interactions in the designed structures were equally involved in the process of allosteric coupling. Rather, the temporal distribution of strain shows that communication between the remote pockets proceeded through specific sub-networks which were apparently critical for allostery in the designed structures. Those networks are formed by a subset of beads connected by springs which undergo major deformations and, hence, contribute essential pathways for the spreading of conformational motions from the allosteric site across the structure to the regulated site. In both designed network structures such remarkably strained springs were found in domain 1 and, in contrast to the random network, in the central domain interface region and the second domain, where they form sparse clusters of important pair interactions between beads.Regions in which the elastic springs remained only marginally populated by strain were rather stiff as compared to the rest of the network. In that regard we observed that the structure of domain 2 in the designed networks is special. In the network with symmetric allosteric cooperativity the two lobes in the tweezer-like domain structure, each harbouring one of the pocket beads of the regulated site, were found to be quite flexible whereas the other parts in domain 2 were stiff (see Fig. 4). In the network with asymmetric allosteric coupling the hinge region connecting both lobes appears to be very flexible instead (see Fig. 4). The specific pattern of flexibility in the designed networks has apparently emerged during the process of evolutionary optimisation in order to make opening/closing motions of the regulated site possible and hence enable the designed network architectures to conduct allosteric communication.The analysis of strain propagation in the designed networks clearly reveals functionally important signatures underlying the investigated types of allosteric coupling. Nonetheless the strain distribution was rather complex and we therefore aimed to deduce simpler pathways which could characterise communication between the pockets, by focusing only on those network springs which carry major strain (see Methods). Results are shown in Fig. 5. In both networks simple chains formed by significantly deformed adjacent network springs were found to meander from the allosteric pocket in domain 1 via the domain interface to the regulated pocket in domain 2. Such chains are composed of a series of stretched springs in domain 1, potentially caused by the closing motion of the allosteric site, connected to a linkage of compressed springs located in domain 2 and triggering the respective allosteric response there.The identified communication chains are obviously critical for the transmission of conformational changes between the two domains and therefore may represent a functional skeleton, critical for allosteric activity in the designed network architectures. Immediately one may pose the question whether, and to what extend, allosteric coupling can be maintained if structural changes would be applied to such motifs. §.§ Robustness of allosteric communication With regard to the robustness of allosteric communication in the two designed networks the intention was not to perform a systematic screening of the effect of structural perturbations. Instead, the aim was rather to present a demonstration. Therefore the analysis of robustness was limited to a few exemplary structural mutations, those applied to beads belonging to the identified communication pathways, and on examining their influence on allosteric interactions. In a first set of simulations mutations which consisted in the deletion of only a single elastic link between two network beads were considered. In the realm of proteins such kind of perturbation would roughly correspond to the mutation of a single amino acid residue, resulting in specific local interactions to disappear. For each specific mutant network a single simulation starting with ligand binding to the allosteric pocket was carried out and after completion the response in the regulated pocket was quantified and the level of allosteric communication determined. The results are listed in Supplementary Table T1. It is found that allostery in both designed networks is generally robust with respect to the removal of single interactions. To understand how long-range coupling between both pockets is maintained in the mutant networks we have visualised the propagation of strain and determined communication pathways for some exemplary cases, similar to what has been performed for the designed wildtype networks (see previous section).For the network with symmetric cooperativity three cases were focused on in more detail, all of which maintained allosteric activity (see Supplementary Table T1). They corresponded to a mutant with one major interaction at the central domain interface having been removed and another mutant with a link deleted in domain 2, closer to the regulated pocket. The computed communication chains in those two mutants are shown in Supplementary Fig. S3. Movies of strain propagation in those two mutant networks are provided as Supplementary Movies V4 and V5. To highlight robustness in the first designed network, a third example consisting in a double mutant network in which both selected links were deleted, was considered too (see Supplementary Movie V6). For all three mutant networks we find that the strain propagation networks as well as the extracted communication chains are very similar to that of the wildtype networks, except in the vicinity of the respective mutation site (see Fig. 6A,B and Fig. S3). There it is found that in the neighbourhood of the deleted link a few bead interactions, which in the wildtype network have not played a major role in the transmission of strain, became important, forming a bridge through which strain was able to side-track the mutation site (see Figs. 6B and S3). This result demonstrates that in the mutant structures similar communication pathways are available which, together with the bridge motifs that compensate the mutation defect, are employed for the propagation of conformational changes and hence may provide the foundation for the robustness of allosteric coupling in this designed network.For the designed network with asymmetric allosteric coupling one mutant structure with a single deleted link located at the domain interface has been selected for illustration. Despite the mutation it revealed full allosteric activity. Propagation of strain subsequent to ligand binding to the allosteric pocket is shown in Supplementary Movie V7 and the extracted communication pathways are depicted in Fig. 6b. It is found thatthe network of bead interactions through which communication between both pockets was transmitted is rather complex, with the strain being populated in large parts of the mutant structure. Hence, in contrast to the wildtype network, allosteric coupling in this mutant does not proceed via simple communication chains.While generally allosteric communication in both designed networks was found to be robust with respect to the deletion of a single bead interaction, there were still few critical mutations which involved a drastic decrease in the coupling between both pockets. They can correspond to perturbations in the vicinity of either pocket or located at the domain interface (see Supporting Figs. S2A,B). In the final step of investigations the effect of stronger mutations was considered, where an entire bead was deleted in a simulation of each network. Those perturbations typically led to a remarkable drop or even complete knockout of allosteric communication (see Supporting Table T1). §.§ Summary and Discussion The phenomenon of allosteric coupling between different functional regions within a macromolecular structure is ubiquitously present in proteins and therefore raises important questions of the fundamental nature of the underlying mechanisms. In contrast to typically employed structure-based modelling of protein dynamics, a different methodology towards approaching such essential problems is presented in this paper. Instead of considering real protein structures, artificial analog structures which encode pronounced long-range allosteric coupling between two spatially remote pockets were designed. The structures were represented by elastic networks, similar to coarse-grained models widely used to describe conformational dynamics of real proteins.A force-probe scheme consisting in ligand-binding to the allosteric pocket, following conformational motions spreading over the structure, and detecting the response generated in the regulated pocket was implemented to evaluate the ability of the network to conduct allosteric communication. Initially starting with a random elastic-network, which did not reveal any allostery, a scheme of evolutionary optimisation was iteratively applied to design two prototype elastic-network structures with perfected allosteric coupling, one with symmetric and the second with asymmetric cooperativity.In the designed networks well-defined pathways and simple chains of important interactions, established by only a few network beads, were identified to constitute the signatures which underly allosteric communication. Remote interactions were robust even if minor structural perturbations were applied. However, a single critical mutation could knockout completely allosteric communication in the designed networks.While the first descriptions of allosteric systems, namely the MWC and the KNF models <cit.>, were formulated more than 50 years ago in the absence of any structural data and were thus of phenomenological nature, the development of very sophisticated experimental techniques and the vast amount of protein structures becoming available in the last decades have ever since allowed to investigate allosteric communication in proteins on the molecular level and led to a decent understanding of the mechanism underlying allostery in several role model proteins. In the recent past structure-based computational modelling, aiming to follow the conformational motions as the underpinning of allosteric effects in proteins, has gained a lot of attention. In particular, due to the time-scale gap present in atomistic-level molecular dynamics simulations, coarse-grained elastic network models which are limited to the mechanical aspect of protein operation became very popular to investigate slow allosteric transitions with timescales beyond the microsecond range. In those studies the analysis of conformational changes is typically based on the computation of normal modes and allosteric effects are discussed in terms of short- and long-ranged correlations of amino-acid residue displacements.The aim in the present study was to present a model which emphasises the mechanical picture of allosteric systems. Therefore, also the elastic network model was used. Here, however, the full elastic dynamics of the network was considered by always numerically integrating the non-linear equations of motion and monitoring processes of conformational relaxation; no linearisation was performed and the conformational dynamics beyond the limiting normal mode approximation could be followed.As a consequence, this model has the obvious advantage that it can resolve the temporal order of events which eventually establish allosteric communication in the network structures, starting from forces and strains which were first generated locally at the allosteric site as a consequence of ligand binding, followed by the subsequent propagation of conformational motions via the domain interface, to finally induce structural changes in the remote regulated pocket. The current model therefore naturally includes causality as the guiding principle to manifest allosteric communication and therefore is richer in its explanatory power compared to correlation-based analyses.The presented model emphasises the structural viewpoint of allostery (as described in <cit.>) in which allostery is regarded as a consequence of optimised communication between the remote functional sites, established through the propagation of strain along pathways which are formed by a set of interacting residues. The employed elastic network description clearly implies limitations. All network particles are of the same kind and physical interactions between them are incorporated by empirical effective potentials; thermal fluctuations were also neglected for simplicity. Despite such gross simplifications the dynamical properties of the designed prototype allosteric structures reveal remarkable similarities with those found in real allosteric proteins: i) specific parts of the structure are flexible whereas other regions form stiff clusters; ii) the propagation of conformational changes which results in the long-range coupling of the remote sites proceeds through communication pathways involving only a few of the many intra-structural interactions; iii) single critical mutations can knockout allosteric coupling. In summary, the designed elastic network structures can provide a general physical model for the mechanics of complex allosteric biomolecules, such as proteins.It should be remarked that the evolutionary pressure imposed in the design process consisted only in magnifying changes in the regulatory pocket in response to ligand-induced changes in the allosteric pocket; no other requirements were imposed and the dynamical properties which actually improved allosteric communication in the evolving structures emerged autonomously. However, in future studies design algorithms which involve coevolution can also be implemented and multiple optimisation criteria can be imposed. Moreover, in future studies the design and analysis of a larger number of allosteric elastic networks can be undertaken. That would allow to compare properties such as communication pathways among them and possibly draw conclusions on the generality of such dynamical motifs.Previously the relaxational elastic-network approach employed in this study was already applied to investigate allosteric coupling in helicase motor proteins <cit.> and in the myosin-V molecular motor <cit.>. In those studies, however, allosteric coupling was only qualitatively discussed. The analysis of strain propagation performed in this paper for the designed artificial protein structures, and the methods to quantify and visualise communication pathways, can easily be applied in the structure-based modelling of real allosteric proteins and allow to investigate intramolecular communication in dynamical simulations.Added note After finishing the manuscript I became aware of two quite recently appeared works in which also the design of mechanical networks with allosteric coupling was undertaken (published in PNAS <cit.>). There, however, different computational algorithms were used and optimisation of allosteric coupling was performed within the linear response limitation; the underlying network architectures were also very different (e.g. 2D on-lattice models <cit.>). Both works principally demonstrate the applicability of design processes to obtain desired responses. The design process introduced in my manuscript was developed completely independent. Besides other differences, its principal distinction is that during the design process the allosteric response was optimised by always considering the full nonlinear elastic dynamics of the networks. The importance of nonlinearities for protein function involving allostery has previously been evidenced (e.g. in <cit.>). Secondly, the network architectures used in the present study resemble more closely the three-dimensional compact fold of real proteins; in fact the designed networks can be regarded as coarse-grained representations of fictitious protein structures. Most importantly this work goes beyond developing the design process. In fact, the main emphasis here was put on understanding the mechanistic principles underlying long-range communication in the designed networks, which was achieved by visualising and analysing propagation of strain, extracting communication pathways and chains which were critical for allosteric coupling, and, investigating robustness of the designed functional properties.§ METHODS§.§ Construction of random elastic networkThe initial two-domain elastic network was constructed by first randomly folding two chains of linked beads, then bringing them into contact and finally determine the network connectivity. One chain consisted of 100 identical beads each and its folded form was constructed as follows. After fixing the position of the first bead, each next bead was placed at random around the position of the previous bead, with the following restrictions: i) the distance to the preceding bead had to lie within the interval between l_min and l_max, ii) the new bead had to be separated from each previous bead by at least the distance l_min, and, iii) the distance from the new bead to the geometric centre of all previous beads should not exceed the threshold r_max. In the simulations prescribed values l_min=4.0Å, l_max=5.0Å, and r_max=20.0Å were used in order to generate a compactly folded backbone chain. After constructing two such chains they were merged in such a way that they came into tight contact but still did not overlap. Positions of the 200 beads in the initial two-domain random structure are denoted by R⃗_i^(0) and their spatial coordinates are provided as Supporting Data. To complete the network construction we have checked all distances d_ij^(0)=|R⃗_i^(0)-R⃗_j^(0)| between beads i and j and introduced an elastic spring between those pairs of beads for which the distance d_ij^(0) was below a prescribed interaction radius of r_int=9Å. With this choice the initial two-domain elastic network had 1467 links. It should be noted that in this model all length units are in principle arbitrary. When data from real protein structures are used, the distances between amino acids have the scale of Angstroms. Hence, throughout the paper this notion is adopted. §.§ Elastic conformational dynamicsThe total elastic energy of the network is the sum of contributions of all elastic links, i.e. U=∑_i<j^NκA_ij/2(d_ij-d_ij^(0))^2. Here, N=200 is the number of beads, κ is the spring stiffness constant (equal for all springs), d_ij=|R⃗_i-R⃗_j| is the actual length of a spring connecting beads i and j in some deformed network conformation, with R⃗_i being the actual position vector of bead i, and d_ij^(0) is the corresponding natural spring length. Coefficients A_ij have value 1 if beads i and j are connected by a spring (i.e. when d_ij^(0)<r_int), and equal 0, otherwise.The dynamics of the elastic network is governed by Newton's equation of motion in the over-damped limit, where the velocity of each network bead is proportional to the forces applied to it. The equation for bead i wasd/dtR⃗_i =-∂/∂R⃗_iU+f⃗_i=-∑_j^NA_ijd_ij-d_ij^(0)/d_ij(R⃗_i-R⃗_j)+f⃗_i.On the right side are the elastic forces exerted by network springs which are connected to bead i, they only depend on the change in the distance between two connected beads. Additionally, an external force f⃗_i could be applied to bead i, which was used in probing allostery (see next section). In the above equations a rescaled time was used to remove dependencies of the beads' friction coefficient (equal for all beads) and κ. To obtain the positions of network beads, and hence the conformation of the network at any time moment, the set of equations of motion was numerically integrated. In the simulations a first order scheme with a time step of 0.1 was employed. §.§ Pocket sites and probing of allostery In the network model the allosteric site and the regulates site were, for simplicity, each represented by two beads. Those beads have been selected in such a way that the two pockets were sufficiently remote from each other. At the same time the two beads forming one pocket should be adjacent, but not directly connected by an elastic spring. In the constructed two-domain network the allosteric pocket was defined by beads with indices 38 and 81, belonging to the first domain, and for the regulated pocket beads from the second domain, with indices 149 and 189, were chosen.To probe allosteric communication between the two pockets a simple force-probe scheme in which conformational dynamics in the allosteric site was initiated through the application of additional forces and the subsequent response generated in the regulated site was probed by evaluating structural changes there. In the simulations pair forces have been applied to the beads of the allosteric pocket. The forces were always acting along the direction given by the actual positions of the two pocket beads. Specifically, the force applied to pocket bead 38 was f⃗_38=0.5·u⃗, with the unit vector u⃗=(R⃗_81-R⃗_38)/|R⃗_81-R⃗_38|. The same force, but with the different sign, was acting on the second pocket bead, i.e. f⃗_81=-0.5·f⃗_38. Such pair forces would induce only internal network deformations and result in a decrease in the distance between the two pocket beads, thus leading to closing of the allosteric pocket. Since this situation is apparently equivalent to assuming that an additional network bead becomes bound to the centre of the two pocket beads, where it generates attractive forces between itself and each pocket bead, we can also say that the chosen force scheme mimics binding of a fictitious ligand bead to the allosteric pocket. The action of the additional forces generated deformations of the network which were first localised in the vicinity of the substrate pocket but gradually spread over the entire network structure. The corresponding process of conformational relaxation was followed by integrating the equations of motion until a steady state of the elastic network, in which all motions were terminated, was reached (at final time T, see SI text). In the resulting conformation of the network, the effect of allosteric coupling was examined by evaluating the distance between the beads corresponding to the regulated site. This distance d_149,189(T)=|R⃗_149(T)-R⃗_189(T)|=: A is termed the allosteric parameter. §.§ Evolutionary Optimisation To design networks with perfected allosteric communication a process of evolutionary optimisation, consisting of mutations followed by selection, was applied iteratively starting from the random network. In particular the following sequence was carried out. First the allosteric response of the elastic network before the mutation was determined and the allosteric parameter A stored. Then a single structural mutation was performed by randomly selecting one network bead (except for one of the four pocket beads) and changing its equilibrium position. The new equilibrium position was chosen to be randomly oriented within a sphere of radius 2.0Å around the old equilibrium position. To preserve the backbone chain of each domain, it was additionally required that, after the mutation, the distance between the mutated bead and its left and right neighbour in the chain still lie within the interval between l_min and l_max. Furthermore, distances from the mutated bead to all other network beads should not be smaller than l_min. After the mutation the network connectivity A_ij was updated by reexamining distances between all beads and the mutated bead; only those pairs which were separated by a distance less than r_int were linked by a spring. After a mutation the elastic network may posses internal free rotations originating from loosely coupled network parts. They can lead to local movements free of energetic cost which was to be prevented. Therefore, when the number of non-zero eigenvalues in the spectrum of the elastic network was smaller than 3N-6 (indicating the occurrence of internal rotation zero modes), the mutation was rejected. Once a mutation which fulfilled all criteria was found, probing of allostery in the new elastic network was proceeded as described in the previous section, the allosteric parameter after the mutation A^mut was determined, and the mutation was evaluated by comparing the allosteric parameter before the mutation A with that after the mutation A^mut. Only mutations which were favourable, i.e. those which improved the allosteric response in the network, were selected. Two situations were distinguished. In the evolution process where symmetric coupling between the allosteric and regulated pocket was to be optimised, a mutation was accepted only if A^mut<A, and rejected otherwise. In the second independent evolutionary process corresponding to the asymmetric situation, the acceptance criteria for the mutations was A^mut>A. As a termination condition for the two evolution processes we imposed (A-A^random)<-2.0Å for the design of the network with symmetric coupling and (A-A^random)>2.0Å for the design of the network with asymmetric coupling. A^random denotes the allosteric parameter of the initial random network. During both design processes the improvement of allosteric response in the evolving networks was recorded and it shown in Supplementary Fig. S1. §.§ Strain propagation and pathways In the initial random, in the two designed networks, and in the selected mutant networks, the propagation of strain after ligand binding to the allosteric pocket was monitored. The strain of an elastic link connecting beads i and j was defined as s_ij(t)=d_ij(t)-d_ij^(0). In the employed model conformational changes corresponded to relaxation processes of the elastic network structure (see Equ.'s (1)). Therefore, the energy injected locally at the allosteric site as a consequence of ligand-binding there would be not only converted into deformations of elastic bead connections but also dissipate. In particular deformations of elastic springs become significantly damped the farther away they are located from the pocket. Since we still wanted to discuss the anisotropy of strain distribution in the network, a method in which the strain was normalised with respect to the distance from the allosteric site (in terms of the minimal path) was employed. Details are described in the Supplementary Information. For the visualisation of strain a link was coloured blue (if s_ij>0) or red (s_ij<0) and the width corresponded to |s̃_ij| (superscript ∼ refers to the normalised strain). To determine the communication chains shown in Fig. 5 the maximum absolute strain of each link during the simulation was stored and only those links whose normalised strain exceeded a prescribed threshold s̃_t were considered (see SI text). For both designed networks a threshold value of s̃_t=60% was imposed. To obtain the communication skeleton in the 52-196-197 double mutant of the designed network with symmetric cooperativity (shown in Fig. 6B) a threshold of s̃_t=45% was used. For the 52-185 mutant of the designed network with asymmetric cooperativity (Fig. 6b)a threshold of s̃_t=25% was used.§.§ Robustness All performed mutations are listed in SI Table T1. For each mutant network a robustness coefficient was computed as the ratio of the change in the pocket size of the regulated pocket in the considered mutant network and the pocket size change in the wildtype network, i.e. (d̃_149,189^(0)-d̃_149,189^final) /(d_149,189^(0)-d_149,189^final). 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To follow conformational motions in a network subsequent to ligand-binding to the allosteric site, the set of equations (Equ.'s (1), main text) was numerically integrated, always until a steady state of that network in which motions were sufficiently terminated was reached (the time needed was referred to as T). As a termination condition for the process of numerical integration a requirement for the instantaneous bead velocity (averaged over all beads) to drop below a prescribed threshold was imposed in the simulations. The condition was 1/N∑_i^N|v_i|<10^-6. During the process of evolutionary optimisation in the two designed networks the improvement of of the allosteric response was recorded. It is shown in Fig. S1. Details are given in the figure caption.§ STRAIN PROPAGATION AND COMMUNICATION PATHWAYSTo discuss the anisotropy of the strain distribution in the random and designed networks we have introduced a method to normalise the strain of elastic links with respect to the distance from the allosteric site. Therefore, we introduced the shortest graph distance between a network bead with index i and a bead of the allosteric pocket with index p as the minimal number of links of a path connecting the two beads, denoted by D_ip. The shortest graph distance of an elastic spring connecting beads i and j to the allosteric pocket was then defined as D_ij:=min{D_ip1,D_ip2,D_jp1,D_jp2}, where p1 and p2 were the indices of beads of the allosteric pocket and p3 and p4 were those of the regulated pocket. Next we defined shells S_n of links, each of which contained all those elastic links (ij) that had the same shortest graph distance n (n=1,2,…) to the allosteric pocket, i.e. S_n:={all links (ij) for which D_ij=n}. Hence, each elastic network link (ij) was uniquely assigned to one shell S_n and its elastic strain at time t (defined in the Methods section of the main text) was denoted by s_ij(t,n). After this procedure, obviously all elastic links had been sorted into their respective shells; all links with shortest graph distance 1 were in shell S_1, those with shortest graph distance 2 were in shell S_2, etc. Now, during a first simulation of the allosteric operation of a designed network - following ligand binding to the allosteric site until the steady conformation was reached - for each shell S_n a maximum absolute strain value m_n=max{|s_ij(t,n)|} of all links belonging to the same shell S_n was determined and stored. Then, in a repeated simulation of the same network, the strain of each elastic link (ij) was normalised by dividing its value by the maximum absolute strain value m_n of the shell n, the particular link belonged to, i.e. s̃_ij(t)=s_ij(t,n)/m_n. The normalised strain s̃_ij(t) was used to visualise the strain propagation in both designed networks in a time-resolved fashion, presented in Supplementary Movies V2 and V3. Corresponding snapshots are shown in the main text Fig. 4. In the same simulation we have memorised for each link (ij) the maximum absolute value m_ij of its normalised strain, i.e. m_ij=max|s̃_ij(t)|. Those values were employed to determine the communication pathways (shown in the main text Fig. 5), which were constructed by considering only those links that were significantly involved in the strain propagation, imposed by the condition m_ij>s̃_t, where s̃_t was a prescribed threshold value for the normalised strain (with its values given in the main text Methods section).In the random elastic networks a similar procedure of strain normalisation was undertaken. However, to compare the propagationof strain to that in the two designed networks, the link strain was normalised with respect to thecoefficients m_n from the designed network with symmetric allosteric coupling.For the selected mutant networks the procedure of strain normalisation was also applied to visualise strain propagation during the allosteric transition (Movies V4-V7) and to construct communication pathways (Fig. 6 main text and Fig. S3).It should be noted that for the normalisation procedure the shortest graph distance between a network bead and the a bead of the allosteric pocket was determined with a standard algorithm using the powers of the adjacency matrix.§ ROBUSTNESS OF ALLOSTERIC COMMUNICATIONRobustness of allosteric communication with respect to exemplary single structural mutations applied to each of the two designed prototype networks was analysed. To determine the communication chains for the two mutants of the designed network with symmetric allosteric communication shown in Fig. S3, a threshold of s̃_t=55% was used for the 52-197 mutant, and s̃_t=60% was used for the 196-197 mutant.§ SUPPORTING MOVIES AND DATA Time-dependent propagation of strain in the random network, the two designed networks, and the selected mutant networks are provided as Supporting Movies V1-V7. In each of the movies the frame rate is not constant and has been adapted such the fast dynamics inside the allosteric pocket, the inter-domain propagation, as well as the slow motions inside the regulated pocket, can be conveniently followed. The actual time during the simulation is always given at the bottom right in the movies.* Supporting Movie V1 Conformational motions and strain propagation in the allosterically inactive initial random elastic network.* Supporting Movie V2 Conformational motions and strain propagation in the designed network with symmetric allosteric coupling.* Supporting Movie V3 Conformational motions and strain propagation in the designed network with asymmetric allosteric coupling.* Supporting Movie V4 Conformational motions and strain propagation in the 52-197 mutant of the designed network with symmetric allosteric coupling.* Supporting Movie V5 Conformational motions and strain propagation in the 196-197 mutant of the designed network with symmetric allosteric coupling.* Supporting Movie V6 Conformational motions and strain propagation in the 52-197-196 double-mutant of the designed network with symmetric allosteric coupling.* Supporting Movie V7 Conformational motions and strain propagation in the 52-185 mutant of the designed network with asymmetric allosteric coupling. As Supplementary Data the set of spatial coordinates R⃗_i^(0) of the initial random network and the two designed networks are provided in the respective equilibrium conformation.* random_network.dat: text file containing spatial coordinates of the initial random network. 1st column: bead index (0 to 199); 2nd,3rd,4th columns: x,y,z coordinate of the bead position. * designed_network1.dat: text file containing spatial coordinates of the designed network with symmetric allosteric coupling. 1st column: bead index (0 to 199); 2nd,3rd,4th columns: x,y,z coordinate of the bead position. * designed_network2.dat: text file containing spatial coordinates of the designed network with asymmetric allosteric coupling. 1st column: bead index (0 to 199); 2nd,3rd,4th columns: x,y,z coordinate of the bead position.

http://arxiv.org/abs/1702.08317v1

3.5cm -1.75cm 2cm -1cm

http://arxiv.org/abs/1702.08482v2

The description of quantized collective excitations stands as a landmark in the quantum theory of condensed matter. A prominent example occurs in conventional magnets, which support bosonic magnons - quantized harmonic fluctuations of the ordered spins. In striking contrast is the recent discovery that strongly spin-orbital coupled magnets, such as α-RuCl_3, may display a broad excitation continuum inconsistent with conventional magnons. Due to incomplete knowledge of the underlying interactions unraveling the nature of this continuum remains challenging.The most discussed explanation refers to a coherent continuum of fractional excitations analogous to the celebrated Kitaev spin liquid. Here we present a more general scenario. We propose that the observed continuum represents incoherent excitations originating from strong magnetic anharmoniticity that naturally occurs in such materials. This scenario fully explains the observed inelastic magnetic response of α-RuCl_3 and reveals the presence of nontrivial excitations in such materials extending well beyond the Kitaev state.winter@physik.uni-frankfurt.de Institut für Theoretische Physik, Goethe-Universität Frankfurt, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, GermanyInstitut für Theoretische Physik, Goethe-Universität Frankfurt, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, GermanyDepartment of Physics and Astronomy, University of California, Irvine, California 92697, USADepartment of Physics and Astronomy, University of California, Irvine, California 92697, USALaboratoire de Physique Théorique et Modélisation, CNRS UMR 8089, Université de Cergy-Pontoise, 95302 Cergy-Pontoise Cedex, FranceInstitut für Theoretische Physik, Goethe-Universität Frankfurt, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, GermanyBreakdown of Magnons in a Strongly Spin-Orbital Coupled Magnet Roser Valentí December 30, 2023 ============================================================== From magnons in ordered magnets tophonons in periodic crystals, the appearance of bosonic collective excitations is ubiquitous in condensed phases of matter <cit.>. For this reason, special attention is given to those states that support more exotic collective modes, for which the conventional paradigm breaks down. In the context of magnetic phases, the breakdown of magnons is commonly thought to require closeness to an unconventional state such as a quantum spin liquid <cit.>. A notable example occurs in Kitaev's exactly solvable honeycomb model <cit.>, for which strongly anisotropic and bond-dependent interactions fractionalize conventional spin excitations into Majorana spinons and fluxes. This Kitaev state has recently risen to prominence due to the suggestion that it may be realized in heavy metal 4d^5 and 5d^5 insulators via a specific interplay between the crystal field and strong spin-orbit coupling <cit.> and, consequently, a variety of candidate materials based on Ir^4+ and Ru^3+ have been intensively explored <cit.>. Encouragingly,evidence of a continuum of magnetic excitations that is inconsistent with conventional magnons was found in the majority of such materials, including the two-dimensional (2D) honeycomb Na_2IrO_3 <cit.> and α-RuCl_3 <cit.>, as well as the 3D analogues β-,γ-Li_2IrO_3 <cit.>, despite all of them having magnetically ordered ground states. While the observed excitation continua in these systems have been interpreted in terms of signatures of the Kitaev state, the low-symmetry crystalline environment of the real materials also allows various additional interactions beyond Kitaev's model <cit.>, which are thought to be large based on both experimental <cit.> and theoretical <cit.> considerations. In this sense, understanding the mechanism for the breakdown of magnons and the appearance of a broad continuum of magnetic excitations remains a key challenge.In this work, we study a representative case α-RuCl_3, which forms a layered 2D honeycomb lattice and displays zigzag magnetic order below T_N∼ 7 K <cit.>. We specifically address the recent inelastic neutron scattering (INS) measurements, which have revealed low-energy magnons <cit.> coexisting with an intense excitation continuum <cit.>. The latter continuum possesses a distinctive six-fold star shape in momentum space, and large intensity at the 2D Γ-point over a wide energy range E = 2 - 15 meV <cit.>.To resolve the nature of this continuum, we take two complementary approaches. We first theoretically investigate the neutron spectra over a range of relevant magnetic interactions in order todetermine the correct spin Hamiltonian for α-RuCl_3, which has been a subject of intense recent discussion <cit.>. Second, we identify the conditions that lead to the breakdown ofconventional magnons in the presence of strongly anisotropic and frustrated interactions, revealing that nontrivial excitations naturally persist well beyond the Kitaev spin liquid.Results The model. Based on previous ab initio studies <cit.>, the largest terms in the spin Hamiltonian of α-RuCl_3 are generally expected to include nearest neighbour Heisenberg J_1, Kitaev K_1, and off-diagonal Γ_1 couplings, supplemented by a 3rd neighbour Heisenberg J_3 term:ℋ =∑_⟨ i,j⟩ J_1𝐒_i ·𝐒_j + K_1 S_i^γ S_j^γ + Γ_1 (S_i^α S_j^β + S_i^β S_j^α)+∑_⟨⟨⟨ i,j⟩⟩⟩ J_3 𝐒_i ·𝐒_jwhere ⟨ i,j⟩ and ⟨⟨⟨ i,j⟩⟩⟩ refer to summation over first and third neighbour bonds, respectively (see Fig. 1). The bond-dependent variables {α,β,γ} distinguish the three types of first neighbour bonds, with {α,β,γ} = {y,z,x}, {z,x,y}, and {x,y,z} for the X-, Y-, and Z-bonds, respectively. The third neighbour interactions are bond-independent. The phase diagram of this model has been discussed previously <cit.>, and is further detailed in Supplementary Note 1; here we review the key aspects. In the limit J_1= Γ_1=J_3=0, the ground state is a gapless Z_2 spin liquid for either positive or negative K_1, as demonstrated in Kitaev's seminal work <cit.>. Small perturbations from the pure K_1 limit may induce various magnetically ordered states, such as the zigzag antiferromagnetic state observed in α-RuCl_3 andshown in Fig. 1. The simplest perturbation is the introduction of a finite J_1, which yields the well-studied (J_1,K_1) nearest neighbour Heisenberg-Kitaev (nnHK) model. This model hosts zigzag order in the region K_1 > 0, J_1 <0, as discussed in Supplementary Note 1. Accordingly, previous analysis of the powder INS experiments within the context of the nnHK model <cit.>, suggested that K_1 ∼ +7 meV, and |J_1/K_1| ∼ 0.3-0.7 for α-RuCl_3. On this basis, the excitation continua observed experimentally were initially interpreted in terms of proximity to the antiferromagnetic (AFM) K_1 > 0 spin liquid <cit.>.However, the further consideration of finiteΓ_1 and J_3 interactions in Eq. (<ref>) significantly expands the experimentally relevant region, as both interactions generally stabilize zigzag order. Indeed, recent ab initio studies <cit.> have suggested that the zigzag order in α-RuCl_3 emerges from J_1 ∼ 0, K_1 < 0, Γ_1 > 0, and J_3 >0, with |Γ_1/K_1| ∼ 0.5 - 1.0, and |J_3/K_1| ∼ 0.1-0.5, as reviewed in Supplementary Note 2. That is, K_1 is ferromagnetic, and supplemented by significant Γ_1 and J_3 interactions. Such interactions would represent large deviations from both Kitaev's original model and the region identified by initial experimental analysis. Before discussing the origin of the excitation continua, it is therefore crucial to first pinpoint the relevant interaction parameters.In order to address this issue directly, we have computed the neutron scattering intensity ℐ(𝐤,ω) for a variety of interactions within the zigzag ordered phase via both linear spin-wave theory (LSWT) and exact diagonalization (ED). For the latter case, we combine results from various periodic 20- and 24-site clusters compatible with the zigzag state in order to probe a wider range of 𝐤-points (see Methods section). Full results for the complete range of models are presented in Supplementary Note 5. Here, we highlight the key results for two representative sets of interactions. Within the (J_1,K_1) nnHK model, we focus on(J_1 = -2.2,K_1=+7.4 meV; |J_1/K_1|=0.3), which lies on the border of the region initially identified in Ref. <cit.>, close to the spin liquid.Beyond the nnHK model, we consider (J_1 = -0.5,K_1=-5.0,Γ_1=+2.5,J_3=+0.5 meV) for which parameters have beenguided by recent ab initio studies <cit.>, and further optimized to improve agreement with the experimental spectra. Results for Models 1 and 2 are first presented in Fig. <ref> and Fig. <ref>, which show detailed ω- and 𝐤-dependence of ℐ(𝐤,ω), along with the evolution of the spectra upon changing parameters towards the K_1 >0 or K_1 < 0 spin liquid regions. Nearest neighbour Heisenberg-Kitaev (nnHK) model. We begin by analyzing the spectraℐ(𝐤,ω) within the zigzag phase of the (J_1,K_1) nnHK model, starting with Model 1 (Fig. <ref>).Despite proximity to the spin liquid, the ED calculations on Model 1 (Fig. <ref>b) show sharp dispersive modes appearing over the majority of the Brillouin zone that are consistent with the conventional magnons of LSWT (Fig. <ref>a). Indeed, for energies below the main spin-wave branch (ω = 1.3-2.3 meV), intensity is localized around the M- and Y-points, corresponding to the pseudo-Goldstone modes of the zigzag order (Fig. <ref>c). ED calculations show clear spin-wave cones emerging from such points and extending to higher energies. Large deviations from LSWT are observed only for the highest energy excitations, which appear near the 2D Γ-point for energies ω > 5 meV. Here, the ED calculations display a broad continuum-like feature centred at ω∼ K_1 that resembles the response expected for the K_1 > 0 Kitaev spin liquid, as shown in Fig. <ref>d. However, comparison with the experimental ℐ(Γ,ω) shows poor agreement; while the experimental intensity extends from 2-15 meV, the ED results for Model 1 predict intensity only at high energies > 5 meV. Indeed, the evolution of the Γ-point intensity with |J_1/K_1| is shown in Fig. <ref>e. On approaching the K_1 >0 spin liquid by decreasing |J_1/K_1|, excitations at the Γ-point shift to higher energy, such that none of the parameters in the vicinity of the spin liquid reproduce the experimental ω-dependence of ℐ(Γ,ω). Similar conclusions can also be drawn from recent DMRG studies of the nnHK model <cit.>. We therefore conclude thatthe broad features observed experimentally in ℐ(Γ,ω) at relatively low energies <cit.> are incompatible with the nnHK model with J_1 < 0 and K_1>0. Extended ab initio guided model. In order to treat the effect of interactions beyond the nnHK model, we consider now the ab initio guided Model 2. In contrast to Model 1, ED calculations on Model 2 (Fig. <ref>b), show large deviations from standard LSWT (Fig. <ref>a) over a wide range of 𝐤 and ω. This model reproduces many of the experimental spectral features <cit.>.In particular, sharp single-magnon-like peaks appear only near the pseudo-Goldstone modes at the M- and Y-points. Elsewhere in the Brillouin zone, broad continuum-like features are observed within the ED resolution. As demonstrated in Fig. <ref>c, we find significant intensity at low energies (ω < 2.3 meV), at both the Γ- and (M,Y)-points. For the intermediate energy region (ω = 5.5 - 8.5 meV), ℐ(𝐤) resembles the six-fold star shape observed in Ref. <cit.>. At higher energies (ω > 10.5 meV) scattering intensity is mainly located at the Γ-point, also in accord with experiment. Furthermore, the ED results for the Γ-point intensity ℐ(Γ, ω) show a broad range of excitations peaked around 4 and 6 meV, and extending up to ∼ 15 meV (Fig. <ref>d). Therefore, ED calculations on Model 2 reproduce all of the main experimental spectralfeatures, validating the range of interactions indicated by ab initio calculations. The only aspect that is not quantitatively reproduced within the Model 2 is the magnitude of the gap at the M-point (∼ 0.8 meV at the level of LSWT vs. ∼ 2 meV experimentally <cit.>). This discrepancy may result from deviations from C_3 symmetry, which are allowed within the C2/m space group <cit.>, but not considered here for simplicity (see Supplementary Figure 11). Interestingly, the spectral features at the Γ-point become dramatically sharper on approaching the K_1<0 spin liquid, as shown in the evolution of ℐ(Γ,ω) with the ratio |Γ_1/K_1| (Fig. <ref>e). This result reveals that the broad continuum may not be directly associated with a proximity to the Kitaev state.Magnon stability beyond LSWT. To gain further insight into the reason for such a drastic contrast between the stability of magnons in Models 1 and 2, it is useful to consider possible magnon decay channels in the zigzag ordered phase. At the level of LSWT, the spin-wave Hamiltonian is truncated at quadratic order, and can be written ℋ_2 = ∑_𝐤,𝐦ϵ_𝐤,m a_𝐤,m^† a_𝐤,m in terms of magnon creation (annihilation) operators a^† (a), where ϵ_𝐤,m denotes the dispersion of the mth magnon band. In this harmonic approximation, magnons represent sharp, well-defined excitations. However, when higher order anharmonic terms are included, the total magnon number N_tot=∑_𝐤,m a_𝐤,m^† a_𝐤,m is typically not a conserved quantity, such that the stability of magnons is not guaranteed beyond quadratic order. Quantum fluctuations associated with the higher-order anharmonic decay terms may mix sharp single-magnon modes with the multi-magnon continuum <cit.>. Similar considerations also apply to the breakdown of other collective modes, such as phonons in anharmonic crystals <cit.>. From this perspective, a large decay rate is expected for any single magnon mode that is energetically degenerate with the multi-particle continuum, unless there are specific symmetries guaranteeing that the two do not couple. It is therefore useful to consider the prerequisites for magnon breakdown in the presence of the strongly anisotropic interactions of Eq. (<ref>). Magnon decay channels for the nnHK model. We first examine the stability of magnons in the nnHK model. For pure J_1 and K_1 interactions, the total spin projections S_tot^γ = ∑_i S_i^γ are conserved along the cubic axes γ = {x,y,z} modulo two. Since the ordered moment also lies along one of the cubic axes in the zigzag phase <cit.> (see Fig. <ref>c), the possible magnon decay channels are restricted. In the local picture, the relevant quantum fluctuations are local singlet S_i^x S_j^x |↑↓⟩ = |↓↑⟩ and triplet S_i^x S_j^x |↑↑⟩ = |↓↓⟩ fluctuations shown in Fig. <ref>a, with Δ S_tot^z = 0 and 2, respectively. In the magnon picture, the Hamiltonian can only contain even order terms (i.e. ℋ = ℋ_2+ℋ_4+...), analogous to conventional Heisenberg antiferromagnets with collinear ordered spins <cit.>. For example, the fourth order decay process due to ℋ_4 mixes the one-magnon states with the three-magnon continuum (Δ N_tot = ± 2), whereℋ_4 = ∑_1-4 V^4_123 a^†_1 a^†_2a^†_3a_4 δ(𝐤_1+𝐤_2+𝐤_3-𝐤_4)+H.c.Here, the bold index (𝐧≡𝐤_n,m_n) labels both momentum and magnon band. This process is pictured in Fig. <ref>b. As noted above, the effect of such terms depends crucially on the availability of low-energy three-magnon states in which to decay.The density of three-magnon states for Model 1 isshown in Fig. <ref>d, based on the one-magnon dispersions obtained in LSWT. At each 𝐤-point, the lowest energy three-magnon state a_𝐪_1^† a_𝐪_2^† a_𝐪_3^† |0⟩, (with 𝐪_1+𝐪_2+𝐪_3 = 𝐤) is obtained by placing two particles in the pseudo-Goldstone modes at opposite M-points (𝐪_1 + 𝐪_2 = 0), and the third particle at 𝐪_3 = 𝐤, with total energy E_3^min (𝐤) = ϵ_𝐤,1 + 2ϵ_M,1. This implies E_3^min (𝐤)≥ϵ_𝐤,1. That is, the three-magnon states lie above the lowest one-magnon band at every 𝐤-point. As a result, every magnon in the lowest band remains kinetically stable, due to the absence of low-energy three-particle states in which to decay. Precisely this condition ensures the stability of low-energy magnons in conventional isotropic antiferromagnets, and explains the sharp magnon-like peaks observed in Fig. <ref>b for Model 1.Strong spectral broadening in the nnHK model can occur only for high-lying excitations with ϵ_𝐤,m > ϵ_𝐤,1=E_3^min, where the density of three-magnon states is finite, such as at the 2D Γ-point. On approaching the spin liquid (at J_1/K_1 = 0), this condition is relaxed due to the vanishing dispersion of the lowest magnon band (i.e. ϵ_𝐤,1→ 0), which corresponds to a vanishing energy cost the singlet fluctuations shown on the left of Fig. <ref>a. The relevant fluctuations in the limit J_1/K_1 → 0 therefore correspond to Δ N_tot = ± 2. For other values of J_1/K_1, the majority of magnons are expected to remain stable due to the absence of low-energy three-magnon states. Magnon decay channels for the extended model. In Model 2, the character of the quantum fluctuations away from zigzag order is notably different (Fig. <ref>). The finite Γ_1 interaction reduces the local symmetry and leads to rotation of the ordered moments away from the cubic axes <cit.> (Fig. <ref>c).In the local picture, this allows additional single-spin fluctuations S_i^xS_j^z|↑↑⟩ = |↓↑⟩ (Fig. <ref>a), which correspond to odd-order anharmonic terms ℋ_3,ℋ_5,... in the magnon Hamiltonian, where <cit.>:ℋ_3 =∑_1-3Λ_12^3 a^†_1 a^†_2a_3δ(𝐤_1+𝐤_2-𝐤_3) + H.c.At lowest order, such terms mix the single-magnon states with the two-magnon continuum (Δ N_tot = ± 1), via the scattering process depicted in Fig. <ref>b. The density of two-magnon states is shown in Fig. <ref>d, for the zigzag domain with 𝐐 = Y. In this case, at each 𝐤-point the lowest energy two-magnon state a_𝐪_1^† a_𝐪_2^† |0⟩is obtained by placing one particle in the pseudo-Goldstone mode at an M-point, and the second particle at 𝐪_2 = 𝐤-M, with total energy E_2^min(𝐤) = ϵ_𝐤-M + ϵ_M≠ E_3^min.It should be emphasized that this condition differs from that of a conventional Heisenberg antiferromagnet (for which E_2^min = E_3^min) <cit.>. In the case of Model 2, the difference is directly related to the strong anisotropic K_1 and Γ_1 interactions, which shift the pseudo-Goldstone modes to the M-points, such that only high energy magnons remain at the Γ-point or ordering wavevector 𝐐 <cit.>. This shift therefore leads to an offset of the low-energy even and odd magnon states in 𝐤-space such that E_2^min(𝐤) < ϵ_𝐤,1 over a wide region of the Brillouin zone; there are many two-magnon states with equal or lower energy than the one-magnon states.Provided there is a finite Γ_1, the spontaneous decay of single magnons into the two-particle continuum is therefore allowed even for the lowest magnon band. The decay rate is expected to be particularly large near the zone center, which represents a minimum in E_2^min.Similar kinematic conditions may also occur in other systems <cit.>. For Model 2, the pseudo-Goldstone magnons near the M-points remain coherent due to the absence of low-energy two particle states in which to decay (Fig. <ref>d). This explains the experimental observation of sharp magnon-like modes near the M-points <cit.>. In contrast, the magnon bands in the remainder of the Brillouin zone directly overlap with the two-particle continuum. It is therefore natural to anticipate a large decay rate even for the lowest magnon bands. To confirm this idea, we have computed the three-magnon interactions and decay rates for all magnon bands for Model 2 using the self-consistent imaginary Dyson equation (iDE) approach <cit.>. Within this approach, it is assumed that the real part of the magnon self-energy is already captured by the LSWT parameters, while the imaginary part is obtained self-consistently (see Methods and Supplementary Note 3). The iDE approach therefore represents an extension of LSWT, in which the one-magnon excitations are broadened according to the momentum and band-dependent decay rate γ_𝐤,n, while other contributions to the neutron intensity from multi-magnon excitations are also absent <cit.>. As a result, comparison of LSWT, ED, and iDE results (Fig. <ref>) allows for the identification of the origin of different contributions to the spectra. The predicted neutron scattering intensity within the iDE approach (Fig. <ref>b) captures many of the most notable features that are observed in the ED and experimental data, showing a significant improvement over the LSWT results (Fig. <ref>a). First, there is an almost complete washout of the two high-energy one-magnon modes due to strong decays. This implies that the higher-energy features > 4 meV appearing in ED are primarily multi-magnon in character (including the 6 meV peak at the Γ-point). The appearance of these higher energy features in the inelastic neutron response may arise partly from direct contributions from the broadened two-magnon continuum via the longitudinal component of the structure factor, which is not included in the iDE approach (see Supplementary Note 3). Second, the broadening of the two lower magnon bands in the iDE results and the resultant variation of their intensities are in a close agreement with the exact diagonalization – particularly in a wide region near the Γ point (see also Supplementary Figure 5). These are precisely the features with which the LSWT results were most incompatible. Over much of the Brillouin zone – and especially for the higher magnon bands – the computed γ_𝐤,n is on the same scale as the one-magnon bandwidth, confirming the absence of coherent magnons. Discussion.The general requirements for strong two-magnon decays are less restrictive than a proximity to a spin-liquid state. Indeed, a large decay rate is ensured by the following three conditions: large anisotropic interactions, deviation of the ordered moments away from the high-symmetry axes, and strong overlap of the one-magnon states with the multi-magnon continuum (see Supplementary Note 3). Of these, the first two conditions ensure that the scattering vertex Λ_12^3 is large –of the order of the underlying interactions, i.e. Λ_12^3∼𝒪(K_1,Γ_1). For α-RuCl_3, the strong overlap with the multi-magnon continuum is ensured by shifting of the low-energy magnons away from the Γ-point. Since the bottom of the two-magnon continuum must always have an energetic minimum at the Γ-point, the shifting of the pseudo-Goldstone modes to a finite momentum ensures the remaining higher-energy magnons are degenerate with the continuum near the zone center. Experimentally, these conditions are also likely to be satisfied by the zigzag ordered Na_2IrO_3 <cit.>, and spiral magnets α-, β-, and γ-Li_2IrO_3 <cit.>. This picture is also consistent with recent indications that the magnetically disordered phase observed at high pressure in β-Li_2IrO_3 <cit.> is driven primarily by large Γ_1 interactions <cit.>. With this in mind, there are two general scenarios that can explain the observed continuum excitations in α-RuCl_3 and the iridates A_2IrO_3. In the first scenario, which has been advanced by several studies, the excitations can be treated as free particles with a small number of flavours. Such excitations are weakly interacting and have well-defined dispersions, but possess quantum numbers (e.g. Δ S_tot = ± 1/2) or topological properties inconsistent with the experimental neutron scattering selection rules (i.e. Δ S_tot = 0, ± 1). The appearance of the broad continuum in energy therefore results only from the fact that these fractional excitations must be created in multiples. If they could have been created individually, they would have represented long-lived and coherent quasiparticles with sharply peaked energies.This scenario indeed describes the Kitaev spin liquid, where the special symmetries of the Hamiltonian allow an exact description in terms of two flavours of particles: non-interacting Majorana spinons and localized fluxes <cit.>. Such excitations are long-lived, but belong to nontrivial topological sectors, and therefore cannot be created individually by any local operations. For the Kitaev spin liquid, the predicted continuum therefore represents coherent multiparticle excitations. In contrast, upon moving away from the pure Kitaev point, the relevant symmetries that protect the spinons and fluxes are lifted both by additional magnetic interactions and by spontaneous symmetry breaking of the magnetic order. This tends to confine spinons into gauge neutral objects such as magnons <cit.>. Despite this lattertendency, we have argued that coherent magnons are unlikely to appear at large Γ_1 due to the strong anharmonicity in the magnon Hamiltonian. While this leaves open the possibility that nearly free Majorana spinons persist into the zigzag ordered phase in some regions of the Brillouin zone, a more general scenario is that the observed continua represent fully incoherent excitations. In this second scenario, the excitations are not describable in terms of any type of free particles with small decay rates and well defined dispersions. The broad continua therefore reflects the absence of coherent quasiparticles altogether, rather than particular experimental selection rules related to fractionalization.At present, it is not clear which of these scenarios applies to the iridates and α-RuCl_3, although a key role must be played by both the Kitaev K_1 and off-diagonal couplings such as Γ_1. In any case, the study of these materials calls into question the stability of magnetic quasiparticles in the presence of strongly anisotropic interactions.In summary, we have shown that all main features of the magnetic excitations in α-RuCl_3 <cit.> are consistent with strongly anisotropic interactions having signs and relative magnitudes in agreement with ab initio predictions. The ferromagnetic Kitaev coupling (K_1 <0) is supplemented by a significant off-diagonal term (Γ_1 >0) that plays a crucial role in establishing both the zigzag order and the observed continua. In the presence of such interactions, the conventional magnon description breaks down even deep in the ordered phase, due to strong coupling of the one-magnon and two-magnon states. This effect is expected to persist over a largerange of the phase diagram suggesting that the observed continua in α-RuCl_3 and the iridates A_2IrO_3 represent a rich and general phenomenon extending beyond the Kitaev spin liquid. For this class of strongly spin-orbital coupled magnets, the presence of complex and frustrated anisotropic interactions leads naturally to dominant anharmonic effects in the inelastic magnetic response. Fully describing the dynamics of these and similar materials therefore represents a formidable challenge that is likely to reveal aspects not found in conventional isotropic magnets.Methods Exact diagonalization. The neutron scattering intensity was computed via: ℐ(𝐤,ω) ∝ f^2(𝐤)∫ dt∑_μ,ν (δ_μ,ν - k_μ k_ν/k^2)× ×∑_i,j⟨ S_i^μ (t) S_j^ν (0)⟩ e^-i𝐤· (𝐫_i-𝐫_j)-iω twhere f(𝐤) is the atomic form factor of Ru^3+ from Ref. <cit.>. ED calculations were performed using the Lanczos algorithm <cit.>, on several 20- and 24-site clusters with periodic boundary conditions. Such periodic clusters are detailed in Supplementary Note 4. Excitations were computed using the continued fraction method <cit.>. Further details and additional results are presented in the supplemental material; these extensive calculations go beyond previous ED studies <cit.>, which focused mainly on the static properties, or a limited portion of the phase diagram. ED results shown for the high-symmetry Γ, M, Y, X, and Γ^' points were averaged over all clusters. TheED 𝐤-dependence of ℐ(𝐤,ω), integrated over theenergy windows E = 1.3-2.3, 5.5-8.5, and 10.5+ meV (Fig. <ref>c and <ref>c), was obtained froma single 24-site cluster respecting all symmetries of the model.The discrete ED spectra were Gaussian broadened by 0.5 meV, consistent with the width of experimental features <cit.>. The intensities were also averaged over the same range of out-of-plane momentum as in the experiment <cit.>.Linear spin-wave theory. LSWT results shown in Fig. 1 and 2 were obtained with the aid of SpinW <cit.>. Following the approach with the ED data, the discrete LSWT spectra were as well Gaussian broadened by 0.5 meV and the intensities were also averaged over the same range of out-of-plane momentum as with ED and in the experiment <cit.>. Imaginary self-consistent Dyson equation approach. In order to calculate magnon decay rates γ_ k,n, we have evaluated three-magnon interaction vertices by performing rotation to local reference frames of spins. The obtained value of the real-space interaction is quite large, about ∼ 3 meV. Next, the Born approximation calculation of the decay rates results in unphysical divergencies <cit.>, thus the self-energy Σ_ k,n needs to be regularized. We have used the so-called iDE approach:a self-consistent solution on the imaginary part of the Dyson's equation, Σ_ k,n(ϵ_𝐤,n+iγ_ k,n)= -iγ_ k,n, see Ref. <cit.>. We have obtained theregularized broadening for the magnon spectrum and have calculated the transverse part of the dynamical structure factor, shown in Fig. <ref>, by adding the calculated decay rates to experimental resolution of 0.25 meV.The spectral function is approximated as a Lorentzian. More technical details can be found in the Supplementary Note 3.Code availabilty Custom computer codes used in this study are available from the corresponding author upon reasonable request. Documentation of the codes is not available.Data availability Data is available from the corresponding author upon reasonable request.Acknowledgements The authors acknowledge useful discussions with J. Chaloupka, A. Banerjee, S. E. Nagler, A. A. Tsirlin, R. Moessner, F. Pollmann, and M. Zhitomirsky. S. M. W. acknowledges support through an NSERC Canada Postdoctoral Fellowship. R. V. and K. R. acknowledge support by the Deutsche Forschungsgemeinschaft through grant SFB/TR 49. 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Matter volume27, pages166002 (year2015).§ FIGURES[pages=1]Supplemental_v12.pdf [pages=2]Supplemental_v12.pdf [pages=3]Supplemental_v12.pdf [pages=4]Supplemental_v12.pdf [pages=5]Supplemental_v12.pdf [pages=6]Supplemental_v12.pdf [pages=7]Supplemental_v12.pdf [pages=8]Supplemental_v12.pdf [pages=9]Supplemental_v12.pdf [pages=10]Supplemental_v12.pdf [pages=11]Supplemental_v12.pdf [pages=12]Supplemental_v12.pdf [pages=13]Supplemental_v12.pdf [pages=14]Supplemental_v12.pdf [pages=15]Supplemental_v12.pdf [pages=16]Supplemental_v12.pdf [pages=17]Supplemental_v12.pdf [pages=18]Supplemental_v12.pdf [pages=19]Supplemental_v12.pdf [pages=20]Supplemental_v12.pdf [pages=21]Supplemental_v12.pdf [pages=22]Supplemental_v12.pdf [pages=23]Supplemental_v12.pdf [pages=24]Supplemental_v12.pdf [pages=25]Supplemental_v12.pdf

http://arxiv.org/abs/1702.08466v3

^1  ^2  By using Fourier-transform scanning tunneling spectroscopy we measure the interference patterns produced by the impurity scattering of confined Dirac quasiparticles in epitaxial graphene nanoflakes. Upon comparison of the experimental results with tight-binding calculations of realistic model flakes, we show that the characteristic features observed in the Fourier-transformed local density of states are related to scattering between different transverse modes (sub-bands) of a graphene nanoflake and allow direct insight into the electronic spectrum of graphene. We also observe a strong reduction of quasiparticle lifetime which is attributed to the interaction with the underlying substrate. In addition, we show that the distribution of the onsite energies at flower defects leads to an effectively broken pseudospin selection rule, where intravalley back-scattering is allowed.73.20.At, 73.22.Pr, 68.37.Ef, 72.10.FkImpurity scattering and size quantization effects in a single graphene nanoflake Mikhail Fonin^1[Email address: mikhail.fonin@uni-konstanz.de] December 30, 2023 ================================================================================Following the exfoliation of graphite monolayers in 2004 <cit.>, graphene has attracted considerable interest as a prospective material for electronic <cit.> and spintronic <cit.> applications. Upon fabrication of graphene nanoribbons <cit.> or nanoislands <cit.> with different edge terminations, new properties can be introducedsuch as tunable bandgaps <cit.> or edge-induced magnetism <cit.>. With regard to the electronic transport through graphene nanoribbons, the effect of impurity scattering and edge disorder becomes an important issue. A powerful tool to examine the quasiparticle interference (QPI) effects in graphene due to scattering at defects and edges is scanning tunneling microscopy and spectroscopy <cit.>. The observed QPI is directly related to modulations in the local density of states (LDOS) <cit.> and provides access to the present scattering vectors and thus to the electronic structure of graphene <cit.>. Recently, the influence of local scattering centers on the local density of states in graphene nanoribbons has been studied theoretically <cit.>. The interplay between single impurity scattering and size quantization was shown to generate characteristic spectral features in the Fourier Transform (FT) LDOS that can be related to the transverse modes of the nanoribbon. Here we present a comprehensive study of size quantization in epitaxial graphene nanoflakes (GNFs) on Ag(111) upon analysis of QPI by STM and tight-binding simulations of realistic model flakes. We indeed find the characteristic features in the FT-LDOS related to scattering between different transverse modes of a GNF as predicted by theory. Detailed analysis of the scattering features allows to gain a profound insight into the behavior of charge carriers in graphene flakes, including discrete electronic spectrum, quasiparticle lifetimes as well as effects of pseudospin.Graphene nanoflakes were initially grown on Ir(111) and decoupled by noble metal intercalation as described elsewhere <cit.>. STM and STS measurements were carried out in an Omicron cryogenic STM setup in ultra-high vacuum at T=5–10 K. Differential conductance (dI/dV) maps were obtained using a standard lock-in technique with modulation voltages V_mod = 3 mV(rms) and at frequencies f_mod = 600–800 Hz. Tight-binding calculations were performed using an atomistic recursive Green's function formalism including the effects of trigonal warping in order to account for the relatively large doping level of graphene on Ag(111). AB sublattice symmetry breaking was introduced via rotational grain boundary defects modeled as a local σ_z (mass) potential in addition to a scalar potential. Tight-binding parameters were inferred from DFT simulations of comparable topological defects. The limited quasiparticle lifetime in modeling realistic flakes was achieved by an increase of the imaginary part of the energy above the energy level separation resulting in the loss of transverse quantization within the flake.Figure <ref> (a) showsa typical elongated graphene nanoflake sitting on top of an at least 7 nm high epitaxial Ag(111) island on Ir(111). Prominent LDOS modulations, modifying the honeycomb appearance of graphene are observed in dI/dV mappings [Fig. <ref> (b)] leading to pronounced features in the fast Fourier transform (FFT) image [Fig. <ref>(c)]. The LDOS patterns at the edges are identical to those previously reported for GNFs on Au(111) <cit.>, suggesting single H-terminated graphene edges. The analysis of the atomic contrast and Moiré structure [Fig. <ref>(d)] suggests the R0 adsorption configuration of the GNF, meaning that the metal <112̅> direction is parallel to the graphene <11̅00> direction. Atomic reconstruction reveals an alignment of the long flake axis roughly parallel to the armchair (ac) direction and a predominant termination with zigzag (zz) edges for both sides of the flake. Besides the edges, point defects within the flake interior also give rise to strong LDOS modulations. The most abundant defect type observed in the experiments are flower defects <cit.> with the atomic appearance shown in Fig. <ref> (e).Figure <ref> (b) displays a dI/dV map on the GNF recorded at 10 mV. As the electronic band minimum of the Ag(111) surface state is shifted upwards to about 200 meV due to the presence of graphene <cit.> as well as due to strain effects in the Ag thin film <cit.>, we can unambiguously assign the observed LDOS modulations to graphene standing waves superposed by the Moiré superstructure. The corresponding FFT of the dI/dV map [Fig. <ref> (c)] reveals intervalley and intravalley scattering features characteristic for graphene <cit.>. The observed intravalley feature stems from the elastic scattering between the states on a constant energy contour (CEC) within a single Dirac cone, whereas the intervalley features correspond to the scattering vectors q⃗_inter connecting the states of the two neighboring cones <cit.> as displayed in Fig. <ref> (f). Due to the charge transfer leading to the shift of the Dirac point to -470 mV [Fig. <ref> (f)], scattering features for graphene on Ag(111) already show the onset of the trigonal warping. The appearance of the intervalley features deviates substantially from the ones reported for a perfect infinite graphene sheet, where an intensity modulated ring-like contour is expected <cit.>. Instead, our measurements show a ring-like feature with considerable intensity inside the ring, which is located on a line through the center of the scattering ring [inset in Fig. <ref> (c)]. The alignment of this feature reflects the direction of the long axis of the GNF in real space. In contrast to that no inner structure within the intervalley features is observed in measurements on extended graphene on Ag(111) underlining the confinement nature of the observed features.In order to understand the effect of quantum confinement on the electron scattering, we first discuss the simplified case of an infinitely long armchair graphene nanoribbon (AGNR) of width W [Fig. <ref> (a)]. With x corresponding to the confined direction, the transverse wave vector k_x is quantized and can be written as k_x → k_dn = nπ/W-K_dx, where n is an integer and d enumerates valley pairs in the first Brillouin zone <cit.>. More precisely, the values of k_x involved in a scattering event at a point defect in the ribbon are located at intersections of lines in the k_y direction with the CEC of graphene at positions k_dn [Fig. <ref> (b)]. In a ribbon, scattering vectors are observed connecting two points on the circular CEC with initial values k_dn = nπ/W-K_dx and final values k_d'm=mπ/W-K_d'x (specific selection rules must also be obeyed) <cit.>. Thus, scattering intensity is observed only at a finite number of q⃗ points, which lie inside the circle of radius 2√(2mE/ħ^2) and constitute a scattering feature as shown in Fig. <ref> (c). When comparing these characteristic patterns with the experimentally observed scattering features of the GNF along the armchair direction [inset in Fig. <ref> (c)], there is already a striking resemblance for both contour and interior.Going back to scattering in realistic graphene structures, we extend the analysis to a direct comparison of the experimental data from a compact GNF with the results of a theoretical treatment of this particular flake geometry. Fig. <ref> (a) shows a R0 GNF on Ag(111) with a lateral size of 70×170 nm^2. The long edges of the experimentally measured flake are virtually parallel to the zigzag direction, resembling the shape of a zigzag graphene nanoribbon with rough edges. The size of the flake allows for a reasonable atomistic tight-binding modeling of graphene π-electrons, utilizing a numerical recursive Green's function approach.In this approach, we assume a free-standing graphene flake with about 3.4· 10^5 atoms [Fig. <ref> (b)] reproducing the overall size and shape of the experimental GNF, with the atomic structure precisely adjusted to the experimentally observed one including flower defects and edges. Since the edges are hydrogen passivated, each edge carbon atom is simply modeled by a single π-orbital as in the flake interior. The results of this approach are in very good agreement with previous experimental observations and theoretical calculations for the edges <cit.>.The tight-binding model used is defined by the Hamiltonian H= ∑_iϵ_ic_i^†c_i +∑_ij t_ijc_i^†c_j, where c_i^† and c_i are creation and annihilation operators for site i, ϵ_i is the onsite energy of site i, and t_ij is the hopping amplitude between sites j and i. We set the tight-binding hopping parameters according to the simple formula for π orbital overlap <cit.>t_ij = t(r) = -t exp[-λ(r-a_cc)],where r is the distance in the plane between carbon atoms i and j, a_cc=1.42 Å is the graphene carbon-carbon distance, and λ≈ 3/a_cc. The nearest-neighbor hopping integral -t is related to the Dirac electron Fermi velocity ħ v_F=3a_cct/2. From the measured Fermi velocity [Fig. <ref> (f)], we get t≈ 2.6 eV. The above formula is applied for inter-atomic distances r less than a cut-off R_c≈ 1.8 a_cc, in order to take into account next-nearest neighbor hopping as well as reasonable hopping parameters in the pentagons and heptagons in the flower defects. We set onsite energies ϵ_i to zero throughout the flake, except in the flower defects where they vary between -0.6t and +1.2t. This serves to model the charge transfer between pentagons and heptagons in the defect. The onsite energies are unknown for the flower defect and should in principle be computed from density functional theory. Here the parameters have been taken in analogy to the Stone-Wales defect, for which this has been done thoroughly by Amara et al. <cit.>. For the FT-LDOS, the exact values of these parameters actually do not matter, as long as they are non-zero. For zero onsite energies across the flower defect, the scattering is too weak to match the experiment. We note that the distribution of onsite energies in the flower defect leads to an effectively broken pseudospin selection rule, where intravalley back-scattering is allowed. This leads to the restoration of the intravalley scattering ring in the FT-LDOS (feature i in Fig. <ref> (e), which can also be observed experimentally in both armchair and zigzag GNFs (see Fig. <ref> and <ref> for a comparison). The presence of flower defects in the flake also yields additional intensities besides scattering and atomic features in the FFT related to the defect geometry, which are found for both the experimental and theoretical flake, however, they are more pronouncedly visible in the tight-binding FFT due to the absence of measurement noise. Taking into account next-nearest neighbor hopping t(r=√(3)a_cc)=-t', we have in the above model E_D=3t'≈ 0.33t. For the simulations we use E_F=0.61t, which is more electron-doped than observed experimentally. This serves to enlarge the scattering features, see Fig. <ref>. It should be noted that this puts us further into the trigonal warping regime due to the substantial shift of E_D with respect to E_F.The LDOS is computed via the imaginary part of the retarded Green's function G=(E_F+iη - H)^-1.Broadening of energy levels can lead to loss of visibility of the scattering features (c.f. Fig. <ref>) associated with size quantization. Broadening can be due to electron-electron interaction, electron-phonon interaction or, for instance, weak coupling of the graphene π-electron system to the underlying metallic substrate. Here we introduce it phenomenologically as an imaginary part of the energy E→ E+iη when computing the Green's function for electron propagation in the flake. For example, for a rectangular flake of dimension 70 nm × 170 nm, with the zigzag direction along the long axis, we have different estimates for energy level separations in the two directions. We get Δ E_ac≈ 10 meV and Δ E_zz≈ 25 meV. Thus, at an imaginary part η∼Δ E_ac, we will no longer see quantization along the long axis, only along the short axis. Indeed, the simulated FT-LDOS images show complicated flake levels for small η, while these are broadened in favor of clean zigzag ribbon like levels for the chosen η=3· 10^-3t≈ 8 meV, see Fig. <ref> (e). This translates into a quasiparticle lifetime of about 80 fs and, taking into account the experimentally obtained Fermi velocity, a mean free path of about 70 nm, which corresponds to the short axis length of the flake. In comparison to extended graphene sheets, the obtained quasiparticle lifetime is approximately one order of magnitude smaller<cit.>. Upon examination of the intervalley scattering features in the experimentally obtained FFT [Fig. <ref> (f)], in addition to graphene bulk features (circles), a rich inner structure due to transverse confinement is visible in the FT-LDOS. The calculated FT-LDOS [Fig. <ref> (e)] displays very pronounced trigonally warped intervalley scattering contours enclosing additional features aligned with the flake's long axis. These fine structures are very narrow at the center of the ring, while fanning out towards the rim. This is in perfect agreement with the experimentally obtained data as can be readily seen from the juxtaposition of the magnified experimental and theoretical scattering features compiled in Fig. <ref> (g). Also the intravalley scattering ring observed in the measured FT-LDOS is restored in the tight-binding calculation upon the inclusion of AB sublattice symmetry breaking at the flower defects. Thus a correct description of the scattering features is only possible when taking into account the size and exact atomic structure of the nanoflake along with the symmetry-breaking at point defects and substantial energy level broadening. In summary, clear footprints of confinement can be observed in quasiparticle scattering of epitaxially grown graphene nanoflakes. We have demonstrated that the experimentally observed scattering features are in very good agreement with tight-binding calculations of well-defined graphene nanoribbons, which besides conventional intra- and intervalley scattering display quasiparticle wave vectors arising from additional scattering channels between the ribbons' transverse modes.However, certain interactions and imperfections of the crystal lattice in realistic flakes have to be accounted for. Modeling flakes with grain boundary defects allows us to distinguish the effects of rough edges, defects and quasiparticle lifetime. While we have shown that the introduced grain boundary defect breaks the AB sublattice symmetry thus restoring the intravalley scattering contour at q⃗=0, it can not reproduce the alignment of the scattering feature interior with the long axis of the calculated flakes. 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http://arxiv.org/abs/1702.07887v1